mm-1058
Interesting!
12=2^2*3 and 56=2^3*7 and 992=2^5*31
These are of the form 2^n*(2^n-1) which if 2^n-1 is prime are
perfect
numbers!
BF
> hi,
> im currently writing a text about perfect numbers and 
many
things related
> to
> it for my funal school exam. in order to get an idea of how
the whole
> thing
> isnt good enough) which are supposed to tell me how many
numbers are
> perfect, abundant, defucient, and similar things. what i got
was a
> frighteningly long list of numbers, so i visualized it
using gnuplot.
> the results were surprinsing for me. i knew from books that
there is just
> little amount of perfect numbers, there are several numbers
with
> defuciency 1
> and none with abundance 1 (plz correct me if i use wrong
terms, english
is
> not my mother tongue).
> but it struck me how many numbers there are with an
abundance of 12 and
> 56.
> observing integers from 1 to 100,000, i found 505 numbers
abundant by 56
> and
> even 1929 numbers abundant by 12. this is especially
astonishing as the
> next
> frequent abundance is 992, found in just 47 numbers in the
observed
range,
> all the others only occur 21 times or even rarer.
> i tried to fund out why i get such extreme values, but
didnt fund
> anything.
> if anyone wants to have a look at it, i can post the scripts
> (unfortunately,
> not knowing that i would ask here, variable and function
names are german)
> or
> gnuplot plots (please tell me which format you prefer).
> if you have any idea why these things are as they are,
please let me
know!
> btw: i also let gnuplot draw a diagram with N on the x-axis
and sigma(N)-n
> on
> the y-axis. drawing everything from 1 to 50,000, the
accumulations of
> points
> on y=x+12 and y=x+56 are invisible, but several other
Ôlines, i.e.,
> accumulations of abundancies as a linear function of n,
are, which
> resemble functions of simple fractions of x. these are just
special
> ones -- i couldnt fund out which until now; 
y=1/2*x and
y=3/4*x are such
> lines, y=2/3*x isnt. i could send images of that as well.
> greetings
> webograph
===
Subject: Re: astonishingly many numbers with abundace 12 and
56
format=ßowed;
reply-type=response
Sorry i should have said these numbers are double the perfect
numbers.
Since perfect numbers are 2^(n-1)*(2^n-1) ...
BF
> Interesting!
> 12=2^2*3 and 56=2^3*7 and 992=2^5*31
> These are of the form 2^n*(2^n-1) which if 2^n-1 is prime
are perfect
> numbers!
> BF
>> hi,
>> im currently writing a text about perfect numbers and
many things
>> related to
>> it for my funal school exam. in order to get an idea of how
the whole
>> thing
>> c
>> isnt good enough) which are supposed to tell me how many
numbers are
>> perfect, abundant, defucient, and similar things. what i
got was a
>> frighteningly long list of numbers, so i visualized it
using gnuplot.
>> the results were surprinsing for me. i knew from books
that there is just
>> a
>> little amount of perfect numbers, there are several
numbers with
>> defuciency 1
>> and none with abundance 1 (plz correct me if i use wrong
terms, english
>> is
>> not my mother tongue).
>> but it struck me how many numbers there are with an
abundance of 12 and
>> 56.
>> observing integers from 1 to 100,000, i found 505 numbers
abundant by 56
>> and
>> even 1929 numbers abundant by 12. this is especially
astonishing as the
>> next
>> frequent abundance is 992, found in just 47 numbers in the
observed
>> range,
>> all the others only occur 21 times or even rarer.
>> i tried to fund out why i get such extreme values, but
didnt fund
>> anything.
>> if anyone wants to have a look at it, i can post the
scripts
>> (unfortunately,
>> not knowing that i would ask here, variable and function
names are
>> german) or
>> gnuplot plots (please tell me which format you prefer).
>> if you have any idea why these things are as they are,
please let me
>> know!
>> btw: i also let gnuplot draw a diagram with N on the
>> sigma(N)-n on
>> the y-axis. drawing everything from 1 to 50,000, the
accumulations of
>> points
>> on y=x+12 and y=x+56 are invisible, but several other
Ôlines, i.e.,
>> accumulations of abundancies as a linear function of n,
are, which
>> resemble functions of simple fractions of x. these are
just special
>> ones -- i couldnt fund out which until now; 
y=1/2*x and
y=3/4*x are such
>> lines, y=2/3*x isnt. i could send images of that as 
well.
>> greetings
>> webograph
===
Subject: Re: astonishingly many numbers with abundace 12 and
56
monsieur Webograph,
there is nothing unusual about this seeming abundancy
of such abundancies, although Im not perfectly clear
on the meaning of the term from numbertheory, since
it hasnt struck my interest of yet.
you could prove the infunitude of both abundancies,
for instance, or ratios to other abundancies
at arbitrary limits.
> observing integers from 1 to 100,000, i found 505 numbers
abundant by 56
and
> even 1929 numbers abundant by 12. this is especially
astonishing as the
next
> frequent abundance is 992, found in just 47 numbers in the
observed
range,
> all the others only occur 21 times or even rarer.
> btw: i also let gnuplot draw a diagram with N on the x-axis
and sigma(N)-n
on
> the y-axis. drawing everything from 1 to 50,000, the
accumulations of
points
> on y=x+12 and y=x+56 are invisible, but several other
Ôlines, i.e.,
> accumulations of abundancies as a linear function of n,
are, which
resemble
> functions of simple fractions of x. these are just special
ones -- i
couldnt
> fund out which until now; y=1/2*x and y=3/4*x are such
lines, y=2/3*x
isnt.
> i could send images of that as well.
--Chairman George and Trickier Dick at Watergate!
http://tarpley.net/bush12.htm
===
Subject: Re: astonishingly many numbers with abundace 12 and
56
> the results were surprinsing for me. i knew from books that
there is just
a
> little amount of perfect numbers, there are several numbers
with
defuciency 1
> and none with abundance 1 (plz correct me if i use wrong
terms, english
is
> not my mother tongue).
> but it struck me how many numbers there are with an
abundance of 12 and
56.
> observing integers from 1 to 100,000, i found 505 numbers
abundant by 56
and
> even 1929 numbers abundant by 12. this is especially
astonishing as the
next
> frequent abundance is 992, found in just 47 numbers in the
observed
range,
> all the others only occur 21 times or even rarer.
> btw: i also let gnuplot draw a diagram with N on the x-axis
and sigma(N)-n
on
> the y-axis. drawing everything from 1 to 50,000,
I dont have anything helpful to say on the abundance of
numbers
with abundance 12, but I think theres something a bit
non-standard
about your terminology. sigma(n) is generally taken to mean
the sum
of all the positive divisors of n, including n itself.
Abundance
of n would be measured by sigma(n) - 2n, not sigma(n) - n.
--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
===
Subject: Re: astonishingly many numbers with abundace 12 and
56
> sigma(n) is generally taken to mean the sum
> of all the positive divisors of n, including n itself.
Abundance
> of n would be measured by sigma(n) - 2n, not sigma(n) - n.
although i know that there is a huge possibility for errors
in my
terminology, in this special case i think i didnt express
myself clearly.
sigma(n)-n is what i meant in the last paragraph. i drew a
diagram of
sigma(n)-n, that is, perfect numbers show up as points
P(n|n), primes as
P(n|1) and so on (i hope this is clearer).
> you could prove the infunitude of both abundancies,
> for instance, or ratios to other abundancies
> at arbitrary limits.
i played around a bit and found the following proof that
there are
infunitely
many numbers with abundance 12 (im afraid this will become
formally ugly,
but my former mathematical activities were all restricted to
school maths,
please correct me):
let n be any number 6*p with p being prime
then n can be factorized 2*3*p
this means that sigma(n) is 2+3+p+6+2p+3p+6p=12p+12
the abundance sigma(n)-2n of n is therefor 12p+12-2*6p=12
since there is an infunite number of primes, there is an
infunite number of
numbers with abundance 12.
there are more numbers with abundance 12, though (24 -
sigma(24)-48=36,
abundance 12).
the same proof can be done for abundance 56 using n=28p
until now, i dont have an idea of how to prove ratios to
other abundancies,
because as far as i know there is no rule for the amount of
primes in a
certain area.
webograph
===
Subject: Re: astonishingly many numbers with abundace 12 and
56
> sigma(n) is generally taken to mean the sum
>of all the positive divisors of n, including n itself.
Abundance
>of n would be measured by sigma(n) - 2n, not sigma(n) - n.
> although i know that there is a huge possibility for errors
in my
> terminology, in this special case i think i didnt express
myself clearly.
> sigma(n)-n is what i meant in the last paragraph. i drew a
diagram of
> sigma(n)-n, that is, perfect numbers show up as points
P(n|n), primes as
> P(n|1) and so on (i hope this is clearer).
OK. Then if p is prime you get the point (2p, p + 3); if you
plot just
these points for lots of p they resemble the line y = x/2,
which may
explain one of the observations you made in your original
post.
> you could prove the infunitude of both abundancies,
> for instance, or ratios to other abundancies
> at arbitrary limits.
> i played around a bit and found the following proof that
there are
infunitely
> many numbers with abundance 12 (im afraid this will 
become
formally ugly,
> but my former mathematical activities were all restricted
to school maths,
> please correct me):
> let n be any number 6*p with p being prime
> then n can be factorized 2*3*p
> this means that sigma(n) is 2+3+p+6+2p+3p+6p=12p+12
typo - right side correct, left side missing 1.
> the abundance sigma(n)-2n of n is therefor 12p+12-2*6p=12
> since there is an infunite number of primes, there is an
infunite number
of
> numbers with abundance 12.
> there are more numbers with abundance 12, though (24 -
sigma(24)-48=36,
> abundance 12).
> the same proof can be done for abundance 56 using n=28p
> until now, i dont have an idea of how to prove ratios to
other
abundancies,
> because as far as i know there is no rule for the amount of
primes in a
> certain area.
There are very good estimates for the number of primes in an
interval.
Youll fund them in texts on analytic number 
theory, and
possibly in
chapters on analytic number theory in more elementary texts,
and
possibly even in one or more of the recently published popular
treatments of the Riemann Hypothesis.
Are there numbers not of the form 6p, p prime, with abundance
12?
Lots of them?
--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
===
Subject: Re: astonishingly many numbers with abundace 12 and
56
> Are there numbers not of the form 6p, p prime, with
abundance 12?
> Lots of them?
24 = 2^3*3
54 = 2*3^3
304 = 2^4*19
Seems to be all of them.
Phil
--
... one Marine noticed one of the prisoners was still
breathing. A Marine
can be heard saying on the pool footage provided to Reuters
Television:
Hes ing faking hes dead. He faking 
hes ing dead.
The Marine then raises his riße and fures into the 
mans head.
The pictures are too graphic for us to broadcast, Sites said.
===
Subject: Unknown operation
Let A be the product ring ZxZ_20. Consider the following
function F:A-->Z_20
:
F((x,[y]_20))= [9xy]_20.
If * is the usual multiplicative law in Z_20, fund the
operation .
such
that F is a homomorphism from A(.) to Z_20(*).
TIA
===
Subject: Re: Unknown operation
fake <fakkicini@invalid.com> ha scritto nel messaggio
> Let A be the product ring ZxZ_20. Consider the following
function
> F:A-->Z_20 :
> F((x,[y]_20))= [9xy]_20.
> If * is the usual multiplicative law in Z_20, fund the
operation .
> such that F is a homomorphism from A(.) to Z_20(*).
> TIA
Maybe (a,[x]).(b,[y]) =def (ab,9[xy]) ????
===
Subject: Re: Probability Question -- Polymer Chain Breaking
Originator: jeyadev@kaveri
Ah! I did win my bet. I had put money down on you responding
:-)
>>The polymeric chains can be seen a chains made up of links,
which are
>>broken to produce smaller chains. Links that are broken
once cannot
>>be broken again. Let the average number (or, fraction)
>>of chains with Ôi links (i=0, 1, 2, ...) at 
time t_k be
given by
>>n_i(t_k). The initial distribtion at t = 0, {n_i(0)} is
given.
>>Assume that
>> 1) there is a constant ßux of entities that attack the
links
>> and break them
>> 2) At any time, any of the links is equally likely ot be
broken
>> 3) The number of links broken at at time t_k is
proportional to
>> the total number of links present at that time (this just
takes
>> into account that as the number of unbroken links reduces,
the
>> breaking entities will become less 
Ôeffucient as there are
>> fewer links to break due to Ôdilution)
>>What we seek is the time evolution of the {n_i(t_k)}.
>One way to do this, I think, is to use a continuous-time
Markov chain.
>In time interval t to t+dt, any given unbroken link breaks
>with probability f dt + O(dt^2), independent of all other
links.
>The probability of an initially unbroken link being still
unbroken
>at time t is then exp(-f t), and the expected number of
unbroken
>links is C exp(-f t) where C is the number of unbroken links
at time 0.
Got that. I had framed it in the discrete time version just
because I
am already thinking of Monte Carlo .... :-) What is really
bothering
me is expected number of the broken bits.
>If you take an initial chain of N unbroken links and break
links
>independently with probability p, if I understand you
>correctly the subchains of unbroken links that remain
correspond
>to runs of successes in a sequence of N Bernoulli trials with
>success probability 1-p. There should be well-known results
>about these, I think.
I thought about it along similar lines, but I think that it
is not
correct. If we assume that all links are equivalent, then, I
believe
that it is not possible to think in terms of individual
chains. Consdier
the n_i chains. One can now make zero link chains by breaking
1) the
furst link of a chain, 2) the last link or 3) adjacent links
anywhere
in any of n_i chains ... Thus, one has to look at the entire
i*n_i
links, even though the fact that each chain has only i links
is
important. In other words, the ÔN that you have 
is itself a
random
variable and there are n_i values such that they add up to
f*i*n_i,
the total number of links broken in the i link chains.
Now, I hope that I am understanding *you* correctly :-)
--
Surendar Jeyadev jeyadev1@wrc.xerox.com
Remove 1 for email address
===
Subject: Re: Probability Question -- Polymer Chain Breaking
Originator: jeyadev@kaveri
>Howdy, Surendar,
>Just curious... any chance the polymers are genetic material?
>Bob H
Nope. Very mundane. Mylar, actually.
This is just to see if some data that we have on mylar
degradation
makes sense. Physical intuition says it does not!
--
Surendar Jeyadev jeyadev1@wrc.xerox.com
Remove 1 for email address
===
Subject: Re: Skolems Paradox and why is math the way it is?
[snip: I think I understand that now, thank you.]
> | I think its just that the
> |teachers I had before werent clear, it seems like
something has to
> |come furst.
> It can be diffucult to be clear without being 
philosophically
> heavy-handed. This happens when people teach quantum
mechanics
> in a similar way. One has these several different
interpretations.
> One could pick one and teach as if it were so, which can be
somewhat
> misleading. On the other hand, one could try to teach the
various
> interpretations along with the more essential material,
which could
> make it confusing. An instructor could try to stick to just
the
> most essential material-- that part you need to understand
to be
> able to compute the probability of observed outcomes.
I think the best way to teach quantum mechanics is to assume
that the
wave-function is real (exists), and that the equations
describe how it
moves, and thats it, in practise thats all 
you need and
every
interpretation takes that seriously to the extent that the
interpretation takes anything seriously at all.
> To be completely unbiased and transparent with regard to
such
> possible qualms as doubting that the natural numbers really
exist
> probably seems like too much of a distraction. So the usual
approach
> is basically not to worry about it. The platonist and the
formalist
> will tend to sound the same as they are developing a
theory, since
> deducing consequences from some assumptions typically
sounds just
> like you believe the assumptions to be actually true in some
> domain. And then one can divert discussion of qualms to
such venues
> as sci.math.
This seems rather ahistorical, the problem is that at one
point people
took the existance of classes for granted and it created
problems.
And the point of the modern theory is to avoid those
problems, so
you have to be clear that everybody is doing the same thing
so that if
someone gets a problem its clear that the system is to
fault, not the
person.
> | We can assume induction in the langauge and then latter
> |show there there is an induction INSIDE the theory as
well, so that we
> |dont have to use induction outside the theory, but 
thats
very very
> |different than proving induction without proving
induction. Thats
> |proving induction in a theory using induction outside the
theory.
> Yes, *if* you assume induction for your original concept of
string,
> its an informal assumption that cant come 
from inside the
theory.
> Having proven mathematical induction from the axioms, to
conclude that
> it applies to actual strings is a further step, requiring
either
> believing the axioms are correct in some sense or something
like that.
As I mentioned before, the PURPOSE of the formal theory was
to avoid
problems, if you actually are depending on the informal
theory (Im
reading Quine now and so far it looks like formulas will be
built out
of philosophical statements and not strings and that
statements of
set theory will be based on other statements, and that the
axioms will
likely be about assuming the truth of some statements (which
is like
interpreting a schemata), I havent funished it 
yet, so dont
think
thats how Im characterizing Quine, 
its just my expectation
based on
where I am so far and it seems at least not to be circular at
this
point.
> | Set
> |theory cant do its own model theory and 
have every
existentionally
> |possible set be in the model,
> Note that I only have a vague idea of what existentionally
possible
> set means. Existentionally looks like a cross between
existentially
> and extentionally.
> I also dont really know what you think do 
its own model
theory
> should mean. I dont consider any particular kind of
self-application
> (do your own model theory, defune your own truth-predicate,
and
> the like) to be an important criterion for a theory. I can
see how if
> one were looking for a theory to be the be-all and end-all
of theories,
> one would need for it to be able to do to itself anything
that any
> other theory could do to it. But the quest for the be-all
and end-all
> of theories seems a bit quixotic. One surely would need to
quit
> focussing on particular formal theories to serve as the
be-all and
> end-all, and look instead at such things as the process by
which we
> develop theories, and try to fund the ultimate
theory-development
> process.
Set theory was billed to me as the type-free be-all theory,
and Im
not sure if you are refuting that as a misrepresentation that
my
teachers made or if yoy are agreeing with them, I cant 
tell.
But IF
logic avoids having infunite regresses into higher-order
logics so
that we CAN sit down and discuss how you make theories, so
isnt that
worth considering? Consistent theories imply strategies.
Thats a
REAL implication. But people complain against IF-logic exactly
because its the right size to do that because it 
isnt the
right
size for NOT doing that (not the right size for formal
deduction).
And thats silly because IF-logic has ordinary FOL as a part
of it
anyway, so anything you do with oFOL you can do with
IF-logic, just
stop using the / or // symbols.
> For theories with more realistic goals, I would say
self-application
> is a bit like being able to lift all the rocks that one can
make.
> It could be a sign that one is strong. Or it could be a
sign that one
> has a limited ability to make rocks. IF logic can defune its
own
> truth-predicate, yes. But thats a combination of being
strong in some
> ways, and being weak in others.
In what way do you think it is weak?
> | but that doesnt mean there isnt a
> |strong theory that *can* do its own model theory that 
has
set theory
> |(and hence everything based on it) as a component. Thats
what Im
> |looking for now, and I think the excluded middle is the
only thing in
> |the way really. There is a subsection of the universe
where the
> |excluded middle holds, and thats what we call set 
theory,
but its
> |intended models (if it has any) live outside that
subsection.
> Why?
Every model of set theory lacks a set that should exist as
much as the
alleged uncounted real should exist. The proof depends on the
excluded middle, that every set A either has to belong to a
set B or
not belong to B. That assumption has no basis real except for
mantaining an excluded middle. You could still have SOME sets
with
that property (that everything is either in them or not in
them), but
they simply wouldnt be all sets. And if your goal is to 
have
every
set that should be, then it looks like you have to have those
nonexclusive sets too because any axiomatic way to carve them
out
takes other (normal) sets with it.
> I mean, I have no big problem with abandoning LEM. Perhaps
its a
> bit of a big step to propose that it be abandoned
generally... but
> constructive mathematics proceeds without making any blanket
> assumption of LEM.
> It is consistent to assume that there exists a function
from some
> subset of the integers onto the reals, e.g. the function
taking the
> indices of Turing machines that compute real numbers to the
real
> numbers that they compute. (Compute is in the sense of
computing
> rational approximations to them.) Perhaps something of that
nature
> would be more agreeable to you. It still doesnt mean that
the reals
> are countable, however.
And once you extend the defunition of set to have non-excluded
middles, the standard proof about the lack of a set with a
specifuc
property goes away, the theorem becomes the graph of the
bijection
between a set and its power set does not have an excluded
middle,
even if it exists. And yes we can make a hierarchy based on
equivalnce classes of sets based on graphs with excluded
middles,
itll be just like cardinality theory if we do it right.
> |The math classes I took assumed that for every sentance
T(x) such that
> |for every set X such that for all x (x in X) => ((T(x) is
true ) or
> |(T(x) is false)) there exists a set Y such that for all y
(y in Y) <= |((x in X) and T(x)), where set and sentance were
both undefuned, then
> |the standard interpretation of set and sentance were
assumed outside
> |the theory, but maybe the set part is fune, but they should
have been
> |more honest about what was a valid sentance, some
professors actually
> |it is true.
> Well, you do realize that a lot of people consider it
rather artifucial
> to go around talking about what is true inside the theory
and what
> is true outside the theory as if they were two very
different spheres
> of discourse? It just sounds like a course being taught as
if
> a standard realist point of view were valid.
> I dont see any problem with sets of natural numbers such 
as
> {n : there exists an Oscar award winner who has had n
divorces}.
> To the extent that the sky being blue or actually being
divorced have
> some degree of grey area, these may be somewhat fuzzily
defuned. But
> its only when you decide to formalize the theory that you
need to
> specify what kind of formulas are allowed.
> The problem only comes in when you want to formalize the
axioms. If
> your instructor stated the selection axiom as above without
noting
> that its an axiom schema, and that the axioms generated 
by
the
> schema can only have T(x) that is expressible in the
language, then
> theyve made a rather technical inaccuracy in the
presentation. But
> if for some specifuc formula T(x) the above is an axiom in
the language
> of ZFC, it obviously must be a T expressible in the
language of ZFC.
I can grant that no deception was intended if you think it
was just a
technical inaccuracy, Im a bit scarred (as in maimed, not 
as
in
afraid) in that I still do not know the order in which to
resolve
things, but Im still hoping this book of 
Quines Im reading
will get
everything in the right order.
> Judging by the kinds of questions people ask on sci.math,
imposing
> this kind of technical detail right away tends to be
confusing to
> many students. Moreover, the informal assumption that this
axiom
> continues to be true for any predicate is the more basic
assumption
> than the axiom schema is.
Students should study a funitely axiomized version of ZFC 
furst
(IMO)
with clearly stated and defuned terms of language 
(theres
only a few
formulas in this baby-set-theory) and satisfaction and
everything, and
then later you bring in the whole schemata, but why not take
the
IF-logic furst order translation of the negation of a second
order
axiom, specifucally the IF-logic furst order 
translation of:
EF ~ (Ax Ey Az (zey) <-> (zex and 0=F[z]))
Then to say T is a theorem you can state that it is true that
either the above true, or the some other axiom is false, or T
is
true. I dont know why we need informal sentances or
metalanguages
or schema at all. Just If-logic furst order sentances, and
assertions
(hypotheticals) that the universe of discourse is such as to
make some
sentances false (like the f.o. translation of EF ~ (Ax Ey Az
(zey)
<-> (zex and 0=F[z]))).
> | But this sneaks a truth predicate into set theory that
> |isnt supposed to be there, and I know they were smart
enough to know
> |better, so Im left to conclude that they did it on
purpose.
> Youre a mighty suspicious guy.
> Including the phrase is true, as you have it above, is
often just
> a kind of verbal ßourish. If someone defunes A or B to 
mean
either
> A is true or B is true, that does not mean that theyre
intending to
> defune or in terms of a truth-predicate. Failing to restrict
to
> formulas in the language of ZF is a matter of allowing set
theoretical
> language to intermix with ordinary language.
I am totally familiar with how to translate either A is true
or B is
true into (A or B) is true by using ordinary language, I
could even
evalute (A or ~A) as being true in the case that there is no
excluded
middle without someone saying A is true or A is false to me, I
know that excluded middle is assumed to mean that one of
those holds
regardless of what A is AND that ~A is true when A is false is
true. I dont think this is about colloquialism. I think 
this
is
about describing separation badly, I dont think 
its about
logic or
language at all, so is EF ~ (Ax Ey Az (zey) <-> (zex and
0=F[z])) a
good descrpition of what is intended by separation or not? I
still
dont know.
> Dont confuse smartness with being 
suffuciently persnickety
to avoid
> making technical slips on the order of the things youre
describing
> here, or caring a lot about them. I realize that to you,
the fact that
> the axiom schema in ZFC only guarantees selection for
formulas in ZFC
> seems very important, but thats because of the kind of
concern you
> have for explaining to yourself how models of ZFC can
manage to be
> countable and things of that kind.
Smartness is related to knowing things, if someone asks if
youre sure
and you say you are sure when in fact you are wrong, then
either (1)
you were dishonest when you replied yes Im sure or (2) you
are not
a master of your material because you WERE sure and yet you
were
wrong. Does that make sense yet? I tend to assume (1) because
if I
assume (2) to everyone who makes an error then I have to
reinvent
absolutely everything myself, and force everyone to use *my*
defunitions and so on, which is socially too akword. Honestly,
people
can be un-persnickety in general, but to persist when someone
asks
if you are sure is what makes people be capable of calling you
dishonest, especially when later conversations reveal that
not only do
you know it was wrong, but if you claim you knew better at
the time.
I agree that it could still just be a slip of the tongue, and
a
mistake, but how many books and how many teachers before this
persistant slip seems to be a concerted case of either (1) or
(2).
> NB: the technical error here is ONLY an issue if the
instructor in
> question was stating the selection axiom schema this way
*as a part
> of ZFC*. If theyre just stating it as an axiom, then
theyre just
> giving an informal axiom.
The only thing the instructor talk about was ZFC, he never
mentioned
schemas or formulas, he said sentance and he gave the sky is
blue
as an example and he assumed truth and falseness of the
sentance T(x)
for every x in a set X. And he said this was set theory, not
we-havent-gotten-to-set-theory-yet.
> | And I
> |shouldnt have to wait for weeks to get a book that 
defunes
formulas
> |without assuming set theory furst, its a bit 
sad that so
many people
> |do this in a non-rigorous way.
> Dont confuse rigor with formality. Only a formalist needs
to have
> formula defuned *inside* of a formal theory separately from
outside.
Im unfamiliar with your defunitions of realist, 
formalist and
so
on, it just wasnt covered in my education. Some people on
this
Usenet group have told me to take a set theory class, I have,
they
didnt cover those terms. Are they covered inmost classes 
and
was I
just unlucky?
Im fune with IF-logic saying that some string 
represent
well-defuned
games and that some games have winning strategies for one
side, and
some for the other, and some dont. Im 
fund with someone
making a
claim that the set theoretical universe is such as to make
the string
(~A1)or(~A2)or...(~An)or(T) true (when ßeshed out with the
right
axioms for A1 through An and the theorem for T, and if they
say that
its true for all theorems, then that just leads to the
question what
are the theorems, but that isnt confusing because we are
forever
talking about actual games and these questions are about what
elements
can be selected for substitution in the games and which atomic
sentances AeB are going to be true, and which are going to be
false.
Its forever a discussion about the rules of the game, no
infunite
regress into types.
This is FINE for physics because in physics we ALSO play
verifucation
and falsifucation games in the laboratory, so I can make them
match
universe is such that when I do this experiment I get this
kind of
results, and one can make DIFFERENT models to help CHOOSE new
things
to TEST (in both math and physics). And based on the results,
you
might decide to change the rules of the game (new axioms), or
just
make new defunitions to make existing questions easier (for
people) to
ask, verify, or falsify.
> |Then there is the whole colloquialness of truth, theorem,
theory,
> |model, proof, that people use. I dont think they were
trying to be
> |dishonest there, but its very very very 
diffucult for
students to
> |learn when people are using the words different ways.
> The responsibility for making an author-reader or
teacher-student
> relationship work is a shared one. I dont think your
teachers or
> the authors of the books youve read have failed you to 
the
degree
> youre suggesting. Most students, although they may have
some
> diffuculties, tend not to get so hung up on the particular
issues
> that youve described. I dont think you 
need to have
regarded it
> as such an impediment.
The physics classes Ive taken I can teach myself, why is
math so into
hiding things? I couldnt honestly teach set theory today,
even a
basic one, because I havent seen a logical presentation. My
physics
teachers would answer questions when the students got
together and
demanded resolution (like when we asked to know how you know
when to
treat a stick as single object versus each molecule like a
separate
object, versus each atom as an object versus electrons and
nucleuses
versus electons and quarks and gluons), and they did so in
non-circular ways. But the math proffessors just say why are
you so
interested in foundations, I thought you liked physics?, Im
just
simply tired of being discriminated with, Im certain that
they talk
not in circles amongst themselves, and I think its down
right rude to
hide the actual logical developement from people just for not
being
in the club, this isnt middle school this is science. I
consider
math to be science.
One of the differences is that as a physicists Im rewarded
and
respected for being skeptical of everything, my work, the
owrk of
others, results I see, etc., but in math thats praised 
until
you ask
about ZFC, and then its like surely you dont 
doubt ZFC, but
how
can I either doubt or trust it if no one tells me what it is?
> |> Mathematicians seem generally, even the ones who are not
formalists,
> |> to treat the job of deducing consequences from axioms as
playing a
> |> special role in doing mathematics. It is supposed to be
what we can
> |> all agree on. I certainly hope that there is no
circularity in your
> |> set theory books in that part! Your set theory books
should contain
> |> many theorems that follow defunitely from one of the
usual sets of
> |> axioms for set theory.
> |If the axioms arent described clearly enough, 
its not
much an
> |exercise in anything.
> Clearly enough for *what*? I dont think you can name any
exercise
> where you are asked to prove a result, and where the reason
why it
> is diffucult for you to complete the exercise is that it
wasnt
> clear enough what the axioms were.
There is the translation from English to set theory and back
constantly, there is no way for me to tell that Im doing it
the same
way. The point is that someone can say X is a theorem: Y, QED
but
if you dont know WHAT it is a thoerem of, then I 
cant turn
around to
the guy next to me and say X is a theorem because if Im 
asked
theorem of what then I cant answer, so the theoremhood of 
the
statement isnt really proven (I cant carry 
it around with
me or
apply it outside of the set theory class), it was only stated
as a
theorem of SOMETHING. Only after we know the axioms can we
know that
X is IN FACT a theorem of THAT axiom system.
> [...]
> |Lost you on the defunitions again. Is a theorem a truth of
all models
> |or a provable statement of a language (assuming a fuxed
standard of
> |proof)?
> If a system of axioms is a formal system, then theorem
means a well-
> formed formula that follows from the axioms by the rules of
the system.
> If we simply give a set of statements as axioms, the
theorems are the
> statements that logically follow from the axioms. If the
language of
> a system is understood as being statements about a
(variable) model,
> then this becomes true in all models, since in that case,
for a
> statement to follow logically from a collection of other
statements
> simply means that it holds for all models in which the
premises do.
With you so far then.
> I had in mind the common situation where one has a
furst-order theory.
> In that case, we have the Goedel completeness theorem that
says the
> logical consequences of a set of axioms are the same as the
consequences
> that can be deduced using standard furst-order logic.
Are you sure about how you stated that? Im assuming 
standard
furst-order logic is ordinary furst order logic 
(so not
IF-logic or
SOL), but with IF-logic you can make furst order statements
that
arent statements of ordinary furst order logic, 
so you claim
seems,
... a bit sensational. If its true, then 
thats great, but I
want to
know if thats what you meant.
> |If the latter, then what does it MEAN to be interested in
> |whether a theorem follows from the axioms, since they all
do?
> What I should have written was whether a statement follows
from
> the axioms, or whether a statement is a theorem.
> In any case, those of us who are not formalists seldom care
whether
> the Riemann hypothesis is a theorem of ZFC or PA or
whatever, or
> whether the twin prime conjecture is a theorem of ZFC. We
do care
> about whether theyre true, however. The formalist thinks
somehow
> that these questions are not well enough defuned, but
everybody
> else aside from Essenin Volpin as far as I know disagrees.
You totally lost me here, people care about whether a
statement is
true when its true in some models and false in others? Just
consider the models where its true, now its 
true. Or
consider the
ones where its false, now its false. Why 
would anyone care
about
this? Am I therefore a formalist to fund this silly?
> [...]
> |The dependancy was in defuning the axioms. What I think you
call
> |formal (what Im used to called pure,a s opposed to
applied)
> |mathematics is about propositional relations, like x is a
y, where you
> |dont say (or know) what x is or y is or even is a is or
means, and
> |the statement x is a y is obviously netihre true or false,
its
> |meaningless. But what you do is assume that certain
relations BETWEEN
> |propositional relations hold, like for all x, for all y,
(x is a y)
> |or (y is a x), then you can consider what other
propositional
> |relations must ALSO hold that hold INDEPENDANT of any
meaning ascribed
> |to x, y, or is a. Then later if a model exists, that means
someone
> |can make an interpretation where the xs, 
ys, and is a
> |propositional relations are interpreted to be mean
something, and the
> |model is faithful is the axioms (as propositional
relations) hold true
> |in the model (as meaningful statements), and a theorem of
the axiom
> |system is a statement in the language of the thoery that
is true in
> |all faithful models of the axioms. Thats how it works 
for
group
> |theory,
> I dont think so.
> I just went over to my bookshelf and opened a group theory
> textbook at a random page. The theorem there was that the
center
> of the group GL(n,F) consists of the set of diagonal
matrices.
> GL(n,F) consists of the invertible n by n matrices with
entries
> in the fueld F.
> What are the axioms that supposedly defune GL(n,F)? We all
know
> what natural numbers are, and what invertible n by n
matricies
> are, but not because there are axioms for them. An n by n
> matrix is a function from {1,2,...,n}x{1,2,...,n} to F;
invertibility
> means that there exists another such one that is its
inverse, etc.
> The starting point is arithmetic, i.e., knowing what it
means
> to have a natural number n. What complete axiomatization of
> arithmetic do you have in mind when doing group theory?
Im really confused, I took this course called abstract
algebra and
we didnt assume any axiomatization of arithmetic, if fact
the whole
point was to avoid that, but instead to axiomize a group so
that
later anything that was a group would have the group theory
theorems
true of it. The group GL(n,F) satisfues the group axiom with
the
multiplication that you would expect for it (any in fact
since F
satisfues the fueld axioms by hypothesis, you can 
prove that
GL(n,F)
is a group, which is GOOD because that makes the nomen group
well-defunedish with the fact that GL(n,F) satisifes the group
axioms,
the whole point is that the results we proved about the
center of
abstract groups can then be applied to the subcase of groups
GL(n,F).
Why you think this starts with arithmetic is comletely behind
me, you
have an abstract fueld F, and from it you make a group
GL(n,F), where
does arthemetic come in?
> | fueld theory,
> I dont really have a book just on fueld 
theory as far as I
know,
> but it occurs to me that one fueld being algebraic over
another
> isnt furst-order defunable. The 
property of a fueld extension,
> that the overfueld is a *funite* dimensional 
vector space over
> the subfueld, comes up often. The funiteness 
intended is what
we
> (foolishly?) understood as just plain funiteness, not
funiteness
> relative to a model of [something].
The common axiomization of fueld include the term set, hence
the
importance of the question what is a set. If you threw in F
is a
fueld iff (FA1, and FA2, ... FAn, and STA1, STA2, ... , and
STAn)
(where FAk is a fueld axiom and STAk is a set theory axioms)
then
youd know what the models of fuelds are, but 
without, you
have to beg
the question over to set theory and ask is this a set to know
if
something is a model of the fueld axioms.
> | geometry, etc.
> How many colors are needed to color the points inside a unit
> square, so that no two points of the same color are a
distance
> of 1/2 apart?
> When people doing Euclidean geometry talk about the
Euclidean
> plane, they are talking about the one thats isometric to
R^2,
> pairs of real numbers, not an arbitrary model of some
furst-order
> axiomatization of it.
Classical geometry is both complete (every statement with the
uninterpreted primatives or geometry is either a theorem or
the
negation of a theorem) and categorical (all models are
isomorphic) as
an axiomatic theory, so what is arbitrary about its models?
And
geometry doesnt have a primative color. 
Youve lost me
completely
again. Isnt that really a question about functions from
manifolds
into integers?
> Generally, your statement comes much closer to correct if we
> are considering relationships between second-order
statements.
> But theres no formal deductive system for second-order
statements
> that captures all the valid deductions that can be made in
> second-order logic. Second-order logic also involves
referring
> to arbitrary subsets of the domain, which is the usual
bugaboo
> of set theory.
Formal deductive methods dont work for set theory correct,
but they
do for geometry I dont know why you think they 
dont. I
dont know
why you keep bringing up formal deduction anyway, we dont 
use
deduction to design theories, so what is this prima facie
reason for
such a hubbub about it? Work at funding the valid deductions
in SOL
or IF-logic if you care about set theory, use formal
deduction if you
care about geometry. How does this relate to anything we are
talking
about?
And if someone is concerned about arbitrary subsets, dont
quantify
over them, just stick to the IF-f.o. translations all the
time, and
then it becomes a question of models and strategies the way
it ALWASY
is. You dont have to consider arbitrary subsets, just
strategies of
games based on formulas, and the ZFC games are the ones that
start a
particular way, and the theorems are the one where you win
regardless
of the model (because either the model fails to satisfy the
axioms, or
you satisfy the statement of the theorem).
> [...]
> |> A formalist considers everything above the bottom line
to be just a
> |> kind of rhetorical ßourish. A Platonist will tend to
regard the
formal
> |> side as being just another technique for refuning
informal reasoning.
> |> Not many mathematicians are very much interested in
either refuning
> |> our explanation of what the undefuned terms like set
mean, or
justifying
> |> the truth of the axioms in those terms, however. Whether
a given
> |> mathematician believes the axiom of choice tends to be
treated as a
> |> matter of personal belief.
> |Thats VERY annoying. I took a class in functional
analysis where the
> |professor actually changed whether the axiom of choice was
true
> |halfway through the semester, I basically had to go redo
everything.
> Thats a great story, and I agree that 
thats annoying, at
least if
> he did it in a way that forced the class to redo work. If
he had
> meant to do this, he should at least have started with the
neutral
> theorems (ones not needing choice) and then added the
additional ones
> that can be proven with choice.
> But whether to believe the axiom of choice is really true
or not has
> no necessary connection with whether someone works using it
or not. One
> formalist who doesnt believe that 10^n exists for each
natural number
> n has proven results in set theory using all the usual
highfalutin
> assumptions.
The PROBLEM is using ONE word set when people ARBITRARILY
CHOOSE to
have the word essentially MEAN different things, its bad 
bad
bad. If
the word electron meant something different depending on who
said
it, physics would get nowhere fast. I agree that someone
could write
out a proof (so could a Turing machine) without believing the
theorem
to be true (so could a Turing machine), but if we are going
to use the
word set it should mean something, not whatever the speaker
imagines
in his head, thats one of my biggest problems, 
Im trying to
get a
straight answer about what is and is not a set, and Im not
getting
one.
> |Hintikka gives a justifucation about why mathematicians
like the axiom
> |of choice because it translates the standard
interpretation second
> |order formulas into equivalent furst order formulas, but
then he shows
> |that that doesnt work in general.
> Mathematicians do tend to like instances of
quantifuer-elimination,
> even when they are unaware of the concept of
quantifuer-elimination.
Im not sure how thats a response to my 
statement, there is
NO model
of set theory where all such translations are valid. Thats
what the
proof-predicate at the furst-order level shows. And if you
start
dancind around the issue by taking a non-standard
interpretation of
SOL, thats just silly to then insist on a standard
interpretation of
ordinary FOL theories. Something seems to have to give here.
> | I think thats because he was
> |holding onto an excluded middle for atomic sentances when
he didnt
> |need to, but thats his problem, there is no reason I
cant assume no
> |excluded middle.
> Some properties of structures seem simply to be
second-order and
> not furst-order. Whether a graph is connected or not, for
example.
> I dont see how one can get around it.
> Keith Ramsay
Set theory is supposed to be type free, e.g. you have a set of
points AND a set of edges AND they are both sets, everything
is a set.
So let A be the set of points and let B be a the set of
unordered
pairs of elements of A such that {a,b}eB iff there is an edge
between
a and b. Now let f: N->P(A) be a function such that f(0) = B,
and
such that f(n+1)={CeP(A):ExEyEz (yeS(n)) & (zeS(n)) & (xey) &
(xez) &
C=U({y,z})}, then the funite graph A is connected iff En
Aef(n),
thats all f.o. set theory. If you had an 
infunite graph, then
youd
need a better defunition of course, but Id have 
to know what
the
exact defunition of connected is to get an iff going, for
instance
A is disconnected iff ErEs An ~ Et (tef(n)) & (ret) & (set)
might be
correct, I dont know. I think its obvious 
that Im failing
to get
your point though, sorry.
===
Subject: Eqautions for generating a sphere?
This is proabably a simple question, but does anyone have a
set
of equations that could be used to generate a set of vertex
data
to model a sphere for a computer graphics application?
===
Subject: Re: Eqautions for generating a sphere?
>This is proabably a simple question, but does anyone have a
set
>of equations that could be used to generate a set of vertex
data
>to model a sphere for a computer graphics application?
Do you mean you want to generate a set of N (pseudo-)random
points
uniformly distributed on a sphere? One way is this.
Generate three random numbers x_i with uniform distribution
in the
interval [-1,1].
If x_1^2 + x_2^2 + x^3^2 > 1 reject these,
otherwise take the point [x_1/r, x_2/r, x_3/r] on the sphere
x^2 + y^2 + z^2 = 1, where r = sqrt(x_1^2 + x_2^2 + x_3^2).
Repeat until you have N points.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
===
Subject: Re: Eqautions for generating a sphere?
>This is proabably a simple question, but does anyone have a
set
>of equations that could be used to generate a set of vertex
data
>to model a sphere for a computer graphics application?
> Do you mean you want to generate a set of N (pseudo-)random
points
> uniformly distributed on a sphere? One way is this.
> Generate three random numbers x_i with uniform distribution
in the
> interval [-1,1].
> If x_1^2 + x_2^2 + x^3^2 > 1 reject these,
> otherwise take the point [x_1/r, x_2/r, x_3/r] on the sphere
> x^2 + y^2 + z^2 = 1, where r = sqrt(x_1^2 + x_2^2 + x_3^2).
> Repeat until you have N points.
> Robert Israel israel@math.ubc.ca
> Department of Mathematics http://www.math.ubc.ca/~israel
> University of British Columbia Vancouver, BC, Canada
Actually although a Pusedo random approach to a sphere would
work for some applications, for the one in the original
enquiry, Perhaps a polyhedrial approximation
to a sphere in which face was a quadrilatrel
could be considered?
The reason for needing 4 vertexs to from a face is to do
with the way the grpahics engine does texture mapping.
Essentialy the texture mapping is defuned as a series of u,v
pairs
for a given vertex. (U and V being the number of
Ôtexture-diatnces
to display.
for a simple face the code might look like this..
Vertex 0,0,0
Vertex 0,1,0
Veretx 0,1,1
Vertex 0,0,1
Face,0,1,2,3
Texture Grass,Bmp
TextureCoordinate 0,0,0,
TextureCoordinate 1,1,0,
TextureCoordinate 2,1,1,
TextureCoordinate 3,0,1
Texture mappings across a mesh are more complicated in that
they can
use % values for U and V but in general follow a simmilar
pattern, Of
course a
u,v pair has to be defuned for each vertex 
defuned.
I knows this probably doesnt help with spheres but it helps
explain
why
faces with at least 2 parrallel sides would be useful..
(Thinking about the parrallel sides wil;l probably be
Ôlongditude;
or Ôlatitude in an approximation but obviously 
not both at
the same
time across the whole sphere...)
Generating an approximation to a donut(ie ring torus) should
be
eaiser...
Alex
===
Subject: Re: Eqautions for generating a sphere?
hi,
there are basically two approaches: uv-spheres and subdivded
icospheres.
which one do you want?
> This is proabably a simple question, but does anyone have a
set
> of equations that could be used to generate a set of vertex
data
> to model a sphere for a computer graphics application?
===
Subject: Re: Eqautions for generating a sphere?
> hi,
> there are basically two approaches: uv-spheres and
subdivded icospheres.
> which one do you want?
> This is proabably a simple question, but does anyone have
a set
> of equations that could be used to generate a set of
vertex data
> to model a sphere for a computer graphics application?
Do you want to fut data to a spherical surface? Is radius
known? Are
points same or near to Platonic/Archimedean solid vertices?
===
Subject: Re: Eqautions for generating a sphere?
>hi,
>there are basically two approaches: uv-spheres and subdivded
icospheres.
>which one do you want?
This is proabably a simple question, but does anyone have a
set
> of equations that could be used to generate a set of vertex
data
> to model a sphere for a computer graphics application?
> Do you want to fut data to a spherical surface? Is radius
known? Are
> points same or near to Platonic/Archimedean solid vertices?
This is not about futting a data set.
The aim is to have some kind of polyhedron approximation to a
spehere,
that appears speherical by means of a texture mapping (or
shading...)
UV Spheres and icospheres sound intresting, please say more
:-)
===
Subject: New Model of Computation
Ive got an idea for a new model of computation using
2-colored binary
trees. It has probably been thought of already or whatever
but check it
out. The paper is now just a draft sketch. Ill get it into 
a
funal form
sooner or later if people think its worth pursuing.
See the paper at:
http://arxiv.org/abs/cs.CC/0411064
Comments appreciated.
===
Subject: Re: New Model of Computation
> See the paper at:
> http://arxiv.org/abs/cs.CC/0411064
you dont want to tell us how to access it, do you?
webograph
===
Subject: Re: New Model of Computation
>> See the paper at:
>> http://arxiv.org/abs/cs.CC/0411064
> you dont want to tell us how to access it, do you?
> webograph
In the bottom of the page with the abstract is a link to the
pdf document.
Here is the link explicitly.
http://arxiv.org/pdf/cs.CC/0411064
===
Subject: Re: New Model of Computation
Originator: tchow@lagrange.mit.edu.mit.edu (Timothy Chow)
> See the paper at:
> http://arxiv.org/abs/cs.CC/0411064
>> you dont want to tell us how to access it, do you?
>In the bottom of the page with the abstract is a link to the
pdf document.
Is the problem that you just submitted it and its not
publicly viewable
yet? After a day that problem should disappear.
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the
artillery---however great, will
never exceed four of those miles of which as many thousand
separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two
New Sciences
===
Subject: Re: New Model of Computation
> Is the problem that you just submitted it and its not
publicly viewable
> yet? After a day that problem should disappear.
You are probably right. I am not familiar with the system yet.
===
Subject: Re: what is the unit of the Matlab 2D FFT output
matrix?
>Hi all,
I just have a question about using Matlabs FFT2 to compute
the 2D DFT
of an
>image...
An image is input as MxN pixels, what should be the unit of
the
frequency
>domain, thats to say, what should be the unit of the 2D 
DFT
matrix
returned
>by the Matlab FFT2 command? What does each grid of those MxN
output
matrix
>represent?
> inches per pixel (I think) - but the grid spacing is
dependent on M
> and N
>If I attach some physical units to the input image, say,
each input
pixel is
>in fact the sampled value of a physical image at sampling
rate 96
DPI(=96
>dot or samples per inch)... In this case, what should each
grid of
those
>MxN DFT output matrix represent?
In one direction they are units of 1/(N*96) inches per pixel
(I
> think) and in the other 1/(M*96) inches per pixel.
>I guess the FFT2 output matrix has no meaning if the MxN
input image
matrix
>is not associated with a physical units...
Too deep for me - if you do the inverse transform to get back
into
> some space where you are confudent of the meaning of your
output I
> dont see that you need to worry.
> Best of Luck - Mike
Aaah! Inches per pixel - I must be going barmy. Take no
notice of me
kiki - Armans quite right! You can see it more or less
immediately by
:
ft=zeros(27,41);
ft(2,40)=1;,ft(2,2)=-1;,ft(26,40)=1;,ft(40,40)=-1;
plot3d(1:27,1:41,abs(ifft(ft)));
playing around with the values at the corners lets you put in
1/2
cycles or DC levels and its all very pretty.
Best of luck - Mike
===
Subject: Constructing analysis counterexample
I know the following theorem is true:
If {f_n} is sequence in L which converges uniformly on X to a
function f, and if mu(X) < +infunity, then the integral of f
with
respect to mu is the same as the limit of the integral of f_n
with
respect to mu. (L is the set of Lebesgue integrable functions
defuned
on X.)
Im trying to prove that the theorem fails if the
hypothesis mu(X) < +infunity is dropped. Any ideas?
D. Dewey
===
Subject: Re: Constructing analysis counterexample
> I know the following theorem is true:
> If {f_n} is sequence in L which converges uniformly on X to
a
> function f, and if mu(X) < +infunity, then the integral of f
with
> respect to mu is the same as the limit of the integral of
f_n with
> respect to mu. (L is the set of Lebesgue integrable
functions defuned
> on X.)
> Im trying to prove that the theorem fails if the
> hypothesis mu(X) < +infunity is dropped. Any ideas?
> D. Dewey
Let f_n = { 1/n if 0 < x < n, 0 otherwise.
===
Subject: Re: Help with Diagonal subgroup problem
days. My association with the Department is that of an
alumnus.
[.snip.]
Once you have proven this, try your understanding by proving
the
generalization known as Goursats Lemma:
DEF. A subdirect product of two groups H and K is a subgroup
G of
the direct product H x K, such that the canonical projections
p_1:G->H
and p_2:G->K are both surjective (that is, for every h in H
there
exists y in K such that (h,y) in G, and for every k in K
there exists
x in H such that (x,k) is in G).
GOURSATS LEMMA. Let H and K be two groups. There exist 
normal
subgroup M of H and N of K such that H/M is isomorphic to K/N
if and
only if there exists a subdirect product G of H and K such
that (G
intersect H) = M and (G intersect K)=N.
--
Its not denial. Im just very selective 
about
what I accept as reality.
--- Calvin (Calvin and Hobbes)
Arturo Magidin
magidin@math.berkeley.edu
===
Subject: Re: Help with Diagonal subgroup problem
days. My association with the Department is that of an
alumnus.
===
>>Subject: Re: Help with Diagonal subgroup problem
>D is a subgroup of G=M x N (INTERNAL direct product) is a
diagonal
subgroup
>>if
>(D intersect M) = 1 = (D intersect N) and DM=G=DN.
>Show that G has a diagonal subgroup IFF M is isomorphic to N.
><== ?????????????
>>Let D be the graph of the isomorphism from M to N: fux
f:M->N which
>>is an isomorphism, and let D = {(m,f(m)) : m in M}.
>I had asked another student about this and they basically
said something
>similar -- (m, f(m)) and they claimed it was just 
Ôtrivial
after that. I
>guess that I just dont Ôsee 
anything. It says diagonal
subgroup, so
I
>assume that means on a Ôgraph it would be the 
diagonal from
left to right
>across only.
As someone pointed out, it is called diagonal because in the
special
case when M=N (which you can assume up to isomorphism if M is
isomorphic to N) and f=id, you get the literal diagonal.
>I need to show that DM=G=DN and (D intersect M)=1=(D
intersect N). Unless
D
>itself is just the identity (doesnt make sense to me), it
would seem that
M
>and N could full up the entire graph and equal G.
G is not the graph. G would be more like the plane: G=MxN.
So, assume f:M->N is an isomorphism. let G = MxN, and let
D = {(m,f(m)) in G: m in M}.
Notice that for each point on the x-axis M, there is one and
only
one point on the y-axis, N; thats why this is sometimes
called
the graph of the isomorphism.
M is identifued with the subgroup {(m,1) in G: m in M} of G; N
corresponds to the subgroup {(1,n) in G: n in N}.
Then D intersect M = {(m,f(m)) in G: m in M, f(m)=1}.
Since f is an isomorphism, it is injective, so the only m for
which
f(m)=1 is m=1. Thus, D intersect M = {(1,1)}, the trivial
element of
G.
Likewise, D intersect N = {(m,f(m)) in G: m in M, m=1}. Since
f is a
group homomorphism, f(m)=1, so D intersect N = {(1,1)}, the
trivial
element of G.
Now we want to show that DM = G. That is, that the set of all
products
of the form
(r,f(r))(m,1) = (rm, f(r)1) = (rm, f(r)).
with m, r in M, cover all elements of G.
Well, let (x,y) be an element of G. We want to express it in
the form
(r,f(r))(m,1) = (rm, f(r)) for some r,m in M.
Since f is an isomorphism, it is surjective, so there exists
r in M
such that f(r)=y. And then if we let m = r^{-1}x, which lies
in M, we
have
(r,f(r))(m,1) = (rm,f(r)) = (rr^{-1}x,y) = (x,y). Thus, G is
contained
in DM, and hence (since DM is certainly contained in G),
equal to DM.
Now do something similar to show that G = DN.
--
Its not denial. Im just very selective 
about
what I accept as reality.
--- Calvin (Calvin and Hobbes)
Arturo Magidin
magidin@math.berkeley.edu
===
Subject: Two new math jokes (Physicists vector space, and
compact math
curriculum)
Ok, here we go with 2 new math jokes! The furst one...
Q: What is the physicists defunition of a 
vector space?
A: A set V satisfying the axiom that for any x in V, x has a
little
arrow drawn over it
Ok now for the second one!
I was talking with a friend in my graph theory class and he
pointed
out that a lot of the material is review from his geometric
topology
class. I thought about this for a moment and said, Ah!
Mathematics
curriculum is compact! He asked me what I meant and I
answered:
Any material covered by an infunite number of math courses can
be
covered by some funite subset of those courses
Ok lets see some other original jokes on this thread... :-)
Snis Pilbor
===
Subject: Re: Two new math jokes (Physicists vector space,
and compact
math
curriculum)
> Ok, here we go with 2 new math jokes! The furst one...
New? Oh, I get it. The redundant new.
> Q: What is the physicists defunition of a 
vector space?
> A: A set V satisfying the axiom that for any x in V, x has
a little
> arrow drawn over it
> Ok now for the second one!
For the second time.
> I was talking with a friend in my graph theory class and he
pointed
> out that a lot of the material is review from his geometric
topology
> class. I thought about this for a moment and said, Ah!
Mathematics
> curriculum is compact! He asked me what I meant and I
answered:
> Any material covered by an infunite number of math courses
can be
> covered by some funite subset of those courses
> Ok lets see some other original jokes on this thread... 
:-)
Have you consider getting a metaphysical? ;-)
===
Subject: Re: Two new math jokes (Physicists vector space,
and compact math
curriculum)
In reply:
There are three kinds of people:
those who understand mathematics
and
those who dont!
--
Casey
===
Subject: Re: Two new math jokes (Physicists vector space,
and compact
math
curriculum)
> In reply:
> There are three kinds of people:
> those who understand mathematics
> and
> those who dont!
> --
> Casey
I think arithmetic works better than mathematics.
===
Subject: Re: Two new math jokes (Physicists vector space,
and compact
math
curriculum)
> In reply:
> There are three kinds of people:
> those who understand mathematics
> and
> those who dont!
> --
> Casey
I prefer this old joke:
There are 10 kinds of people, those who understand binary,
and those who
dont.
===
Subject: Re: Geometrodynamics of inertial force in ßat
spacetime
> this case
> {LC}^100 = F(non-gravity 
reaction force)/mc^2
> Therefore
> -g ~ F(non-gravity reaction force)/m
> QED!
Jack has neatly disassembled a VW in spaceport One,
but the work order calls for servicing the Rolls
in spaceport Two.
Have a look at Tolmans Eq.(103.1), and sub
the RHS Lorentz force/m for Jacks,
F(non-gravity reaction force)/m
Jacks using Tolmans manual, unless there is
another way to depart from the conventional
geodesic , besides EMF.
Im going to work on the Rolls.
For clarity, Im resetting notation for
an absolute derivative of the 4 velocity to,
DU^u/ds = dU^u/ds + Gamma^u_ab U^a U^b.
Then Tolmans (103.1) becomes, (m=1),
DU^u - q*F^u_a dx^a = 0 .
I expect that an outer with g_uv gets,
DU_v - q*F_va dx^a = 0
and I expect an outer with dx^v gets us
DU - q*F_va dx^a dx^v = 0 = power (F*V)
the RHS vanishes by antisymmetry then
DU = 0
is the geodesic equation in the presence of
EMF.
DU/ds = (dU^u/ds + Gamma^u_ab) U^a U^b U_u = 0
Now watch this, do an outer using U^v and get,
0 = (dU^v/ds + Gamma^v_ab) U^a U^b = 0
i.e. DU^v = 0 in the presence of EMF.
Jacks old VW in Spaceport One has a
non-zero DU^v. My Rolls in Spaceport Two
has a DU^v =0 all the time.
Ken S. Tucker
Let us not forget the Tucker Automobile,
that occupies Spaceport Three.
===
Subject: Joel shifting ignored
>this case
{LC}^100 = F(non-gravity 
reaction force)/mc^2
Therefore
-g ~ F(non-gravity reaction force)/m
QED!
> Jack has neatly disassembled a VW in spaceport One,
> but the work order calls for servicing the Rolls
> in spaceport Two.
> Have a look at Tolmans Eq.(103.1), and sub
> the RHS Lorentz force/m for Jacks,
> F(non-gravity reaction force)/m
> Jacks using Tolmans manual, unless there 
is
> another way to depart from the conventional
> geodesic , besides EMF.
> Im going to work on the Rolls.
> For clarity, Im resetting notation for
> an absolute derivative of the 4 velocity to,
> DU^u/ds = dU^u/ds + Gamma^u_ab U^a U^b.
> Then Tolmans (103.1) becomes, (m=1),
> DU^u - q*F^u_a dx^a = 0 .
> I expect that an outer with g_uv gets,
> DU_v - q*F_va dx^a = 0
> and I expect an outer with dx^v gets us
> DU - q*F_va dx^a dx^v = 0 = power (F*V)
> the RHS vanishes by antisymmetry then
> DU = 0
> is the geodesic equation in the presence of
> EMF.
> DU/ds = (dU^u/ds + Gamma^u_ab) U^a U^b U_u = 0
> Now watch this, do an outer using U^v and get,
> 0 = (dU^v/ds + Gamma^v_ab) U^a U^b = 0
> i.e. DU^v = 0 in the presence of EMF.
> Jacks old VW in Spaceport One has a
> non-zero DU^v. My Rolls in Spaceport Two
> has a DU^v =0 all the time.
> Ken S. Tucker
> Let us not forget the Tucker Automobile,
> that occupies Spaceport Three.
Ooont Billy Joel Das singin Uptown girl dah dah dah diddy doo
daaaah
===
Subject: How to solve Diff eq. dx/dt*(f(x)+h(t))=0
Hi Every one
Can Some one give me a clue?
Oren
===
Subject: Re: How to solve Diff eq. dx/dt*(f(x)+h(t))=0
>Can Some one give me a clue?
Hint: There are two ways you can have a*b = 0.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
===
Subject: a tiling egroup has been formed a yahoo: you are
invited to join.
http://groups.yahoo.com/group/true_tile/
This group is about tiles in Euclidean n space, and
Non-Euclidean n space.
Discusions of crystalography, quasicrystalography,
substitution groups,
L systems, aperiodic tiles, Wang tiles.
Pisot numbers and space fulling curves are all welcome here.
Send us your tile defunitions and a picture.
We are trying to get a registry of new tiles.
Respectfully, Roger L. Bagula
tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca
92040-2905,tel:
619-5610814 :
alternative email: rlbtftn@netscape.net
URL : http://home.earthlink.net/~tftn
===
Subject: Re: Root Finder vii.
> Root Finder vii.
> by Jon Giffen
> [...]
> t^3+2t^2+t-4=0 t=1
> a[0]= -4
> a[1]= 1 a[2]=2 a[3]=1 N=(1,2,1) |N|^2=6 Q=(2/3)(1,2,1)
> |Q|=(2/3)6^(1/2)
> D=(1,2,3) S=(1,4,3) S*N=(1,4,3)*(1,2,1)=1+8+3=12
> D*N=(1,2,3)*(1,2,1)=1+4+3=8
>
C=12(1,2,3)+8(1,4,3)=(12+8,24+32,36+24)=(20,56,60)=4(5,14,15)
> |C|=(4)446^(1/2)
> (2/3)6^(1/2)
> (t,t^2,t^3)=(2/3)(1,2,1)+/- ------------ (5,14,15)
> 446^(1/2)
> (t,t^2,t^3)=(2/3)(1,2,1)+/- 0.07732(5,14,15)
> (t,t^2,t^3)=(0.6666,1.3333,0.6666)+/-(0.3866,1.08254,1.1599)
> Selecting the furst component,
> t = 0.6666+0.3866 = 1.0532 ~ 1
Strike Seven. Approximations are not acceptable.
> then
> t^2 + 3t + 4
> ------------------------------
> t-1/ t^3 + 2t^2 + t - 4
> t^3 - t^2
> --------------------
> 3t^2 + t - 4
> 3t^2 -3t
> -------------
> 4t - 4
> and the remaining roots are
> -3+/-{9-16|^(1/2)
> t = ----------------- = -1.5+/-1.3229i
> 2
> Dividing t^3 by t^2,
> t^3 = 0.6666+1.1599 = 2.2656
> -----------------------------
> t^2 = 1.3333+1.0825 = 2.4158
> or
> t = 0.9378 ~ 1
Strike Seven(A). If t^2 > 1, then we must have t > 1 or t <
-1. But you
have t = 0.9378 < 1 (and t > -1). So all this has shown is
that your
method fails (again).
-- Christopher Heckman
===
Subject: Re: Root Finder vii.
The fact that the method you present consistently produces
wrong
answers would have prompted most researchers to go back to the
drawing board until the problem was understood and solved or
to
abandon the method. Doesnt that sound appropriate?
--
There are two things you must never attempt to prove: the
unprovable -- and the obvious.
--
Democracy: The triumph of popularity over principle.
--
http://www.crbond.com
===
Subject: Please help with problem
Find a cubic fcn f(x)=9x^3 +bx^2+cx+d that has a local max
value of 3 at -2
and a local min value of 0 at 1.
===
Subject: Re: Please help with problem
>Find a cubic fcn f(x)=9x^3 +bx^2+cx+d that has a local max
value of 3 at -2
>and a local min value of 0 at 1.
Review how you fund local max and min, using the second
derivative
test. You know f(-2) = f(1) = 0; you know 
f(-2) = 3 and
f(1) = 0;
and you know the signs of f(-2) and 
f(1).
Unfortunately the problem is overdetermined, and (if my
calculations are correct) there is no cubic curve that futs
all the
conditions. Are you sure you copied the problem correctly?
--
Stan Brown, Oak Road Systems, Tompkins County, New York, USA
http://OakRoadSystems.com/
And if youre afraid of butter, which many people are nowa-
days, (long pause) you just put in cream. --Julia Child
===
Subject: Re: Please help with problem
>>Find a cubic fcn f(x)=9x^3 +bx^2+cx+d that has a local max
value of 3
>>at -2
>>and a local min value of 0 at 1.
> Review how you fund local max and min, using the second
derivative
> test. You know f(-2) = f(1) = 0; you know 
f(-2) = 3 and
f(1) = 0;
> and you know the signs of f(-2) and 
f(1).
> Unfortunately the problem is overdetermined, and (if my
> calculations are correct) there is no cubic curve that futs
all the
> conditions. Are you sure you copied the problem correctly?
Something is wrong.......
27x^2 + 2bx + c = 0 when x = -2
27x^2 + 2bx + c = 0 when x = 1
108 -4b + c = 0.............(1)
27 + 2b + c = 0.............(2)
Subtracting (1) from (2)
6b = 81
b = 13.5
c = 54
-72 + 54 - 108 + d = 3 .............(3)
9 + 13.5 + 54 + d = 0................(4)
d = 129.......from (3)
d = -76.5 ......from (4)
PH
===
Subject: Re: Please help with problem
hgl <HGL@COX.NET> escribi.97:
> Find a cubic fcn f(x)=9x^3 +bx^2+cx+d that has a local max
value of 3
> at -2 and a local min value of 0 at 1.
That conditions are inconsistent. Or the coeffucient of x^3 is
a an mot 9?
--
Ignacio Larrosa Ca.96estro
A Coru.96a (Espa.96a)
ilarrosaQUITARMAYUSCULAS@mundo-r.com
===
Subject: correlation functions of uncorrrelated random
variables
Suppose x and y are two uncorrelated random variables with
funite
expectation values.
Then, by defunition, cov(x,y)=0.
Is it true then that
cov(x^2,y)=0?
cov(x^2,y^2) = E[x^2] E[y^2] ?
Roger
===
Subject: Re: correlation functions of uncorrrelated random
variables
>Suppose x and y are two uncorrelated random variables with
funite
>expectation values.
>Then, by defunition, cov(x,y)=0.
>Is it true then that
>cov(x^2,y)=0?
No. For example, it could be that y = x^2.
>cov(x^2,y^2) = E[x^2] E[y^2] ?
No. For example, it could be that x and y are
independent (and neither is ever 0).
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
===
Subject: SORRY, I corrected the equation
Find a cubic fcn f(x) = ax^3 +bx^2+cx+d that has a local max
value of 3
at -2
and a local min value of 0 at 1.
===
Subject: Re: SORRY, I corrected the equation
> Find a cubic fcn f(x) = ax^3 +bx^2+cx+d that has a local
max value of 3
> at -2
> and a local min value of 0 at 1.
===
Subject: Re: SORRY, I corrected the equation
max/min at -2,+1 implies constant times (x+2)(x-1)=0=dy/dx
Integrate, interpret this constant as 3a and last as d, to
give form
y= a[x^3 +(3/2)x^2 -6x] +d which leads to 2 eqns with 2
unknowns.
1,0] 0=(-7/2)a +d
-2,3] 3= 10a +d,
so a =2/9, d=3-20/9=7/9 and the required cubic is
y= (2/9)x^3+(1/3)x^2 -(4/3)x +7/9. Check it and see.
Hope this helps
Ian Hutcheson
===
Subject: Re: SORRY, I corrected the equation
hgl <HGL@COX.NET> escribi.97:
> Find a cubic fcn f(x) = ax^3 +bx^2+cx+d that has a local
max value of
> 3 at -2
> and a local min value of 0 at 1.
Well, you know that f(-2) = 3 and f(1) = 0. Do you know the
value of f(x)
at -2 an 1?
You have a system of 4 equations in 4 indetermined, it is
easy. To check
your result, 9*f(x) has all coeffucients integers.
--
Ignacio Larrosa Ca.96estro
A Coru.96a (Espa.96a)
ilarrosaQUITARMAYUSCULAS@mundo-r.com
===
Subject: Re: SORRY, I corrected the equation
just to check ur answer
a = 2/9
b = 1/3
c = -4/3
d = 7/9
> hgl <HGL@COX.NET> escribi.97:
>> Find a cubic fcn f(x) = ax^3 +bx^2+cx+d that has a local
max value of
>> 3 at -2
>> and a local min value of 0 at 1.
> Well, you know that f(-2) = 3 and f(1) = 0. Do you know the
value of f(x)
> at -2 an 1?
> You have a system of 4 equations in 4 indetermined, it is
easy. To check
> your result, 9*f(x) has all coeffucients integers.
> --
> Ignacio Larrosa Ca.96estro
> A Coru.96a (Espa.96a)
> ilarrosaQUITARMAYUSCULAS@mundo-r.com
===
Subject: RE generalized algebraic laws
This is a followup on a posting of the same title. The initial
posting, in august of last year, by Elaine, starts, No matter
where I
look I cant fund a... rigorous statement of the 
laws of
associativity... in... general form. Everybody knows them...
And whos
put a proof in writing? ...on my own, Ive discovered that
the only
way I can produce a... proof is by
disregarding meaning... and proving theorems about properties
of
formal
languages...
I was thinking about the same problem, that is, defuning
mathematically n-ary products of an operation that is not
apriori
associative. I guess it could be said that the description of
generalized associativity is metamathematical in most algebra
texts.
The mathematical proof needs to be done in a way that
coincides with
the intuitive idea, or with the formal generalized inductive
defunition of the class of valid brackettings (sic?):
the class of terms A satisfying either
a) A is a letter
b) A is a combination BC for elements B and 
C
of the class, where B is the term (B) if B is not
a letter and B is the term B when B is a letter, and
the same for C.
(here AB means combining the two assemblies of symbols in the
suggested order.)
I will give one construction. Please comment if it seems
ineffucient
in applications, if a standard defunition already exists, or
etc. It
is a set of functions on X*, i.e., the union of X^n for the
naturals
n, where X is the algebraic structure, with operation u. In
particular, the set P of functions phi for each of which
exists for
every n an m<n such that
(a_1,a_2,...,a_n) -->
(a_1,a_2,...,a_m-1,u(a_m,a_m+1),a_m+2,...,a_n),
(with some abuse of notation in the image) noting in this
case this
image is in X^n-1. Then given any funite sequence a in X, all
the
different possible ways to apply u to as elements are
represented by
some function phi in particular, the set of products of
a_1,a_2,...,a_n is the set {phi^n-1(a)}_(phi in P).
Some simple observations about such functions involves a
construction
like parents Im taking this word since these functions
could be
imagined as an overly simplistic model of a single-child
baring
population [they technically would have to be hemaphrotides
(sic?) but
Ill leave a margin for non-logical thinking in the
imagination, and I
recommend imagining them as being of opposite sexes].
According to
this model, for instance, if n is the population of China,
then in n-1
generations there will be exactly 1 Chinese person.
The jth generation parents of a term a_i in the sequence
phi^k+j(a)
would then be the elements of
phi^k(a) who either are already j-1th gen. parents or whose
child
is a j-1th gen. parent, the latter meaning that the value m
described
above equals the index of the parent with the smaller index.
The 1st
gen. parent is a_i if it remains from phi^k+j-1(a) or
otherwise
a_i and 
a_i+1, where 
a=phi^k+j-1(a).
Easy observations include, that the jth generation parents
are always
consecutive in the sequence in which they appear. The jth
gen. parents
of a_i+1 appear immediately after 
a_is jth gen. parents.
Furthermore, one should be able to apply the standard
inductive
proof of generalized associativity to the construction P,
with the
aid of these and several other observations, the hypothesis
being that
phi^n(a)=phi^n(a) for all phi, phi in P and 
a in X^n+1. I am
confudent of that, although I havent gone 
through all the
details,
which I must admit can loose my interest.
I will consider this construction a little more, but thats
what I
wish to say about it now and in this connection. I dont 
know
that
there are no mistypes. Many liberties have been taken for the
sake of
===
Subject: RE generalized algebraic laws
This is a followup on a posting of the same title. The initial
posting, in august of last year, by Elaine, starts, No matter
where I
look I cant fund a... rigorous statement of the 
laws of
associativity... in... general form. Everybody knows them...
And whos
put a proof in writing? ...on my own, Ive discovered that
the only
way I can produce a... proof is by
disregarding meaning... and proving theorems about properties
of
formal
languages...
I was thinking about the same problem, that is, defuning
mathematically n-ary products of an operation that is not
apriori
associative. I guess it could be said that the description of
generalized associativity is metamathematical in most algebra
texts.
The mathematical proof needs to be done in a way that
coincides with
the intuitive idea, or with the formal generalized inductive
defunition of the class of valid brackettings (sic?):
the class of terms A satisfying either
a) A is a letter
b) A is a combination BC for elements B and 
C
of the class, where B is the term (B) if B is not
a letter and B is the term B when B is a letter, and
the same for C.
(here AB means combining the two assemblies of symbols in the
suggested order.)
I will give one construction. Please comment if it seems
ineffucient
in applications, if a standard defunition already exists, or
etc. It
is a set of functions on X*, i.e., the union of X^n for the
naturals
n, where X is the algebraic structure, with operation u. In
particular, the set P of functions phi for each of which
exists for
every n an m<n such that
(a_1,a_2,...,a_n) -->
(a_1,a_2,...,a_m-1,u(a_m,a_m+1),a_m+2,...,a_n),
(with some abuse of notation in the image) noting in this
case this
image is in X^n-1. Then given any funite sequence a in X, all
the
different possible ways to apply u to as elements are
represented by
some function phi in particular, the set of products of
a_1,a_2,...,a_n is the set {phi^n-1(a)}_(phi in P).
Some simple observations about such functions involves a
construction
like parents Im taking this word since these functions
could be
imagined as an overly simplistic model of a single-child
baring
population [they technically would have to be hemaphrotides
(sic?) but
Ill leave a margin for non-logical thinking in the
imagination, and I
recommend imagining them as being of opposite sexes].
According to
this model, for instance, if n is the population of China,
then in n-1
generations there will be exactly 1 Chinese person.
The jth generation parents of a term a_i in the sequence
phi^k+j(a)
would then be the elements of
phi^k(a) who either are already j-1th gen. parents or whose
child
is a j-1th gen. parent, the latter meaning that the value m
described
above equals the index of the parent with the smaller index.
The 1st
gen. parent is a_i if it remains from phi^k+j-1(a) or
otherwise
a_i and 
a_i+1, where 
a=phi^k+j-1(a).
Easy observations include, that the jth generation parents
are always
consecutive in the sequence in which they appear. The jth
gen. parents
of a_i+1 appear immediately after 
a_is jth gen. parents.
Furthermore, one should be able to apply the standard
inductive
proof of generalized associativity to the construction P,
with the
aid of these and several other observations, the hypothesis
being that
phi^n(a)=phi^n(a) for all phi, phi in P and 
a in X^n+1. I am
confudent of that, although I havent gone 
through all the
details,
which I must admit can loose my interest.
I will consider this construction a little more, but thats
what I
wish to say about it now and in this connection. I dont 
know
that
there are no mistypes. Many liberties have been taken for the
sake of
===
Subject: Re: sparse matrices and eigenvalue computation
> I am in the position where I may need to implement an
eigenvalue solver
> myself. The objective is to obtain only the lower n
eigenvalues (in
> magnitude). I do the eigenvalue computation at present (in
C++ using TNT
> few of the lower n eigenvalues will be much faster.
> In MATLAB, I have seen some people use sparse matrices to
compute
> eigenvalues (usually when the input matrix, X, is quite
large).
> First, where do I fund information on what sparse matrices
are and why,
> where and how to use them? I would like to go from easy to
more detailed
> explanations.
> Second, are there any C++ programs that already compute
only a subset of
> eigenvalues of a given input matrix?
> ->HS
Have a look at
http://www.caam.rice.edu/software/ARPACK/arpack++.html
Hans Mittelmann
===
Subject: Modern censorship
You read in history books about individuals harried by mobs,
continually barraged with various attacks, who face outrageous
behavior at the hands of some group, and then we come to a
supposedly
enlightened age with extraordinary tools for sharing
information--and
the same damn thing happens again.
Over the years that Ive posted on PUBLIC forums the thing
that has
stood out is how out there, without shame, and without even
hesitating
to hide what theyre doing--groups of people have made it
their
business to try to censor what I write.
Time after time theyd post to tell me to go away, or to 
tell
others
to ignore me, as these people made it their business to try to
control.
When harassing me on Usenet wasnt enough, they took to
putting up
webpage--and if another one of you tries to claim that I
harassed
David Ullrich, a math professor who after harassing me for
years
dragged race into the picture talking about racial slurs,
because I
complained about him bringing race into the picture to his
university,
then you just make my point that much more forcefully. Race
is not a
tool to be used by some unethical math professor who thinks
its a
neat way to insult someone on Usenet. David Ullrich was wrong
to try
to attack me with race, and I was right to call him on it, and
complain to his school.
You people cheat.
to a journal, so some of you emailed that journal to get the
paper
censored.
You are disgusting cretins who follow no rules, no moral
obligations,
and you are irrational.
I can argue point for point, point by point, explain over and
over
again, as Ive done for years, and one of you will just
disagree to be
disagreeable!
I offer compromise and get spat upon.
History shows that there are always those of the mob, who
take it upon
themselves to try and control the few, or especially, the one.
Im thankful that this plays out over the Internet.
In the past you are the people who would be tying someone to
a stake
to burn them and then, blaming the victim, shout your
morality to the
heavens, as if God listens to loudness above reason.
You are the reason that we have a society of today where so
many
problems will not get addressed because a few people fught for
control
they do not have.
You wish to control others, to dominate the conversation, to
force
your will upon people who might want to say something you
dont want
to hear.
So still the webpages are there--some of you illegally using
my
copyrighted material to insult me--and you people refuse to
give up
your attempts at control despite the years, despite the
stupidity of
it all, despite the immorality.
But you do not control me. I post as I will despite your
webpages,
despite emails you might send, despite the dedication with
which you
try to push me this way or that, or to push others, though,
yes, most
posters bend to your will.
They are cows, and cowards. I watch them come and go over the
years
terrifued to ever say anything at all objective about my work
for fear
that theyll be mobbed, as they will.
But I will not be ruled. I will not be controlled by you. I
will not
be conquered.
And I will win. I have a paper at a major math journal. If
they try
to slide out of publication like others before them, it will
go to
another, and another, and another, as I adjust, shift the
wording,
learn the game, play the politics necessary to get published.
And if it takes a decade I WILL GET PUBLISHED.
And then you will be part of history, part of the sad story of
mobbings, and angry people willing to do so much wrong for
the sake of
their own sense of control. Yet another sad sorry tale among
so many
in a world of people who never learn their history.
In a world where people refuse to learn from the mistakes of
the past,
not only to repeat them, but to wallow in the misery they
create for
others, to celebrate the destruction they wreak, and to pride
themselves until the day they funally fall.
And humanity is so much the worse for all of the stink of it.
James Harris
===
Subject: Re: Modern censorship
James Harris <jstevh@msn.com> scribbled the following:
> You read in history books about individuals harried by mobs,
> continually barraged with various attacks, who face
outrageous
> behavior at the hands of some group, and then we come to a
supposedly
> enlightened age with extraordinary tools for sharing
information--and
> the same damn thing happens again.
(snip further crap)
James, no one cares. When you get published, notify us. Last
I recall,
you were working on proving Fermats Last Theorem and 
funding a
polynomial-time prime funding algorithm. Thats 
great. If you
make any
progress with them, tell us. But dont whine and whine about
how
everyone is against you. The only way to make people know
about your
profuciency in math is to actually show your work, not to
whine about
persecution and brag about how great you are.
--
/-- Joona Palaste (palaste@cc.helsinki.fu) -------------
Finland --------
--------------------------------------------------------
rules! --------/
No, Maggie, not Aztec, Olmec! Ol-mec!
- Lisa Simpson
===
Subject: Re: Modern censorship
> You read in history books about individuals harried by mobs,
[snip lengthy diatribe]
> You people cheat.
[snip whining about cheats]
> They are cows, and cowards.
[snip whining about cows and cowards]
> And I will win. I have a paper at a major math journal. If
they try
> to slide out of publication like others before them, it
will go to
> another, and another, and another, as I adjust, shift the
wording,
> learn the game, play the politics necessary to get
published.
Just curious: the most recent version of your paper on
polynomial
factorization, co-authored by you and A. Beckwith, at:
http://www.ne-plus-ultra.net/index.php?option=content&task=
view&id=46&Itemid
=26
includes the following material toward the end:
Now letting m = 1, f = sqrt(5), where I can let u = 1 as its
value doesnt change the as, I have
(m^3 f^6 - 3 m^2 f^4 + 3 m) x^3 - 3 (-1 + m f^2) x u^2 + u^3
= 65 x^3 - 12 x + 1 ...
But clearly, if you substitute m = 1, f = sqrt(5), and u = 1
into the expresson on the left side, you get:
53 x^3 - 12 x + 1.
So thats obviously wrong.
Not only that, but your original polynomial in that paper is
written as
P(m) = f^2 ((m^3 f^4 - 3 m^2 f^2 + 3 m) x^3 - 3(-1 + m f^2)x
u^2 + u^3
f).
Now, if I substitute m = 1, f = sqrt(5), u = 1 into THIS
expression, I get:
P(m) = 65 x^3 - 60 x + 15 sqrt(5)
which of course ALSO does not equal
65 x^3 - 12 x + 1.
So what I am curious about is: is this the version you have
submitted to a journal?
Andrzej
[snip still more whining]
> And humanity is so much the worse for all of the stink of
it.
> James Harris
===
Subject: Re: Modern censorship
!3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(
5eZ41to5f%E@ELIi
$t^
VcLWP@J5p^rst0+(Ô>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw
> Over the years that Ive posted on PUBLIC forums the thing
that has
> stood out is how out there, without shame, and without even
hesitating
> to hide what theyre doing--groups of people have made it
their
> business to try to censor what I write.
[...]
> So still the webpages are there--some of you illegally
using my
> copyrighted material to insult me--
Oh, they would not want to have you censored.
--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
===
Subject: Re: Modern censorship
>You read in history books about individuals harried by mobs,
>continually barraged with various attacks, who face
outrageous
>behavior at the hands of some group, and then we come to a
supposedly
>enlightened age with extraordinary tools for sharing
information--and
>the same damn thing happens again.
>Over the years that Ive posted on PUBLIC forums the thing
that has
>stood out is how out there, without shame, and without even
hesitating
>to hide what theyre doing--groups of people have made it
their
>business to try to censor what I write.
You continue to make this ridiculous claim, no matter how
many times people point out that they cant possibly censor
what you write.
Hmm. Sounds like Im surprised that you continue to say
something
stupid after the stupidity of it has been pointed out. Never
mind.
>Time after time theyd post to tell me to go away, or to
tell others
>to ignore me, as these people made it their business to try
to
>control.
>When harassing me on Usenet wasnt enough, they took to
putting up
>webpage--and if another one of you tries to claim that I
harassed
>David Ullrich, a math professor who after harassing me for
years
>dragged race into the picture talking about racial slurs,
because I
>complained about him bringing race into the picture to his
university,
>then you just make my point that much more forcefully. Race
is not a
>tool to be used by some unethical math professor who thinks
its a
>neat way to insult someone on Usenet. David Ullrich was
wrong to try
>to attack me with race, and I was right to call him on it,
and
>complain to his school.
If Id done what you say you might be right, perhaps.
[*] But youve _stated_ right here that the reason you
complained
about me to OSU was to get me to stop posting in your threads.
That makes _you_ the _only_ person Im aware of who _has_
taken
actual steps to try to censor someone on usenet.
>You people cheat.
>to a journal, so some of you emailed that journal to get the
paper
>censored.
Nobody here said the journal should yank the paper - the
concensus
here is that they did the wrong thing in doing so.
>You are disgusting cretins who follow no rules, no moral
obligations,
>and you are irrational.
See [*] above.
>I can argue point for point, point by point, explain over
and over
>again, as Ive done for years, and one of you will just
disagree to be
>disagreeable!
>I offer compromise and get spat upon.
The compromise you offered was in regard to mathematical
facts -
it doesnt work that way.
Also, as I pointed out once, that compromise would not have
had
the effect you wanted anyway: It was something about how
youd make
some modifucation to some paper if wed agree 
not to be against
its publication. That wouldnt work, because sci.math has
nothing
to do with why journals reject your papers.
>History shows that there are always those of the mob, who
take it upon
>themselves to try and control the few, or especially, the
one.
>Im thankful that this plays out over the Internet.
>In the past you are the people who would be tying someone to
a stake
>to burn them and then, blaming the victim, shout your
morality to the
>heavens, as if God listens to loudness above reason.
Hint: the previous paragraph is _very_ funny, for at least two
reasons. Can you identify one or both?
>You are the reason that we have a society of today where so
many
>problems will not get addressed because a few people fught
for control
>they do not have.
>You wish to control others, to dominate the conversation, to
force
>your will upon people who might want to say something you
dont want
>to hear.
>So still the webpages are there--some of you illegally using
my
>copyrighted material to insult me--and you people refuse to
give up
>your attempts at control despite the years, despite the
stupidity of
>it all, despite the immorality.
>But you do not control me.
True. So whats this nonsense about censorship? A censor
_does_
control things, by defunition.
>I post as I will despite your webpages,
>despite emails you might send, despite the dedication with
which you
>try to push me this way or that, or to push others, though,
yes, most
>posters bend to your will.
>They are cows, and cowards. I watch them come and go over
the years
>terrifued to ever say anything at all objective about my work
for fear
>that theyll be mobbed, as they will.
>But I will not be ruled. I will not be controlled by you. I
will not
>be conquered.
>And I will win. I have a paper at a major math journal. If
they try
>to slide out of publication like others before them, it will
go to
>another, and another, and another, as I adjust, shift the
wording,
>learn the game, play the politics necessary to get published.
>And if it takes a decade I WILL GET PUBLISHED.
Might happen - its happened before that respectable 
journals
have
published total nonsense.
(I forget the name of the journal, but it was a big one that
published a paper where the main result started by covering
the unit sphere by funtely many disjoint spherical caps. 
People
were talking about how the entire editorial staff should
resign...
Rudin published a well, duh sort of refutation in the same
journal.)
>And then you will be part of history, part of the sad story
of
>mobbings, and angry people willing to do so much wrong for
the sake of
>their own sense of control. Yet another sad sorry tale among
so many
>in a world of people who never learn their history.
>In a world where people refuse to learn from the mistakes of
the past,
>not only to repeat them, but to wallow in the misery they
create for
>others, to celebrate the destruction they wreak, and to pride
>themselves until the day they funally fall.
>And humanity is so much the worse for all of the stink of it.
>James Harris
************************
David C. Ullrich
===
Subject: Re: Modern censorship
Discussion, linux)
> So still the webpages are there--some of you illegally
using my
> copyrighted material to insult me--and you people refuse to
give up
> your attempts at control despite the years, despite the
stupidity of
> it all, despite the immorality.
*Because* theyre using your material to insult you, 
its
probably
perfectly legal. Your posts have been excerpted and
paraphrased for
the purpose of criticism. This is a well-established fair use
exception in U.S. copyright law and likely elsewhere, too.
You keep ignoring this fact. You prefer to pretend that 
youve
obviously been wronged by scofßaws. But it just aint 
so.
--
Jesse F. Hughes
My experience indicates that the people who post on this
newsgroup
are about at the level of a 10 year old in the year 2060.
-- More wisdom from James Harris, time traveler
===
Subject: Re: Modern censorship
[snippage]
> You read in history books about individuals harried by mobs,
> continually barraged with various attacks, who face
outrageous
> behavior at the hands of some group, and then we come to a
supposedly
> enlightened age with extraordinary tools for sharing
information--and
> the same damn thing happens again.
> Over the years that Ive posted on PUBLIC forums the thing
that has
> stood out is how out there, without shame, and without even
hesitating
> to hide what theyre doing--groups of people have made it
their
> business to try to censor what I write.
> Time after time theyd post to tell me to go away, or to
tell others
> to ignore me, as these people made it their business to try
to
> control.
You seem to be confusing Ôdisagreement with 
Ôattempted
censorship.
> You are the reason that we have a society of today where so
many
> problems will not get addressed because a few people fught
for control
> they do not have.
Ooh, political.
> You wish to control others, to dominate the conversation,
to force
> your will upon people who might want to say something you
dont want
> to hear.
> So still the webpages are there--some of you illegally
using my
> copyrighted material to insult me
So you still havent learned anything about copyright law.
Hint: the
relevant bits are the ones dealing with Ôfair 
use.
> I have a paper at a major math journal. If they try
> to slide out of publication like others before them, it
will go to
> another, and another, and another, as I adjust, shift the
wording,
> learn the game, play the politics necessary to get
published.
Your mistake is not failing to Ôplay ... 
politics, its
failing to
Ôlearn mathematics.
> And if it takes a decade I WILL GET PUBLISHED.
Hey, youve already been published, remember? You got
published in
GET A CORRECT RESULT PUBLISHED ?
By the way, what exactly will that achieve?
--
Larry Lard
Replies to group please
===
Subject: Re: Modern censorship
!3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(
5eZ41to5f%E@ELIi
$t^
VcLWP@J5p^rst0+(Ô>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw
> And if it takes a decade I WILL GET PUBLISHED.
> And then you will be part of history, part of the sad story
of
> mobbings, and angry people willing to do so much wrong for
the sake
> of their own sense of control. Yet another sad sorry tale
among so
> many in a world of people who never learn their history.
You are laboring under the delusion that getting published is
equivalent to make an impact. It isnt. It is merely a tiny
furst
step. And it is no substitute for having something worthwhile
to
say. You might trick somebody into publishing you. It wont
make a
difference. Except for the reputation of the publisher.
> In a world where people refuse to learn from the mistakes
of the
> past, not only to repeat them, but to wallow in the misery
they
> create for others, to celebrate the destruction they wreak,
and to
> pride themselves until the day they funally fall.
> And humanity is so much the worse for all of the stink of
it.
No hammer this time?
--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
===
Subject: Re: Modern censorship
> You read in history books about individuals harried by mobs,
> continually barraged with various attacks, who face
outrageous
> behavior at the hands of some group, and then we come to a
supposedly
> enlightened age with extraordinary tools for sharing
information--and
> the same damn thing happens again.
> Over the years that Ive posted on PUBLIC forums the thing
that has
> stood out is how out there, without shame, and without even
hesitating
> to hide what theyre doing--groups of people have made it
their
> business to try to censor what I write.
No-one makes it their business to censor what you write.
<snip>
===
Subject: Re: Modern censorship
Hoorah! A new season of JSH!
===
Subject: Re: Modern censorship
> Hoorah! A new season of JSH!
Yeah, but its just not the same. Im not sure 
what future
Harrisologians
will demark as the Golden Age, but it seems to have been over
for some time
now. The farce has grown stale. Once upon a time James
emulated a
mathematician in the same fascinating way that a Tamagotchi
emulated
something
alive. Now it just emits an annoying series of beeps over and
over: beeps
that used to mean feed me back when it was engaging enough
that we
amused
ourselves by ascribing to it an intelligence, some sense of
purpose.
He hoped for blazing glory; we awaited the blazing crash. But
nothing is
burning. All that remains is an incessant beep.
Beep. Beep. Beep.
===
Subject: Re: Modern censorship
>We awaited the blazing crash. But nothing is
>burning. All that remains is an incessant beep.
>Beep. Beep. Beep.
So sci.math = Wile E. Coyote?
===
Subject: [JSH] Re: Modern censorship
In sci.math, Dave Rusin
<rusin@vesuvius.math.niu.edu>
<cnnvm4$ph0$1@news.math.niu.edu>:
>>We awaited the blazing crash. But nothing is
>>burning. All that remains is an incessant beep.
>>Beep. Beep. Beep.
> So sci.math = Wile E. Coyote?
No, sci.math is the other side of the equation. :-)
JSH has yet to catch us, on, or up.
Note subject change.
--
#191, ewill3@earthlink.net
Its still legal to go .sigless.
===
Subject: Re: Modern censorship
> But you do not control me.
If so, why do you devote an entire lengthy post to the people
who dont
control you, instead of devoting it to the nominal topic of
the
newsgroup?
--
Transpose hotmail and mxsmanic in my e-mail address to reach
me directly.
===
Subject: Re: Modern censorship
>But you do not control me.
> If so, why do you devote an entire lengthy post to the
people who dont
> control you, instead of devoting it to the nominal topic of
the
> newsgroup?
That at least is a good question.
The full answer is that the attempts at control involve
whats in the
subject line: censorship.
That involves limiting or stopping content, like people
telling me not
to post, or to post only on certain subjects or various other
types of
control type requests.
The only such requests I acknowledge as being pertinent are
ones about
limiting to math and topics related to the newsgroup itself.
The
topic here is troubling censorship attempts from newsgroup
members
acting often as a gang determined to control my postings, so
its
topical.
Here you have *mob* behavior as well, with a lot of unfair
tactics as
various groups of people do more than one thing in an attempt
to
control what and how I post.
Its actually kind of bizarre, and really, really strange
given my
history of ignoring such requests and using them instead to
simply
gain more attention.
But these people keep trying. Thats kind of interesting, as
what can
go so wrong in peoples heads that no matter what they just
keep at
the same losing proposition, year after year? Like what
motivates an
Erik Max Francis or a Dik Winter, or a David Ullrich, among
others?
Why do they keep trying given their lack of success so far?
Theyre not only losing, theyre losing big, 
but they keep
going.
Why?
Part of me funds such dedication admirable, but then again,
how do you
justify continuing when you NEVER win? Like if you think Im
deluded
and they do win, name one victory. Name a single victory from
any of
those people.
Not something fake, but some way theyve actually won, a
single time.
Yet here I am, more dominant than ever, waiting on a math
paper likely
to get published, when I will come and really stomp them into
the
ground, so their losses become complete, and more biting, as
well as
more direct.
So, whats their motivation?
James Harris
===
Subject: Re: Modern censorship
> Yet here I am, more dominant than ever
/ ____
<> ( oo )
<>_| ^^ |_
<> @
/~~ . . _ |
/~~~~ | |
/~~~~~~/ _| |
|[][][]/ / [m]
|[][][[m]
|[][][]|
|[][][]|
|[][][]|
|[][][]|
|[][][]|
|[][][]|
|[][][]|
|[][][]|
|[|--|]|
|[| |]|
|[[ ]]|
> James Harris
===
Subject: Re: Modern censorship
> Yet here I am, more dominant than ever,
You must be proud of your earthshaking contributions to
mathematics.
Tell me, are you aware of any buzz about you winning a Fields
Medal?
Oh, I know youre probably not supposed to talk about it, 
but
Im
really looking forward to seeing a transcript of your
acceptance
speech. That is, if you decide to accept. The thing is, when
they
award you a Fields Medal, I guess youre supposed to feel
humbled to
be included in such an elite group. Douglas, Kodaira, Roth,
Hironaka,
Lions, . . . there are dozens of these jokers! Adding Harris
to the
list doesnt really seem appropriate. Rather, there should 
be
a list
like Archimedes, Newton, Gauss, Harris. Period. I mean, maybe
you
should take it -- be in their little club -- just to avoid
pissing
them off even more than you already have. Or maybe you just
deliver
an acceptance speech that *really* pisses them off. Like I
said, Id
love to hear it.
> waiting on a math paper likely
> to get published,
This is something you have in common with most, if not all,
of the
other Fields Medalists. They had math papers which were
likely to
get published. Indeed, later many of their papers were
published. Oh,
maybe a few of these guys just sort of espoused their
theories on the
internet or whatever preceded it, but most did, in fact,
publish their
work. So youre in good company there, when you do get it
published,
that is. But only *good* company. Not the great company you
deserve.
> when I will come and really stomp them into the
> ground, so their losses become complete, and more biting,
as well as
> more direct.
Well thats a good reason to accept your Fields Medal right
there. It
is well documented that Fields Medalists typically go on a
rampage of
revenge within a few weeks of accepting their prize. Its as
if the
Medal were some powerful talisman out of Dungeons and Dragons
-- one
which delights in being wielded by a powerful, chaotic evil
mathematician.
It is said that the math departments of seven universities
were decimated
by the wrath of Grothendieck. In the hands of a Harris, it
would wreak a
veritable mathematical apocalypse. Mathematics as we know it
would not
survive.
Of course, the committee is probably waiting for Jamess
paper to get
published before awarding him the medal.
===
Subject: Re: Modern censorship
===
>Subject: Re: Modern censorship
>>But you do not control me.
>> If so, why do you devote an entire lengthy post to the
people who dont
>> control you, instead of devoting it to the nominal topic
of the
>> newsgroup?
>That at least is a good question.
>The full answer is that the attempts at control involve
whats in the
>subject line: censorship.
>That involves limiting or stopping content, like people
telling me not
>to post, or to post only on certain subjects or various
other types of
>control type requests.
>The only such requests I acknowledge as being pertinent are
ones about
>limiting to math and topics related to the newsgroup itself.
The
>topic here is troubling censorship attempts from newsgroup
members
>acting often as a gang determined to control my postings, so
its
>topical.
>Here you have *mob* behavior as well, with a lot of unfair
tactics as
>various groups of people do more than one thing in an
attempt to
>control what and how I post.
>Its actually kind of bizarre, and really, really strange
given my
>history of ignoring such requests and using them instead to
simply
>gain more attention.
>But these people keep trying. Thats kind of interesting, 
as
what can
>go so wrong in peoples heads that no matter what they just
keep at
>the same losing proposition, year after year? Like what
motivates an
>Erik Max Francis or a Dik Winter, or a David Ullrich, among
others?
>Why do they keep trying given their lack of success so far?
Surely _you_ of all people know the answer to that.
>Theyre not only losing, theyre losing big, 
but they keep
going.
>Why?
>Part of me funds such dedication admirable, but then again,
how do you
>justify continuing when you NEVER win? Like if you think 
Im
deluded
>and they do win, name one victory. Name a single victory
from any of
>those people.
>Not something fake, but some way theyve actually won, a
single time.
How about getting your paper yanked from that journal?
That was pretty impressive to me, given that journals dont
normally do that. But then, most papers with errors arent
submitted by people comitting fraud.
>Yet here I am, more dominant than ever, waiting on a math
paper likely
>to get published, when I will come and really stomp them
into the
>ground, so their losses become complete, and more biting, as
well as
>more direct.
>So, whats their motivation?
Whats yours?
>James Harris
--
Mensanator
Ace of Clubs
===
Subject: Re: Modern censorship
Discussion, linux)
> Part of me funds such dedication admirable, but then again,
how do you
> justify continuing when you NEVER win? Like if you think
Im deluded
> and they do win, name one victory. Name a single victory
from any of
> those people.
> Not something fake, but some way theyve actually won, a
single
> time.
Golly. That *is* an interesting observation. What would make
someone
go on and on and on for *years* without accomplishing a damn
thing?
--
Jesse F. Hughes
Thats whats brutal about mathematics! When 
youre wrong,
you can
have spent years, and lots of effort, and come out at the end
with
nothing. -- James S. Harris on the path of self-discovery (?)
===
Subject: Re: Modern censorship
> Thats whats brutal about mathematics! When 
youre wrong,
you can
> have spent years, and lots of effort, and come out at the
end with
> nothing. -- James S. Harris on the path of self-discovery
(?)
My son, you are on the path to enlightenment. Its back
thataway.
--
Chris Henrich
God just doesnt fut inside a single religion.
===
Subject: Re: Modern censorship
>Thats whats brutal about mathematics! When 
youre wrong,
you can
>have spent years, and lots of effort, and come out at the
end with
>nothing. -- James S. Harris on the path of self-discovery (?)
> My son, you are on the path to enlightenment. Its back
thataway.
Hey, Im taking the supposedly proper path to being
recognized as I
send papers to math journals.
So whats wrong with the picture here?
Oh, wait, you people send emails to math journals attacking
the
journal process by cowing editors. I play by the rules. You
people
cheat, and then come back talking as if youre on the level.
You CHEAT, and then you go on as if its nothing, and 
youre
in the
right, when Im the researcher writing up papers and sending
them in
for review.
Posting is just a sideline now, though IN THE PAST, Id 
hoped
itd be
a short-cut to introducing new ideas, but faced a mob, which
is still
continuing but has muted effect now, until the mob can fund
another
journal to target, I guess.
The reality of publishing papers is that you dont
necessarily get a
journal that will publish on the furst try, as 
theres
politics in the
process.
Usually the editors have been nice, and Ive had thoughtful
replies
that dont claim error, but basically come down to politics.
Ive learned THATS NORMAL in the publication 
process and Im
NOT
being singled out here, which is nice :-).
In any event, you people can be as negative as you like, and
keep
playing your old games but the real story is not here--its
with the
journals.
So now Im just kind of goofung off, as I admit 
it, its fun
to post.
And I am not going to be censored by you people. So why are
you
replying then, eh?
Whats your motivation if youre not here to 
try and stop me
from
posting, and Im not interested in being stopped, and 
Im
sending
papers to journals anyway?
Why are you dimwits hounding me now? Eh?
James Harris
===
Subject: Re: Modern censorship
...
> Oh, wait, you people send emails to math journals attacking
the
> journal process by cowing editors. I play by the rules. You
people
> cheat, and then come back talking as if youre on the 
level.
Eh? It was not the journal process that was attacked in the
email, it
was the paper. The proper way for the journal would have been
to
publish the rebuttal of your paper in a later issue.
> You CHEAT, and then you go on as if its nothing, and
youre in the
> right, when Im the researcher writing up papers and
sending them in
> for review.
Sending in rebuttals is not cheating.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam,
nederland,
+31205924131
home: bovenover 215, 1025 jn amsterdam, nederland;
http://www.cwi.nl/~dik/
===
Subject: Re: Modern censorship
>> But you do not control me.
>If so, why do you devote an entire lengthy post to the
people who dont
>control you, instead of devoting it to the nominal topic of
the
>newsgroup?
> That at least is a good question.
> The full answer is that the attempts at control involve
whats in the
> subject line: censorship.
> That involves limiting or stopping content, like people
telling me not
> to post, or to post only on certain subjects or various
other types of
> control type requests.
I think you should post to sci.math every day. It keeps me
and people like
me interested in the group.
> The only such requests I acknowledge as being pertinent are
ones about
> limiting to math and topics related to the newsgroup
itself. The
> topic here is troubling censorship attempts from newsgroup
members
> acting often as a gang determined to control my postings,
so its
> topical.
> Here you have *mob* behavior as well, with a lot of unfair
tactics as
> various groups of people do more than one thing in an
attempt to
> control what and how I post.
> Its actually kind of bizarre, and really, really strange
given my
> history of ignoring such requests and using them instead to
simply
> gain more attention.
> But these people keep trying. Thats kind of interesting,
as what can
> go so wrong in peoples heads that no matter what they 
just
keep at
> the same losing proposition, year after year? Like what
motivates an
> Erik Max Francis or a Dik Winter, or a David Ullrich, among
others?
> Why do they keep trying given their lack of success so far?
> Theyre not only losing, theyre losing big, 
but they keep
going.
> Why?
> Part of me funds such dedication admirable, but then again,
how do you
> justify continuing when you NEVER win? Like if you think
Im deluded
> and they do win, name one victory. Name a single victory
from any of
> those people.
> Not something fake, but some way theyve actually won, a
single time.
was good and published and unretractable, and then published
their own
> Yet here I am, more dominant than ever, waiting on a math
paper likely
> to get published, when I will come and really stomp them
into the
> ground, so their losses become complete, and more biting,
as well as
> more direct.
> So, whats their motivation?
Truth, justice, and the American way!
===
Subject: Re: Modern censorship
===
>Subject: Re: Modern censorship
>Message-id: <vlgop0thlctv1c7amsolefu0c995b9smcu@4ax.com> But
you do not control me.
>If so, why do you devote an entire lengthy post to the
people who dont
>control you, instead of devoting it to the nominal topic of
the
>newsgroup?
At a meta-level, hes controlling us by forcing us to try to
control him.
But what he doesnt realize is that at a
meta-meta-level...oh, never mind.
>--
>Transpose hotmail and mxsmanic in my e-mail address to reach
me directly.
--
Mensanator
Ace of Clubs
===
Subject: Re: Modern censorship
You are a vile human being!
===
Subject: Re: Modern censorship
===
>Subject: Modern censorship
>You read in history books about individuals harried by mobs,
>continually barraged with various attacks, who face
outrageous
>behavior at the hands of some group, and then we come to a
supposedly
>enlightened age with extraordinary tools for sharing
information--and
>the same damn thing happens again.
Oh, youre back again. I thought you said you 
didnt need us
anymore.
>Over the years that Ive posted on PUBLIC forums the thing
that has
>stood out is how out there, without shame, and without even
hesitating
>to hide what theyre doing--groups of people have made it
their
>business to try to censor what I write.
Yeah, so what? Its a free country. Theres no 
law against
censoring you.
>Time after time theyd post to tell me to go away, or to
tell others
>to ignore me, as these people made it their business to try
to
>control.
Go away.
>When harassing me on Usenet wasnt enough, they took to
putting up
>webpage--and if another one of you tries to claim that I
harassed
>David Ullrich, a math professor who after harassing me for
years
>dragged race into the picture talking about racial slurs,
because I
>complained about him bringing race into the picture to his
university,
>then you just make my point that much more forcefully. Race
is not a
>tool to be used by some unethical math professor who thinks
its a
>neat way to insult someone on Usenet. David Ullrich was
wrong to try
>to attack me with race, and I was right to call him on it,
and
>complain to his school.
You were wrong. You were an asshole then and youre still an
asshole.
>You people cheat.
Tough titty for you, isnt it?
>to a journal, so some of you emailed that journal to get the
paper
>censored.
So what are you going to do about it?
>You are disgusting cretins who follow no rules, no moral
obligations,
>and you are irrational.
<blows raspberryI can argue point for point, point by point,
explain over and over
>again, as Ive done for years, and one of you will just
disagree to be
>disagreeable!
You can argue the points Ôtil youre blue in the 
face, youre
still wrong.
>I offer compromise and get spat upon.
P-tui!
>History shows that there are always those of the mob, who
take it upon
>themselves to try and control the few, or especially, the
one.
>Im thankful that this plays out over the Internet.
Were not. Whats the matter, not getting any 
hits on your
blog?
>In the past you are the people who would be tying someone to
a stake
>to burn them and then, blaming the victim, shout your
morality to the
>heavens, as if God listens to loudness above reason.
>You are the reason that we have a society of today where so
many
>problems will not get addressed because a few people fught
for control
>they do not have.
Most people just blame the Republicans.
>You wish to control others, to dominate the conversation, to
force
>your will upon people who might want to say something you
dont want
>to hear.
So?
>So still the webpages are there--some of you illegally using
my
>copyrighted material to insult me--and you people refuse to
give up
>your attempts at control despite the years, despite the
stupidity of
>it all, despite the immorality.
Boo-hoo.
>But you do not control me. I post as I will despite your
webpages,
Weve already noticed that.
>despite emails you might send, despite the dedication with
which you
>try to push me this way or that, or to push others, though,
yes, most
>posters bend to your will.
>They are cows, and cowards. I watch them come and go over
the years
>terrifued to ever say anything at all objective about my work
for fear
>that theyll be mobbed, as they will.
As they should. Anyone who supports you deserves to have a
new asshole
reamed.
>But I will not be ruled. I will not be controlled by you. I
will not
>be conquered.
And you wont get published, either. Guess what matters.
>And I will win. I have a paper at a major math journal. If
they try
>to slide out of publication like others before them, it will
go to
>another, and another, and another, as I adjust, shift the
wording,
>learn the game, play the politics necessary to get published.
Sure, if one journal got suckered by your fraud, theres
probably another.
Not that it matters. It wont survive being made public.
>And if it takes a decade I WILL GET PUBLISHED.
Care to make a wager on that?
>And then you will be part of history, part of the sad story
of
>mobbings, and angry people willing to do so much wrong for
the sake of
>their own sense of control. Yet another sad sorry tale among
so many
>in a world of people who never learn their history.
You mean there was another math crank who turned out to be
right all along?
>In a world where people refuse to learn from the mistakes of
the past,
>not only to repeat them, but to wallow in the misery they
create for
>others, to celebrate the destruction they wreak, and to pride
>themselves until the day they funally fall.
Please, please tell us what happened to The Hammer. My theory
is that
you accidentally deleted it off your computer. Am I right?
>And humanity is so much the worse for all of the stink of it.
Gee, well just have to muddle through, deprived of your
insights.
>James Harris
--
Mensanator
Ace of Clubs
===
Subject: Re: Modern censorship
> You read in history books about individuals harried by mobs,
> continually barraged with various attacks, who face
outrageous
> behavior at the hands of some group, and then we come to a
supposedly
> enlightened age with extraordinary tools for sharing
information--and
> the same damn thing happens again.
And the one thing they all have in common: They are dead.
Perhaps that is
your problem, you are still alive. Maybe we will see your
brilliance only
in your death.
- Tim
--
Timothy M. Brauch
NSF Fellow
Department of Mathematics
University of Louisville
email is:
news (dot) post (at) tbrauch (dot) com
===
Subject: Re: Modern censorship
> I can argue point for point, point by point, explain over
and over
> again, as Ive done for years,
But you dont. You just start a new thread and repeat
yourself.
===
Subject: Re: Modern censorship
> You read in history books about individuals harried by mobs,
I think theres a movie called Married to the Mob. Your 
movie
could be
Harried by the Mob!
===
Subject: Re: Modern censorship
>Time after time theyd post to tell me to go away, or to
tell others
>to ignore me, as these people made it their business to try
to
>control.
When people brush away annoying ßies, theyre not trying 
to
control
anything; they just want to be left in peace.
--
Stan Brown, Oak Road Systems, Tompkins County, New York, USA
http://OakRoadSystems.com/
And if youre afraid of butter, which many people are nowa-
days, (long pause) you just put in cream. --Julia Child
===
Subject: Help on a probabilty problem, HELP
Consider a particular Ph.D. student; we know that after she
begins her
Ph.D.
program, the number of years it takes her to complete her
studies is a
random
variable with distribution exp(1/4) (independent of when she
started).
Suppose
you know that the student completed her Ph.D. today. We wish
to estimate
how
long ago she started. Assume that the a priori distribution
of X is uniform
on
[3, 6].
a. Let X represent the number of years ago the student
started. Let Y
represent the observation of when she completes, relative to
today (so the
given observation is Y = 0).
Find fY | X(y | x) for x >= 0.
can not understand the meaning very precisely ,Can not
understand the
===
Subject: Re: meta-proof that p is not np
Hi Torkel,
>The halting probability Omega is the chain of irreducible
mathematical
>facts which he refers to. If these facts are computationally
>irreducible, it follows that you, as a computational agent,
cannot
>arrive at them using simpler (shorter) facts, that is to use
reasoning
>to fund a solution, etc. etc.
> Chaitin is indeed referring to Omega. What is the argument
for the
> claim that the only way to prove the logically irreducible
true
> statements of the form the n-th bit of Omega is i is to
directly
> asume them as a new mathematical axiom, without using
reasoning at all?
Well, I presented at least two arguments, one of which was
identical
to Chaitins own argument, but you dont seem 
to like them.
Let me try
a slightly different approach, simpler hopefully. (But alas,
you do
not seem to like any argument that you think would be wrong
wrt your
point of view, so...)
Take any bit of Omega, there is no way to know _in general_
this
single bit of information from something else, in the sense of
computing it from something, or more mathematically speaking
proving it from some axioms (which are not distinct things, of
course). We can however, use some clever procedures to
compute some
bits of Omega exactly below, as in Caludes paper.
http://www.cs.auckland.ac.nz/CDMTCS/researchreports/
167omega.pdf
The answer to your question is contained in the Theorem 5 and
Theorem
6 of this paper. There are no philosophical points to be made
about
these theorems. If you are unsure of their relevance, you can
ask me
furst. If youre not satisfued with 
my explanation of this
paper or
your comprehension of it, you can always ask either Calude or
Chaitin,
and Im sure as respectable scholars they would gladly 
answer
your
inquiries.
That is, we are basing our argument on Chaitins version of
Godelian
incompleteness. We say that no (funite) formal axiomatic
system can
ever settle arbitrarily many bits of Omega. In fact, for the
case of
ZFC, this is quite bad already, it can _know_ only very few
bits of
Omega (a reminder: all Omegas have essentially identical
information
content due to universality of computation!). Why? Because
ZFC is a
really simple system. It knows so little!
If reasoning is proving or computation, as is implicitly
assumed in
Chaitins writings, then his conclusions easily follow.
However, he
does not make the kind of exact metaphysical arguments that I
like. I
would have talked much more precisely, as was the case with my
previously presented arguments.
Now, this is a _very_ arguable assumption. Perhaps, there are
other
paths to knowledge, other than empirical trials, but my
magic-ball
says no. (So, if I were to write a philosophical paper, I
would try to
rest this on some principle which is easier to accept, like
some
all-sweeping empiricist statement or physicalism which was
how I
argued for Chaitins thesis before. ) How about yours?
--
Eray Ozkural
PS: Im hoping that you will actually read the referred
theorems.
PS2: It should not come as a surprise that the shakey part of
Raatikainens arguments have no place in this discussion.
There is no
such thing as a computer that can simply compute any n-bits
of Omega
in funite time.
===
Subject: Re: meta-proof that p is not np
> Take any bit of Omega, there is no way to know _in general_
this
> single bit of information from something else, in the sense
of
> computing it from something, or more mathematically speaking
> proving it from some axioms (which are not distinct things,
of
> course).
There is indeed no computational procedure which proves all
true
statements of the form the n-th bit of Omega is i. The same
is true
of any undecidable set of true statements - nothing special
about
Omega here. What I am asking about is the claim that
statements
of the form the n-th bit of Omega is i are true for no reason
and can only be postulated, not proved.
> That is, we are basing our argument on Chaitins version 
of
Godelian
> incompleteness. We say that no (funite) formal axiomatic
system can
> ever settle arbitrarily many bits of Omega.
And how does this fact imply that the truth of a statement of
the
form the n-th bit of Omega is i can only be postulated, not
arrived
at by reasoning?
Your rambling is nothing to the point.
===
Subject: Re: infunitesimal calculation ?
>I am trying to get addtionnal data on infunitesimal numbers
dx.
A Mathematical perspective or a historical perspective?
Historically, you are talking about what Newton called
ßuxions, and
the whole concept is inconsistent. However, there are two
modern[1]
concepts using similar nomenclature.
1. Differential forms, which are defuned in terms of germs of
functions, that is, dF_X is the set of all continuous[2]
functions that agree with F in a neighborhood of F.
2. Infunitesimals in nonstandard analysis (NSA), and axiom
systems
that produce the same types of number systems with different
machinery.
>I think (memory) that the defunition may be something like
whatever
>e>o, 0<|dx|<e where dx is the object satisfying such
defunition.
For that to be consistent, you have to say something like
whatever
standard e>o, 0<|dx|<e
>Therefore, I lately (a shame for me) discover that we can
extend
>consistently (and simply) the axiomatic set theory
>(zermelo-fraenkel) with a new property standard and just 3
axioms
That isnt strictly necessary; its just a 
convenience to
allow you to
avoid dealing with ultrfailters. If youre willing to do the
work, you
can defune everything that you need within ZFC.
[1] But not particularly new.
[2] Generally with additional differentiability conditions
imposed.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT
<http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply
to
domain Patriot dot net user shmuel+news to contact me. Do not
===
Subject: Re: infunitesimal calculation ?
>I am trying to get addtionnal data on infunitesimal numbers
dx.
> A Mathematical perspective or a historical perspective?
mathematics (with possible application to physics). I am
almost sure
that this topic has been already handled since 1960, but I
must admit
that I have missed this aspect.
>Therefore, I lately (a shame for me) discover that we can
extend
>consistently (and simply) the axiomatic set theory
>(zermelo-fraenkel) with a new property standard and just 3
axioms
> That isnt strictly necessary; its just a 
convenience to
allow you to
> avoid dealing with ultrfailters. If youre willing to do
the work, you
> can defune everything that you need within ZFC.
This is one of the points I want to quickly know/understand.
I do not
know the ultra fulters, but I think, it may be a way to
construct the
non standard objects based on the ZFC axioms.
I also understand that the 3 added axioms (Idealisation,
transfer and
standardisation: it is a translation, so I do not know if it
is the
correct terms in English ;) of the non standard analysis are
~logically independent (they can be added or not to ZFC).
Therefore, the introduction of these objects should require
the
addition of new defunitions covering at least a part of these
axioms.
Does the ultra fulters theory add new 
axioms/defunitions
equivalent to
a part of the non standard analysis axioms?
An almost identical point concerns the possible construction
of the
non standard (and also standard) real numbers through infunite
countable real number sequences (ZFC). Now, If I assume that
the
collection of all infunite countable real sequence is a ZFC
set (I
havent tried to demonstrate it, so may be It is a wrong
hypothesis
;), I do not see why we need the idealisation axiom to
introduce the
non standard reals (we just need the ZFC without any new
axioms/objects).
Can anyone help me?
Seratend.
===
Subject: Re: do I understand co-variant vs. contra-variant?
<4190db6a$12$fuzhry+tra$mr2ice@news.patriot.net>
<cmtc77$f9q@netnews.proxy.lucent.com>
>Some people use different notations, I hope
>these are clear.
Not really, because you seem to be mixing some very different
things.
> S^u = (&S^u / &X^v) X^v (contravariant)
Thats fune as long as youre 
dealing with affune coordinates.
It
breaks down when you have curvilinear coordinates.
> S_u = (&X^v / &S^u) X_v (covariant)
You havent defuned S_u. Presumably 
youre assuming a metrix
and
setting S_u to g_uv S^v.
>Do a bit of algebra on the contravariant,
Things get trick once you drag in differential forms or
operators. The
form dX^v is covariant, but the system of forms {dX^v} is
contravariant.
> S^u dX^v = (&S^u / &X^v) X^v dX^v
> = dS^u X^v
>Multiply by g_uv and get
> S dX = X dS (contravariant).
ITYM S.dX = X.dS (scalar)
--
Shmuel (Seymour J.) Metz, SysProg and JOAT
<http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply
to
domain Patriot dot net user shmuel+news to contact me. Do not
===
Subject: Re: do I understand co-variant vs. contra-variant?
>Some people use different notations, I hope
>these are clear.
> Not really, because you seem to be mixing some very
different things.
> S^u = (&S^u / &X^v) X^v (contravariant)
> Thats fune as long as youre 
dealing with affune
coordinates. It
> breaks down when you have curvilinear coordinates.
Thats why it was presented as basic example.
> S_u = (&X^v / &S^u) X_v (covariant)
> You havent defuned S_u. Presumably 
youre assuming a metrix
and
> setting S_u to g_uv S^v.
Yes, if the g_uv cannot be transformed to unity
then the covariant-contravariant difference
cannot be transformed away.
I initially solved S dS = X dX as
S^2 = X^2 + k^2 (covariant)
and the k being invariant cannot be transformed
away. In GR this is sometimes refered to as a
Generally Covariant relation because SdS = XdX
is obviously invariant.
>Do a bit of algebra on the contravariant,
> Things get trick once you drag in differential forms or
operators. The
> form dX^v is covariant, but the system of forms {dX^v} is
> contravariant.
Well, thats a terminology I wont use.
> S^u dX^v = (&S^u / &X^v) X^v dX^v
= dS^u X^v
>Multiply by g_uv and get
> S dX = X dS (contravariant).
> ITYM S.dX = X.dS (scalar)
What happened, I dont understand that???
Ken S. Tucker
===
Subject: Re: do I understand co-variant vs. contra-variant?
> Edward, I think I can speak for the mathematical
> community this way, (by defunition)
> S*dS = X*dX == covariant
> S*dX = X*dS == contravariant
> Starting with that would any SOB like to argue
> mathematics with me?
Fellow traveller Ken:
It would seem scurilous of me not to reply to your
painstakingly
contructed follow up posts here, even if I cant understand
just what
you are talking about. So let me say that I fund the above
mathematical haiku very beautiful, and I am quite sure it
applies to
_something_!
However, let me take this opportunity to gratuitously
synopsize the
results of my own heady researches into covariant and
contravariant vectors.
Consider X,Y, the column vector 
representations of a vector
X and a
covector Y in a furst coordinate system, X,Y, the
representations in
a second, and a matrix A such that X = AX. Further consider
possible distinct applications of the matrix operators t and i
(transpose and inverse) to A.
Then:
X = AX ; Y = A^it Y ; X = 
A^i X ; Y = A^t Y
So (A,A^it) forms a pair giving the forward transformation of
vectors
and covectors, (A^i,A^t) a second pair giving the reverse
transformations. Which matrix we write unadorned is completely
arbitrary, as none of the four matrices is more direct
inverse or
transpose than any of the others; a similar remark applies to
vectors vs. covectors.
===
Subject: Re: do I understand co-variant vs. contra-variant?
>Edward, I think I can speak for the mathematical
>community this way, (by defunition)
S*dS = X*dX == covariant
S*dX = X*dS == contravariant
Starting with that would any SOB like to argue
>mathematics with me?
> Fellow traveller Ken:
> It would seem scurilous of me not to reply to your
painstakingly
> contructed follow up posts here, even if I cant 
understand
just what
> you are talking about. So let me say that I fund the above
> mathematical haiku very beautiful, and I am quite sure it
applies to
> _something_!
Ha, if math is beautiful, make it look pretty.
> However, let me take this opportunity to gratuitously
synopsize the
> results of my own heady researches into covariant and
> contravariant vectors.
> Consider X,Y, the column vector 
representations of a
vector X and a
> covector Y in a furst coordinate system, X,Y, the
representations in
> a second, and a matrix A such that X = AX. Further 
consider
> possible distinct applications of the matrix operators t
and i
> (transpose and inverse) to A.
> Then:
> X = AX ; Y = A^it Y ; X = 
A^i X ; Y = A^t Y
> So (A,A^it) forms a pair giving the forward transformation
of vectors
> and covectors, (A^i,A^t) a second pair giving the reverse
> transformations. Which matrix we write unadorned is
completely
> arbitrary, as none of the four matrices is more direct
inverse or
> transpose than any of the others; a similar remark applies
to
> vectors vs. covectors.
Looks good. (I usually use component notation
as that is more often used in physics but this is
cross-posted to sci.math, the bright guys).
What youve written is true in algebra, but
is it true in geometry. The basis of covariant
and contravariant objects is there relation.
May I ask you to relate 1 cm to 1 inch?
Ken S. Tucker
===
Subject: Re: do I understand co-variant vs. contra-variant?
> Consider X,Y, the column vector 
representations of a
vector X and a
> covector Y in a furst coordinate system, X,Y, the
representations in
> a second, and a matrix A such that X = AX. Further 
consider
> possible distinct applications of the matrix operators t
and i
> (transpose and inverse) to A.
> Then:
> X = AX ; Y = A^it Y ; X = 
A^i X ; Y = A^t Y
> So (A,A^it) forms a pair giving the forward transformation
of vectors
> and covectors, (A^i,A^t) a second pair giving the reverse
> transformations. Which matrix we write unadorned is
completely
> arbitrary, as none of the four matrices is more direct
inverse or
> transpose than any of the others; a similar remark applies
to
> vectors vs. covectors.
Sure, as far as that goes.
But vectors are fundamentally different from covectors,
because a vector
points and a covector copoints. That is, vectors are the base
objects,
and covectors are defuned as real functions on vectors. And
while there
is an isomorphism between them that can be used to
interchange vectors
with covectors, that is not natural (this is one type of
duality
transform).
A vector can be visualized as a little arrow with an obvious
interpretation of points. A covector can be visualized as
a set of nested surfaces which copoint in the direction of
a vector normal to the surfaces. The covector is a function of
a vector, and that can be visualized as a count of the number
of surfaces pierced by the arrow.
Vectors and covectors behave in a fundamentally different
manner when
one applies a mapping to the underlying manifold: a vector
behaves
naturally under the push-forward of the mapping, and a
covector behaves
naturally under the pull-back, so vectors are naturally
covariant and
covectors are naturally contravariant. This is intrinsic to
their
defunitions, and the only ambiguity is the historical
confusion in the
defunitions of covariant and contravariant.
Beware: dont confuse mappings of the manifold with
transforms among coordinate systems applied to the manifold!
The latter affect your representations of vectors and
covectors, but dont affect the vectors or covectors
themselves. Given suitable conditions, the mappings of
the manifold dont affect your representations but do
affect the vectors and covectors themselves. The existence
of a commuting category diagram here permits (sloppy)
physicists to ignore the distinctions, and historically
most have done so (as have many mathematicians). The search
for a theory of Quantum Gravity has required much more care
in this area....
Tom Roberts tjroberts@lucent.com
===
Subject: Re: do I understand co-variant vs. contra-variant?
Great post Tom, and Edward, Ill keep that as
a ref.
[snip good stuff]
> A vector can be visualized as a little arrow with an obvious
> interpretation of points. A covector can be visualized as
> a set of nested surfaces which copoint in the direction of
> a vector normal to the surfaces. The covector is a function
of
> a vector, and that can be visualized as a count of the
number
> of surfaces pierced by the arrow.
The word surface, while conventional may exclude
covariant and contravariant defunitions in 1 dimension.
For example, in GR we are entitled to analyse unit
lengths in the direction of radius by employing only
ds^2 = g_11 dx^1 dx^1 = g^11 dx_1 dx_1 , (dt=0).
and we know g_11 ~ 1/g^11, hence g_11 =/= g^11, so
the covariant and contravariant measurements differ
in 1D.
Along that 1D line, a small unit length x is transformed
two ways,
x = (&x/&x)*x == contravariant
x = (&x/&x)*x == covariant
and provides,
x dx = x dx == contravariant
x dx = x dx == covariant
(I previously expressed that using S and dS
for x and dx, sorry for the confusion).
[...]
> The search
> for a theory of Quantum Gravity has required much more care
> in this area....
Youve mentioned that before, may we ask why you
think that.
Ken S. Tucker
> Tom Roberts tjroberts@lucent.com
===
Subject: Re: do I understand co-variant vs. contra-variant?
> But vectors and tensors are the natural language of
physics. One key
> observation is that physical phenomena are utterly
independent of
> humans, or of their concepts; so physical phenomena must be
independent
> of coordinate system, and tensors are the natural
mathematical objects
> to model this.
Yahbut. Tensors are multi-linear mappings from a cartesian
product of n
copies of a vector space and m copies of the dual vector
space into the
real numbers. All of which are abstractions cooked up by
humans to
-model- some aspects of nature. Tensors and vectors live up
in our
heads, not Out There.
The co-ordinate free formulation of tensors is a matter of
simplifying
and cleaning up the math.
Bob Kolker
===
Subject: Re: do I understand co-variant vs. contra-variant?
>> But vectors and tensors are the natural language of
physics. One key
>> observation is that physical phenomena are utterly
independent of
>> humans, or of their concepts; so physical phenomena must be
>> independent of coordinate system, and tensors are the
natural
>> mathematical objects to model this.
> Yahbut. Tensors are multi-linear mappings from a cartesian
product of n
> copies of a vector space and m copies of the dual vector
space into the
> real numbers. All of which are abstractions cooked up by
humans to
> -model- some aspects of nature. Tensors and vectors live up
in our
> heads, not Out There.
Yes, of course. Thats what I said (...to model this). 
Thats
what
physics IS.
> The co-ordinate free formulation of tensors is a matter of
simplifying
> and cleaning up the math.
Based on observations of myself over several decades, and
dozens of
people in this newsgroup, the simplifying involved gives a
qualitatively different feel for and understanding of the
mathematical
and physical concepts involved. I simply did not understand
GR from
component-based books like Weinberg; but when I started
studying MTW a
lot of things fell into place quite quickly. And most of the
people
around here who are wedded to component notation simply 
dont
have a
clue....
Tom Roberts tjroberts@lucent.com
===
Subject: Re: Sum of an infunite series ....
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iAI29pE24373;
>Could someone let me know how to calculate the sum of the
following
series:
>1 + 4/7 + 9/49 + 16/343 + 25/2401 + .
>> I assume you mean the infunite series
>> 1+ (2^2)/7 + (3^2)/7^2 + (4^2)/7^3 +...,
>> in other words,
>> sum_1^infty (n^2)/7^(n-1).
>> You could start with
>> 1/(1-x)=sum_0^infty x^n
>> Differentiate with respect to x, multiply by x, then
differentiate
>> again. Finally, put x=1/7.
>> --Dan Grubb
>Wow. Thats amazing! That is sooo indirect but elegant. How
did you
>ever thought of that approach?
>Kira
Its a pretty well-known technique. Get hold of Concrete
Mathematics by Graham, Knuth, & Patashnik and read about
generating functions, and have fun.
Todd Trimble
===
Subject: Conversion of (x,z) 2D-space coordinates to (u,c)
2D-space
coordinates
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iAI29ox24319;
Hello there,
Im a Computing Science student from the Eindhoven 
University
of Technology
in the Netherlands and Im currently writing my 
masters
thesis on texture
synthesis for data visualization.
Ive created a texture synthesis model that uses u, v and c
(spatial
frequency, regularity and physical contrast) as its input
parameters. The
output is the texture itself and the (x,y,z)-coordinates of
the texture in a
perceptually uniform texture space.
X, y, and z are functions of u, v, and c and my UVC-To-XYZ
routines are
working perfectly. I am now trying to solve my XYZ-To-UVC
routines based on
the functions that I use to convert from u, v, and c to x, y,
and z. If v and
y are left out and I try to solve for u and c, the problem
boils down to the
following equations:
x = 1.11 + (-1.11 + 2.22 * u) * c^(0.5)
z = 2.13 * c^(0.87) - 1.14 * c * u^(2.46)
I simply want to solve for u and c (i.e. u = ... and c = ...
in
terms of only x and y), so I tried to use Mathematica
(5.0.1.0) on this:
NSolve[
{
x == 1.11 + (-1.11 + 2.22 * u) * c^(0.5),
z == 2.13 * c^(0.87) - 1.14 * c * u^(2.46)
},
{u,c}]
Strangely enough, Mathematica simply goes on and on (even on
powerful PCs)
but doesnt come up with a solution.
Danny Holten.
===
Subject: Simply connected, analytic
The following question occurred to me : Is it possible to map
a non-simply
connected domain in C to a simply connected domain in C via
an analytic
function? I cant think of any examples.
Isaac
===
Subject: Re: Simply connected, analytic
>The following question occurred to me : Is it possible to
map a non-simply
>connected domain in C to a simply connected domain in C via
an analytic
>function? I cant think of any examples.
Simnple examples have been _given_ in posts right here, in
replies
to _your_ posts.
C is simply connected. What is exp(C)?
>Isaac
************************
David C. Ullrich
===
Subject: Re: Simply connected, analytic
> The following question occurred to me : Is it possible to
map a non-simply
> connected domain in C to a simply connected domain in C via
an analytic
> function? I cant think of any examples.
> Isaac
What do you mean by to a simply connected domain? One can
simply
include a non-simply connected domain to a simply connected
one, so I
presume you at least mean surjection, but you should be more
precise
what you mean.
===
Subject: Re: Simply connected, analytic
>> The following question occurred to me : Is it possible to
map a
>> non-simply connected domain in C to a simply connected
domain in C via an
>> analytic function? I cant think of any examples.
>> Isaac
> What do you mean by to a simply connected domain? One can
simply
include
> a non-simply connected domain to a simply connected one, so
I presume you
> at least mean surjection, but you should be more precise
what you mean.
Yes Im sorry, I mean surjection. Is this possible?
===
Subject: Re: Simply connected, analytic
>> The following question occurred to me : Is it possible to
map a
>> non-simply connected domain in C to a simply connected
domain in C via
an
>> analytic function? I cant think of any examples.
>>
Lets see... C - {1}, the plane minus a point, is not simply
connected.
The image under f(z) = z^2 is C, which is simply connected.
Is that what you wanted?
--
G. A. Edgar
http://www.math.ohio-state.edu/~edgar/
===
Subject: Re: Simply connected, analytic
> The following question occurred to me : Is it possible to
map a
> non-simply connected domain in C to a simply connected
domain in C via
> an
> analytic function? I cant think of any examples.
Lets see... C - {1}, the plane minus a point, is not simply
connected.
> The image under f(z) = z^2 is C, which is simply connected.
> Is that what you wanted?
> --
> G. A. Edgar
> http://www.math.ohio-state.edu/~edgar/
Yes thank you.
===
Subject: Number theory reference
In some number theory book the following assertion is proved.
Consider eqn.(1) under the given conditions.
Aa^(1/2) + Bb^(1/2) = Cc^(1/2) + Dd^(1/2) (1)
Condition: A, B, C, D are integers each > 1
and a, b, c, d are square free positive integers.
Assertion:
If (1) is satisfued then either a = c and b = d or a = d and b
= c.
I would greatly appreciate if someone can kindly provide me
with a
reference.
===
Subject: Re: Number theory reference
> In some number theory book the following assertion is
proved.
> Consider eqn.(1) under the given conditions.
> Aa^(1/2) + Bb^(1/2) = Cc^(1/2) + Dd^(1/2) (1)
> Condition: A, B, C, D are integers each > 1
> and a, b, c, d are square free positive integers.
> Assertion:
> If (1) is satisfued then either a = c and b = d or a = d and
b = c.
> I would greatly appreciate if someone can kindly provide me
with a
reference.
This is a *special case* of the general fact that
the set off all square-free positive integers
is linearly independent over the rational numbers.
Proof by induction over the number of distinct prime factors
of the square-free numbers.
The proof is similar to, and only slightly more involved than
the proof that sqrt(2) is irrational.
Thomas
===
Subject: Question on e^(1/z)...
e^(1/z) is analytic on C {0}. However, on any circle around 0
of
arbitrary radius, the integral over that circle of e^(1/z) is
2*pi*i since
the residue is 1. Does this mean that e^(1/z) does not have a
primitive on
C {0} ? For if it did, then the integral around the closed
curve would
be
zero. Is my thinking logically correct?
Is this a good way to check that a general complex valued
function does
not have a primitive somewhere?
Isaac
===
Subject: Re: Question on e^(1/z)...
>e^(1/z) is analytic on C {0}. However, on any circle around
0 of
>arbitrary radius, the integral over that circle of e^(1/z)
is 2*pi*i since
>the residue is 1. Does this mean that e^(1/z) does not have
a primitive on
>C {0} ? For if it did, then the integral around the closed
curve would
be
>zero. Is my thinking logically correct?
>Is this a good way to check that a general complex valued
function does
>not have a primitive somewhere?
Yes, this is the best way to give a clear and concise proof
of things
that would otherwise be proved by a somewhat roundabout
method.
>Isaac
************************
David C. Ullrich
===
Subject: Re: Question on e^(1/z)...
> e^(1/z) is analytic on C {0}. However, on any circle around
0 of
> arbitrary radius, the integral over that circle of e^(1/z)
is 2*pi*i since
> the residue is 1. Does this mean that e^(1/z) does not have
a primitive
on
> C {0} ? For if it did, then the integral around the closed
curve would
be
> zero. Is my thinking logically correct?
Yes, an analytic function f on an open set V has a primitive
in V iff
int_gamma f(z) dz = 0 for every closed contour gamma in V.
===
Subject: Re: Question on e^(1/z)...
>e^(1/z) is analytic on C {0}. However, on any circle around
0 of
>arbitrary radius, the integral over that circle of e^(1/z)
is 2*pi*i since
>the residue is 1. Does this mean that e^(1/z) does not have
a primitive on
>C {0} ? For if it did, then the integral around the closed
curve would
be
>zero. Is my thinking logically correct?
Yes, of course.
>Is this a good way to check that a general complex valued
function does
>not have a primitive somewhere?
Yes.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
===
Subject: Re: Question on e^(1/z)...
the path integral of an analytic function defuned on a set D
over a
closed curve is zero only if the WHOLE interior of the curve
lies in D.
This is not the case here.
Karl
> e^(1/z) is analytic on C {0}. However, on any circle around
0 of
> arbitrary radius, the integral over that circle of e^(1/z)
is 2*pi*i since
> the residue is 1. Does this mean that e^(1/z) does not have
a primitive
on
> C {0} ? For if it did, then the integral around the closed
curve would
be
> zero. Is my thinking logically correct?
> Is this a good way to check that a general complex valued
function
does
> not have a primitive somewhere?
> Isaac
===
Subject: Re: Question on e^(1/z)...
>the path integral of an analytic function defuned on a set D
over a
>closed curve is zero only if the WHOLE interior of the curve
lies in D.
>This is not the case here.
>Karl
This is not true; it is true with if, rather than only if.
The function e^(1/x^2)) is analytic on C {0}, and has
residue 0 at 0. So its integral over a closed curve in
the domain is 0; however, if we multiply it by x, this
is no longer the case.
>> e^(1/z) is analytic on C {0}. However, on any circle
around 0 of
>> arbitrary radius, the integral over that circle of e^(1/z)
is 2*pi*i
since
>> the residue is 1. Does this mean that e^(1/z) does not
have a primitive
on
>> C {0} ? For if it did, then the integral around the closed
curve
would be
>> zero. Is my thinking logically correct?
>> Is this a good way to check that a general complex valued
function
does
>> not have a primitive somewhere?
>> Isaac
--
This address is for information only. I do not claim that
these views
are those of the Statistics Department or of Purdue
University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: Question on e^(1/z)...
> the path integral of an analytic function defuned on a set D
over a
> closed curve is zero only if the WHOLE interior of the
curve lies in D.
False: The path integral of 1/z^2 over the unit circle is 0.
===
Subject: Re: Question on e^(1/z)...
> the path integral of an analytic function defuned on a set D
over a closed
> curve is zero only if the WHOLE interior of the curve lies
in D.
> This is not the case here.
> Karl
I understand that. What I was saying, though is that it is
zero as well IF
it has a primitive defuned and analytic on a domain containing
the curve.
That is why I was trying
to conclude that e^(1/z) doesnt have a derivative. Am I
right?
>> e^(1/z) is analytic on C {0}. However, on any circle
around 0 of
>> arbitrary radius, the integral over that circle of e^(1/z)
is 2*pi*i
>> since the residue is 1. Does this mean that e^(1/z) does
not have a
>> primitive on C {0} ? For if it did, then the integral
around the
>> closed curve would be zero. Is my thinking logically
correct?
>> Is this a good way to check that a general complex valued
function
does
>> not have a primitive somewhere?
>> Isaac
===
Subject: Set theory
I have been asked to solve the following problem. Suppose X
is an infunite
set. Show that the cardinal number of X is less than the
cardinal number of
its power set. Now I have a problem with this. Suppose the
cardinality of X
is c. Now I thought that the only infunite cardinalities are c
and aleph
null which would mean that the cardinality of X is not less
than the power
set of X. What is going on here?????
===
Subject: Re: Set theory
> I have been asked to solve the following problem. Suppose X
is an
infunite
> set. Show that the cardinal number of X is less than the
cardinal number
of
> its power set. Now I have a problem with this. Suppose the
cardinality of
X
> is c. Now I thought that the only infunite cardinalities are
c and aleph
> null which would mean that the cardinality of X is not less
than the
power
> set of X. What is going on here?????
There are infunitely many infunite cardinals.
If PX is the power set of X and #X is the cardinality of X
then
#X < #PX = 2^{#X}.
===
Subject: Re: Set theory
> I have been asked to solve the following problem. Suppose X
is an
infunite
> set. Show that the cardinal number of X is less than the
cardinal number
of
> its power set.
You dont need the hypothesis that X is an 
infunite set. The
conclusion
holds for every X, whether funite or infunite.
> Now I have a problem with this. Suppose the cardinality of X
> is c. Now I thought that the only infunite cardinalities are
c and aleph
> null which would mean that the cardinality of X is not less
than the
power
> set of X. What is going on here?????
You were misinformed. There are infunitely many different
infunite
cardinalities.
--
Dave Seaman
Judge Yohns mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&
bookid=228>
===
Subject: Re: Set theory
>I have been asked to solve the following problem. Suppose X
is an
infunite
>set. Show that the cardinal number of X is less than the
cardinal
number
of
>its power set.
> You dont need the hypothesis that X is an 
infunite set. The
conclusion
> holds for every X, whether funite or infunite.
>Now I have a problem with this. Suppose the cardinality of X
>is c. Now I thought that the only infunite cardinalities are
c and
aleph
>null which would mean that the cardinality of X is not less
than the
power
>set of X. What is going on here?????
> You were misinformed. There are infunitely many different
infunite
> cardinalities.
> --
> Dave Seaman
> Judge Yohns mistakes revealed in Mumia Abu-Jamal ruling.
>
<http://www.commoncouragepress.com/index.cfm?action=book&
bookid=228>
Dr. Seaman,
Goodnight,
Steven
===
Subject: Re: Set theory
>I have been asked to solve the following problem. Suppose X
is an infunite
>set. Show that the cardinal number of X is less than the
cardinal number
of
>its power set.
Classic diagonalization. The subsets of X correspond to
functions from X to {0,1}, depending on whether or not a
given member is included or excluded in the subset. Assume
that there is a surjection f:X->P(X), and construct a subset
of X that is not the image of any member of X; this is done
is analogous manner as the diagonalization proof of the
uncountability of reals, except that instead of a sequence
of Ôdigits, one has the 
Ôdigits (0 and 1, or
exclusion/inclusion) indexed by the set X.
Requiring X to be infunite is actually irrelevant.
>Now I have a problem with this. Suppose the cardinality of X
>is c. Now I thought that the only infunite cardinalities are
c and aleph
>null which would mean that the cardinality of X is not less
than the
power
>set of X. What is going on here?????
No, the power set operation always builds a larger
cardinality.
The cardinality of the reals is c (by defunition), and the
cardinality of the power set of the realsis 2^c > c.
---
Stan Liou
===
Subject: Re: Set theory
> I have been asked to solve the following problem. Suppose X
is an
infunite
> set. Show that the cardinal number of X is less than the
cardinal number
of
> its power set. Now I have a problem with this. Suppose the
cardinality of
X
> is c. Now I thought that the only infunite cardinalities are
c and aleph
> null which would mean that the cardinality of X is not less
than the
power
> set of X. What is going on here?????
Not so. You can get larger and larger cardinalities by using
the power
set operator. Let P{A) be the power set of A. The |A| <
|P(A)| <
|P(P(A))| ... etc
Bob Kolker
===
Subject: Re: Set theory
>I have been asked to solve the following problem. Suppose X
is an
infunite
>set. Show that the cardinal number of X is less than the
cardinal number
of
>its power set. Now I have a problem with this. Suppose the
cardinality
of X
>is c. Now I thought that the only infunite cardinalities are
c and
aleph
>null which would mean that the cardinality of X is not less
than the
power
>set of X. What is going on here?????
Not so. You can get larger and larger cardinalities by using
the power
> set operator. Let P{A) be the power set of A. The |A| <
|P(A)| <
> |P(P(A))| ... etc
True, but that is the thing to be proved!
I wonder who asked a student to come up with this proof on
their own.
Its not the kind of thing one would think of ... in fact
most people
need to go through it many times before they really start to
understand
and believe it.
===
Subject: Re: Set theory
>>I have been asked to solve the following problem. Suppose X
is an
infunite
>>set. Show that the cardinal number of X is less than the
cardinal
number of
>>its power set. Now I have a problem with this. Suppose the
cardinality
of X
>>is c. Now I thought that the only infunite cardinalities are
c and
aleph
>>null which would mean that the cardinality of X is not less
than the
power
>>set of X. What is going on here?????
>> Not so. You can get larger and larger cardinalities by
using the power
>> set operator. Let P{A) be the power set of A. The |A| <
|P(A)| <
>> |P(P(A))| ... etc
>True, but that is the thing to be proved!
>I wonder who asked a student to come up with this proof on
their own.
>Its not the kind of thing one would think of ...
Depends on what was covered recently.
>in fact most people
>need to go through it many times before they really start to
understand
>and believe it.
Really? I dont see why that would be.
************************
David C. Ullrich
===
Subject: Re: Set theory
>I have been asked to solve the following problem. Suppose X
is an
infunite
>set. Show that the cardinal number of X is less than the
cardinal
number
of
>its power set. Now I have a problem with this. Suppose the
cardinality
of X
>is c. Now I thought that the only infunite cardinalities are
c and
aleph
>null which would mean that the cardinality of X is not less
than the
power
>set of X. What is going on here?????
> Not so. You can get larger and larger cardinalities by
using the power
> set operator. Let P{A) be the power set of A. The |A| <
|P(A)| <
> |P(P(A))| ... etc
> Bob Kolker
Bob,
So are you saying that there are more than 2 infunite
cardinalities? If so
(and it seems that you are saying just that) are there a
countable or
uncountable number of infunite cardinalities?
===
Subject: Re: Set theory
>> Not so. You can get larger and larger cardinalities by
using the power
>> set operator. Let P{A) be the power set of A. The |A| <
|P(A)| <
>> |P(P(A))| ... etc
>> Bob Kolker
>Bob,
>So are you saying that there are more than 2 infunite
cardinalities? If so
>(and it seems that you are saying just that) are there a
countable or
>uncountable number of infunite cardinalities?
Mr. Kolker is correct. There are defunitely more than 
countably
many infunite cardinalities, but as for the 
Ônumber of infunite
cardinalities, there is no such thing, in the sense that the
set of all infunite cardinal numbers does not exist, so we
cannot take its cardinality. In some sense, there are too many
cardinals to form a set (analogous to the situations like set
of all sets, etc.)
---
Stan Liou
===
Subject: Re: irreducible components
> f(x,y,z) is defuned by y^2=xz & z^2=y^3
> does this give me only these two irreducible components?
> 1) z=0, y=0 & x=x
> 2) z=1, y^2=x, 1=xy
> I think I goofed bigtime
y=0 <=> z=0 so your furst component is correct.
Otherwise y and z are both nonzero hence
yy = xz, zz = yyy
<=> yy = xz, zz = yxz
<=> yy = xxy, z = xy
<=> y = xx, z = xxx
Therefore (x, xx, xxx) is the second component.
The above deduction holds in any integral domain,
--Bill Dubuque
===
Subject: Re: irreducible components
>> f(x,y,z) is defuned by y^2=xz & z^2=y^3
>> does this give me only these two irreducible components?
>> 1) z=0, y=0 & x=x
>> 2) z=1, y^2=x, 1=xy
>> I think I goofed bigtime
> y=0 <=> z=0 so your furst component is correct.
> Otherwise y and z are both nonzero hence
> yy = xz, zz = yyy
> <=> yy = xz, zz = yxz
> <=> yy = xxy, z = xy
> <=> y = xx, z = xxx
> Therefore (x, xx, xxx) is the second component.
> The above deduction holds in any integral domain,
Above should be commutative semigroup.
The curve (t,t^2,t^3) is called the TWISTED CUBIC.
A web search will turn up much more about it,
e.g. here is a 3-d graph of the twisted cubic
which you can interactively manipulate
http://www.math.rutgers.edu/courses/535/535-f02/pictures/
twistedcubic.html
--Bill Dubuque
===
Subject: Post-sci-math Popular software at low low prices
boundary=--1951814562350648
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
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===
Subject: ANNO: new yahoo forum for curves and surfaces
Ive created a yahoo mailing list for discussing curves and
surfaces.
Please see:
http://xahlee.org/SpecialPlaneCurves_dir/
specialPlaneCurves.html
http://xahlee.org/surface/gallery.html
http://groups.yahoo.com/group/curves_surfaces/
Xah
xah@xahlee.org
http://xahlee.org/PageTwo_dir/more.html
===
Subject: suggest an easier text ??
The following book seems useful for what Im doing 
currently.
Author Freidlin, M. I. (Mark Iosifovich)
Title Random perturbations of dynamical systems / M.I.
Freidlin, A.D.
Wentzell ; translated by Joseph Sz.9fcs
Published New York : Springer-Verlag, c1984
However the level seems to be over my head.
can someone suggest a text that deals with similar topics
that is
geared towards undergraduate/ lower graduate levels?
===
Subject: Re: suggest an easier text ??
> The following book seems useful for what Im doing
currently.
> Author Freidlin, M. I. (Mark Iosifovich)
> Title Random perturbations of dynamical systems / M.I.
Freidlin, A.D.
> Wentzell ; translated by Joseph Sz.9fcs
> Published New York : Springer-Verlag, c1984
> However the level seems to be over my head.
> can someone suggest a text that deals with similar topics
that is
> geared towards undergraduate/ lower graduate levels?
I am not familiar with this text. Can you elaborate a little
bit on
what you are trying to do/learn? Are you trying to get over a
specifuc hump, or are you trying to learn a broad section of
stochastic processes?
--
===
Subject: Re: The construction of a tree saver from rabbits
relating to
tokamaks
(most snipped)
> So what I do is get 6 diameter black plastic drain piping
and use a
> very sharp high quality knife to cut it. I use the same
piping year
> after year. But when I furst began to use it I cut a line
down the
I am not very good at estimating a distance length without
measuring it
with a meter stick. There are some that can look at a piece
of board or
pipe and tell instantly whether it is 4 or 6. I am not.
I made a mistake in that the plastic pipe I am using is 4
diameter. When
I
make a mistake like this I go back and change it in my other
posts with the
symbol of (sic) after making the change. So it would look
like this in my
old post--
4 (sic). I used to use brackets instead of paranthesis but
found out that
some computer protocol looks at brackets as a command rather
than plain
text and so have gone from brackets to paranthesis. I used to
have trouble
with the reverse symbol of > in that the protocol on websites
does not
treat that symbol as plain text.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom where dots
of the electron-dot-cloud are galaxies
===
Subject: The decimal system is outlawed now ! ! !
We have to use the hexadecimal system now or the math-cops
will arrest
us!
So start using it now , because in $07D5 , all offenders
still using
the decimal system will be executed !
===
Subject: Re: The decimal system is outlawed now ! ! !
Hans-Marc Olsen <mallet_riveter@yahoo.co.in> scribbled the
following:
> We have to use the hexadecimal system now or the math-cops
will arrest
> us!
> So start using it now , because in $07D5 , all offenders
still using
> the decimal system will be executed !
If the decimal system is so evil, why are you writing the
year as $07D5?
That shows an underlying decimal bias. Why not simply write
it as 7D5?
Actually even the name hexadecimal itself has a decimal bias.
Hexa
is Greek for six and deca is Latin for ten, yet when you add
those two
numbers together, the hexadecimal number you get is 10, not
16.
--
/-- Joona Palaste (palaste@cc.helsinki.fu) -------------
Finland --------
--------------------------------------------------------
rules! --------/
Holy Banana of this, Sacred Coconut of that, Magic Axolotl of
the
other.
- Guardian in Jinxter
===
Subject: Re: The decimal system is outlawed now ! ! !
> Hans-Marc Olsen <mallet_riveter@yahoo.co.in> scribbled the
following:
>>We have to use the hexadecimal system now or the math-cops
will arrest
>>us!
>>So start using it now , because in $07D5 , all offenders
still using
>>the decimal system will be executed !
> If the decimal system is so evil, why are you writing the
year as $07D5?
> That shows an underlying decimal bias. Why not simply write
it as 7D5?
> Actually even the name hexadecimal itself has a decimal
bias.
Hexa
> is Greek for six and deca is Latin for ten, yet when you
add those
two
> numbers together, the hexadecimal number you get is 10, not
16.
You are understanding! A+6=10.
--
Will Twentyman
email: wtwentyman at copper dot net
===
Subject: Re: The decimal system is outlawed now ! ! !
>We have to use the hexadecimal system now or the math-cops
will arrest
>us!
>So start using it now , because in $07D5 , all offenders
still using
>the decimal system will be executed !
Please let this threat become a thread in a different
newsgroup.
This is against the current trend of using more and more
paper year by
year!
Paper manufacturers have been advocating the binary system
since times
immemorable.
To put an end to joking for now:
In the 1950s chess champion Max Euwe made an issue of
teaching binary
arithmetic in primary schools.
Johan E. Mebius
===
Subject: Re: expectation value for ODE?
>does that exist?
>it that like funding a solution for a given ode?
>like when you have,
>dy/dt = m y
>where m is a random parameter.
>what is expected value of above?
What do you mean by value? Suppose you have a dynamical
system in
IR^n given by x = f(x,m), where m is an n-dimensional
parameter. You
may consider the initial value problem x=f(x,m0), x(0)=x0
and compute
x(T). Here x0,m0,T are fuxed.
OK. Lets assume you want to fund the 
distribution of x(T)
given a
distribution of (x0,m0,T) in IR^(4n+1). If f is a nice,
suffuciently
smooth function of x and m, then the standard theorems of ODE
theory
guarantee smooth dependendence of x(T) depending on T, x0 and
m0.
So the distribution of x(T) is just the image of the
distribution of
(x0,m0,T) under the ßow of your equation.
>is that same as the deterministic solution given an m?
>so...
>y(t) = e^(mt)* C
In this case you have three random variables m, C and t (but
better
think of t as fuxed, randomly distributed value T) and you
have to
defune what the right distribution of them are.
HTH, Thomas
>where C is the integration constant.
>I would really appreciate someone helping me with this,
please.
>how will method of obtaining expectation values change, if I
have a system
of odes?
>ames
===
Subject: Re: expectation value for ODE?
>>does that exist?
>>it that like funding a solution for a given ode?
>>like when you have,
>>dy/dt = m y
>>where m is a random parameter.
>>what is expected value of above?
>What do you mean by value? Suppose you have a dynamical
system in
>IR^n given by x = f(x,m), where m is an n-dimensional
parameter. You
>may consider the initial value problem x=f(x,m0), x(0)=x0
and compute
>x(T). Here x0,m0,T are fuxed.
Correction: ... where m is a k-dimensional parameter.
Thomas
===
Subject: what is the method for disproving limit in analysis?
Hi all,
I am wondering what is the method for disproving limits in
analysis.
I mean: for proving limits, I just need to state the
following:
Given any epsilon > 0, I can fund a N0(epsilon), such that for
all n>N0,
|X_n - X_limit|<epsilon.
But now I want to disprove: thats to say, prove that the
series does not
converge to that limit.
Do I say:
Given any epsilon > 0, I can not fund a N0(epsilon), such that
for all n>N0,
|X_n - X_limit|<epsilon?
Sometimes the above is diffucult to formulate, can I say the
following
instead?
There exists an epsilon, such that |X_n - X_limit| > epsilon?
===
Subject: Re: what is the method for disproving limit in
analysis?
>Hi all,
>I am wondering what is the method for disproving limits in
analysis.
Which method is easiest (or even applicable) depends on the
sequence.
>I mean: for proving limits, I just need to state the
following:
>Given any epsilon > 0, I can fund a N0(epsilon), such that
for all n>N0,
>|X_n - X_limit|<epsilon.
Alright, as long as you prepend that with there exists an
X_limit such that [given any epsilon > 0...].
If I use L for X_limit, A and E for universal and existential
quantifuers, then this is [EL][Ae>0][EN][An>N][|X_n - L|<e].
>But now I want to disprove: thats to say, prove that the
series does not
>converge to that limit.
>Do I say:
>Given any epsilon > 0, I can not fund a N0(epsilon), such
that for all
n>N0,
>|X_n - X_limit|<epsilon?
There does not exist an X_limit such that for all eps>0 ... .
>Sometimes the above is diffucult to formulate, can I say the
following
>instead?
>There exists an epsilon, such that |X_n - X_limit| > epsilon?
Well, as long as you intend to mean
[AL][Ee>0][AN][En>N][ |X_n-L| >= e ],
then yes.
Another way of proving a sequence does (or doesnt) have a
limit
is to prove that it is (or isnt) Cachy:
(Def.) A sequence X_n is Cauchy iff for all eps>0, there
exists
N such that for all n,m>N, |x_n-x_m|<eps.
Intuitively, the difference between a term and subsequent
terms
becomes arbitrarily small. The defunitions of `converging
sequence and `Cauchy sequence are equivalent 
in every
complete
metric space (e.g., real numbers), and the Cauchy criterion is
easier to work with in many situations.
Sequences with specifuc properties are easier to deal with. 
For
example, if the sequence is monotone, it has a limit iff it is
bounded. If one can isolate a _subsequence_ that does not
converge, or two subsequences that converge to a different
value, then this is also a proof of non-convergence. A good
example of that is X_n = (-1)^n[1 - 1/n].
---
Stan Liou
===
Subject: Re: what is the method for disproving limit in
analysis?
>>Hi all,
>>I am wondering what is the method for disproving limits in
analysis.
> Which method is easiest (or even applicable) depends on the
> sequence.
>>I mean: for proving limits, I just need to state the
following:
>>Given any epsilon > 0, I can fund a N0(epsilon), such that
for all n>N0,
>>|X_n - X_limit|<epsilon.
> Alright, as long as you prepend that with there exists an
> X_limit such that [given any epsilon > 0...].
> If I use L for X_limit, A and E for universal and
existential
> quantifuers, then this is [EL][Ae>0][EN][An>N][|X_n - L|<e].
>>But now I want to disprove: thats to say, prove that the
series does not
>>converge to that limit.
>>Do I say:
>>Given any epsilon > 0, I can not fund a N0(epsilon), such
that for all
>>n>N0,
>>|X_n - X_limit|<epsilon?
> There does not exist an X_limit such that for all eps>0 ...
.
>>Sometimes the above is diffucult to formulate, can I say the
following
>>instead?
>>There exists an epsilon, such that |X_n - X_limit| >
epsilon?
> Well, as long as you intend to mean
> [AL][Ee>0][AN][En>N][ |X_n-L| >= e ],
> then yes.
> Another way of proving a sequence does (or doesnt) have a
limit
> is to prove that it is (or isnt) Cachy:
> (Def.) A sequence X_n is Cauchy iff for all eps>0, there
exists
> N such that for all n,m>N, |x_n-x_m|<eps.
> Intuitively, the difference between a term and subsequent
terms
> becomes arbitrarily small. The defunitions of `converging
> sequence and `Cauchy sequence are 
equivalent in every
complete
> metric space (e.g., real numbers), and the Cauchy criterion
is
> easier to work with in many situations.
> Sequences with specifuc properties are easier to deal with.
For
> example, if the sequence is monotone, it has a limit iff it
is
> bounded. If one can isolate a _subsequence_ that does not
> converge, or two subsequences that converge to a different
> value, then this is also a proof of non-convergence. A good
> example of that is X_n = (-1)^n[1 - 1/n].
> ---
> Stan Liou
could you please write the statement of disproving the
existence of limits
in plain English...
I am even more confused by the [AL], [EL], etc...
===
Subject: Re: what is the method for disproving limit in
analysis?
<cnins9$9dk$1@news.Stanford.EDU>There exists an epsilon, such
that |X_n - X_limit| > epsilon?
>Well, as long as you intend to mean
>[AL][Ee>0][AN][En>N][ |X_n-L| >= e ],
>then yes.
> could you please write the statement of disproving the
existence of
limits
> in plain English...
Did you not see my post where I did exactly that?
> I am even more confused by the [AL], [EL], etc...
Yes, its hard to read.
===
Subject: Re: what is the method for disproving limit in
analysis?
> I am wondering what is the method for disproving limits in
analysis.
> I mean: for proving limits, I just need to state the
following:
> Given any epsilon > 0, I can fund a N0(epsilon), such that
for all n>N0,
> |X_n - X_limit|<epsilon.
> But now I want to disprove: thats to say, prove that the
series does not
> converge to that limit.
> Do I say:
> Given any epsilon > 0, I can not fund a N0(epsilon), such
that for all
n>N0,
> |X_n - X_limit|<epsilon?
No, and please use a < b for easier reading.
lim(n->oo) x_n = a when
for all e > 0, some n in N with for all j > n, |x_j - a| < e
The negation is (by DeMorgan rules for quantifuers and
connectives) is
some e > 0 with for all n in N, some j > n with |x_j - a| >= e
In otherwords fund an e and a subsequence (x_nj)_j of (x_n)_n
with
for all j, |x_nj - a| >= e
> Sometimes the above is diffucult to formulate, can I say the
following
> instead?
> There exists an epsilon, such that |X_n - X_limit| >
epsilon?
No. Its a toughie thats best slogged out 
using symbolic
logic approach.
===
Subject: A Geometry question NEED HELP
ABCD is a paralleloram. E is a point on BC and F a point on
CD. AE cuts BF
at G.
AF cuts DE at H and BF cuts DE at K. Prove that
[AGKH]=[BEG]+[CEKF]+[DFH]
===
Subject: Re: A Geometry question NEED HELP
Kessie <byfkhm@hotmail.com> escribi.97:
> ABCD is a paralleloram. E is a point on BC and F a point on
CD. AE
> cuts BF at G. AF cuts DE at H and BF cuts DE at K. Prove
that
> [AGKH]=[BEG]+[CEKF]+[DFH]
[ABF] = [AED] = (1/2)[ABCD]
Then
[BEG] + [CEKF] + [DFH] =
[ABCD] - ([ABF / AED]) = [ABCD] - ([ABF] + [AED] - ([ABF /
AED])) =
[ABCD] - ([ABF] + [AED]) + ([ABF] / [AED])) =
[ABF] / [AED] = [AGKH]
--
Ignacio Larrosa Ca.96estro
A Coru.96a (Espa.96a)
ilarrosaQUITARMAYUSCULAS@mundo-r.com
===
Subject: Re: A Geometry question NEED HELP
> ABCD is a paralleloram. E is a point on BC and F a point on
CD. AE cuts
BF
at G.
> AF cuts DE at H and BF cuts DE at K. Prove that
> [AGKH]=[BEG]+[CEKF]+[DFH]
Youd better choose unit such that [ABCD] = 1.
Compute [AED], [ABF]
Compute [ADF] + [FBC]
Write DAE as disjoint union of polygons.
Same with ADF U FBC.
Conclude.
===
Subject: Re: A Geometry question NEED HELP
> ABCD is a paralleloram. E is a point on BC and F a point on
CD. AE cuts BF
at G.
> AF cuts DE at H and BF cuts DE at K. Prove that
> [AGKH]=[BEG]+[CEKF]+[DFH]
i can prove it wrong:
take a square as ABCD and set E and F equal to C. points G, H
and K will be
on C as well. [AGKH] is therefor sqrt(2), while [BEG] is 1,
[CEKF] is 0 and
[DFH] is 1. you get sqrt(2)=2 and thats wrong.
===
Subject: Re: sci.math.moderated?
>> Quite likely. What is practically a certainty is that
unless moderated
>> these narrower groups would soon contain all sorts of
stuff having
>> nothing to do with the narrow focus, and there would be
pointless
> Imagine a mathematics without points!
Its called point-free.
http://www.cs.bham.ac.uk/events/seminars/seminar_details.html
?seminar_id=45
>> complaints about this. And there would of course still be
a need for
>> a group without any narrow focus.
--
http://hertzlinger.blogspot.com
===
Subject: Re: Surprising Pattern of Floridas Election 
Results
I cannot say anything about US politics, but isnt it 
obvious
that the
*only* way to prevent vote fraud is by using machines that
produce
cryptographic paper trails, which can be publicly verifued
afterwards?
Its a no-brainer, but why cant US, arguably 
the biggest IT
industry
in the world, manage it? After all, youve got some of the
brightest
crypto researchers in the world. They could solve the theory
part in
one day. And IBM could design the machine in two months. These
discussions are quite unfortunate.
>to certify that the code theyre reviewing
>is the same as the code actually being used on every machine,
>etc.
>>While it isnt impossible,
>>it is highly improbable that tampering could take place on
any sort
> of
>>wide-spread basis without being caught.
>Roughly what percentage of people attempting to steal money
by
> hacking
>into banks, credit card companies, etc, are eventually
caught?
>Hint: You cant possibly know, because the ones that remain
>undetected remain undetected.
>************************
>David C. Ullrich
> Anyone who thinks that 1) vote fraud has not occurred
(quite often)
> in the past knows nothing of history
> or 2) thinks that for some reason it cant happen now
> Is being very shortsighted and naive. Think of what is at
stake,
> the huge amounts of money involved. There is certainly the
will
> to get in offuce by fraud, and Ôwhere there is a 
will there
is a way.
> I agree that far more open, transparent, and regulated
voting is
> needed,
> and after the year 2000, with all the irregularities that
came to
> light,
> something should have been done.
> Nothing ever gets done since those who are 
Ôelected dont
want
> anything
> to change that might cause them not to be elected.
> The complex voting registration, which is just a way of
keeping
> large numbers of people from voting must be changed also.
> Everyone should be able to vote by virtue of their SS #.
> There is not need for any of this other BS.
===
Subject: Re: Surprising Pattern of Floridas Election 
Results
> I cannot say anything about US politics, but isnt it
obvious that the
> *only* way to prevent vote fraud is by using machines that
produce
> cryptographic paper trails, which can be publicly verifued
afterwards?
> Its a no-brainer, but why cant US, 
arguably the biggest
IT industry
> in the world, manage it? After all, youve got some of the
brightest
> crypto researchers in the world. They could solve the
theory part in
> one day. And IBM could design the machine in two months.
These
> discussions are quite unfortunate.
Doesnt that open the door to someone (either a hacker or
insider) being
able to determine how every person voted? How about we make a
deal: we pay
you $500,000 to set up the system, but if any votes by
individuals are
revealed within the next decade you die. Are you willing to
sign up for
that?
So what would you do, print out a paper receipt with an
encrypted vote
receipt? Then if there was a question about votes, would you
have all
voters bring in their receipts so that they could be compared
to the
recorded count? What if some voters (lets say poor people)
discarded or
lost their receipts more frequently than other groups (lets
say party
activists), how would you make the recount fair?
--
Phil Sherrod
(phil.sherrod Ôat sandh.com)
http://www.dtreg.com (decision tree modeling)
http://www.nlreg.com (nonlinear regression)
http://www.NewsRover.com (Usenet newsreader)
http://www.LogRover.com (Web statistics analysis)
===
Subject: Re: Surprising Pattern of Floridas Election 
Results
> So what would you do, print out a paper receipt with an
encrypted vote
> receipt? Then if there was a question about votes, would
you have all
> voters bring in their receipts so that they could be
compared to the
> recorded count? What if some voters (lets say poor 
people)
discarded or
> lost their receipts more frequently than other groups
(lets say party
> activists), how would you make the recount fair?
Why not use paper ballots in the furst place. While such a
system can be
tampered with (as anyone from Chicago knows) it is the least
worst
system of secret voting devised. In Chicago, the dead voted
early and
often, and troublesome ballots from Downstate ended up in the
Chicago
River or Lake Michagan.
The Legend comes down, from Chicago on down
The voters, it is said, vote often when dead
when the nights of November come early
Apologies to Gordon Lightfoot.
Bob Kolker
===
Subject: Re: Surprising Pattern of Floridas Election 
Results
>So what would you do, print out a paper receipt with an
encrypted vote
>receipt? Then if there was a question about votes, would you
have all
>voters bring in their receipts so that they could be
compared to the
>recorded count? What if some voters (lets say poor people)
discarded
>or
>lost their receipts more frequently than other groups 
(lets
say party
>activists), how would you make the recount fair?
> Why not use paper ballots in the furst place. While such a
system can be
> tampered with (as anyone from Chicago knows) it is the
least worst
> system of secret voting devised. In Chicago, the dead voted
early and
> often, and troublesome ballots from Downstate ended up in
the Chicago
> River or Lake Michagan.
The only problem with paper ballots is the time it takes for
them to be
counted. My favorite voting system is the old, mechanical
lever machines
which were virtually impossible to rig. What really scares me
is the idea
of voting through the Internet.
I believe the person I posted my response to was suggesting
some sort of
printed receipt that would be given to each voter showing who
the voter
voted for in an encrypted form. Or maybe he wants to record
each
individual
vote in the machine separately but encrypt the identifucation
of the voter.
In either case, I believe that it would only be a matter of
time before
someone fugured out how to crack the code and reveal the votes
cast by
voters. Im opposed to any system which provides the
capability of
determining the votes of individuals. I think the possibility
that votes
would be revealed would skew the votes and scare off some
voters. How
would
you feel if you had been a voter in Iraq and Saddam Hussein
had the
potential of fuguring out that you and your family voted
against him?
--
Phil Sherrod
(phil.sherrod Ôat sandh.com)
http://www.dtreg.com (decision tree modeling)
http://www.nlreg.com (nonlinear regression)
http://www.NewsRover.com (Usenet newsreader)
http://www.LogRover.com (Web statistics analysis)
===
Subject: Re: Surprising Pattern of Floridas Election 
Results
> What really scares me is the idea of voting through the
Internet.
It scares the hell out of me too. In some cultures it is
traditional for
every member of a household to vote as directed by the head.
The UK
government is experimenting with internet voting as a
possible answer to
low turnouts (as opposed to cleaning up government and
bringing back the
notion of public service instead of Yippee, Im an MP so 
Im
now an
instant millionnaire with my own private gravy train). But
theres
absolutely no provision to ensure people get a private vote
without
duress from the head of the household.
NigelH
===
Subject: Re: Surprising Pattern of Floridas Election 
Results
>What really scares me is the idea of voting through the
Internet.
> It scares the hell out of me too. In some cultures it is
traditional for
> every member of a household to vote as directed by the
head. The UK
> government is experimenting with internet voting as a
possible answer to
> low turnouts (as opposed to cleaning up government and
bringing back the
> notion of public service instead of Yippee, Im an MP so
Im now an
> instant millionnaire with my own private gravy train). But
theres
> absolutely no provision to ensure people get a private vote
without
> duress from the head of the household.
Well, you guys should not really think of your country as
under a
current or potential dictators rule. Remember, 
youre still
seen as
one of the bastions of individual freedom and human rights.
Or, youve lost already. Cryptography cant 
save you then.
If you believe that there is still some chance for freedom,
then you
should seriously think about getting the best researchers out
there to
design a voting machine that
* cannot be rigged
* can be securely verifued without revealing 
voters identity
Needless to say, the source code of the machine should be
open, and
all of its hardware design should be open, and there should
be ways
to ensure secure operation. This part is harder than the
application
software, which is kind of trivial to design for a
cryptography
researcher. There seems to be an easy way, but not wholly
secure. The
machine runs linux. The application works only by creating a
secure
virtual machine, completely closed to the outside world
except for
only and only the required I/O channels, and runs itself in
this
sandbox. There is an input/output verifucation procedure which
ensures
the integrity of the hardware, by processing cryptographic
challenges
physically. (Basically, representatives of *every* party
print some
papers at home, and try them at the machine, which the secure
virtual
machine tries. There is still the possibility of an
intelligent hack
that knows just when and how to fake I/O signals, but this
wont work,
because there will be a *mandatory* verifucation process for
each
vote. Needless to say, Im still assuming that it can be 
done
without
revealing voter identity and that your people are smart
enough to use
a simple machine and follow simple procedures.)
At any rate, you can do much much better than those infamous
Diebold
machines, which apparently still use weak DES encryption with
a
*fuxed* key so that some spooks can change the votes.
I imagine that such an effort will face great opposition from
your
current administration, because it can be done.
--
Eray Ozkural
===
Subject: Re: Surprising Pattern of Floridas Election 
Results
> Well, you guys should not really think of your country as
under a
> current or potential dictators rule. Remember, 
youre
still seen as
> one of the bastions of individual freedom and human rights.
I dont see the USA as being under a current or potential
dictators rule,
but I object to putting in place a voting system that
violates a basic
premise which is that it is impossible for anyone to fund out
how you
voted.
Suppose I collect 500 homeless people, put them on a bus and
take them to a
site with some computers. I offer each person $5 if they vote
the way I
want. I then watch as they sit at the computer and check the
desired box
on
the Internet voting screen. If anyone checks the wrong box,
they dont get
the money. I can get a lot of votes for a reasonable amount
of money this
way. Compare that to the current system where they go into a
booth where I
cant watch them, vote and then come out to collect their 
$5.
How do I
know
how they voted? To paint a more malevolent picture, imagine
some
arm-twisting thugs who threaten people and their families if
they dont
cast
their votes the way they want while they watch. Some unions
come pretty
close to that.
> Or, youve lost already. Cryptography cant 
save you then.
> If you believe that there is still some chance for freedom,
then you
> should seriously think about getting the best researchers
out there to
> design a voting machine that
> * cannot be rigged
> * can be securely verifued without revealing 
voters identity
Would you bet your life that a voters identity 
cant be
revealed?
Is the intention of your cryptographic system to make it
possible to later
match up voters with the votes they cast? If it is, then the
security of
the voters is in the hands of the people who hold the
cryptographic keys.
And, despite your faith in Linux, I am not willing to open a
system of this
importance to the world of hackers. Imagine the prestige some
hacker would
gain when he proudly announced that he was responsible for
electing the
President of the United States. Thats a pretty strong
motivating factor
for some people.
> At any rate, you can do much much better than those
infamous Diebold
> machines, which apparently still use weak DES encryption
with a
> *fuxed* key so that some spooks can change the votes.
What information are they encrypting? My understanding is
that they just
tabulate the total vote count. I dont think they record 
voter
identifucation information. I dont care if the 
total vote is
encrypted or
not.
Dont local voting offucials set up the Diebold 
machines for a
particular
election -- that is, decide that the top button is for Bush,
the second one
for Kerry, etc.? If thats the case, then how would Diebold
have any
inßuence since they have no idea when they build the machine
which button
goes with each candidate for each election. What exactly is
your complaint
against Diebold machines?
--
Phil Sherrod
(phil.sherrod Ôat sandh.com)
http://www.dtreg.com (decision tree modeling)
http://www.nlreg.com (nonlinear regression)
http://www.NewsRover.com (Usenet newsreader)
http://www.LogRover.com (Web statistics analysis)
===
Subject: Re: Surprising Pattern of Floridas Election 
Results
Diebold makes ATMs, which use Asynchornous Transfer Mechanism;
this may or may not be relavent. anyway,
the problem is exposed by the terminology
of blackboxvoting.org. like you said,
hwat if you cant actually know *that* votes were made,
then tallied, as opposed to merely calculated
according to some demographical scheme?
anywya,
the optically scanned ballots have the same problem,
of tabulation, and this is said to be apparent
in Florida, compared to the touchscreen crap -- more-so, and
its the same companies (Diebold, ESS, primarily,
I think).
> Dont local voting offucials set up the 
Diebold machines for
a particular
> election -- that is, decide that the top button is for
Bush, the second
one
> for Kerry, etc.? If thats the case, then how would 
Diebold
have any
> inßuence since they have no idea when they build the
machine which
button
> goes with each candidate for each election. What exactly is
your
complaint
> against Diebold machines?
--Give Earth a Trickier Dick Cheeny -- out of offuce, after
gigayears!
http://tarpley.net/bush12.htm
http://www.benfranklinbooks.com/
http://members.tripod.com/~american_almanac
http://www.wlym.com/pdf/iclc/howthenation.pdf
http://www.rand.org/publications/randreview/issues/rr.12.00/
http://www.rwgrayprojects.com/synergetics/plates/fugs/plate02.
html
===
Subject: Re: Surprising Pattern of Floridas Election 
Results
> Is the intention of your cryptographic system to make it
possible to
later
> match up voters with the votes they cast? If it is, then
the security of
> the voters is in the hands of the people who hold the
cryptographic keys.
> And, despite your faith in Linux, I am not willing to open
a system of
this
> importance to the world of hackers. Imagine the prestige
some hacker
would
> gain when he proudly announced that he was responsible for
electing the
> President of the United States. Thats a pretty strong
motivating factor
> for some people.
You seem to know a great deal about open source software, and
how it
is *less* secure than closed software. Really practical
thought you
have here. We are opening a system of this importance to the
world of
hackers. Thats not too bad, because hacker means gifted
programmer.
We want to do exactly that, such that we eliminate any bugs,
security
ßaws or intentional backdoor efforts.
In addition, if the software is open, we dont use crappy C
code. We
can write the application software in a much more reliable
language,
and we can set up the software in the most secure way (We can
even use
secure memory, secure fulesystems etc. They already exist!) We
can be
100% sure that this guy is not doing anything stupid. And it
is tested
worldwide by thousands of users and hackers. It is constantly
audited
for security leaks. Its run against every known system
anomaly. Test
suites are written for it. Strong cryptographic attacks are
tried on
the crypto-system. This is how cryptography research advances,
actually!
PS: You mentioned the place of the buttons are decided later
on. But
what the buttons are, many people can know, they can phone up
the
spooks, and do their remote magic. Not too hard to imagine.
--
Eray Ozkural
===
Subject: Re: Surprising Pattern of Floridas Election 
Results
>Is the intention of your cryptographic system to make it
possible to
later
>match up voters with the votes they cast? If it is, then the
security
of
>the voters is in the hands of the people who hold the
cryptographic
keys.
>And, despite your faith in Linux, I am not willing to open a
system of
this
>importance to the world of hackers. Imagine the prestige
some hacker
would
>gain when he proudly announced that he was responsible for
electing the
>President of the United States. Thats a pretty strong
motivating
factor
>for some people.
> You seem to know a great deal about open source software,
and how it
> is *less* secure than closed software.
Where did I say open source software is *less* secure than
closed? I am
saying
that I dont share your belief that it is secure.
> hacker means gifted programmer.
> We want to do exactly that, such that we eliminate any
bugs, security
> ßaws or intentional backdoor efforts.
Some hackers may be gifted programmers, but the name
certainly does not
imply
that. The majority of the hackers Ive know have been
anti-social
losers who
cant hold real jobs as software developers, so they get
their joy from
bragging about how they are able to crack someone elses
creative work.
> In addition, if the software is open, we dont use crappy 
C
code.
Oh really? What is Linux written in? What is the Java core
run-time
written
in?
You still havent explained your original contention that
cryptographic
methods
were going to eliminate voter fraud. Nor have you explained
why you think
Diebold machines are particularly vulnerable to fraud.
--
Phil Sherrod
(phil.sherrod Ôat sandh.com)
http://www.dtreg.com (decision tree modeling)
http://www.nlreg.com (nonlinear regression)
http://www.LogRover.com (web traffuc analysis)
http://www.NewsRover.com (Usenet newsreader)
===
Subject: Re: Surprising Pattern of Floridas Election 
Results
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<30448rF2q28a5U1@uni-berlin.de>
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<419d2fe9$0$3989$ed2619ec@ptn-nntp-reader01.plus.net>
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5eZ41to5f%E@ELIi
$t^
VcLWP@J5p^rst0+(Ô>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw
> You still havent explained your original contention that
> cryptographic methods were going to eliminate voter fraud.
Thats a puzzle for me, too.
> Nor have you explained why you think Diebold machines are
> particularly vulnerable to fraud.
Because we have a company with a track record of
a) funancially and vocally supporting one political party
b) public statements by company offucials on Republican
conventions
that they intend to deliver Ohios electoral votes to Bush.
c) doing last-minute replacements of certifued software with
hacks of
their own.
d) having had several previous incidents where there were
manipulations as transparent as more Republican votes than
total
votes in some districts, leading to negative votes cast for
Democratic candidates.
e) have stalled delivering paper trails for years citing
technical
reasons, even though their whole other business (ATM machines)
creates paper trails.
In short: Diebold machines are particularly vulnerable to
fraud since
they are delivered by a company that is on the record as being
politically one-sided, of having manipulated software
maliciously in
the past, of having had vast errors (all happening on one
side by
incident) in the past and a ßippant attitude towards that in
internal
communication, and of refusing to provide measures of
verifucation
that they have the perfect technical ability to install, as
shown by
their other business.
In short: the company has a track record that clearly
indicates that
they are not to be trusted; and in strict violation of normal
democratic principles, there are no mechanisms in place that
would
ensure that trust is not needed.
The printer for paper ballots might be as radically
Democratic or
Republican as he wants to: he cant single-handedly thwart 
an
election. Diebold can, and they have shown that they have
little
qualms about committing errors and diverting or ignoring
requests
for more verifuable procedures.
--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
===
Subject: Re: Surprising Pattern of Floridas Election 
Results
>Nor have you explained why you think Diebold machines are
>particularly vulnerable to fraud.
> Because we have a company with a track record of
> a) funancially and vocally supporting one political party
> b) public statements by company offucials on Republican
conventions
> that they intend to deliver Ohios electoral votes to 
Bush.
I understand you dont like the Diebold management because
they are
Republican,
but this is a technical newsgroup, so I would like you to
explain on a
technical level why you think Diebold machines are more
subject to fraud
than
other brands of electronic voting machines.
I have no connection with Diebold or any other voting machine
company. In
fact, I dont like black box voting machines and prefer a 
more
transparent
system such as paper ballots or mechanical machines. But if
youre going
to
make public charges in a technical group, I would like to
hear the
technical
basis of the charge.
Are you saying that the Diebold machines fugure out which
button is
programmed
to record a vote for a Republican candidate, and they add
some extra votes?
Or
do you think Diebold is somehow connecting to the machines
during the
election
and adding votes?
Wouldnt machine tampering for a particular election have to
be done by the
people setting up the machine at the local district? How
would Diebold, who
may
have built the machine a year earlier, have rigged it without
knowing
whats
going to be on the ballot?
I think youre allowing your political emotions to cloud 
your
logical
thinking.
--
Phil Sherrod
(phil.sherrod Ôat sandh.com)
http://www.dtreg.com (decision tree modeling)
http://www.nlreg.com (nonlinear regression)
http://www.LogRover.com (web traffuc analysis)
http://www.NewsRover.com (Usenet newsreader)
===
Subject: Re: Surprising Pattern of Floridas Election 
Results
>>Nor have you explained why you think Diebold machines are
>>particularly vulnerable to fraud.
>> Because we have a company with a track record of
>> a) funancially and vocally supporting one political party
>> b) public statements by company offucials on Republican
conventions
>> that they intend to deliver Ohios electoral votes to 
Bush.
>I understand you dont like the Diebold management because
they are
Republican,
>but this is a technical newsgroup, so I would like you to
explain on a
>technical level why you think Diebold machines are more
subject to fraud
than
>other brands of electronic voting machines.
the machines away from people. Somebody forgot that voters
are also people. I know quite a few people who had a knack
of bringing any system down to its knees; we called them
testers
and were valuable to us because they did unintentional things
that we had never thought to guard against.
This is strictly about code, design, and testing and has
nothing
to do with intentions of businesses.
The only secure computer is one that has no wires nor people
going in or out of the computer room. This also produces
a computing device that can only be used as a boat anchor.
All bets are off now that wireless tech is common.
I also have a problem with installing code for a crucial
application
with the restriction of no sources, especially for code
review.
Instead of being paranoid about the opposing party, you should
be worrying about national security.
<snip>
/BAH
Subtract a hundred and four for e-mail.
===
Subject: Re: Surprising Pattern of Floridas Election 
Results
>>Nor have you explained why you think Diebold machines are
>>particularly vulnerable to fraud.
>> Because we have a company with a track record of
>> a) funancially and vocally supporting one political party
>> b) public statements by company offucials on Republican
conventions
>> that they intend to deliver Ohios electoral votes to 
Bush.
>I understand you dont like the Diebold management because
they are
> Republican,
>but this is a technical newsgroup, so I would like you to
explain on a
>technical level why you think Diebold machines are more
subject to fraud
> than
>other brands of electronic voting machines.
> the machines away from people. Somebody forgot that voters
> are also people. I know quite a few people who had a knack
> of bringing any system down to its knees; we called them
testers
> and were valuable to us because they did unintentional
things
> that we had never thought to guard against.
I worked once for a company that makes subway fare collection
machines.
They thought about someone urinating in the gate ticket slot
(a little
plumbing ends in a hidden copper tube aimed approximately
where the
urinators shoes would be) but a customer was able to 
fugure
out within
a
few days of one installation that a torn farecard could be
used to get new
ones (up to $20 value) for free.
===
Subject: Re: Surprising Pattern of Floridas Election 
Results
Phil Sherod says
> I think youre allowing your political emotions to cloud
your
> logical thinking.
I shall pretend that this is all an innocent misunderstanding,
otherwise there is no point in discussing.
> I would like you to explain on a technical level why you
think
> Diebold machines are more subject to fraud than
> other brands of electronic voting machines.
Your question would be relevant *before* the elections.
Now the burning question is do we trust the result we have;
not would we trust the result if another company had been
contracted.
>In fact, I dont like black box voting machines and prefer 
a
>more transparent system such as paper ballots or mechanical
machines.
May I ask you, if Diebold machines are beyond suspicion, why
would
you prefer paper ballots? If you are of two minds on this
issue,
I can understand, but please present both arguments clearly
and in
separate lines. As it is, you do not even make sense. Btw it
happens
to me, too, sometimes, but then I apologise and post a
better-written
message.
>Are you saying that the Diebold machines fugure out which
button is
>programmed to record a vote for a Republican candidate, and
they add
>some extra votes? Or do you think Diebold is somehow
connecting to
>the machines during the election and adding votes?
Good Grief. Every voter in the precinct knows which button is
for
each candidate. There is no need for a conspiracy theory
*yet*.
Ftm, I share your disagreement with a super-encryption scheme,
which would be opaque to common folk. If that scheme were
applied,
every challenge of its reliability would be automatically a
conspiracy theory, since nothing could ever be proven.
> I think youre allowing your political emotions to cloud
your
> logical thinking.
As I pointed out above, the logic in *your* writing leaves
much to be desired. Until you fux this problem, you cannot
criticize the coherence of another argument. Please, do make
sense
furst.
~PS~ On the relation between emotions and thinking, I remember
a sentence by David Hume: accuracy is, in every case,
advantageous to beauty, and just reasoning to delicate
sentiments.
I just had to end this message on a positive note.
~ George Kahrimanis
===
Subject: Re: Surprising Pattern of Floridas Election 
Results
<90b9p0dtd91jl0t2nht8gt6niom4tehuaj@4ax.com>
!3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(
5eZ41to5f%E@ELIi
$t^
VcLWP@J5p^rst0+(Ô>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw
> I cannot say anything about US politics, but isnt it
obvious that
> the *only* way to prevent vote fraud is by using machines
that
> produce cryptographic paper trails, which can be publicly
verifued
> afterwards?
Cryptographic trails? What is that supposed to mean? No, you
need an
ordinary paper trail which the voter can look at, and which
ends up in
the ballot box. In case of doubt, it can be recounted. And if
there
is a discrepancy, it can be corrected and the responsible
people can
get sued for it.
With the current setup, fraud will go undetected, so nobody
needs to
fear retribution.
> Its a no-brainer, but why cant US, 
arguably the biggest IT
> industry in the world, manage it?
Because the voting machine company heads are big friends with
the
current ruling party, and have sponsored them.
--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
===
Subject: Re: Surprising Pattern of Floridas Election 
Results
<90b9p0dtd91jl0t2nht8gt6niom4tehuaj@4ax.com>
<x5k6sjwjgg.fsf@lola.goethe.zz>
Discussion, linux)
>> I cannot say anything about US politics, but isnt it
obvious that
>> the *only* way to prevent vote fraud is by using machines
that
>> produce cryptographic paper trails, which can be publicly
verifued
>> afterwards?
> Cryptographic trails? What is that supposed to mean? No,
you need an
> ordinary paper trail which the voter can look at, and which
ends up in
> the ballot box. In case of doubt, it can be recounted. And
if there
> is a discrepancy, it can be corrected and the responsible
people can
> get sued for it.
> With the current setup, fraud will go undetected, so nobody
needs to
> fear retribution.
>> Its a no-brainer, but why cant US, 
arguably the biggest
IT
>> industry in the world, manage it?
> Because the voting machine company heads are big friends
with the
> current ruling party, and have sponsored them.
Perhaps we dont have to be so conspiratorial. There are
somewhat
less devious reasons for the companies to oppose paper trails.
For one thing: with paper trails, a botched election could
yield messy
and nasty recounts as in Florida. During those messy and nasty
recounts, it is conceivable that the software in the machines
is shown
to be buggy or faulty. Without recounts, there is
considerably less
chance that ßaws in the software would be discovered.
Now, even ignoring political affuliation, isnt 
easy to see
why paper
trails arent attractive to the company heads?
Personally, I really dont make much of the political ties. 
I
*am*
concerned about the move to complicated polling machines with
closed
source software and no paper trail, but Im more concerned
about
incompetence and avoidance of responsibility than conspiracy.
--
Damn John Jay.
Damn everyone who wont damn John Jay.
Damn everyone who wont put lights in his windows and sit up
all night
damning John Jay. -- Political graffuti from late 18th c.
Boston
===
Subject: Re: Surprising Pattern of Floridas Election 
Results
>>I cannot say anything about US politics, but isnt it
obvious that
>>the *only* way to prevent vote fraud is by using machines
that
>>produce cryptographic paper trails, which can be publicly
verifued
>>afterwards?
> Cryptographic trails? What is that supposed to mean?
It means you have a paper trail but no one can follow it.
Never mind
IBM, Enron seem to be the experts in this area. :-)
Bob
--
Bob OHara
Dept. of Mathematics and Statistics
P.O. Box 68 (Gustaf H.8allstr.9amin katu 2b)
FIN-00014 University of Helsinki
Finland
Telephone: +358-9-191 51479
Mobile: +358 50 599 0540
Fax: +358-9-191 51400
WWW: http://www.RNI.Helsinki.FI/~boh/
Journal of Negative Results - EEB: http://www.jnr-eeb.org
===
Subject: Re: Surprising Pattern of Floridas Election 
Results
<90b9p0dtd91jl0t2nht8gt6niom4tehuaj@4ax.com>
<x5k6sjwjgg.fsf@lola.goethe.zz>
<419CAE0B.6030500@SOD.OFF.Spammers.helsinki.fu>
!3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(
5eZ41to5f%E@ELIi
$t^
VcLWP@J5p^rst0+(Ô>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw
>I cannot say anything about US politics, but isnt it
obvious that
>the *only* way to prevent vote fraud is by using machines
that
>produce cryptographic paper trails, which can be publicly
verifued
>afterwards?
>> Cryptographic trails? What is that supposed to mean?
> It means you have a paper trail but no one can follow it.
Well, then it is without purpose. The problem is that with
something
like a cryptographic trail, you again have to trust the
machine
manufacturers. With paper (of course, visible to the voter!),
you can
always verify the votes afterwards. At least if you take a
headcount
of the voters manually (to guarantee that the machine is not
passing
more votes into the ballot box than actually taken).
--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
===
Subject: Re: Surprising Pattern of Floridas Election 
Results
<90b9p0dtd91jl0t2nht8gt6niom4tehuaj@4ax.com>
<x5k6sjwjgg.fsf@lola.goethe.zz>
<419CAE0B.6030500@SOD.OFF.Spammers.helsinki.fu>
<x5brdvjdsb.fsf@lola.goethe.zz>
Discussion, linux)
>
>>I cannot say anything about US politics, but isnt it
obvious that
>>the *only* way to prevent vote fraud is by using machines
that
>>produce cryptographic paper trails, which can be publicly
verifued
>>afterwards?
> Cryptographic trails? What is that supposed to mean?
>> It means you have a paper trail but no one can follow it.
> Well, then it is without purpose.
It is possible that bob.ohara (Anon.?) knew this. You removed
his
reference to Enron, but its conceivable that it was 
relevant
for his
meaning.
--
Ive been thinking about my problems with getting any kind 
of
admission that my math arguments showing the core error in
mathematics
are correct, so Ive gone to marketing books.
-- James S. Harris, on when mathematics isnt enough
===
Subject: Re: Surprising Pattern of Floridas Election 
Results
> Well, then it is without purpose. The problem is that with
something
> like a cryptographic trail, you again have to trust the
machine
> manufacturers. With paper (of course, visible to the
voter!), you can
> always verify the votes afterwards.
Only if paper ballot are guarded and there is assurance that
no genuine
ballot has been stolen or destroyed, or invalid ballots have
been
stuffed into the ballot box.
At least if you take a headcount
> of the voters manually (to guarantee that the machine is
not passing
> more votes into the ballot box than actually taken).
This is done when the voters check in. They present i.d. and
the i.d. is
checked off against the registration list. So it is known
modulu error
how many voters voted. The total number of ballots equal
(modulo small
error) the number of voters checked off against registration.
This does
not preculude ballot tampering which maintains the total
count.
Bob Kolker
===
Subject: Re: Surprising Pattern of Floridas Election 
Results
thus it is that the alleged inability of Diebold
to provide a paper trail is such a hoot, although
I realize that most folks just toss their ATM reciepts
into the nearest receptacle or Ma Earth.
> have you ever seen a Diebold ATM?... see below,
> for more comments, but the gist is that
> the optical scanning is also tabulated by computer, and
> they were where the real problem was in Florida e.g.
>>80% of all votes in America are counted by only two
>>companies: Diebold and ES&S.
> thus:
> maybe you can *describe* some of those irregularities,
> monsieur Bone. anyway, it seems to have happenned,
> as Id been warning at city council meetings & so forth,
> that the touchscreens were a shill, since *all*
> of the systems are tabulated by computer; apparently,
> 80% of the votes were tabulated by two companies,
> ESS and Diebold, run by two brothers.... wish that
> Id read about that before the election.
> re Diebold, they sell Automated Teller Machines;
> now, it isnt really a coincidence, that
> these machines use the Asynchronous Transfer Mechanism,
> or they used to. I had forgotten this,
> til I happenned to notice one in front of a bank,
> last week.
--Chairman George and Trickier Dick at Watergate!
http://tarpley.net/bush12.htm
===
Subject: Re: More on the simple group of order 168
> and Jyrki Lahtonen for their help and patience.
> This stuff is great for learning how to apply Sylow thms.
and
> using the normalizers.
> ---------
> I did fund a way of showing that if n_2 = 7, then
> N_2 = normalizer P_2 has an element of order 6, so n_2 = 21.
> This language is more familiar to me than the language used
> by others. (I thought I posted this but cant 
fund it).
> |N_2| = 24 ==> there is an element of order 3 in N_2.
> Apply the thm. that the # fuxed points |P_0| = = |P_2| mod p
> = 8 mod 3 = 2, for an element of order 3 acting by
conjugation
> on P_2. Thus there is a non identity element in P_2 which
is fuxed.
> It must be the element in the center of the (non-Abelian)
grp P_2.
> P_2 is thus either D_4 or Q. I can show it must be
> D_4, but it is brute force and there should be an better
> way to show that P_2 = D_4.
Youre making this too hard. A C_3 normalizing a Sylow
2-subgroup P_2, of order 8, must normalize each of its
characteristic subgroups, by the defunition of a 
characteristic
subgroup. All groups of order 8 except the elementary abelian
group E_8 have an obvious characteristic subgroup K of order
2,
which the C_3 must thus normalize, so for these groups the C_3
must centralize the (single) generator of K, so P_2 x| C_3
has a
subgroup isomorphic to C_6. The remaining case is P_2 =~ E_8,
which has 7 involutions, which the C_3 must thus permute among
themselves, so since 7 == 1 (mod 3), the C_3 must centralize
at
least 1 (or 4, or all 7) of them, so again P_2 x| C_3 has a
subgroup isomorphic to C_6.
> -------
> I have managed to follow things up to this point,
>so n_2 = 21. Only 63 elements left
> but I am having diffuculties with the Sylow 2-subgrps.
>so n_2 = 21. Only 63 elements left so some P_2s intersect
>nontrivially; if theres an intersection I with |I| = 4
>then |N(I)| = 24 so N(I) =~ S_4 so P_2 is dihedral; if the
>only Is have |I| = 2 then again P_2 is dihedral since
otherwise
>|N(I)| > 24. An element x of order 4 cant be in 2 
P_2s
since
>otherwise |N(<x>)| > 24, so 42 elements of order 4. Each P_2
has
>a normal involution y that cant be normal in 2 
P_2s since
>otherwise |N(<y>)| > 24, so 21 elements of order 2.
> -------
> I also found this;
>The simple group is relatively easy to construct abstractly
as
>G = AB, where A is S_4 and B is C_7 -- just using Sylows
>Theorems, intersections of normalizers, counting arguments,
>etc.
> ----
> Here is what I can say about this.
> Clearly I am just screwing around and dont really know 
how
> to proceed.
> ---------
> Work on the 24 cosets of a subgroup of order 7,
> though they arent a grp.
> Knowing ahead of time that these would relate to S_4,
??? G acting on the set of 24 cosets of a subgroup of order 7
gives a homomorphism f: G -> S_{24}. I have no idea how you
get
S_4 from that.
> there are (in a sense) 7 copies of S_4.
> Each S_4 has 3 Sylow 2-subgps iso to D_4, for a total of
> 21 copies of D_4 = the Sylow 2 subgrps P_2. These 3 D_4
intersect
> in the normal subgrp usually called N = (1,n_i), where the
> n_i are the 3 even elements of order 2, like (12)(34).
> This N corresponds to the 7 Is in Jim 
Heckmans post,
> giving the 21 elements a_i of order 2, and thus if x_i
> has order 4, there are 42 elements x_i and (x_i)^3 = a_i x_i
> of order 4, giving the remaining 63 elements of the group.
Since I dont know where you got S_4 from, I 
dont really
follow
any of the above.
> ---------
> But this is all I could do. I cant see why the following
is true;
>I with |I| = 4
>then |N(I)| = 24 so N(I) =~ S_4 so P_2 is dihedral;
> and so on as quoted above.
A subgroup I of order 4 of a Sylow 2-group P_2 (of order 8) is
necessarily normal, so if I is contained in 2 P_2s, its
normalizer N(I) has order >= 8*8/4 = 16 and a multiple of 8.
The
only possibility is |N(I)| = 24, since 16 doesnt divide |G| 
=
168 and G cant have any subgroup of index less than 7, 
since
168 = 2^3 * 3 * 7, so A_7 is the smallest alternating group in
which G can be contained. We already know N(I) cant have a
normal Sylow 2-subgroup or 3-subgroup, so N(I) =~ S_4, the
only
such group of order 24.
--
Jim Heckman
===
Subject: Re: More on the simple group of order 168
<10pp3hvhlilbd33@corp.supernews.com>
posting-account=jcZk7AwAAADXpPEyHtVyWC264SxtppRB
> A subgroup I of order 4 of a Sylow 2-group P_2 (of order 8)
is
> necessarily normal,
I had forgotten about this. I recently did the problem that a
subgroup of order p^(n-1) in a group of order p^n must be
normal,
but did not think to apply it here.
> so if I is contained in 2 P_2s, its
> normalizer N(I) has order >= 8*8/4 = 16 and a multiple of
8. The
> only possibility is |N(I)| = 24, since 16 doesnt divide
|G| =
> 168
Wonderful. If |I| = 4, then |N(I)| = 24 is now clear.
>and G cant have any subgroup of index less than 7,
<==> |H| <= 24 for any subgroup H.
> since
> 168 = 2^3 * 3 * 7, so A_7 is the smallest alternating group
in
> which G can be contained.
I recall the theorem
that if G is simple, and H < G with [G:H] = p, then by left
multiplication.
of of G on the p cosets G/H, G =~ subgp of S_p, so |G| | p!.
Since 168 = 2^3 * 3 * 7 | p!, p >= 7 so |H| <= 24. OK.
And you say A_7 rather than S_7 since if G < S_7 and not =~
S_7,
then G < A_7. Or, one could take the elements in G iso to the
even elements, which would be 1/2 the elements, forming a
subgroup H such that |H| = |G|/2, so H would we normal in G,
but G is simple so G < A_7. Is this right?
I is a normal subgroup of at least 2 P_2s, so at least 2
P_2s
are Sylow subgroups of N(I), since |N(I)| = 2^3 * 3 and |P_2|
= 2^3,
so N(I) cant have a
> normal Sylow 2-subgroup or 3-subgroup,
(since n_3 = 1 by sylow thm.)
> so N(I) =~ S_4, the only such group of order 24.
> Jim Heckman
I didnt know this was true, but I believe you.
I will have to verify this.
Van
===
Subject: Re: More on the simple group of order 168
>A subgroup I of order 4 of a Sylow 2-group P_2 (of order 8)
is
>necessarily normal,
> I had forgotten about this. I recently did the problem that
a
> subgroup of order p^(n-1) in a group of order p^n must be
normal,
> but did not think to apply it here.
More generally, a proper subgroup of a funite p-group is
properly contained in its normalizer. Another generalization:
If
p is the smallest prime dividing |G|, then any subgroup of
index
p is normal.
[...]
> And you say A_7 rather than S_7 since if G < S_7 and not =~
S_7,
> then G < A_7.
No. Simplest counterexample: S_n contains subgroups each
generated by a 2-cycle.
> Or, one could take the elements in G iso to the
> even elements, which would be 1/2 the elements, forming a
> subgroup H such that |H| = |G|/2, so H would we normal in G,
> but G is simple so G < A_7. Is this right?
Yes, thats the correct argument.
[...]
--
Jim Heckman
===
Subject: Re: More on the simple group of order 168
<10pp3hvhlilbd33@corp.supernews.com>
<10prdejast58lf5@corp.supernews.com>
posting-account=jcZk7AwAAADXpPEyHtVyWC264SxtppRB
I dont see why n_3 != 1 for N(I).
-----------------
(some hand-waving and babbling follows)
If N(I) had a normal subgroup, this would
not mean that G did, so this doesnt help.
Is there some counting to do here?
n_3 = 28 from above, and in N(I) there must
be 3 P_2s since n_2 = 1 mod 2, and n_2 = 21
in G, so perhaps there
is a way to show that there are 7 Is and 7 
N(I)s,
and that n_3 = 4 for N(I), which is true since it is S_4,
but I dont see it.
Van
===
Subject: Re: More on the simple group of order 168
<10pp3hvhlilbd33@corp.supernews.com>
<10prdejast58lf5@corp.supernews.com>
posting-account=jcZk7AwAAADXpPEyHtVyWC264SxtppRB
If P_3 is normal in N(I), since I normal in N(I),
P_3 I = I P_3 (ay = ya if a in I and y in P_3).
Above, we had N(P_3) = S_3. But I < N(P_3),
which is not possible (if |I| = 4).
Thus P_3 cant be normal in N(I).
I havent considered |I| = 2 yet.
Van
===
Subject: Algebra counterexample
Is there a simple example of a ring homomorphism f: A -> B
where
A and B are both associative K-algebras with 1, such that f is
not an algebra homomorphism, i.e., f is not K-linear, i.e.,
f(ka) <> k*f(a) for some k in K and a in A?
--
Jim Heckman
===
Subject: Re: Algebra counterexample
days. My association with the Department is that of an
alumnus.
>Is there a simple example of a ring homomorphism f: A -> B
where
>A and B are both associative K-algebras with 1, such that f
is
>not an algebra homomorphism, i.e., f is not K-linear, i.e.,
>f(ka) <> k*f(a) for some k in K and a in A?
Complex conjugation for C as a C-algebra?
--
Its not denial. Im just very selective 
about
what I accept as reality.
--- Calvin (Calvin and Hobbes)
Arturo Magidin
magidin@math.berkeley.edu
===
Subject: Re: Algebra counterexample
> Is there a simple example of a ring homomorphism f: A -> B
where
> A and B are both associative K-algebras with 1, such that f
is
> not an algebra homomorphism, i.e., f is not K-linear, i.e.,
> f(ka) <> k*f(a) for some k in K and a in A?
With K=A=B= the complex numbers,
take f to be complex conjugation.
===
Subject: Re: Algebra counterexample
>> Is there a simple example of a ring homomorphism f: A -> B
where
>> A and B are both associative K-algebras with 1, such that
f is
>> not an algebra homomorphism, i.e., f is not K-linear, i.e.,
>> f(ka) <> k*f(a) for some k in K and a in A?
> With K=A=B= the complex numbers,
> take f to be complex conjugation.
In fact, any automorphism of a fueld extension.
--
Timothy Murphy
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2,
Ireland
===
Subject: Re: Algebra counterexample
>> Is there a simple example of a ring homomorphism f: A -> B
where
>> A and B are both associative K-algebras with 1, such that
f is
>> not an algebra homomorphism, i.e., f is not K-linear, i.e.,
>> f(ka) <> k*f(a) for some k in K and a in A?
>With K=A=B= the complex numbers,
>take f to be complex conjugation.
> In fact, any automorphism of a fueld extension.
--
Jim Heckman
===
Subject: Re: Banach limits and ultrafulters
Now let F be any free ultrafulter. If we put
> x_1+x_2+...+x_n
>(1) LIM (x_n) = F-lim -----------------
> n
>then LIM is a Banach limit.
Question: Is each Banach limit of the form (1)?
...or maybe the answer is at the opposite extreme... the set
of
> Banach limits is a convex set, and those of the form (1)
> are the extrememe points. What do you think?
> To disprove the original conjecture, you would only have
> to show the set of Banach limits of the form (1) is
> not convex. Take two different free ultrafulters, then
> use the midpoint of their corresponding Banach limits.
> Show that there is no third ultrafulter that achieves that
limit.
I should apologize for 2 things. The furst one is late
response. The
second one is that I forgot to put references in my posting.
[1] and
[2] are books I used for Banach limit (but they can be found
elsewhere). [3] is a paper I found after sending the question
to
Google Groups.
[1] B. Balcar, P. Stepanek: Teorie Mnozin (Set Thoery, in
Czech)
[2] K.P.S. Bhaskara Rao, M. Bhaskara Rao: Theory of Charges
[3] J. Connor: Almost none of the sequences of 0s and 
1s
are almost
convergent, Internal. J. of Math. and Math. Sci. 13(4) (1990),
775-778.
I tried furst to follow your hint and to prove that the set of
linear
functionals of the form (1) is not convex. I tried choosing
two
appropriate sequences and use them to show this. But this
method was
rather cumbersome and led to lot of computation. I wasnt
able to
think up a simpler proof.
Your note that each linear combination of functionals of the
form (1)
is Banach limit saggest to use infunite linear combinations.
I.e. if
L_1, L_2,... are of the form (1) and Sum (c_i) =1, then
L := Sum (c_i L_i) (2)
is also Banach limit.
Afterwards I came across the paper [3] of J. Connor (from Ohio
University at that time). I learned from it that the set of
all Cesaro
convergent sequences has Lebesgue measure 1. (A sequence is
Cesaro
convergent to L if the arithmetic means of furst n temrs
conveges to L
as n tends to infunity.) On the other hand, the set of all
almost
convergent sequences has Lebesgue measure 0. (A sequence is
almost
convergent if each Banach limit of this sequence has the same
value.)
Each almost convergent sequence is Cesaro convergent.
So there exists a sequence of 0s and 1s 
which is Cesaro
convergent
but not almost convergent. For such a sequence each
functional of the
form (2) has the same value, but the value of Banach limit is
not
determined uniquely. Hence there are Banach limits which
arent of the
form (2).
So at last my question is:
Is some characterization of all Banach limits known?
(My feeling is that this question isnt easy.)
TIA
Martin
===
Subject: New Web Site Dealing with Theoretical Physics(now is
under
construction)
This is the scientifuc place for discussion about some Quantum
theories like Quantum Electrodynamics(QED), Quantum
Chromodynamics
(QCD), Quantum Field Theories(QFT), Quantum Gravity
Theories(QGT)
and some other new topics in theoretical high energy
physics(hep)
like String and Superstring theories, M-Theory (11 dimensional
supergravity at low energies), F-Theory (12 spacetime
dimensions)
and Grand Unifued Theories (GUTs),....
Hope to see you as an active and useful member in this club.
K.Niknejad
-------------------------------------------------------------
--------
For funding this site, you can see below URL:
http://www.kiarash-niknejad.com/
-------------------------------------------------------------
--------
===
Subject: Diophantian equations
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iAID3wk11785;
Let p be a prime, p>3. Is it true, that the equation
x_1+2x_2...+(p-1)x_{p-1}
+px_0(x_0-1)/2+px_1(x_1-1)/2+...+px_{p-1}(x_{p-1}-1)/2
-x(x-1)/2=n, where x=x_0 +x_1+ ... +x_{p-1},
has a solution for every positive integer n with integer x_0,
x_1, ... ...,
x_{p-1}? It is true if p=5.
===
Subject: Re: Algebra counterexample
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iAID3sX11607;
>Is there a simple example of a ring homomorphism f: A -> B
where
>A and B are both associative K-algebras with 1, such that f
is
>not an algebra homomorphism, i.e., f is not K-linear, i.e.,
>f(ka) <> k*f(a) for some k in K and a in A?
>--
>Jim Heckman
How about A = B = C, the complex numbers, viewed as
C-algebras,
and f the conjugation map?
Todd Trimble
===
Subject: Re: Math is a SIN ! ! ! !
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iAID3sn11615;
>> Jesus Christ does not want you to waste your time with
silly numbers
>> or you wont go to heaven !
>Wasnt it Weierstrass who said, God created integers, all
the rest is
>mans handiwork.
>Anyway, in my bible, God tells Noah to build an ark and
gives dimensions.
>I think we have proof God created a coordinate grid system.
Oh, I almost
>forgot, Jesus taught us to multiple using manipulatives,
loaves and
fushes.
Didnt Jesus use loaves and fushes to teach us 
the
Banach-Tarski
paradox?
Todd Trimble
===
Subject: Re: Math is a SIN ! ! ! !
>Anyway, in my bible, ...
Anyway,now maths is IN !! ..that we all are in.. thats in
how you see.
===
Subject: Re: Math is a SIN ! ! ! !
Yup. Especially trigonometry.
Thomas
===
Subject: Re: Algebra counterexample
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iAID3sr11621;
>Is there a simple example of a ring homomorphism f: A -> B
where
>A and B are both associative K-algebras with 1, such that f
is
>not an algebra homomorphism, i.e., f is not K-linear, i.e.,
>f(ka) <> k*f(a) for some k in K and a in A?
>--
>Jim Heckman
A=B=complex numbers as an algebra over itself
f=complex conjugation
H
===
Subject: Re: Set theory
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iAID3ts11637;
>I have been asked to solve the following problem. Suppose X
is an infunite
>set. Show that the cardinal number of X is less than the
cardinal number
of
>its power set. Now I have a problem with this. Suppose the
cardinality of
X
>is c. Now I thought that the only infunite cardinalities are
c and aleph
>null which would mean that the cardinality of X is not less
than the
power
>set of X. What is going on here?????
There are infunitely many infunite cardinalities. 
Aleph_0 and c
are two of them, and P(c), PP(c), ... give more. I dont 
know
where you got the thought from.
Todd Trimble
===
Subject: Re: NEAT PRODUCTS OF TRIG FUNCTIONS
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iAID3ts11628;
>I put problems like this (fun with rational values of
trigonometric
>expressions) on our local mathematical competitions:
>Prove (and where appropriate, evaluate):
>sec(4*pi/9) + sec(2*pi/9) - sec(pi/9) is an integer.
>(The angles in Fahrenheit being 80, 40, 20)
>cos(pi/7) + cos(2*pi/7) - (3*pi/7) is rational.
>8*cos(arccos(7/128)/3) is an integer.
>sin(pi/18) * sin(5*pi/18) * sin(7*pi/18) is rational.
>cos(pi/5) - cos(2*pi/5) is rational.
>Let A=arctan(2); show that for all positive integer numbers
n,
>the numbers 5^(n/2)*sin(n*A) are even integers.
>Finally:
>For fun, plot the curve and fund the area inside it:
>x^2 + (y-(x^2)^(1/3))^2=1
I have found these fascinating, but have a question about
your last problem.
Is the correct relation equivalent to
x^2 + [y - x^(2/3)]^2 = 1?
===
Subject: Diophantian equation
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iAID3tD11656;
Let p be a prime, p>3. Is it true, that the equation
0x_0+1x_1+...+(p-1)x_{p-1}
+px_0(x_0-1)/2+px_1(x_1-1)/2+...+px_{p-1}(x_{p-1}-1)/2
-x(x-1)/2=n,
where x is x_0+x_1+...+x_{p-1}, has a solution
(x_0,x_1,...,x_{p-1})
with integer x_0,x_1,...,x_{p-1}? It is true if p=5.
===
Subject: Re: Multidimensional 
Abels/Schroders functional
equations
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iAID3ta11661;
Peter, thank you very much for the assisstance!
===
Subject: Re: Mahlers equation
cordialy, lolo
===
Subject: Applications of proper classes?
Has the notion of a proper class been necessary to solve any
problems in
number theory or real analysis?
(See Wikipedia entry on Classes at
http://en.wikipedia.org/wiki/Class_%28set_theory%29 )
Dan
Download DC Proof 1.0 at http://www.dcproof.com
===
Subject: Re: Applications of proper classes?
> Has the notion of a proper class been necessary to solve
any problems in
> number theory or real analysis?
> (See Wikipedia entry on Classes at
> http://en.wikipedia.org/wiki/Class_%28set_theory%29 )
> Dan
> Download DC Proof 1.0 at http://www.dcproof.com
No. ZFC has no proper classes, and (so far) all advances in
number theory and real analysis can be formalized in ZFC
(plus, perhaps, some large cardinal axioms).
===
Subject: Re: Applications of proper classes?
>>Has the notion of a proper class been necessary to solve
any problems in
>>number theory or real analysis?
>>(See Wikipedia entry on Classes at
>>http://en.wikipedia.org/wiki/Class_%28set_theory%29 )
> No. ZFC has no proper classes, and (so far) all advances in
> number theory and real analysis can be formalized in ZFC
> (plus, perhaps, some large cardinal axioms).
Why are large cardinal axioms better than classes?
I presume the reference to large cardinal axioms is to cover
the uses of
Grothendieck universes. But then, there are theories of
sets+classes
that are conservative extensions of ZFC (for example
Ackermanns, or
ZFC+reßection principle a la Fefermans proposal with
terminology
change) that can do all the things that Grothendieck
universes are
called upon to do.
And is the elimination of classes by using Grothendieck
universes really
work as advertised? Even those who use Grothendieck universes
talk of
proper classes when nobody is looking: They refer to >the<
category of
sets, meaning all sets, rather than the category of sets in a
fuxed
universe V; the French usage of Ôsmall means 
that the
category of
V-small sets is still a proper class, even if mathematicians
in the
trenches are completely unaware of it. [I am reminded of one
who thought
that the set of all sets of rank < 2omega, a pretty small
topos, is
same as the Ôset of all sets of cardinality < 
2omega,
whatever the
latter means!] The only way out is to defune V-small to mean
Ôis an
element of V. It seems to be the case that all uses of
universes need
only one (in any case, a funite number): 
Ôsmall means is in
V, large
means in not in V, and Ôhuge sets needed are 
only class
abstracts of
the kind usable even in ZFC. Now replace Ôsmall 
by Ôset and
Ôlarge by
class, and it looks a lot like Ackermanns theory of sets 
and
classes,
except that we have a strong version of replacement whose
necessity for
theorems actually being proven is doubtful.
===
Subject: Re: Applications of proper classes?
>>Has the notion of a proper class been necessary to solve
any problems in
>>number theory or real analysis?
>>(See Wikipedia entry on Classes at
>>http://en.wikipedia.org/wiki/Class_%28set_theory%29 )
>>Dan
>>Download DC Proof 1.0 at http://www.dcproof.com
> No. ZFC has no proper classes, and (so far) all advances in
> number theory and real analysis can be formalized in ZFC
> (plus, perhaps, some large cardinal axioms).
The class of all sets would be a proper class. Its just 
that
sets are
more interesting.
--
Will Twentyman
email: wtwentyman at copper dot net
===
Subject: Re: Applications of proper classes?
>>Has the notion of a proper class been necessary to solve
any problems
in
>>number theory or real analysis?
>No. ZFC has no proper classes, and (so far) all advances in
>number theory and real analysis can be formalized in ZFC
>(plus, perhaps, some large cardinal axioms).
> The class of all sets would be a proper class. Its just
that sets are
> more interesting.
Furthermore, it can be shown that anything you can do with
proper classes
(including super-classes of THESE, etc), can in a suitable
sense already
be done with sets plus large cardinals.
-------------------------------------------------------------
---------------
--
Bill Taylor W.Taylor@math.canterbury.ac.nz
-------------------------------------------------------------
---------------
--
Mathematics runs under set theory just as Freecell runs under
windows.
-------------------------------------------------------------
---------------
--
===
Subject: Re: Applications of proper classes?
>>Has the notion of a proper class been necessary to solve
any problems
in
>>number theory or real analysis?
>>(See Wikipedia entry on Classes at
>>http://en.wikipedia.org/wiki/Class_%28set_theory%29 )
>>Dan
>>Download DC Proof 1.0 at http://www.dcproof.com
>
>No. ZFC has no proper classes, and (so far) all advances in
>number theory and real analysis can be formalized in ZFC
>(plus, perhaps, some large cardinal axioms).
> The class of all sets would be a proper class. Its just
that sets are
> more interesting.
I was under the impression that you cant build the class of
all sets
using ZFC, even in principal.
Ôcid Ôooh
===
Subject: Re: Applications of proper classes?
>Has the notion of a proper class been necessary to solve any
problems in
>number theory or real analysis?
>(See Wikipedia entry on Classes at
>http://en.wikipedia.org/wiki/Class_%28set_theory%29 )
>Dan
>Download DC Proof 1.0 at http://www.dcproof.com
>> No. ZFC has no proper classes, and (so far) all advances in
>> number theory and real analysis can be formalized in ZFC
>> (plus, perhaps, some large cardinal axioms).
> The class of all sets would be a proper class. Its just
that sets are
> more interesting.
The class of all sets would be a proper class if ZFC had
proper classes,
but it doesnt.
--
Dave Seaman
Judge Yohns mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&
bookid=228>
===
Subject: Re: Algebra counterexample
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iAIE6ZN17641;
Mathematicians seem to be a bit lazy ...
===
Subject: Best method of combining estimates
I have inherited a processing scheme that looks for features
of a
certain size in an image. This is done by applying a series of
fulters, each of which has a maximum output at a certain
feature size
and whose output drops off at bigger and smaller sizes. At
present,
size detection is carried out by simply selecting the fulter
with the
greatest output.
The outputs of the fulters are fairly smooth, and overlap
signifucantly. The outputs are proportional to image contrast,
which
can vary signifucantly from image to image.
I would like to try to get a bit more accuracy from the
existing
processing scheme. I thought of some method of looking at the
ratios
of the output of the Ômaximum 
fulter to those of the fulters on
either side. This gives me two numbers that should allow some
form of
Ôinterpolation for a more accurate result.
I could run the fulters with features of different lengths and
get
enough information for a 2-D lookup table for each fulter (1-D
for the
ones at the ends), but I cant help thinking that 
furst, there
should
be a more elegant (and storage-effucient) way of doing this;
second,
it might be useful to be able to extend the scheme to more
than just
the ones either side, and that would mean a many-D lookup
table; and
third, someone must have done this before.
Does anyone have any suggestions, references, links, hints,
etc?
Jon
===
Subject: Re: Sum of an infunite series ....
schrieb Kira Yamato <no@mail.com>:
>> I assume you mean the infunite series
>> 1+ (2^2)/7 + (3^2)/7^2 + (4^2)/7^3 +...,
>> in other words,
>> sum_1^infty (n^2)/7^(n-1).
>> You could start with
>> 1/(1-x)=sum_0^infty x^n
>> Differentiate with respect to x, multiply by x, then
differentiate
>> again. Finally, put x=1/7.
> Wow. Thats amazing! That is sooo indirect but elegant. 
How
did you
> ever thought of that approach?
Generatingfunctionology!
http://www.math.upenn.edu/~wilf/DownldGF.html
===
Subject: Re: Sum of an infunite series ....
I do know Geometric series but am not aware of how to
Ôdifferentiate.
Is there any way this can be solved just by using concepts of
Geometric series?
TIA
Raquel.
===
Subject: Re: Sum of an infunite series ....
> I do know Geometric series but am not aware of how to
Ôdifferentiate.
> Is there any way this can be solved just by using concepts
of
> Geometric series?
Yes. Instead of differentiating
1/(1-x) = 1 + x + x^2 + x^3 + ...
you can square it. You get
1/(1-x) = (1 + x + x^2 + x^3 + ...)(1 + x + x^2 + x^3 + ...)
= 1 + 2x + 3x^2 + 4x^3 + ...
Now try cubing it :-)
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square
root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Re: Sum of an infunite series ....
>I do know Geometric series but am not aware of how to
Ôdifferentiate.
>Is there any way this can be solved just by using concepts of
>Geometric series?
> Yes. Instead of differentiating
> 1/(1-x) = 1 + x + x^2 + x^3 + ...
> you can square it. You get
> 1/(1-x) = (1 + x + x^2 + x^3 + ...)(1 + x + x^2 + x^3 + ...)
> = 1 + 2x + 3x^2 + 4x^3 + ...
> Now try cubing it :-)
What amounts to the same thing, you can start with
1 + 4x + 9x^2 + 16x^3 + 25x^4 + ...
and multiply it by 1 - x,
and again by 1 - x,
and if necessary even again by 1 - x.
--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
===
Subject: Help with analysis of complex function
Math folks,
I am trying to show that the magnitude of the following
function achieves
its maximum at z = R + Pi I.
E^(3 z)/(1 + E^z) where z ranges from z = R (>0) to z = R + 2
Pi I.
When z = R, the value of the function is E^(3 R)/(1 + E^R)
When z = R + Pi I, the value of the function is E^(3 R)/(-1 +
E^R)
When z = R + 2 Pi I, the value of the function is E^(3 R)/(1
+ E^R) again.
Help would be appreciated.
Diana
--
God made the integers, all else is the work of man.
L. Kronecker, Jahresber. DMV 2, S. 19.
===
Subject: Re: Help with analysis of complex function
> I am trying to show that the magnitude of the following
function achieves
> its maximum at z = R + Pi I.
> E^(3 z)/(1 + E^z) where z ranges from z = R (>0) to z = R +
2 Pi I.
|e^(3z)/(1 + e^z)| = |e^(3z)|/|1 + e^z| = e^(3R)|/|1 + e^z|.
You maximize
the last expression by minimizing the denominator. But 1 +
e^(R+it)
describes a circle of radius e^R, centered at 1, as t goes
from 0 to 2Pi.
Where does that cirlce have minimum modulus?
===
Subject: Re: Help with analysis of complex function
>I am trying to show that the magnitude of the following
function
achieves
>its maximum at z = R + Pi I.
E^(3 z)/(1 + E^z) where z ranges from z = R (>0) to z = R + 2
Pi I.
> |e^(3z)/(1 + e^z)| = |e^(3z)|/|1 + e^z| = e^(3R)|/|1 +
e^z|. You maximize
> the last expression by minimizing the denominator. But 1 +
e^(R+it)
> describes a circle of radius e^R, centered at 1, as t goes
from 0 to 2Pi.
> Where does that cirlce have minimum modulus?
===
Subject: Need Letters Journal for Chaos Theory
Does anybody know of a mathematical letters journal that
would be
appropriate for publishing a short latter on chaos in
planetary orbits?
TIA.
===
Subject: Re: Cantors diagonal proof wrong?
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iAIFPLW24454;
>> I dont think this is true at all. In my experience, if
you ask a
>> roomful of non-mathematicians whether i, the square root
of -1,
>> exists they will mostly claim it doesnt.
>Nonsense. The imaginary number i is 1 turned 90 degrees
>counterclockwise on the complex plane. It has as much
existence as any
>integer or real number.
>Bob Kolker
He said *non-mathematicians*. Im guessing he himself is
a mathematician who already knew that. (Although Galois
theory teaches us that we could just as well say i is
the 90 degree clockwise turn!)
Todd Trimble
===
Subject: Computer language and category theory
Is there any work done one computer languages and category
theory? To
clarify, what I am looking into is something like the
following:
Given the language
S ::= A | B
A ::= aa | ba | A-aa | B-ba
B ::= ab | bb | A-ab | B-bb
Thus, the set of sentences in S is sequences
{a,b}{a,b}-{a,b}{a,b}-... such that the letter in front of
the dash
(-) is the same as the letter behind it.
I think this language can be described as a category as
follows:
C=<Obj,Morph,*>
Obj={A,B}
Morph=Hom(A,A) U Hom(A,B) U Hom(B,A) U Hom(B,B)
Hom(x,y) = any sentence s in S such that it starts with an x
and
ends with an y.
E.g. bb-ba, ba, ba-aa-ab-ba in Hom(B,A)
The operator * is defuned as follows:
Given x in Hom(X,Y) and y in Hom(Y,Z) such that neither x nor
y
is a unit morphism. Then x*y = x-y
The unit elements of Hom(A,A) and Hom(B,B) is just an empty
sentence
handled specially:
x * e = e * x = x
(a) Does this look sensible?
However, I dont have such a simple language, but rather a
slightly
more complicated one:
T ::= A | B
A ::= aa | ba | A-aa | B-ba
B ::= ab | bb | A-ab | B-bb | A-K
K ::= ( A )-B | ( B )-B
The K, introduced in order to be able to create a tree-like
structure,
makes the elements of T unsuited to form a set of morphisms,
as
each morphism in a category have excactly one source object
and one
destination object.
(b) What kind of mathematical tools should I study to handle
language
constructs like the language T?
In reality, our objects are a little different, but not in
essence.
My collegue wants to defune something like a partial
parametriced
monoid, an animal I never before have encountered. Partiality
follows from the fact that you cannot take two arbitrary
elements of S
(or T) and form a new element of S (or T). Parametrization
does
occure in some sense in the language T, where we must choose
whether
to insert an A sentence into the ( A )-B or a B into the same
meta
sentence. (Was this understandable?)
However, I dont like such animals. So I wonder if there is
some use
of category theory or something else that have been used to
model
language like constructs.
--
Jon Haugsand
Dept. of Informatics, Univ. of Oslo, Norway,
mailto:jonhaug@ifu.uio.no
http://www.ifu.uio.no/~jonhaug/, Phone: +47 22 85 24 92
===
Subject: 3D Model for allocation of total capacity
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iAIFVxL25040;
Can anyone help? I have an interesting issue. Ive chosen to
study maths and
computer science and one of the problems Ive been asked to
solve,
potentially using a 3D Modeling formula is:
In a Shared Server environment, where total capacity of a
server is measured
by total CPU capacity, total disk capacity and total memory
capacity, I need
a formula to help me understand how much each user is
requesting of the total
capacity.
By example:
On a single server total CPU is 2.88 Ghz, 16Gb Memory and
146Gb of Disk
Space.
I have a user who only wants to use 1Ghz CPU, 2Gb Mem and 2Gb
of Disk Space.
What formula can I use that tells me what percentage of the
total capacity of
the machine this person is requiring?
Nick
===
Subject: Re: 3D Model for allocation of total capacity
Possible the maximum of the 3 percentages.
If you need more than 100% of the available disk space you
are done for.
But then again the same may not true of the CPU. If a user is
doing
something real-time they may need x Ghz but if they are not
they may get by
with less.
Something similar could be said of memory as this can be
paged, but with
consequences for CPU and disk.
> Can anyone help? I have an interesting issue. Ive chosen
to study maths
and computer science and one of the problems Ive been asked
to solve,
potentially using a 3D Modeling formula is:
> In a Shared Server environment, where total capacity of a
server is
measured by total CPU capacity, total disk capacity and total
memory
capacity, I need a formula to help me understand how much
each user is
requesting of the total capacity.
> By example:
> On a single server total CPU is 2.88 Ghz, 16Gb Memory and
146Gb of Disk
Space.
> I have a user who only wants to use 1Ghz CPU, 2Gb Mem and
2Gb of Disk
Space. What formula can I use that tells me what percentage
of the total
capacity of the machine this person is requiring?
> Nick
===
Subject: Re: Russia Developing New Nuclear Missile
>Russia Developing New Nuclear Missile
>>Russias new nuclear missile can carry up to 10 nuclear
warheads
>>weighing a total of 4.4 tons,compared with the Topol-Ms
1.32-ton
>>combat payload.
Is this an offensive or a defensive weapon?
>>Super-defensive ... actually defensive^3. (Easy math: Plain
nukes =
>>anti-conventional defense, hence anti-ballistic defense =
defense^2,
>>ergo this new Russian guy = anti-defensive^2 defense =
defense^3.)
>Seems quite logical. What comes after defense^3?
>> No, Topolski (rex, etc) missed a solution. He is saying
this is designed
>> to penetrate Star Wars Shield, with which I absolutely
agree, but his
>> maths is pure rubbish. Defensive and offensive are related
by defensive
>> = - offensive. Is not this the logic ?! Then surely
>> defensive^3=(-offensive)^3=-offensive^3! and the missile
is both
>> affensive and defensive in equal amounts!!!
> Blasphemy aside, how come that the alleged new Russian
missile is
> a[o]ffensive? In my deduction it is defensive^3, ergo
defensive (see
> above). Now by your algebra the guy is -offensive^3, ergo
still defensive.
> Q.E.D.
Q.E.D. aside, if the solution for offensiveness-defensiveness
is y=x^3 to
put it in plain kindergarten maths that even Brookski can
understand, then
you do not know whether y is positive or negative until you
use a particular
x. Ergo, logic sez that if you use missile offensively, then
it will be
offensive (negative x, thus negative y), and vice versa. This
is logic.
===
Subject: Re: Russia Developing New Nuclear Missile
>Russia Developing New Nuclear Missile
Russias new nuclear missile can carry up to 10 nuclear
warheads
>weighing a total of 4.4 tons,compared with the Topol-Ms
1.32-ton
>combat payload.
>>Is this an offensive or a defensive weapon?
Super-defensive ... actually defensive^3. (Easy math: Plain
nukes =
>anti-conventional defense, hence anti-ballistic defense =
defense^2,
>ergo this new Russian guy = anti-defensive^2 defense =
defense^3.)
>>Seems quite logical. What comes after defense^3?
> No, Topolski (rex, etc) missed a solution. He is saying
this is designed
> to penetrate Star Wars Shield, with which I absolutely
agree, but his
> maths is pure rubbish. Defensive and offensive are related
by defensive
> = - offensive. Is not this the logic ?! Then surely
> defensive^3=(-offensive)^3=-offensive^3! and the missile is
both
> affensive and defensive in equal amounts!!!
>> Blasphemy aside, how come that the alleged new Russian
missile is
>> a[o]ffensive? In my deduction it is defensive^3, ergo
defensive (see
>> above). Now by your algebra the guy is -offensive^3, ergo
still
>> defensive. Q.E.D.
> Q.E.D. aside, if the solution for
offensiveness-defensiveness is y=x^3 to
> put it in plain kindergarten maths that even Brookski can
understand, then
> you do not know whether y is positive or negative until you
use a
> particular x. Ergo, logic sez that if you use missile
offensively, then it
> will be offensive (negative x, thus negative y), and vice
versa. This is
> logic.
Hey Homo,
Use Russian math...furst I smack you up side your head. If you
get up, I
smack you a second time.
1+1=2
This is what MTRP understands. It is a more elegant solution
anyway. No
decimal points are required. We only smack you up side the
head in whole
numbers.
===
Subject: Re: Russia Developing New Nuclear Missile
>Russia Developing New Nuclear Missile
>
>>Russias new nuclear missile can carry up to 10 nuclear
warheads
>>weighing a total of 4.4 tons,compared with the Topol-Ms
1.32-ton
>>combat payload.
Is this an offensive or a defensive weapon?
>>Super-defensive ... actually defensive^3. (Easy math: Plain
nukes =
>>anti-conventional defense, hence anti-ballistic defense =
defense^2,
>>ergo this new Russian guy = anti-defensive^2 defense =
defense^3.)
Seems quite logical. What comes after defense^3?
>> No, Topolski (rex, etc) missed a solution. He is saying
this is
>> designed to penetrate Star Wars Shield, with which I
absolutely agree,
>> but his maths is pure rubbish. Defensive and offensive are
related by
>> defensive = - offensive. Is not this the logic ?! Then
surely
>> defensive^3=(-offensive)^3=-offensive^3! and the missile
is both
>> affensive and defensive in equal amounts!!!
> Blasphemy aside, how come that the alleged new Russian
missile is
> a[o]ffensive? In my deduction it is defensive^3, ergo
defensive (see
> above). Now by your algebra the guy is -offensive^3, ergo
still
> defensive. Q.E.D.
>> Q.E.D. aside, if the solution for
offensiveness-defensiveness is y=x^3 to
>> put it in plain kindergarten maths that even Brookski can
understand,
>> then you do not know whether y is positive or negative
until you use a
>> particular x. Ergo, logic sez that if you use missile
offensively, then
>> it will be offensive (negative x, thus negative y), and
vice versa. This
>> is logic.
> Hey Homo,
> Use Russian math...furst I smack you up side your head. If
you get up, I
> smack you a second time.
> 1+1=2
> This is what MTRP understands. It is a more elegant
solution anyway. No
> decimal points are required. We only smack you up side the
head in whole
> numbers.
Hey SCRMC,
I did not know Brookski maths was still alive in Russia, I
though it
disappeared in stone age when they fugured out how to count on
fungers ;)
===
Subject: Re: Russia Developing New Nuclear Missile
>Russia Developing New Nuclear Missile
>Russias new nuclear missile can carry up to 10 nuclear
warheads
>weighing a total of 4.4 tons,compared with the Topol-Ms
1.32-ton
>combat payload.
>>Is this an offensive or a defensive weapon?
Super-defensive ... actually defensive^3. (Easy math: Plain
nukes =
>anti-conventional defense, hence anti-ballistic defense =
defense^2,
>ergo this new Russian guy = anti-defensive^2 defense =
defense^3.)
>>Seems quite logical. What comes after defense^3?
>No, Topolski (rex, etc) missed a solution. He is saying this
is designed
>to penetrate Star Wars Shield, with which I absolutely
agree, but his
>maths is pure rubbish. Defensive and offensive are related
by defensive
>= - offensive. Is not this the logic ?! Then surely
>defensive^3=(-offensive)^3=-offensive^3! and the missile is
both
>affensive and defensive in equal amounts!!!
>>Blasphemy aside, how come that the alleged new Russian
missile is
>>a[o]ffensive? In my deduction it is defensive^3, ergo
defensive (see
>>above). Now by your algebra the guy is -offensive^3, ergo
still defensive.
>>Q.E.D.
> Q.E.D. aside, if the solution for
offensiveness-defensiveness is y=x^3 to
> put it in plain kindergarten maths that even Brookski can
understand, then
> you do not know whether y is positive or negative until you
use a
particular
> x. Ergo, logic sez that if you use missile offensively,
then it will be
> offensive (negative x, thus negative y), and vice versa.
This is logic.
No. Cuz U add one wrong assumption that plain nukes are
offensive,
whereas even Brookski knows they are not. Thus the sign of
our x, and
hence y, is already determined - ponimayesh?
===
Subject: Re: Russia Developing New Nuclear Missile
>>Russia Developing New Nuclear Missile
>>Russias new nuclear missile can carry up to 10 nuclear
warheads
>>weighing a total of 4.4 tons,compared with the Topol-Ms
1.32-ton
>>combat payload.
Is this an offensive or a defensive weapon?
>>Super-defensive ... actually defensive^3. (Easy math: Plain
nukes =
>>anti-conventional defense, hence anti-ballistic defense =
defense^2,
>>ergo this new Russian guy = anti-defensive^2 defense =
defense^3.)
Seems quite logical. What comes after defense^3?
>>No, Topolski (rex, etc) missed a solution. He is saying
this is designed
>>to penetrate Star Wars Shield, with which I absolutely
agree, but his
>>maths is pure rubbish. Defensive and offensive are related
by defensive
>>= - offensive. Is not this the logic ?! Then surely
>>defensive^3=(-offensive)^3=-offensive^3! and the missile is
both
>>affensive and defensive in equal amounts!!!
>Blasphemy aside, how come that the alleged new Russian
missile is
>a[o]ffensive? In my deduction it is defensive^3, ergo
defensive (see
>above). Now by your algebra the guy is -offensive^3, ergo
still
>defensive. Q.E.D.
>> Q.E.D. aside, if the solution for
offensiveness-defensiveness is y=x^3 to
>> put it in plain kindergarten maths that even Brookski can
understand,
>> then you do not know whether y is positive or negative
until you use a
>> particular x. Ergo, logic sez that if you use missile
offensively, then
>> it will be offensive (negative x, thus negative y), and
vice versa. This
>> is logic.
> No. Cuz U add one wrong assumption that plain nukes are
offensive, whereas
> even Brookski knows they are not. Thus the sign of our x,
and hence y, is
> already determined - ponimayesh?
No, even Brookski knows that though offensive/defensive split
of nuclear
weapons use is about 1/99, this does not in any way limit
the offensive
effectiveness of these weapons. Kapish?
===
Subject: Re: Russia Developing New Nuclear Missile
>Russia Developing New Nuclear Missile
>Russias new nuclear missile can carry up to 10 nuclear
warheads
>weighing a total of 4.4 tons,compared with the Topol-Ms
1.32-ton
>combat payload.
>>Is this an offensive or a defensive weapon?
Super-defensive ... actually defensive^3. (Easy math: Plain
nukes =
>anti-conventional defense, hence anti-ballistic defense =
defense^2,
>ergo this new Russian guy = anti-defensive^2 defense =
defense^3.)
>>Seems quite logical. What comes after defense^3?
No, Topolski (rex, etc) missed a solution. He is saying this
is
designed
>to penetrate Star Wars Shield, with which I absolutely
agree, but his
>maths is pure rubbish. Defensive and offensive are related
by defensive
>= - offensive. Is not this the logic ?! Then surely
>defensive^3=(-offensive)^3=-offensive^3! and the missile is
both
>affensive and defensive in equal amounts!!!
>>Blasphemy aside, how come that the alleged new Russian
missile is
>>a[o]ffensive? In my deduction it is defensive^3, ergo
defensive (see
>>above). Now by your algebra the guy is -offensive^3, ergo
still
>>defensive. Q.E.D.
>Q.E.D. aside, if the solution for
offensiveness-defensiveness is y=x^3 to
>put it in plain kindergarten maths that even Brookski can
understand,
>then you do not know whether y is positive or negative until
you use a
>particular x. Ergo, logic sez that if you use missile
offensively, then
>it will be offensive (negative x, thus negative y), and vice
versa. This
>is logic.
>>No. Cuz U add one wrong assumption that plain nukes are
offensive, whereas
>>even Brookski knows they are not. Thus the sign of our x,
and hence y, is
>>already determined - ponimayesh?
> No, even Brookski knows that though offensive/defensive
split of nuclear
> weapons use is about 1/99, this does not in any way limit
the offensive
> effectiveness of these weapons. Kapish?
Of course it does - even by your Brookski logic the
probability would be
merely 0.01010101010. But let us stop here. First I was
kidding and
second nobody can match Our Majesty in a pointless discussion
anyway. So
sez sci.
===
Subject: Re: Russia Developing New Nuclear Missile
>>Russia Developing New Nuclear Missile
>>Russias new nuclear missile can carry up to 10 nuclear
warheads
>>weighing a total of 4.4 tons,compared with the Topol-Ms
1.32-ton
>>combat payload.
Is this an offensive or a defensive weapon?
>>Super-defensive ... actually defensive^3. (Easy math: Plain
nukes =
>>anti-conventional defense, hence anti-ballistic defense =
defense^2,
>>ergo this new Russian guy = anti-defensive^2 defense =
defense^3.)
Seems quite logical. What comes after defense^3?
>>No, Topolski (rex, etc) missed a solution. He is saying
this is designed
>>to penetrate Star Wars Shield, with which I absolutely
agree, but his
>>maths is pure rubbish. Defensive and offensive are related
by defensive
>>= - offensive. Is not this the logic ?! Then surely
>>defensive^3=(-offensive)^3=-offensive^3! and the missile is
both
>>affensive and defensive in equal amounts!!!
>Blasphemy aside, how come that the alleged new Russian
missile is
>a[o]ffensive? In my deduction it is defensive^3, ergo
defensive (see
>above). Now by your algebra the guy is -offensive^3, ergo
still
>defensive. Q.E.D.
>> Q.E.D. aside, if the solution for
offensiveness-defensiveness is y=x^3 to
>> put it in plain kindergarten maths that even Brookski can
understand,
>> then you do not know whether y is positive or negative
until you use a
>> particular x. Ergo, logic sez that if you use missile
offensively, then
>> it will be offensive (negative x, thus negative y), and
vice versa. This
>> is logic.
> No. Cuz U add one wrong assumption that plain nukes are
offensive, whereas
> even Brookski knows they are not.
They are neither offensive or defensive but they are both.
There is nothing to say that nuclear weapons could not be
used offensively.
It is only the naive assumption that they will not be used
offensively
(furst strike) that defunes them as defensive 
weapons.
The US anti-missile star wars program is a true defensive
system though.
It is only designed to shoot down missiles from an agressor
nation.
Thus the sign of our x, and
> hence y, is already determined - ponimayesh?
===
Subject: Re: Russia Developing New Nuclear Missile
>Russia Developing New Nuclear Missile
>Russias new nuclear missile can carry up to 10 nuclear
warheads
>weighing a total of 4.4 tons,compared with the Topol-Ms
1.32-ton
>combat payload.
>>Is this an offensive or a defensive weapon?
Super-defensive ... actually defensive^3. (Easy math: Plain
nukes =
>anti-conventional defense, hence anti-ballistic defense =
defense^2,
>ergo this new Russian guy = anti-defensive^2 defense =
defense^3.)
>>Seems quite logical. What comes after defense^3?
No, Topolski (rex, etc) missed a solution. He is saying this
is
designed
>to penetrate Star Wars Shield, with which I absolutely
agree, but his
>maths is pure rubbish. Defensive and offensive are related
by defensive
>= - offensive. Is not this the logic ?! Then surely
>defensive^3=(-offensive)^3=-offensive^3! and the missile is
both
>affensive and defensive in equal amounts!!!
>>Blasphemy aside, how come that the alleged new Russian
missile is
>>a[o]ffensive? In my deduction it is defensive^3, ergo
defensive (see
>>above). Now by your algebra the guy is -offensive^3, ergo
still
>>defensive. Q.E.D.
>Q.E.D. aside, if the solution for
offensiveness-defensiveness is y=x^3 to
>put it in plain kindergarten maths that even Brookski can
understand,
>then you do not know whether y is positive or negative until
you use a
>particular x. Ergo, logic sez that if you use missile
offensively, then
>it will be offensive (negative x, thus negative y), and vice
versa. This
>is logic.
>>No. Cuz U add one wrong assumption that plain nukes are
offensive, whereas
>>even Brookski knows they are not.
> They are neither offensive or defensive but they are both.
Hi Brookski. Welcome back - your new ID is approved.
> There is nothing to say that nuclear weapons could not be
used
offensively.
> It is only the naive assumption that they will not be used
offensively
> (furst strike) that defunes them as defensive 
weapons.
> The US anti-missile star wars program is a true defensive
system
though.
> It is only designed to shoot down missiles from an agressor
nation.
>> Thus the sign of our x, and hence y, is already determined
- ponimayesh?
===
Subject: Re: Russia Developing New Nuclear Missile
>The US anti-missile star wars program is a true defensive
system
though.
>It is only designed to shoot down missiles from an agressor
nation
Well Putin couldnt have given GW a better Christmas 
present.
Now GW
can push hard for his Star Wars anti missile shield program.
If I were a conspiracy theorist I would have said GW got
Putin to make
the announcement to.......................
===
Subject: Re: Russia Developing New Nuclear Missile
>>The US anti-missile star wars program is a true defensive
system
though.
>>It is only designed to shoot down missiles from an agressor
nation
> Well Putin couldnt have given GW a better Christmas
present. Now GW
> can push hard for his Star Wars anti missile shield program.
> If I were a conspiracy theorist I would have said GW got
Putin to make
> the announcement to.......................
you may be onto something there :)
===
Subject: Re: Russia Developing New Nuclear Missile
>>The US anti-missile star wars program is a true defensive
system
though.
>>It is only designed to shoot down missiles from an agressor
nation
> Well Putin couldnt have given GW a better Christmas
present. Now GW
> can push hard for his Star Wars anti missile shield program.
> If I were a conspiracy theorist I would have said GW got
Putin to make
> the announcement to.......................
Not to worry! Putin will only nuke European rogue nations.
Russia just
signed a major agreement with Boeing on 7E7 production.
Russia needs a new
civilian airliner ßeet. Now they get latest US technology,
part of the pie
and new ßeet of aircraft. You dont nuke somebody 
whos
giving you all this
stuff.
Airbus sucks!
===
Subject: Coset codes of a Reed-Solomon code?
I need some help on a problem regarding coset codes of
Reed-Solomon
codes (using GF(2^m)). I will quickly describe the problem.
Defune the same_symbol_distance of a codeword c as the maximum
number
of same symbols contained in c.
Defune the same_symbol_distance of a codebook C as the number
of same
symbols of c_s, where c_s is a codeword in C containing the
maximum
number of same symbols.
As an example, the same_symbol_distance of the all-zero
codeword of an
(n,k) RS code is equal to n. Also, the same_symbol_distance
of the
all e codeword (e in GF(2^m)) is equal to n. Thus the
same_symbol_distance of an (n,k) RS codebook is equal to n.
Given an (n,k) Reed-Solomon code C_rs, what is the minimum
same_symbol_distance that can be obtained for C = C_rs + h,
using the
appropriate coset h?
How can I determine the cosets that will give a minimum
same_symbol_distance?
I have done a simulation using a (7,4) RS code, with g(x) =
(x-alpha^1)(x - alpha^2)(x-alpha^3).
I used the following polynomials as cosets: a_0 + a_1*x +
a_2*x^2,
where a_i in GF(2^m). (Thus, all polynomials with degree <
degree(g(x)) was used). This is from the fact that any
polynomial
p(x) with degree >= degree(g(x)) will be in C_rs + h_1 where
h_1 =
p(x) mod g(x).
I have also found that for the above code, the
same_symbol_distance is
4. Will the same_symbol_distance always equal k?
Another result is that the cosets generating a minimum
same_symbol_distance have weight equal to (n-k). However,
this is a
necessary condition but not a suffucient condition. To see
that this
is necessary, consider the same_symbol_distance of c_0 + h,
where c_0
is the all zero codeword and h is an arbitrary coset.
Clearly, the
same_symbol_distance of (c_0 + h) = n - weight(h). Then
same_symbol_distance(C_rs + h) <= n-weight(h). (The above is
only
true if it can be shown that the minimum same_symbol_distance
of (C_rs
+ h) is equal to k, for some h.)
Also, from the simulation, I found that there are more than
one coset
h_min generating the minimum same_symbol_distance.
In C_rs + h_min, more than one codeword have a
same_symbol_distance =
same_symbol_distance(C_rs+h_min).
Is there maybe a mathematical way to prove the above?
Any help and/or suggestions will be greatly appreciated
Jaco Versfeld
===
Subject: Re: Mahlers equation
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iAIGbnm31173;
>cordialy, lolo
I would like your looking at some recent threads
on functional equation solving:
7 nov int f(x)*g(x,y) ,
15 nov solving f(h(x),y,z-x)=....
15 nov multidimensional Abels
Moroever, studying Ôforms is very interesting 
:
phi(x) | phi(h(x))=phi(x)+1
properties of f(x,y,z)=m^[y.phi(x)+z+c](L) ?
Bracket is invariant for (x,y,z) ->(h(x),y,z-x) f->f
for z -> z+1 f-> m(f)
Friendly,Alain.
===
Subject: Re: Center of Mass of the Universe?
>[hanson to EL]
>You have of course seen the lights behind the lightwalls.
> [EL]
> what we see by looking into the sky is the sum of all
light-histories?
[hanson]
No, of course not, as can bee seen from some of the responses
in
this thread. Furthermore, re: the light history, summed or
otherwise,
....that story about this issue is far from being suffuciently
known.
The current state of the art in astronomy and astrophysics
gives
us merely some EM data sets from these ßickering dots in the
night sky. This gives us then the empirical anchors around and
about which we tell a heuristic story that the professional do
proudly call a theory.......ahahaha.....and from which its
disciples
and adorers pick out the buzzwords & proselytize all over
town,
hoping that listeners think they are smart
too......ahahahaha....
Once, when we funally do understand what gravity is all about,
beyond the crude and feeble Einstein interpretation, perhaps
from LIGO or VIRGO, or via neutrino physics, then the next
edition
of upgraded cosmological stories will be told.
Mind you that even then the mystery of the universe will not
be solved. I must repeat my Caveat:
Remember, ALL these theories are just stories about nature.
**STORIES** that more or less confurm what we (like to)
believe to have experienced. But, the actual reality of it all
may very well be a very different story whose borne is hidden
from us by fuat......or by the limitations of our own
make-up...
> [EL]
> The farthest is the oldest, but some light have passed by
us already,
> that was sourced from nearer lights at more recent times
than the
> oldest.
> This means that much older yet and farther lights might
arrive soon
> and our knowledge of the size of the universe expands.
[hanson]
Yes. That is one way of looking at it. Equally, another way
may be that
due to **whatever** reason these light/photons, when
traveling for
so long and so far they do... (get stretched, Dopplered, or
converted
or do recycle their EM energy into gravitational energy or
into mass,
get absorbed, get reßected or swallowed by dark mass/energy,
or
manipulated in an unknown fashion by a yet undiscovered
Planck/Mach
domain Aether, or by a million of other yet unregistered
mechanisms)...
.... they do experience SOMETHING, such that they funally
reach the
stage where their initial color/energy is shed to the point,
down to
the level of the cosmic background energy level. Hence, they
are
becoming a ripple in a cosmic sea of like ripples, and
therefore
indistinguishable.
IOW, at this point that dot of initially brilliant star light
in the night
sky
has the same characteristics as the cosmic background
radiation and
hence IS NO LONGER VISIBLE TO US FROM OUR LOCATION. This
is what and where the lightwall is, ~ a sph. shell ~ 15BLY
away from
where you, I, we and all others egos sit and watch, even if
they are
15BLY away from you, (from where they see essentially the same
large scale pix, but they wont see not us any longer, but
they will
see ~ 30 BLY away from us into that additional 15 BLY
forbidden to us)
.....ahahahaha.......
.
Einstein, LeMaitre et. al. knew/saw all this already back
almost a
century ago. But some web-lice and net-ßeas prefer to cite
Einsteinian
buzz words instead, not realizing that this is the most
fundamental
aspect of any
relativity.........AHAHAHAHAHA......ahahahaha..........
There is a wonderful simple, chained equation for this
phenomena.
Some characteristics of the lightwall shell are described by
the
beautiful cosmic 1234 envelope:
:::: c = (G*M/R)^(1/2) = (G*M*H)^(1/3) = (G*M*br)^(1/4) :::
wherein
c= lightspeed, G= Newton, M= total mass in a sphere of R,
H= 1/Travel time from any center to R at the speed of c and
br = the deceleration caused by G & M fuxing the value of c,
or visa versa. Plug in the numbers, use the Hubble value for
H, and then see for instance the Pioneer 11 anomaly thru
effects of [br] ........check it out. Its
fun......ahahaha....
C1234 makes no claim to any particular center, doesnt 
require
higher dimensions, doesnt even require space-time nor 
curved
space. It simply states that the events played by G & M are
giving the travel time H for the plays to get from ANY CENTER
to distance R conditioned by the br of the background that
determines the max velocity being c. The magnitude of all
these
constants are dependent on each other. **INTERPLAY** is the
name of the game. There is no absolute, nor anything fuxed
here,
except with a few manipualtions you can arrive at a dimension
less
Number a ratio of how much is in the bag of the size R. It
turns
out to be N_A ~ 6E23 units/bag ........ahahahaha.......but,
its
too early to tell more about the N_A.....its an orphan,
hated!
[EL]
> I do agree that our local universe must have an age but I
refuse to
> accept that it is the funal mother-set.
[hanson]
I agree on this one too, EL. The 1/H time span for our
visible universe
section of ~ 10-15 BY is hard to accept for many popular
reasons.
My main objection is that empirically nature does not produce
things
that have reservoirs with large excess capacity. So, why
should nature
of 10^40+ years, to exist in a universe that lasts only 10^10
years?
Nature makes building stones that outlast its assumed BB
construct
10^20 times?.. thats 100 billion billion times
longer!....AHAHAHAHA..
[EL]
> If I may say, there must be billions of universes like
ours, separated
> by unthinkable distances.
[hanson]
I do agree with your pov, EL. This is of course only/also the
way I do IMAGINE it to be, and the discussed issue that
*** we are at the center everywhere ****.........
...... DOES NOT MEAN THAT WE ARE AT THE CENTER OF THE
THIS PLANETARY SYSTEM, NOR THE CENTER OF THE GALAXY,
but that we are at the center of THAT portion of the universe
that
is observable by us. THAT portion is ~ a sphere that is
embedded
in a much larger, perhaps spatially infunite cosmos, to which
we
with our current state of the art are not privy to have
access to.
This cosmos is simply the extension of our visible universe
beyond
the time and distance of the light walls ~ 15 BLY away from
us in
every direction, or in all 4 sterrandians.
We, you, me, the sun, every dust speck in a distant galaxy is
at the
center of ITS ~ 15 BLY sphere, each a portion of the 
~infunite
cosmos.
[EL]
> My mind accepts that an infunite chaos might condense
orderly into an
> existence, while it refuses the fantasy of a Big Bang that
comes out
> of a single point for no logical reason.
[hanson]
I agree here too on both counts. If there was chaos, then it
certainly
organized into people who do their very best in organizing to
bring
back the original conditions of
chaos........AHAHAHAHA.........
.....and the fantasy of the Big Bang had logical reasons
of/for a
very different sort, as this replay here shows:
> [hanson]
>> In this light, I think that the all time best story that
was ever
>> concocted was when Gamov et.al conjured up the Bang Bang.
>> It was a stroke of pure genius. Not for scientifuc reasons,
oh, no.
>> It was genius because its slogan, *Let there be light*,
resonated
>> with the religious folks and proved to them that their holy
scriptures
>> were right and that science had funally proven their
spiritual
beliefs.
>> And so the coffers opened and, Hallelujah, the shekels
were doled out
>> with the blessings of the Vatican from the public
treasuries into R&D
>> in astronomy and astrophysics which in turn spawned
funding for all
>> kinds of fuelds in physics even down to industrial
activities.
> [Don]
>Which is the basic plot of the book Angles and Demons by Dan
Brown.
>Don in Upstate NY
> [hanson]
> and I was bitterly attacked for it by the web-lice and
net-ßeas in
> but it goes back further to my *@quick.net archive that I
couldnt fund.
> Anyway, I probably wasnt the furst one with 
this notion
neither. After
> all, Gamov/Gamows BB was cooked up back in the 
40s and
dubbed
> Big Bang in 1950 by Fred Holye.
........ahahahaha.......AHAHAHAHA.........
> So, at least for somebody it has already been A MUST READ!,
Id say.
> Wish Dan Brown luck from me.
> ahahaha.....ahahahanson
[EL]
> I also think that the dispute between you and Richard Perry
is nothing
> more than a misunderstanding.
> Please do not waste your precious time (both of you) in
other than a
> logical and objective discussion.
> You are both equation-able and have logical minds, so
please refrain
> from entering pissing contests.
> [EL]
> {If I could piss in the outer space I could have used it
for my
> self-propulsion} :-) .......
[hanson]
......and with a well aimed turd you could annihilate your
enemies
followed by an after-fart that makes you glow in a sphere of
holiness .....
But, my friend, why should you be the only one allowed to do
that?
There are no pissing contests between me and Rich, nor anybody
else. --- I do administer to these people whenever they ask
me.
Most of the time they ask me in very convoluted ways, for
whatever
their motives may be. One poster, some time back, observed
rather
harshly: hanson likes to with idiots........ahahaha... I
wouldnt
go that far but I have to admit that the bumper sticker on
the Eldo
of Big Boobed Esmeralda, the human resources directress, from
the nanoXXdiamond division, has something to do with it, for
its says:
----- Hire the handicapped. They are fun to watch ---------
All that piss, turds, farts, centers, 4D tesseracts etc. can
be avoided
if you do subscribe to the notions in one of the great 4 hrs.
lecture I
heard as graduate student. It was titled: Phenomena in a
universe
with less then one dimension.......dont ask me about that
now, for it
contained highly classifued instructions, so top secret that
they are best
left either unknown or unacknowledged, whereby it must be
stressed
that I do remember one thing and I do and am still
wondering....
...whether it was the audience or the august professor that
was
more gassed, drunk, loaded.... yep, wondering which was and
still
is the main fun in all such
exercises............AHAHAHAHAHAHA.....
ahahahaha........ahahahahanson
===
Subject: order of the absolute Galois group of Q
If Q^- is the algebraic closure of Q, which I think can be
identifued with the fueld of algebraic numbers, 
then
Q^- is countably infunite.
What is the order of Gal(Q^- / Q)?
David Bernier
===
Subject: Re: order of the absolute Galois group of Q
|If Q^- is the algebraic closure of Q, which I think can be
|identifued with the fueld of algebraic numbers, 
then
|Q^- is countably infunite.
|
|What is the order of Gal(Q^- / Q)?
Its a continuum. Enumerate the algebraic numbers
u1,u2,u3,.... Once weve chosen where to send u1,...,un,
there are only funitely many places to send u_{n+1}. On
the other hand, for each n there is always some u_m
for m>n that has two possible places to go. We can
construct a mapping from sequences of 0s and 1s to
elements of the Galois group by encoding each bit using
the next element for which we have a choice of where to
send it, given the previous choices that have been made.
It doesnt take too much work to show that 
its homeomorphic
to the Cantor set. Compact, totally disconnected, no
isolated points... I forget how many conditions are needed
to characterize it, but its a fairly commonplace
homeomorphism
class of space. The topology is the one with the basis
consisting
of those subsets of the form {g : g(a1)=b1,...,g(a_n)=b_n},
the
natural topology for it as the inverse limit of Gal(K/Q) for
funite
Galois extensions K of Q. The fundamental theorem of Galois
theory applies in the case of infunite extensions if one 
states
the correspondence as being between subextensions and the
*closed* subgroups of the Galois group.
Keith Ramsay
===
Subject: Re: order of the absolute Galois group of Q
> |If Q^- is the algebraic closure of Q, which I think can be
> |identifued with the fueld of algebraic numbers, 
then
> |Q^- is countably infunite.
> |What is the order of Gal(Q^- / Q)?
> Its a continuum. Enumerate the algebraic numbers
> u1,u2,u3,.... Once weve chosen where to send u1,...,un,
> there are only funitely many places to send u_{n+1}. On
> the other hand, for each n there is always some u_m
> for m>n that has two possible places to go. We can
> construct a mapping from sequences of 0s and 1s to
> elements of the Galois group by encoding each bit using
> the next element for which we have a choice of where to
> send it, given the previous choices that have been made.
> It doesnt take too much work to show that 
its homeomorphic
> to the Cantor set. Compact, totally disconnected, no
Someone posting in sci.math once characterized the Cantor set
as
the product of countably many copies of Z/(2Z) (Fred Galvin).
If X_i = Z/((i+1)*Z), i >=1, is PI_{i=1 to oo} [ X_i]
homeomorphic
to the Cantor set?
> isolated points... I forget how many conditions are needed
> to characterize it, but its a fairly commonplace
homeomorphism
> class of space. The topology is the one with the basis
consisting
> of those subsets of the form {g : g(a1)=b1,...,g(a_n)=b_n},
the
> natural topology for it as the inverse limit of Gal(K/Q)
for funite
> Galois extensions K of Q. The fundamental theorem of Galois
> theory applies in the case of infunite extensions if one
states
> the correspondence as being between subextensions and the
> *closed* subgroups of the Galois group.
Yes. This goes beyond what I saw in grad school...
I have a question on category theory, a holdover from
previous months.
For any set Y, the bijective maps from Y to Y form a group.
So the extent of the category of Groups would seem to depend
on the extent of the category of Sets.
AFAIK, the existence of a strongly inaccessible cardinal is
independent of ZFC; If the (unknown?) extent of the class V
of all sets
has no bearing on categorical proofs, then I must be missing
something.
Is it the case that no one is saying how large V is, but
dont worry
because it doesnt matter anyway ?
David Bernier
===
Subject: Re: order of the absolute Galois group of Q
> If Q^- is the algebraic closure of Q, which I think can be
> identifued with the fueld of algebraic numbers, 
then
> Q^- is countably infunite.
> What is the order of Gal(Q^- / Q)?
I dont know.
If x is a real number in [0, 1) then you can associate to x
an automorphism that takes the square root of the nth prime
to its additive inverse if the nth bit of x is one and fuxes
that square root if the nth bit of x is zero. So that should
give you continuum many elements of the group.
--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
===
Subject: Re: order of the absolute Galois group of Q
> If Q^- is the algebraic closure of Q, which I think can be
> identifued with the fueld of algebraic numbers, 
then
> Q^- is countably infunite.
> What is the order of Gal(Q^- / Q)?
> David Bernier
It is a compact group, and has c=2^aleph0 elements.
You can think of it as projective limit of the
funite groups Gal(K/Q) as K ranges over all funite 
extensions
of Q.
--
G. A. Edgar
http://www.math.ohio-state.edu/~edgar/
===
Subject: Re: order of the absolute Galois group of Q
ezcos@yahoo.com
>If Q^- is the algebraic closure of Q, which I think can be
>identifued with the fueld of algebraic numbers, 
then
>Q^- is countably infunite.
>What is the order of Gal(Q^- / Q)?
2^ {aleph_0}.
Clearly it cannot be more than that.
Let P be be the set of prime numbers. Then for any subset S
of P, there is
an
automorphism taking sqrt(p) to sqrt(p) for p in S and sqrt(p)
to -sqrt(p)
for p not in S.
Derek Holt.
===
Subject: Re: order of the absolute Galois group of Q
> If Q^- is the algebraic closure of Q, which I think can be
> identifued with the fueld of algebraic numbers, 
then
> Q^- is countably infunite.
> What is the order of Gal(Q^- / Q)?
2^{aleph_0}
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square
root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Re: r.b^2 congruent to a^3 modulo p
> Hi all,
> Im implementing an cryptographic algorithm based on
elliptic curves.
> The standard my work is based on wants me to generate an
elliptic
> curve verifuably at random.
> The protocol is as follows :
> 1) generate a fairly big integer
> 2) do some weird things on it using a (suposedly) one-way
function
> (SHA-1)
> 3) use the results of those weird things to generate an
unsigned
> integer r
> 4) choose integers a, b such that r.b^2 is congruent to a^3
modulo p
> where p is another know unsigned integer generated before.
> Everything goes well until step 3, but I cant 
fund an
effective way
> to choose a and b (there are no particular requirements on
them).
> Does someone know and effucient algorithm to generate such
integers ?
> Pierre
usually, when you see this, it is for speed improvements.
a is typically fuxed at -3.
This should help answer your questions.
You can see the wonderful book Guide to ECC or see IEEE-P1363
and
P1363a
for how that can be used to speed things up.
Make sense?
HTH, Flip
===
Subject: Re: r.b^2 congruent to a^3 modulo p
...
>3) use the results of those weird things to generate an
unsigned
>integer r
>4) choose integers a, b such that r.b^2 is congruent to a^3
modulo p
>where p is another know unsigned integer generated before.
>Everything goes well until step 3, but I cant 
fund an
effective way
>to choose a and b (there are no particular requirements on
them).
>Does someone know and effucient algorithm to generate such
integers ?
Assuming gcd(r,p) = 1, your equation is equivalent to
(b/r)^2 == (a/r)^3 mod p. So generate another integer c, and
take b = c^3 r mod p and a = c^2 r mod p.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
===
Subject: Re: r.b^2 congruent to a^3 modulo p
> Assuming gcd(r,p) = 1,
Ok, thats something I forgot to tell you. p is prime.
> your equation is equivalent to
> (b/r)^2 == (a/r)^3 mod p. So generate another integer c, and
> take b = c^3 r mod p and a = c^2 r mod p.
I trust you on that, but could you elaborate a little more,
or give a
pointer to a document that could help me understand ?
Pierre DOUCY
===
Subject: Re: r.b^2 congruent to a^3 modulo p
Assuming gcd(r,p) = 1,
> Ok, thats something I forgot to tell you. p is prime.
>your equation is equivalent to
>(b/r)^2 == (a/r)^3 mod p. So generate another integer c, and
>take b = c^3 r mod p and a = c^2 r mod p.
> I trust you on that, but could you elaborate a little more,
or give a
> pointer to a document that could help me understand ?
> Pierre DOUCY
r (c^3 r)^2 = c^6 r^3 = (c^2 r)^3.
What is there you dont understand?
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
===
Subject: Re: hanging cable with discrete loads
>>I have a need to compute the height of loads (specifucally,
traffuc
>>signals and signs) hung from a suspension cable. Googling
found a
lot
>>of catenary stuff, which I was already aware of, and some
about
>>uniform loading, but (AFAICT) nothing helpful.
>>Id appreciate any clues.
>>TIA,
>>George
> See section 1.3 of M. Irvine, Cable Structures, Dover, 1981;
>> subsection Response to Many Point Loads
>Also from way back in the Dark Ages I seem to recall that
Hildebrand
(F.B.)
>in his ÔMethods of Applied Math covered this 
problem in
some detail.
>Cal
> Are you pretty sure of the title? The TOC for that,
> (http://web.doverpublications.com/cgi-bin/toc.pl/0486670023
> doesnt look that promising?
> George
Perhaps he reorganized the text in later editions or perhaps
Dover
blew the TOC.
3.6 and the following sections treat the various cases. In
the original
edition, Chapter 3 is devoted to difference equations.
Cal
===
Subject: Re: hanging cable with discrete loads
>> I have a need to compute the height of loads (specifucally,
traffuc
>> signals and signs) hung from a suspension cable. Googling
found a lot
>> of catenary stuff, which I was already aware of, and some
about
>> uniform loading, but (AFAICT) nothing helpful.
> Id appreciate any clues.
> TIA,
>> George
>See section 1.3 of M. Irvine, Cable Structures, Dover, 1981;
>subsection Response to Many Point Loads
> Alas - apparently, a collectable. I saw one listed for $570.
> George
Yeah, the original hardcover MIT is hard to fund, so there is
price
gauging.
The Dover reprint is listed at about $10 in
http://www.volume5.com/cbookbag/html/structure.html
Failing that your local university engineering library should
have it.
By far the best book on cable structures from an engineering
viewpoint. Frei Ottos (also MIT) is also good but from the
conceptual (architectural) side.
===
Subject: Re: hanging cable with discrete loads
> I have a need to compute the height of loads (specifucally,
traffuc
> signals and signs) hung from a suspension cable. Googling
found a
lot
> of catenary stuff, which I was already aware of, and some
about
> uniform loading, but (AFAICT) nothing helpful.
>
> Id appreciate any clues.
>
> TIA,
> George
>>See section 1.3 of M. Irvine, Cable Structures, Dover, 1981;
>>subsection Response to Many Point Loads
>> Alas - apparently, a collectable. I saw one listed for
$570.
>> George
>Yeah, the original hardcover MIT is hard to fund, so there is
price
>gauging.
>The Dover reprint is listed at about $10 in
> http://www.volume5.com/cbookbag/html/structure.html
Just FYI: for that book, they just link you to Amazon. 
Thats
where
the $570 copy, and some at $150.
George
===
Subject: Re: hanging cable with discrete loads
>> I have a need to compute the height of loads (specifucally,
traffuc
>> signals and signs) hung from a suspension cable. Googling
found a lot
>> of catenary stuff, which I was already aware of, and some
about
>> uniform loading, but (AFAICT) nothing helpful.
>There are too many variables, you may not fund a readymade
solution
>that can be comfortably handled by calculations. I have 3
suggestions.
>1a) Experiment is easier and quick/convincing results are
obtained.
> Well, this would need to be done time and again. Lots of
> intersections, smallish contractor.
>1b) If Pcable << P1+P2+P3+.. then a funicular polygon is
determined by
>engineering graphic statics using a force polygon, and an
arbitrary
>pole.
> I dont yet know how the cable weight compares to the load
weights.
> (Things are a little preliminary at the moment.) Assuming
this
> condition is more or less satisfued, ..
In this case you neglect cable weight and take funicular
polygon,
which has the advantage of being an upper bound for
deßections.
> I found a solution of this sort (ch 4, Peterson, ÔApplied
Engineering
> Mechanics). But, Mr. Peterson seems to insist that the
coordinates
> (x,y) of one of the loads be known in advance.
Errors in assumptions are not of great consequence as cable is
shallow.But I cannot say for sure without looking at that
book.
> This is not out of the question, but it would be easier if
the total
cable length could be provided instead. It seems like that
would be an
equivalent problem?
You could look at 2 extreme cases. Neglect light loads,
consider
deßection of pure catenary, lower bound. Lump cable weights at
proportinately at nearest light positions and draw funicular
polygon
with assumed cable length for maximum possible deßection,
upper
bound.
===
Subject: Re: Uniqueness of physical objects in the universe.
>So, I would like to prove that statement. It is not possible
to
>compare every single physical object in the universe in a
physics
>lab, so it must be proved mathematically.
It cant be proven Mathematically.
>it must be assumed
Then youre not using Mathematics to prove it. At best you
may have
experimental data to justify your assumptions, but those are
not
Mathematics.
>So, lets assume that objects really exist in the physical
universe,
A rather strong assumption, but still not adequate.
>There are 2 possible cases,
>1) O1 and O2 have different names
>2) O1 and O2 have the exact same name
There you have the fallacy. You have not shown that it is
possible to
assign names to objects at all. For that matter, you have not
given a
defunition of object, physical universe or exists. Those
terms, as
you use them, are not Mathematical.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT
<http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply
to
domain Patriot dot net user shmuel+news to contact me. Do not
===
Subject: Re: Uniqueness of physical objects in the universe
(proved /
disproved)
<0lBhd.34354$HA.1344@attbi_s01>
<G9gid.361970$D%.77983@attbi_s51>
at 01:52 PM, studentwalden@yahoo.com (Student) said:
>I have another way of proving this theory:
No.
>Proof:
No.
>I will name objects in physical universe. I name them in
such a way
>that no two names are identical.
Thats begging the question. You are presupposing that the
objects are
unique. You cannot, in fact, name them in such a way that no
two names
are identical.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT
<http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply
to
domain Patriot dot net user shmuel+news to contact me. Do not
===
Subject: a problem about coding theory and threshold schemes
i think my question is suitable here, it is about coding
theory and
threshold schemes.
i want to fund a way to construct a matrix with n lines and m
columns, take
the lines as vectors and add arbitary k lines of the all n
lines, one get a
vector with which the hamming weight is at least h (h is
between 1 and m).
i
have got some examples: for a 3*3 matrix,
((1,0,0),(1,0,1),(0,1,0))the
hamming weight of the sum of any two of the 3 lines is at
least 1. so what
is the general (k,n)-threshold scheme?
===
Subject: Re: a problem about coding theory and threshold
schemes
no one has interest?
liufeng <feng_liu@mails.gscas.ac.cn>
.91.9e[Cent
].81
:cms4n4$13p$1@news.yaako.com...
> i think my question is suitable here, it is about coding
theory and
> threshold schemes.
> i want to fund a way to construct a matrix with n lines and
m columns,
take
> the lines as vectors and add arbitary k lines of the all n
lines, one get
a
> vector with which the hamming weight is at least h (h is
between 1 and
m).
i
> have got some examples: for a 3*3 matrix,
((1,0,0),(1,0,1),(0,1,0))the
> hamming weight of the sum of any two of the 3 lines is at
least 1. so
what
> is the general (k,n)-threshold scheme?
===
Subject: Re: a problem about coding theory and threshold
schemes
> no one has interest?
> liufeng <feng_liu@mails.gscas.ac.cn>
.91.9e[Cent
].81
> :cms4n4$13p$1@news.yaako.com...
>>i think my question is suitable here, it is about coding
theory and
>>threshold schemes.
>>i want to fund a way to construct a matrix with n lines and
m columns,
> take
>>the lines as vectors and add arbitary k lines of the all n
lines, one get
>>vector with which the hamming weight is at least h (h is
between 1 and
m).
>>have got some examples: for a 3*3 matrix,
((1,0,0),(1,0,1),(0,1,0))the
>>hamming weight of the sum of any two of the 3 lines is at
least 1. so
what
>>is the general (k,n)-threshold scheme?
------------------------
> no one has interest?
After doing a search with Google for threshold scheme, I
found many
hits related to cryptology ( science/art of secret
communication).
You mention coding theory; in coding theory, the general
objective
is to have data transmission (or storage) that is as error
free as
possible, in the presence of noise.
I dont know what you mean by the general (k,n)-threshold
scheme ...
David Bernier
===
Subject: The latest stories from Mathforge.net
Im hoping that posting this here will be mutually 
benefucial.
If
there is any widespread disagreement with this sort of
advertising
on public newsgroups, please voice your opinion, although I
included
some content and links as well, and you will never be asked
for money
to read the news on Mathforge.net. Enjoy!
The Latest Math News from Mathforge.net
http://mathforge.net
****The quasi-empiricist view of mathematical development****
Normal science operates on pure empiricism events are
measured and a
mathematical model is found which best predicts the behavior
of new
events. If new events stray from the predictions of the
model,...
http://mathforge.net/index.jsp?page=seeReplies&messageNum=885
****MathML and scientifuc e-content gurus meet in Finland****
The The Second European Workshop on MathML and Scientifuc
E-content
took place on September 16th through the 18th in Kuopio,
Finland.
Keynote speakers included David Carlisle, editor of the MathML
Recommendation...
http://mathforge.net/index.jsp?page=seeReplies&messageNum=883
****A statistical prediction of the U.S. election (and how to
beat the
stock market)****
Rosa Karapandza and Mike Bozovic of the University of Pompeu
Fabra
have discovered a new indicator to predict presidential
election
winners (the popular vote, as in the case of Al Gore) as well
as
stock...
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****The History of the Data Encryption Standard (DES)****
registration is technically required. Its a pain.) in the
Sydney
Morning Herald explains the history of DES and the rise of
cryptography as...
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****MathAction Wiki and Portal for Axiom and Reduce****
ÔEditor:[Capit
alOTilde] Ôreader billpage 
submitted some snippets
regarding the
MathAction project. Its a Wiki-intensive project with lots 
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explore: Axiom and Reduce are mature systems 
for doing
mathematics...
http://mathforge.net/index.jsp?page=seeReplies&messageNum=877
****We dont even know how much we know that we 
dont know we
know****
The total number of pages published in mathematics so far
makes up a
stack of about 60km height, said Michiel Hazelwinkel in his
keynote
http://mathforge.net/index.jsp?page=seeReplies&messageNum=876
****If youre write technical documents in Microsoft
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OpenOffuce is a high-quality budget-responsible 
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As users
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****No Child Left Behind -- Discuss the Effect on Mathematics
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The U.S. presidential election is now one week away.
Mathforge would
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http://mathforge.net/index.jsp?page=seeReplies&messageNum=872
****Science News on Gauss****
Carl Friedrich Gauss offers some perspective on telling
legendary
stories in the classroom. The whole problem of Newtons 
apple
comes...
http://mathforge.net/index.jsp?page=seeReplies&messageNum=871
****Ed Pegg Jr. on Zeta Zeros****
of the search for zeta zeros. He describes deeply how zeta
zeros
relate to the Riemann Hypothesis. The idea is that, without a
proof...
http://mathforge.net/index.jsp?page=seeReplies&messageNum=866
===
Subject: Re: Replicating A Result In Cohen And Harcort
=
> o The department of economics at the University of Rochester
> is dominated by right wingers.
Idelogue alert!!! Idelogue ALERT!!! Rob does not like the
models or the
empirical result because they do not agree so he calls them
right
wingers. Rob accuses almost the ENTIRE ECONOMICS DEPARTMENT
OF THE
UNIVERSITY OF ROCHESTER TO BE RIGHT WINGERS. Rob can not see
what is
scientifuc and what is politically motivated. There is a
simple reason.
He is a Marxist. They believe economics is not a science but
a political
statement. So it is easy for Rob to say that anything that
shows a
result contrary to his is politically motivated. By the
Marxist logic
it must be politically motivated because economics is really
ONLY about
politics. To Rob economics is not a science that has
implications for
policies. It is a political theory.
Do not get me wrong I am willing to look at Marxist models
from an
objective viewpoint. I have read a few of them when I had an
interest in
development. In fact I still have a couple of publications
from VI Lenin
collecting dust with the other papers I collect that deal with
development. What I take objection to is taking the science
out of the
argument and suddenly making the model a political statement.
At that
point Rob can no longer objectively see what the model says
or if the
model is correct. His ideology kicks in and the results must
be rejected
simply because they were found by right wingers. Which to Rob
by
defunition are stupid and illiterate because they disagree
with him. Rob
can not see how an educated person would reject Marxism.
Rochester has produced some fune theoritical and empirical
work. Some of
the seminal papers in the literature on the product life
cycle come from
a professor at Rochester. Rob tries to discredit them by
simple laying a
blanket statement that they are right wingers. This is
equivalent to
Pravda prior to the 1990s calling anyone who disagreed an
imperial
capitalist.
I have already wasted way to much time responding to this
loony. It is a
shame I will have to look at Cohen and Harcourts references
to get a
clear and unbiased reading of the ideas Rob believes he is
propogated.
Strangely enough he invokes Samuelson on this model which is
misleading
Samuelson never applied the model to labor economics nor to
my knowledge
did he ever make statements about employment and wages
similar to Robs
statement. Until Rob learns how to put words between
equations and
equations,
I would suggest anyone interested in the subject to look to
Cohen and
Harcourts references.
===
Subject: Flaw in a Proof of Tychonoffs Theorem?
Im following a book by John D. Baum (the Dover Thrift book
Elements
of Point Set Topology) and after some great labor I think
Ive nearly
understood the proof of Tychonoffs Theorem (that the 
product
of
compact spaces is compact) but Ive hit a snag. 
Theres one
step that
seems to be unsupported, and in fact I think Ive found a
counterexample. Im sure I must be missing something, or
perhaps
theres an error in this proof. Im hoping 
someone whos seen
the book
can set me on the right track.
The proof starts with a family F of subsets of X (X is the
product),
such that F has the funite intersection property. It
ultimately will
show that this family as a whole has a nonvacuous
intersection. If
thats the case, then I understand that X will be compact.
But I dont
see that F must have a nonvacuous intersection.
Baum creates a second family M (superset of F) which is
maximal for
the f.i.p. and for which funite intersections of 
Ms elements
are
again in M. Then he projects the elements of M down to the
various
coordinate (factor?) spaces X_alpha (alpha a member of some
indexing
set); since the coordinate spaces are compact, and the f.i.p.
is
inherited to each one, each family of projections must have a
nonvacuous intersection. Great. For each alpha, choose a
coordinate
x_alpha which lies in that nonvacuous intersection, and
construct the
point x in X which has these coordinates. The goal now is to
show that
x lies in the intersection of the family M.
Now the maximality of M is used to show that a basic set B,
which
contains x, is already in M (Fine and good; Im sketching
this part)
and since M has the f.i.p., any member N in M necessarily
intersects
B. Agreed. The next line in the Baum book says ...whence x in
N for
each N in M, which leads immediately to the conclusion that x
is, in
fact, in the intersection of the family M (and also of F).
But I dont
see why x must be in such an N.
Heres my simple counterexample, in the product I x I. Let
the family
F be:
{ [0, 1/4] x [3/4, 1] u [3/4, 1] x [0, 1/4],
[0, 1/4] x [0, 1/4] u [3/4, 1] x [3/4, 1],
[0, 1] x [0, 1] }
When we project these two regions into the respective
coordinate
spaces, we see that they all contain [3/4, 1] and [0, 1/4].
The choice
of x might then be the point (0, 0). But its not the case
that all
the sets in F (let alone some superfamily M) contain that
point.
Baum seems to be claiming that, just because every set in M
intersects
B, they must intersect it at the point x. Is there another
way to
interpret his argument? Is it possible this is a faulty proof
of the
theorem?
Any illumination would be greatly appreciated.
Ezra
===
Subject: Re: Flaw in a Proof of Tychonoffs Theorem?
> F does not have the F.I.P: the furst two sets do not
intersect.
> KP
Ah, of course--thank you KP.
the essential ingredient I missed was that M is closed under
going to
supersets.
for tempting me with the short-short proof.
Ezra
===
Subject: laplace transform involving matrix
Suppose that the Laplace transform of the function U(t) is
U*(s). For n*n
square matrix A, I hope to know the following integral
results from zero to
positive infunite;
Integral[0..INFINITE] U(t) exp( - A t) dt
I can understand that this is not equal U*(A). And got the
following
results:
Assume that A is diagonalizable and expressed as
A=P^(-1) B P where P is the matrix with eigenvectors and B is
the diagonal
matrix with eigenvalue lambda_i(i=1,2...n).
So:
Integral[0..INFINITE] U(t) exp( - A t) dt = P^(-1)
diag(U*(lambda_1),
U*(lambda_2)... U*(lambda_n)) P
I am wondering any other techniques to get more beautiful
results for
--
Yan ZHANG
http://www.ntu.edu.sg/home5/pg01308021
===
Subject: Re: laplace transform involving matrix
>Suppose that the Laplace transform of the function U(t) is
U*(s). For n*n
>square matrix A, I hope to know the following integral
results from zero
to
>positive infunite;
>Integral[0..INFINITE] U(t) exp( - A t) dt
>I can understand that this is not equal U*(A). And got the
following
>results:
Actually it is U*(A), under appropriate assumptions. Im
assuming
U(t) is, say, a piecewise-continuous scalar function with
|U(t)| <= b exp(c t) for t on the axis for some positive
constants b and
c, and Re(lambda) > -c for all eigenvalues of A.
Then U*(s) is analytic in G = {s: Re(s) > -c}. Since
this is a neighbourhood of the set of eigenvalues of A,
U*(A) is defuned by the holomorphic functional calculus, and
using the continuity of that calculus (if f_n -> f uniformly
on
compact subsets of G, f_n(A) -> f(A)), it wont be hard to
show
that the integral int_0^infty U(t) exp(-At) dt exists and
equals U*(A).
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
===
Subject: Re: laplace transform involving matrix
> Suppose that the Laplace transform of the function U(t) is
U*(s). For n*n
> square matrix A, I hope to know the following integral
results from zero
to
> positive infunite;
> Integral[0..INFINITE] U(t) exp( - A t) dt
> I can understand that this is not equal U*(A). And got the
following
> results:
> Assume that A is diagonalizable and expressed as
> A=P^(-1) B P where P is the matrix with eigenvectors and B
is the
diagonal
> matrix with eigenvalue lambda_i(i=1,2...n).
> So:
> Integral[0..INFINITE] U(t) exp( - A t) dt = P^(-1)
diag(U*(lambda_1),
> U*(lambda_2)... U*(lambda_n)) P
This could be written as P^(-1) U*(B) P. I doubt there is
anything more
beautiful than this.
-Michael.
===
Subject: Re: sum of primitive roots mod p
!3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(
5eZ41to5f%E@ELIi
$t^
VcLWP@J5p^rst0+(Ô>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw
> can somebody tell me how to tackle this problem :
> prove that, for any prime p, the sum of all the distinct
primitive
> roots(mod p)is congruent to mu(p-1) (mod P), where mu is
the mobius
> function
When can k be a primitive root while -k is not?
--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
===
Subject: Re: possible....
<cmqlpd$qja$1@news.hananet.net>
===
>Subject: possible....
>> if
>> g(s-t)=f(s^2 - t^2, t^2 - s^2), f is differentiable.
>> fund the t*(dg/ds) + s*(dg/dt).
>= t.g(s-t) - s.g(s-t) = (t - s) 
g(s-t) **
>> dg/ds = df/ds = (df/dx)(dx/ds)+(df/dy)(dy/ds)
>> = fx(2s) + fy(-2s)
>> (x = s^2 - t^2 , y = t^2 - s^2)
>g(s - t) = f_x 2s + f_y (-2s)
>-g(s - t) = f_x (-2t) + f_y 2t
>Solve for g(s - t) and use in **
>g(s-t) is a function of one variable
>that has been assigned the value s - t.
> um.......i think...
> g(s-t) is not well-defuned look like.
> because, g(1) = g(1-0) = f(1,-1) and g(1) = g(2-1) =
f(3,-3).
> if f(1,-1) =/= f(3,-3), then g(s-t) is not function.
> how do you think about it ?
Neither g nor f are defuned beyond
g:R -> R and f:R^2 -> R
and being differentiable. g and f can be many different
functions.
Now you are told that g and f relate in a particular way.
This will limit the choices of g and f, but yet there are
still many different choices for them.
Alternative you may proclaim s-t to be a typo and that
g(s,t) is actually defuned by the equation. Then look
to the errata sheet or web site and fund if that is so.
Ask your teacher if its a typo. Does the text give an
answer? Then compare the answer with g(s-t), g(s,t) and
the book answer to decide which one is the intended problem.
> g(s,t) = f(s^2 - t^2, t^2 - s^2), f is differentiable.
> fund the t*(dg/ds) + s*(dg/dt).
g_s = 2s.f_x - 2s.f_y
g_t = -2t.f_x + 2t.f_y
t.g_s + s.g_t = 2st(oh my, call a nurse)
===
Subject: Re: possible....
===
>Subject: possible....
>> if
>> g(s-t)=f(s^2 - t^2, t^2 - s^2), f is differentiable.
>g(s-t) = f(...
>> fund the t*(dg/ds) + s*(dg/dt).
>= t.g(s-t) - s.g(s-t) = (t - s) 
g(s-t) **
>> dg/ds = df/ds = (df/dx)(dx/ds)+(df/dy)(dy/ds)
>> = fx(2s) + fy(-2s)
>> (x = s^2 - t^2 , y = t^2 - s^2)
>g(s - t) = f_x 2s + f_y (-2s)
>-g(s - t) = f_x (-2t) + f_y 2t
>Solve for g(s - t) and use in **
>> dg/dt = similar~
>Notice minus sign.
>> is this right method ?
>Yes
>> and is g(s-t)=f(s^2 - t^2, t^2 - s^2) possible function ?
>g(s-t) = f(...
>> g(s-t) => g(s,t) ...is this right ??
>Confusing
>g(s,t) is a function of two variables.
>g(s-t) is a function of one variable
>that has been assigned the value s - t.
> um.......i think...
> g(s-t) is not well-defuned look like.
> because, g(1) = g(1-0) = f(1,-1) and g(1) = g(2-1) =
f(3,-3).
> if f(1,-1) =/= f(3,-3), then g(s-t) is not function.
> how do you think about it ?
Agreed. But since you do not know what function f is, in this
problem
assume it is defuned in such a way that causes g to be
well-defuned.
> thank you very much.
===
Subject: Tricky group theory question
G is a funite group.
Let A be an abelian subgroup of G.
Then the index | G : A | = p^n, where p is a prime number.
Then show that the commutator group [ G, G ] is a subgroup of
G.
===
Subject: Re: Tricky group theory question
> G is a funite group.
> Let A be an abelian subgroup of G.
> Then the index | G : A | = p^n, where p is a prime number.
> Then show that the commutator group [ G, G ] is a subgroup
of G.
Huh? Isnt it true that [ G, G ] is a subgroup of G
whether or not G is funite, and whether or not G has an 
abelian
subgroup of whatever index?
Maybe the tricky thing about this question is fuguring out
what exactly it was that you meant to ask?
--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
===
Subject: Re: Tricky group theory question
>> G is a funite group.
>> Let A be an abelian subgroup of G.
>> Then the index | G : A | = p^n, where p is a prime number.
>> Then show that the commutator group [ G, G ] is a subgroup
of G.
>Huh? Isnt it true that [ G, G ] is a subgroup of G
>whether or not G is funite, and whether or not G has an
abelian
>subgroup of whatever index?
>Maybe the tricky thing about this question is fuguring out
>what exactly it was that you meant to ask?
Well, we might never know for sure, but my guess is that he
meant
show that [G,G] is a proper subgroup of G.
Derek Holt.
===
Subject: Re: Tricky group theory question
>Maybe the tricky thing about this question is fuguring out
>what exactly it was that you meant to ask?
> Well, we might never know for sure, but my guess is that he
meant
> show that [G,G] is a proper subgroup of G.
> Derek Holt.
Yes it is a character theory question.
I have to do something with degrees of irreducible characters
of G.
But i still couldnt fund the standard procedure 
you mentioned.
Steve
===
Subject: Re: Tricky group theory question
>>Maybe the tricky thing about this question is fuguring out
>>what exactly it was that you meant to ask?
>> Well, we might never know for sure, but my guess is that
he meant
>> show that [G,G] is a proper subgroup of G.
>> Derek Holt.
>Yes it is a character theory question.
>I have to do something with degrees of irreducible
characters of G.
>But i still couldnt fund the standard 
procedure you
mentioned.
Try looking for a proof of the Burnside Theorem, that groups
of order
p^a q^b (p,q prime) are solvable. What the character theory
actually
proves
is that a group with a conjugacy class of size a nontrivial
prime power
cannot be simple. A group with an abelian subgroup of prime
power index
does have such a conjugacy class, or else it has a nontrivial
centre.
In either case the group cannot be simple, and then you can
solve the
original problem by induction on the group order.
Derek Holt.
===
Subject: Re: Tricky group theory question
>G is a funite group.
>Let A be an abelian subgroup of G.
>Then the index | G : A | = p^n, where p is a prime number.
That is not necessarily true. For example, let G be a group
of order 6,
and A the subgroup of order 1.
>Then show that the commutator group [ G, G ] is a subgroup
of G.
The commutator subgroup of any group G whatsoever is a
subgroup of G.
Maybe you need to rephrase the problem?
There is a standard application of character theory to prove
a result which
says that a funite group having a conjugacy class of prime
power order
greater
than one cannot be simple. You might be able to use that to
prove whatever
you
are trying to prove.
Derek Holt.
===
Subject: Re: is there any
<x54qjz6ten.fsf@lola.goethe.zz>
<x5r7n35dmf.fsf@lola.goethe.zz>
posting-account=QD46ZQwAAADVt8kxGXXiBHrSZxPB6Op6
You are not good at stating something clearly.
===
Subject: Re: is there any
> David Kastrup <dak@gnu.org> schreef in bericht
>> Sorry I should have stated it more clearly. Is there any
solution for
>> x^2 - n^2 * y^2 = 1? for all x,y,n > 0
>x=2, n=1, y = sqrt(3).
> This is still not a solution for all x,y,n>0, but for some
x,y,n>0.
>You are not good at stating something clearly. If you
instead wanted
>to be all of them in integers, then trivially this is false,
since x^2
>is a square, n^2*y^2 is a square, and their difference can
hardly be
>just 1, as x^2 - (x-1)^2 is already 2x-1, so we get this
only for
>x=1, and n*y=0.
>--
>David Kastrup, Kriemhildstr. 15, 44793 Bochum
> There cannot be such a solution for all x,y,n>0. There
could be a
solution
> for all x,n > 0, but you need at least one variable to
solve it for.
> Otherwise the equation does not make sense, i.e. 0 = 0 has
a solution for
> all x.
Give him/her/it a break. Change for to with and the question
makes perfect sense and patches the loophole pointed out in
the furst
response in the thread:
Is there any solution for x^2 - n^2 y^2 = 1 with all x, y, n
> 0.
Of course, David Kastrups answer to this is simple & 
correct
(assuming, as seems likely, that OP had integers in mind).
--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
===
Subject: Re: is there any
<x54qjz6ten.fsf@lola.goethe.zz>
<x5r7n35dmf.fsf@lola.goethe.zz>
<10p1jreh72n3nad@corp.supernews.com>
posting-account=QD46ZQwAAADVt8kxGXXiBHrSZxPB6Op6
Give him/her/it a break.
(Can confurm at least that I am not 
Ôit. )
===
Subject: Re: is there any
kalikinkar <kalikinkar@gmail.com> schreef in bericht
> Give him/her/it a break.
> (Can confurm at least that I am not 
Ôit. )
No offense meant by the way. I just did not expect that you
meant the
question that way, i.e. that you were looking for just one
solution. I
thought you were looking for some formula, and knowing for
which variable
you want to solve it is clearly crucial then.
Paul
===
Subject: Missing something
Hi there,
I dont know whether Im missing something but 
I have a
probability
question that seems easy, yet I cant quite get my head
wrapped round
it.
There are 53 different trains that go from London to
Nottingham. If I
was to turn up to the station at London at random times to
take the
train, how many journeys would I have to make so that the
probability
of travelling on at least half the trains is 0.5.
Any help or guidance would be greatly appreciated.
Anthony
===
Subject: Re: Missing something
> Hi there,
> There are 53 different trains that go from London to
Nottingham.
that many!
> Any help or guidance would be greatly appreciated.
> Anthony
Anthony,
If you want a >50% chance, go by car!
OK, the next bit is even less relevant...
This is England, not the European mainland. Weve just
managed to reach the
125mph limit here in Blitey once again after a break of 30
years. We did it
in the 70s, then lost it. We did Steam 100mph plus early in
the 20th
century. 100 or so years later, weve managed to regain what
we could do
then.
The continent manage what can only be almost relativistic
speeds to us
Brits
Good luck with your real problem
===
Subject: Re: Missing something
>There are 53 different trains that go from London to
Nottingham. If I
>was to turn up to the station at London at random times to
take the
>train, how many journeys would I have to make so that the
probability
>of travelling on at least half the trains is 0.5.
To re-phrase with the required assumptions:
At each journey you are making an independent choice of one
of the
53 trains, each with equal probability. You want to fund the
least N so that in N journeys the probability of choosing at
least
27 different trains is at least 0.5.
The size of the sample space for N journeys is 53^N.
Given a subset S of {1,...,53}, the number of outcomes where
all trains chosen are in S is |S|^N (where |S| is the number
of
members of S). The number of outcomes where the set of trains
chosen is exactly S is |S|! S_2(N,|S|), where S_2(n,k) is a
Stirling number of the second kind (the number of ways to
partition
a set of n things into k nonempty subsets). Since there are
(53 choose k) = 53!/(k! (53-k)!) sets S with |S|=k, the
probability
of choosing exactly k trains in N journeys is
53!/(53-k)! S_2(N,k)/53^N. Thus you want the least N so that
53!/53^N sum_{k=27}^53 S_2(N,k)/(53-k)! >= 0.5. According to
Maple, the answer is N=37, for which the probability is
approximately
0.5634187598 (for N=36 it would be approximately
0.4637932964).
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
<e.a.rodland@medisin.uio.no>
===
Subject: Re: Missing something
> There are 53 different trains that go from London to
Nottingham. If I
> was to turn up to the station at London at random times to
take the
> train, how many journeys would I have to make so that the
probability
> of travelling on at least half the trains is 0.5.
I have an exact solution to this, and an approximation which
proves to
do quite well. Lets take the approximation 
furst.
** Approximate solution **
Let M=53 be the number of trains (enumerated 1,2,...,M) and
assume
that each time you pick one train by random, each with
likelihood 1/M.
You want the likelihood that after N trips, you have traveled
by all M
trains.
The likelihood that after N trips, you have traveled with
train j at
least once for a specifuc j is
P[train j used] = 1 - (1-1/M)^N
as (1-1/M)^N is the likelihood of having picked N trips each
time
missing train j. Since there are M trains, if we assume as an
approximation that Ôtrain j used and 
Ôtrain i used are near
independent, we get
P[all] ~ [1-(1-1/M)^N]^M.
Solving in N gives
N ~ ln[1-P^(1/M)]/ln[1-1/M]
which, for P=0.5 and M=53 gives N ~ 228.018...
** Exact solution **
Let S_i be the statement that you have not traveled by train
number i.
What you want to know is the probability of having traveled
with all
trains after N trips,
P[all] = P[not(S_1 or S_2 or ... or S_M)],
which can be assessed using the inclusion-exclusion principle:
P[all] = 1 - sum_{i} P[S_i] + sum{i1<i2} P[S_i1 and S_i2] -
...
= sum_{J subset of {1,...,M}} (-1)^|J| P[S_J]
where |J| is the number of elements in |J| and S_J says that
S_j is
true for all j in J, i.e. S_J says that you have not traveled
with any
of the trains in J.
For any set J with |J|=k, the likelihood of S_J is
P[S_J] = (1-k/M)^N
simply because each of the N trips you have a likelihood
1-k/M of
picking one of the trains not on the list J. The set J can be
picked
with |J|=k in binomial(M,k) different ways. Thus,
P[all] = sum_{k=0..M} (-1)^k binomial(M,k) (1-k/M)^N.
For M=53 and N=229, this gives P=0.49668...; for N=230 it
gives
P=0.50348... If you want to be 99% certain of having taking
all the
trains, P>0.99, you need to do the trip at least 450 times
(P=0.990011...); the approximation gives N~499.9... for
P=0.99, i.e.
the same N to get P>0.99.
Einar
<e.a.rodland@medisin.uio.no>
===
Subject: Re: Missing something
>> There are 53 different trains that go from London to
Nottingham. If I
>> was to turn up to the station at London at random times to
take the
>> train, how many journeys would I have to make so that the
probability
>> of travelling on at least half the trains is 0.5.
> I have an exact solution to this, and an approximation
which proves to
> do quite well. Lets take the approximation 
furst.
Im sorry. I just realised Id missed the 
clause Ôon at least
*half*
the trains. What I solved was for having traveled by all 
the
trains.
Einar
===
Subject: Re: Missing something
> Hi there,
> I dont know whether Im missing something 
but I have a
probability
> question that seems easy, yet I cant quite get my head
wrapped
round
> it.
> There are 53 different trains that go from London to
Nottingham. If
I
> was to turn up to the station at London at random times to
take the
> train, how many journeys would I have to make so that the
probability
> of travelling on at least half the trains is 0.5.
> Any help or guidance would be greatly appreciated.
> Anthony
Are you assuming that turn up to the station at London at
random
times is equivalent to choosing one of the trains at random?
If not
(and it typically isnt true), then any answer probably
requires one
to know the times of trains, unless ou sare assuming that
they leave
at random times. There are other problems of terminology ....
by
different trains, do you mean the physical engine-coaches-set
(ie
train) or simply the point in the timetable. In the furst
instance
you would need to know how the sets were allocated to the
timetable
points.
David Jones
===
Subject: Re: Missing something
>Hi there,
>I dont know whether Im missing something 
but I have a
probability
>question that seems easy, yet I cant quite get my head
wrapped
> round
>it.
>There are 53 different trains that go from London to
Nottingham. If
> I
>was to turn up to the station at London at random times to
take the
>train, how many journeys would I have to make so that the
> probability
>of travelling on at least half the trains is 0.5.
>Any help or guidance would be greatly appreciated.
>Anthony
> Are you assuming that turn up to the station at London at
random
> times is equivalent to choosing one of the trains at
random? If not
> (and it typically isnt true), then any answer probably
requires one
> to know the times of trains, unless ou sare assuming that
they leave
> at random times. There are other problems of terminology
.... by
> different trains, do you mean the physical
engine-coaches-set (ie
> train) or simply the point in the timetable. In the furst
instance
> you would need to know how the sets were allocated to the
timetable
> points.
> David Jones
Sorry, when I said turn up to the station at random times I
meant
pick one at random. By the term different trains I meant
purely the
engine-coaches-set. The same problem could be described by
saying 53
different numbered counters in a bag. If I was replacing the
picked
counter after each pick, how many times would I have to draw
from the
bag to get a probability of 0.5 of picking at least half of
them. I
dont know whether that is any clearer. Sorry for the poor
terminology.
Anthony.
===
Subject: Re: Missing something
> Hi there,
> I dont know whether Im missing something 
but I have a
probability
> question that seems easy, yet I cant quite get my head
wrapped
>> round
> it.
> There are 53 different trains that go from London to
Nottingham.
If
>> I
> was to turn up to the station at London at random times to
take
the
> train, how many journeys would I have to make so that the
>> probability
> of travelling on at least half the trains is 0.5.
> Any help or guidance would be greatly appreciated.
> Anthony
>> Are you assuming that turn up to the station at London at
random
>> times is equivalent to choosing one of the trains at
random? If
not
>> (and it typically isnt true), then any answer probably
requires
one
>> to know the times of trains, unless ou sare assuming that
they
leave
>> at random times. There are other problems of terminology
.... by
>> different trains, do you mean the physical
engine-coaches-set (ie
>> train) or simply the point in the timetable. In the furst
instance
>> you would need to know how the sets were allocated to the
timetable
>> points.
>> David Jones
> Sorry, when I said turn up to the station at random times I
meant
> pick one at random. By the term different trains I meant
purely
the
> engine-coaches-set. The same problem could be described by
saying
53
> different numbered counters in a bag. If I was replacing
the picked
> counter after each pick, how many times would I have to
draw from
the
> bag to get a probability of 0.5 of picking at least half of
them. I
> dont know whether that is any clearer. Sorry for the poor
> terminology.
> Anthony.
I see that others have given some approaches for a direct
solution. An
alternative is to derive some recursive expressions. An
outline of
this:
(i) defune P(k,n) to be the probability of have seen exactly k
unique
items (trains) up to and including the nth trial.
(ii) express P(k,n) in terms of values at n-1: this involves
only
P(k,n-1) and P(k-1,n-1).
(iii) Either solve analytically for P(k,n) in general, or use
for
recursive calculations, remembering that at the gfunal step
you need
to form a summation of the P(k,n) over the appropriate range
of k.
David Jones
===
Subject: Re: Missing something
> Sorry, when I said turn up to the station at random times I
meant
> pick one at random. By the term different trains I meant
purely the
> engine-coaches-set. The same problem could be described by
saying 53
> different numbered counters in a bag. If I was replacing
the picked
> counter after each pick, how many times would I have to
draw from the
> bag to get a probability of 0.5 of picking at least half of
them. I
> dont know whether that is any clearer. Sorry for the poor
> terminology.
I assume you mean a probability of at least 0.5 of picking at
least
half of them. It may not be possible to pick any integer
number that
gives exactly 0.5 probability of getting at least half of
them.
Glen
===
Subject: Re: Missing something
> Hi there,
> I dont know whether Im missing something 
but I have a
probability
> question that seems easy, yet I cant quite get my head
wrapped round
> it.
> There are 53 different trains that go from London to
Nottingham. If I
> was to turn up to the station at London at random times to
take the
> train, how many journeys would I have to make so that the
probability
> of travelling on at least half the trains is 0.5.
If the trains depart at equally spaced times, then it is not
that
diffucult.
However, if the two trains depart with, say, ten minutes
between them, then
it is very unlikely that you end up on the furst of these
trains
(probability 10/(60*24)). So you need to take account of the
distribution
of
the train departures.
-Michael.
===
Subject: Re: A question about common understanding of big
sigma
Although it is obvious to all, should I use two equations to
range of b
seperately
sum(j=a,b) x_j = 0 + x_a + x_(a+1) +..+ x_b (where b>=a)
sum(j=a,b) x_j = 0 (where b< a)
Or just one, like following
sum(j=a,b) x_j = 0 + x_a + x_(a+1) +..+ x_b (where b is any
number)
===
Subject: Re: A question about common understanding of big
sigma
> Although it is obvious to all, should I use two equations
to range of b
seperately
> sum(j=a,b) x_j = 0 + x_a + x_(a+1) +..+ x_b (where b>=a)
> sum(j=a,b) x_j = 0 (where b< a)
Ok. Better is to drop the 0 + .
> Or just one, like following
> sum(j=a,b) x_j = 0 + x_a + x_(a+1) +..+ x_b (where b is any
number)
No, you still need to explain when b < a.
The furst is best, otherwise:
defune
sum(j=a,b) x_j = sum(j in [a,b]/N) x_j
sum empty set is 0
sum { x1,.. xj } = x1 +..+ xj
When b < a, [a,b] / N = nulset
as no x with a <= x <= b
Riddle of the day: Whats the product of no numbers?
===
Subject: Re: A question about common understanding of big
sigma
situation now. I am going to either put a note or leave both
there.
Have a good day!
Charlie
===
Subject: Re: A question about common understanding of big
sigma
posting-account=Glvc4AwAAADzVCZ73XnxpzMhXir6xVzs
> Although it is obvious to all, should I use two equations
to range of
b seperately
> sum(j=a,b) x_j = 0 + x_a + x_(a+1) +..+ x_b (where b>=a)
> sum(j=a,b) x_j = 0 (where b< a)
> Or just one, like following
> sum(j=a,b) x_j = 0 + x_a + x_(a+1) +..+ x_b (where b is any
number)
You have the right to introduce any notation that will
simplify
the comprehension. Id write an explicit note up front or in
an appendix, in a section called notation, that explains that
sum (j=a,b) where b<a will be taken to have value 0.
Of course, if youre only going to use this idea once, then
just add the explanatory note where you use it.
- Randy
===
Subject: Re: Simple group of order 60
[...]
>Theyre non-abelian because A_8 has no cyclic subgroup of
order
>21. Also because otherwise the normalizer of an intersection
>would have order >= 21*21/3 = 147.
> I understand the statement re A_8, though this leads to
questions
> about possible orders of elements of A_n and S_n. e.g., I
know
> that products of disjoint cycles have order = lcm(orders of
the
> cycles). Is that enough to determine the order of elements
of A_n
> and S_n?
Of course. How could it not be? So an element of order 21 must
have at least one cycle of length 21, or at least one cycle of
length 7 and at least one of length 3, and thus cant occur
before A_{10}.
> Also, I need to get some notes and/or a text, but later
> for that.
> I dont understand the statement about the order of the
intersection
> of the normalizer, but later for that too.
This follows from the elementary result that if H and K are
subgroups of a funite group, then |HK| = |H| |K| / |H ^ K|. 
And
if H and K both normalize M, then clearly |N(M)| >= |HK|.
[...]
> n_2 != 3 since
> then G would a subgroup of S_3, and n_2 != 7 since then
thered
> be a C_6,
>> I dont follow why this statement is true.
>> n_2 = 7 ==> |N(P_2)| = 3 x 8. Why would there be a C_6?
>Because every group of order 24 with a normal Sylow
2-subgroup
>has one: All groups of order 8, except the elementary abelian
>one E, have a characteristic involution, and E has 7
involutions
>so one of them must be normalized by a C_3 acting on E.
> I am still not clear on this.
Jyrki posted a detailed explanation.
> I recently posted a question on the
> group SL(2,3) and it turned out that all the groups I
looked at
> has an element of order 6, but I still dont see how to
prove that
> all groups of order 24 have an element of order 6, in spite
of
> what you say above.
Not all groups of order 24 -- you yourself later noticed that
S_4 has no such element -- just those with a normal Sylow
2-subgroup. (Actually, also those with a normal Sylow
3-subgroup. Why?)
[...]
--
Jim Heckman
===
Subject: Inverse Laplace Transform
Can anyone help me with this.
[j*(f/c)]/[1+j*(f/c)] = ?
c - some constant
f - frequency in Hz
I need to do inverse Laplace transform on this, but I have no
idea how.
Tilen.
===
Subject: Element of?
As a layman I wonder if the basic meanings of all symbols
used in axioms
of set theory have already been completely and unambiguously
clarifued.
For instance, I doubt that it would be reasonable to include
an integer
in IR as a constituting element because integers have the
quality to
denote e. g. a position in an absolutely reproducible manner
while real
as well as rational numbers are always somewhat uncertain in
practical use.
Given, zero does not exist as a rational number.
Given, the concept of real numbers covers zero just in case
of the
potential infunity.
Is there any justifucation for including or excepting a
rational or real
zero in physics?
Couldnt reals be interpreted as integers divided by an
denominator of
infunite size? I would conclude from that: Reals are of quite
different
quality.
Incidentally, are there examples for explicit use of complex
integer
numbers, in the sense of discrete ones? If so, is the
classifucation
into IZ, IQ, IR, and IC suffucient?
Eckard Blumschein
===
Subject: Re: Element of?
> As a layman I wonder if the basic meanings of all symbols
used in axioms
> of set theory have already been completely and
unambiguously clarifued.
No they havent: set theory has more than one model. Nor can
this be
fuxed, one will always get models which are not intended.
===
Subject: Re: Element of?
sci.math:
>Given, zero does not exist as a rational number.
If you assume that, you can prove pretty much anything you
want to.
Perhaps your thought is too subtle for me to understand, but
as far
as I can see 0 is every bit as good a rational number as
11/17.
--
Stan Brown, Oak Road Systems, Tompkins County, New York, USA
http://OakRoadSystems.com/
And if youre afraid of butter, which many people are nowa-
days, (long pause) you just put in cream. --Julia Child
===
Subject: Re: Element of?
> As a layman I wonder if the basic meanings of all symbols
used in axioms
> of set theory have already been completely and
unambiguously clarifued.
No. The axioms frequently serve to *help* clarify the
meanings of the
symbols. Also, any grammar rules *help* clarify the meanings.
Ultimately, however, the meanings only exist unambiguously
for a model
of the axioms/symbols.
> For instance, I doubt that it would be reasonable to
include an integer
> in IR as a constituting element because integers have the
quality to
> denote e. g. a position in an absolutely reproducible
manner while real
> as well as rational numbers are always somewhat uncertain
in practical
use.
I have no idea what this is supposed to mean. Real and
rational numbers
are no more uncertain than integers.
> Given, zero does not exist as a rational number.
Zero *is* a rational number. What made you think it is not?
> Given, the concept of real numbers covers zero just in case
of the
> potential infunity.
Huh? What do you mean by covers zero and potential infunity?
> Is there any justifucation for including or excepting a
rational or real
> zero in physics?
I thought we were talking about math, which is a tool used in
physics.
> Couldnt reals be interpreted as integers divided by an
denominator of
> infunite size? I would conclude from that: Reals are of
quite different
> quality.
You cant have a denominator of infunite size, 
under any normal
interpretation. You think the reals of quite different
quality from
*what*?
> Incidentally, are there examples for explicit use of
complex integer
> numbers, in the sense of discrete ones? If so, is the
classifucation
> into IZ, IQ, IR, and IC suffucient?
Do you mean Gaussian Integers? What are you referring to with
IZ, IQ,
IR, and IC?
--
Will Twentyman
email: wtwentyman at copper dot net
===
Subject: Re: Element of?
> I thought we were talking about math, which is a tool used
in physics.
Of course, mathematics is much, much more than a tool used in
physics.
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square
root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Re: Element of?
>> As a layman I wonder if the basic meanings of all symbols
used in axioms
>> of set theory have already been completely and
unambiguously clarifued.
> No. The axioms frequently serve to *help* clarify the
meanings of the
> symbols. Also, any grammar rules *help* clarify the
meanings.
> Ultimately, however, the meanings only exist unambiguously
for a model
> of the axioms/symbols.
A mathematics that is not just a self-satisfying game has to
be
carefully futted into the whole knowledge of menkind. This
requires at
furst properly chosen most elementary basics. The Atoms of
mathematics
are not the axioms but what one intends to express for
instance with the
symbol for element of.
> I have no idea what this is supposed to mean. Real and
rational numbers
> are no more uncertain than integers.
If physicists and engineers prefer integers, they often do so
because
these genuine numbers are absolutely precise. Calculation
with reals or
more precisely rationals always depends on the chosen
accuracy.
>> Given, zero does not exist as a rational number.
> Zero *is* a rational number. What made you think it is not?
I gave several reasons in de.sci.mathematics
Let me add a further one:
Zero is supposed to be a neutral number without any sign. Can
you
imagine to divide a by b and yield a result without sign?
>> Given, the concept of real numbers covers zero just in
case of the
>> potential infunity.
> Huh? What do you mean by covers zero and potential infunity?
As did Weyl, I consider a continuum a sauce.
The term potential infunity was introduced by Aristoteles and
means
infunity is a fuction outside the wealth of 
numbers. There is
actually
no infunite number.
>> Is there any justifucation for including or excepting a
rational or real
>> zero in physics?
> I thought we were talking about math, which is a tool used
in physics.
I respect mathematics, but I am an engineer who demands
ßawless tools.
>> Couldnt reals be interpreted as integers divided by an
denominator of
>> infunite size? I would conclude from that: Reals are of
quite different
>> quality.
> You cant have a denominator of infunite size, 
under any
normal
> interpretation. You think the reals of quite different
quality from
> *what*?
The entity of reals as a sauce is quite different from the
notion of a
number.
What are you referring to with IZ, IQ,
> IR, and IC?
Sorry for my awkward letters. I meant integer, rational,
real, and
complex numbers.
Eckard
===
Subject: Re: Element of?
I got a private reply telling me that the axioms must be the
very basis
of mathematics because the numbers are introduced by means of
axioms.
Well, I did not directly refer to the meaning od numbers but
to element
of. The devil is in the detail. I argue, one cannot have a
fenced
border without having a fence.
Eckard
===
Subject: Re: Element of?
> I got a private reply telling me that the axioms must be
the very basis
> of mathematics because the numbers are introduced by means
of axioms.
> Well, I did not directly refer to the meaning od numbers
but to element
> of. The devil is in the detail. I argue, one cannot have a
fenced
> border without having a fence.
> Eckard
In that case, one cannot have the physics without the
mathematics, and
the mathematics takes preeminence.
===
Subject: Re: Element of?
!3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(
5eZ41to5f%E@ELIi
$t^
VcLWP@J5p^rst0+(Ô>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw
> As a layman I wonder if the basic meanings of all symbols
used in
> axioms of set theory have already been completely and
unambiguously
> clarifued.
Layman is not synonymous with moron.
> Given, zero does not exist as a rational number.
I recommend that you take a healthy dose of moonshine from
the same
person that gave you that. Might stop you from bothering us.
--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
===
Subject: Re: Element of?
<x5sm7h3oqa.fsf@lola.goethe.zz>
posting-account=JcYCWQ0AAAClVTkvdB5nDNBgM_6Ir5RQ
I recommend that you take a healthy dose of moonshine from
the same
person that gave you that. Might stop you from bothering us.
Hope you have tested the dose on yourself before recommending
it to
others. ;)
===
Subject: Re: Element of?
>I recommend that you take a healthy dose of moonshine from
the same
>person that gave you that. Might stop you from bothering us.
>Hope you have tested the dose on yourself before
recommending it to
>others. ;)
Whats this about attributing properly?
http://www.xs4all.nl/%7ewijnands/nnq/nquote.html#Q6
--
Stan Brown, Oak Road Systems, Tompkins County, New York, USA
http://OakRoadSystems.com/
And if youre afraid of butter, which many people are nowa-
days, (long pause) you just put in cream. --Julia Child
===
Subject: Re: Element of?
>I recommend that you take a healthy dose of moonshine from
the same
>person that gave you that. Might stop you from bothering us.
>Hope you have tested the dose on yourself before
recommending it to
>others. ;)
Well, a healthy dose might be the zero-dose.
Thomas [SCNR]
===
Subject: Re: Element of?
The discussion continues in de.sci.mathematik: Null rational?
I will inform you about the outcome here.
Eckard Blumschein
===
Subject: Re: Element of?
<419216EF.7090800@et.uni-magdeburg.de>
!3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(
5eZ41to5f%E@ELIi
$t^
VcLWP@J5p^rst0+(Ô>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw
> The discussion continues in de.sci.mathematik: Null
rational? I
> will inform you about the outcome here.
Until such a time, every mathematician is well-advised to
stop doing
any algebraic operations that would require the existence of a
rational number zero.
--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
===
Subject: [Probability] Lp-Norm of Min(X,a)-E[Min(X,a)]
I have asked the following on fr.sci.maths but without
success:
Let X be a real-valued p-integrable random variable,
where p in [1,+oo[. Let a be a real number. I am trying
to determine whether the following inequality is true:
||Min(X,a)-E[Min(X,a)]||_p <= || X - E[X]||_p
where ||.||_p denotes the Lp-Norm. I have seen what I believe
is a valid proof for p=2k (k positive integer) (essentially
based on differentiating w.r. to a, and showing that we have a
non-decreasing mapping in a).
Any help is appreciated.
Noel.
===
Subject: Generalized eigenspace
Hello.
Let T be an endomorphism of a funite dimensional vector space
V. For an
irreducible polynomial f let V(f)={v in V | f(T)^n(v) = 0 for
all n>>0}.
Assume that V(f)=V(g) is non-zero. Why must f and g differ
only by a
non-zero scalar?
--
Michael Knudsen
===
Subject: Re: Generalized eigenspace
> Hello.
> Let T be an endomorphism of a funite dimensional vector
space V. For an
> irreducible polynomial f let V(f)={v in V | f(T)^n(v) = 0
for all n>>0}.
> Assume that V(f)=V(g) is non-zero. Why must f and g differ
only by a
> non-zero scalar?
Consider the ideal { h in k[X] | h(T) acts nilpotent on V(f)
}.
Marc
===
Subject: Re: Generalized eigenspace
> Consider the ideal { h in k[X] | h(T) acts nilpotent on
V(f) }.
--
Michael Knudsen
===
Subject: Re: group theory fraud
As the OP I was musing upon an attempt as an undergraduate to
introduce
probability of outcome into group theory. I didnt have the
ability to get
very far with it and still dont but it is interesteing to
read that
others have also investigated the idea.
I assumed the sum of outcome probabilites over the set would
be unity so
the binary operator would remain a mapping of the set onto
itself. It
never occured to me that it might be less than unity. Thats
also an
interesting idea which would fund application in the real
world, for
example a laundromat function or whatever happened to that
other sock?.
--
-------------------------------------------------------------
---------------
--
William R Watt National Capital FreeNet Ottawas free
community
network
homepage: www.ncf.ca/~ag384/top.htm
its
returned
===
Subject: Re: group theory fraud
>> At an unnamed university in the late 1960s I was
introduced to
>> the underlying concept of mathematics called a group. I
recall
>> it was a set of objects and an operation you could perform
on the
>> objects such that performing the operation on any two
objects in
>> the set always produced an object in the set. What they
didnt
>> specifucally say, but acted as if they had, was that it
always
>> produced the same object. How did they know what object the
>> operation would produce? Suppose they didnt. Suppose 
when
the
>> operation was performed on two objects it would produce
some other
>> object in the set completey at random. Although that might
not be
>> very useful it is how the world actually works, as Ive
come to
>> learn in succeeding years.
> You forgot to take your meds again.
:-)
Maybe youll bite your fungers in a few years, 
as Time
magazine will post
A silly remark in a math group led to te most defunite success
of
mathematics for the past millenium
Interview of William R. Watt, the author of the now famous
silly
message
;-)))
bye
Never say never !
--
Nicolas FRANCOIS
http://nicolas.francois.free.fr
We are the Micro$oft.
Resistance is futile.
You will be assimilated.
===
Subject: Re: This Weeks Finds in Mathematical Physics (Week
208)
> [snip good stuff]
>>For those of you hungering for technical details, Ill 
just
say that
>>the topological theory in question is BF theory with the
symmetry group
>>of deSitter spacetime, namely SO(4,1), as the gauge group.
General
>>relativity can be regarded as a perturbation of this BF
theory by
>>borrowing some ideas from the MacDowell-Mansouri
formulation of
>>general relativity. If you havent heard of that, well,
neither had I.
>>Its a sort of old idea:
>>8) S. W. MacDowell and F. Mansouri, Unifued geometric theory
of gravity
>>and supergravity, Phys. Rev. Lett. 38 (1977), 739-742.
> ^^^^^^^^^^^^^^
> Does anyone have an online reference to that paper?
You could try looking here:
<http://prola.aps.org/>
But I fear one needs a subscription for using that archive...
Bye,
Bjoern
===
Subject: Re: This Weeks Finds in Mathematical Physics (Week
208)
>[snip good stuff]
>>For those of you hungering for technical details, Ill 
just
say that
>>the topological theory in question is BF theory with the
symmetry group
>>of deSitter spacetime, namely SO(4,1), as the gauge group.
General
>>relativity can be regarded as a perturbation of this BF
theory by
>>borrowing some ideas from the MacDowell-Mansouri
formulation of
>>general relativity. If you havent heard of that, well,
neither had I.
>>Its a sort of old idea:
>8) S. W. MacDowell and F. Mansouri, Unifued geometric theory
of gravity
>>and supergravity, Phys. Rev. Lett. 38 (1977), 739-742.
^^^^^^^^^^^^^^
>Does anyone have an online reference to that paper?
> You could try looking here:
> <http://prola.aps.org/ But I fear one needs a subscription
for using that archive...
> Bye,
> Bjoern
supergravity in unitivity. I was hoping to get an
online ref. thereby providing the historical basis
of the concept.
Ken S. Tucker
===
Subject: analysis........-
hello.....doctor~
f: R-> R , f(x) =/= 0 for all x in R.
and f(x+y) = f(x)f(y) for all x, y in R.
show that f(x) > 0
---------------------------------------------
um......i think.......A condition is missing look like.
f is continuous at x=0
right ??
so, f(0+0) = f(0)f(0) => f(0) = 0 or 1 => f(0) = 1(by
assumption)
lim f(x) = lim f(a+h) = lim f(a)f(h) = f(a)
because, lim f(h) = f(0) = 1 (f is continuous at x=0)
h->0
so, f is continuous on R.
if there is c such that f(c) < 0 (c>0),
since f(0) = 1, there is d such that f(d) = 0, 0<d <c.
by Intermediate Value Theorem.
so, by assumption(f(x) =/= 0), contradiction.
right ??
thank you very much for your advice.
===
Subject: Re: analysis........-
> hello.....doctor~
> f: R-> R , f(x) =/= 0 for all x in R.
> and f(x+y) = f(x)f(y) for all x, y in R.
> show that f(x) > 0
> ---------------------------------------------
> um......i think.......A condition is missing look like.
> f is continuous at x=0
> right ??
You dont need that. Simply note that x = x/2 + x/2, and 
that
the
squares of reals are >= 0, and a proof by contradiction is
not far
off.
--
Larry Lard
Replies to group please
===
Subject: Re: analysis........-
> f: R-> R , f(x) =/= 0 for all x in R.
> and f(x+y) = f(x)f(y) for all x, y in R.
> show that f(x) > 0
f(x) = f(2(x/2)) = f(x/2)^2
> um......i think.......A condition is missing look like.
> f is continuous at x=0
> right ??
Wrong.
===
Subject: Re: analysis........-
>f: R-> R , f(x) =/= 0 for all x in R.
>and f(x+y) = f(x)f(y) for all x, y in R.
>show that f(x) > 0
> f(x) = f(2(x/2)) = f(x/2)^2
>um......i think.......A condition is missing look like.
>f is continuous at x=0
>right ??
> Wrong.
Bloombergs quote of the day from Thomas Paine:
To be nobly wrong is more important than to be meanly right
Noel.
===
Subject: dipole coordinates
Are there any orthonormal 2D coordinates, appropriate for
electrostatic
dipole, in the sense that radial coordinate (R) ends up in
one of poles
(charges) ( i.e. at each pole position R=0). I can construct
such
coordinate system using equipotential surfaces but these can
not be
inverted, to get x=x(R,Theta), y=y(R, Theta).
Or, rephrased, if poles are at z-axes what would 3D metrics
look like, in
such coordinates.
B.
===
Subject: Re: dipole coordinates
ETAsAhRtO6p4dMhgqLNJ/jkCKYzgnky9fQIUeS3B9HmDTLomaYbqLIZy+
r8dzfI=
Let N be the point (1,0), S be the point (-1,0, and P be any
point in
the plane. Defune
u = ln(NP/SP) where NP an SP are the distances from N or S to
P.
v = @_N minus @_S where @_N is the angle from the positive
and @_S is the angle from the positive x-axis to SP.
Use elementary algebra and trig to fugure these out in terms
of x and y.
Coordinate curves form two orthogonal sets of circles with
the constant
v curves passing through the poles N and S.
--OL
===
Subject: Re: dipole coordinates
note: poles are not of equal charge and both are negative
===
Subject: Re: Countable Choice
I will try to clarify how the axiom of countable choice
implies that
every infunite set contains a countable subset.
Let X be an infunite set.
Let U_k be the set of injective functions from {1, ..., k} to
X.
U_k is never empty since X is infunite (the percise argument
is simple
using induction on k).
We will use our axiom to choose an element u_k out of each
U_k.
For every k, u_k is an injective function from {1, ..., k} to
X.
Now defune v_k inductively as follows:
v_1 = u_0
Say we have defuned v_k.
Let m<=k+1 be the smallest positive integer such that
u_(k+1)(m) is
not in the range of v_k (well soon see why 
its possible)
And defune v_(k+1) = v_k U (k+1, u_(k+1)(m))
We thus see by induction that each v_k is an injective
function from
{1, ..., k} to X, thats why it is possible to choose the m
above,
because u_(k+1) is a function defuned on a bigger 
funite domain
(we
apply a variant of the piggeonhole principle).
We have constructed a sequence of nested
injective functions from {1, ..., k} to X.
Its union is a function from the positive integers to X as
required.
*-----------------------*
www.GroupSrv.com
*-----------------------*
===
Subject: Re: Countable Choice
===
Subject: Re: Countable Choice
> I will try to clarify how the axiom of countable choice
> implies that every infunite set contains a countable subset.
> Let X be an infunite set.
> Let U_k be the set of injective functions from {1, ..., k}
to X.
> U_k is never empty since X is infunite (the percise argument
is
> simple using induction on k).
> We will use our axiom to choose an element u_k out of each
U_k.
Of countable choices instead of countable dependent choices.
> For every k, u_k is an injective function from {1, ..., k}
to X.
> Now defune v_k inductively as follows:
Now defune injective map v_k from { 1,.. k } into X as 
follows.
> v_1 = u_0
(is an injective map from { 1 } into X)
> Say we have defuned v_k.
Say we have defuned the injective map v_k.
> Let m<=k+1 be the smallest positive integer such that
u_(k+1)(m)
> is not in the range of v_k (well soon see why 
its
possible)
Later? Proof is top down, future loans not accepted. ;-)
By the induction hypothesis
|range v_k| = k < k+1 = |range u_(k+1)|
Thats why. **
> And defune v_(k+1) = v_k U (k+1, u_(k+1)(m))
Which is an injective map from { 1,.. k+1 } into X.
> We thus see by induction that each v_k is an
> injective function from {1, ..., k} to X,
QED.
> thats why it is possible to choose the m above, because
u_(k+1)
> is a function defuned on a bigger funite domain 
(we apply a
> variant of the piggeonhole principle).
Explains ** above.
> We have constructed a sequence of nested
> injective functions from {1, ..., k} to X.
> Its union is a function from the positive
> integers to X as required.
Yes, very clear.
----
===
Subject: Re: Countable Choice
===
>Subject: Re: Countable Choice
>Assuming the countable axiom of choice, how to prove that
every
>infunite set has a countable subset?
> Let X be the given set, and let Y_k be the set of all
> ordered k-tuples of elements of X. Let u_k be the chosen
> element from Y_k; then there is a natural equivalence
> between the integers and the union of the u_k.
>>Would you elaborate?
>>Each u_k is an ordered k-tuple in Y_k = X^k.
>>What is for example, u_k / u_(k+1) ?
>>What are the elements of the union of two different size
tuples?
>>Now from the union of all u_k, how is there a countable
subset of X?
>>What for example, if some x in X with for all k in N, u_k =
{ x }^k?
>> Here are two proofs. One is easy, but obscures the power
>> of what can be done.
>> Defune x_k, w_k recursively as follows.
>> w_0 is the empty set.
>> For k >= 1,
>> x_k is the furst element of u_k not in w_{k-1};
>> w_k = w_{k-1} union {x_k}:
>u_k is an ordered k-tuple and by x in u_k = (x1,.. xk)
> you simply mean x in { x1,.. xk } ?
No. The set u_k is an ordered k-element set, and
w_{k-1} is a(n ordered) k-1-element set. So there
has to be an element of u_k not in w_{k-1}.
>> Then the xs form an explicit infunite 
sequence, as
>> u_k has k elements, and w_{k-1} has only k-1.
>> Another method is to form H = U u_k,
>> and then order this set as follows:
>> If x, y in H, let i be the furst k with x in u_k,
>> and j be the furst k with x in u_k. Then
>> x <= y if i < j or (i = j and x <=_i y),
>> where <=_i is the ordering of u_i. It is easy to
>> show that H so ordered is order-isomorphic to the
>> set of positive integers.
>----
--
This address is for information only. I do not claim that
these views
are those of the Statistics Department or of Purdue
University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: Countable Choice
<cmr5lr$6d4s@odds.stat.purdue.edu>
===
Subject: Re: Countable Choice
>> Assuming the countable axiom of choice, how to prove that
every
>> infunite set has a countable subset?
> Let X be the given set, and let Y_k be the set of all
> ordered k-tuples of elements of X. Let u_k be the chosen
> element from Y_k; then there is a natural equivalence
> between the integers and the union of the u_k.
Thats a new usage of k-tuple as set of k
elements with linear order relation on them.
Its a different and not equivalent defunition 
to the usual
k-tuple
of a map from n, ie { j in Z | 0 <= j < n } into a set of
elements.
> Defune x_k, w_k recursively as follows.
> w_0 is the empty set.
> For k >= 1,
> x_k is the furst element of u_k not in w_{k-1};
> w_k = w_{k-1} union {x_k}:
This isnt a union, its an upward extension 
of a linear
order.
> The set u_k is an ordered k-element set, and
> w_{k-1} is a(n ordered) k-1-element set. So there
> has to be an element of u_k not in w_{k-1}.
Named x_k.
> Then the xs form an explicit infunite 
sequence, as
> u_k has k elements, and w_{k-1} has only k-1.
Ok.
Clipping from above,
> then there is a natural equivalence
> between the integers and the union of the u_k.
makes no sense for the u_k arent nested and how one would
adjoin
funite chains is puzzling especially in event some 
u_ks have
common
elements. It would make more sense to observe theres a
natural
equvalence between the intergers_<= with the union of all w_k
(and the union of all <=_k).
However as |u_k| = k, we see the _set_ union of all u_k
cannot be
funite. So Ônaturally we could 
enumerate the elements,
elements
of u_1, of u_2 skipping and duplicates, etc. However the fuss
with
the w_ks has the potential of a simpler enumeration of a
smaller
subset of X.
--
> Another method is to form H = U u_k,
> and then order this set as follows:
> If x, y in H, let i be the furst k with x in u_k,
> and j be the furst k with x in u_k. Then
> x <= y if i < j or (i = j and x <=_i y),
> where <=_i is the ordering of u_i. It is easy to
> show that H so ordered is order-isomorphic to the
> set of positive integers.
What when different u_k intersect? Say x in u_j / u_k with j
< k.
Then if x <_j z: x <= z <= x, making <= into a preorder
instead of the
lexigraphical order. I suppose in the usual way one could
factor out the
loops L, to produce an order <=/L on H/L which most likely
would be an
infunitely ascending linear order, order isomorphic to N.
----
===
Subject: Re: Countable Choice
===
>Subject: Re: Countable Choice
> Assuming the countable axiom of choice, how to prove that
every
> infunite set has a countable subset?
>> Let X be the given set, and let Y_k be the set of all
>> ordered k-tuples of elements of X. Let u_k be the chosen
>> element from Y_k; then there is a natural equivalence
>> between the integers and the union of the u_k.
>Thats a new usage of k-tuple as set of k
>elements with linear order relation on them.
This is standard terminology in set theory.
>Its a different and not equivalent defunition 
to the usual
k-tuple
>of a map from n, ie { j in Z | 0 <= j < n } into a set of
elements.
It makes no difference which defunition of ordered
k-tuple one uses. It was not explicitly stated, but
should have been, that the elements are distinct.
Your representation is as good as any.
>> Defune x_k, w_k recursively as follows.
>> w_0 is the empty set.
>> For k >= 1,
>> x_k is the furst element of u_k not in w_{k-1};
>> w_k = w_{k-1} union {x_k}:
>This isnt a union, its an upward extension 
of a linear
order.
The set w_k is a union. One can keep the linear
order in mind.
>> The set u_k is an ordered k-element set, and
>> w_{k-1} is a(n ordered) k-1-element set. So there
>> has to be an element of u_k not in w_{k-1}.
>Named x_k.
>> Then the xs form an explicit infunite 
sequence, as
>> u_k has k elements, and w_{k-1} has only k-1.
>Ok.
>Clipping from above,
>> then there is a natural equivalence
>> between the integers and the union of the u_k.
>makes no sense for the u_k arent nested and how one would
adjoin
>funite chains is puzzling especially in event some 
u_ks have
common
>elements. It would make more sense to observe theres a
natural
>equvalence between the intergers_<= with the union of all w_k
>(and the union of all <=_k).
This is a standard operation in set theory. If one has a
well-ordered collection of ordered sets, their union can be
ordered, and if the individual sets are well-ordered, the
union will be well-ordered.
So let {h_alpha: alpha < beta } be a well-ordered
collection of ordered sets. For each x in the union,
defune I(x) to be the smallest alpha such that x is an
element of h_alpha. Then x < y if I(x) < I(y), or if
I(x) = I(y) and x < y in the I(x)-ordering.
>However as |u_k| = k, we see the _set_ union of all u_k
cannot be
>funite. So Ônaturally we could 
enumerate the elements,
elements
>of u_1, of u_2 skipping and duplicates, etc. However the
fuss with
>the w_ks has the potential of a simpler enumeration of a
smaller
>subset of X.
That is what the operation is.
--
This address is for information only. I do not claim that
these views
are those of the Statistics Department or of Purdue
University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: Countable Choice
<cmtd7k$4vk6@odds.stat.purdue.edu>
===
Subject: Re: Countable Choice
> Assuming the countable axiom of choice, how to prove that
every
> infunite set has a countable subset?
>> Let X be the given set, and let Y_k be the set of all
>> ordered k-tuples of elements of X. Let u_k be the chosen
>> element from Y_k; then there is a natural equivalence
>> between the integers and the union of the u_k.
>Thats a new usage of k-tuple as set of k
>elements with linear order relation on them.
> This is standard terminology in set theory.
Ive never seen it before.
>Its a different and not equivalent defunition 
to the usual
k-tuple
>of a map from n, ie { j in Z | 0 <= j < n } into a set of
elements.
> It makes no difference which defunition of ordered
> k-tuple one uses. It was not explicitly stated, but
> should have been, that the elements are distinct.
> Your representation is as good as any.
In the event they are distinct, the two defunitions are
equivalent.
However, a diagonal point (x,x), (x,x,x), ... is an
ordered pair, triplet, ... without distinct elements.
>> Defune x_k, w_k recursively as follows.
>> w_0 is the empty set.
>> For k >= 1,
>> x_k is the furst element of u_k not in w_{k-1};
>> w_k = w_{k-1} union {x_k}:
>> Then the xs form an explicit infunite 
sequence, as
>> u_k has k elements, and w_{k-1} has only k-1.
--
> This is a standard operation in set theory. If one has a
> well-ordered collection of ordered sets, their union can be
> ordered, and if the individual sets are well-ordered, the
> union will be well-ordered.
> So let {h_alpha: alpha < beta } be a well-ordered
> collection of ordered sets. For each x in the union,
> defune I(x) to be the smallest alpha such that x is an
> element of h_alpha. Then x < y if I(x) < I(y), or if
> I(x) = I(y) and x < y in the I(x)-ordering.
Ah ha! I was confusing this with the lexicographical order of
a well
ordered product of ordered sets.
>However as |u_k| = k, we see the _set_ union of all u_k
cannot be
>funite. So Ônaturally we could 
enumerate the elements,
elements
>of u_1, of u_2 skipping and duplicates, etc.
> That is what the operation is.
Indeed, thats a new one on me.
Does it have a particular name or description?
--
> This address is for information only.
And occasional information overload. ;-)
----
===
Subject: 169
posting-account=QD46ZQwAAADVt8kxGXXiBHrSZxPB6Op6
Adding the factorials of the digits of 169: 1! + 6! + 9! =
363601.
Repeat the process with 363601 to get 1454. Repeat again to
get back
169.
===
Subject: Re: 169
posting-account=JcYCWQ0AAAClVTkvdB5nDNBgM_6Ir5RQ
What the F is this you moron. Dont you have any other work?
===
Subject: Re: 169
> What the F is this you moron. Dont you have any other 
work?
Oh come on. The OP discovered an unexpected fuxed point of an
iteration
and was (implicitly) asking about further examples and
generalizations.
Lighten up.
===
Subject: Re: 169
<8Xskd.5$zo1.0@dfw-service2.ext.ray.com>
!3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(
5eZ41to5f%E@ELIi
$t^
VcLWP@J5p^rst0+(Ô>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw
>> What the F is this you moron. Dont you have any other
work?
> Oh come on. The OP discovered an unexpected fuxed point of 
an
> iteration and was (implicitly) asking about further
examples and
> generalizations.
Unexpected? Dont be silly. His rule was one for which a
number with
n digits cant produce more than n*9! as a result. That 
means
that
any number that has more than 7 digits must be reduced to a
smaller
number. And that obviously means that _every_ starting value
has to
lead to a fuxed point or cycle, and it is absolutely trivial
to write
a program to look for them.
And he did not even discover a fuxed point (that would be at
least
mildly amusing) but merely a cycle. , you can take _any_
starting
value and youll discover one (fuxed point being 
a 1-cycle).
--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
===
Subject: Re: 169
David Kastrup <dak@gnu.org> escribi.97:
> What the F is this you moron. Dont you have any other 
work?
>> Oh come on. The OP discovered an unexpected fuxed point of
an
>> iteration and was (implicitly) asking about further
examples and
>> generalizations.
> Unexpected? Dont be silly. His rule was one for which a
number with
> n digits cant produce more than n*9! as a result. That
means that
> any number that has more than 7 digits must be reduced to a
smaller
> number. And that obviously means that _every_ starting
value has to
> lead to a fuxed point or cycle, and it is absolutely trivial
to write
> a program to look for them.
> And he did not even discover a fuxed point (that would be at
least
> mildly amusing) but merely a cycle. , you can take _any_
starting
> value and youll discover one (fuxed point 
being a 1-cycle).
[1, 2, 145, 40585]
No others below 10^6. Or perhaps 0 ... How many digits has 0?
--
Ignacio Larrosa Ca.96estro
A Coru.96a (Espa.96a)
ilarrosaQUITARMAYUSCULAS@mundo-r.com
===
Subject: Re: 169
<8Xskd.5$zo1.0@dfw-service2.ext.ray.com>
<x5vfcdwnv4.fsf@lola.goethe.zz>
<2vßeiF2itm8nU1@uni-berlin.de>
!3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(
5eZ41to5f%E@ELIi
$t^
VcLWP@J5p^rst0+(Ô>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw
> David Kastrup <dak@gnu.org> escribi.97:
>> What the F is this you moron. Dont you have any other
work?
> Oh come on. The OP discovered an unexpected fuxed point of 
an
> iteration and was (implicitly) asking about further
examples and
> generalizations.
>> Unexpected? Dont be silly. His rule was one for which a
number with
>> n digits cant produce more than n*9! as a result. That
means that
>> any number that has more than 7 digits must be reduced to
a smaller
>> number. And that obviously means that _every_ starting
value has to
>> lead to a fuxed point or cycle, and it is absolutely
trivial to write
>> a program to look for them.
>> And he did not even discover a fuxed point (that would be
at least
>> mildly amusing) but merely a cycle. , you can take _any_
starting
>> value and youll discover one (fuxed point 
being a 1-cycle).
> [1, 2, 145, 40585]
Please use { ... } for sets. I confused this with a cycle and
was
going to blow my top about it. In particular since the fuxed
point
was just a parenthetical remark and so it was unclear that
you were
referring to _that_ (perhaps you should have not let it close
like
this
> No others below 10^6. Or perhaps 0 ... How many digits has
0?
here before referring to it).
We are talking about the non-zero digits here, obviously
(since we
dont want to consider 01 different from 1) and 0 has none 
of
those.
So 0 would fut the bill here, too.
--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
===
Subject: Re: 169
David Kastrup <dak@gnu.org> escribi.97:
> Ignacio Larrosa Ca.96estro
<ilarrosaQUITARMAYUSCULAS@mundo-r.com And he did not even
discover a fuxed point (that would be at least
> mildly amusing) but merely a cycle. , you can take _any_
> starting value and youll discover one (fuxed 
point being a
> 1-cycle).
>> [1, 2, 145, 40585]
> Please use { ... } for sets. I confused this with a cycle
and was
> going to blow my top about it. In particular since the fuxed
point
> was just a parenthetical remark and so it was unclear that
you were
> referring to _that_ (perhaps you should have not let it
close like
> this
This is the output of Derive, it is a Ôlist.
>> No others below 10^6. Or perhaps 0 ... How many digits has
0?
> here before referring to it).
> We are talking about the non-zero digits here, obviously
(since we
> dont want to consider 01 different from 1) and 0 has none
of those.
> So 0 would fut the bill here, too.
Yes, I think so.
Other curiosities:
There are two 2cycles, with the smaller term less than
5*10^5, and they are
contiguous:
[871, 45361] and [872, 45362]
and [169, 363601, 1454] is the only 3-cycle with the smaller
term less than
5*10^5.
Trivialities, of course ...
--
Ignacio Larrosa Ca.96estro
A Coru.96a (Espa.96a)
ilarrosaQUITARMAYUSCULAS@mundo-r.com
===
Subject: Ascolis Theorem
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iAAGdT230223;
I really need the proof of this theorem provide by
E.J.McShane (1944) p.336
(Ascolis theorem)
I dont have access to that book so i would be really
thankful if someone
email me with the proof.
-Mario
===
Subject: Re: Functional analysis: space of bounded varation
separable?
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iAAHh7j03690;
>The unit sphere of BV[0,1] is separable in the weak-*
>topology as the dual of C[0,1].
>>Does it hold always?
>>I mean,is it true that the unit sphere of the dual of a
separable
>>Banach space is separable in the weak-*topology?
>Yes. Let X be a separable Banach space. The unit ball of X*
is
>compact and metrizable in the weak-* topology (with one
metric being
>d(f,g) = sum_n 2^(-n) |f(x_n) - g(x_n)| where x_n is a dense
>sequence in the unit ball of X), and any compact metric space
>is separable. Given any weak-* dense sequence {f_n} in the
unit
>ball of X*, let h_n = f_n/||f_n|| (removing any cases where
f_n = 0).
>Then its not hard to show {h_n} is weak-* dense in the 
unit
sphere of
>X*.
I see now that the same argument shows that
the unit sphere of X* is sequentially compact in weak*-
topology.
Ian
===
Subject: Re: Functional analysis: space of bounded varation
separable?
>The unit sphere of BV[0,1] is separable in the weak-*
>topology as the dual of C[0,1].
>>Does it hold always?
>>I mean,is it true that the unit sphere of the dual of a
separable
>>Banach space is separable in the weak-*topology?
>Yes. Let X be a separable Banach space. The unit ball of X*
is
>compact and metrizable in the weak-* topology (with one
metric being
>d(f,g) = sum_n 2^(-n) |f(x_n) - g(x_n)| where x_n is a dense
>sequence in the unit ball of X), and any compact metric space
>is separable. Given any weak-* dense sequence {f_n} in the
unit
>ball of X*, let h_n = f_n/||f_n|| (removing any cases where
f_n = 0).
>Then its not hard to show {h_n} is weak-* dense in the 
unit
sphere of
>X*.
> I see now that the same argument shows that
> the unit sphere of X* is sequentially compact in weak*-
topology.
I think that historically, this is how the sequential
compactness was
DISCOVERED for the bw* topology (when X is separable). Its
the
obvious thing to try (parallelling the Ascoli-Arzela proof).
--Ron Bruck
===
Subject: Re: Beginers Abstract Algebra - Three Questions
Im just trying to put together a proof following your
recomendations
and Im stuck at (3)
> (3) Conclude that there must be a prime of the form 4k+3
not on the
> original list.
How do I prove that?
===
Subject: Re: Beginers Abstract Algebra - Three Questions
days. My association with the Department is that of an
alumnus.
>Im just trying to put together a proof following your
recomendations
>and Im stuck at (3)
>> (3) Conclude that there must be a prime of the form 4k+3
not on the
>> original list.
>How do I prove that?
We know that the number you computed must have some odd prime
factor
which is not of the form 4k+1. And we know that none of the
primes in
our original list divides this number. What is the only
possible
conclusion?
--
Its not denial. Im just very selective 
about
what I accept as reality.
--- Calvin (Calvin and Hobbes)
Arturo Magidin
magidin@math.berkeley.edu
===
Subject: Re: Beginers Abstract Algebra - Three Questions
> Im just trying to put together a proof following your
recomendations
> and Im stuck at (3)
> (3) Conclude that there must be a prime of the form 4k+3
not on the
> original list.
How do I prove that?
k has a prime factor not of the form 4k+1
no prime in the original list is a factor of k
therefore....
===
Subject: CFP: CP-AI-OR05 - furst call for 
papers
Content-Length: 6025
Originator: rusin@vesuvius
.
F I R S T C A L L F O R P A P E R S
International Conference on Integration of AI and OR
Techniques
in Constraint Programming for Combinatorial Optimisation
Problems
CP-AI-OR05
May 30-June 1, 2005, Prague, Czech Republic
http://cpaior05.mff.cuni.cz/
cpaior05@cpaior05.mff.cuni.cz
After a successful series of fuve CP-AI-OR international
workshops
(Ferrara,
Paderborn, Ashford, Le Croisic, and Montreal) devoted to
integration of
Constraint Programming, Artifucial Intelligence, and
Operations Research
in Nice (France) with more than 100 participants. In 2005,
the Second
CP-AI-OR Conference will be held in Prague (Czech Republic),
a beautiful
city in the heart of Europe.
The aim of the conference is to bring together interested
researchers from
AI and OR, and to give them the opportunity to show how the
integration of
techniques from AI and OR can lead to interesting results on
large scale
and
complex problems. We explicitly welcome new ideas and methods
for
integrating OR and AI techniques that have arisen from
real-world
applications.
CP-AI-OR is intended primarily as a forum to focus on the
integration and
hybridization of the approaches of CP, AI, and OR
technologies. A secondary
aim is to provide an opportunity for researchers in one area
to learn about
techniques in others. Therefore, papers that actively
combine, integrate or
contrast approaches from more than one of the areas are
solicited. High
quality pure papers from a single area are eligible provided
that they are
of interest to other communities involved.
As in previous years, CP-AI-OR05 will be preceded by a 
Master
Class where leading researchers give introductory and
overview talks
in a given area. This year, the topic of the Master Class
will be
Metaheuristics and Constraint Programming. In the morning, an
overview
of the main metaheuristics will be given by leading
researchers in the
area and the afternoon will be devoted to some combinations
of constraint
programming and metaheuristics. The Master Class is intended
for PhD
students, researchers, and practitioners.
The program committee invites submissions that include but
are not limited
to the following topics:
- Integration of constraint relaxation methods, e.g.
Constraint
propagation,
Cutting planes, Reduced costs, Global constraints, Graph
algorithms,
Dynamic programming, Lagrangean and convex relaxations,
Heuristic
functions
based on constraint relaxation.
- Integration of search and solving methods, e.g. Branch and
bound,
Intelligent backtracking, Incomplete search, Randomized
search,
Column generation and other decomposition methods, Local
search,
Meta-heuristics.
- Forms of integration, e.g. Static/dynamic problem
decomposition,
Linking variables and constraints in different solvers,
Transformations between models and solvers,
Methods using information derived by other solving methods,
Collaboration between concurrent methods, models, and solvers.
- Problems, modeling, and applications.
IMPORTANT DATES FOR AUTHORS
(THE DEADLINES ARE STRICT DUE TO PUBLISHER CONSTRAINTS)
Deadline for abstracts January 10, 2005
Deadline for paper submissions January 16, 2005
Notifucation of acceptance February 21, 2005
Final paper due March 7, 2005
Master Class May 29, 2005
CP-AI-OR05 May 30-June 1, 2005
The length of a standard technical paper is 15 pages. The
conference
proceedings will be published in the Springer Lecture Notes
in Computer
Science series (http://www.springer.de/comp/lncs/index.html).
Authors are
requested to prepare their papers according to the Springer
instructions
(http://www.springer.de/comp/lncs/authors.html). All papers
are to be
submitted electronically in a PDF or PS format by following
the
instructions
at the URL http://cpaior05.mff.cuni.cz/.
Roman Bartak, Charles University, Czech Republic
Michela Milano, Universita di Bologna, Italy
MASTER CLASS CHAIR
Gilles Pesant, Ecole Polytechnique de Montreal, Canada
Abderrahmane Aggoun, Cosytec, France
Philippe Baptiste, Ecole Polytechnique, France
Roman Bartak, Charles University, Czech Republic
Chris Beck, University of Toronto, Canada
Mats Carlsson, SICS, Sweden
Ondrej Cepek, Charles University, Czech Republic
Hani El Sakkout, CISCO, UK
Bernard Gendron, CRT and Univ. of Montreal, Canada
Carmen Gervet, IC-Parc, UK
Carla Gomes, Cornell University, USA
John Hooker, Carnegie Mellon University, USA
Narendra Jussien, Ecole des Mines de Nantes, France
Stefan Karisch, Carmen Systems, Canada
Francois Laburthe, Bouygues, France
Andrea Lodi, Univ. of Bologna, Italy
Michela Milano, Univ. of Bologna, Italy
George Nemhauser, Univ. of Georgia Tech, USA
Gilles Pesant, CRT and Ecole Polytechnique de Montreal, Canada
Jean-Francois Puget, ILOG, France
Jean-Charles Regin, ILOG, France
Michel Rueher, Univ. of Nice-Sophia Antipolis, France
Meinolf Sellmann, Brown University, USA
Helmut Simonis, IC-Parc, UK
Gilles Trombettoni, Univ. of Nice-Sophia Antipolis, France
Michael Trick, Carnegie Mellon University, USA
Pascal van Hentenryck, Brown University, USA
Mark Wallace, Monash University, Australia
Weixiong Zhang, Washington University, USA
PUBLICITY CO-CHAIRS
Petr Vil.92m, Charles University, Czech Republic
SPONSORSHIP CO-CHAIRS
Ondrej Cepek, Charles University, Czech Republic
Michel Rueher, Univ. of Nice-Sophia Antipolis, France
LOCAL ARRANGEMENTS
Ondrej Cepek, Charles University, Czech Republic
ORGANIZED BY
Charles University, Faculty of Mathematics and Physics
Prague, Czech Republic
CONFERENCE VENUE
The conference will take place in the historical city centre
of Prague
close
to the local attractions like the Charles Bridges and the
Prague Castle.
===
Subject: Integral equations
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id iAAIHvO08677;
In a integral equation of the second kind. let k(s+t)=s(1+t)
and a=0,b=1.
What are the eigenvalues and eigenfunctions?
<nomail@noaddress.not>
===
Subject: about integral equations
In my book I found the use of this result, but I dont know
where to
fund the proof...
Let Omega subseteq R^k, and J:L^2(Omega) --> L^2(Omega) be
the linear operator generated by the kernel K(s,t); i.e.
Jx = y, where y(s) = Int_Omega k(s,t)x(t) dt a.e s in Omega
Let also J_r: L^2(Omega) --> L^2(Omega) be the linear operator
generated by the kernel K_{r-1}(s,t).
How to prove that the operator
J_lambda = sum_{r=1}^inf J^r/(lambda^r)
with |lambda|> ||K||, is generated by:
sum_{r=1}^inf K_{r-1}/(lambda^r)
in L^2(Omega times Omega) ?
===
Subject: Question about limit
Let z_n=(1^k+2^k+...+n^k)/(n^(k+1))
Find lim(n-->infunity)z_n.
My problem is to fund this limit if k=-1 (in rest of cases I
found
this limit).
I dont know if for k=-1 this limit doesnt 
exist or equals to
infunity.
*-----------------------*
www.GroupSrv.com
*-----------------------*
===
Subject: Re: Question about limit
posting-account=OyMMlAwAAADyhoVhXYX4Bw0T-1IatpYa
I found in my old notices another interesting sequence, namely
X_n=(1^n+2^n+...+n^n)/(n^n) .
What about lim_{n-->infty}X_n ?
This sequence appear as a problem proposed by J. Wolstenholme
[in
Thomas Bromwich ,, An Introduction to the Theory
of Infunite series ] and later by I.J.Schoenberg [Problem 640,
Niew
Arch. Wiskd. (1982) III Ser.30,p.116.].
I remember that the desired limit is 1/(e-1) .
===
Subject: Re: Question about limit
> I found in my old notices another interesting sequence,
namely
> X_n=(1^n+2^n+...+n^n)/(n^n) .
> What about lim_{n-->infty}X_n ?
> This sequence appear as a problem proposed by J.
Wolstenholme [in
> Thomas Bromwich ,, An Introduction to the Theory
> of Infunite series ] and later by I.J.Schoenberg [Problem
640, Niew
> Arch. Wiskd. (1982) III Ser.30,p.116.].
> I remember that the desired limit is 1/(e-1) .
This cant be right. X_n > 1 so lim X_n >=1 if it exists and
1/(e-1)<1.
===
Subject: Re: Question about limit
<g5fvd.2$yT4.0@fe08.lga>
posting-account=OyMMlAwAAADyhoVhXYX4Bw0T-1IatpYa
===
Subject: re:Question about limit
Hmmm.... You state that for K different from -1 answer is
1/(k+1).
However for k<-1 I got that lim=+infunity.
For example for k=-2
1^(-2)+2^(-2)+...+n^(-2)=(pi^2)/6
and
n^(-1) --> 0
so
lim [1^(-2)+2^(-2)+...+n^(-2)]/(n^(-1)) = lim
{n[1^(-2)+2^(-2)+...+n^(-2)]}=infunity
*-----------------------*
www.GroupSrv.com
*-----------------------*
===
Subject: Re: Question about limit
escribi.97:
> Let z_n=(1^k+2^k+...+n^k)/(n^(k+1))
> Find lim(n-->infunity)z_n.
> My problem is to fund this limit if k=-1 (in rest of cases I
found
> this limit).
> I dont know if for k=-1 this limit doesnt 
exist or equals
to
> infunity.
For any k
z_n = Sum((i/n)^k, k, 1, n)*(1/n)
Then
z = Lim(z_n, n, inf) = Int(x^k, x, 0, 1) (Riemann sum)
If k =/= - 1,
z = 1/(k+1)
But for k = - 1 you get the improper integral (at 0)
Int(1/x, x, 0, 1) = Lim(Int(1/x, x, r, 1), r, 0+) = Lim(Ln(1)
- Ln(r), r,
0+) = + inf
--
Ignacio Larrosa Ca.96estro
A Coru.96a (Espa.96a)
ilarrosaQUITARMAYUSCULAS@mundo-r.com
===
Subject: Re: Question about limit
Ignacio Larrosa Ca.96estro
<ilarrosaQUITARMAYUSCULAS@mundo-r.com>
escribi.97:
escribi.97:
>> Let z_n=(1^k+2^k+...+n^k)/(n^(k+1))
>> Find lim(n-->infunity)z_n.
>> My problem is to fund this limit if k=-1 (in rest of cases
I found
>> this limit).
>> I dont know if for k=-1 this limit doesnt 
exist or
equals to
>> infunity.
> For any k
> z_n = Sum((i/n)^k, k, 1, n)*(1/n)
> Then
> z = Lim(z_n, n, inf) = Int(x^k, x, 0, 1) (Riemann sum)
> If k =/= - 1,
> z = 1/(k+1)
> But for k = - 1 you get the improper integral (at 0)
> Int(1/x, x, 0, 1) = Lim(Int(1/x, x, r, 1), r, 0+) =
Lim(Ln(1) -
> Ln(r), r, 0+) = + inf
That is true for k > 0 and k = -1, but not for k < 0 and k
=/= - 1 ...
Then Int(x^k, x, 0, 1) is improper at 0. It converge to 1/(k
+ 1) if k in
(-1, 0) and diverge if k < - 1.
I dont know in what I was thinking ...
--
Ignacio Larrosa Ca.96estro
A Coru.96a (Espa.96a)
ilarrosaQUITARMAYUSCULAS@mundo-r.com
===
Subject: Re: Question about limit
> Let z_n=(1^k+2^k+...+n^k)/(n^(k+1))
> Find lim(n-->infunity)z_n.
> My problem is to fund this limit if k=-1 (in rest of cases I
found
> this limit).
> I dont know if for k=-1 this limit doesnt 
exist or equals
to
> infunity.
The harmonic series diverges. (A proof could be to consider
S_n = sum(1/k,
k=1..n), then S_(2n)-S_n >= n*1/(2n)=1/2, which does not
converge to 0,
whence the result).
--
Julien Santini
===
Subject: Re: Question about limit
> Let z_n=(1^k+2^k+...+n^k)/(n^(k+1))
> Find lim(n-->infunity)z_n.
> My problem is to fund this limit if k=-1 (in rest of cases I
found
> this limit).
> I dont know if for k=-1 this limit doesnt 
exist or equals
to
> infunity.
In case when k = -1,
z_n = (1/1 + 1/2 + 1/3 + ... 1/n)/(n^0)
= 1/1 + 1/2 + 1/3 + ... 1/n.
Now the question is reduced to whether the series
1/1 + 1/2 + 1/3 + 1/4 + ...
converges or not.
But this series is the harmonic series, which is known to
diverge.
One can use the integral test to show the harmonic series
diverges.
--
Dae-jung Yoo (IGNJSA YOO) (Delete DELETE to reply by e-mail)
===
Subject: JSH: But what if it works?
posting-account=Q2zO6wwAAABSLuGzZIjG0efOtB9n8fUY
Ive been posting about some new research where 
Im
investigating this
REALLY SIMPLE idea for factoring integers, and Im doing my
usual
process, which involves posting as I work through an idea
looking for
errors.
Usually there are errors, and with something like this,
usually there
are serious errors that kill the idea.
But what if this idea works?
I keep telling myself that itd be really odd if it were 
this
simple,
as how could I fund it when no one has for hundreds if not
thousands of
years?
But what if, despite all of that, it works?
My usual process is to toss out new ideas, and see if posting
reveals
to me a fatal ßaw, as often it does. Over time I work and
re-work an
idea, and my best, most solid works took YEARS in this
process to come
together.
Like advanced polynomial factorization? Over two years of
effort to
get that fully worked out, and more like 4, and if you count
out all
the antecedents work on it kind of goes back to 1995.
Prime counting was much faster but it took MONTHS to work
that out, and
even to clear out all the errors, and get a good hold on how
it all
worked out. Id say it was over two years, from May 2002 
when
I made
the discovery, until I felt I had a good handle on it, and
had cleared
out all the errors in how I looked at it.
This factoring idea is about, oh, three days old. Its a
baby. It
just doesnt fut the pattern for me to just bam, 
get it, right
off the
bat.
So Im looking to be playing with this for months, maybe 
even
years,
down the road, still looking for some basic factoring idea,
but, what
if works?
Well, if it works, then theres a fairly good chance that no
one will
know it, which is kind of hilariously funny. So, even if it
works,
theres probably no early concerns.
If it does work though, I should probably stop talking about
it. But
its early in the process. Time will tell.
James Harris
===
Subject: Re: JSH: But what if it works?
posting-account=sAS5-AwAAABlKnmtMjBbYHvhxI6W0cAg
<snip But what if this idea works?
Well, that would be exciting. So why not fund out whether it
works?
===
Subject: Re: JSH: But what if it works?
posting-account=Q2zO6wwAAABSLuGzZIjG0efOtB9n8fUY
> <snipBut what if this idea works?
> Well, that would be exciting. So why not fund out whether it
works?
I get LOTS of ideas. Most of them are in areas where despite
all the
arguing in places like sci.math, the real world impact is
relatively
minor.
I have a PROCESS for how I work through my ideas, which
includes a lot
of discussion, as talking out ideas helps me understand them.
However, in this area, the potential real-world impact would
seem to be
higher, so Im also talking out that side.
I LIKE doing mathematical research. Im doing that research
as I like
to do it, and with a baby idea that is now 4 days old,
theres no rush
as far as Im concerned.
But what if it works?
Well then, that would be a development that might interest
others, I
would suppose.
As for me, Im doing what I enjoy: researching out ideas, 
and
talking
about them.
Doing this is exciting as it has been for years.
James Harris
===
Subject: Re: JSH: But what if it works?
: I get LOTS of ideas.
No doubt about that.
: I have a PROCESS for how I work through my ideas, which
includes a lot
: of discussion, as talking out ideas helps me understand
them.
Im curious where this process goes on, because 
its
certainly not here in
sci.math. Here in sci.math you post bad mathematics, make
undergraduate-level errors, insult people, and whine like a
fuve-year-old.
: I LIKE doing mathematical research.
Im not sure if you really know what it is.
Best,
Justin
===
Subject: Re: But what if it works?
Also, if you have an actual factoring algorithm, then I
suggest that once
you
get it working, no matter how many YEARS it takes, I want you
to factor the
following number (RSA-2048):
25195908475657893494027183240048398571429282126204
03202777713783604366202070759555626401852588078440
69182906412495150821892985591491761845028084891200
72844992687392807287776735971418347270261896375014
97182469116507761337985909570009733045974880842840
17974291006424586918171951187461215151726546322822
16869987549182422433637259085141865462043576798423
38718477444792073993423658482382428119816381501067
48104516603773060562016196762561338441436038339044
14952634432190114657544454178424020924616515723350
77870774981712577246796292638635637328991215483143
81678998850404453640235273819513786365643912120103
97122822120720357
It will be equal to the product of two primes. When your
revolutionary
WOOHOO! algorithm is done, tell me what it gave for the two
prime number factors, which are of roughly equal bit length.
Even if it
takes
so many YEARS to make, ONCE you have a WORKING algorithm, then
FACTOR the above number with it and tell me what you get. If
you do,
(and the two factors multiply together to give that exact
number) then you
will have proven that you have a REVOLUTIONARY factoring
algorithm,
even if you do NOT disclose it.
Therefore, if you still wish to keep it secret, then just
provide the two
prime
factors of that number to prove your capability.
> Ive been posting about some new research where 
Im
investigating this
> REALLY SIMPLE idea for factoring integers, and Im doing 
my
usual
> process, which involves posting as I work through an idea
looking for
> errors.
> Usually there are errors, and with something like this,
usually there
> are serious errors that kill the idea.
> But what if this idea works?
> I keep telling myself that itd be really odd if it were
this simple,
> as how could I fund it when no one has for hundreds if not
thousands of
> years?
> But what if, despite all of that, it works?
> My usual process is to toss out new ideas, and see if
posting reveals
> to me a fatal ßaw, as often it does. Over time I work and
re-work an
> idea, and my best, most solid works took YEARS in this
process to come
> together.
> Like advanced polynomial factorization? Over two years of
effort to
> get that fully worked out, and more like 4, and if you
count out all
> the antecedents work on it kind of goes back to 1995.
> Prime counting was much faster but it took MONTHS to work
that out, and
> even to clear out all the errors, and get a good hold on
how it all
> worked out. Id say it was over two years, from May 2002
when I made
> the discovery, until I felt I had a good handle on it, and
had cleared
> out all the errors in how I looked at it.
> This factoring idea is about, oh, three days old. Its a
baby. It
> just doesnt fut the pattern for me to just 
bam, get it,
right off the
> bat.
> So Im looking to be playing with this for months, maybe
even years,
> down the road, still looking for some basic factoring idea,
but, what
> if works?
> Well, if it works, then theres a fairly good chance that
no one will
> know it, which is kind of hilariously funny. So, even if it
works,
> theres probably no early concerns.
> If it does work though, I should probably stop talking
about it. But
> its early in the process. Time will tell.
> James Harris
===
Subject: Re: But what if it works?
> Also, if you have an actual factoring algorithm, then I
suggest that once
> you
> get it working, no matter how many YEARS it takes, I want
you to factor
the
> following number (RSA-2048):
> 25195908475657893494027183240048398571429282126204
> 03202777713783604366202070759555626401852588078440
> 69182906412495150821892985591491761845028084891200
> 72844992687392807287776735971418347270261896375014
> 97182469116507761337985909570009733045974880842840
> 17974291006424586918171951187461215151726546322822
> 16869987549182422433637259085141865462043576798423
> 38718477444792073993423658482382428119816381501067
> 48104516603773060562016196762561338441436038339044
> 14952634432190114657544454178424020924616515723350
> 77870774981712577246796292638635637328991215483143
> 81678998850404453640235273819513786365643912120103
> 97122822120720357
> It will be equal to the product of two primes. When your
revolutionary
> WOOHOO! algorithm is done, tell me what it gave for the two
> prime number factors, which are of roughly equal bit
length. Even if it
> takes
> so many YEARS to make, ONCE you have a WORKING algorithm,
then
> FACTOR the above number with it and tell me what you get.
If you do,
> (and the two factors multiply together to give that exact
number) then
you
> will have proven that you have a REVOLUTIONARY factoring
algorithm,
> even if you do NOT disclose it.
> Therefore, if you still wish to keep it secret, then just
provide the two
> prime
> factors of that number to prove your capability.
Good post. An empirical test. A challenge.
I bet there is an excuse for not being able to fund the prime
pair (such as
my Unix system does not support reading of integers that
long, or
the integer has <cr><lf> between the digits, which corrupts
the
algorithm.
What if it works? If so, I will be impressed. If not, ...
Tomasso.
===
Subject: Re: But what if it works?
> Ive been posting about some new research where 
Im
investigating this
> REALLY SIMPLE idea for factoring integers, and Im doing 
my
usual
> process, which involves posting as I work through an idea
looking for
> errors.
> Usually there are errors, and with something like this,
usually there
> are serious errors that kill the idea.
> But what if this idea works?
Youll get really famous.
> I keep telling myself that itd be really odd if it were
this simple,
> as how could I fund it when no one has for hundreds if not
thousands of
> years?
> But what if, despite all of that, it works?
Youll get really famous -- IF you divulge it AND it is
proven to work.
Youll also break most public key cryptography as that
depends on
factoring integers being hard. If your algorithm is
polynomial time, ie.
the number of operations is a polynomial (instead of
exponential)
function, then you will have broken the RSA public key
cryptography
system, the most common system of encryption out there. On the
other hand, if your simple algorithm is still exponential
time it
wont be anything new. What is your basic idea?
> My usual process is to toss out new ideas, and see if
posting reveals
> to me a fatal ßaw, as often it does. Over time I work and
re-work an
> idea, and my best, most solid works took YEARS in this
process to come
> together.
And it took YEARS for Wiles to prove FLT.
> Like advanced polynomial factorization? Over two years of
effort to
> get that fully worked out, and more like 4, and if you
count out all
> the antecedents work on it kind of goes back to 1995.
> Prime counting was much faster but it took MONTHS to work
that out, and
> even to clear out all the errors, and get a good hold on
how it all
> worked out. Id say it was over two years, from May 2002
when I made
> the discovery, until I felt I had a good handle on it, and
had cleared
> out all the errors in how I looked at it.
> This factoring idea is about, oh, three days old. Its a
baby. It
> just doesnt fut the pattern for me to just 
bam, get it,
right off the
> bat.
Post it, please. Oh, pretty please. If you are researching a
new method
for SOMETHING, ANYTHING, you should share it so we can
evaluate
its potential. Or if its ßawed (like 
your proofs of FLT).
> So Im looking to be playing with this for months, maybe
even years,
> down the road, still looking for some basic factoring idea,
but, what
> if works?
IF, and *IF* it works, *AND* its polynomial time, you will
be written
into the history books of mathematics as one of the greatest
mathematicians
of all time, get a Nobel Prize, etc.!
> Well, if it works, then theres a fairly good chance that
no one will
> know it, which is kind of hilariously funny. So, even if it
works,
> theres probably no early concerns.
Then we wont know it works. If you cannot rpovid ethe
algorithm that
YOU claim works, then we could just as well say youve been
lying.
> If it does work though, I should probably stop talking
about it. But
> its early in the process. Time will tell.
> James Harris
Now, PROVIDE it.
===
Subject: Re: JSH: But what if it works?
!3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(
5eZ41to5f%E@ELIi
$t^
VcLWP@J5p^rst0+(Ô>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw
> Ive been posting about some new research
Uh, look up research in a dictionary of your choice. You are
just
doodling around. There are people that actually do this for a
living,
but they have
a) the capacities to know where and how it is worth to doodle
around
b) the capacities to know when they have hit something.
> where Im investigating this REALLY SIMPLE idea for
factoring
> integers, and Im doing my usual process, which involves
posting as
> I work through an idea looking for errors.
> Usually there are errors, and with something like this,
usually
> there are serious errors that kill the idea.
Yes, and there were, and they killed the idea. More than a
year ago
IIRC. Since then youve been dragging its carcass around,
buying new
clothes for it and putting lipstick on.
It still stinks.
> But what if this idea works?
Have you tried applying for a research grant at the
university of
Laputa?
> My usual process is to toss out new ideas, and see if
posting
> reveals to me a fatal ßaw, as often it does. Over time I
work and
> re-work an idea, and my best, most solid works took YEARS
in this
> process to come together.
You mean the prime counting stuff? Yes, you needed years and
lots of
corrections to rediscover something that was somewhat
interesting
several centuries ago.
This is the _only_ solid work in that you have something to
show
that actually ended up working after a lot of help and
prodding. A
result that is a pretty straightforward application of ßoored
division and that most grad students should be able to arrive
at in
about two days of trying.
All your other work has been falsifued. At a quote of 7 years
to
two days, you cant hope to achieve more in mathematics in
the rest of
your life time than a typical grad student could do in a week.
> Like advanced polynomial factorization? Over two years of
effort to
> get that fully worked out, and more like 4, and if you
count out all
> the antecedents work on it kind of goes back to 1995.
Too bad it turns out bunk, with explicit counterexamples
given.
--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
===
Subject: Re: JSH: But what if it works?
posting-account=2E3AcgwAAAB1okjPhMSvjoXLX3tfDcM9
> Ive been posting about some new research where 
Im
investigating
this
> REALLY SIMPLE idea for factoring integers, and Im doing 
my
usual
> process, which involves posting as I work through an idea
looking for
> errors.
> Usually there are errors, and with something like this,
usually there
> are serious errors that kill the idea.
> But what if this idea works?
> I keep telling myself that itd be really odd if it were
this simple,
> as how could I fund it when no one has for hundreds if not
thousands
of
> years?
> But what if, despite all of that, it works?
If mathematics is an infunite structure, then there still is 
an
infunite amount of information yet to be discovered. The
ostensibly
obvious structures can still yield new information and new
insights
Einstein had the happiest thought of his life thinking about
the
obvious:
http://phyun5.ucr.edu/~wudka/Physics7/Notes_www/node85.html
===
Subject: Re: JSH: But what if it works?
> If it does work though, I should probably stop talking
about it.
If it does *not* work, you should certainly stop talking
about it. Those
possibilities are collectively exhaustive -- so stop talking
about it!
--
There are two things you must never attempt to prove: the
unprovable -- and
the obvious.
--
Democracy: The triumph of popularity over principle.
--
http://www.crbond.com
===
Subject: Re: JSH: But what if it works?
<41BC7A03.6D5BC6F4@ix.netcom.com>
posting-account=Q2zO6wwAAABSLuGzZIjG0efOtB9n8fUY
>If it does work though, I should probably stop talking about
it.
> If it does *not* work, you should certainly stop talking
about it.
Those
> possibilities are collectively exhaustive -- so stop
talking about
it!
Which just shows that youre not a mathematician, as
actually, if it
doesnt work, its of some interest to see 
why.
Now if you can fugure out why thats true, then 
maybe Ill
consider
that theres an ounce of hope for you but I think 
youre just
another
loudmouth on sci.math, with delusions of self-importance
based on your
ability to reply to my posts.
The format allows you to live your delusion. But you see,
anyone can
just reply to my posts, but it takes someone of rare caliber
to reply
intelligently with even a basic understanding of the
mathematics
involved.
You betray your lack of even a basic mathematical
understanding.
James Harris
===
Subject: Re: JSH: But what if it works?
> Which just shows that youre not a mathematician, as
actually, if it
> doesnt work, its of some interest to see 
why.
Does it work? Does it not work? Can you provide an example of
either? See Mike3s RSA-2048 number as a valid checkpoint.
> James Harris
Tomasso - old enough to know better...
===
Subject: Re: JSH: But what if it works?
<41BC7A03.6D5BC6F4@ix.netcom.com>
<et5vd.70500$K7.67397@news-server.bigpond.net.au>
posting-account=Q2zO6wwAAABSLuGzZIjG0efOtB9n8fUY
>Which just shows that youre not a mathematician, as
actually, if
it
>doesnt work, its of some interest to see 
why.
> Does it work? Does it not work? Can you provide an example
of
> either? See Mike3s RSA-2048 number as a valid checkpoint.
>James Harris
> Tomasso - old enough to know better...
You didnt read my post carefully. I just discovered the 
idea
about
three days ago. Its a baby idea. Experience has shown me
that baby
ideas usually arent worth much time and effort, as usually
they dont
pan out, and I gave examples of some of my mature ideas and
how long
they took.
Ive tossed my baby idea out there already, as 
Ive made two
posts, one
deriving some formulas, and another stepping through an
algorithm from
it.
Ive tested neither, as Im not interested in 
investing the
time at
this point, with a baby idea.
If it works though, then maybe thats not a good thing, but
hey, its
my process.
James Harris
===
Subject: Re: JSH: But what if it works?
<et5vd.70500$K7.67397@news-server.bigpond.net.au>
!3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(
5eZ41to5f%E@ELIi
$t^
VcLWP@J5p^rst0+(Ô>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw
> You didnt read my post carefully. I just discovered the
idea about
> three days ago. Its a baby idea. Experience has shown me
that
> baby ideas usually arent worth much time and effort, as
usually
> they dont pan out, and I gave examples of some of my
mature ideas
> and how long they took.
Well, if I have something that matured like that in the
kitchen, it is
high time to take out the garbage.
--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
===
Subject: Re: JSH: But what if it works?
>>Which just shows that youre not a mathematician, as
actually, if
> it
>>doesnt work, its of some interest to see 
why.
>> Does it work? Does it not work? Can you provide an example
of
>> either? See Mike3s RSA-2048 number as a valid 
checkpoint.
>>James Harris
>> Tomasso - old enough to know better...
> You didnt read my post carefully.
Of course I read the post carefully. You said it was REALLY
SIMPLE, but
then
you refer to (possible) errors, then funally fatal ßaws. 
Then
(as
is typical) a
sequence of self-reßection about how JSH achieves things.
If the idea is REALLY SIMPLE, then how about a simple test.
If JSHs progress and achievement is dependent on criticism
from
sci.math,
then JSH is conjoined to sci.math.
> I just discovered the idea about three days ago. Its a
baby idea.
Experience
> has shown me that baby ideas usually arent worth much 
time
and effort, as
> usually they dont pan out, and I gave examples of some of
my mature
ideas
> and how long they took.
A discovered idea? Well, Ill accept idea, but leave the
notion of
discovery
to one side at least until its revealed, and/or 
verifued or
refuted.
Really Simple Tomasso.
===
Subject: Re: JSH: But what if it works?
>> If it does work though, I should probably stop talking
about it.
>If it does *not* work, you should certainly stop talking
about it.
> Those
>possibilities are collectively exhaustive -- so stop talking
about
> it!
> Which just shows that youre not a mathematician, as
actually, if it
> doesnt work, its of some interest to see 
why.
To whom? Its *your* job to determine the validity of your
proposed
method, and *your* job to identify any area of interest.
> Now if you can fugure out why thats true, 
then maybe Ill
consider
> that theres an ounce of hope for you but I think 
youre
just another
> loudmouth on sci.math, with delusions of self-importance
based on your
> ability to reply to my posts.
I have posted nothing which implies I harbor 
Ôdelusions of
self-importance. No claims, no outrageous unsupported
assertions, no
banging drums, no insults -- only a simple, logical
observation.
But, hey, its only algebra. Yup, yup, yup.
--
There are two things you must never attempt to prove: the
unprovable --
and the obvious.
--
Democracy: The triumph of popularity over principle.
--
http://www.crbond.com
===
Subject: Re: JSH: But what if it works?
>> If it does work though, I should probably stop talking
about it.
>If it does *not* work, you should certainly stop talking
about it.
> Those
>possibilities are collectively exhaustive -- so stop talking
about
> it!
> Which just shows that youre not a mathematician, as
actually, if it
> doesnt work, its of some interest to see 
why.
> Now if you can fugure out why thats true, 
then maybe Ill
consider
> that theres an ounce of hope for you but I think 
youre
just another
> loudmouth on sci.math, with delusions of self-importance
based on your
> ability to reply to my posts.
> The format allows you to live your delusion. But you see,
anyone can
> just reply to my posts, but it takes someone of rare
caliber to reply
> intelligently with even a basic understanding of the
mathematics
> involved.
Amazing. A new thread, and already we have the expected
insults...
> You betray your lack of even a basic mathematical
understanding.
> James Harris
...and the familiar projection.
===
Subject: Re: mathematical language
>>[. . .]
>> Let me see if I can explain why the idea of empirical
observations
>> works out the way it does. Lets suppose we have 
something
we call an
>> empirical observation based on conventional ideas of
experience,
>> observation, or whatever. Then we have something else
called a
>> tautology according to conventional ideas on tautologies
of the
>> general form t:[subject][not subject].
>As before, you lost me right there because your special
syntax
>t:[blah1][blah2] conjures up no connections for me
whatsoever. You
>might have a lot of thought behind that compressed syntax,
but I have
>not been able to fugure that out from your other words.
>>A tautology (t) is merely a statement which excludes no
logical
>>possibility such as it is a car or not a car
>Hm. Your example corresponds to what I think is a tautology.
But a
>statement which excludes no logical possibility took me a
while to parse,
>and now that I have, I stil fund it diffucult to 
handle. TO
remove one
>level of negation and still maintain , could it mean:
>a statement enjoining all logical possibilities or a
statement which is
>true under all logical possibilities? or a statement which
is true under
>every logical possibility of its logical variables? (this
latter one is
>closer to the wording that I am familiar with)
The defunition I used is just the dictionary 
defunition. I
dont see
that adding terms like logical variables enhances the
meaning, but you
seem to have the basic idea.
>>and so is considered
>>always true when both halves ([subject][not subject]) are
taken
>>together.
>this is way out of my experience. what are the two halves
halves of? what
>is subject? Are the tautologies you are talking about
sentences in some
>language (propositional calculus) or something else? Do you
also have an
>example of something that is not a tautology?
Intended as just plain language. Given a particular
proposition, p, is
p true or false? If we take a proposition such as p car,
meaning it
is a car, is the proposition true or false?If we combine
propositions
to form t:[car][not car] we form a tautological proposition
which
covers all possibilities. The halves referred to are just
parts of the
tautological proposition, positive and negative parts. I call
the
positive half or part an empirical observation.
>>(Conversely, circular logic such as it is a car because it
>>is a car includes no logical possibility and is considered
always
>>false for this reason.)
>the commonly accepted way to rewrite this sentence is as P->P
>which is (also commonly accepted to be) a tautology.
For what its worth, Im trying to avoid 
specialized notation
whether
commonly accepted or not. I realize the notation Ive 
adopted
seems
pretty specialized, but its easy enough to explain in plain
language
without resort to commonly accepted formalisms.
As to whether circular logic is tautological, I can only say
that a
proposition such as p car is non inclusive of anything that I
can
tell. People may want to consider such a thing tautological;
however,
I think theyre just being sloppy. Maybe they 
defune
tautologies as
useless and anything which is useless is a tautology and
circular
logic is certainly useless. The difference is that circular
logic is
never true because it contains no proposition. It just
contains an
observation. And there is no way simple observations are
always true.
They certainly can be false or we wouldnt be investigating
them
logically. In any event tautologies and circular logic
represent polar
opposites in terms of meaning apart from being considered
useless.
>> And, further, according to conventional ideas we fund that
tautologies
>> are considered always true.
>I would not be going out on a limb by saying that the word
tautology
>refers to a thing that -is- true (for a particular defunition
of
>truth, i.e. consideration has nothing to do with it).
>>Well, a statement which excludes no logical possibility
would always
>>be true when considered in its entirety because there can
be no true
>>alternatives.
>That just doesnt follow.
Well, if a statement or proposition includes all
possibilities and
excludes no possibilities its diffucult to see 
how it could
be false.
>...
>> And this in turn makes every positive
>> part of any tautology empirical whatever its source.
>And here Im lost. what does anything that came before have
to do with
>positive what does positive mean for you? For all the
reasonable
>meanings I can imagine for that word, none of them bridge
the gap.
>>Well, here, admittedly Im just retrofutting 
the idea of
empirical to
>>the positive half of a tautology because it futs very
conveniently.
>>Positive just means not negative or not the not half of a
tautology.
>>What Im saying is that here we have this thing called a
tautology
>>that is always true.
>Yes.
>>And we have observations called empirical which
>>conform exactly to the non negative part of tautologies
>What? What is the nonnegative part of a tautology? how does
that conform
>to being empirical observations (as opposed to nonempirical
observations?)
Im not sure what the objection is here. For a given
tautology of the
form t:[subject][not subject] there is a negative part [not
subject]
and a non negative or positive part [subject]. I consider the
non
negative or positive part to be the same as what we call an
empirical
observation and the negative part to be a logical observation.
(Technically there is also a third part to particular
tautologies
consisting of a self contradictory part as well,
t:[subject][not
subject][subject not subject]. But I have omitted
consideration of it
because this third part is always false.)
>...
>2) a community has already explored there, found what works
well (for
>lots of people in the community) and named things for
themselves, and
>since they were there furst, they get dibs.
>...
>>The math community can have any dibs they want as
>>long as they get it right, and when they dont 
defunitions
like any
>>other analytical situation need to be corrected. I dont
see working
>>with inexact or incorrect defunitions as of any merit
regardles of
>>pedigree.
>Sure. You just happen to be in a place were theres little
or no room for
>correction/inexactness. I have no justifucation for that
other than my
>fallible memory which (my memory tells me) is similar to
many other
>peoples observations.
>3) mathematical defunitions are stipulative in that a
defunition is a
>matter of -arbitrarily- naming a concept, such that the name
now
>refers only to such a concept, no matter what connotations
and prior
>associations anyone might have had for that name.
>>But stipulation is only a provisional substitute for
correctness and
>>truth whether in student problems or the foundations of
mathematics.
>Sure. But it makes things easier to discuss. Everybody knows
what things
>are supposed to mean.
I agree if we are discussing the consequences of certain
defunitions
instead of their causes. Then we need to understand exactly
what it is
we are discussing so when we say certain conclusions follow
from
certain defunitions, were all on the same page. 
My objections
to
conventional defunitions arent necessarily 
intended to impact
math
conclusions in general, only to clarify the basis on which
defunitions
have to be drawn in terms of essential or primary
characteristics.
In the case of cardinality, I dont see how we can get away
from equal
differences because the alternative would be unequal
differences. Just
as in the case of the term frame of reference in Special
Relativity
we cant get away from common velocity as the 
defuning
determinant.
===
Subject: Re: mathematical language
Originator: harris@tcs.inf.tu-dresden.de (Mitchell Harris)
>The defunition I used is just the dictionary 
defunition.
A mathematical dictionary? Regular defunitions are notoriously
confusingly
different from the mathematical ones.
However you seem to have placed a lot of extra nontraditional
semantics on
the word tautology, just like has already been done by
logicians. So
youre struggling against a (very successful and much older)
tradition.
>Given a particular proposition, p, is
>p true or false? If we take a proposition such as p car,
meaning it
>is a car, is the proposition true or false?If we combine
propositions
>to form t:[car][not car] we form a tautological proposition
which
>covers all possibilities. The halves referred to are just
parts of the
>tautological proposition, positive and negative parts. I
call the
>positive half or part an empirical observation.
So, for you, are all propositions of the form t:[x][not x]?
How do you combine propositions?
Can you give some more examples of propositions (as you think
of them)?
>(Conversely, circular logic such as it is a car because it
>is a car includes no logical possibility and is considered
always
>false for this reason.)
>>the commonly accepted way to rewrite this sentence is as
P->P
>>which is (also commonly accepted to be) a tautology.
>For what its worth, Im trying to avoid 
specialized
notation whether
>commonly accepted or not.
Why? It makes things clearer. Everybody agrees on the
notation so then
everybody can speak without being misunderstood.
>I realize the notation Ive adopted seems
>pretty specialized, but its easy enough to explain in 
plain
language
>without resort to commonly accepted formalisms.
It should be easy enough, but so far in the case of
discussions using your language, it is obviously -not- easy
in plain
language.
>For a given tautology of the
>form t:[subject][not subject] there is a negative part [not
subject]
>and a non negative or positive part [subject]. I consider
the non
>negative or positive part to be the same as what we call an
empirical
>observation and the negative part to be a logical
observation.
Nothing here matches up with anything I can conjure in may
imagination.
How can positive correspond to empirical and negative to
logical?
If true is positive and false is negative, arent these both
logical?
Mitch
===
Subject: Re: mathematical language
>>The defunition I used is just the dictionary 
defunition.
>A mathematical dictionary? Regular defunitions are
notoriously confusingly
>different from the mathematical ones.
Well, dictionaries can be confusing in this respect. But I
think the
idea of tautologies as inclusive of all possibilities is
reasonably
ancient. Alternatively we could describe some formalism such
as
t:[subject][not subject] in which the t stands for always
true so as
to eliminate any conßict with established usage for tautology.
I dont see that use of the term tautology is essential to
analysis of
the general problem, but I think that my interpretation of
standard
usage is reasonably accurate.
>However you seem to have placed a lot of extra
nontraditional semantics on
>the word tautology, just like has already been done by
logicians. So
>youre struggling against a (very successful and much 
older)
tradition.
Yes, but also a very much less illuminating tradition. The
only reason
I got interested in tautologies is that they are always true
and yet
almost universally considered useless yet they
serendipitously connect
with some of my own ideas related to empiricism and universal
truth.
The truly fascinating things is that we can discover
knowledge which
is demonstrably true for all times and places and not just in
some
mathematical valhalla. The formalized statement of these
truths is
what really interests me in connection with the tautology.
They turn
out to represent a quite simple and instructive formalization
method.
>>Given a particular proposition, p, is
>>p true or false? If we take a proposition such as p car,
meaning it
>>is a car, is the proposition true or false?If we combine
propositions
>>to form t:[car][not car] we form a tautological proposition
which
>>covers all possibilities. The halves referred to are just
parts of the
>>tautological proposition, positive and negative parts. I
call the
>>positive half or part an empirical observation.
>So, for you, are all propositions of the form t:[x][not x]?
>How do you combine propositions?
>Can you give some more examples of propositions (as you
think of them)?
You know, the only examples Ive formulated so far represent
tautologies themselves and their properties in general in
connection
with empirical observation, logical interconnection, and self
contradiction. Given the general form, empirical observations
can be
interconnected through continuum boolean logic in every
imaginable
way, at least that I can imagine, to reach what I consider
universal
conclusions, or conclusions drawn through negation and the
presence or
absence of differences . But the extrapolation remains to be
seen in
terms of examples. Its my contention that every aspect of
the brains
operation in terms of what we consider mental effects
represents
differences between differences and that the tautological
formalism I
suggest shows how and why this occurs mechanically.
>>(Conversely, circular logic such as it is a car because it
>>is a car includes no logical possibility and is considered
always
>>false for this reason.)
>the commonly accepted way to rewrite this sentence is as P->P
>which is (also commonly accepted to be) a tautology.
>>For what its worth, Im trying to avoid 
specialized
notation whether
>>commonly accepted or not.
>Why? It makes things clearer. Everybody agrees on the
notation so then
>everybody can speak without being misunderstood.
Yeah, but there are often disadvantages offsetting advantages
if
people misunderstand the terminology. Im trying to avoid 
any
sense or
suggestion of arcana or mystic mumbo-jumbo. Im convinced
these things
need to be explainable and explained in plain language as a
precursor
to formalization. If these ideas are correct, theyll be
correct as
well in plain language. Believe it or not all of my
preceeding posts
on the subjects of differences and differences between
differences
have been in plain language just to avoid the intellectual
overhead of
a readers having to come to terms with specialized
formalisms and
terminology. It just so happens I developed tautological
formalisms
recently as a succinct expression of these general ideas and
decided
to use it for readers more used to formalisms.
>>I realize the notation Ive adopted seems
>>pretty specialized, but its easy enough to explain in
plain language
>>without resort to commonly accepted formalisms.
>It should be easy enough, but so far in the case of
>discussions using your language, it is obviously -not- easy
in plain
>language.
I recently developed the notation in conjunction with the
analysis of
properties of tautologies in the post Tautologies and
Categories and
will append a copy at the end so you can see which, if
either, makes
better sense. The posts are brief and to the point.
>>For a given tautology of the
>>form t:[subject][not subject] there is a negative part [not
subject]
>>and a non negative or positive part [subject]. I consider
the non
>>negative or positive part to be the same as what we call an
empirical
>>observation and the negative part to be a logical
observation.
>Nothing here matches up with anything I can conjure in may
imagination.
>How can positive correspond to empirical and negative to
logical?
>If true is positive and false is negative, arent these 
both
logical?
True isnt positive and false isnt negative. 
Empirical is
positive
and logical negative. True in universal terms is achieved
through the
mechanization of empirical observations in negative terms.
----------------------
> Tautologies and Empirical Truth
> --------------
>In a frank discussion with Wolf Kirchmeir yesterday
concerning whether
>tautologies constitute empirical evidence he took occasion
to remind
>me quite candidly that tautologies are always true. And the
moral he
>drew from this was that tautological truths cant be
empirical because
>empirical observations are always problematic and
tautologies are not.
>Then I got to pondering. It seemed a shame to have something
that was
>always true and not be able to draw some useful information
from it.
>Here was this beacon of universal truth, and we had no use
for it. I
>understood that philosophers and scientists consider
tautologies
>useless despite their universal truth. However, I decided
that the
>funal chapter on usefullness of the tautology had yet to be
written.
>Lets suppose we have a tautology, any tautology. And we
recognize the
>universal truth of that tautology. What conclusions can we
draw from
>this?
>If a tautology is universally true, alternatives to the
tautology
>cannot be true and must be universally false. And, further,
this
>must be true of all tautologies.
>Consequently, everything including empirical evidence
represents a
>tautology or it cannot be true and must be false.
>Thus any empirical observation which is problematic must
represent
>part of a tautology. For example, three inches and not three
inches or
>blue and not blue. These are empirical observations and form
parts of
>tautologies or they cannot be problematic and must be false.
>In point of fact each part of a tautology is an empirical
observation,
>and this is what we mean by an empirical observation despite
the
>conventional interpretation of empirical observations as
inherently
>problematic.
>Further each part of the tautology is subject to evaluation
either in
>terms of problematic correctness or in terms of self
contradiction. If
>either part of a tautology is self contradictory, it must be
false and
>the other part must be universally true whether empirical in
>conventional problematic terms or not.
>In other words, even though tautoligies in themselves are not
>problematic and cannot represent empirical observations, the
reverse
>is not true and empirical observations can and do represent
parts of
>tautologies.
>And funally we conclude that all this must be true because 
the
>combination of tautology and not tautology itself forms a
tautology
>and must always be true.
>Fascinating. Absolutely fascinating.
>The tautology has funally proven useful after all.
> Tautologies and Ultimate Truth
> ---------
>We know that every empirical observation must form part of a
tautology
>or it cannot be true (Tautologies and Empirical Truth).
However, what
>can we make of the other part, the non empirical part, of
tautologies?
>If we consider any given tautology t:[subject][not subject]
we fund
>that the only difference between components is the term not.
Now, we
>draw from this the empirical observation that every
tautology is
>formed through the same mechanism and that the term not or
its
>functional equivalent is present of necessity in every
tautology.
>This empirical observation we then use to derive another
tautology
>T:[not][not not] consistent with formation of tautologies in
general.
>Alan Jones asked what alternatives there could be to any
particular
>tautology t:[subject][not subject]. And the only possibility
which
>woud not invalidate the truth of t:[subject][not subject]
would have
>be self contradictory and could not be true.
>Hence the alternative to t:[subject][not subject] would have
to be
>u:[subject not subject] such that combined with tautology t,
u would
>preserve the character and truth of the original tautology.
Tautology
>t would thus remain always true because u is always false
and together
>they could be used to form a new tautology.
>Thus the alternative to any tautology t:[subject][not
subject] would
>have to be of the form t2:[subject][not subject][subject not
subject].
>However, in the case of the empirical observation P:[not] it
is not
>possible to regress T in the same way because T already
contains its
>self contradiction. And we fund that there are no
alternatives to T
>not already implicit in T:[not][not not]. And from this we
conclude
>that P:[not] must necessarily be empirically universal
because
>Q:[not not] is self contradictory, and, as Wolf Kirchmeir
reminds us,
>tautologies are always true.
> Tautologies and Universal Truth
> ----------
>Throughout history tautologies have been considered devoid
of value.
>However, if one looks closely at the structure of
tautologies, one can
>see a defunite utility to them in strictly mechanical terms.
>All tautologies are of a common structure t:[subject][not
subject].
>And all empirical observations form one component of
tautologies
>(Tautologies and Empirical Truth). The only problem is how
we can
>employ tautologies in constructive and useful ways.
>Now the obvious answer is that we cant. Yet we must ask 
the
question:
>how is it exactly we put empirical observations together in
mechanical
>terms? We have isolated empirical observations like car
color and
>red, but we dont just weld them together with a blowtorch.
There
>has to be some mechanism at work that produces in
combination the
>universal truth that car color is red.
>This is where the tautology comes in handy. Given the
tautologies
>t1:[car color][not car color] and t2:[red][not red], we can
combine
>these in different ways to produce the difference between
car color
>and red to produce some universal and not empirical truth.
The
>result is universal and non empirical in the sense of not
relying on
>empirical observation for its truth. Thus car color and red
each
>represent empirical observations. However their combination
does not
>because it is reached independently of those observations.
>In other words, we do not observe car color is red
empirically. We
>have to put that conclusion together and thereby knit up the
raveled
>sleeve of reality lying amid the welter of empirical
observations that
>surround us. But to do that we use and need to use
tautological
>mechanics and mechanisms.
>Tautological mechanisms are thus composed of two parts:
empirical
>observations red and logical deductions not red and universal
>truth is constructed piecemeal of these dual threads in
combination.
>Now, I dont know if this insight is original or not. I
might scan the
>web with the search argument tautologies mechanically
useful, but
>for the time being I would rather dwell in ignorance and
ponder the
>elegance and beauty of a heretofore supposedly useless ugly
duckling.
>I will say one thing though. It is tempting to draw a
speculative
>parallel between this double stranded tautological mechanism
and
>the double helix of DNA. However, I have no defunite reason
to say
>that either is or represents any kind of manifestation of
the other.
> Tautologies and Categories
> -----------
>Every empirical observation is capable of forming a
tautology, and
>there are several types or categories of tautology worth
noting:
>Particular - any tautology of the form
> t:[subject][not subject]
>suitable for further regression through self contradiction
of the form
> t:[subject][not subject][subject not subject]
>General - any tautology not subject to further regression of
the form:
> T:[not][not not]
>Imperfect - any tautology of the form
> t:[subject][not subject]
>suitable for perfection through the addition of self
contradiction. A
>tautology is said to be imperfect if it only includes
everything true.
>Perfect -
>Any perfected imperfect tautology of the form
> t:[subject][not subject][subject not subject]
>A tautology is said to be perfect if it not only includes
everything
>true but everything false as well.
===
Subject: An interesting matrix
Here is an interesting matrix at www.guffy.net/matrix.htm.
It is constructed from a very simple algorithm with the
result being that
the matrix is upper triangular with the appearance of
diagonals that have
the same pattern as the rows from which the matrix is
constructed.
I have no conclusions about the matrix but simply wanted to
share it with
others.
David
===
Subject: Re: An interesting matrix
> Here is an interesting matrix at www.guffy.net/matrix.htm.
> It is constructed from a very simple algorithm with the
result being that
> the matrix is upper triangular with the appearance of
diagonals that have
> the same pattern as the rows from which the matrix is
constructed.
> I have no conclusions about the matrix but simply wanted to
share it with
> others.
> David
Will you call it the Matrix of Eratosthenes?
--
John
jmatthews at wright dot edu
www dot wright dot edu/~john.matthews/
===
Subject: Re: An interesting matrix
posting-account=Sny74g0AAADy66iGh6ZMdSlIFta_KAXh
Thats certainly an interesting picture
So your algorithm is saying this, essentially?
If Row and Column start at 1
For all Cell inside Matrix_a_b:
Where Row(Cell) = 1
Value(Cell) = 1 regardless of Column(Cell)
otherwise
Where Row(Cell) = some m ; Such that m > 1
if (Colum(Cell)) congruent to 0 Modulo m
then Value(Cell)=1 else Value(Cell)=0
First cell in a row n thats 1 is m. The next is 2*m, 3*m,
etc?
===
Subject: Re: An interesting matrix
An algorithm for creating the matrix is the following (in C++
or C#):
for (int r = 1; r <= n; r++)
{
for (int c = 1; c <= n; c++)
{
if (c % r == 0)
m(r,c) = 1;
else
m(r,c) = 0;
}
}
The algorithm is rediculously simple. Yet it yields this very
interesting
matrix with lots of interesting structures as depicted in the
picture.
David
> Thats certainly an interesting picture
> So your algorithm is saying this, essentially?
> If Row and Column start at 1
> For all Cell inside Matrix_a_b:
> Where Row(Cell) = 1
> Value(Cell) = 1 regardless of Column(Cell)
> otherwise
> Where Row(Cell) = some m ; Such that m > 1
> if (Colum(Cell)) congruent to 0 Modulo m
> then Value(Cell)=1 else Value(Cell)=0
> First cell in a row n thats 1 is m. The next is 2*m, 3*m,
etc?
===
Subject: Fourier Series/Complex analysis
posting-account=MBqP4A0AAADmTXsiNvaGzTW4Fx5uIecd
Ive been struggling with the following problem for the last
week, and
I just cant make any advances on it. Can anybody help me
with this?
Given the Fourier series of a real-valued continuous function
u(e^i*theta) on the circle of radius 1, fund a real-valued
continuous
function v(e^i*theta) such that the composite function f = u
+ i*v is
extendable to the entire unit disk as a holomorphic function.
GB
===
Subject: Re: Fourier Series/Complex analysis
>Ive been struggling with the following problem for the 
last
week, and
>I just cant make any advances on it. Can anybody help me
with this?
>Given the Fourier series of a real-valued continuous function
>u(e^i*theta) on the circle of radius 1, fund a real-valued
continuous
>function v(e^i*theta) such that the composite function f = u
+ i*v is
>extendable to the entire unit disk as a holomorphic function.
In general this is not possible - you need to assume that u is
somewhat better than just continuous (assuming for example
that
u is continuously differentiable is much more than enough.)
If you stated the problem _exactly_ as given then the problem
is wrong, cant be done no way.
Ill give you a hint regarding a formula for v that works
if any v does: In general if f is a continuous function on
the circle with Fourier coeffucients c_n then the _harmonic_
function in the disk that extends f is given by
F(r e^(it)) = sum_-infunity^infunity c^n r^|n| 
e^(int).
So writing z* for the complex conjugate of z,
F(z) = sum_0^infunity c_n z^n + sum_-infunity^-1 
c_n (z*)^n.
Now what condition on c_n do you need to make F holomorphic?
Then, given u, what should v be so that c_n satisfues that
condition?
************************
David C. Ullrich
===
Subject: Re: Fourier Series/Complex analysis
<f45pr09di836cd1m3pgbnnas5sk829b3i3@4ax.com>
posting-account=MBqP4A0AAADmTXsiNvaGzTW4Fx5uIecd
restrict the coeffucients to make sure that v is real-valued
etc? I
guess that just about solves the problem, so maybe I
shouldnt be
asking it but I do not know if I can do this =)
===
Subject: Re: Fourier Series/Complex analysis
>restrict the coeffucients to make sure that v is real-valued
etc?
Yes, there is.
>guess that just about solves the problem, so maybe I
shouldnt be
>asking it but I do not know if I can do this =)
Indeed.
Ok, another hint: The sum of that series is real-valued if
and only if the sum is equal to its complex conjugate.
Im curious what book this problem came from. Because
it really is wrong, you know.
************************
David C. Ullrich
===
Subject: Dedekind sets and AC
Is the following statement known to be true?
If ZF is consistent, then so is ZF + not AC + Every infunite
set is
Dedekind infunite.
--
Stephen J. Herschkorn sjherschko@netscape.net
===
Subject: Re: Dedekind sets and AC
>Is the following statement known to be true?
>If ZF is consistent, then so is ZF + not AC + Every infunite
set is
>Dedekind infunite.
Yes, and much more. It is easiest to prove with individuals.
--
This address is for information only. I do not claim that
these views
are those of the Statistics Department or of Purdue
University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: Dedekind sets and AC
Stephen J. Herschkorn a .8ecrit :
> Is the following statement known to be true?
> If ZF is consistent, then so is ZF + not AC + Every infunite
set is
> Dedekind infunite.
Ok, I bite. In any infunite set S, denumerable AC permit to
exhibit a
subset in bijection with N, and from there it is easy to
construct a
bijection of S with a proper subset of S. And of course,
denumerable AC
is strictly weaker than AC. So where is the problem?
===
Subject: Re: Dedekind sets and AC
> Stephen J. Herschkorn a .8ecrit :
>> Is the following statement known to be true?
>> If ZF is consistent, then so is ZF + not AC + Every
infunite set is
>> Dedekind infunite.
> Ok, I bite. In any infunite set S, denumerable AC permit to
exhibit a
> subset in bijection with N, and from there it is easy to
construct a
> bijection of S with a proper subset of S. And of course,
denumerable
> AC is strictly weaker than AC. So where is the problem?
of ZF + not CC + Every infunite set is Dedekind 
infunite? Or is
countable choice a theorem of ZF + inifunte implies Dedekind
infunite?
Have I asked these questions before? I was googling sci.math
and
looking through saved e-mail, and I couldnt 
fund it in either
place.
--
Stephen J. Herschkorn sjherschko@netscape.net
===
Subject: Re: Dedekind sets and AC
<41bc94e6$0$3407$8fcfb975@news.wanadoo.fr>
<41BC99F8.3030907@netscape.net>
posting-account=E9O4Ow0AAABTbmswJLdhcZbfGhxYuyCA
> Next question: Does the consistency of ZF imply the
consistency
> of ZF + not CC + Every infunite set is Dedekind 
infunite? Or
is
> countable choice a theorem of ZF + inifunte implies Dedekind
infunite?
The following statements are equivalent in ZF:
(1) every Dedekind-funite set is funite;
(2) every countable family of nonempty Dedekind-funite sets
has a
choice function.
===
Subject: Re: Jesus christ Whats going on?
posting-account=Jngi7wwAAAD2WLn2V2E6Gh2GXydPdCaE
>I just got on Gooleg where I use to post mesasges and
theyre thing
> is
>all fuceked up now! and i cant nkow how to vie waticles
properly
>For god sake are they trying tko Kikc us off or what!!
>Andrew Usher
> I must admit that this is my favorite thread currently
running on
> sci.math.
> I too am having problems with the new Google Groups format.
> But what concerns me the most
> (except probably the Google-specifuc--message-ID issue)
> is that messages posted through Google seem to not be
leaving the
> Google system
> for the rest of Usenet.
> For example, I posted 3 messages (posted them repeatedly,
actually,
> since
> I kept getting an error message, but funally posed without
the error
> message)
> to sci.math and rec.puzzles a few days ago.
> Now, the messages (and their annoying duplicates) did
indeed all
appear
> on Google, but none reached MathForum.
> Actually, this has been a problem in the past with some
posts
> made through Google, and may be an issue with Mathforum
alone.
> But since the message-ID system has just changed for posts
made
through
> Google,
> and since another contributer to this thread has noticed
posts made
> through Google are not appearing on his newsreader (even
though these
> posts were appearing on someone elses newsreader),
> I conclude that, as of few days ago at least, posts made
through
Google
> are not leaving
> their system.
> Have they fuxed this problem? Do they want to 
fux this
problem?
> By the way,this post was made through Google. Has it
propagated
> correctly through Usenet?
> Leroy Quet
Apparently, from meager evidence I conclude, some Usenet
portals and
newsreaders
(In what proportion, I do not know.)
One problem I have emailed them about, they replied that they
will look
into it,
but the problem persists, is that old-style Google Group
(Google
Groups-alpha,
if you will) URLs (in webpages and old posts) which refer to
older
do not work properly.
I get a Not Found error when clicking on such a URL.
You can still click on the authors name on the far left 
part
of the
browswer window
but it would be nice if old-style URLs still worked
automatically.
Leroy Quet
PS: I wonder if this post can be read by most readers of this
thread,
or if there is still
some issue with posts made through Google leaving the Google
system.
===
Subject: Re: Jesus christ Whats going on?
posting-account=CfSJ5AwAAAD1yt3VP50q913IBHikxMCd
Sometimes the sponsored links ads get superimposed on
posting, making
parts of it unreadable..
but postings appear immediately.
Once I made a posting and hit Ôno frame before 
preview and
it went
out!
Narasimham
===
Subject: Set Theory: Rado-Milner Paradox so diffucult?
I use w to represent the smallest infunite ordinal, omega.
type(x)
denotes the (ordinal) order type of x.
The Rado-Milner Paradox: For any ordinal a, there is a
function f: w
--> P(a) such that a = U img(f) and for all n in w,
type(f(n)) <=
|a|^n (ordinal exponentiation).
Kunen marks the proof of this as a diffucult exercise, but it
doesnt
seem so diffucult to me. Am I missing something? Perahps Kunen
thinks
applications of cofunality are diffucult for the 
beginning grad
student.
We proceed by induction on a. (Kunen even gives this as a
hint!) Note
that if a is a cardinal (in particular, if a is 0 or 1), we
may
take f(n) = a for all n.
If a is the ordinal succesor of b >= 1, let g be such a
function
for b, and defune
f(0) = 0
f(n+1) = g(n) U {b}
If a is a limit ordinal and not a cardinal, let k be the
cofunality
of a, and let g: k --> a be increasing and cofunal. For each b
in
k, let h(b) = g(b) U{g(c): c < b}. For each b in k, since
type(h(b)) <= g(b) < a, there exists a function f_b: w -->
P(h(b))
such that U img(f_b) = h(b) and type(f_b(n)) <= |g(b)|^n for
all n in
w. Defune f: w --> P(a) by
f(0) = 0
f(n+1) = U{f_b(n): b in k}
Then U img(f) = a, and for n in w
type(f(n+1)) = sum(b in k, type(f_b(n)))
<= sum(b in k, |g(b)|^n)
<= (|a|^n) k
<= |a|^(n+1)
where all arithmetic is ordinal.
Q.E.D.
--
Stephen J. Herschkorn sjherschko@netscape.net
===
Subject: re: Jack Sarfatti and Leonard Susskind at Cornell
1963
This pretty much sums up Jacks position: any person who
proposes an
interpretation of GR that is not fully consistent with the
Einstein
equivalence hypothesis -- whatever that may have been -- is
ipso facto
off his rocker.
Indeed, Paul your writing the Einstein equivalence hypothesis
--
whatever that may have been says it all. It is very
revealing. You have
not connected the dots properly. You are missing what Michael
Polyani
calls tacit knowing a kind of subconscious heuristic probably
connected with signal nonlocality - what Bierman measures as
presponse.
GODD (I.J. Goods version of P.K. Dicks 
VALIS) is subtle but
not
malicious and moves in strange ways such as my discovery of
Martin
Nietos 1993 paper from Los Alamos describing my role with
Lenny
Susskind and Johnny Glogower (who I brought to Cornell) in
the discovery
of the C & S operators for micro-quantum phase.
Click NOW on the very important
http://arxiv.org/pdf/hep-th/9304036
Thats an order soldier! ;-)
I am now able to look on the furst really important problem I
worked 40
years ago with a new perspective! The issue is what the
nonunitarity of
the time evolution operator for quantum phase is really
telling us -
especially in the macro-quantum case of spontaneous broken
ground state
symmetry with the Goldstone phase and the Higgs fueld that
controls the
dark energy of the universe! Signal nonlocality is the clue.
More on
that anon.
BTW Nietos history is incomplete. I was also working on the
quantum
phase operator problem BEFORE returning to Cornell in the
Autumn of 1963
when, with Phil Morrisons help, I brought Johnny Glogower
with me and
rightaway The Three Stooges Lenny Susskind, Johnny and me
became a
team. We were all equally dysfunctional megalomaniacs who
complimented
each other with the whole greater than the sum of its parts.
I had been
in George Parrents Jr Tech/Ops group on spy satellites,
either NSA or
CIA, in a building shared with Mitre on Route 2, Burlington,
Mass in
early 1963. George was a student of Emil Wolfs and we also
spent a lot
of time in the Boston University physics department. Roy
Glauber was
developing quantum coherence theory at Harvard and George
assigned me to
learn all that stuff to develop the quantum version of 
Wolfs
classical
partial coherence theory. Also lasers were still in their
infancy right
then. George was more a hands on guy and I was his resident
boy genius
theorist. They did not want me to leave Tech/Ops and go back
to Cornell.
I was guaranteed a fast PhD at BU if I stayed and high pay in
defense
work. But then I never would have met Lenny Susskind. Indeed,
Carruthers
probably posed the problem because of me. I think Nieto put
the cart
before the horse.
===
Subject: partitioning algorithm
The problem:
Given a set of N members, you divide the members into 4
groups, if a
group (or groups) have less than 30 members, you divide the
members
into 3 groups, if you still have a group(s) with less than 30
members
divide into 2 groups, etc, the algorithm is funished when you
have at
least 30 members in a group, or 1 group left.
The question:
Is there a fast way of doing this by devising an algorithm
that will
NOT brute force it? if so, how?
===
Subject: Re: partitioning algorithm
===
>Subject: partitioning algorithm
>The problem:
>Given a set of N members, you divide the members into 4
groups, if a
>group (or groups) have less than 30 members, you divide the
members
>into 3 groups, if you still have a group(s) with less than
30 members
>divide into 2 groups, etc, the algorithm is funished when you
have at
>least 30 members in a group, or 1 group left.
>The question:
>Is there a fast way of doing this by devising an algorithm
that will
>NOT brute force it? if so, how?
Take N mod 30. If the the result is >=4, then all partions
can have
at least 30 members.
If the result is <4, then that is the number of partions. Put
30 in each
and divide up the remainder among the partiions.
--
Mensanator
Ace of Clubs
===
Subject: Solution of one of a Polish M.O. problems
Hey,
Just in case anybody else (except F.G.) found it interesting:
Problem: Let f(x)=2^x, g(x)=f(f(f(f(f(f(f(x))))))) (7th
iteration of f).
Determine whether g(3)-g(0) is divisible by g(2)-g(0)
Solution (not mine): First one has to prove that 2^n-1|2^m-1
iff n|m.
This we left to the reader as an excercise :-)
With the help of above lemma we prove that g(3)-g(0) does not
divide
g(2)-g(0). Its very inconvinient to write this without
introducing
special notation so I only sketch the proof:
Start with g(3)-g(0)|g(2)-g(0) then take biggest possible
power of 2
before brackets and use the lemma. After doing so 7 times one
gets
2^2-2^0 |2^3-2^0, that is 3|7, which is contradiction.
sirix.
===
Subject: Nonunitarity in Nietos paper
On M.M.Nietos historical paper on quantum phase and quantum
phase and
time operators.
http://arxiv.org/pdf/hep-th/9304036
The Question is: What is The Question? John A. Wheeler
Nietos discussion starts out a bit loosely at the 
beginning.
For example
U = e^i(phase operator)
by itself is not really a time evolution operator.
For example, given
[P,X] = -ih
Treat P as the operator and a as the c-number
Then
e^iPa/hbar generates the displacement a in the quantum wave
Psi(x), i.e.
e^iPa/hbarPsi(x) = Psi(x + a)
Similarly in the momentum Fourier transform space
e^-iXkPsi(p) = Psi(p - hbark)
therefore, treating PHASE as the operator and n as an 
integer
e^iPHASEnPsi(n) = Psi(n + n)
Psi(n) is a basis in Fock space.
Nieto considers quantized phase eigenvalues in the conjugate
problem
which correspond to things like quantized ßuxes for phase
difference
experiments like the Bohm-Aharonov electron interferometer
showing the
quantum reality of the connection fueld even in a region where
the
curvature vanishes! Of course the connection fueld is only
defuned
locally mod a gauge transformation, or a general coordinate
transformation if we are doing quantum gravity.
===
Subject: Nonunitary time evolution and quantum phase
Of course, the idea is to use these phase operators in some
kind of
Hamiltonian when you consider the time evolution - is it
unitary or not?
For example in the Josephson and Bohm-Aharonov effects
The Hamiltonian H ~ C12, i.e. the cosine phase difference
operator in
Nietos paper.
Is C12 self-adjoint?
If not, then there is signal nonlocality in e^Ht/hbar
operating on some
initial state.
So the math in Nietos paper is concerned with 
funding the
appropriate
phase operators to put into some Hamiltonian.
===
Subject: what is maximal rank of this matrix?
posting-account=kncgtA0AAAAjvHVPgFKTn4k0K3euxgh0
Let n > 2.
Let v(n) and w(n) be any two n-tuplets of real numbers.
Let v(n) and w(n) denote the transposes of 
these n-tuplets.
Let a, b, and c be any 3 real numbers.
Construct the nxn matrix
M(n) = a * v(n) v(n) + b * w(n) w(n) + c * { 
w(n) v(n) +
v(n)
w(n) }
What is the maximum rank of the nxn matrix M(n)?
[I know the answer is 2, but what is a clever way to PROVE
this?]
===
Subject: Re: what is maximal rank of this matrix?
posting-account=nnH_0gwAAAARIDd_i5XyTPj4LuckzgGi
> Let n > 2.
> Let v(n) and w(n) be any two n-tuplets of real numbers.
> Let v(n) and w(n) denote the transposes of 
these
n-tuplets.
> Let a, b, and c be any 3 real numbers.
> Construct the nxn matrix
> M(n) = a * v(n) v(n) + b * w(n) w(n) + c * 
{ w(n) v(n) +
v(n)
> w(n) }
> What is the maximum rank of the nxn matrix M(n)?
> [I know the answer is 2, but what is a clever way to PROVE
this?]
The space spanned by (v,w) is at most 2-dimensional.
Mu = 0 for any vector u that is orthogonal to the space;
i.e., for which uv = uw = 0.
===
Subject: Re: what is maximal rank of this matrix?
> Let n > 2.
> Let v(n) and w(n) be any two n-tuplets of real numbers.
> Let v(n) and w(n) denote the transposes of 
these
n-tuplets.
> Let a, b, and c be any 3 real numbers.
> Construct the nxn matrix
> M(n) = a * v(n) v(n) + b * w(n) w(n) + c * 
{ w(n) v(n) +
v(n)
> w(n) }
> What is the maximum rank of the nxn matrix M(n)?
> [I know the answer is 2, but what is a clever way to PROVE
this?]
For short, write the given n x 1 matrices as
v(n) = V and w(n) = W.
Then (aV + cW, cV + bW) and (V, W) are both n x 2 matrices.
Multiply the furst by the transpose of the second
to express M(n) as (an n x 2 matrix).(a 2 x n matrix)
whose rank is at most 2.
Ken Pledger.
===
Subject: Re: what is maximal rank of this matrix?
posting-account=kncgtA0AAAAjvHVPgFKTn4k0K3euxgh0
this does not explain why the rank of an (nx2)x(2xn) matrix
is 2.
===
Subject: Re: what is maximal rank of this matrix?
>this does not explain why the rank of an (nx2)x(2xn) matrix
is 2.
Theorem (or perhaps defunition, depending how you organize
things): the
rank of a matrix is the dimension of its column space.
Corollary 1: the rank of a matrix is at most its row
dimension, and at
most its column dimension.
Corollary 2: the rank of a product of matrices is at most the
maximum of the ranks of the matrices.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
===
Subject: Re: Golf Competition
>no obvious and simple answer but equally disappointed. I
have had one
>other by e-mail which almost meets the criteria - every
player plays
>every other player once but two players meet in the same
group 5
>times!
Being in no mood to grade funal exams, I took another look at
your
scheduling problem. I ran my random matching routine a little
longer,
and it found some more matchings producing 64 of the possible
66 pairs;
once (after about 10 million trials) it even hit 65. I tried
to swap a few
positions in some of these, but it looked like most of these
would lead
again to confugurations with this same problems (one pair
always together).
But one of the random near-successes caught my eye and I was
able to
massage that one into a schedule that included all 66 pairs,
with no pair
ever playing together more than three times. This one is a
very pretty
solution, with a lot of symmetry. For example, EVERY player
now has an
opposite, with whom s/he plays three matches. Every such pair
of
opposites corresponds with another such pair, in such a way
that the
two pairs share the same set of (four) other players whom
they meet
more than once. And so on and so on; the golfers dont care 
a
whit
about the symmetry, Im sure, but this is the sort of thing
thats
sure to tickle a mathematicians fancy!
Mathematicians: is this arrangement easily related to a
classical
funite geometry?
I also ran through all 6^5 permutations of how to schedule
the foursomes.
Its impossible to arrange it so that no one tees off in the
early slot
more than twice. But there are some schedules which are,
according to
one measure or another, sort of fair to all twelve players.
I will leave it to you to pick the schedule you like best.
The foursomes I found are
[A, B, C, D], [E, F, G, H], [I, J, K, L]
[C, G, K, L], [A, B, H, I], [D, E, F, J]
[B, G, H, J], [C, D, F, I], [A, E, K, L]
[C, E, I, J], [A, B, F, K], [D, G, H, L]
[B, E, F, L], [C, D, H, K], [A, G, I, J]
This alternative scheduling of those same foursomes
[E, F, G, H], [A, B, C, D], [I, J, K, L]
[A, B, H, I], [D, E, F, J], [C, G, K, L]
[A, E, K, L], [C, D, F, I], [B, G, H, J]
[C, E, I, J], [D, G, H, L], [A, B, F, K]
[C, D, H, K], [A, G, I, J], [B, E, F, L]
might strike the golfers as a little more fair since the two
players
(here, E and H) who get the morning time-slot three times
also get the
last time-slot (once). But I think the worst schedules are
now worse.
I dont know what to suggest about how to split the 
foursomes
into pairs.
Consider the pairs E-J, D-F, C-I, each of which play together
twice in
weeks 2,3,4 (and none of those pairs play together in weeks 1
and 5).
Since the pairs come together twice, I suppose youd like
e.g. E&J to
be paired in week 2 and then unpaired in week 4. But that
would make D&F
paired in week 2 and C&I unpaired in week 4, so no matter how
you split
the week-3 foursome CIDF youre going to have to 
sacrifuce the
notion
that pairs of players who encounter each other exactly twice
play
once as a pair and once as opposites.
dave
PS -- Ive never actually played golf, but what difference
does the
grouping make, except as a forum for conversation? I mean, at
the end
of the day there are 12 individual scores and the low score
wins, right?
===
Subject: Re: Golf Competition
>But one of the random near-successes caught my eye and I was
able to
>massage that one into a schedule that included all 66 pairs,
with no pair
>ever playing together more than three times. This one is a
very pretty
>solution, with a lot of symmetry. For example, EVERY player
now has an
>opposite, with whom s/he plays three matches. Every such
pair of
>opposites corresponds with another such pair, in such a way
that the
>two pairs share the same set of (four) other players whom
they meet
>more than once. And so on and so on; the golfers dont care
a whit
>about the symmetry, Im sure, but this is the sort of thing
thats
>sure to tickle a mathematicians fancy!
Here is a description that even the golfers might like.
Write the twelve names on a die, every face getting a left
name and
a right name. Color every other vertex of the die blue. (That
is,
four corners are colored, no two of them adjacent.) Now you
can read off
the partitions of the players as follows.
In the furst week, the foursomes consist of the names on
opposite sides
of the die.
For each of the other weeks, grab the die and stare at one of
the
blue vertices. It is ringed by three faces. Each of these
faces will
be half a foursome. To complete a foursome, look across that
face;
theres another blue vertex at the opposite corner. Go over
the
back left edge of the face and read off the left name there;
go
over the back right edge and read off the right name there.
Add those to the two names on the face; thats your 
foursome.
You want to break the foursome into teams of two? Theres 
now
a natural
way to do it: the lefties are a team and the righties are a
team.
(In the furst week, the teams are those on a single face.)
This description even suggests a natural method to randomly
pick
which foursome gets the furst tee each week ...
dave
(I thought surely that TWELVE players meeting FIVE times in a
symmetric way would lead to another Platonic solid, but
apparently not.)
===
Subject: Re: Golf Competition
<cpikmd$k11$1@news.math.niu.edu>
posting-account=
y3wZYhMAAABYsCtaDBjCWE5oFd14ElQZbfvQjxC1czdFUKdrfKUl4g
A resolvable (12,4,2) covering design describes these
foursomes (so
that three blocks of four partition the players, and every
pair of
players is covered).
I dont see much about these on the Web for block size 4.
This paper
may have introduced the topic, but I dont have access:
Lamken & Mills, Resolvable Coverings
Congressus Numerantium, 96 (1993), pp. 21-26
===
Subject: The consise Cantors proof ?
1. L(a,a) = L(a,a) (obvious)
2. exists b, L(a,b) = L(b,b) (provable from 1, with b=a)
3. forall a, exists b, L(a,b) = L(b,b) (generalization of 2)
4a. not(exists a, forall b, L(a,b) != L(b,b) (negation of 3)
4b. not(exists a, forall b, L(a,b) = !L(b,b) (! is some
suitable digit
change function)
5. not(exists a, forall b, L(a,b) = !L(b)(b) (curry)
5. exists r, not(exits a, forall b, L(a,b) = r(b)) (r= !L(b))
There exists a real that no member of list L matches at every
digit.
Of course the free b in (r = !L(b)) puts doubt on such an
algebraic
derivation of Cantors proof.
Herc
--
i dont know where, i dont know how
i only know that some day well be together again
===
Subject: Re: The consise Cantors proof ?
posting-account=9wVPIwwAAAAonf5Dj39AQaTL2sJYvErF
> 1. L(a,a) = L(a,a) (obvious)
> 2. exists b, L(a,b) = L(b,b) (provable from 1, with b=a)
> 3. forall a, exists b, L(a,b) = L(b,b) (generalization of 2)
> 4a. not(exists a, forall b, L(a,b) != L(b,b) (negation of 3)
> 4b. not(exists a, forall b, L(a,b) = !L(b,b) (! is some
suitable
digit change function)
> 5. not(exists a, forall b, L(a,b) = !L(b)(b) (curry)
> 5. exists r, not(exits a, forall b, L(a,b) = r(b)) (r=
!L(b))
> There exists a real that no member of list L matches at
every
digit.
> Of course the free b in (r = !L(b)) puts doubt on such an
algebraic
derivation of Cantors proof.
> Herc
> --
> i dont know where, i dont know how
> i only know that some day well be together again
Thats it! I just noticed this and havent 
checked it (but
assume it
is correct or can be corrected.) You are talking about
formalizing
Cantors proof. This is what MetaMath should do, not its
phoney-baloney proof.
C-B
In my formalization, you are proving -~YES(x,x) based on
manipulating
defunitions, which I then use as an Incompleteness Axiom.
===
Subject: Re: The consise Cantors proof ?
posting-account=sAS5-AwAAABlKnmtMjBbYHvhxI6W0cAg
> 1. L(a,a) = L(a,a) (obvious)
> 2. exists b, L(a,b) = L(b,b) (provable from 1, with b=a)
> 3. forall a, exists b, L(a,b) = L(b,b) (generalization of 2)
> 4a. not(exists a, forall b, L(a,b) != L(b,b) (negation of 3)
> 4b. not(exists a, forall b, L(a,b) = !L(b,b) (! is some
suitable
digit change function)
> 5. not(exists a, forall b, L(a,b) = !L(b)(b) (curry)
> 5. exists r, not(exits a, forall b, L(a,b) = r(b)) (r=
!L(b))
> There exists a real that no member of list L matches at
every
digit.
> Of course the free b in (r = !L(b)) puts doubt on such an
algebraic
derivation of Cantors proof.
> Herc
> --
> i dont know where, i dont know how
> i only know that some day well be together again
You dont set r= !L(b). You let r be such that r(b)= !L(b,b)
for all b.
===
Subject: Re: The consise Cantors proof ?
posting-account=sAS5-AwAAABlKnmtMjBbYHvhxI6W0cAg
> 1. L(a,a) = L(a,a) (obvious)
> 2. exists b, L(a,b) = L(b,b) (provable from 1, with b=a)
> 3. forall a, exists b, L(a,b) = L(b,b) (generalization of 2)
> 4a. not(exists a, forall b, L(a,b) != L(b,b) (negation of 3)
> 4b. not(exists a, forall b, L(a,b) = !L(b,b) (! is some
suitable
digit change function)
> 5. not(exists a, forall b, L(a,b) = !L(b)(b) (curry)
> 5. exists r, not(exits a, forall b, L(a,b) = r(b)) (r=
!L(b))
> There exists a real that no member of list L matches at
every
digit.
> Of course the free b in (r = !L(b)) puts doubt on such an
algebraic
derivation of Cantors proof.
> Herc
> --
> i dont know where, i dont know how
> i only know that some day well be together again
You dont set r= !L(b). You let r be such that r(b)= !L(b,b)
for all b.
===
Subject: Homeomorphism Btw Compact Subset of C^0 and Compact
Subset of L_p
Given a compact subset A of the metric space of continuous
function over
the real line with the sup norm:
A < C^0 = { f : sup |f(x)| < infty },
and a compact subset B of the metric space of p-integrable
function L^p:
B < L^p = { f : int |f|^p < infty }.
Does there exist a homeomorphsim between A and B?
Ill explain why Im asking this. 
Im trying to prove the
condition for
compactness in L^p. Since Ascoli gives me the condition for
compact
subsets of the space of continuous functions, if I can show
the
existence of such homeomorphism, then the proof comes out
immediately
since compactness is preserved through homeomorphisms.
Kira.
===
Subject: Re: Homeomorphism Btw Compact Subset of C^0 and
Compact Subset of
L_p
>Given a compact subset A of the metric space of continuous
function over
>the real line with the sup norm:
> A < C^0 = { f : sup |f(x)| < infty },
>and a compact subset B of the metric space of p-integrable
function L^p:
> B < L^p = { f : int |f|^p < infty }.
>Does there exist a homeomorphsim between A and B?
Given two compact subsets of the real line, does there exist a
homeomorphism between them?
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
===
Subject: Re: Homeomorphism Btw Compact Subset of C^0 and
Compact Subset of
L_p
>>Given a compact subset A of the metric space of continuous
function over
>>the real line with the sup norm:
>> A < C^0 = { f : sup |f(x)| < infty },
>>and a compact subset B of the metric space of p-integrable
function L^p:
>> B < L^p = { f : int |f|^p < infty }.
>>Does there exist a homeomorphsim between A and B?
> Given two compact subsets of the real line, does there
exist a
> homeomorphism between them?
> Robert Israel israel@math.ubc.ca
> Department of Mathematics http://www.math.ubc.ca/~israel
> University of British Columbia Vancouver, BC, Canada
Um... no. A={0}, B={0,1}.
How about *connected* compact sets?
===
Subject: Re: Homeomorphism Btw Compact Subset of C^0 and
Compact Subset of
L_p
> Um... no. A={0}, B={0,1}.
> How about *connected* compact sets?
A = {0}, B = [0,1]. You need to think a bit more about this.
===
Subject: Re: Homeomorphism Btw Compact Subset of C^0 and
Compact Subset of
L_p
>>Um... no. A={0}, B={0,1}.
>>How about *connected* compact sets?
> A = {0}, B = [0,1]. You need to think a bit more about this.
Yea... Sorry for posting questions without thoughts.
===
Subject: Laplaces equation on a wedge
posting-account=Eo4xeQ0AAAAl6b2gPp3aP27vjEXGlEVO
Does anybody know a book where I can fund the solution to
Laplaces
equation on a wedge? Im trying to do it but 
its rather
complicated.
KH
===
Subject: Re: Laplaces equation on a wedge
ETAsAhRXQtoVP2SjIoUbU8XhMwfC2E9JFgIUe8BGyp8SIu+
3jBBmfB1idbHD9VU=
Use the solution for cylindrical coordinates. The eigenvalue
can be
adjused
to fut the conditions on the edges of the wedge.
--OL
===
Subject: Re: Cantor reloaded
<41b577cb$1_2@Usenet.com>
<cp49lj$7m4$1@mailhub227.itcs.purdue.edu>
>So there is no way to understand the set of reals in a
logical and
>also intuitiv sense?
If course there is; thats what Real analysis is all about.
>I thought, if there are distinguishable objects (the reals)
so I can
>put them in a row.
No; being distinguishable has nothing to do with being
countable.
>The set of reals is a realy paradox thing.
Not really; its just not what you were expecting.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT
<http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply
to
domain Patriot dot net user shmuel+news to contact me. Do not
===
Subject: Differentiable Manifolds texts question
I am interested in taking Differentiable Manifolds at the PhD
level as a
reading course in the spring. Since I will have some leeway
with a reading
course, I was wondering what would you use as a primary
textbook? That is,
if you were to go back to when you furst learned
Differentiable Manifolds,
what would your primary textbook be, knowing what you know
now? As
background, I have taken Complex Analysis and Algebra at the
grad level, and
Differential Geometry at the undergrad level (using the
textbook Elementary
Differential Geometry by Barrett ONeil). I have listed 
below
a few
textbooks that I found online that deal with this stuff, but
please if you
know of others that you would use as your primary textbook to
learn this
stuff out of, please let me know.
John Lee Introduction to Smooth Manifolds
Michael Spivak Calculus on Manifolds
Frank Warner Foundations of Differentiable Manifolds and Lie
Groups
Ralph Abraham/J Marsden/T Ratiu Manifolds, Tensor Analysis,
and
Applications
Tony
===
Subject: Re: Differentiable Manifolds texts question
> Michael Spivak Calculus on Manifolds
Excellent book on what is often called advanced calculus. It
is a
really good book for bright seniors to master. It may be too
tame for a
PhD student.
> Ralph Abraham/J Marsden/T Ratiu Manifolds, Tensor Analysis,
and
> Applications
First rate. Almost anything by Marsden is worth reading.
Bob Kolker
===
Subject: Re: Differentiable Manifolds texts question
^OX9W/.#XpUmm`>TD2zNE-t}emfPkFR.Z5`ßY:3QYT$>dUwN^sm;MBV:
F7aL9x*q!`
ln!l}>Y6_45$%R|P7DSrBkEph@1-;P*s~F_28vO@e4p/>}Pc?@rl8cz]
d9RXOt<s9qM
3N1Z7?1!eq-#Wv[)^xf2
Michael Spivak Calculus on Manifolds
> Excellent book on what is often called advanced calculus.
It is a
> really good book for bright seniors to master. It may be
too tame for a
> PhD student.
Maybe he meant Spivaks fuve book series on 
Differential
geometry.
===
Subject: Defunition of limsup help?
Hi all,
Im having a slight diffuculty with a 
defunition, since I
havent seen it
before. If f is a function from C to C, the complex numbers,
I am trying to
think of
limsup |f(z)|
z--> a
in a geometric way.
I would defune it as : Since the actual limit might not exist,
we take all
sequences z_n converging to a, such that |f(z_n)| converges.
Then the
largest of these limits is the limsup. Is this correct?
If I have got this right, my question is : does there have to
exist a
convergent subsequence |f(z_n)|? That is, what if there is no
sequence z_n
converging to a such that |f(z_n)| converges? Is this
possible? i.e. Does
limsup have to exist?
Tony
===
Subject: Re: Defunition of limsup help?
>Hi all,
>Im having a slight diffuculty with a 
defunition, since I
havent seen it
>before. If f is a function from C to C, the complex numbers,
I am trying
to
>think of
>limsup |f(z)|
>z--> a
>in a geometric way.
>I would defune it as : Since the actual limit might not
exist, we take all
>sequences z_n converging to a, such that |f(z_n)| converges.
Then the
>largest of these limits is the limsup. Is this correct?
Yes (assuming of course that we allow infunity as a limit of a
sequence of positive numbers).
>If I have got this right, my question is : does there have
to exist a
>convergent subsequence |f(z_n)|? That is, what if there is
no sequence z_n
>converging to a such that |f(z_n)| converges? Is this
possible? i.e. Does
>limsup have to exist?
The space [0, infunity] is compact.
>Tony
************************
David C. Ullrich
===
Subject: Re: Defunition of limsup help?
> I would defune it as : Since the actual limit might not
exist, we take
all
> sequences z_n converging to a, such that |f(z_n)|
converges. Then the
> largest of these limits is the limsup. Is this correct?
Yes, as long as you replace converges by has a limit (you must
allow
for the possibility that the lim is +oo). Thats not the
*defunition* of
limsup but, yes, its correct.
> If I have got this right, my question is : does there have
to exist a
> convergent subsequence |f(z_n)|? That is, what if there is
no sequence
z_n
> converging to a such that |f(z_n)| converges? Is this
possible? i.e.
Does
> limsup have to exist?
Theres always some subsequence of (z_n)_n such that the
sequence
(|f(z_n)|)_n has a limit. Thats rather easy to establish:
i) if the sequence (|f(z_n)|)_n is bounded, you use the
Bolzano-Weierstrass theorem;
ii) otherwise, theres some subsequence whose limit is +oo.
Jose Carlos Santos
===
Subject: Re: Defunition of limsup help?
>Im having a slight diffuculty with a 
defunition, since I
havent seen it
>before. If f is a function from C to C, the complex numbers,
I am trying
to
>think of
>limsup |f(z)|
>z--> a
>in a geometric way.
>I would defune it as : Since the actual limit might not
exist, we take all
>sequences z_n converging to a, such that |f(z_n)| converges.
Then the
>largest of these limits is the limsup. Is this correct?
By defunition, isnt it inf {sup{|f(z)| : |z - 
a| < e}: e >
0}, which
is the same as the limit of the suprema as e decreases to 0?
--
Stephen J. Herschkorn sjherschko@netscape.net
===
Subject: Re: Defunition of limsup help?
> If I have got this right, my question is : does there have
to exist a
> convergent subsequence |f(z_n)|? That is, what if there is
no sequence
z_n
> converging to a such that |f(z_n)| converges? Is this
possible? i.e.
Does
> limsup have to exist?
What if a is a singularity or pole of f(z)?
Bob Kolker
===
Subject: Re: Defunition of limsup help?
>> If I have got this right, my question is : does there have
to exist a
>> convergent subsequence |f(z_n)|? That is, what if there is
no sequence
>> z_n converging to a such that |f(z_n)| converges? Is this
possible?
>> i.e. Does limsup have to exist?
> What if a is a singularity or pole of f(z)?
> Bob Kolker
If a is a pole of f(z) wouldnt that mean limsup = 
infunity?
===
Subject: Question about simple connectedness, logarithm
I know that if G is simply connected in C, f : G --> C is
analytic such
that f(z) doesnt equal 0 for all z in G, then 
theres an
analytic
function g such that f(z) = e^(g(z)).
Does this mean that one cannot map (via an analytic function)
a simply
connected region G to C whose image is an annulus
about 0? Since if one could, then f(z) would not be 0 for all
z in G, but
it is not possible to defune a branch of log on an annulus
about 0?
If Im right, is there a better argument here for showing
that one cant
analytically map a simply connected region to C whose image
is an annulus
about 0?
Tony
===
Subject: Re: Question about simple connectedness, logarithm
> I know that if G is simply connected in C, f : G --> C is
analytic such
> that f(z) doesnt equal 0 for all z in G, then 
theres an
analytic
> function g such that f(z) = e^(g(z)).
> Does this mean that one cannot map (via an analytic
function) a simply
> connected region G to C whose image is an annulus
> about 0?
Well, no. You can map the open unit disk D analytically and
1-1 onto the
strip O < Re z < 1. Follow that with the map e^z and youve
mapped D onto
the annulus {1 < |z| < e}. (This is a many-to-one mapping,
but its
conformal in the sense that the derivative never vanishes.)
===
Subject: Re: Question about simple connectedness, logarithm
>> I know that if G is simply connected in C, f : G --> C is
analytic such
>> that f(z) doesnt equal 0 for all z in G, then 
theres an
analytic
>> function g such that f(z) = e^(g(z)).
>> Does this mean that one cannot map (via an analytic
function) a simply
>> connected region G to C whose image is an annulus
>> about 0?
> Well, no. You can map the open unit disk D analytically and
1-1 onto the
> strip O < Re z < 1. Follow that with the map e^z and 
youve
mapped D onto
> the annulus {1 < |z| < e}. (This is a many-to-one mapping,
but its
> conformal in the sense that the derivative never vanishes.)
So what was wrong with my original line of reasoning?
Something must be
wrong...
Tony
===
Subject: Re: Question about simple connectedness, logarithm
>>Well, no. You can map the open unit disk D analytically and
1-1 onto the
>>strip O < Re z < 1. Follow that with the map e^z and 
youve
mapped D onto
>>the annulus {1 < |z| < e}. (This is a many-to-one mapping,
but its
>>conformal in the sense that the derivative never vanishes.)
> So what was wrong with my original line of reasoning?
Something must be
> wrong...
Yes. You have assumed implicitely that the only way of
obtaining a
function g such that f = exp o g was this: to take some
branch log of
the logarithm whose domain was the image of f and then to
defune g as
log o f. Well, no, it isnt. For instance, if f is the
restriction of
the exponential function to the half-plane { z | Re z < 0 },
then the
image of f is D(0,1){0} an no, you cant have a branch of 
the
logarithm
with that domain. However, there is a function g defuned there
such that
f = exp o g, such as, for instance, the one defuned by g(z) =
z.
Jose Carlos Santos
===
Subject: Re: Question about simple connectedness, logarithm
>Well, no. You can map the open unit disk D analytically and
1-1 onto the
>strip O < Re z < 1. Follow that with the map e^z and youve
mapped D
onto
>the annulus {1 < |z| < e}. (This is a many-to-one mapping,
but its
>conformal in the sense that the derivative never vanishes.)
>> So what was wrong with my original line of reasoning?
Something must be
>> wrong...
> Yes. You have assumed implicitely that the only way of
obtaining a
> function g such that f = exp o g was this: to take some
branch log of
> the logarithm whose domain was the image of f and then to
defune g as
> log o f. Well, no, it isnt. For instance, if f is the
restriction of
> the exponential function to the half-plane { z | Re z < 0
}, then the
> image of f is D(0,1){0} an no, you cant have a branch of
the logarithm
> with that domain. However, there is a function g defuned
there such that
> f = exp o g, such as, for instance, the one defuned by g(z)
= z.
> Jose Carlos Santos
===
Subject: Another Galois group question
Hi everyone,
Im trying to fund the Galois group of f(x) = 
x^4 + x^2 + x +
1. I have
found that there are no real roots. Thus, the Galois group
contains a
product of two disjoint transpositions. Also, I have found
that f is
irreducible. Thus, the Galois group contains a 4-cycle.
But I couldnt fund anything else. How can I 
funish this
problem? Also, is
there a different way (other than calculating discriminant)
that I could go
about this problem? (Just out of curiosity)
Tony
===
Subject: Re: PROOF that 0.99999... = 1
posting-account=I8cafwwAAACoOYfL9BiKocZY4Rsgl4L7
I made a typing error here,
10x = 9 + x
===
Subject: Re: PROOF that 0.99999... = 1
In sci.math, John Schoenfeld
<j.schoenfeld@programmer.net I made a typing error here,
> 10x = 9 + x
Thats not a proof:
x = .999...
10x = 9.999...
10x - x = 9.000... or 8.999..., depending on whether one
assumes borrow
or not from the last digit. (Since there is no such,
one can go either way here.)
A more rigorous proof would simply use limit principles
(or partial sums, which S. Enterprize seems to have
troubles with on his limited-precision calculator):
Let x_n = .999...9 = 1 - 10^(-n).
Then x_1 = .9, x_2 = .99, x_3 = .999, etc.
Since (1 - x_n) = 10^(-n), for any epsilon > 0, I can pick
an N = ceil(-log10(epsilon)) and for any n > N, (1 - x_n) <
epsilon.
Hence x = lim (n->+oo) x_n = 1.
Another method involves the usual series:
Y_n = y_1 + y_2 + ... + y_n
where y_n = 9 * 10^(-n). Since this is a geometric series,
Y_n = .9 * (1 - 10^(-n)) / (1 - 10^(-1)). At the limit,
Y_n = .9 / .9 = 1. Note that lim(n->+oo) y_n = 0, but
lim(n->+oo) Y_n = 1. Size ... erm, case ... matters. :-)
There is a discipline of mathematics called nonstandard
analysis,
however, which involves hyperreals. A hyperreal number is
assumed
to be less than any 1/n (n in N), but greater than 0. I know
little more about it than that, though Mathworld
http://mathworld.wolfram.com/HyperrealNumber.html
does contain an entry for it.
--
#191, ewill3@earthlink.net
Its still legal to go .sigless.
===
Subject: Re: PROOF that 0.99999... = 1
>I made a typing error here,
>10x = 9 + x
Its still useless.
let x = 2
10( 2) = 9 + 2
20 =/= 11
Smarts Alt. Physics News Group
http://pub39.bravenet.com/forum/show.php?usernum=3320272813&
cpv=1
S. Enterprize (Science Journal)
http://smart1234.s-enterprize.com/
===
Subject: Re: PROOF that 0.99999... = 1
In sci.math, S. Enterprize Company
<smart1234@aol.com>I made a typing error here,
>>10x = 9 + x
> Its still useless.
> let x = 2
> 10( 2) = 9 + 2
> 20 =/= 11
Congratulations! Youve proven that x=2 is not a root of
the above equation.
Care to try another number, though? :-)
[.sigsnip]
--
#191, ewill3@earthlink.net
Its still legal to go .sigless.
===
Subject: Re: Degree of Curvature
Logan:
>I am doing a 8th grade science fair project that tests the
strength of an
arch. However, I need to fugure the degree of curvature of
each arch i
create. I asked my algebra teacher, but she said she didnt
know how to do
it.
> My science teacher didnt know either.......However, she
said I have to
have the degrees listed in my procedures and results.
> Ive gotten myself into a corner,now I dont 
know how to
fugure this
out......any suggestions anyone? THANKS.....LOGAN J.
Radius of curvature for f(x):
rho(x) = (1 + (f Ô(x)^2)^(3/2) / f 
Ô (x).
If you have a parabolic arch (y = -Ax^2+Bx+C, f Ô (x) =
-2Ax+B, f Ô(x) =
-2A.
Curvature = 1/rho.
If you dont know f, fut a smooth curve to it 
(eg, polynomial
fut with Excel
would be reasonable, with maybe 15+ sample points, and poly
of degree 5 or
so).
Tomasso.
===
Subject: Re: Wonderings
> A defunition for irrationals which Ive read 
repeatedly goes
something
> like a number whose decimal expansion continues forever
without
> repeating or terminating. Is that accurate? (Yes, I know an
> irrational is just anything that cant be expressed as a
ratio, but if
> I have a decimal expression, how do I know?)
How do you know sqrt2 is irrational? You dont have the
decimal
expansion, in any useful sense, but its easy to use the
not-a-ratio
defunition to give a proof. The non-terminating, non-repeating
bit
is best thought of as a theorem about irrationals, not as a
defunition.
> If non-repeating & non-terminating is accurate, then the
number
> 0.101001000100001000001000000100000001...
> would be an irrational number, right? (number of zeros
between
> consecutive 1s increases by 1 each time).
Right.
> That leads to a whole (I guess countably infunite) class of
similarly
> constructed numbers. (use other numbers in place of zeros
and ones,
> use more than 2 numbers as digits, increment number of 0s
by different
> values, etc..)
Uncountably infunite, if youre clever enough in 
funding ways
of incrementing the number of zeros.
> Does this class of numbers have a name? What is it? Are
they useful?
No, not applicable, and not particularly.
> Are there any examples of irrational numbers which can be
observed in
> nature?
Are there examples of fractions that can be observed in
nature?
Are there examples of negative integers that can be observed
in nature?
> I know there are physical constants, pi, e, etc..., but
> theyre calculated, not measured, right?
I dont know what makes pi a physical constant. 
Its a
mathematical
constant, and its neither calculated nor measured, but
defuned.
> It occurred to me that a quasar, could be an seen as
generating an
> infunite list of irrational numbers, if it had an 
infunite
life time.
> But... quasars die, so eventually the sequence of digits
terminates.
> Therefore, they generate rationals, not irrationals.
I dont see how quasars generate digits. If you do, take 
them
to be
digits to some transcendental base. Then your funite lifetime
quasars
will joyfully generate irrational numbers.
> So, if we assume that the universe lasts forever, are there
places
> where we can see irrationals being generated?
You can see an irrational any time you look at a (model)
square and
its diagonal.
--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
===
Subject: Floor function for rational numbers
posting-account=I8cafwwAAACoOYfL9BiKocZY4Rsgl4L7
Does anyone know an equivalent representation of the ßoor
function for
the strict case of rational numbers?
More precisely, solve for ßoor(x) such that x = a/b where a
and b are
integers.
The wolfram site provides such a representation but it fails
when x is
an integer and unnecessarily accounts for when x is
irrational.
http://functions.wolfram.com/IntegerFunctions/Floor/27/02/
Any information is greatly appreciated.
===
Subject: Re: Floor function for rational numbers
> Does anyone know an equivalent representation of the ßoor
function for
> the strict case of rational numbers?
> More precisely, solve for ßoor(x) such that x = a/b where a
and b are
> integers.
How about ßoor(a/b) = a/b - (a mod b)/b.
-Michael.
===
Subject: Re: Floor function for rational numbers
>Does anyone know an equivalent representation of the ßoor
function for
>the strict case of rational numbers?
>More precisely, solve for ßoor(x) such that x = a/b where a
and b are
>integers.
> How about ßoor(a/b) = a/b - (a mod b)/b.
> -Michael.
Thats great when a,b > 0 but if either or both are less 
than
0?
What does (-12) mod 5 mean?
If it means -2 then
ßoor (-12/5) = -12/5 - (-2)/5 = -2; a wong answer
So it must mean 3 (from the least positive residues)
= -12/5 - 3/5 = -3.
With a=12 and b = -5 since 2 = 12 mod (-5) then 12/(-5) -
2/(-5) = -2.
===
Subject: Re: Floor function for rational numbers
<41bd369b$0$216$edfadb0f@dread12.news.tele.dk>
posting-account=I8cafwwAAACoOYfL9BiKocZY4Rsgl4L7
>Does anyone know an equivalent representation of the ßoor
function
for
>the strict case of rational numbers?
>More precisely, solve for ßoor(x) such that x = a/b where a
and b
are
>integers.
> How about ßoor(a/b) = a/b - (a mod b)/b.
> -Michael.
Michael,
a mod b = a - ßoor(a/b).b, thus your solution reduces to
ßoor(a/b) = a/b - (a - ßoor(a/b).b)/b
= (a - a + ßoor(a/b).b) / b
= ßoor(a/b)
===
Subject: Mathematica question
I recently purchased a copy of mathematica from someone who
gave me a
license numebr and a math-id number, saying this is what is
needed
to make the software work. I have at times bought used/old
software
from people who didnt need it for whatever reason and,
possibly
luckily, never had problems.
When I tried to install mathematica, fulling the license # I
had, it
gave me a different math-id number than what I had. Moreover,
it
wouldnt let me change it! I also notice that a passwd was
needed,
something the seller had not mentioned. When I clicked to
register
anyway, it said license number was not valid.
Frankly I dont know what is going on and what I should do.
Could
===
Subject: Re: Mathematica question
> Frankly I dont know what is going on and what I should 
do.
Could
sci.math newsgroups? Try the comp.soft-sys.math.mathematica
newsgroup
instead.
Jose Carlos Santos
===
Subject: Re: Mathematica question
: sci.math newsgroups? Try the comp.soft-sys.math.mathematica
: newsgroup instead.
Maybe OP didnt know about the mathematica ng. However, his
choice of physics and math groups was reasonable; thats 
where
one would expect to fund many mathematica users.
===
Subject: Re: Mathematica question
> : sci.math newsgroups? Try the
comp.soft-sys.math.mathematica
> : newsgroup instead.
> Maybe OP didnt know about the mathematica ng. However, 
his
> choice of physics and math groups was reasonable; thats
where
> one would expect to fund many mathematica users.
Jose Carlos Santos
===
Subject: Re: Mathematica question
: > Maybe OP didnt know about the mathematica ng. However,
his
: > choice of physics and math groups was reasonable; thats
where
: > one would expect to fund many mathematica users.
:
Why, sci.physics is primarily about politics, elections,
racism,
holocaust, Islam, terrorism, UFOs, underground railroad,
etc? :-)
I see no reason to believe that one is less likely to fund
mathematica
users in s.p.p than in s.p . Maybe OP is a regular visitor to
s.p.p;
given several reasonable options I would tilt in favor of the
groups I
read regularly anyway.
I am sure youd have chosen differently. So would I (I would
have added
sci.math.symbolic). But OP made reasonable choices.
===
Subject: Re: Please please help with log(z)
|R is simply connected if and only if the
|complement in the extended plane (ie including infunity)
|is connected, so the complement must be a minimal connected
|compact subset of the extended plane containing 0 and
infunity.
We were assuming R was a region.
Obviously if R is not simply connected, then the complement
of R is disconnected; the closed curve in R not homotopic in R
to a point divides the complement into two nonempty parts.
The converse is less obvious to me. It seems that there exist
disjoint open sets in the sphere not separated by a curve;
they
could be separated by something like the topologists sine
curve.
But your only if part seems to be equivalent to the statement
that for any disconnected closed set in the sphere there
exists
a curve in its complement separating it into two nonempty
parts.
That sounds plausible but I dont see how to prove it.
|A curve is the obvious example of such a thing but surely
|there are others. Oh: for example the complement could be
|a topologists sine curve thingie: Thats 
connected,
|not a curve, and its a minimal connected compact set,
because
|it consists of the closure of a simple curve - omit a point
|of the curve and its not connected, omit one of the other
|points and its not closed.
Thats nice; it hinges on the distinction between connected
and arcwise connected which seems to arise naturally in
the problem.
So my favorite branch of the log function is now the one
with log(1)=0 and a branch cut given by a logarithmic
spiral around the origin, t->(-e^t*cos(t), -e^t*sin(t)).
Keith Ramsay
===
Subject: Re: Please please help with log(z)
posting-account=BjC-YAwAAADQ91Zm3XkS3aGs3XlaqZ4X
Keith Ramsay gave a vivid picture
in
branch of log z
in sci.math.
on 14th of February this year
May be this link works:
Have fun
Hero
===
Subject: Re: Please please help with log(z)
>|R is simply connected if and only if the
>|complement in the extended plane (ie including infunity)
>|is connected, so the complement must be a minimal connected
>|compact subset of the extended plane containing 0 and
infunity.
>We were assuming R was a region.
Sorry, I was assuming that R was connected.
>Obviously if R is not simply connected, then the complement
>of R is disconnected; the closed curve in R not homotopic in
R
>to a point divides the complement into two nonempty parts.
That may be obvious, but I dont see that its 
obvious
unless we assume something obvious like the Jordan Curve
Theorem or something. One can give a simpler proof by complex
analysis: If (R is connected and) the complement of R is
connected then every function holomorphic in R can be
approximated by polynomials (Runges theorem).
This implies that integrals of holomorphic functions over
closed curves are always 0, hence non-vanishing holomorphic
functions have logarithms, and hence the standard proof of
the Riemann Mapping Theorem shows that R is conformally
equivalent to a disk, hence is simply connected.
(The standard proof of RMT doesnt have simple connectedness
as a hypothesis, the hypothesis is that every non-vanishing
holomorphic function has a square root...)
>The converse is less obvious to me. It seems that there exist
>disjoint open sets in the sphere not separated by a curve;
they
>could be separated by something like the topologists sine
curve.
>But your only if part seems to be equivalent to the statement
>that for any disconnected closed set in the sphere there
exists
>a curve in its complement separating it into two nonempty
parts.
>That sounds plausible but I dont see how to prove it.
Thats actually easy. Well, at least its easy 
if you settle
for a cycle, ie an algebraic sum of closed curves, which
separates the two sets in the sense that the winding numbers
are different:
Say K1 and K2 are disjoint compact subsets of the extended
plane; say K1 does not include infunity. Cover K1 by a
square grid fune enough that a square intersecting K1
cannot intersect K2. Start with the cycle consisting of
the sum of the boundaries of the squares which intersect
K1. Of course those curves intersect K1. But note that
if you consider the directed graph that has corners
of theses squares as vertices and edges as edges then
every vertex has order 0. Now discard the edges that
intersect K. Every vertex still has order 0, since
each edge that intersects K occurs twice, once in
each direction.
Now an easy induction shows that in a directed graph
where every vertex has order 0 you can partition the
edges into a union of cycles, or circuits, or whatever
the term is - say a circuit is <v1, ... vn> where
there is an edge from vj to v{j+1} and and edge from
v1 to vn, and say two circuits are disjoint if they
have no _edges_ in common. Then in a directed graph
where every vertex has order 0 you can partition the
edges into disjoint circuits.
Apply that lemma to the situation above, and youve
joined the remaining edges into a funite bunch of
closed curves in the plane. At least for points not
on any of the original edges, the total winding
number about any point is the same as the total
winding number of the original collection of squares
intersecting K1. It follows that the net winding
number about any point of K1 is 1 and the net
winding number about any point of K2 is 0.
Ok, that was a few paragraphs, but its actually a
sketch of a _complete_ proof where you can full
in all the details, doesnt use any big theorems.
>|A curve is the obvious example of such a thing but surely
>|there are others. Oh: for example the complement could be
>|a topologists sine curve thingie: Thats 
connected,
>|not a curve, and its a minimal connected compact set,
because
>|it consists of the closure of a simple curve - omit a point
>|of the curve and its not connected, omit one of the other
>|points and its not closed.
>Thats nice; it hinges on the distinction between connected
>and arcwise connected which seems to arise naturally in
>the problem.
>So my favorite branch of the log function is now the one
>with log(1)=0 and a branch cut given by a logarithmic
>spiral around the origin, t->(-e^t*cos(t), -e^t*sin(t)).
>Keith Ramsay
************************
David C. Ullrich
===
Subject: Re: Please please help with log(z)
>>Obviously if R is not simply connected, then the complement
>>of R is disconnected; the closed curve in R not homotopic
in R
>>to a point divides the complement into two nonempty parts.
>That may be obvious, but I dont see that 
its obvious
>unless we assume something obvious like the Jordan Curve
>Theorem or something.
Why not go whole hog and assume something obvious like
Alexander Duality for Cech (Co)homology? Then you get
theorems in all dimensions...
>One can give a simpler proof by complex
>analysis:
Spoilsport.
Lee Rudolph
===
Subject: Recurision/Array problem sal Tia
Greets All,
Im having a little trouble in creating dynamic arrays /
recursion
formula Ive had some help
in getting this formula but Im not sure how to make the
array dynamic
enough in vb.net to include Cn.....Xn in example 1.
Example 1:
User Input for four numbers 15,17,8,12,...N (Note: user can
input any
amount
of numbers)
(15+17) /25/2 ->b1
(b1+8) /25/2 ->b2
(b2+12) /25/2 ->b3
(b1+b2) /25/2 ->c1
(c1 +b3)/25/2 ->c2
(c1 +c2) /25/2 ->c3
What I have so far not funished though having problems.
Store user input in array [a0,...,aN]
Create new array [b0,...,bN]
a[0]-->b[0]
for k=1 to N {
b[k]<--((a[k]+b[k-1])/25)/2
}
TIA
PS. im using vb.net but any programming help in any 
language
would be
helpful
===
Subject: [OT] Re: Recurision/Array problem sal Tia
> Greets All,
Hi.
NOTE: this post imho is slightly OT in
comp.soft-sys.math.scilab, so
Ill add the OT tag, but I wont eliminate the 
x-post because
I dont
know where does the message come from.
> Im having a little trouble in creating dynamic arrays /
recursion
formula
> Ive had some help
> in getting this formula but Im not sure how to make the
array dynamic
> enough in vb.net to include Cn.....Xn in example 1.
> Example 1:
> User Input for four numbers 15,17,8,12,...N (Note: user can
input any
> amount
> of numbers)
> (15+17) /25/2 ->b1
> (b1+8) /25/2 ->b2
> (b2+12) /25/2 ->b3
> (b1+b2) /25/2 ->c1
> (c1 +b3)/25/2 ->c2
> (c1 +c2) /25/2 ->c3
We have an input of N numbers, so we can create a vector of N
real numbers
(assuming that the values arent simply integers), right?
I dont get the problem... Wheres the 
recursion?
(a1 + a2) / 25 / 2 -> b1
(b1 + a3) / 25 / 2 -> b2
(b2 + a4) / 25 / 2 -> b3
(b1 + b2) / 25 / 2 -> c1
(c1 + b3) / 25 / 2 -> c2
(c1 + c2) / 25 / 2 -> c3
The furst term of every new array (call the array x_n, and the
previous
x_(n-1) is:
x_n[1] = (x_(n-1)[1] + x_(n-1)[2]) / 25 / 2
that is, the sum of the furst two elements of the previous
array divided
by 25/2.
The second term is:
x_n[2] = (x_n[1] + x_(n-1)[3]) / 25 / 2
the sum of the furst element of the array and the third
element of the
previous array divided by 25/2
But this schema gets broken on the third element of the
array..
Maybe if you explain more clearly the algorithm someone may
be able to
help you.. Personally, I dont get the algorithm (maybe
because Im a
little tired, Ive been all the day at the university and
Ive been
working in the last two hours).
HTH
--
[ Andrea Spadaccini a.k.a. Lupino / Lupin85 - from Catania ]
[ MAIL: lupin85(at)email.it - ICQ: 91528290 - Linux 2.4.22 ]
[ Linux Registered User #313388 - Powered by Slackware 9.1 ]
===
Subject: Re: Recurision/Array problem sal Tia
(b1+b2) /25/2 ->c1
(c1 +b3)/25/2 ->c2
(c1 +c2) /25/2 ->c3
can u explain more whats going in there ^^^, i dont see how
ur proceeding
from one to another.
> Greets All,
> Im having a little trouble in creating dynamic arrays /
recursion
> formula Ive had some help
> in getting this formula but Im not sure how to make the
array dynamic
> enough in vb.net to include Cn.....Xn in example 1.
> Example 1:
> User Input for four numbers 15,17,8,12,...N (Note: user can
input any
> amount
> of numbers)
> (15+17) /25/2 ->b1
> (b1+8) /25/2 ->b2
> (b2+12) /25/2 ->b3
> (b1+b2) /25/2 ->c1
> (c1 +b3)/25/2 ->c2
> (c1 +c2) /25/2 ->c3
> What I have so far not funished though having problems.
> Store user input in array [a0,...,aN]
> Create new array [b0,...,bN]
> a[0]-->b[0]
> for k=1 to N {
> b[k]<--((a[k]+b[k-1])/25)/2
> TIA
> PS. im using vb.net but any programming help in any
language would be
> helpful