mm-223
===
Subject: Differential topology (Sard's Theorem),does
anybody know an example (or a reference) for the following:1)
a (non-differentiable) homeomorpsm R^n -> R^n wch maps a set
with Lebesgue-measure zero to a set with positive measure?2)
interesting but easy corollaries or applications of Sard's
===
Theorem)> ,> does anybody know an example (or a reference) for
the following:> 1) a (non-differentiable) homeomorpsm R^n ->
R^n wch maps> a set with Lebesgue-measure zero to a set with
positive measure?> 2) interesting but easy corollaries or
applications of Sard's Theorem?For (2) there is ts great 
book
by Milnor called sometng like Topology from a differentiable
point of view.For (1) I have some memory that sometng related
===
measurements 610 by support1.mathforum.org (8.11.6/8.11.6/The
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1.9 $, proapp) id i1QNDi825116;triple x is mine610 is the
measurement triangle. reply i'll tell
uapocalypsis610@hotmail.com><pre>caj@baker.math.niu.edu
says...> Whoa. There you go AGAIN!! God, yer slippery. The
slope>is *close* to a whole-number multiple of a popular
approximation to >Pi. NOT AN EGYPTIAN VALUE. As one other
poster has already noted,>no evidence seems to exist for 22/7
being used until much later.>Except, of course, your claims
that 22/7 as Pi was deliberately >encoded into the Great
Pyramid. >>Pi wasn't encoded into anytng it was incorporated
>>in the standards of measure.> Ts is your claim, and ts is
what you will have to prove.> Particularly, do you have any
evidence that the Egyptians>used 22/7 for Pi, *other* than
that you can find it in the Great>Pyramid?>>Petrie measured
the perimeter of the wall and noted the>>relationsp of the
length to the circuit of the wall.> So does Petrie say the
ratio is 1:1+1/7 or does he>say it's 1:1.141592654... as you
claimed earlier?> Don't you tnk that a culture capable of
writing tngs>down would have just said, We tnk Pi is ...
instead of >building a chamber>>Libraries full of books are
rotting away, the paper crumbling to dust,>>How much data is
stored on obsolete systems like punch cards that>>eventually
won't have a functioning reader, or for that matter>>on
electronic media subject to the millenium bug.>>Their pyramid
has lasted 5 millenia. > And what does that prove? The
Egyptians carved heiroglypcs>into rock, and could do ts if
they wanted information to last 5,000>years. Besides, we have
surviving works like the RMP wch do >mention ways to
approximae Pi like 256/81. Does the RMP mention 22/7?>with
dimensions such that if a pyramidologist>takes twice its
length plus twice its height *after removing the >ßoor* and
divides by the length he/she'll get 22/7????? >>Its really
much simpler than that. Respected archaeologists
>>specializing in egyptology like Petrie and Lauer noted the
>>relationsp in many forms of the Egyptians art.> Such as? 
I'm
sorry, but there's sometng very suspicious>about purported
archaeologists who, trying to find Pi, find it >in 
places where
others don't.>>It isn't just any random 
figure thats repeatedly
arrived at here,>>That was part of our criteria that there be a
regular repeated>>consistent relationsp with Egyptian standards
of measure.> You are the only one here who claims that 22/7 was
an >Egyptian approximation for Pi. So far you have found 22/7
in>two ways. The first seems to be due to the fact that the
pyramid's>slope has a factor of 11 in the denominator and a
factor of 7 in >the numerator; the second one is still iffy,
since you have so>far given three different values for the
perimeter of the wall.>Someone measured it as 2x1x1, someone
else found 2x1xPi-2, and>now you're saying 
it's 2x1x8/7, wch
suddenly allows you to get>the 22/7 you got by messing about
with the side length and height.> Further, if the circuit of
the wall divided by the side>length approximates 22/7, why
would that be any reason to suggest>that the Egyptians knew
that 22/7 was a good approximation for Pi??!?>Or had anytng to
do with Pi????> It doesn't matter how many big words 
you've
learned to say.>What you're talking about does not reduce 
the
uncertainties of>a measurement, whether you call it correction
or calibration. >>Any desired precision can be specified. Once
specified the >>required precision provides constraints. Those
constraints are>>recognized in the process of calibrating the
instruments.> Not after the measurements are taken. If a
measurement>is 200 meters +/- 1 mm, no corrections after the
measurements are>taken will reduce the margin of error to
better than +/- 1 mm.>Except perhaps for LYING.>
Okaaaaaaaaaaaaaay. First of all, you said that the ratio>of
side length to circuit was 1.141592653..., that is, Pi-2
all>the way down to your calculator's round-off. NOW 
you're
saying>it's 3+1/7 ???>>I said that Petrie had noted a Pi 
ratio
between the length of>>the wall and its circuit, If you convert
the reported dimensions >>(17'2, 34'-4) to 
Egyptian units you
get even palms and cubits.> Oh, must we do ts
rectangles. One is the space>inside the walls 1 x 2 x
1.41592654... and the other is the space>inside the walls
Did Petrie get 8/7 or Pi-2? > Secondly, don't you tnk 
it's a
tad contrived, finding Pi>by ripping up the ßoor so? >>It 
isn't
necessary to rip up the ßoor, its inset from the wall.>>In fact
it might be better described as a dias, or raised ßoor>>wch
comes close to the wall. > Okay. That's much clearer. So did
Petrie really get 8/7:1:2>for ts room or sometng else? And why
does that disagree with >other people's measurements?> Only
provided that the initial measurements possess the
desired>precision!>>What do you tnk the purpose of supporting
the tape during>>measuring is?> Okay, so first Cole was
correcting for the sag in the tape,>and now he was actually
supporting the tape so that it wouldn't sag!>Does he 
actually
say he did ts in s report? >>The archaeologist says No
problem, we will choose the one wch >>best fits what we know
about how the Egyptians laid out their >>measures from our
studies of other Egyptian structures and look>>for consistancy
and repetition of similar dimensions..> And you're saying 
that
there's no uncertainty there.>>Yes; look, If I lay out a
number of increments with a ruler>>and you come along and
measure them later, it ought to be>>pretty obvious whether or
not I used a ruler divided into>>inches or one divided into
mm... don't you agree? > Your point being what? That the
archaeologists picked>the points so that the side lengths were
as close as possible to >even multiples of Egyptian measures?
Such a process doesn't >magically eliminate the uncertainty 
in
the choice of endpoints.>Archaeologists can only estimate the
lengths of Egyptian standards>of measure to witn a small
margin of error. Ts error propagates>just like every other. On
the scale of the Great Pyramid you>bet that's going to be
sometng big enough to cause trouble!> Look, ts all boils down
to you not wanting to incorporate>the uncertainty values into
your calculations. I don't know if you>are just lazy, or
terrified that you'll do it wrong and make a 
fool>of yourself
again. But it's not worth making a bigger fool of >yourself 
by
using every possible dodge to claim that uncertainties>don't
matter. You say that Cole's survey was accurate to 1 part>in
100000, and still haven't actually given any evidence for
ts>(apart from saying that that level of accuracy is
typical!); you say>that the big uncertainties that Cole *did*
publish don't count, >because they were uncertainties in 
where
to measure FROM, wch >magically go away in your universe; You
have said that the GPOG's>dimensions were measured to better
accuracy than the pyramid itself>could possibly *HAVE*,
claiming that corrections were made to>fix ts small problem. >
Having taught and tutored mathematics, I can say that I
have>seen people go to great lengths to avoid doing a little
math>(Me: You're in a dark room. There's a 
dresser here with
10 red> socks and 10 blue socks... Student: I turn on the
light. Me:> It's burnt out. Student: I get a ßashlight. 
Me:
You don't have> a ßashlight. Student: Then I 
don't want to
wear any socks)> but ts is ridiculous. Redo your calculations
showing actual>margins of error, or tell me where I can find a
published work >where someone does.> Okay. But that means that
it doesn't have a well-defined>length all the 
way down to, say,
a tenth of a millimeter. Its >actual length ßuctuates by a
tiny amount over time, and you can't>>The size of the
individual blocks varies with temperature, not time.> Stevie,
temperature varies with time. I just said that.>Maybe not much
at all in the King's chamber, but certainly over the>entire
pyramid.> You've claimed ts a number of times, that the
pyramid>only expands and contracts as much as a single block.
Do you have>any evidence for ts whatsoever?>>It's a standard
principle of structural engineering. Each block>>expands and
contracts independently of the other blocks. The formula >>is
in the back of the steel book. The coefficient of expansion
for>>Limestone is.00042 Fahrenheit; wch is the change in
length per >>unit of length per degree of temperature.> But
you're saying that the entire pyramid only expands
and>contracts as much as a single block. Wouldn't the
expansion of each>individual block add up to sometng more than
===
that? Subject: re:about neglecting small terms in
deposition comes indiscrete units (you can't have a fraction
===
calculus> all,> the question may appear vulguar or stupid but
let me assure that its> my genuine confusion or doubt.>
calculus is notng but the rate of change of a function with
the> variable when the variable approaches to zero.and wle
calculating it> we often eliminate the smaller term..> eg. let
y=x**3;> the > y + dy = x**3 + 3 x **2 dx + 3x dx**2 + dx **3>
so we neglect dx**2 and dx **3 term> hence dy/dx = 3x**2.> i
understand it mechanically but i am not convinced why to
neglect the> above two terms however small they may be..Tnk
about it ts way:What are you doing when when you find the rate
of change? How is therate of change, or dy/dx defined? I bet
it's defined to be the slopeof the best linear 
approximation,
and that's precisely what you aredoing when you neglect the
===
then , as long there is no conßict with :>Re(e^[iPi]) =
Re(-1+i[0]) = -1 >>That has never been in doubt. Neither has
the observation that>>Im(e^[i pi]) = Im(-1+i[0]) = 0.>>The
statement that caused all the problem was the statement
that>>e^[i pi] = 0.>Actualy stated e^[ipi]=i0=0>>Wch is
exactly the same statement in the long run. And I have
not>>yet seen you retract the statement, as you should have by
now.>[REF:> A) e^[ipi]=-1 the real part solution and > B)
e^[ipi]=i[0] , or e^[ipi]=0 the imaginary part
solutio.]>>Arguments wch you have now acknowledged as invalid
(they are either>>both valid or both invalid, and the false
conclusion in B from the >>true premise that exp[i pi] =
-1+i[0] demonstrates that the argument>>used in B is
invalid).>> McAnally>The following was agreed:>Re(e^[iPi]) =
Re(-1+i[0]) = -1 AND>Im(e^[i pi]) = Im(-1+i[0]) =
0.True.>However,let me point out that ts theory of>complex
notation>being ,admitedly ,helpful for solving >complicated
engineeringIt is FAR MORE IMPORTANT than just being helpfulfor
solving complicated engineering. And it isnot just a theory of
complex notation. It isa theory of complex NUMBERS. You appear
to be drastically underestimating the importance ofcomplex
numbers.> [not a panacea however ]problems>[taking care simply
angles of unity]>in engineering it should not interfere with
>Eucledian Geometry.>The Classical Plosophers were very strict
,in >specifying that the only tools that should be allowed >to
be used,in solvinf Geometric Problems ,should be
the>ungraduated straight edge and the compass [maximum].Ts is
just a completely irrelevant comment wch looks designed to
stop you having to retract your previous statement that exp[i
pi] = 0.Just what do you tnk the relevance of your
statementabove to complex numbers is?>I, shall add ts
too.>When the resultant of the two components of the
complex>number is calculated ,then -i , and +i are ignored
.>[I, would rather say that only +i is ignored ]So what? What
relevance does ts have to do with the fact the you should
retract your statement that exp[i pi] = 0?Why don't you just
retract your statement that exp[i pi] = 0, and be done with
it? McAnally Despite anytng you may have heard to the
contrary, the rain in Spain stays almost invariably in the
===
anticlassicalist }{ i: linguistic negation)> Shouldn't that 
be
G.9fnther von Arschloch?> Wow, to be insulted by THE authority
in insults mself ! I feel> honoured, really. It's not a 
great
deal of course, I just got two> assholes and one emetic but
it's a start after all. How does one> get promoted to 
sometng
more fancy, more sopsticated?You will have to become more
sopsticated to deserve more advancedverbal abuse. But so that
you won't feel too bad, I've changed your_nom 
de guerre_ from
G.9fnther von Knakspott to G.9fnther von Kackpott. For those
of you who have not acquired colloquial German
(likemonolingual snippy little bitch Petey Daniels), it means
chamber pot;literally, st pot. It kind of fits your primitive
personality.[...]> Er... on second thoughts... we might have
here a pot-bellied old> geezer with snot in s moustache, half
s breakfast still on> s srt front (don't egg yolks make for 
a
lovely colour?)...> That sounds like G.9fnther.> No buddy
no.You need to learn when to use commas: No, buddy, no.> I
don't mean to start a ßame war against you, you are> the
expert in ts tngs, but take a look at yourself at:>
http://www.sonic.net/maledicta/aman.htmlSuch a cute bugger,
eh? Bugger in the non-Daniels meaning,_bien s.9er_.> and
picture yourself in a ragged yellowed t-srt and with> a snotty
moustache, it fits the description perfectly.Have I mentioned
that you're primitive? All right, make thattroglodytic. I
don't wear ragged yellowed T-srts; my neat moustacheis 
always
snot-free; and I have no potbelly.> In my view, satire and
ridicule is the most effective way> of dealing with s, her, or
its, kind. And satyre too.> Vive l'esprit fran.8dais!> Would
you talk metaphysics with a broken 78rpm record?> Or even...
strawberry-pie recipes, or baby-bum wiping> techniques? I
don't.> Mais pourquoi parlez-vous avec un trou du cul comme
G.9fnther?> There you goComma, dammit, comma!> ladies and
gentlemen, thatsApostrophe, dammit.> your run of the
millHyphens, dammit, as run-of-the-mill is used adjectivally.>
kraut naziAs a Nazi Kraut, I'd prefer seeing these words
capitalized. (Don't everlet bitchy Petey see you mangle your
acquired language! He'll rip youapart like a mouth-foaming
affenpinscher with a crusty tampon up s ass.)>
sczophreniaComma, dammit, comma!> if you mark my meaning. One
second he is celebrating> the libertarian frenchFrench>
spirit, the next he is censoring s frenchDitto. Vide supra.>
chum and questioning s actions.Well, Herr von Kackpott,
there's notng wrong with asking M. Guy why helowers mself to
talking with a Cro-Magnon like you. I mean, you'res very
antithesis: he's brilliant, extremely learned,
andfantastically witty -- whereas you are dull you. M. Guy has
explainedwhy; it has sometng to do with _cer_ and
intellectually slumming, Iunderstand. Or, perhaps, as we say
in Bavaria, it's a matter of _lesextr.8fmes se touchent_.>
What was that tng about the first amendmentCaps, cabr.97n!
First Amendment.> on your homepage?> Since I don't even
understand what bug bit you,> I respectfully bid my farewell
for now.It's notng personally, Herr von Kackpott. I just
didn't take myanti-misanthropy pills ts week and see all 
these
assholes in variouslanguage newsgroups; those who strike me as
being particularly assholeyget a swift kick in the keister.>
===
Advice for future math majors?I'm a gh school senior 
planning
on studying pure math as a majorts fall in college. What I'm
wondering is, what kind of advicecould those of you who have
already been through the experience offerto people in my
position? What kind of background is necessary (orrecommended)
to begin college mathematics courses, and what could afuture
math major do in order to get a head start? Are there anybooks
or other information sources that would be helpful? Any
===
math majors?I completed a self-taught college calculus course
when JR in HS (got it fromthe teacher), did it outside of
class at home, it got me way ahead, scoredin top 1% of math
SAT. Just take enough contact time to be with thematerial, and
defer other time for HS social stuff till later. Math leadsto a
lot of places too, engineering, sciences, physics, can't 
loose
withMath.///////////////////////////////////////////> I'm a 
gh
school senior planning on studying pure math as a major> ts
fall in college. What I'm wondering is, what kind of advice>
could those of you who have already been through the
experience offer> to people in my position? What kind of
background is necessary (or> recommended) to begin college
mathematics courses, and what could a> future math major do in
order to get a head start? Are there any> books or other
information sources that would be helpful? Any advice> is
===
majors?> I completed a self-taught college calculus course
when JR in HS (got itfrom> the teacher), did it outside of
class at home, it got me way ahead, scored> in top 1% of math
SAT. Just take enough contact time to be with the> material,
and defer other time for HS social stuff till later. Math
leads> to a lot of places too, engineering, sciences, physics,
can't loose with> Math.Deferred the Spelling classes,
===
school senior planning on studying pure math as a major> ts
fall in college. What I'm wondering is, what kind of advice>
could those of you who have already been through the
experience offer> to people in my position? What kind of
background is necessary (or> recommended) to begin college
mathematics courses, and what could a> future math major do in
order to get a head start? Are there any> books or other
information sources that would be helpful? Any advice> is
greatly appreciated.If you haven't already, get some of the
easier topics out of the way bystudying on your own: linear
algebra, probability, number theory, settheory, as much
calculus as you can stand. If you don't find 
these topicseasy,
===
majors?>>I'm a gh school senior planning on studying pure
math as a major>>ts fall in college. What I'm wondering is,
what kind of advice>>could those of you who have already been
through the experience offer>>to people in my position? What
kind of background is necessary (or>>recommended) to begin
college mathematics courses, and what could a>>future math
major do in order to get a head start? Are there any>>books or
other information sources that would be helpful? Any advice>>is
greatly appreciated. >If you haven't already, get some of 
the
easier topics out of the way by>studying on your own: linear
algebra, probability, number theory, set>theory, as much
calculus as you can stand. If you don't find 
these topics>easy,
change to accounting.My advice is to ignore above advice,
particularly the last sentence.You have already completed most
of gh school. If you really like mathematics, you *can* do some
reading on your own (have you looked at Courant and Robbins,
_What_is_ Mathematics?_ ?), but you will see when you get
there if you need to fill any holes in your background. I tnk
the existence of such holes is rare on the undergraduate level
for people who already have a pronounced interest in math.Also,
you may find that your exposure to other fields 
(e.g.,
engineering, economics, or statistics) wch use mathematics is
quite limited and that you are really interested in the
application of mathematics in these fields. Keep an open mind 
-
you don't have to declare a major upon entry.-- Stephen J.
===
for future math majors?Stephen J. Herschkorn>I'm a gh school
senior planning on studying pure math as a major>ts fall in
college. What I'm wondering is, what kind of advice>could
those of you who have already been through the experience
offer>to people in my position? What kind of background is
necessary (or>recommended) to begin college mathematics
courses, and what could a>future math major do in order to get
a head start? Are there any>books or other information sources
that would be helpful? Any advice>is greatly appreciated.If
you haven't already, get some of the easier topics out of 
the
way by>studying on your own: linear algebra, probability,
number theory, set>theory, as much calculus as you can stand.
If you don't find thesetopics>easy, change to 
accounting. My
advice is to ignore above advice, particularly the last
sentence.mathematical vocation (from the Latin for calling or
summons) arealways learning in advance of the curriculum.> You
have already completed most of gh school. If you really like>
mathematics, you *can* do some reading on your own (have you
looked at> Courant and Robbins, _What_is_ Mathematics?_ ?),
but you will see when> you get there if you need to fill any
holes in your background. I tnk> the existence of such holes
is rare on the undergraduate level for> people who already
have a pronounced interest in math.> Also, you may find that
your exposure to other fields (e.g.,> engineering, economics,
or statistics) wch use mathematics is quite> limited and that
you are really interested in the application of> mathematics
in these fields. Keep an open mind - you don't 
have to> declare
a major upon entry.I agree with SJH ts time. And even if 
you're
sure your major will be inmath, don't be too quick to
concentrate your courseload in math and its nearrelatives. In
university you get a chance to learn some story, a languageor
two, and so forth, from people who have devoted their lives to
thosesubjects. You won't meet many such people later on,
whereas you can alwaysbring yourself up to speed in some
mathematical specialty that you omittedat school.Just my 2
===
fi[CapitalEth]zmachava@hotmail.com (Achava
Nakhash, the Loving Snake)n264G> Ts is a skew-symmetric
matrix. It has the property that all eigenvalues> are purely
imaginary.> As an added nt, you might take note that iA is
Hermitian, and> Hermitian matrices necessarily have
non-negative eigenvalues. What is>
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Let A be nxn Hermitian maatrix.> If c is an eigenvalue of A,
=> there is x != 0 such that Ax=cx.> c<x,x> = <cx,x>= <Ax,x> =
<x,A*x> = <x,Ax> = <x,cx> = conjugate(c)<x,x => c =
conjugate(c).> => c is real.> Hou do you show that c is
non-negative?> You don't, because it's false.> 
Example: A =
[-1].> I tnk Achava is confused with Ôpositive 
definite'.>
WilbertWilbert, Achava is not confused with positive definite.
He is justconfused. However, the fact that iA has real
eigenvalues shows that Ahas purely imaginary eigenvalues.
Since they are the solutions ofpolynomials with real
coefficients, they must come in conjugate pairs. Hence their
product is non-negative, and that solves the problem asChapman
was nting originally. Sorry for the
===
the position of 4 points on a 2D plane. The points
areunequally spaced.Is there anyway I could fit an ellipse (or
any other circular shape)to these points (it has to pass
through the 4 points)?FYI, ts is for an image processing
algorithm. I have tried to useHermite Interpolation, but I
can't seem to find a way to get thetangent 
values at each of
the 4 points so the curve looks like anellipse/circle.Any help
===
from 4 pointsI believe that there are an infinite number of
elipses that go throughfour points. For a circle, three points
are sufficient since thereare only three independent variables
required to define a circle in aplane. (The x and y coordinate
of the center and the diameter define acircle.) For an elipse,
there are two more independent variables, soit should take 
five
points to define the elipse. (An elipse has twofoci instead of
one center.) You need to define an additional limit onthe
elipse in order to get a solution.> I have the position of 4
points on a 2D plane. The points are> unequally spaced.> Is
there anyway I could fit an ellipse (or any other circular
shape)> to these points (it has to pass through the 4
points)?> FYI, ts is for an image processing algorithm. I have
tried to use> Hermite Interpolation, but I can't seem to 
find a
way to get the> tangent values at each of the 4 points so the
curve looks like an> ellipse/circle.> Any help would be
===
pointsGenerally Noa single circle requires 3 points, and
ellipse requires 3 also.with 4 points you can have a set of 3
such circles and average between themto smooth itsame with
ellipses.Cant run a circle/ellipse through 4 arbitrary points,
as after the firstthree are picked the circle is 
defined, and I
can place the 4th far away orinside the edge, and it is not on
the edge.Or if you are trying to get one circle/ellipse near
all the points use anerror squared approach to distance from
each point to the circle/ellipseedge and minimize it.> I have
the position of 4 points on a 2D plane. The points are>
unequally spaced.> Is there anyway I could fit an ellipse (or
any other circular shape)> to these points (it has to pass
through the 4 points)?> FYI, ts is for an image processing
algorithm. I have tried to use> Hermite Interpolation, but I
can't seem to find a way to get the> tangent 
values at each of
the 4 points so the curve looks like an> ellipse/circle.> Any
===
ellipse from 4 points> I have the position of 4 points on a 2D
plane. The points are> unequally spaced.> Is there anyway I
could fit an ellipse (or any other circular shape)> to these
points (it has to pass through the 4 points)?> FYI, ts is for
an image processing algorithm. I have tried to use> Hermite
Interpolation, but I can't seem to find a way to 
get the>
tangent values at each of the 4 points so the curve looks like
an> ellipse/circle.> Any help would be appreciated greatly. >
,> TejasYou can put a conic through any _five_ points
(admittedly sometimes adegenerate conic). The idea is to take
the equation Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,plug in your
n points, and get n linear equations for A,B,C,D,E,F. Sofor
four more-or-less random points in the plane, you'll get
4homogeneous equations for the six variables, yielding a
onedimensional family of solutions.These will be (usually)
either ellipses or hyperbolas. Since you wantan ellipse,
you'll need to check wch of the equations in yourfamily, if
any, are ellipses. You can write down a general condition,or
simply complete the squares to get an equation of the form
aX^2 + bY^2 = c.If a,b,c all have the same sign, voila, an
ellipse. If not, you have ahyperbola. (You won't end up with
the empty set from an equation likeX^2+Y^2=-1, since you
===
the position of 4 points on a 2D plane.... (The usual kind of
plane. :-)> Is there any way I could fit an ellipse ....> to
pass through the 4 points?.... Yes. I'll outline a method
(pure algebra, no calculus), then go through an example. In
general, 5 points in a plane determine a unique conic (ellipse
or hyperbola or whatever). Specifying only 4 points allows you
an infinite family called a pencil of conics. If you know the
equations of any two members of the pencil, in the form S = 0
and S' = 0, then all the others have equations S + 
tS' = 0
where t can be any real number. For the two basic conics 
it's
a good idea to use line-pairs (degenerate conics), because
their equations are easy to find. For a simple example 
I'll
take the 4 given points to be (0,0), (1,0), (0,1), (3,1). Sort
them into two pairs, say {(0,0), (1,0)} and {(0,1), (3,1)}. The
line through the first pair is y = 0, and the line through the
second pair is y - 1 = 0, so these two lines together make up
the degenerate conic y(y - 1) = 0. Similarly, the line through
{(0,0), (0,1)} and the line through {(1,0), (3,1)} make up the
line-pair x(x - 2y - 1) = 0. Similarly, the line through
{(0,0), (3,1)} and the line through {(1,0), (0,1)} make up the
line-pair (x - 3y)(x + y - 1) = 0. I've calculated all three
line-pairs to show you the idea, but you need only two of
them. I'll choose the two simplest.x(x - 2y - 1) = 0 can be
the conic S = 0, andy(y - 1) = 0 can be the conic S' = 
0.Then
all other conics through the four points have equationsx(x -
2y - 1) + ty(y - 1) = 0 for various real numbers t.Multiplying
out givesx^2 - 2xy + t(y^2) - x - ty = 0. (*) Various values of
t give conics of various kinds, but you're interested in
ellipses only. The condition for a conica(x^2) + 2hxy + b(y^2)
+ 2gx + 2fy + c = 0to be an ellipse is ab - h^2 > 0. In our
example (*), a = 1, b = t, h = -1, soab - h^2 = t - 1. Hence
every real number t > 1 makes (*) the equation of an ellipse
through the original 4 points. In cases wch interest you, you
should be able to use the same method to find 
infinitely many
ellipses through your four points. Among all those you can
choose wch you like best, perhaps by imposing some other
===
4 points on a 2D plane....> (The usual kind of plane. :-)> Is
there any way I could fit an ellipse ....> to pass through the
4 points?....> Yes....> In cases wch interest you, you should
be able to use the same > method to find 
infinitely many
ellipses through your four points.... Since Lynn Kurtz and
Virgil have raised the issue, I'll add that if no ellipses 
are
possible, then the inequality for t will have no real
solutions. I didn't mention ts complication before, partly 
to
avoid making the explanation even longer, and partly because I
guessed that Tejas (the OP) would be starting from feasible
sets of four points in s image-processing application. Hence
my words In cases wch interest you above. Ken
===
position of 4 points on a 2D plane. The points are> unequally
spaced.> Is there anyway I could fit an ellipse (or any other
circular shape)> to these points (it has to pass through the 4
points)?> FYI, ts is for an image processing algorithm. I have
tried to use> Hermite Interpolation, but I can't seem to 
find a
way to get the> tangent values at each of the 4 points so the
curve looks like an> ellipse/circle.> Any help would be
appreciated greatly. > ,> TejasIt is possible to put some
conic section through any 4 distinct points, but not always
ellipse.Consider the case where one point is inside the
triangle formed by the other 3. Since the interiors of
ellipses are convex, any ellipse through those 3 must contain
all but the vertices of that triangle as interior points, thus
there cannot be an ellipse through all 4 such
===
position of 4 points on a 2D plane. The points are>unequally
spaced.>Is there anyway I could fit an ellipse (or any other
circular shape)>to these points (it has to pass through the 4
points)?>FYI, ts is for an image processing algorithm. I have
tried to use>Hermite Interpolation, but I can't seem to 
find a
way to get the>tangent values at each of the 4 points so the
curve looks like an>ellipse/circle.>Any help would be
appreciated greatly. >,>TejasThat is clearly too much to hope
for. For example what if your fourpoints are the three
vertices and the centroid of an equilateraltriangle. You 
can't
even draw a free-hand approximation that is
===
all! Just wondering...(not homework) Supose we have a bag full
of numbered balls. 4 balls numbered 1, 4 balls> numbered 2 and
so on. So we have 4n balls. Now supose we extract balls from
the bag 1 by 1. How many balls do we have to extract so that
the probability of having> extracted 2 balls with the same
number is 50%? and please forgive my english! N.> Let x =
number of balls extracted. Then there will be 3*x balls>
remaining wch match the balls drawn.Not always true for x > 1.
Consider, for instance, the case where the firsttwo balls 
drawn
===
asked me a question, and since I know notng aboutWebTV, I
can't answer. The problem is that the person isinvalid and
connects to the outside world mostly throughposting to the
internet and her favorite USENET group insists that she post
in plain text. But she uses WebTVnow, wch by default uses some
sort of fancy text, andso the moderator won't pass her 
posts.So
the question is, does anyone know how to change thesettings on
WebTV so that postings to USENET are inplain text?(I'm 
helping
her over e-mail, so I can't do any experimentingon my
===
question is, does anyone know how to change the> settings on
WebTV so that postings to USENET are in> plain text?Post via
interface -- all it requires is an email address.-- My email
address has an extra @ (spell it out) and an extra invalid.
===
transcendentals than e and pi got my quriosity perched.> So
besides the obvious generalization to Lousville's
construction, whereone> has all the digits after the decimal
point equal to k except the digits at> positions n! wch might
equal m=/=k, are there any famous decimals thatare>
constructed by using digits taken from a particular sequence
a_n?I replied earlier about how the factorial base can
generate a lot oftranscendental numbers.There have been
several proofs, given in ts newsgroup, that every
rationalnumberhas a finite representation in base !..111...
(base !) = 1/2 + 1/6 + 1/24 + 120 + ... = e-2 is
transcendentalAnother product base that has ts property is the
root of prime powers.(Sloane A051451). Every rational has a
finite representation.The base series is 1, 2, 3, 2, 5, 7, 2,
3, 11, 13, 2, 17, 19, 23,...The product series is 1, 2, 6, 12,
60, 420, 840, 2520, 27720, 360360,720720, ....111... (base rpp)
= 1/2 + 1/6 + 1/12 + 1/60 + ... should be transcendental.A new
product base I have come up with is the roots of prime powers
ofprimes.Ts is similar to the series above, except numbers
like 2^4 and 2^6 are notincluded.The base series is 1, 2, 3,
2, 5, 7, 2, 3, 11, 13, 17, 19, 23, 25, ...The product series
is 1, 2, 6, 12, 60, 420, 840, 2520, 27720, 360360,6126120,
...Is .111... (base rppp) transcendental?Does every rational
number have a finite representation in ts base?If not, can
someone give an example of a rational number that has
aninfiniterepresentation.I tnk all of these series can be
turned into Lousville numbers:base ! -> .110001000000000...
(every n! position is 1).base rpp -> .11000100000100... (every
position in Sloane A051451 is 1).base rppp ->
===
coordinates on a unit sphere> If you circumscribe a cylinder
with radius r and height 2*r about the> sphere, then the
horizontal projection from the cylinder's lateral> surface
onto the sphere preserves area. Hence there is an>
area-preserving map from a rectangle with width 2*pi*r and
height 2*r> onto the surface of the sphere with radius r.>
There is still the objection that the mapping is not a
bijection, but it> does at least preserve area, if not
cardinality.Bijection - new word to me, but (after looking it
up) exactly what I meant. Cardinality however in ts context
doesn't make much sense to me... I'm only 
familiar with it's
use in terms of the size of a
set.http://mathworld.wolfram.com/CardinalNumber.htmldidn't
===
Re: Clean coordinates on a unit sphere>> If you circumscribe a
cylinder with radius r and height 2*r about the>> sphere, then
the horizontal projection from the cylinder's lateral>>
surface onto the sphere preserves area. Hence there is an>>
area-preserving map from a rectangle with width 2*pi*r and
height 2*r>> onto the surface of the sphere with radius
r.There is still the objection that the mapping is not a
bijection, but it>> does at least preserve area, if not
cardinality.> Bijection - new word to me, but (after looking
it up) exactly what I > meant. Cardinality however in ts
context doesn't make much sense to > me... I'm 
only familiar
with it's use in terms of the size of a set.>
http://mathworld.wolfram.com/CardinalNumber.html> didn't 
help
me much either. Can someone explain?You said that bijection
was exactly what you meant. Cardinality isdefined in terms of
bijections.You were the one who objected that the
representation of the poles isnon-unique. That is, the north
pole is represented by (x,pi/2) for any xin [0,2pi). So there
are infinitely many points on the edge of therectangle that 
all
map to a single point at the north pole. That is, themapping
from edge to pole is not a bijection, and in fact the sets
havedifferent sizes (cardinalities). The fact that cardinality
is is notpreserved by ts mapping is exactly what you were
objecting to.You could restrict the mapping to [0,2pi) x
(-pi,pi), and throw in thesingle points at (0, +/- pi). That
would make the mapping a bijection,but somehow I have the
feeling that it still won't satisfy you.-- Dave SeamanJudge
Yohn's mistakes revealed in Mumia Abu-Jamal
ruling.<http://www.commoncouragepress.com/index.cfm?action=
===
sphere> Your mistake is with your infinite tight winding idea.
I don't tnk that> the idea can be made rigorous enough to
work. Either it goes north by ZERO> from one turn to the
next... wch is not very useful. Or, it goes north by> some
positive amount... and we miss some points.If it goes north by
an infinitesimal amount, do we miss any 
points?I've also been
banging my head against the wall for the 2d analoge problem of
filling a rectangle with a single thread. Imagine the thread
starts in the top left corner, then wraps in a zig zag over
the page. It finally ends in the bottom right. The parameter
value at the top left is 0, bottom right 1. Centre of the
rectangle is 0.5. In fact, all points corresponding to 1/(a*2
+ b*2^2 + c*2^3 + ...) where a, b, c... are either 0 or 1, lie
on the diagonal line.Shouldn't there then be a mapping that
associates each real number with a 2d coordinate? Is ts
effectively the same as interleaving the two parameters of
cartesian coordinates into a single parameter (or
concatenating, or any other jumbling of digits)? If yes,
===
However, if you _really_ want to do it with just a single
parameter... just> do it with two parameters somehow... and
then interleave their decimal> digits. So that a point such as
(2.345678..., 0.987654....) is mapped to the> number
20.394857667584...:) I was tnking about interleaving too
(doesn't necessarily have to be base 10 of course). It made 
me
realize that striving towards using less parameters is probably
useless if all you are doing is combining the information of a
gher dimensional representation (there's no real reduction 
of
information happening). So I'm tending to believe the nature
of my ideal solution would be _very_ different from the
===
on a unit sphere> - I'm a computer science student working 
in
the field of vision and> grapcs. I'm looking for 
a clean way to
represent a _direction_ in 3d> space (ts can also be imagined
as a position on a unit sphere).> I don't like the 
traditional
methods of using a 3d vector or spherical> coordinates for the
following reasons:> - 3d Vector e.g. (x, y, z): Ts is of
course very common - can be> rotated with matrices,
quaternions... but it also describes length,> wch I don't
need.> - Spherical coordinates e.g. (theta, p): Ts is more
appropriate as> it does not describe length. But it still
appears ugly to me since> multiple coordinate pairs can
describe the same point e.g. (x, pi/2).> Is there a better
===
coordinates on a unit sphere<snip>Is there a better way? >
Direction cosines?After looking up the term... direction
cosines are just the components of the normalized vector, so
that's not what I'm after. As JHS pointed out, 
only two of the
components are really necessary, and they definitely 
don't meet
my (vaguely defined) density criterion.But for your
===
specific properties> Generate a series of pseudo-random 
numbers.
Then use a filter to throw out> those that do not satisfy your
condition.>Is ts considered the standard general approach to
creating data with>arbitrary parameters -- the one to fall
back on when there isn't a>more direct generator algorithm?Are there other such general approaches, or is ts one by far
the>best (again, in cases where there isn't a more direct
approach)?> One should use random numbers if possible. But the
> acceptance-rejection process is not what is stated
above.Sorry, but now I'm a bit more confused. The poster I
responded tosuggested that I generate random numbers then
filter them to create adesired distribution. 
You're telling me,
no, use random numbers, butI don't know what that means in 
ts
context, nor do I understand whatyou're saying about the
acceptance-rejection process.How does one filter s way to a
random data set with desiredstatistical properties? Could
someone provide a bit more detailregarding whether ts should
===
QuestionLet S be the real numbers (0,1)>Since it is a set of
real numbers it is partially ordered and everychain
is>obviously bounded by 1>Yet (0,1) does not have a maximal
element.What am I missing here?The chain (0,1), for example,
does *not* have a least upper bound *in S.*> More relevant to
a question about Zorn's lemma is the fact that (0,1)> does 
not
have an upper bound in S.I don't know if ts is relevant to 
the
original poster's reason forconfusion, but in 
Kaplansky's _Set
Theory and Metric Spaces_ (1972), hedefines 
Zorn's Lemma asLet L
be a partially ordered set in wch every chain has an upper
bound.Then L contains a maximal element.Our teacher instructed
us to write in the word nonempty before partiallyordered, and
in L after upper bound. To be fair, Kaplansky notes ins
remarks on the lemma that the upper bound must be in
===
Lebesgue measure on R (reals) and let p be in (1, infinity).
Construct a function that is in L^p(mu) but not in L^q(mu) for
===
Re: Fundamental group of the circle - question on uniqueness
of liftsFirst, why does ~F(1,t) lift the constant path at x0?
All b) specifies> is>that F(0,t) = x0, not that F(1,t) = x0 as
well. Assume F(1,t) = x2, for>arguments sake.>I looked over
the definitions in Hatcher (the book you are using).Homotopies
of paths preserve endpoints. Thus, F(1,t) is constant, sayx2.
Now, ~F(1,t) is a lift of F(1,t). Another lift is going to be
theconstant map at ~F(1,0). By uniqueness, ~F(1,t) must be a
constantmap. I agree that F(1,t) need not be the constant map
at x0. Hatcherseems to be in error at ts point, but for what
he wants to prove,he's OK.> Thus, t->po~F(0,t) is the 
constant
path at x0., I.e. ~F(0,t) lifts> the constant path at x0. Do
you know a path that *also* lifts the> constant path at x0?
What does uniqueness then tell you?>> - I agree F(0,t) =
p(~F(0,t)) = x0. Ts is because I see that b) directly> defines
F and ~F ts way.> - By uniqueness, ts other path must be equal
to ~F(0,t) then. I just> haven't gotten to the proof of that
yet.Do you see that ts other path must be the constant one at
~x0?Clearly ts is a lift of the constant path at x0. By
uniqueness,~F(0,t) must be the constant path at ~x0, hence
constant.> As for ~F(1,t), are you sure that F isn't a
homotopy of *loops* at x0> rather than of paths starting at
x0?>> I'm pretty sure it is, but I can't seem 
to see how it's
implied. b) defines> F simply as a homotopy of paths starting
at x0. So in order to prove b)> from c), one must show c)
implies b) works for more than just F's that are> homotopies
of loops. Perhaps the other properties of b) force F to be a>
loop homotopy? Check out page 29, Section 1.1, theorem 1.7, of
the> following text for the full treatment. Maybe (probably) I
have left some> a) b) and c):> a) For each path f: I -> S^1
starting at a point x0 in S^1 and each ~x0 in> p^(-1)(x0)
there is a unique lift ~f : I -> R starting at ~x0.> b) For
each homotopy f_t : I -> S^1 of paths starting at x0 and each
~x0 in> p^(-1)(x0) there is a unique lifted homotopy ~f_t : I
-> R of paths starting> at ~x0.> c) Given a map F : Y x I ->
S^1 and a map ~F : Y x {0} -> R lifting> F|Y x {0}, then there
is a unique map ~F: Y x I -> R lifting F and> restricting to
the given ~F on Y x {0}.It seems to me that you have been
missing that a constant map will bea lift of a constant map.
Uniquess will say that it is the *only* liftsubject to certain
===
Interesting problem>>I'm still not convinced that a 
non-empty
A can exist...>Neither am I. But I'm also not convinced that
it can't exist.>Robert Israel israel@math.ubc.ca>Department 
of
Mathematics http://www.math.ubc.ca/~israel >University of
British Columbia >Vancouver, BC, Canada V6T 1Z2I'm not
proposing an answer here, just asking a question. What
goeswrong if A is the set of all transcendentals and B is
===
>I'm still not convinced that a non-empty A can exist...>Neither am I. But I'm also not convinced that it 
can't
exist.>>I'm not proposing an answer here, just asking a
question. What goes>wrong if A is the set of all
transcendentals and B is everytng else?I believe you lost
sight of the original problem:>A and B are two disjoint sets
whose union is |R+. Both A and B are closed>under sum and
multiplication. Is it possible that neither A nor B is
the>VOID set?-- Stephen J. Herschkorn
===
stuck-togetherness> So the earth's surface is about
one-quarter land. That land could> be distributed across the
surface of the globe in different ways -> I can imagine at one
extreme, all the land stuck together in one> roundish
continent, and at the other extreme, all the land spread> out
evenly, in equally-spaced tiny islands. Is there a measure,>
preferably just one number, that captures these differences?>
If so, how is it calculated?One measure of fragmentation is
entropy from information theory:H = -Sum(p log(p)) where p is
the fraction of land area in each part.But ts doesn't 
consider
the positions of the land areas.For another possibility, 
define
a function over the sphere to be onefor land and zero for
water, then compute the spectrum in terms ofspherical
harmonics. The zeroeth order term is the amount of land.There
are 2n+1 modes of order n. Their oscillation frequency
squaredis proportional to n(n+1). Terms of the same order can
be combinedby summing their squares. Summing the squares of
all terms accountsfor all of the variance of the original
===
distance between two points uniformly and > independently
distributed over the land area.That would be E dist(x,y) in my
earlier posting.> For the more general case, somehow add in:>
The total land area.> The radius of the earth.For distance
measures like ts, it would make sense to form a ratio by
dividing by its maximum possible value wrt to a given total
proportion of land area. I tnk the maximum value occurs if the
total land area is divided equally between two polar
caps.ObPuzzle:If x,y are iid uniform on a two-polar-cap region
thatcovers a total proportion P of the surface of a sphereof
radius R, what is the value of E dist(x,y)? Is ts the maximum
value of E dist(x,y) among all the possible ways to distribute
===
does Inverse of a Matrix mean?>I have been using matrix
formulas for calculating regression coefficients>etc.>It is
simple to understand what we are doing when we add,
multiply,>transpose a matrix (in terms of the underlying
data). However, what exactly>are we doing when we take the
inverse of a matrix? Can someone explain in>simple terms - if
possible with a practical example.>Suppose you have n
equations in n unknowns. In matrix form:AX = B where A is the
nxn matrix of coefficients, X is the columnvector of unknowns,
and B is a column vector of constants.Given that the inverse
inv(A) has the property that inv(A)*A = I, theidentity, you
can multiply your system by inv(A) to get:inv(A)*(A*X) =
(inv(A)*A)*X = I*X = X on the left side and inv(A)*B onthe
right side soX = inv(A) * B, solving the system of equations.
It is similar to solving ax = b by multiplying both sides by
a^(-1) =1/a to get x = a^(-1)b and that is why inv(A) is
called what it is.The above assumes, of course, that A has an
inverse and uses the factthat matrix multiplication is
===
magic e-voting booth/bus & the symm. of the swastika)the
literature from the tourbus on campus says thatthe filmed hype
(wch I missed, along with the touchscreen demos)includes Mel
son, the Liberal Media's favourite British Israeliterunning
what is hopefully s very last temptation --the LAtribcoTimes
has been adoring m for months -- beforehe makes the final
sacrifice. that's Mel and s Dawg,Tops Ltd., 
ak.a. Jesus H.
Whom? another key backer of ts tripe is Norman Lear,who we
know from MoveOn's phoney primary,wishes to obliterate the
fact that LaRouche is on the ballot,here in California (and
probably 35 states en toto; more, ifyou wake up and smell the
fungus .-) aside from Swift's clinical 
definition of the main
sponsor's name,in _Gulliver's Travels_, the 
driver has a
British ßag plate,not affixed to the bumper but in the 
window!
its rotational symmetry, although dystorted becausethe ßag is
not a square, is the very same (accountingfor the mirrored
image, if req'd). it may be happenstance:the application of
the colour of one of the 3 consitutents,the red cross (of St.
(Bruce?); Scotland, andfor Betty Dos' lineage) was sinmply
applied with a skew,over the wte cross (of St. Geroge; #1 of
England?)to differentiate it. (the vertical (red?) cross is
St. Andrew's;don't ask me why he was so God-am 
holy, buthe
must represent Wales ... and the Pretty Scarab Beetles,Harry
Potter's Lonely Hearts military-marcng band .-) perhaps, the
driver is hoping to be ka-nighted,as a striving member of the
Royal Order of Bards,Druids and Ovates, or the first
degreeabove freezing in freemasonry (in s free time,he would
preside over faery-dances to make cropcircles .-)they're
hauling it to USC, i guess for tomorrow, andonly a total of 18
===
campuses -- Subject: Re: Saxon Math critiquemon Drieux! I
should have known,what a dysservice I have done, all these
yearsof going to schoolboard meetings,without investigating ts
Scholastic ****. maybe they can get Dame Jo and Company [*] to
investin Agnlosaxon Math Futures!--* Industrial Smoke &
Mirrors, Ltd. Parents Left Bend, Inc. Mel son and s Dawg, Tops
Corp.>unusual punishment. I finally took an entire week-end 
and
edited the idiot>tng's exercises to remove the problems 
based
on lessons about a hundred>pages back from that set of
problems.>Now, don't mistake me: As a mathematician, I was
handed a shock the like>of wch I'd not felt since 
ÔNam when I
first ran into that--Give Earth a Trickier Dick Cheeny --
OBNOXICO, out of
office?http://www.benfranklinbooks.com/http://www.rand.org/
publications/randreview/issues/rr.12.00/http://
===
#1,I have a scholarsp exam in less than 2 weeks and need help
with somepast paper questions. The questions are more logic
questions thanmath. Any help will be appreciated. I do not
only want the answers,but also the explanations so that I can
understand it myself. -----Poy Rott, the great Thai detective,
receives a call from InspectorGoofball. Mr. Neil Diamond, the
manager of Timpani's, the famousjewelry store, had reported
that he had had a massive robbery duringthat morning. The
police had rounded up the only three people who hadcome in for
the day, Messrs Agassi, Butler and Cooper.Agassi is blind, and
would have had to have an accomplice. But he is aknown crook
who would never work with more than one person.Butler swears
that Cooper is innocent, so if Butler is innocent, thenso is
Cooper.The police have found out that if exactly two of them
are guilty,Agassi would have to be one of them.They also know
that if Cooper is innocent, Butler is innocent.Poy Rott turns,
and points to.....(A) Agassi (B) Butler (C) Cooper (D) Agassi
and Cooper (E) Agassi and Butler (F) Inspector Goofball (G)
===
a scholarsp exam in less than 2 weeks and need help with some>
past paper questions. The questions are more logic questions
than> math. Any help will be appreciated. I do not only want
the answers,> but also the explanations so that I can
understand it myself. > -----> Poy Rott, the great Thai
detective, receives a call from Inspector> Goofball. Mr. Neil
Diamond, the manager of Timpani's, the famous> jewelry 
store,
had reported that he had had a massive robbery during> that
morning. The police had rounded up the only three people who
had> come in for the day, Messrs Agassi, Butler and Cooper.>
Agassi is blind, and would have had to have an accomplice. But
he is a> known crook who would never work with more than one
person.> Butler swears that Cooper is innocent, so if Butler
is innocent, then> so is Cooper.> The police have found out
that if exactly two of them are guilty,> Agassi would have to
be one of them.> They also know that if Cooper is innocent,
Butler is innocent.How do they know that?> Poy Rott turns, and
points to.....> (A) Agassi (B) Butler (C) Cooper (D) Agassi and
Cooper> (E) Agassi and Butler (F) Inspector Goofball (G) Neil
===
across the following questions that I'm having trouble
cracking:1. Let f,g be in L^p. Let ||_p be the L^p norm. How
can I show that|f+g|_p = |f|_p + |g|_p only if f or g are zero
almost everywhere or theyare multiples of each other? I really
don't know where to start.2. Let f be in 
L^infinity and L^p for
some finite p. I'm trying to showthat lim as p 
-> infinity of
|f|_p = |f|_infinity. I can show thatlim sup |f|_p <=
|f|_infinityso all I need now is that lim inf |f|_p >=
|f|_infinity. Is there a quickway to do ts? Should I be tnking
about ts in another way?3. Ts one uses Lebesgue measure on R.
Let p be in (1, infinity). Find afunction f in L^p but not in
L^q for q in [1,infinity) not equal to p. 
I'mhaving trouble
even believing that ts is possible. For some function f,if
|f(x)| > 1 for all x, then |f|^q <= |f|^p for q <= p so if f
is in L^p,then it will be in L^q, right? So the function 
we're
looking for must beone whose absolute value jumps back and
forth above 1 and below 1 Iconclude. I also know that it 
can't
be essentially bounded, right? Anynts on what the function f
===
Re: Farey series= equivalent statement to Riemann Hypothesis
Huh?> In various recent popular mathematics books they write
of the equivalent> statement to RH that relates to the
difference in the order of the fractions> in the farey series
and the ordered value of the fractions.> Could someone
translate ts?At
http://www.maths.ex.ac.uk/~mwatkins/zeta/
RHreformulations.htmit saysThe Riemann Hypothesis can also be
reformulated in terms of a problem involving Farey sequences.
Ts is dealt with in the following:J. Franel, Les suites de
Farey et les problemes des nombres premiers. Gottinger
Nachrichten, 198-201 (1924)E. Landau, Bemerkungen zu der
vorstehenden Abhandlung von Herrn Franel. Gottinger
Nachrichten, 202-206 (1924)A. Fujii, A remark on the Riemann
hypothesis. Comment. Math. Univ. St. Pauli 29 (1980),
195-201A. Fujii, Some explicit formulae in the theory of
numbers. A remark on the Riemann Hypothesis. Proc. Japan
Acad., Ser. A 57(1981), 326-330S. Kanemitsu, and M. Yosmoto,
Farey series and the Riemann hypothesis. Acta Arith. 75
(1996), no. 4, 351-374S. Kanemitsu and M. Yosmoto, Farey
series and the Riemann hypothesis. III. Ramanujan J. 1 (1997),
no. 4, 363-378J. Kopriva, Contribution to the relation of the
Farey series to the Riemann hypothesis on the zeros of the
zeta function (Czech), Casopis Pest. Mat. {bf 78} (1953),
49-55J. Kopriva, Contribution to the relation of the Farey
series to the Riemann hypothesis (Czech), Casopis Pest. Mat.
79 (1954), 77-82M. Mikolas, Sur l'hypothese de Riemann. C. 
R.
Acad. Sci. Paris 228 (1949), 633-636M. Mikolas, Farey series
and their connection with the prime number problem. I. Acta
Univ. Szeged. Sect. Sci. Math.13 (1949), 93-117M. Mikolas,
Farey series and their connection with the prime number
problem. II. Acta Univ. Szeged. Sect. Sci. Math.14 (1951),
5-21M. Mikolas, On the asymptotic behaviour of Franel's sum
and the Riemann hypothesis. Results Math 21(1992) no. 3-4,
368-378M. Yosmoto, Farey series and the Riemann hypothesis.
II. Acta Math.Hungar. 78 (1998), no. 4, 287-304 and it gives
ts linkwch gives a brief statement of a Farey series
equivalent to RH and also gives ts
linkhttp://www.math.jussieu.fr/~miw/telecom/
biblio-Amoroso.htmlwhere you'll find ts list of
references:Amoroso, F.``On the heights of a product of
cyclotomic polynomials.Number theory, I (Rome, 1995). Rend.
Sem. Mat. Univ. Politec. Torino 53 (1995), no. 3,
183--191.Amoroso, F.``Algebraic numbers close to 1 and
variants of Mahler's measure.J. Number Theory 60 (1996), no.
1, 80--96.Codeca, P.``Some properties of the local discrepancy
of Farey sequences.Atti Accad. Sci. Istit. Bologna Cl. Sci.
Fis. Rend (13) 8 (1980/81), no. 1-2, 163-173.Codeca, P. and
Perelli, A.``On the uniform distribution (mod 1) of the Farey
fractions and l^p spaces.Math. Ann. 279 (1988), no. 3,
413-422.Franel, J.``Les suites de Farey et les problemes des
nombres premiers.Gottinger Nachrichten, 198-201 (1924).Fujii,
A.``A remark on the Riemann hypothesis.Comment. Math. Univ.
St. Pauli 29 (1980), 195-201 .Fujii, A.``Some explicit
formulae in the theory of numbers. A remark on the Riemann
Hypothesis.Proc. Japan Acad., Ser. A 57 (1981), 326-330
.Huxley, M.N.``The distribution of Farey points, I.Acta Arith.
18 (1971), 281-287 .Kanemitsu, S. and Yosmoto, M.``Farey series
and the Riemann hypothesis.Acta Arith. 75 (1996), no. 4,
351-374.Kanemitsu, S. and Yosmoto, M.``Farey series and the
Riemann hypothesis. III.Ramanujan J. 1 (1997), no. 4, 363-378
.Kopriva, J.``Contribution to the relation of the Farey series
to the Riemann hypothesis on the zeros of the zeta
function(Czech), Casopis Pest. Mat. 78 (1953), 49-55 .Kopriva,
J.``Contribution to the relation of the Farey series to the
Riemann hypothesis(Czech), Casopis Pest. Mat. 79 (1954), 77-82
.Landau, E.``Bemerkungen zu der vorstehenden Abhandlung von
Herrn Franel.Gottinger Nachrichten, 202-206 (1924);.Mikolas,
M.``Sur l'hypothese de Riemann.C. R. Acad. Sci. Paris 228
(1949), 633-636.Mikolas, M.``Farey series and their connection
with the prime number problem. I.Acta Univ. Szeged. Sect. Sci.
Math. 13 (1949), 93-117.Mikolas, M.``Farey series and their
connection with the prime number problem. II.Acta Univ.
Szeged. Sect. Sci. Math. 14 (1951), 5-21.Mikolas, M.``On the
asymptotic behaviour of Franel's sum and the Riemann
hypothesis.Results Math 21 (1992), no. 3--4,
368-378.Niederreiter, E.``The distribution of Farey
points,Math. Ann. 201 (1973), 341-345.Yosmoto, M.``Farey
series and the Riemann hypothesis. II.Acta Math. Hungar. 78
(1998), no. 4, 287-304.Zulauf, A.``The distribution of Farey
Numbers.J. Reine Angew. Math. 289 (1977), 209-213.That should
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statement to Riemann Hypothesis Huh?> The Farey series of
order n is the fractions a/b between zero and one, >
inclusive, in lowest terms, with b not exceeding n, in their
natural > order. E.g., the Farey series of order 5 is > 0/1,
1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1, > in that
order. there is an extremely simple 1-1 map of the rationals
ontonon negative integers.nt continued fraction
surreals.10-millionth rational in mcdonald scheme.
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multiplicative order in general and mersenne numbers> For
example I would be very interested in documents wch tell>
anytng about factors of mersenne numbers.OK, I believe I gave
you the URL for the homepage of the Cunningham project. > I
tnk that most people concentrate on mersenne numbers whose>
exponent is prime, but for me also the mersenne numbers with
composite> exponents are interesting.The Cunningham project
deals with prime and composite exponents equally. > 2^p |
2^(p*q)-1> 2^q | 2^(p*q)-1> where p and q are prime.I tnk you
mean 2^p - 1 divides 2^(pq) - 1, etc.> but 2^(p*q)-1 has some
factors besides the ones in 2^p and 2^q.> What is known about
ts factors?> Of course I am not just interested in exponents
wch are composed of> exactly two exponents, but in all
composed exponents in general,> because i know now that the
divisors of mersenne numbers with prime> exponents are
Cunningham numbers, right?Sorry, never heard the expression,
Cunningham number.> But I hardly know anytng about the factors
of mersenne numbers wch> are composite but have no gcd with one
of the mersenne numbers wch> are divisors of the first 
mersenne
number with the composite exponent> (described above).Ts will
have to be translated into English before I attempt to deal
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exist?>Computational complexity not only involves the number
of operations, but>also the complexity of operations.
Operations on large numbers are more>complex than operations
on small numbers. When you look at it from a pure>mathematical
viewpoint it is a bit different. But when you are
actually>implementing stuff the largeness of numbers gets to
play a big role.Well, here's the whole computation. 
Complexity
is in the eye of the beholder,but ts doesn't look too 
complex
to me. Formulas for the iterations arebelow. Use fixed-width
font to view. Robertson i P_i Q_i a_i A_i B_i -2 0 1 -1 1 0 0
0 1 20 20 1 1 20 9 4 81 4 2 16 17 2 182 9 3 18 5 7 1355 67 4
17 24 1 1537 76 5 7 15 1 2892 143 6 8 23 1 4429 219 7 15 8 4
20608 1019 8 17 15 2 45645 2257 9 13 16 2 111898 5533 10 19 3
13 1500319 74186 11 20 3 13 19616045 969951 12 19 16 2
40732409 2014088 13 13 15 2 101080863 4998127 14 17 8 4
445055861 22006596 15 15 23 1 546136724 27004723 16 8 15 1
991192585 49011319 17 7 24 1 1537329309 76016042 18 17 5 7
11752497748 581123613 19 18 17 2 25042324805 1238263268 20 16
9 4 111921796968 5534176685 21 20 1 P_i = a_{i-1} Q_{i-1} -
P_{i-1}Q_i = (409 - P_i^2)/Q_{i-1}a_i =int((P_i + 20)/Q_i)A_i
= a_i A_{i-1} + A_{i-2}B_i = a_i B_{i-1} + B_{i-2}Note that
the 20 above is the largest integer less than
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