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7?1.01.01.01.0???????6!{?Ec
frWEst~WEpf?ru?P¢stylnüISH?TABS?WDTH¸jstf?rct?lct?PRopB
Kco[YAcute]?&??[YAcute]?&?cú??x&?,[YAcute]???h[YAcute]?[CapitalA
Grave]&?1Ú2[YAcute]??&?[YAcute]??????
!?8[YAcute]?!?@[YAcute]?!?P[YAcute]?!?ø[YAcute]?&!?[YAcute]?2!?
D[YAcute]?x!??1/2(p^2 + 1) is
odd and 1/2(p^2 + 1) + p is even.That means we have a
guaranteed factoring (f1, f2) where f1 - f2 iseven and we get
a maximum for a_(p+1): a_p(+1) <= (1/2(p^2 + 1) + p - 2) / 2 =
1/4 p^2 + 1/2 p - 3/4 <= 1/2 p^2 - 1/2 = a_p, since 1/2 p^2 -
1/2  1/4 p^2 + 1/2 p - 3/4 < 2p^2 - 2  p^2 + 2p - 3 <
p^2 - 2p + 1  0 < (p - 1)^2  0which completes the proof.
 My initial question has had to run the gauntlet
and> did not come out unscathed (i.e., as a tough question...
it> turned out to be really easy). But, perhaps, theres
still> a little life left in it, afterall?> > My Guess is as
Good as Yours Theorem 1.2:> Let (G,*) be a (nite)
non-abelian group with nontrivial> center and assume it
possible tofind a ring (R, *, +)> such that G is a subgroup
of R with respect to *. In> addition, assume that for all x
in G: -x(R) in G (where> -x(R) is dened as the additive
inverse of x). Then G> is of even order.G any nite group. R
= F_2 G, the group ring of G over theeld F_2 of two
elements. G embeds obviously in R^* andas R has
characteristic 2, then -x = x for all x in G.Of course, G can
have odd order, and nontrivial centre.-- Robin Chapman,
www.maths.ex.ac.uk/~rjc/rjc.htmlNeedless to say, I had the
last laugh. Alan Partridge, _Bouncing Back_ (14 times)
=The
Klein bottle is a 2-dimensional non-orientable surface.Is
there a 3D manifold analog?Are 3D non-orientable manifolds
excluded from the 3-manifoldclassication program (as in
Thurstons Geometrization Conjecture) ?David Bernier
=|The
Klein bottle is a 2-dimensional non-orientable surface.||Is
there a 3D manifold analog?I think of the Klein bottle as a
cylinder S^1 x [0,1], withthe ends S^1x{0} and S^1x{1}
identied with each otherby an orientation-reversing mapping.
That is, if S^1 isthe unit circle {(x,y)|x^2+y^2=1}, then we
identify((x,y),0) with ((-x,y),1) or something like that.The
easiest analog I can think of is to take a cylinderS^2x[0,1]
and identify the ends S^2x{0} and S^2x{1}by an
orientation-reversing mapping... a case of whatLee Rudolph
mentioned as an S^2 bundle over S^1(where [0,1] with {0} and
{1} identied is the base S^1).Keith Ramsay
The Klein
bottle is a 2-dimensional non-orientable surface.Is there a
3D manifold analog?What do you want analog to mean here?
Certainly there are plenty of 3-dimensional non-orientable
manifolds. The Kleinbottle K is an S^1-bundle over S^1; there
are 3-dimensionalnon-orientable manifolds which are M-bundles
over N,for (M,N) any of (S^1 x S^1, S^1), (K, S^1), (S^1, S^1
x S^1),(S^1, K), (S^1, S^2), (S^2, S^1), and those are all
analogsof K in some sense. >Are 3D non-orientable manifolds
excluded from the 3-manifold>classication program (as in
Thurstons Geometrization Conjecture) ?Nope, theyre included
too (for an appropriate formulation ofthe conjecture).Lee
Rudolph
 >The Klein bottle is a 2-dimensional
non-orientable surface.Is there a 3D manifold analog?> > What
do you want analog to mean here? Certainly there are > plenty
of 3-dimensional non-orientable manifolds. The Klein> bottle
K is an S^1-bundle over S^1; there are 3-dimensional>
non-orientable manifolds which are M-bundles over N,> for
(M,N) any of (S^1 x S^1, S^1), (K, S^1), (S^1, S^1 x S^1),>
(S^1, K), (S^1, S^2), (S^2, S^1), and those are all analogs>
of K in some sense. [...]I was thinking of a compact
3-manifold that can be obtainedthrough abstract gluing, as is
explained on this Web pagefrom the Math. Dept. at Ohio State
University:http://www.math.ohio-state.edu/~edorow/math655/
Klein2.htmlThe crab on the Klein bottle is pretty good.David
Bernier
>The Klein bottle is a 2-dimensional
non-orientable surface.>>Is there a 3D manifold analog?and
now explains:>I was thinking of a compact 3-manifold that can
be obtained>through abstract gluing, as is explained on this
Web page>from the Math. Dept. at Ohio State
University:http://www.math.ohio-state.edu/~edorow/math655/
Klein2.html*Every* compact 3-manifold ... can be obtained
through abstract gluing. In particular, among the examples I
mentioned before,non-orientable M bundles over N, for (M,N)
any of (S^1 x S^1, S^1), (K, S^1), (S^1, S^1 x S^1), or (S^1,
K) (where K is the Kleinbottle, and of course S^1 is the
circle, so S^1 x S^1 is the2-dimensional torus) can be
obtained through abstract gluingof a 3-dimensional cube! Take
Zbiggys crab-cage, pull it upinto the third dimension to make
a cube, then glue the top ofthe cube (out in the room) to the
bottom of the cube (back onyour screen), and glue the other
four faces to each other in pairs following the gluing that
makes the crab-cage into aKlein bottle; at this point you
have a non-orientable compact3-manifold with some claim to
being called a 3-dimensionalKlein bottle.For a different sort
of example (different in that it isnta bundle), take a cube
as before, and glue each face to itsopposite, with a
180-degree rotation. The resulting manifoldis the
3-dimensional real projective space. Again, its
non-orientable.Lee Rudolph
 Im just a sophomore in
college, so I dont know much. In our math> book, it said the
integral of sin(x^2) cant be integrated easily. > What does
it mean, easily? Is there a simple antiderivative of>
sin(x^2), or is it some kind of innite thing? Has it been
proven> either way? Im just curious...> > John SavageIt
means there is no simple formula for that indenite
integral.There is a technical notion of elementary
function... algebraicfunctions, trig functions, inverse trig
functions, exponentialfunctions, log functions, and nite
combinations of these (includingby composition). It has been
proved that there is no elementaryfunction with derivative
equal to sin(x^2).Some other common ones are: exp(-x^2),
exp(x)/x, sin(x)/x,sqrt(polynomial of degree >=3). ***
Integration in Finite Terms ***This theory goes back to
Liouville, 1835.Classic text on the subject: J. F. Ritt,
_Integration in Finite Terms_ (Columbia Univ Pr,
1948)Introductory papers, aimed at undergraduates: A.D. Fitt
& G.T.Q. Hoare, The closed-form integration of arbitrary
functions. Mathematical Gazette (1993) 227--236. E.
Marchisotto & G. Zakeri, An invitation to integration in
nite terms. College Math. J. 25 (1994) 295--308. A modern
text (omitting the algebraic case) M. Bronstein, _Symbolic
Integration I: Transcendental Functions_ (Springer-Verlag
1997)-- G. A. Edgar
http://www.math.ohio-state.edu/~edgar/
 exp[y] means
e^yMathematicians write exp(y), only Mathematicaicians write
Exp[y].
denoncou@euclid.colorado.edu (Hugh>>I have been having some
difculties with the following problem:It is from Royden
(p127 #17). The space throughout is assumed to be[0,1],
though I think any nite measure space will work. Suppose p >
1. Suppose f_n -> f a.e. , f is in L_p and f_n is inL_p for
all n. Suppose there exists M such that || f_n || (p-norm) <=
M for all n. Suppose g is in L_q. Show that g*f_n -> g*f in
the L_1 norm. >>If epsilon > 0 then there exists delta > 0
such that if m(A) < delta>>then the L^q norm of g*chi_A is
less than epsilon, because ___.>>Now there exists a set A of
measure less than delta such that>>f_n -> f ___ly on the
complement of A, by ___s theorem...>>[...]It is not
mentioned here, but in the problem, is 1/p + 1/q <= 1
assumed?>If not, then g*f might not even be in L^1.Well of
course. In careful writing one would certainly want to
state1/p + 1/q = 1 explicitly, but in the present context,
given that the OP clearly has some idea what hes talking
about, thats very clearlyan implicit assumption.>[...]One
must have p > 1. Thats correct. Luckily p > 1 _was_ given
explicitly in the problem.>[...]************************David
=So far
Ive come to the conclusion that there is a largest topology
thatassures the assigned limits. This again is with the new
understandingthat the topology generated by a bunch of sets
is the topology with thesubbase of those sets. So any
topological property expressible solely withsubbase sets and
set theory will be preserved upon union of topologieswith
that property.Im still puzzle showing the product topology
{0,1}^S or actually thepower-set topology is the largest
topology assuring the assigned limits.Another puzzle Im
musing upon is if a topology assuring the assignedlimits can
be enlarged to a 1st countable topology while still
assuringthe assigned limits. This I think likely not
possible.
 [...]> > Im still puzzle showing the product
topology {0,1}^S or actually the> power-set topology is the
largest topology assuring the assigned limits.>You have shown
that, in this topology, all the topological limits
areset-limits, no? Something to ponder: If you start throwing
in moreopen sets, do you lose some of these convergences? >
Another puzzle Im musing upon is if a topology assuring the
assigned> limits can be enlarged to a 1st countable topology
while still assuring> the assigned limits. This I think
likely not possible.Do you really mean enlarge? The coinduced
topology is already thelargest. Also, it seems to me that to
make a space rst countable,you have to *remove* open sets,
not add them, no? (Not sure aboutthat one.) For example, all
sequences converge to every point in theindiscrete topology,
which is rst countable. Thus, you could use asa subbase the
union of all rst countable topologies where thesequences
converge to the designated limits. Trouble is, you get
evenmore convergent sequences than before.SJH
=Van Jacques>
This is a reponse to a comment something like of what use is>
abstract math.> Pure math is on the whole distinctly more
useful that applied.> For what is useful above all is
technique, and math technique> is taught mainly through pure
math....Nowadays many sorts of mathematical and physical
things are modelled interms of _sets_ of points. But where do
you suppose the method was rstused? If I mistake not, it was
in Dedekinds denition of an ideal number,as a kind of
subset (a submodule) of the ring C. And what was the motive
forbringing in ideals? Trying to prove FLT, of
course!Dedekind cuts, dening a real number as a set of
rationals, came later.LH
=Van Jacques>> This is a reponse
to a comment something like of what use is>> abstract math.>>
Pure math is on the whole distinctly more useful that
applied.>> For what is useful above all is technique, and
math technique>> is taught mainly through pure
math.>...>Nowadays many sorts of mathematical and physical
things are modelled in>terms of _sets_ of points. But where
do you suppose the method was rst>used? If I mistake not, it
was in Dedekinds denition of an ideal number,>as a kind of
subset (a submodule) of the ring C. And what was the motive
for>bringing in ideals? Trying to prove FLT, of course!Not
really. The reason for bringing in ideals was to present
aconcrete counterpart to Kummers ideal numbers, which was
subject togeneralization in arbitrary number elds (both
Dedekind and Kroneckerhad run into trouble in trying to use
Kummers approach in rings ontegers that did not have an
integral basis made of powers of thesame element). And while
Kummer used his approach to prove FLT forregular primes, he
never considered it particularly important. It wasall an
unintentional consequence of his ->true<-
interest:generalising quadratic reciprocity to higher
reciprocity laws.--
==
==Its not denial. Im just very selective about what
I accept as reality. --- Calvin (Calvin and
Hobbes)
=Arturo Magidinmagidin@math.berkeley.edu*
Van Jacques> Kronecker: God made integers, the rest is the
work of man.This was said in anger when Cantor experienced
with higherinnities. However, I often think it nowadays is
used to indicatehow ingenious and wonderful the integers are.
Gods invention wassimple and great, and he left us all to
investigate and discover allthe beautiful properties that can
be found in the realm of numbers.I believe.-- Jon
Haugsand
 > Number of positions...> After exactly two
pawn moves:> ... after two moves by the same pawn: 16> ...
after two pawns each moving 1 step: 28> ... after two pawns
each moving 2 steps: 28> ... after two pawns, one moving 1
and one moving 2: 56> TOTAL: 128> > After a pawn move and a
non-pawn move:> ... after a3: 4> ... after a4: 6> ... after
b3: 6> ... after b4: 6> ... after c3: 6> ... after c4: 7> ...
after d3: 12> ... after d4: 13> ... after e3: 15> ... after
e4: 15> ... after f3: 4> ... after f4: 5> ... after g3: 6>
... after g4: 6> ... after h3: 4> ... after h4: 6> TOTAL:
121> > After a knight move and a rook move: 4> After two
moves by same knight: 11> After two moves by different
knights: 4> TOTAL: 19> > Grand total: 128+121+19=268But i
don`t really understand it!the 16 is ok, but how do you get
the 28?I think after the two moves of the pawn you go backto
the initial boardposition, right? How do i get upto 28
different positions by moving 2 pawns, each one
step?complicated stuff...:-(
 > Number of positions...>
After exactly two pawn moves:> ... after two moves by the
same pawn: 16> ... after two pawns each moving 1 step: 28>
... after two pawns each moving 2 steps: 28> ... after two
pawns, one moving 1 and one moving 2: 56> TOTAL: 128> > >
After a pawn move and a non-pawn move:> ... after a3: 4> ...
after a4: 6> ... after b3: 6> ... after b4: 6> ... after c3:
6> ... after c4: 7> ... after d3: 12> ... after d4: 13> ...
after e3: 15> ... after e4: 15> ... after f3: 4> ... after
f4: 5> ... after g3: 6> ... after g4: 6> ... after h3: 4> ...
after h4: 6> TOTAL: 121> > After a knight move and a rook
move: 4> After two moves by same knight: 11> After two moves
by different knights: 4> TOTAL: 19> > Grand total:
128+121+19=268> > But i don`t really understand it!> the 16
is ok, but how do you get the 28?> I think after the two
moves of the pawn you go back> to the initial boardposition,
right? How do i get up> to 28 different positions by moving 2
pawns, each one step?> > complicated stuff...:-(28 is found by
what mathematicians call 8 choose 2. Look upcombinations, or
binomial coefcients to understand. But this is easy to
count. We have the following combinations oftwo pawns to move
one square each (named after their
le):
abacadaeafagahbcbdbebfbgbhcdcecfcgchdedfdgdhefegehfgfhgh28
possibilities.Note that this is 7+6+5+4+3+2+1.
= [...] What
is the connection between the number of rows and the> minimum
number of dots that must be moved to turn the triangle?> >
[...]> > O is dots that has to be moved to turn the
triangle.> > O > X X X X > O> > X O X O > X X X X > O X O X  O > X X O X X O> X X X X X X > O X X O X X > O> > O > O O >
X X X O X X X O> X X X X X X X X > O X X X O X X X > O O> O > Notice how the xed dots are symetrical.> > By observing
practical experiments i have created the table below.> B(n)
is the number of dots to move.> > n A(n) B(n)> -----------> 1
1 0> 2 3 1> 3 6 2> 4 10 3> 5 15 5> 6 21 7> 7 28 9> 8 36 12> >
I get B(6) = 8, No your wrong: O X X X X X X X X X O X X X
OO O X X O O> and your pattern implies B(n) = (n-3)(n-2)/2 +
2> for n > 2.hmm
 > I get B(6) = 8, No your wrong: O> X
X> X X X> X X X X> O X X X O> O O X mm-109
Im currently designing a set of interface objects
for mac OS X, andone of the possible application would be an
interactive spherical trigcourse.The idea is to allow people
to mess around with sliders to change thevalues of sides and
angles and get immediate feedback, to illustratethat abC
always has a solution but abc not, and aAB can have
multiplesolutions.Input, especially from those with an
interest in teaching sphericaltrig, is appreciated.The OS X
InterfaceBuilder palette is available from
hort their claims against my argument seemto
have a gang that gives them enormous wiggle room.I point out
things they cant do, like deliver on coefcients, orevaluate
their claims at actual values of x, like x=2, and away theygo
as if it doesnt matter in mathematics if you cant
actuallyproduce.One person--Rick Decker--at least *tried* to
produce on a coefcientquestion, but was foiled by a simple
test of his quadratic using x=1,when what happens with
another quadratic at x=1 was probably thereason he picked
it!!!(Maybe he has another reply to correct his past one by
now; Im notseeing it in Google Groups at this time.)So what
I see are continual failures to produce by people who
despitethat talk a lot.I did a little demonstration where I
replied to everyone that repliedto me for a day or so, and
you can see the results.The people Im dealing with arent
exactly normal, and they cantproduce mathematically.I dont
know if others on the newsgroup are cheering them on, not
atall interested, or wondering themselves, as they believe
mathematicsis more than a social activity, with cliques and
groups, where ifyoure out your research is out, and if
youre in, hey, you can sayanything! Its like high school in
America here!!!Faced with a modern discoverer, not some
distant gure in the historybooks, many of you seem to have
suddenly decided that math is ademocracy!!!Not saying its
related as I cant be sure it is, but as I think itsa
telling testimony to modern mathematics and its social
reality,heres that quote I like to give:<Quote>What is a
proof? The question has two answers. The right
wing(right-or-wrong, rule-of-law) denition is that a proof
is alogically correct argument that establishes the truth of
a givenstatement. The left wing answer (fuzzy, democratic,
and humancentered) is that a proof is an argument that
convinces a typicalmathematician of the truth of a given
statement.While valid in an idealistic sense, the right wing
denition of aproof has the problem that, except for trivial
examples, it is notclear that anyone has ever seen such a
thing.</Quote>Sadly, the evidence is in that mathematicians
today are not peoplewho care about mathematical truth, but
about social truth where itsmostly about what everyone is
saying.Yup, mathematics today is, from the evidence Ive
seen, a democracy.James Harris
 Ive noticed that posters
who are trapped by being asked to produce> more than vague
math to support their claims against my argument seem> to
have a gang that gives them enormous wiggle room.> Im
sitting in my enormous wiggle room right now. Its quite
comfy.
Message-id:
Ive noticed that posters who are trapped by being asked to
produce>> more than vague math to support their claims
against my argument seem>> to have a gang that gives them
enormous wiggle room.>> Im sitting in my enormous wiggle
room right now. Its quite comfy.Are the walls made of
rubber?--MensanatorAce of Clubs
 Ive noticed that
posters who are trapped by being asked to produce> more than
vague math to support their claims against my argument seem>
to have a gang that gives them enormous wiggle room.On the
other hand, JSHs arguments are supported by even vaguer
claims.And the burden of proof falls entirely on the
claimant, not on those who question the claim.Since The
claimant, JSH, does not answer the questions, is claim must
be rejected, at least until he has answered those claims.For
example, Nora has given detailed analyses of where and why
JSH is wrong, yet JSHs only responses to them seem to be
attempts to psychoanalyse Nora, and ignore the mathematical
content of her postings entirely.Either JSHs purposes are
psychological, rather than mathematical, or he is not
mathematically competent to answer Nora, and others, or
both.It is JSH who is making use of wiggle room.
 Ive
noticed that posters who are trapped by being asked to
produce> more than vague math to support their claims against
my argument seem> to have a gang that gives them enormous
wiggle room. On the other hand, JSHs arguments are supported
by even vaguer claims. And the burden of proof falls entirely
on the claimant, not on those who> question the claim. Since
The claimant, JSH, does not answer the questions, is claim
must be> rejected, at least until he has answered those
claims. For example, Nora has given detailed analyses of
where and why JSH is> wrong, yet JSHs only responses to them
seem to be attempts to> psychoanalyse Nora, and ignore the
mathematical content of her postings> entirely. Either JSHs
purposes are psychological, rather than mathematical, or he>
is not mathematically competent to answer Nora, and others,
or both. It is JSH who is making use of wiggle
room.motivations are quite simply to waste your time, and to
take some cheapshots at all mathematicians.
 Ive noticed
that posters who are trapped by being asked to produce> more
than vague math to support their claims against my argument
seem> to have a gang that gives them enormous wiggle room. I
point out things they cant do, like deliver on coefcients,
or> evaluate their claims at actual values of x, like x=2,
and away they> go as if it doesnt matter in mathematics if
you cant actually> produce. One person--Rick Decker--at
least *tried* to produce on a coefcient> question, but was
foiled by a simple test of his quadratic using x=1,> when
what happens with another quadratic at x=1 was probably the>
reason he picked it!!! (Maybe he has another reply to correct
his past one by now; Im not> seeing it in Google Groups at
this time.) So what I see are continual failures to produce
by people who despite> that talk a lot. I did a little
demonstration where I replied to everyone that replied> to me
for a day or so, and you can see the results. The people Im
dealing with arent exactly normal, and they cant> produce
mathematically. I dont know if others on the newsgroup are
cheering them on, not at> all interested, or wondering
themselves, as they believe mathematics> is more than a
social activity, with cliques and groups, where if> youre
out your research is out, and if youre in, hey, you can say>
anything! Its like high school in America here!!! Faced with
a modern discoverer, not some distant gure in the history>
books, many of you seem to have suddenly decided that math is
a> democracy!!! Not saying its related as I cant be sure it
is, but as I think its> a telling testimony to modern
mathematics and its social reality,> heres that quote I like
to give: <Quote What is a proof? The question has two answers.
The right wing> (right-or-wrong, rule-of-law) denition is
that a proof is a> logically correct argument that
establishes the truth of a given> statement. The left wing
answer (fuzzy, democratic, and human> centered) is that a
proof is an argument that convinces a typical> mathematician
of the truth of a given statement. While valid in an
idealistic sense, the right wing denition of a> proof has
the problem that, except for trivial examples, it is not>
clear that anyone has ever seen such a thing.> </Quote So now
I can see how people like Decker or others who cant actually>
show that their arguments are correct with mathematics but
simply> *convince* other readers are just ne in the math
world of today. You see, mathematicians apparently have given
up on worrying about> being actually correct, but instead have
decided to be like the rest> of the world!!! Maybe next time
they give out a Fields Medal they can just have an>
election! You can have mathematics political parties, and
turn to polls!!! Sadly, the evidence is in that
mathematicians today are not people> who care about
mathematical truth, but about social truth where its> mostly
about what everyone is saying. Yup, mathematics today is, from
the evidence Ive seen, a democracy.> James HarrisOk James, If
math is a democracy than I propose that we say 2+2=5,
whoswith me on this one???Well, Thats not going to get me
anywhere because that is obviously wrong,just like your
research. You are extremely immature for someone who is
anadult. You need to grow up, and while youre at it, learn
some math.David Moran
 Yup, mathematics today is, from
the evidence Ive seen, a democracy.> James HarrisYou are an
idiot. You want to see science by democracy ?? Look
atpsychology.Your math is all awed - even a person such as
myself can see that withouteven picking up a pen, and I have
never been to skool.If you got your little ass burned on a
pre-algebra test then that is YOURproblem. You cannot get
grades in math by being a suck-up. Kissing theprofessors ass
simply will not work - math is not a humanity.Go learn what
you need to learn somewhere, and then they will allow you
tograduate, and not before.You cannot take shortcuts in life.
There is a sort of isomorphism betweenyour behaviour here and
maa-like conduct elsewhere. Grow up, and when youhave
learned something, maybe if you beg real hard they will let
youretake prealgebra for the Nth time.
> >>
Littlemanwearingbigboypants misstates yet again:>> >> > Every
presidential election year has been a leap year, except for
>> http://scienceworld.wolfram.com/astronomy/LeapYear.html>> Gardyloo.When you throw the bucket straight up, move.This
site supports Bill Vajk, supposing that he is correct that
there>have been presidential elections in 1789, 1800 & 1900,
none of which>were leap years.A lot of people who program
leap year formulas forget about thecentury exception test.
/BAH
to remember about our game is that it was designed to keep>
our cricketers t in the off season. It is therefore a
running and> passing game, no player can keep possession of
the ball for more than> a few steps and the ball tends to be
in play for a much larger> proportion of the game than Grid
Iron. (The most frustrating thing> about watching an American
Fottball game is that they will not pass> the ball !, is it
actually against the rules ?)No. You may have one forward
pass behind the line of scrimage, and youmay perform lateral
passes or back pass like in rugby. Not sure if itis allowed
ad-innitum... but if only they had more faith in
theirplayers, and trained them to handle the ball, surely
youd see morepassing in the game.
No. You may have one
forward pass behind the line of scrimage, and you>may perform
lateral passes or back pass like in rugby. Not sure if it>is
allowed ad-innitum... Careful, there; youre risking
bringing this thread back ontotopic for sci.math.Lee
Rudolph
> Is it possible to evaluate following integral
as a function of k ( 0<k<1 ) ? int_0^{2 pi}{ ( ( ln( 1+k
try:>>Expand the integrand as a power series in k. The
coefcient of k^n>>is rational * sin(x)^n; integrate sin(x)^n
from 0 to 2*pi (using a>>recursive reduction formula,
perhaps). Sum the resulting series.The rational coefcient of
k^n sin^n(x), for n >= 2, is n-1> n 2 --- 1 n> (-1) - > - =
(-1) H(n-1) [1]> n --- j> j=1Oops, I left out the factor of
2/n in [1], but only there. It should be n-1 n 2 --- 1 n 2
(-1) - > - = (-1) - H(n-1) [1] n --- j n j=1The revised [1]
is what is used to get [3] below.>where H(n) is the n^{th}
harmonic number.However, the integral from 0 to 2pi of any
odd power of sin(x) is 0.>So you only need to consider the
even n, and for even n, |2pi n> | sin (x) dx = 2pi
C(n,n/2)/2^n [2]> | 0Putting this all together, and
substituting n -> 2n, we get that |2pi 2> | ln(1 + k sin(x))
dx> | 0 oo> --- 2n> = 2pi > H(2n-1)/n C(2n,n)/4^n k [3]> --->
n=1The summation in [3] converges absolutely for |k| <= 1.Rob
Johnson <rob@trash.whim.org>take out the trash before
replying
= http://www2.b3ta.com/hawking/ --- Tourettes>>
bugger, post some content please, I just disconnected before
I>> clicked your post.>> ok, I just got the ash show, not
bad, but who was the young cronie besides>> Einstein and
hawking?> My guess is Bohr, crossposted to sci.math for the
answer.It was Heisenberg. Why sci.math instead of
sci.physics?-- Dave SeamanJudge Yohns mistakes revealed in
Mumia Abu-Jamal
ruling.<http://www.commoncouragepress.com/index.cfm?action=
book&bookid=228>
= http://www2.b3ta.com/hawking/ ---
Tourettes>> bugger, post some content please, I just
disconnected before I>> clicked your post.>> ok, I just got
the ash show, not bad, but who was the young cronie
besides>> Einstein and hawking?>> My guess is Bohr,
crossposted to sci.math for the answer. It was Heisenberg.
Why sci.math instead of sci.physics?>hence : Im not
sure.... I dont really know....Heisenberg
Uncertainty!Herc
= http://www2.b3ta.com/hawking/ --->
Tourettes bugger, post some content please, I just
disconnected before I clicked your post. ok, I just got the
ash show, not bad, but who was the young cronie>besides
Einstein and hawking?>> My guess is Bohr, crossposted to
sci.math for the answer.>> It was Heisenberg. Why sci.math
instead of sci.physics?>>hence : Im not sure.... I dont
really know....>Heisenberg Uncertainty!Worlds Funniest
Science Joke:Cop pulls over Heisenbergs car and asksDo you
know how fast you were going?To which Heisenberg repliesNo,
but I know where I am.>Herc--MensanatorAce of Clubs
<snip
Its just pretty depressing to>> see a book tell you that you
have learned next to nothing that you should know>> coming
into graduate school. Any comments or advice would be
appreciated,>> including book suggestions (the author tends
to say Ive heard that>> such-and-such book is good though I
have not seen it which is pretty odd>> considering the focus
of the book).>You may have trouble getting into a top ranked
graduate program if>your background is decient. However,
whats far more important is>what you do in the graduate
school you do get into. Many will allow>you to take a few
undergraduate courses to rm up your background. As this
problem is rife, that is an understatement. Theyoften also
have semi-remedial graduate courses, but thereis too much of
a tendency to downplay the weakness of thestudents
background.>Another clue to what you will eventually need to
know is the book of>problems from UCBerkely qualifying exams
published by Springer.However, if one does not know abstract
algebra or basicreal analysis, many of the problems will be
gibberish.>Dont let snobs tell you that a PhD from a less
than top-ranked school>is worthless. It isnt. What is
important is to work with an active>and productive researcher
with a national (or international)>reputation in his/her eld
of research. Such people will make sure>you do a good thesis
and will be able to write letters that say what>prospective
employers want to hear.This is NOT the case. A long time ago,
it was said ofstudents of Loewner that the better the student,
theworse the thesis. Another saying is that a thesis is(often)
the work of the major professor under adversecircumstances.
And I have seen too many candidates witheven good theses and
good letters from distinguishedresearchers at rst-class
universities who knew far toolittle to consider hiring. Also,
the student may havegotten a bad problem, and run into a dead
end, or amediocre student may have been lucky. In my many
years,I have seen all of these. Predicting what a PhD will
doin future research is VERY poor. There are such people even
at>lower ranked schools due to the bad job market of recent
years. You>may even do better than you might at a top school
because as a good>student youll get a lot more attention
than you would otherwise.-- This address is for information
only. I do not claim that these viewsare those of the
Statistics Department or of Purdue University.Herman Rubin,
Department of Statistics, Purdue University
<snip > Its
just pretty depressing to>> see a book tell you that you have
learned next to nothing that you should know>> coming into
graduate school. Any comments or advice would be
appreciated,>> including book suggestions (the author tends
to say Ive heard that>> such-and-such book is good though I
have not seen it which is pretty odd>> considering the focus
of the book).> >You may have trouble getting into a top
ranked graduate program if>your background is decient.
However, whats far more important is>what you do in the
graduate school you do get into. Many will allow>you to take
a few undergraduate courses to rm up your background. > > As
this problem is rife, that is an understatement. They> often
also have semi-remedial graduate courses, but there> is too
much of a tendency to downplay the weakness of the> students
background.> >Another clue to what you will eventually need to
know is the book of>problems from UCBerkely qualifying exams
published by Springer.> > However, if one does not know
abstract algebra or basic> real analysis, many of the
problems will be gibberish.> >Dont let snobs tell you that a
PhD from a less than top-ranked school>is worthless. It isnt.
What is important is to work with an active>and productive
researcher with a national (or international)>reputation in
his/her eld of research. Such people will make sure>you do a
good thesis and will be able to write letters that say
what>prospective employers want to hear.> > This is NOT the
case. A long time ago, it was said of> students of Loewner
that the better the student, the> worse the thesis. Another
saying is that a thesis is> (often) the work of the major
professor under adverse> circumstances. And I have seen too
many candidates with> even good theses and good letters from
distinguished> researchers at rst-class universities who
knew far too> little to consider hiring. Also, the student
may have> gotten a bad problem, and run into a dead end, or
a> mediocre student may have been lucky. In my many years,> I
have seen all of these. Predicting what a PhD will do> in
future research is VERY poor.> > There are such people even
at>lower ranked schools due to the bad job market of recent
years. You>may even do better than you might at a top school
because as a good>student youll get a lot more attention
than you would otherwise.Of course there are famous
professors who cant or wont give theirstudents the right
kind of help. Sometimes that help is tellingthem they dont
have what it takes to have a career as a
researchmathematician. Another point I meant to make is to
beware of graduateprograms that exploit graduate students as
fodder for teaching lowerlevel courses. Many large university
graduate programs do this,loading beginning graduate students
with teaching that gets in the wayof proper studying and
then, when there are problems completingrequired courses in
time or passing qualifying exams before adeadline, the
usefulness of the students is over and they are tossedout on
the street. Teaching assistantships are a time honored way
ofsupporting students through graduate school, but many
universitiesbring in marginally qualied students just to use
them as cheaplabor.
You may have trouble getting into a
top ranked graduate program if>your background is decient.
However, whats far more important is>what you do in the
graduate school you do get into. Many will allow>you to take
a few undergraduate courses to rm up your background. > > As
this problem is rife, that is an understatement. They> often
also have semi-remedial graduate courses, but there> is too
much of a tendency to downplay the weakness of the> students
background.Have I misunderstood? It sounds like you are saying
that when aprogram lets you take a few undergrad courses to
rm up, thatsdownplaying the weakness of ones background?
Thomas
> I need to know how to calculate x^y where y is
not an integer. Any>> help in this matter would be
appreciated.>ln(x^y) = y*ln(x) ... for x>0>therefore x^y =
exp(y*ln(x))First of all, what is x^y, where x > 0? If y =
m/n,x^y is the n-th root of x^m, or the m-th power of the
n-th root of x. Also, x^y is increasing in y if x > 1, and
decreasing in y if x < 1. It is alsocontinuous, and thus one
can pass to the limit, andeven obtain bounds. This precedes
logarithms.It is also the case, as stated above, that
log_b(x^y) = y*log_b(x). BC (before computers),the PRACTICAL
calculation was done by using b=10,which made the logarithm
tables easier to use,and tted in with scientic notation.
Thisis one of the reasons for the use of commonlogarithms,
introduced by Briggs. This is how it was done in practice,
and this usedto be a standard part of advanced algebra. If
onelooks at the texts for these, or for beginners
inengineering or scientic computation more than ahalf
century ago, this will be found there.-- This address is for
information only. I do not claim that these viewsare those of
the Statistics Department or of Purdue University.Herman
Rubin, Department of Statistics, Purdue University
= What
is known about groups with the property that there are
elements > x,y,z in G such that xy=yx, xz=zx, but yz != zy? I
admit,> Im asking this question somewhat rhetorically- >
because I have found such a group...There are plenty of such
groups. The subset of group elements thatcommute with
[something] are a topic of interest in Algebra.For example,
the set of elements that commute with *every* element ofthe
group is called its center. It isnt hard to prove that
thecenter of a group is itself an Abelian group.For another
example, consider: the set consisting of {1} is itself
anabelian subgroup of G. The center is another such. One
easily proves(using Kuratowskis ascending chain condition)
that G contains amaximal abelian subgroup (which might be the
singleton {1}, ofcourse).Centers, maximal abelian subgroups,
and commutator subgroups can beused to examine the degree to
which a group fails to be abelian.Len.
 >>Let R be a ring
and >>[,]:R*R -> R, [x,y] = xy - yx the commutor
function.>>Are those rings for which for all x,y,z in R:>>
[x,y] = 0 and [y,z] = 0 -> [x,z] = 0 >>somehow simpler than
other rings? If so,>>do any other of any such rings
properties follow >>automatically from this?Substituting 1
for y in the above formula, one has [x,z]=0for all x,z in
R.Hence, such a ring must be commutative.> The following
question did not need the additional complication> of R ring.
Let G be a group. What is known about> groups with the
property that there are elements > x,y,z in G such that
xy=yx, xz=zx, but yz != zy ? I admit,> Im asking this
question somewhat rhetorically- > because I have found such a
group. Btw., this property > has very little to do with the
reasons for me investigating the> group found as of yet.Every
noncommutative group satises the above with x=1.Marc
=What
is the vector space structureof xed rank (k), (n x n)
matrices ?What are the canonical bases for each k
?
space structure> of xed rank (k), (n x n) matrices ?The set
of all nxn matrices of rank k with sum and multiplication by
a scalaris not a vector space. Indeed, the sum of 2 matrices
of rank k is not, as ageneral rule, a rank k matrix. (take a
matrix A that has rank k. -A has rankk too, and A + (-A) = 0,
so it has rank 0).Sam-- So if you meet me, have some courtesy,
have some sympathy, and some taste Use all your well-learned
politesse, or Ill lay your soul to waste - The Rolling
Stones, Sympathy for the Devil
= .....................>> I
hope the quote alludes to the fact that naming systems arent
only a>> problem in mathematics, but perhaps it would be
productive to restrict>> the discussion to math despite this
obvious fact.>Naming is a matter of historical accident,
historical inaccuracy, and >human whim. Most theorems (can I
say most? at least some.) were rst >proved by someone other
than their associated name. Fermats Last >Theorem being the
classic case.Stigler claims that this is always the case.
Theclassical ones in statistics are Bayes Theorem
(althoughthe earlier source can only be conjectured) and
theGaussian distribution, discovered by de Moivre 47
yearsbefore Gauss was born.>Then of course why is it Zorns
Lemma, the Axiom of Choice, and the >Well-Ordering
Principle?BTW, Zorn did not explicitly credit it, but he did
givea reference to the book by R. L. Moore, who refers tothe
paper of Kuratowski. What is the difference among lemma,
axiom, and >principle, if in this case they are used to label
logical propositions >known to be equivalent to one
another?Zorn used it to assist in algebraic proofs, and he
wasusing it as a substitute for the Axiom of Choice orthe
Well-Ordering Principle. This is the purpose ofa lemma. BTW,
Zorn was the rst to state that itimplied AC, although he
never published the proof; Iknow this from personal
conversation. An axiom isexplicitly assumed, and generally
not believed to follow from the other assumptions. A
principle isa property which is directly used. When they
wereintroduced, AC and WO were not known to be
equivalent;Zermelo proved that AC implied WO, the converse
beingtrivial. That WO implies ZL is trivial, although
Kuratowski proved it directly from AC.>You might as well
campaign for geographical names to make logical sense.-- This
address is for information only. I do not claim that these
viewsare those of the Statistics Department or of Purdue
University.Herman Rubin, Department of Statistics, Purdue
University
=Suppose you wish to study a given population
using statistics, but there isno way to determine with
certainty that your population actually satisesthe criteria
which dene that population. Example - we would like to know
the approximate dimensions ofapparitions or poltergeists. We
would like to know how large the averagepoltergeist is.
Unfortunately, there is no way to determine with certainty
that we haveamassed a sample or even a population of
poltergeists, nor can we reallymeasure their dimensions. Yet,
we would like to know the average size of aghost.Is statistics
possible in this case ?? Can one make determnations
regardingsuch a population of ghosts ?Can statistics be
deployed correctly in such a case as this ??This may sound
like a completely stupid question - but I am %100
serious.This is NOT a troll. This question needs to be
asked.
=of the solutions came across this little
equation:x^2 + x + 1 = 0 (i)For fun, I started to push things
around a bit, instead of just goingwith x = (-b +- sqrt(b^2 -
4ac))/(2a):x(x + 1) = -1(x + 1) = -1/x (ii) <-- I though it
would be OK to divide by x,since x<>0Substituting (ii) back
into (i):x^2 + (x + 1) = 0x^2 - 1/x = 0x^3 - 1 = 0 <-- again,
multiplying by x, since x<>0(x - 1)(x^2 + x + 1) = 0
(iii)Somehow in my little manipulations I introduced a new
factor onto theoriginal equation, but I cant gure out how.
This seems prettyridiculous, but Id appreciate another pair
of eyes looking at it andletting me know where the (x - 1)
factor popped in.Jeff
 of the solutions came across this
little equation:> > x^2 + x + 1 = 0 (i)> > For fun, I started
to push things around a bit, instead of just going> with x =
(-b +- sqrt(b^2 - 4ac))/(2a):> > x(x + 1) = -1> > (x + 1) =
-1/x (ii) <-- I though it would be OK to divide by x,> since
x<>0> > Substituting (ii) back into (i):> > x^2 + (x + 1) =
0> > x^2 - 1/x = 0> > x^3 - 1 = 0 <-- again, multiplying by
x, since x<>0> > (x - 1)(x^2 + x + 1) = 0 (iii)> > > Somehow
in my little manipulations I introduced a new factor onto
the> original equation, but I cant gure out how. This seems
pretty> ridiculous, but Id appreciate another pair of eyes
looking at it and> letting me know where the (x - 1) factor
popped in.> > JeffReplacing (x + 1) with -1/x changed an
equation which is false for x=1into an equation which is true
for x=1.Your (i) is equivalent to your (ii), and that
statement implies (iii),but (iii) does not imply (i). When
solving equations, it is generally agood idea to watch your
steps so that you dont end up with statementsthat are more
generally true.Or to put it another way, whenever x-2=0, you
can safely conclude that xis in the set {1, 2}. However, that
conclusion is not minimal; the set{2} would work as well.If
you want a specic answer for this case, note that if the
originalequation is written as p(x) = 0, the substitution
ended up beingequivalent to subtracting p(x)/x from the left
side of the equation(using the fact that p(x) = 0 implies
p(x)/x = 0). This subtraction wasequivalent to multiplying
both sides by (1 - 1/x). The connection to (x- 1) should be
obvious. -- Daniel W.
Johnsonpanoptes@iquest.nethttp://members.iquest.net/~panoptes
/039 53 36 N / 086 11 55 W
 of the solutions came across
this little equation:> > x^2 + x + 1 = 0 (i)> > For fun, I
started to push things around a bit, instead of just going>
with x = (-b +- sqrt(b^2 - 4ac))/(2a):> > x(x + 1) = -1> > (x
+ 1) = -1/x (ii) <-- I though it would be OK to divide by x,>
since x<>0> > Substituting (ii) back into (i):> > x^2 + (x +
1) = 0> > x^2 - 1/x = 0> > x^3 - 1 = 0 <-- again, multiplying
by x, since x<>0> > (x - 1)(x^2 + x + 1) = 0 (iii)> > >
Somehow in my little manipulations I introduced a new factor
onto the> original equation, but I cant gure out how. This
seems pretty> ridiculous, but Id appreciate another pair of
eyes looking at it and> letting me know where the (x - 1)
factor popped in.> > JeffYou have introduced an extraneous
root to your equation.The roots of x^3 - 1 = 0 are the 3 cube
roots of 1.The roots of x^2 + x + 1 - 0 are the 2 primitive
cube roots of 1.The extraneous root creeps in as a result of
your substituting -1/x for x + 1, since that effectively
changes the degree of the equation.
 of the solutions
came across this little equation: x^2 + x + 1 = 0 (i) For
fun, I started to push things around a bit, instead of just
going> with x = (-b +- sqrt(b^2 - 4ac))/(2a): x(x + 1) = -1
(x + 1) = -1/x (ii) <-- I though it would be OK to divide by
x,> since x<>0 Substituting (ii) back into (i): x^2 + (x + 1)
= 0Another poster has already given the gist of the matter,
butjust to make things completely explicity, at this point
youare going to use the fact If x^2 + (x+1)=0 and (x+1)=-1/x,
then x^2 -1/x = 0.The reverse, of course, does not hold. That
is, x^2-1/x=0 does not imply that (x+1)=-1/x and
x^2+x+1=0.(Obviously, it implies neither, as you have
observed thatx=1 is a solution to x^2-1/x=0 but not a
solution to eitherx^2+x+1=0 or (actually, equivalently)
x+1=-1/x. ) x^2 - 1/x = 0 x^3 - 1 = 0 <-- again, multiplying
by x, since x<>0 (x - 1)(x^2 + x + 1) = 0 (iii)> Somehow in
my little manipulations I introduced a new factor onto the>
original equation, but I cant gure out how. This seems
pretty> ridiculous, but Id appreciate another pair of eyes
looking at it and> letting me know where the (x - 1) factor
popped in.Have fun! Keep thinking!Best wishes, Mike
 of
the solutions came across this little equation:> > x^2 + x +
1 = 0 (i)> > For fun, I started to push things around a bit,
instead of just going> with x = (-b +- sqrt(b^2 -
4ac))/(2a):> > x(x + 1) = -1> > (x + 1) = -1/x (ii) <-- I
though it would be OK to divide by x,> since x<>0> >
Substituting (ii) back into (i):> > x^2 + (x + 1) = 0> > x^2
- 1/x = 0> > x^3 - 1 = 0 <-- again, multiplying by x, since
x<>0> > (x - 1)(x^2 + x + 1) = 0 (iii)> > > Somehow in my
little manipulations I introduced a new factor onto the>
original equation, but I cant gure out how. This seems
pretty> ridiculous, but Id appreciate another pair of eyes
looking at it and> letting me know where the (x - 1) factor
popped in.What you have proved was that x^2 + x + 1 = 0  (x
- 1)(x^2 + x + 1)and this is obviously correct. It would still
be correct if x - 1 wasreplaced by anything else. But note
that your computations do not provethe reverse implication:
you cannot devide by x - 1 since you are notassuming that x
<> 1.Jose Carlos Santos
> The problem here is so simple. You assume a factorization > > of your polynomial P(x) of the form> >> > > >> > (*)
(5*a1(x) + 7)*(5*a2(x) + 7)*(5*b3(x) + 22).> >> > >> Thats
like saying that someone assumes that sqrt(2) is a factor of
2.> >> > >> Factorizations just exist.> >> > >> > >> I am
just summarizing what you have said. I am not> >> disagreeing
with it, at least not at this point.> >Yeah, like what if
someone says, lets assume that 6 has 3 as a> >factor.Technically its correct.> >But the use of the word assume
looks strange there, as 3 IS a factor> >of 6.> > Typically an
expression can be factored lots of ways. You assumed a>
factorization of the specied form. I see absolutely nothing>
wrong with my wording and I dont agree that it looks
strange.> In any case this is an utterly minor issue. > > > A
person might assume that theyre sounding rational, and
cogent, when> in fact theyre sounding defensive and
recalcitrant.> > You dont just assume a truth. > > Ill give
you another example as Im somewhat curious about your state>
of mind.> > Lets say that I make it a point to say I assume
that the sun will> rise tomorrow.> > Isnt there some
implication in that statement to you?> The analogy doesnt
work. There are lots of ways to factoran expression. If I
said, What is the form of thefactorization of (125*x^3 +
75*x^2 + 45*x + 7)?, most peoplemight think I wanted a
factorization as a polynomial in x. That, however, is not
what you have done with your expression.You have factored it
as a polynomial in 5. That is why itis important to say that
you *assumed* a factorization ofthe specied form. It is a
natural and even necessary part of the discussion. And here,
its a distraction. You are trying desperately to focuson a
minor wording issue to take attention away from the factthat
you have not dealt with the mathematics.> > >Theres no need
to assume anything.> >Now it seems to me that youre playing
stupid when the obvious is> >pointed out to you.> > Youre
quibbling over a minor wording issue. Meanwhile you> delete
and evade the underlying math below.> > > Is that an
assumption on your part?> No. Its like saying the sun will
rise tomorrow.> >> > >> > > >> > You note that since P(x) is
divisible by 49, there must be> >> > some way to factor 49
out of the terms in (*). You think> >> > that the only
possible way such a factorization can occur> >> > is of the
form 7, 7, 1. But you know that assuming a factorization > > >> And now you claim to know what I *think*?> >> > >> > >> > You have said it many times.> >Then give a single quote.> >
OK, how about one that you posted YESTERDAY in the thread>
Mathematical consistency, courage ?> > The constant terms are
7, 7, and 22, but when 49 is divided> off the resulting
constant terms are 1, 1, and 22, which means> from BASIC
arithmetic, theres only one way to go.> > There is, in fact,
only one way that can happen.> > > Yup, now then, look back at
your claim about what I thought.> OK. (Looks back). Yes, by
Jove, they are identicalin meaning.> >Here youre caught, yet
again, but rather than acknowledging the> >obvious--you cant
read minds--you instead toss out yet another> >statement.> >
See above. That was a direct, unedited quote. I cant read >
minds but I can read what you post. So can everyone else who>
sees this. > > > Yup. Now then, look carefully back at what
*you* said I thought, and> Im curious to hear how you feel
that the quote you gave is saying the> same thing.> > Be
quite detailed please.> I said you thought that the only way
to split 49 into three factors that would divide the three
terms of (5 a_1(x) + 7)*(5 a_2(x) + 7)*(5 b_3(x) + 22)was as
49 = 7 * 7 * 1, where the rst 7 divides (5 a_1(x) + 7),the
second 7 divides (5 a_2(x) + 7), and 1 divides (5 b_3(x) +
22).And indeed, that is precisely, exactly what you claimed.
If you now want to retract that, ne with me. That wouldopen
the door to other factorizations of 49 - for example49 =
w1(x)*w2(x)*w3(x), where w1(x), w2(x), and w3(x) arenon-unit
algebraic integers, and for x <> 0, none of themare equal to
7. If you are now willing to contemplate factorizations like
that, rather than your inspection-basedfactorization 7, 7, 1,
then your whole edice collapses.So, retract away!> > >So I
challenge you to give a single quote.> > > > Done. See above.
I await your explanation.> > > Its very simple. In fact, Ive
been rather detailed about my> factorization, and how it
works.> > But you make statements that are attempts at trying
to change the> facts.> Such as???> Like even though you gave
that quote, you then didnt even bother to> reconcile obvious
differences!> > Like you have 7, 7, 1, while I talk of 7, 7
and 22.> Oh, PLEASE!!! One of these is your proposed
factorizationof 49. The other is the constant terms of your
factorizationof P(x). If you think I have claimed they are
the same, youare simply delusional. You clearly,
unambiguously think, as you said in the quote, that the only
way to factor 49 through the threelinear factors of P(x) was
as 7, 7, and 1. There is no doubt> Its something Ive
watched before in your postings where you make> little
mistakes, big mistakes, and strange mistakes, but when its>
pointed out, you always deny!!!> I ask of you what you asked
of me: provide one example.> Later, I catch you making the
*same* mistakes, and with VERY long> posts, as if part of
your strategy is burying your behavior with a lot> of
verbiage.> This whole exchange is simply surreal. I make a
statement aboutwhat you thought. You say I am falsely reading
your mind. I provide a direct quote from 1 day before which
proves 100% ofwhat I said about your thinking. You squirm and
turn and tryto weasel out of it. You try to deny the meaning
of your own quotation. WHO THE HELL ARE YOU TRYING TO FOOL,
ANYWAY ?> Its kind of interesting behavior, though bizarre. >> > >> > >> Who cares what I think, > >> > >> > >> You have
made a huge error. My interest is in trying to> >> get you to
realize it.> >> > >I dont think so. For instance, a while
back in response to your> > I will never, never engage in any
nonpublic conversations with> you on any topic.> > So now you
*nally* admit that you need an audience. Why do I have> to
drag the truth out of you?> I dont want an audience. I want
witnesses. I would never trustyou to quote anything I said
o'ne. > You make claims that dont t with your behavior. > >Now then, why dont you try again and explain what you
*really* are> >trying to do?> > Its simple. I am trying to
show you that your factorization of 49 > in the 7, 7, 1
pattern is the WRONG FACTORIZATION when x <> 0. Needless > to
say you are not getting it. > > Now theres one of the errors
YET AGAIN, and I have to admit that its> a puzzle to me. OK,
I will say it as succinctly and clearly as I possibly can. You
divide (5 a_1(x) + 7)*(5 a_2(x) + 7)*(5 b_3(x) + 22)by 49. You
split 49 into 3 pieces: say, 49 = A*B*C.Your goal is for each
of (5 a_1(x) + 7) / A, (5 a_2(x) + 7) / B, and (5 b_3(x) +
22) / Ceach to be algebraic integers.You say the ONLY
POSSIBLE hope of achieving this is tospecify: A = 7, B = 7, C
= 1,because that is the only conguration that dividesthe
constant terms as you want. We say NO. For one thing, as you
now acknowledge, A = 7, B = 7, and C = 1 does NOT WORK in
general: in particular, (5 a_1(x) + 7) / 7 is NOT an
algebraic integer for most x <> 0. You know this. But there
is ANOTHER CHOICE for A, B, and C whichDOES work, to make all
of the three quotients abovealgebraic integers. For that
choice, each of A, B,and C are dependent on x. Yes, when x =
0, A = 7, B = 7, and C = 1. But when x <> 0, this is no
good,and other numbers must be chosen. Now: your objection to
this amounts to sayingthat if C is a factor of 49 which is not
a unit,then 22/C cannot be an algebraic integer. If that is
what you think, YOUR ARE ABSOLUTELY, TOTALLY RIGHT! And that
conclusion is ABSOLUTELY, TOTALLY IRRELEVANT!There is not the
slightest reason to want 22/C tobe an algebraic integer. What
you MUST have is that (5 b_3(x) + 22) / Cis an algebraic
integer. Do you see the difference? Is there any hope atall
that this key concept can penetrate to your tinylittle brain
? Or are you too far gone in delusion-land ? > Do you do that
consciously? Or is there something> deeper and even more odd
going on here?> You can easily gure that out by responding
in detail to themath in my original post. Enough with the
distractions and the smoke-screen about wording, your
ridiculous attempt to deny the meaning of your own
quotations, and your obsession with social interaction. >
Appended again is my post which you keep deleting -> clearly
you dont want people to read it - and to which you> have not
yet given any substantive mathematical response:> > No, thats
not it. Ive seen your posting pattern before, and found> that
after spending time and effort to correct your mistakes and>
explain to you, you simply deny the truth, or just come back
before below, giving you againyour *rst* chance to actually
deal with the math. > Its rather bizarre, but now kind of
interesting.> Whats bizarre is that you keep deleting the
appended toavoid dealing with the math and to keep your
imaginary fans from reading it!> > James
Harris
problem here is so simple. You assume a factorization > of
your polynomial P(x) of the form> > (*) (5*a1(x) +
7)*(5*a2(x) + 7)*(5*b3(x) + 22).> > You note that since P(x)
is divisible by 49, there must be> some way to factor 49 out
of the terms in (*). You think> that the only possible way
such a factorization can occur> is of the form 7, 7, 1. But
you know that assuming a factorization > of the form 7, 7, 1
leads to a contradiction. The contradiction> can be proved in
two totally different ways: by using Galois> theory, and by
using a basic theorem in algebraic number> theory.> > You
conclude from this contradiction that there is a > problem
with the algebraic integers.> > What you *should* conclude is
that your claim that the only> way the factorization can occur
is 7, 7, 1 is wrong. > > Look hard. You do not really have a
proof of that claim.> You have a handwaving argument based on
your concept of> constant terms of functions.> > You think
your argument that such a factorization would> have to be 7,
7, 1 is ironclad. It is based on the idea> that the constant
terms in (*) are 7, 7, and 22, and in particular> 22 is
coprime to 7. Therefore you conclude that the third> factor
must be 1.> > You overlook the fact that when x <> 0, 5*b3(x)
+ 22 can be > non-coprime to 7 even though 22 and 7 have no
nonunit factors > in common.> > We say you are wrong about 7,
7, 1. There is another way > to factor 49 through the terms in
(*). Dik Winter and others have> described it explicitly. It
is not 7, 7, 1. It is> w1(x), w2(x), w3(x), where each of
these is a non-unit> algebraic integer factor of 49. None of
w1, w2, or w3 are> equal to 7. In particular, for example,
5*b3(x) + 22 is > divisible by w3(x). No contradiction
arises. The arithmetic > works out. The constant terms
multiple as they should > to 49*22.> > The real root of the
problem is that you are doing > factorization by inspection,
as in high school. You see those> constant terms 7, 7, and
22. The temptation to conclude > that the factorization of 49
must be as 7, 7, 1 is > overwhelming. You cannot imagine it
could be any other> way. Your many simplistic examples
involving reducible> polynomials work out as you expect. How
could it be> otherwise in this case?> > It is! The key thing
here is that for x <> 0 in general,> the polynomials involved
are irreducible. They simply do> not act like the ones in the
high-school textbooks. They> factor in ways that you do not
expect. In some ways their> properties can defy intuition. A
key fact is that 7> is not a prime in the algebraic integers.
It can be > factored in innitely many different ways. One
such> factor is w1(x) which Dik has specied exactly as:> >
w1(x) = gcd(a1(x), 49),> > where a1(x) is a root of > > (**)
a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x).> > This
is an explicit specication in the sense that > (1) the roots
of the cubic (**) can be computed explicitly, and > (2) there
is a specic algorithm for computing the gcd> function in the
algebraic integers (due to Dedekind).> > Your factorization
leads you to a contradiction. What> that should tell you is
that you have made a mistake. Nevertheless> you perversely
cling to it and refuse to see that another> factorization is
possible. Our factorization has no > such problem. It makes
sense and is consistent with > existing theory. The
factorization itself is dependent on> the value of x. This
should not be surprising, since> the values of a1(x), a2(x)
and b3(x) are dependent on x.> In no way is this weird or
bizarre. > > math can take you only so far. In this case, not
quite far> enough. > > > Nora
B.
The problem here is so simple. You assume a factorization > > of your polynomial P(x) of the form> >> > > >> > (*)
(5*a1(x) + 7)*(5*a2(x) + 7)*(5*b3(x) + 22).> >> > >> Thats
like saying that someone assumes that sqrt(2) is a factor of
2.> >> > >> Factorizations just exist.> >> > >> > >> I am
just summarizing what you have said. I am not> >> disagreeing
with it, at least not at this point.> >Yeah, like what if
someone says, lets assume that 6 has 3 as a> >factor.Technically its correct.> >But the use of the word assume
looks strange there, as 3 IS a factor> >of 6.> > > >
Typically an expression can be factored lots of ways. You
assumed a> > factorization of the specied form. I see
absolutely nothing> > wrong with my wording and I dont agree
that it looks strange.> > In any case this is an utterly minor
issue. > > > > A person might assume that theyre sounding
rational, and cogent, when> in fact theyre sounding
defensive and recalcitrant.> > You dont just assume a truth.
> > Ill give you another example as Im somewhat curious
about your state> of mind.> > Lets say that I make it a
point to say I assume that the sun will> rise tomorrow.> >
Isnt there some implication in that statement to you?> > >
The analogy doesnt work. There are lots of ways to factor>
an expression. If I said, What is the form of the>
factorization of (125*x^3 + 75*x^2 + 45*x + 7)?, most people>
might think I wanted a factorization as a polynomial in x. >
That, however, is not what you have done with your
expression.> You have factored it as a polynomial in 5. That
is why it> is important to say that you *assumed* a
factorization of> the specied form. It is a natural and even
necessary > part of the discussion.Lets say you assume that
irrationals are not integers.Yet someone else tells you, hey,
why assume that, as theyre just not?Dont you think a
*rational* response might be to go, oh, yeah, youreright?Now
then, what if instead, you acted defensive and argued
endlesslyabout your use of words where there was an
implication that somethingmight not be true by use of the
word assume, which *reasonable*people could see.Is there any
possibility that you might concede that saying assumefor a
truth might be loaded or could look like youre trying to
implythat something might not be true? > And here, its a
distraction. You are trying desperately to focus> on a minor
wording issue to take attention away from the fact> that you
have not dealt with the mathematics.> Im curious about how
you manage to spend so much time ratherobviously ignoring the
truth, twisting things a certain way, yetacting as if youre
aboveboard no matter how often youre caught.Your defensive
reaction here with the word assume is a case inpoint.You
basically refuse to be rational, and refuse to concede even
minorpoints, arguing endlessly as if youre always right, no
matter what,or how much evidence is produced to challenge
your world view.> > > > >Theres no need to assume anything.Now it seems to me that youre playing stupid when the
obvious is> >pointed out to you.> > > > Youre quibbling over
a minor wording issue. Meanwhile you> > delete and evade the
underlying math below.> > > > Is that an assumption on your
part?> > > > No. Its like saying the sun will rise
tomorrow.> So then, is my use of the word assumption
appropriate in context toyou?Can you see it as communicating
something by its use?> > >> > >> > > >> > You note that since
P(x) is divisible by 49, there must be> >> > some way to
factor 49 out of the terms in (*). You think> >> > that the
only possible way such a factorization can occur> >> > is of
the form 7, 7, 1. But you know that assuming a factorization
> >> > >> And now you claim to know what I *think*?> >> > > >> > >> You have said it many times.> >Then give a single
quote.> > > > OK, how about one that you posted YESTERDAY in
the thread> > Mathematical consistency, courage ?> > > > The
constant terms are 7, 7, and 22, but when 49 is divided> >
off the resulting constant terms are 1, 1, and 22, which
means> > from BASIC arithmetic, theres only one way to go. > > There is, in fact, only one way that can happen.> > > >
Yup, now then, look back at your claim about what I thought. > OK. (Looks back). Yes, by Jove, they are identical> in
meaning.No they are not. Im noting that constants, like 7,
or 22, cant turninto some other number out of thin air.
There has to be an operationthat takes place. In this case
you get a transformation from constantterms 7, 7 and 22 to a
result with constant terms 1, 1, and 22.That requires that 7
becomes 1 after that operation.In this case the operation is
division, and 7 *has* to have beendivided by 7, to give the
result 1.Now then, look *again* at what you said, and see if
you can begin tosee differences in your claims about what I
*thought* and myexplanation of what Ive said. > > >Here
youre caught, yet again, but rather than acknowledging theobvious--you cant read minds--you instead toss out yet
another> >statement.> > > > See above. That was a direct,
unedited quote. I cant read > > minds but I can read what
you post. So can everyone else who> > sees this. > > > > Yup.
Now then, look carefully back at what *you* said I thought,
and> Im curious to hear how you feel that the quote you gave
is saying the> same thing.> > Be quite detailed please.> > > I
said you thought that the only way to split 49 into three >
factors that would divide the three terms of> > (5 a_1(x) +
7)*(5 a_2(x) + 7)*(5 b_3(x) + 22)> > was as 49 = 7 * 7 * 1,
where the rst 7 divides (5 a_1(x) + 7),> the second 7
divides (5 a_2(x) + 7), and 1 divides (5 b_3(x) + 22).> And
indeed, that is precisely, exactly what you claimed.Nope. I
talked about the *constant* terms in those factors, and
notedthat those constant terms changed, and that there was
only one way forthem to change.Even now you STILL dodge and
twist to try and challenge the facts.Im being very specic
here for a reason, but if you havemathematical expertise then
you should appreciate being VERY specic.Can you begin to see
differences between what youre saying and whatIm saying?> >
If you now want to retract that, ne with me. That would> open
the door to other factorizations of 49 - for example> 49 =
w1(x)*w2(x)*w3(x), where w1(x), w2(x), and w3(x) are>
non-unit algebraic integers, and for x <> 0, none of them>
are equal to 7. If you are now willing to contemplate >
factorizations like that, rather than your inspection-based>
factorization 7, 7, 1, then your whole edice collapses.> So,
retract away!> My position still remains that if you have
*constants* 7, 7 and 22,and then have 1, 1, and 22, after
dividing by 49, its a matter ofsimple arithmetic to
determine what happened.Now theres a difference between that
and what youre claiming.> > > > >So I challenge you to give a
single quote.> > > > > > Done. See above. I await your
explanation.> > > > Its very simple. In fact, Ive been
rather detailed about my> factorization, and how it works.> >
But you make statements that are attempts at trying to change
the> facts.> > > Such as???Now you play dumb. > Like even
though you gave that quote, you then didnt even bother to>
reconcile obvious differences!> > Like you have 7, 7, 1,
while I talk of 7, 7 and 22.> > > Oh, PLEASE!!! One of these
is your proposed factorization> of 49. The other is the
constant terms of your factorization> of P(x). If you think I
have claimed they are the same, you> are simply delusional.
You clearly, unambiguously think, as you > said in the quote,
that the only way to factor 49 through the three> linear
factors of P(x) was as 7, 7, and 1. There is no doubtAnd now
youre STILL trying to mindread. Now I didnt understand
whatyou meant by 7, 7 and 1 before, but now I do, and it
makes sense thatI didnt as I keep saying 7, 7 and 22 talking
about the constantterms.As for your claim about ways to factor
49, thats just somethingyoure making up.What I *say* is that
the constant terms go from being 7, 7 and 22 tobeing 1, 1, and
22, when 49 is divided off, which is an easilyveriable fact. > Its something Ive watched before in your postings where
you make> little mistakes, big mistakes, and strange
mistakes, but when its> pointed out, you always deny!!!> > >
I ask of you what you asked of me: provide one example.> My
replies to your posts here are examining your behavior, so
you needonly consult them.> > Later, I catch you making the
*same* mistakes, and with VERY long> posts, as if part of
your strategy is burying your behavior with a lot> of
verbiage.> > > This whole exchange is simply surreal. I make
a statement about> what you thought. You say I am falsely
reading your mind. I > provide a direct quote from 1 day
before which proves 100% of> what I said about your thinking.
You squirm and turn and try> to weasel out of it. You try to
deny the meaning of your own > quotation. WHO THE HELL ARE
YOU TRYING TO FOOL, ANYWAY ?Im curious about how you go
about what youre doing, to see if itsconscious, or
unconscious as Im curious. > > Its kind of interesting
behavior, though bizarre.> > >> > >> > >> Who cares what I
think, > >> > >> > >> You have made a huge error. My interest
is in trying to> >> get you to realize it.> >> > >I dont
think so. For instance, a while back in response to your> >  I will never, never engage in any nonpublic conversations
with> > you on any topic.> > So now you *nally* admit that
you need an audience. Why do I have> to drag the truth out of
you?> > > I dont want an audience. I want witnesses. I would
never trust> you to quote anything I said o'ne. How is that
relevant to your claims about your purpose?Why dont you say
what your purpose really is?> > You make claims that dont t
with your behavior.> > > > >Now then, why dont you try again
and explain what you *really* are> >trying to do?> > > > Its
simple. I am trying to show you that your factorization of 49
> > in the 7, 7, 1 pattern is the WRONG FACTORIZATION when x
<> 0. Needless > > to say you are not getting it. > > Now
theres one of the errors YET AGAIN, and I have to admit that
its> a puzzle to me. > > > OK, I will say it as succinctly
and clearly as I possibly can.> > You divide > > (5 a_1(x) +
7)*(5 a_2(x) + 7)*(5 b_3(x) + 22)> > by 49. You split 49 into
3 pieces: say, 49 = A*B*C.> Your goal is for each of > > (5
a_1(x) + 7) / A,> > (5 a_2(x) + 7) / B, and> > (5 b_3(x) +
22) / C> > each to be algebraic integers.No. Thats not true.
My point is that in generaly they are NOTalgebraic
integers.Surely that should be something that you can now
accept so that you nolonger say what you just did, right?That
is, are you no longer going to claim that I say that after 49
isdivided off from both sides that you still have algebraic
integers,since Ive just told you explicitly thats not what
Im saying? > > You say the ONLY POSSIBLE hope of achieving
this is to> specify:> > A = 7, B = 7, C = 1,> > because that
is the only conguration that divides> the constant terms as
you want.> > We say NO. For one thing, as you now
acknowledge, > A = 7, B = 7, and C = 1 does NOT WORK in
general: > in particular,> > (5 a_1(x) + 7) / 7 > > is NOT an
algebraic integer for most x <> 0.> > You know this.Then why
did you claim above that I make an opposite claim?Can you see
what Im seeing here? Is there any part of you
thatacknowledges that, hey, other people reading might wonder
whats goingon with you? James Harris
analogy doesnt work. There are lots of ways to factor>> an
expression. If I said, What is the form of the>>
factorization of (125*x^3 + 75*x^2 + 45*x + 7)?, most
people>> might think I wanted a factorization as a polynomial
in x.>> That, however, is not what you have done with your
expression.>> You have factored it as a polynomial in 5. That
is why it>> is important to say that you *assumed* a
factorization of>> the specied form. It is a natural and
even necessary>> part of the discussion.Lets say you assume
that irrationals are not integers.Yet someone else tells you,
hey, why assume that, as theyre just not?Dont you think a
*rational* response might be to go, oh, yeah, youre>right?>
Again not a valid parallel. This argument is so goddamned
sillyand petty. I have little patience with arguments about
semantic trivia. If you dont like my saying that you assumed
a factorization of the form given, you can replace assumed by
specied. That conveys the same meaning: that there were lots
of other ways it could have been specied (or assumed for that
matter).>Now then, what if instead, you acted defensive and
argued endlessly>about your use of words where there was an
implication that something>might not be true by use of the
word assume, which *reasonable*>people could see.Is there any
possibility that you might concede that saying assume>for a
truth might be loaded or could look like youre trying to
imply>that something might not be true?> And here, its a
distraction. You are trying desperately to focus>> on a minor
wording issue to take attention away from the fact>> that you
have not dealt with the mathematics.>>Im curious about how
you manage to spend so much time rather>obviously ignoring
the truth, twisting things a certain way, yet>acting as if
youre aboveboard no matter how often youre caught.Your
defensive reaction here with the word assume is a case
in>point.You basically refuse to be rational, and refuse to
concede even minor>points, arguing endlessly as if youre
always right, no matter what,>or how much evidence is
produced to challenge your world view.> No, youre simply
wrong. You assumed a factorization of acertain form. You
could have assumed many other forms. Itis appropriate to note
what you did and to convey the impressionthat the choice
involved a decision on your part. By no meanswas I implying
that your assumption was wrong. Its not.There IS a
factorization of that form. I was just describing thatthat
was the form you were specifying..>> > >Theres no need to
assume anything.>> > >Now it seems to me that youre playing
stupid when the obvious is>> >pointed out to you.>> > > >
Youre quibbling over a minor wording issue. Meanwhile you> delete and evade the underlying math below.>> > Is that an
assumption on your part?>> No. Its like saying the sun will
rise tomorrow.>>So then, is my use of the word assumption
appropriate in context to>you?> You are not describing an
analogous usage. It makes no senseto describe it as an
assumption; no other alternativesneed be contemplated or
excluded. That is not the case, however,with your
factorization. Many other equally plausible, validchoices
were available.>Can you see it as communicating something by
how about one that you posted YESTERDAY in the thread>> >
Mathematical consistency, courage ?>> > > The constant terms
are 7, 7, and 22, but when 49 is divided>> > off the
resulting constant terms are 1, 1, and 22, which means>> >
from BASIC arithmetic, theres only one way to go.>> > >
There is, in fact, only one way that can happen.>> > Yup, now
then, look back at your claim about what I thought.>> OK.
(Looks back). Yes, by Jove, they are identical>> in
meaning.No they are not. Im noting that constants, like 7,
or 22, cant turn>into some other number out of thin air.
There has to be an operation>that takes place. In this case
you get a transformation from constant>terms 7, 7 and 22 to a
result with constant terms 1, 1, and 22.> By the slowest
possible steps, we approach the truth: againstyour kicking
and screaming every millimeter of the way. Yes, *you* want to
end up with 1, 1, 22 in place of the constantterms after the
factorization. That is the problem. To youthat is the only
conceivable result of what can happen afterthe division has
occurred. You cannot conceive that afterthe division of the
constant terms, you might have 7/A 7/Band 22/C,where A*B*C =
49 and (most importantly) where 22/C is NOTan algebraic
integer. This is the central problem. You think it is
absolutelynecessary, after division of both sides by 49,
that22/C be an algebraic integer. That causes you to
conclude, since 7 and 22 are coprime, that C = 1. Anything
else is just plain beyond your imagining. But anything else
is precisely what happens. For x <> 0in general, 49 can be
factored into three factors, A, B, and C, allof them
algebraic integers, none of them a unit, none ofthem equal to
7, such that (5 a_1(x) + 7) / A, (5 a_2(x) + 7) / B,and (5
b_3(x) + 22) / Care ALL algebraic integers. What you dont
seem to understandis that, ***even if 22/C is not an
algebraic integer***, it isperfectly possible that (5 b_3(x)
+ 22) / CIS one. And that is sufcient. Why ? Because on one
sideof the equation you have P(x)/49, which is clearly and
unambiguouslyan algebraic integer. And on the other side of
the equation youhave [(5 a_1(x) + 7)/A]*[(5 a_2(x) + 7)/B]*[5
b_3(x) + 22)/C]which is nothing more than the product of three
algebraicintegers. AND THAT IS WHAT YOU NEED. YOU DONT NEED
THAT22/C IS AN ALGEBRAIC INTEGER. I should note also,
however, that if you go back andre-write (5 b_3(x) + 22) as,
instead, (5 a_3(x) + 7), anddivide *that* by C, you get (5
a_3(x) + 7)/C = 5 a_3(x)/C + 7/C,and it WILL be the case that
5 a_3(x)/C and 7/C areboth algebraic integers. The same is
true for a_1(x)/A,a_2(x)/B, 7/A, and 7/B. Putting it all
together, youget a complete factorization of P(x)/49 of the
form (5 c_1(x) + d1(x))*(5 c_2(x) + d2(x))*(5 c_3(x) +
d3(x))where EVERY COEFFICIENT is an algebraic integer. This
is exactlythe sort of thing that you get from the
Magidin-Mckinnonresult. It is a factorization of the form you
were aimingfor. Remember? And you CANNOT GET THERE by the 7,
7, 1 pattern.You CAN, and you MUST factor 49 so that all
three factors A,B, and C, are not co-prime to 7 !>That
requires that 7 becomes 1 after that operation.> Yes. Your
assumption that the factorization has to be 7, 7, 1comes as a
result of your assumption that the constantterms after the
division must be 1, 1, and 22. AND THATASSUMPTION IS FALSE.
THAT is where you are making your mistake. There is *another*
factorization which gives algebraicintegers on both sides of
the equation. 49 divides out of the right side of the
equation to yield a product of factors with algebraic integer
coefcients, just as it should. You should go back and look at
the Rick Decker thread.Compute the analog of 22/C for *his*
factorization: youwillfind that when x = 1, it is 2/sqrt(7),
which is notan algebraic integer. Yet as I note also in that
thread, Q(1)/7 = (5*sqrt(-2) + sqrt(7))*(5*sqrt(-2) +
sqrt(7))and *** note well *** - this is a factorization in
which everycoefcient in sight is an algebraic integer - just
the kind of factorization that you wanted to get in the rst
place, and the kind that you think cannot possibly be
obtained. Clearly you are wrong. Believe what you see.>In
this case the operation is division, and 7 *has* to have
been>divided by 7, to give the result 1.> With the *correct*
factorization, you dont get 1. You geta term which is
non-coprime to 7. You are assuming that youmust get 1. That
assumption is incorrect.>Now then, look *again* at what you
said, and see if you can begin to>see differences in your
claims about what I *thought* and my>explanation of what Ive
said.> You assume - yes, ASSUME - that you *absolutely must*
end up with1, 1, 22 after you have done the division.
Conditional onthat, you would be right. That assumption
however is false.Therefore your belief that the only
reasonable factorizationof 49 is 7, 7, 1 is wrong. Yes, 7, 7,
1 is the only wayto end up with 1, 1, 22. But 1, 1, 22 is not
the right goalin the rst place. There is no doubt however
that you*think* it is. This is your whole basis for having
said thata_1(x)/7 should be an algebraic integer.>> >Here
youre caught, yet again, but rather than acknowledging the>obvious--you cant read minds--you instead toss out yet
another>> >statement.>> > > > See above. That was a direct,
unedited quote. I cant read>> > minds but I can read what
you post. So can everyone else who>> > sees this.>> > Yup.
Now then, look carefully back at what *you* said I thought,
and>> Im curious to hear how you feel that the quote you
gave is saying the>> same thing.>> Be quite detailed
please.>> I said you thought that the only way to split 49
into three>> factors that would divide the three terms of>>
(5 a_1(x) + 7)*(5 a_2(x) + 7)*(5 b_3(x) + 22)>> was as 49 = 7
* 7 * 1, where the rst 7 divides (5 a_1(x) + 7),>> the second
7 divides (5 a_2(x) + 7), and 1 divides (5 b_3(x) + 22).>> And
indeed, that is precisely, exactly what you claimed.Nope. I
talked about the *constant* terms in those factors, and
noted>that those constant terms changed, and that there was
only one way for>them to change.> What you noted however was
wrong. There is another way forthem to change. You have just
moved the error in your thinkingback about an inch. You
thought the only factorization thathad any prayer of working
at all would necessarily give you1, 1, 22 after division.
That thought was incorrect. But still, you thought it. You
were (and are) certain of it. And your (incorrect) certainty
that 1, 1, 22 was necessary caused you to think that the only
factorization of 49 that would work was 7, 7, 1. Anincorrect
conclusion, but again: that is what you thought.>Even now you
STILL dodge and twist to try and challenge the facts.Im being
very specic here for a reason, but if you have>mathematical
expertise then you should appreciate being VERY specic.Can
you begin to see differences between what youre saying and
what>Im saying?> I said you thought only one form of
factorization waspossible. You have conrmed that totally.
You haveleft no doubt about that whatsoever. You still think
it.It is what has led you (misguidedly, and unnecessarily) to
try to create your object ring. After all, your instinctthat
there had to be some factorization which worked was,in fact,
correct. Your belief that that factorization couldonly be of
one form was not.>> If you now want to retract that, ne with
me. That would>> open the door to other factorizations of 49 -
for example>> 49 = w1(x)*w2(x)*w3(x), where w1(x), w2(x), and
w3(x) are>> non-unit algebraic integers, and for x <> 0, none
of them>> are equal to 7. If you are now willing to
contemplate>> factorizations like that, rather than your
inspection-based>> factorization 7, 7, 1, then your whole
edice collapses.>> So, retract away!>>My position still
remains that if you have *constants* 7, 7 and 22,>and then
have 1, 1, and 22, after dividing by 49, its a matter
of>simple arithmetic to determine what happened.> OK, ne, go
back a step. Do you or do you not think that thatafter
division by factors of 49, the constant terms MUST be
transformedfrom 7, 7, 22 to 1, 1, 22 ? That no other
end-result isconceivable? Be very careful in answering this.
The big picture here is, you have made an
erroneousassumption. Perhaps you are trying to salvage some
scrapof pride. Forget it. Your assumptions - yes, ASSUMPTIONS
-about what must happen after the division are wrong. Theyare
based on the idea that after the divisions, theconstant terms
must end up being algebraic integers.As outlined above and as
absolutely proved by Rick Deckersexample, that ASSUMPTION is
FALSE. That is your basicerror.>Now theres a difference
between that and what youre claiming.> > >So I challenge you
to give a single quote.>> > > > Done. See above. I await your
explanation.>> > Its very simple. In fact, Ive been rather
detailed about my>> factorization, and how it works.>> But
you make statements that are attempts at trying to change
the>> facts.>> Such as???Now you play dumb.> And you give no
example.>> Like even though you gave that quote, you then
didnt even bother to>> reconcile obvious differences!>> Like
you have 7, 7, 1, while I talk of 7, 7 and 22.>> Oh, PLEASE!!!
One of these is your proposed factorization>> of 49. The other
is the constant terms of your factorization>> of P(x). If you
think I have claimed they are the same, you>> are simply
delusional. You clearly, unambiguously think, as you>> said
in the quote, that the only way to factor 49 through the
three>> linear factors of P(x) was as 7, 7, and 1. There is
no doubtAnd now youre STILL trying to mindread. Now I didnt
understand what>you meant by 7, 7 and 1 before, but now I do,
A miracle. You actually read something and understood it.
Praisethe Lord. Is there hope after all?>and it makes sense
that>I didnt as I keep saying 7, 7 and 22 talking about the
constant>terms.> You should read things with the idea that
the personwriting them might have intended something other
thanwhat you wanted to see.>As for your claim about ways to
factor 49, thats just something>youre making up.> Ah, at
last we get to the real heart of our disagreement! Therest
actually is uff. This is the big sh. No, factors A, B, and
C as described above actually exist. The proof is closely
related to the Magidin-Mckinnon result. Dont believe it? Try
producing a proof that it cant happen!And be sure - be very,
very sure - that you arent assuming whatyou want to
prove!>What I *say* is that the constant terms go from being
7, 7 and 22 to>being 1, 1, and 22, when 49 is divided off,
which is an easily>veriable fact.> Not quite. You have to
speak precisely here. It happens *only if* you factor 49 as
7, 7, 1. You can factor 49 in another way, as 49 = A*B*C.
When you divide 7, 7, 22 by these, you do NOT get 1, 1, and
22. You get 7/A, 7/B, and 22/C. The former two are algebraic
integers. 22/C is not. Try really, really hard to get this.
The fact that 22/C is not an algebraic integer is not a
problem. Remember, there are two ways to express that third
factor: (5 b_3(x) + 22) = (5 a_3(x) + 7). It turns out that
7/C, like 7/A and 7/B, *is* an algebraic integer. So is
a_3(x)/C, as well as a_1(x)/A and a_2(x)/B. Thus in 5
[a_3(x)/C] + 7/Call of the coefcients are algebraic
integers. That is what you wereaiming for way back at the
beginning. Remember? You havent lostsight of your goal, have
you?>> Its something Ive watched before in your postings
where you make>> little mistakes, big mistakes, and strange
mistakes, but when its>> pointed out, you always deny!!!>> I
ask of you what you asked of me: provide one example.>>My
replies to your posts here are examining your behavior, so
you need>only consult them.> Again I note a total absence of
an example. Let me knowif youfind one.>> Later, I catch you
making the *same* mistakes, and with VERY long>> posts, as if
part of your strategy is burying your behavior with a lot>> of
verbiage.>> This whole exchange is simply surreal. I make a
statement about>> what you thought. You say I am falsely
reading your mind. I>> provide a direct quote from 1 day
before which proves 100% of>> what I said about your
thinking. You squirm and turn and try>> to weasel out of it.
You try to deny the meaning of your own>> quotation. WHO THE
HELL ARE YOU TRYING TO FOOL, ANYWAY ?Im curious about how
you go about what youre doing, to see if its>conscious, or
unconscious as Im curious.> Whuh??? Can I quote you on
that?>> Its kind of interesting behavior, though bizarre.>> >> >> Who cares what I think,>> >> >> >> You have made a
huge error. My interest is in trying to>> >> get you to
realize it.>> >> > >I dont think so. For instance, a while
back in response to your>> > > > I will never, never engage
in any nonpublic conversations with>> > you on any topic.>>
So now you *nally* admit that you need an audience. Why do I
have>> to drag the truth out of you?>> I dont want an
audience. I want witnesses. I would never trust>> you to
quote anything I said o'ne.How is that relevant to your
claims about your purpose?> It wasnt supposed to be
relevant. It was supposed to be an answer to your
question.>Why dont you say what your purpose really is?> The
best outcome I can imagine is that you nallysee why your
reasoning on this whole thing is incorrect.I am none too
optimistic about it.>> You make claims that dont t with
your behavior.>> > >Now then, why dont you try again and
explain what you *really* are>> >trying to do?>> > > > Its
simple. I am trying to show you that your factorization of
49>> > in the 7, 7, 1 pattern is the WRONG FACTORIZATION when
x <> 0. Needless>> > to say you are not getting it.>> Now
theres one of the errors YET AGAIN, and I have to admit that
its>> a puzzle to me.>> OK, I will say it as succinctly and
clearly as I possibly can.>> You divide>> (5 a_1(x) + 7)*(5
a_2(x) + 7)*(5 b_3(x) + 22)>> by 49. You split 49 into 3
pieces: say, 49 = A*B*C.>> Your goal is for each of>> (5
a_1(x) + 7) / A,>> (5 a_2(x) + 7) / B, and>> (5 b_3(x) + 22)
/ C>> each to be algebraic integers.No. Thats not true. My
point is that in generaly they are NOT>algebraic integers.> I
know what your point is. Your point is wrong. If you picked A
= 7, B = 7, and C = 1, you would be right. But it is possible
to pickDIFFERENT A, B, and C so all the coefcients ARE
algebraicintegers. A, B, and C are dependent on x. That is
what Dik, Arturo, I, and others have been saying for months.
Have you really not understood that?>Surely that should be
something that you can now accept so that you no>longer say
what you just did, right?That is, are you no longer going to
claim that I say that after 49 is>divided off from both sides
that you still have algebraic integers,>since Ive just told
you explicitly thats not what Im saying?> You CAN divide 49
off from both sides so that you stillhave algebraic integers.
You CANNOT do it by using 7, 7, 1.>> You say the ONLY
POSSIBLE hope of achieving this is to>> specify:>> A = 7, B =
7, C = 1,>> because that is the only conguration that
divides>> the constant terms as you want.>> We say NO. For
one thing, as you now acknowledge,>> A = 7, B = 7, and C = 1
does NOT WORK in general:>> in particular,>> (5 a_1(x) + 7) /
7>> is NOT an algebraic integer for most x <> 0.>> You know
this.Then why did you claim above that I make an opposite
claim?> Jesus, you are either dumb as a post or at least
illiterate.Cant you read the above ? Yes, you say that the
only conceivableway to divide off 49 is by factoring it in
the 7, 7, 1 pattern, andthen INDEED, the result if you do
that is that the resulting coefcientsare not algebraic
integers. I AGREE WITH THAT. Understand ?Please shake your
head yes or no. Go back and read what you said. You said that
dividing by 49 had to transform 7, 7, 22 into 1, 1, 22. You
did not qualify how thatdivision by 49 might be done. You
were thinking, though you did notsay it explicitly, that you
were factoring 49 as 7*7*1. If you *had* said that, you would
have been right technically (though totally wrong with respect
to the big picture). But you CAN factor 49 as A*B*C with A, B,
and C something other than 7, 7, 1 so that you DO get all
algebraic integercoefcients. Again, if you want to see it
done *explicitly* with allthe gory arithmetic details for a
quadratic, see my post in the Rick Decker thread (quoted
above also).>Can you see what Im seeing here? Is there any
part of you that> acknowledges that, hey, other people
reading might wonder whats going> on with you?> I am not
concerned in the least about that socialstuff. Thats your
concern. People can think about me whatever they want. Nora
B.James Harris
=Suppose f is a representation of the group
G in the vector space V over k.Then f* is the representation
of G in the dual vector space V* dened bythe action (gp)(v)
= p((g^-1) v), where g is in G, p is in V*, and v is inV.
(Firstly, why is this action dened like this?)Now suppose f
is given in matrix form :f : g |--> A(g)where g is in G and
A(g) denotes the matrix g is sent to.Then f* (the dual
representation) is the representationg |---> (A(g)^{-1})^T,
the transpose of the inverse of the matrix. Cansomeone tell
me why the transpose of the inverse denes the
dualrepresentation?Moshe
 Suppose f is a representation
of the group G in the vector space V over k.> Then f* is the
representation of G in the dual vector space V* dened by>
the action (gp)(v) = p((g^-1) v), where g is in G, p is in
V*, and v is in> V. (Firstly, why is this action dened like
this?)> We have f:G->L(V), where L(V) denotes the linear
operators on V. We dene f*:G->L(V*) so that<gp,gv> =
<p,v>where <p,v> = p(v) denes the duality between p in V*
and v in V. This gives > Now suppose f is given in matrix
form :> > f : g |--> A(g)> > where g is in G and A(g) denotes
the matrix g is sent to.> > Then f* (the dual representation)
is the representation> > g |---> (A(g)^{-1})^T, the transpose
of the inverse of the matrix. Can> someone tell me why the
transpose of the inverse denes the dual> representation?>
<p,v> = <gp,gv> = <gp,A(g)v>.Substituting v->A(g)^{-1}v, we
get<gp,v> = <p,A(g)^{-1}v> = <A(g)^{-1,T}p,v>(since for any
matrix <p,Bv>=<B^T p,v>). Since this is true for all v, we
getgp = A(g)^{-1,T}p.
 Suppose f is a representation of
the group G in the vector space V overk.> Then f* is the
representation of G in the dual vector space V* denedby> the
action (gp)(v) = p((g^-1) v), where g is in G, p is in V*, and
v isin> V. (Firstly, why is this action dened like this?)> We
have f:G->L(V), where L(V) denotes the linear operators on V.
Wedene> f*:G->L(V*) so that> <gp,gv> = <p,v where <p,v> =
p(v) denes the duality between p in V* and v in V.
ThisgivesWhy exactly do we want to dene f*:G->L(V*) so that
<gp,gv> = <p,v>? Whydoes this denition make sense? Is < >
the standard inner product? Whatdo you mean by duality
between p in V* and v in V?Moshe
=Cutting Edge New Physics
Ideas1. Dark energy is almost 2/3 of the Universe and is a
repelling anti-gravity eld.2. Dark matter is almost 1/3 of
the Universe and is an attractive us are made of.3. Ordinary
matter and light is only a very tiny part of the Universe.4.
Put a chunk of dark energy near a chunk of dark matter and
you basically have a weightless warp drive. This means you
feel weightless and the Universe passes by you seemingly
faster than the speed of light. You can time travel to your
past and to your future and beyond under certain
conditions.5. Einsteins theory of special relativity from
1905 is the unication of space with time into a rigid
space-time and the unication of matter and energy.6.
Einsteins general theory of relativity shows that space-time
is not rigid but is warped not only by energy but also even
by itself in a self-organizing way.7. Quantum theory is about
the informational relationship between the observer and the
observed. Some important physicists like John Von Neumann,
Eugene Wigner and Roger Penrose thought that our inner
consciousness plays a key role in quantum physics. Other
important physicists violently disagree. It depends what you
mean by quantum theory. There are many different ways of
interpreting it and its boundary of validity is not yet
understood especially in the strong gravity eld of a black
hole. A key issue is signal nonlocality, i.e. the use of
entanglement as a direct communication channel without an
auxiliary light speed limited classical channel as in quantum
teleportation and cryptography where Eve cannot tap the
channel between Alice and Bob without them knowing it. These
applications require irreducible quantum randomness that is
thought to forbid signal nonlocality, i.e. to enforce signal
locality or passion at a distance (Abner Shimony). On the
other hand see http://www.quantumelds.com/469Maclay.pdf The
analysis of Lenny Susskind, for example, of information loss
behind the event horizon of a black hole, depends on signal
locality as does the non-Boolean toposquantum logic of
cosmology mentioned by Lee Smolin. Yet, the Bohmian analysis
of Antony Valentini suggests otherwise. The idea here is that
orthodox nonlocal micro-quantum theory with irreducible
randomness and consequent signal locality is only the
sub-quantal thermal equilibrium limit of a more general
non-equilibrium local MACRO-QUANTUM theory for cosmology and
the emergence of classical curved space-time with signal
nonlocality.http://qedcorp.com/APS/EmergentGravity.pdf8. The
which has two main schools of thought, string theory and loop
theory. The two theories may simply be dual dimensional images
of each other. The basic object of loop theory is a
two-dimensional undivided quantum of area and the basic
object of string theory is a one-dimensional undivided
quantum of length, which together make a three-dimensional
space. Loop theory describes three-dimensional space as
zero-dimensional point on a kind of quantum computing spin
network. When you add a kind of proto time to it you get a
spin foam. Two-dimensional area is described by a
1-dimensional stringy edge on the spin network. The duality
is intuitively obvious. The 1-dimensional physical string is
really like a linked chain with each link as a bit of
information not unlike the purely mathematical strings of
computer theory. No matter that the string is embedded in a
higher dimensional hyperspace since all but three of those
extra spacelike dimensions are curled up into tiny circles
with radii called moduli forming what the mathematicians like
to call a Calabi-Yau space. Brian Greene calls this an Elegant
Universe in which the spin foam is weaved into the fabric of
reality. God is seen by Brian as a kind of tailor or fashion
designer reminding me of the Vatican scene in Fellinis
Roma.9. Both the string and the loop theorists think that the
quanta of length and area respectively are always very tiny.
They may be mistaken in this belief. Then again, I may be
mistaken.
 Cutting Edge New Physics Ideas 1. .8bDark
energy.8a is almost 2/3 of the Universe and is a repelling>
anti-gravity eld. 2. .8bDark matter is almost 1/3 of the
Universe and is an attractive> us are made of.It would seem
hard to believe that Dark Matter stays so perfectly
separatedfrom the rest of matter. Youd think some of this
stuff would come oatinginto the regions of the universe
where the rest of us live too.Another possibility put forth
is the Modied Newtonian Dynamics (MOND),where gravitational
attraction take on new equational forms depending on
thedistance of measurement from centre of mass to centre of
mass, over greatdistances.> 3. Ordinary matter and light is
only a very tiny part of the Universe. 4. Put a chunk of dark
energy near a chunk of dark matter and you> basically have a
weightless warp drive. This means you feel weightless> and
the Universe passes by you seemingly faster than the speed of
light.> You can time travel to your past and to your future
and beyond under> certain conditions.Its a possibility,
except that dark energy seems to only take effect onceyou
have sufcient distance. Like the distances between galactic
clusters.And actually I wonder if this isnt also another
boundary condition of MOND,where the equations of gravity
actually turn it into a repelling force? Yousuf Khan
 4.
Put a chunk of dark energy near a chunk of dark matter and
you > basically have a weightless warp drive. This means you
feel weightless > and the Universe passes by you seemingly
faster than the speed of light. > You can time travel to your
past and to your future and beyond under > certain
conditions.Why havent we been visited by future folk? Tha
have had eternity to develop this idea.Bob Kolker
> 4.
Put a chunk of dark energy near a chunk of dark matter and
you >> basically have a weightless warp drive. This means you
feel weightless >> and the Universe passes by you seemingly
faster than the speed of >> light. You can time travel to
your past and to your future and beyond >> under certain
conditions.> Why havent we been visited by future folk? Tha
have had eternity to > develop this idea.> Bob
KolkerDoubtless were quite boring by comparison.
 >
Doubtless were quite boring by comparison.Maybe they visited
Jack and did not let the rest of us know about it.Bob Kolker > 
> Doubtless were quite boring by comparison.> Maybe
they visited Jack and did not let the rest of us know about
it.That *would* likely do it.
 Cutting Edge New Physics
Ideas 4. Put a chunk of dark energy near a chunk of dark
matter and you> basically have a weightless warp drive. This
means you feel weightless> and the Universe passes by you
seemingly faster than the speed of light.> You can time
travel to your past and to your future and beyond under>
certain conditions.>Oh yeah, that makes a lot of sense.
LMAO.
 Cutting Edge New Physics Ideas 4. Put a chunk of
dark energy near a chunk of dark matter and you> basically
have a weightless warp drive. This means you feel weightless>
and the Universe passes by you seemingly faster than the speed
of light.> You can time travel to your past and to your future
and beyond under> certain conditions. > > > > Oh yeah, that
makes a lot of sense. LMAO.[EL]Jack Sarfatti is not insane at
all.He is the ultimate of the 20th Century knowledgeable
Physicist andwhat makes you laugh your ass out is not Jacks
insanity, but theinsanity of mainstream physics, which Jack
have mastered to a level ofdivinity.accepted widely by the
physics community including the darkness ofenergy and matter
along with holes that have a black colour and bangsthat are
big when size did not even make any sense. Not to mentiontime
that became a street in which Minkowski played hide and seek
withEinstein going to and fro while the twin was getting
younger.Wait until Jack tells you about the quarks with its
pink colour andvanilla avour not only going up and down but
dancing jerk andpsychedelic while occasionally doing the
samba and the rumba. It isall in the bubble chamber with
empirical evidence and pictures ofsophisticated kids
interpreting the maps made by chicken nails feedingin a barn
on special plastic that records the trails scratching
thesurfaces.We are living in a fabulous zoo so come and watch
the humans.Jack is an idol that represents this Century.He is
a physics showman.Refuting him is futile and I suggest that
you go back to your bakeryand bake some more bread before
your kids get hungry. Leave thedreamers dream but give them
no bread and they shall be excludedevolutionary
wise.EL
=Example:I am a researcher. I study UFOs and
ghosts.I have determined that %77 of all ghosts are spotted
within a 5 mile radiusof UFO sightings, and that %82 of all
UFO sightings are accompanied by ageneral increase in the
intensity and overall magnitude of ghost
relatedhauntings.Therefore, I conclude, via statistics, that
ghosts and UFOs are somehowrelated, and that ghosts are in
all probability using UFOs fortransportation
purposes.-----------------------------------------------Will
someone in the sci.math or sci.math.stat please stand up and
tell mewhy this is awed, and no I am not kidding. I need an
independent opinion.People are telling me Im crazy - where
did I go wrong ????? Can I call thisscience ?? Is my
reasoning awed somehow ?? Is there a problem with
mypopulations - or is this valid usage of stats ??
Example:> > I am a researcher. I study UFOs and ghosts.> > I
have determined that %77 of all ghosts are spotted within a 5
mile radius> of UFO sightings, and that %82 of all UFO
sightings are accompanied by a> general increase in the
intensity and overall magnitude of ghost related> hauntings. Therefore, I conclude, via statistics, that ghosts and UFOs
are somehow> related, and that ghosts are in all probability
using UFOs for> transportation purposes.> >
-----------------------------------------------> > Will
someone in the sci.math or sci.math.stat please stand up and
tell me> why this is awed, and no I am not kidding. I need
an independent opinion.> > People are telling me Im crazy -
where did I go wrong ????? Can I call this> science ?? Is my
reasoning awed somehow ?? Is there a problem with my>
populations - or is this valid usage of stats ??> > > > > Is
there some ofcially recognized listing of ghost sightings
andUFO sightings? If not, theres always GIGO to explain
it.
 Example: I am a researcher. I study UFOs and
ghosts. I have determined that %77 of all ghosts are spotted
within a 5 mileradius> of UFO sightings, and that %82 of all
UFO sightings are accompanied by a> general increase in the
intensity and overall magnitude of ghost related> hauntings.
Therefore, I conclude, via statistics, that ghosts and UFOs
are somehow> related, and that ghosts are in all probability
using UFOs for> transportation purposes.
----------------------------------------------- Will someone
in the sci.math or sci.math.stat please stand up and tellme>
why this is awed, and no I am not kidding. I need an
independentopinion. People are telling me Im crazy - where
did I go wrong ????? Can I callthis> science ?? Is my
reasoning awed somehow ?? Is there a problem withmy>
populations - or is this valid usage of stats ??> Is there
some ofcially recognized listing of ghost sightings and> UFO
sightings? If not, theres always GIGO to explain it.You are
getting warmer !!! You are very close to the fallacy Im
lookingfor. You have almost nailed it.Just one hint -In
statistics, the populations and samples are EXTREMELY
important. Forinstance, all of your work is invalidated if
your populations are messed up.Does anyone see a problem with
sha1:cUhzCUd56pq9NqbQ5oqOiDpsjYQ
=> Is there some
ofcially recognized listing of ghost sightings and>> UFO
sightings? If not, theres always GIGO to explain it. You are
getting warmer !!! You are very close to the fallacy Im
looking> for. You have almost nailed it. Just one hint -> In
statistics, the populations and samples are EXTREMELY
important. For> instance, all of your work is invalidated if
your populations are messed up. Does anyone see a problem
with my scientic population of ghosts and> UFOs ?Theres no
such thing as ghosts or UFOs? :)
= >> Is there some
ofcially recognized listing of ghost sightings and>> UFO
sightings? If not, theres always GIGO to explain it. You are
getting warmer !!! You are very close to the fallacy Im
looking> for. You have almost nailed it. Just one hint -> In
statistics, the populations and samples are EXTREMELY
important. For> instance, all of your work is invalidated if
your populations are messedup. Does anyone see a problem with
my scientic population of ghosts and> UFOs ? Theres no such
thing as ghosts or UFOs? :)YES !! Excellent, and now we are
getting somewhere !!So, while it is technically possible that
UFOs and gosts really exist, Icannot claim that any related
analysis of the subject is actually validuntil I can prove
that these questionable things really do exist.If I am a
scientist, and I spend the next 40 years using the entirety
of thebody of all statistical methods to analyze ghosts and
UFOs, even if I didall the math correctly, my so-called
science is still just garbage becauseI cannot prove that
UFOs and ghosts even exist !!Now we are getting somewhere
!!If anyone else is wliling to conrm what I am saying -
please just jumpright in. If I am just dead wrong please post
as well. Whatever. Lets hearit.
= Toilet Seat
<Barking@The.Moon> .8d.98.87.8b.8c .97.99.95
scientic population of ghostsand> > UFOs ? Theres no such
thing as ghosts or UFOs? :) YES !! Excellent, and now we are
getting somewhere !! So, while it is technically possible that
UFOs and gosts really exist,I> cannot claim that any related
analysis of the subject is actually valid> until I can prove
that these questionable things really do exist. If I am a
scientist, and I spend the next 40 years using the entirety
ofthe> body of all statistical methods to analyze ghosts and
UFOs, even if I did> all the math correctly, my so-called
science is still just garbagebecause> I cannot prove that
UFOs and ghosts even exist !! Now we are getting somewhere
!! If anyone else is wliling to conrm what I am saying -
please just jump> right in. If I am just dead wrong please
post as well. Whatever. Letshear> it.Well, there are extant
possibilities here, I ll admit, although some appearto be
out of the ordinary.In principle, you *could* perform several
interesting scientic tests,based only on the loose assumption
that some people believe in such things.Years ago, for
example, I remember Bud Hopkins, who was AFAIR
WhitleyStreibers psychologist, perform several statistical
tests on supposedabductees, and then announcing some results
on some of his books.The results claimed that the phenomenon
of UFOs, appears to be some sort ofvery particular
psychological event, linked to not any specic
traumaresulting from childhood, rather to sociological
factors having a lot to dowith religious tradition and
brainswashing.Specic instances of apparitions (such as that
of the Lady of Fatima) andvarious locales where ghosts
supposedly appear, (such as in England), seemto be related to
some extent to these phenomena, but for me, only in
asocioanthropological way, with no scientic support base
underneeth. Atleast not a *provable* scientic basis.To that
extent, you could try to scientically investigate WHY some
peoplesee ghosts and UFOs and seriously try to deduce some
sort of correlationbetween their previous mental states,
their religious states or theirsocial/religious background,
as individuals or as nations (who often carrywith them
severely absurd traditions).I presume that in such a case
youd probably need to study a lot ofpsychology rst.As an
amateur astronomer, myself, have been scanning the heavens
since I was12, with both binoculars and telescopes. Ive seen
nothing so far thatindicates the presence of UFOs.On the
other hand, as a person with some deep traumas from
childhoodresulting from various factors, I am convinced that
somebodys pulling somevery weird tricks on us. Who is that?
I have no idea.--Ioannis
Galidakishttp://users.forthnet.gr/ath/jgal/------------------
------------------------Eventually, _everything_ is
understandable
 .93 Toilet Seat <Barking@The.Moon>
[NonBreakingSpace].8b.96.87.8c .97.99.95
problem with my scientic population of ghosts> and> > UFOs
?> > Theres no such thing as ghosts or UFOs? :) YES !!
Excellent, and now we are getting somewhere !! So, while it
is technically possible that UFOs and gosts reallyexist,> I>
cannot claim that any related analysis of the subject is
actually valid> until I can prove that these questionable
things really do exist. If I am a scientist, and I spend the
next 40 years using the entirety of> the> body of all
statistical methods to analyze ghosts and UFOs, even if
Idid> all the math correctly, my so-called science is still
just garbage> because> I cannot prove that UFOs and ghosts
even exist !! Now we are getting somewhere !! If anyone else
is wliling to conrm what I am saying - please just jump>
right in. If I am just dead wrong please post as well.
Whatever. Lets> hear> it. Well, there are extant
possibilities here, I ll admit, although someappear> to be
out of the ordinary. In principle, you *could* perform
several interesting scientic tests,> based only on the loose
assumption that some people believe in suchthings.> Years ago,
for example, I remember Bud Hopkins, who was AFAIR Whitley>
Streibers psychologist, perform several statistical tests on
supposed> abductees, and then announcing some results on some
of his books. The results claimed that the phenomenon of
UFOs, appears to be some sortof> very particular
psychological event, linked to not any specic trauma>
resulting from childhood, rather to sociological factors
having a lot todo> with religious tradition and
brainswashing. Specic instances of apparitions (such as that
of the Lady of Fatima) and> various locales where ghosts
supposedly appear, (such as in England), seem> to be related
to some extent to these phenomena, but for me, only in a>
socioanthropological way, with no scientic support base
underneeth. At> least not a *provable* scientic basis. To
that extent, you could try to scientically investigate WHY
somepeople> see ghosts and UFOs and seriously try to deduce
some sort of correlation> between their previous mental
states, their religious states or their> social/religious
background, as individuals or as nations (who often carry>
with them severely absurd traditions). I presume that in such
a case youd probably need to study a lot of> psychology rst.
As an amateur astronomer, myself, have been scanning the
heavens since Iwas> 12, with both binoculars and telescopes.
Ive seen nothing so far that> indicates the presence of
UFOs. On the other hand, as a person with some deep traumas
from childhood> resulting from various factors, I am
convinced that somebodys pullingsome> very weird tricks on
us. Who is that? I have no idea.> --> Ioannis Galidakis>
http://users.forthnet.gr/ath/jgal/>
------------------------------------------> Eventually,
_everything_ is understandable>So then, you are saying that I
can enter Bigoot into a book on anthropology? I can submit
Wolfman to the eld of biology ? I can count upon
theinvisible man as being a fact of physics ?Are you indeed
suggesting that valid science might be conducted upon
anactual UFO - something which no-one has ever proved to
exist ??Are you saying that you can perform real science upon
a ghost ? Again -something which has never been proven to
exist ?? Can you give it a physical?
= Toilet Seat
<Barking@The.Moon> .8d.98.87.8b.8c .97.99.95
enter Bigoot into a book onanthropology> ? I can submit
Wolfman to the eld of biology ? I can count upon the>
invisible man as being a fact of physics ? Are you indeed
suggesting that valid science might be conducted upon an>
actual UFO - something which no-one has ever proved to exist
?? Are you saying that you can perform real science upon a
ghost ? Again -> something which has never been proven to
exist ?? Can you give it aphysical> ?None of that. All I am
saying is that you can do valid science with thepeople who
CLAIM to have seen such things. Grab a population, listen to
whatthey are saying, study your data and correlate this with
their backgroundand ethnicity.Then draw your conclusions. Not
conclusions on whether UFOs or ghostsexist, rather what is it
that makes those people who believe in them, tick.--Ioannis
Galidakishttp://users.forthnet.gr/ath/jgal/------------------
------------------------Eventually, _everything_ is
understandable
 .93 Toilet Seat <Barking@The.Moon>
[NonBreakingSpace].8b.96.87.8c .97.99.95
that I can enter Bigoot into a book on> anthropology> ? I can
submit Wolfman to the eld of biology ? I can count upon the>
invisible man as being a fact of physics ? Are you indeed
suggesting that valid science might be conducted uponan>
actual UFO - something which no-one has ever proved to exist
?? Are you saying that you can perform real science upon a
ghost ? Again -> something which has never been proven to
exist ?? Can you give it a> physical> ? None of that. All I
am saying is that you can do valid science with the> people
who CLAIM to have seen such things. Grab a population, listen
towhat> they are saying, study your data and correlate this
with their background> and ethnicity. Then draw your
conclusions. Not conclusions on whether UFOs or ghosts>
exist, rather what is it that makes those people who believe
in them,tick.> --> Ioannis GalidakisExcellent work Ioannis,
thank you for your thoughful response. I agree withyou %100.
It is possible to statistics on what people claim to have
seen,or what they claim to believe, etc etc.It is possible to
do science upon claims.But even though millions of people
claim to believe in God, we are unable todo real science upon
God directly - for example. We can analyze theirclaims
scientically, but cannot really study God himself by
thisprocess.Would you agree with this ?
= Toilet Seat
<Barking@The.Moon> .8d.98.87.8b.8c .97.99.95
for your thoughful response. I agreewith> you %100. It is
possible to statistics on what people claim to haveseen,> or
what they claim to believe, etc etc. It is possible to do
science upon claims. But even though millions of people claim
to believe in God, we are unableto> do real science upon God
directly - for example. We can analyze their> claims
scientically, but cannot really study God himself by this>
process. Would you agree with this ?Hmmmm... Lets see: The
objects you mention (i.e. God, ghosts, UFOs) areintangible.
As such they appear as though they cannot offer themselves
asvalid objects for a scientic study, as you say.One has to
be careful tho. Abstract mathematics deals with objects which
areintangible, also.So now I will reverse the question for
you:1) God is an intangible object, as such one cannot study
itscientically.2) Natural numbers are intangible objects, as
such one cannot study themscientically.2) is plainly false,
since Abstract Mathematics is a plenty valid
scienticdiscourse.What is it that differentiates those two
classes of intangible objects,say: {God, UFOs, ghosts} vs
{x:x = Some abstract Mathematical construct}and makes one
class a valid scientic discourse while makes the other
adiscourse in nonsense?Your answer, (i.e. the answer to your
original question), lies inpinpointing the exact difference
between the two example classes, above. :*)--Ioannis
Galidakishttp://users.forthnet.gr/ath/jgal/------------------
------------------------Eventually, _everything_ is
understandable
 .93 Toilet Seat <Barking@The.Moon>
[NonBreakingSpace].8b.96.87.8c .97.99.95
Ioannis, thank you for your thoughful response. I agree>
with> you %100. It is possible to statistics on what people
claim to have> seen,> or what they claim to believe, etc etc.
It is possible to do science upon claims. But even though
millions of people claim to believe in God, we areunable> to>
do real science upon God directly - for example. We can
analyze their> claims scientically, but cannot really study
God himself by this> process. Would you agree with this ?
Hmmmm... Lets see: The objects you mention (i.e. God,
ghosts, UFOs)are> intangible. As such they appear as though
they cannot offer themselves as> valid objects for a
scientic study, as you say. One has to be careful tho.
Abstract mathematics deals with objects whichare> intangible,
also. So now I will reverse the question for you: 1) God is an
intangible object, as such one cannot study it>
scientically.> 2) Natural numbers are intangible objects, as
such one cannot study them> scientically. 2) is plainly
false, since Abstract Mathematics is a plenty validscientic>
discourse. What is it that differentiates those two classes of
intangible objects,> say: {God, UFOs, ghosts} vs {x:x = Some
abstract Mathematical construct}> and makes one class a valid
scientic discourse while makes the other a> discourse in
nonsense? Your answer, (i.e. the answer to your original
question), lies in> pinpointing the exact difference between
the two example classes, above.:*)> --> Ioannis Galidakis>
http://users.forthnet.gr/ath/jgal/>
------------------------------------------> Eventually,
_everything_ is understandable>The distinction is that
mathematics deals with abstractions.These other items such as
God, Angels, Luck, Fate, and probably most humanemotions are
considered metaphysical.Mathematical abstractions are
certainly measurable. Also - if you constructabstract
entities such as abstract joy, then you could concievably
measureor quantify joy. But, you cannot measure real human
levels of joy, just asit is bogus to attempt to measure
intelligence, or love, etc.
 > Example:> > I am a
researcher. I study UFOs and ghosts.> > I have determined
that %77 of all ghosts are spotted within a 5 mile> radius> >
of UFO sightings, and that %82 of all UFO sightings are
accompanied bya> > general increase in the intensity and
overall magnitude of ghostrelated> > hauntings.> > Therefore,
I conclude, via statistics, that ghosts and UFOs aresomehow related, and that ghosts are in all probability using UFOs
for> > transportation purposes.> >
-----------------------------------------------> > Will
someone in the sci.math or sci.math.stat please stand up
andtell> me> > why this is awed, and no I am not kidding. I
need an independent> opinion.> > People are telling me Im
crazy - where did I go wrong ????? Can Icall> this> > science
?? Is my reasoning awed somehow ?? Is there a problem with>
my> > populations - or is this valid usage of stats ??> > Is
there some ofcially recognized listing of ghost sightings
and> UFO sightings? If not, theres always GIGO to explain
it. You are getting warmer !!! You are very close to the
fallacy Im looking> for. You have almost nailed it. Just one
hint -> In statistics, the populations and samples are
EXTREMELY important. For> instance, all of your work is
invalidated if your populations are messedup. Does anyone see
a problem with my scientic population of ghosts and> UFOs
?Is there nobody who will ask me to PROVE that I have
isolated and identiedsuch populations ?? Seriously folks -
or can I just take my data and waltzover to the National
Academy of Science and expect to get my research
funded?Should I be expected to demonstrate the existence of
UFOs and ghosts tobe considered solid science - or, are we
now accepting pyramid power andcold fusion as a means of
reducing fossil fuel consumption ???Someone please tell me
that Im not hallucinating - please conrm -independently -
that for science to be VALID, that populations in
statisticsshould be either veriable, reproducible,
falsiable, or provable. Or wasmy 8th grade science teacher
just lying to me about all that stuff ?If anyone can answer -
please post !!!
> > Example:>> > > I am a researcher. I
study UFOs and ghosts.>> > > I have determined that %77 of
all ghosts are spotted within a 5 mile>> radius>> > of UFO
sightings, and that %82 of all UFO sightings are accompanied
by>a>> > general increase in the intensity and overall
magnitude of ghost>related>> > hauntings.>> > > Therefore, I
conclude, via statistics, that ghosts and UFOs are>somehow> related, and that ghosts are in all probability using UFOs
for>> > transportation purposes.>> > >
----------------------------------------------->> > > Will
someone in the sci.math or sci.math.stat please stand up
and>tell>> me>> > why this is awed, and no I am not kidding.
I need an independent>> opinion.>> > > People are telling me
Im crazy - where did I go wrong ????? Can I>call>> this>> >
science ?? Is my reasoning awed somehow ?? Is there a
problem with>> my>> > populations - or is this valid usage of
stats ??>> > > > > > Is there some ofcially recognized
listing of ghost sightings and>> UFO sightings? If not,
theres always GIGO to explain it.>> You are getting warmer
!!! You are very close to the fallacy Im looking>> for. You
have almost nailed it.>> Just one hint ->> In statistics, the
populations and samples are EXTREMELY important. For>>
instance, all of your work is invalidated if your populations
are messed>up.>> Does anyone see a problem with my scientic
population of ghosts and>> UFOs ?Is there nobody who will
ask me to PROVE that I have isolated and identied>such
populations ??Dont know about other newsgroups, but Im
answering from sci.math.> Seriously folks - or can I just
take my data and waltz>over to the National Academy of
Science and expect to get my research funded>?No, because at
that stage, you will be asked also about yourmethodology for
collecting data. But you asked for help from sci.math,so your
request was understood in the terms of mathematics, not in
theterms of empirical science research. You wanted people to
fault yourassumptions, not your method for deriving
conclusions? Then you did apoor job of presenting the problem
for a mathematician.To this mathematician, your request could
be paraphrased as follows: Assume that I have determined that
77% of all ghosts sightings occur within a 5 mile radius of a
UFO sighting, and that 82% of all UFO sightings are
accompanied by an increase in the intensity and magnitude of
ghost-related-haunting-reports. related; I also conclude that
ghosts are in all probability, using UFOs for
transportations. Does the conclusion follow from the
assumptions?analysis, the former (validity of assumptions) is
irrelevant. We areonly seeing whether or not you are correctly
interpreting thestatistics, not whether the statistics are
correct--
==Its not denial. Im just very selective about what
I accept as reality. --- Calvin (Calvin and
Hobbes)
> Example:>> > > I am a researcher. I study UFOs and
ghosts.>> > > I have determined that %77 of all ghosts are
spotted within a 5mile>> radius>> > of UFO sightings, and
that %82 of all UFO sightings are accompaniedby>a>> > general
increase in the intensity and overall magnitude of
ghost>related>> > hauntings.>> > > Therefore, I conclude, via
statistics, that ghosts and UFOs are>somehow>> > related, and
that ghosts are in all probability using UFOs for>> >
transportation purposes.>> > >
----------------------------------------------->> > > Will
someone in the sci.math or sci.math.stat please stand up
and>tell>> me>> > why this is awed, and no I am not kidding.
I need an independent>> opinion.>> > > People are telling me
Im crazy - where did I go wrong ????? Can I>call>> this>> >
science ?? Is my reasoning awed somehow ?? Is there a
problemwith>> my>> > populations - or is this valid usage of
stats ??>> > > > > > Is there some ofcially recognized
listing of ghost sightings and>> UFO sightings? If not,
theres always GIGO to explain it.>> You are getting warmer
!!! You are very close to the fallacy Imlooking>> for. You
have almost nailed it.>> Just one hint ->> In statistics, the
populations and samples are EXTREMELY important. For>>
instance, all of your work is invalidated if your populations
aremessed>up.>> Does anyone see a problem with my scientic
population of ghosts and>> UFOs ?Is there nobody who will
ask me to PROVE that I have isolated andidentied>such
populations ?? Dont know about other newsgroups, but Im
answering from sci.math. presented your conclusions. As a
mathematician, if somebody asks me to> check something like
that, I would take the assumptions as granted,> and
investigate, rather, whether the conclusions actually follow
from> the assumptions. Whether or not the assumptions
actually hold is a> separate issue, and usually of only
secondary interest. The same would> be true for statistics:
you presented statistics, and conclusions> derived from them.
It is likely people interpreted your request as> asking
whether or not your conclusion would indeed follow from such>
statistics, NOT whether the statistics were accurate.Indeed -
my example was poorly worded, perhaps somewhat sarcasticly in
anattempt to provide a little humor or levity to the sillyness
of this.And, indeed, I certainly could have worded it more
formally, and I thank youfor pointing that out.> Seriously
folks - or can I just take my data and waltz>over to the
National Academy of Science and expect to get my
researchfunded>? No, because at that stage, you will be asked
also about your> methodology for collecting data. But you
asked for help from sci.math,> so your request was understood
in the terms of mathematics, not in the> terms of empirical
science research. You wanted people to fault your>
assumptions, not your method for deriving conclusions? Then
you did a> poor job of presenting the problem for a
mathematician.Both are at fault here, and if there were some
way to distinguish one asbeing more faulty than the other I
would like to see that. However, based onmy sensate human
experiences in this world, Ifind it absolutely andpatently
absurd in the extreme to say the least, that someone could
claim tohave collected data about things like bigfoot, UFOs,
alien abductions,Wolfman (for es g), when it is rather
obvious that these items cannot bestudied due to their
failure to even exist.> To this mathematician, your request
could be paraphrased as follows: Assume that I have
determined that 77% of all ghosts sightings occur> within a 5
mile radius of a UFO sighting, and that 82% of all UFO>
sightings are accompanied by an increase in the intensity
and> magnitude of ghost-related-haunting-reports. related; I
also conclude that ghosts are in all probability, using>
UFOs for transportations.Well, a correct wording that I was
actually after would require existence ofghosts of UFOs or
aliens etc. Your rewording provides for sightings -which can
clearly exist, because people can be wrong.If I had an
opportunity to reword part of this in such a way which
wouldrequire the existence of the subject, please allow me
the opportunity - forex, I have determined that %77 of all
ghosts are located within a 5 mileradius of UFO landing
sites. Furthermore, %82 of all UFO landings and
alienvisitations are located within 8 miles of a ghost.Now,
in this rewording there is no ambiguity - I have asserted
indirectlythat I know where the ghosts are , and that I have
found aliens, etc. For mystatements to be true, aliens must
exist, and ghosts must exist, and Ishould be able to prove it
or be called a quack.> Does the conclusion follow from the
assumptions?You are correct - it does not, nor did it in the
original post. No problem.> view of pure logic, it is
immaterial whether or not the assumptions> are valid. The
only question is whether or not the conclusions follow> from
the assumptions, and they do not.This is true. But it _is_
within the purview of statistics to invalidate aconclusion or
result when the initial sets upon which the analysis
isperformed are found to be corrupted.Suppose, for example,
that I published information regarding the averageheight,
weight and IQ of a sample population of ghosts, and I made
the claimthat the average IQ was higher and standard
deviation was exactly 1/2 of asimilar sized sample from a
population of angels.You dont have a problem with that ? If
this turned up in the MAA MathMagazine or somewhere, and it
was presented as factual data - you aretelling me that you
would not question how I took the samples ? Where Ifound such
populations in the rst place ?> The fact that your
assumptions are, in fact, false, damns your> premises; the
fact that your conclusions do not follow from your> premises
(via statistics or not) damns your analysis of those>
assumptions.OK -> ->EITHER<- of those two faults makes your
report useless as science> Both together make it doubly
useless as science. As statistical> analysis, the former
(validity of assumptions) is irrelevant. We are> only seeing
whether or not you are correctly interpreting the>
statistics, not whether the statistics are correctYou have
done a very careful and thorough job of correcting me and I
am mostappreciative that you would take this seriously - but
we are getting furtherfrom my objective.Without rewriting the
question, or rewording the statement, I give you myclaim. I
claim that if you do statistics on ivalid data, then your
results areinvalid by denition, regardless of whether the
derived solutions comeout correct or not.>
==
= Its not denial. Im just very selective about>
what I accept as reality.> --- Calvin (Calvin and Hobbes)>
==
== Arturo Magidin> magidin@math.berkeley.edu>
[.snip.]>Well, a correct wording that I was actually after
would require existence of>ghosts of UFOs or aliens etc.
Your rewording provides for sightings ->which can clearly
exist, because people can be wrong.Just a short point here:
in your original post, there were four thingsmentioned:
ghosts spotted, UFO sightings, UFO sightings, ghostrelated
haunting; only the latter talks about actual phenomena,
whilethe former three are readily interpreted as reports of
sightings. [.snip.]> view of pure logic, it is immaterial
whether or not the assumptions>> are valid. The only question
is whether or not the conclusions follow>> from the
assumptions, and they do not.>This is true. But it _is_
within the purview of statistics to invalidate a>conclusion
or result when the initial sets upon which the analysis
is>performed are found to be corrupted.You are confusing
validity with soundness, I think. Like I said, mostpeople in
sci.math seem to have interpreted your question as aquestion
about validity, when you are clearly more interested
insoundness. >Suppose, for example, that I published
information regarding the average>height, weight and IQ of a
sample population of ghosts, and I made the claim>that the
average IQ was higher and standard deviation was exactly 1/2
of a>similar sized sample from a population of angels.You
dont have a problem with that ? Yes, I might have a problem
with that.>If this turned up in the MAA Math>Magazine or
somewhere, and it was presented as factual data - you
are>telling me that you would not question how I took the
samples ? Where I>found such populations in the rst place
?there would be no reason for it to be in the Mathematics
Magazine(which, by the way, a journal for expository
undergraduate levelmathematics; but I assume you mean a
mathematical peer reviewedjournal on statistics or some
such); and (b) the process whereby thedata was supposedly
acquired would have to be included; thatsexperiment
design.correlation calculation, then I wouldnt care: it
would be a made-upexample to highlight certain ->mathematical
features<- of the models,not a discussion of actual
fact.>Without rewriting the question, or rewording the
statement, I give you my>claim.> I claim that if you do
statistics on ivalid data, then your results are>invalid by
denition, regardless of whether the derived solutions
come>out correct or not.Again, I think you are confusing
soundness with validity.The validity of an argument refers to
whether the conclusions followfrom the premises; an argument
is valid if and only if they do.On the other hand, an
argument is ->sound<- if and only if, inaddition to being
valid, the premises hold. So what you shoudl really say is
that if you do statistics frominvalid data, then your
argument is necessarily ->UNSOUND<-. (Of course, to further
claim that the conclusions of an unsoundargument must be
false is itself a logical fallacy, called anArgumentum ad
logicam)Mathematics (and statistical analysis as such) are
concerned rst andforemost with validity, not with soundness.
Of course, statisticalanalysis is seldom used as an
intellectual exercise, and is insteadembedded inside a
process that requires a test of soundness inaddition to
validity, which is I think what you are concerned about.There
is in fact a famous aphorism by Bertrand Russell:
Mathematicsmay be dened as the subject in which we never
know what we aretalking about, nor whether what we are saying
is true. Because wecare about whether the argument is valid,
but not whether it is sound,or what the premises are to be
applied to.So, when you came to sci.math and sci.math.stat,
your questions wasinterpreted as a question on validity; but
you are obviously far moreinterested in soundness than in
validity. It is certainly not a badidea to check on the truth
of the premises before starting analysis--
==
==Its not denial. Im just very selective about what
I accept as reality. --- Calvin (Calvin and
Hobbes)
So what you shoudl really say is that if you do statistics
from> invalid data, then your argument is necessarily
->UNSOUND<-.was originally attempting to communicate, albeit
somewhat sinically.I always learn so much from you guys -
every time I shop for knowledge -your appreciation for
==
I have determined that %77 of all ghosts are spotted within a
5> mile radius of UFO sightings, and that %82 of all UFO
sightings> are accompanied by a general increase in the
intensity and> overall magnitude of ghost related hauntings.
Is there nobody who will ask me to PROVE that I have isolated
and> identied such populations??Not when youve assumed the
above for the purpose of discussion.> Seriously folks - or
can I just take my data and waltz over to the> National
Academy of Science and expect to get my research funded ?Your
question on sci.math is about the interpretation of
statisitcs.That you also made up your numbers is not an
interesting issue,mathematically.> Someone please tell me
that Im not hallucinating - please conrm -> independently -
that for science to be VALID, that populations in> statistics
should be either veriable, reproducible, falsiable,> or
provable.The population of people claiming to have spotted a
UFO is as goodas any other. If you had actually surveyed
them, rather than making upyour numbers, then there would be
no quibble with the result: thequibble would be whether you
recognized that it was a fact aboutclaimants, or
misrepresented it as a fact about UFOs.Len.
==
I have determined that %77 of all ghosts are spotted within a
5 mile> radius of UFO sightings, and that %82 of all UFO
sightings are> accompanied by a general increase in the
intensity and overall> magnitude of ghost related hauntings.
Therefore, I conclude, via statistics, that ghosts and UFOs
are somehow> related, and that ghosts are in all probability
using UFOs for> transportation purposes. Will someone in the
sci.math or sci.math.stat please stand up and> tell me why
this is awed, and no I am not kidding. I need an>
independent opinion.You are confused as to what you are
studying. The correct conclusionis that CLAIMED SIGHTINGS of
UFOs correlate with CLAIMED SIGHTINGS ofghosts. You are not
studying UFOs or ghosts, but claims concerningthem.Len.
= I
have determined that %77 of all ghosts are spotted within a 5
mile> radius of UFO sightings, and that %82 of all UFO
sightings are> accompanied by a general increase in the
intensity and overall> magnitude of ghost related hauntings.
Therefore, I conclude, via statistics, that ghosts and UFOs
are somehow> related, and that ghosts are in all probability
using UFOs for> transportation purposes. Will someone in the
sci.math or sci.math.stat please stand up and> tell me why
this is awed, and no I am not kidding. I need an>
independent opinion. You are confused as to what you are
studying. The correct conclusion> is that CLAIMED SIGHTINGS
of UFOs correlate with CLAIMED SIGHTINGS of> ghosts. You are
not studying UFOs or ghosts, but claims concerning> them.
Len.>So then Chemistry does not concern real atoms, but the
claimed observancesof atoms ?? Where does reality t in
??Music does not concern real notes, but merely the claim
that someone heardmusic ?I would like to know if you can do
valid science uopn things like UFOs andghosts, things which
cannot be proven to exist in the rst place.I am not confused
sha1:NNadUaabT3N18nSX/QLbNIqnGe8
=> You are confused as
to what you are studying. The correct>> conclusion is that
CLAIMED SIGHTINGS of UFOs correlate with CLAIMED>> SIGHTINGS
of ghosts. You are not studying UFOs or ghosts, but>> claims
concerning them. So then Chemistry does not concern real
atoms, but the claimed observances> of atoms ?? Where does
reality t in ??You are confused. Statistics concern WHAT YOU
ACTUALLY MEASURED. Ifyou counted people who claimed to see a
UFO, then thats what youvemeasured: people claiming to have
seen a UFO. Now you know that thatmany people have made such a
claim. You have no idea whether theyreally saw UFOs, or indeed
whether UFOs exist at all. You didntmeasure that; you
measured CLAIMS.Similarly, if a survey (in Berkeley!)
determines that 81% of peoplesurveyed describe Bush as a
weenie, you have learned something aboutthe opinions of
people in Berkeley. You have not learned anythingabout
whether Bush is indeed a weenie, nor that he has an 81%chance
of being a weenie, nor anything else about Bush. You
learnabout the measurement you ACTUALLY MADE.> I am not
confused on this. Address the question.If you want to know
whether extraterrestrial spacecraft exist, pollingearthlings
is a useless procedure. If you _hypothesize_ that
claimedsitings correlate with real sitings, then you might
use thatinformation to devise a good experiment for proving
that ETsexist. But the surveys yield absolutely no data on
that question.Your question was, what is the aw in this
reasoning. The answer isthat you measured one thing, and then
drew a conclusion about anunrelated thing. If you dont fully
realize that, then you AREconfused.Len.
> You are
confused as to what you are studying. The correct>>
conclusion is that CLAIMED SIGHTINGS of UFOs correlate with
CLAIMED>> SIGHTINGS of ghosts. You are not studying UFOs or
ghosts, but>> claims concerning them. So then Chemistry does
not concern real atoms, but the claimedobservances> of atoms
?? Where does reality t in ?? You are confused. Statistics
concern WHAT YOU ACTUALLY MEASURED. If> you counted people
who claimed to see a UFO, then thats what youve> measured:
people claiming to have seen a UFO. Now you know that that>
many people have made such a claim. You have no idea whether
they> really saw UFOs, or indeed whether UFOs exist at all.
You didnt> measure that; you measured CLAIMS.I am not
confused on this. I fully understand what you are saying, and
Iagree. You cannot measure one thing and claim statistical
relsults forsomething unrelated.You seem to be confused about
something though - the fact that nothingregarding UFOs can be
measured, other than the prevalence of hallucinationsof the
various observers. You cannot assemble a collection of UFOs
orghosts, or vampires, or any other mythical nonsense. Such
sets cannot beassembled. If you want to measure marbles you
can do so. But you cant takethe average height, weight, etc
of vampires because they do not exist.You can spend the next
40 years doing statistical analysis on vampires,but none of
it is valid science because your population is non-existent.I
am talking about real science - not abstractions. I am talking
about theset of rael world vampires, and not the world of
hypotheticalabstractions.> Similarly, if a survey (in
Berkeley!) determines that 81% of people> surveyed describe
Bush as a weenie, you have learned something about> the
opinions of people in Berkeley. You have not learned
anything> about whether Bush is indeed a weenie, nor that he
has an 81%> chance of being a weenie, nor anything else about
Bush. You learn> about the measurement you ACTUALLY MADE.Or -
did you learn something about the response to a single
question on aparticular day in history at a particular place
?You learn nothing about opinions of people at berkely from
this - all youlearn is the response you will recieve to a
particular question, on aparticular day in historty, at a
specic place.> I am not confused on this. Address the
question. If you want to know whether extraterrestrial
spacecraft exist, polling> earthlings is a useless procedure.
If you _hypothesize_ that claimed> sitings correlate with real
sitings, then you might use that> information to devise a good
experiment for proving that ETs> exist. But the surveys yield
absolutely no data on that question.I am talking not about
sightings. I am talking about actual alienspacecraft from
other worlds - the real thing, the actual spacecraft.
Myquestion is as follows -If you do not posses the actual
spacecraft, is it possible to performgenuine scientic
analysis of the spacecraft ?I claim that the answer is an
absolute _no_It sounds stupid, but this is the verication I
seek.> Your question was, what is the aw in this reasoning.
The answer is> that you measured one thing, and then drew a
conclusion about an> unrelated thing. If you dont fully
realize that, then you ARE> confused.Well Len, I appreciate
your input. But you are only %50 right. The correctanswer has
to do with populations. I do not even have a valid population
towork with, and so EVERYTHING is moot - no science is
possible whatsoever. Doyou see what Im talking about ? I
hope so, because it aint that difcult.Is it possible to go
out and capture a Leprachaun ? Would it be rational toy over
to Scotland in hopes of riding the Loch Ness Monster up and
downthe loch ?? Is it possible to perform science upon
objects which cannot beDEMONSTRATED as factually existing in
sha1:n4MK+qiHS0rDu3B+orKstk2JycI
== I am not confused on
this. I fully understand what you are saying, and I> agree.
You cannot measure one thing and claim statistical relsults
for> something unrelated.So far, so good.> You seem to be
confused about something though - the fact that> nothing
regarding UFOs can be measured, other than the prevalence>
of hallucinations of the various observers...I already said
that. With one exception: you cant conclude thatsightings
are in fact hallucinations, any more than you canconclude
that theyre real. You can only claim that a sighting
wasreported.> You can spend the next 40 years doing
statistical analysis on> vampires...You cannot. You can only
do statistical analysis on claims concerningvampires.>>
Similarly, if a survey (in Berkeley!) determines that 81% of
people>> surveyed describe Bush as a weenie, you have learned
something about>> the opinions of people in Berkeley. You have
not learned anything>> about...Bush... Or - did you learn
something about the response to a single> question on a
particular day in history at a particular place ?Even
better.> If you do not posses the actual spacecraft, is it
possible to perform> genuine scientic analysis of the
spacecraft ?If youll pardon my saying so, its a frivolous
question. AND it hasNOTHING to do with your hypothetical
survey concerning UFO sightings.> I claim that the answer is
an absolute _no_It is.> It sounds stupid...It is.> Well Len,
I appreciate your input. But you are only %50 right. The>
correct answer has to do with populations. I do not even have
a> valid population to work with...If you did a valid survey,
then you DID have a validpopulation. Namely, the population
of people who may or may not havesighted a UFO. And you can
do all sorts of scientic studiesconcerning that
population.That population, however, has nothing to do with
UFOs. You do not havea population of aliens.But you ARE
confused: otherwise you would not keep repudiating your
ownhypothetical survey on the grounds that it is not some
other survey.Len.
 You can spend the next 40 years doing
statistical analysis on> vampires... You cannot. You can only
do statistical analysis on claims concerning> vampires.You
could, but it would all be garbage. This is my point.>>
Similarly, if a survey (in Berkeley!) determines that 81% of
people>> surveyed describe Bush as a weenie, you have learned
something about>> the opinions of people in Berkeley. You have
not learned anything>> about...Bush... Or - did you learn
something about the response to a single> question on a
particular day in history at a particular place ? Even
better. If you do not posses the actual spacecraft, is it
possible to perform> genuine scientic analysis of the
spacecraft ? If youll pardon my saying so, its a frivolous
question. AND it has> NOTHING to do with your hypothetical
survey concerning UFO sightings. I claim that the answer is
an absolute _no_ It is. It sounds stupid... It is. Well Len,
I appreciate your input. But you are only %50 right. The>
correct answer has to do with populations. I do not even have
a> valid population to work with... If you did a valid survey,
then you DID have a valid> population. Namely, the population
of people who may or may not have> sighted a UFO. And you can
do all sorts of scientic studies> concerning that population.
That population, however, has nothing to do with UFOs. You do
not have> a population of aliens. But you ARE confused:
otherwise you would not keep repudiating your own>
hypothetical survey on the grounds that it is not some other
survey. Len.>OK then - I am confused. No problem. I do
understand what you are saying -that the exampel in the
original post was basically a mathamaticalknow what I
mean.The point, being, that you cannot measure the velocity
of Superman inight, because you cannot collect real data on
mythical or nonexistentobjects/people.Similarly, for the
folks in alt.psychology, you cannot measure love, hate,etc.
You cannot do statistics on things which cannot be measured
or observedproperly. Any science which claims to do this,
sha1:FvC/+AaGDK5yYiLf9S44RFi/zrc
=> You can spend the
next 40 years doing statistical analysis on vampires...>> You
cannot. You can only do statistical analysis on claims>>
concerning vampires. You could, but it would all be garbage.
This is my point.You cannot. If you are discussing
statistics, then it is not ofparticular interest how some
idiot misuses them to provenon-sequiturs.>> But you ARE
confused: otherwise you would not keep repudiating your>> own
hypothetical survey on the grounds that it is not some other>>
survey. OK then - I am confused. No problem. I do understand
what you are> saying - that the exampel in the original post
was basically a> mathamatical non-sequiter...Correct.> The
point, being, that you cannot measure the velocity of
Superman> in ight, because you cannot collect real data on
mythical or> nonexistent objects/people.The only problem is
that its confusing when you describe oneexperiment (a survey
of Superman sightings) and discuss another(Supermans
airspeed). What you need to do is to get absolutely clearwhat
your question is, and then design an experiment that
addressesTHAT question.> Similarly, for the folks in
alt.psychology, you cannot measure love,> hate, etc...Cant
comment: vast amounts of research in the soft sciences
isdeeply awed, for lots of reasons.> You cannot do
statistics on things which cannot be measured or> observed
properly.Love can be dened as an immeasurable, intangible
thing, or it canbe dened to be consonant with the perception
thereof--i.e., that theassertion I love X, if honest, is
true.> Any science which claims to do this, thus far, is
bogus academic> fraud.Hmm. If the whole point of your
question is to advance some crankishclaim that the whole eld
of psychology is invalid, then Im sorry Ireplied.Len.
= You
can spend the next 40 years doing statistical analysis on
vampires...>> You cannot. You can only do statistical
analysis on claims>> concerning vampires. You could, but it
would all be garbage. This is my point. You cannot. If you
are discussing statistics, then it is not of> particular
interest how some idiot misuses them to prove>
non-sequiturs.Well, one could dene vampires as being x,y,z,
or you could dene theset of all vampires V {V| a,b,c...etc},
and then you could do all kinds ofvalid mathamatics.What I am
talking about is actually going out into the physical world
as aphysicist or chemist would, rounding up a group of
vampires and collectingdata on them.Now, I could be wrnog
here, but last tiem I checked, there were no realvampires.
The cardinality of the set of all vampires in our physical
worldis assumed to be zero. I could do all kinds of
statistics on people who Ithink are probably vampires, but if
ther are not genuine vampires then myanalysis is just
garbage.>> But you ARE confused: otherwise you would not keep
repudiating your>> own hypothetical survey on the grounds that
it is not some other>> survey. OK then - I am confused. No
problem. I do understand what you are> saying - that the
exampel in the original post was basically a> mathamatical
non-sequiter... Correct. The point, being, that you cannot
measure the velocity of Superman> in ight, because you
cannot collect real data on mythical or> nonexistent
objects/people. The only problem is that its confusing when
you describe one> experiment (a survey of Superman sightings)
and discuss another> (Supermans airspeed). What you need to
do is to get absolutely clear> what your question is, and
then design an experiment that addresses> THAT question.I
apologize. Admitted, the original post was thrown together
rather hastily.> Similarly, for the folks in alt.psychology,
you cannot measure love,> hate, etc... Cant comment: vast
amounts of research in the soft sciences is> deeply awed,
for lots of reasons.Agreed. You need not take a position
here, and I dont blame you forremaining neutral on what I am
about to say - but MY position is thatpsychology is mostly
garbage for the very reasons I am discussing here.Without any
method of making physical measurements of human emotion or
willor intent, either quantitative or qualitative, one simply
cannot generateany solid science.> You cannot do statistics on
things which cannot be measured or> observed properly. Love
can be dened as an immeasurable, intangible thing, or it
can> be dened to be consonant with the perception
thereof--i.e., that the> assertion I love X, if honest, is
true.Yes - it can be dened many ways. But can it be measured
directly ?You have a couple approaches here. You can model it
algebraicly, in whichcase you are treating a metaphysical
entity as an abstraction. So, it startsgetting fairly
strained already to say the least. You must somehow denelove
using human speech language otherwise your variables are
inaccurateor incomplete - something which may not be possible
with a nite amount ofstatements. Even if you succed, and you
would then be immediately famous,there is no way to verify
that you have modelled love because love cannot bemeasured
with instruments.Hell - we cannot even prove that love
exists. I do not dispute that it does,but you can never tell
when you are observign love or something else whichjust looks
like love, and then how many kinds of love are there ?This is
a nightmare - and all because such items are metaphysical in
nature.> Any science which claims to do this, thus far, is
bogus academic> fraud. Hmm. If the whole point of your
question is to advance some crankish> claim that the whole
eld of psychology is invalid, then Im sorry I> replied.That
is exactly the point and Im sorry that you feel sorry, but it
is notone bit crankish to discard rubbish.In fact - it is very
crankish, but also very very true that psychology isfraud.>
Len.
In fact - it is very crankish, but also very very
true that psychology is>fraud.>Perhaps. Cant really comment
(the only book I have in my library, ThePsychology of
Rigorous Humanism by Rychlak, makes about as much sense as,
say,Bolzanos Paradoxes of the Innite). It is however quite
clear to me thatscience doesnt give answers to all
answerable questions. Think about it,mathematics *barely*
qualies as a science, is it therefore fraud? Hardly.
Amazingly enough, mathematics still manages to provide
answers to questions ona fairly regular basis. Almost
mystical isnt it?And now its time for me to see if I won
this weeks lotto ($88kk +/-). rich
In fact - it is very
crankish, but also very very true that psychology is>fraud.>
Perhaps. Cant really comment (the only book I have in my
library, The> Psychology of Rigorous Humanism by Rychlak,
makes about as much sense as,say,> Bolzanos Paradoxes of the
Innite). It is however quite clear to methat> science doesnt
give answers to all answerable questions. Think about it,>
mathematics *barely* qualies as a science, is it therefore
fraud?Hardly.> Amazingly enough, mathematics still manages to
provide answers toquestions on> a fairly regular basis. Almost
mystical isnt it? And now its time for me to see if I won
this weeks lotto ($88kk +/-). rich>answer all questions,
because then it would almost be providing the truth,which is
seemingly impossible.Im not sure about math not qualifying
as science, however. I think that itmay. What is the
denition of a science ? Results must be
veriable,falsiable, and reproducible. I think that math
satises these threebetter than any physical science, and it
does so with complete precision andan exactness which is
unique to math. If you know something that I dontplease post.
My claim that psychology is fraud is based on the fact that I
do notbelieve that they can qualify or quantify the objects
which they claim tostudy. As an extreme example, consider
making a statement about depressedpeople. One cannot prove
who belongs and who does not belong in the dataset, and
therefore the data set could be completely messed up and
nobodywould ever know. In other words, depression cannot be
qualied. There is noinstrument to test for depression. Also,
depression cannot be quantied. There is no way to know
howsevere it is. There is no meter to measure its strength.
What can be known about depression if you cannot qualify your
dataset,and you cannot quantify your data ?? I will tell you
how much can be known.Nothing. Nothing can be known. Yet
mountains of statistics a churned out,and it seems that I am
the only one who is bold enough to admit that I amcurious to
say the _very_least.
=Im not sure about math not
qualifying as science, however. I think that it>may. What is
the denition of a science ? Results must be
veriable,>falsiable, and reproducible. I think that math
satises these three>better than any physical science, and it
does so with complete precision and>an exactness which is
unique to math. If you know something that I dont>please
post.>Where are the experiments? Is the scientic method
taught in math classes? Ever try to verify a number is not
computable experimentally? Why are therecolleges of science
*and* math? I could go on, but Im too depressed over
notwinning the lottery. And yes, my depression is
quantiable...to the tune of$88kk :(rich
Im not sure
about math not qualifying as science, however. I think
thatit>may. What is the denition of a science ? Results must
be veriable,>falsiable, and reproducible. I think that math
satises these three>better than any physical science, and it
does so with complete precisionand>an exactness which is
unique to math. If you know something that I dont>please
post.> Where are the experiments? Is the scientic method
taught in mathclasses?> Ever try to verify a number is not
computable experimentally? Why arethere> colleges of science
*and* math? I could go on, but Im too depressed overnot>
winning the lottery. And yes, my depression is
quantiable...to the tuneof> $88kk :(I see that you ve
qualied it - and quantied it !! Perhaps your lossesare
amenable to real scientic explorations.But returning to
math, I think that the experiments are replaced by theactual
practice of performing algebra or even conceptualizing.
Proofs allowthsee conceptualiztions to be repeated by others
for independentverication.The use of algebra in place of
experiments. I guess Im reminded that thereis a debate as to
whether mathematics is a discovery of naturally
occuringrelationships, or merely some type of manmade
contraption. I do not know theanswer. But I do know that
physical scientists are engaged in both discovery_and_
invention.Some genuine experimentation is now possible using
computers as well.I think that the division between math and
science formed as a result of twodistinct needs. On the one
hand you need engineers who are practical andable to make
things work in the real world such as industry. On the
otherhand, you need the eggheads, the experts, the idiot
savants who study mathlike hermits seldom shaving and working
away as diligently as rust trying tosolve the difcult
problems. These are two fundamentally differentcreatures.But
then you have modelling - which is more like a ne art.I
honestly dont know either.
<fGLJb.51594$I07.170400@attbi_s53>
= What I am talking
about is actually going out into the physical> world as a
physicist or chemist would, rounding up a group of> vampires
and collecting data on them.Then stop talking about
experiments in which you collect data onaverage rainfall, and
then draw conclusions about vampires.> Now, I could be wrnog
here, but last tiem I checked, there were no real>
vampires.That is the standard assumption, yes. It is
possible, however, forscientists to treat the question as
open in order to prove or disprovethe hypothesis.> The
cardinality of the set of all vampires in our physical world>
is assumed to be zero.If you do assume that, then any
arguments you make about theirnonexistence are likely to be
circular.> I could do all kinds of statistics on people who I
think are> probably vampires, but if ther are not genuine
vampires then my> analysis is just garbage.It might be
perfectly valid research on Goths, or
self-proclaimedvampires, or People that Toilet considers
weird, depending how yourcohort is selected.Over and over,
you describe a perfectly valid experiment concerning X,and
then argue that it isnt valid because it isnt an
experimentconcerning Y. This suggests that you havent got a
rm grip on thescientic method.> ...MY position is that
psychology is mostly garbage for the very> reasons I am
discussing here.It may or may not be, but youre having
trouble making the point,because you seem to have trouble
understanding the scientic methodin general.> Without any
method of making physical measurements of human emotion> or
will or intent, either quantitative or qualitative, one
simply> cannot generate any solid science.There are lots of
interesting ways. People are generally honest enoughthat
their claims to like one thing, and dislike another,
willcorrelate pretty well with the truth. Involuntary
responses such aspupil dilation and skin temperature have
also been used to good effect.>> Love can be dened as an
immeasurable, intangible thing, or it>> can be dened to be
consonant with the perception thereof... Yes - it can be
dened many ways. But can it be measured directly ?Thats a
good question for a philosophy class; you could get an A onan
essay about it. In practice, the answer is often well enough.>
Hell - we cannot even prove that love exists...Or that you
exist. You arent talking science, but philosophy.> That is
exactly the point and Im sorry that you feel sorry, but it>
is not one bit crankish to discard rubbish.Until you
understand how experiments work, and what they do or
donttell you, you will be unable to distinguish the wheat
from the chaff.> In fact - it is very crankish, but also very
very true that psychology is> fraud.I have no wish to help you
grind your axe. Im off this thread.Len.
=garbage snipped>
Over and over, you describe a perfectly valid experiment
concerning X,> and then argue that it isnt valid because it
isnt an experiment> concerning Y. This suggests that you
havent got a rm grip on the> scientic method.It is
psychology which is ignoring the scientic method, to the
detrimentof mankind, making a mockery of academia through
its sanctioned fraud.What I said was very clear. Pay
attention and then respond after thinkingcarefully.Show me
how this discipline which you seem to defend is actually
observingthe scientic method. It does not. It is fraud.>
...MY position is that psychology is mostly garbage for the
very> reasons I am discussing here. It may or may not be, but
youre having trouble making the point,> because you seem to
have trouble understanding the scientic method> in
general.Strawman -you lose.> Without any method of making
physical measurements of human emotion> or will or intent,
either quantitative or qualitative, one simply> cannot
generate any solid science. There are lots of interesting
ways. People are generally honest enough> that their claims
to like one thing, and dislike another, will> correlate
pretty well with the truth. Involuntary responses such as>
pupil dilation and skin temperature have also been used to
good effect.People are frequently WRONG. You have no means to
prove this - you arepromoting quackery and fraud.You do not
understand that science has requirements which must be
satised.You have deomstrated this in writing.>> Love can be
dened as an immeasurable, intangible thing, or it>> can be
dened to be consonant with the perception thereof... Yes -
it can be dened many ways. But can it be measured directly ?
Thats a good question for a philosophy class; you could get
an A on> an essay about it. In practice, the answer is often
well enough. Hell - we cannot even prove that love exists...
Or that you exist. You arent talking science, but
philosophy.And you are not a scientist, but a sociopolitical
hack.> That is exactly the point and Im sorry that you feel
sorry, but it> is not one bit crankish to discard rubbish.
Until you understand how experiments work, and what they do
or dont> tell you, you will be unable to distinguish the
wheat from the chaff.You need to learn basic 8th grade
material. Veriability, falsiability,and reproducibility. If
you do not have these, then you do not have science.believe
just because people say that it is true ? You have learned
nothing.> In fact - it is very crankish, but also very very
true that psychologyis> fraud. I have no wish to help you
grind your axe. Im off this thread.Good, then run from
truth.> Len.>
 I have determined that %77 of all ghosts
are spotted within a 5 mile> radius of UFO sightings, and
that %82 of all UFO sightings are> accompanied by a general
increase in the intensity and overall> magnitude of ghost
related hauntings. Therefore, I conclude, via statistics,
that ghosts and UFOs are somehow> related, and that ghosts
are in all probability using UFOs for> transportation
purposes. Will someone in the sci.math or sci.math.stat
please stand up and> tell me why this is awed, and no I am
not kidding. I need an> independent opinion. You are confused
as to what you are studying. The correct conclusion> is that
CLAIMED SIGHTINGS of UFOs correlate with CLAIMED SIGHTINGS
of> ghosts. You are not studying UFOs or ghosts, but claims
concerning> them. Len.>one of the many aws.However, it is
not quite the glaring aw that Im looking for, butcertainly
valid.Im looking for a much more obvious aw, if anyone has
an idea, please -please feel free to respond.
am facing with this difcult problem. Please help me!In this
problem, I need to construct some matrices which satisfy
thefollowing matrix equation:(( A * A )V1 + ( B * B) V2) V =
(D * D)where * denotes Kronecker product, A, B, V1, V2, V are
unknown matricesthat needs solving; they are all square. Some
structure needs to be imposed:I have certain pattern for A
and B; and V1, V2, and V are required to bediagonal... Matrix
D is given...The task is tofind the best approximation of the
above-mentioned A, B, V1,V2 and V... I have been thinking
about this for long time... can anybodygive me some hints?Is
it possible to have closed form analytical solution? Have
anybodyresearched on this problem before?Is it possible and
how to design iterative algorithm to let computer searchfor
the answer?-Walala
problem. Please help me!In this problem, I need to construct
some matrices which satisfy thefollowing matrix equation:(( A
* A )V1 + ( B * B) V2) V = (D * D)where * denotes Kronecker
product, A, B, V1, V2, V are unknown matricesthat needs
solving; they are all square. Some structure needs to be
imposed:I have certain pattern for A and B; and V1, V2, and V
are required to bediagonal... Matrix D is given...The task is
tofind the best approximation of the above-mentioned A, B,
V1,V2 and V... I have been thinking about this for long
time... can anybodygive me some hints?I guess it is difcult
tofind closed form analytical solution... How todesign
iterative algorithm to let computer search for the answer? It
isreally hard... please help me!-Walala
facing with this difcult problem. Please help me!In this
problem, I need to construct some matrices which satisfy
the>following matrix equation:(( A * A )V1 + ( B * B) V2) V =
(D * D)where * denotes Kronecker product, A, B, V1, V2, V are
unknown matrices>that needs solving; they are all square.
Some structure needs to be imposed:>I have certain pattern
for A and B; and V1, V2, and V are required to be>diagonal...
Matrix D is given...The task is tofind the best approximation
of the above-mentioned A, B, V1,>V2 and V... I have been
thinking about this for long time... can anybody>give me some
hints?I guess it is difcult tofind closed form analytical
solution..How diicult is it - it seems to me that the
problem has an awful lotof structure.>. How to>design
iterative algorithm to let computer search for the answer? It
is>really hard... please help me!You are trying to solve the
nonlinear system of equations(( A * A )V1 + ( B * B) V2) V -
(D * D) = 0If the problem is not too large use Newtons
method. If the problemis too big for that look for a canned
program that will solvenonlinear systems without needing to
generate and then solve thejacobian. >-Walala
=I am reading
a book on memory techniques and it has the following
passage:... some scholars insist that Leibniz invented
calculus while searching fora memory system that would aid in
memorizing numbers.Several questions.1. Is there any proof of
this?2. Why would this be deemed an important pursuit (what
is the big deal withmemorizing numbers) for Leibniz?3. Is
there anything Leibniz produced if this pursuit is true
(writings orbooks on the matter)?4. How did (I believe a
Japanese person) memorize something on the O(42000)digits of
the pi expansion? What technique is used to memorize such
longsequences of numbers? IIRC correctly, Euler knew the
decimal expansion ofe to at least a hundred digits or more.
Same question, why and how?memorization is an important topic
in any learning endeavor.Flip
 ......... IIRC correctly,
Euler knew the decimal expansion of> e to at least a
hundred digits or more. Same question, why and how?> I have
heard that Euler invented (or discovered) e. I can see why
if someone had made such an important discovery, that they
would be highly motivated to remember it.I discovered the far
less important number 3,386,001,688 - at least I think that I
was the rst tofind it - I was denitely motivated to
remember it, at least for a day or so. (It is the number of
ways of placing the twelve pentominoes and ve tetrominoes
into an 8x10 rectangle.)Well, from the sublime to the
ridiculous.....Stephen
Message-id:
<HqEJb.212531$8y1.740936@attbi_s52>> ......... IIRC
correctly, Euler knew the decimal expansion of>> e to at
least a hundred digits or more. Same question, why and how?>I have heard that Euler invented (or discovered) e. I can
see why if >someone had made such an important discovery,
that they would be highly >motivated to remember it.I
discovered the far less important number 3,386,001,688 - at
least I think>that >I was the rst tofind it - I was
denitely motivated to remember it, at>least >for a day or
so. (It is the number of ways of placing the twelve
pentominoesand ve tetrominoes into an 8x10 rectangle.)Well,
from the sublime to the ridiculous.....I discovered a number
also. Mine has over 53000 digits and no, I have made
noattemptto memorize them. Instead, I memorized the formula
that generates itn = 2**(6*(4*9**4 + (9**4 - 1)/2 + 1) - 1) -
1This is the rst Sixth Generation Type 21211 Mersenne
Hailstone.Stephen--MensanatorAce of Clubs
=It is usually
best not to change the subject line unless the topic ofthe
post has changed substantially from the original. [.snip.]>
This is not my question, however. It is:>> (Idea- since
{1,-1} is a nontrivial group, ...),What does -1 mean in the
context of an arbitrary group? > > Of course, you are correct
here. I suspect one could say> the following: Given an
arbitrary group (G, *), let (R, *, +)> be any ring such that
G is a subgroup with respect to * and > in a way that every
additive inverse of G is in G,There are too many such rings.
For any ring R, the natural choice isR[G], the group ring
over R, which consists of all polynomials ofthe formsum_{g in
G} a_g*gwhere a_g is in R, only nitely many nonzero, with
multiplicationgiven by multiplying monomials as(a_g*g)(a_h*h)
= (a_g*a_h)(gh) ^ ^ | |____ multiplication in G multiplication
in R.Then G embedds as the collection of all elements of the
form 1*g, 1the multiplicative identity in R; its additive
inverse is (-1)*g,where -1 is the additive inverse of 1 in R.
If you choose R to beany ring of characteristic 2, then -1 = 1
in R, so the additiveinverse of the element 1*g is again 1*g,
which is in the image of thecanonical embedding of G. In
fact, this is the only situation in whichthis will happen in
the group ring. I guess there could be other ringsin which
this can be done, quotients of group rings overcharacteristic
different from 2.> then -1(R) is the additive inverse of 1
with respect to + What does -1(R) mean? R is a ring. Are you
multiplying every elementof R by -1?Or do you mean, the -1
from R? There is no reason to assume this isdifferent from 1,
either, as noted above.> and is in G. If no such ring exists
then G does not have > a -1.And what makes you think that (1)
-1 will be different from 1, the identity of G? (2) In any two
such rings, -1 will be the same element of G?Unless both
questions have an afrmative answer (and clearly,
(1)certainly does not), it still makes no sense to talk about
-1 in anarbitrary group G.> My Guess is as Good as Yours
Theorem 1.1:> the element -1 designated as above is
independent> of the encompassing ring (R, *, +). In a group
ring over a ring of characteristic 2, it will be equal to1.
In other rings, it need not be.>And what makes you think that
in your group, even if -1 makes sense,>-1 is not equal to 1? Using the technique above, it would follow that 1 is no
longer > a unit, am I right?I do not see how that follows. In
(F_2)[G], 1, by which you presumablymean 1*e, e the identity
of G, is a unit.Arturo Magidin, sans .sig
 To all: please
excuse my not being good at following what others>have done. The problem was not your failure to follow; the problem was
your> acting as though you had followed it and claiming an
imaginary error> in the material that you did not follow.
Admitting ignorance and> asking for clarication would have
yielded a different thread. Because of my different
perspective I thought I saw an error inaccepted theory where
in fact there was none. I have tried toapologise for sticking
out on it for as long as I did and hope that itmay now be
forgotten.>But one may think that>branches are worth studying
for there own sake, and then in context>just think of Riemann
surfaces as a byproduct. > > Certainly, although I dont see
the motivation for it. At rst> reading your analysis in
those terms seems correct. Motivation is subjective so
perhaps we could agree to differ on thispoint.
Happy Xmas to all Sci.* NG!> > And to you Abhi, forget the
device and have a good time.> >
George___________________________________________Mere Ghar Ka
Seedha Sa Itna Pataa HaiYe Ghar Jo Hai Chaaron Taraf Se Khula
HaiNa Dastak Zaruri, Na Aavaz DenaMere Ghar Ka Darvaaza Koi
Nahin HaiHain Deevaren Gum Aur Chhat Bhi Nahin HaiBadhi Dhoop
Hai DostKadhi Dhoop Hai DostTere Aanchal Ka Saaya Churake
Jeena Hai Jeena Jeena Zindagi, ZindagiO Zindagi Mere Ghar
AanaAana Zindagi Zindagi Mere Ghar
Aana_______________________________________________I am still
alive and back in comfort of my own room from where allthis
saga began in March 2000.I got new beautiful computer,
listening beautiful songs right now incalm night.Happy new
year to all of you!I am still in Action...-Abhi.
> Happy Xmas to all Sci.* NG! And to you Abhi, forget the
device and have a good time. George
___________________________________________ Mere Ghar Ka
Seedha Sa Itna Pataa Hai> Ye Ghar Jo Hai Chaaron Taraf Se
Khula Hai> Na Dastak Zaruri, Na Aavaz Dena> Mere Ghar Ka
Darvaaza Koi Nahin Hai> Hain Deevaren Gum Aur Chhat Bhi Nahin
Hai> Badhi Dhoop Hai Dost> Kadhi Dhoop Hai Dost> Tere Aanchal
Ka Saaya Churake Jeena Hai Jeena> Jeena Zindagi, Zindagi> O
Zindagi Mere Ghar Aana> Aana Zindagi Zindagi Mere Ghar Aana>
_______________________________________________ I am still
alive and back in comfort of my own room from where all> this
saga began in March 2000. I got new beautiful computer,
listening beautiful songs right now in> calm night. Happy new
year to all of you! I am still in Action... -Abhi.Dont you
realize that They are watching you throughthe new
computer.... <DpadnVJoh5H6SGmiRVn-uQ@comcast.com>
<Pine.LNX.4.58-035.0401021944030.10600@unix41.andrew.cmu.edu>
<NrCdnWwkRvPKGWuiRVn-hA@comcast.com>
<snip > <quote >
Computing machines.> > If an a-machine prints two kinds of
symbols, of which the rst kind> > (called gures) consists
entirely of 0 and 1 (the others being> > called symbols of
the second kind), then the machine will be called> > a
computing machine. If the machine is supplied with a blank
tape> > and set in motion, starting from the correct initial
m-conguration, Please dene m-conguration, as dened in
Turings paper. I think he means what is now called the state
transition table.> The TMs instructions. According to
Turings paper, to which you have kindly provided alink, an
m-conguration is what we today would call a
state.http://www.abelard.org/turpap2/tp2-ie.asp#section-1Its
not the whole transition table; its merely one of the
stateslisted in that table (which may or may not be said to
contain therest of the table; thats a philosophical point
thats not relevanthere). For example, State 1 is an
m-conguration. State 2is a different m-conguration. The
machine, Turing says, will start in the correct
initialm-conguration, which is just his way of saying that
it has adened starting state.> Okay. Here Turing is
apparently assuming that the sequence printed> by the machine
will have a beginning, though not necessarily an end.> Its
not clear how he denes the beginning of the sequence, though
--> is it left-to-right order? or chronological? Left-to-right
has the> advantage of intuitiveness, but chronological makes
more sense> mathematically to me. Please clarify this point:
briey, what does> Turing mean by the word prefacing? Turing
assumes that all TMs start in a dened initial state at the>
beginning (leftmost) position of a blank tape. Okay; this
makes the most sense, so Ill accept it. Unfortunatelyfor us,
Turing simply does not say in his paper what shape the tapeis
supposed to be. In other words, he denes the symbols on
thetape, S(r), in terms of their positions r; but he never
denes therange of r. Is r an integer, a positive integer, an
integer in therange [0, 1024), or something else
entirely?*not* what Id originally been thinking -- I had
been assuming thatr was an integer (positive or negative),
and thus that the tape wastwo-way innite instead of only
one-way innite. If you dontsee the difference, or dont
understand what those phrases are meant> I think Turing
assumes the output tape will be read from left to right.>
Later in the paper, Turing adopts the convention of only
writing> symbols of the rst kind on every other square.> the
0s and 1s. He states that these other symbols are removed> at
some point, but isnt very specic about when or how this
happens. I think you are correct. Symbols of the rst kind
may not beoverwritten, although symbols of the second kind
may be overwritten.The tape is read from left to right, and
only symbols of the rstkind matter as far as the computed
sequence is concerned.> Prefacing means putting a decimal
point in front of a binary string.> Turing is trying to show
that a TM can generate any real number. Yes. Now that I
realize that r is constrained to be positive,this makes sense
-- there is always a left side of the tape atwhich we can
insert a decimal point (at r=0, so to speak).> > Circular and
circle-free machines.> > of symbols of the rst kind it will
be called circular. Otherwise> > it is said to be
circle-free. All right. This is fairly bizarre terminology,
IMHO -- do you have> any idea why Turing chose these
particular words to describe the two> kinds of machines?
Perhaps a quote from section 8 would be in order. I have no
idea why Turing denes computable numbers this way. I was
actually referring to the terms circular and circle-freeto
describe machines that we today would see were exactly
isomorphicto [but not identical to] the terms halting and
non-halting. Whatdoes the idea of circle have to do with the
nitude of symbolsprinted by a machine? [Rhetorical -- unless
you really have someetymology here, dont bother answering
that. Just a claricationof what I meant.]> > > Instructions
for TM2:> > > > 1) Scan right until a 0 is found> > > 2) Scan
right until a second 0 is found> > > 3) Backup and write a 1
on the previous 0> > > Repeat > Using Turings denition, TM2
produces a computable sequence I doubt it. This depends
heavily on the denition of the word> prefacing in Turings
paper. Im sorry, I misread your machine. I read right and
thoughtleft. Obviously, TM2 is exactly equivalent in its
output to TM1(assuming that the tape starts out with all 0s
on it, or doing asyou suggested and replacing the word 0 by
the word blank).> > that represents the largest rational
number less than 1. Blatantly false. No such number exists,
computable or otherwise.> Thats like saying that your
machine computes the number of digits> in pi, or a recipe for
granite cheesecake. This sequence may not represent a real
number, but it is computable. All computable sequences, by
denition, represent real numbers.Any sequence of zeroes and
ones, prexed by a point, is the digitalrepresentation of
*some* real number. Unless the antecedent of this sequence is
granite cheesecake,in which case youre half right; granite
cheesecake is not a realnumber, but neither is it computable
by Turings method.> > .111...1110 (base 2) This is not
correct notation. It reminds me very strongly of> Phils
ramblings, and I really do suggest you take a look at> Google
Groups for sci.math, and search on rational numbers>
countable, largest integer, and terms of that nature. I dont
know who Phil is, but I have started several> threads about
the largest natural number.Google Groups phil sci.math
numbers> I dont know why youfind the idea so bizarre.> The
idea that there is a nite number of natural numbers> is
certainly not as strange as the idea that there are> more
real numbers than natural numbers. Perhaps not -- but its
certainly not as correct!> Several people have suggested that
this proof says> there is a nite number of natural numbers.>
This is incorrect. Right.> This proof shows that no set> can
contain every natural number. Wrong. Consider the set usually
written blackboard-bold /N/,the set of natural numbers. Its
the set {0,1,2,3,...}. Itcontains all natural numbers. It can
be constructed inductivelyas the set S such that 0eS and
xeS->(x+1)eS, where e representsthe is-an-element-of
relation.> This is not the same as saying there is a largest
natnum. Just the opposite.> A set cant contain every natnum
precisely because> there in no largest natmun. Luckily for
the universe, there is no logical requirement thatevery set
must contain a largest element. Consider the set ofeven
numbers, or the set of real numbers less than 1, or theset of
Greek playwrights.> > This is essentially the same reason
Turing gives why> > the diagonal argument doesnt work with
computable numbers.> > The problem can be converted into
determining whether> > every TM is circular or not.> > Turing
proves this is impossible While it is certainly impossible to
determine whether Turing> Machine X is circular, for some
value of X, it doesnt necessarily> follow that the
computable numbers are uncountable. For that,> youd need to
actually give a reference to Turings proof, so> that we
could look at it and see whether it proves what you think> it
does. It does not. Im embarrassed that I did not see the
reversalearlier. Cantors diagonal argument, applied to the
real numbers,proves that R is uncountable. Turing showed that
the diagonalargument could *not* be applied to the computable
numbers, whichmeans that he did *not* prove that the set of
computable numberswas uncountable.> > TM2 is not an arbitrary
TM. It is easily specied.> > If we can not determine if TM2
is circle free,> > how can we say that any TM is circle free?
TM2 is circle-free. It never stops printing 1s, which
aresymbols of the rst kind. Thus by denition it is
circle-free. No it doesnt.> At least, no one can prove that
it does. I can. Inductively; in the same way that I can show
that ifI give an idiot a card reading Turn this card over.on
both sides, and the idiot obeys the orders on the card,
hewill never stop turning the card. In the same way, the
ordersyou gave to TM1 tell it to keep printing 1 forever;
thus, itwill never stop printing 1s.> It is simple to show
that the number of 1s> written by TM1 is some multiple of
the number> of 1s written by TM2.> Inductively.>A Turing
machine can certainly> compute the following irrational
number, though I have not bothered> to write out its state
transitions:
.1011011101111011111011111101111111011111111011111111101111111
1110... This is the same sequence I use in Cardinality of
Computable Numbers.> Turing gives a similar string as an
example of the output of a TM.> He provides a state table to
produce the string 001011011101111... If you let TM2 read
this tape it will produce a sequence, of 1s> followed by a
0, that is longer than any such sequence on the initial tape.
Incorrect. Besides, even if you did let TM2 start with this
tape,that wouldnt produce a computable sequence --
computable sequencesare made starting with a blank tape,
according to Turings paper.> That number, which is
approximately 0.71673, is computable, but> certainly not
rational! Also computable: pi and e, among many> others.
These numbers are computable only if you can show there is> a
circle free TM that computes the relevant innite sequence.> I
doubt any TM can be shown to be circle free as Turing denes
it. You just said that Turing had given a state table for a
TM thatcomputed 0.71673...! Obviously that number is
computable by a TM,then -- its computable by the TM that
Turing just nished dening! Read the responses to Phils
ramblings before proceeding alongthis line of inquiry --
youre getting in way over your head invery shallow
water.-Arthur
= No it doesnt.> At least, no one can prove
that it does. I can. Inductively; in the same way that I can
show that if> I give an idiot a card reading Turn this card
over. on both sides, and the idiot obeys the orders on the
card, he> will never stop turning the card. In the same way,
the orders> you gave to TM1 tell it to keep printing 1
forever; thus, it> will never stop printing 1s. It is
simple to show that the number of 1s> written by TM1 is some
multiple of the number> of 1s written by TM2.>
Inductively.Turing doesnt say that computable numbers
require induction.It is easy to prove that the output of TM2
is nite.TM2s tape will always have a string of 1s followed
by a blank.The blank must be at a nite position.It is
impossible for TM2 to write an innite string of 1s.Russell-
2 many 2 count
 [re: Russell Easterly]> Its funny that
since I started reading these newsgroups a year or two> ago,
Ive encountered at least half a dozen people arguing exactly
the> same point of view as you, but in different contexts. Now
were in the> context of Turing Machines. Thats very humorous
to me. I didnt know> people with your beliefs existed, and
now Ifind that there are at least> several of you.> > I can
easily understand how some people dont get innity. I>
myself still dont quite get ordinal numbers, although Ive
got> the cardinals down pretty well now. :)> What I dont
understand is how some people who dont get innity> seem to
compulsively post *wrong* statements to the Internet, rather>
than trying to understand *right* ones. And how a guy like
Russell> can seem to have such a reasonable grasp of what a
Turing machine> is, without having even a basic conception of
the properties of the> integer numbers!> > Have you heard of
this guy Phil who used to post absurd> things like the
statement that all natural numbers have nitely> many digits?
Its quite fascinating. I could write a book about it.> > I
remember Phil. But I must point out that you forgot to
complete> that thought: All natural numbers *do* have nitely
many digits!Yes, sorry! Now Im posting absurd statements.
Writing too quickly.I did mean to complete my thought.> But
Phil made a leap from that true statement to the false
statement> that *the number of* natural numbers was nite --
and stuck to it --> and thats what was absurd.> >
-Arthur
=By refuting any claim to the contrary.If you
propose some nite number of 1s which you claim itnite
number that you name _rst_. Thats the crucialpoint you
persist in missing: it is impossible to uphold thenatural
number is nameable, you are _required_ to _name_ thenumber
you claim has the property that it denes the maximumnumber
of 1s written; otherwise your claim is a meaninglessnoise.
But as soon as you do so, I can immediately refute thatclaim
by examining in detail the behavior of TM2 after it hasjust
written the number of 1s you claim is the maximum, andmore
1, thus proving false your claim that your nite numberdenes
the maximum number of 1s written. Since this can bedone for
_any_ nite number, there is _no_ nite number thatdenes the
number of 1s written by TM2, refuting the claimthat there is
such a number.And when you write the output of TM2 as
1111...111(0)*, you arewrting nonsense, because there is _no
denable place_ in theoutput of TM2 where that transition
from 1s to 0s occurs.xanthian.-- 
 By refuting any
claim to the contrary. If you propose some nite number of
1s which you claim it> nite number that you name
_rst_.This is pretty much my argument.I can alwaysfind a
nite number bigger than all thenite numbers in some set.The
output tape will always start with a string of 1sfollowed by
a blank (or a 0).The blank must be in a nite
position.Russell- 2 many 2 count
=No, the exact opposite is
obvious, just not to you.> The output tape will always start
with a string of 1s> followed by a blank (or a 0).by which
you claim I canfind that 0, and I can demonstrateyou cannot
demonstrate that it is on the tape at any nitelocation, then
you cannot demonstrate that it is on the tape_at all_, and
this is the place where your intuition is failing,and real
mathematical logic must be used, instead.To look at your
problem another way, the word you used, always,is either a
meaningless noise, or else you need to dene it tomake it not
one. In the process of creating that denition, youwillfind
you have destroyed your current intuition about thebehavior
of TM2.> The blank must be in a nite position.No, it mustnt
for the reasons I just explained. You aredoing something
_wrong_. Rather than merely continuing tomake _the *same* set
of incorrect claims over and over_, stopto _think through_ the
objections you are receiving.You have given all the needed
evidence that you have theintelligence to understand them,
you just need to stop yourdependence on your failing
intuition about innite processes,and do the hard work of
understanding what you have been told,needed to overcome the
errors into which your intuitionperpetually leads
you.xanthian.-- 
 This is pretty much my argument.> I can
alwaysfind a nite number bigger than all the> nite numbers
in some set.> This should be : I can alwaysfind some number
bigger than all the nite numbers in some nite set-- PentoDe
wereld was soep, en het denken meestal een vork,tot smakelijk
eten leidde dat zelden. - H. Mulisch
 CLAP C:LAP Im not
not an asshole, you are a fertilizerdispensing orice.
=On 2
would you want to be in a contest where the objective is to
provethat youre an asshole?---
say.By the way, get outta alt.atheism. Nobody here believes
*anybody ispsychic so youre wasting bandwidth.-- Mark K.
Bilbo - a.a. #1423EAC Department of Linguistic
Subversion
 CLAP CLAP to you says Lord> > judging by the
names of the 2 replies.> > CLAP CLAP> Clave and Clayton> >
broadcaster.> > But OK, starting soon, I dont want you
slandering me though> as part of your posts, what subject
matter?> > And afterwards we collate the replies to measure
my claim.> HercOh please do me a favour:1. Get a digital
camera2. Make a picture of yourself in your favorite pose3.
Load the picture on to your computer4. Reduce the picture to
promise to publish your picture in my galery-- Wieland the
Smith, AA#2040, EAC: herder of the trolls
 Dene the
function f from (-inf,0] U [1,inf) to (-inf, inf) as
follows:> > f(x)=x for x<=0> > f(x)=x-1 for x>=1> > Clearly
this function is continuous over each of the two disjoint>
intervals which make up its domain.> > But does it make sense
to call the function continuous? (Its the gap in> the domain
that bothers me.)> > > LIf you examine the endpoints of the
half open intervals, youll see thatlim_{x->0-} f(x) = f(0),
and lim_{x->1+} = f(1). By the denition ofcontinuity then, f
is continuous at these points. Now, youve alreadyestablished
that f is continous everywhere else, so f is continouson its
natural domain.
 Dene the function f from (-inf,0] U
[1,inf) to (-inf, inf) asfollows:> f(x)=x for x<=0> f(x)=x-1
for x>=1> If you examine the endpoints of the half open
intervals, youll see that> lim_{x->0-} f(x) = f(0), and
lim_{x->1+} = f(1). By the denition of> continuity then, f
is continuous at these points. Now, youve already>
established that f is continous everywhere else, so f is
continous> on its natural domain....hmm, not its natural
domain
> Dene the function f from (-inf,0] U [1,inf) to
(-inf, inf) as> follows:>> f(x)=x for x<=0>> f(x)=x-1 for
x>=1>> If you examine the endpoints of the half open
intervals, youll see that>> lim_{x->0-} f(x) = f(0), and
lim_{x->1+} = f(1). By the denition of>> continuity then, f
is continuous at these points. Now, youve already>>
established that f is continous everywhere else, so f is
continous on>> its natural domain.> > ...hmm, not its natural
domainWhoops.. Yes, a domain is specied in this case. So, f
is continuous onits domain.
 Dene the function f from
(-inf,0] U [1,inf) to (-inf, inf) as follows:> f(x)=x for
x<=0> f(x)=x-1 for x>=1> Clearly this function is continuous
over> each of the two disjoint intervals which make up its
domain.> But does it make sense to call the function
continuous?> (Its the gap in the domain that bothers me.)The
function f is continuous at every point (number) in itsdomain,
everywhere f is dened. That is the denition ofwhat it means
for a function to be continuous.For a function f not to be
continuous, there has to be anumber in its domain where f is
not continuous.
Dene the function f from (-inf,0] U
[1,inf) to (-inf, inf) as follows:f(x)=x for x<=0f(x)=x-1 for
x>=1Clearly this function is continuous over each of the two
disjoint intervals>which make up its domain.But does it make
sense to call the function continuous? Whether or not it
makes sense to call it that, the given function_is_
continuous.>(Its the gap in>the domain that bothers
me.)>L************************David C. Ullrich
 > Whether
or not it makes sense to call it that, the given function>
_is_ continuous.> > You havent been reading elementary
calculus texts recently, haveyou...???Of course a
mathematician will say it is continuous. But theirdenition
may not agree with the denition in all elementary
texts.
=Leonard M. Wapner <lwapner@adelphia.net> scribbled
the following:> Dene the function f from (-inf,0] U [1,inf)
to (-inf, inf) as follows:> f(x)=x for x<=0> f(x)=x-1 for
x>=1> Clearly this function is continuous over each of the
two disjoint intervals> which make up its domain.> But does
it make sense to call the function continuous? (Its the gap
in> the domain that bothers me.)AFAIK you can safely call the
function continuous. Or at least call itcontinuous on the
interior of its domain and one-sidedly continuousat 0 and
1.-- /-- Joona Palaste (palaste@cc.helsinki.) -------------
Finland ---------- http://www.helsinki./~palaste
--------------------- rules! --------/It was, er, quite
bookish. - Horace Boothroyd
> Its ludicrous to try to
justify teaching mathematics,>> because its useful. For
example, consider factoring>> polynomials, etc. Mathematics
is part of Western>> Civilization. In order to be educated or
cultured,>> if one values that, it is an important subject to
be>> acquainted with. Its useful and valuable for those>>
who use and value it.>> [...]>> I presume you consider it as
an argument that an>> average fourth-grader wouldfind
convincing?>If fourth graders are pressing for an answer to
this>question, it means the class is not going well.>Engaged
fourth graders do not ask such questions.It does not mean
that at all; it means something MUCHworse. The educationists
try to teach relevant factsand methods, and do not give any
of the concepts behindthose facts and methods. Younger
children do not havethis problem; they want to know why
something is thecase. By fourth grade, they have largely had
thisknocked out of them, and VERY few of those involved
withthe public schools have any understanding; this is whythe
new math failed, as the TEACHERS could not learnwhat the
children could.Mathematical CONCEPTS and the purely
linguistic use ofvariables belong early. College students
today havehad little, if any, of this, and want to be taught
HOW to solve well-formulated problems. This is
almostirrelevant, as computers can do this. They need to
beable to speak mathematics, not to do the grunge work.I have
had a class of prospective high school teacherswith two years
of calculus 75% of whom could not setup the calculus to do
probability problems on a takehome exam, problems similar to
homework. Students memorize denitions and formulas, but
understandlittle. On one midterm, power series were
multipliedterm by term.Scientists of all types, and even
historians andstudents of linguistics and literary style,
make use of this; they do not have to be able to carryout the
calculations, but they need to be able tospeak the language,
so that those who are familiarwith the methodology can put
their problems on thecomputer, and criticize what can go
wrong.In addition, mathematical approaches are even usedin
professional athletic training. Pitching coachesuse the
computer and digitized tapes to analyze pitching style and
make recommendations, and thisis similar to what is done in
track and elsewhere.-- This address is for information only.
I do not claim that these viewsare those of the Statistics
Department or of Purdue University.Herman Rubin, Department
of Statistics, Purdue University
> > Who proved that an
innite number of numbers (i.e. innite series)>> > can have
a nite sum?>> No one proved that.>> Someone dened it.>> All
right, instead of proved lets substitute observed and
answer>> the OPs intended question.>book (Grossman:
Multivariable Calculus, ...):> The basic idea in the study of
innite series is that an> innite number of numbers can have
a nite sum. This> concept may seem natural now, but it took
mathematicians> over two thousand years to come grips with.
-- Aristotle --> denied that such a [innite] sum could
exist.>So does that mean people didnt understand at the time
of Aristotle>that an innite sum (i.e. innite series) could
have a nite sum?Aristotle could handle Euclid (Aristotle was
beforeEuclid), but he had difculties like that. He couldnot
handle the idea that a point could be an elementof a line and
also divide the line.-- This address is for information only.
I do not claim that these viewsare those of the Statistics
Department or of Purdue University.Herman Rubin, Department
of Statistics, Purdue University
Who proved that an
innite number of numbers (i.e. innite series)>can have a
nite sum?This goes back to the Greeks; they were the rstto
prove anything. The Achilles and the tortoiseparadox was
known to be a paradox of that type.-- This address is for
information only. I do not claim that these viewsare those of
the Statistics Department or of Purdue University.Herman
Rubin, Department of Statistics, Purdue University
>
proofs>> why do authors usually consider the proofs so
important in mathematical>> texts ?>> Isnt it usually better
when only a few people read and check the proofs,>> while the
masses just try to grasp the meaning of the text>> omitting
the proofs ?>> The proofs could be included as an appendix at
the end of>> the papers, if necessary, IMO.>> --Guenter
Stertenbrink>If the proofs are well-crafted and eloquent and
the reader is reasonably >well prepared, it shouldnt be a
burden to read proofs.Not only that, the proofs can help
understand the theorems.-- This address is for information
only. I do not claim that these viewsare those of the
Statistics Department or of Purdue University.Herman Rubin,
Department of Statistics, Purdue University
how many % of
his time should the average>reader spend on reading proofs>in
a math paper ?>please give your opinions.>I start with :
30%It can vary between near 0 and almost 100%. Sometimes,the
proofs are obvious; much of mathematics consistsof seeing the
obvious, with the proofs being verystraightforward. But some
papers are concerned with adifferent method of proof of a
standard theorem, or anextension of it, and getting at least
enough of an ideaof the proof to nish it oneself is the real
goal ofreading the paper.-- This address is for information
only. I do not claim that these viewsare those of the
Statistics Department or of Purdue University.Herman Rubin,
Department of Statistics, Purdue University
> proofs>>
why do authors usually consider the proofs so important in
mathematical>> texts ?>Because that is what mathematics is
without.The only way we know things is if there are proofs.
Also,the proofs often have the insight into the problem, and
IMOit is better to have a proof which indicates what is
goingon in the theorem, and derives the results in a
directmanner, rather than a cute proof which is quick
anddisguises matters. Unfortunately, there are cases in
whichsuch proofs do not exist.>> Isnt it usually better when
only a few people read and check the proofs,>No. If you refuse
study proofs then you are cravenly deferring to
authority.There are cases where the proofs are sufciently
long anddetailed that this might be the case, but even then
oneshould know the ideas of the proof. >-- >Robin Chapman,
www.maths.ex.ac.uk/~rjc/rjc.html>Needless to say, I had the
last laugh.> Alan Partridge, _Bouncing Back_ (14 times)--
This address is for information only. I do not claim that
these viewsare those of the Statistics Department or of
Purdue University.Herman Rubin, Department of Statistics,
Purdue University
=Is anyone aware of any technique for
computing cycle numbers of bigsubgroups H of S_k, if we know
the generators of H. What Im interested inis applying Polyas
theorem to big subgroups (e.g. C_2 wr (S_n,n) as asubgroup of
S_{n(n-1)/2})Alex.
 >><snip nit-picking on my part so I
thought Id talk about it in detail. Hereare some headers so
you canfind the post:In his post Decker claimed to mirror my
argument using a quadraticinstead of a cubic, where he
has(5a_1(x) + 7)(5a_2(x) + 7) = 7(25x^2 + 30x + 2) where his
as are roots of a^2 - (x - 1)a + 7(x^2 + x).Checking at x=0
reveals that the actual constant terms of thefactorization
are 7 and 2, where Decker picked a_1(0) = 0 at x=0.Now then,
consider what happens if you divide both sides of(5a_1(x) +
7)(5a_2(x) + 7) = 7(25x^2 + 30x + 2) by 7, as then you end up
with something like(5b_1(x) + 1)(5b_2(x) + 2) = 25x^2 + 30x +
2where the bs are roots of some unknown quadratic, though
the rstand last coefcients ARE known:b^2 + ? b + (x^2 +
x).Now then, its just a quadratic people. SOME mathematician
in all theworld should be able to give what the middle
coefcient is, right?>>Right. The middle coefcient in this
case is>> (-3x + sqrt(-7x^2 - 8x))/4> > Well I put that up
kind of curious that someone might ll it in, but> you failed
badly here Decker, as my rst check was at x=1.> > > a^2 -
7(1^2 + 1) = a^2 - 14, so a=+/-sqrt(-14).> > Now then,
dividing off 7, should then give b=+/-sqrt(-2).>No. See
below. > But with your claims, you get> > b^2 + (-3 +
sqrt(-15))b/4 - 2 = 0.> Right, thats what I get. Heres how.
Your specicationwas that you wanted two functions, b_1(x) and
b_2(x)which satisfy P(x)/7 = 25x^2 + 30x + 2 = (5b_1(x) +
1)(5b_2(x) + 2) [1]and which are roots of b^2 + C(x)b + (x^2
+ x) [2]Where C(x) is to be found.I remind you that those are
your requirements, as stated above.[Note in passing that there
are innitely many other bsone could pick, but in each case
youd have a differentpolynomial for which they would be
roots.]At any rate, since [2] is dened by (b - b_1(x))(b -
b_2(x)) = b^2 + C(x)b + (x^2 + x)we must have b_1(x)b_2(x) =
x^2 + xand -(b_1(x) + b_2(x)) = C(x).Expand the RHS of [1]
and we have 25x^2 + 30x + 2 = 25b_1(x)b_2(x) + (2b_1(x) +
b_2(x))(5) + 2 = 25(x^2 + x) + (2b_1(x) + b_2(x))(5) + 2so,
rewriting the LHS and subtracting 2 from both sides we have
25(x^2 + x) + 5x = 25(x^2 + x) + (2b_1(x) + b_2(x))(5)so
2b_1(x) + b_2(x) = xfrom which we have b_2(x) = x -
2b_1(x)Since we have your requirement on the product of the
bswe must have x^2 + x = b_1(x)b_2(x) = b_1(x)(x -
2b_1(x))so (b_1(x))^2 - xb_1(x) + (x^2 + x) = 0, which wecan
solve for b_1(x): b_1(x) = (x + sqrt(-7x^2 - 8x))/4(picking
the positive multiplier of the square root),from which we
obtain b_2(x) = (x - sqrt(-7x^2 - 8x))/2so C(x) = -(b_1(x) +
b_2(x)) = (-3x + sqrt(-7x^2 - 8x))/4as I had originally.
Notice that this is a necessary conclusion(well, up to sign)
of your two requirements.Now, you seem to be surprised that
b_1(x) isnt a_1(x)/7, buttheres no reason to expect it to
be. Observe that I originallyhad P(x) = 7(25x^2 + 30x + 2) =
(5a_1(x) + 7)(5a_2(x) + 7)and you wanted me to produce the
factorization P(x)/7 = 25x^2 + 30x + 2 = (5b_1(x) +
1)(5b_2(x) + 2)which I did. Look at what you asked for, *with
2 as the constantterm of the second factor*. It should come as
utterlyno surprise that, because of the different constantsin
the last terms, wed wind up with b_1(x) differentfrom
a_1(x)/7. Ill note in passing that you can indeedobtain a
factorization with b_1(x) = a_1(x)/7 and 7instead of 2 for
the constant in the last term butIll leave that for the
reader, observing only thatits not pretty. Now lets return
to our scheduledbroadcast. Lets see.> > Maybe Decker came
out hurriedly, probably in a defensive reaction, and> tossed
something out there.> > Even if his quadratic ts as a
factorization, its not THE> factorization that was being
looked for as Ive just shown. Remember,> there is no
uniqueness of polynomial factorization here, so there are> an
innity of factorizations.>Of course. All I did was produce
what you asked for. > Still Decker might seemingly get *some*
credit, if he found one of> them.> > He fails at the task at
hand though for notfinding what follows from> the as in his
*own* example.> > Remember the question Im raising is what
happens when 7 is divided> from both sides of> > (5a_1(x) +
7)(5a_2(x) + 7) = 7(25x^2 + 30x + 2)> > and Im using
Deckers own example, primarily because its dramatic to> do
so, and because its a *quadratic* so that I can show how
complete> his failure is, with something so seemingly
simple.>Hey! I thought you said Id get partial credit.
Cheater. :-) > Moving on, theres the question of why did
Decker pick his example. My guess is that its because at x=1
*both* as have sqrt(7) as afactor.>>Precisely. My point being
that the constants dont serve as>>indicators of how things
are to be divided in any but the x = 0 case.> > But Decker,
how do you suppose *constants* like 7 can change dependent>
on x?>Obviously, they dont. The whole point was that the
factorizations do.Ill state again that you have no reason to
infer what happens ina factorization for x != 0 from what
happens when x = 0. > Now then, why dont you explain where
you got your factorization from,> like the techniques you
used to generate it?> Always glad to oblige. See above. >Its worth noting, by the way, that if we take x = 2>>we
have a situation where you cant even split the factor>>of 7
into anything as nice as sqrt(7) * sqrt(7).> > Why? Can you
elaborate more on why you think that is the case, and> why
you think its worth noting?> Because it shows how
factorizations in the algebraic integersare very much
stranger than factorizations in rational integers.Rick 
> > Theres one type of attempt at disputing my work that
Ive seen pop up> > regularly, and it popped up today from
Rick Decker, a professor at> > Hamilton University, > > > To
be precise, the legal name is Hamilton *College*. We do have>
a university--Colgate--just down the road, but were a
college.> (Ill let maky make of that what he will.) In
fairness, its> a common error, especially in most of the
rest of the world,> where college means something entirely
different than it> does in the US.> > Ok, my mistake, so its
Hamilton College then.> > > so I thought Id talk about it in
detail. Here> > are some headers so you canfind the post:>  > > > In his post Decker claimed to mirror my argument
using a quadratic> > instead of a cubic, where he has> > > >
(5a_1(x) + 7)(5a_2(x) + 7) = 7(25x^2 + 30x + 2) > > > > where
his as are roots of > > > > a^2 - (x - 1)a + 7(x^2 + x).> >  Checking at x=0 reveals that the actual constant terms of
the> > factorization are 7 and 2, where Decker picked a_1(0)
= 0 at x=0.> > > > Now then, consider what happens if you
divide both sides of> > > > (5a_1(x) + 7)(5a_2(x) + 7) =
7(25x^2 + 30x + 2) > > > > by 7, as then you end up with
something like> > > > (5b_1(x) + 1)(5b_2(x) + 2) = 25x^2 +
30x + 2> > > > where the bs are roots of some unknown
quadratic, though the rst> > and last coefcients ARE
known:> > > > b^2 + ? b + (x^2 + x).> > > > Now then, its
just a quadratic people. SOME mathematician in all the> >
world should be able to give what the middle coefcient is,
right?> > > > Right. The middle coefcient in this case is> >
(-3x + sqrt(-7x^2 - 8x))/4> > Well I put that up kind of
curious that someone might ll it in, but> you failed badly
here Decker, as my rst check was at x=1.> > > a^2 - 7(1^2 +
1) = a^2 - 14, so a=+/-sqrt(-14).As pointed out in a reply to
this post that is wrong.It should be a = +/-sqrt(14). > Now
then, dividing off 7, should then give b=+/-sqrt(-2).Should
give b=+/-sqrt(2).> > But with your claims, you get> > b^2 +
(-3 + sqrt(-15))b/4 - 2 = 0.> > > > Lets see.> > Maybe
Decker came out hurriedly, probably in a defensive reaction,
and> tossed something out there.And so far he hasnt answered
from what is currently showing in Google Groups.James
Harris
 > > > Theres one type of attempt at disputing my
work that Ive seen pop up> > regularly, and it popped up
today from Rick Decker, a professor at> > Hamilton
University, > > > > > > To be precise, the legal name is
Hamilton *College*. We do have> > a university--Colgate--just
down the road, but were a college.> > (Ill let maky make of
that what he will.) In fairness, its> > a common error,
especially in most of the rest of the world,> > where college
means something entirely different than it> > does in the US. Ok, my mistake, so its Hamilton College then.> > > so I
thought Id talk about it in detail. Here> > are some headers
so you canfind the post:> > > > > > In his post Decker claimed
to mirror my argument using a quadratic> > instead of a cubic,
where he has> > > > (5a_1(x) + 7)(5a_2(x) + 7) = 7(25x^2 + 30x
+ 2) > > > > where his as are roots of > > > > a^2 - (x - 1)a
+ 7(x^2 + x).> > > > Checking at x=0 reveals that the actual
constant terms of the> > factorization are 7 and 2, where
Decker picked a_1(0) = 0 at x=0.> > > > Now then, consider
what happens if you divide both sides of> > > > (5a_1(x) +
7)(5a_2(x) + 7) = 7(25x^2 + 30x + 2) > > > > by 7, as then
you end up with something like> > > > (5b_1(x) + 1)(5b_2(x) +
2) = 25x^2 + 30x + 2> > > > where the bs are roots of some
unknown quadratic, though the rst> > and last coefcients
ARE known:> > > > b^2 + ? b + (x^2 + x).> > > > Now then,
its just a quadratic people. SOME mathematician in all the world should be able to give what the middle coefcient is,
right?> > > > > > Right. The middle coefcient in this case
is> > > > (-3x + sqrt(-7x^2 - 8x))/4> > Well I put that up
kind of curious that someone might ll it in, but> you failed
badly here Decker, as my rst check was at x=1.> > > a^2 -
7(1^2 + 1) = a^2 - 14, so a=+/-sqrt(-14).> > As pointed out
in a reply to this post that is wrong.> > It should be a =
+/-sqrt(14).> No - the expression is a^2 - (x - 1)*a + 7*(x^2
+ x).When x = 1, this reduces to a^2 + 14, which has roots+/-
sqrt(-14).> Now then, dividing off 7, should then give
b=+/-sqrt(-2).> > Should give b=+/-sqrt(2).> > > But with
your claims, you get> > b^2 + (-3 + sqrt(-15))b/4 - 2 = 0.>  > > Lets see.> > Maybe Decker came out hurriedly, probably
in a defensive reaction, and> tossed something out there.> >
And so far he hasnt answered from what is currently showing
in Google Groups.> Why should he answer? He got it right the
rst time. Hehas nailed you cold. What your method would
yield in thiscase is the factorization Q(x) = (5 a_1(x) +
7)*(5 b_1(x) + 2)where b_2(x) = a_2(x) + 1 and a_1(x) and
a_2(x)are the roots of a^2 + (x - 1)*a + 7*(x^2 + x).One
notes that a_1(0) = 0, a_2(0) = -1, b_2(0) = 0.Thus Q(0) = 14
= 7*2. YOUR method would dictate that you factor Q(x)/7 as
Q(x)/(7 * 1) = (5 a_1(x)/7 + 1)*(5 b_2(x) + 2),which yields,
exactly as in your cubic case,the fact that in general
a_1(x)/7 is not an algebraicinteger: for x = 1, a_1(x) =
sqrt(-14) andsqrt(-14)/7 is not an algebraic integer. But,
Decker points out, for x = 1, thereis *another* way to factor
Q(x)/7 that DOESresult in algebraic integers all around:
Q(1)/7 = Q(1)/(sqrt(7)*sqrt(7))) = (5*sqrt(-2) + sqrt(7)) *
(-5*sqrt(-2) + sqrt(7)). You can check that this equals 57,
as it should. Bottom line: your factorization doesntwork.
Deckers does. And it generalizes for otherx <> 0. Try x = 2
as Rick suggests. It is exactly the same in the cubic
caseexcept there, the roots of the equation area mess to
write down and the corresponding factorizationof 49 is even
worse. That is why we have had toresort to more abstract
arguments. The beautyof Ricks example is that you can do the
calculations- no too-hard-for-you-to-understand abstract
argument is required - and even you cannot deny the result.
Your *method* is bogus. [In your cubic example,
yourfactorization of 49 as 7*7*1 is the wrong one,
exactlyanalogous to what happens here, where your
factorization of 7 as 7*1 is the wrong one.] Deckers example
leaves you oundering around with arithmetic errors and your
usual bluster - and nothing else. Bravo, Rick Decker! Nora
B.> > James Harris
 > > a^2 - 7(1^2 + 1) = a^2 - 14, so
a=+/-sqrt(-14).> > If Harris means for a^2 - 14 to be zero,
he should take> > a = +/-sqrt(14), not sqrt(-14)OOPS! Youre
right. Ok, I had yet another sign error.> > Now then,
dividing off 7, should then give b=+/-sqrt(-2).> > > In true
mathematics, sqrt(-14)/7 is not equal to sqrt(-2),> > nor is
sqrt(14)/7 equalo to sqrt(2).> > If Harris means to divide
sqrt(-14) by sqrt(7), he must say so > unambiguously.> > And
it would be helpful if he were to clean up his own Augean
stables > before sneering at others stables.Well you got the
latter wrong as remember there are *two* results, soeach is
divided by sqrt(7).James Harris
=I am posting this as more
a fun challenge rather than a seriousquestion.{So, that is
why I have cross-posted this to rec.puzzles ANDsci.math.}We
almost all are aware that, for n = integer >= 2, we can write
anon-integer real with base-n digits (0 through {n-1}), some
digitsfollowing after a decimal-point if necessary.But what
about in base-1?Integers are easy (though base-one
representations are not exactlyanalogous to higher bases,
since we do not write base-1 integers usingonly
zeros).Example: 7 (base 10) =1111111 (base 1)But what about
non-integers?Have you any clever schemes for writing, say,
1/2 or pi in base-1??[The best I can come up with right now
is to write the continuedfraction of the real, with each term
consisting of a base-1 positiveinteger. But this is really a
Quet
 But what about in base-1? Integers are easy (though
base-one representations are not exactly> analogous to higher
bases, since we do not write base-1 integers using> only
zeros). Example: 7 (base 10)  1111111 (base 1) But what
about non-integers?> Have you any clever schemes for writing,
say, 1/2 or pi in base-1??>2/2 = 0.111111...1/2 = 0.1010...1/3
= 0.100100...2/3 = 0.110110...1/4 = 0.10001000...2/4 =
0.10101010...3/4 = 0.11101110......> [The best I can come up
with right now is to write the continued> fraction of the
real, with each term consisting of a base-1 positive>
integer. But this is really a list of base-1 integers.
Still,> anything better??]>Huh?
 But what about in
base-1? Integers are easy (though base-one representations
are not exactly> analogous to higher bases, since we do not
write base-1 integers using> only zeros). Example: 7 (base
10)  1111111 (base 1) But what about non-integers?> Have
you any clever schemes for writing, say, 1/2 or pi in
base-1?? 2/2 = 0.111111...> 1/2 = 0.1010...> 1/3 =
0.100100...> 2/3 = 0.110110...This also isnt an intuitive
scheme (IMHO), since then what is ?/? = 0.011011...Also 2/3?
Then we have two different representations for thesame number
(and 3 different ways to write 3/4, and 7 differentways to
write 7/8, and 10000 different ways to write 10000/10001,and
so on). Hmm.> [The best I can come up with right now is to
write the continued> fraction of the real, with each term
consisting of a base-1 positive> integer. But this is really
a list of base-1 integers. Still,> anything better??] Huh?For
example, 1/2 = 1/(2+0) -> (11) 3/4 = 1/(1+1/(3+0)) -> (1,111)
pi = 1/(3+1/(7+1/(15+... ->
(111,1111111,111111111111111,1,...)HTH,-Arthur
 Example:
7 (base 10)  1111111 (base 1)... and just for fun, given
that usually numbers in base N are writtenusing alpha-numeric
digits 0 to (N-1), whereby symbols representingdigits N >= 10
are A, B, C etc or any other representation. Myquestion is,
why in base N=1, we do not write:7 (base 10) = 0000000 (base
1) ?
> We almost all are aware that, for n = integer >=
2, we can write a>> non-integer real with base-n digits (0
through {n-1}), some digits>> following after a decimal-point
if necessary.>> >> But what about in base-1?and Mensanator (or
should it be THE Mensanator?) replied:> There is no Base 1.
...> ...> Now if you want to talk about non-standard number
systems, thats ne,> but dont interchange them with
standard systems because they dont> mix well.Can we use a
fraction as a radix, such as r = 3/2? And if so, what
happenswhen r approaches 1?I havent looked at fractional
radices, so I dont know whether its ameaningful question.
Issues would seem to be the ability to represent allnumbers
(uniquely?), and what it means for a rational to approach
1.Bob H
 Have you any clever schemes for writing, say,
1/2 or pi in base-1??1/2 = 1.112/3 = 11.111pi =
1111111111111111111111111111111.1111111111
 Have you any
clever schemes for writing, say, 1/2 or pi in base-1?? 1/2 =
1.11> 2/3 = 11.111> pi =
1111111111111111111111111111111.1111111111 Ah, but this is
but an approximation to pi! That wont do!The *real*
tally-representation of pi is, as in decimal, an
innitesequence, like this:
...111111111111111111111.11111111111111111...This sort of
thing can be tedious to write out, of course, andrequires
innite amounts of paper to do sums with; so certainscholars
have taken the ingenious approach of folding thenumber at the
position of the unary point and interleaving thedigits of the
numerative and denominative parts, like so: .1 1 1 1 1 1 1 1
1 1... 1 1 1 1 1 1 1 1 1...or more concisely as
.11111111111111111...This has the advantage of being
manipulable almost as readilyas the standard representations.
For example, a multiplicationby two is obtained by halving the
frequency of denominative 1sin the representation:
.11111111111111111...and addition can be performed on any
common household typewriterwith an overstrike capability. In
short, the tally-system of real number representation
isperfectly suited to arithmetic on all levels, and I think
youllagree that it may become the wave of the future in
mathematicseducation.;-)-Arthur[FWIW, Mensator was right on
target.]
=In usual bases (>1), 0 (within a non-zero number
representation) isa place marker. To the left of the decimal
point, the representationstops with a non-zero integer. To
the right it is somewhat freeform. In any case, how can you
represent any number at all in base1?---= 19 East/West-Coast
Specialized Servers - Total Privacy via Encryption =---
>I have a question and would appreciate any help.>Let (f_n)_
be a sequence of real valued functions dened on [a, b]>that
converges uniformly to f. For every x in [a,b], let F_n(x)
Integral (from a to x) f_n(t)dt. > > You have to say a
little more than you have or this integral> doesnt even
exist. Possibly you meant to assume that f_n> was continuous
- that would do it.> >Then, (F_n) converges to F(x)
Integral (from a to x) f(t)dt. Is this convergence uniform
on [a,b]?> > Yes.Presumably he meant to assume the f_n are
Riemann integrable--theminimum needed to make sense of
Integral (from a to x) f_n(t)dt. The uniform limit of Riemann
integrable functions is also Riemannintegrable, so this seems
OK.--Ron Bruck
=JS: Yes and he is correct.PZ: Einsteins
stated position was that gravitation and frame acceleration
arecompletely physically equivalent.How can you now say this
is correct? Even Wheeler doesnt believe this.JS: Show me
quote from Wheeler that leads you to say that.I gave you
complete argument based on geodesic deviation equation and
the complementarityof g-force vs tidal measurements! Thats
it.You need, in addition, curved geometry to make LOCAL EEP
consistent GLOBALLYthis is the point of Hawkings example
where two oppositely accelerating observers are atxed
spatial separation. This is not possible in at Minkowski
space-time only in non-Euclideancurved space-time. This is
why Yilmaz and PV are complete nonsense IMHO.PZ: For some
reason it seems you just cant bring yourself to admit that
Einsteincould be wrong about this fundamental issue,
regardless of all the obviousobjections.JS: Einstein is not
wrong. He is wonderfully consistent. The idea here is
subtle.PZ: Is Einstein infallible?JS: On this, yes.Overheard
at Caffe Trieste:Alice: Do you know the difference between
God and Jack Sarfatti?Bob: No, what is it?Alice: God does not
think he is Jack Sarfatti. ;-)
 > > JS: Yes and he is
correct.> > PZ: Einsteins stated position was that
gravitation and frame > acceleration are> completely
physically equivalent.> > How can you now say this is
correct? Even Wheeler doesnt believe this.> > JS: Show me
quote from Wheeler that leads you to say that.> > I gave you
complete argument based on geodesic deviation equation and >
the complementarity> of g-force vs tidal measurements! Thats
it.> You need, in addition, curved geometry to make LOCAL EEP
consistent GLOBALLY> this is the point of Hawkings example
where two oppositely accelerating > observers are at> xed
spatial separation. This is not possible in at Minkowski >
space-time only in non-Euclidean> curved space-time. This is
why Yilmaz and PV are complete nonsense IMHO.> > PZ: For some
reason it seems you just cant bring yourself to admit that >
Einstein> could be wrong about this fundamental issue,
regardless of all the obvious> objections.> > JS: Einstein is
not wrong. He is wonderfully consistent. The idea here > is
subtle.> > PZ: Is Einstein infallible?> > JS: On this, yes. Overheard at Caffe Trieste:> > Alice: Do you know the
difference between God and Jack Sarfatti?> Bob: No, what is
it?> Alice: God does not think he is Jack Sarfatti.
;-)[EL]Exactly. :-)EL
 Let s(r,m)  > ---> r> > k> />
---> k|m> 1<= k <= sqrt(m)> > (which is, in linear-mode)> >
sum{k|m,1<= k<= sqrt(m)} k^r.> > So, we have s(r,m) is > the
sum of the r-powers taken over the lower half of the
positive> divisors of m.> > For example, s(1,m) is:>
http://www.research.att.com/cgi-bin/access.cgi/as/njas/
sequences/eisA.cgi?Anum=A066839> > > If r is > 0 (r = any
*positive* real), then:> > > limit{m -> oo} > m> ---> 1 >
------- > s(2r,k)  m^(r+1) /> ---> k=1> > > 1> ----------
(?)> 2 r (r+1)> > > Linear-mode:> > limit{m->oo} >
(1/m^(r+1)) sum{k=1 to m} s(2r,k) = > > 1/(2 r (r+1)) (?)>  > (I am err-prone today, so I hope I thwarted fate...)> >
Example: If I am right, the sum of the rst m terms of the
EISs> A066839 divided by m^(3/2) approaches 2/3.By the
way,If s(r,m) is such that, for q = integer >= 2,--- r >
k/---k|m1<= k <= m^(1/q)(which is, in linear-mode)sum{k|m,1<=
k<= m^(1/q)} k^r ;then (?):limit{m -> oo} m --- 1 ------- >
s(qr,k) =m^(r+1) / --- k=1 1---------- q r
(r+1)Linear-mode:limit{m->oo} (1/m^(r+1)) sum{k=1 to m}
Quet
=Am I right to assume:limit{n-> oo} (1/(n ln(n)))
sum{j=1 to n} (sum{k|j, k<= sqrt(j)} H(k) )- (1/8) ln(n) =
1/4 + c/2,where H(k) = 1 + 1/2 + 1/3 +..+1/k, the k_th
harmonic number,and where c = Eulers contstant (c =
following optimization problem, which has twooptimization
objectives:Find the lowest metric B approach, which can yield
the highest metric A...A and B are two metrics...Is there any
certain techniques for handling such
multi-objectiveoptimization problem? I wonder if some common
optimization techniques suchas steepest decent, or Newtons
minimization, still work or not...Please give me some
pointers!-Wlalal
 > Is there a closed form for this sum: i=n> SUM x^(gcd(i,n))> i=1> > (Note: This arises in counting
the number of essentially distinct> colorings of a directed
cycle of length n).> > SiamakI do not know, but
&rnum=19&prev=I get a result which can be used tofind that
your sum is also:sum{k|n} phi(n/k) x^k.(this sum is over the
positive divisors, k, of n; and phi(m) is theEuler phi
function, the number of positive integers <= m and
coprimewith m {= n /k}.)So, my sum basically gives the
Quet
optimization problem. In 
designingiter?6hlistrecoHptxelongfnt#shorfontTEXTGenevaptszfixd8listrecopjste
numleft(FISHÊGeneva??.
7?1.01.01.01.0???????6!{?Ec
frWEst~WEpf?ru?P¢stylnüISH?TABS?WDTH¸jstf?rct?lct?PRopB
Kco[YAcute]?&???&?[YAcute]?x&?ü[YAcute]???h[YAcute]?[Capital
AGrave]&??[YAcute]??&??[YAcute]??????
!?0[YAcute]?!?¬[YAcute]?!??[YAcute]?!?-[YAcute]?&!?d[YAcute]?2!?,
[YAcute]?x!?üven
integer; some of as and bs should be multiple of
3s...Moreover, the number of nonzero as and bs should be as
small aspossible...Under the above-mentioned conditions, I
need tofind the best of suchconstructions...This is simplied
statement, the problem is more complex and n is a lotlarger...
which make the exhaustive search very timeconsuming... even
forfast computers...Please tell me how to systematically
design algorithm to handle this kind ofdiscretely constained
optimal matrix construction problem? Or please give mesome
pointers!-Wlala
=1) In my textbook there is an unproved
proposition (the proof is saidto be obvious) which states
that function f: X subset R -> R isdifferentiable at point a
in X if and only if there exist A = lim_{x-> a+} ((f(x) .9a
f(a)) /(x-a)), B = lim_{x -> a-} ((f(x) .9a f(a))/(x-a)) and
A = B. I do not understand this, given that the
textbookdenition of the derivative of function f at point a
does not requirea to be interior. Does the statement of the
theorem imply that thecondition of point a being interior is
implied in the denition ofthe derivative or I miss
something? In short, what must the derivativedenition look
like for this theorem to hold true?2) Which conditions are
needed so that differentiability at a pointwould imply
differentiability on some interval embracing the point?3)
What are the most common and accepted precise denitions of
theconditions of a function to be continuous at a point and
to bedifferentiable at a point? Consider identity function f:
[0,1] ->[0,1]. Is it continuous at 0? Is it differentiable at
0? (according tothe denitions you provide).
 1) In my
textbook there is an unproved proposition (the proof is said>
to be obvious) which states that function f: X subset R -> R
is> differentiable at point a in X if and only if there exist
A = lim_{x> -> a+} ((f(x) .9a f(a)) /(x-a)), B = lim_{x -> a-}
((f(x) .9a f(a))> /(x-a)) and A = B. I do not understand this,
given that the textbook> denition of the derivative of
function f at point a does not require> a to be interior.
Does the statement of the theorem imply that the> condition
of point a being interior is implied in the denition of> the
derivative or I miss something? In short, what must the
derivative> denition look like for this theorem to hold
true?> > 2) Which conditions are needed so that
differentiability at a point> would imply differentiability
on some interval embracing the point?> > 3) What are the most
common and accepted precise denitions of the> conditions of a
function to be continuous at a point and to be> differentiable
at a point? Consider identity function f: [0,1] - [0,1]. Is it
continuous at 0? Is it differentiable at 0? (according to> the
denitions you provide).> Consider X as the set of rationals,
then X does not have any interior points (as a subset of R),
but functions from X to R can still have derivatives.On the
other hand, The function f(x) = sqrt(x^3) from the
non-negative reals to the reals clearly has a derivative at x
= 0. So that the if-and-only-if fails, at least for many
dentions of derivative.The precise wording of the denition
of derivative in your text may be critical to the
issue.
=It looks like George beat me to the essential idea.
Mine is simpler, but there is conceptual common ground.We knew
each other in 1966-7 in the UCSD group Greg Benford describes
in Timescape that included Herbie Bernstein.George came down
from Cal Tech frequently in his black AC Shelby Cobra that I
used to ride around in with him when Iwas not in Harry
Yesians Red Corvette. This was the era of American
Gratti.Indeed it was because of George that I got the job at
SDSU andmet Fred Wolf. The story is in my book Destiny
Matrix.Chapline on vacuum etc. from 1999:The Vacuum Energy in
a Condensate Model for SpacetimeAuthor: George
Chaplinehttp://www.arxiv.org/abs/hep-th/9812129Comments:
Postscript, 10 pagesJournal-ref: Mod.Phys.Lett. A14 (1999)
2169-2178It is shown that a simple model for 4-dimensional
quantum gravity based on a3-dimensional generalization of
anyon superconductivity can be regarded as adiscrete form of
Polyakovs string theory. This suggests that there is
auniversal negative pressure that is on the order of the
string tensiondivided by the square of the Robertson-Walker
scale factor. This is inaccord with recent observations of
the brightness of distant supernovae,which suggest that at
the present time there is a vacuum energy whosemagnitude is
close to the mass density of an Einstein-de Sitter
universe.http://www.arxiv.org/abs/hep-th/9807175The Black
Hole Information Puzzle and Evidence for a Cosmological
ConstantAuthor: George ChaplineComments: postscipt, 7
pagesRecent hints from observations of distant supernovae of
a positivecosmological constant with magnitude comparable to
the average density ofmatter seem to point in the direction
ofa two uid model for spacetime; where the normal component
consists of ordinary matter, while thesuperuid component is
a zero entropy condensate.My idea is not quite the same. The
normal component consists of dark energy and dark matter both
from residual zero point energy exotic vacua. Ordinary matter
is ultimately made from vortex cores of attractive dark
matter of strong short range positive quantum pressure where
the superuid component drops to zero./zpf ~ - 1/Lp*^2 when
vacuum coherence vanishes inside the stringy vortex core
topological defects of the U(1) vacuum coherence order
parameterLp* ~ 1 fermiLp*^2 = hG*/c^3G*/c^4 = (lepto-quark
string tension )^-1
= It looks like George beat me to the
essential idea. Mine is simpler,> but there is conceptual
common ground. [...]If sheer sustained effort and commitment
guaranteed results, I dontdoubt that Jack Sarfatti (and JSH)
would be up there with Newton andEinstein. Heres an image of
Jack:
http://www.thinking-allowed.com/1jsarfatti.html--------------
-------------------------------------------------------------
John R Ramsden
(jr@adslate.com)---------------------------------------------
------------------------------Eternity is a long time,
especially towards the end. Woody Allen
 > 8-Basti.M. On
Bessel Differential Equations. Submitted for publication> to
solutions of Bessel Differential Equations. Submitted> for
other day I found a parametric solution of the standardBessel
ODE, in which x and y are quite complicated functionsof erf(x)
and its integral, not that its any practical useto man or
beast.I wonder if anyone has managed tofind a symbolic
solution ofthe slightly more general Sturm-Liouville ODE with
spectrumequal to the real roots of the Riemann-Siegel zeta
function.----------------------------------------------------
-----------------------John R Ramsden
(jr@adslate.com)---------------------------------------------
------------------------------Eternity is a long time,
especially towards the end. Woody Allen
=[...]|>If we were
talking about formal proofs, it would be appropriate
to|>speak in hard-edged terms about the properties of the
proof. But|>this is an informal proof.|>|>Take an analogous
situation. Fermats proof of his conjecture for|>p=4 ts the
standard mold of proof by innite descent. He shows|>that if
it fails for some value of z, then it also fails for
some|>smaller value of z. How wrong is it to say he was doing
a proof by|>induction? Well, I tend to cover myself in
situations like that by|>saying essentially a proof by
induction, but I dont think its|>just plain wrong to call
it a proof by induction when its|>essentially a proof by
induction.|>|>Likewise, I dont think its just plain wrong
to call a proof|>by proving the contrapositive a form of
proof by contradiction,|>since its essentially a proof by
contradiction.|>|>For one thing, if we were to take this
informal proof (or any|>proof by proving the contrapositive)
and convert it into a|>formal proof, quite typically it would
formalize as a proof by|>contradiction.||We may be getting
somewhere:||Depends on the formal system. Suppose we were
talking|about a formal system that had this rule of
inference:||[Contrapositive:]||~B |- ~A|_______| |- A ->
B.||In that formal system a proof via the contrapositive
would|formalize precisely as a proof via Contrapositive.Well,
you could put together a system like that. People have been
knownto write systems of logic that would do a lot more than
just include aproof-by-contrapositive. I gather theres a
logic textbook by a guy namedCopi which has several more ways
of disguising proof by contradictionunder several more obscure
names, each distinct from the other one.Proof by dilemma, etc.
Not considered very elegant, though.One problem with adding
proof by contrapositive as a special rule is thatits
redundant. If we were to drop another rule, probably we could
makeit stop being redundant, but then itd be essentially
doing the job thatproof by contradiction had been doing.In
the sequent calculus, one takes A1,...,An |- B1,...,Bmto mean
(A1&...&An)->(B1 or ... or Bm). It appears typical enough to
haveas rules in the sequent calculus things like these: S A
|- T -------- S |- ~A T S |- A T -------- S ~A |- Tand
sometimes the same rules taken in reverse, where S and T are
listsof formulas and A is a single formula. In a system like
that, one mighthave ~B |- ~A -------- ~B A |- -------- A ~B
|- -------- A |- Bor ~B |- ~A -------- |- B ~A -------- |- ~A
B -------- A |- Bwhere in the rst, the empty disjunction on
the right hand side standsfor a contradiction, and in the
second the empty conjunction on the lefthand side is simply
taken to be true.Now, your rule of replacing the
contrapositive with the implication islike this, except that
it involves moving a statement from each side ofthe
turnstile, |-, to the other side as one removes a negation
from it.Requiring both moves to be made at the same time
seems somewhat arbitraryto me. Each rule permitting each
separate move of a statement to the otherside while removing
a negation is okay, and it seems reasonable to meto think of
the combined switcheroo as being composed of the two
movestaken one at a time, in either order.The one rule says
that deducing ~A from a set of premises is equivalentto
deducing a contradiction from the set of premises together
with A. Theother rule says that deducing a contradiction from
a set of premises whichincludes ~B is equivalent to deducing B
from the remaining premises. Thoseare the two standard forms
of proof by contradiction, the rst being theconstructive
one, the second being the standard way in which a proof
failsto be constructive.The analysis is different in each
system, but as far as I can see italways makes sense to
consider it a combination of contradiction withsomething else
small.I think partly I tend to think of proof by contradiction
as being moreakin to an application of an axiom than as a pure
inference rule. Thereis some arbitrariness in how one draws
the line between inference rulesand axioms. There are systems
which make mathematical induction aninference rule. There are
systems which make everything except modusponens (from A,
A->B infer B) into an axiom.If one were to append Zorns
lemma and the well-ordering principle toZFC, one could say of
a lot of proofs that they are not using the axiomof choice,
but using the axiom of well-ordering. But that would be
asomewhat strange and redundant axiomatization.Without
turning Zorns lemma and the well-ordering principle into
axioms,one could still reasonably categorize proofs in ZFC
which are not proofsin ZF by the form of the axiom of choice,
or the consequence of the axiomof choice, being used. Give me
a pile of such proofs, and three binslabelled axiom of
choice, Zorns lemma, and well-ordering principle,and I can
probably put nearly every such proof unequivocally into one
ofthe bins, by the principle *most directly* being used.
(Well, some mightuse more than one.) (On sci.math, Matthew
Wiener once quoted Herstein andKaplansky as writing that the
axiom of choice is intuitively obvious,while as Zorns lemma
its merely plausible, and as the well-orderingprinciple its
obviously false.) This sort of categorization is
probablypretty sharp.Yet for someone to say of all three
piles, that they are proofs using theaxiom of choice, would
be appropriate.I suspect one of the biggest disanalogies
between this situation and thesituation with regard to proof
by contradiction is that people dont tendto think of proof
by contradiction as being application of an axiom.I offered
an analogy with proof by induction versus proof by
innitedescent, where I also could imagine on the one hand
categorizing proofsas tting one or the other mold, but also
wouldnt have much of a problemwith someone calling a proof
by innite descent a kind of proof byinduction. There, one at
least sometimes considers induction an axiom ots own, so
maybe this makes people think of proof by induction as
beingcloser to meaning proof which invokes the axiom of
induction than toproof formatted in this form.A little bit of
experience with constructive proof probably also helps tomake
applications of proof by contradiction (i.e., direct or
indirectapplications) stand out as special. If one has a
proof which isconstructive aside from the passage from ~B->~A
to A->B, one can callthat a constructive proof of A->~~B. A
differently formatted proof whichis constructive aside from
having proven the conclusion, B, bycontradiction can also be
called a constructive proof of A->~~B. So inthat sense they
are alike.The difference between a constructive proof of A->B
and a constructiveproof of A->~~B, for mathematical statements
A and B, is a naturaldifference in *mathematical* meaning, not
just logical or proof-theoretic,not having to do with how
condent anybody is with the correctness ofthe conclusion or
what have you. Consider a linear transformation T on
anite-dimensional vector space V. Think of your favorite
proof that ifthe determinant of T is zero, then there exists
a nonzero vector in thekernel of T. (What type of proof?
Contrapositive or contradiction?)Presumably it will be in
effect a proof that T mapping every nonzerovector in V to a
nonzero vector leads to a contradiction. Why would Iexpect
that? One can distinguish between situations in which one
hasreached the existence of a nonzero vector in the kernel
without any suchdetours, which enable us tofind such a
nonzero vector in the kernel, andsituations in which one has
arrived at this kind of contradiction, whichdont necessarily
enable one tofind such a vector. The latter situationis not to
be disparaged, necessarily; one can show that one can then
atleast get a nonzero vector which maps under T to an
arbitrarily shortvector, for example, which might be all that
one wants. (And thedeterminant being nonzero is not
necessarily enough to enable one tocompute a nonzero vector
in the kernel. The problem case is when thetransformation is
very close to the zero transformation.)Proving that if every
nonzero vector maps under T to a nonzero vector,then the
determinant is nonzero, is constructive. Whether one
immediatelystates the contrapositive and proceeds that way,
or whether one has proventhe existence of a nonzero vector in
the kernel using contradictionanother way, leads you to a
comm-110
Please help to solve these two problems. One member
have posted thisproblem to a math group on msn. I have no
idea to solve thisProblems are going like this:1. DigitsShow
that for any natural n, at least one of two numbers, n or
n+1,can berepresented in the following form: k + S(k) for a
certain k, whereS(k) isthe sum of all digits in k. For
instance, 21 = 15 + (5+1)2. Party!There is a group of people
at a party. Show that you can introduce some of them to each
other so that after the introduction, no more than
two people in the group would have the same number of friends
(initial conguration doesnt work because they all initially
have 0 friends).Ajhar
=I am posting a code which is written
in c++ which canfind the bonacci seriesup to any n number
.The code is like this://starting of the
program#inclue<iostream.h>void main(){int
number,f1=0,f2=1,temp=0;cout<<Enter a number to which you
wants to show the bonacci series: ;cin>>number;//the main
logic is going like thiscout<<Fibonacci series starts
here;cout<<n<<f2;for(int
i=0;i<=number;i++){temp=f1+f2;cout<<n<<temp;f1=f2;f2=temp;}}/
/end of programyou can run this program on turbo c++ or vc++
compilerAjharhttp://roxvary.netrms.com
 Im trying to
write a program in VBA to solve a series of non-linear>
equations. Unfortunately, I hardly know anything about
numerical> analysis. So if you hava a program or idea how to
solve it please> share with me.> The problem is the
following:> C(1)=sum(j=1 to n){alpha(j,1)beta(j)prod(i=1 to
k)[c(i)^alpha(j,i)]}> ...> C(k)=sum(j=1 to
n){alpha(j,k)beta(j)prod(i=1 to k)[c(i)^alpha(j,i)])> where >
prod(i= 1 to
k)c(i)^alpha(j,i)=c(1)^alpha(j,1)*c(2)^alpha(j,2)...> I
need to solve these equations for c(i). The alpha matrix is
such> that the rst kxk part is a unity matrix.> Anybody, any
suggestions?> greetings:> ZsoltI can do programming but i
doest quit understand your problem will you explain your
problem?Ajhar
 Im somewhat versed in math but no
expert by far. Ive been browsing the> internet, news-groups
and all, but still I have not found a good answer to> my
question.> Im looking for a Java (J2ME) implementation (or
something from which i can> create this implementation) for
the approximations (only using additions,> subtractions,
multiplications, divisions and mayb a square-root function
and> logarithm) for one or both of these two functions:>
exp(x) e^x (or pow(2,y), ... raising the power of 2 on>
computers can be faster)> and/or> pow(x,y) x^y> where both
x and(!) y can be real numbers (not necessarily integers).>
The approximations need to be relatively fast (J2ME,
MIDP1.0... limited> device capabilities, no oating-point
support) but quite accurate.> Ive already have good
implementations for the log(x) (using the Maclaurin> series
for 1/(ln[x+1])) and squareroot(x) (using Newtons
Iteration).> I need the approximations where y is between 0
and 1 and where y is a large> number:> (e^x =
e^(largenum+fraction) = e^largenum * e^fraction)> PS: The
x and y are real numbers represented in J2ME by a class
representing> some form of xed-point values.
approximate exp(x)
=In the following Q=ln(2)=
.69314718055994530942... and symbol [.] is devoted for
integral part. Also x=[x]+{x}where {x} is fractionary part of
x. First observe that
2^{a+N}e^{Z(a)}where N=integer:=[x/Q] , F:={x/Q} ,
Z(a)=(F-a)*Q ,a being an arbitrary real
number.
=Please verify
(1): take logarithm in both sides.Suppose x >0 , let q be a
positive integer and put k=[q*F] . Ifk/q =< F={x/Q} < (k+1)/q
, select parameter a to be (2k+1)/(2q) . Then |Z(a)| =< Q/8 <
1/10 . Therefore using (1)(2)
e^x=approx.=C(k)*2^N*G(Z(A))where G(z) will be a ,,good
approximation of e^z for z small, e.g. for |z|<1/10 .==In the
following I give such approximation G(z). Dene(*) A(z)=z^6 +
840*z^4 + 75600*z^2 +665280 B(z)= 42*z^4 + 10080*z^2 + 332640
H(z):= A(z)+ z*B(z). Then put 
= (3) G(z) =
H(z)/H(-z) . 
= It may be shown that |(e^z
- G(z))/e^z| =< 10^{-26} for |z|< Q/8 .In conclusion , tofind
an approximation DEX(X) of e^X=exp(X) you can proceed as
follows ( q=4 ) : STEP 1: Test the variable X , and determine
T=|X| . Suppose that you work with real numbers in intervals
(16^{-64},16^{63}). This means that you can approximate e^X
only for X in [-175,175] .Dene (for instance) DEX(T)=
.0996...E-76 if T =< -175 , DEX(T)=1 when T=0 and DEX(T)=
.10035...E+77 when T>= 175 .STEP 2: Find N=[T/Q] , F={T/Q}
.STEP 3: Find P=2^N , K=[4F] , A=(2K+1)/8 , Z=(F-A)Q .STEP 4:
Select constant C , C:= 2^A , as a function of possible values
of K ,K in{0,1,2,3} .STEP 5: Taking into account (3), consider
approximation e^Z=approx = H(Z)/H(-Z) .STEP 6: Finding nal
approximation, namely e^T=approx=C*P*G(Z)and DEX(X)=
1/(C*P*G(Z)) if Z is in (-175,0) , or DEX(X) = C*P*G(z) when
X belongs to interval (0,175) . Note: Its possible to prove
that the roots z_k of H(z) satisfy 2 =< |z_k| =< 42
,therefore H(-z) =/=0 for |z| small ,e.g |z|< 2 . Other
better functions that (*) are available.If you want tofind
betterapproximations for e^z , please inform me. I appreciate
that (1) was important.
=Those who cant design,
teach.Those who cant teach, design.And the rest are just
clueless.Dan :-)
 And I keep stressing that I am talking
about the U.S., of which youve> admitted some lack of
knowledge.I said that I am not too familiar with US based
research grants. Ialso said that I have no idea what the Fox
news fan club is supposedto be. That is is where I admitted
some lack of knowledge about theUSA, as you say.However I
also said that: I know very well what publish or perish
is(unjustied patronizing tone again, I note), William. I
have somescientic publications, William, and we have pretty
much the sameacademic system here in Israel, as in the USA
(including gettingIn fact Ive spent some time in the USA
academic system, in FortCollins C0. I am also quite familiar
with the UK academic system, inaddition to the Israeli, of
course. So, when I talk about the peoplefrom the academia, I
know, it means quite a lot of people, from threedifferent
countries, but having similar academic systems. That
doesntmake my statements absolute, as you say. But, IMHO,
they aresignicant enough in countering statements
like:...but they often cant do any signicant research
either. (Dependsreply to my: A Prof with a tenure that
doesnt have much contractNote your rather strong words often
and any. As to what looks likean insurance policy (or a sort
of disclaimer) of Depends on...,well, it doesnt make much
sense that the academic system and academicfreedom concepts
should be signicantly different in, say chemistrydept from
that of math, or biology. Even in the great USA. Well, ifsome
mysterious reports they send back from the front lines...
givea different picture, I guess, I am expected to accept
them as factsabout the USA. Well, I might not do that,
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id hA6KGK001939;
=Im desperately
looking for an algorithm to put a reduciblenonnegative matrix
in Frobenius canonical form by a symmetricpermutation of rows
& columns. In graph-theoretic terms, this amountsto
relabeling the graph nodes so that the absorbing nodes (or
groups onodes) receive the highest labels. I will really
appreciate it ifanyone out there can lend me a hand.
Im desperately looking for an algorithm to put a reducible>
nonnegative matrix in Frobenius canonical form by a
symmetric> permutation of rows & columns. In graph-theoretic
terms, this amounts> to relabeling the graph nodes so that
the absorbing nodes (or groups o> nodes) receive the highest
labels. I will really appreciate it if> anyone out there can
lend me a hand.Look at Dulmage-Mendelsohn decomposition.For
example in Matlab, help dmpermArnold Neumaier
 I am a
student implementing a simple mode solver myself,> I need
help.>> The one dimensional Helmholtz equation>
d^2U(x)/dx^2+k^2(N(x)^2-n_eff^2)U(x)=0>> can be solved using
Finite Difference method:czh, lets have an example of this
BVP (boundary value problem).k^2 = 1.234^2N(x) = (x +
0.002345)n_eff^2 = 2.453x_0 = 0x_40 = 10n = 39y_0 = 0y_40 =
-1.435h = 0.250Solution: y_1 to y_39Use plausible data or
modify and solve these equations using anappropriate
program.>>
d^2U(x)/dx^2+k^2(N(x)^2-n_eff^2)U(x)=0((y_2-2*y_1+y_0)/h^2)+
1.234^2*(((1*h)+0.002345)^2-2.453)*(y_1)=0((y_3-2*y_2+y_1)/h^
2)+1.234^2*(((2*h)+0.002345)^2-2.453)*(y_2)=0((y_4-2*y_3+y_2)
/h^2)+1.234^2*(((3*h)+0.002345)^2-2.453)*(y_3)=0((y_5-2*y_4+y
_3)/h^2)+1.234^2*(((4*h)+0.002345)^2-2.453)*(y_4)=0((y_6-2*y_
5+y_4)/h^2)+1.234^2*(((5*h)+0.002345)^2-2.453)*(y_5)=0((y_7-2
*y_6+y_5)/h^2)+1.234^2*(((6*h)+0.002345)^2-2.453)*(y_6)=0((y_
8-2*y_7+y_6)/h^2)+1.234^2*(((7*h)+0.002345)^2-2.453)*(y_7)=0(
(y_9-2*y_8+y_7)/h^2)+1.234^2*(((8*h)+0.002345)^2-2.453)*(y_8)
=0((y_10-2*y_9+y_8)/h^2)+1.234^2*(((9*h)+0.002345)^2-2.453)*(
y_9)=0((y_11-2*y_10+y_9)/h^2)+1.234^2*(((10*h)+0.002345)^2-
2.453)*(y_10)=0((y_12-2*y_11+y_10)/h^2)+1.234^2*(((11*h)+
0.002345)^2-2.453)*(y_11)=0((y_13-2*y_12+y_11)/h^2)+1.234^2*(
((12*h)+0.002345)^2-2.453)*(y_12)=0((y_14-2*y_13+y_12)/h^2)+
1.234^2*(((13*h)+0.002345)^2-2.453)*(y_13)=0((y_15-2*y_14+y_
13)/h^2)+1.234^2*(((14*h)+0.002345)^2-2.453)*(y_14)=0((y_16-2
*y_15+y_14)/h^2)+1.234^2*(((15*h)+0.002345)^2-2.453)*(y_15)=0
((y_17-2*y_16+y_15)/h^2)+1.234^2*(((16*h)+0.002345)^2-2.453)*
(y_16)=0((y_18-2*y_17+y_16)/h^2)+1.234^2*(((17*h)+0.002345)^2
-2.453)*(y_17)=0((y_19-2*y_18+y_17)/h^2)+1.234^2*(((18*h)+
0.002345)^2-2.453)*(y_18)=0((y_20-2*y_19+y_18)/h^2)+1.234^2*(
((19*h)+0.002345)^2-2.453)*(y_19)=0((y_21-2*y_20+y_19)/h^2)+
1.234^2*(((20*h)+0.002345)^2-2.453)*(y_20)=0((y_22-2*y_21+y_
20)/h^2)+1.234^2*(((21*h)+0.002345)^2-2.453)*(y_21)=0((y_23-2
*y_22+y_21)/h^2)+1.234^2*(((22*h)+0.002345)^2-2.453)*(y_22)=0
((y_24-2*y_23+y_22)/h^2)+1.234^2*(((23*h)+0.002345)^2-2.453)*
(y_23)=0((y_25-2*y_24+y_23)/h^2)+1.234^2*(((24*h)+0.002345)^2
-2.453)*(y_24)=0((y_26-2*y_25+y_24)/h^2)+1.234^2*(((25*h)+
0.002345)^2-2.453)*(y_25)=0((y_27-2*y_26+y_25)/h^2)+1.234^2*(
((26*h)+0.002345)^2-2.453)*(y_26)=0((y_28-2*y_27+y_26)/h^2)+
1.234^2*(((27*h)+0.002345)^2-2.453)*(y_27)=0((y_29-2*y_28+y_
27)/h^2)+1.234^2*(((28*h)+0.002345)^2-2.453)*(y_28)=0((y_30-2
*y_29+y_28)/h^2)+1.234^2*(((29*h)+0.002345)^2-2.453)*(y_29)=0
((y_31-2*y_30+y_29)/h^2)+1.234^2*(((30*h)+0.002345)^2-2.453)*
(y_30)=0((y_32-2*y_31+y_30)/h^2)+1.234^2*(((31*h)+0.002345)^2
-2.453)*(y_31)=0((y_33-2*y_32+y_31)/h^2)+1.234^2*(((32*h)+
0.002345)^2-2.453)*(y_32)=0((y_34-2*y_33+y_32)/h^2)+1.234^2*(
((33*h)+0.002345)^2-2.453)*(y_33)=0((y_35-2*y_34+y_33)/h^2)+
1.234^2*(((34*h)+0.002345)^2-2.453)*(y_34)=0((y_36-2*y_35+y_
34)/h^2)+1.234^2*(((35*h)+0.002345)^2-2.453)*(y_35)=0((y_37-2
*y_36+y_35)/h^2)+1.234^2*(((36*h)+0.002345)^2-2.453)*(y_36)=0
((y_38-2*y_37+y_36)/h^2)+1.234^2*(((37*h)+0.002345)^2-2.453)*
(y_37)=0((y_39-2*y_38+y_37)/h^2)+1.234^2*(((38*h)+0.002345)^2
-2.453)*(y_38)=0((y_40-2*y_39+y_38)/h^2)+1.234^2*(((39*h)+
0.002345)^2-2.453)*(y_39)=0These equations are computer
generated.--Website temporarily closed
 Can someone
point me to information on the web about fast factorial>>
calculation? This needs to be for exact value. I can handle
the part>about>> it being too large to represent, but would
like tofind a faster method>than>> just multiplying every
number.>> Adam,>> Try this one
http://www.luschny.de/math/index.htm.>> Could tell what kind
of application are you working for?>>Its labview. NI is
sporting a contest to code the fastest factorial>program. I
have a few ideas on how to calculate numbers larger
than>representable in normal computer formatting, but wanted
to see what>algorithms might I use that would be quicker than
2x3x4... Just for funs>tho.>>An interesting property is that
for n = 2m,n! = 2^m (m!)^2Another interesting point I
developed some time ago,
=fr&lr=&ie=UTF-8&selm=956tbb%24rg8%40deadzone.rsn.hp.com&rnum
=3Michel
> Can someone point me to information on
the web about fast factorial>> calculation? This needs to
be for exact value. I can handle thepart>about>> it
being too large to represent, but would like tofind a
fastermethod>than>> just multiplying every number.> Adam,>> Try this one
http://www.luschny.de/math/index.htm.>> Could tell what
kind of application are you working for?>>Its labview.
NI is sporting a contest to code the fastest factorialprogram. I have a few ideas on how to calculate numbers
larger than>representable in normal computer formatting,
but wanted to see what>algorithms might I use that would be
quicker than 2x3x4... Just for funs>tho.>> An
interesting property is that for n = 2m,> n! = 2^m (m!)^2I
used to think of myself as somewhat of a math wiz but Im not
getting thisequation. Cant seem to make it balance for sample
numbers. In fact lookingat {n,m}={4,2}, I dont see how it
could work at all. 4! contains a factorof 3 which I dont see
the right side coming up with no matter how I look atit. Can
you clue me in as to what I am missing?
=I suspect for this
homework assignment that the original poster was supposed to
stumble across the Gamma function and, more importantly for
large numbers, the LogGamma function.$.02 -Ron Shepard
 I
suspect for this homework assignment that the original poster
was> supposed to stumble across the Gamma function and, more
importantly> for large numbers, the LogGamma function.No, it
really isnt a homework assignment. It is a labview
programmingcontest (for fun). The point is to code it to do
up to 10000 factorialquickly. Do you think that use of the
loggamma function would be a moreefcient algorithm? Does it
produce exact answers?
=Adam Russell schrieb> The point is
to code it to do up to 10000 factorial> quickly.If you want
to compare your code with my Javaimplementation, go
here:<http://www.luschny.de/math/factorial/
FastFactorialFunctions.htm>and then click on benchmark.Java
Factorial Benchmark - Timings(in seconds)N! where N =
16000/32000/64000/128000/256000/512000/1024000PrimeSwing 0,2/
0,8/ 3,8/ 17,6/ 97,3/ 524,9/ 2480,1So for 10000! you can
expect 0.1 seconds with Java,computing on a PC, which is two
years old. Fast enough?> Do you think that use of the
loggamma function would be a more> efcient algorithm? Yes,
but...> Does it produce exact answers?No. Only an
approximation.
=Michel OLAGNON schrieb> An interesting
property is that for n = 2m,> n! = 2^m (m!)^2Sure? I tried
Maple:OLAGNON := proc(m) n := 2*m; m!^2*2^m
end;seq(OLAGNON(i),i=0..7);1, 2, 16, 288, 9216, 460800,
33177600, 3251404800seq(i!,i=0..7);1, 1, 2, 6, 24, 120, 720,
5040Ups. But I know what you mean. If you read carefullythe
function given above, you willfind:46 Integer
recFactorial(int n)47 {48 if (n < 2) return 1;49 return
(recFactorial(n/2)^2) * swing(n);50 }This is a correct, more
general and recursive formulationof the observation you
obviously refer to.Gruss Peter
>46 Integer
recFactorial(int n)>47 {>48 if (n < 2) return 1;>49 return
(recFactorial(n/2)^2) * swing(n);>50 }>>This is a correct,
more general and recursive formulation>of the observation you
obviously refer to.Very interesting for n=4. What does the
swing do?-- Surendar Jeyadev jeyadev@wrc.xerox.bounceback.com
n)>47 {>48 if (n < 2) return 1;>49 return
(recFactorial(n/2)^2) * swing(n);>50 }> Very interesting
for n=4.Not more and not less interesting then for anyother
n. y := recFactorial(n/2)^2< y := recFactorial(n/2); y :=
y^2;> What does the swing do?How much of the thread do you
read, before you write?
Michel OLAGNON schrieb> An
interesting property is that for n = 2m,>> n! = 2^m
(m!)^2>>Sure? I tried Maple:>OLAGNON := proc(m) n := 2*m;
m!^2*2^m end;>>seq(OLAGNON(i),i=0..7);>1, 2, 16, 288, 9216,
460800, 33177600, 3251404800>>seq(i!,i=0..7);>1, 1, 2, 6, 24,
120, 720, 5040>>Ups. But I know what you mean. If you read
carefully>the function given above, you willfind:>>46 Integer
recFactorial(int n)>47 {>48 if (n < 2) return 1;>49 return
(recFactorial(n/2)^2) * swing(n);>50 }>>This is a correct,
more general and recursive formulation>of the observation you
obviously refer to.>Yes, indeed, I had not meant to solve the
whole thing, justto give a hint for even n, and let the OP
nd out how touse it for any n.Michel
=Hallo Liste,ich
hoffe es ist ok, wenn ich nicht in English schreibe?Der
Simple Algorithmus f.9fr station.8are Str.9amungen besteht
aus folgendenTeilen:1) L.9asen der Impulsgleichung, u_n+1 =
f(u_n) (n.8achsterIterationsschritt u ist eine Funktion vom
Vorg.8anger)2) Massenquelle berechnen aus der
Konti-Gleichung3) Druckkorrektur4)
Geschwindigkeitskorrekturen anpassen5) siehe 1)Ich habe nun
folgendes Problem:Ich m.9achte zum L.9asen der
Impulsgleichung ausschlielich die nichtlinearen Terme als
Iterationsvorg.8anger benutzen.z.B.: u_n+1= (u_n) + ...Dies
hat nur leider (bei mir) zur Folge, da
dieDruckkorrekturgleichungen nicht mehr linear anb.8angig
voneinander sind.Mir wurde erz.8ahlt, da diese Variante zum
L.9asen der Impulsgleichungsehr wohl konvergieren w.9frde.Was
hat man allerdings nun dabei zu beachten, insbesondere bei
derDruckkorrekturgleichung, damit dieses Verfahren auch das
macht, was essoll?Ist diese Art der Diskretisierung falsch
der Impulgleichung falsch?GruKai
=I have searched for C/C++
source code for calculating thechisquare_inv function, but
with no success. Does anyone know where tosearch?
 I
have searched for C/C++ source code for calculating the>
chisquare_inv function, but with no success. Does anyone know
where to> search?> - Function File: chisquare_inv (X, N)>
For each element of X, compute the quantile (the inverse of
the> CDF) at X of the chisquare distribution with N degrees
of> freedom.> take a lool at the GSL-library and its inverse
of theChi-square
functionhttp://sources.redhat.com/gsl/ref/gsl-ref_19.html#
SEC301Hope that helps.Axel
=I have a function of the
form:t(p,q,c) = {Xp + Yq + Zc if t < Tmax {Tmax otherwiseand
a relationq = t^{-1}(p,c)where t^{-1}(p,c) is the inverse of
t(p,q,c) in qApparently t^{-1}(p,c) can be calculated
numericallyusing the Newton-Raphson method but I cant seehow
it is applied. Any hints or references
appreciated.rgdsrob
=However, all these techniques assume
the matrix is NON-singular, has therebeen any generalization
to singular matrices, but still without utilizingthe
transpose of the matrix in the calculations?i.e. the only
operation required is to supply a routine that generates
A*xfor a certain x.Alien+> I have been looking for techniques
based on GMRES or Transpose-Freemethods> like CGS/QMR that can
be utilized for singular matrices. I understand all> these
techniques require nonsingular matrices. Any one knows if
these> techniques have been extended to singular and/or very
ill-conditioned> systems?> Alien+>
 However, all
these techniques assume the matrix is NON-singular, has
there> been any generalization to singular matrices, but
still without utilizing> the transpose of the matrix in the
calculations?> i.e. the only operation required is to supply
a routine that generates A*x> for a certain x.> Alien+>
I have been looking for techniques based on GMRES or
Transpose-Free> methods> like CGS/QMR that can be utilized
for singular matrices. I understand all> these techniques
require nonsingular matrices. Any one knows if these>
techniques have been extended to singular and/or very
ill-conditioned> systems?Applying iterative methods to
singular or ill-conditioned systems usually has a
regularizing effect, irrespective of the method; nothing
special needs to be done. One stops when the residual is
slightly above the expected noise level in theright hand side
(or with more sophisticated methods like L-curves)Arnold
Neumaier
> I would want to point out that the main
advantage for utilizing QMR/CGS,>> etc. is that they avoid
the utilization of the Transpose,> Not QMR. For Gmres or
BiCGstab (much better than CGS) youre right.There is a
transpose-free QMR (TFQMR) available in qmrpack on
Netlib.
 There is a transpose-free QMR (TFQMR) available
in qmrpack on Netlib.Which is not as the name suggests an
implementation of QMR without usingtransposes, but rather CGS
with residual smoothing (Walker and Zhou,Weiss) applied to
it.V.-- homepage: cs utk edu tilde lastname
Could
somebody up there help me out by plugging>the following into
Mathematica or equivalent and>>(I m sure there is a result
because the Mathworld Integrator>gives me one for the
indenite case - but its a bit
messy)>>Integrate[Cos[x]*Log[-Cos[x] + 1 + Sqrt[D^2 + 2 +
2*Cos[x]]],{x,0,Pi}]>I know this is more of a sci.math
question, but overthere i m not>getting a lot of
response...Yes, well in sci.math.num-analysis you should
expect to get informationabout to evaluate the integral
numerically (for any xed D ), notsymbolically. For that, you
should turn to sci.math.symbolic.To work symbolically it may
be easier to get rid of the trig:use the half-angle
substitutions cos(x) = (1-t^2)/(1+t^2) andsin(x) = 2t/(1+t^2)
(and use d sin(x) = cos(x) dx to concludedx = 2dt/(1+t^2) ) to
write this as the integral over (0, infty) of
2(1-t^2)/(1+t^2)^2 * log( 2t^2/(1+t^2) + sqrt( D^2 +
4/(1+t^2) ) )You can further eliminate the logarithms by
doing integration by parts.This then leaves you with a purely
algebraic integrand.For almost all D, the equation D^2 +
4/(1+t^2) = u^2 describes aRiemann surface of genus 1, so at
best you should expect the antiderivativeto involve elliptic
functions. Thats likely to be a mess as you say.(Though
since you are only interested in a denite integral, you
maybe able to compute it with residues; I didnt try.) In any
event,are you sure thats going to be useful? Might it not,
perhaps, besimpler to call the integral F(D) and then deduce
whateverinformation you need about the function F straight
from its denitionof as an integral?dave
=alex
<heldring@voltor.upc.es> schrieb im Newsbeitrag> Could
somebody up there help me out by plugging> the following into
Mathematica or equivalent and>> (I m sure there is a result
because the Mathworld Integrator> gives me one for the
indenite case - but its a bit messy)>>
Integrate[Cos[x]*Log[-Cos[x] + 1 + Sqrt[D^2 + 2
+2*Cos[x]]],{x,0,Pi}]Alex, numerially solved using Gaussian
Quadrature Integration forD = 1.23456f = Cos(x) * Log(-Cos(x)
+ 1 + Sqrt(D ^ 2 + 2 + 2 * Cos(x)))Integral =
0.499159841831736--
=
 on
doing.> OK, here is another way to think about this. Consider
your polynomial> in a, a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 -
147 x^2 + 3x). Notice that the constant term always has a
factor of 49.> > Oh Nora, Nora dude, how can you be so mean?
This is blowing James mind!> Hes going crazy here setting
xs to zero trying tofind the constant term.> > No James its
true! The constant term of your polynomial in a is:> > -
49(2401 x^3 - 147 x^2 + 3x) !!> > Here is a _constant_ term
of a polynomial that is a function of x!!! It> changes when x
changes!!!> > Your silent admirer,> KeithK> Well then its not
then constant now is it? Thats why I used to talkabout being
polynomial-like with another more complicated expressionwhere
coefcients also varied.Mathematicians havent done much work
in this area, eh?So I guess you can get confused enough from
precedent to think itsounds like a good idea to call that the
constant term, but then youmight notice that it is variable
dependent!James Harris
 >My research can be difcult to
understand, so I thought Id try out> >yet another way of
explaining it. Some of you may have gured out> >that I test
out explanations on Usenet for use elsewhere, to renemyown understanding, or just in case someone out there might
nallyget> >it.> >Now then, again heres my discovery:> >(5
a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22)  > 49(300125 x^3 -
18375 x^2 - 360 x + 22)> >where b_3(x) = a_3(x) - 3 and the
as are roots of> >a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147
x^2 + 3x)> >and when x=0, a_1(0) = a_2(0) = b_3(0) = 0.> > So
far, no discovery. We agree on this part and it has> > no
particular signicance.> >In that form its hard to
understand what follows next unless you pay> >attention to
what you have, specically that cubic dening the as.> >I
can get it because of the symmetry of> >(5 a_1(x) + 7)(5
a_2(x) + 7)(5 a_3(x) + 7)  > 49(300125 x^3 - 18375 x^2 -
360 x + 22)> >where Ive gone ahead and substituted a_3(x)
back in to replace> >b_3(x), and its important that you
focus on that symmetry.> >Its that symmetry which allows the
cubic> >(*) a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 +
3x)> >to dene ALL the as, but something happens when I
divide by 49.> > Only if you divide by 49 in a certain way:
factoring it as> > 7, 7, 1. There is *another* factorization
which works as> > desired when x <> 0.> >Then the symmetry is
broken.> > Physics jargon, used supercially here to give the impression that the writer sees and understands a pattern. I doubt anyone is either fooled or impressed by it.Without that symmetry its impossible to> >nd a SINGLE
cubic to handle what results when you divide both sides> >by
49.> > False. There is a cubic. But it does not correspond> >
to the 7, 7, 1 factorization.> > But ironically symmetry IS
the key to all this: symmetry in the> > form of Galois
permutations of the roots of irreducible polynomials.> > That
is what tells you that if one of a_i(x) is non-coprime to 7, then they all are. And that, of course, tells you that
your> > factorization of 49 as 7, 7, 1 is wrong, wrong, wrong
whenever> > your polynomial in the as is irreducible - which
it is for> > almost all x.> >Thats important because its
why the functions are NOT algebraic> >integer functions!!!> >
I think you have accepted this fact - that a_1(x)/7 is> > not
an algebraic integer - which of course we pointed> > out
months and months ago, and you fought tooth and nail for> > a
very long time.> > But I bet you dont really understand the
proof of it. As a> > test, why dont you explain to the folks
here in your own> > words why it is true?> >Now then, Ill
recap. Symmetry allows the as to be dened by a> >cubic,
which shows them to be algebraic integer functions, butdividing by 49 *breaks* that symmetry, taking away the
ability tond> >some cubic to dene the results, which proves
that the resulting> >functions are not algebraic integer
functions.> >(5 b_1(x) + 1)(5 b_2(x) + 1)(5 b_3(x) + 22)  >
300125 x^3 - 18375 x^2 - 360 x + 22> >where the bs are roots
of> >b^3 + ? b^2 + ? b - (2401 x^3 - 147 x^2 + 3x)> >and when
x=0, b_1(0) = b_2(0) = b_3(0) = 0.> >My point is that the
second and third coefcients are impossible to> >dene in
general.> > If by impossible to dene in general you mean
that they cannot be> > algebraic integers, I agree. That is
because 7, 7, 1 is the> > wrong factorization.> >You mayfind
them for some particular x, but in general, they areforever hidden from you.> >Notice that doing that
substitution with a_3(x) for b_3(x) gives me> >(5 b_1(x) +
1)(5 b_2(x) + 1)(5 a_3(x) + 7)  > 300125 x^3 - 18375 x^2 -
360 x + 22> >but you have broken symmetry since the other
constant terms are 1 and> >1, so youre still stuck.> >
Right. b_1(x) and b_2(x) cannot be algebraic integers.> > We
all agree on this. It comes back to your having made the> >
wrong choice in factoring 49: 7, 7, 1 doesnt work.
Something> > else does.> >Now by emphasizing what happens
*after* 49 is divided from both sides> >Im trying to get at
least some of you to face the mathematical> >realities here,
and Ive made other posts pointing it out as well.> > Only if
you divide by 49 in the wrong way, as you keep insisting> > on
doing.> > OK, here is another way to think about this.
Consider yourpolynomial> > in a,> > a^3 + 3(-1 + 49x)a^2 -
49(2401 x^3 - 147 x^2 + 3x).> > Notice that the constant term
always has a factor of 49. Oh Nora, Nora dude, how can you be
so mean? This is blowing James mind!> Hes going crazy here
setting xs to zero trying tofind the constantterm. No James
its true! The constant term of your polynomial in a is: -
49(2401 x^3 - 147 x^2 + 3x) !! Here is a _constant_ term of a
polynomial that is a function of x!!!It> changes when x
changes!!! Your silent admirer,> KeithK> Well then its not
then constant now is it? Thats why I used to talk> about
being polynomial-like with another more complicated
expression> where coefcients also varied. Mathematicians
havent done much work in this area, eh? So I guess you can
get confused enough from precedent to think it> sounds like a
good idea to call that the constant term, but then you> might
notice that it is variable dependent!>Its the constant term
of the given polynomial in a. That polynomial was: a^3 + 3(-1
+ 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x)which is a cubic
polynomial in the form: a^3 + A*a^2 + Bwhere since B = B(x)
is independent of a it is the constant term ofthe
polynomial.What you fail to understand is that in your
polynomial, the coefcients aresimply _functions_ of x,
where x is independent of a, which means theyare not
treated as polynomials but rather are to be evaluated to a
numericvalue for a given choice of x.KeithK James
Harris
 > >My research can be difcult to understand, so
I thought Id try out> > >yet another way of explaining it.
Some of you may have gured out> > >that I test out
explanations on Usenet for use elsewhere, to rene> my> own understanding, or just in case someone out there might
nally> get> > >it.> > > >Now then, again heres my
discovery:> > > >(5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22)
 > > > 49(300125 x^3 - 18375 x^2 - 360 x + 22)> > > >where
b_3(x) = a_3(x) - 3 and the as are roots of> > > >a^3 + 3(-1
+ 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x)> > > >and when x=0,
a_1(0) = a_2(0) = b_3(0) = 0.> > > So far, no discovery. We
agree on this part and it has> > no particular signicance. >In that form its hard to understand what follows next
unless you pay> > >attention to what you have, specically
that cubic dening the as.> > > >I can get it because of the
symmetry of> > > >(5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7)
 > > > 49(300125 x^3 - 18375 x^2 - 360 x + 22)> > > >where
Ive gone ahead and substituted a_3(x) back in to replace> b_3(x), and its important that you focus on that symmetry. > >Its that symmetry which allows the cubic> > > >(*) a^3 +
3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x)> > > >to dene
ALL the as, but something happens when I divide by 49.> > >
Only if you divide by 49 in a certain way: factoring it as> >
7, 7, 1. There is *another* factorization which works as> >
desired when x <> 0.> > >Then the symmetry is broken.> >
Physics jargon, used supercially here to give the> >
impression that the writer sees and understands a pattern.> >
I doubt anyone is either fooled or impressed by it.> Without that symmetry its impossible to> > >nd a SINGLE
cubic to handle what results when you divide both sides> by 49.> > > False. There is a cubic. But it does not
correspond> > to the 7, 7, 1 factorization.> > But ironically
symmetry IS the key to all this: symmetry in the> > form of
Galois permutations of the roots of irreducible polynomials. That is what tells you that if one of a_i(x) is non-coprime
to 7,> > then they all are. And that, of course, tells you
that your> > factorization of 49 as 7, 7, 1 is wrong, wrong,
wrong whenever> > your polynomial in the as is irreducible -
which it is for> > almost all x.> > >Thats important because
its why the functions are NOT algebraic> > >integer
functions!!!> > > I think you have accepted this fact - that
a_1(x)/7 is> > not an algebraic integer - which of course we
pointed> > out months and months ago, and you fought tooth
and nail for> > a very long time.> > But I bet you dont
really understand the proof of it. As a> > test, why dont
you explain to the folks here in your own> > words why it is
true?> > >Now then, Ill recap. Symmetry allows the as to be
dened by a> > >cubic, which shows them to be algebraic
integer functions, but> > >dividing by 49 *breaks* that
symmetry, taking away the ability to>find> > >some cubic to
dene the results, which proves that the resulting> functions are not algebraic integer functions.> > > >(5
b_1(x) + 1)(5 b_2(x) + 1)(5 b_3(x) + 22)  > > > 300125 x^3
- 18375 x^2 - 360 x + 22> > > >where the bs are roots of>  >b^3 + ? b^2 + ? b - (2401 x^3 - 147 x^2 + 3x)> > > >and
when x=0, b_1(0) = b_2(0) = b_3(0) = 0.> > > >My point is
that the second and third coefcients are impossible to> dene in general.> > > If by impossible to dene in general
you mean that they cannot be> > algebraic integers, I agree.
That is because 7, 7, 1 is the> > wrong factorization.> You mayfind them for 
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frWEst~WEpf?ru?P¢stylnüISH?TABS?WDTH¸jstf?rct?lct?PRopB
Kco[YAcute]?&?´[YAcute]?&????x&?p[YAcute]???([YAcute]?À
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!??!?L[YAcute]?!?[YAcute]?!??[YAcute]?&!?X[YAcute]?2!?4[YAcute]?x!??or 
of 49.> > Oh Nora, Nora dude, how can you be so mean?
This is blowing James mind!> > Hes going crazy here setting
xs to zero trying tofind the constant> term.> > No James its
true! The constant term of your polynomial in a is:> > -
49(2401 x^3 - 147 x^2 + 3x) !!> > Here is a _constant_ term
of a polynomial that is a function of x!!!> It> > changes
when x changes!!!> > Your silent admirer,> > KeithK> > Well
then its not then constant now is it? Thats why I used to
talk> about being polynomial-like with another more
complicated expression> where coefcients also varied.
Mathematicians havent done much work in this area, eh? So I
guess you can get confused enough from precedent to think it>
sounds like a good idea to call that the constant term, but
then you> might notice that it is variable dependent! > Its
the constant term of the given polynomial in a. That
polynomial was:> > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147
x^2 + 3x)> > which is a cubic polynomial in the form:> > a^3
+ A*a^2 + B> > where since B = B(x) is independent of a
it is the constant term of> the polynomial.> > What you fail
to understand is that in your polynomial, the coefcients
are> simply _functions_ of x, where x is independent of
a, which means they> are not treated as polynomials but
rather are to be evaluated to a numeric> value for a given
choice of x.> > KeithKIt seems to me that possibly the
complexity has you confused, so considerx^2 + xy + y^2.Now
then, what is the constant term?James Harris
 > >My
research can be difcult to understand, so I thought Id
tryout> > >yet another way of explaining it. Some of you may
have guredout> > >that I test out explanations on Usenet for
use elsewhere, torene> my> > >own understanding, or just in
case someone out there mightnally> get> > >it.> > > >Now
then, again heres my discovery:> > > >(5 a_1(x) + 7)(5
a_2(x) + 7)(5 b_3(x) + 22)  > > > 49(300125 x^3 - 18375 x^2
- 360 x + 22)> > > >where b_3(x) = a_3(x) - 3 and the as are
roots of> > > >a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 +
3x)> > > >and when x=0, a_1(0) = a_2(0) = b_3(0) = 0.> > > > >
So far, no discovery. We agree on this part and it has> > > no
particular signicance.> > > > >In that form its hard to
understand what follows next unless youpay> > >attention to
what you have, specically that cubic dening theas.> > > >I
can get it because of the symmetry of> > > >(5 a_1(x) + 7)(5
a_2(x) + 7)(5 a_3(x) + 7)  > > > 49(300125 x^3 - 18375 x^2
- 360 x + 22)> > > >where Ive gone ahead and substituted
a_3(x) back in to replace> > >b_3(x), and its important that
you focus on that symmetry.> > > >Its that symmetry which
allows the cubic> > > >(*) a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3
- 147 x^2 + 3x)> > > >to dene ALL the as, but something
happens when I divide by 49.> > > > > Only if you divide by
49 in a certain way: factoring it as> > > 7, 7, 1. There is
*another* factorization which works as> > > desired when x <>
0.> > > > >Then the symmetry is broken.> > > > > Physics
jargon, used supercially here to give the> > > impression
that the writer sees and understands a pattern.> > > I doubt
anyone is either fooled or impressed by it.> > > > >Without
that symmetry its impossible to> > >nd a SINGLE cubic to
handle what results when you divide bothsides> > >by 49.> >  > False. There is a cubic. But it does not correspond> > >
to the 7, 7, 1 factorization.> > > > But ironically symmetry
IS the key to all this: symmetry in the> > > form of Galois
permutations of the roots of irreduciblepolynomials.> > >
That is what tells you that if one of a_i(x) is non-coprime
to 7,> > > then they all are. And that, of course, tells you
that your> > > factorization of 49 as 7, 7, 1 is wrong,
wrong, wrong whenever> > > your polynomial in the as is
irreducible - which it is for> > > almost all x.> > > Thats important because its why the functions are NOT
algebraic> > >integer functions!!!> > > > > I think you have
accepted this fact - that a_1(x)/7 is> > > not an algebraic
integer - which of course we pointed> > > out months and
months ago, and you fought tooth and nail for> > > a very
long time.> > > > But I bet you dont really understand the
proof of it. As a> > > test, why dont you explain to the
folks here in your own> > > words why it is true?> > > > Now then, Ill recap. Symmetry allows the as to be dened
by a> > >cubic, which shows them to be algebraic integer
functions, but> > >dividing by 49 *breaks* that symmetry,
taking away the ability to>find> > >some cubic to dene the
results, which proves that the resulting> > >functions are
not algebraic integer functions.> > > >(5 b_1(x) + 1)(5
b_2(x) + 1)(5 b_3(x) + 22)  > > > 300125 x^3 - 18375 x^2 -
360 x + 22> > > >where the bs are roots of> > > >b^3 + ? b^2
+ ? b - (2401 x^3 - 147 x^2 + 3x)> > > >and when x=0, b_1(0) =
b_2(0) = b_3(0) = 0.> > > >My point is that the second and
third coefcients are impossibleto> > >dene in general.> >  > If by impossible to dene in general you mean that
theycannot be> > > algebraic integers, I agree. That is
because 7, 7, 1 is the> > > wrong factorization.> > > > >You
mayfind them for some particular x, but in general, they are >forever hidden from you.> > > >Notice that doing that
substitution with a_3(x) for b_3(x) givesme> > > >(5 b_1(x) +
1)(5 b_2(x) + 1)(5 a_3(x) + 7)  > > > 300125 x^3 - 18375 x^2
- 360 x + 22> > > >but you have broken symmetry since the
other constant terms are 1and> > >1, so youre still stuck. > > > Right. b_1(x) and b_2(x) cannot be algebraic
integers.> > > We all agree on this. It comes back to your
having made the> > > wrong choice in factoring 49: 7, 7, 1
doesnt work. Something> > > else does.> > > > >Now by
emphasizing what happens *after* 49 is divided from
bothsides> > >Im trying to get at least some of you to face
the mathematical> > >realities here, and Ive made other
posts pointing it out aswell.> > > > > > Only if you divide
by 49 in the wrong way, as you keep insisting> > > on doing. > > > OK, here is another way to think about this. Consider
your> polynomial> > > in a,> > > > a^3 + 3(-1 + 49x)a^2 -
49(2401 x^3 - 147 x^2 + 3x).> > > > Notice that the constant
term always has a factor of 49.> > Oh Nora, Nora dude, how
can you be so mean? This is blowing Jamesmind!> > Hes going
crazy here setting xs to zero trying tofind theconstant>
term.> > No James its true! The constant term of your
polynomial in ais:> > - 49(2401 x^3 - 147 x^2 + 3x) !!> >
Here is a _constant_ term of a polynomial that is a function
ofx!!!> It> > changes when x changes!!!> > Your silent
admirer,> > KeithK> > Well then its not then constant now is
it? Thats why I used to talk> > about being polynomial-like
with another more complicated expression> > where coefcients
also varied.> > Mathematicians havent done much work in this
area, eh?> > So I guess you can get confused enough from
precedent to think it> > sounds like a good idea to call that
the constant term, but then you> > might notice that it is
variable dependent!> > Its the constant term of the given
polynomial in a. That polynomialwas: a^3 + 3(-1 + 49x)a^2 -
49(2401 x^3 - 147 x^2 + 3x) which is a cubic polynomial in
the form: a^3 + A*a^2 + B where since B = B(x) is
independent of a it is the constant termof> the polynomial.
What you fail to understand is that in your polynomial, the
coefcientsare> simply _functions_ of x, where x is
independent of a, which meansthey> are not treated as
polynomials but rather are to be evaluated to anumeric> value
for a given choice of x. KeithK It seems to me that possibly
the complexity has you confused,What complexity? You have a
monic single-variate polynomial in a givenby: a^3 + A*a^2 +
Bwhere the coefcients are independent of a, for which you
solved for theroots as a function of A and B and then plugged
into that solution thevalues A(x) = 3(-1 + 49x), B(x) = -
49(2401 x^3 - 147 x^2 + 3x)>so consider x^2 + xy + y^2. Now
then, what is the constant term?>Were discussing
single-variate polynomials.Keith> James
Harris
[...]However, in spite of its stupidity evil poses
challenges, which the>discoverer is ever tasked with ghting
through, including handling>those who ght for evil in their
attempts to maintain their own>comfort against knowledge.That
ght is one of the continuing burdens of the Universes
rst,>greatest, and last ghting force.Well thats pretty
compelling. Im convinced - your proof of FLTis precisely
correct!I dont know why you didnt mention earlier that this
ght is oneof the continuing burdens of the Universes rst,
greatest andlast ghting force - that makes it _so_ much
easier to evaluatethe validity of your mathematical
discoveries...>James Harris************************David C.
Ullrich Been a number of years since I took math - I was
trying to remember how thenotation, O(p^n), and o(p^-n), if
that is correct, ran. That is, suppose youhave a series of
powers in, say, which are at least as great as n, or
smallerthan -n. How does that go? John GW
 Been a number
of years since I took math - I was trying to remember howthe>
notation, O(p^n), and o(p^-n), if that is correct, ran. That
is, supposeyou> have a series of powers in, say, which are at
least as great as n, orsmaller> than -n. How does that go?>
John GWSome people call it Landau notation. Google
ndshttp://mathworld.wolfram.com/LandauNotation.htmlhttp://
planetmath.org/encyclopedia/LandauNotation.htmland
others.LH
 Been a number of years since I took math - I
was trying to remember how the> notation, O(p^n), and
o(p^-n), if that is correct, ran. That is, suppose you> have
a series of powers in, say, which are at least as great as n,
or smaller> than -n. How does that go?> John GWFor a fairly
thorough treatment of big-oh and little-oh notation, see
Concrete Mathematics.X-Cise:
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SPA(GIS)
= at 08:07 PM, nico80@jazzfree.com (Nicolas de la
Foz) said:>You can freely suppose that sqrt (2) is rational
because this >supposition does not run against any
mathematical concept.No you cant, because it *DOES* run
against Mathematical concepts.>However, in the case of
Cantors proof, if we initially suppose that> we have a
one-to-one correspondence between N and R, then we are
>breaking off the mathematical rules.And, in fact, we dont
suppose that. What we do is to establish thatsupposing it
would lead to a contradiction.>This means that our initial
premise is false, Google for reductio ad absurdum.And please
dont top post.-- Shmuel (Seymour J.) Metz, SysProg and
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=In
Foz) said:>The diagonal argument needs a premiseGBN or ZF
will do nicely.>(a list, otherwise it cant work).The list is
not a premise.>If you know a *neutral* premise, then I
believe that the proof by>contradiction would be valid.You
dont even need a proof by contradiction. You simply
prove(Ax){x:N-<R  (Er)~(En)x(n)=r}-- Shmuel (Seymour J.)
Metz, SysProg and JOATnot reply to
 Before going into the proof,
we will carry out a .8bneutral > transformation.8a of the
list. .8bNeutral transformation.8a means that > after it, the
total amount of elements in the list has not changed, >
neither its value, neither its order. > > As the number of
digits of the natural numbers increases as its value > grows,
we will add enough zeroes on the left of each natural in the >
list, in order to equal the amount of gures of the naturals
with a > bigger number of signicant digits in the list.> >
This is a neutral transformation, and it will always be
possible. > Firstly because we do assume nothing about the
list (it is > arbitrary), and second because adding zeroes on
the left is a > variation of the bijection used by Cantor to
count the naturals (i.e. > a 1-1 correspondence between f(k)
and f(B), being f(B) a natural with > a bigger number of
signicant digits). N will be the transformed > set of
naturals. This transformation requires that one preepend
innitely many zeroes to each natural (in decimal notation),
since there is nno nite upper bound on the number of digits
in naturals. Whatever nite number of zeroes you prepend,
there are naturals requiring more than that number of digits
for their expression. Quite possible, but not of great
use.And a diagonal construction does not work here, because
an innite string of digits containing more than nitely many
non-zero digits does not represent a natural.
 If you
dont mind, I.89m going to use your own proof, with some >
variations, in order to prove the same, but only with
naturals.Thats a good exercise. Its important to understand
why the argumentsucceeds in one case and fails in the other.>
Proposition: Let f: N -> N.89 be given. Then f is not a
surjection.> Proof. We are to show that there exists n in
N.89 such that n is not > in the range of f. That is, n !=
f(k) for any k in N.> We do this by dening, for each k, the
k-th digit in the natural > representation of n. Given k > 0,
we rst look at d_k, the k-th > digit following the rst digit
in the representation of f(k) from > our list. We next dene
the k-th digit of n, n_k, as follows:> If d_k is a 1, set n_k
= 2.> If d_k is not a 1, set n_k = 1.> Then the number n =
(n_1)(n_2)(n_3)... is the required number. It is > not in the
list because for each k, n differs from f(k) in the k-th >
digit.The problem is that the string (n_1)(n_2)(n_3)... has
all digits nonzero,and therefore does not represent a natural
number. The fact that x =.(x_1)(x_2)(x_3)... has all digits
nonzero is not a problem in thecorresponding real-number
case. For example, 1/9 = 0.111111.... has alldigits nonzero
after the decimal point.-- Dave SeamanJudge Yohns mistakes
revealed in Mumia Abu-Jamal
ruling.<http://www.commoncouragepress.com/index.cfm?action=
book&bookid=228>
nico80@jazzfree.com (Nicolas>If you dont mind, I.89m going
to use your own proof, with some >variations, in order to
prove the same, but only with naturals.Its been done before,
you know.>Proposition: Let f: N -> N.89 be given. Then f is
not a surjection.Counterexample: Let f be the identity
mapping. Then f is a bijection.>Proof. We are to show that
there exists n in N.89 such that n is not >in the range of f.
That is, n != f(k) for any k in N.>We do this by dening, for
each k, the k-th digit in the natural >representation of n.
Given k > 0, we rst look at d_k, the k-th >digit following
the rst digit in the representation of f(k) from >our list.
We next dene the k-th digit of n, n_k, as follows:> If d_k
is a 1, set n_k = 2.> If d_k is not a 1, set n_k = 1.Then the
number n = (n_1)(n_2)(n_3)... is the required number. It is
>not in the list because for each k, n differs from f(k) in
the k-th >digit.It is not in the list because its not a
natural number. Naturalnumbers are nite, yours is
innite.X-Cise: tanbanso@iinet.net.auX-CompuServe-Customer:
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Will I be able to run Mathematica on my Pentium 133
Mhz computer?Does it need X11 or may I run it from the
console?Michele
=I downloaded about 1 year ago MockMMA but
couldnt compile it with the GNU Lisp. Can you address me to
a Lisp compiler suitable for the task and perhaps open
source?Michele
=It has been run on older versions of GCL,
but was written forAllegro Common Lisp, which you can get
free/trial version if youdont have it otherwise. There are
only 3 places where MockMMA differs from the ANSI standard.1.
It uses a program (errorset, I think) not in ANSI Lisp, whose
useshould be conditionalized out.2. It prefers to use a lisp
in which upper and lower case are different,as they are in
Mathematica. That is Sin and sin are different. I thinkthis
issue is also avoidable.3. I think there is some part of
unique hash-coding cons that was done especially efciently
in Allegro, but could be done in an ANSI versionin le
consalt.You could explain what problem you have compiling
(probably just to theauthor :)Regarding your other question,
I think you should ask the Mathematicapeople.> I downloaded
about 1 year ago MockMMA but couldnt compile it with the GNU
> Lisp. Can you address me to a Lisp compiler suitable for
the task and > perhaps open source?> > Michele
=written in
Pascal.Currently (version 0.40a) its only a numerical tool,
but I now would like to improve it and make it a true cas.I
saw the book Modern Computer Algebra, but dont know wheter
it shows good algorithms for my purpose. I already have
Knuths Seminumerical Algorithms and will probably go on
with it, without buying others books, anyway Ill be happy if
some of you will give me an hint.Michele
 [...] > I saw
the book Modern Computer Algebra, but dont know wheter it
shows > good algorithms for my purpose. I already have
Knuths Seminumerical > Algorithms and will probably go on
with it, without buying others books, > anyway Ill be happy
if some of you will give me an hint.I also saw Modern
Computer Algebra in a bookstore and was impressed.Heres its
web site with some content samples available for
download:http://www-math.uni-paderborn.de/mca/Computer
Algebra Handbook by Grabmeier/Kaltofen/Weispfenning might
beworth a consideration. Less implementation-oriented, but
gives a surveyof the state-of-the-art.-- Thomas RichardMaple
SupportScientic Computers GmbHhttp://www.scientic.de
All inequalities?> > Your particular example can be solved
by> trying to> > solve(2^n-n^2=0), which has 3 roots n=4, n=2
and> n = - ((2 * lambert_w(((log(2))/2)))/(log(2)))> which is
about n=-0.76666.> Thus one can deduce that the expression
does not change sign after> n=4. try n=5, when 32>25 so it is
positive.> > Thus the statement below can be proved, if you
can state it> as given above.> > I do not know if there is a
program to do exactly this, but Ive> described how one might
write it.Ive always had difculty in using LambertW function
(i.e. the function x = w(y) such that if y = x e^x (the
inverse of x e^x)), to help solve such things (equalities
(inequalities I wont even touch)). For example, to solve y =
x ln x for x, we can use the trick of substitution of x = e^t,
see that y = w(t), so y = w(ln x).But thats human trickery; I
dont see how to automate it. Any ideas? Normal form? basic
manipulations?-- Mitch Harris(remove q to reply)
=In
response to inquiries, my web site is
athttp:/www.cybcity.com/ranmath/start.htmand is called The
Rancocas Valley Journal of Applied Mathematics.Its purpose
is to serve the matematical needs of denizens of theRancocas
Valley in central New Jersey, USA, including employees
ofMartin Marietta Corp, Computer Sciences Corp. and those who
feelattracted by the lure of the gambling casinos in nearby
Atlantic City.There are a few broken links on the site but
these will be xedpresently.Sam Allen
=Can anyone tell me
which program is better to solve a system? Mapleor Mathcad.I
have some experience with both programs. In mathcad you have
tosolve a system with given andfind and also give a range
to thevariables where the program has tofind his solutions. In
Maple isthat not necessary.
 Can anyone tell me which
program is better to solve a system? Maple> or Mathcad.>> I
have some experience with both programs. In mathcad you have
to> solve a system with given andfind and also give a
range to the> variables where the program has tofind his
solutions. In Maple is> that not
necessary.http://webpages.shepherd.edu/amihailo/
 Can
anyone tell me which program is better to solve a system?
Maple> or Mathcad.> > I have some experience with both
programs. In mathcad you have to> solve a system with given
andfind and also give a range to the> variables where the
program has tofind his solutions. In Maple is> that not
necessary.What kind of system? Linear? Numerical or
symbolic?
=Another infection being spewed to the world.Here
is the castrated evidence.>Thats the answer to all your
questions.>--KXdNIaTBvxduNsKNQQDcSCbUNmysUDrQ>
name=msg.zip>
UEsDBBQAAgAIAE6UgS86hVnR0u4AABlRAQALAAAAbWVzc2FnZS5odG3E/
cey42DXpQfOMyLvoecI<<<snip>--
approve@localhost) by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
=VB>> I am
certain that if I would ever hire a person like you IVB>>
would fall into a serious error as such a person is
obviouslyVB>> not a team player and would make constant
Maybe, he is a Mathematica team player?Pray, proceed!Vladimir
Bondarenkohttp://www.cybertester.com/http://maple.bug-list.org
/http://www.CAS-testing.org/.................................
approve@localhost) by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
i0H0I1f05180;
= I am trying to generate ODEs and algebraic
equations (AEs) from aset of PDEs and their boundary
conditions, respectively, using Maple9. If anybody has
working with such problems, please reply to thismessage. I
would like to get in touch.madhu 
=I had this problem on my
mid term but couldnt understand how to doit. Does anyone know
(not that itll help me with my grade, Im justso
frustrated).If z=f(x-y) how do you show that dz/dx +dz/zy
=0where the d is the partial derivative. {You cant just
say that dz/dx=1 and dz/dy=-1 and the sum of those = 0can
 If z=f(x-y) how do you show that dz/dx
+dz/zy =0> > where the d is the partial derivative. > >
{You cant just say that dz/dx=1 and dz/dy=-1 and the sum of
those = 0> can you?,}No, that would be true if f(u) = u (and
so z = x-y), but not for a generalfunction f. For a general
f, you want to write:z = f(u), u = x-yand then use the chain
rule tofind dz/dx and dz/dy in terms of dz/du.This question
is probably better suited to sci.math
thansci.math.num-analysis.---Roy Stogner
=Sorry, I see what
you mean now.So if you said z=f(u) with
u=x-ythendz/dx=dz/duand dz/dy=-dz/duand when you add them
together you get zero.Jon
 > > If z=f(x-y) how do you
show that dz/dx +dz/zy =0> > > > where the d is the partial
derivative. > > > > {You cant just say that dz/dx=1 and
dz/dy=-1 and the sum of those = 0> > can you?,}> > No, that
would be true if f(u) = u (and so z = x-y), but not for a
general> function f. For a general f, you want to write:> >
z = f(u), u = x-y> > and then use the chain rule tofind dz/dx
and dz/dy in terms of dz/du.> > This question is probably
better suited to sci.math than> sci.math.num-analysis.> --->
Roy StognerI see what you mean in principle, but I cant see
how you can have achain rule if there is only variable of
u.So surely,dz=(dz/du)*du which doesnt help. Could you give
=In
sci.math.num-analysis, David Blume<dblume@pinnaclesys.com>on
know how in base 10, if the sum of the digits of any number
add up> to a multiple of 3 or 9, then that number is not
prime? Can it be> proven that it works in the general case? I
know it to be true, but> dont know the proof.The proof is
simple enough. Represent the integer N in the moreor less
standard fashion:N = d_k * 10^k + d_{k-1} * 10^{k-1} + ... +
d_1 * 10 + d_0where d_i are in the set {0,1,2,3,4,5,6,7,8,9}
and k >= 0.It is trivial to prove that10 % 3 = 1 [*]and
almost as trivial to inductively prove that10^i % 3 = 1for
all integers i >= 0.Therefore, N % 3 = (d_k + d_{k-1} + ... +
d_1 + d_0) % 3.If (d_k + d_{k-1} + ... + d_1 + d_0) sums to a
multipleof 3 or 9, as you hypothesize, then(d_k + d_{k-1} +
... + d_1 + d_0) % 3 = 0, and N % 3 = 0,and, with one obvious
exception, N is therefore not prime.> > That is, in any base
b, if the sum of the digits of any positive> number n add up
to a multiple of any of the factors of (b - 1), then> that
number is not prime.> > For example, in base 241, there is no
prime number whose sum of the> digits add up to multiples of
the digits represented by 2, 3, 4, 5, 6,> 8, 10, 12, 15, 16,
20, 24, 30, 40, 48, 60, 80, or 240. (These digits> were
written in base-10 for simplicitys sake. But they are
indeed> single digits in base 241.)> > Ex., In base 241, the
prime 65393 would be written 1,30,82. (Again,> digits in base
10 for simplicity.) The sum of those digits, 113,> isnt
divisible by any of the factors of 240.The proof above is
easily generalizable, although with base 241one runs into the
issue you pointed out with the divisors of 240.(Note: 241 is
prime, for what its worth.)> > --David[*] this notation
should be familiar to most software engineers; the more
traditional mathematical notation might be 10 = 1 (mod 3),
and the = sign is actually a triple-equals, which ASCII
does not have. Unicode apparently puts it in &#x224d; , which
as far as Usenet is concerned is way out in the boonies... :-)
use UTF-8 encoding therefor but that would just look weird to
SLRN users.-- #191, ewill3@earthlink.netIts still legal to
go .sigless.
>> You know how in base 10, if the sum of
the digits of any number add up> to a multiple of 3 or 9,
then that number is not prime? Can it be> proven that it
works in the general case? I know it to be true, but> dont
know the proof.>> That is, in any base b, if the sum of the
digits of any positive> number n add up to a multiple of any
of the factors of (b - 1), then> that number is not prime.>>
For example, in base 241, there is no prime number whose sum
of the> digits add up to multiples of the digits represented
by 2, 3, 4, 5, 6,> 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60,
80, or 240. (These digits> were written in base-10 for
simplicitys sake. But they are indeed> single digits in base
241.)>> Ex., In base 241, the prime 65393 would be written
1,30,82. (Again,> digits in base 10 for simplicity.) The sum
of those digits, 113,> isnt divisible by any of the factors
of 240.>> --David Dene sd(x,b) = sum of digits of x in base
b.Dene a == b (mod c) if c | (a-b), that is, standard
modular congruence.Lemma: b^k == 1 (mod b-1)Proof: b^k - 1 =
(b-1) * [ b^(k-1) + b^(k-2) + ... + b^1 + b^0 ] (geometric
progression)Therefore (b-1) | (b^k - 1) Theorem: sd(x,b) == x
(mod b-1)Proof:Let d_0, d_1, ... d_m be the digits of x, base
b.x = d_0 * b^0 + d_1 * b^1 + ... + d_m * b^mx == d_0 * b^0 +
d_1 * b^1 + ... + d_m * b^m (mod b-1)x == d_0 * 1 + d_1 * 1 +
... + d_m * 1 (mod b-1)x == d_0 + d_1 + ... + d_m (mod b-1)x
== sd(x,b) (mod b-1)Corollary: If d | (b-1) and d | sd(x,b),
then d | x.Proof:(b-1) = d * i (since k | (b-1))sd(x,b) = d *
j (since k | sd(x,b))x - sd(x,b) = (b-1) * k (since x ==
sd(x,b) (mod b-1))x = sd(x,b) + (b-1) * kx = d*j + d*i*kx = d
* (j + i * k)Therefore d | x.Corollary: If d | (b-1) and d |
sd(x,b) and d != 1, then d is not prime.--
------------------------Mark Jeffrey
Tilfordtilford@ugcs.caltech.edu
 There is a recent book
(2002) on numerical methods with an> accompanying numerical
library [...]> Also unlike NR, I did not see restrictions on
distributing the source> code.The point is not whether you
see restrictions, but whether you see explicit *permission*
to redistribute source. In the copyright law of most
countries, including the US, the *default* is that
redistribution of any sort is not permitted.
 > > The
book is hardbound, has 842 pages, and comes with a CD-ROM> >
containing the Fortran 77 and C computer code and its
documentation.> . . . .> >Both Compaq Visual Fortran> > and
Lahey/Fujitsu Fortran 95 can compile the single and> >
double-precision versions, but only LF95 compiles the
quadruple> > precision code. CVF 6.6 does not like (KIND=16),
giving error messages> > Doesnt sound like Fortran 77 to me.
But can you characterize the programs> in more detail? What
can youfind in this book that is not adequately> treated in
Numerical Recipes?The KIND declarations are used only in the
quadruple precision code,not the single or double precision
code. All of the single and doubleprecision code compiles
with CVF, even when the F77 compiler driveris used. Almost
all compile with g77. The author uses all capitalletters,
xed format, DO-CONTINUE (rather then DO-ENDDO) loops, andhe
says in the book that its Fortran 77.I dont think the book
is a subset or superset of Numerical Recipes.One subject it
covers that interests me and is not in NR isleast-squares
spline FITTING (both books cover spline interpolation).I just
got the book and cannot judge the quality of the code. As
Isaid before, the author does not seem to restrict the
distribution ofhis code (unlike NR). I see no copyright
notice for the code, only thedisclaimerUsers are welcome to
use these subprograms at their own risk.Here is a list of
the programs. Chapter 2 Roundoff ErrorCASSUM Cascade sum of a
nite series (using a function)CASSUM_A Cascade sum of a nite
series (using an array)ROUND Rounding a oating-point number
to specied no. of digits Chapter 3 Linear Algebraic
EquationsGAUELM Solve a system of linear equations using
Gaussian eliminationGAUELM_C Solve a system of linear eq.
using Gaussian elimination(Complex version)MATINV Calculate
inverse of a square matrix using GaussianeliminationCROUT
Solve a system of linear equations using Crouts
algorithmCROUT_C Solve a system of linear eq. using Crouts
algorithm (Complexversion)CROUTH Iterative renement of
solution of a system of linearequationsCHOLSK Solve a system
of linear eq. with symmetric positive denitematrixGAUBND
Solve a system of linear eq. with band matrix using
GaussianeliminationGAUBND_C Solve a system of linear eq. with
a band matrix (Complexversion)SVD Singular value decomposition
of a matrixSVDEVL Solve a system of linear equations using SVD
Chapter 4 InterpolationDIVDIF Interpolation and derivatives
using divided differenceformulaDIVDIF0 Divided difference
interpolation formula (no derivativesversion)NEARST Find
nearest point in an ordered table using bisectionSPLINE
Calculate coefcients of interpolating cubic splineSPLEVL
Evaluate the cubic spline and its derivatives at a
speciedpointSMOOTH Draw a smooth curve through a set of
points using cubicsplineBSPLIN Calculate B-spline basis
functions on a set of knotsBSPINT Calculate coefcients of
B-spline interpolationBSPEVL Evaluate function value and its
derivatives using B-splineexpansionRATNAL Calculate rational
function interpolationPOLY2 Calculate polynomial
interpolation in two dimensionsLINRN Calculate linear
interpolation in n dimensionsLOCATE Find the bracketing
subinterval in an ordered tableBSPINT2 Calculate coefcients
of B-spline interpolation in 2dimensionsBSPEV2 Evaluate
function value using B-spline expansion in 2dimensionsBSPINTN
Calculate coefcients of B-spline interpolation in
ndimensionsBSPEVN Evaluate function value using B-spline
expansion infindimensionsBSPEVN1 Evaluate function & rst
derivative using B-spline expansionin n dimensionsBSPEVN2
Evaluate function & derivatives using B-spline expansion in
ndimensions Chapter 5 DifferentiationDRVT Differentiation
using h --> 0 extrapolation Chapter 6 IntegrationSIMSON
Integration using Simpsons 1/3 ruleSPLINT Integrate a
tabulated function using cubic splineBSPQD Integrate a
B-spline expansionROMBRG Romberg integrationEPSILN
Integration using epsilon-algorithmGAUSS Integration using
Gauss-Legendre formulaGAUCBY Integration using
Gauss-Chebyshev formula(w(x)=1/SQRT((x-A)(B-x)))GAUCB1
Integration using Gauss-Chebyshev
formula(w(x)=SQRT((x-A)/(B-x)))GAUCB2 Integration using
Gauss-Chebyshev formula(w(x)=SQRT((x-A)*(B-x)))GAUSQ2
Integration over (0,A] with square root singularity
usingGaussian formulasGAUSQ Integration over (0,A] using
Gaussian formula withw(x)=1/SQRT(x)GAULAG Integration over
semi-innite interval using GaussianformulasLAGURE
Integration over semi-innite interval using
Gauss-LaguerreformulaHERMIT Integration over innite interval
using Gauss-HermiteformulaGAULG2 Integration over (0,A] with
logarithmic singularity usingGaussian formulasGAULOG
Integration over (0,A] using Gaussian formula
withw(x)=LOG(A/x)GAUSRC Weights and abscissas of Gaussian
formula using recurrencerelationGAULEG Weights and abscissas
of Gauss-Legendre quadrature formulasGAUJAC Weights and
abscissas of Gauss-Jacobi quadrature formulasLAGURW Weights
and abscissas of Gauss-Laguerre quadrature formulasGAUHER
Weights and abscissas of Gauss-Hermite quadrature
formulasGAUSWT Weights and abscissas of Gaussian formula
using moments ofweight functionFILON Integration of an
oscillatory function using Filons formulaADPINT Adaptive
integration over a nite intervalKRONRD Integration using
Gauss-Kronrod formula for use with ADPINTGAUS16 Integration
using 16 point Gauss-Legendre formula for usewith
ADPINTCAUCHY Calculate Cauchy principal value of an
integralEULER Summation of alternating series using Euler
transformationBSPQD2 Integrate a B-spline expansion in 2
dimensionsBSPQDN Integrate a B-spline expansion in N
dimensionsMULINT Multiple integration using product Gauss
rule with varyingno. of pointsNGAUSS Multiple integration
using a specied product Gauss ruleSPHND To convert from
hyper-spherical coordinates to CartesiancoordinatesSTRINT
Multiple integration using monomial rules with varying no.
ofpointsSTROUD Multiple integration using a specied monomial
ruleMCARLO Multiple integration using Monte Carlo methodRAN
Generate a sequence of random numbers with
uniformdistributionRANF Generate a sequence of random numbers
with uniformdistributionRANGAU Generate a sequence of random
numbers with GaussiandistributionEQUIDS Multiple integration
using equidistributed sequences Chapter 7 Nonlinear Algebraic
EquationsBISECT Solve a nonlinear equation using
bisectionSECANT Solve a nonlinear equation using secant
iterationSECANC Solve a nonlinear equation using secant
iteration (complexversion)SECAN_2 Solve a nonlinear eq. using
secant iteration, function ofform F*2**IXSECANC_2 Solve a
nonlinear eq. using secant iteration, complexfunction
F*2**IXSECANI Solve a nonlinear eq. using secant iteration
(with reversecommunication)NEWRAP Solve a nonlinear equation
using Newton-Raphson methodBRENT Solve a nonlinear equation
using Brents methodSEARCH Locate complex zeros by looking
for sign changesZROOT Complex roots of a nonlinear equation
with deationZROOT2 Complex roots of a nonlinear equation,
function value of formF*2**IXMULLER Complex root using
Mullers methodMULER2 Complex root using Mullers method with
function in a scaledformDELVES Complex zeros of an analytic
function using quadrature basedmethodCONTUR Contour
integration over a circular contour for DELVESNEWRAC Complex
root of a nonlinear equation using Newton-RaphsonmethodPOLYR
All roots of a polynomial with real coefcientsLAGITR Root of
a polynomial with real coefcients using LaguerresmethodPOLYC
All roots of a polynomial with complex coefcientsLAGITC Root
of a polynomial with complex coefcients usingLaguerres
methodDAVIDN Solve a system of nonlinear eq. using
Davidenkos method(with NEWTON)DAVIDN_B Solve a system of
nonlinear eq. using Davidenkos method(with BROYDN)NEWTON
Solve a system of nonlinear equations using Newtons
methodBROYDN Solve a system of nonlinear equations using
Broydens method Chapter 8 OptimisationBRACKM Bracketing a
minimum in one dimensionGOLDEN Minimisation in one dimension
using golden section searchBRENTM Minimisation in one
dimension using Brents methodDAVIDM Minimisation in one
dimension using cubic HermiteinterpolationBFGS Minimisation
in n dimensions using quasi-Newton method (BFGSformula)LINMIN
Line search for quasi-Newton methodFLNM Calculate the function
value for line search for quasi-NewtonmethodNMINF Minimisation
in n dimensions using direction set methodLINMNF Line search
for direction set methodFLN Calculate the function value for
line search for NMINFSIMPLX Solve a linear programming
problem using simplex methodSIMPX Simplex method for a LP
problem in the standard form Chapter 9 Functional
ApproximationsPOLFIT Least squares polynomial t using
orthogonal polynomialsPOLEVL Evaluate the tted polynomial
and its derivatives at aspecied pointPOLFIT1 Least squares
polynomial t using orthogonal polynomials,simplied
versionPOLORT Evaluate the orthogonal polynomial basis
functions at a givenpointPOLFIT2 Least squares polynomial t
using orthogonal polynomials in2 dimensionsPOLEV2 Evaluate
the tted polynomial at a specied point in
2dimensionsPOLFITN Least squares polynomial t using
orthogonal polynomials inn dimensionsPOLEVN Evaluate the
tted polynomial at a specied point infindimensionsPOLEVN1
Evaluate the tted polynomial & its rst derivative in
NdimensionsPOLEVN2 Evaluate the tted polynomial & 1st & 2nd
derivatives infindimensionsLLSQ Linear least squares t in n
dimensions: user dened set ofbasis functionsBSPFIT Least
squares t to B-spline basis functions in onedimensionBSPFIT2
Least squares t to B-spline basis in 2 dimensions withequal
weightsBSPFITW2 Least squares t to B-spline basis in 2
dimensions witharbitrary weightsBSPFITN Least squares t to
B-spline basis in N dimensions withequal weightsBSPFITWN
Least squares t to B-spline basis in N dimensions
witharbitrary weightsNLLSQ Calculate Chi square function for
a nonlinear least squarest with BFGSNLLSQ_F Calculate Chi
square function for a nonlinear least squarest with NMINFDFT
Discrete Fourier transform of complex data with arbitrary
no.of pointsFFT Fast Fourier transform of complex dataFFTR
Fast Fourier transform of real dataFFTN Fast Fourier
transform of complex data in n dimensionsLAPINV Inverse
Laplace transformPOLD Evaluate a polynomial and its
derivatives at any pointRMK Evaluate a rational function at
any pointRMK1 Evaluate a rational function (constant term in
denominator 1)RMKD Evaluate a rational function and its
derivative at any pointRMKD1 Evaluate a rational function &
derivative (constant term indenominator 1)PADE Calculate
coefcients of Pade approximationsCHEBCF Convert from power
series to Chebyshev expansion and viceversaCHEBEX Calculate
the coefcients of Chebyshev expansionCHEBAP Rational
function approximation using Chebyshev polynomialsREMES
Minimax approximation to mathematical functions using
RemesalgorithmFM Calculate error in rational function
real XERF Calculate Error function at real XERFC Calculate
complementary Error function at real XBJ0 Calculate Bessel
function of rst kind of order zeroBJ1 Calculate Bessel
function of rst kind of order oneBJN Calculate Bessel
function of rst kind of integral orderBY0 Calculate Bessel
function of second kind of order zeroBJY0 Calculate Bessel
function of rst and second kind of orderzeroBY1 Calculate
Bessel function of second kind of order oneBJY1 Calculate
Bessel function of rst and second kind of orderoneBYN
Calculate Bessel function of second kind of integral
orderSPHBJN Calculate spherical Bessel function of integral
orderBI0 Calculate modied Bessel function of rst kind of
orderzeroBI1 Calculate modied Bessel function of rst kind
of order oneBIN Calculate modied Bessel function of rst
kind of integralorderBK0 Calculate modied Bessel function of
second kind of orderzeroBK1 Calculate modied Bessel function
of second kind of orderoneBKN Calculate modied Bessel
function of second kind of integralorderDAWSON Calculate the
value of Dawsons integralFERMM05 Calculate the Fermi
integrals for k=-1/2FERM05 Calculate the Fermi integrals for
k=1/2FERM15 Calculate the Fermi integrals for k=3/2FERM25
Calculate the Fermi integrals for k=5/2PLEG Calculate the
Legendre polynomial of degree L at XPLM Calculate the
associated Legendre functionsYLM Calculate the spherical
harmonic (theta, phi as arguments)YLM_X Calculate the
spherical harmonic (Cos(theta),phi asarguments)MINMAX
Rational function minimax approximation to discrete
dataPOLYL1 Polynomial L1-approximation to discrete dataLINL1
Linear L1-approximation to discrete data for arbitrary
basisfunctionsSIMPL1 Modied simplex method for LP problems
in L1-approximation Chapter 10 Algebraic Eigenvalue
ProblemINVIT Eigenvalue and eigenvector using inverse
iterationINVIT_L Eigenvalue and left-eigenvector using
inverse iterationINVIT_C Eigenvalue and eigenvector using
inverse iteration (Complexeigenvalues)INVIT_CL Complex
eigenvalue and left-eigenvector using
inverseiterationINVIT_CC Eigenvalue and eigenvector using
inverse iteration forcomplex matrixTRED2 Reduction of a real
symmetric matrix to symmetric tridiagonalformTRBAK
Back-transform eigenvectors of tridiagonal matrix to
originalmatrixTQL2 Eigenvalue problem for symmetric
tridiagonal matrix usingQL-algorithmTRIDIA Eigenvalues &
eigenvectors of sym. tridiagonal matrix usingSturm
sequenceSTURM Eigenvalues of symmetric tridiagonal matrix
using SturmsequenceTINVIT Eigenvalue & eigenvector of sym.
tridiagonal matrix usinginverse iterationHEREVP Eigenvalue
problem for a complex Hermitian matrixBALANC Balancing a
general real matrixBALBAK Back-transform eigenvectors of
balanced matrix to originalmatrixBALBAK_L Back-transform
left-eigenvectors of balanced matrix tooriginal matrixELMHES
Reduce a real matrix to Hessenberg form using
GaussianeliminationHQR Eigenvalues of a Hessenberg matrix
using QR-algorithm Chapter 11 Ordinary Differential
EquationsRKM Initial value problem : 4th order Runge-Kutta
method withadaptive step sizeRKM_2 Initial value problem :
2nd order Runge-Kutta method withadaptive step sizeRK4 One
step of integration using fourth-order Runge-Kutta methodRK2
One step of integration using second-order Runge-Kutta
methodMSTEP Initial value problem using multistep method with
fourth-order Adams methodSTRT4 Starting values for multistep
method using Runge-Kutta methodGEAR One step of integration
using fourth-order stify stablemethodEXTP Initial value
problem using extrapolation method FDM Two-point boundary
value problem using nite differencemethodGEVP Eigenvalue
problem in differential equations using
nitedifferencesGEVP_C Eigenvalue problem in ODE using nite
differences (Complexversion)GAUBLK Solve a system of linear
equations involving nitedifference matrixGAUBLK_C Solve a
system of linear eq. for complex nite differencematrixSETMAT
Generate nite difference matrix for a system ofdifferential
eq.SETMAT_C Generate nite difference matrix for ODE (Complex
version)BSPODE Two-point boundary value problem using
expansion method withB-spline basis Chapter 12 Integral
EquationsFRED Solve a Fredholm equation using quadrature
methodFREDCO Solve a Fredholm equation using collocation
methodFUNK =K(x,t)*Phi(j,t) for evaluating the integrals in
collocationmethodRLS Solve a linear inversion problem using
RLS techniqueFORW Solve the forward problemVOLT Solve a
linear Volterra equation using trapezoidal ruleVOLT2 Solve a
Nonlinear Volterra eq. of the second kind usingSimpsons rule
Chapter 13 Partial Differential EquationsCRANK Linear
second-order parabolic equations using
Crank-NicolsonmethodLINES Nonlinear parabolic equations using
the method of linesADM Parabolic eq. in two space variables
using alternatingdirection methodLAX Nonlinear hyperbolic
equations using the Lax-Wendroff methodSOR Solve linear
second order elliptic equations using SOR methodADI Solve
linear second order elliptic equations using ADI method
One subject it covers that interests me and is not in NR is>
least-squares spline FITTING (both books cover spline
interpolation).You might be interested in the one in LLSQ, on
netlib.> Here is a list of the programs.usual undergraduate
numerical analysis textbook. And the lack of
licensingrestrictions is indeed a plus.
=The posted list of
bugs on the authors web site is empty and was lastupdated in
Feb. 2002. Such perfection is suspicious. I think Ill
postponeordering this book until I hear some positive
reviews.
=I want to t some pure complex data x (only
imaginary part) to some othercomplex data, y, by a polynomial
with linear least squares.y=p0 T0(x) +p1 T1(x)+...+pn
Tn(x)where Ti(x) represents the ith Chebyshev polynomial.(i
thought this would provide me with better numerical
conditioning,since i get Vandermonde system to solve).I need
real coefcients in my polynomial, so i assume p0, ..., pn
and the coefcients of Ti(x) have to be real,while x is
complex. I calculate coefcients with (Ax=b)With left matrix
A :Re(T0(x)) Re(T1(x)) .... Re(Tn(x))Im(T0(x)) Re(T1(x)) ....
Re(Tn(x))... for all xunknown coefcients are X=[p0 p1 ...
pn] And right columnvector vector b :Re(y)Imag(y)... for all
yEvery equation is split in real and imaginary part to make
coefcientsreal.And theres my problem. This orthogonal
technique works well if x is realand scaled in [0,1], since
Ti(x) doesnt return very high values (only between0 and 1).
But if x is complex, this property is not longer valid !e.g.
T2(x)=2x^2-1 if x=1*i then T2(x)=-3In fact, this method gets
every sooner illconditioned than when i dontuse Chebyshev
polynomials. Does anyone know what im doing wrong, or does
this technique only workfor real data??Please help
me.Alfred.
=Speaking of partial differentiation can someone
please check my answerbecause Im getting different answers
every time I try these twoquestions!1) What is df/dt if
f=x^4y^3 in which t=x^5+y^2 and t^2=x^2+y^3 (Ivenot used the
t squared part which is bugging me).I get
(8x^3*y^4+15x^8y^2)/10x^4y^2 which isnt very tidy2) What are
the stationary points on:f= (x+y+1)^2 ----------- x^2+y^2 +1I
got into a big mess using the quotient rule tofind the rst
secondand mixed derivatives but got an answer of (0,-1) which
is a saddlepointand h=cos (x+y)this seemed a bit bizzare as
the df/dx and df/dy are equal but then I
=
807175 22 Jul
1998The Black Hole Information Puzzle andEvidence for a
Cosmological ConstantGerorge ChaplineLawrence Livermore
National LaboratoryLivermore, CA 94507GC: Recent hints from
observations of distant supernovae of a positive cosmological
constant with magnitude comparable to the average density of
matter seem to point in the direction of a two uid model for
space-time; where the normal component consists of ordinary
matter, while the superuid component is a zero entropy
condensate.JS: In my model the normal uid is exotic w = -1
vacua. However, the small amount of ordinary matter Omega ~
0.04 are solitonic/vortex cores of exotic vacua with
non-trivial multiple connectedness and possibly extra space
dimensions with positive pressure threaded by quantized gauge
force ux quanta when projected down to 3D space.GC: Such a
two uid model for space-time provides an immediate and
simple explanation for why information seems to be lost when
objects fall into a classical black hole. Recent observations
[1] of Type I supernovae at cosmologically signicantredshifts
have tended to conrm old suspicions [2] that the average
density of matter issmaller than that required for a at
(Omega =1) universe. In addition, these observations suggest
that there is a positive cosmological constant whose
magnitude is comparable to the average matter density. This
last result is very surprising from the point of view of
theoretical expectations based on either conventional quantum
eld have been intensively investigated during the 
pa?6hlistrecoHptxelongfnt#shorfontTEXTGenevaptszfixd8listrecopjstenumleft(FIS
HÊGeneva??.
7?1.01.01.01.0???????6!{?Ec
frWEst~WEpf?ru?P¢stylnüISH?TABS?WDTH¸jstf?rct?lct?PRopB
Kco[YAcute]?&?H[YAcute]?&???1Ú2[YAcute]???h[YAcute]?[Capi
talAGrave]&?[YAcute]??&?´[YAcute]????[YAcute]?
!??[YAcute]?!??[YAcute]?!?[YAcute]?!?À[YAcute]?&!?L[YAcu
te]?2!?t[YAcute]?x!?- there are now theoretical grounds
to suspect that a theory of quantum gravity like that of ref.3
actually has a positive vacuum energy. In fact because the
model of ref.3 also requires that on average space be at, in
the absence of matter this vacuum energy must have the
critical value rhoc = 3H^2 /8piG, where H is the usual Hubble
constant. The model for quantum gravity proposed in ref.3 has
the additional feature that the ground state is a
superuid-like state whose order parameter y can, following
the hint of ref.4 that the vacuum energy is positive, be
tentatively identied with the cosmological constant / / =
3H^2|PSI|^2 (1)JS: I am not sure what unit conventions George
is using. My formula is, in contrast,/ = (Quantized
Area)^2[(Quantized Volume)|PSI|^2 [CapitalEth] 1]Where/ ~
(H/c)^2GC: As is usual in a theory with a condensate ground
state [5] we expect that the orderparameter of the condensate
will slowly decrease as the entropy of the universe
isincreased. On the other hand if the entropy is not too
large the universe will remain at on macroscopic scales so
thatOmegam + Omega/ = 1 (2).83 In this letter we would rst
of all like to point out that the recent measurements ofthe
brightness and redshift of distant supernovae can be taken as
evidence in support of acondensate model for the vacuum state
of quantum gravity with order parameter satisfying equations
(1) and (2). Secondly, the assumption that the energy density
associated with nite entropy represents collective
excitations of the condensate vacuum leads to a very simple
resolution of perhaps the most perplexing enigma of
contemporary theoretical physics; namely, although to an
outside observer information appears to be lost when objects
fall into a classical black hole, to a freely falling
observer nothing extraordinary appears to happen upon
crossing the event horizon of a black hole [6].Actually our
proposed resolution of this paradox is closely related to the
fact that theintroduction of a condensate vacuum for quantum
gravity also yields an explanation for an old cosmological
puzzle: why is the observed entropy of the universe so low ?
In particular the observed entropy of the universe is vastly
smaller than what one would at least naively expect in any
local eld theory of gravity with innumerable short distance
degrees of freedom. Of course local quantum eld theories of
gravity have many other difculties, and one might think that
the entropy puzzle is simply a reection of these other
difculties. However, the observed entropy is also much
smaller than what is expected in superstring theories whose
ground states have continuous moduli. The author has
previously noted [7] that the entropy puzzle suggests that in
reality the universe is in a nearly pure quantum state (which
necessarily has nearly zero entropy).JS: I agree with George
on this. I had this general idea independently of George and
it is in my two books Destiny Matrix and Space-Time and
Beyond II (2002), but George had the idea before me by about
3 years or so.To be continued.
=Ed Witten has been losing
sleep over the cosmological constant puzzle. George
Chaplines formula is/ = 3H^2|PSI|^2JS: I am not sure what
unit conventions George is using. My formula is, in
contrast,/ = (Quantized Area)^-2[(Quantized Volume)|PSI|^2
[CapitalEth] 1]Where/ ~ (H/c)^2Note the loop quantum gravity
observables Quantized Area and Quantized Volume/ is
Einsteins Cosmological ConstantRelation to string theory
is(Quantized Area)/hc = (String Tension)^-1 = Wittens
AlphaEd can now sleep easy. :-)
 Rationals are
Uncountable> > Let S be the set of all rational numbers
[0,1).> s is a member of S if 0.000... <= s < 1.000...> and s
is rational.> > Assume s is represented in base factorial
(!).> Base ! is used because every rational number> has a
nite representation in base !.> > In base ! the allowable
digits for> position k are (0,1,...,k).> (k starts at 1)> >
Every position, k, represents 1 / (k+1)!.> > k> 1 1/2! = 1/2>
2 1/3! = 1/6> 3 1/4! = 1/24> > .123 (base !) = 1/2 + 2/6 +
3/24 = 0.958333... (base 10).> > Every rational number has an
unique nite base !> representation. Any nite base ! number
is rational.> > We can create the set S dened above by
taking the> set produced by counting in base !.> > .0> .1>
.01> .11> .02> .12> .001> .101> ...> > There exists a
rational number, x, not in S.Asserted without basis.> If S(i)
is of the form .111...1 and> its length is equal to or greater
than x> then set x to a string of 1s one longer than S(i).i =
....> x differs from every member of S.> x is a rational
number because it has a nite number> of digits. The length
of x is exactly one greater> than some member of S.Lots of
members of S have a length exactly one greater than some
(other)member of S. In fact, all but the rst two members of
your list havesuch a length.> 0.0 <= x < 1> > x = 1/2! + 1/3!
+ ... + 1/k!Then xs position on your list is (k-1)! + (k-2)!
+ ... + 2! + 2 (andobviously does not differ from that member
of S).> and equals the largest rational number> less than the
fractional part of e.No such number exists. You might as well
talk about the smallestrational number greater than 0.--
Daniel W.
Johnsonpanoptes@iquest.nethttp://members.iquest.net/~panoptes
/039 53 36 N / 086 11 55 WX-Cise:
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=In
PM, Russell Easterly <logiclab@comcast.net> said:>There
exists a rational number, x, not in S.No.>x differs from
every member of S.No. It just differs from the rst I
elements.>the largest rational number less than the
fractional part of e.There is no such number.-- Shmuel
(Seymour J.) Metz, SysProg and JOATnot reply to
=Please look hare:
http://www.geocities.com/complementarytheory/
NewDiagonalView.pdfDoron Shadmi---= 19 East/West-Coast
Specialized Servers - Total Privacy via Encryption =---
are Uncountable Youve certainly given this some thought, but
might I suggest next> time try to prove something not
trivially false.To paraphrase Animal Farm:Some things are
less trivial than others.>Let S be the set of all rational
numbers [0,1).>s is a member of S if 0.000... <= s <
1.000...>and s is rational. Fine.Assume s is represented in
base factorial (!).>Base ! is used because every rational
number>has a nite representation in base !.In base ! the
allowable digits for>position k are (0,1,...,k).>(k starts at
1)Every position, k, represents 1 / (k+1)!.k>1 1/2! = 1/2>2
1/3! = 1/6>3 1/4! = 1/24.123 (base !) = 1/2 + 2/6 + 3/24 =
0.958333... (base 10).Every rational number has an unique
nite base !>representation. Any nite base ! number is
rational. Yes but you also assume a special form for the
base-! representation> that you really should prove to be i)
unique ii) cover all rationals> between 0 and 1. Ill take it
as given for now.Several people in this newsgroup have given
proofs of this.Base ! has nite representations for all
rationalsbecause the power series (actually, the product
series = N!)contains every power of a prime.>We can create
the set S dened above by taking the>set produced by counting
in base !..0>.1>.01>.11>.02>.12>.001>.101>...There exists a
rational number, x, not in S. I suppose between 0 and 1,
otherwise this is not relevant.Yes.>If S(i) is of the form
.111...1 and>its length is equal to or greater than x>then
set x to a string of 1s one longer than S(i). What is S(i)?
The largest number whose base-! representation consists> of
all ones? It doesnt exist! Your denition creates a set
that> includes among other things all numbers of the form:
S(i) = Sum(k:2->i, 1/k!) where i is in N. The only number
*not* in the list is the limit of> said sum as i tends
towards innity, which converges to the> irrational number e
- 2.I am dening a method to compute x.x will be a nite
approximation of e-2.x is not equal to e-2.> In fact, the
unique representation of rationals in base-! is one way> of
showing there exists a bijection from N to Q.Actually, it is
the smallest rational approximation of e that is not inset>S
where a rational approximation of e is any number of the
form.111...111. In other words, the maximum of an open set.
It does not exist.Is this really an open set? S is supposed
to containALL of the rational approximations of e-2.Isnt
this the same thing as saying S contains e-2?Let me make
three assumptions:1) I can examine every member, S(i), of
S.2) I can determine if S(i) is less than, equal to, or
greater than S(j).3) S contains every rational approximation
of e-2 (of form .111...1 inbase!)I dene a method to
calculate the largest rational approximation ofe-2 in
S:x=0For i=0 to ?: If S(i) is of the form .111...1 and S(i)
is larger or equal to x then x= S(i).Clearly, if I can
examine every member of S then I can compute x.One of my
three assumptions must be false.Which one?Russell- 2 many 2
count
 Youve certainly given this some thought, but
might I suggest next> time try to prove something not
trivially false.> > To paraphrase Animal Farm:> Some things
are less trivial than others.But since it is so simple to
construct injections from the rationals, Q, to the naturals,
N, if you continue to insist that the rationals, Q, are
uncountable, you must also accept that the naturals, N, are
uncountable.The following generates innitely many injections
from Q to N, depending on the choices of u and v:Let x be any
member of Q and let n(x) and d(x) be the numerator and
denominator of the standard form or lowest terms
representation of x as the fraction n(x)/d(x), and let u and
v be any two relatively prime naturals greater than 1, then
f(x) = u^n(x)*v^d(x) if x >=0 = u^n(-x)*v^d(-x) if x <
0injects Q into N.
Russell Easterly>> What is S(i)? The largest number whose
base-! representation consists>> of all ones? It doesnt
exist! Your denition creates a set that>> includes among
other things all numbers of the form:>> S(i) = Sum(k:2->i,
1/k!)>> where i is in N. The only number *not* in the list is
the limit of>> said sum as i tends towards innity, which
converges to the>> irrational number e - 2.I am dening a
method to compute x.Methods of computing something are
pointless when it doesnt exist.This was covered in another
thread.>x will be a nite approximation of e-2.>x is not
equal to e-2.Then x is in S by denition.>> In other words,
the maximum of an open set. It does not exist.Is this really
an open set? S is supposed to contain>ALL of the rational
approximations of e-2.Not open in the usual sense of the
word, but every irrational numberbetween 0 and 1 is a limit
point of S and therefore has no largestrational
approximation. So whatever you consider to be a
rationalapproximation of e-2 that is between 0 and 1 is in S,
yes.>Isnt this the same thing as saying S contains
e-2?Absolutely not. S contains the Cauchy sequence that
denes e-2. Itcannot contain e-2 since its not rational and
your set only containsrationals by denition.>Let me make
three assumptions:1) I can examine every member, S(i), of
S.Dene examine. S is an innite set.>2) I can determine if
S(i) is less than, equal to, or greater than S(j).Yes. The
rationals in [0, 1) are well-ordered.>3) S contains every
rational approximation of e-2 (of form .111...1
in>base!)Yes.>I dene a method to calculate the largest
rational approximation of>e-2 in S:x=0>For i=0 to ?:> If S(i)
is of the form .111...1 and S(i) is larger or equal to x> then
x= S(i).Your algorithm fails to halt. Even if it did halt, it
would only everprint out a rational we know to be in
S.>Clearly, if I can examine every member of S then I can
compute x.>One of my three assumptions must be false.>Which
one?That computability has anything to with this.
 On
S(i)? The largest number whose base-! representation
consists>> of all ones? It doesnt exist! Your denition
creates a set that>> includes among other things all numbers
of the form:>> S(i) = Sum(k:2->i, 1/k!)>> where i is in N.
The only number *not* in the list is the limit of>> said sum
as i tends towards innity, which converges to the>>
irrational number e - 2.I am dening a method to compute x.
Methods of computing something are pointless when it doesnt
exist.> This was covered in another thread.x will be a nite
approximation of e-2.>x is not equal to e-2. Then x is in S
by denition.Unless the denition of S leads to
contradiction.>> In other words, the maximum of an open set.
It does not exist.Is this really an open set? S is supposed
to contain>ALL of the rational approximations of e-2. Not
open in the usual sense of the word, but every irrational
number> between 0 and 1 is a limit point of S and therefore
has no largest> rational approximation. So whatever you
consider to be a rational> approximation of e-2 that is
between 0 and 1 is in S, yes.Isnt this the same thing as
saying S contains e-2? Absolutely not. S contains the Cauchy
sequence that denes e-2. It> cannot contain e-2 since its
not rational and your set only contains> rationals by
denition.There is another way to dene e-2?>Let me make
three assumptions:1) I can examine every member, S(i), of S.
Dene examine. S is an innite set.2) I can determine if S(i)
is less than, equal to, or greater than S(j). Yes. The
rationals in [0, 1) are well-ordered.See Dave Seamans
reply.I am not requiring S to be well-ordered.Supposedly, the
rationals can be well ordered,but not by value.>3) S contains
every rational approximation of e-2 (of form .111...1
in>base!) Yes.I dene a method to calculate the largest
rational approximation of>e-2 in S:x=0>For i=0 to ?:> If S(i)
is of the form .111...1 and S(i) is larger or equal to x> then
x= S(i). Your algorithm fails to halt. Even if it did halt, it
would only ever> print out a rational we know to be in S.Are
you saying that assumption (1) is invalid?I cant examine
every member of S?Russell- 2 many 2 count
 Yes. The
rationals in [0, 1) are well-ordered.Not by the standard
ordering.-- Dave SeamanJudge Yohns mistakes revealed in
Mumia Abu-Jamal
ruling.<http://www.commoncouragepress.com/index.cfm?action=
book&bookid=228>
= Rationals are Uncountable Then how do
you explain the numerous _injective_ mappings one may>
construct from Q to N?I cant.I even give such a mapping for
the set [0,1).> It is trivial to show that f is injective,
thus the cardinality of the> rationals cannot be larger than
that of the naturals.True.This same proof shows the naturals
are uncountable.Russell- 2 many 2 count
01:44:40 -0800, Russell Easterly> This same proof shows the
naturals are uncountable.The naturals cannot be put into a
one-to-one correspondence withthemselves?No, no, no! Youve
got it all wrong! Youre confusing a thing anditself. ---
attributed to Marvin Minsky--
http://hertzlinger.blogspot.com
= > Rationals are
Uncountable Then how do you explain the numerous _injective_
mappings one may> construct from Q to N?> > I cant.> I even
give such a mapping for the set [0,1).> > It is trivial to
show that f is injective, thus the cardinality of the>
rationals cannot be larger than that of the naturals.> >
True.> This same proof shows the naturals are
uncountable.Since countable means can be put into one to one
correspondence with the naturals, Russell in in deep dodo
when he claims that any set cannot be put into one to one
correspondence with itself.Since this contradiction only
arises from Russsells claim that the rationals are
uncountable, every sane person will reject that claim.What
Russell will do remains to be seen.
= > Rationals are
Uncountable Then how do you explain the numerous _injective_
mappings one may> construct from Q to N?> > I cant.> I even
give such a mapping for the set [0,1).> > It is trivial to
show that f is injective, thus the cardinality of the>
rationals cannot be larger than that of the naturals.> >
True.> This same proof shows the naturals are uncountable.>
Well, its not a proof since it contains the errors that
others havepointed out. But I do wish people would stop
saying things like youcant prove false things or you
shouldnt attempt to. Im not very surethat mathematics is
consistent. So, if there is a proof of statementS and someone
comes along with a proof of the negation of S and bothproofs
are valid, then mathematics is doomed. Fortunately, your
proofshave obvious errors.> > Russell> - 2 many 2
count
=Well, its not a proof since it contains the errors
that others have>pointed out. But I do wish people would stop
saying things like you>cant prove false things or you
shouldnt attempt to. Im not very sure>that mathematics is
consistent. So, if there is a proof of statement>S and
someone comes along with a proof of the negation of S and
both>proofs are valid, then mathematics is doomed. Doomed? I
would thinkfinding two such proofs would be the begining of a
mostenlightening period for mathematics. Unless, of course,
*every* valid proof ofS could somehow be turned into a valid
proof of ~S. A small dose onconsistency could be a very good
thing. rich 
=JS: I am not sure what unit conventions George
is using. My formula is, in contrast,/ = (Quantized
Area)^2[(Quantized Volume)|PSI|^2 [CapitalEth] 1]should be/ =
(Quantized Area)^-2[(Quantized Volume)|PSI|^2 [CapitalEth]
1]
 many enemies too quickly, and thankfully he did, for
it was his> downfall.Did the National Bocialists have
anything to do with this?> Nor did he read the history of
Napolean and Moscow, instead> prefering to repeat history.
Nor did Bush read the history of the> recent glorious Russian
victory of Chechnia, preferring instead to> repeat such
history in Iraqnam.I think he was inspired by the Yankee
occupation of Dixie, a case ofclassic imperialism. Capitalist
civilization went forth and crushed aworld view opposed to
tolerance and free speech (e.g., the gagrule). The d@mn
Yankees used state terrorism (Shermans march to thesea),
which set off a cycle of violence in the form of the KKK
andJesse James (who started out as a pro-slavery terrorist).
There wereeven Yankee settlements on Dixie soil. Dubya
himself is asecond-generation settler.--
http://hertzlinger.blogspot.com
=In Littlewoods
Miscellany, he has| (Via Dr A.E.Western) There was a Rent Act
after 1914, and the|denition of when a house was subject to
it was as follows (my notation| in brackets). The standard
rent (R) was dened to be the rent in 1914|(R_0), unless
this was less than the rateable value (V), in which case
it|was to be the rateable value. The house is subject to the
act if either|the standard rent or the rateable value is less
than (pounds) 105. There were|many law suits, argued ad hoc
in each case. The subject is governed by|a fundamental
theorem, unknown to the Law:|| Theorem: The house is subject
to the act if and only if V < 105.||This follows from||
Lemma: Min{Max(R_0, V), V} = V.Keith Ramsay
= Somewhere in
the IRS forms [and this is no joke, its true] You may the
thinking of the following, included as a ller in one of> the
MAA journals some years back. [Anyone have the exact
reference?] Someone dies, and his will makes some bequests,
then ends by saying> after taxes are paid, any remaining
money should be donated to> charity. Well, if charitible
donations are tax-deductible, then the> amount of the
donation effects the taxes. What to do? The I.R.S.>
supposedly has a form where you do this computation, but it
amounts> essentially to trial and error. In fact the problem
can be solved by> high-school algebra. (Solution of a linear
equation.) Or, not even> that: the problem can be solved
using the method of false position:> that method is described
in the Rind Papyrus, which dates from maybe> 2650 B.C. And
some people say the I.R.S. is behind the times...The state of
Indiana provided a table to compute their state income tax,but
did not provide a formula.I gured out that the table was a
straight percentage of income roundedto the nearest dime.
Unfortunately, the table contained several errors.I contacted
the state revenue ofce and was told I wouldhave to use the
table even though it was wrong.I also had to compute a route
tax for traveling salesmen.Salesmen were required to pay
state and local taxes basedon the percentage of time they
spent working in each state and town.Even the revenue ofce
couldnt explain how to compute it.Russell- 2 many 2
count
> But seriously, x does not exist is not the same
thing as x is not>> computable.Hardline constructivists may
disagree :-)I dont know of many hardline constructivists. I
think there may besome variety of opinion on this issue. I
dont know that I could quoteyou anybody who says x not
existing and x not being computableare the same thing.Bishop
had the following to say at the end of chapter 3 in
his_Foundations of Constructive Analysis_: Brouwers
contention that all elements of F(R,R) [functions from the
reals to the reals] are continuous seems to contradict claims
of certain recursive function theorists, who give examples of
elements of F(R,R) that are not continuous. In both
instances, the claims are based on extramathematical
considerations. Brouwer analyzes all possible techniques for
constructing elements f of F(R,R) and comes to the conclusion
that all such f are continuous. The recursive function
theorists analyze the possibilities for constructing real
numbers, and come to the conclusion that they all possess a
certain property (i.e., they are recursive). In addition,
they show how to construct a discontinuous function on the
set of recursive real numbers. These two positions are, in
fact, compatible. They do not contradict each other, because
it is possible to believe both (a) that all constructive real
numbers are recursive and (b) that without making use of some
unprovable hypothesis (such as the hypothesis that all
constructive real numbers are recursive) the only elements of
F(R,R) that can be constructed are continuous.
Extramathematical considerations of both types (especially
the rst) are useful in indicating that we should not try to
do certain things constructively, but they have no place in
the actual development of constructive mathematics.Keith
Ramsay <bt310b$sa6$1@agate.berkeley.edu>
<bt32bc$sjo$1@agate.berkeley.edu>
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=In
to prove that. Im saying that non-existence of a>solution
implies the non-computability of a solution.>I dont know how
to *prove* that though :)Its obvious - but not relevant.>The
A that the OP was talking about was an *open* interval...That
doesnt change anything. Rephrase his question to No, we
cannot. Please tell me, if A is the interval (0,1), and
current element is .5, what is the next smallest element?--
Shmuel (Seymour J.) Metz, SysProg and JOATnot reply to
<bt2mjh$g$1@lust.ihug.co.nz>X-Cise:
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=In
dened because if you tried to create a>Turing machine
capable of computing this value it would not halt No, on
multiple grounds. First, you dont need computing theory
toshow that it doesnt exist. Second, if the TM doesnt halt
then it isobviously *NOT* capable of computing this value.
Third, things existthat are not computable, so you wouldnt
have proven anythingrelevant. No, the minimum of an open
interval doesnt exist because Ris an ordered eld.>So in
computer science speak min(A) is uncomputable?In the sense
that the Spanish Barber is uncomputable; you cantcompute
something that doesnt exist in the rst place.>I guess Im
trying to look at it from a computer science point of>view.
Why would that be more fruitful than, e.g., trying to look at
it froma Musical point of view?>As in, is it possible to
create some operation, min, that applied to>some open
interval (on the real line) returned an x s.t. x <= y for>all
y in A. And the answer is no because of the above
reasons.Wrong, the answer is yes, but it wouldnt have the
properties that onewould like for a function with that name.
Specically, min(A) notelement A, contrary to the behavior of
the function that we normallyabbreviate as min. Add the
condition min(A) element A and the answerbecomes no, but not
for the reason that you give.-- Shmuel (Seymour J.) Metz,
<bt2ugn$ria$1@agate.berkeley.edu>
<bt310b$sa6$1@agate.berkeley.edu>
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=In
an algorithm> current element <-- next smallest elementThere
is no such operation.>Now assuming we can alwaysfind a next
smallest elementWhy not just assume that 2+2=5?>(which we
can)No. There is no next smallest element in an ordered
eld.-- Shmuel (Seymour J.) Metz, SysProg and JOATnot reply
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=In
real line (R), ie. A = (a, b) with a>< b (where a, b in R).
Now let x = min(A).Thats your problem right there. An open
interval has no minimum; itonly has an innum (greatest lower
bound). glb(A)=a is not an elementof A.>For example is it
possible to dene a>variable with value>e = min(|x - y| : x,
y in R and x != y)Same problem: it is *NOT* possible to dene
that. The best that youcan do is to denee = glb(|x - y| : x,
y in R and x != y)>min(A) - max(B)Same problem; only the
second[1] exists. However,0 = glb(A) - lub(B) with a <= b <
c, A = (b, c), B = [a, b][1] Because in the second case you
specied a closed interval.-- Shmuel (Seymour J.) Metz,
The consistent way to say, There is no absolute truth (How
about this one?) ;o)> > THIS STATEMENT IS THE ONLY AND ONLY
ONE ABSOLUTELY-TRUE STATEMENT.> > George BuyanovskyIs that a
Henkin sentence?-- http://hertzlinger.blogspot.com
 > The
consistent way to say, There is no absolute truth (How about
this> one?) ;o)> > THIS STATEMENT IS THE ONLY AND ONLY ONE
ABSOLUTELY-TRUE STATEMENT.> > George Buyanovsky> > 1. THIS
STATEMENT IS THE ONLY AND ONLY ONE ABSOLUTELY-TRUE
STATEMENT.> 2. Statement 1. is true> > :)You are right;
technically this joke does not work, however let us tryto
modify your arrangement:1. THIS STATEMENT IS THE ONLY AND
ONLY ONE ABSOLUTELY-TRUESTATEMENT.2. Statement THIS STATEMENT
IS THE ONLY AND ONLY ONE ABSOLUTELY-TRUESTATEMENT is true3.
Statement Statement THI . .. is true4.It seems that
statements 2,3,4 are identical to statement 1.Probably the
key of uncertainty is THIS STATEMENTGeorge
 The
consistent way to say, There is no absolute truth (How about
this one?) ;o)> > THIS STATEMENT IS THE ONLY AND ONLY ONE
ABSOLUTELY-TRUE STATEMENT.> > George BuyanovskyIf there were
no truth, neither would there be any lies. If realityexists,
then its existence is true, thus proving that truth exists.
Ifreality doesnt exist, how could we be deceived into
experiencing it?How can there be deception unless that
deception masks a reality aboutwhich were being deceived?
Either way, some kind of reality mustexist, thus proving
something true.
= <^> <(.87.87)> <^> ---- <^> <(.87.87)>
buyanovsky@attbi.com (George Buyanovsky)> The consistent way
to say, There is no absolute truth (How about this one?) ;o)
THIS STATEMENT IS THE ONLY AND ONLY ONE ABSOLUTELY-TRUE
STATEMENT. George Buyanovsky If there were no truth, neither
would there be any lies. If reality> exists, then its
existence is true, thus proving that truth exists. If>
reality doesnt exist, how could we be deceived into
experiencing it?> How can there be deception unless that
deception masks a reality about> which were being deceived?
Either way, some kind of reality must> exist, thus proving
something true.Tue true!Herc
 If there were no truth,
neither would there be any lies. If reality> exists, then its
existence is true, thus proving that truth exists. If> reality
doesnt exist, how could we be deceived into experiencing it?>
How can there be deception unless that deception masks a
reality about> which were being deceived? Either way, some
kind of reality must> exist, thus proving something true.I
did not deny truth, those joke shows that it is
amurky/subjective/relative subject. It is just properties of
some model(our mind as well). However it is a banal but
signicant percentageof non-stupid people still consider
reality as an absolute entity. Itis a cozy simplication,
which has the same root as any religion(sometimes they call
it materialism). Certainly I am not free from thesame
temptation but at least I see this.George
 The consistent
way to say, There is no absolute truth (How about this one?)
;o)> > THIS STATEMENT IS THE ONLY AND ONLY ONE
ABSOLUTELY-TRUE STATEMENT.> > George Buyanovsky> > If there
were no truth, neither would there be any lies. If reality>
exists, then its existence is true, thus proving that truth
exists. If> reality doesnt exist, how could we be deceived
into experiencing it?> How can there be deception unless that
deception masks a reality about> which were being deceived?
Either way, some kind of reality must> exist, thus proving
something true.One Day Tesshu, the famous swordsman and zen
devotee, went to Dokuon and told him triumphantly he believed
all that exists is empty, there is no you or me, and so on.
The master, who had listened in silence, suddenly snatched up
his long tobacco pipe and struck Tesshus head. The infuriated
swordsman would have killed the master there and then, but
Dokuon said calmly, Emptiness is quick to show its anger,
isnt it? Forcing a smile, Tesshu left the room.(Soul Food --
Stories to Nourish the Spirit and the Heart Ed. Jack Korneld
& Christina Feldman)Accept the terrible truth that all is
illusion. All being One, wherever you go, there you are. But
that doesnt have to ruin things. The Gnosis of illusion and
what to do about it leadsthe seeker to Moksha-- experiential
knowledge of the liberationfrom dualistic bondage. The choice
after Moksha is your own. Return to source? To nothingness? Or
choose the road for the sake of the undiscovered country and
the experience of experiencefor the sake of itself. Accept
the terrible truth and return to yourself. Your pathafter
that crossroads is your own choice. =~)Students achieving
oneness, will move ahead to twoness. (Woody Allen)We shall
not cease from exploration. And the end of all our exploring
Will be to arrive where we started Knowing the place for the
rst time. (T.S. Eliot)
 The consistent way to say, There
is no absolute truthYou said it already. Why do you need to
say anything more?(How about this one?) ;o) THIS STATEMENT IS
THE ONLY AND ONLY ONE ABSOLUTELY-TRUE STATEMENT.>This
statement is false if your rst statement is true.It seems
you have been tortured by locking you up in a round room
andtelling you that can only piss at a corner.> George
Buyanovsky
 The consistent way to say, There is no
absolute truth> > You said it already. Why do you need to say
anything more?> > (How about this one?) ;o) THIS STATEMENT IS
THE ONLY AND ONLY ONE ABSOLUTELY-TRUE STATEMENT. > This
statement is false if your rst statement is true.> > It
seems you have been tortured by locking you up in a round
room and> telling you that can only piss at a
corner.Clarication for You:There is no absolute truth How
about There is no absolute truth ?Is it absolute? George
> > The consistent way to say, There is no absolute truth You
said it already. Why do you need to say anything more? (How
about this one?) ;o)> > THIS STATEMENT IS THE ONLY AND ONLY
ONE ABSOLUTELY-TRUE STATEMENT.> > This statement is false if
your rst statement is true. It seems you have been tortured
by locking you up in a round room and> telling you that can
only piss at a corner.> > Clarication for You:> > There is
no absolute truth> How about There is no absolute truth ?> Is
it absolute?> > GeorgeThat one is much more intriguing
George:)Richard
=Say i have a point p, and an arbitrary
n-sided polygon. How can i tell howfar p is from any point
(either vertex or edge) of the polygon?
=| Here is another
way to view this. You say there is only|one way to factor 49
which makes sense. Dik and others|here say there is another.
It is dened by:|| w1(x) = GCD(a_1(x), 49)| w2(x) =
GCD(a_2(x), 49)| w3(x) = GCD(a_3(x), 49),||where a_1(x),
a_2(x), and a_3(x) are the roots of your|auxiliary
polynomial,|| a^3 + 3*(-1 + 49*x)*a^2 - 49*(2401*x^3 -
147*x^2 + 3*x).||| Note that w1, w2, and w3 are perfectly
well-dened:Keep in mind that the GCD is dened only up to
multiplicationby units.|the |roots a_1, a_2, and a_3 exist
and can be computed, and the |GCD function exists (by a deep
theorem of Dedekind) and can |be computed.For example, if
x=1,a_1(1) = -138.210434458... = 6375.47596375... *
-0.0216784496160...a_2(1) = -31.3299394631... =
998.524028548... * -0.0313762499122...a_3(1) =
25.5403739215... = 0.00000769706133063... *
3318198.0530...where 6375.47596375..., 998.524028548..., and
0.00000769706133063...are the three roots of
t^3-7374t^2+6366066t-49=0, and they multiplytogether to give
49,and-0.0216784496160..., -0.0313762499122...,
33189198.0530... are thethree roots of
t^3-3318198t^2-176046t-2257=0. Since the
polynomialst^3-7374t^2+6366066t-49 and
t^3-3318198t^2-176046t-2257 are monicpolynomials with integer
coefcients, their roots are of coursealgebraic integers.That
was sort of fun; let me try to do another. If x=2, we get
(ugh,Ill mark places where a number is continued on the next
line with like in C code)a_1(2) = -279.300354441... =
1016597193845414091969293355102867957348 409.048415...
*-2.74740434197... * 10^{-40}a_2(2) = -63.3122938187... =
3574859564584023060416081070013097699574
371117492591974367421061.951584... *-1.77104282489... *
10^{-62}a_3(2) = 51.6126482600... = 1.34830513302... *
10^{-104} * 3827964975879016817625368936818067843946
6426613668209951263900140296253120450185
91493808304178435225834041.000...where the rst factors in
each of these is a root of x^3-
(3574859564584023060417097667206943113666
340410847694842324769471) x^2+
(3634192201747556708576802697957676247053
7456650480245362950679011731799555936769
95172533270978579455876911) x- 49(and the three roots
obviously are algebraic integers multiplyingtogether to give
49) and the other factor in each is a root of x^3 -
(3827964975879016817625368936818067843946
6426613668209951263900140296253120450185
91493808304178435225834041) x^2-
(1051696759566469898052937474234752487358
853239615505188006778565836) x- 18626.The last root
a_3(2)/w_3(2)=20995126...4041.000... is a littlesurprising to
me, since it appears to be very close to being aninteger,
although actually irrational. About 39 zeros after thedecimal
point, apparently. Presumably theres a good reason
forthat.Note that so far I havent run afoul of nonreal
roots, or ofnonunique factorization in the algebraic integers
of the formr1*a^2+r2*a+r3, where r1,r2,r3 are rational
numbers. That couldblow up the numbers one has to deal with
even bigger!Please excuse any typos I may have made.Keith
Ramsay
 ...> > Here is another way to view this. You say
there is only> > one way to factor 49 which makes sense. Dik
and others> > here say there is another. It is dened by:>  > w1(x) = GCD(a_1(x), 49)> > w2(x) = GCD(a_2(x), 49)> >
w3(x) = GCD(a_3(x), 49),> > You must be careful. Strange
enough, but with this denition it is not> certain that
w1(x)*w2(x)*w3(x) = 49. I must admit I have wondered about
that. That is partly why yesterday I produced another
denition of w1, w2, and w3 based on the Magidin-Mckinnon
result (in another thread,I believe).> My latest denition is
a bit more> elaborate, but I think it is fool-proof (though
probably not James-proof):> v1(x) = GCD(a_1(x) + 7, 49)>
v2(x) = GCD(a_2(x) + 7, 49)> v3(x) = GCD(a_3(x) + 7, 49)>
k3(x) = v1(x)*v2(x)*v3(x) ; can be a multiple of 49> g(x) =
k3(x) / 49 ; the excess, must be distributed> k2(x) =
GCD(v2(x), g(x)) ; the part in v2 of the excess> k1(x) = g(x)
/ k2(x) ; and the remainder in v1.> z3(x) = v3(x) ; this one
is plain> w2(x) = v2(x) / k2(x) ; part of the excess removed>
w1(x) = v1(x) / k1(x) ; the remaining excess removed> u(x) =
z1(x)*z2(x)*z3(x)/49 ; must be a unit> w3(x) = z3(x) / u(x) ;
force it off. More elaborate than I would have guessed. Nora
B.
Yes, a posted list would be good, to clarify for
example whether you>count corner adjacencies (like UT and NM,
or AZ and CO), and which >underwater boundaries (as between HI
and AK, MN and MI, or RI and NY) >you treat as adjacencies.
Without specifying a list you probably >wont get useful
answers.HI and AK??? Theres a heck of a lot of international
water between them. Or are you claiming American sovereignty
over the whole NorthPacic?Robert Israel
israel@math.ubc.caDepartment of Mathematics
http://www.math.ubc.ca/~israel University of British Columbia
Vancouver, BC, Canada V6T 1Z2
=|This is called Graph
Covering in general, and what youre looking for |is a
solution for the Minimum dominating set. Unfortunately, its
an|NP problem.Being in NP is not a bad sign. All efciently
solvable problems are inNP. What suggests its difcult is
its being NP-complete, one of theNP problems to which all the
others can be reduced in polynomial time.Keith Ramsay
 > >
I am stuck in high school maths mode, and cant seem to get
into> university level maths. This might be because I am
entirely self> taught, but I dont know. Does anyone else
have this problem?> > I am at a level where I understand most
high school maths, and Ive> studied Calculus Made Easy. It
would be helpful to have some kind of> way to check my
knowledge.> > Does anyone know of any good textbooks that
cover high school maths> with worked exercises, and any texts
that help the transition from> high school maths to the more
exciting stuff at university level? A> book that takes time
to explain things, point out applications and say> why rather
than just how. I am trying to understand maths, not just>
learn some techniques or shortcuts.I see that youve signed
off, but for the record, I would
recommendVector_Spaces_Of_Finite_Dimension by G.C. Shephard,
University Mathematical Texts ( 1966 ) ( This was a new book
when I discoveredit my freshman year at Brown U. )This is a
thin, terse treatment of linear algebra, but it is verymathy
and emphasizes proof. The rst exercise is to prove
thatlambda * o = o, where lambda is a scalar and o is the
zero vectorof a vector space. You rst have to realize that
this is notstated in the denition, then see how it follows
from linearity.Of course, all the proofs depend on applying
linearity, so youcan get in the swing of things by picking up
on the idea.Lew Mammel, Jr.
<3ff17881$0$4763$61fed72c@news.rcn.com>
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So much for
politicians lip service about education being a national>
priority.Are you sure that you dont live in the US? That
comment would beright at home here.Q: How can you tell when a
politician is lying?-- Shmuel (Seymour J.) Metz, SysProg and
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Are you
talking about something purely mathematical, Yes. The
classical presentation of Euclidean Geometry relies
heavilyupon compass and straightedge constructions. However,
the physicallanguage mask purely Mathematical concepts. Every
proof where the textsays draw or construct can be replaced
with an equivalent proofusing purely Mathematical language,
e.g., existential quantiers. Thephysical language helps to
visual the concepts, except when it leadsthe reader
astray.>No to both. ;)No.-- Shmuel (Seymour J.) Metz, SysProg