mm-471
===
Subject: Re: Schauder basisskip...>This means that, for
every>x in X and for every eps>0, if n is sufficiently large,
then ||(Sum>(k=1,n) ex_k * x_k) - x||<eps and, consequently,
Sum (k=1,n) ex_k *>x_k) is in the open ball of radius eps and
center x. The letter x in>ex is to emphasize that the scalars
of the infinite linear combination>of the x_k's that leads to
x depends on x.>If we define D = {ex_1 * x_1 ...+ ex_k * x_k |
k in N (the naturals),>x in X}, then, for every x in X and
every eps>0, the open ball>centered at x and of radius eps
intersects D. Therefore, D is dense in>X. It remains to show
(if it's true, of course) that D is countable. >Well, if X is
countable, this is immediate (in fact, if X is countable>we
have nothing to prove), but I'm still wondering how I can
prove D>is countable or prove it contains a countable subset
that's also dense>in X.> D is not countable. That's because
the ex_j can be any real numbers,> and the reals are
uncountable. But...>But we can take the set of all rational
linear combinations of the x_n>(that is, the set of all linear
combinations with rational>coefficients). Since the rationals
are countable and are dense in R,>we get a countbale set dense
in X and we are done. Right?> Yes!>If X is a vector space over
the complexes, we can take the complexes>with rational real
and rational imaginary parts.> Yes.>The thing gets more
complicated if X is vector space over an arbitrary>field F...>
If we're talking about a Banach space then we're talking about>
real or complex scalars.>But, we didn't use the fact that X is
Banach. The completeness of X>was not considered at all>
Amazing. We've proved more than was required: If X is a
normed> vector space and there exist e_1, ... such that every
x in X can> be written x = a_1 e_1 + ... then X is
separable.Yes. And now that the problem is solved, it doesn't
look so hard.
===
Subject: Re: Schauder basis>[...]>Yes. And now
that the problem is solved, it doesn't look so hard.>AmandaIt's
not so hard - your biggest mistake in all this was when you
saidyou thought it was probably beyond you.
===
Subject: Re:
Antidiagonal, InfinityAgain, in binary there is one and
exactly and only one antidiagonal,if that.If you demand the
flexibility of selecting any of the b-1 numbers notthe element
of the list expansion, or in other cases a value basedupon two
or more consecutive values of the expansion, I said to
modifythat to get a rational number. That's as simple as an
infiniteprocess that guarantees some repeating terminus of the
antidiagonal. That is not the case.Yet, where I am considering
that, it is somewhat counter to myconsideration that the
antidiagonal of a list of all elements of thereals would have
dual representation because the reals comprise allinfinite
bitstrings with a beginning, for convenience's sake the
unitinterval. That is to say, where the list is of all
possiblepermutations, there are no others, and at least one
permutation is oneof those.Anyways in approaching that I
consider what the diagonal is andvarious ways to construct an
antidiagonal. For example if the decimaldiagonal is 5..., then
the antidiagonal is 4..., a rational number,that simple
everywhere-different sequence is always representing arational
when the sequence represents a rational. When the
diagonalsequence represents an irrational, then an algorithm
is modified toproduce a rational sequence. Perhaps that is not
possible. Forexample the diagonal might be a sequence normal to
decimal, with eachof 0-9 equally probable to be the next
element at any random point inthe sequence. There it would be
necessary for me to show that somealgorithm outputs a sequence
that is everywhere different from thenormal sequence and also
represents a
rational.31830988618379067153776752674502872406891929148091___
_______________5_____5____5____________________
55555555555555555545555545555455555555555555555555____________
________________4______4_________4____
55555555555555555545555545555455554555545555455554You can see
how that finite sequence has a repeating terminus of 55554but
also perhaps how it could not continue infinitely. You
canconstruct a non-normal irrational, but not necessarily a
rational,from the naturally normal diagonal. Is it also not
necessarily sothat a rational can be constructed from a
non-normal sequence, or evenfrom artificially normal
sequences?be constructed. That concurs with that for a given
irrationalsequence another irrational sequence everywhere
different can beconstructed, but a rational sequence may not,
unless the irrationalsequence is not normal, and from a
rational sequence a rationalsequence can be constructed that
is everywhere different. So I dropmy argument that the
everywhere non-diagonal of a list of rationalscan be made
rational without reference to forcing. That's basicallybecause
for a normal sequence no finite repeating terminus would
beeverywhere different than the randomly distributed in
occurrenceelements of the normal sequence.The antidiagonal of
some (ir)rationals is not necessarily(ir)rational, in binary.
Kovarik's continued fraction might be of arational. The
antidiagonal of a list of irrationals might berational.I
consider more the consideration of the set of all bitstrings.
Theredoes not exist a sequence different from each of those
bit strings, orit would be an element of that set.Consider not
the list, but the set of all bitstrings. There does notexist a
sequence different from each of those bitstrings, else itwould
be an element of that set. By various fiats (axioms) or as
Iclaim theorems of a null-axiom theory, the set of all reals
is a set,and enumerable. Thus, where the set of bitstrings
represents allreals, there does not exist a sequence different
than each. A rose isa rose is a rose.That leads into a concern
about the list of all infinite bit strings,formed by
enumerating the abovementioned set, that the
constructionitself of some other bitstring would fail to be
everywhere different.Because of the representation of such a
set, necessarily inclusive ofall possible bit strings, the
antidiagonal process simply generates aset with the same value
yet a differing representation. This is whatwe see in the case
of the reals and the singular antidiagonal for anyenumeration,
with dual representation of rationals.In a sense the conclusion
of the antidiagonal process leads to dualrepresentation.For the
Equivalency Function only the binary antidiagonal is allowed.
The nested interval conditions do not hold. Present another
reasonwhy you claim no mappings exist between the naturals and
unit intervalof reals, indeed, any other. Good luck.
===
Subject:
Re: Antidiagonal, Infinity> Again, in binary there is one and
exactly and only one antidiagonal,> if that.Wrong! There are
at least a countable number of anti-diagonal
algorithms(1)Assume an anti-diagonal binary algorithm, such as
that based on the two digit operations that guarantee a result
with unique binary representation not already listed.(2) Apply
it to your list and make a new list with this number first
followed by the rest of the list is its originally given
order.(3) go to (1)Each iteration will produce a number not in
any previous lists, and countably many iterations are
possible.
===
Subject: Re: Antidiagonal, InfinityYou say there
are multiple antidiagonals because you generate
anantidiagonal, and then add that to the list and diagonalize
a newlist. If you keep generating antdiagonals that way, they
aren't ofthe original list, which has one and exactly one
binary antidiagonal.I think more about the antidiagonal of an
infinite list ofparticularly all possible sequences of binary
elements.This concept is that if a set contains every possible
permutation,then no method exists to get a different
permutation, because it wouldinstead be one of the possible
permutations instead of different.With something like the
binary sequences that represent reals, aset-wise
antidiagonalization method is something like this:R[0,1]
existsthere exists a, a is an element of R[0,1]there exists
x_1, x_1 is an element of R[0,1]a is different than x_1 in the
1'st integral modulus after zerothere exists x_2, x_2 is an
element of R[0,1] x_1a is different than x_2 in th 2'nd
integral modulus after zeroThen, so on and so forth for each
n'th element of R[0,1]there exists x_n, x_n is an element of
R[0,1] x_1 U ... U x_n-1a is different than x_n in the n'th
integral modulus after zeroThen for each element x of R[0,1],
some n'th element of x is not equalto the n'th element of a.
Where a is an element of R[0,1], that leadsto a problem that
some n'th element of a is not equal to itself. Oneway to
resolve that contradiction is to say that a doesn't exist as
anelement of R[0,1].In extending from R[0,1] to R, the unit
interval of reals to allreals, much the same procedure and
conclusion results, except wherebefore a could be some real
outside of the unit interval, in extensionto all reals a can
not be a real.Another possible conclusion is to allow that a,
while being differentthan each as described, is yet the same,
for example via dualrepresentation of rationals whose
denominator is a power of the base.Thus, either a does not
exist or a is dually represented. Why wouldthat not be so?
Alternatively, as the reals are uncountable, youmight say, or
infinite, x_n, is never the last element of the reals toaffect
the antidiagonal, yet x_n is always itself.One special feature
of the reals is that each bit string represents areal on the
unit interval, yes even that one.Many reals have multiple
representations, the antidiagonal always hasone of the unused
representations, else it would not be a real. Inthat sense a
list of rationals might have a dually represented value aa
rational or be irrational, or the antidiagonal of all
irrationals isnecessarily rational, which is somewhat
surprising, unless someirrational has dual
representation.Select an integer between one and ten that is
not 1, 2, 3, 4, 5, 6, 7,8, 9, or 10. Is your answer oo+4?In
examining the process itself, it's necessary to explain
howsomething can be different than everything, including
itself, orperhaps more agreeably, that it's not.
===
Subject:
Calculus BooksI'm looking for a good book on calculus. I'm
interested in a bookthat's sort of the equivalent to Feynman
Lectures of Physics in theCalculus world. I currently have
books by Stein, Edwards, andStewart. This books are good in a
practical sense; however, I wantsomething more insightful. It
has to be rigourous, but at the sametime have good
explanations. I don't mind if it's wordy. I've heardApostol
and Spivak's books are a good choice. I've actually
seenApostol and it looks promising; however, this opinion is
based onreading the preface and a quick look through the
chapters (no morethan a half-hour). Any
suggestions?
===
Subject: Re: Calculus Books> I'm looking for a
good book on calculus. I'm interested in a book> that's sort
of the equivalent to Feynman Lectures of Physics in the>
Calculus world. I currently have books by Stein, Edwards, and>
Stewart. This books are good in a practical sense; however, I
want> something more insightful. It has to be rigourous, but
at the same> time have good explanations. I don't mind if it's
wordy. I've heard> Apostol and Spivak's books are a good
choice. I've actually seen> Apostol and it looks promising;
however, this opinion is based on> reading the preface and a
quick look through the chapters (no more> than a half-hour).
Any suggestions?Calculus by Michael
Spivakhttp://www.amazon.com/exec/obidos/ASIN/0914098896/qid=
1082162810/sr=2-3/ref=sr_2_3/102-1387591-6151359
===
Subject:
Re: y=tan(x)-x => x=?>y=tan(x)-x => x=? (in terms of y
...)>Not an elementary function.Of course there are series
solutions>such as x = v - (2/15) v^3 + (3/175) v^5 - (2/1575)
v^7 ->(16/202125) v^9 + ... where v = (3 y)^(1/3). You can
also use>Newton's Method and the like.>For this case, due to
x-shift and other branches, when not added that>way, is there
a way to get all solutions?>Yes. All solutions are given by> x
= k*pi + v - 2/15*v^3 + 3/175*v^5 - 2/1575*v^7 - 16/202125*v^9
+ ...>where v = (3*(y + k*pi))^(1/3).> As RI has pointed out
below, the series above has a small radius of> convergence.
Nonetheless, the answer to GLN's question is still Yes.> I
should have said:> Let F(x) = tan(x) - x, -pi/2 < x < pi/2,
and let G be its inverse function.> Then all the solutions of
y = tan(x) - x are given by> x = k*pi + G(y + k*pi), k
integer.>The trouble with this method is that the series does
not seem to>converge rapidly unless v is small.> I wouldn't
expect it to converge at all unless |v| < (3 pi)^(1/3).> So my
comment above was, at best, a gross understatement of the
trouble.> Note that if f(x) = tan(x)-x, f'(x) = tan^2(x) = 0
when x = Pi,> and f(Pi) = -Pi.> Hmm. I had thought, in getting
the series, that you had been thinking of a> function with
domain (-pi/2, pi/2), in other words, my F(x) above. Then> x =
pi would not be pertinent.> So an inverse for f should have a
singularity at y = -Pi.> ??? G(y) has no singularity at y =
-pi.It depends on which branch. Note that (the complex version
of) f is a three-to-one function in a deleted neighbourhood of
0, so f^(-1) is a multivalued function. On one of the sheets
there's a singularity (actually a branch point) at y=-Pi, and
on one there's a branchpoint at y=Pi.Robert Israel
israel@math.ubc.caDepartment of Mathematics
http://www.math.ubc.ca/~israel University of British Columbia
Vancouver, BC, Canada V6T 1Z2
===
Subject: Re: Q: Orbital
Intercept by support1.mathforum.org (8.11.6/8.11.6/The Math
Forum, $Revision: 1.9 primary) id i3H21qc08209;Is there any
chance you could help me out with this problem?
===
Subject: Re:
Resistance to Change by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
i3H21sB08270;They laughed at Einstein. They laughed at the
Wright Brothers. But they> also laughed at Bozo the Clown. --
Carl Sagan>It seems that the correct quotation is:But the fact
that some geniuses were laughed at does not imply that all >who
are laughed at are geniuses. They laughed at Columbus, they
laughed >at Fulton, they laughed at the Wright brothers. But
they also laughed at >Bozo the Clown.>Jose Carlos SantosMight
we take this one step futher?But the fact that some normal
people get laughed at because theychoose to act as if they
were stupid does notimply that all who are laughed at are not
bonafied idiots. They laughed at Bozo the Clown, they laughed
at the Three Stooges. But they also laughed at G.W.
Bush.
===
Subject: Re: PDE literature by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
i3H21ov08151;>I'm doing some work concerning numerical
solution of PDE problems,>mainly problems for parabollic
equations (heat, convection-diffusion).>Also interested in
singularly perturbed problems.>Can anyone give a hint, where
can I download some general literature>on these topics
(actually i need free downloads): problems>formulation,
correctness conditions, physical sense etc. ?>Any hint would
be appreciated.when you first get on math forum, go to
internet mathematics librarysome of the subcategories under
differential equationshave links to on line
journals
===
Subject: Re: another boring critisism of Cantor's
Theorem> Could you explain why V can't be countable?
Certainly you can't> prove that in any consistent first order
formalism for which the> Loewenheim Skolem theorem
applies.>It's a triviality to prove V can't be countable in
ZFC. The>Loewenheim-Skolem theorem says that if ZFC is
consistent, it has a>countable model. But that's not
V.>Philosophically speaking, if we are discorsing in the
first-order>language of set theory and uttering theorems of
ZFC, we can always>suppose, without making any of our theorems
false, that our discourse>is relative to a countable model. But
as I say, that's different to>saying V can be
countable.>Somehow, I don't think you explained anything. You
merely restated>what you said the first time.>If you can prove
V exists, you can prove that it is countable>relative to some
meta-language. But you claim otherwise. Why?>I suppose I
should admit that I don't know a great deal about>this topic,
but you have hardly increased my knowledge of it.> If ZFC is
consistent, and we are uttering theorems of ZFC in the>
first-order language of set theory, we can always entertain
the> possibility, without making any of our theorems false,
that our> discourse, let's call it the object language, is
relative to some> countable model. Then a metalanguage will be
available in which what> we referred to as V in our object
language, will now in fact be> countable. This possibility
*may* hold of our language, I don't think> it *has* to hold,
this may be a point of difference between us.> But in any
case, you have to decide what language you're talking in.>
Whether you're talking in the object language or the
metalanguage, it> won't be true that V is countable. In both
the object language and the> metalanguage, all theorems of ZFC
are true, and it's a trivial theorem> of ZFC that V is not
countable.> But then, just like the original thing only
_seemed_ uncountable,> but in reality countable, your 'meta
world' could also be only> seemingly uncountable, but be
'meta-meta' countable.> Or do i miss something?> Herman
JurjusWe can always entertain the *possibility* of any
language that itssemantics is relative to a model which is
correctly called countablein some metalanguage. I don't think
this possibility always has tohold.
===
Subject: lim inf (-1,
1/n)if A_n = (-1/n,1] if n is odd, and A_n = (-1, 1/n] if n is
even thenlim sup A_n is (-1,1) and lim inf A_n is 0where lim
sup is defined as intersection of unions and lim inf isdefined
as union of intersections.
===
Subject: Re: lim inf (-1, 1/n)>if
A_n = (-1/n,1] if n is odd, and A_n = (-1, 1/n] if n is even
then>lim sup A_n is (-1,1) and lim inf A_n is 0>where lim sup
is defined as intersection of unions and lim inf is>defined as
union of intersections.Almost. The lim sup and lim inf are
respectively (-1, 1] and {0}. I suspect your answer for the
lim inf is just a matter of notation. Do you see why 1 is in
the lim sup? (Or was there a typo in your proposed
answer?)Cite sources of help on submitted homework.
===
Subject:
A question on third order linear differential equationsLet x_1,
x_2 be linearly independent sols. of a third order
homogenouslinear differential equation. Is there a method for
finding a third sol. x_3such that x_1, x_2 and x_3 are
linearly independent?
===
Subject: Re: A question on third order
linear differential equations>Let x_1, x_2 be linearly
independent sols. of a third order homogenous>linear
differential equation. Is there a method for finding a third
sol. x_3>such that x_1, x_2 and x_3 are linearly
independent?Yes. See reduction of order in just about any book
on differentialequations. (Not a math book, an elementary text
- might not bein a theoretical math book but it's in all the
texts.) That's usuallypresented as a way to get a second
solution to a second-orderequation given one solution, but you
should have no troubleextending it to your problem.
===
Subject:
Re: A question on third order linear differential
equations>Let x_1, x_2 be linearly independent sols. of a
third order homogenous>linear differential equation. Is there
a method for finding a third sol.x_3>such that x_1, x_2 and
x_3 are linearly independent?> Yes. See reduction of order in
just about any book on differential> equations. (Not a math
book, an elementary text - might not be> in a theoretical math
book but it's in all the texts.) That's usually> presented as a
way to get a second solution to a second-order> equation given
one solution, but you should have no trouble> extending it to
your problem.Well, I have trouble extending it. It ends up
with a second order linearequation in an unknown function
which I don't know how to solve.
===
Subject: Re: A question on
third order linear differential equations>Let x_1, x_2 be
linearly independent sols. of a third order homogenous>linear
differential equation. Is there a method for finding a third
sol. x_3>such that x_1, x_2 and x_3 are linearly
independent?It suffices to find linear independent initial
conditions:Let P1 = (x_1(0),x_1'(0),x_1''(0)) and P2 =
(x_2(0),x_2'(0),x_2''(0))just find a point P3 =
(x_3(0),x_3'(0),x_3''(0)) that does not lie inthe Span of P1
and P2. That should be easy using methods from linearalgebra.
P3 uniquely determines a solution that satisfies
yourcondition.
===
Subject: Re: A question on third order linear
differential equationsBut I want a 'formula' in terms of x_1,
x_2 and the coefficients of theequation, just like the formula
in second order case.>Let x_1, x_2 be linearly independent
sols. of a third order homogenous>linear differential
equation. Is there a method for finding a third sol.x_3>such
that x_1, x_2 and x_3 are linearly independent?> It suffices
to find linear independent initial conditions:> Let P1 =
(x_1(0),x_1'(0),x_1''(0)) and P2 = (x_2(0),x_2'(0),x_2''(0))>
just find a point P3 = (x_3(0),x_3'(0),x_3''(0)) that does not
lie in> the Span of P1 and P2. That should be easy using
methods from linear> algebra. P3 uniquely determines a
solution that satisfies your> condition.> Thomas
===
Subject:
new st. mary's college moraga, ca observatory 21 pics. this is
not a observatory it's a silo for 2 nuke missilesnew st. mary's
college moraga, ca observatory 21 pics.
http://community.webshotshttp://community.webshots.com/user/
nightstarsmoragahttp://tinyurl.com/2k5zslat 37.8404881 long
-12210355 this is not a observatory it's a silo for 2 nuke
missiles
===
Subject: Re: new st. mary's college moraga, ca
observatory 21 pics. this is not a observatory it's a silo for
2 nuke mis
===
>Subject: new st. mary's college moraga, ca
observatory 21 pics. this is not a>observatory it's a silo for
2 nuke mis>new st. mary's college moraga, ca observatory 21
pics.
>http://community.webshots>http://community.webshots.com/user/
nightstarsmoraga>http://tinyurl.com/2k5zs>lat 37.8404881 long
-12210355 >this is not a observatory it's a silo for 2 nuke
missilesOne missle per silo, idiot.Although it could have
multiple warheads.
===
Subject: Re: Angular
measure3/10
===
Subject: Re: functions that halt
<9o04l1-jgu.ln1@lexi2.athghost7038suus.net>
<407f780e$0$4548$afc38c87@news.optusnet.com.au
<4080091c$0$567$b45e6eb0@senator-bedfellow.mit.edu>Tell me
what the 'next' real is in any set of reals. You can't.> I
don't understand your argument. Surely it is not:> 1. In the
natural ordering of the reals, there is no next real after>
any given real. Therefore the reals are uncountable.> That's
certainly fallacious, as may be seen by replacing real
byrational. So maybe you mean: He's not talking about a 'next'
real in a total ordering of real values, but talking about the
'next' real that exists in the enumeration of the reals (in
the case that they are listable/countable) For the rationals,
there is definitely a next rational when one orders them in
that standard way (in which I'm sure you are familiar... it is
almost always used when showing the rationals are
countable.)
===
Subject: Re: functions that haltOriginator:
tchow@lagrange.mit.edu.mit.edu (Timothy Chow)> He's not
talking about a 'next' real in a total ordering of real
values, >but talking about the 'next' real that exists in the
enumeration of the >reals (in the case that they are
listable/countable)My point is that Cantor's diagonal argument
does *not* show that thereals cannot be ordered in such a way
that there is always a 'next'real after any given real. By
saying that you can't name the 'next'real after pi in this
context, the writer sounds like he's confusingthe
well-orderability of the reals with their uncountability.Tim
Chow tchow-at-alum-dot-mit-dot-eduThe range of our
projectiles---even ... the artillery---however great,
willnever exceed four of those miles of which as many thousand
separate us fromthe center of the earth. ---Galileo, Dialogues
Concerning Two New Sciences
===
Subject: Re: functions that halt
<4080091c$0$567$b45e6eb0@senator-bedfellow.mit.edu
<Pine.LNX.4.44.0404162321520.26350-100000@tees.cs.ualberta.ca
<40815bf6$0$575$b45e6eb0@senator-bedfellow.mit.edu> My point
is that Cantor's diagonal argument does *not* show that the>
reals cannot be ordered in such a way that there is always a
'next'> real after any given real. By saying that you can't
name the 'next'> real after pi in this context, the writer
sounds like he's confusing> the well-orderability of the reals
with their uncountability. That wasn't my interpretation at
all... it sounded like the author was relating the
countability/uncountability of a set with its property of
being listable (with a first element), which is entirely
acceptable. (But I do agree that asking what number comes next
'after pi' is kind of irrelevant... as there will exist a
one-to-one correspondence of a set like the rationals with any
rational at all after some given rational, a/b. The only
possible answers to his question were any real you want or no
real at all.)
===
Subject: Re: functions that halt>Tell me what
the 'next' real is in any set of reals. You can't.> I don't
understand your argument. Surely it is not:> 1. In the natural
ordering of the reals, there is no next real after> any given
real. Therefore the reals are uncountable.> That's certainly
fallacious, as may be seen by replacing real byrational. So
maybe you mean:> He's not talking about a 'next' real in a
total ordering of real values,> but talking about the 'next'
real that exists in the enumeration of the> reals (in the case
that they are listable/countable)> For the rationals, there is
definitely a next rational when one orders> them in that
standard way (in which I'm sure you are familiar... it is>
almost always used when showing the rationals are countable.)I
think the problem is that you can't distinguish a computable
real froman uncomputable one.You can write down some
specifications (for reals) that areuncomputable, but I think
you can't decide whether the real thatfulfills that spec just
might happen to be a fraction of square roots orsome such
(precisely because you can't compute it and so check). Atleast
all the examples I have seen were of that kind.So if someone
does not accept the proof that the cardinality of reals
islarger than the cardinality of computable numbers (which can
be listedby listing all programs), what can you use to convince
him?The diagonalisation proof attempts to construct such a
number which iscomputable outside of the formal system that is
the Turing Machine it isapplied to.(Re)reading
http://www.abelard.org/turpap2/tp2-ie.asp might be a goodidea;
I did, and was surprised that computable numbers must be
computedby Turing machines that keep on producing output
infinitely...MichaelFeel the stare of my burning hamster and
stop smoking!
===
Subject: Re: functions that halt E29yc_kQC&^>
I think the problem is that you can't distinguish a>
computable real from an uncomputable one.I'm not sure what you
mean here. If you can describe anumber and prove that it's not
computable, that counts asdistinguishing it in my book. If
it's one of the uncountablenumber of reals that we can't even
describe, you'd probablybest appeal to Wittgenstein.> You can
write down some specifications (for reals) that> are
uncomputable, but I think you can't decide whether the> real
that fulfills that spec just might happen to be a> fraction of
square roots or some such (precisely because> you can't compute
it and so check). At least all the> examples I have seen were
of that kind.No: all the proofs of uncomputability I can think
of provethat a number really isn't computable. For example the
realB defined as 0.bbbbb... where the n'th bit is 1 iff
theprogramme with G.9adel number n halts is not computable. If
itwere, then one could use another programme that
tookprogrammes as argument, G.9adel encoded them and indexed B
asa solution to the halting problem. Note that the proof
saysnothing to specify the reason behind computing B;
thecontradiction would arise if /any/ programme could compute
Bin any fashion. So B isn't a fraction of the square root
ofanything computable.> So if someone does not accept the
proof that the> cardinality of reals is larger than the
cardinality of> computable numbers (which can be listed by
listing all> programs), what can you use to convince him?It
depends on why they don't accept it. If they are arguingthat
numbers such as B above have no meaning in nature, orthat the
universe we inhabit is constructed entirely on acountable
basis there is no way to refute that. I don'tthink we know a
single physical constant to more than a fewbillion digits.
Besides, every axiomatisation of the realshas a countable
model; in a sense we cannot really escapefrom our discrete
means of expressing mathematics.Also, from a practical point
of view, if the someone is tooobtuse there may be nothing that
will convince him.[1] FSVO the. We have to pick a particular
notion ofcomputation and a particular G.9adel encoding -- so
in factthere's a (countable) infinity of Bs...[2] I'm allowing
for a certain amount of future scientificeffort here. J.97n
Fairbairn Jon.Fairbairn@cl.cam.ac.uk
===
Subject: FW: Smart way
to l0se the 1bs.. boundary=--5453756934327096 by
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
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$Revision: 1.6 secondary) with SMTP id i3H7DJt26596;X-CS-IP:
220.116.200.78------------------------------------------------
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===
Subject: Re: Commutative rings aren't physical spaces
(This Week's Finds 205)
<c5l6tr$94a$1@news.Stanford.EDUWhitehead's cool; Russell's
kook. but I think his presumption was pretty good, althoughI
didn't follow it too closely. after all,commutativity is just
math, two. > space as in general relativity. Whitehead showed
you could do> relativity in flat space. It's all a device for
thinking about things--Give Earth a Trickier Dick Cheeny --
out of office, after GIGA
years.http://www.benfranklinbooks.com/http://www.rand.org/
publications/randreview/issues/rr.12.00/http://
members.tripod.com/~american_almanachttp://www.wlym.com/pdf/
iclc/howthenation.PDF
===
Subject: Re: Yao Ziyuan's Conjecture
:-)> Yao Ziyuan's Conjecture> After a simple search within
100,000, Yao Ziyuan from Fudan University> found that only 4,
6, 8, 12 (even number) can be represented as the> sum of one
and only one pair of primes. So he made another conjecture>
one representation of different pairs of primes. LOL.> This
practice demonstrated that we can easily make as seemingly>
beautiful conjectures as the Goldbach one, as many as
possible. This> lowers the uniqueness of the Goldbach
conjecture and makes it much> less significant to prove merely
one such conjecture, even if it is> eventually proven.nbr could
be represented as the sum of two prime nbrs. I seem toremember
that the largest nbr which could only be represented in
twoways was 68.But more interestingly, apart from a long term
rising trend inrepresentations, I found that the nbr of
representations rose and fellwith a period of 3, so that 400
might have 5 representations, say,then 402 7, then 404 8, then
406 back to say 5, 408 8 , 410 9. Thesefigures are purely made
up just by way of an illustration.Unfortunately I threw the
details out a long time ago.
===
Subject: sheaf of functions
using certain schemes....I need help on the following
algebraic geometry exercise:Describe the points and the sheaf
of functions on the followingschemes:(1) X = Spec
C[x]/(x^2)(2) X = Spec C[x]/(x^2-x)(3) X = Spec R[x]/(x^2+1)I
think I can probably handle any two of these given a
discussion ofhow to solve the other one, but my class is so
abstract that I'mhaving a hard time applying the definitions
and actually computingthese sheaves and points.....
===
Subject:
Re: (phi(m)+1) divides m> Let phi(m) > be the number of
positive integers <= m and relatively prime with m.> I get, by
hand, that the sequence of m's where> (phi(m)+1)|m> begins> 2,
3, 5, 6, 7, 10, 11, 13, 14, 17, 19,..> (Not in the EIS,
apparently.)> What can be said about this sequence, such as
its asymptotics, the> best way to determine if an integer is
in it, etc ?Note that m = r (phi(m) + 1) can be rewritten m/r
- 1 = (phi(m) / phi(m/r)) phi(m/r)If there are solutions other
than m = p, m = 2p for p prime,phi(m) / phi(m/r) will be an
integer greater than 1, and thenyou have a solution to the
famous Lehmer's problem.---J K
Hauglandhttp://www.neutreeko.com
===
Subject: Re: (phi(m)+1)
divides m>According to Maple, these are all the members of the
sequence up to 10000.According to Excel, there are no
exceptions up to 20,000,000.John Robertson
===
Subject: Re:
Yikes! I've Broken Incommensurability!> [snippity snip]> A>
_______> | /> | /> | /> B | / C> | /> | /> |/> O> [Hope it
displays correctly, I keep> changing my newsreader font to>
fixed, but it doesn't stay, and I'm> 'tired' of changing it
:-]> Imagine lower vertex at the center> of the circle, and
the upper two ver-> tices on the circumference of the> Circle
- all as the central angle is> varied 0 - 360 [0 to 2pi
radians].> Errr...> Nope.> As you've described it, side B is a
radius.side C is the radius.
===
Subject: Re: Yikes! I've Broken
Incommensurability!> [snippity snip]> A> _______> | /> | /> |
/> B | / C> | /> | /> |/> O>[Hope it displays correctly, I
keep>changing my newsreader font to>fixed, but it doesn't
stay, and I'm>'tired' of changing it :-]>Imagine lower vertex
at the center>of the circle, and the upper two ver->tices on
the circumference of the>Circle - all as the central angle
is>varied 0 - 360 [0 to 2pi radians].> Errr...> Nope.> As
you've described it, side B is a radius.> side C is the
radius.Sorry, in my earlier post, I wasn't clear. WhatI meant
was that side A and side C share avertex that is a point on
the Circle.At the 'instant' of the diagram, above, side Ais
the semi-chord. Side B extends from thecenter of the Circle to
meet side A at 90.Forgive me, please.
===
Subject: Re: Simple
differential equation query >Given the following equation
eliminate the arbitrary constant: >1. x^3 - 3*(x^2)*y = c >Ok,
differentiate both sides giving: 3x^2 - 6xy -3(x^2)y' = 0
>DIVIDE THROUGH BY 3X: x - 2y - xy' = 0 >Rearrange: (x-2y)dx -
x dy = 0 (= answer given in the book) >[Q] But why can we be
sure x != 0 ?! (and divide through by it?)If x can ever take
the value 0, then c = 0. So handle c = 0 as aspecial case,
which makes y = x/3 when x /= 0 and anthing when x = 0.In the
event that y is continuous then y = x/3 and again x - 2y - xy'
= 0which is the desired result for is it not stipulated for
theproblem series that y is differentiable, hence continuous?
>2. cy^2 = x^2 + y >DIVIDE THROUGH BY Y^2: c = (x^2 + y)/y^2
>Differentiate both sides and simplify: [ 2xy dx - (y+2x^2) dy
] / >y^4 = 0 >Now multiply both sides by y^4 and we're done.
>[Q] But why can we be sure y != 0 ?!cy^2 = x^2 + y2cyy' = 2x
+ y'2cyyy' = 2xy + yy'2(x^2 + y)y' = 2xy + yy'2x^2 y' + yy' =
2xyLook, no division!----
===
Subject: Limit PointsIn a metric
space what are the limit points of {x}?Is the definition
different for topological spaces?----
===
Subject: Re: Limit
PointsA point x is a limit point of a set if every open ball
centered at xcontains a point in that set different from xit
can be proved that a finite set doesn't have any limit points,
so {x}doesn't have any limit points> In a metric space what are
the limit points of {x}?> Is the definition different for
topological spaces?> ----
===
Subject: Re: Limit Points> A point
x is a limit point of a set if every open ball centered at x>
contains a point in that set different from x> it can be
proved that a finite set doesn't have any limit points, so
{x}> doesn't have any limit points> In a metric space what are
the limit points of {x}?> Is the definition different for
topological spaces?> ----Interestingly, a nonempty finite set
is closed, meaning it containsall its limit points. Therefore,
it contains all its limit pointssince it does not have
any!
===
Subject: Re: Limit Points> In a metric space what are
the limit points of {x}?The set {x} has no limit points. To
say that y is a limit pointof {x} is to say that y belongs to
the closure of {x} y, whichis the empty set if y = x and it is
equal to {x} otherwise.> Is the definition different for
topological spaces?Perhaps that some textbooks say that in a
metric space a pointx is a limit point of a set S if, for
every r > 0, the setB(x,r) x has some element of S, but this
is equivalent tothe general definition.
===
Subject: Re: Limit
Points> In a metric space what are the limit points of {x}?>
The set {x} has no limit points. To say that y is a limit
point> of {x} is to say that y belongs to the closure of {x}
y, which> is the empty set if y = x and it is equal to {x}
otherwise.> Is the definition different for topological
spaces?> Perhaps that some textbooks say that in a metric
space a point> x is a limit point of a set S if, for every r >
0, the set> B(x,r) x has some element of S, but this is
equivalent to> the general definition.Exactly. However at
Ask-a-Topologist web site, students there actas tho limit
point is same as adherence point, ie closure point.Arggg, are
they getting that errant notion from their text
book?
===
Subject: Re: Limit Points>In a metric space what are
the limit points of {x}?> The set {x} has no limit points. To
say that y is a limit point> of {x} is to say that y belongs
to the closure of {x} y, which> is the empty set if y = x and
it is equal to {x} otherwise.>Is the definition different for
topological spaces?> Perhaps that some textbooks say that in a
metric space a point> x is a limit point of a set S if, for
every r > 0, the set> B(x,r) x has some element of S, but this
is equivalent to> the general definition.> Exactly. However at
Ask-a-Topologist web site, students there act> as tho limit
point is same as adherence point, ie closure point.> Arggg,
are they getting that errant notion from their text
book?Unfortunately, different texts have different definitions
for limit point.Personally, I prefer to use accumulation point,
boundary point,or cluster point (depending on what I mean) and
avoid the ambiguous term.G. A. Edgar
http://www.math.ohio-state.edu/~edgar/
===
Subject: Re: Limit
PointsI thought there were inconsistent definitions out there.
For some, a limit point and an accumulation point are the same
thing; for others not. For some, x is a limit point of {x};
for others, not. Am I wrong in this impression?
===
Subject: Re:
Limit Points> I thought there were inconsistent definitions out
there. For some, a > limit point and an accumulation point are
the same thing; for others > not. For some, x is a limit point
of {x}; for others, not. Am I > wrong in this
impression?Perhaps not. BTW, did you see the definition of
limit point atMathWorld?
See:http://mathworld.wolfram.com/LimitPoint.htmlIt has only
two sentences. The first one gives the same definition
formetric spaces that I gave in my previous post. But the
second one, forgeneral topological spaces, seems to be in
contradiction with the firstone; I write seems because the
expression open set around it looksambiguous to me (but then,
I may be wrong; english is not even my
secondlanguage).
===
Subject: Re: Limit PointsOK, I have
surveyed thre analysis and topology books (list of the nine of
them available on request) on my shelf. Some of these limit
themselves to Euclidean space and others to metric spaces; the
remainder refer to general topological spaces.All texts that
mention the terms say that an accumulation point, also called
a cluster point, is a point approached by a sequence of points
that differ from the point in question. All but two also define
a limit point as the same thing. Amongst the other two, one
does not mention the term limit point at all. The other,
Goldberg, Methods of Real Analysis, defines a limit point as
one approached by *any* sequence of points in the set.
Goldberg limits his discussion to metric spaces.Thus, for
Goldberg, x is a limit point, but not a cluster point, of {x}
For the others, in a T1 space, no point is a limit point of
{x}.
===
Subject: Descending setsLet { A_xi } be a descending
transfinite series of subsets ofA = A_0 with the property for
all eta, xi < eta, A_eta subset A_xiLet |A| = omega_nu and mu
= omega_(nu+1)Without loss of generality we can not have A_eta
= /{ A_xi | xi < eta }for limit ordinals eta? / = cap =
intersectionShow A_mu = A_(mu+1).Often there's some eta < mu
with A_eta = A_(eta+1).Dare one conjecture that will always be
possible?In particular if I and the transcendent transfinite
spirit of Zenostubbornly persist to remove single elements
from A, how soon will it beemptied? At A_mu or before?Then
after lunch, one by one we'll put the elements back and for
dinnerconservation, talk about how we reversed an ordinal.
Tomorrow we'll usethis process to smuggle a set across the
universe. ;-)----
===
Subject: Re: Formal Languages Question
<c5oo13$1m2a$1@ulysses.noc.ntua.gr>
<c5opsu$mdp$1@ares.cs.utwente.nl> Now I am having trouble
expressing the language S->aSSb | e with an> expression such
as a^n b^k so that i can use the lemma.> I doubt whether you
need the whole language (as a set defined by some> predicate).
Is some useful characteristic, single sentence such as> e.g.,>
z = a^k b^k a^k b^k> (where k is the number provided by the
pumping lemma) not enough?> Of course, you have to show that z
can be generated by S->aSSb | e.> But that is not a serious
problem.Not serious? Then you make jest?Nay, show how to
produce a^k b^k a^k b^k.
===
Subject: Re: Formal Languages
QuestionI just found out that the language i gave is actually
the language of balanced parenthesis. So the question now is
to prove that the language of balanced parenthesis is not a
linear cfl one.
===
Subject: Show that a polynimial is
reducible. How?i have two question: 1 - show that 2 x^9 -2 x^6
is reducible in Z5 2 - show that x^3 + 2 x^2+ x +2 is reducible
in Z5[x]does anyone know how to solve these to question
?/Peter
===
Subject: Re: Show that a polynimial is reducible.
How?1 - 2x^9-2 x^6 = (2 x^6) (x^3-1)2 - x^3+2
x^2+x+2=(x^2+1)(x+2)Bye.
===
Subject: Factorisation!I'm very
interessted in factorisation algorithms, but as pupil
inhighschool it's very difficult to get information to this
topic Iunderstand. Can somebody explain following algorithms
to me:[Eth] Brent algorithm[Eth] Pollard method[Eth] Williams
method[Eth] Lenstra algorithm[Eth] Quadratic sieve method[Eth]
...If somebody has source code [C,...] I would like to get
it.
===
Subject: Re: Factorisation> !> I'm very interessted in
factorisation algorithms, but as pupil in> highschool it's
very difficult to get information to this topic I> understand.
Can somebody explain following algorithms to me:On top of
Riesel and C&P, as mentioned by others, you'll wantDavid
Bressoud, Factorization and Primality TestingDonald Knuth, The
Art of Computer Programming, Vol 2. Seminumerical Algorithms> ?
Brent algorithm'Rho'? Best covered in Riesel. C&P and Knuth OK
too.> ? Pollard method'P-1'? Well covered in all of them.> ?
Williams method'P+1'? Best covered in C&P> ? Lenstra
algorithm'ECM'? Best covered in C&POne hint - become fluent
with P-1 first, and P+1 and ECM will simply drop out as
logical progressions. > ? Quadratic sieve methodBest covered
in Bressoud and C&P> ? ...C&P and Riesel.> If somebody has
source code [C,...] I would like to get it.There's pseudocode
in all of them.However, get Mike Scott's Miracl bignum library
as it has fairly competant C implementations of all of the
above. www.indigo.ie? They aren't just good implementations,
they are ones that you can learn from -- totally lucid
code.Phil1st bug in MS win2k source code found after 20
minutes: scanline.cpp2nd and 3rd bug found after 10 more
minutes: gethost.cBoth non-exploitable. (The 2nd/3rd ones
might be, depending on the CRTL)
===
Subject: Re: Factorisation>
!> I'm very interessted in factorisation algorithms, but as
pupil in> highschool it's very difficult to get information to
this topic I> understand. Can somebody explain following
algorithms to me:> .89 Brent algorithm> .89 Pollard method>
.89 Williams method> .89 Lenstra algorithm> .89 Quadratic
sieve method> .89 ...> If somebody has source code [C,...] I
would like to get it.A good book is Prime Numbers and Computer
Methods for Factorization by Hans Riesel,which covers all these
I believe. There is also a more recent bookPrime Numbers A
Computational Perspective by Crandall and Pomerance, which is
excellent.Chris
===
Subject: Re: FactorisationBTW, I am sure
that your high school library could easily get these booksfor
you through an inter-Library loan, and Librarians usually love
helpingwith this sort of thing.> !> I'm very interessted in
factorisation algorithms, but as pupil in> highschool it's
very difficult to get information to this topic I> understand.
Can somebody explain following algorithms to me:> . Brent
algorithm> . Pollard method> . Williams method> . Lenstra
algorithm> . Quadratic sieve method> . ...> If somebody has
source code [C,...] I would like to get it.> A good book is
Prime Numbers and Computer Methods for Factorization by> Hans
Riesel,> which covers all these I believe. There is also a
more recent bookPrime Numbers A Computational Perspective by
Crandall and Pomerance,> which is excellent.> Chris
===
Subject:
Re: who is captain in here?hot-girl (math2050@yahoo.co.kr)asked
one of the hottest questions, not only for sci.math. :who is
captain in here? where is sci.math server? who is a
administrator? I wonder about how to operate sci.math.here is
miraculousIt's question is still not properly answered, and
important for all ofus !What made me angry - and frightened,
was the letter frommensanator@aol.com.But first: hot-girl - is
it a girl, or a boy, does this matter?(I like science fiction,
so just a thought, may be it's acomputer-being,feeling lonely,
asking :anybody there ...??, no reply - Sorry, iwasn't online
that time.)It thinks, all other beings in sci-math. are male,
writing:Sir...perfect genius...Female contributed much to math
and science, often propagated by malesas their own ideas.(It's
now under investigation, what the mother ofEinsteins first
child gave to him - and what he made out of this, andwhat he
did to her and his child).hot-girl, You are one of my
teachers, You teach and Your questionsare even more educative
for me. (My level: after more then 30 years ijust finished the
basics of complex numbers, by finding that theimaginary axis is
only another word for the second axis or y-axis,that there is
no mathematical definition for it, it's not more and notless
imaginary than the first or x-axis. So now i proceed
toelementary functons in 2D. I'm like a snail reaching the
first flooron a ramp of a parking garage). You are ahead of
me, but may You canlearn from me too, as i learned some basic
facts.What i know is, that math should never lead to
numerology, like thismensanator did (he is either bad or a
very poor-soul orboth).Mensanator is ignorant and uneducated.
He wants to impress us orto frighten us with his words about a
beast and the number 666.He doesn't know, that 6 is a lucky
number for many sin-people.In the western sphere, You have the
bible, the Christians have asecond part , the New Testament and
in its last chapter, theApocalypsis Ioannis
(http://apokalypse.de.ki) there is this beast= one
manifestation of the devil or an evil person with
supernaturalpowers. You can compare it to the white-bone-ghost
in Sun-Wu-Kung(sometimes a beautiful young girl and sometimes a
death-bringingsceleton). And 666 is the name of the beast
encoded in a number.Inthis chapter is a lot of numerologie.
Would You be impressed, ifsomeone tells You, that the number
in Your address: 2050 has themeaning of the year, when You are
going to die ?No, that's rubbish! Nobody can know these things.
May be its just theyear, in which some of your question are
answered.When people can not talk free, they use encoded
language. Like with I Ging a military-general can give his son
information, how toarrrange troops in the next battle.One can
frighten people, who never learned to count, with numbers.Most
probably women invented counting, by associating their menses
tothe moon-cycles and to the cycle of the year - counting up
to twelveor one further on, and predicting the birth of their
babys by countingup to nine moon-cycles from conception. With
math one can see into thefuture. It's not a bit miraculous for
us any more.administrator, I was helping a guy with an e-mail
problem and noticedhe used the salutation chow. Sometimes
there just aren't anyexplanations. And Stephen replied: I
think the explanation for thatnose high up :Mensanator ( now
under: aol.compost)Yeah, i meant an explanation for how
someone can be that dumbWhat kind of esteem this man (or
woman) must have of itself ?May be in some languages chow has
an explanation.Most people are very kind, just like George
Cox.Now i challenge You, mensanator !An admin will know, if
there is any captain on top of him, so tellus(may be it's a
white Lady?). Considering the importance of theinternet, i
would expect that leading figures in powerfulorganisations
would like to have power over the net, or really haveit, an
admin should know.And what is meant by you have to except the
mark of the beast on yourhead or your hand ? - just writing
nonsense ?! Why one has to disable a sandbox-protection of his
computer (like finjan-safe-zone) inSometimes there just aren't
any explanations. But at other times,there are.And now is the
time to answer. I give mensanator 24+1 hour for it.(and may be
rickO and A,N.Niel can tell us, what they mean by talkingabout
a fire-theater , not every english-writing persons isfamiliar
with Your personal education).I challenge mensanator - if he
doesn't respond, i'll reveal somemathematical truth about the
Ace of clubs, he is using like atitle.Just like his pseudonym
mensanator:In Rom a senator was one of the older and supposed
to be wiserpeople and with some powers to censorship and
control, even about warand peace. (Transfer this thought to an
admin of the net). And menssana in corpore sano is an latin
expression, meaning a healthy soulin a healthy body.Up to now
mensanator doesn't seem to live up to his choosen name, maybe
he is not allowed to, may be he is a very poor soul. Even the
adminof North koreahttp://www.dprkorea.com is more free,
opening up to us a bit, allowing his people to read AnneFrank,
even keeping his name as second son (and showing in this
waythat a number in a hierarchie or family is not a burden for
hispersonality)- and fulfilling his duty of preserving the
preciousbiosof his peoples Internet from flashing with
US-software. Doesmensanator have a name (sometimes there are
reasons for anonymposting, i respect it) or is he lacking the
powers to reveal it to us,this powerful admin?Does mensanator
really wants to be someone, who helps, does hereally have this
capacity? Mensanator, this hot-girl deserves betteranswers, be
miraculous.....please.....Hero
===
Subject: Re: who is captain
in here?
===
>Subject: Re: who is captain in here?>hot-girl
(math2050@yahoo.co.kr)>asked one of the hottest questions, not
only for sci.math. :who is captain in here? where is sci.math
server? >who is a administrator? >I wonder about how to
operate sci.math.>here is miraculous>It's question is still
not properly answered, and important for all of>us !>What made
me angry - and frightened, was the letter
from>mensanator@aol.com.?>But first: hot-girl - is it a girl,
or a boy, does this matter?>(I like science fiction, so just a
thought, may be it's a>computer-being,>feeling lonely, asking
:anybody there ...??, no reply - Sorry, i>wasn't online that
time.)>It thinks, all other beings in sci-math. are male,
writing:Sir...perfect genius...>Female contributed much to
math and science, often propagated by males>as their own
ideas.(It's now under investigation, what the mother
of>Einsteins first child gave to him - and what he made out of
this, and>what he did to her and his child).>hot-girl, You are
one of my teachers, You teach and Your questions>are even more
educative for me. (My level: after more then 30 years i>just
finished the basics of complex numbers, by finding that
theimaginary axis is only another word for the second axis or
y-axis,>that there is no mathematical definition for it, it's
not more and not>less imaginary than the first or x-axis. So
now i proceed to>elementary functons in 2D. I'm like a snail
reaching the first floor>on a ramp of a parking garage). You
are ahead of me, but may You can>learn from me too, as i
learned some basic facts.>What i know is, that math should
never lead to numerology, like this>mensanator did (he is
either bad or a very poor-soul or>both).Or neither.>Mensanator
is ignorant and uneducated. Neither. Check out
http://members.aol.com/mensanator>He wants to impress us or>to
frighten us with his words about a beast and the number 666.Or
maybe entertain. You didn't consider that possibility, did
you?>He doesn't know, that 6 is a lucky number for many
sin-people.And you call _me_ the numerologist? While you're on
my Home Page,be sure to check out The Joy of Six.>In the
western sphere, You have the bible, the Christians have
a>second part , the New Testament and in its last chapter,
theApocalypsis Ioannis (http://apokalypse.de.ki) there is this
beast>= one manifestation of the devil or an evil person with
supernatural>powers. You can compare it to the
white-bone-ghost in Sun-Wu-Kung>(sometimes a beautiful young
girl and sometimes a death-bringing>sceleton). And 666 is the
name of the beast encoded in a number.In>this chapter is a lot
of numerologie. Would You be impressed, if>someone tells You,
that the number in Your address: 2050 has the>meaning of the
year, when You are going to die ?I would be very depressed if
I knew I was going to live that long.>No, that's rubbish!
Whew!>Nobody can know these things. May be its just the>year,
in which some of your question are answered.>When people can
not talk free, they use encoded language. Like with I Ging a
military-general can give his son information, how to>arrrange
troops in the next battle.>One can frighten people, who never
learned to count, with numbers.Good. That gives them
motivation to learn, doesn't it?>Most probably women invented
counting, by associating their menses to>the moon-cycles and
to the cycle of the year - counting up to twelve>or one
further on, and predicting the birth of their babys by
counting>up to nine moon-cycles from conception. With math one
can see into the>future. It's not a bit miraculous for us any
more.>administrator, I was helping a guy with an e-mail
problem and noticed>he used the salutation chow. Sometimes
there just aren't any>explanations. And Stephen replied: I
think the explanation for that>nose high up :>Mensanator ( now
under: aol.compost)Yeah, i meant an explanation for how someone
can be that dumb>What kind of esteem this man (or woman) must
have of itself ?>May be in some languages chow has an
explanation.Maybe since I actually knew the person, I knew
what his intended use was.>Most people are very kind, just
like George Cox.>Now i challenge You, mensanator !>An admin
will know, if there is any captain on top of him, so tell>us
(may be it's a white Lady?).Just because I was a network
administrator at some unspecified periodin the past doesn't
mean I am still a network administrator. I havegone on to more
important and productive work.>Considering the importance of
the>internet, i would expect that leading figures in
powerful>organisations would like to have power over the net,
I used to tell people that I had god-like powers over time and
space,but that was just in reference to my local network. The
Internet wasbeyond my domain.>or really have it, an admin
should know.>And what is meant by you have to except the mark
of the beast on your>head or your hand ? - just writing
nonsense ?! Ok, I'll explain it so even you can understand
it.The story goes that you must have the mark of the beast
(666) toengage in transactions. It isn't soecifically stated
what this mark actually is. Now assume that a transaction is a
math question you wantanswered. Your question must be addressed
to sci.math if you want itanswered. As I've shown, scimath=666,
so you must stamp your postingswith the mark of the beast. Was
that so hard to figure out?>Why one has to disable> a
sandbox-protection of his computer (like finjan-safe-zone)
inSometimes there just aren't any explanations. But at other
times,>there are.>And now is the time to answer. I give
mensanator 24+1 hour for it.Or what?>(and may be rickO and
A,N.Niel can tell us, what they mean by talking>about a
fire-theater , not every english-writing persons is>familiar
with Your personal education).>I challenge mensanator - if he
doesn't respond, i'll reveal some>mathematical truth about the
Ace of clubs, he is using like a>title.Go ahead. I'm sure your
lame attempt at humor won't compare
tohttp://members.aol.com/mensanator666/2ofclubs/2ofclubs.htmI
haven't had a chance to update that page since being promoted
fromTwo to Ace (for the amazing feat of getting two usenet
trolls ( Harris and T_o_m_P_o_t_t_e_r) to foam at the mouth in
one day!>Just like his pseudonym mensanator:>In Rom a senator
was one of the older and supposed to be wiser>people and with
some powers to censorship and control, even about war>and
peace. (Transfer this thought to an admin of the net). And
mens>sana in corpore sano is an latin expression, meaning a
healthy soul>in a healthy body.Instead of wildly wrong
speculation, you could simply ask:mensanator - n. Old Norse 1.
slayer of the MensaIn the autumn of 1996, the Mensa were
holding theirannual All Hallows Eve Festival, when,
unbeknownstto them, a traitor in their midst raised the
portcullis of their fortress and allowed a Vikingraiding party
to slip in, whereupon they slew theentire host with the jawbone
of an ass and carriedoff the great prize.Upon the great warrior
chieftan, Paul the Berserk, was bestowed the title Mensantor in
honor of that great victory.>Up to now mensanator doesn't seem
to live up to his choosen name, may>be he is not allowed to,
may be he is a very poor soul. Stupid conclusion based on
false premise.>Even the admin of North korea
http://www.dprkorea.com >is more free, opening up to us a bit,
allowing his people to read Anne>Frank, even keeping his name
as second son (and showing in this way>that a number in a
hierarchie or family is not a burden for his>personality)- and
fulfilling his duty of preserving the preciousbios>of his
peoples Internet from flashing with US-software.
Does>mensanator have a name Paul the Berserk.>(sometimes there
are reasons for anonym>posting, i respect it) or is he lacking
the powers to reveal it to us,>this powerful admin?Didn't you
just get done saying you respect why some people don't wantto
have their real names posted? I have enough trouble with spam
to myemail address (which is why you see some messages labeled
.compost.Do I have to spell it out that .compost is a feeble
attempt at spamblocking?)>Does mensanator really wants to be
someone, who helps, does he>really have this capacity? Try
doing a Google search on thank+mensanator.>Mensanator, this
hot-girl deserves better answers, So if mensanator doesn't
deserve respect for using a psuedonym,why should hot-girl
_deserve_ better answers? And did I preventanyone else from
answering?>be miraculousI was. You wouldn't have been able to
figure out how to derive666 from
scimath.>.....please.....>HeroIs that you real
name?
===
Subject: Re: who is captain in here?> hot-girl
(math2050@yahoo.co.kr)> asked one of the hottest questions,
not only for sci.math. :who is captain in here? where is
sci.math server? > who is a administrator? > I wonder about
how to operate sci.math.> here is miraculous> It's question is
still not properly answered, and important for all of> us !>
What made me angry - and frightened, was the letter from>
mensanator@aol.com.WHOOSH!Wayne Brown (HPCC #1104) | When your
tail's in a crack, you improvisefwbrown@bellsouth.net | if
you're good enough. Otherwise you give | your pelt to the
trapper.e^(i*pi) = -1 -- Euler | -- John Myers Myers,
Silverlock
===
Subject: Re: who is captain in here?===>Subject:
Re: who is captain in here?>Message-id:
<Wobgc.39968$UC4.38756@bignews2.bellsouth.net> hot-girl
(math2050@yahoo.co.kr)> asked one of the hottest questions,
not only for sci.math. :who is captain in here? where is
sci.math server? > who is a administrator? > I wonder about
how to operate sci.math.> here is miraculous> It's question is
still not properly answered, and important for all of> us !>
What made me angry - and frightened, was the letter from>
mensanator@aol.com.>WHOOSH!I guess I was a fool for biting on
the troll bait, but at least myreply allowed me to engage in a
little shameless self promotion.>-- >Wayne Brown (HPCC #1104) |
When your tail's in a crack, you
improvise>fwbrown@bellsouth.net | if you're good enough.
Otherwise you give> | your pelt to the trapper.e^(i*pi) = -1
-- Euler | -- John Myers Myers, Silverlock
===
Subject: Re:
Resistance to Change>Please substantiate your assertion that
the Boyer-Moore paper formally>expresses a proof of the
unsolvability of the halting problem. Please>make your
explanation self-contained and contain at least a little
bit>of detail e.g. what are the formal axioms, rules of
inference and>theorem, and the steps that lead from the axioms
to the theorem?> I'm beginning to suspect that this is all a
big troll - you're making> a fool of yourself just for fun. If
not you'd read the friggin paper> before asking questions like
this.>Charlie Volkstorf> ************************>Mr.
Ullrich,>I am very interested in the subject of axiomaic
systems and also the>halting problem. I have read here that
several times Charley posted a>link to his paper that shows an
axiomatic proof of this theorem and>several other theorems. He
also seems to have copied parts of it into>messages and given
additional explanation.>Your answer seems to be that other
people have already done that, and>someone posted the above
paper. I looked at that paper, and all I see>are a lot of LISP
programs and I don't really see a proof of the>halting problem.
Charlie seems to be saying the same thing, that this>paper
doesn't do what its title says.>I would like to ask if you
could please explain a little how these>other people you talk
about were able to create an actual proof of>this theorem.
Charlie asked you for the axioms and rules and the>proof
itself, but your only answer seems to be that it is silly and
he>is being foolish.>However, he has laid out his axioms and
rules, and given at least one>proof, and talked about several
others (as opposed to just the halting>problem proof.) I
looked at his paper and system and it seems valid. >Everything
is spelled out, although the syntax is not completely>familiar.
But it is still explained in the paper and a little in
his>postings, although he tends to be a little scattered in
his>explanations in postings.>I am asking if you could explain
(1) the axioms, rules, etc that you>say other people already
did, and (2) what is wrong with Charley's>proofs. In the
interest of an intelligent exchange of ideas on logic,>I think
that is a reasonable request. Again, I don't see any
actual>proof or axioms etc in the above paper, so maybe you
can let us all>know what they are.>The only explanation that I
can think of is that you really don't have>an answer and
instead just call it foolish. But that is no answer.>You have
said at least once or more that you think that Charley
is>being foolish. But so far, he has laid out his work and you
have>refused to explain your answers of how it was done
elsewhere and what>is wrong with his work. It seems to me that
he is not the one who is>being foolish at all.When I read this
I immediately suspected that you must be Charley,just because
the idea that there could be two people in the worldso
confused about things that are so utterly simple seemed
implausible. Someone else has noted that the two of you
havethe same IP address...On the off chance<giggle> that
you're a real person:Charley is being incredibly foolish when
he suggests thatthere's some problem with the standard proofs
of theunsolvability of the halting problem. Astonishingly
foolishwhen he suggests that nobody until him ever _thought
of_formalizing the proof, and hilariously foolish when hesays
that in fact nobody has done so, in the face ofdirect evidence
to the contrary (you just see a bunchof Lisp code in that
paper? Huh. That's very curious,that the paper would consist
of a bunch of code, giventhat the aim of the paper is to
present a program thatconstructs a certain proof...)No, I
haven't stated any of the things you say I haven'tstated.
That's because all those things are both verysimple and very
well-known. I mean it _is_ utterly simple - I haven't given a
referenceto the proof because I don't know that I've ever
_read_the proof - I don't work in this field, many years ago
whenI read about the problem I had no problem finding theproof
myself, so I never tried to look it up. Maybe thatmeans I'm an
amazing genius - I don't think that's whatit means, I think it
means that the proof is more or lessobvious if you're familiar
with the idea of how diagonalarguments work.But to show that
the proof _is_ well-known, I triedit because it might not
contain a proof, and thesecond hit looked like it contained a
proof. And sure enough it
did:http://www.cs.pitt.edu/~kirk/cs1510/notes/halt/
halt.htmlsays<quoteThe Halting Problem is: INPUT: A string P
and a string I. We will think of P as a program. OUTPUT: 1, if
P halts on I, and 0 if P goes into an infinite loop onI.
Theorem (Turing circa 1940): There is no program to solve the
HaltingProblem. Proof: Assume to reach a contradiction that
there exists a programHalt(P, I) that solves the halting
problem, Halt(P, I)returns True if and only P halts on I. The
given this program for theHalting Problem, we could construct
the followingstring/code Z: Program (String x)If Halt(x, x)
then Loop ForeverElse Halt.End.Consider what happens when the
program Z is run with input Z Case 1: Program Z halts on input
Z. Hence, by the correctness of theHalt program, Halt returns
true on input Z, Z. Hence,program Z loops forever on input Z.
Contradiction. Case 1: Program Z loops forever on input Z.
Hence, by the correctnessof the Halt program, Halt returns
false on input Z, Z.Hence, program Z halts on input Z.
Contradiction. End Proof. </quoteThat's a complete, correct,
utterly simple proof of the unsolvabilityof the halting
problem. And a very well-known proof to boot.Charley seems to
think that the proof above is somehow deficientbecause it's
not expressed in a formal language. This is simplystupid. If
he wants to say that writing the proof in a formal language
would be interesting he can say that. But suggesting,as he has
many times, that the informal proof is not a realproof just
because it's written in English is, well utterly stupidis the
only phrase that springs to mind.(Hmm, looking at the proof I
might quibble with the notationused: Halt is the name of the
hypothetical procedure thatwe're showing does not exist and
also the name of aninstruction in the programming language.
I'd change thename of the procedure to Halts:Program (String
x)If Halts(x, x) then Loop ForeverElse Halt.End.Wow, I've
discovered a problem with the standard proofof the
unsolvability of the halting problem. I better writea paper on
this. Guffaw.(Uh, note that the problem is not necessarily a
problem;in some languages different procedures can have
thesame name if they have different signatures. But
Halts()seems like a better name for the procedure
anyway.))>sincerely,
===
Subject: Re: Resistance to ChangeMr.
Ullrich,appreciate it.> When I read this I immediately
suspected that you must be Charley,> just because the idea
that there could be two people in the world> so confused about
things that are so utterly simple seemed > implausible.No, it's
because there aren't two people as smart! Proof: Nobody elsehas
axiomatized the theory of computation (or program synthesis
ordatabase query processing.)> Someone else has noted that the
two of you have the same IP address...Yikes! Oh No! Someone
must have stolen my IP address without myknowledge or
permission. Must've been when I used the computer arKinko's. >
Charley is being incredibly foolish when he suggests that>
there's some problem with the standard proofs of the>
unsolvability of the halting problem.Never said that.>
Astonishingly foolish> when he suggests that nobody until him
ever _thought of_> formalizing the proof,Didn't say that
either. You said that earlier (suggested that I saidit.)> and
hilariously foolish when he says that in fact nobody has done
so,suggesting that you weren't so sure.)> in the face of>
direct evidence to the contrary (you just see a bunch> of Lisp
code in that paper? Huh. That's very curious,> that the paper
would consist of a bunch of code, given> that the aim of the
paper is to present a program that> constructs a certain
proof...)Does this paper formally prove the unsolvability of
the HaltingProblem? Then where's the proof? What are the
axioms, rules ofinference and steps in the proof? That is what
you have notsubstantiated.> No, I haven't stated any of the
things you say I haven't> stated. That's because all those
things are both very> simple and very well-known. Yes, the
proof is fairly simple. BTW: Still, it's not simple enoughfor
Gregory Chaitin to understand it.
Seehttp://www.cs.auckland.ac.nz/CDMTCS/chaitin/amsci.pdf top
of lastcolumn on the 5th. page (# 168). He still hasn't gotten
it right,after writing about it (over and over again) for 30
years! (Who seeshis mistake?)> Charley seems to think that the
[standard] proof is somehow deficient> because it's not
expressed in a formal language. This is simply> stupid. If he
wants to say that writing the proof in a formal > language
would be interesting he can say that.1. Is it better to
axiomatize a theory than to carry it out by hand? (What did
Hilbert say about it?)2. What are the advantages of formal
methods?3. Who has done it before?
http://citeseer.ist.psu.edu/boyer82mechanical.html ? No, there
isno proof: no axioms, rules of inference or derivation of the
theorem.4. How about the idea of reducing Peano's 5 axioms to
a single simplewff that is demonstrably equivalent? Would that
be an improvement? And what if that simple wff turned out to be
one of only a handful ofaxioms that are used to generate
theorems from the theory ofcomputation, such as Turing 1937?
Would that be kind of cool, tounite the work of Peano and
Turing into one formal system?5. What if this system also
formalized (generated) theorems fromGodel's 1931 paper
concerning recursive functions and relations (thatnobody has
ever done before) - would that be of value?6. How about
journals that have published attempts to formalize
theseresults, but were only informal  do you think they should
publish apaper IF it did indeed formalize them?7. Does my paper
formalize (axiomatize) these
results?http://www.arxiv.org/html/cs.lo/0003071http://
www.mathpreprints.com/math/Preprint/CharlieVolkstorf/
20021008.1/18. How about if the same rules of inference were
used to synthesizecomputer programs that compute functions
from Number Theory and alsoDatabase Query Processing? Would
that be useful? Who has done thatbefore?> But suggesting,> as
he has many times, that the informal proof is not a real>
proof just because it's written in English is, well utterly
stupid> is the only phrase that springs to mind.I never said
that. (Where?)The $100.00 is for anyone who can show that
someone else axiomatizedthe Theory of Computation (or show
that my paper doesn't)! That iswhat is unsubstantiated.Charlie
Volkstorf
===
Subject: Re: Resistance to Change> That's a
complete, correct, utterly simple proof of the unsolvability>
of the halting problem. [...]I would prefer the following
(slight) variant
though:-------------------------------------------------------
------The Halting Problem is: INPUT: A string P and a string
I. We will think of P as a program withinput I. OUTPUT: 1 (or
True), if P halts on input I, and 0 (or False) if P goesinto
an infinite loop on input I. Theorem: There is no program to
solve the Halting Problem. Proof: Assume to reach a
contradiction that there exists a programHalt(P, I) that
solves the halting problem, Halt(P, I) returns True ifand only
if P halts on input I, False otherwise. Then given this
programfor the Halting Problem, we could construct the
following string/code Z: Program (String x) If Halt(x, x) then
Loop Forever Else Halt. End.Consider what happens when the
program Z is run with input Z.Case 1: Program Z halts on input
Z. Hence, by the correctness of theHalt program, Halt returns
True on input Z, Z. Hence, program Z loopsforever on input Z.
Contradiction. Case 2: Program Z loops forever on input Z.
Hence, by the correctnessof the Halt program, Halt returns
False on input Z, Z. Hence, program Zhalts on input Z.
Contradiction. End Proof.
------------------------------------------------------I think,
it's important to note that the solution does not depend on
aspecific program language used for program Z (and/or P). It
should justbe the same language the program Halt(P,I) assumes
for its input P, Iguess.> Hmm, looking at the proof I might
quibble with the notation> used: Halt is the name of the
hypothetical procedure that> we're showing does not exist and
also the name of an> instruction in the programming
language.Hmmm... Actually, not really a problem. 'Halt' and
'Halt(.,.)' *should*certainly denote different
functions/procedures/commands (whatever),since 'Halt' doesn't
take any arguments as input, but 'Halt(.,.)' takestwo
arguments as input.On the other hand, we might certainly as
well adopt some differentnotation. Another variant: Program
(String x) If Halt(x, x) then Loop Forever Else Stop. End.-
with the comment: The command/instruction 'Stop' just halts
theprogram.F.P.S.Turing's original argumentation a good deal
more complicated, and muchharder to understand.
===
Subject: Re:
Resistance to
Change>[...]>http://www.cs.pitt.edu/~kirk/cs1510/notes/halt/
halt.html>says><quote>The Halting Problem is: >INPUT: A string
P and a string I. We will think of P as a program. >OUTPUT: 1,
if P halts on I, and 0 if P goes into an infinite loop on>I.
>Theorem (Turing circa 1940): There is no program to solve the
Halting>Problem. >Proof: Assume to reach a contradiction that
there exists a program>Halt(P, I) that solves the halting
problem, Halt(P, I)>returns True if and only P halts on I. The
given this program for the>Halting Problem, we could construct
the following>string/code Z: >Program (String x)>If Halt(x, x)
then> Loop Forever>Else Halt.>End.>Consider what happens when
the program Z is run with input Z >Case 1: Program Z halts on
input Z. Hence, by the correctness of the>Halt program, Halt
returns true on input Z, Z. Hence,>program Z loops forever on
input Z. Contradiction. >Case 1: Program Z loops forever on
input Z. Hence, by the correctness>of the Halt program, Halt
returns false on input Z, Z.>Hence, program Z halts on input
Z. Contradiction. >End Proof. ></quote>That's a complete,
correct, utterly simple proof of the unsolvability>of the
halting problem. And a very well-known proof to boot.>Charley
seems to think that the proof above is somehow
deficient>because it's not expressed in a formal language.
This is simply>stupid. If he wants to say that writing the
proof in a formal >language would be interesting he can say
that. But suggesting,>as he has many times, that the informal
proof is not a real>proof just because it's written in English
is, well utterly stupid>is the only phrase that springs to
mind.>(Hmm, looking at the proof I might quibble with the
notation>used: Halt is the name of the hypothetical procedure
that>we're showing does not exist and also the name of
an>instruction in the programming language. I'd change
the>name of the procedure to Halts:>Program (String x)>If
Halts(x, x) then> Loop Forever>Else Halt.>End.>Wow, I've
discovered a problem with the standard proof>of the
unsolvability of the halting problem. I better write>a paper
on this. Guffaw.>(Uh, note that the problem is not necessarily
a problem;>in some languages different procedures can have
the>same name if they have different signatures. But
Halts()>seems like a better name for the procedure
anyway.))Oops. I just noticed another problem with the proof
above.Both cases are labelled Case 1. I have no idea how tofix
this one, looks to me like the proof is simple Wrong.I guess I
have to take back everything I've said.<giggle
===
Subject: Re:
Resistance to Change> sincerely,> Carl ManningHey, you and
Charlie have the same IP address! What an
astonishingcoincidence! You must be posting from the same
computer lab...perhaps you could discuss this brilliant
discovery amongst yourselves?Nathan
===
Subject: Re: Resistance
to Change> sincerely,> Carl Manning> Hey, you and Charlie have
the same IP address! What an astonishing> coincidence! You must
be posting from the same computer lab...> perhaps you could
discuss this brilliant discovery amongst yourselves?> NathanOh
really? More sarcasm without mathematical content! I guess
weboth have each other pegged. :) LOL How about if I offer a
$100.00reward* for anyone who can substantiate (elucidate)
David Ullrich'sclaims? (or find a flaw in my 4 theorems) I
realize that pales incomparison to the Clay Mathematical
Institute rewards - and theproblem seems just as difficult!So
don't worry. I won't hold it against you that you can't find
thatelusive proof that David knows so much about. Nobody else
can either.C & C(separated at birth)* US Postal money order or
PayPal payment!
===
Subject: Re: Resistance to Change>
sincerely,> Carl Manning> Hey, you and Charlie have the same
IP address! What an astonishing> coincidence! You must be
posting from the same computer lab...> perhaps you could
discuss this brilliant discovery amongst yourselves?>
Nathan>Oh really? More sarcasm without mathematical content!
Giggle. You manufacture support from an imaginary playmate,
andwhen someone points this out it has no mathematical
content.>I guess we>both have each other pegged. :) LOL How
about if I offer a $100.00>reward* for anyone who can
substantiate (elucidate) David Ullrich's>claims? You really
have no idea what a gibbering moron this makes yousound like,
suggested that it's _my_ claim that there's no problemwith the
well-known proof of the unsolvability of the halting
problem?See
http://www.cs.pitt.edu/~kirk/cs1510/notes/halt/halt.htmlEither
point out an error in the proof or decide whether tosend the
$100 to me or to the author of the page.>(or find a flaw in my
4 theorems) I realize that pales in>comparison to the Clay
Mathematical Institute rewards - and the>problem seems just as
difficult!>So don't worry. I won't hold it against you that you
can't find that>elusive proof that David knows so much about.
Nobody else can either.>C & C>(separated at birth)>* US Postal
money order or PayPal payment!
===
Subject: Topic for Proof
Treatise: Prove/Disprove Manure Really Stinks. by
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id i3HChMc08048;I have often pondered
the signifcance of Mathematical proofs. I tend to think some
math is true if it has a usefull and duplicable
application(s). Otherwise proofs are just academic
exercises....How Would a mathematician go about
proving/disproving Manure really Stinks. How do you define
Stinks. Or would your really want tobother with a definition.
What kind of definition would be usefull and appropriate. All
definitions I guess are not equal. Probably noneof them. Which
is significant on its own. But then if there was someother
criteria to frame the proof, such as a proposed problem to
solve maybe a proof and defintion(s) would make more
sense....I think Manure really does stink, myself. And I guess
I can think of some proofs that I won't get into. Which should
probably done for some Mathematical proofs I have heard of.Zim
Olsonhttp://www.zimmathematics.com
===
Subject: Re: Topic for
Proof Treatise: Prove/Disprove Manure Really Stinks.>How
>Would a mathematician go about proving/disproving Manure
really >Stinks. Where I come from, the smell of manure is
often refered to as the smell ofmoney. QED, I
suppose.Rich
===
Subject: Re: Topic for Proof Treatise:
Prove/Disprove Manure Really Stinks.
===
>Subject: Topic for
Proof Treatise: Prove/Disprove Manure Really Stinks.>I have
often pondered the signifcance of Mathematical proofs. I tend
>to think some math is true if it has a usefull and duplicable
>application(s). Otherwise proofs are just academic
exercises....How >Would a mathematician go about
proving/disproving Manure really >Stinks. How do you define
Stinks. Or would your really want to>bother with a definition.
What kind of definition would be usefull >and appropriate. All
definitions I guess are not equal. Probably none>of them.
Which is significant on its own. But then if there was
some>other criteria to frame the proof, such as a proposed
problem to >solve maybe a proof and defintion(s) would make
more sense....I >think Manure really does stink, myself. And I
guess I can think of >some proofs that I won't get into. Which
should probably done for >some Mathematical proofs I have
heard of.>Zim Olson>http://www.zimmathematics.comOff Topic.
You should address this to sci.chem.
===
Subject: Re:
Non-Noether Conserved Quantity?P.S. ...I was on the point of
apologizing to you, out of some old time fellowfeeling. But I
note in this thread, Daryl McCullough, a now rare andserious
poster, injects a rare gem of a question in sci.physics,
andyou immediately responded with some nonsense which hijacked
theenergies of Bjoern Feuerbacher -- through his own
inexperience.Even without recognizing your persona I was
disappointed that apromising thread was diverted into nonsense
so quickly.
===
Subject: Re: Non-Noether Conserved Quantity?> In
message <c537n3$r2r$2@news.urz.uni-heidelberg.de>, Bjoern
Feuerbacher > [...]>Why don't you actually answer my questions
for a change, instead of >ignoring most of them, and evading
the rest?> DYOC. Johns Hopkins seem to have let him go.>
Hopkins doesn't let many people go. Since to > begin with,
don't they hire many people who actually> believe in quantum
mechanics. To end with > they work a lot more with > Admiral
Rickover and Company than they do with Einstone.Ohmygod ...
the tone, Johns Hopkins ...Oh. I see Richard Herring was there
first. I had to Google on 'tard,with a confirmatory search on
goober.
===
Subject: Re: Non-Noether Conserved Quantity?>In
message <c537n3$r2r$2@news.urz.uni-heidelberg.de>, Bjoern
Feuerbacher >[...]>Why don't you actually answer my questions
for a change, instead of >ignoring most of them, and evading
the rest?>DYOC. Johns Hopkins seem to have let him go.>
Hopkins doesn't let many people go. Since to > begin with,
don't they hire many people who actually> believe in quantum
mechanics. To end with > they work a lot more with > Admiral
Rickover and Company than they do with Einstone.> Ohmygod ...
the tone, Johns Hopkins ...> Oh. I see Richard Herring was
there first. I had to Google on 'tard,> with a confirmatory
search on goober. It matters less to me what Richard Herring
does, that what a retard like Feynmann does, if you haven't
noticed.
===
Subject: Re: Is calculus wrong?[[ This message was
both posted and mailed: see the To, Cc, and Newsgroups headers
for details. ]]> I feel> that there is an inconsistency here.
What do you think?No, it is quite consistent.Doesn't model the
real world? You could argue that, but doit in sci.physics, not
sci.math ...G. A. Edgar
http://www.math.ohio-state.edu/~edgar/
===
Subject: Re: Is
calculus wrong?> Is our Calculus wrong?> The concept of limits
was visualized using a figure drawn on paper. In> describing
this figure we have used two contradictory notions. On one>
hand we said we are approaching a limit and on other we still
used the> same gross level view of the figure.The 19-th
century was the time when purely visual understanding of
functions, derivatives and integrals were replaced by
rigourous formal definitions of limits. Calculus is helped and
aided by pictures, but the reasoning does not depend on
pictures in any way. One of the jokes we had when I was a kid,
was the German textbook on differential and integral calculus
mit acht figuren.The analysis of real and complex functions is
quite alive and well. It serves physics, engineering, economics
and other disciplines effectively.Bob Kolker
===
Subject: Re: Is
calculus wrong?An isomorphism is when we use one system as an
example of anothersystem. For example, (no pun intended) if I
talk about apples witharithmetic, telling students that since
2 apples and 3 apples make 5apples, to help explain 2+3=5,
this is a useful tool. This is muchlike using geometry, with
its lines and tangents and slopes, toexplain calculus. It
works quite well for the most part. However,remember that
apples are not exactly like numbers: for example,numbers never
rot, or decompose.So isomorphisms, like apples to arithmetic,
work well to explainthings, but we have to remember that they
don't always work in theexact same fashion.> Is our Calculus
wrong?> The concept of limits was visualized using a figure
drawn on paper. In> describing this figure we have used two
contradictory notions. On one> hand we said we are approaching
a limit and on other we still used the> same gross level view
of the figure. When we approach a limit we must> also magnify
the figure to see exactly what is happening. If we> approach
100 times closer to a point we should magnify the figure by>
100 times or a thousand times to see the result. To observe>
microscopic things we need a microscope. It is the same thing
as> saying that we are moving closer and closer to a distant
star and> still thinking that the star will maintain the same
small size. I feel> that there is an inconsistency here. What
do you think? Please let me> know at subhendu.das@excite.com>
Now consider how we proved that the derivative is a tangent.
Imagine> the figure, a circle, and a line OP, starting at O on
the circle,> intersecting at Q and extending beyond to P. If we
keep the point O> fixed and move the point Q closer and closer
to O along the> circumference, then the line OP will move and
become a tangent at O.> We see the same fallacy here; we are
not using a microscope. As the> point Q moves towards O we
must magnify the figure. If you do that> then you will see
that the figure is basically not changing, only> becoming
bigger and bigger, and the line will never become tangent. Do>
you agree with me? Is our calculus wrong? Please let me know
your> opinion at subhendu.das@excite.com
===
Subject: Re: Is
calculus wrong?* Subhendu Das> If you do that> then you will
see that the figure is basically not changing, only> becoming
bigger and bigger, and the line will never become tangent.This
is actual true. The line never becomes tangent. That is why
weuse words like 'approaching' and not 'reaching'.However, as
others have pointet out. Use epsilon-delta if you try
tocritisize limits, not gemoetry.Jon Haugsand Dept. of
Informatics, Univ. of Oslo, Norway, mailto:jonhaug@ifi.uio.no
http://www.ifi.uio.no/~jonhaug/, Phone: +47 22 85 24
92
===
Subject: Re: Is calculus wrong?No.===Subject: Re: Is
calculus wrong?| Is our Calculus wrong?|| The concept of
limits was visualized using a figure drawn on paper. In|
describing this figure we have used two contradictory notions.
On one| hand we said we are approaching a limit and on other we
still used the| same gross level view of the figure. When we
approach a limit we must| also magnify the figure to see
exactly what is happening. If we| approach 100 times closer to
a point we should magnify the figure by| 100 times or a
thousand times to see the result. To observe| microscopic
things we need a microscope. It is the same thing as| saying
that we are moving closer and closer to a distant star and|
still thinking that the star will maintain the same small
size. I feel| that there is an inconsistency here. What do you
think? Please let me| know at subhendu.das@excite.com||| Now
consider how we proved that the derivative is a tangent.
Imagine| the figure, a circle, and a line OP, starting at O on
the circle,| intersecting at Q and extending beyond to P. If we
keep the point O| fixed and move the point Q closer and closer
to O along the| circumference, then the line OP will move and
become a tangent at O.| We see the same fallacy here; we are
not using a microscope. As the| point Q moves towards O we
must magnify the figure. If you do that| then you will see
that the figure is basically not changing,Exsqeeze me?!? Try
actually drawing something like this and then see if you
wantto make this claim. only| becoming bigger and bigger, and
the line will never become tangent. Do| you agree with me? Is
our calculus wrong? Please let me know your| opinion at
subhendu.das@excite.com
===
Subject: Re: 3^n + 130532We can
prove that there are infinitely many 3-Sierpinski numbersin
the same way that we can prove the infinitude of
Sierpinskinumbers. If I am not mistaken, k can be chosen such
that 3^n + k(or k 3^n + 1) is always divisible by one of the
odd prime factorsof 3^48 - 1, namely 5, 7, 13, 17, 41, 73, 97,
193, 577, 769 and6481 (we can even leave out one of 97, 577 and
769). But I did notfind any small numbers with this
property.---J K Hauglandhttp://www.neutreeko.com
===
Subject:
Re: 3^n + 130532> We can prove that there are infinitely many
3-Sierpinski numbers> in the same way that we can prove the
infinitude of Sierpinski> numbers. If I am not mistaken, k can
be chosen such that 3^n + k> (or k 3^n + 1) is always divisible
by one of the odd prime factors> of 3^48 - 1, namely 5, 7, 13,
17, 41, 73, 97, 193, 577, 769 and> 6481 (we can even leave out
one of 97, 577 and 769). But I did not> find any small numbers
with this property.I'll check your working:(set fonts to
sane...) 17 1/16 _ 1/8 193 1/16 / >- 1/4 41 1/8 / >- 1/2 5 1/4
/ 2 from 97 1/48 _ 1/24 _ 1577 1/48 / _ 1/12 /769 1/48 / / _
1/6 / 6481 1/24 / / / 73 1/12 / >- 1/2 / 7 1/6 / 13 1/3 /If
you check mine:a) k==4 (mod 5) places 5 in the n==0 (mod 4)
slotsb) k==9 (mod 41) places 41 in the n==2 (mod 8) slotsc)
k==9 (mod 193) places 193 in the n==6 (mod 16) slotd) k==8
(mod 17) places 17 in the n==14 (mod 16) slotsSo that's all
even powers
covered.a.b.a.c.a.b.a.d.a.b.a.c.a.b.a.d.a.b.a.c.a.b.a.d.e)
k==2 (mod 7) places 7 in the n==1 (mod 6) slotsf) k==12 (mod
13) places 13 in the n==3 (mod 6) slotsg) k==3 (mod 73) places
73 in the n==5 (mod 12) slotsh) k==3 (mod 6481) places 6481 in
the n==11 (mod 24) slotsi) k==3 (mod 97) places 97 in the
n==23 (mod 48) slotj) k==574 (mod 577) places 577 in the n==47
(mod 48)
slotaebfagceafbhaedfagbeafciaebfagdeafbhaecfagbeafdjRun the
CRT to getMod(1581339422187635434, 1620742097628261335)So
1581339422187635434 should work.Double-check:(00:11) gp >
for(i=0,47,print1( ,factor(1581339422187635434+3^i)~[1,1])) 5
577 17 13 5 7 13 73 5 13 193 7 5 23 41 13 5 7 13 73 5 13 41 7
5 79 193 13 5 7 13 73 5 13 17 7 5 6481 19 13 5 7 13 73 5 13 23
7There are a large number of choices -- a was from 4; b,c were
from 2; d was forced; e was from 3; f,g,h from 2; i,j combined
from 6 => total# = 4*2*2*3*2*2*2*6 = 2304 -- therefore I expect
a value <10^15 works.Phil1st bug in MS win2k source code found
after 20 minutes: scanline.cpp2nd and 3rd bug found after 10
more minutes: gethost.cBoth non-exploitable. (The 2nd/3rd ones
might be, depending on the CRTL)
===
Subject: Re: 3^n + 130532>
Is this always composite, does anybody know, please?> PS I'd
like to christen this a 3-Sierpinskiesque number, if it doesn't
> already have a name?> PPS I think I can show that there are
an infinite number of Sierpinski > numbers - does this make
sense/is this new?One thing we can rule out is primes of the
form 6x+13^n + 130532 = 6x+13^n + 130531 = 6x(3^n + 130531)
mod 6 = 4Hence x will never be an integerTherefore them are no
primes of the form 6x+1 in the 3^n + 130531 listUnfortunately,
6x-1 is still open.
===
Subject: Re: 3^n + 130532>very simple)
probably is right...> But why break the habit of a lifetime?
:-)Indeed - all my other proofs are _certainly_ right!
:-)
===
Subject: Re: 3^n + 130532>Is this always composite, does
anybody know, please?>PS I'd like to christen this a
3-Sierpinskiesque number, if it doesn't >already have a
name?>PPS I think I can show that there are an infinite number
of Sierpinski >numbers - does this make sense/is this new?>
again!> I ran a check on it last night using a program called
PrimeForm, and> it found a probable prime with n=17411. This
isn't proving that> 3^17411+130532 is prime, just that it is
probably prime. The only> thing about this number is that it
has 8308 digits(!), so it will be> very difficult to prove it
prime, without resorting to a distributed> computing attack on
it.> AnthonyExcellent, excellent, excellent!You're calculating
way quicker than me - I wouldn't have found this for
yonks!range for Francois Morain's ECPP, so I'll give that a
try and see if I can set it going...J
===
Subject: Re: 3^n +
130532> Is this always composite, does anybody know, please? PS I'd like to christen this a 3-Sierpinskiesque number, if
it > doesn't already have a name?> PPS I think I can show that
there are an infinite number of > Sierpinski numbers - does
this make sense/is this new?> again!> I ran a check on it last
night using a program called PrimeForm, and> it found a
probable prime with n=17411. This isn't proving that>
3^17411+130532 is prime, just that it is probably prime. The
only> thing about this number is that it has 8308 digits(!),
so it will be> very difficult to prove it prime, without
resorting to a distributed> computing attack on it.> Anthony>
Excellent, excellent, excellent!> You're calculating way
quicker than me - I wouldn't have found this for > yonks!>
range for Francois Morain's ECPP, so I'll give that a try and
see if I > can set it going...> JNo :-(, ...Try
Primo...!J
===
Subject: Re: 3^n + 130532> Is this always
composite, does anybody know, please?> PS I'd like to
christen this a 3-Sierpinskiesque number, if it > doesn't
already have a name?> PPS I think I can show that there are an
infinite number of > Sierpinski numbers - does this make
sense/is this new?> again!> I ran a check on it last night
using a program called PrimeForm, and> it found a probable
prime with n=17411. This isn't proving that> 3^17411+130532 is
prime, just that it is probably prime. The only> thing about
this number is that it has 8308 digits(!), so it will be> very
difficult to prove it prime, without resorting to a
distributed> computing attack on it.> Anthony> Excellent,
excellent, excellent!> You're calculating way quicker than me
- I wouldn't have found this for > yonks!> range for Francois
Morain's ECPP, so I'll give that a try and see if I > can set
it going...> J> No :-(, ...> Try Primo...!> JI might recommend
taking a look at the record-length primes that havebeen
computed with Primo. You can find it at this
link:http://www.ellipsa.net/primo/record.htmlGood luck with
your calculations!Anthony
===
Subject: Re: 3^n + 130532>Is this
always composite, does anybody know, please?>PS I'd like to
christen this a 3-Sierpinskiesque number, if it doesn't
>already have a name?>PPS I think I can show that there are an
infinite number of Sierpinski >numbers - does this make
sense/is this new?> !> I have checked for all numbers n below
11000, and all have turned up composite. > Anthony If this
turns out to always be composite, it will be hard to prove
such simply using congruences on the exponent. For example,
when n = 59 we get the factorization 3^59 + 130532 =
p1*p2*p3where p1 = 1587251 p2 = 2413771 p3 =
3688182084440719We can check that p1 divides 3^n + 130532 when
n == 59 (mod 793625 = 5^3 * 7 * 907) p2 divides 3^n + 130532
when n == 59 (mod 804590 = 2 * 5 * 61 * 1319 p3 divides 3^n +
130532 when n == 50 (mod 1229394028146906 = 2 * 3 * 277 *
54907 * 24572009)One of the exponent congruences must be
modulo a multiple of 793625, 805490, or 1229394028136906 .John
Adams served two terms as Vice President and one as President,
but lostreelection. Later his son became President despite
losing the popular vote.That son lost his reelection attempt
badly. Now history is repeating itself.pmontgom@cwi.nl
Microsoft Research and CWI Home: San Rafael,
California
===
Subject: Re: 3^n + 130532> Is this always
composite, does anybody know, please?> PS I'd like to
christen this a 3-Sierpinskiesque number, if it> doesn't
already have a name?> PPS I think I can show that there are an
infinite number of> Sierpinski numbers - does this make
sense/is this new?> !> I have checked for all numbers n below
11000, and all have turned> up composite.> Anthony> If this
turns out to always be composite, it will be hard> to prove
such simply using congruences on the exponent.And in fact,
classical conjectures then say that it is not always
composite.A counter-exemple should appear somewhere for
10000<n<30000, I would say.> For example, when n = 59 we get
the factorization> 3^59 + 130532 = p1*p2*p3> where> p1 =
1587251> p2 = 2413771> p3 = 3688182084440719> We can check
that> p1 divides 3^n + 130532 when n == 59 (mod 793625 = 5^3 *
7 *> 907) p2 divides 3^n + 130532 when n == 59 (mod 804590 = 2
* 5> * 61 * 1319 p3 divides 3^n + 130532 when n == 50 (mod>
1229394028146906 = 2 *> 3 * 277 * 54907 * 24572009)> One of
the exponent congruences must be modulo a multiple of> 793625,
805490, or 1229394028136906 .
===
Subject: Re: What type series
is this?> Using this formula, I can easily prove what I was
trying to prove:> No positional numbering system has one and
only one> representation for each real number.A good result.
It can also be proved using topology...The association from a
representation to the numberis continuous, but the set of
representations istotally disconnected, while the set of
numbers isconnected, so they are not homeomorphic.Doesn't
require positive digits, just that there isa finite choice of
digits for each place.Another interesting one: A positional
numbering systemfor *complex numbers* ... If every number has
at least onerepresentation, then some number has at least 3
differentrepresentations.G. A. Edgar
http://www.math.ohio-state.edu/~edgar/
===
Subject: Weak and
strong derivativeIf f is a function : A -> R , where A is an
open set of R^n, f isC^{infinity}(A), except one point, f is
L^1(A) and the same for itsderivatives, can I say that its
strong partial derivative is also the weakpartial
derivative?
===
Subject: Re: Weak and strong derivative> If f is
a function : A -> R , where A is an open set of R^n, f is>
C^{infinity}(A), except one point, f is L^1(A) and the same
for its> derivatives, can I say that its strong partial
derivative is also the weak> partial derivative?What have you
tried? At first glance the n = 1 looks simple, and the n > 1
case should follow easily.
===
Subject: Re: Curious quote by
A.N.Whitehead on Shakespeare> I was quite young when I first
realized that I understood some books > better than the
teacher, or, worse, that the teacher was merely > parroting
blather currently fashionable in the academic world.> wonder
if you'd care to perform a public act of recall of an example
of > said parroting of blather... none of my highschool
teachers even > approached such a parroting act... unless man
vs man; man vs society; > man vs himself etc... was ever
remotely fashionable?Not until my first year of college for
that one.I'm afraid I've succeeded in repressing the exact
memories altogether.John W. KennedyCompact is becoming
contract,Man only earns and pays. -- Charles Williams. Bors to
Elayne: On the King's Coins
===
Subject: Re: Curious quote by
A.N.Whitehead on ShakespeareOh great! Yet another for our
collection, a LaRouchie!You know, the possibilities for
psychotic synergy between Oxfordians and LaRouchies are simply
astounding, both being obsessed with British aristocracy and
all.John W. KennedyBut now is a new thing which is very
old--that the rich make themselves richer and not poorer,which
is the true Gospel, for the poor's sake. -- Charles Williams.
Judgement at Chelmsford
===
Subject: Re: Curious quote by
A.N.Whitehead on Shakespeareyou mean, PS#1, unless Cambridge
was founded, firstly. Harry Potter goes to a PS; y'know?...
and, ifyou want to know about the Rhodesian brainwashing
programme-- that Bill Clinton did not totally submit to,in my
vague opinion --you can read about the history of that, as
well.> Oh great! Yet another for our collection, a LaRouchie!
--Give Earth a Trickier Dick Cheeny -- out of office, after
GIGA
years.http://www.benfranklinbooks.com/http://www.rand.org/
publications/randreview/issues/rr.12.00/http://
members.tripod.com/~american_almanachttp://www.wlym.com/pdf/
iclc/howthenation.PDF
===
Subject: Re: Curious quote by
A.N.Whitehead on Shakespeare> Hi. HH here.> I've been quite
curious about this curious quote:... To this day I can't read
King Lear, having had> the advantage of studying it accurately
in school.> -ALFRED NORTH WHITEHEADI found the attribution on
the web to Atlantic, v. 138, pg. 197,and lo and behold, The
College of Dupage Library holds this inbound volume. This
would be late 1926, although I forgot tocheck the exact
issue.It's from _THE EDUCATION OF AN ENGLISHMAN_, an essay
ofseven small pages in two columns. He dwells mostly onlocal
lore and mood, and explains the general tenor ofthe classical
education he was afforded by a school inDorsetshire, at
Sherborne, in the 1870's .The context does not amplify on the
gist of this quotation.The only help is in a closely preceding
mention of theirstudy of the Bible in Greek, where he says, In
this Greekpresentation of religion the passion for accurate
philologysometimes overcame the religious interest. Also
immediately preceding, btw : The Platonizing Jewsof Alexandria
are mixed in my mind with monastery buildingsin Dorsetshire on
warm Sunday afterenoons in May.Lew Mammel, Jr.
===
Subject: Re:
Curious quote by A.N.Whitehead on ShakespeareOops, I see Chris
Green beat me to it. Well, no matter.
===
Subject: Re: Curious
quote by A.N.Whitehead on Shakespeare> | > I'm using the
handle bookburn partly because of Bradbury's novel> | >
Fahrenheit 451, the temperature at which books burn. Society>
burned> | > books that interferred with its agenda, and
refugees collected> outside> | > cities, reading to each other
from memory. bb> |> | You are missing either a comma or a major
plot point.> Can you be clearer? I read the novel, saw the
movie, but could be> missing something interesting that
relates to the point about people> learning. Could be commas
set off bookburn and Fahrenheit 451,> but I don't see how to
add a single comma to gain a major plot point.|_All_ books are
burned in the Fahrenheit 451 world. No discrimination.John W.
KennedyGive up vows and dogmas, and fixed things, and you may
grow like That. ...you may come to think a blow bad, because
it hurts, and not because it humiliates. You may come to think
murder wrong, because it is violent, and not because it is
unjust. -- G. K. Chesterton. The Ball and the Cross
===
Subject:
Re: Curious quote by A.N.Whitehead on Shakespeare> hey hey, hi
hi;> Schroedinger's cat will never die.> But the kitty litter>
isn't very dry;> hey hey, hi hiThat which trickles down is
yellow.
===
Subject: Re: Curious quote by A.N.Whitehead on
Shakespeare> |> | > Education with inert ideas is not only
useless: it is, above all> things,> | > harmful - Corruptio
optimi, pessima.> | > | > Do not teach too many subjects, and
again, What you teach,> teach> | > thoroughly.> | > | Teach
with fervent interest only the subjects that enthrall you.>
Don't> | teach subjects for which you lack paramount interest;
your students> will> | learn your boredom, lack of interest and
ineptness.> The argument goes that teaching is different from
learning, especially> with respect to the controversy
surrounding mastery, or> competency-based education.> Those
who can toss stuff like that around have replaced common sense
with> excessive erudition.> (The NEA justifies its position
against the Voucher System on the basis> of the importance of
cross-cultural learning and schools going to> greater lengths
in helping slow learners, disadvantaged, etc..)> To hell with
those pampered dumb brats sucking up scarce school dollars.>
Put your money on the winners, providing them with resources
and> opportunity to learn as they will instead or boring them
to tears with> standard curriculum and wasting their
potential.> Some favor teaching programmable specifics in a
sequential,> non-repetitive, cumulative curricula and tend to
blame mere learning> about warm and fuzzy stuff as responsible
for low national test scores.> Complusuary education is
oxymoron for boot camp training.> Throw the unruly out of
school. But no, now school has become> a population wide
experiment in prescription drugging young kids> into sombies.
More ill health being piled upon ill health.Are we to believe
that you would have been a winner if you hadn'tbeen bored to
tears by the standard curriculum ? There are excellentfree
dictionaries on-line for when you get the urge to use
difficultwords so you don't have to put yourself out on a
limb:http://dictionary.cambridge.org/define.asp?key=56863&dict
=CALD> This dichotomy is represented in The Dead Poets Society
where the> teacher is artful, but the head master is only
interested in measurable> results. The irony is that those
colleges, like St. Johns, emphasizing> reading the Great Books
and discussing in Socratic dialogue, have the> best GRE test
scores, I hear. bb> Indeed, verifying what I
proclaim.
===
Subject: Re: Curious quote by A.N.Whitehead on
Shakespeare>Are we to believe that you would have been a
winner if you hadn't>been bored to tears by the standard
curriculum ?Well, the quality of teaching makes a great deal
of difference. I didvery poorly in mathematics until I had a
new teacher. No miracles, butfrom 'poor' to 'reasonable'. But
if I'd had Teach 2 from Day 1, maybeI'd have got to 'good'.
Who knows?The good news is that nothing is compulsory.The bad
news is that everything is
prohibited.http://www.jmwa.demon.co.uk Also see
http://www.isce.org.uk 
===
Subject: Re: Curious quote by
A.N.Whitehead on Shakespeare>Are we to believe that you would
have been a winner if you hadn't>been bored to tears by the
standard curriculum ?> Well, the quality of teaching makes a
great deal of difference. I did> very poorly in mathematics
until I had a new teacher. No miracles, but> from 'poor' to
'reasonable'. But if I'd had Teach 2 from Day 1, maybe> I'd
have got to 'good'. Who knows?What level was this? College?
How much do you remember now? Is itstill meaningful to you?
Have you heard of Father Guido Sarducci'sFive Minute
University? ( You can learn in five minutes everythingyou will
remember, so why bother with the rest ? )Lew Mammel,
Jr.
===
Subject: Re: Curious quote by A.N.Whitehead on
Shakespeare <EGx0n$DQ9UgAFwbt@jmwa.demon.co.uk>
<4081730A.92062CD4@worldnet.att.netI read in
sci.lang.translation that Lewis Mammelt>) about 'Curious quote
by A.N.Whitehead on Shakespeare', on Sat, 17>Are we to believe
that you would have been a winner if you hadn't>been bored to
tears by the standard curriculum ?> Well, the quality of
teaching makes a great deal of difference. I did> very poorly
in mathematics until I had a new teacher. No miracles, but>
from 'poor' to 'reasonable'. But if I'd had Teach 2 from Day
1, maybe> I'd have got to 'good'. Who knows?>What level was
this? College? Sixth form at grammar school in England.
Equivalent to 11 and 12 gradein US?>How much do you remember
now? Is it>still meaningful to you? Since I'm an electronics
engineer, I still use a lot of it and it hadbeet be
meaningful!>Have you heard of Father Guido Sarducci's>Five
Minute University? No. Sounds a bit Jesuitical.(;-)>( You can
learn in five minutes everything>you will remember, so why
bother with the rest ? )I can't agree with that. After 66
years I'm still learning. But perhapsI'm just that much slower
than the good cleric. (;-)The good news is that nothing is
compulsory.The bad news is that everything is
prohibited.http://www.jmwa.demon.co.uk Also see
http://www.isce.org.uk 
===
Subject: Re: Curious quote by
A.N.Whitehead on Shakespeare... To this day I can't read King
Lear, having had> the advantage of studying it accurately in
school.> -ALFRED NORTH WHITEHEAD> Um, well, is literature
supposed to be read and understood> *accurately*? Is there
only one way to perceive Shakespeare's King> Lear?In the days
of yore, Shakespeare plays were the soap opera of the
day.People went to them not to learn nor study nor analyse but
for enjoymentand gossip about what sordid life the nobles
live.So the approach to literature in school misses altogether
the essence.I would suppose Shakespeare (an obscene pen name
pun?) would be besttaught with a pleasant dose of history of
those times. Not dry politicalhistory, but juicy how people
lived in those days and how they liked theirkings.Truth is
stranger than fiction.That's why history is so much
fun.
===
Subject: Re: Question about set closure in metric
space>self-learning? Is it worth spending $50 - $75 on it
(Amazon)? Your>advice appreciated.> I find Halmos a little
old-fashioned and perhaps not the best book for > beginners.
An excellent introduction is Bartle, The Elements of >
Integration. This thin volume is quite well-written and easy
to > follow. After that, you might be better prepared to look
at Halmos (or > Rudin's Real Analysis).> Also, do you need to
learn some topology as well? Munkres is a good choice.days is
you've got to be a millionaire to buy texts, nothing appearsto
be for less than $50!Incidentally, on your recommendation I
bought Halmos' Set Theory book. I have read only the beginning
chapters, and upon first reading itappears somewhat
superficial, maybe even trivial. However, every timeI re-read
a passage some new subtlety emerges, and after
repeatedreadings it is anything but superficial and certainly
not trivial. Iguess this is a sign either of my slowness, the
excellence of thebook, or perhaps both.very valuable
assistance.
===
Subject: Re: Product of these fractions never a
power of 2 ?> [...]> 1 1 1> (3 + ---)( 3 + ---)( 3 + ---) =/=
2^m> a b c> You can bound solutions a, b, c as follows:>
[...]I received the following email which wasn't posted in
this thread.I'm not sure what the netiquette is regarding
this; but hopefullythe sender won't mind if I reproduce it
here without mentioningtheir name:Curiously, it's possible to
rule out solutions to(3 + 1/a)(3 + 1/b)(3 + 1/c) = 32 a, b, c
*all odd* positive integersjust by getting some simple bounds
and looking at the factors of 2.Note that (because we're
assuming a,b, and c are all odd)(3a + 1)(3b+1)(3c+1) == 32
(mod 64),and each factor is divisible by 2 at least once. To
cover thepossibilities we have3x + 1 == 2 (mod 4) when x == 3
(mod 4)3x + 1 == 4 (mod 8) when x == 1 (mod 8)3x + 1 == 8 (mod
16) when x == 13 (mod 16)The values of a, b, and c must each
satisfy one of thesecongruences. Now, assuming thata <= b <=
cwe easily obtain the bounds9/5 < a <= 1/(32^(1/3) - 3),
i.e.1.8 < a < 5.72...The only value of a in this interval
satisfying any of thecongruences isa = 3.But a = 3 would force
at least one of (3 + 1/b) or (3 + 1/c)to be a fraction with a
factor of 3 in the numerator when inlowest terms, a condition
which disallows b and c both beingintegers (which is probably
the reason for the OP's restrictionthat none of a, b, and c be
divisible by 3). So there are nosolutions with a, b, and c all
odd.Of course, one has(3 + 1/1)^3 = 64, a power of 2 (a = b =
c = 1)but this solution violates the restriction that a, b, c
all begreater than
1.------------------------------------------------------------
---------------John R Ramsden
(jr@adslate.com)----------------------------------------------
-----------------------------Eternity is a long time,
especially towards the end. Woody Allen
===
Subject: Re: How to
Solve this nonlinear ODE system> everybody! I have a system of
nonliear ODE as what follows:> dx/dt=A1*y^2+B1*y+C1>
dy/dt=A2*X^2+B2*x+c2> x(t=0)=M0, X(t=infinity)=Mf;> y(t=0)=N0,
y(t=infinity)=Nf;> A1,A2,B1,B2,C1,C2, M0,Mf,N0,Nf are
constants;> Could anybody give some suggestions about how to
get a closed> form of x(t), y(t)? You reply will be highly
appreciated!> [...]Seeing again my solution to the above, it
struck me how long-windedit is towards the end (a bit like
Woody Allen's eternity - see .sig),and I was wondering if
anyone can think of a simple[ish] direct wayof obtaining a
parametric solution of: a.(x^3 + 3x) + b.(y^3 + 3y) + c = 0in
terms of Weierstrass functions (or Jacobian elliptic
functionsif more convenient, although that seems unlikely
given the cubicnature of the
equation).----------------------------------------------------
-----------------------John R Ramsden
(jr@adslate.com)----------------------------------------------
-----------------------------Eternity is a long time,
especially towards the end. Woody Allen
===
Subject: Re: How to
Solve this nonlinear ODE system>Seeing again my solution to
the above, it struck me how long-winded>it is towards the end
(a bit like Woody Allen's eternity - see .sig),>and I was
wondering if anyone can think of a simple[ish] direct way>of
obtaining a parametric solution of:> a.(x^3 + 3x) + b.(y^3 +
3y) + c = 0>in terms of Weierstrass functions (or Jacobian
elliptic functions>if more convenient, although that seems
unlikely given the cubic>nature of the equation). You can let
x = 2*sinh(u/3) y = 2*cosh(v/3)where 2*a*sinh(u) + 2*b*sinh(v)
+ c = 0. Perhaps start with two values A, B such that A + B + c
= 0.Solve x^3 + 3*x = A/aby substituting x = w - 1/w, which
yields w^3 - 1/w^3 = A/a.Solve the quadratic for w^3.John
Adams served two terms as Vice President and one as President,
but lostreelection. Later his son became President despite
losing the popular vote.That son lost his reelection attempt
badly. Now history is repeating itself.pmontgom@cwi.nl
Microsoft Research and CWI Home: San Rafael,
California
===
Subject: geophysics/math Ph.D. proposal, comments
desired.I'm currently editing my geophysics/applied math Ph.D.
proposal and would like comments back, preferably to my campus
e-mailaddress n06drd@mun.ca (where that is zero and not Oh) on
itspotential originality and on any literature review
referencesI may have missed.Now the possible disadvantage of
my posting it here is thatsomeone might steal my ideas but I
judge that unlikely,along the lines of my presenting a poster
paper in July andthen someone immediately writing a journal
paper basedon it without referencing me. But if someone does
buildthis post and the web page PDF copy of my proposal.The
advantage of posting is that I might learn of newreferences or
possible duplication by me of an approachof someone else such
that I should shift my focusslightly to ensure origiality.
Also comments Iget, while they may not be in time for me to
editthe proposal before probable April 20 1800 UTCsubmission
to the examining committee, mighthelp me prepare for questions
during the April 27 defense.If you do comment, in exchange I
could proofread apaper of yours at a later date (after the end
of April)and I am a very good proofreader of other
people'smaterial but not quite as good at proofreadingmy own
material.OK, yesterday's draft of the proposal in PDF formis
at http://www.nfld.com/~dalton/proposal/proposal.pdf .And the
title for now isSeismic wave and ray theory in elastic
media:Solution of equations of motion by method of
characteristics,principal symbols, and Fourier integral
operators,withcomparison to numerical results and real
data.and the abstract for now isMy proposed research will
focus on the mathematics of seismic wavetheory in elastic
media. In particular I will solve the three coupledCauchy's
equations of motion in anisotropic inhomogeneous media
usingthe standard method of characteristics, a variant based
on principalsymbol analysis, Fourier-integral operator (FIO)
theory, and numericalanalysis. I have begun and will continue
to apply the method ofcharacteristics to the equations of
motion. I have also begun toapply Principal Symbol theory,
related to Fourier Integral Operators,to the equations of
motion and from that should also get the samecharacteristics.
This would also put the equations in a frameworkamenable to
application of Fourier-integral operators. In parallelwith
that I will try to apply Fourier-integral operator theory
tosimple cases so that solutions can be tested against known
solutions.Then I will extend that theory to the inhomogeneous
anisotropic caseand ideally show that it extends the results
of ray theory. Howeverto handle the general case I expect the
FIO theory will reduce theequations somewhat but some
numerical solution will still be requiredto complete the
solution but less than for full numerical solution ofthe
equations. All that will involve some computer
implementation,and comparison to real data over well known
geology. The FIOmodelling results for the inhomogeneous
anisotropic case could bechecked by e.g. inserting the
isotropic elasticity matrix into thealgorithm and checking
that the solution reduces to that for theisotropic case. I
will also check it against a full numericalsolution for at
least one test case.Followups set to sci.physics
.http://www.nfld.com/~dalton
===
Subject: What is a
Huang-Hilbert transformation?Can anyone explain to me in plain
words what a Huang Hilbert transformationis and what it is used
for?
===
Subject: homologyHiI have this exercise in homology /
algebraic topology, which I can't quitefigure out. I hope
somebody here can help me:Show that H_1(X,A) is not isonorphic
to ~H_1(X/A) if X = [0,1] and A is thesequence 1, 1/2, 1/3, ...
together with its limit 0.Here is what I have done so far:Let Y
be the the space obtained by collapsing all the points in A /in
X to asingle point, Y =X/A this space must be the shrinking
wedge of circles.Let C_n be the circle of Y with radius
1/nConsidering the retractions r_n: Y --> C_n collapsing alle
C_i's except C_n.Each of these retractions r_n induces a
surjection R_n: H_1(Y) --> H_1(C_n).Since the reduced holomogy
H_1 of a circle is Z,I get that R_n: H_1(Y) --> ZNow the
product of all these R_n's is a homomorphism R: H_1(Y) -->
theinfinite direct product Z x Z x ...This product is as far
as I can see uncountable.Now I have some questions:Is it
possible to determine H_(Y) this way?Is it possible to show
that R is surjective? And if so...how?Now I look at the
reduced holomogy H_1(X,A)I know that H_1(X) = 0 and H_0(X) = 0
so I have the short exact sequence:0 --> H_1(X,A) --> H_0(A)
--> 0This mean that H_1(X,A) is isomorphic to H_0(A)I know
that A is a countable set of points, so my guess is that
H_0(A) isthe countably infinite direct sum of Z.But I dont
know if this is correct or how to show it.Can anybody help me
with this or tell me about the relation between theuncountable
direct product and the countable infinite direct sum....Or give
me another hint how to solve this exercise.Anne
===
Subject: Re:
Question about a sequence equation>Or even cleaner> s_i =
2*s_{i-2} + s_{i-1} + 1> Or super clean:> s_i = 2*s_{i-1} + (i
mod 2), with s_1 = 1.> Well, but the even cleaner one is a
linear nonhomogeneous recurrence> with a polynomial in the
nonhomogeneous part (covered by one of the> cases in the web
page I pointed to) whereas (i mod 2) is not polynomial> (not
that it's harder to solve, but there's less guidance from
standard> techniques).Yes, now I see better what the clean
means (not necessary fastercomputation).Michael Taktikos,
Hamburg, Germany> --> David Eppstein
http://www.ics.uci.edu/~eppstein/
===
Subject: Difficult Algebra
ProblemI have a problem that I'm having a tough time with. Let
R be a ring (perhaps without 1). Prove that (S,+,.) is a ring
with unitywhen S=R X Z, (r,m)+(s,n)=(r+s,m+n), and
(r,m)(s,n)=(rs+ms+nr,mn).Z is the integers. The additive
identity is just (0,0) right? My main problem is that I'm not
sure how to deal with the Unity. If R mightnot have 1, then
how can S definitely have 1 since it's R X Z? If R has aunity
1, how does that unity relate to the unity of S? Are they the
same? Would it contradict the uniqueness of unity?Please
Help!
===
Subject: Re: Difficult Algebra Problem> I have a
problem that I'm having a tough time with. Let R be a ring
(perhaps without 1). Prove that (S,+,.) is a ring with unity>
when S=R X Z, (r,m)+(s,n)=(r+s,m+n), and
(r,m)(s,n)=(rs+ms+nr,mn).> Z is the integers. The additive
identity is just (0,0) right? > My main problem is that I'm
not sure how to deal with the Unity. If R might> not have 1,
then how can S definitely have 1 since it's R X Z? If R has a>
unity 1, how does that unity relate to the unity of S? Are they
the same? > Would it contradict the uniqueness of unity?>
Please Help!Do you know what the ring R looks like? I don't
how R could have theform (x,y) for a unity like S does. If the
unity is just 1 then itisn't the same as the unity in S and the
uniqueness of unity wouldhave nothing to do with this.I might
be misunderstanding this problem. It's strange.
===
Subject: Re:
Difficult Algebra Problem> I have a problem that I'm having a
tough time with.Let R be a ring (perhaps without 1). Prove
that (S,+,.) is a ring withunity> when S=R X Z,
(r,m)+(s,n)=(r+s,m+n), and (r,m)(s,n)=(rs+ms+nr,mn).SNIPWe
want a (s,n) such that (r,m)(s,n)=(r,m) therefore r=rs+ms+nr
and m=mnclearly the second part implies n=1 which is feasible
since n e Zso now we know r = rs+ms + r = s(r+m) + rif we let
s=0 (since R is a ring, 0 has to be in it), the above
equationbecomes r = rSo unity in S is (0,1). Notice this does
NOT require unity in R.-Tralfaz
===
Subject: Re: Difficult
Algebra Problem> I have a problem that I'm having a tough time
with. Let R be a ring (perhaps without 1).> Prove that (S,+,.)
is a ring with unity> when S=R X Z, (r,m)+(s,n)=(r+s,m+n), and
(r,m)(s,n)=(rs+ms+nr,mn).>[...]> If R has a unity 1, how does
that unity relate to the unity of S?> Are they the same? Would
it contradict the uniqueness of unity?Since R is not even a
subset of S, the two unities are certainly not equal. Even if
you regard R as a subset of S via the identificationr -->
(r,0) (for r in R)you can check, that (r,0)(0,n) =
(nr,0).Hence no element of the form (r,0) can be the unity of
S.So if you start with a ring unity, the new ring S willhave
another unity.Marc
===
Subject: Re: Difficult Algebra Problem> I
have a problem that I'm having a tough time with. Let R be a
ring (perhaps without 1). Prove that (S,+,.) is a ring with
unity> when S=R X Z, (r,m)+(s,n)=(r+s,m+n), and
(r,m)(s,n)=(rs+ms+nr,mn).> Z is the integers. The additive
identity is just (0,0) right? According to your rule, do you
get (r,m)+(0,0)=(r,m) ?> My main problem is that I'm not sure
how to deal with the Unity. If R might> not have 1, then how
can S definitely have 1 since it's R X Z? If R has a> unity 1,
how does that unity relate to the unity of S? Are they the
same? Did you try (r,m)(s,n) = (s,n) and see what equations
you get?Can you choose r,m to make them always true?> Would it
contradict the uniqueness of unity?> Please Help!Is uniqueness
part of the definition? No, it is a theorem.How do you prove
it?
===
Subject: Re: Bush's dealx-mimeole: Produced By Microsoft
MimeOLE V6.00.2600.0000> taht really sucks!> also, if you
really want to know,> most of us would consider Lewinsky to
have> been a (perhaps not whitting) Israeli mole, although>
Goldberg could pull all of that truck, herself.> just look-up
Get Clinton on larouchepub.com --> and don't move your lips,
when reading American Almanac!Oh please. Dont be so ing
lameMimicZGF0YWZsZXhAY2FubmFiaXNtYWlsLmNvbQ== (
www.hidemyemail.net )Without knowledge you have fear. With
fear you create your own nightmares.Alzheimer's, cheaper than
rohypnolThere are 10 types of people in the world. Those that
understand Binary,and those that dont.He who controls Google,
controls the world.
===
Subject: Re: Bush's deal> George W. Bush
has a heart attack and dies. He goes to Hell where the> devil
is waiting for him.I don't know what to do with you here, says
the devil You're on my> list but I have no room for you. You
definitely have to stay here, so> I'll tell you what I'm going
to do. I've got three folks here who> weren't quite as bad as
you. I'll let one of them go, but you have to> take their
place.> I'll even let YOU decide who leaves.> George thought
that sounded pretty good, so he agreed.> The devil opened the
first room: in it was Richard Nixon and a large> pool of
water. He kept diving in and surfacing empty-handed over and>
over and over. Such was his fate in hell.No ! George said. I
don't think so. I'm not a good swimmer and I> don't think I
could do that all day long.> The devil led him to the next
room: in it was Tony Blair with a> sledgehammer and a room
full of rocks. All he did was swing that> hammer, time after
time after time.No, I've got this problem with my shoulder. I
would be in constant> agony if all I had to do was break rocks
all day! commented George.> The devil opened a third door. In
it, George saw Bill Clinton, lying> on the floor with his arms
folded behind his head, and his legs staked> in a spread eagle
pose. Bent over him was Monica Lewinsky, doing what> she does
best.> George Bush looked at this in disbelief for a while and
finally said:Yeah, I can handle this.> The devil smiled and
said........ OK, Monica, you're free to> go!Oh heavens....
ROTFLMFAO!!!!!!!!!!!! :*) I wasnt expecting that punchline.Im
literally laughing my ass off!!! Good one, and I like Bush,
LOL.
===
Subject: Re: Bush's dealmaybe I should dysinclude the
Mensa list, but*why* do you like Dubya, other than his
wonderful gladhandings? > Im literally laughing my ass off!!!
Good one, and I like Bush, LOL.--Give Earth a Trickier Dick
Cheeny -- out of office, after GIGA
years.http://www.benfranklinbooks.com/http://www.rand.org/
publications/randreview/issues/rr.12.00/http://
members.tripod.com/~american_almanachttp://www.wlym.com/pdf/
iclc/howthenation.PDF
===
Subject: Re: Bush's deal> George W.
Bush has a heart attack and dies. He goes to Hell where the>
devil is waiting for him.I don't know what to do with you
here, says the devil You're on my> list but I have no room for
you. You definitely have to stay here, so> I'll tell you what
I'm going to do. I've got three folks here who> weren't quite
as bad as you. I'll let one of them go, but you have to> take
their place.> I'll even let YOU decide who leaves.> George
thought that sounded pretty good, so he agreed.> The devil
opened the first room: in it was Richard Nixon and a large>
pool of water. He kept diving in and surfacing empty-handed
over and> over and over. Such was his fate in hell.No ! George
said. I don't think so. I'm not a good swimmer and I> don't
think I could do that all day long.> The devil led him to the
next room: in it was Tony Blair with a> sledgehammer and a
room full of rocks. All he did was swing that> hammer, time
after time after time.No, I've got this problem with my
shoulder. I would be in constant> agony if all I had to do was
break rocks all day! commented George.> The devil opened a
third door. In it, George saw Bill Clinton, lying> on the
floor with his arms folded behind his head, and his legs
staked> in a spread eagle pose. Bent over him was Monica
Lewinsky, doing what> she does best.> George Bush looked at
this in disbelief for a while and finally said:Yeah, I can
handle this.> The devil smiled and said........ OK, Monica,
you're free to> go!> Oh heavens.... ROTFLMFAO!!!!!!!!!!!! :*)
I wasnt expecting thatpunchline.> Im literally laughing my ass
off!!! Good one, and I like Bush, LOL.> RobIt is an old
punchline, Rob, I've heard the same story in many
differentways about many different people.jt
===
Subject: Re:
Bush's deal
===
>Subject: Re: Bush's deal>Message-id:
<5Lagc.14642$k05.3148@newsread2.news.pas.earthlink.net> oo>
____|mn> / /_/ / _ - Herc, The Unrecognised Truman> / K-9/ /_/
- Join www.chatty.net -> /____/_____ - Nanotechnology is gonna
be HUGE... (RMF)> --------------> George W. Bush has a heart
attack and dies. He goes to Hell where the> devil is waiting
for him.I don't know what to do with you here, says the devil
You're on my> list but I have no room for you. You definitely
have to stay here, so> I'll tell you what I'm going to do.
I've got three folks here who> weren't quite as bad as you.
I'll let one of them go, but you have to> take their place.>
I'll even let YOU decide who leaves.> George thought that
sounded pretty good, so he agreed.> The devil opened the first
room: in it was Richard Nixon and a large> pool of water. He
kept diving in and surfacing empty-handed over and> over and
over. Such was his fate in hell.No ! George said. I don't
think so. I'm not a good swimmer and I> don't think I could do
that all day long.> The devil led him to the next room: in it
was Tony Blair with a> sledgehammer and a room full of rocks.
All he did was swing that> hammer, time after time after
time.No, I've got this problem with my shoulder. I would be in
constant> agony if all I had to do was break rocks all day!
commented George.> The devil opened a third door. In it,
George saw Bill Clinton, lying> on the floor with his arms
folded behind his head, and his legs staked> in a spread eagle
pose. Bent over him was Monica Lewinsky, doing what> she does
best.> George Bush looked at this in disbelief for a while and
finally said:Yeah, I can handle this.> The devil smiled and
said........ OK, Monica, you're free to> go!> Oh heavens....
ROTFLMFAO!!!!!!!!!!!! :*) I wasnt expecting that>punchline.>
Im literally laughing my ass off!!! Good one, and I like Bush,
LOL.> Rob>It is an old punchline, Rob, I've heard the same
story in many different>ways about many different people.So
Bush didn't really die and go to Hell?>jt
===
Subject: Re:
number of indep. soloutions to diffyqs?I believe your question
is about completeness.The references cited by the other
respondents are very good, butyou might have to consult some
more advanced texts as
well.______________________________________________________>
I've taken some classes in solving differential equations, and
Iunderstand> all the different techniques, and everything, but
am confused as to how we> can be assured that our techniques
yield *all* possible solutions.> Especially when we do things
like, if x1 is a solution, put x2(x) => x1(x)*f(x) into the
differential equation and obtain another solution. Is> there a
general way to tell how many linearly independent solutions a>
differential equation has, (especially regardless of
homogeneity and> linearity)? I understand why these methods
work, of course, but can't see> exactly how we can be sure we
are obtaining all the solutions. I can see> intuitively a
little why it should be the case (especially when using>
eigenvalues of a matrix in the matrix form of a diffyq x'=Ax,
but wouldlike> something more rigorous).> Jeremy
===
Subject:
Re: Question about first-order logic > There seems to be poor
readers in this newsgroup. :-) Yep.
===
Subject: Need some
clarification.This might be a really stupid question to ask
but could someoneexplain to me what I need to do for these
problems. I've read themthrough several times and looked
through my notes and my book and Idon't know what exactly
needs to be done. If someone is willing topush me in the right
direction it would greatly be appreciated.Question: If G is a
group that acts on a set X, then for any x E X we define
theorbit of x to be O(x) = {g  x : g E G} and we also define
thestabilizerof x to be Gx = {g E G : g  x = x}.Each of the
following four parts describes a group action of G on X.Ineach
case compute Ox and Gx for every x E X.(a) Let G = Z and let X
= R^2. We define an action of G on X asfollows: For n E Z and
(x, y) E R^2 we define n  (x, y) to be thepoint in R^2
obtained by rotating (x, y) by an angle of pi(n)/2radiansin a
counterclockwise direction about the origin. (Hint: In orderto
compute G(x,y) for (x, y) E R^2, consider two cases  first
when(x, y) = (0, 0), and then when (x, y) /= (0, 0). )(b) Let
G = R and let X = S1. (Recall that S1 is the circle, which
maybe viewed as S1 = {z E C : |z| = 1} = {e^(i*theda) : 0 <=
theda <2(pi)}.) We define an action of G on X by defining r  z
= e^(ir)z;that is, rotate z by anangle of r radians in a
counterclockwise direction about the origin.(c) Let G = R and
let X = R^2. We define an action of G on X asfollows:For r E R
and (x, y) E R^2 we define r  (x, y) = (x + 2r, y + r).(d) Let
G = Z2 and let X = R. We define an action of G on X
asfollows:For n E Z2 and x E R we definen  x = ( x if n = 0)
(-x if n = 1)
===
Subject: Closest Rank Algorithm?I need an
algorithm similar to amazon.com which would give me n items
that were rated highest by the users that rated item x high.
i.e. people who liked this book also liked those 5..So,N =
number of items.X = number of users.For a given items i and
user u, R(i,u) | {Rmin <= R(i,u) <= Rmax, R(i,u) belongs to
N+} is the rating of the item i by the user u.What i need
is:Given an item i, what are the m (m=5 for example) items
that were ranked highest by users that ranked i high?There are
some ambiguities in this quite informal problem description:1.
What does it mean item ranked high by a user? imho it means
R(i,u) > avg ( R(i,u) for all u, i), i.e. higher than the
average of all ratings by all users.2. What does it mean items
that were ranked highest by users that ranked item i high? No
idea. Help me interpret that please.P.S.If you reply, please
reply to me as well, not just the newsgroup.
===
Subject: Re:
Force and accelerationIn sci.math, Donald G.
Shead<dcshead@charter.net> Cut<> That's no way to talk about
your twin. At least he *knows* that> pounds are units of mass,
even though he is dishonest and pretends> otherwise.> If you
guys would quit fighting each other, and take some time out
to> think about physics:Translation: swallow whole Donald G.
Shead's collective wisdom.You do realize that you're not
exactly the most respected physicisthere, I hope? :-)> That
pounds and newtons are measures of force,That bit, at least,
is correct.> and slugs and kilograms are measures of
inertia;No, they're measures of mass; inertia is merely a
concept,not a measurable quantity. Of course one can equate
thetwo, which more or less makes sense -- but most people
don'tbother and simply write '1 kg'. (Or, if one insists on
usingImperial units, '1 slug' or '1 pound-mass'.)> you'll find
that inertia is the ratio of the force> exerted on and/or by an
object; body or mass of matter,> to the rate of change in
velocity that it causes.So what?At best it gives a method by
which one can measure mass.#191, ewill3@earthlink.netIt's
still legal to go .sigless.
===
Subject: Re: Force and
acceleration>Cut<> Weight never has the meaning given there
when anybody talks about net> weight. That is a term of
commerce, not of physics.>Do you mean commerce buys and sells
mass?No, but people do measure many of the goods they buy and
sell in unitsof mass, such as pounds or kilograms. But you've
known that for manyyears, Dumb Donny.> Weight never has the
meaning given there when anybody talks aboutatomic weight or
molecular weight--terms which are used in> physics.>Those are
chemist's terms, not physicisist's.Nonsense. When I first
learned this, physicists' atomic weights weredifferent from
chemists' atomic weights. Now both are based oncarbon-12.>
Weight never has the meaning given there when anybody talks
abouttroy weight. There is no troy ounce force, and there
never has been> a troy ounce force.>What gives troy weight,
weight?> Weight never has the meaning given there when anybody
talks about the> units called the pennyweight or the
hundredweight (whether long or> short).>What's the difference
between long and short weight?17 millimeters?Short weight is
what you can go to jail for giving. A shorthundredweight is
something entirely different, a unit of mass equal to45.359237
kg.> Weight never has the meaning given there when anybody
talks aboutcarat weight of a diamond--one gram equals 5
carats.>Do you mean a 5 carat diamond is some rock?> Weight
never has the meaning given there when anybody talks
aboutdrained weight of a jar of olives.>That means the weight
of the water was drained off, doesn't it?> Weight never has
the meaning given there when anybody talks about thedeadweight
of a ship.>Does that mean when the engines die a ship is dead
in the water?> Weight never has the meaning given there when
anybody talks aboutflyweight, bantamweight, cruiserweight, or
the 63 kg weight> class in boxing or wrestling or judo.>Does
that mean heavyweights mass only about 90 kg?In some sports,
that's the class without an upper limit.> Weight never has the
meaning given there when anybody talks about the> weight of a
hammer in the hammer throw, the weight of a shot in the> shot
put, the weight of a bowling ball, etc.>Since when I was a kid
too?Since Moby Dick was a minnow.Even though you are older than
dirt, people had been measuring thisweight which is the same
thing as mass in physics jargon for over7000 years before you
were born.> Weight never has the meaning given there when
anybody talks abouttest weights used in calibrating or
certifying a scale.>How about the weights used in calibrating
spring scales?Same is true of them--it's just that in that
case, in addition to thetest weights, you'd also need some
other independent measurement ofthe strength of the local
gravitational field, if you want to testtheir accuracy in
measuring force. You don't need that if you aretesting their
accuracy in measuring mass at a particular location.For all
the spring scales I have seen, there is no calibration thatcan
be done--the most you are able to do on any of them is to
zerothem.> Weight never has the meaning given there when
anybody talks aboutbrass weights or stainless steel weights or
the> often-unadorned-with-an-adjective set of weights used with
a> balance.>Are you saying that weight means mass?You've known
that for many years that it often does, Dishonest Don.You
repeatedly keep playing on that ambiguity, every time you
whineabout kilograms being used as units of weight. You know
that therewill always be some fool reading your messages who
isn't as smart asyou are, someone you can hoodwink by being
deceptive about that.> Weight never has the meaning given
there when anybody talks about the> weight of a variable in
mathematics. (Had to get in something> relevant to the
sci.math newsgroup, one of them in which Dense Donny> started
this thread.)>What ever happened to the 16 ounce pound - the
weight of 2 - 8 oz cups of water?That is, and always was, a
figment of Dumb Donny's imagination.
http://ourworld.compuserve.com/homepages/Gene_Nygaard/
===

Subject: Re: Force and accelerationSo far no known language is
perfect in the sense of lacking ambiguities.Because language
and the uses to which it is put are constantly changing,
nolanguage is ever likely to be completely unambiguous. New
languages evolveprimarily to adapt to the ever-changing needs
of the users and the fact thatthe language they were taught as
children is insufficient to meet the needsthey encounter as
adults.The English language has some very fundamental
ambiguities (such as the lackof distinction between singular
and plural in the second person pronoun, orbetween dative and
accusative phrases). That is a small hindrance indeed tothe
hundreds of millions of people around the world who use the
languageevery day.Having pointed that much out, I now need to
point out that when anindividual finds it impossible to adapt
to broadly accepted socialconventions (such as using a single
word to represent multiple conceptsdistinguishable by
context), such lack of socialization becomes a pathologyof its
own.sHead's intractability on the matters which he consistently
brings up, whichappear to be a problem only to *him*, is
indicative of a deep-rootedpsychological problem which should
receive professional attention.As long as he remains
non-violent, the sociopath is relatively benign. LikeTed
Kazinski, however, there is always a possibility that,
unmonitored, hecan go off the deep end and declare war on them
- those people who arenot like him.sHead is sick and probably
needs 24/7 care.Tom DavidsonRichmond, VA
===
Subject: Re: Force
and acceleration> So far no known language is perfect in the
sense of lacking ambiguities.> Because language and the uses
to which it is put are constantly changing, no> language is
ever likely to be completely unambiguous. New languages
evolve> primarily to adapt to the ever-changing needs of the
users and the fact that> the language they were taught as
children is insufficient to meet the needs> they encounter as
adults.> The English language has some very fundamental
ambiguities (such as the lack> of distinction between singular
and plural in the second person pronoun, or> between dative and
accusative phrases). That is a small hindrance indeed to> the
hundreds of millions of people around the world who use the
language> every day.> Having pointed that much out, I now need
to point out that when an> individual finds it impossible to
adapt to broadly accepted social> conventions (such as using a
single word to represent multiple concepts> distinguishable by
context), such lack of socialization becomes a pathology> of
its own. But since it only a socialization issue between
Novelists and people who make robots, who don't socialize to
begin with, it's not really like a heady problem that people
as stupid as chemistry dictionary dorks get involved in.>
sHead's intractability on the matters which he consistently
brings up, which> appear to be a problem only to *him*, is
indicative of a deep-rooted> psychological problem which
should receive professional attention.> As long as he remains
non-violent, the sociopath is relatively benign. Like> Ted
Kazinski, however, there is always a possibility that,
unmonitored, he> can go off the deep end and declare war on
them - those people who are> not like him.> sHead is sick and
probably needs 24/7 care.> Tom Davidson> Richmond,
VA
===
Subject: Re: martingale inequality, indep. random
variablesDOOB'S INEQUALITY for p=2 gets you the constant 4
that you want>Alright, I'm obviously missing something easy
here, but suppose X_nis>a martingale with sup E[(X_n)^2].
Write X_n = X_0 + Y_1 + ... + Y_n,>with X_0 and the Y_i being
independent random variables.>Define K_n = sup(m>n) |X_m -
X_n|. Show that>E[(K_m)^2] < = 4 SUM(n+1,oo) E[(Y_i)^2].
(*)>Now, I know that>SUM(n+1,oo) E[(Y_i)^2] = E [(
SUM(n+1,oo) Y_i )^2].>If I could somehow take the sup outside
of the expectation in (*),>then I would have the inequality but
in fact I wouldn't need the 4,>and I guess that's what's
puzzling me...any hints?>For general martingales one could
obtain an inequality like this from>Doob's inequality,
although it seems to me that I would get 16 instead>of 4. For
sums of independent random variables there is
Levy's>inequality, which I think really would give you the
constant 4.Neither>of these inequalities are obvious (in the
sense that one is likely to>figure them out by oneself). Look
in the book Some random series of>functions by Kahane - Levy's
inequality is in Chapter 2 or 3, Ithink.> Also, I recall seeing
a rather simpler argument for this, which is partof a> proof of
the law of large numbers. I think it is due to Kolmogorov.
Inany> case, the result is not so obvious, and you need to
find it in a booksomewhere.> OK, I found Levy's inequality:>
Suppose X_i are i.r.v.s such that> P {|X_i + ... + X_n| >=
L/2} <= d> for 1 <= i <=n.> Then> P {sup |X_1 + ... + X_n| > =
L} <= d/(1-d).> But I still don't see how this applies...it
gives results about sups> of the sum, but I want something on
the expectation of the square of> that sup....
===
Subject: Re:
martingale inequality, indep. random variablesDOOB'S
INEQUALITY gives what you want, with the constant 4 in
theL^2-integrable case
===
Subject: Re: stopping time for a
uniformly integrable martingaleThese = the facts stated after
his question? Aren't they true?> Let X be an integral random
variable. Let T be a stopping time for for> a filtration
{F_n}. Define X_n = E(X | F_n). For all these> definitions, n
belongs to 0,1,...,oo (the extended natural numbers).> How
does one show that> E(X | F_T) = SUM(all n) {T=n}X_n = X_T>
almost surely???> I know this definition makes the sequence
{X_n} a uniformly integrable> martingale. Also, I think we can
write> E(X_T) = SUM(all n) E(X_T | {T=n}) = SUM(all n) E(X_n |
{T=n}).> I think it has to do with Doob's optional stopping
theorem, but I> haven't been able to apply it in just the
right way I guess. Also,> it's not given that T is bounded, so
there might be convergence> issues....suggestions are very
welcome, thank you> Why do you think these are
true?
===
Subject: analysis problemdefine f(x)= x + x^2(sin
(1/x^2)) if x =/= 0 and is 0 when x = 0. I need toshow that
there exist no interval containing 0 throughout which f
isincreasing. So I found f ' (x) = 1 + 2xsin (1/x) - (2/x)cos
(1/ x^2) if x=/=0 and 0 elsewhere.assume x is near 0 but
positive. Then the term 2x sin (1/x^2) is negligible.Want to
show 1 - (2/x)cos (1/ x^2) oscillate between +/-
.?????
===
Subject: re:analysis problemI'm not sure what you
need to show, but cos(1/x^2) and sin(1/x^2)oscillate
infinitely often as x->0.http://www.newsfeed.com The #1
Newsgroup Service in the World! >100,000 Newsgroups---= 19
East/West-Coast Specialized Servers - Total Privacy via
Encryption =---
===
Subject: Re: analysis problem> define f(x)=
x + x^2(sin (1/x^2)) if x =/= 0 and is 0 when x = 0. I needto>
show that there exist no interval containing 0 throughout which
f is> increasing.What are the zero's of the function?Clearly
x=0 is one. Check for othersthen check f' at those points. If
it is negative, good. If it is 0, checkf'' to see if it a
point of inflection or a local max/min, althoughtechnically by
your question it doesnt make a difference since if
f'(x)=0,there is not increasing at x. However, your professor
may interpret itdifferently. For example, they may say f(x) =
x^3 is an increasing functionover all
intervals.-Tralfaz
===
Subject: Re: analysis problem> define
f(x)= x + x^2(sin (1/x^2)) if x =/= 0 and is 0 when x = 0. I
need> to> show that there exist no interval containing 0
throughout which f is> increasing.> What are the zero's of the
function?> Clearly x=0 is one. But x = 0 is the only zero of f
in small enough intervals containing 0 (for example, (-1,1)).>
Check for others> then check f' at those points. If it is
negative, good. If it is 0, check> f'' to see if it a point of
inflection or a local max/min,None of this is necessary.>
although> technically by your question it doesnt make a
difference since if f'(x)=0,> there is not increasing at
x.Sorry, that's wrong.> However, your professor may interpret
it> differently. For example, they may say f(x) = x^3 is an
increasing function> over all intervals.It's not a matter of
interpretation. x^3 is strictly increasing on the whole real
line by definition. The definition of increasing does not
mention derivatives - nor should it.
===
Subject: Re: analysis
problem> define f(x)= x + x^2(sin (1/x^2)) if x =/= 0 and is 0
when x = 0. I need to> show that there exist no interval
containing 0 throughout which f is> increasing. So I found f '
(x) = 1 + 2xsin (1/x) - (2/x)cos (1/ x^2) if x> =/=0 and 0
elsewhere.> assume x is near 0 but positive. Then the term 2x
sin (1/x^2) is negligible.> Want to show 1 - (2/x)cos (1/ x^2)
oscillate between +/- .?????Consider the points x =
1/sqrt(nPi), n = 1,2,...
===
Subject: Re: When should I stop
iterative method? by support1.mathforum.org (8.11.6/8.11.6/The
Math Forum, $Revision: 1.9 primary) id i3HKZ3s28029;>I use
iterative methods (Jacobi, Gauss-Seidel, ...) for solving
linear>equation systems and i want to know when should I stop
the iteration?>I know that the condition for stopping has
something to do with Banach's>theorem or something like
this:>d(x*,x^(k+1))<=q/(1-q)[d(x^(k),x^(k+1)], x* is real
solution;>but i dont understand it!>So, what the condition to
stop iterative procedure for solving linear>equation
systems?Stop when you are no longer having fun
===
Subject: Re:
Hungarian method for non-square assignment matrix
(non-bipartite)> To OUP,> I thought the Munkres
algorithm==Hungarian method,> both of which were designed for
two sets of equal size.The Munkres algorithm that I'm familiar
with is not limited to sets of equal size. The book
Multiple-Target Tracking With Radar Applications by Samuel S.
Blackman has a detailed description of this algorithm.There is
also an iterative version of this algorithm called the
Jonker-Volgenant-Castanon (a.k.a. JVC)
algorithm.OUP
===
Subject: Re: Computer based proofs> I'm not
quite sure what is meant by a published version.Appel, K.;
Haken.W. and Koch, J. Every Planar Map is FourColorable. I:
Discharging. Illinois J.. Math. 21 429-490, 1977.
===
Subject:
Re: Computer based proofs<posted & mailed> I'm not quite sure
what is meant by a published version.> Appel, K.; Haken.W. and
Koch, J. Every Planar Map is Four> Colorable. I: Discharging.
Illinois J.. Math. 21 429-490, 1977.Surely this did not
contain 400 microfiche pages?In any case, the document I was
referring to was evidentlya precursor of this - in effect, a
preprint,which appeared to be a complete proof, in some 150
duplicated pages.Timothy Murphy e-mail (<80k only): tim /at/
birdsnest.maths.tcd.ietel: +353-86-2336090,
+353-1-2842366s-mail: School of Mathematics, Trinity College,
Dublin 2, Ireland
===
Subject: Representations of the circleWhy
are the irreducible unitary representations of the circle the
same asthe dual of the circle? I can't see any connection
here:Dual of circle = group of continuous homomorphisms from
circle to circle,which are the functions x ---> exp(2Pi*ikx)
where k is in Z.Irreducible unitary representations : We have
the regular representation ofS^1 on L^2(S^1) by R(y)f(x) =
f(x+y), where f is in L^2(S^1). R : S^1 ---U (L^2(S^1)) where
U (L^2(S^1)) is the unitary representations ofL^2(S^1). R is
irreducible. Are there other irreducible
unitaryrepresentations?Is there something going on here? Are
the characters of a locally compactabelian group the same as
the irreducible unitary representations of thelocally compact
abelian group? Is this true? What if the group is justlocally
compact?I appreciate any insight into this!
===
Subject: Re: A
problem proving a Ring Isomorphism>I am using Gallian's book
and am trying to complete the proof of>THeorem 15.6 Field of
Quotients. I'm trying to prove that the>mapping phi: D -> F
given by x -> x/1 is a ring isomorphism from D to>phi
(D).>I've proven that it's one-to-one and also that it's a
ring>homomorphism. My problem is that I'm not sure how to
prove that it's>onto. It would seem awkward to use a/b since
I'm given x/1 rather than>any arbitrary a and b.>I appreciate
any help completing the theorem.> Reread the problem: you are
not asked to show that phi is an> isomorphism from D to F: you
are merely asked to show that it is an> isomorphism from D to>
phi(D) = { phi(x) : x in D} = {f in F: there exists x in D
with f=phi(x)}.> Every function is surjective onto its image.I
knew that I was having to prove that phi is an isomorphism from
D tophi(D). I just did not understand how to prove it. It just
seemedlike it 'obviously' had to!Jacob>

===
===========================================================

===
=====It's not denial. I'm just very selective about> what I
accept as reality.> --- Calvin (Calvin and Hobbes)>

===
===========================================================

===
========Subject: differential equations teachingOk, so I'm
finally able to settle down enough to take someconventional
classes, and one of them is basic differential
equations.Halfway through the class, I see that it has rapidly
degenerated into:1. Spend 2 minutes doing differential
equations2. Spend 3 hours doing basic algebra to clean up the
mess3. Repeat over and over and overAnd looking at the trend
so far, one can only conclude it will becomemore so before the
end.Please god tell me all math isn't this way.Or if it is, put
a bullet in my head.Also, when people write math problems, do
they just randomly throwwhatever numbers pop in their heads
into a differential operator? Until now, it's usually seemed
like they've taken efforts to minimizethe above nonsense, ie
by having strategic things cancel out,radicands conveniently
being powers, etc.Use the very nice differential equations
method to change thisdifferential equation into a system of 20
equations in 20 unknowns,all quadratic. Then solve. NOOOO!!!