mm-1178
===
Subject: Re: JSH: Typical sci.math behavior
> As an experiment I posted in a rather straightforward and
succinct
> manner with the thread JSH: Equation has no memory and you
can look
> at the thread to see what happened.
Yes. Your ÔexperimentÕ, like everything else you \
attempt, was
a complete
failure.
--
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: Typical sci.math behavior
> And it probably doesnÕt matter to many of you, but I have
an interest
> in showing what sci.math is in actuality, so I continue my
> experiments.
sci.math is a Usenet newsgroup. You have evidence to the
contrary?
===
Subject: Re: JSH: Typical sci.math behavior
>>And it probably doesnÕt matter to many of you, but I have
an interest
>>in showing what sci.math is in actuality, so I continue my
>>experiments.
> sci.math is a Usenet newsgroup. You have evidence to the
contrary?
He believes we are Red Lectroids from Planet 10.
EVIL!!!! PURE AND SIMPLE FROM THE EIGHTH DIMENSION!!!!
Dale.
Whor\[CapitalThorn]n:
Where are we going?
Lectroids:
PLANET 10!!!!
Whor\[CapitalThorn]n:
When?
Lectroids:
REAL SOON!!!!!!!
===
Subject: Re: JSH: Typical sci.math behavior
===
>Subject: Re: JSH: Typical sci.math behavior
>Message-id: <3FDE6549.3010003@farir.comAnd it probably
doesnÕt matter to many of you, but I have an interest
>in showing what sci.math is in actuality, so I continue my
>experiments.
>> sci.math is a Usenet newsgroup. You have evidence to the
contrary?
>He believes we are Red Lectroids from Planet 10.
>EVIL!!!! PURE AND SIMPLE FROM THE EIGHTH DIMENSION!!!!
>Dale.
> Whor\[CapitalThorn]n:
> Where are we going?
> Lectroids:
> PLANET 10!!!!
> Whor\[CapitalThorn]n:
> When?
> Lectroids:
> REAL SOON!!!!!!!
Sealed with a curse, as sharp as a knife. Doomed is your
soul, and damned
is
your life.
--
Mensanator
Ace of Clubs
===
Subject: Re: a request to owners of Mathematics in Western
Culture by M.
Kline
I know you said after 1964... but my copy is from 1954. If it
can be of any
use to you, email me.
> Does anyone have handy, an edition of
> Mathematics in Western Culture by M.
> Kline, published after 1964? I am
> looking
> for such a person, who would be willing
> to look up a paragraph and compare some
> values in it to my 1964 edition.
> Rose Anne Leonard
> --
> --
> __________________________________________
> R.A. Leonard
> Ottawa Canada
> http://www.raleonard.com/
===
Subject: Re: a request to owners of Mathematics in Western
Culture by M.
Kline
through shelves, please stand down. I
have resolved the issue.
RA
> I know you said after 1964... but my copy is from 1954. If
it can be of
any
> use to you, email me.
> Does anyone have handy, an edition of
> Mathematics in Western Culture by M.
> Kline, published after 1964? I am
> looking
> for such a person, who would be willing
> to look up a paragraph and compare some
> values in it to my 1964 edition.
> Rose Anne Leonard
> --
> --
> __________________________________________
> R.A. Leonard
> Ottawa Canada
> http://www.raleonard.com/
--
__________________________________________
R.A. Leonard
Ottawa Canada
http://www.raleonard.com/
===
Subject: Re: JSH: Consider Dik Winter
One question being debated in this thread is:
Given algebraic integer functions a(x) and b(x), does
f(x) = gcd(a(x),b(x)) (*)
amount to a de\[CapitalThorn]nition of f(x)?
At best f(x) is de\[CapitalThorn]ned only up to multiplication by
an algebraic integer unit. For while we know that for every
x there exists a gcd of a(x) and b(x), we also know that
there are in\[CapitalThorn]nitely many (with the ratio of any two being
a unit). Furthermore the choice of gcd has to be made for
every x. If a and b are continuous, then one can
restrict f to continuous functions, however, this does
not specify f uniquely, even if the value f(0) is given [1].
In light of this, some would argue that (*) is
not a de\[CapitalThorn]nition of f(x) but rather a form of existence
proof. Alternately, one could think of (*) as de\[CapitalThorn]ning
an equivalence class of functions (under the
equivalence f-g iff f(x)=w(x)g(x) where w(x) is an
algebraic integer unit for all x). However, as there does
not seem to be an obvious way to choose a canonical
element of each class, there does not seem to be an
obvious way to associate a value to f(x) for every x.
In the end the question of whether (*) amounts to a de\[CapitalThorn]nition
is a question of de\[CapitalThorn]nition[2].
- William Hughes
[1] A similar problem occurs when trying to de\[CapitalThorn]ne a(x) to
be one of the roots of a cubic P whose coef\[CapitalThorn]cients are
continuous functions of x. At every x there
are three possible choices of a(x). However, here continuity
does serve to produce a unique function a(x) (there are some
minor
issues with possible double and triple roots).
[2] Pun intentional.
===
Subject: Re: JSH: Consider Dik Winter
>One question being debated in this thread is:
>Given algebraic integer functions a(x) and b(x), does
> f(x) = gcd(a(x),b(x)) (*)
>amount to a de\[CapitalThorn]nition of f(x)?
>At best f(x) is de\[CapitalThorn]ned only up to multiplication by
>an algebraic integer unit. For while we know that for every
>x there exists a gcd of a(x) and b(x), we also know that
>there are in\[CapitalThorn]nitely many (with the ratio of any two being
>a unit). Furthermore the choice of gcd has to be made for
>every x. If a and b are continuous, then one can
>restrict f to continuous functions, however, this does
>not specify f uniquely, even if the value f(0) is given [1].
>In light of this, some would argue that (*) is
>not a de\[CapitalThorn]nition of f(x) but rather a form of existence
>proof. Alternately, one could think of (*) as de\[CapitalThorn]ning
>an equivalence class of functions (under the
>equivalence f-g iff f(x)=w(x)g(x) where w(x) is an
>algebraic integer unit for all x). However, as there does
>not seem to be an obvious way to choose a canonical
>element of each class, there does not seem to be an
>obvious way to associate a value to f(x) for every x.
The use being given the the values of f has to do with
divisibility
properties. Although, as has been noted, the value of f is
only
well-de\[CapitalThorn]ned up to algebraic integer units, you may pick any
values
for two of them and adjust the third accordingly given the
equation
they are required to satisfy; since divisibility properties
are
invariant under multiplication by units, any choice of gcd
will do.
--
ItÕs not denial. IÕm just very selective \
about
what I accept as reality.
--- Calvin (Calvin and Hobbes)
Arturo Magidin
magidin@math.berkeley.edu
===
Subject: Re: JSH: Consider Dik Winter
...
>In light of this, some would argue that (*) is
>not a de\[CapitalThorn]nition of f(x) but rather a form of existence
>proof. Alternately, one could think of (*) as de\[CapitalThorn]ning
>an equivalence class of functions (under the
>equivalence f-g iff f(x)=w(x)g(x) where w(x) is an
>algebraic integer unit for all x). However, as there does
>not seem to be an obvious way to choose a canonical
>element of each class, there does not seem to be an
>obvious way to associate a value to f(x) for every x.
It does not matter, see below.
> The use being given the the values of f has to do with
divisibility
> properties. Although, as has been noted, the value of f is
only
> well-de\[CapitalThorn]ned up to algebraic integer units, you may pick any
values
> for two of them and adjust the third accordingly given the
equation
> they are required to satisfy; since divisibility properties
are
> invariant under multiplication by units, any choice of gcd
will do.
There is one point where divisibility properties do not play
a role:
w1(x).w2(x).w3(x) = 49.
But due to the way (in my de\[CapitalThorn]nition) in which \
gcdÕs are also
used as
divisors this will be true for *any* choice of units. So
while you
might debate whether the de\[CapitalThorn]nitions are indeed explicit, the
de\[CapitalThorn]nitions
are explicit enough for the purpose. But indeed it is more an
existence
proof.
--
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: JSH: Consider Dik Winter
Adjunct Assistant Professor at the University of Montana.
>...
>In light of this, some would argue that (*) is
>not a de\[CapitalThorn]nition of f(x) but rather a form of existence
>proof. Alternately, one could think of (*) as de\[CapitalThorn]ning
>an equivalence class of functions (under the
>equivalence f-g iff f(x)=w(x)g(x) where w(x) is an
>algebraic integer unit for all x). However, as there does
>not seem to be an obvious way to choose a canonical
>element of each class, there does not seem to be an
>obvious way to associate a value to f(x) for every x.
>It does not matter, see below.
> The use being given the the values of f has to do with
divisibility
> properties. Although, as has been noted, the value of f is
only
> well-de\[CapitalThorn]ned up to algebraic integer units, you may pick any
values
> for two of them and adjust the third accordingly given the
equation
> they are required to satisfy; since divisibility properties
are
> invariant under multiplication by units, any choice of gcd
will do.
>There is one point where divisibility properties do not play
a role:
> w1(x).w2(x).w3(x) = 49.
[Y]ou may pick any values for two of them and adjust the third
accordingly given the equation they are required to satisfy.
--
ItÕs not denial. IÕm just very selective \
about
what I accept as reality.
--- Calvin (Calvin and Hobbes)
Arturo Magidin
magidin@math.berkeley.edu
===
Subject: Re: JSH: Consider Dik Winter
...
>There is one point where divisibility properties do not play
a role:
> w1(x).w2(x).w3(x) = 49.
> [Y]ou may pick any values for two of them and adjust the
third
> accordingly given the equation they are required to satisfy.
Ah, but the interesting part is that in my formulaÕs the
adjustment is
never needed. You can chose any (equivalent) value for any
gcd around,
in the end the product will be 49.
--
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: JSH: Consider Dik Winter
...
> > If I didnÕt I do now. Delete those pages.
> >
> > ItÕs bad manners to keep after someone for failed math
arguments,
> > especially when they concede they failed.
>
> I keep them up because they show your discussing technique.
> However, now youÕve con\[CapitalThorn]rmed knowledge that \
they are past
attempts
> which I have told you failed.
Your discussing technique is still the same.
> I have also speci\[CapitalThorn]cally requested that you remove my failed
attempts
> from public view, and I think I can make the case that you
are
> *deliberately* acting in an attempt to humiliate me in
public.
But even *if* I remove those pages your failed attempts will
still be
I may even put up pages of your later failed attempts (like
the
current one). This is the last I will say about those pages.
> Snipping some more again, because it is irrelevant and
repetition.
> ...
> > > <Quote>
> > > w1(x) = gcd(5 a1(x) + 7, 49)
> > > > w2(x) = gcd(5 a2(x) + 7, 49)
> > > > w3(x) = gcd(5 b3(x) + 22, 49)
> > > > k(x) = w1(x).w2(x).w3(x)/49.
> > > > These three are easily shown to be algebraic integers
for
all x.
> > > > We factor as:
> > > > [k(x).(5 a1(x)+7)/w1(x)] * (5 a2(x)+7)/w2(x) * (5
> b3(x)+22)/w3(x) =
> > > > 300125 x^3 - 18375 x^2 - 360(x) + 22.
> > > </Quote>
> > >
> > > and I ask speci\[CapitalThorn]cally, is it your claim that you have
*de\[CapitalThorn]ned* the
> > > wÕs with those statements Dik Winter?
> >
> > I have de\[CapitalThorn]ned wÕs, and I have \
de\[CapitalThorn]ned a factorisation;
which was
at that
> >
> > Then I have a simple question as x is the only independent
variable,
> > what is w_1(2), w_2(2) or w_3(3)?
>
> I have no idea, and it is completely irrelevant. Perhaps
some program
> like maple, matlab or mathematica can calculate them. But
it is not
> interesting.
> No it canÕt. ItÕs not possible to get \
numerical values, even
> approximations with what youÕve given. \
ThatÕs because you
didnÕt
> explicitly de\[CapitalThorn]ne the wÕs.
I now understand you use a non-mathematical de\[CapitalThorn]nition of
ÔexplicitÕ.
What are the explicit values of a1(2), a2(2) and a3(2), and
which is
which and why? Can you *show* that a3(x)+22 is coprime to 7?
If so
how would you do that? Note that approximations are not
possible in
this case.
> You are behaving in a standard crank way Dik Winter.
I would think the shoe \[CapitalThorn]ts the other foot.
> A simple request, like giving an actual value for functions
you claim
> to have explicitly de\[CapitalThorn]ned is ignored with an excuse.
I have de\[CapitalThorn]ned them explicitly in the mathematical sense.
> A *rational* person might re-think their position, but the
crank is
> not rational.
Indeed. Like considering using terms in their mathematical
sense rather
than using own private meanings.
> > That is, can you or can you not give values--numerical
approximations
> > will do--for these functions you *claim* to have
EXPLICITLY
de\[CapitalThorn]ned?
>
> Numerical approximations will *never* do in such cases
because they
can
> not show whether a number is an algebraic integer or not.
Neither can
> approximations show division criteria. Moreover, the gcd
function
will
> not work when you use approximations.
> I didnÕt ask for excuses Dik Winter. Consider the
polynomial I use to
> de\[CapitalThorn]ne the aÕs, which is
> a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x).
> Numerical approximations can be given for those aÕs.
Yes, but which one is a1, which is a2 and which is a3? And
why?
Note that this is exactly the same ambiguity that you appear
to see
with the square root function on the reals. And those
approximations
tell us nothing about divisibility properties.
> Now then, do you Dik Winter *still* claim to have
explicitly de\[CapitalThorn]ned
> the wÕs?
Yup.
> > For readers note that IÕm impeaching Dik \
WinterÕs claims
by showing
> > that he canÕt produce an actual result, but depends on
vague
claims.
>
> What vague claims? There is only one vague claim above, and
that is
> that a gcd function exists in the algebraic integers, that
delivers a
> Then give a value for w_1(2), w_2(2), or w_3(2).
> Ask for help if you need it from others on the sci.math
newsgroup.
> The essential point here Dik Winter is that your claim of
having
> *explicitly* de\[CapitalThorn]ned those functions fail one of the most
basic tests
> in mathematics: the ability to give an actual result.
The essential point here is that you fail one of the most
basic concepts of mathematics. In mathematics for a function
to be
de\[CapitalThorn]ned explicitly it is *not* necessary to have the ability
to give
an actual result.
> IÕm curious if readers might chime in here on cases where \
a
value
> canÕt be determined for an explicitly de\[CapitalThorn]ned \
function where
there is
> only one independent variable, not even an approximate one.
Try the M.9abius function for suf\[CapitalThorn]ciently large argument. It
has an
explicit (in the mathematical sense) de\[CapitalThorn]nition, but can not \
be
calculated for most of its arguments. See:
<http://mathworld.wolfram.com/MoebiusFunction.html>.
> I \[CapitalThorn]gure there are such functions, but IÕd \
like some actual
examples
> to show what they look like, and *why* they arenÕt even
approximable.
Approximations make *no* sense in number theory .
> ThatÕs called reality testing Dik Winter.
Is it?
> result, unique upto units. However, *any* value returned by
the gcd
> function in each and every case will do because the results
will
> always satisfy my claim. And the claim that a gcd exists
was already
> Quit yapping and give a value for w_1(2), w_2(2), or
w_3(2). If
> youÕre not capable of making the calculation, ask for \
help.
Quit yapping and give a value for a1(2), a2(2) and a3(2) and
tell us
why the one is a1, the other is a2 and the third is a3.
> known to Dedekind (although this result is a bit beyond
highschool
> math).
>
> > v1(x) = gcd(5 a1(x) + 7, 49)
> > v2(x) = gcd(5 a2(x) + 7, 49)
> > v3(x) = gcd(5 b3(x) + 22, 49)
> > k3(x) = v1(x).v2(x).v3(x)/49
> > u3(x) = gcd(v3(x), k3(x))
> > w3(x) = v3(x)/u3(x)
> > k2(x) = k3(x)/u3(x)
> > u2(x) = gcd(v2(x), k2(x))
> > w2(x) = v2(x)/u2(x)
> > k1(x) = k2(x)/u2(x)
> > w1(x) = v1(x)/k2(x)
> ...
> > ItÕs as simple as asking you for values for your
supposedly
explicitly
> > de\[CapitalThorn]ned functions, so again, give w_1(2), w_2(2), or
w_3(2).
> >
> > Admitting defeat here is a simpler course Dik Winter.
>
> This is rubbish. A gcd function exists (known already to
Dedekind),
> but the result is not easily calculated. However, the above
is an
> Then ask for help. Or, why donÕt you *outline* how to make
the
> calculation?
Ok. Here an outline. Calculate a1(2), a2(2) and a3(2)
*exactly*
(approximations will not do). Use the de\[CapitalThorn]nitions above to
calculate all functions required. And there you are. You will
have *exact* values of w1(2), w2(2) and w3(2). Moreover, these
exact values will give you algebraic integers all around.
> Any idiot can claim that something is not easily calculated.
Any idiot can claim that approximations will do when doing
number
theory.
> existence proof of the functions w1 to w3. If you
acknowledge
> existence only when actual numbers can be provided, I allow
defeat.
> Readers note that Dik Winter canÕt give values to support
his claim of
> explicitly de\[CapitalThorn]ning the wÕs.
Note also that in mathematics it is *not* necessary to give
values for
explicitly de\[CapitalThorn]ned functions.
> I will not give actual numbers, nor will I go to the
rigmarole to
> provide actual numbers. Those are completely irrelevant.
> Readers note the typical crank insouciance at being called
on to
> produce and failing.
Ok James. Produce a1(2), a2(2) and a3(2) and tell us *why*
one of
them is a1, the second is a2 and the third is a3. And pray
show
us also that (5 b3(2) + 22) is coprime to 7.
> I may just as well ask you what a(2)/7 is, and note: *not*
an
> approximation, because that will *not* show whether it is an
> algebraic integer or not. In number theory (and other parts
of
> That is a_2(x)/7 the way IÕm currently writing it, and my
point is
> that itÕs NOT an algebraic integer in general for \
algebraic
integer x.
Yes, and it has already been proven suf\[CapitalThorn]ciently often that
that is
indeed not the case. On the other hand you have still *not*
proven
that (5 b3(x) + 22) is coprime to 7; it has been shown
already many
times that that is *not* true.
> However, I *can* give an appoximate value for a_2(x)/7, for
some
> integer x.
That is *completely irrelevant*. Approximate values do not
count in
number theory. For instance, is 0.00000000082592268740... a
unit?
I have no idea, but I know that it is a good approximation
for some
unit. Is 7 close to a unit? What is the algebraic integer unit
closest to 7?
> mathematics) there are many functions that are well-de\[CapitalThorn]ned
but
> where it is not easy to plug in an argument and get a
result.
>
> My point is that Dik Winter made a speci\[CapitalThorn]c claim: that he
had
> *explicit* de\[CapitalThorn]nitions for functions he calls \
wÕs, but when
tasked to
> give even approximate values for his functions at x=2, he
fails.
> And importantly, considering my case that Dik Winter is
behaving as a
> crank, he fails to acknowledge his failure with any sign of
rational
> humbleness or concern.
You are using a non-mathematical version of the word explicit.
> > Pray show in *what* way the functions w I de\[CapitalThorn]ned above
are *not*
> > algebraic integer functions.
> >
> > ThatÕs not necessary. To impeach your crank claim for \
the
wÕs, I
> > simply ask for a value at x=2.
>
> Yes, you really are a crank. But if you really want one,
here some
> Now instead he insults me and gives bogus values.
> approximations:
> w1(2) = 5.13425...
> w2(2) = 9.81346...
> w3(2) = 0.97252...
> If you do not believe that, just try to disprove it.
> Are you claiming that *those* are approximations to your
wÕs for x=2?
The irony is that you do not know, and neither I do know.
Whether they
are good approximations or not entirely depends on the
density of
algebraic integer units in the reals. And that is why
approximations
do not count.
> So being a smart-aleck doesnÕt work in mathematics because
next I ask
> how you made the calculation. Then someone else can check
it as well.
I pulled them out of my hat. Yes, I admit that. The problem
with
approximations is that when the density of units in the
algebraic
integers favour me, the above approximations are right. As I
am
inclined to think that the algebraic integer units are pretty
dense,
they may even be good approximations (as would any
approximation you
wish to state). And that is why number theory is *not*
satis\[CapitalThorn]ed
with approximations.
> In mathematics there are steps Dik Winter. Those steps are
> replicable.
Yes, replicate the step. Pull some approximations out of your
hat
that multiply to 49, and show that they are *not* good
approximations
to the wÕs.
> You canÕt escape by just tossing out numbers, as next you
have to
> justify them.
You are harping on something. Your insistence on
approximation has
*no* value at all in number theory. Approximations are
worthless.
You need exact expressions.
> > Pray *show* that they are not algebraic integer
functions. For
instance
> > w3(x) as de\[CapitalThorn]ned above, in the current \
de\[CapitalThorn]nition. v1(x),
v2(x)
and
> > v3(x) are all algebraic integer functions. k3(x) is an
algebraic
integer
> > function. And so w3(x) = v3(x)/gcd(k3(x), v3(x)) is an
algebraic
integer
> > function.
> >
> > How do you know? You canÕt even give any values or even
approximate
> > values for any of them, yet claim to have explicitly
de\[CapitalThorn]ned them.
>
> How do you show something is an algebraic integer by giving
an
> approximation? By the very de\[CapitalThorn]nition of the gcd function,
those
> are all algebraic integer functions. You appear to be
claiming here
> that the gcd function does not exist. Please take that up
with
> Dedekind, not with me.
> Your failure is not DedekindÕs.
What failure? Failure to calculate the gcdÕs? Pray provide \
me
with
expressions for a1(2), a2(2) and a3(2) so that I can even try
to start.
And, please, no approximations.
> Right now IÕm testing your claims and itÕs \
abundantly clear
that you
> have no clue how to make calculations with your wÕs.
It is abundantly clear that you have no clue how to go about
in
number theory.
> Your de\[CapitalThorn]nitions are useless.
So are your de\[CapitalThorn]nitions.
--
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: JSH: Consider Dik Winter
> ...
> > approximations:
> > w1(2) = 5.13425...
> > w2(2) = 9.81346...
> > w3(2) = 0.97252...
> > If you do not believe that, just try to disprove it.
> >
> > Are you claiming that *those* are approximations to your
wÕs for x=2?
> The irony is that you do not know, and neither I do know.
Whether they
> are good approximations or not entirely depends on the
density of
> algebraic integer units in the reals. And that is why
approximations
> do not count.
You probably knew this already [if it is right], though if
you did, I donÕt see why you would say ... neither I do \
know..
Theorem: Algebraic integer units are dense in the reals.
Proof: First show that algebraic integers are dense in the
reals. This is clear from consideration of the function
s(n) = sqrt(n) - [sqrt(n)],
where [.] denotes the greatest-integer function.
Next, consider the function
f(x) = sqrt(x + 1) - sqrt(x)
where x > 0. This is a continuous function from (0, 1] to
(0, 1]; f(0) = 1 and f(x) --> 0 as x --> in\[CapitalThorn]nity. Restricting
f(x) to the algebraic integers and using the fact that the
algebraic integers are dense in the reals shows that the
range of f(a), a = algebraic integer, is dense in (0, 1].
f(a) is a unit for each algebraic integer a since
f(a) * (sqrt(a + 1) + sqrt(a)) = 1.
Finally, consider the function g(x) = 1/x: This is continuous
on (0, 1] and has range (0, in\[CapitalThorn]nity]. If u is a unit,
certainly
g(u) = 1/u is a unit also. Therefore the set U = {g(u), u an
alg.
int. unit in (0, 1]} is dense in the positive real numbers.
Thus U + (-U) is dense in the reals.
Nora B.
===
Subject: Re: JSH: Consider Dik Winter
...
> > approximations:
> > w1(2) = 5.13425...
> > w2(2) = 9.81346...
> > w3(2) = 0.97252...
> > If you do not believe that, just try to disprove it.
> >
> > Are you claiming that *those* are approximations to your
wÕs for
x=2?
>
> The irony is that you do not know, and neither I do know.
Whether
they
> are good approximations or not entirely depends on the
density of
> algebraic integer units in the reals. And that is why
approximations
> do not count.
> You probably knew this already [if it is right], though if
> you did, I donÕt see why you would say ... neither I do
know..
To be honest, no I did not know whether the algebraic integer
units were
--
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: JSH: Consider Dik Winter
...
> > > approximations:
> > > w1(2) = 5.13425...
> > > w2(2) = 9.81346...
> > > w3(2) = 0.97252...
> > > If you do not believe that, just try to disprove it.
> > >
> > > Are you claiming that *those* are approximations to
your wÕs for
x=2?
> >
> > The irony is that you do not know, and neither I do know.
Whether
they
> > are good approximations or not entirely depends on the
density of
> > algebraic integer units in the reals. And that is why
approximations
> > do not count.
> >
> > You probably knew this already [if it is right], though if
> > you did, I donÕt see why you would say ... neither I do
know..
> To be honest, no I did not know whether the algebraic
integer units were
As a corollary to the denseness of the algebraic integer
units, the
values I gave are indeed proper approximations. So:
*approximations do not count*.
--
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: JSH: Consider Dik Winter
> Theorem: Algebraic integer units are dense in the reals.
> [Nice short proof snipped.]
Very nice!
The units you construct seem to be algebraic integers of
degree 8
over the rationals. So a natural question is:
What is the smallest integer n such that the algebraic
integer units
of degree n are dense in the reals?
-- Dot.
===
Subject: Re: JSH: Consider Dik Winter
> What is the smallest integer n such that the algebraic
integer units
> of degree n are dense in the reals?
Actually, this is not hard. Let K be a totally real cubic
\[CapitalThorn]eld.
Then the unit group of (the ring of integers of) K is a free
abelian
group on two generators -- call them u and v. It is not hard
to
see that under any embedding of K into the reals, the group
generated
by u and v is dense.
So not only will n=3 do, we only need to take the units from
one \[CapitalThorn]eld!
-- Dot.
===
Subject: Re: JSH: Consider Dik Winter
> What is the smallest integer n such that the algebraic
integer units
> of degree n are dense in the reals?
> Actually, this is not hard. Let K be a totally real cubic
\[CapitalThorn]eld.
> Then the unit group of (the ring of integers of) K is a
free abelian
> group on two generators -- call them u and v. It is not
hard to
> see that under any embedding of K into the reals, the group
generated
> by u and v is dense.
> So not only will n=3 do, we only need to take the units
from one \[CapitalThorn]eld!
> -- Dot.
I think what you are considering here are numbers of the form
u^n * v^m,
where n and m are integers (not necessarily positive). While u
and v themselves may be roots of a cubic polynomial of the
form
x^3 + a*x^2 + b*x + 1,
where a and b are integers, the same is not necessarily true
of
u^n and v^m.
Nora B.
===
Subject: Re: JSH: Consider Dik Winter
>
> What is the smallest integer n such that the algebraic
integer units
> of degree n are dense in the reals?
>
> Actually, this is not hard. Let K be a totally real cubic
\[CapitalThorn]eld.
> Then the unit group of (the ring of integers of) K is a
free abelian
> group on two generators -- call them u and v. It is not
hard to
> see that under any embedding of K into the reals, the group
generated
> by u and v is dense.
>
> So not only will n=3 do, we only need to take the units
from one \[CapitalThorn]eld!
>
> -- Dot.
> I think what you are considering here are numbers of the
form
> u^n * v^m,
> where n and m are integers (not necessarily positive).
While u
> and v themselves may be roots of a cubic polynomial of the
> form
> x^3 + a*x^2 + b*x + 1,
> where a and b are integers, the same is not necessarily
true of
> u^n and v^m.
> Nora B.
Arturo points out that u^n * v^m is of degree 1 or 3, so \
DotÕs
argument looks OK.
Nora B.
===
Subject: Re: JSH: Consider Dik Winter
>What is the smallest integer n such that the algebraic
integer units
>of degree n are dense in the reals?
>>Actually, this is not hard. Let K be a totally real cubic
\[CapitalThorn]eld.
>>Then the unit group of (the ring of integers of) K is a
free abelian
>>group on two generators -- call them u and v. It is not
hard to
>>see that under any embedding of K into the reals, the group
generated
>>by u and v is dense.
>>So not only will n=3 do, we only need to take the units
from one \[CapitalThorn]eld!
>>-- Dot.
> I think what you are considering here are numbers of the
form
> u^n * v^m,
> where n and m are integers (not necessarily positive).
While u
> and v themselves may be roots of a cubic polynomial of the
> form
> x^3 + a*x^2 + b*x + 1,
> where a and b are integers, the same is not necessarily
true of
> u^n and v^m.
> Nora B.
Hunh?
An integer u is a unit if there is another integer u^{-1}
such that u
u^{-1} = 1. If u and v are units, then obviously (u^n v^m) is
a unit,
as (u^n v^m)(u^{-n} v^{-m})=1. The fact that tne minimal
polynomial of
U^n v^m has one as the constant term follows from the theorem
on the
minimal polynomials of algebraic integer units.
===
Subject: Re: JSH: Consider Dik Winter
>>What is the smallest integer n such that the algebraic
integer units
>>of degree n are dense in the reals?
>Actually, this is not hard. Let K be a totally real cubic
\[CapitalThorn]eld.
>Then the unit group of (the ring of integers of) K is a free
abelian
>group on two generators -- call them u and v. It is not hard
to
>see that under any embedding of K into the reals, the group
generated
>by u and v is dense.
>So not only will n=3 do, we only need to take the units from
one \[CapitalThorn]eld!
>-- Dot.
>> I think what you are considering here are numbers of the
form
>> u^n * v^m,
>> where n and m are integers (not necessarily positive).
While u
>> and v themselves may be roots of a cubic polynomial of the
>> form
>> x^3 + a*x^2 + b*x + 1,
>> where a and b are integers, the same is not necessarily
true of
>> u^n and v^m.
>> Nora B.
>Hunh?
>An integer u is a unit if there is another integer u^{-1}
such that u
>u^{-1} = 1. If u and v are units, then obviously (u^n v^m)
is a unit,
>as (u^n v^m)(u^{-n} v^{-m})=1. The fact that tne minimal
polynomial of
>U^n v^m has one as the constant term follows from the
theorem on the
>minimal polynomials of algebraic integer units.
No, NoraÕs point is that while each of u and v are roots of
cubics, it
might not necessarily follow that a product u^n*v^m is also
the root
of a cubic.
However, what Nora missed is that u and v are generators of
the free
part of the unit group of the ring of the ring of integers of
K;
therefore, both u and v are in the same K, and since [K:Q]=3,
any
product u^n*v^m lies in K and thus is of degree either 1 or 3
over Q.
--
ItÕs not denial. IÕm just very selective \
about
what I accept as reality.
--- Calvin (Calvin and Hobbes)
Arturo Magidin
magidin@math.berkeley.edu
===
Subject: Re: JSH: Consider Dik Winter
> Theorem: Algebraic integer units are dense in the reals.
>
> [Nice short proof snipped.]
> Very nice!
> The units you construct seem to be algebraic integers of
degree 8
> over the rationals. So a natural question is:
> What is the smallest integer n such that the algebraic
integer units
> of degree n are dense in the reals?
Certainly not n = 2. n = 3 would be a possibility.
===
Subject: Re: JSH: Consider Dik Winter
>
> Theorem: Algebraic integer units are dense in the reals.
>
> [Nice short proof snipped.]
>
> Very nice!
>
> The units you construct seem to be algebraic integers of
degree 8
> over the rationals. So a natural question is:
>
> What is the smallest integer n such that the algebraic
integer units
> of degree n are dense in the reals?
> Certainly not n = 2. n = 3 would be a possibility.
Yes, I think that works. It is enough to show that the
algebraic
integer units of degree 3 are dense in the positive reals -
so it
suf\[CapitalThorn]ces to prove that for any x,y with 0 < x < y then there
exists
integer a,b such that:
x^3 + ax^2 + bx + 1 < 0
y^3 + ay^2 + by + 1 > 0
as then the cubic equation has a root in (x,y).
To prove the claim, it suf\[CapitalThorn]ces to prove that for any K > 0 we
can
\[CapitalThorn]nd integer a,b such that ax+b < -K and ay+b > K (because
then we can
certainly ensure ax+b < -(x^3+1)/x and ay+b > -(y^3+1)/y,
which is
what is required.) This is easy - take a such that a(y-x) >
2K+1 then
select integer b such that K < ay+b <= K+1.
Michael
===
Subject: Re: JSH: Consider Dik Winter
>
> Theorem: Algebraic integer units are dense in the reals.
>
> [Nice short proof snipped.]
>
> Very nice!
>
> The units you construct seem to be algebraic integers of
degree 8
> over the rationals. So a natural question is:
let us see. First Nora starts with sqrt(n) - entier(sqrt(n))
to show
the algebraic integers are dense. This dense set consists
entirely of
algebraic integers of degree 2. Next she uses sqrt(x + 1) -
sqrt(x)
as units, with x from the dense set, this is the difference
of two
algebraic integers of degree 4, so the degree is indeed at
most 8.
> What is the smallest integer n such that the algebraic
integer units
> of degree n are dense in the reals?
> Certainly not n = 2. n = 3 would be a possibility.
I have looked at the units of quadratic \[CapitalThorn]elds. They are not
dense
themselves, but I think that products of such units might
yield a dense
set. If so 4 might be it. The study of cubic \[CapitalThorn]elds is a bit
underdeveloped.
--
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: JSH: Consider Dik Winter
...
> > The units you construct seem to be algebraic integers of
degree 8
> > over the rationals. So a natural question is:
> let us see. First Nora starts with sqrt(n) -
entier(sqrt(n)) to show
> the algebraic integers are dense. This dense set consists
entirely of
> algebraic integers of degree 2. Next she uses sqrt(x + 1) -
sqrt(x)
> as units, with x from the dense set, this is the difference
of two
> algebraic integers of degree 4, so the degree is indeed at
most 8.
That was too fast. The degree is at most 16. (The degree of
the sum
or product of two algebraic integers is at most the *product*
of the
individual degrees.)
--
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: JSH: Consider Dik Winter
> ...
> > > The units you construct seem to be algebraic integers
of degree 8
> > > over the rationals. So a natural question is:
> >
> > let us see. First Nora starts with sqrt(n) -
entier(sqrt(n)) to show
> > the algebraic integers are dense. This dense set consists
entirely of
> > algebraic integers of degree 2. Next she uses sqrt(x + 1)
- sqrt(x)
> > as units, with x from the dense set, this is the
difference of two
> > algebraic integers of degree 4, so the degree is indeed
at most 8.
> That was too fast. The degree is at most 16. (The degree of
the sum
> or product of two algebraic integers is at most the
*product* of the
> individual degrees.)
Maybe it was too fast, but the answer was still correct :)
x is degree 2. Sqrt(x) is quadratic *over Q(x)*. And
sqrt(x+1) is
also quadratic over Q(x). So sqrt(x+1)-sqrt(x) is quartic
over Q(x),
and therefore degree-8 over Q.
-- Dot.
===
Subject: Re: JSH: Consider Dik Winter
[cut]
> > IÕm curious if readers might chime in here on cases \
where
a value
> > canÕt be determined for an explicitly \
de\[CapitalThorn]ned function
where there is
> > only one independent variable, not even an approximate
one.
> Try the M.9abius function for suf\[CapitalThorn]ciently large argument.
It has an
> explicit (in the mathematical sense) de\[CapitalThorn]nition, but can not
be
> calculated for most of its arguments. See:
> <http://mathworld.wolfram.com/MoebiusFunction.html>.
I donÕt think this is an example. Could you clarify. It
appears
that you are saying that the value of the M.9abius function
cannot
be calculated for a large value of its argument since it would
take a long time (like centuries) to factor the argument. But,
that criteria would apply to most functions, even ones like
f(x) = x*x.
I think even James would say that the values of the M.9abius
function
can be determined for all values of its arguments.
I still think there are too many different interpretations of
exactly what is being sought. It would help to clarify what
is desired. Posters can give examples of what they think is
meant and maybe a consensus can be obtained. But, I guess
James would ultimately have to say what he is seeking.
My example would be an explicit function that is
non-computable.
IÕm not aware of any, but my understanding is that do \
exists.
Usually, one invokes the impossibility of a universal Turing
machine to construct such a function (I think).
satisfactory. You could have an explicitly de\[CapitalThorn]ned function
with a non-computable value, yet still give approximate
values for the answer. James seems to be making the false
assumption that if you can give approximate values to any
degree of accuracy, then you can compute the actual value
of the function for that particular argument. The standard
examples involve undecidable propostions. For example, for
sake of agrument, assume that FermatÕs last theorem were
actually undecidable. Let r = 1/2. De\[CapitalThorn]ne the number y
to be a_3*r^3 + a_4*r^4 + a_5*r^5 + ..., where a_n = 0
if x^m+y^m = z^m has no solutions for all m<=n and
for all x,y,z <= n, else a_n = 1. Assuming FermatÕs last
theorem was undecidable, then y would be able to be
calculated to any desired degree of accuracy, but
one could not say whether y was actually equal to zero
or not.
-- Bill Hale
===
Subject: Re: JSH: Consider Dik Winter
> [cut]
> > IÕm curious if readers might chime in here on cases \
where
a value
> > canÕt be determined for an explicitly \
de\[CapitalThorn]ned function
where there
is
> > only one independent variable, not even an approximate
one.
>
> Try the M.9abius function for suf\[CapitalThorn]ciently large argument.
It has an
> explicit (in the mathematical sense) de\[CapitalThorn]nition, but can not
be
> calculated for most of its arguments. See:
> <http://mathworld.wolfram.com/MoebiusFunction.html>.
> I donÕt think this is an example. Could you clarify. It
appears
> that you are saying that the value of the M.9abius function
cannot
> be calculated for a large value of its argument since it
would
> take a long time (like centuries) to factor the argument.
But,
> that criteria would apply to most functions, even ones like
> f(x) = x*x.
But I think it is quite similar. It is possible to express
the aÕs in
an exact way. I think it is also possible to calculate the
gcdÕs that
are used in my expressions; but that is hard work.
--
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: JSH: Consider Dik Winter
> [cut]
> > > IÕm curious if readers might chime in here on cases
where a value
> > > canÕt be determined for an explicitly \
de\[CapitalThorn]ned function
where there
is
> > > only one independent variable, not even an approximate
one.
> >
> > Try the M.9abius function for suf\[CapitalThorn]ciently large argument.
It has
an
> > explicit (in the mathematical sense) de\[CapitalThorn]nition, but can
not be
> > calculated for most of its arguments. See:
> > <http://mathworld.wolfram.com/MoebiusFunction.html>.
> >
> > I donÕt think this is an example. Could you clarify. It
appears
> > that you are saying that the value of the M.9abius
function cannot
> > be calculated for a large value of its argument since it
would
> > take a long time (like centuries) to factor the argument.
But,
> > that criteria would apply to most functions, even ones
like
> > f(x) = x*x.
> But I think it is quite similar. It is possible to express
the aÕs in
> an exact way. I think it is also possible to calculate the
gcdÕs that
> are used in my expressions; but that is hard work.
There are no solutions for the wÕs you gave unless the cubic
de\[CapitalThorn]ning
the aÕs is reducible over Q *and* a_1(x)/7, and a_2(x)/7 are
algebraic
integers, and then two of the wÕs equal 7 while the other
equals 1.
If a_1(x)/7 *and* a_2(x)/7 are not algebraic integers, then
your wÕs
donÕt exist.
So no calculation is necessary. Still I \[CapitalThorn]nd it interesting
that you
STILL tried to rely on the idea that itÕs hard to get values
for
functions you claim you explicitly de\[CapitalThorn]ned.
===
Subject: Re: JSH: Consider Dik Winter
...
> > So no calculation is necessary. Still I \[CapitalThorn]nd it
interesting that you
> > STILL tried to rely on the idea that itÕs hard to get
values for
> > functions you claim you explicitly de\[CapitalThorn]ned.
> Strange. Finding the gcd of two algebraic integers *is*
hard.
As an easy exercise for you, try gcd(3, sqrt(7) - 1).
--
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: JSH: Consider Dik Winter
> ...
> > > So no calculation is necessary. Still I \[CapitalThorn]nd it
interesting that
you
> > > STILL tried to rely on the idea that itÕs hard to get
values for
> > > functions you claim you explicitly de\[CapitalThorn]ned.
> >
> > Strange. Finding the gcd of two algebraic integers *is*
hard.
> As an easy exercise for you, try gcd(3, sqrt(7) - 1).
I donÕt think is going to answer this.
I will give my method below, which is pretty ad hoc,
on calcuting the gcd. After I calculated it, I tried
to check it with the Pari computer program, but it did
not have a gcd function for algebraic integer input.
I would think that methods would have been worked out.
Maybe someone can check if Mathematica or Maple or something
else can compute the gcd.
I want to compute gcd(3, sqrt(7) - 1) in the ring of
algebraic integers.
First, I work in the ring A of integers of the \[CapitalThorn]eld
Q[sqrt(7)], which
is equal to the ring Z[sqrt(7)]. I do this since I know more
about
that ring than the ring of algebraic integers, since it has a
\[CapitalThorn]nite
basis.
Next, I note that 3 splits in A: 3 = (sqrt(7) + 2)*(sqrt(7) -
2),
where the two factors on the right side are primes in A.
I note that (sqrt(7) - 1)*(sqrt(7) + 1) = 3 * 2.
Also, I note that 2 rami\[CapitalThorn]es in A: <2> = <2, 1 + sqrt(7)>^2,
where the brackets <..> denotes the ideal generated by the
numbers inside. This doesnÕt seem to lead anywhere.
Thus, I have:
(sqrt(7) - 1)*(sqrt(7) + 1) = (sqrt(7) + 2) * (sqrt(7) - 2) *
2.
The \[CapitalThorn]rst two factors on the right side are primes in A.
I try to use one of them to divide into (sqrt(7) - 1),
hoping that I may \[CapitalThorn]nd a common factor of 3 and (sqrt(7) - 1).
Trying (sqrt(7) + 2), I get:
(sqrt(7) - 1) / (sqrt(7) + 2) = (sqrt(7) - 1) * (sqrt(7) - 2)
/ 3
which turns out to be equal to 3 - sqrt(7), which is in A.
Thus, gcd(3, sqrt(7) - 1) = (sqrt(7) + 2) * gcd(sqrt(7) - 2,
3 - sqrt(7)).
But, gcd(sqrt(7) - 2, 3 - sqrt(7)) = 1 since 1*(sqrt(7) - 2) +
1 * (3 - sqrt(7)) = 1.
Hence, gcd(3, sqrt(7) - 1) = sqrt(7) + 2.
-- Bill Hale
===
Subject: Re: JSH: Consider Dik Winter
> I donÕt think is going to answer this.
I am pretty sure...
> Hence, gcd(3, sqrt(7) - 1) = sqrt(7) + 2.
I had [sqrt(14) - sqrt(2)]/2, but that is not a real problem,
their
quotient is a unit. Yours is nicer because it stays in
Q[sqrt(7)].
--
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: JSH: Consider Dik Winter
> ...
> > But I think it is quite similar. It is possible to
express the aÕs
in
> > an exact way. I think it is also possible to calculate
the gcdÕs
that
> > are used in my expressions; but that is hard work.
> >
> > There are no solutions for the wÕs you gave unless the
cubic de\[CapitalThorn]ning
> > the aÕs is reducible over Q *and* a_1(x)/7, and a_2(x)/7
are algebraic
> > integers, and then two of the wÕs equal 7 while the \
other
equals 1.
> Well, Dedekind has shown the gcdÕs exist, and so the \
wÕs
exist. What
> is the problem?
At risk of being accused by Harris that I am trying to read
his
mind, I would say: The problem is that still, after all this
time,
he cannot believe that the wÕs can be anything other than 7,
7,
and 1. (The lure of those Ôconstant termsÕ is \
just too great;
Harris
has always relied almost entirely on visual inspection in
thinking about
factorization.) He knows also however that that leads to a
contradiction.
He thinks that eliminates all the possibilities. He doesnÕt
trust
DedekindÕs gcd function; it is a mysterious black box.
Previously he believed that a1(x)/7 was necessarily an
algebraic
integer, and this was a key to his proof of FLT and his
contention
that there is an error in core mathematics. He now recognizes
that this is not true. That leaves him with no proof of FLT,
but
potentially still a problem with core mathematics. Now you
tell
him that gcd(a1(x), 49) must exist because of a theorem of
Dedekind,
and he knows that this gcd cannot be equal to 7. Formerly he
thought
the error in core mathematics is due to an error in the
de\[CapitalThorn]nition
of algebraic integers. That simply makes no sense. Logically
now
he must conclude (but hasnÕt, yet) that \
DedekindÕs theorem
must be
wrong, and that other mathematicians have overlooked this for
the
past 120+ years. Since the proof of DedekindÕs theorem is
deep and
dif\[CapitalThorn]cult it is going to be hard to persuade him that he is
wrong.
Bottom line: no end in sight.
Nora B.
> > If a_1(x)/7 *and* a_2(x)/7 are not algebraic integers,
then your wÕs
> > donÕt exist.
> What nonsense. I have given *de\[CapitalThorn]nitions*, and with those
de\[CapitalThorn]nitions
> the wÕs do exist and satisfy the requirements. If you now
state that
> the wÕs do not necessarily exist you are actually claiming
that in the
> algebraic integers there is no gcd function. Is that your
claim? Was
> Dedekind wrong?
> > So no calculation is necessary. Still I \[CapitalThorn]nd it
interesting that you
> > STILL tried to rely on the idea that itÕs hard to get
values for
> > functions you claim you explicitly de\[CapitalThorn]ned.
> Strange. Finding the gcd of two algebraic integers *is*
hard.
===
Subject: Re: JSH: Consider Dik Winter
...
> But I think it is quite similar. It is possible to express
the aÕs in
> an exact way. I think it is also possible to calculate the
gcdÕs that
> are used in my expressions; but that is hard work.
> There are no solutions for the wÕs you gave unless the
cubic de\[CapitalThorn]ning
> the aÕs is reducible over Q *and* a_1(x)/7, and a_2(x)/7
are algebraic
> integers, and then two of the wÕs equal 7 while the other
equals 1.
Well, Dedekind has shown the gcdÕs exist, and so the \
wÕs
exist. What
is the problem?
> If a_1(x)/7 *and* a_2(x)/7 are not algebraic integers, then
your wÕs
> donÕt exist.
What nonsense. I have given *de\[CapitalThorn]nitions*, and with those
de\[CapitalThorn]nitions
the wÕs do exist and satisfy the requirements. If you now
state that
the wÕs do not necessarily exist you are actually claiming
that in the
algebraic integers there is no gcd function. Is that your
claim? Was
Dedekind wrong?
> So no calculation is necessary. Still I \[CapitalThorn]nd it interesting
that you
> STILL tried to rely on the idea that itÕs hard to get
values for
> functions you claim you explicitly de\[CapitalThorn]ned.
Strange. Finding the gcd of two algebraic integers *is* hard.
--
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: JSH: Consider Dik Winter
> I still think there are too many different interpretations
of
> exactly what is being sought. It would help to clarify what
> is desired. Posters can give examples of what they think is
> meant and maybe a consensus can be obtained. But, I guess
> James would ultimately have to say what he is seeking.
I think the problem is that James, like every successful
crank, now
specialises on producing false statements that take some
effort to
refute.
It starts with using mathematical terms in ways that no
mathematician
uses them - probably every of his statements is true if you
bend your
de\[CapitalThorn]nitions suf\[CapitalThorn]ciently (and differently from one \
statement to
the
next).
Then he moved into the \[CapitalThorn]eld of algebraic integers that many
readers
here donÕt understand very well (James understands less of \
it
than most,
but that is not the point), and even if you do understand it
many things
are just plain hard work and often counterintuitive. That
makes it a lot
easier to produce wrong statements than refuting them.
I think what we should do is use the criterion that will be
used if you
have to pass a maths test: If you cannot express your
thoughts clearly
enough so that anyone understands them, then you have failed.
By that
criterion, is an absolute failure. With one exception he
has never produced anything that wasnÕt complete nonsense.
Next thing he will complain that I called him a failure
because that is
not nice...
===
Subject: Re: JSH: Consider Dik Winter
> Too bad I left him an excuse as x=0 mod 7 is a
fascinatingly simple
> way to blow holes through his claims!!!
In general, the fact that x=0 mod 7 can tell you something
about
the divisiblity of a(x) only if a(x) is a polynomial. In the
present case
the a_iÕs are not polynomials so this argument does not
apply. I
suggest you produce your independent veri\[CapitalThorn]cation.
- William Hughes
===
Subject: Re: JSH: Rationality test, math
> Well, to be honest, heÕs gone at you many many times \
before
and after;
> I think the proliferation of threads with another posterÕs
name is
> more of an indication of a desire to get rid of another
critic
> through sheer obnoxiousness.
I do not think I am going to offer him something similar.
> (But youÕd be surprised how much more time I have to work
now... (-: )
I hope it is not for things I am posting? ;-).
--
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: JSH: Rationality test, math
> Given, where x is in the ring of algebraic integers, IÕve
shown the
> factorization
> (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) =
> 49(300125 x^3 - 18375 x^2 - 360 x + 22)
> where b_3(x) = a_3(x) - 3 and the aÕs are roots of
> a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x)
> so when x=0, a_1(0) = a_2(0) = b_3(0) = 0.
What are the values of Ôa_1(x)Õ, \
Ôa_2(x)Õ, Ôa_3(x)Õ \
and
Ôb_3(x)? Apparently
you do
not accept any criciticism of your argument unless the critic
is able to
explicitly post speci\[CapitalThorn]c values for the variables. You should
do so now. Put
up,
or SHUT UP!
--
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: Rationality test, math
In sci.logic, C. Bond
<cbond@ix.netcom.com>
<3FDE660A.34B75881@ix.netcom.com>:
>> Given, where x is in the ring of algebraic integers, IÕve
shown the
>> factorization
>> (5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) =
>> 49(300125 x^3 - 18375 x^2 - 360 x + 22)
>> where b_3(x) = a_3(x) - 3 and the aÕs are roots of
>> a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x)
>> so when x=0, a_1(0) = a_2(0) = b_3(0) = 0.
> What are the values of Ôa_1(x)Õ, \
Ôa_2(x)Õ, Ôa_3(x)Õ \
and
Ôb_3(x)?
Obviously the aÕs are the roots of his given equation
(which I was able to verify, strangely enough, without
explicitly computing them), which basically means one can
compute them as a function of x.
It would take a little work to explicitly do so (cubics
can be solved, though), and it is far from clear which
attribute of these values JSH \[CapitalThorn]nds important. I can
tell you that, if they are algebraic integers at all,
they are not generally divisible by 7 -- substitute a =
7c in his third equation and divide by 7^3 and youÕll see
that readily, as one gets
c^3 + 3*(-1/7+7*x)*c^2 - 343*x^3 + 21*x^2 - (3/7)*x.
If one restricts x to the rationals thereÕs no way to
massage this equation to one with integer coef\[CapitalThorn]cients.
However, I donÕt know how to prove a wider conclusion:
that there exists no algebraic number x such that the
c roots are all algebraic integers, which means all of
the aÕs are algebraic integers and divisible by 7.
What this all means for JSHÕs grander argument (which led to
this equation), IÕve no idea at present.
[rest snipped]
--
#191, ewill3@earthlink.net
ItÕs still legal to go .sigless.
===
Subject: Re: Typical sci.math behavior
===
Subject: Re: Typical sci.math behavior
LOL! Quit whining, Harris, and take your meds like a good
little raving
nutcase.
===
Subject: Re: Typical sci.math behavior
following:
> LOL! Quit whining, Harris, and take your meds like a good
little raving
> nutcase.
That was not the real .
--
/-- Joona Palaste (palaste@cc.helsinki.\[CapitalThorn]) -------------
Finland --------
-- http://www.helsinki.\[CapitalThorn]/~palaste ---------------------
rules! --------/
===
Subject: Re: Typical sci.math behavior
following:
> LOL! Quit whining, Harris, and take your meds like a good
little
raving
> nutcase.
> That was not the real .
Does it matter? If it sounds like Harris... if it smells like
Harris....
well.... I guess it passes the Harris Turing test!
===
Subject: Re: Rationality test, math
===
Subject: Re: Rationality test, math
I always get confused at this - which comes \[CapitalThorn]rst, the pot or
kettle?
===
Subject: Re: Rationality test, math
>I always get confused at this - which comes \[CapitalThorn]rst, the pot or
kettle?
Look at the headers - that post was not from Harris.
************************
David C. Ullrich
===
Subject: cauchy-schwarz inequality, normed \[CapitalThorn]elds and vector
spaces
Let k be a \[CapitalThorn]eld. Is k(T) a normed \[CapitalThorn]eld?
I was wondering whether the Cauchy-Schwarz inequality holds
over
normed k-vector spaces for k other than R or C.
===
Subject: Absract Algebra- Help
All that I got is what I put below, which is not too much.
Any hints will
be
helpful.
Let a in S_n be a \[CapitalThorn]xed-point involution or inversion where
a^2 = e and a
moves every element.
Ex: (12) (34) in S_4. Note that this is only possible if n is
even. Show
that b in S_n must be an inversion or transposition if the
group of all
elements c in S_n which b communtes with is a maximal
subgroup.
Assume the group of all elements c in S_n which b communtes
with is a
maximal subgroup and n is even.
If b in S_n is an inversion then we are done. Now we assume
that b in S_n
is
not an inversion (and we try to show that b is a
transposition)
Steve
===
Subject: Re: Absract Algebra- Help
>All that I got is what I put below, which is not too much.
Any hints will
be
>helpful.
>Let a in S_n be a \[CapitalThorn]xed-point involution or inversion where
a^2 = e and a
>moves every element.
>Ex: (12) (34) in S_4. Note that this is only possible if n
is even. Show
>that b in S_n must be an inversion or transposition if the
group of all
>elements c in S_n which b communtes with is a maximal
subgroup.
ThatÕs confusing. It took me some to realize that the \
\[CapitalThorn]rst
sentence
Let a in S_n ... is just a de\[CapitalThorn]nition of the term \
\[CapitalThorn]xed-point
involution
and inversion!
>Assume the group of all elements c in S_n which b communtes
with is a
>maximal subgroup and n is even.
The problem does not state that n is even, so you are not
allowed to
assume that!
>If b in S_n is an inversion then we are done. Now we assume
that b in S_n
is
>not an inversion (and we try to show that b is a
transposition)
>Steve
Let b = (a1,b1)(a2,b2)...(ar,br) with \[CapitalThorn]xed points
c1,c2,...,cs, where
r,s > 0 and 2r + s = n. Let C be the centralizer of
(a1,b1)...(ar,br)
in S_{2r}. Then the centralizer of b in S_{2n} is C X S_s,
which is
properly contained in S_{2r} X S_s except when r = 1.
Derek Holt.
===
Subject: Re: Absract Algebra- Help
in message <brmj7t$aq6$1@wisteria.csv.warwick.ac.uk>:
>All that I got is what I put below, which is not too much.
Any hints
will
>be helpful.
>Let a in S_n be a \[CapitalThorn]xed-point involution or inversion where
a^2 = e and
a
>moves every element.
>Ex: (12) (34) in S_4. Note that this is only possible if n
is even. Show
>that b in S_n must be an inversion or transposition if the
group of all
>elements c in S_n which b communtes with is a maximal
subgroup.
[...]
>If b in S_n is an inversion then we are done. Now we assume
that b in
S_n
>is not an inversion (and we try to show that b is a
transposition)
>Steve
> Let b = (a1,b1)(a2,b2)...(ar,br) with \[CapitalThorn]xed points
c1,c2,...,cs, where
> r,s > 0 and 2r + s = n. Let C be the centralizer of
(a1,b1)...(ar,br)
> in S_{2r}. Then the centralizer of b in S_{2n} is C X S_s,
which is
> properly contained in S_{2r} X S_s except when r = 1.
Hmm. I think youÕre leaving out a step where you show that b
must be a product of disjoint 2-cycles, since otherwise
thereÕs
an element x such that xbx^-1 = b^-1 =/= b, so the centralizer
of b is properly contained in the normalizer of <b>.
Also, for the OP, note that the above proof only shows that
*if*
b isnÕt an inversion, then it must be a transposition. In
particular, it doesnÕt show that the centralizer of either \
an
inversion or a transposition is in fact maximal in S_n. (It
turns out that both are, except for a transposition in S_4,
but
I donÕt know an elementary way to show this.)
--
Jim Heckman
===
Subject: Re: Absract Algebra- Help
>in message <brmj7t$aq6$1@wisteria.csv.warwick.ac.uk>:
>>All that I got is what I put below, which is not too much.
Any hints
will
>>be helpful.
>>Let a in S_n be a \[CapitalThorn]xed-point involution or inversion where
a^2 = e and
a
>>moves every element.
>>Ex: (12) (34) in S_4. Note that this is only possible if n
is even.
Show
>>that b in S_n must be an inversion or transposition if the
group of all
>>elements c in S_n which b communtes with is a maximal
subgroup.
>[...]
>>If b in S_n is an inversion then we are done. Now we assume
that b in
S_n
>>is not an inversion (and we try to show that b is a
transposition)
>>Steve
>> Let b = (a1,b1)(a2,b2)...(ar,br) with \[CapitalThorn]xed points
c1,c2,...,cs, where
>> r,s > 0 and 2r + s = n. Let C be the centralizer of
(a1,b1)...(ar,br)
>> in S_{2r}. Then the centralizer of b in S_{2n} is C X S_s,
which is
>> properly contained in S_{2r} X S_s except when r = 1.
>Hmm. I think youÕre leaving out a step where you show that \
b
>must be a product of disjoint 2-cycles, since otherwise
thereÕs
>an element x such that xbx^-1 = b^-1 =/= b, so the
centralizer
>of b is properly contained in the normalizer of <b>.
Right. For some reason I misread the question, and I thought
b was de\[CapitalThorn]ned
to be an element of order 2.
>Also, for the OP, note that the above proof only shows that
*if*
>b isnÕt an inversion, then it must be a transposition.
But that was all that the problem asked for!
>In particular, it doesnÕt show that the centralizer of
either an
>inversion or a transposition is in fact maximal in S_n. (It
>turns out that both are, except for a transposition in S_4,
but
>I donÕt know an elementary way to show this.)
This is not too dif\[CapitalThorn]cult. The centralizer of a transposition
is
S_2 X S_{n-2}. In general, if r + s = n with r != s, then S_r
X S_s is
maximal in S_n. You could show that as follows. Let S_r X S_s
< H <= S_n.
Suppose S_r, S_s act on {1,..,r}, {r+1,...,r+s} with r > s.
Then an element
of H - (S_r X S_s) must map some pair of points i,j <= r to
points k,l
with k<=r, l>r, and this element will conjugate (i,j) to
(k,l). But now
the conjugates of (k,l) under S_r X S_s together with
transpositions in
S_r, S_s include all transpositions of S_n, so we have H =
S_n.
The centralizer of an inversion is a wreath product S_2 wr
S_m with 2m=n.
The wreath products S_r wr S_s with r,s> 1 are always maximal
in S_n with
n=rs. You can show that in a similar way. If S_r wr S_s < H
<= S_n
then an element in H - (S_r wr S_s) will conjugate a
transposition
in the base group of the wreath product to one which
straddles two of
the orbits of the base group, and then once again you get all
remaining
transpositions in S_n as conjugates of this one under S_r wr
S_s.
Derek Holt.
===
Subject: Re: Help with factorials
> IÕll try to help you with your \[CapitalThorn]rst \
question.
> a) How many digits are there in 1000! (generalize, how many
digits in
> n!)
> In 1000!, there are 2568 decimal digits. In general the
number of decimal
> digits in n! is just ceiling( log_10(n!) ),
Oops! Although that is correct for n > 1, it fails for n = 0
and n = 1. As
ßip recently said, the number of decimal digits is
ßoor( log_10(n!) ) + 1
which is correct for all nonnegative integer n. I was seduced
into making
the mistake of using the ceiling function by my attempt to
get an
expression for the number of digits while avoiding a
computation of n!
itself (or a Gamma or logGamma function etc.) See below.
> where log_10 denotes the
> decimal (i.e., base 10) logarithm. Is that OK, or do you
want an
> expression which avoids computing n! (or a gamma function)
perhaps? (If
> the latter, it might be tricky to get the expression
exactly right.)
In fact, it might be worse than tricky. It might not be
possible. But one
can \[CapitalThorn]nd expressions which _seem_ to work very well. My
suggestion for such
an expression is
ceiling(log_10(e)*(1/2*ln(2*pi) + (n + 1/2) (ln(n + 1/2) -
1)))
where ln denotes the natural logarithm. Does that expression
always give
the number of decimal digits in n! exactly? [I certainly
think the answer
is No.] If not, then
What is the smallest n for which it fails to do so?
and
Can anyone \[CapitalThorn]nd an expression which always gives the number of
decimal
digits in n! exactly and which avoids computing n! (or a
Gamma or
logGamma function etc.) ?
David Cantrell
===
Subject: Re: Help with factorials
> How many digits are there in 1000! (generalize, how many
digits in n!
> As ßip recently said, the number of decimal digits is
> ßoor( log_10(n!) ) + 1
> where log_10 denotes the
> decimal (i.e., base 10) logarithm. Is that OK, or do you
want an
> expression which avoids computing n! (or a gamma function)
perhaps? (If
> the latter, it might be tricky to get the expression
exactly right.)
> In fact, it might be worse than tricky. It might not be
possible. But one
> can \[CapitalThorn]nd expressions which _seem_ to work very well. My
suggestion for
> such an expression is
(*) ceiling(log_10(e)*(1/2*ln(2*pi) + (n + 1/2) (ln(n + 1/2)
- 1)))
> where ln denotes the natural logarithm. Does that
expression always give
> the number of decimal digits in n! exactly? [I certainly
think the answer
> is No.]
I may have been too pessimistic. But let me explain the
source of the
pessimism I had. The argument of the ceiling function in (*)
above is
always somewhat larger than log_10(n!), and log_10(n!) can be
made
arbitrarily close to and slight less than an integer by
suitably choosing
n.
[Numerical example:
log_10(e)*(1/2*ln(2*pi) + (8765 + 1/2)(ln(8765 + 1/2) - 1)) =
30753.9999207...
while log_10(8765!) = 30753.9999187...]
Thus it had seemed likely to me that there would be a value
of n for which
the argument of the ceiling function in (*) would be very
slightly larger
than an integer while log_10(n!) would be very slightly less
than that same
integer. Of course, if there were such a value of n, then (*)
would fail
there, giving 1 more than the correct number of decimal
digits in n! .
Numerical investigation, however, shows that (*) gives the
correct number
of decimal digits in n! at least for n <= 10^6. Furthermore,
to get
log_10(n!) to be closer and closer to integers, we must use
larger and
larger values of n, and in so doing the argument of the
ceiling function in
(*) of course gets closer and closer to log_10(n!). And it
does so to such
an extent that it appears that (*) might never fail. OTOH, if
I had a proof
that (*) always worked, IÕd be giving it here now... So I
must be agnostic
regarding whether (*) always works (but if I had to guess
now, IÕd guess
that it does always work).
Does anyone know any expression proven to give the number of
digits in n!
exactly using just elementary functions together with
functions such as
ceiling or ßoor?
David Cantrell
===
Subject: Re: Help with factorials
(*) ceiling(log_10(e)*(1/2*ln(2*pi) + (n + 1/2) (ln(n + 1/2)
- 1)))
as an expression for the number of decimal digits in n! :
> ... So I must be agnostic regarding whether (*) always
works (but if I
> had to guess now, IÕd guess that it does always work).
> Does anyone know any expression proven to give the number
of digits in n!
> exactly using just elementary functions together with
functions such as
> ceiling or ßoor?
I was not careful enough in stating my request. By private
email, I
received a response (which I appreciate!) which said, in part:
Sure! 1 + ßoor(SUM, k = 1 to n, ln(k)/ln(10))
OK, OK, thatÕs not real useful, and certainly not what you
had in mind.
True, thatÕs not what I had in mind. But it does bring up an
interesting
question: Should we say that such a sum is in closed form?
Were the upper
limit of the summation +in\[CapitalThorn]nity, then I suppose that we would
unanimously
say that it is not in closed form. But since n is \[CapitalThorn]nite here,
the
summation is not actually open ended, so to speak, in that it
terminates
when k reaches n. Perhaps some people would then say that
such a sum is in
closed form. However, Graham, Knuth and Patashnik, in their
_Concrete
Mathematics_, would disagree. On page 7, they say
Sums like 1 + 2 + ... + n are not in closed form -- they
cheat by using
Ô...Õ; but expressions like n(n + 1)/2 are. We \
could give a
rough
de\[CapitalThorn]nition like this: An expression for a quantity f(n) is in
closed form
if we can compute it using at most a \[CapitalThorn]xed number of \
Ôwell
knownÕ standard
operations, independent of n.
Thus, I should have also speci\[CapitalThorn]ed that the desired expression
for the
number of decimal digits in n! should be in closed form (in
the sense used
by GK&P).
David Cantrell
===
Subject: Re: Help with factorials
> a) How many digits are there in 1000! (generalize, how many
digits in n!)
> b) What is the digit in the kth position of 1000!. for any
k.
(generalize,
same
> question but for n! instead of 1000!)
for a), you want to look at LegendreÕs Theorem
Thm: If n is a positive integer and p is a prime such that p
divides n,
then
p appears in the canonical representation of n! with exponent
w, where w =
Sum[ [[n/p{k}]], k = 1..In\[CapitalThorn]nty].
Corollary: If n = Product[p{i}^a{i}, then n! =
Product[p{i}^e{p{i}}, i = 1
.. r].
Thus, the number of digits of n to the base b is given by [[
Log[n{b}] ]] +
1, where [[ ]] is the greatest integer function.
So, [[ Log[10, 1000!] ]] + 1 = [[ 2567. 6 ]] + 1 = 2568
Can you take it from here?
HTH
===
Subject: Re: JSH: Discussion with Dik Winter
> IÕve started a thread to go over some statements by Dik
Winter which I
> say are crank statements. IÕm also going to outline some
crank
> behavior by that person.
Hint: Calling someone a crank who quite clearly isnÕt one
will do
nothing to disguise the fact that you *are* a crank.
> I donÕt mind others posting in the thread or in this one,
and I may
> reply to people other than Dik Winter, but I want you to
know where
> the focus is.
ThatÕs mighty nice of you, but no one needs your permission
to post
> Some of you may know that I have independent veri\[CapitalThorn]cation of
the
> argument that he attacks, but IÕve been puzzled both by \
his
> persistence in making his claims against those argument,
and in the
> acceptance of his claims by the sci.math newsgroup.
So you say. We have yet to see any evidence of this
independent
veri\[CapitalThorn]cation, and I donÕt believe for a moment \
that it exists.
(Unless,
of course, youÕre talking about that Mega Society vanity rag
that you
paid your fellow cranks to print.)
> So IÕm doing an experiment. My guess is that despite
hearing that
> thereÕs independent veri\[CapitalThorn]cation of the \
argument Winter
attacks, and
> despite the wackiness of his position the sci.math
newsgroup will
> STILL either show support for Winter or fail to correct him.
Another hint: the sci.math newsgroup doesnÕt support or
correct
anyone, as itÕs not a person. (You really need to get over
this
compulsion to personify everything.) There merely are a lot of
individuals who read this newsgroup who have reached certain
conclusions
about Dik (and about you) independently, and weÕve (each)
concluded
that heÕs right and youÕre wrong. \
(HowÕs *that* for
independent
veri\[CapitalThorn]cation?)
> ThatÕs the hypothesis that IÕm currently \
testing.
No, thatÕs the lie youÕre currently \
promoting.
--
Wayne Brown (HPCC #1104) | When your tailÕs in a crack, you
improvise
fwbrown@bellsouth.net | if youÕre good enough. Otherwise you
give
| your pelt to the trapper.
e^(i*pi) = -1 -- Euler | -- John Myers Myers,
Silverlock
===
Subject: Neo Con Triumph and WMD Physics in Iraq and on UFOs
There is a report that Saddam Hussein let terrrorist A. Nidal
train Atta
for 911 attack back in Baghdad. If that is true it justi\[CapitalThorn]es
our removal
of Saddam. Apparently Saddam murdered A. Nidal about a year
ago to cover
his guilt?
Prime Minister of Italy, an ally BTW, quipped of SaddamÕs
capture Well
at last a WMD has been found in Iraq. :-)
Now if Wolfowitz would change his silly idea that France,
Russia &
Germany cannot be prime contractors in Iraq and not subvert
Rummy who is
doing quite well ...
Rumor in San FranciscoÕs North Beach is that Francis Ford
Coppola is
interrogating Saddam at his North Beach Shelter for the
Homeless on
Columbus and Greenwich St in a sequel to Apocalypse Now. Louis
Dinardi is a Saddam double. :-)
Back to the physics of WMD:
bcc
JS: Hal, I ask you, what would prove PV wrong? ;-)
HP: (1) Black holes really exist (instead of only very dark
gray
holes).
JS: What kind of observational data needed to make this
distinction?
HP: (2) So-called non-black-hole Yilmaz stars (M > 2.8 solar
masses)
found not to exist. (Robertson evidence is that they do.)
JS: Please give complete details on this.
HP: (3) For dense matter SS distribution observation were
found to
match Schwarzschild instead of exponential metric.
JS: Not enough information in that cryptic sentence. What
does it mean?
How do you explain dark energy and dark matter with PV?
HP: Our cosmological modeling under way as we speak. Stay
tuned!
Hal
JS: Does PlanckÕs h make any appearance in your PV math \
model?
Again I ask what does PV mean when your GM/c^2r >> 1.
Do you think super-steel rods exist?
If they do not exist, what physical meaning does your equation
cÕ = c/K
really have?
How does a ßying saucer ßy in your PV model?
How do you metric engineer a Star Gate in your PV model?
Is time travel to the past thinkable in your PV model?
Are parallel universes next door thinkable in your PV model?
Do you think Jacques ValleeÕs Magonia effects mean parallel
universes
next door?
What about Eric DavisÕs MUFON 2001 report of creature
emerging out of
a kind of sphere of light (Star Gate) in sky
at Robert BigelowÕs NIDS Utah Ranch?
Do you plan to extend PV to include extra space dimensions?
PZ: The erroneous conßation of formal covariance with physical
relativity is almost a
founding principle of Einsteinan physics (later abandoned by
Einstein).
JS: There are several legitimate issues here.
PZ: OK.
JS: Why donÕt you make a dictionary of these key terms and
give best
de\[CapitalThorn]nitions you can to avoid confusion.
PZ: The \[CapitalThorn]nal de\[CapitalThorn]nitions will depend on the \
outcome of the
arguments I
am making.
IÕll try to de\[CapitalThorn]ne ambiguous terms as they come \
up.
JS: You should come to London March 8 to March 12 lots of
philosophers
of physics there.
I am urging Hal Puthoff and Eric Davis to come.
That book I mentioned last time is very useful
in this regard BTW. More on that anon.
PZ: OK, IÕll look at it.
JS: My debate with Hal on PV is really in relation to the the
pseudo-group of passive LOCAL coordinate transformations at a
\[CapitalThorn]xed
point P and of world crystal.
This is not same as active diffeomorphisms P -> PÕ =/= P .
The active
and passive transformations must
be made mutually consistent and this may solve the Kretchmann
issue?
PZ: Intuitively I would have thought that the concept of
smoothly
differentiable coordinate maps and their inverses
would be unproblematic. I would have thought the critical
distinction is
between smooth coordinate transformations
that are linear, and those that are non-linear, speci\[CapitalThorn]cally
in the time
coordinate.
Such transformations that are non-linear in the *space*
coordinates
only, have no special physical meaning in GR.
JS: The is supposed to be 4-D spacetime diffeomorphisms so
that time and
space are not really split - no preferred foliation when such
foliations
can be done which is not always. On the other hand the
canonical 3 + 1
split tries to keep the 4D in the form of Dirac constraints.
It gets
into trouble when one tries to quantize the constraints.
Ashtekar made a
non-perturbative advance in 1986 leading to loop quantum
gravity as
in latest Sci Am BTW with spin networks in 3D and spin foam
in 4D. Not
all world crystal lattices can be thought of as tiling
polytopes. There
are curved spin networks that are not tiled polytopes.
Kleinert has a
different approach however. There are still lots and lots of
obscure
issues in all concpetual directions at once, like breakdown
of causality
when light cones quantum ßuctuate in this pure QM gravity
quest as
distinct from M theory string -> membranes which used
hyperspace/supersymmetry ideas to get ALL gauge forces not
gravity
alone. Gravity is emergent in the M theory program with
non-perturbative T and S dualities. None of these approaches
really
solve some of the key problems including the problem of time,
which
has a natural solution in Bohmian realism with a preferred
foliation
substratum and with IT + BIT rather than BIT alone. Problem
is that the
time Dirac constraint is the Wheeler-DeWitt pure BIT equation
HPSI(Universe) = 0
which has no time in it.
Time needs to emerge like pressure and temperature. BIT
nonlocality problem in QM gravity not only is total energy of
gravity
nonlocal ALL observables if active diffeomorph invariants are
NONLOCAL -
this is a disaster of sorts because it violates WheelerÕs
Keep IT local, Stupid! ;-)
i.e. my re-wording of WheelerÕs
Physics is simple when it is local.
BIT is nonlocal. IT is local.
Since all the QM Gravity Pundits eschew Bohmian realism, no
wonder they
get stuck with BIT nonlocality when they try to get IT from
BIT.
Also there is the issue of whether
or not the different points P are distinguishable and what is
an
observable in GR? There is the Einstein hole
problem. BTW Joy Christian is a male.
JS: Active diffeomorphism invariants are NONLOCAL - a problem
in
interpretation. There is no consensus on these deep issues
and others
among the Pundits. ItÕs almost as bad as
the wars over the interpretation of quantum theory.
I have to see whether this is a real substantive issue or
whether itÕs
about how many angels can dance on
a passive diffeomorphism.
Do you mean general coordinate transformations GCT? Be more
precise if you use
plain English.
PZ: The standard terminology is loaded. When I say coordinate
generality, I mean coordinate
generality: the *desideratum* that laws should be formulated
in
such a manner that their form
does not depend on the particular choice of coordinate system.
Hence the use of coordinate-
free devices such as tensors
It is not clear to me why a modern theory of gravitation
*must* be
formulated in such a manner
-- other than as an expression of physical general relativity
of
motion, which I contend does
not exist.
JS: You are still on the classical macro-level I am more
interested in
how this level emerges from micro-quantum or something beyond
even that.
Also I think classical macro is an error. It is a \[CapitalThorn]ction like
The
Unicorn and like Hal PuthoffÕs super-steel in his quasi
measurement
Tables I & II in his PV model that disintegrates when one
asks what
happens when GM/c^2r >> 1?
JS: If GCT is what you mean the answer is NO. What I mean is
that
the manifold looks pathological and unphysical with at least a
countable in\[CapitalThorn]nity of coordinate
patches outside the turning point r* = GM/c^2 for curvature
radial coordinate,
which is analogous to event horizon in EinsteinÕs GR where
there are only
TWO patches outside r* = 2GM/c^2 in that case (Einstein-Rosen
Bridge, i.e. non-traversable wormhole
in non-exotic vacuum case Ruv = 0 everywhere-when. It is clear
to me that Hal is not really thinking
about the topology and differential geometry in his naive
engineering approach.
PZ: OK, so you are saying that there is an unavoidable
pathological
discontinuity in the exponential PV
solution for a point mass?
JS: If by point mass you mean taking the vacuum solution to
the max
yes.
Hal goes into a state of denial pretending there is no
problem. He tries
to solve the cosmological constant problem the same way. It
just will
not do IMHO.
...
JS: In the case of GR I mean vacuum all the way i.e. solutions
of Ruv = 0 with wormhole global topology of source Mass
without mass
(JA Wheeler). And for PV what would correspond to that.
I do not think Dicke knew the differential geometry when he
introduced
the exponential metric ~ 1961?
PZ: Maybe not -- but he should not be underestimated. He had
very strong
physical intuition from what I can see.
I am arguing that the complicated Riemann-Ricci-Levi-Civita
apparatus
can also act as a mathematical smokescreen
and can even be fundamentally misleading in that it serves to
obscure or
even block certain important mathematical
and physical possibilities.
JS: No in the MACRO domain. In the quantum domain - all bets
are off.
Almost anything goes. What is not forbidden is mandatory.
PZ: I wasnÕt even aware that a manifold was \
de\[CapitalThorn]ned in PV. A
physical metric, yes; but a manifold?
This is not a curved spacetime theory as far as I am aware.
The
model is a polarizable vacuum with
physical rubber rods and clocks.
JS: A metric without a manifold is pure mathematical
nonsense. You
can have a manifold without a metric but not the other way
round.
You mean by
The model is a polarizable vacuum with physical rubber rods
and
clocks.
A constructive theory like Lorentz-FitzgeraldÕs approach (as
done e.g.
by JS Bell) to special relativity as opposed to EinsteinÕs
phenomenological geometrodynamic one. Einstein himself
compared
Lorentz-Fitzgerald way to kinetic theory of gases and his to
thermodynamics. He was NOT AGAINST the former as an alternate
POV.
However, Hal has no real dynamical constructive theory in
that deep
sense at all. Indeed, I have one for GR based on
<0|e+(x)e-(x)|0> as a
dynamical QED global ßat vacuum instability in which curved
space-time
vacuum emerges after the lowering of energy and entropy in
the vacuum
phase transition explaining also the origin of inßation.
All Hal ever write is
K = e^2GM/c^2r
where is the dynamical PV in that?
Where does M come from? Hal simply piggybacks on Einstein and
Dicke and
then throws away the rigorous underpinning.
He throws in an action formalism that is mere window
dressing. He also
adds charge and EM \[CapitalThorn]elds. So what? ItÕs all \
black box
phenomenology without any dynamical constructive deep
structure at all.
No QED. No vacuum coherence. Nada.
Also, you forgot something big in HalÕs scheme.
Einstein already has the rubber rods and clocks thatÕs why
itÕs c for
vacuum speed of light all the way for ALL both LIF and LNIF
test
the moment about replacing P with nonlocal extended things.
What Hal has is his mythical super steel rods which would
give NOT c
but c/K! You forgot that!
The problem is that Hal is completely obscure to my mind on
the
fundamental world view of his model. He uses metric notation
after all?
PZ: But it is a physical metric that is simply a mathematical
description of the physical deformation of measuring
devices and the resulting scaling of the measured intervals.
It is not a
theory about the fundamental chronogeometric
structure of the world -- any more than is the description of
the
behavior of metal bars on a heated surface (an
example that Feynman liked to use).
JS: I think you are missing the point here Paul. Hal seems to
think that
there really are super steel rods and clocks that would
measure
c/K when the Einstein rubber rods and clocks measure c. This
after all
would be a real bimetric world with super steel in the
globally ßat
Yilmaz world parallel to to EinsteinÕs rubbery curved world.
This
Yilmazian split is \[CapitalThorn]ction that Hal thinks is fact IMHO.
Again I ask. What does HalÕs PV mean when his GM/c^2r >> 1?
Do you know?
PZ: Although in the alternative paradigm, general
covariance looks more like a
mathematical fetish, since physical general relativity is
absent.
JS: There is a lot about all this in Physics Meets Philosophy
at the Planck Scale Callender & Huggett Cambridge Press
2001. I suggest that we temporarily cease this line of inquiry
until we both digest what is in that book - some really GOOD
STUFF!
PZ: OK, IÕve ordered it. But why stop the press?
JS: Because those guys are pretty smart and have thought
through a lot
of the issues you are interested in. So itÕs time to catch \
up.
PZ: OK, at least it sounds like you are beginning to take
some of these
heretical arguments a little more seriously.
JS: Yes, but in the context of extending classical GR to the
quantum
domain. NO ONE questions GR in its proper domain. I mean no
one at the
cutting edge does like Chris Isham, Penrose, Rovelli, Smolin,
Tegmark
any of them including John Baez to a man they all agree that
YilmazÕs
claims are basically Cargo Cult and Baez has explicitly
mentioned that
HalÕs PV is of the not even wrong class. Do a Google. I am \
on
HalÕs
case on all this because people in the defense military
intelligence
community listen to him because of his high former security
clearances
(USN & NSA background) and Hal is privy to real UFO info and
all of this
has defense implications even WMD implications and the danger
of
intelligence failure like we saw in Iraq in threat assessment
of
technological surprise for advanced space-weapons is very
real. The
Black Ops boys in USG listen to Hal Puthoff because he is one
of their
own. They do not pay attention to John Baez if only because
his Aunt is
Joan Baez! The Old Boy Network is not always rational. I mean
itÕs more
Skull and Bones than Phi Beta Kappa. USG Intelligence Honchos
at the
highest levels, especially in the present Neo Con dominated
Bush
Administration, does not really trust the top rank physicists
who they
consider, like most academics, to be too left dating from the
anti-Teller pro- Oppenheimer -> Bethe-Morrison-Panofsky Era
that I
witnessed \[CapitalThorn]rst-hand at Cornell in late 50Õs and \
early 60Õs
and later at
UCSD (Project Jason people) in mid to late 60Õs. They do not
even trust
Colin Powell and the State Department! They are more prone to
believe
Nick Cook from JaneÕs Defense Weekly in The Hunt For The \
Zero
Point
than Ch 9 of Sir Martin ReesÕs Our Final Hour.
PZ: I should have their book within a week or so.
JS: The book was written prior to the realization of the new
cosmology of dark energy/matter - there is no mention of that.
However, it is good background stuff by hip philosophers and
some top
physicists. Mathematician (in physicistÕs clothing) John \
Baez
also has a
n-categories and how they may make an interesting formal
connection
between GR and QM. On the other hand I see a lot of
conceptual ßaws in
the thinking of the Pundits in both Q Gravity and M-Theory,
one of which
being that they all assume Lp is a constant and not a
variable where
perhaps
Lp*/Lp ~ e^(metric engineering control parameter?)
PZ: Beware of wishful thinking.
JS: Tell that to Hal. ;-) I rather think that what you see
above is
precognitive remote viewing! ;-)
JS: Another is that none of them seem to have read P.W.
AndersonÕs idea More is different and how it applies to
quantum
measurement problem for example. Nevertheless, there are many
good
relevant insights in the book.
PZ: OK.
WhatÕs any of that got to do with what I am talking about?
I am talking about internal tensions within orthodox GR.
JS: We are also talking about HalÕs PV and also the book \
does
deal also
with the internal tensions.
PZ: OK.
JS: Also my focus
is how to combine quantum theory and GR in order to solve the
important
real problems in physics today:
1. What is the Universe made of?
2. What is the physical nature of consciousness?
3. How do we achieve the kind of metric engineering we see in
the UFO
observations?
PZ: Many would be satisi\[CapitalThorn]ed with # 1.
JS: Not me.
Studying the internal consistency of this or that theory is
secondary to
these objectives.
Such study may well be necessary however. That seems to be
so. I have
essentially had
my eyes on this Golden Ring for 50 years and I want to get
some
satisfaction! :-)
http://www.\[CapitalThorn]ndmidis.com/listen.go/589
So does Hal and that is why I am not letting him rest on the
issues.
PZ: Also, the
metric is not the \[CapitalThorn]eld; the tensor potential phi_uv
represents the physical \[CapitalThorn]eld
and the gravitational-inertial metric is derived from it.
Non-linear coordinate
transformations play a fundamentally different role in this
alternative model.
I am talking here about Yilmaz.
JS: I mean there are an in\[CapitalThorn]nity
of isotropic coordinate patches outside the turning point
boundary at
GM/c^2 for a single curvature coordinate. In EinsteinÕs GR
this ratio
is only 2:1, i.e. 2 coordinate patches outside the event
horizon at
2GM/c^2 in the Penrose-Kruskal diagram with 4 coordinate
patches
covering the entire vacuum manifold.
PZ: ItÕs still interesting to me that a coordinate
discontinuity
was originally mistaken for a
physical event horizon.
Even if you are right that the PV solution is pathological,
this
does not necessarily apply
to YilmazÕs phi_uv. In YilmazÕs theory it is \
phi_uv that is
physically fundamental, while
the exponential metric is secondary and derivative.
JS: Perhaps. Just what is the Yilmaz theory in your
understanding? I
mean what is its world view?
What is the physical picture behind the obscure formalism?
PZ: Basically:
(1) Any satisfactory tensor theory of gravitation should have
a precise
static Newtonian correspondence
model and should have good (localizable, frame-independent)
energy-momentum analogs satisfying
Newtonian conservation principles in limiting cases (<-->
Poisson
equation);
JS: Red Herring. EinsteinÕs GR has that already. In weak
curvature slow
speed limit one gets Galilean Newtonian physics. BTW \
NewtonÕs
gravity
does not have localized energy density either!
U(Newton) = - GMm/r
this is action at a distance.
Do you simply mean Poisson eq? Apparently you do. Then you
have no real
point at all here.
Grad^2 U/m ~ Grho(r) ?
EinsteinÕs GR has that except it is
Grad^2(U/m) = G(rho + 3p/c^2) = Grho(1 + 3w)
p is the local pressure.
In ordinary matter
3p/c^2 << rho
For ordinary vacuum rho = 0 and w = -1
For exotic vacuum Grho = c^2/zpf since w = -1.
/zpf = Lp^-1(Lp^3/2|Vacuum Coherence|^2 - 1)
/zpf > 0 is STRONGLY anti-gravitating universally repulsive
exotic
vacuum zero point stress - dark energy density.
/zpf < 0 is STRONGLY gravitating universally attractive
exotic vacuum
zero point stress - dark matter density.
Compare this to AlcubierreÕs weightless warp drive and
Bondi-Terletskii negative matter propulsion and Kip ThorneÕs
exotic matter for traversable wormhole time machines and
\[CapitalThorn]nally to
Ch 9 of Sir Martin ReesÕs Our Final Hour.
http://qedcorp.com/APS/StarGate1.mov
READ MY EQUATIONS!
Look at the formal algebra not only the ambiguous informal
English words.
PZ: (2) EinsteinÕs vacuum stress-energy pseudotensor does \
not
satisfy
these correspondence requirements;
JS: What are you talking about? Where in NewtonÕs gravity
theory do you
have even a stress energy tensor for gravity?
Newton did not even have the idea of a tensor. What object in
NewtonÕs theory should EinsteinÕs vacuum \
stress-energy
pseudotensor
limit to?
You must give exact mathematical examples in making these
sweeping
blanket pronouncements. Extraordinary claims require precise
extraordinary justi\[CapitalThorn]cations.
In EinsteinÕs GR
tuv(Geometry) = (String Tension)Guv(Einstein)
EinsteinÕs local GR geometrodynamic \[CapitalThorn]eld \
equation is simply
the static
equilibrium
tuv(Marble Geometry) + Tuv(Wood Matter) = 0
for ordinary vacuum when /zpf = 0.
End of Part I. Part II anon.
===
Subject: Re: Neo Con Triumph and WMD Physics in Iraq and on
UFOs
>There is a report that Saddam Hussein let terrrorist A.
Nidal train Atta
>for 911 attack back in Baghdad. If that is true it justi\[CapitalThorn]es
our removal
>of Saddam. Apparently Saddam murdered A. Nidal about a year
ago to cover
>his guilt?
>Prime Minister of Italy, an ally BTW, quipped of SaddamÕs
capture Well
>at last a WMD has been found in Iraq. :-)
>Now if Wolfowitz would change his silly idea that France,
Russia &
>Germany cannot be prime contractors in Iraq and not subvert
Rummy who is
>doing quite well ...
>Rumor in San FranciscoÕs North Beach is that Francis Ford
Coppola is
>interrogating Saddam at his North Beach Shelter for the
Homeless on
>Columbus and Greenwich St in a sequel to Apocalypse Now.
Louis
>Dinardi is a Saddam double. :-)
>Back to the physics of WMD:
>bcc
>JS: Hal, I ask you, what would prove PV wrong? ;-)
>HP: (1) Black holes really exist (instead of only very dark
gray
holes).
>JS: What kind of observational data needed to make this
distinction?
>HP: (2) So-called non-black-hole Yilmaz stars (M > 2.8 solar
masses)
>found not to exist. (Robertson evidence is that they do.)
>JS: Please give complete details on this.
>HP: (3) For dense matter SS distribution observation were
found to
>match Schwarzschild instead of exponential metric.
>JS: Not enough information in that cryptic sentence. What
does it mean?
>How do you explain dark energy and dark matter with PV?
>HP: Our cosmological modeling under way as we speak. Stay
tuned!
>Hal
Lets zip ahead to 2036 and see what they have to say...
http://www.anomalies.net/time_travel/john.html
===
Subject: Re: Focuses of hyperbola
>>Let we have a hyperbolic function: y(x)=a/(b*x+c)+d. How to
\[CapitalThorn]nd the
>>coordinates of focuses of this hyperbola through the
coef\[CapitalThorn]cients?
>Since the function can be translated to
> (y-d)(x+c/b) = a/b
>the curve is a right hyperbola, which has an eccentricity of
sqrt(2).
>The axis of this hyperbola has slope 1 and passes through
the center of
>the hyperbola, (-c/b,d). The eccentricity is the ratio of
the distance
>between the foci and the length of the major axis. The
length of the
>major axis is 2 sqrt(a/b);
I would believe 2 sqrt(2a/b)
> therefore, the distance between the foci is
not
>2 sqrt(2a/b). Thus...
>Rob Johnson <rob@trash.whim.orgtake out the trash before
replying
Of course I am relying on my memory and could be wrong. :-)
--Lynn
===
Subject: Re: Focuses of hyperbola
>Let we have a hyperbolic function: y(x)=a/(b*x+c)+d. How to
\[CapitalThorn]nd the
>coordinates of focuses of this hyperbola through the
coef\[CapitalThorn]cients?
>>Since the function can be translated to
>> (y-d)(x+c/b) = a/b
>>the curve is a right hyperbola, which has an eccentricity
of sqrt(2).
>>The axis of this hyperbola has slope 1 and passes through
the center of
>>the hyperbola, (-c/b,d). The eccentricity is the ratio of
the distance
>>between the foci and the length of the major axis. The
length of the
>>major axis is 2 sqrt(a/b);
>I would believe 2 sqrt(2a/b)
>> therefore, the distance between the foci is
>not
>>2 sqrt(2a/b). Thus...
>>Rob Johnson <rob@trash.whim.org>take out the trash before
replying
>Of course I am relying on my memory and could be wrong. :-)
No, you are correct, the coordinates of the ends of the major
axis are
sqrt(a/b)(1,1) and -sqrt(a/b)(1,1), so the length of the
major axis is
2 sqrt(2a/b) as you say. Thus, the distance between the foci
is
4 sqrt(a/b) and the coordinates of the foci are
(-c/b,d) + sqrt(2a/b)(1,1)
(-c/b,d) - sqrt(2a/b)(1,1)
I forgot to multiply by the length of (1,1) when computing
the length of
the major axis. The points I gave in my previous post are the
ends of
Rob Johnson <rob@trash.whim.org>
take out the trash before replying
===
Subject: integral involving tail of Taylor series of sin
Evaluate the integral from 0 to in\[CapitalThorn]nity of
(sin(x) - sum( (-1)^(k-1) x^(2k-1)/(2k-1)!, k=1..n))/x^(2n+1)!
(The numerator is the terms of the Taylor series of sin(x) at
x=0 starting
at
the x^(2n+1)/(2n+1)! term.)
Ted Hwa
===
Subject: Re: integral involving tail of Taylor series of sin
: Evaluate the integral from 0 to in\[CapitalThorn]nity of
: (sin(x) - sum( (-1)^(k-1) x^(2k-1)/(2k-1)!,
k=1..n))/x^(2n+1)
(corrected above: thatÕs x^(2n+1) not x^(2n+1)! ... )
Ted
===
Subject: Re: integral involving tail of Taylor series of sin
> : Evaluate the integral from 0 to in\[CapitalThorn]nity of
> : (sin(x) - sum( (-1)^(k-1) x^(2k-1)/(2k-1)!,
k=1..n))/x^(2n+1)
> (corrected above: thatÕs x^(2n+1) not x^(2n+1)! ... )
Is it!
:-)
--Ron Bruck
===
Subject: Re: integral involving tail of Taylor series of sin
>Evaluate the integral from 0 to in\[CapitalThorn]nity of
>(sin(x) - sum( (-1)^(k-1) x^(2k-1)/(2k-1)!,
k=1..n))/x^(2n+1)!
>(The numerator is the terms of the Taylor series of sin(x)
at x=0 starting
at
> the x^(2n+1)/(2n+1)! term.)
I doubt that you really mean x^((2n+1)!) there, because it
will have a bad
singularity at x=0. More reasonably is (...)/x^(2n+1). Then
the answer
appears to be (-1)^n/(2 (2n)!). It should be provable by
induction.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
===
Subject: Re: integral involving tail of Taylor series of sin
:>Evaluate the integral from 0 to in\[CapitalThorn]nity of
:>(sin(x) - sum( (-1)^(k-1) x^(2k-1)/(2k-1)!,
k=1..n))/x^(2n+1)!
:>(The numerator is the terms of the Taylor series of sin(x)
at x=0 starting
at
:> the x^(2n+1)/(2n+1)! term.)
: I doubt that you really mean x^((2n+1)!) there, because it
will have a
bad
: singularity at x=0. More reasonably is (...)/x^(2n+1).
Yes, thatÕs what I meant.
Then the answer
: appears to be (-1)^n/(2 (2n)!). It should be provable by
induction.
That should be (-1)^n pi/(2 (2n)!). ThatÕs what I suspected
the answer
to be, but couldnÕt really get started on a proof.
Ted
===
Subject: Re: integral involving tail of Taylor series of sin
>:>Evaluate the integral from 0 to in\[CapitalThorn]nity of
>:>(sin(x) - sum( (-1)^(k-1) x^(2k-1)/(2k-1)!,
k=1..n))/x^(2n+1)!
>:>(The numerator is the terms of the Taylor series of sin(x)
at x=0
starting at
>:> the x^(2n+1)/(2n+1)! term.)
>: I doubt that you really mean x^((2n+1)!) there, because it
will have a
bad
>: singularity at x=0. More reasonably is (...)/x^(2n+1).
>Yes, thatÕs what I meant.
>Then the answer
>: appears to be (-1)^n/(2 (2n)!). It should be provable by
induction.
>That should be (-1)^n pi/(2 (2n)!). ThatÕs what I suspected
the answer
>to be, but couldnÕt really get started on a proof.
Let F(n) = int_0^in\[CapitalThorn]nity (sin(x) - P_n(x))/x^(2n+1) dx
where P_n(x) = sum_{k=1}^n (-1)^(k-1) x^(2k-1)/(2k-1)!.
F(0) = int_0^in\[CapitalThorn]nity sin(x)/x dx = pi/2.
Note that P_nÕÕ = -P_{n-1}. Two integrations \
by parts give
F(n) = -1/(2n(2n-1)) F(n-1).
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
===
Subject: Re: Cryptogram from Newton?
James Buddenhagen
LH > The anonymous author of this website
> http://www.mathpages.com/home/index.htm
> seems to be quite sane, but on this subpage
> http://www.mathpages.com/home/quotes.htm
> we \[CapitalThorn]nd a quotation
> 6accdae13eff7i3l9n4o4qrr4s8t12ux.
> Isaac Newton, 1676
> Does anybody here know what this is about?
> LH
> You will \[CapitalThorn]nd some expanation at the same site.
> See http://www.mathpages.com/home/kmath414.htm
> Jim Buddenhagen
> P.S. The now anonymous author (IÕm sure he has
> his reasons) of that site used to post quite
> regularly to sci.math
Thx Omri and James.
ItÕs quite a nice site; lots of interesting odds and ends.
LH
===
Subject: Re: limit problem
>lim as x -> in\[CapitalThorn]nity of (1- 1/(2x))^x?
Another approach: take logarithms; apply lÕHopital. (I can
already
hear the roar.)
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
===
Subject: Re: limit problem
>>lim as x -> in\[CapitalThorn]nity of (1- 1/(2x))^x?
> Another approach: take logarithms; apply lÕHopital. (I can
already
> hear the roar.)
LÕHopital!
Write its logarithm as
x log(1 - 1/2x) = x(-1/2x + O(1/x^2)) etc.
Or just remember what the derivative of the logarithm is :-)
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Needless to say, I had the last laugh.
Alan Partridge, _Bouncing Back_ (14 times)
===
Subject: Re: limit problem
> I am working on a limit problem and seem to be stuck.
Anyone know how to
> take the lim as x -> in\[CapitalThorn]nity of (1- 1/(2x))^x?
1/sqrt e
===
Subject: regression software.
I am looking for regression formulas in order to make
software that
can show the regression curve on screen.
the regressions that IÕm search are from all types:
linear, quadratic, cubic, power, exponential ,logarithmic ,
parabolic...
I am not a statistician, so in order to calculate these
regression I
need only the coef\[CapitalThorn]cients of the formula.
shraga friedman.
===
Subject: Re: Question on the mathematics of an elliptic curve
cryptography
protocol
> I have implemented basic elliptical curve mathematical
functions
elliptic curve
> modulo some number p, as well as the Miller-Rabin test to
generate a
> k-bit prime number, but I am slightly confused about the
generation of
> the two groups G1 and G2 by the BDH Parameter Generator
(page 19).
> In short, I am unclear on how to generate G1 and G2.
> What exactly goes into these groups? I have generated the
q, found
> the smallest prime p such that p=2 mod 3, q divides P+1,
and q^2 does
> not divide p+1. I am unclear on what is meant by the
Ôsubgroup of
> order q of the group of points on the curve over FpÕ.
It means the subgroup of order q of the group of points on
the curve
over F_p.
Let E be the eliptic curve in question. The group E(F_p) is
the
group of all points on E with coordinates in F_p. It is an
abelian
group of order p+1 (the curve E is supersingular). By
construction
q divides p+1. By CauchyÕs theorem E(F_p) has a subgroup (a
subgroup
is closed under addition) of order q. (In fact this subgroup
is unique).
> Is it necessary
> to simply try all integer values of X (0,1,2,...), check if
it
> satis\[CapitalThorn]es the curve equation, then choose the \
\[CapitalThorn]rst q of
these?
That is unlikely to form a group! (I.e., to be closed under
addition).
> Since
> p is less than q, how can there be a group of order q over
the \[CapitalThorn]eld
> Fp?
Really? How can p be less than q if q is cooked up to be a
factor of p + 1?
> On page 23, Ôlet P be some generator of G1Õ, \
how do i go
about \[CapitalThorn]nding
> said P?
Write p+1 = qm. Let Q be a random point on the curve. Compute
mQ.
Then p(mQ) = O. If we are lucky (probability (q-1)/q) then P
= mQ =/= O
and G_1 is the subgroup of E(F_p) generated by P. If we are
unlucky,
try again!
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Needless to say, I had the last laugh.
Alan Partridge, _Bouncing Back_ (14 times)
===
Subject: Re: Question on the mathematics of an elliptic curve
cryptography
protocol
> On page 23, Ôlet P be some generator of G1Õ, \
how do i go
about \[CapitalThorn]nding
> said P?
> Write p+1 = qm. Let Q be a random point on the curve.
Compute mQ.
> Then p(mQ) = O. If we are lucky (probability (q-1)/q) then
P = mQ =/= O
q(mQ)=O ?
===
Subject: Re: decomposition of sl_2 representation
>
> let I=k[x_1,x_2,..,x_n] is polinomial ring over \[CapitalThorn]eld of
char=0 and
> I_n - subspase gomogenius polinomial of power n. Let sl_2 -
3 -
> dimesional simple lie algebra wich act at I_n in usual way.
How \[CapitalThorn]nd a
> irreducible components of decomposition of this
representation? Need
> \[CapitalThorn]nd something like as formulae of (Klebsh-Gordon)for tensor
product.
> You use n in two different ways. Let I_k be the
k-homogeneous part.
> What was the action of sl_2 again?. The way I see it I_k is
naturally
> acted upon by sl_n for all k. But these sl_n-modules are
all simple
> (in characteristic zero) so there is no decomposition.
Obviously
> something is wrong.
> Jyrki Lahtonen, Turku, Finland
No mistake. For example I=k[x_1,x_2], then
I_2=<x_1^2,x_1*x_2,x_2^2>.
sl_2, acts on n-dimensional space in usual way, for example
for
E{-1)(x_i)=x_(i+1),E(0)x_i=(n-2*i)*x_i
,E(1)x_i=i(n-i+1)x_(i-1).
Action of sl_2 on I_k is just extenshion as derivation this
action.
And this sl_2 -modul is not simple, but as sl_n module , of
cource is
simple.
===
Subject: Re: decomposition of sl_2 representation
> > let I=k[x_1,x_2,..,x_n] is polinomial ring over \[CapitalThorn]eld of
char=0 and
> I_n - subspase gomogenius polinomial of power n. Let sl_2 -
3 -
> dimesional simple lie algebra wich act at I_n in usual way.
How \[CapitalThorn]nd
a
> irreducible components of decomposition of this
representation? Need
> \[CapitalThorn]nd something like as formulae of (Klebsh-Gordon)for tensor
product.
> You use n in two different ways. Let I_k be the
k-homogeneous part.
> What was the action of sl_2 again?. The way I see it I_k is
naturally
> acted upon by sl_n for all k. But these sl_n-modules are
all simple
> (in characteristic zero) so there is no decomposition.
Obviously
> something is wrong.
> Jyrki Lahtonen, Turku, Finland
> No mistake. For example I=k[x_1,x_2], then
I_2=<x_1^2,x_1*x_2,x_2^2>.
> sl_2, acts on n-dimensional space in usual way, for example
for
I wouldnÕt call this action usual, but that is just my \
taste:)
> E{-1)(x_i)=x_(i+1),E(0)x_i=(n-2*i)*x_i
,E(1)x_i=i(n-i+1)x_(i-1).
Are you sure there isnÕt an indexing problem here? The
highest weight
(weight space spanned by x_1) seems to be n-2 and the lowest
weight
(weight space spanned by x_n) seems to be -n. May be you want
the
indeterminates to be named x_0,x_1,...,x_n. Anyway, the
highest weight
in the n-dimensional representation is of weight n-1. May be,
it is
enough to make the torus act by E(0)x_i=(n-1-2*i)x_i
I would even describe THIS action as a derived action. What I
had in mind
was a more natural action (let n=2) of sl_2 by the
differential operators
(roots) x_1 (d/dx_2), x_2 (d/dx_1) and (torus) x_1
(d/dx_1)-x_2 (d/dx_2).
This extends naturally to an action on any I_k, (any
non-negative integer
k)
and on I_{n-1} gives you the usual n-dimensional
representation (up to
an isomorphism). Similarly for n>2 - you just get more roots
and a bigger
torus.
> Action of sl_2 on I_k is just extenshion as derivation this
action.
> And this sl_2 -modul is not simple, but as sl_n module , of
cource is
> simple.
If I understand your action correctly, you want the symmetric
tensor power
of the n-dimensional representation of sl_2. Once we agree on
what the
action
is, surely this is easy to compute. If x_i belongs to the
weight space
(n-1-2*i), then a monomial prod_i (x_i^{a_i}) belongs to the
weight
sum_i a_i(n-1-2*i). Using this it is easy to put together the
formal
character
of your representation, and going from there is pretty
straightforward.
Jyrki Lahtonen, Turku, Finland
===
Subject: Re: decomposition of sl_2 representation
>> let I=k[x_1,x_2,..,x_n] is polinomial ring over \[CapitalThorn]eld of
char=0 and
>> I_n - subspase gomogenius polinomial of power n. Let sl_2
- 3 -
>> dimesional simple lie algebra wich act at I_n in usual
way. How \[CapitalThorn]nd
a
>> irreducible components of decomposition of this
representation? Need
>> \[CapitalThorn]nd something like as formulae of (Klebsh-Gordon)for \
tensor
product.
>> You use n in two different ways. Let I_k be the
k-homogeneous part.
>> What was the action of sl_2 again?. The way I see it I_k
is naturally
>> acted upon by sl_n for all k. But these sl_n-modules are
all simple
>> (in characteristic zero) so there is no decomposition.
Obviously
>> something is wrong.
>> Jyrki Lahtonen, Turku, Finland
>> No mistake. For example I=k[x_1,x_2], then
I_2=<x_1^2,x_1*x_2,x_2^2>.
>> sl_2, acts on n-dimensional space in usual way, for
example for
> I wouldnÕt call this action usual, but that is just my
taste:)
Nor would I, itÕs more usual to act on the powers, not the
subscript. itÕs
not just your taste.
===
Subject: Re: Stupid homeomorphism question
> Yes, but your way of proving that g is continuous is not
correct; there
> is no product involved here, unless you call product of f:A
--> B by
> g:A --> C to the function from A to B x C de\[CapitalThorn]ned by x |->
(f(x),g(x)).
> The product did refer to the product of two functions: id:
R^n ->
R^n,
> id(x)=x, and f(x): R^n -> R, given in the problem. Thus g =
id * f =
> (id(x), f(x)). Identity functions are always continuous,
and the
> continuiety of f was given in the problem. Does this prove
g to be
> continuous?
Yes. ItÕs like I said in my previous post (see above): if \
you
de\[CapitalThorn]ne
product
that way, then you have proved correctly that g is continous.
Jose Carlos Santos
===
Subject: Re: Stupid homeomorphism question
Jose Carlos Santos <jcsantos@fc.up.pt> scribbled the
following:
>> Yes, but your way of proving that g is continuous is not
correct;
there
>> is no product involved here, unless you call product of
f:A --> B by
>> g:A --> C to the function from A to B x C de\[CapitalThorn]ned by x |->
(f(x),g(x)).
>> The product did refer to the product of two functions: id:
R^n ->
R^n,
>> id(x)=x, and f(x): R^n -> R, given in the problem. Thus g
= id * f =
>> (id(x), f(x)). Identity functions are always continuous,
and the
>> continuiety of f was given in the problem. Does this prove
g to be
>> continuous?
> Yes. ItÕs like I said in my previous post (see above): if
you de\[CapitalThorn]ne
product
> that way, then you have proved correctly that g is
continous.
--
/-- Joona Palaste (palaste@cc.helsinki.\[CapitalThorn]) -------------
Finland --------
-- http://www.helsinki.\[CapitalThorn]/~palaste ---------------------
rules! --------/
Bad things only happen to scoundrels.
- Moominmamma
===
Subject: Re: No Highest Prime, by de\[CapitalThorn]nition?
> So many computer languages use *, that it has almost become
the
> newsgroup defacto standard for multiplications. It is
certainly less
> liable to be misinterpreted than ..
OK, I think that IÕll use it from now on.
Jose Carlos Santos
===
Subject: Topologies implied by limits of sets
Let X be an arbitrary set, and let I be one of R[0,1], R, or
N U
{infty} under the usual topology. Let F = {f in P(X)^I: f(j)
->
f(i) as j -> i for all i in I}. Is there a simple way to
describe
the topology on P(X) (or a base or subbase for this topology)
coinduced by F?
I suspect that the answer is more interesting when X is
in\[CapitalThorn]nite, and
that the answer probably varies with the cardinality of X (at
least
countable vs. uncountable).
[Since there was confusion about limits of sets in a recent
thread, I
repeat the intended de\[CapitalThorn]nition here: For example,
> Let I indicate the intersection operator. Given a sequence
S of
> sets, by de\[CapitalThorn]nition
> lim inf S = U{I{S_m: m >= n}: n in N}, and
> lim sup S = I{U{S_m: m >= n}: n in N}.
> Iff the limits inferior and superior are equal, this set is
by
> de\[CapitalThorn]nition the limit of S.
> One can generalize this de\[CapitalThorn]nition to nets of sets.
P(X) denotes the power set of X.]
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
===
Subject: Re: Topologies implied by limits of sets
> Let X be an arbitrary set, and let I be one of R[0,1], R,
or N U
> {infty} under the usual topology. Let F = {f in P(X)^I:
f(j) - f(i) as j -> i for all i in I}. Is there a simple way
to describe
> the topology on P(X) (or a base or subbase for this
topology)
> coinduced by F?
.
I see no intuitive reason why to chose F the way you did.
Were P(X) to be given a topology consistent with de\[CapitalThorn]nition of
limits of a sequence of sets, then the question IÕd \
\[CapitalThorn]rst ask:
if for all j in N, Aj in K subset P(X), and Aj -> A, is
A in cl K ?
Has a topology ever been given to P(X), thusly consistent
with set limits?
Could we de\[CapitalThorn]ne a closure operator based upon the limit?
cl F = { A in P(X) | some A1,... in F with lim Aj = A }
LetÕs check it out.
cl nulset = nulset
F subset cl F
F subset G ==> cl F subset cl G.
So far so good. To rap it up, itÕs needed to show
cl cl F = cl F
which is the paramount question, how to show that.
Then cl:P(P(X)) -> P(P(X)) would induce a topology upon P(X)
with closed sets being exactly those sets that include the
limits
of all convergent sequences.
Now cl { A } = { A }, so P(X) would be T1.
Thus if X is \[CapitalThorn]nite, P(X) is \[CapitalThorn]nite hence \
discrete.
What would the topology of P(N) look like?
> Let I indicate the intersection operator. Given a sequence
S of
> sets, by de\[CapitalThorn]nition
> lim inf S = U{I{S_m: m >= n}: n in N}, and
> lim sup S = I{U{S_m: m >= n}: n in N}.
> Iff the limits inferior and superior are equal, this set is
by
> de\[CapitalThorn]nition the limit of S.
> One can generalize this de\[CapitalThorn]nition to nets of sets.
> P(X) denotes the power set of X.]
===
Subject: Re: Topologies implied by limits of sets
>Let X be an arbitrary set, and let I be one of R[0,1], R, or
N U
>{infty} under the usual topology. Let F = {f in P(X)^I: f(j)
->
>f(i) as j -> i for all i in I}. Is there a simple way to
describe
>the topology on P(X) (or a base or subbase for this topology)
>coinduced by F?
If the topology coninduced by F is the same as the topology
generated by F, ie the weakest topology such that all the
elements
of F are continuous, then I think itÕs clear that this is
just the
product topology, or rather the topology that arises by
identifying
P(X) with {0,1}^X in the natural way and considering the
product
topology on {0,1}^X.
(So a typical neighborhood of a set S in P(X) would be
determined
by n elements x_1, ... x_n in X; the neighborhood is the set
of
all SÕ such that x_j is in SÕ if and only if \
x_j is in S, 1
<= j <=
n.)
>I suspect that the answer is more interesting when X is
in\[CapitalThorn]nite, and
>that the answer probably varies with the cardinality of X
(at least
>countable vs. uncountable).
>[Since there was confusion about limits of sets in a recent
thread, I
>repeat the intended de\[CapitalThorn]nition here: For example,
>> Let I indicate the intersection operator. Given a sequence
S of
>> sets, by de\[CapitalThorn]nition
>> lim inf S = U{I{S_m: m >= n}: n in N}, and
>> lim sup S = I{U{S_m: m >= n}: n in N}.
>> Iff the limits inferior and superior are equal, this set
is by
>> de\[CapitalThorn]nition the limit of S.
>> One can generalize this de\[CapitalThorn]nition to nets of sets.
>P(X) denotes the power set of X.]
************************
David C. Ullrich
===
Subject: Re: Topologies implied by limits of sets
>>Let X be an arbitrary set, and let I be one of R[0,1], R,
or N U
>>{infty} under the usual topology. Let F = {f in P(X)^I:
f(j) ->
>>f(i) as j -> i for all i in I}. Is there a simple way to
describe
>>the topology on P(X) (or a base or subbase for this
topology)
>>coinduced by F?
>>
>If the topology coninduced by F is the same as the topology
>generated by F, ie the weakest topology such that all the
elements
>of F are continuous, then I think itÕs clear that this is
just the
>product topology, or rather the topology that arises by
identifying
>P(X) with {0,1}^X in the natural way and considering the
product
>topology on {0,1}^X.
>(So a typical neighborhood of a set S in P(X) would be
determined
>by n elements x_1, ... x_n in X; the neighborhood is the set
of
>all SÕ such that x_j is in SÕ if and only if \
x_j is in S, 1
<= j <=
>n.)
I am not sure what David means by weaker; I am more familiar
with
smaller = coarser vs. larger = \[CapitalThorn]ner. (In fact, Munkres states
that
weaker is used both ways amongst mathematicians.) So letÕs
get our
de\[CapitalThorn]nitions straight.
Let T and U be two topologies on a set. Iff T is a subset of
U,
then T is smaller, or coarser, than U, and U is larger, or
\[CapitalThorn]ner, than T.
Let X be a set, Y a topological space, and F a subset of Y^X.
The topology on X induced by F is the smallest topology such
that
all functions in F are continuous. The prototypical examples
are the
product and relative topologies, which are respectively
induced by
projections and inclusions.
Let X be a topological space, Y a set, and F a subset of Y^X.
The topology on Y coinduced by F is the largest topology such
that
all functions in F are continuous. The prototypical example
is the
quotient topology, coinduced by the cannonical map to
equivalence classes.
That much said, I think David has picked up on the same thing
as William
Elliot, viz., that what I really wanted was some topology on
P(X) such
that the set limits coincided with topological limits. Why
didnÕt I say
that in the \[CapitalThorn]rst place?
Using DavidÕs more workable de\[CapitalThorn]nition of set \
limits from
another thread,
> lim S_n = S if (i) for every x in S there exists N such that
> x is in S_n for all n > N and (ii) for every x not in S
there
> exists N such that x is in S_n for no n > N,
it appears that David is correct that the translation of the
product
topology is what I was looking for. However, doesnÕt David
last
sentence need a little tweaking? It seems to me that a base
for
topology is composed of sets of the form {Y in P(X): A subset
of Y
subset of X B}, where A and B vary over all (disjoint) \[CapitalThorn]nite
subsets of X. That is, you need to specify a \[CapitalThorn]nite number of
elements
*not* in Y as well as elements in Y.
It is not clear to me that this answers my original question;
nor (as
William has pointed out) is it clear that the question is
interesting. What is a little interesting is whether my
original
question is equivalent to my intended question. Or is the
desired
topology equivalent to that induced by those functions f in
[0,1]^P(X) such that lim f(A) = f(lim A) for all sequences A
where
lim A exists? Do we need to consider other nets besides
sequences?
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
===
Subject: Re: Topologies implied by limits of sets
<mb1utvc1f0jpa6bq4v9lekevcjh9dmcs9n@4ax.com>
<oMVDb.235521$655.42920982@news4.srv.hcvlny.cv.net>
>Let X be a topological space, Y a set, and F a subset of
>Y^X. The topology on Y coinduced by F is the largest topology
>such that all functions in F are continuous.
The topology for Y = { U | for all f in F, f-1(U) open }
U open in Y iff for all f in F, f^-1(U) open in X
The smallest topology for Y making all fÕs continuous
is the indiscrete topology
The biggest topology for Y could be like embedding F copies
of X into Y, the disjoint sum of F copies of X.
The smallest topology for X that makes all fÕs continuous
is the topology generated by the subbase
{ f-1(U) | f in F, U open in X }
The bigest topology for X making all fÕs continuous
is the indiscrete topology
The smallest topology for X could be like the product
topology for F
copies of Y, the component spaces.
>Let X be an arbitrary set, and let I be one of R[0,1], R, or
>N U {infty} under the usual topology. Let F = {f in P(X)^I:
>f(j) -> f(i) as j -> i for all i in I}. Is there a simple
way to
>describe the topology on P(X) (or a base or subbase for this
>topology) coinduced by F?
The topology for P(X) is
{ U subset P(X) | for all f in F, f-1(U) open }
U open in P(X) iff for all f in F, f^-1(U) open in R
--
It de\[CapitalThorn]es my imagination how you embed F copies of R into \
P(X).
Perhaps another application of the coinduced topology would
be a
more appropiate comparision or model to use.
To have a concrete example to aid in visualization
present some collection F of maps from R into P(N).
ThatÕs a nice cardinality \[CapitalThorn]t.
What if I asked for maps from R into P(R) or P(P(R)) or N
Again, with aim to topologize, what intuitive justi\[CapitalThorn]cation
have you
for picking maps from the reals? Nor have you depicted F
other than
just any untutored bunch of maps.
--
The approach of closure operator has immediate intuitive
thrust to
topologizing P(X) using set limits. For A subset P(X), de\[CapitalThorn]ne
cl A = { a in P(X) | some a1,a2,... in A with lim aj = a }
Immediatly
cl nulset = nulset
A subset cl A
A subset B ==> cl A subset cl B
The crutial part, if possible, is to show
cl cl A = cl A
With those four properties, the closure operator cl, as \
de\[CapitalThorn]ned
induces a topology upon P(X) consistent for limits with the
set limit.
Open sets U of P(X) would be those subsets of P(X) for which
the
complement of U is closed, ie
U open iff cl P(X)U = P(X)U
Conversely any induced or coinduced topology for P(X) would
have to conclude cl cl A = cl A and would it not produce the
same expression for cl A as IÕve given directly?
To continue with this speculation
cl { A } = { A }, ie P(X) is T1.
Thus if X is \[CapitalThorn]nite, P(X) is \[CapitalThorn]nite hence \
discrete.
What would the topology of P(N) look like? As a starter,
what would cl P(2N), the closure of subsets even integers, be?
Does cl P(A) = P(A) for all A subset N?
--
>It is not clear to me that this answers my original question;
WhatÕs the orginal question? Is that the second quote \
IÕve
included?
>nor (as William has pointed out) is it clear that the
question is
>interesting.
The problem of topologizing P(X) is of interest.
When X is a metric space, then the Hausdorff metric is a
metric for
nonnul closed bounded elements of P(X). Is that metric
consistent
with the set de\[CapitalThorn]nition of limit?
>What is a little interesting is whether my original
>question is equivalent to my intended question.
Aah! What question are you intending?
>Or is the desired topology equivalent to that induced by
those
>functions f in [0,1]^P(X) such that lim f(A) = f(lim A) for
all
>sequences A where lim A exists?
Then P(X) would be the induced topology, akin to the product
topology.
>Do we need to consider other nets besides sequences?
Extend the de\[CapitalThorn]nition of set limit to nets? Maybe if
the results of topologizing P(X) isnÕt separable.
-- set limits
> Given a sequence S_n of sets, by de\[CapitalThorn]nition
> lim S_n = S if (i) for every x in S there exists N such that
> x is in S_n for all n > N and (ii) for every x not in S
there
> exists N such that x is in S_n for no n > N,
> lim inf S = U{I{S_m: m >= n}: n in N}, and
> lim sup S = I{U{S_m: m >= n}: n in N}.
> Iff the limits inferior and superior are equal, this set is
by
> de\[CapitalThorn]nition the limit of S.
> One can generalize this de\[CapitalThorn]nition to nets of sets.
----
===
Subject: Re: Topologies implied by limits of sets
In response to WilliamÕs request for \
clari\[CapitalThorn]cation:
Let X be a set. Consider the following three topologies on
P(X).
U1: that where set-theoretic limits coincide with topological
limits.
U2: that induced by all functions f in [0,1]^P(X) such that
lim f
A = f (lim A) for all sequences (nets?) A such that lim A
exists.
U3(I): that coinduced by all functions f in P(X)^I such that
lim f
A = f(lim A) for all set-convergent nets A indexed by I,
where I
is one of N U {infty}, [0,1], or R.
The question on the table: Is U1 the same as any of the
others?
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
===
Subject: Re: Topologies implied by limits of sets
>>
>Let X be an arbitrary set, and let I be one of R[0,1], R, or
N U
>{infty} under the usual topology. Let F = {f in P(X)^I: f(j)
->
>f(i) as j -> i for all i in I}. Is there a simple way to
describe
>the topology on P(X) (or a base or subbase for this topology)
>coinduced by F?
>
>>If the topology coninduced by F is the same as the topology
>>generated by F, ie the weakest topology such that all the
elements
>>of F are continuous, then I think itÕs clear that this is
just the
>>product topology, or rather the topology that arises by
identifying
>>P(X) with {0,1}^X in the natural way and considering the
product
>>topology on {0,1}^X.
>>[...]
>It is not clear to me that this answers my original
question; nor (as
>William has pointed out) is it clear that the question is
>interesting. What is a little interesting is whether my
original
>question is equivalent to my intended question. Or is the
desired
>topology equivalent to that induced by those functions f in
>[0,1]^P(X) such that lim f(A) = f(lim A) for all sequences A
where
>lim A exists? Do we need to consider other nets besides
sequences?
No, sequences are enough, because [0,1] is metrizable:
Suppose that X is a metric space, Y is a topological space,
and f:X->Y is sequentially continuous. Then f is continuous.
(An open set O in Y such that f^{-1}(O) is not open easily
leads to a sequence x_n -> x in X such that f(x) is in O
but no f(x_n) is in O, hence f(x_n) does not converge to
f(x).)
************************
David C. Ullrich
===
Subject: Re: Topologies implied by limits of sets
[...]
>>Or is the desired
>>topology equivalent to that induced by those functions f in
>>[0,1]^P(X) such that lim f(A) = f(lim A) for all sequences
A where
>>lim A exists? Do we need to consider other nets besides
sequences?
>>
>No, sequences are enough, because [0,1] is metrizable:
>Suppose that X is a metric space, Y is a topological space,
>and f:X->Y is sequentially continuous. Then f is continuous.
But in the case at hand, the range is metrizable, not the
domain.
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
===
Subject: Re: Topologies implied by limits of sets
>[...]
>>
>Or is the desired
>topology equivalent to that induced by those functions f in
>[0,1]^P(X) such that lim f(A) = f(lim A) for all sequences A
where
>lim A exists? Do we need to consider other nets besides
sequences?
>
>>No, sequences are enough, because [0,1] is metrizable:
>>Suppose that X is a metric space, Y is a topological space,
>>and f:X->Y is sequentially continuous. Then f is continuous.
>But in the case at hand, the range is metrizable, not the
domain.
I didnÕt realize youÕd switched questions - \
previously it was
P(X)^[0,1].
************************
David C. Ullrich
===
Subject: Re: Topologies implied by limits of sets
>>
>Let X be an arbitrary set, and let I be one of R[0,1], R, or
N U
>{infty} under the usual topology. Let F = {f in P(X)^I: f(j)
->
>f(i) as j -> i for all i in I}. Is there a simple way to
describe
>the topology on P(X) (or a base or subbase for this topology)
>coinduced by F?
>
>>If the topology coninduced by F is the same as the topology
>>generated by F, ie the weakest topology such that all the
elements
>>of F are continuous, then I think itÕs clear that this is
just the
>>product topology, or rather the topology that arises by
identifying
>>P(X) with {0,1}^X in the natural way and considering the
product
>>topology on {0,1}^X.
>>(So a typical neighborhood of a set S in P(X) would be
determined
>>by n elements x_1, ... x_n in X; the neighborhood is the
set of
>>all SÕ such that x_j is in SÕ if and only if \
x_j is in S, 1
<= j <=
>>n.)
>[...]
>That much said, I think David has picked up on the same
thing as William
>Elliot, viz., that what I really wanted was some topology on
P(X) such
>that the set limits coincided with topological limits. Why
didnÕt I say
>that in the \[CapitalThorn]rst place?
>Using DavidÕs more workable de\[CapitalThorn]nition of set \
limits from
another thread,
>> lim S_n = S if (i) for every x in S there exists N such
that
>> x is in S_n for all n > N and (ii) for every x not in S
there
>> exists N such that x is in S_n for no n > N,
>it appears that David is correct that the translation of the
product
>topology is what I was looking for. However, doesnÕt David
last
>sentence need a little tweaking? It seems to me that a base
for
>topology is composed of sets of the form {Y in P(X): A
subset of Y
>subset of X B}, where A and B vary over all (disjoint) \[CapitalThorn]nite
>subsets of X. That is, you need to specify a \[CapitalThorn]nite number of
elements
>*not* in Y as well as elements in Y.
ThatÕs exactly what I said:
So a typical neighborhood of a set S in P(X) would be
determined
by n elements x_1, ... x_n in X; the neighborhood is the set
of
all SÕ such that x_j is in SÕ if and only if \
x_j is in S, 1
<= j <=
n.
Note thatÕs x_1, ... x_n in X, not x_1, ... x_n in S. Given
x_1, ... x_n in X, let A be the set of all the x_j which are
in
S and B be the set of all x_j which are not in S. Then the
basis element I de\[CapitalThorn]ned, the set of all SÕ such \
that x_j is
in SÕ if and only if x_j is in S, 1 <= j <=n, is exactly the
same as {Y in P(X): A subset of Y subset of X B}.
>It is not clear to me that this answers my original
question; nor (as
>William has pointed out) is it clear that the question is
>interesting. What is a little interesting is whether my
original
>question is equivalent to my intended question. Or is the
desired
>topology equivalent to that induced by those functions f in
>[0,1]^P(X) such that lim f(A) = f(lim A) for all sequences A
where
>lim A exists? Do we need to consider other nets besides
sequences?
************************
David C. Ullrich
===
Subject: Re: Final Rout of Synchronization Clocks in
Relativity
> ABSTRACT. The synchronization of clocks in Relativity has
speculative
> chatter, and this speculative chatter about
Synchronizations of
> clocks in Relativity has not ACTUAL TECHNICAL EMBODYING In
CONCRETE
> TECHNICAL DEVICES.
The deductive Analysis of Surprising paradox of mythical
so-called
pseudo of synchronization of clocks in the Relativity is
given below:
> Sometimes claimants misquote or exaggerate to further their
> own agendas. It is best to keep an open opinion until you
have heard
> from both sides of any story. -|Tom|-
>
>
> Tom Van Flandern - Washington, DC - see our web site on
replacement
> astronomy research at http://metaresearch.org
===
Subject: Re: Final Rout of Synchronization Clocks in
Relativity
> ABSTRACT. The synchronization of clocks in Relativity has
speculative
> chatter, and this speculative chatter about
Synchronizations of
> clocks in Relativity has not ACTUAL TECHNICAL EMBODYING In
CONCRETE
> TECHNICAL DEVICES.
> The deductive Analysis of Surprising paradox of mythical
so-called
> pseudo of synchronization of clocks in the Relativity is
given below:
>
> Sometimes claimants misquote or exaggerate to further
their
> own agendas. It is best to keep an open opinion until you
have heard
> from both sides of any story. -|Tom|-
>
>
> Tom Van Flandern - Washington, DC - see our web site on
replacement
> astronomy research at http://metaresearch.org
The concept of clock synchronization confused me for the
longest time.
I was trying to think of it in absolute terms of the twin
paradox. As
a concept to be thought of deeply.
And the long antenna simulation proved it a far different
thing.
A gps can have the clock synchronized by the input of the
correct
location. Without any satilite message!!!!!!
A special kind of time difference is de\[CapitalThorn]ned by special
relativity.
It is not general relativities kind though!!!
So the clock of special relativity is to be rememebered as
only the
clock of EinstienÕs gedanken experiment.
He de\[CapitalThorn]ned a special dilemma in theory which the answer
caused!!!!
So think carefully of taking the gedanken experiment as a
physical
experiment. It is to be a very, very, very special thought
experiment.
Meaning it is a test of the theory in the school of classical
theory.
So when the train has two times. What causes the
synchronization to
fail?
And here the train has the relative speed of light never
failing as
the cause of the gedanken experiment.
That is its real purpose, while the lack of synchronized
clocks is the
synthetic necessity to cause the relation of, never failing
to reach
the same speed independent of the inertial reference frame.
One relation is the theory, while the other causes the theory.
This is the gedenkan experiment in abstract form. Real odd in
form!!!!
Read the EPR paper. It uses the opportunity to de\[CapitalThorn]ne the
gedanken
experiment abstractly. Never do the experiment, because it
de\[CapitalThorn]nes
the fool.
So when the synchronized clock is introduced in special
relativity it
is natures clock and never the experimenterÕs clock!!!!
NatureÕs as a physical theory would cause to be identical to
require
the solution.
To allow the physical clock to follow this requirement would
require
the synchronized clock to never be measurable! An absolute
outcome of
special relativity is the meaning of clocks once separated
are forever
unsynchronized.
And here the frame of reference is the only means of
inference of the
clockÕs difference.
So, a complete theory is de\[CapitalThorn]ned!!!!!!!!
And back again to the synchronized clocks. All reference
frames are
independent.
Complete in every fashion.
Douglas Eagleson
Gaithersburg, MD USA
===
Subject: Re: Math Joke
You guys should stay in more.
> What can you tell about changes in the measuring techniques
of
> quantities of planetary masses between 1980 and 1990?
> During that period, several outer planet masses were greatly
> improved by visitation by spacecraft. Such ßy-bys permit a
far more
> accurate mass determination that measurements of satellite
> orbits. -|Tom|-
What can you add in this context for Mercury, Venus,
Earth-Moon,
Mars, and asteroid Icarus & so on
Aleksandr
> Tom Van Flandern - Washington, DC - see our web site on
replacement
> astronomy research at http://metaresearch.org
===
Subject: Re: Russell-like paradoxes
> The Theory of Types disallows a set from being a member of
itself.
> Applied to barbers, it disallows a barber from shaving
himself.
> Not so. x shaves x, is not dependent on any theory of types.
> The Theory of Types is an arbitrary prohibition meant to
avoid
> RussellÕs Paradox. As noted by all, there is a parallel
between
> barbers shaving and sets containing, i.e., RussellÕs
Paradox and the
> Barber Paradox.
Yes, they are both instances of Ax(yRx <-> ~(xRx)), ie. they
have R in
common.
There is no y of any type such that Ax(yRx <-> ~(xRx)).
> If we continue this analogy, then the Theory of Types
> translates into a prohibition against barbers shaving
themselves.
No it does not! We cannot continue this analogy.
The theory of types does deny ~(x e x), but, it does not deny
~(x shaves
x).
> If you are saying that barbers can shave themselves, then
one could
> say sets can contain themselves and deny that the Theory of
Types
> applies to sets either.
Of course, many languages deny the theory of types.
> Whether you believe that sets or barbers can be
self-applied, the same
> set of 3 rules of inference are in effect in both cases,
and that
> combination, as I have said, is inconsistent. The Theory of
Types is
> merely arbitrarily prohibiting one of these 3 rules of
inference. The
> point is that you canÕt have those 3 rules simultaneously,
whatever
> system (sets, Turing Machines, Logic, English) you are
using.
What 3 rules are you talking about?
> The predicate ~(x e x) has no extension.
> > ThatÕs what THEY think. ArenÕt predicates \
and sets
intuitively the
> same thing?
> No. It depends on the intuitive assumptions of the
structure of the
language
> in use. For a language that imposes the theory of types,
they are
equal.
See:
> Russell, Carnap.etc.
> This is merely adding an arbitrary restriction, making their
> de\[CapitalThorn]nition of set and predicate different from the intuitive
> de\[CapitalThorn]nition. I am saying that is not necessary. One can keep
the
> de\[CapitalThorn]nition of a set and predicate as being essentially the
same,
> rather than altering these concepts to the point of
contradicting the
> intuitive notions.
You are correct only if you include some ÔTypeÕ \
theory in
your assumed
intuitive theory.
> The problem isnÕt that sets and predicates \
canÕt be the
same thing.
> This just leads to more problems, besides the fact that the
system
> denies that sets and predicates are synonymous. De\[CapitalThorn]ne a tet
to be
> the same thing as a set. Then there is no tet of all tets
that
> donÕt contain themselves, but there is a tet that contains
all sets
> that donÕt contain themselves. But then, a tet is a set, \
so
we have
> another contradiction. Solution: There is no tet of sets
that donÕt
> contain themselves, because that is equivalent to the tets
that donÕt
> contain themselves. Likewise, there is no predicate x is a
set that
> does not contain itself. (This is just applying one more
step in my
> formal proof of inconsistency.)
> SETS = CLASSES = PREDICATES. That is true intuitively and
it is not
> necessary to contradict that fact.
Only if you include some ÔTypeÕ theory in your \
assumed
intuitive theory.
> For systems that do not adopt a theory of types, eg:
Zermelo, von
Neumann,
> Quine, etc., other methods of avoiding the \
ÔparadoxesÕ are
required.
> Right. They have to avoid one of the three rules that I
formalized.
> But they donÕt have to deny that sets, classes and
predicates are all
> the same thing. I KNOW they say that in their systems. But
theyÕre
> wrong. Sorry.
They are not wrong within their intuitive set theory.
They are wrong within your intuitive set theory.
Why do you think that your way is the only correct way?
> The problem isnÕt that a predicate doesnÕt \
have a
> corresponding set. The problem is that there is no
predicate x is
a
> set that does not contain itself, because that would be the
predicate
> x is a predicate that does not hold for itself, which
doesnÕt
exist.
> Not so. ~(x e x) exists for Quine and others, but {x:~(x e
x)} does not
> exist. (NF)
> Some writers claim {x:~(x e x)}exists as a \
ÔproperÕ class
but not as a
set
> ???
> IÕm really not talking about what some people write,
believe and work
> with. They can have their theories, but they are
unnecessarily
> abandoning the real, intuitive notion of a set being
another name for
> a predicate.
> Question: Would you prefer (all else considered equal) a
set theory
> in which sets and predicates are the same thing, or one in
which they
> are different?
I admit that set theories with a type theory are easier to
deal with, but,
I prefer a set theory without the very awkward theory of
types.
y e {x:Fx} <-> Fy is valid for you, and, E!{x:Fx} ->. y e
{x:Fx} <-> Fy
is valid for me.
Your assumption that every predicate determines a class is
false without
the assumption of ÔtypesÕ.
> Witt
> Charlie Volkstorf
> Cambridge, MA
>
http://www.mathpreprints.com/math/Preprint/CharlieVolkstorf/
20021008.1/1
> http://www.arxiv.org/html/cs.lo/0003071
===
Subject: Re: Russell-like paradoxes
> Ax(M(x) -> (~S(x,x) <-> S(y,x)))
> ...one can conclude ~M(y).
Right. Probably one actually should interpret M(x) with x is
a men of
Seville. This way the /pseudoparadox/ would take a quite
reasonable
form.
(Curry uses the term pseudoparadox to describe an apparent
paradox,
such as the catalogue paradox, for which there is no
underlying actual
contradiction. --mathworld.wolfram.com)
F.
===
Subject: Re: Russell-like paradoxes
> I believe the original is a sign in a barber shop that says
I shave
> all those men, and only those men, who do not shave
themselves.
> The sign can be consistently true, provided the barber is
not a man.
> No it cannot.
Of course, it can.
> Ax(y shaves x <-> ~(x shaves x)) is a contradiction.
Right. But your statement is NOT an appropriate translation of
y shaves all those men, and only those men, who do not shave
themselves,
for you dropped the condition men (!). Hence we have to use
the
translation:
Ax(men x -> (y shaves x <-> ~(x shaves x))). (*)
And actually there CAN be a female barber, say Barbara (b),
which shaves
all men [of Seville] who do not shave themselves.
BTW: Actually the statement (*) is much better than the one
without the
condition on man. Since in the latter case the barber would be
condemned to shave ANYTHING which does not shave itself...
well...
actually a supertask NO MAN can perform... :-)
F.
Well, of course, the solution to this conundrum is that our
universe
of discourse is [silently] restricted just to _the man of
Seville_.
Then we may ask:
EyAx(y shaves x <-> ~(x shaves x))?
And the answer certainly will be:
~EyAx(y shaves x <-> ~(x shaves x)).
===
Subject: Re: Russell-like paradoxes
> There is no paradox in the barber who shaves each man that
does not
> shave himself.
> The barber is a woman. Shame on Russell. :)
> Shame on you, the barber cannot exist.
You are wrong (this time), Owen, IF we phrase the paradox as
mentioned
above. There really IS NO paradox. The barber may be any
being _except a
man_.
Only ONE thing is sure (i.e. can be derived from the
statement above):
> it [the barber] cannot be male.
Of course, with the presupposition that ONLY man can be
barbers
[something that certainly w a s true when Russell came up
with his
statement] we get the conclusion:
> the barber cannot exist.
F.
The formalization of the paradox would be now:
Ax(men x -> b shaves x <-> ~(x shaves x))
Now *assume*
men b.
Then we would (immediately) get
b shaves b <-> ~(b shaves b)
and hence a contradiction. Thus
~men b.
===
Subject: Re: Russell-like paradoxes
> I wonder if Russell actually saw such a sign before he
exposed this
> paradox in naive set theory.
You mean in FregeÕs highly non-naive system - not in naive \
set
theory.
===
Subject: Re: Russell-like paradoxes
:
: > I wonder if Russell actually saw such a sign before he
exposed this
: > paradox in naive set theory.
: You mean in FregeÕs highly non-naive system - not in naive
set
: theory.
You mean youÕre a tendentious jackass as usual, Torkel.
The reason WHY RussellÕs paradox occurs in \
FregeÕs system
IS BECAUSE it occurs in naive set theory. The axiom in \
FregeÕs
system that produced this particular paradox is itself
DEFINITIVE
of naive set theory.
--
---
ItÕs dif\[CapitalThorn]cult ... you need to be united to have \
any
strength, but internal issues have to be addressed.
--- E. Ray Lewis, on liberalism in America
===
Subject: Re: Russell-like paradoxes
permission for an emailed response.
cursed
> I wonder if Russell actually saw such a sign before he
exposed this
> paradox in naive set theory.
> You mean in FregeÕs highly non-naive system - not in naive
set
> theory.
Sorry, yes, I should have been more precise.
Naive set theory is an underspeci\[CapitalThorn]ed beast, so whether the
Burali-Forti paradox or the Russell paradox really affect it
is hard
to answer clearly.
Thomas
===
Subject: Re: Russell-like paradoxes
:
: >
: > I wonder if Russell actually saw such a sign before he
exposed this
: > paradox in naive set theory.
: >
: > You mean in FregeÕs highly non-naive system - not in
naive set
: > theory.
:
: Sorry, yes, I should have been more precise.
No, really, you shouldnÕt have.
Torkel should learn to wait until he has more to say than
irrelevant 1-liners before posting.
: Naive set theory is an underspeci\[CapitalThorn]ed beast,
Maybe, but thatÕs not the point. The point is that
unrestricted comprehension is dangerous.
: so whether the
: Burali-Forti paradox or the Russell paradox really affect
it is hard
: to answer clearly.
No, it isnÕt. Even though naive set theory is under-
speci\[CapitalThorn]ed, it is ALWAYS speci\[CapitalThorn]ed well enough for \
people to
know that it includes unrestricted comprehension. That is
enough
for RussellÕs paradox.
--
---
ItÕs dif\[CapitalThorn]cult ... you need to be united to have \
any
strength, but internal issues have to be addressed.
--- E. Ray Lewis, on liberalism in America
===
Subject: Re: Russell-like paradoxes
permission for an emailed response.
in
mourning
> : Burali-Forti paradox or the Russell paradox really affect
it is hard
> : to answer clearly.
> No, it isnÕt. Even though naive set theory is under-
> speci\[CapitalThorn]ed, it is ALWAYS speci\[CapitalThorn]ed well enough \
for people to
> know that it includes unrestricted comprehension. That is
enough
> for RussellÕs paradox.
Well, the reason for my semi-retraction is twofold:
To give a good example to , and
At least one naive set theory text on my bookshelf doesnÕt
have
unrestricted comprehension, but instead waves about with this
is a
dangerous area when it gets near the universal set, or the
largest
ordinal, and whatnot., and says you canÕt just comprehend
anything
without problems.
Thomas
===
Subject: Re: Russell-like paradoxes
: At least one naive set theory text on my bookshelf doesnÕt
have
: unrestricted comprehension,
IÕm sorry, I donÕt believe you.
: but instead waves about with this is a
: dangerous area
Well, WHY is it dangerous, if it restricts comprehension?
If you restrict comprehension, you can eliminate the danger;
that
is the whole reason why you accept to the restriction! It
would certainly
be BAD to accept restrictions and STILL be in danger, would
it not??
: when it gets near the universal set, or the largest
: ordinal, and whatnot.,
Well, at this point, the antecedent of it needs clarifying.
IÕll not wax so pedantic as to demand that you post the \
bookÕs
actual particular axiomatization, but my point is, its
framework
either calls a universal set into existence or it doesnÕt,
and if
it does, it either suffers from RussellÕs paradox or it
doesnÕt.
If it does, then saying it lacks unrestricted comprehension
is almost irrelevant: itÕs got something provably just as
dangerous.
: and says you canÕt just comprehend anything
: without problems.
But to be *aware* of PRECISELY *this* is PRECISELY what it
means NOT to be *naive*, in the relevant sense.
More to the point, why would it *need* to warn, you canÕt \
just
comprehend anything IF it was not (on the theoretical basis
of what was presented before the warning) in fact about to
ALLOW you to comprehend anything? To say you canÕt just
comprehend anything IS, dictionarially, restricting
comprehension.
If the theory-as-presented actually incorporated this
natural-language
prohibition mathematically, into its axioms, then it was not
naive
in the relevant sense. If it didnÕt, then \
thatÕs what it
means for
its comprehension to be unrestricted.
--
---
ItÕs dif\[CapitalThorn]cult ... you need to be united to have \
any
strength, but internal issues have to be addressed.
--- E. Ray Lewis, on liberalism in America
===
Subject: Re: Russell-like paradoxes
> At least one naive set theory text on my bookshelf doesnÕt
have
> unrestricted comprehension,
In what naive set theory do you in fact \[CapitalThorn]nd unrestricted
comprehension?
===
Subject: Re: Russell-like paradoxes
:
: > At least one naive set theory text on my bookshelf
doesnÕt have
: > unrestricted comprehension,
: In what naive set theory do you in fact \[CapitalThorn]nd unrestricted
: comprehension?
All of them, by de\[CapitalThorn]nition.
--
---
ItÕs dif\[CapitalThorn]cult ... you need to be united to have \
any
strength, but internal issues have to be addressed.
--- E. Ray Lewis, on liberalism in America
===
Subject: Re: Russell-like paradoxes
permission for an emailed response.
> At least one naive set theory text on my bookshelf doesnÕt
have
> unrestricted comprehension,
> In what naive set theory do you in fact \[CapitalThorn]nd unrestricted
> comprehension?
As I said, naive set theory as a term is underspeci\[CapitalThorn]ed, and
different texts give different accounts.
Barwise and EtchemendyÕs Language, Proof, & Logic, pp
405-441, in
describing naive set theory, then builds up to RussellÕs
paradox, and
then shows a ZFC axiom. On page 408, in the section titled
Naive Set
Theory:
The second principle of naive set theory is the so-called
Unrestricted Comprehension Axiom. It states, roughly, that
every
determinate property determines a set. That is, given any
determinate property P, there is a set of all objects that
have this
property.... This way of talking about the Axiom of
Comprehension
has a certain problem, namely it talks about properties. We
donÕt
want to get into the business of having to axiomatize
properties as
well as sets. To get around this, we use formulas of
\[CapitalThorn]rst-order
logic. Thus, for each formula P(x) of FOL, we take as a basic
axiom
the following: EaVx[x in a <-> P(x)].
By contrast, the textbook Shen and Vereshchagin Basic Set
Theory,
takes the approach of saying that there are danger areas,
which they
warn about, and describe ZFC as giving safety rules to keep
one out
of the danger areas.
This difference is of course related to the difference in the
point of
these two books. LPL is concerned with showing the value and
importance of FOL axiomatizations, whereas Basic Set Theory is
concerned with introducing potential working mathematicians
to the
necessary basics of set theory.
Thomas
===
Subject: Re: Russell-like paradoxes
permission for an emailed response.
The following page describes naive set theory implying limited
comprehension:
http://www.math.niu.edu/~rusin/known-math/index/03EXX.html
Naive set theory considers elementary properties of the union
and
intersection operators -- Venn diagrams, the DeMorgan laws,
elementary counting techniques such as the inclusion-exclusion
principle, partially ordered sets, and so on. This is perhaps
as
much of set theory as the typical mathematician uses. Indeed,
one
may construct the natural numbers, real numbers, and so on in
this
framework. However, situations such as RussellÕs paradox \
show
that
some care must be taken to de\[CapitalThorn]ne what, precisely, is a set.
So here naive set theory means some kind of subset of
axiomatic
set theory. Lest one get a paradox, vaguely de\[CapitalThorn]ned some care
must
be taken. Moreover, it is clear that this naive set theory is
really used.
Thomas
===
Subject: Re: Russell-like paradoxes
permission for an emailed response.
HereÕs a lecture note page I ran across, as evidence for my
claim that
the phrase naive set theory often refers to a system with
vaguely
limited comprehension:
http://www.cs.odu.edu/~toida/nerzic/content/set/intr_to_
set.html:
Though the concept of set is fundamental to mathematics, it
is not
going to be de\[CapitalThorn]ned rigorously here. Instead we rely on
everyoneÕs
notion of set as a collection of objects or a container of
objects. In that sense set is an unde\[CapitalThorn]ned concept here.
Similarly
we say an object belongs to or is a member of a set without
rigorously de\[CapitalThorn]ning what it means. This approach to set theory
is
called naive set theory as opposed to axiomatic set theory.
The
naive set theory produces paradoxes such as RussellÕs
paradox, hence
it is not consistent, meaning that a statement which should
be true
may not be proven true following the naive set theory.
However, it is
simpler and practically all the results we need can be
derived within
the naive set theory. Thus we shall be following this naive
set theory
in this course.
So the course will use naive set theory, which produces
paradoxes,
but at the same time since practically all the results we
need can be
derived within the naive set theory.
Now if naive set theory produces real antinomies, then of
course all
the results we need can be derived in it. And a lot more
results that
we donÕt want either.
So the authors of that paragraph are dancing a \[CapitalThorn]ne line, and
are
problably saying something strictly incoherent. They are
saying that
naive set theory is inconsistent, and they are saying that it
matters
what results it proves.
But there is another interpretation, in which they mean to
say it
provise practically all the results we need, and then they
say in
the back of their head and we will only use methods that we
know can
be repeated in a proper axiomatized [ZFC, GBN, etc] set
theory.
Imagine if a bright student comes up and says hey, if itÕs
inconsistent, then what does it matter if a result can be
proved in
it---*anything* can be proved in it?! The instructor would
answer by
saying... that uses of comprehension will be limited in
certain
ways...which donÕt produce the paradoxes...
Now if you want naive set theory to refer to a single,
well-de\[CapitalThorn]ned
thing, then sure, it probably must refer to the unrestricted
comprehension system. If there is an axiom system for naive
set
theory, it must be FregeÕs or one like it.
But my whole point is that people very frequently use naive
set
theory in a vague hand-wavy sort of way, which does not
conform to
the expectation that it be a single well-de\[CapitalThorn]ned thing.
Thomas
===
Subject: Re: Russell-like paradoxes
permission for an emailed response.
conviction
http://www.cs.nyu.edu/pipermail/fom/1998-September/002167.html
(which gives an interesting variation on FreilingÕs \
argument,
I
thought), describes there as being mathematicians who accept
naive
set-theory. Now perhaps Soren Riis thinks that these
mathematicians
really accept a system which is inconsistent. But I donÕt
think
thatÕs what he means.
I think he means naive set theory to include a vaguely
restricted
comprehension, without committing to any particular
restriction or any
particular axiomatization.
Thomas
===
Subject: Re: Russell-like paradoxes
> Barwise and EtchemendyÕs Language, Proof, & Logic, pp
405-441, in
> describing naive set theory, then builds up to RussellÕs
paradox, and
> then shows a ZFC axiom. On page 408, in the section titled
Naive Set
> Theory:
> The second principle of naive set theory is the so-called
> Unrestricted Comprehension Axiom.
But where is this naive set theory to be found? Has anybody
ever
used naive set theory?
===
Subject: Re: Russell-like paradoxes
:
: > Barwise and EtchemendyÕs Language, Proof, & Logic, pp
405-441, in
: > describing naive set theory, then builds up to RussellÕs
paradox, and
: > then shows a ZFC axiom. On page 408, in the section
titled Naive
Set
: > Theory:
: > The second principle of naive set theory is the so-called
: > Unrestricted Comprehension Axiom.
: But where is this naive set theory to be found?
Well, last&least, on pp.405-441 of this book.
: Has anybody ever used naive set theory?
Well, since it is inconsistent, maybe not.
Used is too strong a term. People have attempted
to axiomatize set theory and possibly stumbled across it
along the way. People (speci\[CapitalThorn]cally Frege) have assumed that
unrestricted comprehension was legitimate and have used it.
There was a lot of it going on before it was FIGURED OUT that
comprehension needed to be limited. Speci\[CapitalThorn]cally, Russell
of set theory that cured it was published by Zermelo in 1908.
All IÕm saying is that people were forming sets without \
being
careful about how, for several years prior to 1903.
Since you already knew this, the motivation for your
question remains tragically obscure, but while you may
have just barely avoided actual impropriety here, you have
certainly not avoided the appearance of impropriety.
--
---
ItÕs dif\[CapitalThorn]cult ... you need to be united to have \
any
strength, but internal issues have to be addressed.
--- E. Ray Lewis, on liberalism in America
===
Subject: Re: Russell-like paradoxes
> Has anybody ever used naive set theory?
Actually, that question is already moot.
6.5 For an answer which cannot be expressed the question
too cannot be expressed.
(L. Wittgenstein, TLP)
As Thomas Bushnell said: Ônaive set theoryÕ as a \
term is
underspeci\[CapitalThorn]ed.
6.53
The right method of philosophy would be this: To say nothing
except what can be said, i.e. the propositions of natural
science, i.e. something that has nothing to do with
philosophy:
and then always, when someone else wished to say something
metaphysical, to demonstrate to him that he had given no
meaning to certain signs in his propositions. This method
would
be unsatisfying to the other -- he would not have the feeling
that we were teaching him philosophy -- but it would be the
only strictly correct method.
In this case the meaning of the signs
naive set theory
in Torkels question [i.e. in its context], is underspeci\[CapitalThorn]ed.
Still, we might consider the following to be a reasonable
answer:
> Well, since it is inconsistent, maybe not.
But...
> Used is too strong a term. People have attempted
> to axiomatize set theory and possibly stumbled across it
> along the way. People (speci\[CapitalThorn]cally Frege) have assumed that
> unrestricted comprehension was legitimate and have used it.
Right.
> There was a lot of it going on before it was FIGURED OUT
that
> comprehension needed to be limited [somehow].
Well. Actually, CANTOR *himself* knew about that! But for
many others it
w a s a discovery.
> axiomatization of set theory that cured it was published by
Zermelo
> in 1908.
Right. Zermelo actually STATES in his paper that he considers
his theory
to be a cure of (for?) set theory.
> All IÕm saying is that [many] people were forming sets
without being
> careful about how, for several years prior to 1903.
Right.
> Since you already knew this, the motivation for your
> question remains tragically obscure [...]
Not really, I guess. :-)
Probably his point is that CANTOR never (actually) used naive
set
theory (i.e. allowed for unrestricted comprehension). And he
is right.
On the other hand..., itÕs ALSO true, that Cantor never
published
ANYTHING that could be considered a comprehensive description
of his
theory that actually forms a _Set Theory_ in which the more
obvious
antinomies cannot arise. If Torkel thinks otherwise it would
be NICE if
he could describe the (this) PRINCIPLES [axioms] of CantorÕs
theory.
F.
===
Subject: Re: Russell-like paradoxes
permission for an emailed response.
> > Barwise and EtchemendyÕs Language, Proof, & Logic, pp
405-441, in
> > describing naive set theory, then builds up to RussellÕs
paradox, and
> > then shows a ZFC axiom. On page 408, in the section
titled Naive
Set
> > Theory:
> >
> > The second principle of naive set theory is the so-called
> > Unrestricted Comprehension Axiom.
> But where is this naive set theory to be found? Has anybody
ever
> used naive set theory?
What I said was that the term naive set theory is
underdetermined,
and that some texts on my shelf say it has unrestricted
comprehension,
and some instead say it doesnÕt, but are vague (and
non-axiomatic, and
non-rigorous) about what the restrictions are.
texts. (HalmosÕs Naive Set Theory isnÕt at my \
home, so I canÕt
check it as easily, or I would have.)
Now you shift, onto a question of where is this to be found.
Well,
one place it is to be found is in Barwise and EtchemendyÕs
LPL.
Naive set theory just isnÕt a single \
well-de\[CapitalThorn]ned thing.
ThatÕs my
point. Your insistence that it *must* be, thus de\[CapitalThorn]ning out of
existence the texts which take an opposing view on what it
is, does
not demonstrate that, in *fact*, the term naive set theory
refers to
a variety of different things, and simply has no one
universal and
well understood de\[CapitalThorn]nition.
Thomas
===
Subject: Re: Russell-like paradoxes
> texts. (HalmosÕs Naive Set Theory isnÕt at \
my home, so I
canÕt
> check it as easily, or I would have.)
That book presents ZFC.
> Now you shift, onto a question of where is this to be
found. Well,
> one place it is to be found is in Barwise and EtchemendyÕs
LPL.
You mean that the term naive set theory is to be found
there. Sure. But if you say that Russell showed naive set
theory to
be inconsistent, there is a strong suggestion that somebody
had
actually formulated, explicitly or implicitly, such a theory.
Is this
the case?
===
Subject: Re: Russell-like paradoxes
> texts. (HalmosÕs Naive Set Theory isnÕt at \
my home, so I
canÕt
> check it as easily, or I would have.)
> That book presents ZFC.
> Now you shift, onto a question of where is this to be
found. Well,
> one place it is to be found is in Barwise and EtchemendyÕs
LPL.
> You mean that the term naive set theory is to be found
> there. Sure. But if you say that Russell showed naive set
theory to
> be inconsistent, there is a strong suggestion that somebody
had
> actually formulated, explicitly or implicitly, such a
theory. Is this
> the case?
Hi Torkel,
Yes, it is the case that, for example, FregeÕs set theory is
naive.
It is naive to assume that all predicates have an extension.
It is naive to assume that EyAx(x e y <-> Fx) is true for all
predicates F.
~EyAx(x e y <-> ~(x e x)) is a theorem.
FregeÕs axiom V: Ax(Fx <-> Gx) -> {x:Fx}={x:Gx}, is naive,
because it is
not
valid.
It fails if either {x:Fx} or {x:Gx} do not exist.
Ax(~(x e x) <-> ~(x e x)) -> {x:~(x e x)}={x:~(x e x)} fails.
Because,
Ax(~(x e x) <-> ~(x e x)) is tautologous and {x:~(x e
x)}={x:~(x e x)}is
contradictory.
It is naive to assume: y e {x:Fx} <-> Fy is true for all \
FÕs.
Because it
fails if {x:Fx} does not exist.
~Ey(y e {x:~(x e x)}) is a theorem that conßicts with the
naive set
theories of: Frege, Cantor , Quine, etc..
Witt
===
Subject: Re: Russell-like paradoxes
permission for an emailed response.
> Now you shift, onto a question of where is this to be
found. Well,
> one place it is to be found is in Barwise and EtchemendyÕs
LPL.
> You mean that the term naive set theory is to be found
> there. Sure. But if you say that Russell showed naive set
theory to
> be inconsistent, there is a strong suggestion that somebody
had
> actually formulated, explicitly or implicitly, such a
theory. Is this
> the case?
No, I mean that Barwise and Etchemendy explicitly state that
what they
present is naive set theory. This does not prove what naive
set
theory means. But it *does* prove that the term is used, in
print, to
refer to a system with unlimited comprehension.
My point was that there are divergent uses of the term in
print. If
you want to pick one and say thatÕs the right one, go ahead.
Just
be careful, because there are plenty out there in print
saying the
other one--for whichever you print.
Since the term Naive Set Theory may only go back to HalmasÕs
book,
which he titles Naive Set Theory, and then says isnÕt about
naive
set theory, there is something amusing in
1) Insisting that the term is relevant to what happened a
hundred
years ago, and
2) Insisting that it must have a sure rigid meaning.
Thomas
===
Subject: Re: Russell-like paradoxes
> 1) Insisting that the term is relevant to what happened a
hundred
> years ago, and
Given that the term is irrelevant to what happened a hundred
years
ago (e.g. in set theory), why did you wonder if Russell
actually saw
such a sign before he exposed this paradox in naive set
theory?
===
Subject: Re: Russell-like paradoxes
permission for an emailed response.
said
haltingly
> 1) Insisting that the term is relevant to what happened a
hundred
> years ago, and
> Given that the term is irrelevant to what happened a
hundred years
> ago (e.g. in set theory), why did you wonder if Russell
actually saw
> such a sign before he exposed this paradox in naive set
theory?
Good grief, you are an annoying little pedant, arenÕt you?
So far you have screwed up several times, but IÕm sure that
wonÕt stop
you in your continual attempt to make me look bad.
Yes, people did formulate a family of theories which *today*
we call
naive set theory; at RussellÕs time, it was not called naive
set
theory.
Indeed, FregeÕs theory is rightly called naive *today*--and
is, in
print--because it is unaware of the paradoxes and naively
proceeds as
if they wonÕt occur.
Did Russell call FregeÕs theory naive? No.
Is the *term* relevant to what happened a hundred years ago?
No.
Does that mean that the *concept* is not important? No. The
*concept* is different from the *term*.
Perhaps the problem is that your English isnÕt so good.
Regardless,
please take a big step back and stop the stupid attempts to
score
points. Is that really all that fun?
In any case, I hereby award you a zillion points, thus saving
you from
any further need to score any.
Thomas
===
Subject: Re: Russell-like paradoxes
> Yes, people did formulate a family of theories which
*today* we call
> naive set theory; at RussellÕs time, it was not called
naive set
> theory.
Who formulated such a theory? Cantor certainly didnÕt, and
itÕs far
from clear who else you might have in mind.
===
Subject: Re: Russell-like paradoxes
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<87wu8vbyjv.fsf@becket.becket.net>
<vcb8ylbdcxo.fsf@beta13.sm.luth.se> Yes, people did formulate
a family of theories which *today* we call
>> naive set theory; at RussellÕs time, it was not called
naive set
>> theory.
> Who formulated such a theory? Cantor certainly didnÕt, and
itÕs far
> from clear who else you might have in mind.
arguably, naive set theory is not a theory at all, in the
current
logical sense;
(but that doesnÕt stop the term from being meaningful,
IMHO).
--
Alan Smaill email: A.Smaill@ed.ac.uk
School of Informatics tel: 44-131-650-2710
University of Edinburgh
===
Subject: Re: Russell-like paradoxes
Alan Smaill says...
>arguably, naive set theory is not a theory at all, in the
current
>logical sense;
>(but that doesnÕt stop the term from being meaningful,
>IMHO).
The way I understood it, using naive set theory means using
de\[CapitalThorn]nitions
such as the set of all x such that Phi(x) without worrying
too much
about the sorts of formulas Phi(x) for which this de\[CapitalThorn]nition
makes
sense.
--
Daryl McCullough
Ithaca, NY
===
Subject: Re: Russell-like paradoxes
: The way I understood it, using naive set theory means using
de\[CapitalThorn]nitions
: such as the set of all x such that Phi(x) without worrying
too much
: about the sorts of formulas Phi(x) for which this de\[CapitalThorn]nition
makes
: sense.
It always makes sense;
it just doesnÕt always have reference.
It causes whatever trouble it causes in virtue of the
PARTICULAR sense that is made by the inconvenient
applications.
It is not the case that P&~P does not make sense or is
meaningless --
it is contradictory precisely BECAUSE it has the particular
meaning that it
has.
--
---
ItÕs dif\[CapitalThorn]cult ... you need to be united to have \
any
strength, but internal issues have to be addressed.
--- E. Ray Lewis, on liberalism in America
===
Subject: Re: Russell-like paradoxes
> (but that doesnÕt stop the term from being meaningful,
> IMHO).
As a technical term, itÕs perfectly meaningful. In the
context of
discussions of RussellÕs paradox, it is too often used as
though
referring to some set theory used at the time, e.g. by Cantor.
===
Subject: Re: Russell-like paradoxes
permission for an emailed response.
Tom said
randomly
> Yes, people did formulate a family of theories which
*today* we call
> naive set theory; at RussellÕs time, it was not called
naive set
> theory.
> Who formulated such a theory? Cantor certainly didnÕt, and
itÕs far
> from clear who else you might have in mind.
Have you heard of Frege? His set theory certainly naive.
And Cantor certainly did, to the extent he is rightly the \
\[CapitalThorn]rst
serious discoverer of Set Theory.
But the funny thing is that you are trying to distract
attention from
*your* screw up, which was your silly claim implication
nobody thinks naive
set theory includes unrestricted comprehension.
When I make mistakes, I have the decency and honesty to admit
them.
Do you?
Thomas
===
Subject: Re: Russell-like paradoxes
> Have you heard of Frege? His set theory certainly naive.
I donÕt think you want to describe FregeÕs \
Grundgesetze
system as
naive set theory in the sense of Barwise and Etchmendy.
> And Cantor certainly did, to the extent he is rightly the
\[CapitalThorn]rst
> serious discoverer of Set Theory.
Cantor did indeed create set theory. Since he did not
introduce any
unrestricted comprehension axiom there is no basis for the
idea that
he used or introduced naive set theory in your technical
sense.
===
Subject: Re: Russell-like paradoxes
: Naive set theory just isnÕt a single \
well-de\[CapitalThorn]ned thing.
ThatÕs my
: point.
But it is well-enough de\[CapitalThorn]ned to make its comprehension, if \
not
unrestricted, at least overbroad enough to get you into
trouble.
Naive already has a dictionary meaning in natural language,
before
set theory comes along. Even if naive set theory is not well
de\[CapitalThorn]ned, all of the various things it could be \
de\[CapitalThorn]ned as STILL
have to merit being called naive about something. The
importance of
not waxing overbroad in comprehension is usually that
something.
: Your insistence that it *must* be, thus de\[CapitalThorn]ning out of
: existence the texts which take an opposing view on what it
is,
The text you cite DOES NOT take an opposing view on what it
is.
You cannot hope to demonstrate that the text you cited thinks
that
it can restrict comprehension and still call itself naive.
That text
does not in fact do that.
: does
: not demonstrate that, in *fact*, the term naive set theory
refers to
: a variety of different things, and simply has no one
universal and
: well understood de\[CapitalThorn]nition.
It is universally well understood that it is about avoiding
things
like RussellÕs paradox that can arise from unrestricted
comprehension.
The set theory that you were trying to allege was naive but
lacked
unrestricted comprehension does in fact restrict
comprehension, but
it is NOT naive. ThatÕs not due to some choice made by \
Torkel
Franzen;
itÕs due to the communityÕs linguistic \
practice generally,
which,
around this particular term and issue, is in fact more
homogeneous
than you are giving it credit for.
But Torkel is still abusing you.
--
---
ItÕs dif\[CapitalThorn]cult ... you need to be united to have \
any
strength, but internal issues have to be addressed.
--- E. Ray Lewis, on liberalism in America
===
Subject: Re: Russell-like paradoxes
permission for an emailed response.
ROCKAWAY!!
> : existence the texts which take an opposing view on what
it is,
> The text you cite DOES NOT take an opposing view on what it
is.
> You cannot hope to demonstrate that the text you cited
thinks that
> it can restrict comprehension and still call itself naive.
That text
> does not in fact do that.
Huh? The back cover identi\[CapitalThorn]es it as a presentation of naive
set
theory; the text is a translation from Russian, so perhaps
itÕs a
poor guide about usage.
At best, what you are saying is that the authors of the book,
or the
using the term wrongly.
Ok--then \[CapitalThorn]ne, but they are still using it in a certain way,
to refer
to a vaguely and suggestively limited comprehension.
> It is universally well understood that it is about avoiding
things
> like RussellÕs paradox that can arise from unrestricted
comprehension.
> The set theory that you were trying to allege was naive but
lacked
> unrestricted comprehension does in fact restrict
comprehension, but
> it is NOT naive.
IÕm not trying to allege itÕs naive. \
IÕm saying that it is in
print
labelled as naive. That labelling may well be wrong, but it
is there,
nontheless.
I think I would generally agree that the usage of naive to
mean
vaguely limited comprehension is a disappointing usage. I
would
prefer to restrict it just as you do, and in my own usage, I
tend to.
One also hears of informal set theory to be the vaguely
limited
comprehension version; thatÕs probably the best way to
describe the
book I refer to as well as Halmos.
Wikipedia thinks that the name Naive set theory may well have
originated with HalmosÕ book, which is odd, given his \
preface.
> But Torkel is still abusing you.
Well, I made a mistake here not too long ago, which I
corrected after
about two posts. He apparently thinks that if a person makes a
mistake, itÕs fair game from then on to take random pot \
shots
without
limitation.
Thomas
===
Subject: Re: Russell-like paradoxes
> HalmosÕs Naive Set Theory isnÕt at my home, \
so I canÕt
> check it as easily, or I would have.
HalmosÕ book is -despite itÕs title- NOT about \
naive set
theory. :-)
(He concedes that in the preface of the book. :-)
Actually, he just describes good old ZFC in the book.
F.
===
Subject: Re: Russell-like paradoxes
> ... <->x ...
> is just RussellÕs shorthand notation for
>
> (x)(... <-> ...),
>
> It seems strange to refer to a variable and only later in
the
> expression indicate that it is universally quanti\[CapitalThorn]ed.
> Huh?
> Actually, thereÕs NOTHING strange concerning that \
notation.
ItÕs just
> not common these days any more. ThatÕs all.
strange [straengz]: adjective: not common
> (Hint: itÕs just an in\[CapitalThorn]x notation.)
In\[CapitalThorn]x notation refers to functions (often called operators in
this
context), not quanti\[CapitalThorn]ers.
Predicate Calculus puts quanti\[CapitalThorn]ers before any reference to \
the
variable quanti\[CapitalThorn]ed, so that you know when you get to it
(without
lookahead) - sort of like how one would naturally say it, For
all x .
. .
> Look, man, things would be simpler if you werenÕt such an
ignorant
> bonehead.
Whether someone considers something strange or not is a
funnction of
their value system, not their intellect.
> F.
Charlie Volkstorf
Cambridge, MA
http://www.mathpreprints.com/math/Preprint/CharlieVolkstorf/
20021008.1/1
http://www.arxiv.org/html/cs.lo/0003071
===
Subject: Re: Russell-like paradoxes
> In\[CapitalThorn]x notation refers to functions (often called operators
in this
> context), not quanti\[CapitalThorn]ers.
Go away, idiot.
===
Subject: homotopy
Hi all,
Dfn.Maps f,g:X->Y are homotopic if thereÕs a map H: XxI -> Y
s.t.
H(x,0)=f(x) and H(x,1)=g(x) for all x in X.
Now, this seems \[CapitalThorn]ne for spaces like E^n, but what about the
spaces
with bigger cardinality than aleph_1 ? The homotopy is set of
(aleph_1) maps continuously transforming f into g, but in
bigger
spaces dont we need more maps between f and g ?
Jore
===
Subject: Re: homotopy
>Hi all,
>Dfn.Maps f,g:X->Y are homotopic if thereÕs a map H: XxI -> \
Y
s.t.
>H(x,0)=f(x) and H(x,1)=g(x) for all x in X.
>Now, this seems \[CapitalThorn]ne for spaces like E^n, but what about the
spaces
>with bigger cardinality than aleph_1 ?
You should note that the cardinality of E^n is c, or
2^aleph_0, _not_
aleph_1. (More precisely, it can be proved that the usual
axioms
of set theory do not determine whether or not c = aleph_1.)
>The homotopy is set of
>(aleph_1) maps continuously transforming f into g, but in
bigger
>spaces dont we need more maps between f and g ?
I donÕt know why we need this - in the situation \
youÕre
worried
about thereÕs no problem with the de\[CapitalThorn]nition, \
there just may be
fewer homotopies than youÕd like.
A person might de\[CapitalThorn]ne a generalization of homotopy as follows
(I imagine people have already done this): Say
Dfn.Maps f,g:X->Y are topologically-homotopic if there exist
a connected space C, points a, b in C, and a (continuous!) map
H: XxC -> Y s.t. H(x,a)=f(x) and H(x,b)=g(x) for all x in X.
>Jore
************************
David C. Ullrich
===
Subject: Re: homotopy
>Hi all,
>Dfn.Maps f,g:X->Y are homotopic if thereÕs a map H: XxI -> \
Y
s.t.
>H(x,0)=f(x) and H(x,1)=g(x) for all x in X.
>Now, this seems \[CapitalThorn]ne for spaces like E^n, but what about the
spaces
>with bigger cardinality than aleph_1 ?
> You should note that the cardinality of E^n is c, or
2^aleph_0, _not_
> aleph_1. (More precisely, it can be proved that the usual
axioms
> of set theory do not determine whether or not c = aleph_1.)
>The homotopy is set of
>(aleph_1) maps continuously transforming f into g, but in
bigger
>spaces dont we need more maps between f and g ?
> I donÕt know why we need this - in the situation \
youÕre
worried
> about thereÕs no problem with the de\[CapitalThorn]nition, \
there just may
be
> fewer homotopies than youÕd like.
> A person might de\[CapitalThorn]ne a generalization of homotopy as \
follows
> (I imagine people have already done this): Say
> Dfn.Maps f,g:X->Y are topologically-homotopic if there exist
> a connected space C, points a, b in C, and a (continuous!)
map
> H: XxC -> Y s.t. H(x,a)=f(x) and H(x,b)=g(x) for all x in X.
>Jore
> ************************
> David C. Ullrich
There are several interesting questions here. First off, the
original
question is entirely reasonable. For instance, suppose you
try to
de\[CapitalThorn]ne the fundamental group as the group of deck
transformations of
the universal covering space. In that case the long circle
(def
below) will have Z as its fundamental group. If you de\[CapitalThorn]ne it
by
homotopy classes of closed paths, it turns out there are no
closed
paths in the usual sense. (This is not quite the same
question, but
is closely related.) You cannot get a short path around a long
circle. And although I donÕt know any examples offhand, I
would be
surprised if there werenÕt some examples of closed paths \
that
were not
homotopic with the usual line, but were with the long line.
So yes,
you could develop a theory along those lines (no pun
intended).
The long line is gotten by taking the \[CapitalThorn]rst uncountable
ordinal Omega
and inserting a copy of the unit interval between alpha and
alpha + 1,
for each countable ordinal alpha. If you add Omega to that
space, you
get a compact space called the long line. It is compact and
connected. Now identify Omega with 0 and you have the long
circle.
You could use the long circle to de\[CapitalThorn]ne long paths and the
long line
to de\[CapitalThorn]ne homotopies among them. But then you could replace
Omega by
some arbitrarily large ordinal. So maybe you should take the
direct
limit of all the groups you get. Or just use the universal
covering,
assuming the space is locally simply connected.
===
Subject: Re: homotopy
>Dfn.Maps f,g:X->Y are homotopic if thereÕs a map H: XxI -> \
Y
s.t.
>H(x,0)=f(x) and H(x,1)=g(x) for all x in X.
> >Now, this seems \[CapitalThorn]ne for spaces like E^n, but what about
the spaces
>with bigger cardinality than aleph_1 ?
> A person might de\[CapitalThorn]ne a generalization of homotopy as \
follows
> (I imagine people have already done this): Say
> Dfn.Maps f,g:X->Y are topologically-homotopic if there exist
> a connected space C, points a, b in C, and a (continuous!)
map
> H: XxC -> Y s.t. H(x,a)=f(x) and H(x,b)=g(x) for all x in X.
IÕd suggest a linear continuum such as the long line.
> There are several interesting questions here. First off,
the original
> question is entirely reasonable. For instance, suppose you
try to
> de\[CapitalThorn]ne the fundamental group as the group of deck
transformations of
> the universal covering space. In that case the long circle
(def
> below) will have Z as its fundamental group. If you de\[CapitalThorn]ne
it by
> homotopy classes of closed paths, it turns out there are no
closed
> paths in the usual sense. (This is not quite the same
question, but
> is closely related.) You cannot get a short path around a
long
> circle. And although I donÕt know any examples offhand, I
would be
> surprised if there werenÕt some examples of closed paths
that were not
> homotopic with the usual line, but were with the long line.
So yes,
> you could develop a theory along those lines (no pun
intended).
Or any other linear continuum. Give a homotopy based upon a
linear
continuum C, would not a C-contractible space be C-path
connected?
If D is another linear continuum with greater cardinality
than C,
can C be embedded in D? If a space is D-path connected, it may
not be C-path connected. If a space is C-path connected, will
it be D-path connected? Same speculations also fantasized for
homotopy.
Now de\[CapitalThorn]ne shomotopic to be C-homotopic for some linear
continuum C.
Is shomotopic transitive? Yes, if all of my dreams above come
true.
> The long line is gotten by taking the \[CapitalThorn]rst uncountable
ordinal Omega
> and inserting a copy of the unit interval between alpha and
alpha + 1,
The same as lexicographically ordered Omega x [0,1)
> for each countable ordinal alpha. If you add Omega to that
space, you
> get a compact space called the long line. It is compact and
The same as lexicographically ordered Omega x [0,1) /
{Omega}x[0,1]
Is lexicographically ordered Omega x [0,1) / {Omega}x[0,1]
homeomorphic or order ismorphic to
lexicographically ordered Omega x [0,1) / {(Omega,0)} ?
> connected. Now identify Omega with 0 and you have the long
circle.
> You could use the long circle to de\[CapitalThorn]ne long paths and the
long line
> to de\[CapitalThorn]ne homotopies among them. But then you could replace
Omega by
> some arbitrarily large ordinal. So maybe you should take
the direct
> limit of all the groups you get. Or just use the universal
covering,
> assuming the space is locally simply connected.
The longer line is same construction based upon omega_2 + 1
If C is a linear continuum, then is C homeomorphic
to a long line based upon some initial ordinal?
If C is the long Omega+1 line and D is the long Omega+omega+1
line are they homeomorphic?
A more doubtful homeomorphism is when C is the long omega_nu
+ 1 line
and D is the long omega_(nu+1) + omega_nu + 1 line
or the long omega_nu + omega + 1 line.
Perhaps for my thoughts about C-homotopic, C-contractible and
C-path
connected, a modi\[CapitalThorn]cation to C needs be made such as symmetric
(reversible) or homogeneous. This to have a nest of embedded
long lines
as quested for the transitivity of shomotopy.
Easiest thus is just to assume the nest of linear continuums
to be all the
long omega_nu lines or to \[CapitalThorn]t into ZF, all the long omega_nu
lines for nu
in sigma, the mother of all ordinals suf\[CapitalThorn]cient for the topic
at hand.
===
Subject: Re: homotopy
> There are several interesting questions here. First off,
the original
> question is entirely reasonable. For instance, suppose you
try to
> de\[CapitalThorn]ne the fundamental group as the group of deck
transformations of
> the universal covering space. In that case the long circle
(def
> below) will have Z as its fundamental group. If you de\[CapitalThorn]ne
it by
> homotopy classes of closed paths, it turns out there are no
closed
> paths in the usual sense. (This is not quite the same
question, but
> is closely related.) You cannot get a short path around a
long
> circle. And although I donÕt know any examples offhand, I
would be
> surprised if there werenÕt some examples of closed paths
that were not
> homotopic with the usual line, but were with the long line.
So yes,
> you could develop a theory along those lines (no pun
intended).
I believe that SmaleÕs solenoid is such an example. It is \
the
maximal
attractor of the map in S x D, where S is the circle, and D
is the
subset of the complex numbers that have norm less than 2,
de\[CapitalThorn]ned by
f(s,d) = (2s, exp(is) + d/10)
The maximal attractor is a connected subset of S x D. It
wraps around
c-many times.
~ Chris
===
Subject: Re: homotopy
>Hi all,
>Dfn.Maps f,g:X->Y are homotopic if thereÕs a map H: XxI -> \
Y
s.t.
>H(x,0)=f(x) and H(x,1)=g(x) for all x in X.
>Now, this seems \[CapitalThorn]ne for spaces like E^n, but what about the
spaces
>with bigger cardinality than aleph_1 ? The homotopy is set of
>(aleph_1) maps continuously transforming f into g, but in
bigger
>spaces dont we need more maps between f and g ?
Why should we?
Lee Rudolph
===
Subject: Another math pun
You all know the story of ArchimedesÕ death,
trying to protect his circles from these Roman
louts?
ÔTwas the \[CapitalThorn]rst attempt in Discrete Math.
--
Hauke Reddmann <:-EX8
Private email:fc3a501@math.uni-hamburg.de
For our chemistry workgroup,remove math from the address
===
Subject: Re: Idiot discovers large power of 2.
ETAsAhRi2lNgyu0TQFGs+jdA+
YwdWLODQQIUILQcgW1gxq07RJ285YCy7i7LXSY=
How about:
2 is the only prime p such that some integers having residue
1 in Z_p
are not squares of p-adic integers.
--OL
===
Subject: Re: Idiot discovers large power of 2.
<Pine.LNX.4.40L0.0312121611010.9381-100000@cauchy.theorem.ca
> AP is reporting that this genius had 2 gigahertz of memery.
> > 2 billion cycles per second of memory? ThatÕs a rate, \
not
a
quantity.
> Welcome to the genius club.
> Kuinka?
> David Ames
Unfortunately, I donÕt understand Finnish... however, after
extensive use
of Google and a keen eye for context, I have inferred that
Kuinka?
means
How?
You see, the guy who posted the AP comment was poking fun at
the AP
reporter who confused gigahertz as a measurement of quantity
instead of
rate. It was a joke at APÕs expense, you see. However, you
then came in
with your comment explaining the error, which I assume
everyone else had
spotted right away given that it was the punchline to the
joke.
So, by pointing out something very obvious, you have become
the target of
my humourous jibe. Of course, now that IÕve taken the \
trouble
to explain
it, my jibe doesnÕt seem so funny anymore. However, I found
it hilarious
at the time. Sigh.
Iloinen?
-Mike
===
Subject: Re: Idiot discovers large power of 2.
> AP is reporting that this genius had 2 gigahertz of memery.
> > 2 billion cycles per second of memory? ThatÕs a rate, \
not
a
quantity.
> > Welcome to the genius club.
> Kuinka?
> David Ames
> Unfortunately, I donÕt understand Finnish... however, \
after
extensive use
> of Google and a keen eye for context, I have inferred that
Kuinka?
means
> How?
> You see, the guy who posted the AP comment was poking fun
at the AP
> reporter who confused gigahertz as a measurement of
quantity instead
of
> rate. It was a joke at APÕs expense, you see. However, you
then came in
> with your comment explaining the error, which I assume
everyone else had
> spotted right away given that it was the punchline to the
joke.
> So, by pointing out something very obvious, you have become
the target of
> my humourous jibe. Of course, now that IÕve taken the
trouble to explain
> it, my jibe doesnÕt seem so funny anymore. However, I \
found
it hilarious
> at the time. Sigh.
> Iloinen?
> -Mike
Joo. Iloinen.
(For those who donÕt read Zippy and therefore have no \
Finnish,
iloinen means happy.)
===
Subject: Re: Idiot discovers large power of 2.
>Joo. Iloinen.
>(For those who donÕt read Zippy and therefore have no
Finnish,
>iloinen means happy.)
Sounds very odd when used in this context. Replace with the
Finnish
word for pleased.
===
Subject: Re: Idiot discovers large power of 2.
David Ames <worldrecord@juno.com> scribbled the following:
>> Iloinen?
>> -Mike
> Joo. Iloinen.
> (For those who donÕt read Zippy and therefore have no
Finnish,
> iloinen means happy.)
I donÕt read Zippy. Does that mean I donÕt \
have Finnish? Voi
ei, kuinka
kauheaa!
--
/-- Joona Palaste (palaste@cc.helsinki.\[CapitalThorn]) -------------
Finland --------
-- http://www.helsinki.\[CapitalThorn]/~palaste ---------------------
rules! --------/
To doo bee doo bee doo.
- Frank Sinatra
===
Subject: How to code the non-trivial solution of Ax=0 type
simultaneous
equation.
Please, would you let me know how to make or code the
non-trivial
solution of a homogeneous equation(Ax=b, b=0 type).
A is 1473*1473 matrix
x is 1*1473
b is all 0 (zero).
How can I derive x, not Zero by C, Fortran or IDL?
===
Subject: Re: How to code the non-trivial solution of Ax=0
type simultaneous
equation.
> Please, would you let me know how to make or code the
non-trivial
> solution of a homogeneous equation(Ax=b, b=0 type).
> A is 1473*1473 matrix
> x is 1*1473
> b is all 0 (zero).
> How can I derive x, not Zero by C, Fortran or IDL?
You have to \[CapitalThorn]nd 0 kernel of the matrix.
You can do singular value decomposition.
You will get some singular values that are
not 0 and some that are 0. Throw away those
singular vectors that belong to non-zero
singular values. Then make - any - linear
combination of the remaining singular vectors
(suppose you got 2 of these, v1 and v2).
This will solve your problem:
x = c1*v1+c2*v2
where c1 and c2 are arbitrary numbers.
Matlab script:
A = [1 1; 0 0]
[u s v] = svd(A)
s =
1.4142 0
0 0
This means that \[CapitalThorn]rst vector has non-0 singular
value 1.4142, and second has 0 singular value.
So, use second vector in the solution. Lets use
c1=0.06 (whatever):
x = 0.06*v(:,2)
x =
-0.0424
0.0424
Lets check if this is a solution:
A*x
ans =
0
0
Yes , it is.
===
Subject: Re: limits, cardinalities
> ThatÕs a problem, all right.
> LetÕs think about this. Evidently there are at least two
things the
> notion of the limit of a sequence of sets might mean. One
of them
> is in fact the limit of a sequence of sets - the other one
is _not_
> actually a notion of the limit of a sequence of sets, itÕs
the
> limit of a structure involving a bunch of sets together with
> maps between them.
> Now we have a problem about the limit of a sequence of sets.
> With no maps given. DoesnÕt it seem plausible to assume \
that
> this was about the notion that only involves a sequence of
> sets?
> HereÕs what IÕm curious about. Say we \
de\[CapitalThorn]ne a sequence of
reals
> by letting x_n be sqrt(2) to n decimals: x_1 = 1.4, x_2 =
1.41, etc.
> Can we say that lim x_n = sqrt(2), or would that be keeping
special
> meanings for the digits in the decimal expansions?
But are you taking lim in the category of sets with the
categorical notion
of lim? (Should point out I mean direct limit, or colimit in
fact itÕs bad
to talk of lim when i mean colim without being able to put an
arrow under
the word). Obviously not: weÕre limits in the sense of
analysis. But how
would you relate that lim to the lim of a sequence of sets?
What would it
mean for Ôthe indicator function of x_n converges pointwise
to the
indicator function of x.Õ
But it was more sensible to assume that there were no maps
lying around,
in fact any usefulness for this de\[CapitalThorn]nition ceased pretty much
when it was
limited to limits indexed by naturals. ItÕs just an \
algebraic
version of
nets. But they are useful for constructing in\[CapitalThorn]nitely
generated modules
from \[CapitalThorn]nitely generated, sheafs as limits of compactly
supported sheafs,
and if you switch it round and take inverse limits, the
p-adics.
So it might have been useful for constructing some odd sets
from \[CapitalThorn]nite
sets.
Why the hell do I feel like this is an argument? Is this
normal for
usenet? I certainly wouldnÕt wish to argue against anyone \
who
riles JSH as
much as you do over something this trivial. I think IÕd
rather buy them a
beer. (This is me in placatory mode after being bloody
annoying about
something.)
So, yes, it was silly of me to demand maps between sets,
agreed.
===
Subject: Re: limits, cardinalities
>> ThatÕs a problem, all right.
>> LetÕs think about this. Evidently there are at least two
things the
>> notion of the limit of a sequence of sets might mean. One
of them
>> is in fact the limit of a sequence of sets - the other one
is _not_
>> actually a notion of the limit of a sequence of sets, \
itÕs
the
>> limit of a structure involving a bunch of sets together
with
>> maps between them.
>> Now we have a problem about the limit of a sequence of
sets.
>> With no maps given. DoesnÕt it seem plausible to assume
that
>> this was about the notion that only involves a sequence of
>> sets?
>> HereÕs what IÕm curious about. Say we \
de\[CapitalThorn]ne a sequence of
reals
>> by letting x_n be sqrt(2) to n decimals: x_1 = 1.4, x_2 =
1.41, etc.
>> Can we say that lim x_n = sqrt(2), or would that be
keeping special
>> meanings for the digits in the decimal expansions?
>But are you taking lim in the category of sets with the
categorical notion
>of lim?
No - what I canÕt \[CapitalThorn]gure out is why you thought \
that that was
the
notion the OP had in mind.
>(Should point out I mean direct limit, or colimit in fact
itÕs bad
>to talk of lim when i mean colim without being able to put
an arrow under
>the word).
I was going to point this out yesterday, but I looked it up,
and it
seems that there _is_ a standard notion of limit in category
theory - I read that inverse limits are limits but direct
limits
are not limits, for example, not that I have any idea what any
of those things are.
So when I read that I decided not to make the following
comment,
which IÕve now decided to _make_, since you say you were
really
talking about direct limits or colimits: DoesnÕt the fact
that he used
the word limit give further evidence that what youÕve been
talking
about is simply not what he had in mind?
>Obviously not: weÕre limits in the sense of analysis. But \
how
>would you relate that lim to the lim of a sequence of sets?
What would it
>mean for Ôthe indicator function of x_n converges pointwise
to the
>indicator function of x.Õ
The analogy was to convergence of sequences of _functions_,
not
sequences of numbers. If f_n is the indicator function of the
set
S_n (I put the word function in quotes because f_n is a proper
class) then S_n -> S in the sense the OP meant if and only if
f_n(x) -> f(x) for all x; that last is exactly pointwise
convergence
as in analysis.
>But it was more sensible to assume that there were no maps
lying around,
>in fact any usefulness for this de\[CapitalThorn]nition ceased pretty much
when it was
>limited to limits indexed by naturals. ItÕs just an
algebraic version of
>nets. But they are useful for constructing in\[CapitalThorn]nitely
generated modules
>from \[CapitalThorn]nitely generated, sheafs as limits of compactly
supported sheafs,
>and if you switch it round and take inverse limits, the
p-adics.
IÕve got no doubt that these categorical limits are very
useful (in
fact IÕve read that the standard de\[CapitalThorn]nition of \
the topology on
the class of test functions in an open set in R^n is the
inverse
or maybe it was direct limit of the natural topologies on the
test functions supported in a given compact set. So if I
understood the category stuff IÕd have a better idea what \
that
topology was - luckily the topology is irrelevant for
practical
purposes.) That question is totally independent of the
question
of what notion of limit was intended by the OP.
>So it might have been useful for constructing some odd sets
from \[CapitalThorn]nite
>sets.
>Why the hell do I feel like this is an argument? Is this
normal for
>usenet?
You must be new here...
>I certainly wouldnÕt wish to argue against anyone who riles
JSH as
>much as you do over something this trivial. I think IÕd
rather buy them a
>beer. (This is me in placatory mode after being bloody
annoying about
>something.)
If youÕre referring to what youÕve said in \
this thread, I
didnÕt \[CapitalThorn]nd
anything annoying about it, just puzzling.
>So, yes, it was silly of me to demand maps between sets,
agreed.
************************
David C. Ullrich
===
Subject: Re: limits, cardinalities
> No - what I canÕt \[CapitalThorn]gure out is why you \
thought that that
was the
> notion the OP had in mind.
>>(Should point out I mean direct limit, or colimit in fact
itÕs bad
>>to talk of lim when i mean colim without being able to put
an arrow under
>>the word).
> I was going to point this out yesterday, but I looked it
up, and it
> seems that there _is_ a standard notion of limit in category
> theory - I read that inverse limits are limits but direct
limits
> are not limits, for example, not that I have any idea what
any
> of those things are.
OneÕs covariant, the otherÕs a contravariant \
functor.
Roughly speaking, when the maps go in the same direction as
the ordering
youÕve a direct limit (colimit) (obj 1 maps to obj 2 maps to
) and inverse
limit is the other way, along with a bunch of maps to the
direct limit and
from
the inverse limit with some compatability requirements.
You tend to get sloppy with including the inverse or direct
especially as
it is often the case that you put an arrow under the word to
indicate
direction, and just say lim when you read it out or use it on
the board.
And the reason why it is applicable in the original question
is precisely
because when you have inclusion of sets, the direct limit is
exactly what
you get using the ordinary notion, so the *colimit is the
limit*. More
usefully here is that the sequence could then be taken to be
indexed by
something outrageously horrible (all you need is a category,
preferably
whose objects form a set with some partial ordering going
on), and that
might have been able to produce something that satis\[CapitalThorn]ed his
requirement.
Just because something wasnÕt in the original hypothesis
doesnÕt mean you
canÕt change your mind later.
The limit here is pretty much going to be the nested union of
sets,
omitting any elements that donÕt appear in every set
eventually, which
ever way you look at it.
> So when I read that I decided not to make the following
comment,
> which IÕve now decided to _make_, since you say you were
really
> talking about direct limits or colimits: DoesnÕt the fact
that he used
> the word limit give further evidence that what youÕve been
talking
> about is simply not what he had in mind?
Certainly not what he had in mind, canÕt argue with that, \
but
it could
perhaps be used.
>>So it might have been useful for constructing some odd sets
from \[CapitalThorn]nite
>>sets.
>>Why the hell do I feel like this is an argument? Is this
normal for
>>usenet?
> You must be new here...
Guilty
>>I certainly wouldnÕt wish to argue against anyone who \
riles
JSH as
>>much as you do over something this trivial. I think IÕd
rather buy them a
>>beer. (This is me in placatory mode after being bloody
annoying about
>>something.)
> If youÕre referring to what youÕve said in \
this thread, I
didnÕt \[CapitalThorn]nd
> anything annoying about it, just puzzling.
>>So, yes, it was silly of me to demand maps between sets,
agreed.
> ************************
> David C. Ullrich
===
Subject: Re: limits, cardinalities
>> No - what I canÕt \[CapitalThorn]gure out is why you \
thought that that
was the
>> notion the OP had in mind.
>(Should point out I mean direct limit, or colimit in fact
itÕs bad
>to talk of lim when i mean colim without being able to put
an arrow
under
>the word).
>> I was going to point this out yesterday, but I looked it
up, and it
>> seems that there _is_ a standard notion of limit in
category
>> theory - I read that inverse limits are limits but direct
limits
>> are not limits, for example, not that I have any idea what
any
>> of those things are.
>OneÕs covariant, the otherÕs a contravariant \
functor.
>Roughly speaking, when the maps go in the same direction as
the ordering
>youÕve a direct limit (colimit) (obj 1 maps to obj 2 maps \
to
) and inverse
>limit is the other way, along with a bunch of maps to the
direct limit and
from
>the inverse limit with some compatability requirements.
>You tend to get sloppy with including the inverse or direct
especially as
>it is often the case that you put an arrow under the word to
indicate
>direction, and just say lim when you read it out or use it
on the board.
>And the reason why it is applicable in the original question
is precisely
>because when you have inclusion of sets, the direct limit is
exactly what
>you get using the ordinary notion, so the *colimit is the
limit*. More
>usefully here is that the sequence could then be taken to be
indexed by
>something outrageously horrible (all you need is a category,
preferably
>whose objects form a set with some partial ordering going
on), and that
>might have been able to produce something that satis\[CapitalThorn]ed his
requirement.
>Just because something wasnÕt in the original hypothesis
doesnÕt mean you
>canÕt change your mind later.
>The limit here is pretty much going to be the nested union
of sets,
>omitting any elements that donÕt appear in every set
eventually,
Pretty much, sort of. In fact the nested union of sets,
omitting any elements that donÕt appear in every set
eventually
_is_ the lim inf, exactly. (Which equals the limit _when_ the
limit _exists_.)
It occured to me that if we need to say something about the
limit in terms of category theory we could do this: First,
letÕs restrict attention to the subsets of a set X. Now P(X)
is identi\[CapitalThorn]ed with the product {0,1}^X in a natural way, and
with this identi\[CapitalThorn]cation the notion of limit that \
IÕve been
talking about is precisely the product topology. Surely
thereÕs a categorical de\[CapitalThorn]nition of product \
topology?
>which
>ever way you look at it.
>[...]
************************
David C. Ullrich
===
Subject: Re: limits, cardinalities
>>The limit here is pretty much going to be the nested union
of sets,
>>omitting any elements that donÕt appear in every set
eventually,
> Pretty much, sort of. In fact the nested union of sets,
> omitting any elements that donÕt appear in every set
eventually
> _is_ the lim inf, exactly. (Which equals the limit _when_
the
> limit _exists_.)
> It occured to me that if we need to say something about the
> limit in terms of category theory we could do this: First,
> letÕs restrict attention to the subsets of a set X. Now \
P(X)
> is identi\[CapitalThorn]ed with the product {0,1}^X in a natural way, and
> with this identi\[CapitalThorn]cation the notion of limit that \
IÕve been
> talking about is precisely the product topology. Surely
> thereÕs a categorical de\[CapitalThorn]nition of product \
topology?
>>which
>>ever way you look at it.
>>[...]
> ************************
> David C. Ullrich
IÕve no idea if thereÕs a categorical \
de\[CapitalThorn]nition for product
topology.
But I wouldnÕt be surprised to hear one.
If itÕs any interest, the discrete Hilbert cube (countably
in\[CapitalThorn]nite
product of two point sets) is the limit an \
ÔobviousÕ sense of
a
system of \[CapitalThorn]nite products of two point sets (ie we can imagine
it as the
item at the ÔendÕ of an in\[CapitalThorn]nitely \
long list). ThatÕs an
inverse limit
though as the maps in the chain go from larger to smaller
sets, and
thereÕs a map to any \[CapitalThorn]nite product from the \
cube.
Of course the lim inf de\[CapitalThorn]nition doesnÕt work \
here. Though I
would
nominate the empty set as being better than saying there is
no limit.
So in that case youÕve got a \
ÔlimitÕ with all the smaller
sets \[CapitalThorn]nite and
the limit of their cardinalites is going to be aleph_0, th
limit has
2^aleph_0 elements.
IÕve been trying to explain to Doron Shadmi though that this
reasoning
doesnÕt tell you that they are the same cardinal - \
itÕs the
same as the
\[CapitalThorn]nite sets of the power set being countable not implying the
sets of
power set itself are countable.
If you do need to think of categories, then the way to do so
I feel is
best expressed as saying Ôpass to a sequence of \
subsetsÕ
ie, if some element isnÕt in the lim inf, then it might as
well have not
been in any of the sets in the \[CapitalThorn]rst place.
So in each set in the sequence, if an element doesnÕt appear
in the next one, throw it out, and all the times it appeared
in previous
sets. The resulting sequence has exactly the same lim inf,
and it is an
example of a direct limit as I would think of it.
In terms I would understand (limits of vector spaces) itÕs
like adding on
some more vector spaces at each space in the chain, but
de\[CapitalThorn]ning the zero
maps between them. So youÕre just adding redundant
information.
===
Subject: Re: limits, cardinalities
> Is there a sequence S of sets such that lim card S_n < card
lim
> S_n, where card denotes cardinality, both limits exist, and
the
> inequality is strict? Here trans\[CapitalThorn]nite limits are considered
to
> exist as long as they are well-de\[CapitalThorn]ned. Also, feel free to
call on
> the axiom of choice.
> If so, does there exist such a sequence with lim S_n \[CapitalThorn]nite?
In this thread, there seems to be some confusion as to the
meaning of
the limit of sets. I thought the de\[CapitalThorn]nition was standard. Here
was my
intention:
Let I indicate the intersection operator. Given a sequence S
of
sets, by de\[CapitalThorn]nition
lim inf S = U{I{S_m: m >= n}: n in N}, and
lim sup S = I{U{S_m: m >= n}: n in N}.
Iff the limits inferior and superior are equal, this set is by
de\[CapitalThorn]nition the limit of S.
This de\[CapitalThorn]nition is equivalent to DCUÕs, viz.,
>lim S_n = S if (i) for every x in S there exists N such that
>x is in S_n for all n > N and (ii) for every x not in S there
>exists N such that x is in S_n for no n > N.
(N.B.: DCU uses S to denote a set, while I use the letter to
denote a
sequence.) One can generalize this de\[CapitalThorn]nition to nets of sets.
I accept DavidÕs proof of the negative answer to my initial
question.
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
===
Subject: Re: limits, cardinalities
> Now card S_n = 10*n, thus lim card S_n = Aleph_0. However
lim S_n is
> the set of all reals on [0,1), thus card lim S_n is
2^Aleph_0.
I assume lim S_n is the same as the union of S_n over all n.
In that
case it does not contain any no terminating decimal, hence it
does not
contain the square root of 2 for example.
Bob Kolker
===
Subject: seperating std deviation (process and test)
in a process of adding an additive to a plastic compound and
then testing
the level of additive. I need to know what the std deviation
of the
process
is.
ie say my test measurements of a large number or samples show
a result of
105 ppm with a std deviation of 8 ppm. but the test itself if
I measure one
sample many times has a std deviation of 5 ppm., then what is
the std
deviation of the process itself.? ie how accurately are we
actually
physically adding the additive?
in other words the total std deviation is a function of both
the actual
variation and the test variation but what is this function?
do you add std
deviations?
terry
===
Subject: Re: seperating std deviation (process and test)
>in a process of adding an additive to a plastic compound and
then testing
>the level of additive. I need to know what the std deviation
of the
process
>is.
>ie say my test measurements of a large number or samples
show a result of
>105 ppm with a std deviation of 8 ppm. but the test itself
if I measure
one
>sample many times has a std deviation of 5 ppm., then what
is the std
>deviation of the process itself.? ie how accurately are we
actually
>physically adding the additive?
>in other words the total std deviation is a function of both
the actual
>variation and the test variation but what is this function?
do you add std
>deviations?
_If_ you assume that the test error is indendent of the sample
then you can add the _variances_. The variance is the square
of the standard deviation.
So in your example above you would have
8^2 = 5^2 + (actual standard deviation)^2,
making the actual standard deviation sqrt(39).
>terry
************************
David C. Ullrich
>in a process of adding an additive to a plastic compound and
then
testing
>the level of additive. I need to know what the std deviation
of the
process
>is.
>ie say my test measurements of a large number or samples
show a result
of
>105 ppm with a std deviation of 8 ppm. but the test itself
if I measure
one
>sample many times has a std deviation of 5 ppm., then what
is the std
>deviation of the process itself.? ie how accurately are we
actually
>physically adding the additive?
>in other words the total std deviation is a function of both
the actual
>variation and the test variation but what is this function?
do you add
std
>deviations?
> _If_ you assume that the test error is indendent of the
sample
> then you can add the _variances_. The variance is the square
> of the standard deviation.
> So in your example above you would have
> 8^2 = 5^2 + (actual standard deviation)^2,
> making the actual standard deviation sqrt(39).
===
Subject: Re: Vedic Mathematics --- Myth and Reality
> No, that is wrong. The Indian philosophical thought -
Sanatana
> dharma, or the way of life beyond the scope of time - is
completely
> different from the modern and dominant Jewish thinking [...]
>
> This frame of mind, of course, serves to lend additional
credence
> to the otherwise unbelievable notion that the Swastika
actually
> originated in India.
> The sign of the swastika relates to good health and well
being, from
> the Indian perspective.
This only a brahmin perspective. This might be true for
brahmins who
constitue
less than 5% of Indian population. We, Dalits (constitute
more than
20% of Indian popualtion) have no relation to swastika. I do
not know
its relation
to Indian Muslims, Indian Christians, Indian Sikhs, Indian
Buddhists
etc.
Interestingly brahmins are \[CapitalThorn]re worshippers. Fire is
unavidable for
their rituals. In contrast, Dalits do not give importance to
\[CapitalThorn]re like
Muslims and Chrstians, and Dalits do not have \[CapitalThorn]re as
essential thing
for their religious and spritual rituals and duties.
Please note, Mr Arindam Banerjee. You write about your
religion, do
not give
distorted picture of India to others.
Brahmins/hindus have no right to talk about Dalits.
Brahmins/hindus
are not representatives
of Dalits.
===
Subject: Re: Vedic Mathematics --- Myth and Reality
> No, that is wrong. The Indian philosophical thought -
Sanatana
> dharma, or the way of life beyond the scope of time - is
completely
> different from the modern and dominant Jewish thinking [...]
>
> This frame of mind, of course, serves to lend additional
credence
> to the otherwise unbelievable notion that the Swastika
actually
> originated in India.
>
> The sign of the swastika relates to good health and well
being, from
> the Indian perspective.
> This only a brahmin perspective. This might be true for
brahmins who
> constitue
> less than 5% of Indian population. We, Dalits (constitute
more than
> 20% of Indian popualtion) have no relation to swastika. I
do not know
> its relation
> to Indian Muslims, Indian Christians, Indian Sikhs, Indian
Buddhists
> etc.
> Interestingly brahmins are \[CapitalThorn]re worshippers. Fire is
unavidable for
> their rituals. In contrast, Dalits do not give importance
to \[CapitalThorn]re like
> Muslims and Chrstians, and Dalits do not have \[CapitalThorn]re as
essential thing
> for their religious and spritual rituals and duties.
> Please note, Mr Arindam Banerjee. You write about your
religion, do
> not give
> distorted picture of India to others.
> Brahmins/hindus have no right to talk about Dalits.
Brahmins/hindus
> are not representatives
> of Dalits.
Nor is any anonymous coward.
===
Subject: Re: Vedic Mathematics --- Myth and Reality
> No, that is wrong. The Indian philosophical thought -
Sanatana
> dharma, or the way of life beyond the scope of time - is
completely
> different from the modern and dominant Jewish thinking [...]
>
> This frame of mind, of course, serves to lend additional
credence
> to the otherwise unbelievable notion that the Swastika
actually
> originated in India.
>
> The sign of the swastika relates to good health and well
being, from
> the Indian perspective.
> This only a brahmin perspective. This might be true for
brahmins who
> constitue
> less than 5% of Indian population. We, Dalits (constitute
more than
> 20% of Indian popualtion) have no relation to swastika. I
do not know
> its relation
> to Indian Muslims, Indian Christians, Indian Sikhs, Indian
Buddhists
> etc.
Hey asshole evangalist propoganda machine Sawstik is revered
by all
the religions which originated in India hindus , jains, sikhs
and
buddhist have Swastik as a religious symbol.
Buddhist tempels in India, Korea, Japan and China have a
swastik
symbol at enterence.Ask any sikh which symbol their ladies
make on
ßoor during marriage
ceremonies.
If Klu Klax Klan has a cross as symbol does it make cross a
symbol of
racism ? Similarly Nazi Swastika was abuse of ancient Indian
symbol
and
hindu swastik should not be confused with it
===
Subject: Prime Numbers expansion canals
I am researching prime numbers using a visual representation.
IÕd like
to hear some opinions on some things that are emerging, please
give some feedback:)
http://www.night-stars.org/nitro/primes
===
Subject: Re: Prime Numbers expansion canals
> I am researching prime numbers using a visual
representation. IÕd like
> to hear some opinions on some things that are emerging,
please
> give some feedback:)
> http://www.night-stars.org/nitro/primes
IÕm really pleased to learn from your web page that
Stanislaw Ulam was also a Polish-American physician.
I already knew that he was a nuclear physicist [1]
and a Polish-American mathematician [2] and apparently
big in baseball (Stan the Man, according to [3]). He
was not only just an ordinary mathematician [4], he was
also a brilliant Polish mathematician from the University
of Wisconsin! [5] Anyhow, always glad to hear more about
-jiw
[1] http://www.maths.ex.ac.uk/~mwatkins/zeta/ulam.htm
[2] http://mathworld.wolfram.com/PrimeSpiral.html
[3] www.abarim-publications.com/artctulam.html
[4] http://www.maa.org/mathland/mathtrek_05_06_02.html
[5] http://www.airpowermuseum.org/trspcbmb.html
===
Subject: Re: Prime Numbers expansion canals
eheh, I apologize for the typo, but it was late and IÕm not
English :). I looked around and I found heÕs mainly
recognized of\[CapitalThorn]cially as mathematician, so I assumed that
statement for good. Let me know if you think itÕs ok:)
> IÕm really pleased to learn from your web page that
> Stanislaw Ulam was also a Polish-American physician.
> I already knew that he was a nuclear physicist [1]
> and a Polish-American mathematician [2] and apparently
> big in baseball (Stan the Man, according to [3]). He
> was not only just an ordinary mathematician [4], he was
> also a brilliant Polish mathematician from the University
> of Wisconsin! [5] Anyhow, always glad to hear more about
> -jiw
> [1] http://www.maths.ex.ac.uk/~mwatkins/zeta/ulam.htm
> [2] http://mathworld.wolfram.com/PrimeSpiral.html
> [3] www.abarim-publications.com/artctulam.html
> [4] http://www.maa.org/mathland/mathtrek_05_06_02.html
> [5] http://www.airpowermuseum.org/trspcbmb.html
===
Subject: Re: Prime Numbers expansion canals
> eheh, I apologize for the typo, but it was late and IÕm \
not
> English :). I looked around and I found heÕs mainly
> recognized of\[CapitalThorn]cially as mathematician, so I assumed that
> statement for good. Let me know if you think itÕs ok:)
a few minor items to \[CapitalThorn]x: ResearchÕs, simmetry, \
regural,
matematical, develope, expecially.
Another webpage problem -- The html code for your pictures is
broken. When I use Netscape 4.8 to view the page, all the
.gifÕs appear at the beginning, before any text, rather than
inline near their captions. Perhaps replace (eg)
<br>
<img style=width: 1001px; height: 886px; alt=
src=spiral_clean.gif><br>
by
<br clear=left>
<img src=spiral_clean.gif width=1001 height=886><br>
Re the content -- although I donÕt know precisely what you
refer
to in logarithmic expansion of the canals where prime numbers
never fall, I think most empty channels are easily explained
by noting that all the numbers in them are even. Perhaps you
refer to some deeper property, but for major diagonals,
continuing evenness is forced after an even beginning because
the jÕth complete turn of the spiral adds 8j+4 cells, ie,
always
adds an even number of cells.
-jiw
===
Subject: Re: Prime Numbers expansion canals
>
> eheh, I apologize for the typo, but it was late and IÕm \
not
> English :). I looked around and I found heÕs mainly
> recognized of\[CapitalThorn]cially as mathematician, so I assumed that
> statement for good. Let me know if you think itÕs ok:)
> a few minor items to \[CapitalThorn]x: ResearchÕs, \
simmetry, regural,
> matematical, develope, expecially.
> Another webpage problem -- The html code for your pictures
is
> broken. When I use Netscape 4.8 to view the page, all the
> .gifÕs appear at the beginning, before any text, rather \
than
> inline near their captions. Perhaps replace (eg)
> <br <img style=width: 1001px; height: 886px; alt=
src=spiral_clean.gif><br by
> <br clear=left <img src=spiral_clean.gif width=1001
height=886><br Re the content -- although I donÕt know
precisely what you refer
> to in logarithmic expansion of the canals where prime
numbers
> never fall, I think most empty channels are easily explained
> by noting that all the numbers in them are even. Perhaps you
> refer to some deeper property, but for major diagonals,
> continuing evenness is forced after an even beginning
because
> the jÕth complete turn of the spiral adds 8j+4 cells, ie,
always
> adds an even number of cells.
> -jiw
But that doesnÕt apply to the horizontal and vertical \
channels
does it? For the clear horizontal channel starting at 6, I got
6 21 44 75 114 161 216 279 350 429 516 611 714 825 944 1071
1206
1349 1500 1659 1826 2001 2184
which alternates between even and odd numbers and can be
generated
by
4x^2 + 11x + 6 or (x+2)(4x+3)
Is there a way to tell from this that all the results will be
composite?
===
Subject: Re: Prime Numbers expansion canals
> I am researching prime numbers using a visual
representation. IÕd like
> to hear some opinions on some things that are emerging,
please
> give some feedback:)
> http://www.night-stars.org/nitro/primes
ThatÕs some nice graphs. When IÕm having a bit \
spare time for
that, IÕll see for some possible artifacts because of inter-
ferences of divisibility and the given shape of the spiral.
ItÕs a nice and creative idea, IÕd say.
Gottfried Helms
===
Subject: Re: Computers as a tool in foundations research
> I suspect that the foundations of mathematics is perhaps
the only
remaining
> major scienti\[CapitalThorn]c discipline in which computers do not play a
fundamental
> role in research. I am posting this in part to \[CapitalThorn]nd out if I
am mistaken.
The only one? How about philosophy?
> The impression I get is that mathematicians working in this
area believe
> that intuition about large cardinals is the most powerful
way to extend
> foundations and computer models are not relevant.
> No matter how pretentious the cardinals a formal system
claims to deal
with,
> it is still a computer program for enumerating theorems
> and computer
> simulations are, at least in theory, relevant to
understanding its
> structure. I have long suspected that directly attacking the
combinatorial
> content of formal systems will ultimately lead to far more
powerful ways
of
> extending mathematics than large cardinal axioms. This is
true precisely
> because the combinatorial content is something you can do
computer
> experiments on to test your intuition. The result of course
will be
systems
> far more complex and less transparent than existing
mathematics. That is
the
> inevitable price to be paid for more powerful systems.
You might be interested in the ideas of Edward Nelson.
(Homepage via Google).
There are more voices recently who think that computers
will inevitably get greater inßuence on the foundations of
math.
(Or at least on the way we will do and envision math in
practice.)
Some other names that come to mind are Doron Zeilberger
and Gregory Chaitin.
And letÕs not forget the work of Wladimir Sazonov.
Herman Jurjus
===
Subject: Re: Computers as a tool in foundations research
>> I suspect that the foundations of mathematics is perhaps
the only
remaining
>> major scienti\[CapitalThorn]c discipline in which computers do not play
a fundamental
>> role in research. I am posting this in part to \[CapitalThorn]nd out if
I am
mistaken.
>The only one? How about philosophy?
Goodness. I realize that this is cross-posted to sci.logic, so
maybe IÕm about to break a taboo local to a group I \
donÕt
know,
but: who is it who believes that philosophy is a scienti\[CapitalThorn]c
discipline? For a start, do most or even many philosophers
believe this? (I know that many mathematicians/logicians who
study logic and foundations are *employed* in Departments of
Philosophy, and to that extent--even further perhaps--are
philosophers who might fancy themselves scientists with as
much justi\[CapitalThorn]cation as some mathematicians might. But they must
be a small minority of philosophers.)
Lee Rudolph
===
Subject: Re: Computers as a tool in foundations research
> I suspect that the foundations of mathematics is perhaps
the only
remaining
> major scienti\[CapitalThorn]c discipline in which computers do not play a
fundamental
> role in research. I am posting this in part to \[CapitalThorn]nd out if I
am
mistaken.
> The only one? How about philosophy?
Philosophy is not a scienti\[CapitalThorn]c discipline.
> You might be interested in the ideas of Edward Nelson.
> (Homepage via Google).
> There are more voices recently who think that computers
> will inevitably get greater inßuence on the foundations of
math.
> (Or at least on the way we will do and envision math in
practice.)
> Some other names that come to mind are Doron Zeilberger
> and Gregory Chaitin.
> And letÕs not forget the work of Wladimir Sazonov.
--
Paul Budnik
Mountain Math Software
http://www.mtnmath.com
===
Subject: Re: monotonically normal
===
Subject: Re: monotonically normal
>>A space S is monotoniclly normal when for
>>each x in S and each open U nhood x, thereÕs
>> assigned mx(x,U) another open nhood of x,
>>for which for all x,y, open U,V
>> x in U, y in V, mu(x,U) / mu(y,V) not empty
>> ==> x in V or y in U.
>>Problem: to show a linear order space is monotonically
ordered
>>Let S be linear order space
>>Let < be order of S for the order-topology
>>Let <_w well-order S
>>de\[CapitalThorn]ne mu(x,U) as follows: let x in (c,d) be interval inside
U
>You have to be more speci\[CapitalThorn]c here; usually one takes the
maximal
>convex subset of U that contains x, note that, e.g., in the
rationals
>{q:q^2<2} is not an interval, though it is a convex open
set. How
>would you specify c and d in this case?
c and d are chosen by axiom of choice. Chosing the maximal
convex
component of an open set, which is open, doesnÕt seem easier
way to go.
However with your suggestion to use interval base sets, then
we could
chose the maximal convex interval. I suppose that could avoid
the use of
axiom of choice.
>>If (c,x) nonnul, let x_l = <_w-\[CapitalThorn]rst in (c,x),
>> otherwise let x_l = c
>>If (x,d) nonnul, let x_r = <_w-\[CapitalThorn]rst in (x,d),
>> otherwise let x_r = d
>>let mu(x,U) = (x_l, x_r) which is an open nhood of x
>>If mu(x,U) / mu(y,V) nonnul then
>>x in (x_l, x_r); y in (y_l, y_r)
>>some z in I = (x_l,x_r) / (y_l,y_r) = (max x_l,y_l, min
x_r,y_r)
>>Consider when x_l <= y_l. [(y_l <= x_l) similar]
>> if y_r <= x_r: I = (y_l, y_r); y in I subset U
>> if x_r < y_r: I = (y_l, x_r) assume x,y not in I
>>then we have x <= y_l < z < x_r <= y
>>if x < y_l < z < x_r < y, then comes desired contradiction
>> x_r <=_w y_l; y_l <=_w x_r; y_l = x_r; z in I = nulset
>> Otherwise, what to do when x = y_l or x_r = y ?
>In that case youÕd conclude that m(x,U) / m(y,W) is null!
For x_r = y, I was shown to look Ôoutside the \
boxÕ:
x <= y_l < z < y = x_r < d; y in (c,d) subset U
similar for l_r = x
>It would be more convenient to use the fact that you only
have to do
>this for U coming from a base and, in this case, this means
that you
>can indeed assume that U is an interval.
You could, yet still you have to consider when (c,x) empty.
----
===
Subject: [JSH]: Now Rejects Prime Counting Algorithm
has now declared that square roots are
*inherently* ambiguous, which leads to the inevitable
conclusion that his prime counting algorithm (which contains
square roots) must be rejected by his own criteria. After
all, he appears to argue, what if the positive square roots
in his algorithm are replaced with negative values? Will the
result be correct? If not, the algorithm fails.
--
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]: Now Rejects Prime Counting Algorithm
> has now declared that square roots are
> *inherently* ambiguous, which leads to the inevitable
> conclusion that his prime counting algorithm (which contains
> square roots) must be rejected by his own criteria. After
> all, he appears to argue, what if the positive square roots
> in his algorithm are replaced with negative values? Will the
> result be correct? If not, the algorithm fails.
> --
> There are two things you must never attempt to prove: the
> unprovable -- and the obvious.
> --
> Democracy: The triumph of popularity over principle.
and as someone else has pointed out, his own Ôcore \
errorÕ
argument
contains points where it is necessary to take roots.(I wonder
if thats
where the core error comes from then?) (Please, no one take
that
seriously.)
===
Subject: Problem from Optics
Cc: t.albaho@ic.ac.uk
The text quoted below comes from a 1923 text
analysing a lens guide. The transformation is
entirely mathematical - in that I cannot see
any new physics beign injected. Hence I am
posting it here!
The problem relates to a series of lenses lined
up one after another. The power of each lens
is k1, k2, k3 ... etc. They are separated by
distances t such that t1 is the distance between
k1 and k2. The total power of all the lenses
in sequence is:
K = k1 + k2 + k3 + ... + kn
- k1t1k2 - k1(t1 + t2)k3 ....
- k1(t1 + t2 + .... tn-1)kn
- k2t2k3 - k2(t2+t3)k4 ...
+ k1t1k2t2k3 + ....
There is nothing controversial up to this point.
On putting all the kÕs and tÕs equal to one \
another,
the coef\[CapitalThorn]cients depend on the numerical value of
the sums of continued products obtained by
dividing at r-1 points a line whose length varies
from r to n into r parts each of a length
represented by an integer. The result is
K = nk - [(n+1)n(n-1)/3!]k^2t
+ [(n+2)(n+1)n(n-1)(n-2)/5!]k^3t^2 ....
+ (-1)^r{(n+r)!/[(n-r-1)!(2r+1)!]}k^(r+1)t^r ...
and that is it. and this is the \[CapitalThorn]rst time he introduces
the parameter r. I have tried to write it as clearly
as possible - but it is complicated.
Any help in explaining what he means by the
text quoted above and how he arrives at that
result greatly appreciated.
--
_________________________________________________________
Tareq t.albaho@imperial.ac.uk
Quantum Optics & Laser Science
Blackett Laboratory, Imperial College, London.
===
Subject: Re: Problem from Optics
> The text quoted below comes from a 1923 text
> analysing a lens guide. The transformation is
> entirely mathematical - in that I cannot see
> any new physics beign injected. Hence I am
> posting it here!
> The problem relates to a series of lenses lined
> up one after another. The power of each lens
> is k1, k2, k3 ... etc. They are separated by
> distances t such that t1 is the distance between
> k1 and k2. The total power of all the lenses
> in sequence is:
> K = k1 + k2 + k3 + ... + kn
> - k1t1k2 - k1(t1 + t2)k3 ....
> - k1(t1 + t2 + .... tn-1)kn
> - k2t2k3 - k2(t2+t3)k4 ...
> + k1t1k2t2k3 + ....
> There is nothing controversial up to this point.
> On putting all the kÕs and tÕs equal to one \
another,
> the coef\[CapitalThorn]cients depend on the numerical value of
> the sums of continued products obtained by
> dividing at r-1 points a line whose length varies
> from r to n into r parts each of a length
> represented by an integer. The result is
> K = nk - [(n+1)n(n-1)/3!]k^2t
> + [(n+2)(n+1)n(n-1)(n-2)/5!]k^3t^2 ....
> + (-1)^r{(n+r)!/[(n-r-1)!(2r+1)!]}k^(r+1)t^r ...
> and that is it. and this is the \[CapitalThorn]rst time he introduces
> the parameter r. I have tried to write it as clearly
> as possible - but it is complicated.
> Any help in explaining what he means by the
> text quoted above and how he arrives at that
> result greatly appreciated.
> --
> _________________________________________________________
> Tareq t.albaho@imperial.ac.uk
> Quantum Optics & Laser Science
> Blackett Laboratory, Imperial College, London.
LetÕs call each lens ÔkÕ and each \
space ÔtÕ. We have:
k t k t k t k t k .... k with n kÕs and (n-1) \
tÕs.
Now, weÕll take sequences of r kÕs out this \
sequence. There
are n+1-r
such sequences, starting at positions 1, 2, 3, ... and
spanning up to
position 1+r-1, 2+r-1, 3+r-1, ...
So, for example, n = 6, r = 3 we have:
(k t k t k) t k t k t k
k t (k t k t k) t k t k
k t k t (k t k t k) t k
k t k t k t (k t k t k)
total of 6+1-3 = 4 sequences.
For each sequence, we have the \[CapitalThorn]rst and last \
kÕs but maybe
other kÕs
inside as well. LetÕs say we have p internal \
kÕs in the
sequence of r
kÕs. So, for example, with r = 6 and p = 2 we have:
*k t *k t *k t k t k t *k
*k t *k t k t *k t k t *k
*k t *k t k t k t *k t *k
*k t k t *k t *k t k t *k
*k t k t *k t k t *k t *k
*k t k t k t *k t *k t *k
(* denotes internal/external k weÕre interested in).
Our problem is to calculate all possible products of
sequences of tÕs.
For example, with r = 10, p = 2 we might have
*k t k t k t k t *k t k t k t *k t k t *k
So there are 4 tÕs between the \[CapitalThorn]rst pair, 3 \
tÕs between the
second pair
and 2 tÕs between the third pair. The \
coef\[CapitalThorn]cient in this case
is 4*3*2
(because itÕs k(t+t+t+t)k(t+t+t)k(t+t)k which makes the
coef\[CapitalThorn]cient of t
4*3*2).
There are many ways of choosing the internal kÕs, and for
each choice we
have a different product. The easiest way to get something
here I could
think of is recursion.
Let S(r,p) be the coef\[CapitalThorn]cient with a sequence of r \
kÕs and p
internal
kÕs. If p is 0, we know the coef\[CapitalThorn]cient is r-1 \
(the number of
tÕs
between the two kÕs).
S(r,0) = r-1
But if p is not 0, we can choose where the \[CapitalThorn]rst internal k is
going to
be (and so getting the \[CapitalThorn]rst factor of the \
coef\[CapitalThorn]cient), and
then
reducing the problem to a shorter sequence and one less
internal k.
The \[CapitalThorn]rst internal k can come after 1, 2, ... r-2 \
tÕs. LetÕs
call this
number j. This leaves a shorter sequence of r-j kÕs.
So S(r,p) = sum(j=1,r-2) j * S(r-j,p-1)
Since we have n-r+1 such sequences for each r, and r can be
anything
between 1 and n, we get in total, for a given p:
sum(r=1,n) (n-r+1)*S(r,p)
When r goes from 1 to n, (n-r+1) goes from n to 1, So we can
change
indices:
= sum(r=1,n) r*S(n-r+1,p)
But this looks extremely familiar. Yes, itÕs very close to \
our
de\[CapitalThorn]nition of S(r,p). If we play with this expression a bit,
we get:
= S(n+1,p+1)
(because S(1,p) = 0).
Amazingly enough, S(n,p) is exactly (n+p-1) choose (2*p+1).
You can
either trust me on this one, or prove it by induction (on p).
And so we get:
= (n+(p+1)) choose (2*(p+1)+1)
call r=p+1 and you get your formula:
= (n+r) choose (2r+1)
= [(n+r)!/((n-r-1)!*(2r+1)!)]
As for (-1)^r, I think itÕs a physics thing, so I \
canÕt
really comment.
For r=p+1, we have p internal kÕs in the sequence, and so \
p+1
sequences
of tÕs. Therefore this is the coef\[CapitalThorn]cient of \
t^(p+1) = t^r. We
also have
p+2 kÕs (p internal + 2 external), which is exactly r+1, and
so this is
the coef\[CapitalThorn]cient of k^(r+1).
Hope this helps (and still relevant).
===
Subject: generalized Euler angles and Haar measure on SO(n):
2 questions
Before I spend any serious time trying to think this
through by myself, I thought it would be easier to ask
sci.math in congress assembled to do my homework for me.
There are two, related, parts.
(1) Given the standard orthonormal basis e_1,...,e_n of
n-dimensional Euclidian space E_n, for each pair (i,j)
with 1=<i<j=<n one has the 1-parameter subgroup T_{i,j}
of SO(n) consisting of those special orthogonal maps
which \[CapitalThorn]x e_k for k neither i nor j, and which rotate
the plane spanned by e_i and e_j in the standard way.
Fixing an enumeration of the pairs (i,j), we can thus
parametrize a subset of SO(n) by n-choose-2 angles
(i.e., real numbers modulo 2pi). For n=3, this is
essentially the Euler angles parametrization (I
think), and is (therefore) onto. Is it always onto?
(2) Assuming the af\[CapitalThorn]rmative answer to the \[CapitalThorn]rst \
question,
is the Haar measure on SO(n) the product measure induced
from this (nonhomorphic!) map of (S^1)^(n-choose-2) onto
SO(n)? If (as I fear is likely) not, what correcting factor
do I need to put in?
This is all in aid of a cute, though trivial, application
to real linear algebra which shall be revealed to all
in good time. And I know I should do it myself.
Lee Rudolph
===
Subject: Re: generalized Euler angles and Haar measure on
SO(n): 2
questions
>Before I spend any serious time trying to think this
>through by myself, I thought it would be easier to ask
>sci.math in congress assembled to do my homework for me.
>There are two, related, parts.
>(1) Given the standard orthonormal basis e_1,...,e_n of
>n-dimensional Euclidian space E_n, for each pair (i,j)
>with 1=<i<j=<n one has the 1-parameter subgroup T_{i,j}
>of SO(n) consisting of those special orthogonal maps
>which \[CapitalThorn]x e_k for k neither i nor j, and which rotate
>the plane spanned by e_i and e_j in the standard way.
>Fixing an enumeration of the pairs (i,j), we can thus
>parametrize a subset of SO(n) by n-choose-2 angles
>(i.e., real numbers modulo 2pi). For n=3, this is
>essentially the Euler angles parametrization (I
>think), and is (therefore) onto. Is it always onto?
Yes, at least if you choose the enumeration properly:
SO(n) = T(1,2) T(1,3) T(2,3) ... T(1,n) ... T(n-1,n).
Prove by induction.
n=1 is trivial.
Given any member A of SO(n), it maps some unit vector v
to e_n = <0,...,0,1>^T. There is some B in
T(n-1,n) T(n-2,n) ... T(1,n) that maps e_n to v.
Then AB is a member of SO(n) that \[CapitalThorn]xes e_n,
and thus is of the form
[ C 0 ]
[ 0 1 ]
with C in SO(n-1). By the induction hypothesis C is
in T(1,2) ... T(n-2,n-1), and then A = AB B^(-1) is in
T(1,2) ... T(n-2,n-1) T(1,n) ... T(n-1,n).
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
===
Subject: Best Elimination Algorithm for Large System in
Finite Field?
Summary: How well does StrassenÕs Algorithm do?
Keywords: StrassenÕs Gaussian Elimination
Originator: root@precision.moscito.org (root)
I am faced with the prospect of running a large Gaussian
Elimination in over a \[CapitalThorn]nite \[CapitalThorn]eld. There will be \
around 300,000
variables and 600,000 equations. Around half of the equations
are
redundant. The elimination is to terminate once we are down to
the last K variables, where K is a given. The matrix is rather
sparse, starting out with perhaps 500 entries a line. Can
something
like StrassenÕs algorithm, which uses blocks of little
matrices,
even work well in this situation? Obviously, no machine that
I can
lay my hands on has 180 GB of main memory, so I must do such a
problem using sparse matrices, and how does that jibe with my
running time? [Note: \[CapitalThorn]nite \[CapitalThorn]eld elements \
\[CapitalThorn]t in a byte, and
multiplication is done using table lookups.]
B.Y.
===
Subject: Re: Random-Sequence Contest
>> |By randomness, perhaps I mean the minimum order of a
polynomial needed
>> |to duplicate the sequence for a \[CapitalThorn]xed number of terms
(say, 100).
>> |
>> |There might be a much better de\[CapitalThorn]nition of randomness.
Feel free to
>> |post any de\[CapitalThorn]nition you feel might be better.
>> if you donÕt \[CapitalThorn]nd a better \
de\[CapitalThorn]nition of randomness then my
sequence is
>> f(n) = 2^n.
>n! is one character shorter (=
I can duplicate 2^n with a polynomial of order 1. In
Neanderthal
notation, a number is represented as a stick followed by a
series of
circular pebbles, thus 5 in Arabic is 100000 in Neanderthal.
But that
is also 2^5 in binary!
In other words, it all depends on the notation ;-)
Gerry Quinn
--
http://bindweed.com
Screensavers and Games for Windows
Download free trial versions
New arcade-puzzler just out - Volcano
===
Subject: Statistics Questions
Hi there,
I was wondering if anyone could help me with a few practise
exam
questions i have. I have an exam on statistics in a few weeks
and
there are two practise questions which i have come completely
stuck
on. If you understand and can answer these, please help me.
5) Students have to achieve a certain grade (50%) in
programming in
order to be
allowed to proceed to a 3rd year course in games programming.
However,
if a
student is very keen these standards may be lowered. If
students who
have
passed the module in the past, have achieved a set of marks
for
programming
showing a normal distribution with mean 63 and standard
deviation 10,
would
you allow a keen student with a mark in programming of 48% to
take
this
module? How about one with 43%? Explain your reasoning. (10
marks)
6) You have designed and written a program to identify faces
in
pictures
containing many people. In order to evaluate this program,
you test it
against
another program that you have obtained that claims to be the
state of
the art.
The test consists of identifying people from 30 photographs.
The
result of
your program is 72% of faces correctly identi\[CapitalThorn]ed with \
standard
deviation
16%; the results for the other program is 64% of faces
correctly
identi\[CapitalThorn]ed
with standard deviation 19%. Is the apparent improvement
statistically
signi\[CapitalThorn]cant? Justify your answer. (10 marks)
===
Subject: Re: Statistics Questions
>5) Students have to achieve a certain grade (50%) in
programming in
>order to be allowed to proceed to a 3rd year course in games
>programming. However, if a student is very keen these
standards may
>be lowered. If students who have passed the module in the
past, have
>achieved a set of marks for programming showing a normal
distribution
>with mean 63 and standard deviation 10, would you allow a
keen student
>with a mark in programming of 48% to take this module? How
about one
>with 43%? Explain your reasoning. (10 marks)
Maybe look at the tail distribution (Z < 50%) and identify
what
percentage of those students would have scores higher than
48% and 43%
respectively? I guess most have higher than 43% so thereÕs \
no
point in
admitting that student.
I didnÕt know school administrators delegated their decision
making to
statistics students nowadays :)
>6) You have designed and written a program to identify faces
in
>pictures containing many people. In order to evaluate this
program, you
>test it against another program that you have obtained that
claims to be
>the state of the art. The test consists of identifying
people from 30
>photographs. The result of your program is 72% of faces
correctly
>identi\[CapitalThorn]ed with standard deviation 16%; the results for the
other program
>is 64% of faces correctly identi\[CapitalThorn]ed with standard deviation
19%. Is the
>apparent improvement statistically signi\[CapitalThorn]cant? Justify your
answer.
Smith-Satterthwaite mean comparison test?
===
Subject: Re: Short proof of FLT
> Hi I have a short proof of FLT. Does anyone want to check
this for
> me? It looks correct to me. This proof is only 8 lines,
shorter than
> the Wanker (sp?) proof.
> Forgive the other posters for being harsh. The fact is that
there are
> many reasons, historical and mathematical, that we greatly
suspect
> your proof to be incorrect. If you post it here, someone
will be glad
> to tell you whatÕs wrong with it. Obviously you \
donÕt have
to worry
> about anyone stealing it, because the public record will
show that
> you posted it. So just post it here.
> Nathan
HereÕs another one that needs to me posted.
www.math.fsu.edu/Science/Specialized
===
Subject: Math question
If the third term in a geometric sequence is 2x-3 and the
\[CapitalThorn]fth term
is 3x+3 and the recursive rule is that each term is
multiplied by 3,
then \[CapitalThorn]nd the value of x.
===
Subject: Re: Math question
Sometimes, my posts never go through:
>If the third term in a geometric sequence is 2x-3 and the
\[CapitalThorn]fth term
>is 3x+3 and the recursive rule is that each term is
multiplied by 3,
>then \[CapitalThorn]nd the value of x.
x = 2
===
Subject: Re: Math question
jussy1234@hotmail.com want to know:
>If the third term in a geometric sequence is 2x-3 and the
\[CapitalThorn]fth term
>is 3x+3 and the recursive rule is that each term is
multiplied by 3,
>then \[CapitalThorn]nd the value of x.
The way to reach term #5 from term #3 is my 3*3:
(term#3)*3*3=term#5
LetÕs substitute:
(2*x - 3)*3*3 = 3*x + 3
Do the algebraic steps, and ....
\[CapitalThorn]nd that x = 2
G C
===
Subject: Re: Math question
> If the third term in a geometric sequence is 2x-3 and the
\[CapitalThorn]fth term
> is 3x+3 and the recursive rule is that each term is
multiplied by 3,
> then \[CapitalThorn]nd the value of x.
The successive terms of a geometric series can be represented
by
a, a*r, a*r^2, a*r^3 a*r^4, ..., where a is the \[CapitalThorn]rst term and
r is
the ratio of any but the \[CapitalThorn]ret term to the immediately
previous term.
With a little thought, you should be able to derive a system
of two
linear equations in unknowns a and x, from which you can \[CapitalThorn]nd
the
value of x.
===
Subject: triangle question
Can someone tell me how to derive the relationship
L1 / (2*L2) = C / [2*(C-1)]
from the following triangle info:
http://members.cox.net/eckiller/tri.jpg
===
Subject: Re: triangle question
vsgdp
> Can someone tell me how to derive the relationship
> L1 / (2*L2) = C / [2*(C-1)]
> from the following triangle info:
> http://members.cox.net/eckiller/tri.jpg
Something is wrong; C-1 is unde\[CapitalThorn]ned. But I do see
C+d = sqrt(2) L1
and
C = sqrt(2) L2.
To see the second equation, extend the segment of length L2
until it meets
the segment of length C+d.
LH
===
Subject: IT FROM BIT
Part II
(note typo in Part I correct formula is
/zpf = Lp^-2(Lp^3/2|Vacuum Coherence|^2 - 1)
JS: Perhaps. Just what is the Yilmaz theory in your
understanding? I
mean what is its world view?
What is the physical picture behind the obscure formalism?
PZ: Basically:
(1) Any satisfactory tensor theory of gravitation should have
a precise
static Newtonian correspondence
model and should have good (localizable, frame-independent)
energy-momentum analogs satisfying
Newtonian conservation principles in limiting cases (<-->
Poisson
equation);
JS: Not a valid point as shown in Part I.
PZ: (2) EinsteinÕs vacuum stress-energy pseudotensor does \
not
satisfy
these correspondence requirements;
JS: Not well-posed as shown in Part I.
interactive n-body solutions as a result
of the fact that the Einstein-Hilbert \[CapitalThorn]eld equations are
mathematically
overdetermined;
JS: I am not prepared to really respond to this one. I think
(3) is
false since, for example, people
simulate black hole collisions for example. WhatÕs this \
about
mathematically overdetermined? References?
Quantum \[CapitalThorn]eld theory cannot really even solve the 0-body
(vacuum) problem.
PZ: (4) Addition of a true tensor self-gravitating vacuum
stress-energy
term t_uv(vac) to the RHS of the standard
\[CapitalThorn]eld equations, of the form speci\[CapitalThorn]ed by Tupper \
and Yilmaz,
eliminates
this overdetermination, leading to
computable exact n-body solutions exhibiting precise
correspondence with
Newtonian theory;
JS: Can you write an explicit formula for the true tensor
self-gravitating vacuum stress-energy term t_uv(vac) to the
RHS of the
standard
\[CapitalThorn]eld equations in a speci\[CapitalThorn]c toy model case? For \
example,
write it
down for Hal PuthoffÕs SSS solution with K = e^2GM/c^2r. It
seems if
there is any point to all this, you or Hal et-al should be
able to show
all the math in this example?
Remember EinsteinÕs equation is
tuv(Ordinary Vacuum) + Tuv(Matter) = 0
where
tuv(Ordinary Vacuum) = (String Tension)Guv(Einstein)
Guv(Einstein) = Ruv(Ricci) - (1/2)R(Ricci)guv(Curved)
My \[CapitalThorn]eld equation IF there is exotic vacuum dark energy/matter
is
tuv(Ordinary Vacuum) + tuv(Exotic Vacuum) + Tuv(Matter) = 0
Where
tuv(Exotic Vacuum) = (String Tension)/zpfguv(Curved)
/zpf = Lp^-2(Lp^3/2|Vacuum Coherence|^2 - 1)
So now, where are your corresponding equations for the Yilmaz
theory?
PZ: (5) The gravitational conservation principles are then
based on the
*ordinary* divergence of the
total gravitational stress-energy density, as opposed to the
*covariant*
divergence as in GR;
JS: Who ordered that? Why is that a good idea? To me it seems
a bad
crank idea. I suppose the reason is that
\[CapitalThorn]ction about the physical reality of super steel measuring
rods that
unlike EinsteinÕs rubber rods do not
shrink when oriented radially in say the Schwarzschild
solution so that
the measured dR is
dR = (1 - 2GM/c^2r)^-1/2dr
r > 2GM/c^2r
EinsteinÕs r above is NOT same as PuthoffÕs r \
in his
K = e^2GM/c^2r
Super steel rods is HalÕs way of talking about the second
globally
ßat metric in some kind of parallel shadow universe. Problem
is there
appears to be no way to measure its presence?
PZ: (6) There is a physical distinction between inertial and
gravitational \[CapitalThorn]elds: the inertial \[CapitalThorn]eld is a \
kinematical
\[CapitalThorn]ction (as in Newtonian physics), while the permanent
gravitational
\[CapitalThorn]eld is a real physical \[CapitalThorn]eld of the
classic type, except that it is non-linearly self-interacting
and is
represented by a tensor potential phi_uv
with a derivative exponential metric representing *physical*
deformations of measuring instruments;
JS: Again this is meaningless without math formulae to
illustrate each
speci\[CapitalThorn]c allegation.
By inertial \[CapitalThorn]eld I suppose you mean the arbitrary curvilinear
coordinates which are physically realized by a dust cloud of
tiny
observers with transceivers in arbitrary non-geodesic paths
like us on
the rotating surface of Earth when the Coriolis inertial
force is
measured?
The only real point here is EEP that the g-force is locally
eliminated
mod weak tidal effects on a timelike geodesic for a
non-rotating observer.
Since, the inertial forces are independent of the rest mass
of the test
threshold tidal local curvature effects that may or may not
be present.
End of story. These are facts veri\[CapitalThorn]ed beyond doubt in their
proper
macro-domain of validity. No one at the cutting edge of
theoretical
physics worries about that level and rightly so. This is why
Yilmaz and
anyone who follows his idea is rightly IMHO considered cranky
with a
decidedly uninteresting idea. Why, for example, one should
reject
covariant divergences is a good example of the crank mind. I
think I
have just accurately represented say what Charles Misner, for
example,
might say if really pressed by someone he respected on what
he thought
of his, I think, former student Alley who seems to give
YilmazÕs idea
serious plausibility? The cutting edge is at the intersection
of quantum
theory with GR not with YilmazÕs bi-metric fantasy of
super-steel.
;-)
(compensate) at
some point in an LIF, while the
physical gravitational \[CapitalThorn]eld and its real energy content are
still
nevertheless present *at every point*
in the LIF (Newtonian model);
JS: This is GR you do not need Yilmaz for this. I showed the
GR formula
Eq (9) for this before at
http://mathworld.wolfram.com/
ChristoffelSymboloftheSecondKind.html
PZ: (8) Exact Yilmaz solution for a single point mass has no
event
horizons (no black holes).
JS: A false prediction I will wager. So at least YilmazÕs
idea is
falsi\[CapitalThorn]able. ThatÕs good.
Paul you have so far not expressed the philosophical world
view of
Yilmaz. For example, why no
covariant divergences of the stress-energy density tensor
\[CapitalThorn]eld?
PZ: Yet at the same time I think you get rid of all the tricky
properties of event
horizons, since you get a smooth solution for a point mass
with no lightcone
inßection boundaries
JS: There seems to be observational evidence of event
horizons? I am not
up on the latest on this. But I sure get the impression that
competent people like Martin Rees are pretty con\[CapitalThorn]dent on that
score?
PZ: Any such evidence is bound to be malleable and Yilmaz,
Alley, Leiter
et al. have shown how their theory
actually solves some problems associated with the orthodox
account of
the data:
Carroll Alley, Darryl Leiter, Huseyin Yilmaz, et al., Energy
Crisis in
Astrophysics, arXiv: astro-ph/9906458 v1 28 Jun 1999
JS: Leiter, who I knew at Brandeis, and saw brießy recently
at APS,
seems to have given up on this? In any case this issue needs
to
be considered by the pros, e.g. Martin ReesÕs group on
Madingley Road in
Cambridge UK. Has there been any response to that paper?
JS: No because you still have the turning point where
dr(isotropic)/dr(curvature) has a critical point passing
through zero and changing sign. This acts spatially somewhat
like an
event horizon, i.e.
dR ---> in\[CapitalThorn]nity at the turning point.
dR = [1 - GM/c^2r(isotropic)]^-1dr(curvature) TURNING POINT
dT = e^-GM/c^2r(isotropic) dt NO EVENT HORIZON
That is, EinsteinÕs event horizon is replaced by turning
point in HalÕs model.
PZ: But in any case there is no fundamental reason in PV for
insisting that every smooth
coordinate system is good.
JS:ThatÕs relativity locally. ItÕs worse than \
that. Hal and
Ibison seem
to have no understanding that a manifold must
generally be covered by more than one overlapping coordinate
patches
like on the surface of a
sphere. They are completely reckless with their r which they
misapply
in the region GM/c^2r >> 1
where they need a second coordinate patch IMHO independent of
their
metric \[CapitalThorn]eld equation,
This is a matter of global topology of the manifold and you
cannot write
their metric without a
manifold although you can have a manifold without a metric.
Again all
this is independent of
what action one uses at the metrical level!
JS: This is not the key point. I am talking about HalÕs
speci\[CapitalThorn]c SSS PV
model.
PZ: OK.
PZ: Look, once general relativity is out of the picture,
dogmatic
insistence on general covariance
begins to look like a mathematical fetish. I see this as an
example
of irrationality in contemporary
physics.
and dismiss it as crank or crackpot.
That book will show you how to think of general covariance in
a
balanced way. There are legitimate foundational issues,
but you have not expressed them above and such a loaded
remark is not
wise if you want anyone to pay attention
to any real insight you may have.
JS: The problem is deeper than that.
PZ: It is supposed to be -- but it may turn out that it is
not that deep
at all.
Why make it look super\[CapitalThorn]cially as if the physical effects of
acceleration do not mark off inertial frames if
in fact they do?
You cannot throw away differential geometry.
JS: Huh?
PZ: Intrinsic geometry and general covariance are not
interchangeable --
although of course they are related.
JS: There you go again with grand pronouncements out of
context so that
I do not know what you mean without
speci\[CapitalThorn]c examples.
PZ: Did Riemann himself even know that the Riemann curvature
is a tensor
quantity? I dont think Gauss
and Riemann even knew about tensors. I think it was Ricci who
introduced
an absolute geometry
of manifolds?
JS: The problem is that PVÕs rules of the game
are nebulous and shifting.
PZ: Or perhaps it is that they are not the rules of GR? Maybe
you are
trying to understand PV through a
chronogeometric optic?
JS: I am saying it is impossible to understand the rules of
PV because
there arenÕt any. If you think there are what are they?
Where is Euclid? I am saying Hal has not given a coherent
clear
explanation of his world picture that would be acceptable
to any philosopher of physics and any theoretical physics
interested in
foundational issues.
PZ: DidnÕt Cartan produce a general covariant version of
Newtonian
theory? CanÕt you do all the
metric tensor stuff within a purely Newtonian framework? The
metric
tensor description is
a mathematical truism. It doesnÕt apply only to Einsteinian
physics.
JS: ThatÕs why I brought up the distinction between the \
local
pseudo-group of coordinate transformations at a single P and
the
active P -> PÕ =/= P diffeomorphisms. That distinction may \
be
important
in posing the relevant question here.
There is the issue of the relation of map to territory and
even what is
the territory?
PZ: At this point I canÕt claim to understand the relevance
of this
distinction.
JS: One point is that if reality is objective it cannot
depend on how we
perceive it. We must be able to compute invariants that are
the same
numbers for all local observers independent of their local
points of
view or frames of reference. That is, the raw pattern of
detector clicks
speci\[CapitalThorn]c to a local frame must be processed to produce an
invariant
pattern for the actual thing that happened. That is basically
relativity
in its most general form. There is also the complementary
notion of no
action without direct reaction. That is, no non-dynamical
absolutes. Not
only no absolute time, no absolute space, but also no absolute
space-time and also NO ABSOLUTE BIT QUANTUM WAVE that acts on
its IT
(extra variable) without direct reaction of IT back on BIT.
That is not
IT FROM BIT
in isolation but also
BIT FROM IT
at the deepest level.
End of Part II. To be continued.
===
Subject: Re: Math question
===
>Subject: Math question
>If the third term in a geometric sequence is 2x-3 and the
\[CapitalThorn]fth term
>is 3x+3 and the recursive rule is that each term is
multiplied by 3,
>then \[CapitalThorn]nd the value of x.
found it, now what?
adam
===
Subject: Re: Math question
>If the third term in a geometric sequence is 2x-3 and the
\[CapitalThorn]fth term
>is 3x+3 and the recursive rule is that each term is
multiplied by 3,
>then \[CapitalThorn]nd the value of x.
> found it, now what?
Convert it into a lottery number.
===
Subject: Re: Math question
permission for an emailed response.
> >If the third term in a geometric sequence is 2x-3 and the
\[CapitalThorn]fth term
>is 3x+3 and the recursive rule is that each term is
multiplied by 3,
>then \[CapitalThorn]nd the value of x.
> found it, now what?
> Convert it into a lottery number.
I canÕt fathom the lottery. (Except on those rare days when
the
expect gain goes over zero. There were people at MIT when I
was there
that kept track and went out and bought lots of tickets on
those
days. God bless Ôem for consistency.)
But a horse race can be fun; there is at least an athletic
contest
there, and you can cheer, and the atmosphere of the track can
be fun
too. So can I put $2 onto horse number x, rather than buy a
lottery
ticket?
Thomas
===
Subject: Hal Puthoff answers John BaezÕs objections
well indirectly he does
Actually, a good deal of evidence exists to show some do carry
suf\[CapitalThorn]cent Q. But it is classi\[CapitalThorn]ed codeword SAP.
I need more details on that. Note to the others, Kit was in a
very
high position in USG to have access to important REAL UFO
data. Anything
he says on the empirics here must be taken very seriously.
Also latest from Hal:
JS: Hal, I ask you, what would prove PV wrong? ;-)
HP: (1) Black holes really exist (instead of only very dark
gray
holes).
JS: What kind of observational data needed to make this
distinction?
HP: Photon circular orbits are at slightly different radii.
JS: How could we detect such a difference here on Earth?
That is general of course not limited to PV.
HP: (2) So-called non-black-hole Yilmaz stars (M > 2.8 solar
masses)
found not to exist. (Robertson evidence is that they do.)
JS: Please give complete details on this.
HP: See S. L. Robertson, The Astrophys. Jour. 517, L117-L119
(1999);
515, 365-380 (1999).
HP: (3) For dense matter SS distribution observation were
found to
match Schwarzschild instead of exponential metric.
JS: Not enough information in that cryptic sentence. What
does it mean?
HP: For a high mass, spherically symmetric density
distribution object,
the metric in free space surrounding the object would in the
PV model be
exponential rather than Schwarzschild in form, e.g., no event
horizon, etc.
JS: Cliff Will would be the guy to \[CapitalThorn]gure out if any such
difference
could actually be measured today?
The basic issue however is conceptual i.e. GM/c^2r >> 1
region in your
model.
How do you explain dark energy and dark matter with PV?
HP: Our cosmological modeling under way as we speak. Stay
tuned!
JS: Does PlanckÕs h make any appearance in your PV math \
model?
HP: Not overtly at the macroscopic classical level. PV, like
GR, is a
classical theory. Only covertly at the level of the virtual
electron-positron-pairs plasma interactions that form the
underlying
polarizability of the vacuum.
JS: ThatÕs not good enough IMHO. You will note my model for
metric
engineering is quite explicit about the role of PlanckÕs
quantum of
action and what coherence of the virtual electron-positron
pairs
inside the physical vacuum means mathematically. The result
is that the
Josephson effect coupling a
real superconductor to the vacuum is how to metric engineer.
There is a
similarity to Ray ChiaoÕs Gravity Radio. Ray is dealing with
propagating far \[CapitalThorn]elds of guv & Fuv, I am dealing with
non-propagating
near \[CapitalThorn]elds of both. We need to plug far \[CapitalThorn]eld \
leaks like a
dripping faucet.
Again I ask what does PV mean when your GM/c^2r >> 1.
HP: It means the same as when GM/c^2r << 1. It is not a
special
coordinate position in PV as it is in GR. It has no special
signi\[CapitalThorn]cance. As in Yilmaz, there is no event horizon there,
itÕs just
another place on a smoothly varying coordinate map.
JS: Yes, I thought you would say that. This is simply bad
mathematics on
your part IMHO and I bet every Big Pundit in theoretical
physics, if
they gave your claim attention, would essentially agree with
me on this
particular.
Do you think super-steel rods exist?
HP: Not locally, of course not! The remote rods as a
reference play
that role, however.
JS: This is a bait and switch, you do not need PV for that! GR
explains that in for example the
gravity redshift and the gravity bending of light (gravity
lensing
evidence for dark matter exotic w = -1 vacuum regions IMHO).
Two rubber rods and rubber clocks in different regions of
VARIABLY
curved space-time is all one needs.
Super-steel? Who ordered that? You cannot challenge
EinsteinÕs GR on
such a ßimsy pretext.
Also it is not clear if dark energy exotic vacuum may not
also give a
gray hole?
The Hawking-Penrose singularity theorems may break down now
that exotic
vacuum dark energy has been discovered sinced
tuv(Exotic Vacuum)(dX^u/ds)(dX^v/ds) < 0
I think can happen. If so, the basic assumption of the
Hawking-Penrose
theorems are false. I am not sure of this yet.
If they do not exist, what physical meaning does your equation
cÕ = c/K
really have?
HP: It means that from afar the delay time of a light ray
passing by a
mass is seen to be delayed by an amount corresponding to this
slower
speed of light (as in GR, where cÕ = c[1-2phi/c^2]).
JS: OK that is reasonable. Has this effect been found? If so,
does it
not have an orthodox GR explanation? I am sure it does if it
is real.
Give references I am rusty on this off the top of my head.
Are there not
now accurate laser ranging of Moon etc that can detect this
sort of
thing? Is it accurate enough to
tell difference between
goo(PV) = e^-2GM/c^2r
and
goo(GR) = 1 - 2GM/c^2r
when GM/c^2r << 1 ?
How does a ßying saucer ßy in your PV model?
How do you metric engineer a Star Gate in your PV model?
HP: Sorry, proprietary trade secrets! :-)
JS: That is not playing by the rules of good physics since we
all know
of your
serious involvement in the UFO issue since at least the early
1970Õs.
Indeed
I have the tape recording of our 1973 meeting in which you
essentially
admit to reality
of the saucers!
HP: Honestly, still exploring various options for
manipulating the
spacetime metric.
JS: Fine, but rules of good physics demand that you freely
publish these
options so that
I, for example, can vet them. You are free to return the
favor on my
competing ideas.
Is time travel to the past thinkable in your PV model?
HP: No, at least at its present level of development.
JS: IMHO this is further proof that you are on the wrong
path. But here
I may be in the
minority opinion.
Are parallel universes next door thinkable in your PV model?
HP: I donÕt see any barrier to it, though we have not
explored that
option to date.
JS: Do you think Jacques ValleeÕs Magonia effects mean
parallel
universes
next door?
HP: ItÕs a little hard to differentiate between parallel
universes next
door and spacetime warp regions right here. Probably
isomorphic
descriptions.
JS: Plausible reply. It depends if you believe M theory or
not.
What about Eric DavisÕs MUFON 2001 report of creature
emerging out of
a kind of sphere of light (Star Gate) in sky at Robert
BigelowÕs NIDS
Utah Ranch?
HP: Sounds like a wormhole, Krasnikov tube, or some such,
doesnÕt it?
JS: Yes, we agree on that. Important point here is that we
both take
Eric DavisÕs NIDS backed report seriously and you are, were,
on the
NIDS Advisory Board with Jacques Vallee. Robert Bigelow has
put serious Las Vegas money into all this and now has a big
Space Vehicle Company in Vegas, so these are not simply
powerless kooks playing inconsequential games. NIDS at one
time at least not too long ago employed a large number of
ex-military,
FBI, police
investigators on the UFO data.
Do you plan to extend PV to include extra space dimensions?
HP: No. For right now extreme spacetime warps look to give
much of what
might be bought from extra space dimensions, so weÕll stick
to that for
now.
JS: OK
....<snip>...
It is a \[CapitalThorn]ction like
The Unicorn and like Hal PuthoffÕs super-steel in his quasi
measurement Tables I & II in his PV model that disintegrates
when one
asks what happens when GM/c^2r >> 1?
HP: Not so. See above.
JS: OK for now.
It is clear to me that Hal is not really thinking about the
topology and
differential geometry in his naive engineering approach.
HP: Finally you got something right! ThatÕs the beauty of \
PV,
you donÕt
have to drag that baggage around to answer certain
engineering questions.
JS: I do not think the mainstream theoretical physics
community would
\[CapitalThorn]nd your position here professionally acceptable and this is
a serious
issue which lessens the credibilty of UFO research. They say,
look that
Puthoff fellow is professing an obviously crackpot challenge
to
EinsteinÕs GR. Certainly this is what John Baez has
essentially
published and I am sure he speaks for all his friends who
dominate the
\[CapitalThorn]eld. I think your math here is simply bad.
Hal goes into a state of denial pretending there is no
problem. He tries
to solve the cosmological constant problem the same way. It
just
will not do IMHO.
HP: DonÕt know what you mean by this. We have no problem
including a
cosmological constant in the cosmological treatments, and
have a paper
submitted with some nice results (e.g., matching observation
without
missing matter, acceleration comes out naturally, etc.)
JS: Your earlier remark over a year ago about zero point
energy
K = e^2GM/c^2r
HP: Not so. ThatÕs the solution for one problem only, a
spherical
distribution of mass. You know I have published other
solutions for
charged masses, Levi-Civita effects, rotating dumbbells, etc.
Why do
you keep repeating this (false) mantra?
JS: I did not mean that literally. I meant you have no h, you
only have
classical phenomenology. I did mention you added a Q indeed
Kit Green
alludes to that in his remark at begining of this message.
What Hal has is his mythical super steel rods which would
give NOT c
but c/K! You forgot that!
HP: Surely, youÕre joking Mr. Sarfatti. At least \
IÕm joking.
My super
steel rods are a euphemism for unperturbed measurement
instruments far
from metric distortion, providing a virtual background, as it
were.
Were you taking them seriously? Houston, we have a problem.
JS: Yes, because your writing is not clear on this issue and
there is
the notion of Yilmaz of two local metrics. You do not, in
your informal
explanations seem to take into account that the EEP is a
LOCAL principle
so that LIF observers at P should see what your distant rest
LNIF
observer sees.
That is the LIF observer at P corresponding to your r sees
special
relativity locally so that the free ßoat timelike geodesic
LIF rods and
clocks are NOT gravitationally distorted the way the
COINCIDENT LNIF
forces putting them on the timelike non-geodesic! Therefore,
the Lorentz
constructive theory is really there in orthodox GR if you dig
deep
enough!
The problem is that Hal is completely obscure to my mind on
the
fundamental world view of his model. He uses metric notation
after all?
HP: Not really. Only to make comparisons with those like you
who are
addicted to such!
JS: Like 99% of all theoretical physicists in the \[CapitalThorn]eld. This
is exactly
what I mean when I say PV has no rational rules of
engagement. ItÕs
Anything Goes.
> PZ: But it is a physical metric that is simply a
mathematical
> description of the physical deformation of measuring
> devices and the resulting scaling of the measured
intervals. It is not
> a theory about the fundamental chronogeometric
> structure of the world -- any more than is the description
of the
> behavior of metal bars on a heated surface (an
> example that Feynman liked to use).
PZ: Bravo Paul, bravo!
JS: I think you are missing the point here Paul. Hal seems to
think
that there really are super steel rods and clocks that would
measure
c/K when the Einstein rubber rods and clocks measure c.
HP: Tsk, tsk, Jack. See above. There are no super steel rods,
really. DonÕt need them. If you think that I really think
there are,
then IÕll have to slow up the discussion so you can follow
the bouncing
ball more easily.
JS: This after all
would be a real bimetric world with super steel in the
globally
ßat Yilmaz world parallel to to EinsteinÕs rubbery \
curved
world. This
Yilmazian split is \[CapitalThorn]ction that Hal thinks is fact IMHO.
HP: Tsk, tsk.
JS: Again I ask. What does HalÕs PV mean when his GM/c^2r >>
1?
HP: As I said above, same as when GM/c^2r << 1. Not a magic,
mystical
coordinate location of an event horizon as in canonical GR.
[BTW, to
give some clari\[CapitalThorn]cation, the event horizon appears in GR in
the form of,
e.g., a term (1-x) in a metric term denominator. In the
exponential
metric this is seen to be simply a truncated form of the
correct metric
polynomial 1-x+x^2/2! - x^3/3! ..... = exp (-x) .]
Hal
===
Subject: Re: Hal Puthoff answers John BaezÕs objections
my UFO technology allows me to predict taht you wonÕt answer
me!
seriously, why on Earth should we take this Kit clown
seriously
about REAL UFO data -- can you give us a single (unclassi\[CapitalThorn]ed)
example
of his awe-inspiring veracity?
like, did the Cheif Roswell Spook, Corso, give him a piece
of balsa from the Shrine of the Enola GayÕs Bomber Squadron
Base?
surely, youÕre joking, mister Sarfatti!
> I need more details on that. Note to the others, Kit was in
a very
> high position in USG to have access to important REAL UFO
data. Anything
> he says on the empirics here must be taken very seriously.
> HP: See S. L. Robertson, The Astrophys. Jour. 517,
L117-L119 (1999);
> 515, 365-380 (1999).
> HP: (3) For dense matter SS distribution observation were
found to
> match Schwarzschild instead of exponential metric.
> JS: Not enough information in that cryptic sentence. What
does it mean?
> HP: Surely, youÕre joking Mr. Sarfatti. At least \
IÕm
joking. My super
> steel rods are a euphemism for unperturbed measurement
instruments far
> from metric distortion, providing a virtual background, as
it were.
> Were you taking them seriously? Houston, we have a problem.
--Give the Gift of Dick Cheeny -- out of of\[CapitalThorn]ce, at last!
http://www.benfranklinbooks.com/
http://www.wlym.com/pages/music.html
http://www.rand.org/publications/randreview/issues/rr.12.00/
http://members.tripod.com/~american_almanac
===
Subject: Re: Archimedes the Combinatorist
> as the \[CapitalThorn]rst combinatorist. See www.nytimes.com, National
News, or the
url
> Bull, that url is asking for my membership number and
password.
> It isnÕt any reference, itÕs just a way of \
advertizing for
NYT.
is to the Seattle Times, no subscription required:
http://seattletimes.nwsource.com/html/nationworld/2001814654_
archimedes14.ht
ml
===
Subject: Re: Archimedes the Combinatorist
> as the \[CapitalThorn]rst combinatorist. See www.nytimes.com, National
News, or
the url
>
http://seattletimes.nwsource.com/html/nationworld/2001814654_
archimedes14.htm
l
Ok, Ôtwas a pleasant read about a bit of history.
===
Subject: Re: Archimedes the Combinatorist
> ÔCause IÕm not wanting a subscription. \
IÕm want a one time
use to scan
an
> better spent scanning the large volumn of sci.math posts.
Yes, IÕd
\[CapitalThorn]ll
> out the form if I knew that I really really wanted to read
it.
Otherwise,
> such a deterent to scanning posts, puts it last to be read,
to be read
if
> itÕs a slow night and IÕm caught up on \
email,
sci.space.news posts and
> lack for a problem to slove.
> Um, ok, fair enough. But that doesnÕt mean the site is
really just an
> ad.
Perhaps not. However in these days of hyper-commericalism...
Why do they need people to register?
It smells like Safeway where you have to have to be a
registered
plastic card carrying shopper to be given sales prices.
Needless to say, being so insulted by a store that was once
open to the
public, I donÕt shop there. I go where shoppers are openly
welcome.
===
Subject: Re: Archimedes the Combinatorist
permission for an emailed response.
> Perhaps not. However in these days of hyper-commericalism...
> Why do they need people to register?
I can think of three obvious reasons. (Keep in mind that pay
customers have to register too...)
1) They want to know what their geographical reach is;
2) They want to be able to offer demographic information to
their
advertisers;
Now they donÕt *need* people to register, but the New York
Times is a
pretty respectable out\[CapitalThorn]t (in my book, at least), and they
perform a
pretty nice service.
> It smells like Safeway where you have to have to be a
registered
> plastic card carrying shopper to be given sales prices.
> Needless to say, being so insulted by a store that was once
open to the
> public, I donÕt shop there. I go where shoppers are openly
welcome.
Well, I shop at such stores all the time, but I refuse to do
the
plastic card routine, and one reason is because I know that
they track
information on individual shoppers, and have at least once
used it
against the interest of the shopper.
I am very con\[CapitalThorn]dent that this will not occur with the New York
Times,
in part because of their excellent reputation for caring
greatly about
\[CapitalThorn]rst amendment issues.
Thomas
===
Subject: Re: Archimedes the Combinatorist
> Perhaps not. However in these days of hyper-commercialism...
> Why do they need people to register?
> I can think of three obvious reasons. (Keep in mind that pay
> customers have to register too...)
> 1) They want to know what their geographical reach is;
> 2) They want to be able to offer demographic information to
their
> advertisers;
No problem, if they want me to unzip in public, for their
service
theyÕd be welcome to my zip code. Zip, there it is and on to
the
paper age cultivation of a neomalady, formaphobia, has become
a survival
skill.
I hate stores that pump you of all sorts of information
before they
answer a simple question, do have what IÕm looking for. Such
obscene
behavior is appropiate for job interviews, including dates
and mates.
> It smells like Safeway where you have to have to be a
registered
> plastic card carrying shopper to be given sales prices.
> Needless to say, being so insulted by a store that was once
open to the
> public, I donÕt shop there. I go where shoppers are openly
welcome.
> Well, I shop at such stores all the time, but I refuse to
do the
> plastic card routine, and one reason is because I know that
they track
> information on individual shoppers, and have at least once
used it
> against the interest of the shopper.
Whoa! How could that be? IÕve heard stories where students
were given
anonymous surveys which were used to crack down, not on
individual
students, but on the schools where a suf\[CapitalThorn]cient naive percent
of the
students admitted drug use.
> I am very con\[CapitalThorn]dent that this will not occur with the New
York Times,
> in part because of their excellent reputation for caring
greatly about
> \[CapitalThorn]rst amendment issues.
===
Subject: Re: Archimedes the Combinatorist
permission for an emailed response.
> No problem, if they want me to unzip in public, for their
service
> theyÕd be welcome to my zip code. Zip, there it is and on
to the
> paper age cultivation of a neomalady, formaphobia, has
become a survival
> skill.
I donÕt object to this... As it happens, my browser keeps \
the
info,
and I never have to register more than once. I use the same
id for
all the newspapers I read online, so if IÕm on some \
different
web
browser than my usual, I always know what to type.
If you donÕt want to do that, thatÕs \
\[CapitalThorn]ne! IÕm not saying you
is not the same as giving a url for just an advertisement. If
I told
way to get it was to go register at your library, or buy a
copy, would
that mean I had only given you an ad?
> I hate stores that pump you of all sorts of information
before they
> answer a simple question, do have what IÕm looking for.
Such obscene
> behavior is appropiate for job interviews, including dates
and mates.
I donÕt think of dates and mates as a job, but then perhaps \
I
have a
different attitude about such things. Still, I do share the
general
annoyance that you feel here.
> Well, I shop at such stores all the time, but I refuse to
do the
> plastic card routine, and one reason is because I know that
they track
> information on individual shoppers, and have at least once
used it
> against the interest of the shopper.
> Whoa! How could that be? IÕve heard stories where students
were given
> anonymous surveys which were used to crack down, not on
individual
> students, but on the schools where a suf\[CapitalThorn]cient naive
percent of the
> students admitted drug use.
Because in order to register at the shop you have to give
them your
name and such. There was a case where the court issued a
subpoena for
the record, IIRC, and another where the store used the
purchasing
history of a patron against them in a liability lawsuit. IÕm
very
con\[CapitalThorn]dent that the nytimes would not to such a thing.
Thomas
===
Subject: Re: Archimedes the Combinatorist
> Well, I shop at such stores all the time, but I refuse to
do the
> plastic card routine, and one reason is because I know that
they
track
> information on individual shoppers, and have at least once
used it
> against the interest of the shopper.
> > Whoa! How could that be? IÕve heard stories where
students were given
> anonymous surveys which were used to crack down, not on
individual
> students, but on the schools where a suf\[CapitalThorn]cient naive
percent of the
> students admitted drug use.
> Because in order to register at the shop you have to give
them your
> name and such. There was a case where the court issued a
subpoena for
> the record, IIRC, and another where the store used the
purchasing
> history of a patron against them in a liability lawsuit.
IÕm very
> con\[CapitalThorn]dent that the nytimes would not to such a thing.
Safeway didnÕt require identi\[CapitalThorn]cation. However \
lots of not
thinking
people would give info just on reßex. Can you recall any
details about
the subponea and the liability lawsuit?
===
Subject: Re: Archimedes the Combinatorist
permission for an emailed response.
peeringly
> Because in order to register at the shop you have to give
them your
> name and such. There was a case where the court issued a
subpoena for
> the record, IIRC, and another where the store used the
purchasing
> history of a patron against them in a liability lawsuit.
IÕm very
> con\[CapitalThorn]dent that the nytimes would not to such a thing.
> Safeway didnÕt require identi\[CapitalThorn]cation. However \
lots of not
thinking
> people would give info just on reßex. Can you recall any
details about
> the subponea and the liability lawsuit?
Not more than I posted above. Google might have more
information. If
memory serves, the liability lawsuit was Safeway, but I \
canÕt
be
sure. I donÕt remember which company got the subpoena.
Thomas
===
Subject: Re: looking for a formula to derive these numbers
>>Can a formula be found for the numbers on the right given
the value on the
left?
>>4 2
>>8 4
>>9 2
>>12 6
>>16 12
>>18 4
>>20 10
>>24 12
>>25 4
>>27 6
>>28 14
>>32 24
>>36 30
>>40 20
>>44 22
IÕm adding from a posting in atl.math.recreational, that I
just posted:
> Can anyone recognize how the numbers on the right hand side
are derived
from the
> left hand number?
Well, this may be a start.
Call the columns like this
---------------------
Columns
a b
---------------------
> 4 SN 2 = 2*1
> 8 4 2*(4)
> 9 SP 2 = 3-1
First, select all numbers, where b=a/2, and add a column c,
which gives the 4Õth part of a
---------------------
Columns
a b c=a/4
---------------------
4 2 1
8 4 2
xxx 9
12 6 3
xxx 16
20 10 5
xxx 18
24 12 6
xxx 25
xxx 27
28 14 7
xxx 32
xxx 36
40 20 10
44 22 11
xxx 45
xxx 48
52 26 13
xxx 49
xxx 50
xxx 52
56 28 14
60 30 15
68 34 17
76 38 19
84 42 21
88 44 22
92 46 23
Then in column c certain values are missing.
The sequence of c is
c =
[
1,2,3,5,6,7,10,11,13,14,15,17,19,21,22,23,26,29,30,31,33,34,35
,37,38,39,41,4
2,43,46,47]
Now check the missings in c more accurately and de\[CapitalThorn]ne
the sequence d of these missings:
c = [1,2,3, 5,6,7, 10,11, 13,14,15, 17, 19, 21,22,23, 26,
29,30,31, 33,...
d = [ 4, 8,9, 12, 16, 18, 20, 24,25,
27,28, 32,
Compare d to a:
a = [ 4, 8,9, 12, 16, 18, 20, 24,25,
27,28, 32,36,40,44,45,48,49,50,52,
So I think, d is a pretty good prognosis for a. IÕm not
enough engaged to
put this in a formula, but I have seen things similar like
this recently,
and it was de\[CapitalThorn]ned something like:
take any element of N but delete all, which are the double of
n
which allows all odd natural numbers plus all naturals, which
are divisble
by 4.
(But as some other posters pointed out: you also can generate
a polynom covering the two sequences)
Gottfried Helms
> IÕve \[CapitalThorn]lled in some, but a general rule still \
seems dif\[CapitalThorn]cult
to \[CapitalThorn]nd.
> SN = square non-prime
> SP = square prime
> *(n) means multiplied by the right hand side of the line
that contains n
> 4 SN 2 = 2*1
> 8 4 2*(4)
> 9 SP 2 = 3-1
> 12 6 3*(4)
> 16 SN 12 4*3
> 18 4 2*(9)
> 20 10 5*(4)
> 24 12 6*(4)
> 25 SP 4 = 5-1
> 27 6 3*(9)
> 28 14 7*(4)
> 32 24 2*(16)
> 36 SN 30 6*5
> 40 20 10*(4)
> 44 22 11*(4)
> 45 10 5*(9)
> 48 36 3*(16)
> 49 SP 6 = 7-1
> 50 8 2*(25)
> ------------------
> 52 26
> 54 12
> 56 28
> 60 30
> 63 14
> 64 SN 56 = 8*7
> 68 34
> 72 60
> 75 12
> 76 38
> 80 60
> 81 SN 24
> 84 42
> 88 44
> 90 20
> 92 46
> 96 72
> 98 12
> 99 22
> 100 SN 74
> 104 52
> 108 90
> 112 84
> 116 58
> 117 26
> 120 60
> 121 SP 10 = 11-1
> 124 62
> 125 20
> 126 28
> 128 112
> 132 66
> 135 30
> 136 68
> 140 70
> 144 SN 164
> 147 18
> 148 74
> 150 24
> 152 76
> 153 34
> 156 78
> 160 120
> 162 48
> 164 82
> 168 84
> 169 SP 12 = 13-1
> 171 38
> 172 86
> 175 28
> 176 132
> 180 150
> 184 92
> 188 94
> 189 42
> 192 168
> 196 SN 134
> 198 44
> 200 148
> Is there some sum of products formuli for this?
===
Subject: Re: looking for a formula to derive these numbers
> IÕm adding from a posting in atl.math.recreational, that I
just posted:
> > Can anyone recognize how the numbers on the right hand
side are
> derived from the
> > left hand number?
> Well, this may be a start.
> First, select all numbers, where b=a/2, and add a column c,
> which gives the 4Õth part of a
> ---------------------
> Columns
> a b c=a/4
> ---------------------
> 4 2 1
> 8 4 2
> xxx 9
> 12 6 3
> xxx 16
> 20 10 5
> xxx 18
> 24 12 6
> xxx 25
> xxx 27
> 28 14 7
> xxx 32
> xxx 36
> 40 20 10
> 44 22 11
> xxx 45
> xxx 48
It seems, the column c is somehow the basic sequence.
a is then constructed by
4 #* [c] // #* is the operato for elementwise multiplication
9 #* [c]
16 #* [c]
25 #* [c]
...
Where in c are all intergers>0 except that, which are in a.
So c and a must be constructed iteratively in combination, or,
much more simple:
c is the sequence of the square-free numbers.
a is the combined sequence of all partial sequences where
sequence c multiplied
with squares of all integers>1 (call them SQ)
Example:
a is the joint list of all rows below:
4 #*[1,2,3,5,6,7,10,...]
9 #*[1,2,3,5,6,7,10,...]
16 #*[1,2,3,5,6,7,10,...]
...
SQ(i) #*c
--------------------------------------------------
b is also a multiple of c, but it is more dif\[CapitalThorn]cult, since
for each SQ there is another coef\[CapitalThorn]cient.
For instance the entries of b, which correspond to that of
4#*c = 4*[1,2, 3, 5, 6, 7, 10,...]
= [4,8,12,20,24,28, 40,...]
are
2#*c = [2,4, 6,10,12,14, 20,...]
for that of
9#*c = 9*[1, 2, 3, 5, 6, 7,10,...]
= [9,18,27,45,54,63,90,...]
are
2#*c = [2, 4, 6,10,12,14,20,...]
and so on, with a distinct coef\[CapitalThorn]cient f for each SQ.
The underlaying rule is not yet clear. A starting table is
i SQ f
-----------------------
2 4 2
3 9 2
4 16 12 3*4
5 25 4
6 36 30 5*6
7 49 6
8 64 56 7*8
9 81 24
10 100 74 2*37
11 121 10
12 144 164 4*41
13 169 12
14 196 134 2*67
At least one can assume, that for the primes in i, the value
for f is just i-1, but for the other numbers things are
obscure.
------------------------------
The display in more detail:
While a is the joint list of all rows below:
4*[1,2,3,5,6,7,10,...]
9*[1,2,3,5,6,7,10,...]
16*[1,2,3,5,6,7,10,...]
25*[1,2,3,5,6,7,10,...]
36*[1,2,3,5,6,7,10,...]
...
SQ(i)*c
is b the joint list of all
2*[1,2,3,5,6,7,10,...]
2*[1,2,3,5,6,7,10,...]
12*[1,2,3,5,6,7,10,...]
4*[1,2,3,5,6,7,10,...]
30*[1,2,3,5,6,7,10,...]
...
f(i)*c
in the corresponding order to that of a
Gottfried Helms