mm-374
===
Subject: Re: Reconsidering Halton Arp
>> Considering most things fall in air, not a good experiment,
assuming
>> it was done at all.
>Why? If Aristotle is taken to be referring to terminal
velocity, it's
>much better to check in water, since terminal velocity is
more easily
>reached.
>Thomas
And it occured to me today to wonder which he was referring
to. Was much
of a distinction made in Aristotle's day between velocity and
acceleration?
--
Is that plutonium on your gums?
Shut up and kiss me!
-- Marge and Homer Simpson
Subject: Re: Reconsidering Halton Arp
Originator: grubb@lola
>And it occured to me today to wonder which he was referring
to. Was much
>of a distinction made in Aristotle's day between velocity and
>acceleration?
Of course not! The concept of acceleration would have been
quite outside
of the mindset of that time. In particular, you could only
consider
ratios of like things, eg. lengths and lengths, times and
times.
Doing even a velocity (ratio of length and time) was stretching
things. Usually for 'constant velocity' they said something
about
the ratio of the lengths traveled and the ratio of the times
taken
being equal.
It wasn't really until Oresme in the 14th century that even
constant
acceleration was thought about. And Oresme was way ahead of
his time.
--Dan Grubb
Subject: Re: Reconsidering Halton Arp
>>And it occured to me today to wonder which he was referring
to. Was much
>>of a distinction made in Aristotle's day between velocity and
>>acceleration?
>Of course not! The concept of acceleration would have been
quite outside
>of the mindset of that time. In particular, you could only
consider
>ratios of like things, eg. lengths and lengths, times and
times.
>Doing even a velocity (ratio of length and time) was
stretching
>things. Usually for 'constant velocity' they said something
about
>the ratio of the lengths traveled and the ratio of the times
taken
>being equal.
>It wasn't really until Oresme in the 14th century that even
constant
>acceleration was thought about. And Oresme was way ahead of
his time.
Looking back from the 21st century, that seems strange.
Clearly states of
motion change. That must be one of those paradigm things. They
could
determine how far something goes, and how much time it took,
and the thing
that gets there first went faster.
--
I'm giving you the chance to look fate in those pretty eyes of
hers
and say, 'Step off, bitch. This is my party and you're not
invited.'
-- Chris Shugart, _Testosterone Magazine_
Subject: Re: Reconsidering Halton Arp
Originator: grubb@lola
>And it occured to me today to wonder which he was referring
to. Was much
>of a distinction made in Aristotle's day between velocity
and
>acceleration?
>>Of course not! The concept of acceleration would have been
quite outside
>>of the mindset of that time. In particular, you could only
consider
>>ratios of like things, eg. lengths and lengths, times and
times.
>>Doing even a velocity (ratio of length and time) was
stretching
>>things. Usually for 'constant velocity' they said something
about
>>the ratio of the lengths traveled and the ratio of the times
taken
>>being equal.
>>It wasn't really until Oresme in the 14th century that even
constant
>>acceleration was thought about. And Oresme was way ahead of
his time.
>Looking back from the 21st century, that seems strange.
Clearly states of
>motion change. That must be one of those paradigm things.
They could
>determine how far something goes, and how much time it took,
and the thing
>that gets there first went faster.
And they were aware of that, of course. But they had no
mathematical
way of thinking about it. Remember that ratios were *not
fractions* for
these people. They could talk about equal ratios, but did not
consider
ratios to be numbers. Furthermore, the ratio of say, an area
to a
length was just not allowed, let alone a ratio of a length to
a time.
It really was a major advance to be able to think about
velocity
as a separate thing and as a ratio.
--Dan Grubb
Subject: Re: Reconsidering Halton Arp
>>And it occured to me today to wonder which he was
referring to. Was
much
>>of a distinction made in Aristotle's day between velocity
and
>>acceleration?
>Of course not! The concept of acceleration would have been
quite outside
>of the mindset of that time. In particular, you could only
consider
>ratios of like things, eg. lengths and lengths, times and
times.
>Doing even a velocity (ratio of length and time) was
stretching
>things. Usually for 'constant velocity' they said something
about
>the ratio of the lengths traveled and the ratio of the
times taken
>being equal.
>It wasn't really until Oresme in the 14th century that even
constant
>acceleration was thought about. And Oresme was way ahead of
his time.
>>Looking back from the 21st century, that seems strange.
Clearly states of
>>motion change. That must be one of those paradigm things.
They could
>>determine how far something goes, and how much time it took,
and the thing
>>that gets there first went faster.
>And they were aware of that, of course. But they had no
mathematical
>way of thinking about it. Remember that ratios were *not
fractions* for
>these people. They could talk about equal ratios, but did not
consider
>ratios to be numbers. Furthermore, the ratio of say, an area
to a
>length was just not allowed, let alone a ratio of a length to
a time.
>It really was a major advance to be able to think about
velocity
>as a separate thing and as a ratio.
I never really thought about fractions as such a conceptual
advance. And
if the Greeks had thought about fractions, one of them would
have been
sure to have worried about a ratio of distance to time that
can't be given
as a ratio of integers, and drawn conclusions about the
validity of the
method or the nature of motion from that.
--
You're not as dumb as you look. Or sound. Or our best testing
indicates. -- Monty Burns to Homer Simpson
Subject: Re: Reconsidering Halton Arp
permission for an emailed response.
X-Tom-Swiftie: Remember to kill all the child processes, Tom
said on
signal
>> Considering most things fall in air, not a good experiment,
assuming
>> it was done at all.
>Why? If Aristotle is taken to be referring to terminal
velocity, it's
>much better to check in water, since terminal velocity is
more easily
>reached.
>Thomas
> And it occured to me today to wonder which he was referring
to. Was much
> of a distinction made in Aristotle's day between velocity and
> acceleration?
None whatsoever that I can see.
Thomas
Subject: Re: Reconsidering Halton Arp
>Now, I've defined everything for you, and yet you still
refuse to identify
a
>method. I'm done with you in the thread, unless you care to
proffer a
>single counterexample.
Patience, grasshopper. I said in advance that I expected you
would
answer my questions, and that once things were clarified I was
prepared to respond. I also said that I bookmarked a bunch of
sites. I
also said I wanted to take the time to read the methods
thoroughly to
describe them correctly. This process will take, given work,
etc.,
about another 24 hours.
I'm no expert, I just skimmed a bunch of stuff. But again,
it's pretty
pointless to use the Hubble constant to generate an x-axis if
you're
trying to measure the Hubble constant, isn't it? If your x
values are
y/H0, what do you think you'll measure for a in the equation y
= a*x?
- Randy
Subject: Re: Reconsidering Halton Arp
>Now, I've defined everything for you, and yet you still
refuse to
identify a
>method. I'm done with you in the thread, unless you care to
proffer a
>single counterexample.
> Patience, grasshopper. I said in advance that I expected you
would
> answer my questions, and that once things were clarified I
was
> prepared to respond. I also said that I bookmarked a bunch
of sites. I
> also said I wanted to take the time to read the methods
thoroughly to
> describe them correctly. This process will take, given work,
etc.,
> about another 24 hours.
> I'm no expert, I just skimmed a bunch of stuff. But again,
it's pretty
> pointless to use the Hubble constant to generate an x-axis
if you're
> trying to measure the Hubble constant, isn't it? If your x
values are
> y/H0, what do you think you'll measure for a in the equation
y = a*x?
One last time, Mr. Osmium. I never claimed, nor implied, that
anyone is
using the Hubble constant to measure the Hubble constant.
Hopefully this additional post will save you from wasting
effort in
attempting to address a point no one made.
--
greywolf42
ubi dubium ibi libertas
{remove planet for return e-mail}
Subject: Re: Reconsidering Halton Arp
>>Now, I've defined everything for you, and yet you still
refuse to identify
a
>>method. I'm done with you in the thread, unless you care to
proffer a
>>single counterexample.
>Patience, grasshopper. I said in advance that I expected you
would
>answer my questions, and that once things were clarified I was
>prepared to respond. I also said that I bookmarked a bunch of
sites. I
>also said I wanted to take the time to read the methods
thoroughly to
>describe them correctly. This process will take, given work,
etc.,
>about another 24 hours.
>I'm no expert, I just skimmed a bunch of stuff. But again,
it's pretty
>pointless to use the Hubble constant to generate an x-axis if
you're
>trying to measure the Hubble constant, isn't it? If your x
values are
>y/H0, what do you think you'll measure for a in the equation
y = a*x?
What was the evidence for dark energy again?
--
Suppose you were an idiot... And suppose you were a member of
Congress... But I repeat myself. - Mark Twain
Subject: Re: Reconsidering Halton Arp
permission for an emailed response.
X-Zippy-Says: I want another RE-WRITE on my CAESAR SALAD!!
> Which, actually, we learned from Aristotle, the first great
> emiricist.
> Real empiricists check their work carefully. Aristotle did
not. He
> gets a D- in physics.
Where does that put, say, Galileo, who's statements about
pendulums
(false, by the way) can be checked by a simple experiment?
Subject: Re: Reconsidering Halton Arp
permission for an emailed response.
X-Tom-Swiftie: I can't get this knife to cut, Tom said dully
> couldn't they use an arbitrary pendulum, a particular one,
> to measure a few experiments?...
An interesting example. Galileo claimed that the period of a
pendulum
is independent of its amplitude. As any freshman physics
student
knows, this is false--and it can be checked quite simply.
Thomas
Subject: Re: Reconsidering Halton Arp
>>couldn't they use an arbitrary pendulum, a particular one,
>>to measure a few experiments?...
> An interesting example. Galileo claimed that the period of a
pendulum
> is independent of its amplitude. As any freshman physics
student
> knows, this is false--and it can be checked quite simply.
Galileo also got the tides wrong.
However he got enough right. He gets a C+.
The first A goes to Newton. There was no technology for Newton
to get a
whiff of relativistic mass. Newton gets a B- in optics, since
he did not
do the double slit experiment and he could have.
Aristotle got a D- because he got just about everything wrong.
Bob Kolker
Subject: Re: Reconsidering Halton Arp
permission for an emailed response.
X-Tom-Swiftie: We're going to sue you for that window system,
Tom said
inexorably
> However he got enough right. He gets a C+.
Physics is fine. (Of course, it's actually totally wrong, in
the same
was that Newton's physics is totally wrong. But it's right, in
the
way that Newton's is right.)
As for the biology, Charles Darwin said that Aristotle was the
greatest biologist before Linnaeus.
> The first A goes to Newton. There was no technology for
Newton to get
> a whiff of relativistic mass. Newton gets a B- in optics,
since he did
> not do the double slit experiment and he could have.
Newton could have looked at the bending of light around the
sun during
eclipses.
> Aristotle got a D- because he got just about everything
wrong.
You get an F, because you get everything wrong.
Thomas
Subject: GIMPS finds 40th Mersenne Prime
Probably old news to most who have an interest in huge prime
numbers, but it's now official (independently verified):
http://www.mersenne.org/20996011.htm
The new records table now has 5 GIMPS primes at the top again:
http://primes.utm.edu/top20/page.php?id=3
Congratulations to George, Scott, Michael, and all involved
in GIMPS.
(Note, this is the 40th known, there's a chance that there
exist
smaller ones in ranges that haven't been fully tested yet, and
therefore we can't be sure it's the 40th by size yet.)
More info about the GIMPS project can be found from browsing
its
home page at
http://mersenne.org/
Phil
--
Unpatched IE vulnerability: Alexa Related Privacy Disclosure
Description: Unintended disclosure of private information
when using the Related feature
Reference: http://www.secunia.com/advisories/8955/
Reference: http://www.imilly.com/alexa.htm
Subject: Re: Undecidiable number theory
> If ZFC could decide every 9-variable problem, then I
> think the 9-variable problem would be algorithmically
solvable.
> ??? why ???
<< Here's a quote from the Book review:
Matiyasevich's most important contribution
since solving the problem has to be his
introduction of new exponential Diophantine
coding techniques. With such, he improved the
initial Matiyasevich-Robinson
small-number-of-variables result from
the algorithmic unsolvability of the
general 13-variable Diophantine problem to
the algorithmic unsolvability of the 9-variable problem. >
Of course, we must assume that ZFC is consistent...
Suppose we ask: are the non-trivial solutions to x^5 + y^5 +
z^5 = w^5 ?
If the statement there are no non-trivial solutions to x^5 +
y^5 + z^5 =
w^5
is denoted by Q, then if Q is decidable in ZFC, there is a
fixed
algorithm (applying to other diophantine equations) that will
eventually find a proof in ZFC either of Q or of Not(Q):
A formal proof takes a form similar to:
Statement 1 [justification]
Statement 2 [justification]
...
Statment n [justification]
We can use the @ symbol to mean cariage return or new line.
Then the proof can be presented as a string of symbols:
@ symbols of line 1 @ symbols line 2 .... @ symbols line n @
[ The first and last @'s may be useful , in case blank spaces
are allowed
in formal proofs ... ]
So the size of a proof can be measured by the number of
characters
(blanks, variable names, ands, ors, quantifiers, parentheses,
...
the epsilon-set-membership symbol, and @ line separators).
We assume that only finitely many different symbols exist,
which is the
case with ZFC.
Then, for any number M>0, only finitely many prospective
theorems
of ZFC of length at most M exist. Checking whether as string
of symbols
is a proof of something is a mechanical procedure. Also
checking
whether a string of symbols is a proof in ZFC of Q can be done
by
an effective or mindless or mechanical procedure.
Assuming that the statement Q above is decidable in ZFC, then
one of Q, Not(Q) is a theorem of ZFC.
So here's a procedure to determine which of the two is a
theorem:
-------------
Let M = 0;
While no proof of Q or Not(Q) has been found:
{ //begin while block
(i) Let M = M +1
(ii) Check all strings of length <= M for being proofs
of Q; if one is found, make a note proof of Q found.
(iii) Check all strings of length <= M for being proofs
of Not(Q) ; if one is found, make a note proof of Not(Q)
found.
} // end while block
(comment: we have a proof now .... )
Print Proof of Q found or Proof of Not(Q) found, according to
the
result
noted in (ii) or (iii).
Stop.
--------------
We assumed that Q was decidable; following a mindless or
mechanical
procedure, we determined which of Q, Not(Q) is a theorem of
ZFC.
So, assuming that ZFC theorems are true, we know whether the
statement Q is true or not.
Another diophantine problem statement , say Proposition_111,
could be any
9-variable diophantine problem (a statement).
Suppose Proposition_111 is decidable in ZFC....
By applying the same procedure
above to Proposition_111 instead of statement Q, we could get a
True/False answer regarding Proposition_111. This would give us
an algorithm to decide, for any 9-variable diophantine
equation,
whether or not it has at least one solution. According to
the Book review, Matijasevic has shown
that it's impossible.
So this seems to me to imply that , if ZFC is consistent,
some 9-variable problem is undecidable in ZFC.
David Bernier
Subject: Re: Undecidiable number theory
Originator: tchow@lagrange.mit.edu.mit.edu (Timothy Chow)
>Of course, we must assume that ZFC is consistent...
[...]
>So, assuming that ZFC theorems are true, we know whether the
>statement Q is true or not.
Assuming that [arithmetic] ZFC theorems are true (i.e., that
ZFC has an
omega-model) is a stronger assumption than the mere assumption
that ZFC
is consistent.
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the
artillery---however great, will
never exceed four of those miles of which as many thousand
separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two
New Sciences
Subject: Re: What are the constants in the computing sciences?
--
www.StealthHostiing.com You rule Truman.
http://tinyurl.com/iky4
Hey Trueman...love the show. YOU ARE the Truman I heard him.
Very
spooky!
>Is the truman living in Townsville? I've been hearing stuff,
yeah.
Webmasters help the TRUEman by joining www.theBanner.net
Current:1
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--------------------------------------------------------------
--------------
------
> What are the constants in the computing sciences? By
computing
> sciences I mean Set Theory (ZF), the Theory of Computation
(Turing),
> Recursion Theory (Kleene), Proof Theory (Godel), Foundations
of
> Mathematics (Peano), Paradoxes (Russell), etc.
> Here are some examples:
> 1. Set Theory
> a. {}
> b. {{}}
> 2. Theory of Computation
> a. the smallest Turing Machine that never halts
> b. the smallest Universal Turing Machine (simulates any
given Turing
> Machine plus its input)
> 3. Recursion Theory
> a. the smallest program that outputs only itself
> b. the function that maps a 1-input program and literal to
the
> smallest 0-input program that uses the literal in place of
the input
> c. the function that maps any 1-input program that computes
function
> f(x) into the smallest program that computes f(f(x))
> 4. Proof Theory
> a. the smallest wff in PA that is undecidable (neither
provable nor
> refutable)
> 5. Foundations of Mathematics
> a. 0
> 6. Paradoxes
> a. the smallest English sentence that is neither true nor
false
> In each case, we are taking a system that can be used to
model objects
> in the real world, and instead are talking only about
objects created
> within the theory itself. Each constant is created
completely within
> the system.
> What other constants can people think of?
Isn't each a subset of the next? Rearraange Peano arithmetic
before Godels
proof
as Godel uses Peano in his proof. Hence the real primitive is
just those of
set theory,
a TM can be modelled in set theory, and so on.
Herc
Subject: Re: Conjecture regarding the number 12
> (My personal interest relates to the importance of the
number 12 in
> music. I want to support this importance with a formal--but
> simple--mathematical theorem, for some music classes I am
preparing.
> In music 12 is important because it has divisors 2, 3, 4
which are
> important rytmic divisions, and 12 is also the number of
chromatic
> tones.)
Musical scales and the generalized circle of fifths, American
Math.
Monthly 93 (1986) 695-701, and Variety and multiplicity in
diatonic
result is that the familiar diatonic scale is one of a quite
special
class of scales that possess all three of the following
attributes:
the partitioning property, CV, and deepness.
If you want to know what those terms mean, you have to read
the paper,
because I've long since forgotten.
--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
Subject: Some Partial Sums Differing By...(& a limit)
> Let {a(k)} and {b(k)} be two sequences of nonnegative reals,
where
> {a(k)} is either monotonically nondecreasing or
nonincreasing.
> Let m and n be positive integers.
> Let
> A(n) = sum{k=1 to n} a(k),
 B(m) = sum{k=1 to m} b(k),
> and
> C(n) = sum{k=1 to n} a(k) b(k mod m),
> where 1 <= (k mod m) <= m.
> So, I get :
> |A(n*m) *B(m) - m *C(n*m)| <= m *B(m) *max(a),
> where 'max(a)'
> is the largest a(k), among 1 <= k <= n*m.
> (So, max(a) = a(n*m) or a(1), depending on whether {a{k)} is
> nondecreasing or nonincreasing.)
> (A recent post(s) of mine
> uses a specific case of the above.)
> A result derived from above:
> If a(x) is a real -> real function that is, for all x,
defined and
> 1) positive
> 2) finite (no poles)
> 3) monotonic
> then:
 limit{ n-> oo}
> (sum{k=1 to m*n} a(k/n) b(k mod m)) /n
> =
> (1/m) (sum{k=1 to m} b(k)) integral{0 to m} a(x) dx
> First, am I right in all cases I have restricted the results
above to,
> if right in even those cases?
> And do I really need all 3 additional constraints upon a(x)
for the
> limit-result to be true?
> (since the result seems trivial somehow)
>
Well, yes, my integral-limit-result, anyway, IS trivial (and
most of
its restrictions on a(x) unnecessary).
Rewrite:
limit{n->oo}
(sum{k=1 to m*n} a(k/n) b(k mod m)) /n
as
limit{n-> oo}
(sum{j=0 to n-1} sum{k=1 to m} a((jm+k)/n) b(k) ) /n =
=
(sum{k=1 to m} b(k)) integral{0 to 1} a(xm) dx =
(sum{k=1 to m} integral{0 to m} a(x) dx /m ;
so the only condition we need on a(x) is that it be
integratable.
As for the inequality above:
|A(n*m) *B(m) - m *C(n*m)| <= m *B(m) *max(a),
this stands still as not very trivial.. .
Subject: Re: Asymptotics of Averages Involving
Number-of-Divisors
I have generalized the result in copy/pasted original post and
bottom-posted it.
> Let r be a fixed positive integer.
> Let S be a sequence of distinct positive integers
> where, if k is an element of S, then (k+r) is an element as
well.
> Consider a restricted number-of-divisors function, d[m],
> where d[m] = sum{k|m, k is element of S, k <= sqrt(m)} 1,
> = the number of positive divisors of m which are both <= the
> squareroot of m, and are elements of S.
> Now, if q = the number of elements of S which are <= r
> (which is the same number as between any adjacent multiples
of r),
> then:
> limit {n -> oo}
> (r/n) (sum{m=1 to n} d[m]) - ln(n) (q/2)
> always converges to a finite constant.
> (But which constant, depending upon S, I do not know.)
> ( As some of you have already guessed, this result is based
on two
> earlier results of mine {and, given the results' ease of
discovery,
> they are others' results probably as well}.)
>
If {b(j)} is a sequence of nonnegative reals, and r = positive
integer,
limit{n-> oo}
n
--- ---
r
--- > b(k(mod r))
n / /
--- ---
m=1 k|m
k <= sqrt(m)
r
---
-( > b(k) ) * ln(n)/2
/
---
k=1
always = a constant.
In linear-mode:
limit{n -> oo}
(r/n) sum(m=1 to n} sum{k|m, k <= sqrt(m)} b(k(mod r))
-(sum{k=1 to r} b(k)) ln(n)/2
always = a constant.
By the way, k(mod r) is such that
1 <= k(mod r) <= r.
(I forget now what restrictions are necessary to be put on
{b(k)} for
this to be necessarily true.)
Subject: Re: Goedels completeness theorem..
<382qsvk88q452rcg4dmrpmne7gtprv65ft@4ax.com>...
>Hi everyone!
>Is there anyone who can tell me where to find resources on
Goedels
>completeness theorem?
>/Anders
Dave Marker's lecture notes on first-order logic are here
http://www2.math.uic.edu/~marker/math502/mm1-5.pdf
--- Jeff
Subject: Re: Goedels completeness theorem..
> Is there anyone who can tell me where to find resources on
Goedels
> completeness theorem?
At www.google.com:
Goedel completeness theorem gives me 3510 hits.
Rene.
--
Ren.8e Meyer
Student of Physics & Mathematics
Zhejiang University, Hangzhou, China
Subject: Sequences and Applications?
> Hi all,
> shows how numerical sequences are applied? I have heard, and
maybe seen,
> them used in continued fractions and the like, but what else
is there?
> TIA,
> Lurch
Here is the URL of the Journal of Integer Sequences:
http://www.math.uwaterloo.ca/JIS/
which is associated with the Encyclopedia of Integer Sequences
(which
you *must* know about already, right???):
http://www.research.att.com/~njas/sequences/index.html#L
Subject: Re: Sequences and Applications?
Yes, I do know about the Encyclopedia of Integer Sequences.
First place I
looked.
Thanks everyone!
Lurch
> Hi all,
> shows how numerical sequences are applied? I have heard, and
maybe
seen,
> them used in continued fractions and the like, but what else
is there?
> TIA,
> Lurch
> Here is the URL of the Journal of Integer Sequences:
> http://www.math.uwaterloo.ca/JIS/
> which is associated with the Encyclopedia of Integer
Sequences (which
> you *must* know about already, right???):
> http://www.research.att.com/~njas/sequences/index.html#L
>
Subject: Re: Consider Chebyshev
I am just a girl with an interest in maths (did a Comp Sci
degree and I
think I drooled the entire way through the discrete maths
part), note
interest, I have a lot to learn.
Thought I might pick up a bit by tuning into this newsgroups.
I have never been to a newsgroup with such personallity. It is
hilarious.
All the usually stuff goes on, except every second post is
from some guy
claiming to that he knows something really important and he
hates all
mathematicians.
I have never read a funnier rant in my life. Kind of like the
town fool. I
mean even if this guy actually has discovered something
important, he has
zero social skills, and zero abilitiy to promote himself?
The big issue is that he 'is in it for money' which means he
can't go to
his
grave smiling knowing that someone may just reread his papers
and one day
be
proved true to the benefit of society. (Hey that raving guy
was actually
right!).
Nope, he is in it for the money, so although he hasn't
realised it, he must
conform to the same rules that all of us moneymaking people
conform to (ie
awareness of how to 'play the game').
Academics do get a rep for being geeks, but why is it that in
my career I
must be well rounded (ie understand social concepts, know how
to build my
career, know how to promote myself etc) but apparently this
guys doesn't
need any of this, he can be as rude as he likes because he
knows something
very important.
An absolute classic. I look forward to reading more of his
rantings....
Particulary laughed when fuffy posted 'Good morning James. You
are an
idiot.' kind of a calm, yes we know you are here, and do our
best to ignore
you, kind of response.
(First and probably last post, because as interesting as this
all is, it is
way over my head)
> Now I posted about the importance of my partial difference
equation
> that counts primes and pointed out that Riemann failed in
finding why
> the prime distribution and continuous functions like li(x)
were
> connected, which is an important point worth elaborating on,
and an
> important name in that elaboration is Chebyshev.
> You should know who he is, as he is the one who found the
limits for
> the prime distribution by using Euler's zeta function.
> The fact that he isn't lionized can tell you how valuable
that
> information was in answering the question that had intrigued
the likes
> of Gauss.
> Still he's actually a far more important figure in prime
research than
> Riemann, though Riemann today gets all the attention.
> Now you'll need to go do your research to get the
particulars, but
> basically what I've done is possibly find THE reason why the
count of
> prime numbers--the prime distribution--relates at all to
continuous
> functions.
> That is, the answer to WHY the discrete distribution and
continuous
> functions like x/ln x, or li(x) are connected.
> Now that is the case because I found a partial difference
equation,
> which is why it's so important, and also why you shouldn't be
> surprised that no one else in recorded history found it
before me.
> If some big name like Wiles or Ribet had found it, then it'd
be all
> over the freaking news. But nope, I was the one who found
it, and
> yes, I'm in it for the money.
> Told you the Universe has a sense of humor.
> James Harris
My math discoveries, found for profit
> http://mathforprofit.blogspot.com/
Subject: Re: Consider Chebyshev
> I am just a girl with an interest in maths (did a Comp Sci
degree and I
> think I drooled the entire way through the discrete maths
part), note
> interest, I have a lot to learn.
> (First and probably last post, because as interesting as
this all is, it
is
> way over my head)
I don't think it's way over your head, and I hope you post
some more,
because you definitely have something to say, along with a
nice way with words. Cheers!
Skip
Subject: Re: Consider Chebyshev
> (First and probably last post, because as interesting as
this all is, it
is
> way over my head)
What he has done is actually quite simple, James Harris is
just not very
good at explaining things.
We want to calculate how many prime numbers <= N there are.
The function
that counts the prime numbers <= N is named pi. pi (N) is the
number
of primes <= N. For example, pi (10) = 4 because there are
exactly four
primes 2, 3, 5 and 7 <= 10.
Instead of calculating the number of primes, we look for
something that
is related but easier to calculate. We define
S (N, k) = Set of positive integers <= N which are not
divisible by
the k smallest primes and phi (N, k) = Number of elements of S
(N,
k).
As an example, let N = 1000. Every composite number <= 1000
must have a
factor <= 31 (because if it hasn't, then the smallest factor
is >= 32.
Because the number is composite it is a product of some
factors, so it
is at least 32 * 32 = 1024). 31 is the eleventh smallest prime
(2, 3, 5,
7, 11, 13, 17, 19, 23, 29, 31). So lets see what elements are
there in S
(1000, 11):
S (1000, 11) does not contain _any_ composite numbers, because
the
elements of S (1000, 11) are not divisible by the 11 smallest
primes,
that is all primes up to 31.
S (1000, 11) contains all the primes <= 1000 except the 11
smallest
primes, because for example 29 is divisible by the 10th
smallest prime
(which is also 29).
S (1000, 11) contains the number 1 which is not a prime,
because 1
cannot be divided by any prime.
So if we compare the elements of S (1000, 11) and the set of
primes <=
1000, we notice that eleven primes are missing in S (1000, 11)
and it
contains one number that is not a prime. Therefore
phi (1000, 11) = pi (1000) - 11 + 1
or
pi (1000) = phi (1000, 11) + 11 - 1
In general, if there are exactly k prime numbers <= the square
root of
N, then
pi (N) = phi (N, k) + k - 1.
Now how to calculate phi (N, k): phi (N, 0) is simple. All
integers are
not divisible by one of the 0 smallest primes, so phi (N, 0) =
N.
S (N, k) is the set of integers not divisible by the k
smallest primes.
So the elements of S (N, k) are the numbers <= N that are not
divisible
by the (k - 1) smallest primes, minus those that have the
k-smallest
prime as the smallest factor. The number of integers <= N that
are not
divisible by the (k - 1) smallest primes is of course phi (N,
k-1).
If x <= N has for example 31 as its smallest factor, then we
can write x
= 31 * y for some y <= N / 31 which is not divisible by the 11
smallest
primes. So the number of integers <= N that have 31 as their
smallest
factor is phi (N/31, 11). The number of integers <= N that
have the
k-smallest prime p(k) as their smallest factor is phi (N / p
(k), k).
If we put these simple things together, then we get
phi (N, k) = phi (N, k-1) - phi (N / p (k), k) if k > 0.
phi (N, k) = N if k = 0.
In the example above, to calculate pi (1000) we chose k = 11
because
there are 11 primes <= square root of 1000, and calculated phi
(1000,
11) + 11 - 1. That number 11 is pi (sqrt (N)). So here is the
complete
formula to calculate pi (N):
Calculate k = pi (sqrt (N)), then pi (N) = phi (N, k) + k - 1.
Calculate phi (N, k) by using the formulas
phi (N, k) = phi (N, k-1) - phi (N / p (k), k) if k > 0.
phi (N, k) = N if k = 0.
(You probably won't understand a thing when he starts talking
about
partial difference equations. That's fine, nobody except him
understands
that stuff. He is in a world on his own).
Subject: Re: Prime issue challenge
>Well I claim that my prime formula is a great discovery,
while others
>keep posting that it's not important at all!
However, I know exactly how my formula works, so it seems to
me that a
>good check of others is to see if they do as well.
So I have a simple challenge. Posters assertions imply
expertise, and
>at a minimum that expertise should involve understanding
*how* my
>formula works.
So I'll give them a couple of weeks to try and explain it in
this
>thread, then if all goes according to plan I'll explain to
you how it
>works, and you can see who is the real expert.
 Let me see if I have the general idea. First you delete
sci.cognitive
> from your address list and then we all agree that the
algorithm works
> whatever it is.
> The relevance to sci.cognitive is that not only does my
discovery
> represent an intriguing case of a unique find in a
well-worked
> area--as prime numbers are VERY well worked--it also raises
questions
> about how human beings think.
> Why did it take so long before anyone found my formula?
> How readily do most people understand something that I'll
tell you
> relies on VERY simple ideas?
> Why would mathematicians fight even *recording* it,
challenging the
> very values that define them!!! Like, modern mathematicians
make
> claims about beauty and purity in mathematics, where
practical
> applications and practical concerns are secondary to the
purity of
> mathematical knowledge gained for the sake of knowledge
itself.
> There are so many issues for the cognitive sciences that I
could go on
> and on.
> It's fascinating and exciting as it all plays out.
> Oh yeah, lest readers forget, the point of this thread is to
see if
> people claiming my math discovery that can count and *find*
prime
> numbers is unimportant and not worth acknowledging can prove
expertise
> by managing to explain how it works.
> difficult for posters to lie about the value, yet manage to
accurately
> explain how it works!
> You see, it's like there's a switch in people's heads that
goes one
> way or another. If they're going to lie about my work, they
lose
> cognitive function necessary to accurately explain how it
works...or
> at least that's my theory.
> James Harris
My math discoveries, found for profit
> http://mathforprofit.blogspot.com/
fuffy
Subject: Re: Is this an NP complete problem?
> Given an undirected graph. Every edge in this graph will be
in a
> specific color. Now I want to find a subset of edges that
contains the
> least kinds of colors.
> Is it an NP complete problem?
You maybe mean a connected graph with the least number of
colours?
Lucas
[ comp.ai is moderated. To submit, just post and be patient,
or if ]
[ ask your news administrator to fix the problems with your
system. ]
Subject: Re: Is this an NP complete problem?
> Given an undirected graph. Every edge in this graph will be
in a
> specific color. Now I want to find a subset of edges that
contains the
> least kinds of colors.
> Is it an NP complete problem?
> You maybe mean a connected graph with the least number of
colours?
If this is the case, then it is NP complete as SETCOVERING can
be
reduced to it.
/Bjarke
[ comp.ai is moderated. To submit, just post and be patient,
or if ]
[ ask your news administrator to fix the problems with your
system. ]
Subject: two-colourable proof
I read that it is easy to prove that a collection of
(infinite) straight
lines partitions the plane into a 2-colourable map. I actually
only need
the more restricted result that a collection of infinite
straight lines
such that no intersections coincide (i.e. there is no point
where more
than 2 lines intersect) partitions the plane into a
2-colourable map. It
is obvious to me upon drawing the thing, but I have no idea
how such
things are proved. Any help appreciated. Best -Arco
Subject: Re: two-colourable proof
>I read that it is easy to prove that a collection of
(infinite) straight
>lines partitions the plane into a 2-colourable map. I
actually only need
>the more restricted result that a collection of infinite
straight lines
>such that no intersections coincide (i.e. there is no point
where more
>than 2 lines intersect) partitions the plane into a
2-colourable map. It
Just my wild guess, and not a rigorous proof, but an intuitive
(visual) approach to a sketch of proof:
Assuming you mean of course a countable infinity of lines, by
induction you have that a zero-line map is two-colourable (say,
whole plane either black or white). As for the induction step,
assume
you have a two-colourable map: then draw a line, choose one of
the
halfplanes it divides the plane into and switch colours in
*that*
halplane.
Hopefully this can be made rigoruous (think of chi functions),
but
maybe it may help to think by analogy in computer terms of the
plane
as of a monitor and of the operation as of XOR'ing.
HTH,
Michele
--
> Comments should say _why_ something is being done.
Oh? My comments always say what _really_ should have happened.
:)
- Tore Aursand on comp.lang.perl.misc
Subject: Re: two-colourable proof
Hmmm ... infinite number of lines; therefore no proof by
induction ...
Pick a point within any region (ie not on a line) as an
origin. Colour this
region white.
Consider any other point (within any region). Consider any
path from this
point to the origin that it doesn't go through any
intersection points (ie
where 2 or more lines intersect). This path will always cross
an even or
odd
number of lines, independent of the route.
(Proof: There are either an odd or even number of lines that
cross the
direct path between the two points. Each of these lines must
be crossed an
odd number of times in any path between the two points. All
other lines
must
be crossed an even number of times - mostly zero - by the path
between the
2
points. So whether there is an odd or even number of crossings
is
independent of the actual path between the points).
Colour the regions which are accessed by an even number of
crossings white,
colour the odd regions black. Adjacent regions have a path
with one
crossing, therefore they must be different colours.
This proof does not assume that there are no pints where more
than 2 lines
coincide - its the stronger result.
It does, however, assume that there are only a finite number
of lines
between any two points. So it doesn't work for grids like x =
0.9, 0.99,
0.999, ... 1.9, 1.99, 1.999 ... 2.9, 2.99, 2.999 .... and the
same for y. I
am not sure that the question is reasonable for these
pathological cases,
because the two regions that join at x=1 are not well defined
....
> I read that it is easy to prove that a collection of
(infinite) straight
> lines partitions the plane into a 2-colourable map. I
actually only need
> the more restricted result that a collection of infinite
straight lines
> such that no intersections coincide (i.e. there is no point
where more
> than 2 lines intersect) partitions the plane into a
2-colourable map. It
> is obvious to me upon drawing the thing, but I have no idea
how such
> things are proved. Any help appreciated. Best -Arco
Subject: Re: two-colourable proof
>Hmmm ... infinite number of lines; therefore no proof by
induction ...
First, learn how to quote properly. This would increase very
much
readability and attribution of claims to respective authors.
Second, the OP actually spoke of infinite number of lines:
this is
definitely vague. I guess he means a countable infinity of
lines, in
which case...
...Third: induction *IS* a (determining) charachteristic of
countable
infinity. Period!
Thus no no proof by induction...
Michele
--
> Comments should say _why_ something is being done.
Oh? My comments always say what _really_ should have happened.
:)
- Tore Aursand on comp.lang.perl.misc
Subject: Re: two-colourable proof
>>Hmmm ... infinite number of lines; therefore no proof by
induction ...
>First, learn how to quote properly. This would increase very
much
>readability and attribution of claims to respective authors.
>Second, the OP actually spoke of infinite number of lines:
this is
>definitely vague. I guess he means a countable infinity of
lines, in
>which case...
>...Third: induction *IS* a (determining) charachteristic of
countable
>infinity. Period!
D'Oh! I APOLOGIZE!! I'm terribly sorry for this utterly
nonsensical
idiocy. I'd like to cancel my post, but it must have been
already
propagated to other news servers...
As far as the second remark is concerned, some care is actually
required in any case with an infinite number of lines.
And I still agree completely with myself on the first one!
Sorry,
Michele
--
> Comments should say _why_ something is being done.
Oh? My comments always say what _really_ should have happened.
:)
- Tore Aursand on comp.lang.perl.misc
Subject: Re: two-colourable proof
>>Hmmm ... infinite number of lines; therefore no proof by
induction ...
> First, learn how to quote properly. This would increase very
much
> readability and attribution of claims to respective authors.
In Peter's defense, he quoted accurately and didn't
misattribute
anything (though he did top-post... is that your complaint?)
> Second, the OP actually spoke of infinite number of lines:
this is
> definitely vague. I guess he means a countable infinity of
lines, in
> which case...
Actually I said a collection of (infinite) lines, and my
intended
interpretation was Robert Israel's--a collection of infinitely
extended
lines, i.e. not segments. Sorry for the confusion, though it
would seem
more natural, to me, to speak of an infinite collection if I
had meant
what you thought I meant.
Anyway I liked your proof in the other post, and it is
rigorous enough for
my purposes. Thank you.
Regards --Arco
Subject: Re: two-colourable proof
Hmmm ... infinite number of lines; therefore no proof by
induction ...
> First, learn how to quote properly. This would increase
very much
>> readability and attribution of claims to respective authors.
>In Peter's defense, he quoted accurately and didn't
misattribute
>anything (though he did top-post... is that your complaint?)
Yes, more or less. Top-posting is not properly quoting...
generally
people are not much strict about that here, but encouraging it
doesn't
help attaining an effective communication either.
>> Second, the OP actually spoke of infinite number of lines:
this is
>> definitely vague. I guess he means a countable infinity of
lines, in
>> which case...
>Actually I said a collection of (infinite) lines, and my
intended
>interpretation was Robert Israel's--a collection of
infinitely extended
>lines, i.e. not segments. Sorry for the confusion, though it
would seem
>more natural, to me, to speak of an infinite collection if I
had meant
>what you thought I meant.
post. And I have already apologized. Also, to this effect I am
doing
something that I *never* do: answering on-line (I *still* pay
per
minute of connection!).
But I had understood what you meant with lines (i.e. not
segments),
as you can understand by my sketch of proof (in which I speak
of
halfplane, that would not exists, if we were dealing with
segments!)
>Anyway I liked your proof in the other post, and it is
rigorous enough for
>my purposes. Thank you.
about induction...
Michele
--
> Comments should say _why_ something is being done.
Oh? My comments always say what _really_ should have happened.
:)
- Tore Aursand on comp.lang.perl.misc
Subject: Re: two-colourable proof
>Hmmm ... infinite number of lines; therefore no proof by
induction ...
I read the question as talking about infinite lines, i.e. lines
rather than line segments.
>It does, however, assume that there are only a finite number
of lines
>between any two points. So it doesn't work for grids like x =
0.9, 0.99,
>0.999, ... 1.9, 1.99, 1.999 ... 2.9, 2.99, 2.999 .... and the
same for y.
I
>am not sure that the question is reasonable for these
pathological cases,
>because the two regions that join at x=1 are not well defined
....
If there are infinitely many lines intersecting a compact set,
there may
not be any regions to talk about. Assuming each compact set
intersects
only finitely many lines, the result follows from the version
for finitely
many lines plus a compactness argument: if for every R there's
a way to
2-colour the disk of radius R centred at the origin, then
there's a way
to 2-colour the whole plane.
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: Normal distribution and convergence help please
Dear Math experts,
How can I prove this?
Suppose that {X_i : 1 <= i < infinity} is a sequence of
independent random
variables with the standard normal distribution. Let S_n = X_1
+ X_2 + ...
+ X_n and let
Z_n = exp(a*S_n - bn)
Show that Z_n ---> 0 with probability 1 if and only if b > 0,
and show
that
for p >=1 we have E[(Z_n)^p] ---> 0 if and only if p < 2b/a
Thank you very very much for your help,
Steven Rossi
Subject: Re: Normal distribution and convergence help please
> Dear Math experts,
> How can I prove this?
> Suppose that {X_i : 1 <= i < infinity} is a sequence of
independent
random
> variables with the standard normal distribution. Let S_n =
X_1 + X_2 +
...
> + X_n and let
> Z_n = exp(a*S_n - bn)
S_n has the normal distribution with mean 0 and variance n.
Therefore,
E[Z_n] = exp(a^2*n/2 - bn),
and more generally,
E[(Z_n)^p] = exp(a^2p^2*n/2 - bpn).
> Show that Z_n ---> 0 with probability 1 if and only if b >
0, and show
that
> for p >=1 we have E[(Z_n)^p] ---> 0 if and only if p < 2b/a
It should now be clear that E[(Z_n)^p) --> 0 if and only if
a^2p < 2b.
Also, that Z_n --> 0 w.p. 1 if and only if a^2 < 2b.
--
A.
Subject: Re: Normal distribution and convergence help please
>> Dear Math experts,
> How can I prove this?
> Suppose that {X_i : 1 <= i < infinity} is a sequence of
independent
random
>> variables with the standard normal distribution. Let S_n =
X_1 + X_2 +
...
>> + X_n and let
>> Z_n = exp(a*S_n - bn)
>S_n has the normal distribution with mean 0 and variance n.
>Therefore,
> E[Z_n] = exp(a^2*n/2 - bn),
>and more generally,
> E[(Z_n)^p] = exp(a^2p^2*n/2 - bpn).
>> Show that Z_n ---> 0 with probability 1 if and only if b >
0, and show
that
>> for p >=1 we have E[(Z_n)^p] ---> 0 if and only if p < 2b/a
>It should now be clear that E[(Z_n)^p) --> 0 if and only if
>a^2p < 2b.
Yes,
>Also, that Z_n --> 0 w.p. 1 if and only if a^2 < 2b.
That last bit is not at all clear to me. Is there some reason
it's
clear that Z_n -> 0 a.s. if and only if E[Z_n] -> 0?
Didn't work out the details this morning, but I did just now,
and if I'm not dropping something it seems clear to me
from the CLT (or rather, duh, from the fact that S_n has a
normal distribution with mean 0 and variance n) that Z_n -> 0
a.s.
if and only if b > 0:
Suppose wlog that a > 0. Let eps > 0. Now Z_n > eps
if and only if
a S_n / sqrt(n) > b sqrt(n) + log(eps)/sqrt(n),
and if b > 0 this shows that sum_n P(Z_n > eps)
is finite, while if b < 0 then P(Z_n > eps) -> 1
(and P(Z_N > eps) -> 1/2 if b = 0).
???
and
************************
Subject: Re: Normal distribution and convergence help please
>Dear Math experts,
>How can I prove this?
>Suppose that {X_i : 1 <= i < infinity} is a sequence of
independent
random
>variables with the standard normal distribution. Let S_n =
X_1 + X_2 +
...
>+ X_n and let
>Z_n = exp(a*S_n - bn)
>Show that Z_n ---> 0 with probability 1 if and only if b > 0,
If I recall what the Law of the Iterated Logarithm says
correctly
then this follows immediately from that. (No doubt it also
follows
by easy arguments from the Central Limit Theorem, but if I'm
recalling LIL correctly then there's no argument needed.)
>and show that
>for p >=1 we have E[(Z_n)^p] ---> 0 if and only if p < 2b/a
Well, you have a product of independent random variables;
the expected value of the product is the product of the
expected values...
>Thank you very very much for your help,
>Steven Rossi
************************
Subject: Re: Normal distribution and convergence help please
>Dear Math experts,
>How can I prove this?
>Suppose that {X_i : 1 <= i < infinity} is a sequence of
independent
random
>variables with the standard normal distribution. Let S_n =
X_1 + X_2 +
...
>+ X_n and let
>Z_n = exp(a*S_n - bn)
>Show that Z_n ---> 0 with probability 1 if and only if b > 0,
> If I recall what the Law of the Iterated Logarithm says
correctly
> then this follows immediately from that. (No doubt it also
follows
> by easy arguments from the Central Limit Theorem, but if I'm
> recalling LIL correctly then there's no argument needed.)
I guess my problem is that I don't know the Law of Iterated
Logarithms and
would much rather
solve this using CLT arguments....how could I go about this?
It seems like
a variation on CLT but
what kind of variation? I just don't see it...
Steven
>and show that
>for p >=1 we have E[(Z_n)^p] ---> 0 if and only if p < 2b/a
> Well, you have a product of independent random variables;
> the expected value of the product is the product of the
> expected values...
>Thank you very very much for your help,
>Steven Rossi
> ************************
>
Subject: Re: Normal distribution and convergence help please
>>Dear Math experts,
>>How can I prove this?
>>Suppose that {X_i : 1 <= i < infinity} is a sequence of
independent
random
>>variables with the standard normal distribution. Let S_n =
X_1 + X_2 +
...
>>+ X_n and let
>>Z_n = exp(a*S_n - bn)
>>Show that Z_n ---> 0 with probability 1 if and only if b > 0,
>If I recall what the Law of the Iterated Logarithm says
correctly
>then this follows immediately from that. (No doubt it also
follows
>by easy arguments from the Central Limit Theorem, but if I'm
>recalling LIL correctly then there's no argument needed.)
>>and show that
>>for p >=1 we have E[(Z_n)^p] ---> 0 if and only if p < 2b/a
>Well, you have a product of independent random variables;
>the expected value of the product is the product of the
>expected values...
Was there a typo in your post? When I work out the integrals
the condition I get is p < 2b/a^2.
(And come to think of it 2b/a _can't_ be right, because the
value of E[(Z_n)^p] is unchanged if you replace a with -a.)
>>Thank you very very much for your help,
>>Steven Rossi
>************************
>
************************
Subject: Re: Normal distribution and convergence help please
>>Dear Math experts,
>How can I prove this?
>Suppose that {X_i : 1 <= i < infinity} is a sequence of
independent
random
>>variables with the standard normal distribution. Let S_n =
X_1 + X_2 +
...
>>+ X_n and let
>>Z_n = exp(a*S_n - bn)
>Show that Z_n ---> 0 with probability 1 if and only if b > 0,
>If I recall what the Law of the Iterated Logarithm says
correctly
>then this follows immediately from that. (No doubt it also
follows
>by easy arguments from the Central Limit Theorem, but if I'm
>recalling LIL correctly then there's no argument needed.)
>>and show that
>>for p >=1 we have E[(Z_n)^p] ---> 0 if and only if p < 2b/a
>Well, you have a product of independent random variables;
>the expected value of the product is the product of the
>expected values...
> Was there a typo in your post? When I work out the integrals
> the condition I get is p < 2b/a^2.
> (And come to think of it 2b/a _can't_ be right, because the
> value of E[(Z_n)^p] is unchanged if you replace a with -a.)
I tried to solve this, and am not getting anything.........for
E[(Z_n)^p] =
[E(Z_n)]^p = [E(exp(aS_n - bn))]^p =
[E(exp(aX_1 + aX_2 + ... + aX_n - bn))]^p = [ (E(exp(aX_1)))^n
E(exp(-bn)) ]^p = [ (sqrt(2)exp(a^2/2))^n * exp(-bn) ] ^p =
(sqrt(2))^p exp(-bnp + npa^2/2)....
Now I want -bnp + npa^2 < 0 and the p's cancel out...let's
make sure that
we
are both getting that E[exp(aX_1)] = sqrt(2)exp(a^2/2), right
?(a simple
substitution I think...)
How are you getting p < 2b/a^2?
>>Thank you very very much for your help,
>Steven Rossi
************************
 ************************
>
Subject: Re: Normal distribution and convergence help please
>Dear Math experts,
>How can I prove this?
>Suppose that {X_i : 1 <= i < infinity} is a sequence of
independent
>random
>variables with the standard normal distribution. Let S_n =
X_1 + X_2
+
>...
>+ X_n and let
>Z_n = exp(a*S_n - bn)
>Show that Z_n ---> 0 with probability 1 if and only if b >
0,
>>If I recall what the Law of the Iterated Logarithm says
correctly
>>then this follows immediately from that. (No doubt it also
follows
>>by easy arguments from the Central Limit Theorem, but if I'm
>>recalling LIL correctly then there's no argument needed.)
>and show that
>for p >=1 we have E[(Z_n)^p] ---> 0 if and only if p < 2b/a
>>Well, you have a product of independent random variables;
>>the expected value of the product is the product of the
>>expected values...
>> Was there a typo in your post? When I work out the integrals
>> the condition I get is p < 2b/a^2.
>> (And come to think of it 2b/a _can't_ be right, because the
>> value of E[(Z_n)^p] is unchanged if you replace a with -a.)
>I tried to solve this, and am not getting
anything.........for E[(Z_n)^p]
=
>[E(Z_n)]^p =
??? Why would E[(Z_n)^p] = [E(Z_n)]^p ? That can't be right.
What's true is that E[(Z_n)^p] = E[(Z_1)^p]^n, because (Z_n)^p
is the product of n independent random variables, each with the
same distribution as (Z_1)^p. So you need to find E[(Z_n)^p],
which is essentially the same as finding E[(Z_n)] (with a
different value of a and b.) ArtlfDogr was able to find
E[(Z_n)]
using only the word therefore<g>; since I don't know any
probability I had to work it out...
>[E(exp(aS_n - bn))]^p =
>[E(exp(aX_1 + aX_2 + ... + aX_n - bn))]^p = [ (E(exp(aX_1)))^n
>E(exp(-bn)) ]^p = [ (sqrt(2)exp(a^2/2))^n * exp(-bn) ] ^p =
>(sqrt(2))^p exp(-bnp + npa^2/2)....
>Now I want -bnp + npa^2 < 0 and the p's cancel out...let's
make sure that
we
>are both getting that E[exp(aX_1)] = sqrt(2)exp(a^2/2), right
?(a simple
>substitution I think...)
No, it seems to me that E[exp(aX_1)] = exp(a^2/2) (again,
setting
a = 0 shows that what you got there can't be right.)
>How are you getting p < 2b/a^2?
>Thank you very very much for your help,
>Steven Rossi
>>************************
>> ************************
>>
************************
Subject: logic, triple systems and designs
In the link
http://plato.stanford.edu/entries/qt-quantlog/#1
you will find the remark
Whereas logicians have usually assumed that
properties  of negation were the ones least
able to withstand a critical analysis, the study of
mechanics points to the distributive identities
as the weakest link in the algebra of logic.
[1937, p. 839]
There is nothing particularly authoritative about the link. It
is just
that I have heard this particular statement about negation
before and
wished to provide some source.
If you scroll down to where you find quasigroup products such
as
5*0=4
| 1 | | | | 0 |
| | | N | | |
| 0 | | | | 1 |
| | * | O | = | |
| 0 | | | | 1 |
| | | T | | |
| 1 | | | | 0 |
you can decide if you are interested in this post.
I am tired of arguing.
:-)
mitch
---
I will start this post presenting the incidence relations for
the
trivial affine geometry consisting of 4 points and 6 lines.
[begin fixed width]
a b c d e f
A x x x
B x x x
C x x x
D x x x
[end fixed width]
This starting point was explained somewhat in
revisited as needed in this post.
As was done at that time, comparison is made to the
presentation,
[begin fixed width]
a b c d e f
(1,1) 1 0 0 1 1 0
(1,0) 1 1 0 0 0 1
(0,1) 0 1 1 0 1 0
(0,0) 0 0 1 1 0 1
[end fixed width]
So that it can be seen that a fixed specification for six
connectives of
the sixteen propositional connectives relate to one another in
a manner
comparable to the components of the geometric model.
actual intuitions involved here come from rotational
symmetries in solid
geometry. The 2.3.4 symmetry family consists of six 2-fold
symmetries,
four 3-fold symmetries, and three 4-fold symmetries.
Relative to a cube, a 2-fold symmetry is a 180 degree
rotation. For a
given 2-fold rotation, the axis of symmetry is the line from
the
midpoint of a given edge to the midpoint of the opposite edge.
The six
2-fold symmetries segregate into pairs according to their
relation to
the three 4-fold symmetries.
To see this, consider the diagram,
[begin fixed width]
/|
/ |
------/--|----------
/ / | /
/ / | /
/ / / /
/ | / /
/ | / /
--------|---/--------
| /
| /
|/
[end fixed width]
The line corresponding to the intersection of the two planes
would
correspond to one of the 4-fold symmetries. The following
diagram
attempts to show axes for the 4-fold symmetries. The '*' symbol
represents the intersection of the three axes in the center of
the cube
and the '+' symbols represent the center of the three faces
through
which these axes pass.
Obviously, the diagram lacks perspective given the nature of
the
presentation in ASCII.
[begin fixed width]
|
|
-----|------------------
/ | /|
/ | / |
/ | / |
/ + / |
/ | / |
/ | / |
/ | / |
------------------------ |
| | | |
| | | |
| |/ | /
| --*---------|--+---------
| /| | /
| / | /
| + | /
| / | /
| / |/
-------/----------------
/
/
[end fixed width]
So, each 4-fold symmetry corresponds with four of the 2-fold
symmetries.
The reason the 3-dimensional perspective becomes important is
that there
are 24 possible ways to orient a truth table using columns
chosen from
among
[begin fixed width]
1 0 0 1 1 0
1 1 0 0 0 1
0 1 1 0 1 0
0 0 1 1 0 1
[end fixed width]
A specific selection of two vectors that accomplishes an
orientation
must be chosen so that they are not complementary--that is,
such a
selection must result in 4 distinct rows when the two columns
are placed
into a 4x2 matrix. So, while
[begin fixed width]
a b c d e f
(1,1) 1 0 0 1 1 0
(1,0) 1 1 0 0 0 1
(0,1) 0 1 1 0 1 0
(0,0) 0 0 1 1 0 1
[end fixed width]
had been used to begin this discussion, it is certainly
comparable to
something like
[begin fixed width]
a b c d e f
(1,0) 1 0 0 1 1 0
(0,0) 1 1 0 0 0 1
(0,1) 0 1 1 0 1 0
(1,1) 0 0 1 1 0 1
[end fixed width]
where columns d and c are used rather than colums a and e.
That is, the
resultant interpretation still yields the projection
connectives, their
complements, logical equivalence, and exclusive disjunction.
But, the reason for the 3-dimensional perspective is that such
a
selection constitutes a deselection of a complementary pair.
For the
introductory example, that pair would be
[begin fixed width]
1 0
0 1
0 1
1 0
[end fixed width]
and for the second example it would be
[begin fixed width]
1 0
0 1
1 0
0 1
[end fixed width]
In both cases, the deselected pair ends up being interpreted
as logical
equivalence and exclusive disjunction.
It is this deselection that corresponds to the relationship
with the
4-fold rotational symmetry. One manifestation of this
relationship can
be seen in
http://citeseer.nj.nec.com/feigelson97forbidden.html
where Lemma 4 concludes with
Therefore, there are precisely two assignments
that are justifying for both x and y
followed by the footnote,
Before proving Lemma 4, note that it is stated only
for |V|>2. If |V|=2 then there are only four possible
assignments to the variables of V and f must be either
XOR or ~XOR. In either case, all four assignments
are justifying for both x and y. Thus the conclusion of
the lemma does not hold for |V|=2.
But, it has proven somewhat fruitless to attempt establishing
this
4-fold relationship using the recognition problem for unate
switching
functions.
Hopefully, the above remarks suffice to show reasonable cause
for
investigating the relation between logic, the trivial affine
geometry,
and the symmetry families in solid geometry.
I would like to now direct attention to the diagram in
http://phil240.tamu.edu/LectureNotes/6.3.pdf
As noted in a prior posting, this paper shows a picture of a
Venn
diagram labeled with 8 regions. The paper discusses the use of
Venn
diagrams for testing the validity of categorical syllogisms.
The method
described in the paper is vaguely comparable to what is being
discussed
here to the extent that representing the conclusion involves
only two of
the three terms. Our purpose is somewhat more subtle.
In a prior post, we noted that the mapping
[begin fixed width]
1 -> 3
2 -> 4
3 -> 6
4 -> 2
5 -> 0
6 -> 1
7 -> 5
8 -> #
[end fixed width]
would align the diagram with the multiplication table,
[begin fixed width]
* | 0 1 2 3 4 5 6
--|----------------------------
0 | 0 3 6 1 5 4 2
|
1 | 3 1 4 0 2 6 5
|
2 | 6 4 2 5 1 3 0
|
3 | 1 0 5 3 6 2 4
|
4 | 5 2 1 6 4 0 3
|
5 | 4 6 3 2 0 5 1
|
6 | 2 5 0 4 3 1 6
[end fixed width]
For the moment ignore region 8 and its mapping to the
octothorpe
(Previously, I had used an ampersand; but, 8 and octo- seem to
go
together better.).
The multiplication given here is for a Steiner quasigroup. By
definition, a Steiner quasigroup is a commutative quasigroup
whose
binary quasigroup satisfies
x*x=x
(x*y)*y=x
The multiplication table given above is for the Steiner
quasigroup of
order seven.
With respect to the incidence matrixes given above, we need to
extend
our mapping to a new set of labels. Once again, the first
number comes
from the diagram in
http://phil240.tamu.edu/LectureNotes/6.3.pdf
while the second number is associated with the multiplication
table
above. So, the new labels are given by
[begin fixed width]
| 1 |
| |
| 1 |
1 -> 3 -> | |
| 0 |
| |
| 0 |
| 0 |
| |
| 1 |
2 -> 4 -> | |
| 1 |
| |
| 0 |
| 1 |
| |
| 0 |
3 -> 6 -> | |
| 1 |
| |
| 0 |
| 0 |
| |
| 1 |
4 -> 2 -> | |
| 0 |
| |
| 1 |
| |
| N |
| |
5 -> 0 -> | O |
| |
| T |
| |
| 0 |
| |
| 0 |
6 -> 1 -> | |
| 1 |
| |
| 1 |
| 1 |
| |
| 0 |
7 -> 5 -> | |
| 0 |
| |
| 1 |
[end fixed width]
What now follows is a long listing verifying the sense of the
quasigroup
products for these labels. We ignore the idempotence of the
products,
x*x=x
The first grouping involves interpretation of the quasigroup 0
as the
complementation operation.
[begin fixed width]
0*1=3
| | | 0 | | 1 |
| N | | | | |
| | | 0 | | 1 |
| O | * | | = | |
| | | 1 | | 0 |
| T | | | | |
| | | 1 | | 0 |
3*1=0
| 1 | | 0 | | |
| | | | | N |
| 1 | | 0 | | |
| | * | | = | O |
| 0 | | 1 | | |
| | | | | T |
| 0 | | 1 | | |
3*0=1
| 1 | | | | 0 |
| | | N | | |
| 1 | | | | 0 |
| | * | O | = | |
| 0 | | | | 1 |
| | | T | | |
| 0 | | | | 1 |
1*0=3
| 0 | | | | 1 |
| | | N | | |
| 0 | | | | 1 |
| | * | O | = | |
| 1 | | | | 0 |
| | | T | | |
| 1 | | | | 0 |
1*3=0
| 0 | | 1 | | |
| | | | | N |
| 0 | | 1 | | |
| | * | | = | O |
| 1 | | 0 | | |
| | | | | T |
| 1 | | 0 | | |
0*3=1
| | | 1 | | 0 |
| N | | | | |
| | | 1 | | 0 |
| O | * | | = | |
| | | 0 | | 1 |
| T | | | | |
| | | 0 | | 1 |
0*2=6
| | | 0 | | 1 |
| N | | | | |
| | | 1 | | 0 |
| O | * | | = | |
| | | 0 | | 1 |
| T | | | | |
| | | 1 | | 0 |
6*2=0
| 1 | | 0 | | |
| | | | | N |
| 0 | | 1 | | |
| | * | | = | O |
| 1 | | 0 | | |
| | | | | T |
| 0 | | 1 | | |
6*0=2
| 1 | | | | 0 |
| | | N | | |
| 0 | | | | 1 |
| | * | O | = | |
| 1 | | | | 0 |
| | | T | | |
| 0 | | | | 1 |
2*0=6
| 0 | | | | 1 |
| | | N | | |
| 1 | | | | 0 |
| | * | O | = | |
| 0 | | | | 1 |
| | | T | | |
| 1 | | | | 0 |
2*6=0
| 0 | | 1 | | |
| | | | | N |
| 1 | | 0 | | |
| | * | | = | O |
| 0 | | 1 | | |
| | | | | T |
| 1 | | 0 | | |
0*6=2
| | | 1 | | 0 |
| N | | | | |
| | | 0 | | 1 |
| O | * | | = | |
| | | 1 | | 0 |
| T | | | | |
| | | 0 | | 1 |
0*5=4
| | | 1 | | 0 |
| N | | | | |
| | | 0 | | 1 |
| O | * | | = | |
| | | 0 | | 1 |
| T | | | | |
| | | 1 | | 0 |
4*5=0
| 0 | | 1 | | |
| | | | | N |
| 1 | | 0 | | |
| | * | | = | O |
| 1 | | 0 | | |
| | | | | T |
| 0 | | 1 | | |
4*0=5
| 0 | | | | 1 |
| | | N | | |
| 1 | | | | 0 |
| | * | O | = | |
| 1 | | | | 0 |
| | | T | | |
| 0 | | | | 1 |
5*0=4
| 1 | | | | 0 |
| | | N | | |
| 0 | | | | 1 |
| | * | O | = | |
| 0 | | | | 1 |
| | | T | | |
| 1 | | | | 0 |
5*4=0
| 1 | | 0 | | |
| | | | | N |
| 0 | | 1 | | |
| | * | | = | O |
| 0 | | 1 | | |
| | | | | T |
| 1 | | 0 | | |
0*4=5
| | | 0 | | 1 |
| N | | | | |
| | | 1 | | 0 |
| O | * | | = | |
| | | 1 | | 0 |
| T | | | | |
| | | 0 | | 1 |
[end fixed width]
The remainder of this listing will give products whose factors
could be
interpreted as orienting columns for a truth table. The reader
is asked
to observe the consistency with which the constant rows are
matched to
the same value (0).
[begin fixed width]
2*1=4
| 0 | | 0 | | 0 |
| | | | | |
| 1 | | 0 | | 1 |
| | * | | = | |
| 0 | | 1 | | 1 |
| | | | | |
| 1 | | 1 | | 0 |
4*1=2
| 0 | | 0 | | 0 |
| | | | | |
| 1 | | 0 | | 1 |
| | * | | = | |
| 1 | | 1 | | 0 |
| | | | | |
| 0 | | 1 | | 1 |
2*4=1
| 0 | | 0 | | 0 |
| | | | | |
| 1 | | 1 | | 0 |
| | * | | = | |
| 0 | | 1 | | 1 |
| | | | | |
| 1 | | 0 | | 1 |
1*4=2
| 0 | | 0 | | 0 |
| | | | | |
| 0 | | 1 | | 1 |
| | * | | = | |
| 1 | | 1 | | 0 |
| | | | | |
| 1 | | 0 | | 1 |
4*2=1
| 0 | | 0 | | 0 |
| | | | | |
| 1 | | 1 | | 0 |
| | * | | = | |
| 1 | | 0 | | 1 |
| | | | | |
| 0 | | 1 | | 1 |
1*2=4
| 0 | | 0 | | 0 |
| | | | | |
| 0 | | 1 | | 1 |
| | * | | = | |
| 1 | | 0 | | 1 |
| | | | | |
| 1 | | 1 | | 0 |
6*1=5
| 1 | | 0 | | 1 |
| | | | | |
| 0 | | 0 | | 0 |
| | * | | = | |
| 1 | | 1 | | 0 |
| | | | | |
| 0 | | 1 | | 1 |
5*1=6
| 1 | | 0 | | 1 |
| | | | | |
| 0 | | 0 | | 0 |
| | * | | = | |
| 0 | | 1 | | 1 |
| | | | | |
| 1 | | 1 | | 0 |
6*5=1
| 1 | | 1 | | 0 |
| | | | | |
| 0 | | 0 | | 0 |
| | * | | = | |
| 1 | | 0 | | 1 |
| | | | | |
| 0 | | 1 | | 1 |
1*5=6
| 0 | | 1 | | 1 |
| | | | | |
| 0 | | 0 | | 0 |
| | * | | = | |
| 1 | | 0 | | 1 |
| | | | | |
| 1 | | 1 | | 0 |
1*6=5
| 0 | | 1 | | 1 |
| | | | | |
| 0 | | 0 | | 0 |
| | * | | = | |
| 1 | | 1 | | 0 |
| | | | | |
| 1 | | 0 | | 1 |
5*6=1
| 1 | | 1 | | 0 |
| | | | | |
| 0 | | 0 | | 0 |
| | * | | = | |
| 0 | | 1 | | 1 |
| | | | | |
| 1 | | 0 | | 1 |
3*2=5
| 1 | | 0 | | 1 |
| | | | | |
| 1 | | 1 | | 0 |
| | * | | = | |
| 0 | | 0 | | 0 |
| | | | | |
| 0 | | 1 | | 1 |
5*2=3
| 1 | | 0 | | 1 |
| | | | | |
| 0 | | 1 | | 1 |
| | * | | = | |
| 0 | | 0 | | 0 |
| | | | | |
| 1 | | 1 | | 0 |
3*5=2
| 1 | | 1 | | 0 |
| | | | | |
| 1 | | 0 | | 1 |
| | * | | = | |
| 0 | | 0 | | 0 |
| | | | | |
| 0 | | 1 | | 1 |
2*5=3
| 0 | | 1 | | 1 |
| | | | | |
| 1 | | 0 | | 1 |
| | * | | = | |
| 0 | | 0 | | 0 |
| | | | | |
| 1 | | 1 | | 0 |
2*3=5
| 0 | | 1 | | 1 |
| | | | | |
| 1 | | 1 | | 0 |
| | * | | = | |
| 0 | | 0 | | 0 |
| | | | | |
| 1 | | 0 | | 1 |
5*3=2
| 1 | | 1 | | 0 |
| | | | | |
| 0 | | 1 | | 1 |
| | * | | = | |
| 0 | | 0 | | 0 |
| | | | | |
| 1 | | 0 | | 1 |
6*3=4
| 1 | | 1 | | 0 |
| | | | | |
| 0 | | 1 | | 1 |
| | * | | = | |
| 1 | | 0 | | 1 |
| | | | | |
| 0 | | 0 | | 0 |
4*3=6
| 0 | | 1 | | 1 |
| | | | | |
| 1 | | 1 | | 0 |
| | * | | = | |
| 1 | | 0 | | 1 |
| | | | | |
| 0 | | 0 | | 0 |
6*4=3
| 1 | | 0 | | 1 |
| | | | | |
| 0 | | 1 | | 1 |
| | * | | = | |
| 1 | | 1 | | 0 |
| | | | | |
| 0 | | 0 | | 0 |
3*4=6
| 1 | | 0 | | 1 |
| | | | | |
| 1 | | 1 | | 0 |
| | * | | = | |
| 0 | | 1 | | 1 |
| | | | | |
| 0 | | 0 | | 0 |
3*6=4
| 1 | | 1 | | 0 |
| | | | | |
| 1 | | 0 | | 1 |
| | * | | = | |
| 0 | | 1 | | 1 |
| | | | | |
| 0 | | 0 | | 0 |
4*6=3
| 0 | | 1 | | 1 |
| | | | | |
| 1 | | 0 | | 1 |
| | * | | = | |
| 1 | | 1 | | 0 |
| | | | | |
| 0 | | 0 | | 0 |
[end fixed width]
Once again, we shall eventually discuss region 8 from our
diagram (our
octothorpe). However, at this point we consider what has been
done so
far. The exposition began by discussing how the list of columns
[begin fixed width]
1 0 0 1 1 0
1 1 0 0 0 1
0 1 1 0 1 0
0 0 1 1 0 1
[end fixed width]
relate to the manner by which the system of propositional
connectives in
logic associate with one another via truth tables. In this
analysis, we
noted that the orientation choices segregate logical
equivalence and
exclusive disjunction from the projections and their
complements. This
aspect of the orientation was related to the 2.3.4 family of
rotational
symmetries from solid geometry, and, an algebraic system has
been
presented so that context specific labels relate complements
to one
another while simultaneously associating labels so that
possible
orientations of the propositional system enforce the
segregating
relationship with logical equivalence and exclusive
disjunction.
Now we can look at what all of this has accomplished with
respect to
invariance.
The Steiner triple system that results from the quasigroup
above
consists of the triples,
[begin fixed width]
{0,1,3}
| | | 0 | | 1 |
| N | | | | |
| | | 0 | | 1 |
| O | | | | |
| | | 1 | | 0 |
| T | | | | |
| | | 1 | | 0 |
{0,2,6}
| | | 0 | | 1 |
| N | | | | |
| | | 1 | | 0 |
| O | | | | |
| | | 0 | | 1 |
| T | | | | |
| | | 1 | | 0 |
{0,4,5}
| | | 0 | | 1 |
| N | | | | |
| | | 1 | | 0 |
| O | | | | |
| | | 1 | | 0 |
| T | | | | |
| | | 0 | | 1 |
{1,2,4}
| 0 | | 0 | | 0 |
| | | | | |
| 0 | | 1 | | 1 |
| | | | | |
| 1 | | 0 | | 1 |
| | | | | |
| 1 | | 1 | | 0 |
{1,6,5}
| 0 | | 1 | | 1 |
| | | | | |
| 0 | | 0 | | 0 |
| | | | | |
| 1 | | 1 | | 0 |
| | | | | |
| 1 | | 0 | | 1 |
{2,3,5}
| 0 | | 1 | | 1 |
| | | | | |
| 1 | | 1 | | 0 |
| | | | | |
| 0 | | 0 | | 0 |
| | | | | |
| 1 | | 0 | | 1 |
{3,4,6}
| 1 | | 0 | | 1 |
| | | | | |
| 1 | | 1 | | 0 |
| | | | | |
| 0 | | 1 | | 1 |
| | | | | |
| 0 | | 0 | | 0 |
[end fixed width]
If we recall our original incidence matrix,
[begin fixed width]
a b c d e f
A x x x
B x x x
C x x x
D x x x
[end fixed width]
we may compare it with the relabeling,
[begin fixed width]
a/1 b/5 c/3 d/4 e/2 f/6
{1,2,4} x x x
{1,6,5} x x x
{2,3,5} x x x
{3,4,6} x x x
[end fixed width]
It should be obvious what is going on here:
a corresponds with 1 because {1}={1,2,4} cap {1,6,5}
b corresponds with 5 because {5}={2,3,5} cap {1,6,5}
c corresponds with 3 because {3}={3,4,6} cap {2,3,5}
d corresponds with 4 because {4}={1,2,4} cap {3,4,6}
e corresponds with 2 because {2}={1,2,4} cap {2,3,5}
f corresponds with 6 because {6}={3,4,6} cap {1,6,5}
It should also be clear that our construction has reversed the
sense of
our original comparison. Whereas we translated the original
incidence
matrix as
[begin fixed width]
a b c d e f
A 1 0 0 1 1 0
B 1 1 0 0 0 1
C 0 1 1 0 1 0
D 0 0 1 1 0 1
[end fixed width]
this relabeling corresponds with
[begin fixed width]
a b c d e f
A 0 1 1 0 0 1
B 0 0 1 1 1 0
C 1 0 0 1 0 1
D 1 1 0 0 1 0
[end fixed width]
But, we note that our quasigroup was formulated without any
semantic
interpretation of 0 and 1, whence the juxtaposition is an
artifact of
the introductory exposition rather than any failure on the
part of the
formal structure being discussed.
It is important to understand what is going on here. First of
all,
there is nothing particularly special about the triples
{1,2,4}
{1,6,5}
{2,3,5}
{3,4,6}
Relative to the triple system with which they are associated,
they form
a Pasch configuration (also known as a fragment or a
quadrilateral).
Such a configuration is characterized by the relations
{a,w,x}
{a,y,z}
{b,w,z}
{b,x,y}
so that, for example,
{0,1,3}
{0,2,6}
{5,1,6}
{5,3,2}
is also a Pasch configuration for our system. What is
important,
however, is that the Pasch configuration used to label the
incidence
matrix is the single Pasch configuration whose triples are
composed only
of those labels referring to the columns.
Unfortunately, we are not yet in a position to understand this
labeling
as independent from the assignment
[begin fixed width]
| |
| N |
| |
0 = | O |
| |
| T |
| |
[end fixed width]
The reason for this is that a Steiner triple system specified
by
{0,1,3}
{0,2,6}
{0,4,5}
{1,2,4}
{1,6,5}
{2,3,5}
{3,4,6}
is nested in a (7,4,2)-design given by
[begin fixed width]
|-------|---|
| 0 1 3 | 6 |
| | |
| 1 2 4 | 0 |
| | |
| 2 3 5 | 1 |
| | |
| 3 4 6 | 2 |
| | |
| 4 5 0 | 3 |
| | |
| 5 6 1 | 4 |
| | |
| 6 0 2 | 5 |
|-------|---|
[end fixed width]
By a (7,4,2)-design we mean that 7 symbols can be described by
blocks of
4 symbols so that any given pair of symbols will occur in 2
blocks. The
Steiner triple system of our quasigroup is a (7,3,1)-design.
To understand why this (7,4,2)-design interferes with our
objectives,
first consider the triple system that is of interest in the
sense that
it only refers to labels for the columns in our incidence
matrix:
{1,2,3}
{1,3,4}
{1,4,5}
{1,5,6}
{1,2,6}
{2,4,6}
{2,3,5}
{3,4,6}
{2,4,5}
{3,5,6}
This is a (6,3,2)-design--or, in other words, a TS(6,2) triple
system
since the block size is 3. Interest in this design is
specifically
motivated by the desire to recover a purely geometric
interpretation for
the incidence matrix.
To see how the (7,4,2)-design interferes with the geometric
intepretation, we recast the explanation of the newly labeled
incidence
matrix to see what is being obscured by the set-theoretic
intersection
operation. The presentation is given in outline form so that
the
relationship between the (7,4,2)-design and the (6,3,2)-design
is made
clear. Also, for similar reasons, there is an abuse of notation
involving ordered pairs.
So, now we write:
[begin fixed width]
I. a corresponds with 1 because {1}={1,2,4} cap {1,6,5}
A. (7,4,2)-design
1. {<1,4>,0,2}
2. {<1,4>,5,6}
B. (6,3,2)-design
1. {<1,4>,3}
2. {<1,4>,5}
II. b corresponds with 5 because {5}={2,3,5} cap {1,6,5}
A. (7,4,2)-design
1. {<1,5>,2,3}
2. {<1,5>,4,6}
B. (6,3,2)-design
1. {<1,5>,4}
2. {<1,5>,6}
III. c corresponds with 3 because {3}={3,4,6} cap {2,3,5}
A. (7,4,2)-design
1. {<2,3>,4,6}
2. {<2,3>,1,5}
B. (6,3,2)-design
1. {<2,3>,1}
2. {<2,3>,5}
IV. d corresponds with 4 because {4}={1,2,4} cap {3,4,6}
A. (7,4,2)-design
1. {<2,4>,0,1}
2. {<2,4>,3,6}
B. (6,3,2)-design
1. {<2,4>,5}
2. {<2,4>,6}
V. e corresponds with 2 because {2}={1,2,4} cap {2,3,5}
A. (7,4,2)-design
1. {<1,2>,0,4}
2. {<1,2>,3,5}
B. (6,3,2)-design
1. {<1,2>,3}
2. {<1,2>,6}
VI. f corresponds with 6 because {6}={3,4,6} cap {1,6,5}
A. (7,4,2)-design
1. {<4,6>,1,5}
2. {<4,6>,3,2}
B. (6,3,2)-design
1. {<4,6>,2}
2. {<4,6>,3}
[end fixed width]
Note that the (6,3,2)-design triples do not convey the
properties of the
geometric interpretation directly. Rather, the outline above is
presented to emphasize the fact that the (7,4,2)-design
emulates the
(6,3,2)-design triples in a definite way. Specifically, the
relation
between the designs is such that a naive use of set-theoretic
operations
obtains the same result.
Later on, we will have cause to return to the assignment,
[begin fixed width]
| |
| N |
| |
0 = | O |
| |
| T |
| |
[end fixed width]
so that we can understand how it chromatically generates the
Steiner
triple system of our quasigroup.
However, we are now in a position to begin discussing region 8
from our
diagram and the octothorpe symbol to which we mapped it.
Recall the
multiplication table for our Steiner quasigroup,
[begin fixed width]
* | 0 1 2 3 4 5 6
--|----------------------------
0 | 0 3 6 1 5 4 2
|
1 | 3 1 4 0 2 6 5
|
2 | 6 4 2 5 1 3 0
|
3 | 1 0 5 3 6 2 4
|
4 | 5 2 1 6 4 0 3
|
5 | 4 6 3 2 0 5 1
|
6 | 2 5 0 4 3 1 6
[end fixed width]
Now, given a Steiner triple system, it is possible to extend
it to a
totally symmetric loop. This loop is called a Steiner loop.
A totally symmetric loop is obtained from a Steiner triple
system by
extending the symbol set by one symbol,
<0,1,2,3,4,5,6> -> <0,1,2,3,4,5,6,#
The symbols are given a product satisfying:
1. #*x = x*# = x
2. x*y=z whenever {x,y,z} is a block of the triple system
It follows that the new symbol set is a totally symmetric loop
because
the following axioms are satified,
1. x*y = y*x
2. #*x = x
3. x*x = #
4. x*(x*y) = y
where # is the loop identity.
These products are expressed in the multiplication table for
the
unipotent quasigroup,
[begin fixed width]
* | 0 1 2 3 4 5 6 #
--|--------------------------------
0 | # 3 6 1 5 4 2 0
|
1 | 3 # 4 0 2 6 5 1
|
2 | 6 4 # 5 1 3 0 2
|
3 | 1 0 5 # 6 2 4 3
|
4 | 5 2 1 6 # 0 3 4
|
5 | 4 6 3 2 0 # 1 5
|
6 | 2 5 0 4 3 1 # 6
|
# | 0 1 2 3 4 5 6 #
[end fixed width]
The importance of this extension comes from factorization.
Let us begin with the original quasigroup. For a symbol set V,
a
Steiner near-1-factorization is defined as
{{{x,y}: x,y e V, ~(x=y), x*y=z} : z e V}
So, the near-1-factorization obtained for the labeling of our
quasigroup, for example, is given by
{{1,3},{2,6},{4,5}}
{{0,3},{2,4},{6,5}}
{{0,6},{1,4},{3,5}}
{{0,1},{2,5},{4,6}}
{{0,5},{1,2},{3,6}}
{{0,4},{2,3},{1,6}}
{{0,2},{3,4},{1,5}}
For the Steiner loop and its associated unipotent quasigroup
we can
define the Steiner 1-factorization,
{{{#,z}}cup {{x,y}: x,y e V, ~(x=y), x*y=z} : z e V}
so that our corresponding Steiner 1-factorization is
{{#,0},{1,3},{2,6},{4,5}}
{{#,1},{0,3},{2,4},{6,5}}
{{#,2},{0,6},{1,4},{3,5}}
{{#,3},{0,1},{2,5},{4,6}}
{{#,4},{0,5},{1,2},{3,6}}
{{#,5},{0,4},{2,3},{1,6}}
{{#,6},{0,2},{3,4},{1,5}}
---
I ended here. It is just too long.
Subject: symmetries of a cube
I had to cancel a previous reply due to failing to
understand the intended meaning of opposite.
: Relative to a cube, a 2-fold symmetry is a 180 degree
rotation. For a
: given 2-fold rotation, the axis of symmetry is the line from
the
: midpoint of a given edge to the midpoint of the opposite
edge.
That would be the DIAGONALLY opposite edge.
: The six 2-fold symmetries segregate into pairs according to
their
relation to
: the three 4-fold symmetries.
:
: To see this, consider the diagram,
:
: [begin fixed width]
:
: /|
: / |
: ------/--|----------
: / / | /
: / / | /
: / / / /
: / | / /
: / | / /
: --------|---/--------
: | /
: | /
: |/
:
:
: [end fixed width]
This diagram is completely irrelevant to identifying the axes
of symmetry in question, WHICH ARE DIAGONAL.
Mitch drew another diagram involving axes through
the centers of faces, but nobody needed the help
on that one. The diagram he SHOULD have drawn
shows 3 of the 2-fold symmetries as:
:
: ----------2------------
: / / /|
: / / |
: / / / |
: 1 / |
: / / / |
: / / 3
: / / / |
: ----------------------- |
: | / | |
: | | |
: | / | /
: | | /
: 3 / | /1
: | | /
: | / | /
: | | /
: | / |/
: ----------2------------
I didn't draw the line connecting the 3's because the
diagram gets unparseably busy;it is pointless to argue over
whose
ascii art manages to accurately communicate. His point was
that there are 6 of these mid-edge diagonals and
: So, each 4-fold symmetry corresponds with four of the 2-fold
symmetries.
This is just ridiculous. I mean, if you are yourself observing
and
DEFINING a correspondence then you can allege any old
correspondence
you like. But 4 is not a divisor of 6 and if there are three
4-fold
symmetries, they will not have any natural correspondence with
any 4 of
the six 2-fold ones. What WILL be natural (since there 6 of one
and of the other) is for each of the 3
lines-through-the-opposite-face-centers
to correspond to TWO of the 6
lines-through-the-diagonally-opposite-edge-midpoints.
You can then get a natural 4 by just taking the other 4 of
those 2.
But it's all an awful lot of work for, basically, nothing,
until you
can come up with an explanation that is shorter than 1537
lines.
Subject: quantum mechanics as an attack on classical logic
: In the link
:
: http://plato.stanford.edu/entries/qt-quantlog/#1
:
: you will find the remark
:
: Whereas logicians have usually assumed that
: properties of negation were the ones least
: able to withstand a critical analysis, the study of
: mechanics points to the distributive identities
: as the weakest link in the algebra of logic.
: [1937, p. 839]
This is rather dramatically over-stating the case.
> In the 1960s and early 1970s, this thesis was advanced rather
> more aggressively by a number of authors, including
especially
> David Finkelstein and Hilary Putnam, who argued that quantum
> mechanics requires a revolution in our understanding of logic
> per se. According to Putnam [1968], Logic is as
> empirical as geometry. We live in a world with a
> non-classical logic.
If you were Franz Fritsche, you could say that this proves
was written by someone with better sense, who continues,
> For Putnam, the elements of L(H) represent categorical
> properties that an object possesses, or does not,
independently
> of whether or not we look. Inasmuch as this picture of
physical
> properties is confirmed by the empirical success of quantum
> mechanics, we must, on this view, accept that the way in
which
> physical properties actually hang together is not Boolean.
Since
> logic is, for Putnam, very much the study of how physical
> properties actually hang together, he concludes that
classical
> logic is simply mistaken: the distributive law is not
> universally valid.
This is, of course, simply ridiculous. The distributive law
is only a law in the context of formal systems where it occurs
as an axiom or a theorem; in other contexts, it is simply
irrelevant; nobody has ever tried to claim any sort of
physical,
let alone universal validity for it. It is simply not a
physical
thing.
> Classically, if S is the set of states of a physical system,
> then every subset of S corresponds to a categorical property
of
> the system, and vice versa. In quantum mechanics, the state
> space is the (projective) unit sphere S = S(H) of a Hilbert
> space. However, not all subsets of S correspond to
> quantum-mechanical properties of the system. The latter
> correspond only to subsets of the special form S intersect M,
> for M a closed linear subspace of H. In particular, only
subsets
> of this form are assigned probabilities. This leaves us with
two
> options. One is to take only these special properties as
real or physical or meaningful, regarding more general
> subsets of S as corresponding to no real categorical
properties
> at all. The other is to regard the quantum
> properties as a small subset of the set of all physically
(or at
> any rate, metaphysically) reasonable, but not necessarily
> observable, properties of the system. On this latter view,
the
> set of all properties of a physical system
> is entirely classical in its logical structure,
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
proving that Putnam was just being ridiculous. This
despite the fact that he's very highly respected and won a lot
more honors than anyone reading (let alone writing) this
message ever
will. This is a common occurrence, not a surprising one as FF
seems to
think.
Just in case anybody wants to use QM to attack logic in
general,
> If we put aside scruples about measurement
> as a primitive term in physical theory, and accept a
principled
> distinction between testable and
> non-testable properties, then the fact that L(H) is not
Boolean
> is unremarkable, and carries no implication about logic per
se.
> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
> Quantum mechanics is, on this view, a theory about the
> possible statistical distributions of outcomes of certain
> measurements, and its non-classical logic
> simply reflects the fact that not all observable phenomena
can
> be observed simultaneously. Because of this, the set of
> probability-bearing events (or propositions) is less rich
than
> it would be in classical probability theory, and the set of
> possible statistical distributions, accordingly, less tightly
> constrained. That some non-classical probability
distributions
> allowed by this theory are actually manifested in nature is
perhaps
> surprising,
> but in no way requires any deep shift in our understanding
> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
> of logic or, for that matter, of probability.
> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Well, I hope that clears THAT up.
Subject: Re: Logic missing from Proofs from THE BOOK
<3fba8c5c$8$fuzhry+tra$mr2ice@news.patriot.net
<3FBC7EA7.5090709@tcs.inf.tu-dresden.de
<3fc01fe6$7$fuzhry+tra$mr2ice@news.patriot.net
<3fcbabc3$0$577$b45e6eb0@senator-bedfellow.mit.edu
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at 08:59 PM, tchow@lsa.umich.edu said:
>It is? Surely the cohomology (or cohomologies) of projective
spaces
>of finite fields is central to their study?
Algebraic Geometry is not Topology. Which is not to say that
methods
and results of one are not useful in the other, but they are
distinct
fields. The same words occur in both fields with very different
meanings.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT
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Subject: Re: Logic missing from Proofs from THE BOOK
Originator: tchow@lagrange.mit.edu.mit.edu (Timothy Chow)
> at 08:59 PM, tchow@lsa.umich.edu said:
>>It is? Surely the cohomology (or cohomologies) of projective
spaces
>>of finite fields is central to their study?
>Algebraic Geometry is not Topology. Which is not to say that
methods
>and results of one are not useful in the other, but they are
distinct
>fields. The same words occur in both fields with very
different meanings.
Hmmm...if you want to play that game, then I'm going to claim
that the
study of projective spaces over finite fields is not geometry.
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the
artillery---however great, will
never exceed four of those miles of which as many thousand
separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two
New Sciences
Subject: Re: Logic missing from Proofs from THE BOOK
<3fc01fe6$7$fuzhry+tra$mr2ice@news.patriot.net
<3fcbabc3$0$577$b45e6eb0@senator-bedfellow.mit.edu
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at 06:35 PM, tchow@lsa.umich.edu said:
>Hmmm...if you want to play that game, then I'm going to claim
that
>the study of projective spaces over finite fields is not
geometry.
That'll never sell in Erlangen.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT
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Subject: Re: Short Fermat Proof
> Now you are complaining that when
> I answer your question, the answer must be relevant to
everything else
> I say?
No, I'm just baffled to learn that your statement The natural
numbers are *defined* by PA below was not intended to be
relevant to
the question of the provability of FLT in PA.
|
|> I didn't think that Wiles restricted himself to the Peano
axioms.
|
|> Indeed not. It is assumed (on the basis of experience) that
the proof
|> will turn out to have a counterpart in PA, which does not
mean that we
|> could actually use only PA in proving the theorem.
|
|The natural numbers are *defined* by PA. Of course a proof
using only
|PA is untenably huge, but that's not relevant here.
Subject: Re: Short Fermat Proof
permission for an emailed response.
> Now you are complaining that when
> I answer your question, the answer must be relevant to
everything else
> I say?
> No, I'm just baffled to learn that your statement The natural
> numbers are *defined* by PA below was not intended to be
relevant to
> the question of the provability of FLT in PA.
That statement was tied up with my mistake about whether we
know if
FLT can be proved within PA.
My later definition of natural number model was an answer to
your
question, and it was my usage of the term natural number model
which
you are apparently still upset at, even though I have told you
about a
jillion times that it was a strictly stipulative definition.
Thomas
Subject: Re: Short Fermat Proof
> That statement was tied up with my mistake about whether we
know if
> FLT can be proved within PA.
And that of course was the statement I was referring to in my
comment that you should have realized immediately its
irrelevance
to the question of the provability of FLT in PA.
Subject: Re: Axiomatic Set Theory and Foundation
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>Huh? Can you give me the proof please?
Sorry, I misread the statement. Thomas was correct and, in
fact, Quine
has extensionality but the statement is false.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT
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Subject: Re: Axiomatic Set Theory and Foundation
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at 11:58 PM, jesse@phiwumbda.org () said:
>No, it doesn't.
You're right; I misread the statement. You're describing what
Quine
called atoms, and in his theory there can be more than one of
them.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT
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Subject: (un)stable fixed points
f:R-->R differentiable, continuous derivative, and f(a)=a for
some a
in R.
if |f'(a)|<1, then the sequence x_n=f(x_n-1) converges to a
when x_0
is sufficiently close to a.
if |f'(a)|>1, then there exists c_0>0 s.t. for all x_0=/=a,
|x_N-a|>c_0 for some positive integer N.
the first part was coming along nicely. it seems that i am
close to
showing that f is a contraction from [a-d,a+d] for some d>0.
the other
part seems more difficult. i think i just need a hint. thanks
Subject: Re: (un)stable fixed points
>f:R-->R differentiable, continuous derivative, and f(a)=a for
some a
>in R.
>if |f'(a)|<1, then the sequence x_n=f(x_n-1) converges to a
when x_0
>is sufficiently close to a.
>if |f'(a)|>1, then there exists c_0>0 s.t. for all x_0=/=a,
>|x_N-a|>c_0 for some positive integer N.
>the first part was coming along nicely. it seems that i am
close to
>showing that f is a contraction from [a-d,a+d] for some d>0.
Not true. But it will have the property that, for some c with
0 < c < 1, |f(x) - a| < c |x - a| if x is sufficiently close
to a.
> the other
>part seems more difficult. i think i just need a hint. thanks
Hint: There is C > 1 such that |f(x) - a| > C |x - a| if x is
sufficiently
close to a.
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: (un)stable fixed points
>>f:R-->R differentiable, continuous derivative, and f(a)=a
for some a
>>in R.
>>if |f'(a)|<1, then the sequence x_n=f(x_n-1) converges to a
when x_0
>>is sufficiently close to a.
>>if |f'(a)|>1, then there exists c_0>0 s.t. for all x_0=/=a,
>>|x_N-a|>c_0 for some positive integer N.
>>the first part was coming along nicely. it seems that i am
close to
>>showing that f is a contraction from [a-d,a+d] for some d>0.
>Not true.
Counterexample? It seems true to me - I hesitate to post what
seems like the easy proof since we're just giving hints.
Possibly I'm being stupid again (or possibly you missed
the fact that f is _continuously_ differentiable?)
>But it will have the property that, for some c with
>0 < c < 1, |f(x) - a| < c |x - a| if x is sufficiently close
to a.
>> the other
>>part seems more difficult. i think i just need a hint. thanks
>Hint: There is C > 1 such that |f(x) - a| > C |x - a| if x is
sufficiently
>close to a.
>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: (un)stable fixed points
>f:R-->R differentiable, continuous derivative, and f(a)=a
for some a
>in R.
>if |f'(a)|<1, then the sequence x_n=f(x_n-1) converges to a
when x_0
>is sufficiently close to a.
>if |f'(a)|>1, then there exists c_0>0 s.t. for all x_0=/=a,
>|x_N-a|>c_0 for some positive integer N.
>the first part was coming along nicely. it seems that i am
close to
>showing that f is a contraction from [a-d,a+d] for some d>0.
>>Not true.
>Counterexample? It seems true to me - I hesitate to post what
>seems like the easy proof since we're just giving hints.
>Possibly I'm being stupid again (or possibly you missed
>the fact that f is _continuously_ differentiable?)
Oops, yes of course I missed that the derivative is continuous
(which is
not necessary for the result).
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: (un)stable fixed points
>>f:R-->R differentiable, continuous derivative, and f(a)=a
for some a
>>in R.
>>if |f'(a)|<1, then the sequence x_n=f(x_n-1) converges to
a when x_0
>>is sufficiently close to a.
>>if |f'(a)|>1, then there exists c_0>0 s.t. for all x_0=/=a,
>>|x_N-a|>c_0 for some positive integer N.
>>the first part was coming along nicely. it seems that i am
close to
>>showing that f is a contraction from [a-d,a+d] for some
d>0.
>Not true.
>>Counterexample? It seems true to me - I hesitate to post what
>>seems like the easy proof since we're just giving hints.
>>Possibly I'm being stupid again (or possibly you missed
>>the fact that f is _continuously_ differentiable?)
>Oops, yes of course I missed that the derivative is
continuous (which is
>not necessary for the result).
And since that hypothesis is not needed it must not have been
given. Much better than the excuses I usually come up
with<g>...
>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: (un)stable fixed points
> f:R-->R differentiable, continuous derivative, and f(a)=a
for some a
> in R.
> if |f'(a)|<1, then the sequence x_n=f(x_n-1) converges to a
when x_0
> is sufficiently close to a.
> if |f'(a)|>1, then there exists c_0>0 s.t. for all x_0=/=a,
> |x_N-a|>c_0 for some positive integer N.
x = a + e; f(a + e) ~ f(a) + ef'(a)
e_n = f'(a)e_(n-1)
This is a linear recurrence which can be solved:
e_n = e_0*[f'(a)]^n
--
P.A.C. Smith
The vast majority of Iraqis want to live in a peaceful, free
world.
And we will find these people and we will bring them to
justice.
Subject: cylinder equation
I wonder if someone can help me with a problem.
In the paper of l.piegl geometric method of intersecting
natural
quadrics represented in trimmed surface form (CAD 21, 1989). I
found
the following implicit equation for conic curves
Au^2v+Bu^2+Cuv+Du+Ev+F=0.
Its said this equation can be derived by substituting the
cylinder
resp. cone's parametric equation into the implicit plane
equation.
But nowhere in the paper is given a parametric equation for
the cone
resp. cylinder.
Does someone know an parametric equation for the cylinder
resp. cone
to yield the formula above?
Is the formula above similar to the general equation for
planar conic
curves?
In the same paper is mentioned the possibility to convert
monomial
equations into bezier or bspline form. Does someone know an
approach
(maybe some papers) for that?
Thanks in advance for your help
Stephan
Subject: Re: cylinder equation
>I wonder if someone can help me with a problem.
>In the paper of l.piegl geometric method of intersecting
natural
>quadrics represented in trimmed surface form (CAD 21, 1989).
I found
>the following implicit equation for conic curves
>Au^2v+Bu^2+Cuv+Du+Ev+F=0.
>Its said this equation can be derived by substituting the
cylinder
>resp. cone's parametric equation into the implicit plane
equation.
It's the other way around: you substitute the parametric
equation
of the plane into the implicit equation for the cone or
cylinder.
The plane is V1 u + V2 v = V0 where V0, V1 and V2 are vectors,
V1 and
V2 linearly independent. The cylinder can be taken as x^2 +
y^2 = 1
and the cone as x^2 + y^2 = c z^2.
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: cylinder equation
> But nowhere in the paper is given a parametric equation for
the cone
> resp. cylinder.
If you have to ask _that_, odds are you don't have the
necessary
background knowledge to understand the rest of that paper,
either.
You should probably retract your path a couple of steps and
brush up
on your analytic geometry before proceeding.
But since you asked: the general equation for a cone is
P(u,v) = P0 + u * (D + cos(v) * N + sin(v) * (D x N))
for a cone with apex P0, axis D (|D|=1), and opening angle
alpha
given by |N|= tan(alpha)), where N.D=0. The cylinder is
similar,
except that the orthogonal pieces aren't multiplied by u, and N
encodes the radius instead of the opening angle in its length.
> Does someone know an parametric equation for the cylinder
resp. cone
> to yield the formula above?
It doesn't really yield the formula above all that easily. It
takes
quite some algebra to arrive at the final solution, and it has
quite a
zoo of special cases to take care of. These conic sections
were a
favourite subject of classical mathematical education in the
past
which frustrated students no end by the abstract way they were
usually
taught.
> Is the formula above similar to the general equation for
planar conic
> curves?
The formula you show *is* the general equation for planar conic
curves, with the sole change that the independent variables
are named
u and v instead of the more usual x and y.
--
Hans-Bernhard Broeker (broeker@physik.rwth-aachen.de)
Even if all the snow were burnt, ashes would remain.
Subject:
=?ISO-8859-1?B?
UmU6IEluY29tcGxldGVuZXNzIFRoZW9yZW1zIG9mIEt1cnQgR/ZkZWw=?=
|I would like to know if this is the right newsgroup for
asking questions
about
|the Incompleteness Theorems of Kurt G.9adel.
sci.logic is more specifically for logic.
Subject: Re: Irrationality / transcendence of these numbers.
How to find
out?
>Neat, thanks a lot. Just a quick follow-up question: is there
an
>extension of this concept which applies to say, continuity or
>differentiability? (Putting constraints on the error term, I
mean).
I'm afraid I don't see what you're getting at. Continuity of
the
error term as a function of what?
> Also, are there any good references for this subject?
I'm afraid I don't know one offhand. The general area is called
diophantine approximation.
Keith Ramsay
Subject: How to improve?
Hi All,
I've just completed my first college level math subject...
It included such topics as
proofs
induction
complex numbers
limits
continuity
sequences
series
derivatives
linear algebra -determinates, vectors, orthogonal projection,
finding
inverses etc
I felt that I grasped most of the material covered. I'm
confident that
I've passed the subject but without much distinction...
I'm interested in finding out how those of you with some
mathematical
flair approach this business of doing math. What do you
recommend as
those essential general areas and techniques that will help a
plodder
like me improve!?
Some suggestions on texts would also be appreciated.
Thanks in advance
Bruce.
Subject: Re: How to improve?
What kind of course included all that? What were the course's
prerequisites? You say it was your first math subject, huh?
Lurch
> Hi All,
> I've just completed my first college level math subject...
> It included such topics as
> proofs
> induction
> complex numbers
> limits
> continuity
> sequences
> series
> derivatives
> linear algebra -determinates, vectors, orthogonal
projection, finding
> inverses etc
> I felt that I grasped most of the material covered. I'm
confident that
> I've passed the subject but without much distinction...
> I'm interested in finding out how those of you with some
mathematical
flair approach this business of doing math. What do you
recommend as
> those essential general areas and techniques that will help
a plodder
> like me improve!?
> Some suggestions on texts would also be appreciated.
> Thanks in advance
> Bruce.
Subject: Re: How to improve?
<ZXrzb.2683$Oe5.300@newsread2.news.atl.earthlink.net
> What kind of course included all that?
First eight weeks of the Cambridge course includes all that,
plus
Riemann integration and differential equations.
(Of course, it's carved up into three courses of 48, 24 and 24
1 hour
lectures and the exam isn't until June, but nevertheless.)
> Lurch
> Hi All,
> I've just completed my first college level math subject...
> It included such topics as
> proofs
> induction
> complex numbers
> limits
> continuity
> sequences
> series
> derivatives
> linear algebra -determinates, vectors, orthogonal
projection, finding
> inverses etc
> I felt that I grasped most of the material covered. I'm
confident that
> I've passed the subject but without much distinction...
> I'm interested in finding out how those of you with some
mathematical
flair approach this business of doing math. What do you
recommend
as
> those essential general areas and techniques that will help a
plodder
> like me improve!?
Practise.
> Some suggestions on texts would also be appreciated.
Analysis texts:
W. A. Sutherland, Introduction to Metric and Topological Space
(OUP
1975)
Applied texts:
H. Schey, Div, Grad, Curl and All That (Norton 1996)
--
P.A.C. Smith
The vast majority of Iraqis want to live in a peaceful, free
world.
And we will find these people and we will bring them to
justice.
Subject: Re: How to improve?
>> I'm interested in finding out how those of you with some
mathematical
>> flair approach this business of doing math. What do you
recommend
as
>> those essential general areas and techniques that will help
a
plodder
>> like me improve!?
> Practise.
OK!
>> Some suggestions on texts would also be appreciated.
> Analysis texts:
> W. A. Sutherland, Introduction to Metric and Topological
Space (OUP
1975)
> Applied texts:
> H. Schey, Div, Grad, Curl and All That (Norton 1996)
Thanks. I'll check these out.
Subject: Re: How to improve?
> What kind of course included all that? What were the course's
> prerequisites? You say it was your first math subject, huh?
> Lurch
See http://turing.une.edu.au/dept/units/undergrad/MATH101.html
There were no prerequisites for the subject. I'm assuming most
people
attempting the subject had at least taken some advanced math
at high
school level (not me!).
I actually did this subject externally by distance
education. Correspondence with the lecturer was by email and I
sat the
final exam at a centre here in Brisbane (Australia).
Bruce.
>> Hi All,
>> I've just completed my first college level math subject...
It included
>> such topics as
>> proofs
>> induction
>> complex numbers
>> limits
>> continuity
>> sequences
>> series
>> derivatives
>> linear algebra -determinates, vectors, orthogonal
projection, finding
>> inverses etc
>> I felt that I grasped most of the material covered. I'm
confident that
>> I've passed the subject but without much distinction...
>> I'm interested in finding out how those of you with some
mathematical
>flair approach this business of doing math. What do you
recommend as
>> those essential general areas and techniques that will help
a
plodder
>> like me improve!?
>> Some suggestions on texts would also be appreciated.
>> Thanks in advance
>> Bruce.
Subject: Re: [HS] Re: Proof by induction
> Notice in passing
> I'd be flabbergasted if this were real English :)
No, no, you ARE flabbergasted :)
(Robert & Collins Senior, 5th Edition).
I remember my being astonished when our English teacher (a
former curator
at
the British Museum) said that phrase which I thought was
typically French.
BTW, a Google search of in passing (exact search) lets me
think that
phrase is popular enough.
Subject: Re: [HS] Re: Proof by induction
> The reason is different. We are from far the best
mathematicians of the
> world.
Perhaps you meant to say: We are by far the best
mathematicians in the
world.? Your sentence has almost the opposite meaning.
Gib
Subject: Re: Comment welcome -- martian mounds anomaly
> first of all, there is no reason for the geographical
features
> of Mars to be random, if we can assume some sort
> of analog of plate tectonics (see Euler poles). after that,
> there will always be some minimum of patterns,
> which can become quite involved, as witness [place in France
> that's associated with Xian spooks], but it hardly means that
> they were built by folks on Mars.
Who talks about folks on Mars? About mounds not being random,
THAT
is exactly the initial hypothesis. We test against a random
homogeneous continuous distribution (same density of
probability for
every point).
> on the other hand,
> you can take a picture of any thing -- clouds, stucco, what
ever --
> and you can always manage to find close approximations
> to pi, phi, gamma, the dihedral of the tetrahedron etc.,
> by drawing and measuring enough lines & angles.
That is pseudoscience. Provide the necessary images,
calculations,
etc. Just talking that way is like saying, oh, it is easy that
I can
win the lottery, I only have to buy enough numbers.
> it is of no consequence,
> no matter what Art Bell's guests say about it!
I am spanish, I don't listen to Art Bell. And mainly because I
have
good taste.
I recommend that you learn to read before writing such stupid
critics.
http://www.thequantummachine.com/mounds.php
Find where exactly I talk of folks on Mars. And please, get a
life.
>> Nothing. What do the mounds have to do with math?
> Well, take a look at the polygon BAGED. It displays the
maximum number
> of DIFFERENT parallel and perpendicular directions of lines
drawn
> among their vertex. The only other polygon that display this
property
> is a square with a fifth point being (as D in BAGED) in the
> intersection of the diagonals of the square (parallelogram
for BAGED)
> drawn from the adjacent vertex.
> It gets very much complex to say than to draw it.
 I am not talking of little green martians. I am saying (and
that is
> objectable, but I will just consider scientific objections)
that the
> mounds distribution is not random. Nothing else.
> And this pattern of maximality of parallel/perpendicular may
suggest
> some unknown geological mechanism. Of course, in order to
discover it,
> maths have to be applied (maths is applied in astrophysical
alignments
> of galaxies, for example, in a way similar to the
Kolmogorov-Smirnov
> test that was applied to the mounds. Seeing that the pattern
may be
> non-random, you go further in your analysis).
> --ils duces d'Enron!
Subject: Help / ideas with fixed income analysis problem
Hello everybody,
Although the context of my problem isn't important, it has to
do with
the accretion of original issue discount for fixed income
securities.
I have been given the following iterative procedure by the
business/tax experts to calculate a particular point on a
curve:
(I)Y - C = p1 (delta first period)
(I+p1)Y - C = p2 (delta second period)
(I+p1+p2)Y - C = p3 (delta third period)
Given these equations, one can draw a curve from point(I) to
point(I+p1) to point(I+p1+p2) etc. However, in order to find
point(I+p1+p2+...+pn), one must iteratively solve n equations
and then
take their sum.
I would like to know if there is any way to create a closed
equation
to describe this function? Something along the lines of y =
nIY^(n-1) - nC, where n = number of periods (this is a totally
fabricated solution, I'm just worried that my terminology
isn't clear,
so I'm trying to give an example of what I'm looking for).
It seems to me that I have a curve, therefore I have a
continuous
function. And if I have a continuous function, I should be
able to
find an equation to represent it. However, all the financial
analysis
textbooks point to the iterative solution.
Can anyone help me?
Thank you!
Anthony
Subject: references for an efficient algorithm for extended
euclidean
algorithm?
I want to implement an extended Euclidean algorithm in MATLAB
that can
compute the following: GCD(A(x),B(x)) = u(x)A(x) + v(x)B(x),
where
A(x), B(x), u(x) and v(x) are from GF(2^m).
I have a basic understanding how this works for integers, but
don't
want to re-invent the wheel for the above case. I did a search
on
IEEEXPLORE, and the closest hit that I got was from Mandelbaum,
although this was a hardware implementation. Does anyone have
perhaps
pointers to literature where this is clearly explained?
(I once got a glimpse of such a technique in a textbook, but I
spent
the whole morning trying to find the book in the library
without
success. It did something with a matrix)
I will greatly appreciate any suggestions and help
Jaco
Subject: Re: references for an efficient algorithm for
extended euclidean
algorithm?
Hello Jaco,
Try looking at Prime Numbers by Crandall and Pomerance. They
present
the
basic algo with some improvements. Are you trying to find
inverses or GCDs?
Clay
> I want to implement an extended Euclidean algorithm in
MATLAB that can
> compute the following: GCD(A(x),B(x)) = u(x)A(x) + v(x)B(x),
where
> A(x), B(x), u(x) and v(x) are from GF(2^m).
> I have a basic understanding how this works for integers,
but don't
> want to re-invent the wheel for the above case. I did a
search on
> IEEEXPLORE, and the closest hit that I got was from
Mandelbaum,
> although this was a hardware implementation. Does anyone
have perhaps
> pointers to literature where this is clearly explained?
> (I once got a glimpse of such a technique in a textbook, but
I spent
> the whole morning trying to find the book in the library
without
> success. It did something with a matrix)
> I will greatly appreciate any suggestions and help
> Jaco
Subject: Re: references for an efficient algorithm for
extended euclidean
algorithm?
X-ID: VItBUOZpoeX3qQAVrN-HU-uRgwoTOFLV0K7mlRUrWJth-a4MH7gjYg
Jaco Versfeld schrieb:
> I want to implement an extended Euclidean algorithm in
MATLAB that can
> compute the following: GCD(A(x),B(x)) = u(x)A(x) + v(x)B(x),
where
> A(x), B(x), u(x) and v(x) are from GF(2^m).
You could use gcd from PARI-GP
hth
Klaus
> I have a basic understanding how this works for integers,
but don't
> want to re-invent the wheel for the above case. I did a
search on
> IEEEXPLORE, and the closest hit that I got was from
Mandelbaum,
> although this was a hardware implementation. Does anyone
have perhaps
> pointers to literature where this is clearly explained?
> (I once got a glimpse of such a technique in a textbook, but
I spent
> the whole morning trying to find the book in the library
without
> success. It did something with a matrix)
> I will greatly appreciate any suggestions and help
> Jaco
Subject: Re: Vedic Mathematics --- Myth and Reality
> By the by, we understood all modern sceince discoveries
> (like special theory of relativity) are explained in Vedas
> in detail.
> 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 [...]
> all this time, the other shoe has finally dropped.
> But could you enlight us discoveries yet to be happened
> (Like design of perpetual machine.) from Vedas?
> Vimans - which should work upon that principle - are
mentioned in the
> ancient Indian epics. People seeking enlightenment should
first get
> their grammar and spelling correct. Knowledge is wasted upon
> low-minded and deliberate fools.
> Well, 5000 years without flight in the presence of all that
> knowledge: clearly it must have been wasted on the whole
> subcontinent all those millennia.
mortals. Ravana,
who defeated the Gods, had the Pushpak Vimana.
Mortals neither designed nor made vimans, though some aspects
of their
design were known
according to certain sources.
Mortals managed on foot, chariots, etc. till the invention of
the bicycle
and the
internal combustion engine.
> Meanwhile, the first airplanes used by Deccan Airways were
made
> and purchased from whom?
We are talking about the past, with reference to the future.
The present
is
just a bad idea.
Arindam Banerjee.
Subject: Re: Hilbert's 16th
> can someone tell me the status of the proof of the 2nd part
of Hilbert's
> 16th problem? According to news reports a 22-year old
student from Sweden
> (Elin Oxenhielm) has managed to prove it. However, Grigori
Rozenblum
> (Chalmers Univ. of Techn, Goeteborg, Sweden) has apparently
found an
> error.
> Thanks,
> George Szpiro
> --------
> George Szpiro
> Neue Z.9frcher Zeitung (Switzerland)
> POB 6278
> Jerusalem 91060
> Israel
as a computer algebraist/number theorist, I am no expert for
dynamical
systems, and I normally would never have dared to answer, but
it seems that
the experts wont, so here is my general impression.
1. An exact argument can never go like
Noticing that the state variable x of the Li.8enard equation
(1) behaves
approximately like a sine function in simulations (see Fig. 1)
because this means that the author gives no mathematical proof
at all, but
just shows some nice pictures and trusts her computer
simulations. Such
dubious computer simulations appear again in a later part of
the
proof.
In short: she can not simulate infinitely many values of k and
infinitely
many different values of the coefficients, so her proof
is a proof by example.
2. The proof contains many places where things are
approximately
true,
some terms can be discarded etc. I did not understand what the
author
means by dominated (just the problem Rozenblum had).
etc.
I think the problem is (just as with amateur mathematics) that
there is no
single error at a particular point, but the whole paper
contains no
rigorous argument. As far as I understand, this is what
Rozenblum wanted to
say.
Now come on, you experts in this field, and correct me if I'm
wrong ...
--
Stefan Wehmeier
stefanw@mupad.de
Subject: Isometries of closed disks
Hi all,
Each closed disk in the plane has the following properties:
1) it's compact;
2) its interior is non-empty;
3) its boundary is connected;
4) its group of isometries is infinite.
My question is: are closed disks the only plane sets with these
properties. My guess is that the answer is yes. Does anyone
know
how to prove it?
Best regards
Subject: Re: Isometries of closed disks
En el mensaje: 184bc135.0312030304.193e44f0@posting.google.com,
Jose Carlos Santos <jcsantos@fc.up.pt> dijo:
> Hi all,
> Each closed disk in the plane has the following properties:
> 1) it's compact;
> 2) its interior is non-empty;
> 3) its boundary is connected;
> 4) its group of isometries is infinite.
> My question is: are closed disks the only plane sets with
these
> properties. My guess is that the answer is yes. Does anyone
know
> how to prove it?
Following D. Ulrich idea, take straight segments of length 1
from origin,
at
any angle that be a rational multiple of pi, from OX axis. Add
a closed
disk
of centre O and radius less than 1.
--
Best regards,
Ignacio Larrosa Ca.96estro
A Coru.96a (Espa.96a)
ilarrosaQUITARMAYUSCULAS@mundo-r.com
Subject: Re: Isometries of closed disks
>En el mensaje:
184bc135.0312030304.193e44f0@posting.google.com,
>Jose Carlos Santos <jcsantos@fc.up.pt> dijo:
>> Hi all,
>> Each closed disk in the plane has the following properties:
>> 1) it's compact;
>> 2) its interior is non-empty;
>> 3) its boundary is connected;
>> 4) its group of isometries is infinite.
>> My question is: are closed disks the only plane sets with
these
>> properties. My guess is that the answer is yes. Does anyone
know
>> how to prove it?
>Following D. Ulrich idea, take straight segments of length 1
from origin,
at
>any angle that be a rational multiple of pi, from OX axis.
Add a closed
disk
>of centre O and radius less than 1.
Following my ideas on _this_ problem is probably not wise...
************************
Subject: Re: Isometries of closed disks
> En el mensaje:
184bc135.0312030304.193e44f0@posting.google.com,
> Jose Carlos Santos <jcsantos@fc.up.pt> dijo:
>> Hi all,
>> Each closed disk in the plane has the following properties:
>> 1) it's compact;
>> 2) its interior is non-empty;
>> 3) its boundary is connected;
>> 4) its group of isometries is infinite.
>> My question is: are closed disks the only plane sets with
these
>> properties. My guess is that the answer is yes. Does anyone
know
>> how to prove it?
> Following D. Ulrich idea, take straight segments of length 1
from origin,
at
> any angle that be a rational multiple of pi, from OX axis.
Add a closed
disk
> of centre O and radius less than 1.
i had the same thought. however, that's not compact: pick some
point at
the end of a line segment with argument an irrational multiple
of pi. it
is the limit point of some sequence lying in your set
(approximate the
argument by rationals). hence it isn't closed, and isn't
compact in the
standard topology. (compact iff closed and bounded.)
matt
Subject: Re: Isometries of closed disks
matt grime <mattgrime@o2.co.uk> dijo:
> i had the same thought. however, that's not compact: pick
some point
> at the end of a line segment with argument an irrational
multiple of
> pi. it is the limit point of some sequence lying in your set
> (approximate the argument by rationals). hence it isn't
closed, and
> isn't compact in the standard topology. (compact iff closed
and
> bounded.)
Yes, you are right ... :^((
--
Best regards,
Ignacio Larrosa Ca.96estro
A Coru.96a (Espa.96a)
ilarrosaQUITARMAYUSCULAS@mundo-r.com
Subject: Re: Isometries of closed disks
>Following D. Ulrich idea, take straight segments of length 1
from origin,
at
>any angle that be a rational multiple of pi, from OX axis.
Add a closed
disk
>of centre O and radius less than 1.
>matt grime <mattgrime@o2.co.uk> dijo:
>i had the same thought. however, that's not compact: pick
some point
>>at the end of a line segment with argument an irrational
multiple of
>>pi. it is the limit point of some sequence lying in your set
>>(approximate the argument by rationals). hence it isn't
closed, and
>>isn't compact in the standard topology. (compact iff closed
and
>>bounded.)
>Yes, you are right ... :^((
So instead of rational mutilples of pi, how about {2 pi c: c
in the
Cantor set}?
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
Subject: Re: Isometries of closed disks
>Hi all,
>Each closed disk in the plane has the following properties:
>1) it's compact;
>2) its interior is non-empty;
>3) its boundary is connected;
>4) its group of isometries is infinite.
>My question is: are closed disks the only plane sets with
these
>properties. My guess is that the answer is yes. Does anyone
know
>how to prove it?
Seems like it may be true if you add
5) it is the closure of its interior.
But as stated it's false. For example, imagine a binary tree,
drawn as
a bunch of straight line segments in the plane, with branches
converging to points of a Cantor set; include the Cantor set to
make the set closed. That has infinitely many isometries; it
does everything you ask except for (2). So add a disk touching
the top node.
>Best regards
************************
Subject: Re: Isometries of closed disks
>Hi all,
>Each closed disk in the plane has the following properties:
>1) it's compact;
>2) its interior is non-empty;
>3) its boundary is connected;
>4) its group of isometries is infinite.
>My question is: are closed disks the only plane sets with
these
>properties. My guess is that the answer is yes. Does anyone
know
>how to prove it?
> Seems like it may be true if you add
> 5) it is the closure of its interior.
> But as stated it's false. For example, imagine a binary
tree, drawn as
> a bunch of straight line segments in the plane, with branches
> converging to points of a Cantor set; include the Cantor set
to
> make the set closed. That has infinitely many isometries;
I don't think so. Any isometry of a set in the plane extends
to an
isometry of the whole plane.
> it
> does everything you ask except for (2). So add a disk
touching
> the top node.
>Best regards
> ************************
>
--
G. A. Edgar
http://www.math.ohio-state.edu/~edgar/
Subject: Re: Isometries of closed disks
>Hi all,
>>Each closed disk in the plane has the following properties:
>>1) it's compact;
>>2) its interior is non-empty;
>>3) its boundary is connected;
>>4) its group of isometries is infinite.
>>My question is: are closed disks the only plane sets with
these
>>properties. My guess is that the answer is yes. Does anyone
know
>>how to prove it?
> Seems like it may be true if you add
> 5) it is the closure of its interior.
> But as stated it's false. For example, imagine a binary
tree, drawn as
>> a bunch of straight line segments in the plane, with
branches
>> converging to points of a Cantor set; include the Cantor
set to
>> make the set closed. That has infinitely many isometries;
>I don't think so. Any isometry of a set in the plane extends
to an
>isometry of the whole plane.
Aargh. Of course you're right - the maps I had in mind are not
isometries.
(Well, they're isometries in a sort of different sense of the
word -
if we measure the distance between two points of K as the
length
of the shortest path _in K_ connecting the two points. But of
course that's not what the word meant here. I guess I'm
assuming
here that people are curious what I was thinking, which is
probably unlikely. Never mind...)
>> it
>> does everything you ask except for (2). So add a disk
touching
>> the top node.

>>Best regards
> ************************
>
************************
Subject: Re: Isometries of closed disks
> Hi all,
> Each closed disk in the plane has the following properties:
> 1) it's compact;
> 2) its interior is non-empty;
> 3) its boundary is connected;
> 4) its group of isometries is infinite.
> My question is: are closed disks the only plane sets with
these
> properties. My guess is that the answer is yes. Does anyone
know
> how to prove it?
Look at the group of isometries, and prove it is compact.
Then deduce that the proper isometries project onto SO(2).
Also prove that the isometries have a common fixed point.
Deduce that the set has circular symmetry about that point.
--
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: Isometries of closed disks
> Each closed disk in the plane has the following properties:
> 1) it's compact;
> 2) its interior is non-empty;
> 3) its boundary is connected;
> 4) its group of isometries is infinite.
> My question is: are closed disks the only plane sets with
these
> properties. My guess is that the answer is yes.
Do closed filled squares, rectangles and ellipses have those
properties?
Subject: Re: Isometries of closed disks
William Elliot <marsh@privacy.net> escribi.97 en el mensaje
>> Each closed disk in the plane has the following properties:
>> 1) it's compact;
>> 2) its interior is non-empty;
>> 3) its boundary is connected;
>> 4) its group of isometries is infinite.
>> My question is: are closed disks the only plane sets with
these
>> properties. My guess is that the answer is yes.
> Do closed filled squares, rectangles and ellipses have those
> properties?
No, its groups of isometries are finite.
--
Best regards,
Ignacio Larrosa Ca.96estro
A Coru.96a (Espa.96a)
ilarrosaQUITARMAYUSCULAS@mundo-r.com
Subject: quadratic solution
A colleague from my pre-retirement workplace asked me the
following:
Solve:
(5 + sqrt(x))^2 - 9*(5 + sqrt(x) ) + 20 = 0
This was a test question she had used previously asking for a
solution over
the reals. No problem.
But this year she put this on a test covering quadratics over
the complex
#'s with coeff's possibly in C.
By the way, sqrt() is traditional check mark-overscore thingy.
This led to needing a solution of sqrt(x) = -1. A root of
course was
discarded on the real number test, but what about here.
What is the solution to this over the complex numbers? What
the rules
are
for applying sqrt() to complex radicand actually seems to be
what the
problem comes down to.
I a little embarassed to say that I don't know the rules for
principle
values or branching here.
Can someone help me (us) out.
Thanks,
Ken
Subject: Re: quadratic solution
> A colleague from my pre-retirement workplace asked me the
following:
> Solve:
> (5 + sqrt(x))^2 - 9*(5 + sqrt(x) ) + 20 = 0
> This was a test question she had used previously asking for
a solution
over
> the reals. No problem.
> But this year she put this on a test covering quadratics
over the complex
> #'s with coeff's possibly in C.
> By the way, sqrt() is traditional check mark-overscore
thingy.
> This led to needing a solution of sqrt(x) = -1. A root of
course was
> discarded on the real number test, but what about here.
> What is the solution to this over the complex numbers? What
the rules
are
> for applying sqrt() to complex radicand actually seems to be
what the
> problem comes down to.
> I a little embarassed to say that I don't know the rules for
principle
> values or branching here.
Depends how you define sqrt(z).
If you make a cut along the +ve real axis
so 0 < arg(z) <= 2*pi and define sqrt(1.exp(i*0)) = 1, then
sqrt(z) = -1 => sqrt(z) = exp(i*pi) => z = exp(2*pi*i) - the
other side of
the cut.
Alternatively, you could take -pi < arg(z) <= pi and sqrt(1) =
1; then
sqrt(z) = -1 => arg(z) = 2*pi; no solution.
--
P.A.C. Smith
The vast majority of Iraqis want to live in a peaceful, free
world.
And we will find these people and we will bring them to
justice.
Subject: Re: quadratic solution
> A colleague from my pre-retirement workplace asked me the
following:
> Solve:
> (5 + sqrt(x))^2 - 9*(5 + sqrt(x) ) + 20 = 0
> This was a test question she had used previously asking for
a solution
over
> the reals. No problem.
> But this year she put this on a test covering quadratics
over the
complex
> #'s with coeff's possibly in C.
> By the way, sqrt() is traditional check mark-overscore
thingy.
> This led to needing a solution of sqrt(x) = -1. A root of
course was
> discarded on the real number test, but what about here.
> What is the solution to this over the complex numbers? What
the
rules
are
> for applying sqrt() to complex radicand actually seems to be
what the
> problem comes down to.
> I a little embarassed to say that I don't know the rules for
principle
> values or branching here.
> Depends how you define sqrt(z).
> If you make a cut along the +ve real axis
> so 0 < arg(z) <= 2*pi and define sqrt(1.exp(i*0)) = 1, then
> sqrt(z) = -1 => sqrt(z) = exp(i*pi) => z = exp(2*pi*i) - the
other side
of
> the cut.
> Alternatively, you could take -pi < arg(z) <= pi and sqrt(1)
= 1; then
> sqrt(z) = -1 => arg(z) = 2*pi; no solution.
> --
> P.A.C. Smith
The vast majority of Iraqis want to live in a peaceful, free
world.
> And we will find these people and we will bring them to
justice.
Let u=5+sqrt(x)
u^2-9u+20=0
(u-4)(u-5)=0
u=4 or u=5
4=5+sqrt(x)
No solution
5=5+sqrt(x)
x=0
David Moran
Subject: Opposing points of contact in a circle
Is there a mathematical term for the opposite point of contact
of a
circle's diameter?
Or, to put it another way, any point on a circle's
circumference 180
degrees opposite to another point on the same circle's
circumference.
Thanks
Daryl
Subject: Re: Opposing points of contact in a circle
> Is there a mathematical term for the opposite point of
contact of a
> circle's diameter?
> Or, to put it another way, any point on a circle's
circumference 180
> degrees opposite to another point on the same circle's
circumference.
These phrases are perhaps even more used in common language,
than in
mathematics. The two points are said to be
diametrically opposite (or opposed)
in diametric opposition
etc.
David
Subject: Re: Opposing points of contact in a circle
>> Is there a mathematical term for the opposite point of
contact of a
>> circle's diameter?
>> Or, to put it another way, any point on a circle's
circumference 180
>> degrees opposite to another point on the same circle's
circumference.
> These phrases are perhaps even more used in common language,
than in
> mathematics. The two points are said to be
> diametrically opposite (or opposed)
> in diametric opposition
> etc.
> David
antipodal.
and it's used for higher dimensions
Subject: Re: Is a retract an identification map?
Subject: Is a retract an identification map?
>well, I don't know if the term identification map is
>the right one in English, but it means the following:
>It is a continuous surjective map between topological spaces
X and Y,
>namely
> f:(X,Tx) -> (Y,Ty)
>where Ty is the final topology w.r.t. f.
We call it quotient map.
>I am now trying to prove that a retract, namely a continuous
map
> r:X -> A, A subset X, r|A = I_A:A -> A
>is such an identification map, but I don't get it.
Yes.
>3) The same with: f:X->Y is an identification map iff ( U
>is open in Y iff f^{-1}(U) is open in X). The direction => is
>no problem as r is continuous, but with <= I got the same
problem
>as above in showing that r is open.
for U subset A, / intersect
if r^-1(U) open, then U = r^-1(U) / A open in A
>I didn't used the property that r restricted to A is the
>identity map and I am really not shure where to use it.
Now do you know where?
----
Subject: Re: Is a retract an identification map?
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X-Treme: C&C,DWS
at 01:24 PM, Rene Meyer <meyr2@web.de> said:
>where Ty is the final topology w.r.t. f.
What do you mean by final topology? Is that a term that you
made up?
What does it have to do with f being a continuous surjective
map?
>1) Show that the subspace topology on A and the final
topology on A
> w.r.t. r are identical, but after some tries I gave up.
The definition of retract does not involve anything called a
final
topology. Try working directly from the definition. Take into
account
that F(X) C A (You don't need to take into account that
F(X)=A.)
>2) Use: If a map f is a continuous open surjection, then it
is an
> identification map. Continuous surjection is given, but I
can't
> show that r is necessarily open, especially for open subsets
> of X that lie partly in A and partly in X-A.
Why do you care whether r is open?
>3) The same with: f:X->Y is an identification map iff ( U
> is open in Y iff f^{-1}(U) is open in X).
F:R->R, F(x)=0 for all x is not an identification map but
f^{-1}(U) is
open in R for U open.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT
Unsolicited bulk E-mail will be subject to legal action. I
reserve
the right to publicly post or ridicule any abusive E-mail.
Reply to domain Patriot dot net user shmuel+news to contact
me. Do
not reply to spamtrap@library.lspace.org
Subject: Re: Is a retract an identification map?
>>where Ty is the final topology w.r.t. f.
> What do you mean by final topology? Is that a term that you
made up?
> What does it have to do with f being a continuous surjective
map?
I thought it to be the correct term for the following: Take a
map
f:(X,T)->Y. The finest topology that makes f continuous is the
final
topology w.r.t. f.
>>1) Show that the subspace topology on A and the final
topology on A
>> w.r.t. r are identical, but after some tries I gave up.
> The definition of retract does not involve anything called a
final
> topology. Try working directly from the definition. Take
into account
> that F(X) C A (You don't need to take into account that
F(X)=A.)
I know the definition of retract. And the definition of an
identification map. And I want to prove that a retract is an
identification map.
>>2) Use: If a map f is a continuous open surjection, then it
is an
>> identification map. Continuous surjection is given, but I
can't
>> show that r is necessarily open, especially for open subsets
>> of X that lie partly in A and partly in X-A.
> Why do you care whether r is open?
Because I want to show that r is an identification map. I know
the
theorem f is a continuous open surjection. Then f is an
identification map.
--
Ren.8e Meyer
Student of Physics & Mathematics
Zhejiang University, Hangzhou, China
Subject: build array given eigenvalues
Given a vector V1, n= length(V1), and a real a>1,
i need to build an array M (nxn)
- det(M)>0
- M==M'
- eigenvalues of M are a, 1,1,.... 1
- eigenvector of M are V1, V2, ... Vn;
-- Vs' * Vt = 0 if s~= t ; Vs' * Vt = 1 if s=t
Could anyone please let me know how can I solve this problem?
Any suggestion is greatly welcome.
thanks
Subject: Re: build array given eigenvalues
>Given a vector V1, n= length(V1), and a real a>1,
I assume you mean that V1 is in R^n. In mathematics the length
of a
vector in R^n is usually taken to mean its norm, i.e. sqrt(V1'
V1)
say below, V1' V1 = 1.
>i need to build an array M (nxn)
>- det(M)>0
>- M==M'
>- eigenvalues of M are a, 1,1,.... 1
>- eigenvector of M are V1, V2, ... Vn;
>-- Vs' * Vt = 0 if s~= t ; Vs' * Vt = 1 if s=t
Look up Gram-Schmidt. Once you have the orthonormal basis
V1, V2, ..., Vn, take M = a V1 V1' + sum_{j=2}^n Vj Vj'.
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: build array given eigenvalues
Thanks for your help!
>>Given a vector V1, n= length(V1), and a real a>1,
>I assume you mean that V1 is in R^n.
Yes, sorry..
Subject: Re: Consider John Nash, math awards
>Remember John Nash, the subject of the movie A Beautiful Mind?
>Famous mathematician, now, who gets a major prize as he won a
Nobel
>prize, but remember, it was in Economics.
>Out of curiousity I did a search on the Internet to see if
he'd won
>any awards from *math* society, and found not one.
>I'd be interested to hear if anyone can track down any awards
that
>Nash's fellow mathematicians gave him, as I haven't found
any, and I
>don't think they've given him any because math society is
weird.
>Now consider Andrew Wiles, who received *multiple* math
awards for
>purportedly proving Fermat's Last Theorem, and he didn't even
get what
>used to be *the* big math award--the Field's Medal.
>If you've never heard of it, that's ok. Mathematicians have a
society
>that is strangely separate from the rest of the world, which
seems to
>follow its own rules, like how it has to my knowledge *still*
not
>given John Nash a single award.
>Wiles actually made several hundred thousand dollars just
from awards,
>but was too old to get the Field's Medal which is awarded for
>significant work done I think before the age of 35.
>John Nash got an Economics Nobel, which is more presitigious
than the
>math awards anyway, but I think it telling that
mathematicians,
>snubbed him.
>James Harris
************************
Subject: Re: Consider John Nash, math awards
Nevermind!
Charlie
>Remember John Nash, the subject of the movie A Beautiful Mind?
>Famous mathematician, now, who gets a major prize as he won a
Nobel
>prize, but remember, it was in Economics.
>Out of curiousity I did a search on the Internet to see if
he'd won
>any awards from *math* society, and found not one.
>I'd be interested to hear if anyone can track down any awards
that
>Nash's fellow mathematicians gave him, as I haven't found
any, and I
>don't think they've given him any because math society is
weird.
>Now consider Andrew Wiles, who received *multiple* math
awards for
>purportedly proving Fermat's Last Theorem, and he didn't even
get what
>used to be *the* big math award--the Field's Medal.
>If you've never heard of it, that's ok. Mathematicians have a
society
>that is strangely separate from the rest of the world, which
seems to
>follow its own rules, like how it has to my knowledge *still*
not
>given John Nash a single award.
>Wiles actually made several hundred thousand dollars just
from awards,
>but was too old to get the Field's Medal which is awarded for
>significant work done I think before the age of 35.
>John Nash got an Economics Nobel, which is more presitigious
than the
>math awards anyway, but I think it telling that
mathematicians,
>snubbed him.
>James Harris
> ************************
>
Subject: Re: Consider John Nash, math awards
Hi David,
IIRC, in Sylvia Nasser's book, A beautiful mind, she
interviewed
several
mathematicians who knew him and/or his work well. Most of them
said that
he
was seriously considered for the Field's medal, but the board
members felt
that he was so prolific for a young mathematician that they
held off
awarding the medal to him. Assuming he would do more. But,
alas, a year
or
two after the awarding of the Field's medal he began to suffer
his
delusions. I have heard that some of his proofs were quite
astonishing,
but
I am not ready to read them yet. I hope soon.
Charlie R. Johnson
>Remember John Nash, the subject of the movie A Beautiful Mind?
>Famous mathematician, now, who gets a major prize as he won a
Nobel
>prize, but remember, it was in Economics.
>Out of curiousity I did a search on the Internet to see if
he'd won
>any awards from *math* society, and found not one.
>I'd be interested to hear if anyone can track down any awards
that
>Nash's fellow mathematicians gave him, as I haven't found
any, and I
>don't think they've given him any because math society is
weird.
>Now consider Andrew Wiles, who received *multiple* math
awards for
>purportedly proving Fermat's Last Theorem, and he didn't even
get what
>used to be *the* big math award--the Field's Medal.
>If you've never heard of it, that's ok. Mathematicians have a
society
>that is strangely separate from the rest of the world, which
seems to
>follow its own rules, like how it has to my knowledge *still*
not
>given John Nash a single award.
>Wiles actually made several hundred thousand dollars just
from awards,
>but was too old to get the Field's Medal which is awarded for
>significant work done I think before the age of 35.
>John Nash got an Economics Nobel, which is more presitigious
than the
>math awards anyway, but I think it telling that
mathematicians,
>snubbed him.
>James Harris
> ************************
>
Subject: Re: [Set Theory] Class of Ordinals well-ordered ?
Subject: [Set Theory] Class of Ordinals well-ordered ?
>I am assuming ZF (no choice, but regularity), I call a class
any
>well-formed statement R(X) with one-free variable (I am not
able to
>be more formal than that). I define the class of ordinals
Ord(X):
>Ord(X):
>(i) for all a in X , a <X (set inclusion)
>(ii) for all a,b in X, (a=b) or (a in b) or (b in a)
If that were a class, it'd be a constant.
As stated, it's a predicate with X as variable,
best understood as 'X is ordinal'
>I can work out from my textbook that Ord(X) is linearly
ordered by
>set inclusion (denoted <).
It's straight forward to show that if Ord(X), then X itself
is linearly ordered by 'in' or equivalently 'subset'
>I would like to confirm that it is in fact well-ordered,
>that if R(X) is a subclass of Ord(X) [i.e. R(X) =
>Ord(X)] which is not empty [i.e. there is X with R(X)], then
R(X)
>has a smallest element, i.e.
Furthermore any ordinal X is well ordered
as a direct consequence of regularity.
>I hope I haven't said anything silly.
Your notation needs complete overhaul.
>So is it true that Ord(X) is well-ordered by inclusion
>and that you don't require choice to show it?
To show that any set of ordinals, not just any subset of an
ordinal X, has
a least element by 'in' or 'subset' order requires more than
the
definition ord(X). Do you have this theorem?
If A,B are well-ordered sets, then only one follows:
(i) A simeq B
(ii) exists b in B [A simeq O(b)]
(iii) exists a in A [O(a) simeq B]
simeq is order isomorphic to
----
Subject: Re: [Set Theory] Class of Ordinals well-ordered ?
<bqikoe$a9$1@mozo.cc.purdue.edu
Subject: Re: [Set Theory] Class of Ordinals well-ordered ?
>If R is nonempty, then consider the intersection of all the
members
>of R. There are several things to show, but all are fairly
easy:
I'll presume R a set.
> (i) The intersection is well defined.
> (ii) The intersection is a set.
Results of ZF.
> (iii) The intersection is an ordinal.
Straight forward
(iv) The intersection is a member of R.
Whoops. Can I use R is well ordered? That with (v) makes (iv)
easy.
What's your proof for (iv)?
(v) The intersection is a lower bound for the members of R.
Basic set theory.
>I don't see where either regularity or choice comes into play
here.
>Ordinals are transitive sets with the property of being
well-ordered
>by the set membership relation. This is a matter of
definition, not
>of regularity. There may be other sets (non-ordinals) that
violate
>the regularity axiom, but ordinals are ordinals whether you
have
>regularity or not.
If R is bounded, then regularity not needed as you define
ordinal as
well ordered set.
Now if R is a set, then Union R is ordinal bound for R.
However, for union of ordinals to be ordinal, regularity is
used.
If R is unbounded, then pick any r in R and consider first
element of
{ s in R | s <= r }. Will that do?
Futhermore if R(x) is a predicate about ordinals, to find the
least
ordinal s with R(s), then I can do the same with { s <= r |
R(s) } ?
No, regularity is used for manipulating the order of
arbitrarly large
ordinals.
But therein is the essence of ZF, set theory within bounds.
NBG allows the unspeakably large while requiring ordinary
discourse
to be somewhat limited. It's like an exclusive club open only
by referal.
To be a member of our club, you have to be a member of some
club.
Much like insurance agencies, to get insurance, they always
demand who
your previous insurer was.
----
Subject: Re: [Set Theory] Class of Ordinals well-ordered ?
> Subject: Re: [Set Theory] Class of Ordinals well-ordered ?
>If R is nonempty, then consider the intersection of all the
members
>of R. There are several things to show, but all are fairly
easy:
> I'll presume R a set.
> (i) The intersection is well defined.
> (ii) The intersection is a set.
> Results of ZF.
> (iii) The intersection is an ordinal.
> Straight forward
> (iv) The intersection is a member of R.
> Whoops. Can I use R is well ordered? That with (v) makes
(iv) easy.
> What's your proof for (iv)?
Point (iv) is really out of order. I should have exchanged
(iv) and (v).
First, notice that < for ordinals means is an element of, and
<=
for ordinals means is a subset of. Suppose alpha = intersect R.
Then
by (v) we have alpha <= x for each x in R. Suppose the
inequality is
strict: alpha < x for each x in R. Then it follows that
(alpha+1) <= x
for each x in R, and therefore (alpha+1) is a subset of
intersect R,
contrary to the assumption that alpha = intersect R.
Therefore, we must
have alpha = x for some x in R, Q.E.D.
> (v) The intersection is a lower bound for the members of R.
> Basic set theory.
>I don't see where either regularity or choice comes into play
here.
>Ordinals are transitive sets with the property of being
well-ordered
>by the set membership relation. This is a matter of
definition, not
>of regularity. There may be other sets (non-ordinals) that
violate
>the regularity axiom, but ordinals are ordinals whether you
have
>regularity or not.
> If R is bounded, then regularity not needed as you define
ordinal as
> well ordered set.
> Now if R is a set, then Union R is ordinal bound for R.
> However, for union of ordinals to be ordinal, regularity is
used.
If R is not bounded, then choose x in R and note that the
intersection of
all members of R is identical to the intersection of all
members bounded
by x.
> If R is unbounded, then pick any r in R and consider first
element of
> { s in R | s <= r }. Will that do?
Yes.
> Futhermore if R(x) is a predicate about ordinals, to find
the least
> ordinal s with R(s), then I can do the same with { s <= r |
R(s) } ?
Yes.
> No, regularity is used for manipulating the order of
arbitrarly large
> ordinals.
I don't see where regularity is needed to define the
intersection.
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&
bookid=228
Subject: Re: [Set Theory] Class of Ordinals well-ordered ?
<bqkp63$1pb$1@mozo.cc.purdue.edu
Subject: Re: [Set Theory] Class of Ordinals well-ordered ?
> Subject: Re: [Set Theory] Class of Ordinals well-ordered ?
>>If R is nonempty, then consider the intersection of all the
members
>>of R. There are several things to show, but all are fairly
easy:
> I'll presume R a set.
>> (i) The intersection is well defined.
>> (ii) The intersection is a set.
> Results of ZF.
>> (iii) The intersection is an ordinal.
> Straight forward
>> (iv) The intersection is a lower bound for the members of R.
> Basic set theory.
>> (v) The intersection is a member of R.
>I should have exchanged (iv) and (v).
Change made.
>First, notice that < for ordinals means is an element of, and
><= for ordinals means is a subset of.
For ordinals a,b, a in b iff a proper subset b
>Suppose alpha = intersect R. Then by (iv) we have alpha <= x
for
>each x in R. Suppose the inequality is strict: alpha < x for
each x
>in R. Then it follows that (alpha+1) <= x for each x in R, and
>therefore (alpha+1) is a subset of intersect R, contrary to
the
>assumption that alpha = intersect R. Therefore, we must have
alpha =
>x for some x in R, Q.E.D.
Ok, interesting, well ordering of an ordinal without
regularity.
>>I don't see where either regularity or choice comes into
play here.
>>Ordinals are transitive sets with the property of being
well-ordered
>>by the set membership relation. This is a matter of
definition, not
>>of regularity.
Have you not shown that a transitive set that's only linearly
orderd
by 'in' is an ordinal?
>> Futhermore if R(x) is a predicate about ordinals, to find
the
>> least ordinal s with R(s), then I find least of { s <= r |
R(s) ?
>Yes.
>I don't see where regularity is needed to define the
intersection.
I've checked to see where regularity is used.
It's used to show if eta in ordinal beta, then eta ordinal.
S is transitive when for all x in S, x subset S ?
Ok, I've amended the proof so it doesn't use regularity.
Now without regularity, can you show for ordinals eta,beta
eta in beta ==> eta /= beta ? I'll even settle for a proof of
ordinal eta ==> eta not in eta
without regularity.
If not, then I can not use
For ordinals a,b, a in b or a = b iff a subset b
instead of
For ordinals a,b, a in b iff a proper subset b
Likely I can, but then how do I show eta < S(eta) instead of
eta <= S(eta). Again I need show eta not in eta otherwise
eta / {eta} = eta.
----
Subject: Re: [Set Theory] Class of Ordinals well-ordered ?
>I don't see where either regularity or choice comes into
play here.
>Ordinals are transitive sets with the property of being
well-ordered
>by the set membership relation. This is a matter of
definition, not
>of regularity.
> Have you not shown that a transitive set that's only
linearly orderd
> by 'in' is an ordinal?
I don't think I have shown that.
>> Futhermore if R(x) is a predicate about ordinals, to find
the
>> least ordinal s with R(s), then I find least of { s <= r |
R(s) ?
>Yes.
>I don't see where regularity is needed to define the
intersection.
> I've checked to see where regularity is used.
> It's used to show if eta in ordinal beta, then eta ordinal.
See my other post. We are using different definitions.
> S is transitive when for all x in S, x subset S ?
> Ok, I've amended the proof so it doesn't use regularity.
> Now without regularity, can you show for ordinals eta,beta
> eta in beta ==> eta /= beta ? I'll even settle for a proof of
> ordinal eta ==> eta not in eta
> without regularity.
That's a consequence of the fact that each ordinal is well
ordered by set
membership.
> If not, then I can not use
> For ordinals a,b, a in b or a = b iff a subset b
> instead of
> For ordinals a,b, a in b iff a proper subset b
Hypothesis violated.
> Likely I can, but then how do I show eta < S(eta) instead of
> eta <= S(eta). Again I need show eta not in eta otherwise
> eta / {eta} = eta.
I take it you mean S(eta) = successor of eta = eta / {eta}.
But this
shows that eta in S(eta), and that's what < means.
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&
bookid=228
Subject: Re: [Set Theory] Class of Ordinals well-ordered ?
> I don't see where regularity is needed to define the
intersection.
With regularity, we can give a slightly simpler definition of
the von
Neumann ordinals (a transitive set every member of which is
transitive) but nothing in the theory of von Neumann ordinals
depends
on regularity.
Subject: Re: [Set Theory] Class of Ordinals well-ordered ?
<bqkp63$1pb$1@mozo.cc.purdue.edu>
<vcbk75et3c2.fsf@beta13.sm.luth.se
Subject: Re: [Set Theory] Class of Ordinals well-ordered ?
>> I don't see where regularity is needed to define the
intersection.
>With regularity, we can give a slightly simpler definition of
the
>von Neumann ordinals (a transitive set every member of which
is
>transitive) but nothing in the theory of von Neumann ordinals
>depends on regularity.
A transitive when for all x in A, x subset A?
A ordinal when A transitive and for all x in A, x transitive.
Does x,y in A ==> x in y or x = y or y in x ?
----
Subject: Re: [Set Theory] Class of Ordinals well-ordered ?
> Subject: Re: [Set Theory] Class of Ordinals well-ordered ?
>> I don't see where regularity is needed to define the
intersection.
>With regularity, we can give a slightly simpler definition of
the
>von Neumann ordinals (a transitive set every member of which
is
>transitive) but nothing in the theory of von Neumann ordinals
>depends on regularity.
> A transitive when for all x in A, x subset A?
> A ordinal when A transitive and for all x in A, x transitive.
No, look again. That's not the definition I have been using in
this
thread. An ordinal is a transitive set that is well ordered by
the set
membership relation.
> Does x,y in A ==> x in y or x = y or y in x ?
We are given that A is well ordered by set membership and that
{x,y} is a
nonempty subset of A. Therefore {x,y} has a least element with
respect
to set membership.
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&
bookid=228
Subject: Re: Suggestions for Applied Maths
>>I consider this quite unfortunate; I actually started my
degree doing
>>maths with physics, but Cambridge doesn't allow that
option past the
>>first year so I had to choose between maths and physics;
as I'm
>>interested in both pure maths and science, and the
cambridge maths
>>degree has a reasonably large applied maths component, I
chose to do
maths.
>Should Cambridge be this shortsighted? I mean modern
physics in every
>part is a great deal of mathematics, especially geometry,
functional
>calculus and topology.
>Rene.
>>I agree completely. Unfortunately the Cambridge
undergraduate course
>>isn't really all it's cracked up to be. It has a lot of
points in its
>>favour of course; some of it is genuinely very good, but it
has some
>>fairly major problems as well.
>>In the Maths with Physics case, there is a lot of maths
available to do
>>in the Natural Sciences tripos, but done from an applied and
not very
>>rigorous at all perspective (or at least that's my
impression of it).
>>Pure maths is fairly lacking in it as far as I know. So it's
not like
>>choosing to do physics cuts you off entirely from the maths,
just the
>>pure maths.
>>I believe there have at various points been attempts to
allow a joint
>>maths with physics degree, but it's never gone through.
>>David
>>(E-mail address spam-blocked in the obvious way).
> You may do worse than looking into the area of fluid
mechanics. There
> are interesting problems people are working on to do with the
> singularities that form when droplets form and leave the
bulk of the
> fluid. There are also interesting problems in the area of
> viscoelasticity. You may have to get into numerical analysis
and
> computer programming to follow this up. Problems in fluid
mechanics
> may also have industrial relevance. The rheology of grease,
paint,
> suspensions, polymers, etc flowing through pipes etc may not
be
> particularly exciting but is quite challenging
mathematically and of
> relevance to many industries.
Eep. But fluid mechanics is all scary and horrible. :)
Nah, just kidding. I was rather put off by the subject, as I
didn't find
the maths course on it very good, but I'm willing to give it
another
try. It actually looks quite interesting, given a good
treatment of the
subject.
My computer programming is passable, and should improve a lot
as I work
on the computer projects over the holiday. Numerical analysis
less so,
as I couldn't handle last year's lecturer (I know, I know;
this is a
common theme. I don't respond well to bad teaching, and it's
another
course which is universally agreed to be badly lectured) and
have yet to
get around to properly learning the subject myself. It's on my
to do list.
> I did Natural Sciences at Cambridge, graduating in 1984, and
chose
> Physics & Theoretical Physics. There was a lot of maths in
the course
> at the time, although it wasn't taught in a rigourous
fashion at all
> (more in the spirit of the book Mathematical Methods of
Physics by
> Matthews and Walker - a good book if you can get hold of a
copy).
Contrary to what I may have implied earlier, I don't mind the
maths in
the science itself being done non-rigorously. I want to have
the
rigorous maths to fall back on if neccesary, but I'm quite
happy to skip
details and hand-wave when doing science. The problem I have
is when all
worthwile science content has been cut out *and* the maths is
done badly
and non-rigorously, because by that point all interest and use
has been
removed from the material.
Do you have any suggestions for good books on the subject?
> which is online. The link is:
I've read so far looks very interesting and I will be sure to
give it a
look in greater detail later.
> Hope this helps
> Ian Taylor
Thanks very much,
David
(E-mail address spam-blocked in the obvious way)
Subject: Re: electrons cling more tightly as universe expands?
>> I like Richards approach but disagree with
>> his conclusion. I do agree with Richard, it
>> is a fact that
>> INFO = frequency * duration = invariant.
>> where Richard calls INFO,. A cycle is an
>> absolute event and the information...
>> But the duration is expressed by (N = cycles)
>> duration = N * wavelength, then
>Wrong
>time interval source = N / frequency
>frequency = c/wavelength
>Thus
>time interval source = N * wavelength/ c
>> INFO = N*c, (c=freq*wavelength).
>INFO = N
> Ok, thats fine, INFO and N are invariant.
>> Because the wavelength increases there is
>> no loss of information carried by the photon.
>True, but how are you going to shift the wavelength?
> Excatly the same way a photon emitted
> at the surface of the sun is red-shifted by
> gravitation as it moves up.
You are suffering here from a false impression. In the case of
gravitational redshift the photon at the surface is emitted at
the lower
frequency, it never changes in flight.
>Each end of the
>wave represents the information of the beginning of two
separate
>objective events, e.g. cycles, both of these information
'bits' is
> Richard seems to be adopting a photon *model*
> that carries 2 bits of information. I thinks it's best
> to trust the invariants we've agreed upon,
> c, N = INFO, and of course
> c =wavelength*frequency, then these must always
> remain reciprical analogies in any model.
>propagating at the same speed wrt the observer, and thus their
>instantaneous displacement in space wrt each other is fixed
wrt the
>observer from the time of its emission to its absorption, i.e.
>throughout its entire trip. The only way to change a
wavelength in
>flight is to change the speed of the wave, OTOH, this will
not change
>the frequency. This is simple Jr. High level optics.
> (Simple for you maybe :-)
> This question remains : No information is lost
> when a photon is reflected from a receding mirror.
> A receding mirror will red-shift the photon on
> reflection (doppler effect) but no INFO is lost.
In order to provide a workable analogy you would need two
partially
silvered mirrors thus reflecting the photon back again along
its
original path. OTOH, this will not change the photon, it will
instead
generate new photons, which again are shifted due to
'relative' motion
of source and detector. If OTOH you could show that photons do
in fact
take a step back for every two steps forward, and that space
itself is
expanding, then your analogy would provide for the observed
red shift.
OTOH this requires space expansion, and thus your tired light
theory
becomes equivalent to the expanding universe explanation. Your
version
is redundant in that if the source is receding, then you don't
need the
oscillatory reflected motion of the photon.
> N and c remain invariant but frequency and
> wavelength are altered.
>> As a photon propagates across the empty voids
>> of inter-galatic space, it is being deflected by all
>> the matter that gravitationally influences that
>> location, and gravitational deflection sucks photon
>> momentum, and frequency, but INFO remains
>> invariant.
>The frequency never changes wrt a given inertial frame.
> That's a very good point, and produces a tired
> light analogy. Suppose a Spalding golf ball is struck
> very hard in some distant galaxy, and is moving near c.
> As it moves through intergalatic space it deaccelerates
> imparts acceleration to, such as galaxies. It slowly gives
> up momentum but the information (SPALDING) remains
> invariant., (written on the ball).
> Rather like a bullet moving threw water.
>Richard Perry
> Regards
> Ken S. Tucker
Subject: Re: electrons cling more tightly as universe expands?
> PRESS RELEASE
> Northeastern University
> As Universe Comes Undone, Electrons Cling More Tightly to
Protons
> http://www.spaceref.com/news/viewpr.html?pid=13101
> Northeastern scientists question the fundamental constants
of nature
> BOSTON, Mass. In this topsy-turvy world of changing trends
and stormy
> alliances, two Northeastern University scientists propose an
answer to
why
> even the fundamental constants of nature don't seem constant
anymore. The
> bond between electrons and protons, called the fine
structure constant,
or
> alpha, may not be constant and may have been 200,000 times
weaker about
ten
> billion years ago. This is a recent astronomy finding that
is hotly
debated
> because it departs from the standard model of physics and
may point to
> modifications introduced by string theory -- the modern
Theory of
> Everything which attempts to unify all forces in nature.
> According to Drs. Luis Anchordoqui and Haim Goldberg of the
Department of
> Physics at Northeastern University in Boston, Mass., this
apparent tiny
> change in alpha through the years may mirror the apparent
accelerating
> expansion rate of the Universe, as if electrons and protons
clung ever
more
> tightly together as the Universe began to fly apart. The
scientists
describe
The apparent change in the fine structure constant remains
controversial,
> partly because it stands in contrast to standard field
theory, the basis
of
> all the successes in atomic and nuclear physics, in which
this constant
is
> an unvarying input to all calculations, said Anchordoqui. We
find,
> however, that the apparent change agrees with a variety of
different
types
> of observations.
> Light signals from exceedingly bright and distant galaxies
called quasars
> seem to indicate that the bond between electrons and protons
was weaker
in
> the early universe. Light left these galaxies about 10
billion years ago
and
> thus reflects the state of matter (and the laws of nature)
from that
epoch.
> This apparent change in the fine structure constant has been
observed in
> several independent measurements.
> On Earth, however, studies of a natural nuclear fission
reactor which
> operated in Gabon two billion years ago reveal no change in
the fine
> structure constant, down to an accuracy of one part in ten
million. Thus,
if
> the fine structure constant has changed, it did not do so
evenly through
the
> years. Anchordoqui and Goldberg attempt to reconcile this
discrepancy.
> They propose that the apparent change in the fine structure
constant is
> coupled to quintessence. This is a theory of dark energy in
which a
> mysterious universal repulsive force, once weaker long ago,
now dominates
> over the force of gravity and is causing the universe to fly
apart at an
> ever-expanding rate. Anchordoqui and Goldberg worked with
one particular
> model of quintessence proposed by Drs. Andreas Albrecht and
Constantinos
> Skordis of the University of California, Davis, in 2000.
They found that
> their own theory of the fine structure constant, when viewed
in the
context
> of this quintessence model, provides agreement between the
quasar data
and
> the Gabon data.
> That is, the fine structure constant was measurably weaker
ten billion
years
> ago, but as quintessence assumed dominance about eight
billion years ago,
> the force between electrons and protons became stronger and
more
constant.
> The strength of the electron-proton bond from any matter
created anytime
> within the last several billion years is essentially
indistinguishable.
> The reason for this lies in the peculiar behavior of the
Albrecht-Skordis
> model, in which the quintessence field has all but ceased
its variation
> during the present era. The model is also consistent with
landmark data
> collected by the NASA Wilkinson Microwave Anisotropy Probe,
which has
> determined fundamental properties of the universe, such as
its age and
> analyzing the light from even more distant quasars will
reveal a steady
> decrease in electron-proton binding strength.
> Also, they said their theory could be tested soon with just
a ten-fold
> improvement in sensitivity in measuring the acceleration of
different
> objects in free fall. This is because a variation in the
fine structure
> constant would imply a variation of this type of
acceleration as the
> chemical makeup varied, a violation in the equivalence
principle
introduced
> by Albert Einstein in his general theory of relativity. Two
proposed
> space-based mission will have this sensitivity: the
MICROSCOPE mission
from
> France's Centre National d'Etudes Spatiales, expected to fly
in 2005; and
a
> NASA-ESA mission called STEP, Satellite Test of the
Equivalence
Principle.
We may be able to test this model of a 'changing' fine
structure
constant
> within a couple of years with instruments on satellites,
said Goldberg.
Or, we could continue observing alpha in lab experiments for
another
> several billion years to see changes on the order of the
quasar values.
I'm
> counting on the satellites. For more information, refer to
Anchordoqui
and
> Driven by Quintessence, available at
http://arXiv.org/abs/hep-ph/0306084.
Has any of you listed all assumptions on which above findings
are based on
?
I think that you could find some errors from there, for example
I would like to ask on what measurements or assumptions the
used different quintessence potentials V are based on etc. ?
Hannu
Subject: Re: drawing math functions as postscript files
>> What's wrong with gnuplot? Type help data to learn how to
>> do what you want in gnuplot. For analytic functions, just
do a
>plot sin(x) or something like that. Last but not least, you
>> can generate EPS from gnuplot.
Gernot> Just to plot sin(x) requires:
Gernot> Definition of the origin e.g. in mm. Definition of the
Gernot> axes lengths in mm. Definition of scaleX and scaleY.
Gernot> Definition of xmin and xmax at axis minimum and
maximum.
Gernot> Definition of ymin and ymax at axis minimum and
maximum.
Gernot> Definition of dx for the function (number of steps).
Gernot> Definition of axes line widths and colors. Definition
of
Gernot> graph line width and color. Definition of tic marks by
Gernot> divisions and heights/widths. Definition of tic mark
line
Gernot> widths. Definition of tic mark colors. Definition of
Gernot> graph visible bounding box (coordinate system frame).
Gernot> Definition of graph visible bounding box line width.
Gernot> Definition of graph visible bounding box color.
Gernot> Definition of graph visible bounding box miter mode.
Gernot> Writing numbers at tic marks with fixed format (not
Gernot> overlapping). Writing numbers for the y-axis with
Gernot> appropriate x-offsets depending on the fixed format.
Gernot> Definition eventuallly graph line-gap-line-gap pattern.
Gernot> Definition of end caps for axes. Definition of end caps
Gernot> for graph. Definition of clipping area for graph. Maybe
Gernot> thats not all ...
Gernot> GnuPlot handles this automatically ?
Have you tried?
Of course, no program or machine can be intelligent enough to
read
your mind. If you don't like the defaults, you have to
fine-tune it.
I can't see any alternatives, whether you use 'gnuplot',
METAPOST,
hand-code Postscript or however.
Gernot> I prefer direct programming by PS.
You could of course reimplement everything of gnuplot in
Postscript.
The latter is a complete programming language. I was talking
about
using existing wheels to get the job done quickly.
Gernot> There are still some problems left. Mainly the rounding
Gernot> which is handled differently by different interpre-
Gernot> ters.
So, emulate all real number operations with fixed-point
numbers. You
can even implement infinite-precision numbers (a la GNU 'bc').
Postscript is a language general enough to allow you to
implement
them. If you like, you could even implement surreal numbers!
GernotJust do a plot sin(x) is wishful thinking.
Have you tried it already? I was quite happy to see that
gnuplot can
do most things satisfactorily for me.
Gernot> Illustrations & Examples are much appreciated.
Download and install gnuplot (if your X11-enabled unix system
hasn't
already got one) and:
$ gnuplot
gnuplot> plot sin(x)
And see the results. For EPS:
gnuplto> set terminal postscript eps
gnuplot> set output sin.eps
gnuplot> replot
The built-in help system of gnuplot is excellent and clearly
documents
almost everything.
--
Lee Sau Dan
.be.8c.be[Eth][CapitalI
DoubleDot](Big5) ~{@nJX6X~}(HZ)
E-mail: danlee@informatik.uni-freiburg.de
Home page: http://www.informatik.uni-freiburg.de/~danlee
Subject: Re: drawing math functions as postscript files
Have a look at GLE (Graphics Layout Engine), it may do what
you want.
http://glx.sourceforge.net/
Subject: Re: relative topology in R^2
>Can someone help for this;
>Let B is the class of
>{(x,y)| a <= x < b, c <= y < d}
>in R^2, then B is the base for a topology T on R^2.
1) Show that the relative topology T_A on the line
> A={(x,y)| x + y = 0 } (ie. x = -y )
> is a discrete topology.
2) Show that the relative topology T_B on the line
> B={(x,y)| x = y }
> is not discrete.
 Sure. First of all, what does the phrase 'relative topology'
> mean? What will be a base for the relative topologies on
> those sets? After considering that question, you may want to
look
> at intersections of your sets A and B with elements of your
> original base. Drawing a picture may help.
> What does it mean to be the discrete topology? What are the
> smallest sets you can get to be open in A? What type of sets
> are basis sets for B?
> --Dan Grubb
I can see that I have to show there is open singleton set in
Base/A.
But Base/A = {(x,-x) | a<=x<b, -b<-x<=-a}
I don't know how to extract singleton set from this set.
Is (a,-a) a singleton set we are looking for?
Subject: Re: relative topology in R^2
Originator: grubb@lola
>> Sure. First of all, what does the phrase 'relative topology'
>> mean? What will be a base for the relative topologies on
>> those sets? After considering that question, you may want
to look
>> at intersections of your sets A and B with elements of your
>> original base. Drawing a picture may help.
> What does it mean to be the discrete topology? What are the
>> smallest sets you can get to be open in A? What type of sets
>> are basis sets for B?
> --Dan Grubb
>I can see that I have to show there is open singleton set in
Base/A.
>But Base/A = {(x,-x) | a<=x<b, -b<-x<=-a}
Not quite. The base elements are {(x,y):a<=x<b,c<=y<d}. If you
have
a point (r,-r) in A, can you find a basis element that only
intersects A in that one point? Hint: Draw a picture.
>I don't know how to extract singleton set from this set.
>Is (a,-a) a singleton set we are looking for?
--Dan Grubb
Subject: Re: relative topology in R^2
>Let B is the class of
>{(x,y)| a <= x < b, c <= y < d}
>in R^2, then B is the base for a topology T on R^2.
>1) Show that the relative topology T_A on the line
> A={(x,y)| x + y = 0 } (ie. x = -y )
> is a discrete topology.
{ (x,-x) } = ([x,x+1) x [-x,-x+1)) / A is open singleton in A
>2) Show that the relative topology T_B on the line
> B={(x,y)| x = y }
> is not discrete.
A base set for B is
B / {(x,y)| a <= x < b, c <= y < d}
= { (x,x) | max a,c <= x < min b,d }
So B, the diagonal of the Sorgenfrey square,
is homeomorphic to the Sorgenfrey line.
----
Subject: Difference equations, calculus, prime numbers
I've been talking a lot about a partial difference equation I
found
that sums over a certain range to give the count of prime
numbers, but
it occurs to me that your calculus may be rusty, so you might
be hazy
on some of what I mean.
Remember the following?
As dx approaches 0, f'(x) = (f(x+dx) - f(x))/dx
Now the idea is that you start with some dx and shrink it
towards 0,
so let's say you start with dx=1, then you have
df(x) = f(x+1) - f(x)
which is a difference equation.
It's kind of obvious why it's called a *difference* equation as
well!!!
Now then, let's say you were given
df(x) = f(x+1) - f(x)
then it's not complicated how you get back to the differential
equation as you assume that the 1 is from the delta. So you
have
df(x) = f(x+dx) - f(x), and dividing by dx, you have
df(x)/dx = (f(x+dx) - f(x))/dx,
and as dx approaches 0 in the limit you have
f'(x) = (f(x+dx) - f(x))/dx,
which completes the circle.
Now then, what I found is
dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) -
p(y-1,
sqrt(y-1))],
where p(x, y) = floor(x) - S(x, y) - 1.
Taking the first steps in approaching the differential, I have
dS(x,y)/dy = [p(x/y, y-dy) - p(y-dy, sqrt(y-dy))][ p(y,
sqrt(y)) -
p(y-dy, sqrt(y-dy))]/dy,
where p(x, y) = x - S(x, y) + C.
In the limit as dy approaches 0, I have
S'_y(x,y)= [p(x/y, y) - p(y, sqrt(y))] p'_x(y, sqrt(y))
and looking at
p(x, y) = x - S(x, y) + C,
I can differentiate with respect to y to get
p'_y(x,y) = - S'_y(x,y),
and making the substitution gives
p'_y(x,y)= -[p(x/y, y) - p(y, sqrt(y))] p'_x(y, sqrt(y)),
which is the partial differential equation.
There's only one step in that process which might be
confusing, which
is how I went from
[ p(y, sqrt(y)) - p(y-dy, sqrt(y-dy))]/dy to p'_x(y, sqrt(y)),
as dy approached 0, and the answer has to do with the
directional
derivative as dy approaches 0, which has a vector that
approaches that
of p'_x, and equals it in the limit (I think).
So what I just showed you is something that no one in recorded
human
history has managed to do with any discovery related to prime
numbers
before me.
It gives a *why* for the connection between the distribution
of prime
numbers and continuous functions and opens up an unimaginably
huge
area for future research.
James Harris
My math discoveries, found for profit
http://mathforprofit.blogspot.com/
Subject: Re: Difference equations, calculus, prime numbers
> I've been talking a lot about a partial difference equation
I found
> that sums over a certain range to give the count of prime
numbers, but
> it occurs to me that your calculus may be rusty, so you
might be hazy
> on some of what I mean.
Prime Counting Function
http://mathworld.wolfram.com/PrimeCountingFunction.html
Prime Difference Function
http://mathworld.wolfram.com/PrimeDifferenceFunction.html
Crank Information
http://www.crank.net/harris.html
http://www.crank.net/usenet.html
http://www.google.com/search?q=harris+site%3Awww.crank.net
http://www.google.com/search?q=%22james+harris%22+site%
3Ausers.pandora.be
Subject: Re: Difference equations, calculus, prime numbers
<snip
> In the limit as dy approaches 0, I have
> S'_y(x,y)= [p(x/y, y) - p(y, sqrt(y))] p'_x(y, sqrt(y))
> and looking at
> p(x, y) = x - S(x, y) + C,
> I can differentiate with respect to y to get
> p'_y(x,y) = - S'_y(x,y),
> and making the substitution gives
> p'_y(x,y)= -[p(x/y, y) - p(y, sqrt(y))] p'_x(y, sqrt(y)),
> which is the partial differential equation.
Are you concerned by the fact that p has rather a lot of points
of discontinuity? Perhaps you should be. Derivatives of
discontinuous functions can be (ahem) somewhat ill-behaved,
as I'm sure you're aware.
<snip
Rick
Subject: Re: Difference equations, calculus, prime numbers
> <snip
>> In the limit as dy approaches 0, I have
>> S'_y(x,y)= [p(x/y, y) - p(y, sqrt(y))] p'_x(y, sqrt(y))
>> and looking at
>> p(x, y) = x - S(x, y) + C,
>> I can differentiate with respect to y to get
>> p'_y(x,y) = - S'_y(x,y),
>> and making the substitution gives
>> p'_y(x,y)= -[p(x/y, y) - p(y, sqrt(y))] p'_x(y, sqrt(y)),
>> which is the partial differential equation.
> Are you concerned by the fact that p has rather a lot of
points
> of discontinuity? Perhaps you should be. Derivatives of
> discontinuous functions can be (ahem) somewhat ill-behaved,
> as I'm sure you're aware.
Here's a bit more. First, as has been pointed out already,
using a difference equation as a basis for passing to a
differential equation is in general wrong. Second, in the
case you presented, you should realize that p(x, y) is
just a 2-variable step function. In fact, if you look at
p(y, sqrt(y)) - p(y - dy, sqrt(y - dy))
and perform your derivative calculation, you'll find that
the derivative at any y value is either 0 or undefined.
Not, IMO, a very fruitful basis for further work.
By the way, your use of p'_x(y, sqrt(y)) above is completely
wrong. Review directional derivatives and you'll see why.
Rick
Subject: Re: Difference equations, calculus, prime numbers
> I've been talking a lot about a partial difference equation
I found
> that sums over a certain range to give the count of prime
numbers,
Idiot. It has been variously shown that you couldn't pull the
lint
out of your own bellybutton without causing bleeding. You know
near
nothing about mathematics, and pervert it to suit your
psychosis.
Hey stooopid loud troll James Harris, put up or shut up,
http://www.rsasecurity.com/rsalabs/challenges/factoring/
faq.html
http://www.rsasecurity.com/rsalabs/challenges/factoring/
numbers.html
http://www.crank.net/harris.html
It's not every braying jackass that gets a whole page at
crank.net
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
Quis custodiet ipsos custodes? The Net!
Subject: Re: Difference equations, calculus, prime numbers
<snip calculus
> There's only one step in that process which might be
confusing, which
> is how I went from
> [ p(y, sqrt(y)) - p(y-dy, sqrt(y-dy))]/dy to p'_x(y,
sqrt(y)),
> as dy approached 0, and the answer has to do with the
directional
> derivative as dy approaches 0, which has a vector that
approaches that
> of p'_x, and equals it in the limit (I think).
suppose p(x,y) = y^2 so that the partial derivative wrt x
vanishes.
and consider, changing dummy variables to avoid notational
confusion,
lim {p(u,sqrt(u)) - p(u-e,sqrt(u-e))}/e
e ---> o
= lim {u - (u -e)}/e = 1.
e--->0
bugger, eh?
> So what I just showed you is something that no one in
recorded human
> history has managed to do with any discovery related to
prime numbers
> before me.
> It gives a *why* for the connection between the distribution
of prime
> numbers and continuous functions and opens up an
unimaginably huge
> area for future research.
nope, still no why going on here that wasn't already understood
> James Harris
My math discoveries, found for profit
> http://mathforprofit.blogspot.com/
Subject: Re: Difference equations, calculus, prime numbers
>I've been talking a lot about a partial difference equation I
found
>that sums over a certain range to give the count of prime
numbers, but
>it occurs to me that your calculus may be rusty, so you might
be hazy
>on some of what I mean.
>Remember the following?
>As dx approaches 0, f'(x) = (f(x+dx) - f(x))/dx
>Now the idea is that you start with some dx and shrink it
towards 0,
>so let's say you start with dx=1, then you have
>df(x) = f(x+1) - f(x)
>which is a difference equation.
>It's kind of obvious why it's called a *difference* equation
as
>well!!!
>Now then, let's say you were given
>df(x) = f(x+1) - f(x)
>then it's not complicated how you get back to the differential
>equation as you assume that the 1 is from the delta. So you
have
>df(x) = f(x+dx) - f(x), and dividing by dx, you have
>df(x)/dx = (f(x+dx) - f(x))/dx,
>and as dx approaches 0 in the limit you have
>f'(x) = (f(x+dx) - f(x))/dx,
>which completes the circle.
>Now then, what I found is
>dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) -
p(y-1,
>sqrt(y-1))],
>where p(x, y) = floor(x) - S(x, y) - 1.
>Taking the first steps in approaching the differential, I have
>dS(x,y)/dy = [p(x/y, y-dy) - p(y-dy, sqrt(y-dy))][ p(y,
sqrt(y)) -
>p(y-dy, sqrt(y-dy))]/dy,
>where p(x, y) = x - S(x, y) + C.
>In the limit as dy approaches 0, I have
>S'_y(x,y)= [p(x/y, y) - p(y, sqrt(y))] p'_x(y, sqrt(y))
>and looking at
>p(x, y) = x - S(x, y) + C,
>I can differentiate with respect to y to get
>p'_y(x,y) = - S'_y(x,y),
>and making the substitution gives
>p'_y(x,y)= -[p(x/y, y) - p(y, sqrt(y))] p'_x(y, sqrt(y)),
>which is the partial differential equation.
>There's only one step in that process which might be
confusing, which
>is how I went from
>[ p(y, sqrt(y)) - p(y-dy, sqrt(y-dy))]/dy to p'_x(y, sqrt(y)),
>as dy approached 0, and the answer has to do with the
directional
>derivative as dy approaches 0, which has a vector that
approaches that
>of p'_x, and equals it in the limit (I think).
>So what I just showed you is something that no one in
recorded human
>history has managed to do with any discovery related to prime
numbers
>before me.
>It gives a *why* for the connection between the distribution
of prime
>numbers and continuous functions
No, it does nothing of the sort. You've explained where your
pde
came from. You have not given _any_ reason to think that the
solution has anything to do with pi(x).
And you've continually ignored an explanation for why it
seems most likely that the pde has nothing whatever to do
with pi(x). Consider the precisely analogous process of
converting a difference equation to a differential equation,
in a much simpler case where we can see what the answer
is:
Start with the difference equation
f(x+1) - f(x) = f(x), f(0) = 1.
The solution is f(x) = 2^x.
Now consider the corresponding differential equation:
f'(x) = f(x), f(0) = 1.
The solution is f(x) = e^x.
So. If you've explained what you claim to have explained
then _I_ have just explained why e^x is an approximation
to 2^x.
Guffaw.
>and opens up an unimaginably huge
>area for future research.
>James Harris
>My math discoveries, found for profit
>http://mathforprofit.blogspot.com/
************************
Subject: Re: Difference equations, calculus, prime numbers
>It gives a *why* for the connection between the distribution
of prime
>numbers and continuous functions
> No, it does nothing of the sort. You've explained where your
pde
> came from. You have not given _any_ reason to think that the
> solution has anything to do with pi(x).
> And you've continually ignored an explanation for why it
> seems most likely that the pde has nothing whatever to do
> with pi(x). Consider the precisely analogous process of
> converting a difference equation to a differential equation,
> in a much simpler case where we can see what the answer
> is:
> Start with the difference equation
> f(x+1) - f(x) = f(x), f(0) = 1.
> The solution is f(x) = 2^x.
> Now consider the corresponding differential equation:
> f'(x) = f(x), f(0) = 1.
> The solution is f(x) = e^x.
> So. If you've explained what you claim to have explained
> then _I_ have just explained why e^x is an approximation
> to 2^x.
> Guffaw.
;-))
Dirk Vdm
Subject: Re: Difference equations, calculus, prime numbers
<SNIP
> Guffaw.
LOL! Well said David!
Subject: Re: Difference equations, calculus, prime numbers
> I've been talking a lot
Yes, and you are approaching the status of a newsgroup vandal.
Your
continued posting of repetitive material and diatribes has
elevated your
status from that of a simple troll or crank to that of a
spammer, to put
it kindly. If you continue your tirades and constant reposts,
you are
certainly using up bandwidth and degrading the SNR.
> about a partial difference equation I found
> that sums over a certain range to give the count of prime
numbers, but
> it occurs to me that your calculus may be rusty, so you
might be hazy
> on some of what I mean.
Your calculus is not only rusty and hazy, but completely
wrong. (See
below.)
> Remember the following?
> As dx approaches 0, f'(x) = (f(x+dx) - f(x))/dx
> Now the idea is that you start with some dx and shrink it
towards 0,
> so let's say you start with dx=1, then you have
> df(x) = f(x+1) - f(x)
> which is a difference equation.
> It's kind of obvious why it's called a *difference* equation
as
> well!!!
> Now then, let's say you were given
> df(x) = f(x+1) - f(x)
> then it's not complicated how you get back to the
differential
> equation as you assume that the 1 is from the delta. So you
have
> df(x) = f(x+dx) - f(x), and dividing by dx, you have
> df(x)/dx = (f(x+dx) - f(x))/dx,
> and as dx approaches 0 in the limit you have
> f'(x) = (f(x+dx) - f(x))/dx,
> which completes the circle.
> Now then, what I found is
> dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) -
p(y-1,
> sqrt(y-1))],
> where p(x, y) = floor(x) - S(x, y) - 1.
> Taking the first steps in approaching the differential, I
have
> dS(x,y)/dy = [p(x/y, y-dy) - p(y-dy, sqrt(y-dy))][ p(y,
sqrt(y)) -
> p(y-dy, sqrt(y-dy))]/dy,
You are simply replaced unity with a differential. This time,
however, in
a flash of brilliance you realized you must divide the right
side by 'dy',
which you didn't do in previous posts. But your equation is
still wrong!
You have taken a difference equation which solves a given
problem, and
converted it to a finite difference equation which may have no
relevance
whatsover wrt the original problem. Your results will be
different for
every value of 'dy' and will only be correct when 'dy = 1.
> where p(x, y) = x - S(x, y) + C.
> In the limit as dy approaches 0, I have
> S'_y(x,y)= [p(x/y, y) - p(y, sqrt(y))] p'_x(y, sqrt(y))
> and looking at
> p(x, y) = x - S(x, y) + C,
> I can differentiate with respect to y to get
> p'_y(x,y) = - S'_y(x,y),
> and making the substitution gives
> p'_y(x,y)= -[p(x/y, y) - p(y, sqrt(y))] p'_x(y, sqrt(y)),
> which is the partial differential equation.
> There's only one step in that process which might be
confusing, which
> is how I went from
> [ p(y, sqrt(y)) - p(y-dy, sqrt(y-dy))]/dy to p'_x(y,
sqrt(y)),
> as dy approached 0, and the answer has to do with the
directional
> derivative as dy approaches 0, which has a vector that
approaches that
> of p'_x, and equals it in the limit (I think).
Hahahaha. What a joke! This is the most spectacular example of
handwaving
humbug I've ever seen. Go ahead and try to produce meaningful
results with
your formulation. You programmed the original difference
equation, didn't
you? Now do this one and post your results.
Hahahaha. I dare you!
> So what I just showed you is something that no one in
recorded human
> history has managed to do with any discovery related to
prime numbers
> before me.
No, you didn't. You haven't shown that your bizarre
formulation actually
has any relevance to the prime counting function. You have
shown *no*
results whatsoever.
> It gives a *why* for the connection between the distribution
of prime
> numbers and continuous functions and opens up an
unimaginably huge
> area for future research.
It does no such thing. Go back to your playpen.
--
There are two things you must never attempt to prove: the
unprovable --
and the obvious.
--
--
http://www.crbond.com
Subject: Re: Difference equations, calculus, prime numbers
>Yes, and you are approaching the status of a newsgroup vandal.
Please remove approaching the status of in the sentence above.
This
is not a limit problem; the function exists and *is* a vandal
at that
point.
Doug
Subject: Re: Difference equations, calculus, prime numbers
>Yes, and you are approaching the status of a newsgroup vandal.
> Please remove approaching the status of in the sentence
above. This
> is not a limit problem; the function exists and *is* a
vandal at that
point.
> Doug
LOL!
The proof goes as follows:
For every epsilon about the point.....
Subject: Re: Difference equations, calculus, prime numbers
> I've been talking a lot
> Yes, and you are approaching the status of a newsgroup
vandal.
He already reached that status years ago.
Subject: Re: Difference equations, calculus, prime numbers
> I've been talking a lot
Yes, you have. Shut up already and adjust your medications.
Subject: Re: is the sci.math board moderated ? how do I
de-post ?
>I do not know if it generates cancel messages that are
accepted by
>other servers, but
I *thought* that messages could be canceled *only* from your
news
server, provided it supports such an operation. I *thought*
nothing
can be done after the posts propagate over the net...
Michele
--
> Comments should say _why_ something is being done.
Oh? My comments always say what _really_ should have happened.
:)
- Tore Aursand on comp.lang.perl.misc
Subject: Re: is the sci.math board moderated ? how do I
de-post ?
>I *thought* that messages could be canceled *only* from your
news
>server, provided it supports such an operation. I *thought*
nothing
>can be done after the posts propagate over the net...
Anybody who has seen the effects of a cancelstorm can tell you
otherwise.
Subject: Re: is the sci.math board moderated ? how do I
de-post ?
X-Cise: tanbanso@iinet.net.au
X-CompuServe-Customer: Yes
X-Coriate: admin@interspeed.co.nz
X-Ecrate: tanandtanlawyers.com
X-Pose: george_cox@btinternet.com
X-Punge: Micro$oft
X-Sanguinate: themvsguy@email.com
X-Terminate: SPA(GIS)
X-Tinguish: Mark Griffith <markgriffith@rocketmail.com
X-Treme: C&C,DWS
>Subject: is the sci.math board moderated ?
It's a news group, not a board. And no, it is not moderated.
>how do I de-post ?
I assume that you mean cancel. Many news readers have a cancel
function, but there is no guaranty that the news servers will
honor
don't know whether it issues cancels, and even if it does they
may not
be honored.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT
Unsolicited bulk E-mail will be subject to legal action. I
reserve
the right to publicly post or ridicule any abusive E-mail.
Reply to domain Patriot dot net user shmuel+news to contact
me. Do
not reply to spamtrap@library.lspace.org
Subject: Re: is the sci.math board moderated ? how do I
de-post ?
<l79psvs8qdam0heucun26m8jl4ok6ikjke@4ax.com
<bqi99i$19oi$1@agate.berkeley.edu
X-Cise: tanbanso@iinet.net.au
X-CompuServe-Customer: Yes
X-Coriate: admin@interspeed.co.nz
X-Ecrate: tanandtanlawyers.com
X-Pose: george_cox@btinternet.com
X-Punge: Micro$oft
X-Sanguinate: themvsguy@email.com
X-Terminate: SPA(GIS)
X-Tinguish: Mark Griffith <markgriffith@rocketmail.com
X-Treme: C&C,DWS
at 02:57 PM, magidin@math.berkeley.edu (Arturo Magidin) said:
>My guess would be cancelling a post. Many newsreaders allow
that
>option,
And some news servers don't honor it, do to forged cancels.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT
Unsolicited bulk E-mail will be subject to legal action. I
reserve
the right to publicly post or ridicule any abusive E-mail.
Reply to domain Patriot dot net user shmuel+news to contact
me. Do
not reply to spamtrap@library.lspace.org
Subject: Re: is the sci.math board moderated ? how do I
de-post ?
> ...
>His newsreader is evidently Google. I know nothing about
posting
>through Google, so I don't know how to cancel posts that
way.
 I do not know if it generates cancel messages that are
accepted by
> other servers, but
> There are many servers around that do not accept all cancel
messages.
> So even if you issue a cancel message, and your service
provider allows
> that, the probability that the message is still around at a
number of
> places is huge. (Some ISP's allow only cancelling by some
well-known
> pronouncing your error. Tough luck if what you did was
illegal. (I once
> posted something to comp.sources.mac, I think, but received
a message
> seconds later by somebody who accused me of plagiarism. I
immediately
> posted a cancel, but had to let it remove by hand at
WU-archive.
Although
> I do not think the accuser had something real, I posted a
completely
> different version ten minutes later... Finding all archives
can be a
> pain. And that was back in 1989 or something like that.)
Supersedes tend to work better than cancels. Supercedes,
however, don't
work at all, which it took some Co$ member quite a while to
fathom, if
I remember my a.r.s lore.
Phil
--
Unpatched IE vulnerability: Click hijacking
Description: Pointing IE mouse events at non-IE/system windows
Reference:
http://safecenter.net/liudieyu/HijackClick/
HijackClick-Content.HTM
Exploit:
http://safecenter.net/liudieyu/HijackClick/HijackClick2-
MyPage.HTM
Subject: Partial difference equation, alternate form
I've been giving the partial difference equation that I
discovered
which sums to count prime numbers for a while in one way, but
for
completeness I figured I should throw down another slightly
different
form.
dS(x,y) = [p(x/y, y-1) - p(y-1, y-1)][ p(y, y) - p(y-1, y-1)],
S(x,1) = 0.
And p(x, y) = floor(x) - S(x, y) - 1, and you get S as the sum
of dS
from dS(x,2) to dS(x,y).
Well, I think that form works as I haven't tested it, but it
*should*
work, so I'll record it here.
James Harris
Subject: Re: Partial difference equation, alternate form
> Well, I think that form works as I haven't tested it, but it
*should*
> work, so I'll record it here.
> James Harris
For over 7 years James has been posting things that he claims
should work.
For the record, how many of them have, James?
Subject: What is this property called?
Apologies for loose terminology. If f:X->Y, what is the name
of the
property of f whereby,
f(a1x1 + a2x2 + a3x3 + ...) = a1 f(x1) + a2 f(x2) + a3 f(x3) +
...
where the a's are real coefficients to form a linear
combination,
and the x's are in X.
Thanks,
Toby.
Subject: Re: What is this property called?
>Apologies for loose terminology. If f:X->Y, what is the name
of the
>property of f whereby,
>f(a1x1 + a2x2 + a3x3 + ...) = a1 f(x1) + a2 f(x2) + a3 f(x3)
+ ...
>where the a's are real coefficients to form a linear
combination,
>and the x's are in X.
Assuming that we're dealing with finite number of a's and x's,
then this
is the vector-space definition of linearity.
If you want an infinite number of a's/x's, then I'm not sure
what it should
be called, but maybe linearish or linearlike. Or just linear if
you
want to be crass. :-)
Doug
Subject: Re: What is this property called?
> Apologies for loose terminology. If f:X->Y, what is the name
of the
> property of f whereby,
> f(a1x1 + a2x2 + a3x3 + ...) = a1 f(x1) + a2 f(x2) + a3 f(x3)
+ ...
> where the a's are real coefficients to form a linear
combination,
> and the x's are in X.
For finite sums that property of f is simply called linearity.
For infinite sums (series) I guess the property could be
called linear convergence compatibility or something ;-)
Dirk Vdm
Subject: Re: Why Good Will Hunting is fantasy
What if I don't know how to write a paper for a math journal?
Isn't that something that is taught to math students in school?
Why can't I just show what I have like the guy on the movie
Good Will
Hunting?
He didn't write a paper, now did he?
> Sounds a lot like the 'creation scientists' tirade against
getting
published
> because 'their research falls outside of accepted beliefs.'
I've worked
in
> science long enough to know the competitive nature of
research and
> publishing. If you have anything of importance, whether it's
proof that
the
> earth is 6000 years old or a new math proof, journal editors
will fight
to
> be the first one to publish. I know why creation scientists
don't get
> published - not sure about your work. Write it up, submit
it, and see
what
> happens.
> D Mitton
> The big deal in Good Will Hunting is that some smart kid
while a
> janitor manages to write out an answer to a difficult
problem, and
> then he gets acknowledgement, and other wonderful things.
> I'm there to tell you that what would have really happened
is that the
> math professor would have been excited, until he found out
it wasn't
> one of his students. Then he would have been angry when he
found out
> it was a janitor, and would have erased the board.
> How do I know?
> Because I made my own discovery of a partial difference
equation that
> can be used to count PRIME NUMBERS, and it turns out that
the big deal
> has to do with that phrase partial difference equation as no
one in
> recorded history has managed such a feat:
> dS(x,y)= [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) -
p(y-1,
> sqrt(y-1))],
> S(x,1) = 0, and p(x, y) = floor(x) - S(x, y) - 1,
> while you get S(x,y) by summing dS from dS(x,2) to dS(x,y).
> Here p(x,sqrt(x)) gives the count of primes up to and
including x.
> That I found a partial difference equation is of major
importance
> because it has a partial differential equation analog, and
even a bit
> of testing reveals that they match each other well, which
may solve an
> over one hundred year old mystery that intrigued the likes
of Gauss
> and Riemann.
> Remember? Riemann was looking for a reason why. He wanted to
find
> out the connection between the prime distribution and
continuous
> functions like li(x), so he did his analysis, but didn't
find the
> connection.
> So I may have succeeded where Riemann, Gauss and others
failed.
> I say may because that's not proven yet, but the questions
raised
by
> my discovery are IMMENSE, and if Goodwill Hunting were at all
> accurate to the way mathematicians behave, they'd be all
over it.
> I'd be cheered instead of derided. Marveled at as a real,
live
Good Will Hunting, versus maligned as an annoying crank.
> You see, there is no Good Will Hunting because
mathematicians won't
> allow it, as they live in a world where non-mathematicians
aren't
> supposed to be capable of making major research finds, so if
one does,
> like I have, mathematicians simply *decide* that it can't be
important
> as a mathematician didn't find it.
> Oh, and after erasing the board, the mathematician would
have probably
> tried to get the janitor fired for interfering with his
class.
> James Harris
My math discoveries, found for profit
> http://mathforprofit.blogspot.com/
Subject: Re: Why Good Will Hunting is fantasy
> What if I don't know how to write a paper for a math journal?
> Isn't that something that is taught to math students in
school?
> Why can't I just show what I have like the guy on the movie
Good Will
Hunting?
> He didn't write a paper, now did he?
Yes he did. He was in his prof's office, and the prof looked
at it, and
said
But have you considered... to which Will replied, It's right.
Trust
me.
Then to show contempt for the prof (played by Skarsgaard) he
burnt the
paper.
This was shortly before the end of the movie, maybe 20-30 mins
from the
end.
GREG
Subject: Re: Why Good Will Hunting is fantasy
> What if I don't know how to write a paper for a math journal?
> Isn't that something that is taught to math students in
school?
> Why can't I just show what I have like the guy on the movie
Good Will
> Hunting?
> He didn't write a paper, now did he?
He showed that he had an understanding of mathematics. You
haven't.
Subject: Re: Why Good Will Hunting is fantasy
> What if I don't know how to write a paper for a math journal?
> Isn't that something that is taught to math students in
school?
Most students of math learn how to write math papers by
reading math
papers. Something that JSH has never tried.
Subject: Re: Why Good Will Hunting is fantasy
<3c65f87.0312030713.1a2f3670@posting.google.com
Discussion, linux)
> Why can't I just show what I have like the guy on the movie
Good
> Will Hunting?
Because (1) that's a movie and (2) you're not Will Hunting.
> He didn't write a paper, now did he?
--
To assert otherwise is to say that mathematical operations and
symbols
mean whatever we want them to mean, varying from context to
context [...] It in essence introduces post-modernism into
mathematics.
--Paul Lutus on absolute notation and mathematical humanism
Subject: Re: Why Good Will Hunting is fantasy
>> Why can't I just show what I have like the guy on the movie
Good
>> Will Hunting?
> Because (1) that's a movie and (2) you're not Will Hunting.
He *does* seem to have Will's obnoxious personality and
obscenity, though.
Now if he only had a bit of Will's math knowledge to go with
it...
--
Wayne Brown (HPCC #1104) | When your tail's in a crack, you
improvise
fwbrown@bellsouth.net | if you're good enough. Otherwise you
give
| your pelt to the trapper.
e^(i*pi) = -1 -- Euler | -- John Myers Myers,
Silverlock
Subject: Re: Why Good Will Hunting is fantasy
> What if I don't know how to write a paper for a math journal?
Ask around how to go about it. And heed the advise.
> Isn't that something that is taught to math students in
school?
In principle, no.
> Why can't I just show what I have like the guy on the movie
Good Will
> Hunting?
Because the way you show it makes it ununderstandable.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam,
nederland,
+31205924131
home: bovenover 215, 1025 jn amsterdam, nederland;
http://www.cwi.nl/~dik/
Subject: Brian Greene's PBS NOVA Elegant Universe
Never has so much tax money been spent on a physics theory (M
-Theory in
Elegant Universe on NOVA), aside from hot fusion, with so
little
contact with experimental fact. Nevertheless, M Theory may be
on the
right track to my program of Making Star Trek real. Note how
Brian
Greene talks to ET on the telephone like in 1953 in my book
Destiny
Matrix (2002) and how the idea of time travel to parallel
brane worlds
is taken seriously - as the UFO facts suggest it should be.
Brian has
made my ideas on the potential consequences of the New Physics
much more
mainstream.
JS: Locally gauge the globally flat Poincare group to get the
curved
spacetime
guv(curved) - nuv(global flat) = huv(curved)
as the compensating field.
PZ: [I suppose your nuv(global flat) is my nuv(flat level) --
the
Minkowski metric?]
JS: Yes, obviously. There is, like God, only one globally flat
metricthough it is unstable. Beware False Metrics!
PZ: Yes, indeed. Let us not worship idols. :-)
At the same time, one man's god is another's idol.
I have that on the authority of no less than Gen. Boykin.
JS: Whose Bodkin?
...
PZ: In order to understand the full Freud-Yilmaz decomposition
at a
fundamental level, I am arguing that there is no need to split
the
metric g_uv itself -- only the gravitational and inertial
contributions
to the *gradients* of the metric g_uv, w that appear in the
Christoffel
symbols. The beauty of this is that it allows an interpretive
gestalt
switch without disturbing the mathematical elegance of the
Einsteinian
unified metric g_uv. It
leaves the unified metric intact, while it merely decomposes
the g_uv,
w and thus the connection field.
JS: I do not see you doing anything other than the trivial and
obvious
entirely inside Einstein's paradigm
Connection g-force = Tensor + Non-Tensor
In an LIF Connection = 0
PZ: Of course I would prefer to spin this as true and correct
and
mathematically unassailable. :-)
JS: Physically accurate. Astronauts float.
PZ: I seem to recall that it has taken me a more than a year
to get you
to agree
even to this.
JS: False. I always knew that. It's one of Wheeler's key
teachings. I
did not realize
that was all you were talking about in your obscure use of
ordinary
language. Hadyou simply written the math as I did above I
would have
instantly understood what you were trying to say. The fault
Dear Brutus
was in the lack of clarity of using too many English words
with their
seven types of ambiguities to cloud a very simple formal
issueand
informal or physical concept and you call that compensation
fine.
You do not need excess metaphysical baggage, which is what Hal
clouds
the issue with in his infamous heretical Tables I & II as if
he were
Moses - The Real Sarfatti. ;-)
PZ: In another context, you and your friends say that you have
decomposed
components:
Psi(r,t) = R(r,t) EXP [iS/hbar]
and then you define Q as
Q = (-hbar^2/2m) {(DEL_1^2 + DEL_2^2) R} /R
Fine, but so what?
I do not see that this is any more than a trivial mathematical
rearrangement.
However, if you want to call the Q derived from real part of
this
decomposition a quantum potential, go ahead. I doubt that this
will
shake the marble temple of Copenhagenism. So let's not take
any of that
too seriously. After all, who needs all this excess
metaphysical baggage? Who ordered that?
JS: A read of The Undivided Universe shows why that is not a
valid
comparison.
In no sense is huv small in this local gauging, which is not
perturbative relative to flat background. Unfortunately the
same
notation is used in perturbation theory.
PZ: As far as I can see this has nothing to do with
perturbation theory
or linearized GR. The Freud decomposition is exact in a broad
class of
internally consistent geometrodynamic theories of which
Einstein GR is
supposed to be an example.
JS: Sure. I never implied otherwise.
PZ: OK, so this is now clear and agreed?
JS: Yes, but it is a peripheral issue.
A Poincare group local stress-energy density tensor for the
first order
perturbative gravity where h'uv is small can be defined of
course, so
that may be
at least one of the pseudotensors in the Freud identity?
PZ: Again, the Freud theorem is exact and quite general, and
so is the
underlying decomposition I am proposing.
JS: But it's trivial in the orthodox formalism.
PZ: Excellent.
Again trivially even in the full non-perturbative case
Tuv(Gravity) = (String Tension)Guv(Geometry)
and in the non-exotic vacuum where the Diff(4) divergence-free
Tuv(Source Mass-Energy) = 0
Tuv(Gravity) = 0
PZ: OK.
JS: because the local Einstein field equation in that case is
Tuv(Gravity) + Tuv(Source Mass-Energy) = 0
Note that the supreme objective of our guerrilla mission on
this
primitive planet is metric engineering which the ordinary
Bianchi
identities strictly forbid. It's like orthodox QM strictly
forbids
signal nonlocality. All the really juicy stuff is forbidden
fruit.
Theoretical physics has been taken over by Vegans on
Macrobiotics! ;-)
I mean when Bianchi identities hold
Tuv(Gravity)^;v = 0
and that stops us dead in the water for metric engineering
star gates
and weightless warp drives.
You need to intermingle the underground stream of the Vacuum
Einstein
Current with the Matter Einstein Current. You do not want
Tuv(Matter)^;v = 0
all by itself. The vacuum propeller demands only that the SUM
VANISHES
not the individual terms in
Tuv(Marble)^;v + Tuv(Wood)^;v = 0
Marble = Geometry
Wood = Matter in general sense.
Again note
Tuv(Geometry) = (String Tension)Guv(Einstein)
I am not aware that Hal Puthoff or anyone else has made this
point
before as essential to the metric engineering tricks of Once
and Future
Men Like Gods? Perhaps there is something to Picknett's and
Prince's
The Star Gate Conspiracy after all?
http://www.cassiopaea.org/cass/stargate.htm
http://www.hiddenmysteries.com/item300/item359.html
Caveat: It has wacko pseudo-archaeology of course.
PZ: But then you have to argue that the vacuum stress-energy
is non-local,
as Einstein did. Which makes it very difficult to make sense
out of
gravitational waves -- among many other things.
JS: NO! NO! NO! You have garbled two different problems.
Tuv(Gravity) here is LOCAL.
PZ: I was assuming that by Tuv(Gravity) you were referring to
a vacuum
stress-energy.
If not, then what exactly is Tuv(Gravity)?
JS:
Tuv(Gravity) = (String Tension)Guv(Einstein)
Gravity = Geometry = Non-Exotic Vacuum = Marble
Guv(Einstein) = Ruv(Ricci) - (1/2)R(Ricci)guv
Tuv(Exotic Vacuum) = (String Tension)/zpfguv
Total LOCAL Diff(4)stress-energy density tensor equation is
balancing
the Scales of Cosmic Justice
Tuv(Exotic + Non-Exotic Marble) + Tuv(Wood) = 0
where, in general the separate terms on LHS DO NOT HAVE
VANISHING
DIVERGENCE
only their sum's divergence vanishes.
This is Action-Reaction compensation in your favorite sense
for the
Marble-Wood Einstein stress-energy current densities.
These are all bona-fide local Diff(4) tensors. No Freud
identity needed.
No computation of global far field gravity wave Pu and Muv in
asymptotic flat geometries needed for the strictly near field
metric
engineering application to Make Star Trek with Q real. Note
LIGO not
seeing gravity waves. Gravity waves no good for metric
engineering they
are like a leaky toilet. You want to trap energy not let it
fade away to
distant places and times and even other brane worlds perhaps.
PZ: The Einstein pseudotensor u_uv was originally supposed by
Einstein
to be a
stress-energy density that would account for the
energy-momentum stored in
the gravitational field, in order to specify the total
energy-momentum of a
closed system of gravitating masses.
The real LOCAL tensor here is simply
Tuv(Gravity) = (String Tension)Guv
which is zero in the non-exotic vacuum limit.
In general exotic vacuum
Tuv(Gravity) + (String Tension)/zpfguv = 0
The zero point energy density for any observer with 4 velocity
V^u is
(String Tension)/zpfguvV^uV^v
where, for h = c = 1
/zpf = (String Tension)[(String Tension)^-3/2|Vacuum
Coherence|^2 - 1)
Non-Exotic Vacuum is when /zpf = 0.
Vacuum Coherence obeys Diff(4) covariant LOCAL nonlinear
Landau-Ginzburg
type equation.
PZ: It was Schrodinger, and then Bauer, that showed (in 1918)
that this
quantity was
locally frame-dependent, since it could be made to locally
vanish or
appear at will
by a suitable choice of coordinates.
JS: They may be asking the wrong question?
The Question is: What is The Question? (Wheeler)
PZ: Einstein in the same year showed how one could integrate
the u4 part
of this vacuum
density over a large volume of space in order to obtain an
*approximately*
invariant measure of the energy-momentum in a spatial region,
even while
the local
field energy density u_uv itself was entirely frame-dependent:
J_u = INT[T_uv(matter) + u_u4(field)] dx_1 dx_2 dx_3
JS: Yes, the key word is missing it is global. And invariant
means a
sloppy kind of quasi-Poincare invariance
only in the asymptotically flat regions. It's a clumsy idea to
begin
which showing that MACRO-QUANTUM EMERGENT
relativity physics is only simple when its local.
PZ: What is really strange about this from my and others' POV
is that
these non-local
integrals are never *really* frame-independent over finite
volumes
except in certain
artificial limiting conditions -- and thus cannot generally
provide any
*objective*
measure of field energy-momentum -- even while the covariant
metric
field g_uv and its
derivatives are perfectly definite *at every point in
spacetime*.
JS:, OK but that's from asking the wrong kinds of questions.
MTW like
to generalize the Stoke's/Gauss flux integral theorems to the
pre-metrical boundary of a boundary is zero, but obviously
curvature
collides with topology and one cannot think in terms of
topology alone.
PZ; Considering that in Einstein's theory g_uv is supposed to
*define
the entire
physical reality* of the g-field, I and others can only
suspect that
this points to a
fundamental conceptual problem in the Einsteinian model.
The question is: What is it?
JS: It is ZERO in non-exotic vacuum.
PZ: How do you define this quantity Tuv(Gravity)? With respect
to
which theory?
*What* is zero?
JS: In orthodox GR
Ruv = 0
is the non-exotic local vacuum field equation.
R = 0
Therefore Guv = 0
Therefore
Tuv(Gravity) = (String Tension)Guv = 0
only in non-exotic vacuum.
Metric engineering is its pure form is from
Tuv(Gravity) + (String Tension)/zpfguv = 0
with
/zpf^,v =/= 0.
This is a violation of Bianchi identities like presponse signal
nonlocality is a violation of micro-quantum reality!
God is subtle, but not malicious.
God would never settle for a boring universe without star gate
time
travel and weightless warp drive IMHO. ;-)
Computing the nonlocal flux integrals for Pu and Muv in an
asymptotic
space-time is a horse of a different color.
PZ: This has nothing specifically to do with asymptotically
flat
spacetime, so is it
really me who is confused here? Of course the situation is
*similar* in
some
respects when we want to compute these approximately invariant
integrals in
the approximately flat spacetime region far from any source --
but the
question
of non-locality of u_uv and the *approximate* invariance of
*any* such
integrals
formed from it over finite spatial volumes is quite general in
scope.
JS: Oh no? What about Ch 20 of MTW? My understanding of your
invariance is that it is a kind of asymptotic Poincare
group not a local Diff(4) group.
The reason for the need for these pseudo-tensors is, I
suspect, the
collision between metric and topology. They are not
independent although
I suppose local metric geometry is the slave of global
topology.
JS: : You have confounded apples with oranges
PZ: : I have? Are you sure?
JS: Pretty much. We are talking about different problems. I am
not
interested right now in gravity waves which is what all your
stuff is
really about. The Pu and Muv Fourier transforms are the power
spectra of
the gravity waves from the source geometry. It's all far field
radiation in the asymptotically flat regions. I mean that's
what those
Flux Integrals are all about. The Flux is gravity waves
through the
bounding surfaces.
PZ: Strictly speaking, there are no proper analogs in Einstein
GR to the
energy and
momentum integrals of Newtonian theory. There is no question
about this.
JS: Fine, but that is not my problem. My problem is LOCAL
metric
engineering where leakage into gravity waves and EM waves is an
annoyance especially for stealth cloaking of spacecraft.
PZ: Obviously Einstein thought that this was important in
1918, since he
struggled to
find such analogs. Schrodinger also regarded the issue as
fundamentally
significant
for Newtonian correspondence.
JS: I am not denying the importance of the problem for
foundational
studies. I am only trying to distinguish that your problem is
not the
one that interests me, which is metric engineering.
PZ: If one can find a simple way to fix this problem in a
manner that is
fully consistent
with the fundamental theorems of geometrodynamics, that makes
intuitive
physical sense, and that is consistent with all available
empirical
data, then I would
have thought that such an opportunity would be welcomed with
open arms.
Apparently not.
JS: When I have time I will go back to MTW Ch 20 and try to
say what
they are doing in simple terms.
like when you
thought that string theory's removal of UV renormalization
infinity
explained the
smallness of the cosmological constant. Witten is very clear
that is
not the case.
PZ: Well, you may have a point there -- but that's another
kettle of fish.
JS: Right it is the Moby Dick of Physics Today. You are
fishing for
minnows with this problem.
In any case you would need to come up with something amazing
and precise
that explained some
mysterious data of profound importance.
Pu is not a simple flux integral of Tuv(Geometry)= 0 since
Pu(Geometry)
=/ = 0 in general.
PZ: Did I say any of this was simple in standard GR? I thought
I was
saying that it is
surprisingly complicated.
I am still not clear as to what exactly your Tuv(Geometry) is.
JS: By now I hope you are. It is exceedingly straight forward.
Tuv(Gravity) = (String Tension)Guv(Einstein)
Subject: Re: Brian Greene's PBS NOVA Elegant Universe
> Never has so much tax money been spent on a physics theory
(M -Theory in
Elegant Universe on NOVA), aside from hot fusion, with so
little
> contact with experimental fact.
And your tripe is grounded in experimental basis?
Hahahahahahahahahahaha!
Subject: Re: I WILL GET MY MONEY

> |> Hi James. I checked your math - it's wrong. I will
> |> be cancelling all of your posts later this evening
> |> so that no one will have to read them.
> |
> |what, exactly, gives you the authority to do such thing?
 i said he could.
 and, who or what exactly, gives you the authority to do
> such thing?
 I said he could.
> and who made you the authority???
> Sorry, but I don't have the authority to tell you that.
but, who does?
> Fortunately, I do have the authority to tell you I don't have
> the authority to tell you that.
and who gave you that authority?
> Have a nice day.
> Jim Burns
Subject: JSH: Not even close
What makes the story here *extremely* pathetic is that
mathematicians
can't in *any* way approach what I showed you in going from a
partial
difference equation that counts primes, as they don't have
one, to a
continuous function.
If they can, I'd like to see *anyone* post a demoonstration
like mine.
I wouldn't be surprised if rank and arrogant posters try to
attack
that posting, but pay careful attention to what they say, and
remember, despite it's gloried history, nothing even close to
that has
been possible with anything that mathematicians have
discovered that
counts primes in recorded human history.
That more than anything else is what you can use to realize
that I
have a first-find and a significant one, as nothing they have
can
approach what can be done with my discovery.
Mathematicians are here flat on their backs and not even in the
ballpark.
So why would they fight such a discovery?
Good question.
Any ideas?
Mathematicians aren't even close, yet they keep posturing and
fighting
against mathematics itself.
If they continue, mathematics will destroy them.
Isn't that ironic, don't you think?
James Harris
Subject: Re: JSH: Not even close
>What makes the story here *extremely* pathetic is that
mathematicians
>can't in *any* way approach what I showed you in going from a
partial
>difference equation that counts primes, as they don't have
one, to a
>continuous function.
Huh. First I saw the subject line: JSH: Not even close, and I
assumed that it was a JSH thread started by someone else.
Then I saw the from line and I assumed it was one of your
periodic retractions. Imagine my surprise when the thread
turned out to be the same old crap.
>[...]
>Isn't that ironic, don't you think?
I see there's another new thread that mentions something about
rank denial in the subject line. Now _that's_ ironic.
>James Harris
************************
Subject: Re: JSH: Not even close
> What makes the story here *extremely* pathetic is that
mathematicians
> can't in *any* way approach what I showed you in going from
a partial
> difference equation that counts primes, as they don't have
one, to a
> continuous function.
On the other hand, they have a number of better ways to count
primes!
Subject: Re: Not even close
<snip junk
I've found that insulting a group of people is the best way to
get them to
believe you. Really.
--
KB - MCNGP #26
first initial last name AT hotmail DOT com
Subject: Re: Not even close
> <snip junk
> I've found that insulting a group of people is the best way
to get them
to
> believe you. Really.
Of course. It's called Marketing.
Subject: Re: JSH: Not even close
> What makes the story here *extremely* pathetic is that
...you might actually do something useful with your life if
you'd get some counseling and maybe some medication, but
that's not likely to happen.
> mathematicians
> can't in *any* way approach what I showed you in going from
a partial
> difference equation that counts primes, as they don't have
one, to a
> continuous function.
Since this has been proven false many time, I can only
guess that your plan is to repeat it so many times that
everyone
will get bored and stop responding, at which point you'll
declare victory.
> If they can, I'd like to see *anyone* post a demoonstration
like mine.
Sane people realize that the only differences between
your method and the Legendre method
http://mathworld.wolfram.com/LegendresFormula.html
are notational.
> I wouldn't be surprised if rank and arrogant posters try to
attack
> that posting, but pay careful attention to what they say, and
> remember, despite it's gloried history, nothing even close
to that has
> been possible with anything that mathematicians have
discovered that
> counts primes in recorded human history.
Since this has been proven false many time, I can only
guess that your plan is to repeat it so many times that
everyone
will get bored and stop responding, at which point you'll
declare victory.
> That more than anything else is what you can use to realize
that I
> have a first-find and a significant one, as nothing they
have can
> approach what can be done with my discovery.
And what would that be, exactly? You seem to have a
greatly inflated view of the importance of prime
counting algorithms in the overall scheme of things.
> Mathematicians are here flat on their backs and not even in
the
> ballpark.
> So why would they fight such a discovery?
> Good question.
> Any ideas?
> Mathematicians aren't even close, yet they keep posturing
and fighting
> against mathematics itself.
> If they continue, mathematics will destroy them.
Did I mention there are good medications on the market
these days?
-E
> Isn't that ironic, don't you think?
> James Harris
Subject: Re: JSH: Not even close
> What makes the story here *extremely* pathetic is that
mathematicians
> can't in *any* way approach what I showed you
Hey stooopid loud troll James Harris, put up or shut up,
http://www.rsasecurity.com/rsalabs/challenges/factoring/
faq.html
http://www.rsasecurity.com/rsalabs/challenges/factoring/
numbers.html
http://www.crank.net/harris.html
It's not every braying jackass that gets a whole page at
crank.net
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
Quis custodiet ipsos custodes? The Net!
Subject: Re: Not even close
You can't spell, and you can't do math. You are a
sad, pathetic, mentally ill loser. Eat a bullet, dick-weed.
> What makes the story here *extremely* pathetic is that
mathematicians
> can't in *any* way approach what I showed you in going from
a partial
> difference equation that counts primes, as they don't have
one, to a
> continuous function.
> If they can, I'd like to see *anyone* post a demoonstration
like mine.
[moon this, face!]
> I wouldn't be surprised if rank and arrogant posters try to
attack
> that posting, but pay careful attention to what they say, and
> remember, despite it's [LEARN TO ING SPELL
THREE-LETTER ENGLISH PRONOUNS, ASSHOLE]
gloried history, nothing even close to that has
> been possible with anything that mathematicians have
discovered that
> counts primes in recorded human history.
> That more than anything else is what you can use to realize
that I
> have a first-find and a significant one, as nothing they
have can
> approach what can be done with my discovery.
> Mathematicians are here flat on their backs and not even in
the
> ballpark.
> So why would they fight such a discovery?
> Good question.
> Any ideas?
> Mathematicians aren't even close, yet they keep posturing
and fighting
> against mathematics itself.
> If they continue, mathematics will destroy them.
> Isn't that ironic, don't you think?
> James Harris
Subject: Re: JSH: Not even close
> What makes the story here *extremely* pathetic is that
mathematicians
> can't in *any* way approach what I showed you in going from
a partial
> difference equation that counts primes, as they don't have
one, to a
> continuous function.
A continuous function which does exactly what? You have not
shown that
it has any bearing whatsoever on the prime counting function.
You have
shown *no* results!
> If they can, I'd like to see *anyone* post a demoonstration
like mine.
Look at Ullrich's recent post under one of your recent threads.
> I wouldn't be surprised if rank and arrogant posters try to
attack
> that posting, but pay careful attention to what they say, and
> remember, despite it's gloried history, nothing even close
to that has
> been possible with anything that mathematicians have
discovered that
> counts primes in recorded human history.
Puleez! You have made this self-aggrandizing claim countless
times. Once
should be enough, at least it would be for a rational person.
[snip umpteenth plus umpteenth diatribe against mathematicians]
James, you are an incredible, hilarious joke. Your gray matter
has
decidedly brown tinge to it. Please shut up and go away. Come
back when
you crack RSA encryption.
--
There are two things you must never attempt to prove: the
unprovable --
and the obvious.
--
--
http://www.crbond.com
Subject: Re: Not even close
> What makes the story here *extremely* pathetic is that
you continue to bleat about your PNT and FLT even though you
have been
shown
to be incorrect in both.
Shut up and adjust your medications James.
Subject: Re: Proof by induction
1) it's misleading to have the subject line in English and the
message in
French
2) &#8804 conveys absolutely nothing to me
3) given the Fundamental Theorem of Arithmetic, it is trivial
that kt/n is
an
integer iff n/pgcd(t,n) divides k; hence I don't think your
problem is the
best
one to assign to a class (I assume that's what you're
considering doing with
it)
| On a un n-gone (polygone .88 n c.99t.8es) r.8egulier dont
les sommets
sont
| num.8erot.8es 0, 1, ... , n-1, o.9d le sommet i suit le
sommet i-1
dans
| l'ordre anti-horaire.
| Pour tout entier positif t &#8804; n, une rotation
anti-horaire par un
| angle de 360t/n degr.8es envoie le sommet 0 vers le sommet t.
|
| D.8emontrez que le plus petit nombre strictement positif de
rotations
| anti-horaires par 360t/n degr.8es qui envoie le sommet 0 vers
lui-m.90me
| est n/pgcd(t,n).
|
| En d.8eduire que n rotations anti-horaires par 360t/n
degr.8es sont
| n.8ecessaires pour envoyer le sommet 0 vers lui-m.90me si et
seulement
si
| pgcd(t,n)=1.
|
| pgcd = plus grand commun diviseur
|
| j'aimerai avoir vos opinions.....
Subject: Re: Proof by induction
> Michel <mikechico75@hotmail.com> grava .88 la saucisse et au
marteau:
> j'aimerai avoir vos opinions.....
> The first one is that you must not post in French (enven
though I did it
> once) in this newsgroup. So ask your question in english or
post it on
> fr.education.entraide.maths
I'd say that you should feel free to post in whatever language
you like.
If I don't understand it, I won't reply. If I reply, I might
reply in
any language I like. Using English most likely increases your
chances to
be understood and get an answer, but that is up to the poster.
Subject: Re: Proof by induction
>Pour tout entier positif t &#8804; n,
French I can live with; but HTML codes are _not_ up to spec!
(You can read all about this at
http://abcdrfc.free.fr/rfc-vf/rfc1036.html
which, as an HTML document, can be written with ISO8859
encoding for
single characters, and can refer to these multi-digit
entities; but
the (French) text seems to refer only to ASCII characters as
being
permissible in key fields of a USENET post!)
dave
Subject: Re: Proof by induction
> On a un n-gone (polygone .88 n c.99t.8es) r.8egulier dont
les sommets
sont
> num.8erot.8es 0, 1, ... , n-1, o.9d le sommet i suit le
sommet i-1
dans
> l'ordre anti-horaire.
> Pour tout entier positif t &#8804; n, une rotation
anti-horaire par un
> angle de 360t/n degr.8es envoie le sommet 0 vers le sommet t.
> D.8emontrez que le plus petit nombre strictement positif de
rotations
> anti-horaires par 360t/n degr.8es qui envoie le sommet 0 vers
lui-m.90me
> est n/pgcd(t,n).
> En d.8eduire que n rotations anti-horaires par 360t/n
degr.8es sont
> n.8ecessaires pour envoyer le sommet 0 vers lui-m.90me si et
seulement
si
> pgcd(t,n)=1.
> pgcd = plus grand commun diviseur
> j'aimerai avoir vos opinions.....
> ....
> The first one is that you must not post in French (even
though I did it
> once) in this newsgroup. So ask your question in English or
post it on
> fr.education.entraide.maths
Beaucoup d'entre les lecteurs de sci.math sont des anglophones,
mais qu'importe?
Il me semble que proof by induction ne sert pas bien .88 ce
probl.8fme, parce que les diviseurs de n+1 (ou de t+1) sont
fort
diff.8erents des diviseurs de n (ou de t). Mais la rotation par
k(360t/n) degr.8es envoie 0 vers lui-m.90me ssi n|kt, n'est-ce
pas?
Comment d.8emontre-t-on que n|kt ssi (n/pgcd(t,n))|k ?
(N'oublier pas
B.8ezout!)
Ken Pledger.
Subject: Re: Proof by induction
> Comment d.8emontre-t-on que n|kt ssi (n/pgcd(t,n))|k ?
(N'oublier pas
> B.8ezout!)
Hey, I learned something today: ssi is French for iff!
(Ouais, on apprend quelque chose tous les jours: ssi, c'est iff
en Francais!)
--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
Subject: Re: Proof by induction
Ken Pledger <Ken.Pledger@vuw.ac.nz> grava .88 la saucisse et
au marteau:
> Beaucoup d'entre les lecteurs de sci.math sont des
anglophones,
> mais qu'importe?
Ok, I may be wrong. I thought we MUST post in english, but
looks like
it isn't the case. My mistake.
--
Nicolas
Subject: Probability generating functions : Poisson help
Dear all,
I am having trouble with this problem, and if anyone can
assist I would
highly appreciate it.
We know that the probability generating function of an integer
valued
random
variable X is defined to be
f(s) = E(s^X) = sum (over all k) of (s^k)*P(X=k)
Suppose that E((s^X)(t^Y)) =
exp(a(s-1))exp(b(s-1))exp(c(st-1)). How can I
calculate P(X=k) and P(Y=k) for k = 0,1,2,...
and, what is a necessary and sufficient condition on c in
order for X+Y to
have a Poisson distribution?
Sincerely,
-Henrique Carlos
Subject: Re: Probability generating functions : Poisson help
>We know that the probability generating function of an
integer valued
random
>variable X is defined to be
>f(s) = E(s^X) = sum (over all k) of (s^k)*P(X=k)
>Suppose that E((s^X)(t^Y)) =
exp(a(s-1))exp(b(s-1))exp(c(st-1)).
Let's call this g(s,t). I assume the second factor should be
exp(b(t-1)).
> How can I
>calculate P(X=k) and P(Y=k) for k = 0,1,2,...
>and, what is a necessary and sufficient condition on c in
order for X+Y to
>have a Poisson distribution?
g(s,1) and g(1,s) are the respective pgf's for the marginal
seems to me that X+Y is Poisson iff c = 0.
If this is homework, please cite your sources in submitted
work.
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
Subject: Another integer-like transcendental number
Let a=1/10, b=sqrt(2)/20. Then
cos(a)+cosh(a)+2cos(b)cosh(b)=4.000000000000992.....
Can anyone explain this one? Or give a reference.
Subject: Re: Another integer-like transcendental number
> Let a=1/10, b=sqrt(2)/20. Then
> cos(a)+cosh(a)+2cos(b)cosh(b)=4.000000000000992.....
> Can anyone explain this one? Or give a reference.
cos(a) + cosh(a) = 2 (1 + a^4/4! + a^8/8! + ...)
cos(b)cosh(b) = 1 - b^4/6 + b^8/2520 + ... .
If a^4 = 4 b^4 (as it is here) then
cos(a) + cosh(a) + 2 cos(b)cosh(b) = 4 + b^8/1008 + ...
is very close to 4 if b (and so a) is small.
--
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: Another integer-like transcendental number
> Let a = 1/10, b = sqrt(2)/20. Then
>> cos(a)+cosh(a)+2cos(b)cosh(b) = 4.000000000000992...
>> Can anyone explain this one? Or give a reference.
> cos(a) + cosh(a) = 2 (1 + a^4/4! + a^8/8! + ...)
> cos(b) * cosh(b) = 1 - b^4/6 + b^8/2520 + ... .
> If a^4 = 4 b^4 (as it is here) then
> cos(a) + cosh(a) + 2 cos(b)cosh(b) = 4 + b^8/1008 + ...
> is very close to 4 if b (and so a) is small.
Simpler: f(x) := 2 (cos(x) + cosh(x)) + cos(x) cosh(x)
= 5 + x^8/2016 + O(x^12)
e.g. f(0.1) = 5.00000000000496 ~= 5 + 5*10^-12
For deeper numerical coincidences see my prior post below.
-------------
sci.math post of 1998/06/25: Subject: Re: Can e & pi be easily
related?
http://google.com/groups?threadm=y8zlnqlmzwl.fsf%
40nestle.ai.mit.edu
|
| Is there a reasonably simple real-numbers equation connecting
| e & pi, that can be used to calculate either one from the
other?
One of the most beautiful and mysterious approximations is
pi 3 1/sqrt(163) -30
e = ( 640320 + 744 ) - 5.177 10
Check it on your calculator, then see my prior post [1]
for references to the deep and fascinating explanation
involving class fields, complex multiplication, modular
functions, Kronecker's Jugendtraum (youthful dream), etc.
Or behold Shanks' [2] amazing approximation
6 24 -r pi -163
pi = - log(2abcd) + -- e + 6.69 10
r r
a = A+sqrt(A^2-1), b = B+sqrt(B^2-1), c = C+sqrt(C^2-1), d =
D+sqrt(D^2-1),
A = 1/2 (1071 + 184 sqrt(34)), B = 1/2 (1553 + 266 sqrt(34))
C = 429 + 304 sqrt(2), D = 1/2 (627 + 442 sqrt(2)), r =
sqrt(3502)
To improve an approximation P to pi use P + sin(P) or P + 2
cos(P/2),
which triples the accuracy, or use P + (2 sin P - tan P)/3,
which
quintuples the accuracy, cf. [3]. For example, applying these
to the
pi approximation from the first equation above triples the
accuracy
from 10^-31 to 10^-93, and quintuples the accuracy to 10^-155.
Quintupling Shanks' approximation yields over 800 digits of pi.
-Bill Dubuque
[1] sci.math post of 1996/09/27 titled: Re: (pi+20)^i ~= -1,
explain why
http://google.com/groups?selm=y8zohirmj49.fsf%
40martigny.ai.mit.edu
[2] Shanks, Daniel. Dihedral quartic approximations and series
for pi.
J. Number Theory 14 (1982), no. 3, 397-423.
MR 83k:12010 (Reviewer: Harvey Cohn) 12A70
[3] Shanks, Daniel. Improving an approximation for pi.
Amer. Math. Monthly 99 (1992), no. 3, 263.
MR 94a:11197 (Reviewer: W. W. Adams) 11Y60
Archived as the last page of the following JStor document:
http://links.jstor.org/sici?sici=0002-9890(199203)99:3%3C259%3E
Subject: Re: Group character...silly question
> Your subject line is a little different, though: when you
are looking
> at a character of a (finite) group, you're looking at a
_function_
> defined on the group, namely chi(g) = trace( R(g) ) where
R(g) is
> a representation matrix. That is, you haven't got just one
matrix
> but lots of them. In fact, one is not usually so interested
in _a_
> character of a group but rather the whole character table,
which sheds
> light on the structure of the group. These pieces of
information make
> a difference in your question: it's not just linear algebra
now but
> rather group theory. For example, we have some theorems:
I must confess this is a bit confusing. R(g) is a matrix, each
R
(representation) gives me a set of matrices R(g) for each g,
and there
are several R's possible. That's OK by me. But the whole
character
table...of course we are not interested in ALL the R's (if R
is a
rep., so is R+R, and so on as far as I get it), so we only
talk about
irreducible ones. So far so good. But the next step...there
are as
many irreps as the conjugacy classes...is mindblowing. I am
still
trying to understand the proof. Could you offer something
alternative
to what the text books say? I mean, not the *proof*, it's
there all
right, but hints to make it look plausible to myself?
> 1. If R_1 and R_2 are two complex representations of a
finite group
> whose characters are equal, then the representations are
equivalent
> (that is, there is a single invertible P with P R_1(g) =
R_2(g) P for
> every P; you might say the R1's and R2's are uniformly
similar.)
> In other words: the trace is all you need to distinguish two
> representations anyway.
This one I understand. Linear Algebra cannot get simpler than
this
(without reducing to something more trivial).
> 2. Any complex-valued function on G which is constant on
conjugacy
> classes is a linear combination of characters. In other
words, the
> traces of all the characters already give you all the kinds
of functions
> you could make out of the similarity classes anyway.
So what we are really after is the partition of the conjugacy
classes?
> Taken together, (1) and (2) sort of tell us that traces are
the only
> similarity invariant we need in classical representation
theory.
Got it. Thanks for your (easy to swallow) reply.
> dave
Subject: Re: Group character...silly question
>So far so good. But the next step...there are as
>many irreps as the conjugacy classes...is mindblowing. I am
still
>trying to understand the proof. Could you offer something
alternative
>to what the text books say? I mean, not the *proof*, it's
there all
>right, but hints to make it look plausible to myself?
Yes, it's a little strange, in part because there's no natural
one-to-one correspondence between the sets which are proved to
have
the same number of elements.
Maybe it's easier to see in the case of finite abelian groups.
There
all the irreducible characters are 1-dimensional, and all the
conjugacy
classes have cardinality 1, so the fact that the numbers are
equal is
equivalent to the statement that there are as many characters
as
elements. Again, there's no natural correspondence, but now
you can
sort of see why: the characters are DUAL to the elements. If,
for
example, the group is (Z/2Z)^3, then your characters are
determined
by which of the generators is sent to +1 and which to -1. You
can
identify those two values with the elements of Z/2Z, so that
the
set of characters is naturally isomorphic to the dual space
Hom( G, Z/2Z ). Now you need to remember some linear algebra;
do you
consider it mindblowing to know that V and Hom(V,F) are
isomorphic but no naturally so? (You should -- it's not true
for a
perfectly general vector space! But after a while it feels
pretty normal.)
The actual proof that there are k irreps is not all that
mysterious
anyway. You really, really need to get used to the idea that a
representation is not just a homomorphism of group G -->
GL_n(C) ; it's
a homormophism of rings C[G] --> M_n(C). That's crucial when
you move
to other aspects of representation theory, e.g. modular
representations,
representations of algebras, or non-finite groups. It helps a
lot
in this context to know that the ring C[G] is just a direct
sum of
matrix rings, and that matrix rings are simple (no 2-sided
ideals),
so that a representation of a group simply becomes a
projection onto
one of the summands. Keep that picture in mind, because it's
what
makes the magic of character theory work: you use Cayley's
theorem
to turn group elements into matrices, you choose a new basis
to make
the matrices into a block sum, you throw in linear
combinations of
these matrices to get full matrix rings, and then you project
onto
one or the other isomorphism class of summands. That's what a
representation is. That means there are as many of them as
there
are block summands. On the other hand, those projections onto a
summand are accomplished by multiplying by a minimal
idempotent (a matrix
filled with 0's except for some 1's along the diagonal of one
of the
block summands in the matrix), and linear combinations of those
idempotents give you all there is in the center of the matrix
ring.
Then you just observe that the center of C[G] can be described
in
a different but very natural way, using conjugacy classes, and
you
say Duh!: the counts are equal because they're just two ways to
count the size of the center of the ring.
>> 2. Any complex-valued function on G which is constant on
conjugacy
>> classes is a linear combination of characters. In other
words, the
>> traces of all the characters already give you all the kinds
of functions
>> you could make out of the similarity classes anyway.
>So what we are really after is the partition of the conjugacy
>classes?
Not sure I understand what you're asking. Conjugate elements
of the
group will map to conjugate, hence similar, matrices in the
representations
(see above model), so the traces will be equal. So traces are
class
functions. After you have the equality of dimensions in the
above
discussion, you know that the traces provide all the class
functions
there can be. I don't think people were desperately trying to
describe
class functions before the invention of representation theory;
but they're
easily understood and help describe what it is we're doing
with the
numerical tricks which come out of character theory.
(Orthogonality
relations and all that.)
dave
Subject: Re: all math problems reduce to linguistics
Yea sure, and all other problems too.
kribaddak semore alegudla sianabbe dodisok
Thats the building instructions for a free energy device.
abbeleda saibondego rumlavonde soustarik
this is a proof that William Shakespeare was in truth queen
Elizabet.
And now for the best one:
conebik yemulka sichonde emndaden
This means: Guenther is Emperor of the universe by the grace
of God
and all humans must obey him.
Guenther says please be silent!
Subject: Re: [OT] Fields Medal VS Abel Prize
> Field Medal is awarded for one brilliant work (in
particular), while Abel
> Prizes is awarded for one brilliant career.
When these prizes are awarded? In July?
Thanks
--
Noixe
Subject: Re: [OT] Fields Medal VS Abel Prize
> Both Fields Medal that Abel Prize are the Nobel prize for
the math.
But,
> what's the criterion to assign one or other? One of they is
more
> prestigious?
Are you trying to decide which one to go for?
Gib
Subject: Re: [OT] Fields Medal VS Abel Prize
> Are you trying to decide which one to go for?
Nothing, was only for curiosity.
Thanks
Subject: Max/Min Value Problem
Suppose I have a standard 3-dimensional egg-shaped ellipsoid
with the
following equation:
(x^2)/(a^2) + (y^2)/(b^2) + (z^2)/(c^2) = 1
If I rotate the ellipsoid around the y-axis, x-axis, then
z-axis I am
left with an equation of the form:
Ax^2 + By^2 + Cz^2 + Dx + Ey + Fz + Gxy + Hxz + Iyz + J = 0
,where A,B,C,D,E,F,G,H,I,J are constants resulting from
rotation
matrices.
How do I find the max/min values of x and y ?
Subject: Re: Max/Min Value Problem
>Suppose I have a standard 3-dimensional egg-shaped ellipsoid
with the
>following equation:
>(x^2)/(a^2) + (y^2)/(b^2) + (z^2)/(c^2) = 1
>If I rotate the ellipsoid around the y-axis, x-axis, then
z-axis I am
>left with an equation of the form:
>Ax^2 + By^2 + Cz^2 + Dx + Ey + Fz + Gxy + Hxz + Iyz + J = 0
>,where A,B,C,D,E,F,G,H,I,J are constants resulting from
rotation
>matrices.
>How do I find the max/min values of x and y ?
Lagrange multipliers?
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
Subject: Re: Max/Min Value Problem
Hi Machine,
did you already try to solve your problem? Or are you just
finding someone,
doing your housework?
I will not give a complete solution, just some ideas how I
would try to
solve for it.
1) Rotating the mentioned ellipsoid will not have D,E or F
unequal to zero.
2) See your resulting Ellipsoid as scalar function of three
unknowns:
u = F(x,y,z). Now calculate the gradient vector: (Fx, Fy, Fz).
This
will
be
a linear system in x,y,z. If looking for the max y value, the
gradient
will need
to be in parallel with the y-axis. Solve for this. You will
get the
equation of
a line. Intersect it with your ellipsoid. The intersection
points y,
coordinate is
your max y value.
3) Prodeed with the remaining coordinates in a simmilar
fashion.
Hope this helps.
roland
> Suppose I have a standard 3-dimensional egg-shaped ellipsoid
with the
> following equation:
> (x^2)/(a^2) + (y^2)/(b^2) + (z^2)/(c^2) = 1
> If I rotate the ellipsoid around the y-axis, x-axis, then
z-axis I am
> left with an equation of the form:
> Ax^2 + By^2 + Cz^2 + Dx + Ey + Fz + Gxy + Hxz + Iyz + J = 0
> ,where A,B,C,D,E,F,G,H,I,J are constants resulting from
rotation
> matrices.
> How do I find the max/min values of x and y ?
Subject: Re: Max/Min Value Problem
Hi again,
> to be in parallel with the y-axis. Solve for this. You will
get the
> equation of
> a line.
> Intersect it with your ellipsoid. The intersection points y,
> coordinate is your max y value.
> 3) Prodeed with the remaining coordinates in a simmilar
fashion.
Fogive me, if my results are false, I did not veryfy them for
the rotated
case,
but the results are simple enough, so I couldn't withstand,
posting them.
When R is the rotation and L the diagonal matrix of the
original ellipsoid
the min/max values are:
xmax = +/- sqrt((trans(R)*L^-1*R)[1,1])
ymax = +/- sqrt((trans(R)*L^-1*R)[2,2])
zmax = +/- sqrt((trans(R)*L^-1*R)[3,3])
i.e the sqrt's of the diagonal elements. of trans(R)*(L^-1)*R.
regards,
roland
Subject: The upper triangular group (T,x) is solvable
I've read the proof that the group of upper triangular
invertible matrices
is solvable; I had no problem to get it but Lang introduces
the notation
N^r
(N like nilpotent), say the set of upper triangular matrices
with all the
coefficients on the r first upper diagonals equal to 0. For
instance, N^1
is
the set of upper triangular matrices with the main diagonal
containing
only
0s.
This notation tends to suggest that for any r positive
integer, the map:
f: N^1 -> N^r, f(A) = A^r is surjective. Is that true -I guess
this can be
checked easily for small dimensions, so maybe a generalization
can be
found,
but I didn't give it a try-?
--
Julien Santini
Subject: Re: The upper triangular group (T,x) is solvable
>I've read the proof that the group of upper triangular
invertible matrices
>is solvable; I had no problem to get it but Lang introduces
the notation
N^r
>(N like nilpotent), say the set of upper triangular matrices
with all the
>coefficients on the r first upper diagonals equal to 0. For
instance, N^1
is
>the set of upper triangular matrices with the main diagonal
containing
only
>0s.
>This notation tends to suggest that for any r positive
integer, the map:
>f: N^1 -> N^r, f(A) = A^r is surjective. Is that true -I
guess this can be
>checked easily for small dimensions, so maybe a
generalization can be
found,
>but I didn't give it a try-?
Counterexample for r = 2:
[ 0 0 0 1 0 ]
[ 0 0 0 1 1 ]
[ 0 0 0 0 0 ]
[ 0 0 0 0 0 ]
[ 0 0 0 0 0 ]
is not the square of an upper triangular matrix.
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: Method of proof witch one?
On a un n-gone (polygone .88 n c.99t.8es) r.8egulier dont les
sommets
sont
num.8erot.8es 0, 1, ... , n-1, o.9d le sommet i suit le sommet
i-1 dans
l'ordre anti-horaire.
Pour tout entier positif t < = n, une rotation anti-horaire
par un
angle de 360t/n degr.8es envoie le sommet 0 vers le sommet t.
D.8emontrez que le plus petit nombre strictement positif de
rotations
anti-horaires par 360t/n degr.8es qui envoie le sommet 0 vers
lui-m.90me
est n/pgcd(t,n).
En d.8eduire que n rotations anti-horaires par 360t/n degr.8es
sont
n.8ecessaires pour envoyer le sommet 0 vers lui-m.90me si et
seulement si
PGCD(t,n)=1.
Subject: JSH: Smell of rank denial
My point is that what I've discovered isn't something that
should just
be totally ignored, while there are posters who seem to have a
career
of trying to claim that it's not importat *at all* against all
evidence.
Think about it.
I'm just saying, hey, record the discovery in some math text or
journal, while you can see posters jumping up and down like
angry
monkeys claiming that it's worthless!!!
Funny, eh?
Mathematicians are an interesting lot. After having spent some
time
studying you as a group, I think I have you figured out.
Yup, I'm saying that I have mathematicians as a group figured
out.
You need to feel like you're in control. You want to believe
that you
dominate mathematics, and not the other way around.
If you see something that challenges your *belief* of control,
you
will fight it, fight it, fight it, against all reason, against
mathematics itself, as you descend to a baser level of human
evolution.
And I can trigger your descent at will.
You're weak, mathematicians, and you might as well deal with
your
weakness, or it will destroy you.
James Harris
Subject: Re: Smell of rank denial
Since it appears you do read responses made occassionally (I
had you
figured
for your average usenet crank who crossposts all over the
place and then
never reads the replies) I will just reply to the group for
now.
Now let me help you out here. You aren't getting through to
these
`mathematician bozos'. I'll translate what you are saying into
mathspeak
(although amateurish mathspeak to reflect your current
standing within the
mathematical community).
> My point is that what I've discovered isn't something that
should just
> be totally ignored, while there are posters who seem to have
a career
> of trying to claim that it's not importat *at all* against
all
> evidence.
I'd like to undertake a rebuttal of recent remarks made in
this forum about
my recent work on ????, which I have exposed to the same.
It appears that a number of my critics have sought to supress
my point of
view. Some of my learned colleagues appear to have undertaken
a campaign of
publicly downplaying my results. This distresses me for a
number of
reasons.
Firstly, it is my opinion that one should not make a value
judgement with
regards to mathematics which has not as yet fully bloomed and
reject it out
of hand as though it may not eventually flower and be
attractive to some.
Although I confess that I have been unable to demonstrate the
usefulness of
my work to date, it is yet a tiny bud which has yet to even
sprout a leaf,
much less produce a pungent odour which should produce the
vile response of
the gardener with his weed killer. I ask that my colleages
allow the
results
which I have enumerated to come to flower before they judge
their worth.
Secondly, I believe that my results have already shown some
promise. Though
they are simple and perhaps ill conceived they are yet
attractive to me due
to the fact that they are easily manipulated. Without the
heavy machinery
of
analysis I have constructed some sort of discrete system which
somehow
mirrors something of the structure of the primes. Whether this
will lead to
insight or further results I do not know. But at any rate,
even though they
are extremely modest results, perhaps of little import, they
ought not in
my
opinion be decried without a fair hearing. That fair hearing
is precisely
what I intend to give them over the coming days as I allow
them to speak to
me of any secrets they may have. If at that point they appear
to remain
impotent, I will discard them and apply my powers elsewhere.
(Now if you were an attention getter and wished to really rub
people the
wrong way, and I think you are, then you would continue as
follows:)
> Think about it.
I would encourage my critics to examine the results I have
given. I am not
so much asking that they examine the results for maturity and
fragrance,
for
as yet they are too young for this. But instead I have
presented them here
to share with others a problem on which I should like to have
their
mathematical opinion. Not so much that I require them to act
as wine
tasters
or cake judges, but that I should like them to apply their
considerable
powers to the problem at hand and find whether this lead
actually heads
somewhere or whether it is merely a curiosity of the simplest
order, hardly
worthy of note.
> I'm just saying, hey, record the discovery in some math text
or
> journal, while you can see posters jumping up and down like
angry
> monkeys claiming that it's worthless!!!
Instead, I find my colleagues refusing to even allow my
seedling to see the
light of day, hoping, I suppose, that by refusing to allow it
to come to
light that it might somehow die an unnatural death in the
darkness. However
it has come to light and although I am unsure that it has
never been
conceived before by someone of greater stature than I who
investigated it
fully and tossed it aside as a weed without publication, I
feel that it
might better be presented in a note even if only in an obscure
source
somewhere that it may be referenced by those who may again
discover it and
wonder whether it may be of value. I would save them the
fruitless journey
if indeed it might be so.
It seems to me that the establishment aping and hyperacting,
in a manner of
speaking, more befitting of those close cousins of mathematics
than of the
stayed establishments which their long histories purport them
to be, is in
this case telling.
> Funny, eh?
It shows in an amusing way how hyperactive mathematics can be
when the
subject is contentious. But contentious as it may be I still
feel justified
in personally seeing it through at a steady pace given that it
was I who
was
tripped by this plant of unknown origin which has appeared in
the garden of
mathematical delights. Of course I too will be keen to dispose
of it should
it be a weed which has appeared where it does not belong, but
I fear
strangling it yet, lest it be a flower of some beauty not yet
fully ripe.
(Now if you were rude and wanted to do a little aping of your
own, you
might
continue:)
> Mathematicians are an interesting lot. After having spent
some time
> studying you as a group, I think I have you figured out.
My colleages are an interesting lot. After many years of
noting the
response
to innovation I fear that I have seen all too often the
results of what I
would term `academic shortsightedness'. Not that I am saying
that
skepticism
should be eschewed in this instance, I am more skeptical than
you all
despite my convictions. But simply that skepticism all too
often becomes
atheism. Not only are others not convinced that this or that
line of
enquiry
will lead somewhere, but they are certain it will not. Too
often we humans
allow skepticism to be replaced by certainty. Certainty, I
say, which we
have no right to. Having seen this response before and being
warned by the
outcome I feel I know you too well atheist.
> Yup, I'm saying that I have mathematicians as a group
figured out.
Yes I know you too well shortsighted. I have seen you play out
your foolish
game to its end in the past. But you could not see even when
you stumbled.
> You need to feel like you're in control. You want to believe
that you
> dominate mathematics, and not the other way around.
But `victory is not always to the srong, nor the race to the
swift'. The
control you have of your senses and of the destiny of ideas
does not come
in
your rush to agree with others who are as atheistic as you,
but in your
recognition that ideas in fact control you. It is the humbler
spirit who
fears spurning wisdom lest he perish who is prepared to be
ridiculed for
ideas sake, who will be justified. Even though he may be
trampled, he does
not care. Though all the wild horses of nonsense ride over
him, he feels
more privileged to be in the company of those who have opened
the gate to
allow the passage of one white stallion than to be in the
company of those
who sit in the seat of mockers.
> If you see something that challenges your *belief* of
control, you
> will fight it, fight it, fight it, against all reason,
against
> mathematics itself, as you descend to a baser level of human
> evolution.
But believing that you are in control of ideas as a skillful
jockey on a
horse, you will challenge anyone who claims he is not. But
without the
skill
of wisdom and without explanation for purpose and order you
descend only to
the baser primates and become nothing more than an ape. Your
evolution has
been downward.
> And I can trigger your descent at will.
I might say you have entered the path of `common descent.'
> You're weak, mathematicians, and you might as well deal with
your
> weakness, or it will destroy you.
You are weak and only the strong will survive. Your atheism
will destroy
you. I would counsel you to deal with your atheism lest it
destroy you.
(Now supposing you wished to come off sounding somewhat sane
(which often I
believe you do not) and rescue some credibility after taking
moral high
ground, you might be quick to add:)
But I wax lyrical. What is it that I propose. Merely this,
that ideas be
heard before they are derided. Not that I personally wish to
be heard, for
I
am not an idea. Only that we attend if we have the time to
this problem and
in concert either dispose of it rapidly or perhaps (and I
should be
suprised
if this even rarely happens) we find something of note where
the water laps
at our feet on the shore of the infinite sea of knowledge.
Only let us not
reject those things that are close by simply because they are
just that.
For
the prettiest shells have not all been found very deep.
> James Harris
Note this does NOT mean I support your crusade to have your
ideas heard.
They are your ideas, and although I am willing to hear of them
if you
should
show their worth, I do think that you are approaching the
mathematical
community the wrong way. You simply must not carry on about
how great you
are and how fantastic your ideas are when that is not the
case. You too
must
be humble and set an example if you feel that you have the
moral high
ground. Otherwise you are just crowing about your own
achievements,
blundering about making mistakes and telling others in no
uncertain terms
that you are a crank, a rank amateur and someone who does not
listen to
correction when they make mistakes. That is not a worthy
position to be in
but a ridiculous one, worse than an ass's ass.
Bill Hart.
Subject: Re: JSH: Smell of rank denial
> My point is that what I've discovered isn't something that
should just
> be totally ignored, while there are posters who seem to have
a career
> of trying to claim that it's not importat *at all* against
all
> evidence.
And I was thinking that your stuff is *not* totally ignored in
this
newsgroup.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam,
nederland,
+31205924131
home: bovenover 215, 1025 jn amsterdam, nederland;
http://www.cwi.nl/~dik/
Subject: James Harris == Jack Sarfatti ?
[snipped same old drivel]
James' tendency to never reply to any of the replies offered
to his
postings
these days, reminds me of Jack Sarfatti. James, stop using
Usenet as a
write-only medium!
Subject: Re: JSH: Smell of rank denial
>[...]
>If you see something that challenges your *belief* of
control, you
>will fight it, fight it, fight it, against all reason, against
>mathematics itself, as you descend to a baser level of human
>evolution.
>And I can trigger your descent at will.
>You're weak, mathematicians, and you might as well deal with
your
>weakness, or it will destroy you.
Oops. Slipping into that mode where we sound like a raving
lunatic
again. You should really watch that.
>James Harris
************************
Subject: Re: JSH: Smell of rank denial
> My point is that what I've discovered isn't something that
should just
> be totally ignored, while there are posters who seem to have
a career
> of trying to claim that it's not importat *at all* against
all
> evidence.
What you have discovered hasn't been ignored. It is part of the
mathematical literature for more than two hundred years, under
the name
Legendre's formula.
Subject: Re: JSH: Smell of rank denial
> I'm just saying, hey, record the discovery in some math text
or
> journal, while you can see posters jumping up and down like
angry
> monkeys claiming that it's worthless!!!
Things aren't published until some qualified individuals deem
it worth
publishing. You'd rather journals be full of arbitrary
submitted stuff,
this one was good, ...?
Btw, your subject lines have been very accurate lately. Not
even
close, Rank denial. Your self-knowledge is impressive.
V.
--
email: lastname at cs utk edu
homepage: cs utk edu tilde lastname
Subject: Re: JSH: Smell of rank denial
Adjunct Assistant Professor at the University of Montana.
[.snip.]
>Things aren't published until some qualified individuals deem
it worth
>publishing.
You are also forgetting a very important step: things are not
format. And that individual is, with only extremely rare
exceptions
normally due to very special circumstances, always the author.
This is true even when the thing is of major importance. The
second
part of Goedel's famous incompleteness paper was never
published,
because Goedel never got around to writing it.
To be sure, the reason he never got around to writing it is
because
that paper was supposed to contain the detailed proofs of
several of
his claims in Part I, and the consensus upon publication of
Part I was
that the details were more or less unnecessary: it was 'clear'
that
the arguments were correct.
But Goedel did not say Here's my idea, it is brilliant, so it
should
be published and waited for someone to write it up for him.
The vetting process of having qualified individuals assess the
item's
worth can only take place after that important first stage has
been
completed.
--
==============================================================
========
It's not denial. I'm just very selective about
what I accept as reality.
--- Calvin (Calvin and Hobbes)
==============================================================
========
Arturo Magidin
magidin@math.berkeley.edu
Subject: Re: JSH: Smell of rank denial
>Things aren't published until some qualified individuals deem
it worth
>publishing.
> You are also forgetting a very important step: things are not
> format.
Good point. The rare times that I see JSH write lemma or so,
the
verbage following is nothing like a normally expressed lemma.
You think in six (seven? eight?) years he would occasionally
have opened
a math paper and absorbed its style.
V.
--
email: lastname at cs utk edu
homepage: cs utk edu tilde lastname
Subject: Re: JSH: Smell of rank denial
...
> You are also forgetting a very important step: things are not
> format.
> Good point. The rare times that I see JSH write lemma or so,
the
> verbage following is nothing like a normally expressed lemma.
> You think in six (seven? eight?) years he would occasionally
have opened
> a math paper and absorbed its style.
he has not absorbed the style. He tries to mimic, but fails
horribly.
--
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: Smell of rank denial
> My point is that what I've discovered isn't something that
should just
> be totally ignored, while there are posters who seem to have
a career
> of trying to claim that it's not importat *at all* against
all
> evidence.
What evidence? You have produced *no evidence whatsoever* that
your
discovery is of any value to anyone, except you, and then only
in your own
imagination. Put up or shut up. Where's the evidence that your
discovery
is important? So far we have only your *testimony* that it is
important.
(see signature line below)
> Think about it.
> I'm just saying, hey, record the discovery in some math text
or
> journal, while you can see posters jumping up and down like
angry
> monkeys claiming that it's worthless!!!
Monkeys? In a recent post you decried that they were only
human. Why
cry
for someone to record your discovery for you? Publish it
yourself, like
any decent discoverer would.
I reserve the right to determine what has value to me. It is
not *your*
prerogative to determine *my* values. Your discovery has NO
VALUE TO ME,
i.e. it is worthless.
> Funny, eh?
Hysterical. You are, that is.
> Mathematicians are an interesting lot. After having spent
some time
> studying you as a group, I think I have you figured out.
> Yup, I'm saying that I have mathematicians as a group
figured out.
> You need to feel like you're in control. You want to believe
that you
> dominate mathematics, and not the other way around.
You are in danger of practicing psychology without a license
here. That is
a punishable offense.
> If you see something that challenges your *belief* of
control, you
> will fight it, fight it, fight it, against all reason,
against
> mathematics itself, as you descend to a baser level of human
> evolution.
You're now adding psychology/sociology to your list of
failures. Isn't
failing in technology enough? You've already established that
you are a
lousy mathematician and a worse human being. Trying for the
Guiness Book
of Records?
> And I can trigger your descent at will.
The only descent you have triggered is the descent required to
deal with
you at your level.
> You're weak, mathematicians, and you might as well deal with
your
> weakness, or it will destroy you.
Thanks for the warning. Now go back to your playpen. (Remember
you flunked
out of sandbox 101.)
Wacky, isn't it? But hey, it's just basic math. Yup, yup, yup!
--
The second greatest error in reasoning is mistaking evidence
for proof.
The greatest error is mistaking testimony for evidence.
--
--
http://www.crbond.com
Subject: Re: Smell of rank denial
> My point is that what I've discovered isn't something that
should just
> be totally ignored
It is hard to ignore the smell, thus we flush your discovery.
Subject: Help!!!
There is a n-gone (polygon with N sides) regular whose tops are
numbered 0, 1..., n-1, where top I follows the top i-1 in the
anti-clockwise order. For all positive T < = N, an
anti-clockwise
rotation by an angle of 360t/n degrees sends top 0 towards the
top T.
Show that the smallest strictly positive number of
anti-clockwise
rotations by 360t/n degrees which sends to the top 0 to itself
is
n/pgcd(t, n).
Deduct that N rotations anti-clockwise by 360t/n degrees are
necessary
to send to the top 0 worms itself if and only if PGCD(t, n)=1.
Subject: Re: logistic map - periodic locations beyond critical
point
>Thanks - so you are saying that between 3.7-4 its ALL
periodic windows
>?? Thats news to me.. Can you name a parameter value that is
periodic
>and has LOTS of iterates/points ?? I will try 3.900000001
> Are you asking about periodic orbits, or about stable
periodic orbits?
> There should be periodic orbits all over the chaotic region.
> For example, for r=3.9 the logistic map f: x -> r x (1-x)
has two
> unstable 3-cycles (approximately (.1326525274, .4487177896,
.9647439150)
> and (.1809860054, .5780971634, .9512126972). If you want
stable
> periodic orbits, one way is to look for values of r where
1/2 will
> be in a periodic orbit. For example, at r=3.9, (f@@15)(1/2)
< 1/2
> while at r=3.9001, (f@@15)(1/2) > 1/2 (where I'm using f@@n
for f
> iterated n times). By the Intermediate Value Theorem
> there is some r between 3.9 and 3.9001 at which (f@@15)(1/2)
= 1/2,
> and thus there is a stable periodic orbit of period 15 (or a
divisor
> of 15). It turns out to be approximately
r=3.9000686655875783473 which
> has a stable periodic orbit of period 15.
> Robert Israel israel@math.ubc.ca
> Department of Mathematics http://www.math.ubc.ca/~israel
> University of British Columbia
> Vancouver, BC, Canada V6T 1Z2
Dr. Israel,
I am looking for a periodic parameter value that has:
1- the most number of iterates in the map (out of curiosity,
how many
iterates would this parameter generate?)
2- where I can reproduce the exact same iterates every time
3- is finite (not infinite, and not chaotic).
That being the case is 3.99 for example this parameter value,
or are
any of the parameters between say 3.9 -3.9999999999 periodic
and
therefore would work ?
Subject: Re: logistic map - periodic locations beyond critical
point
> Thanks - so you are saying that between 3.7-4 its ALL
periodic
> windows ?? Thats news to me.. Can you name a parameter
value that
> is periodic and has LOTS of iterates/points ?? I will try
> 3.900000001
>> Are you asking about periodic orbits, or about stable
periodic
>> orbits? There should be periodic orbits all over the
chaotic region.
>> For example, for r=3.9 the logistic map f: x -> r x (1-x)
has two
>> unstable 3-cycles (approximately (.1326525274, .4487177896,
>> .9647439150) and (.1809860054, .5780971634, .9512126972).
If you
>> want stable periodic orbits, one way is to look for values
of r
>> where 1/2 will be in a periodic orbit. For example, at
r=3.9,
>> (f@@15)(1/2) < 1/2 while at r=3.9001, (f@@15)(1/2) > 1/2
(where I'm
>> using f@@n for f iterated n times). By the Intermediate
Value
>> Theorem there is some r between 3.9 and 3.9001 at which
(f@@15)(1/2)
>> = 1/2, and thus there is a stable periodic orbit of period
15 (or a
>> divisor of 15). It turns out to be approximately
>> r=3.9000686655875783473 which has a stable periodic orbit
of period
>> 15.
>> Robert Israel israel@math.ubc.ca
>> Department of Mathematics http://www.math.ubc.ca/~israel
>> University of British Columbia
>> Vancouver, BC, Canada V6T 1Z2
> Dr. Israel,
> I am looking for a periodic parameter value that has:
> 1- the most number of iterates in the map (out of curiosity,
how many
> iterates would this parameter generate?)
??? For any n, there are (infinitely many) value of the
parameter with n
iterates
This one has n=15. For godsake, can't you read?
> 2- where I can reproduce the exact same iterates every time
???? How come you couldn't. This is deterministic, after all.
> 3- is finite (not infinite, and not chaotic).
> That being the case is 3.99 for example this parameter value,
No. Read again. The value given is 3.9000686655875783473 . 3.9
is not
clearly periodic, and anyway would only be at the limit
or are
> any of the parameters between say 3.9 -3.9999999999 periodic
and
> therefore would work ?
No, not any. Most of them. To be precise, for any n, any x and
any eps,
there exist y periodic , with period >n, in the interval
]x-eps,x+eps[. Dr
Israel gave you a way of constructing them (and even a proof
that this
method works. Now do your own homework.
Subject: Re: Special Kauffman polynome values
> I wonder whether there are interesting parameter values
> for the Kauffman polynome K(A) (with the writhe variable set
> to 1 as the most basic invariants should be achiral).
> E.g. A=golden mean
> All knots/links have the value +-sqrt(5)^n
> A=-2
^ Maybe the sign is wrong?!
> All knots/links are integer squares
> Surely this has been investigated?
I guess, your version of Kauffman polynomial K(A)
is what other people call the Brandt-Lickorish-Millett-Ho
polynomial
(or sometimes the $Q$-polynomial)
Q_L(x) := F_L(1,x)
(see
http://mathworld.wolfram.com/BLMHoPolynomial.html
http://mathworld.wolfram.com/KauffmanPolynomialF.html )
In this case you can find what you want in the original paper
Brandt, R. D.; Lickorish, W. B. R.; and Millett, K. C.
A Polynomial Invariant for Unoriented Knots and Links.
Invent. Math. 84, 563-573, 1986
where amongst other properties you find that
Q(1)=1
Q(-1)=(-3)^d where d is the dimension of the mod 3 homology
of the 2-fold branched cover of L
Q(2)=delta_L^2 where delta_L is the determinant of L
and Q(-2)=(-1)^{c-1} where c s the number of components of L
The value at the golden mean follows of course from the more
general,
simple formula for the Kauffman polynomial of disjoint unions
F_{L1 cup L2} =[(a^{-1} + a)x^{-1} - 1] F_L1 F_L2
Jones, V. F. R.
On a certain value of the Kauffman polynomial.
Comm. Math. Phys. 125 (1989), no. 3, 459--467.
Stoimenow, A.
Branched cover homology and $Q$ evaluations.
Osaka J. Math. 39 (2002), no. 1, 13--21.
Thomas
Subject: Algebra problem?
If I know that:
3m + 2.5 = 2a + 1.5w = m + 3w = m + a + 1.5 = 2m + w = 4w + a
Is there a way to get the approximate values of the letters?
This is for a point system in a game I am designing and I set
the
equivalencies above as rough guidelines.
I realize there may be conflicts in this formula, but if there
could
be a way to average out the values to get the closest possible?
Thanks
Sonja Elen Kisa
www.kisa.ca
Subject: Re: Algebra problem?
> If I know that:
> 3m + 2.5 = 2a + 1.5w = m + 3w = m + a + 1.5 = 2m + w = 4w + a
> Is there a way to get the approximate values of the letters?
> This is for a point system in a game I am designing and I
set the
> equivalencies above as rough guidelines.
> I realize there may be conflicts in this formula, but if
there could
> be a way to average out the values to get the closest
possible?
> Thanks
There are conflicts in your expression which make any exact
solution
impossible.
I am not sure what you mean by closest possible. Do you mean
that
you want each of the 6 expressions { 3m + 2.5, 2a + 1.5w, m +
3w,
m + a + 1.5, 2m + w,4w + a} to be as close as possible to the
same
value?
Subject: Re: Algebra problem?
>If I know that:
>3m + 2.5 = 2a + 1.5w = m + 3w = m + a + 1.5 = 2m + w = 4w + a
>Is there a way to get the approximate values of the letters?
>This is for a point system in a game I am designing and I set
the
>equivalencies above as rough guidelines.
>I realize there may be conflicts in this formula, but if
there could
>be a way to average out the values to get the closest
possible?
You have five independent equations in three variables, so
your system
is overdetermined. The solution to the first three equations
will not
satisfy the last two. Also, this solution yields a negative
value of
m; is that acceptable?
Let e = (3m + 2.5, 2a + 1.5w, m + 3w, m + a + 1.5, 2m + w, 4w
+ a).
You want to define some criterion function f(e) and values of
a,
m, and w which minimizes f(e). Some possibilities:
1) f(e) = sum (e_i - e_j)^2
2) f(e) = sum |e_i - e_j|
3) f(e) = max |e_i - e_j|
where the sums and maximum are taken over all pairs (i,j) with
1 <= i <
j <= 6. (1) has the advantage that it is easily minimized via
calculus,
and it is easy to do in Mathematica(R) or the like. However,
it will
yield negative values for some of the variables;
Mathematica(R) gives me
2 1159 61
Out[23]= {{a -> -(-), m -> -(----), w -> ---}}
3 1278 639
if I entered everything correctly. If you want to keep all
variables
nonnegative, these contraints add complications.
Standard methods allow the formulation of (2) and (3) as linear
programs. These have the advantage that you cam easily add the
constraints that the variables be nonnegative. In fact, these
constraints simplify the problem in these cases.
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
Subject: Re: Algebra problem?
>> If I know that:
>> 3m + 2.5 = 2a + 1.5w = m + 3w = m + a + 1.5 = 2m + w = 4w +
a
>> Is there a way to get the approximate values of the letters?
>> This is for a point system in a game I am designing and I
set the
>> equivalencies above as rough guidelines.
>> I realize there may be conflicts in this formula, but if
there could
>> be a way to average out the values to get the closest
possible?
> You have five independent equations in three variables, so
your system
> is overdetermined. The solution to the first three equations
will not
> satisfy the last two. Also, this solution yields a negative
value
> of m; is that acceptable?
> Let e = (3m + 2.5, 2a + 1.5w, m + 3w, m + a + 1.5, 2m + w,
4w +
> a). You want to define some criterion function f(e) and
values of
> a, m, and w which minimizes f(e). Some possibilities:
> 1) f(e) = sum (e_i - e_j)^2
> 2) f(e) = sum |e_i - e_j|
> 3) f(e) = max |e_i - e_j|
> [...]
> Standard methods allow the formulation of (2) and (3) as
linear
> programs. These have the advantage that you cam easily add
the
> constraints that the variables be nonnegative. In fact, these
> constraints simplify the problem in these cases.
It is not hard to implement these linear programs in
Mathematica(R),
either. (I do need to get a life.) For (2), I get a = 15/38, m
= 0,
and w = 10/19. For (3), I get a = m = 0, w = 5/8. I assumed
these
must be nonnegative. All results are modulo my entry of the
commands
and data.
If you want strictly positive values, you could choose some
positive
constant c and add the constraints a >= c, m >= c, w >=c.
Also you could modifiy the criterion function by multipliying
terms by
positive weights (if some equlaites are more important than
others.)
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
Subject: Re: Integrability of over an interval
> I found the following exercise in a book:
> If f is bounded on [a, b] and integrable over [c,b] for
every c in
> (a,b), then f is integrable over [a,d] and lim c->a Int
(from c to b)
> f(x) dx = Int (from a to b) f(x) dx . For the first part it
was
> sugested we show that, for every eps>0, there's a partition
P of [a,b]
> such that U(f,p) - L(f,P) <eps, but I found it simpler to use
> Lebesgue's criterion. According to it, for every c in (a,b),
the set
> Dc of discontinuites of f in [c,b] is null (has measure 0).
Therefore,
> for every c we can cover Dc with a countable collection of
intervals
> whose total length is < eps. If 0< eps<b-a is given, make c
= a
> +eps/3, I_0 = (c -eps/2 , c+eps/2) and choose a countable
collection
> I ={I1, I2...} of open intervals that covers Dc and has
total length <
> eps. Since I_0 contains [a,c], it contains all the
discontinuities of
> f on [a,c]. If D is the set of discontinuities of f on
[a,b], then
> adjoining I_0 to I gives a countable collection J = {I_0,
I-1, I-2...}
> of open intervals that covers D and has total length < eps +
> Length(I_0) = eps + eps = 2eps. Since eps is arbitrary, we
conclude D
> is null and Lebesgue's criterion ensures f is integrable
over [c,d].
> Is this proof correct?
> It seems to be correct (I must admit I didn't check all the
details of
> of the proof), but I do have two problems with it.
> Firstly, you're rather reinventing the wheel as you prove
it. All you
> need to show that the set of discontinuities of f on [a, d]
is null is
> countable subadditivity of the measure.
> Secondly it's a bit of a 'hammer to crack an eggshell'
proof. Lebesgue's
> criterion requires a fair bit of work to prove, and there
are much much
> more elementary proofs of this result which would give you a
better
> understanding of what's going on. For example:
> Fix e > 0. Pick y > a 'quite close' to a (where how close
will depend on
> e and an upper bound for |f|). Then by choosing an
appropriately fine
> partition D of [y, d] (which will exist by integrability on
[a, d]), you
> can make the difference in the upper and lower sums on D
union{a} less
> than e. But D union {a} is a partition of [a, d]. Hence
you've found
> such a partition, so the function is integrable on [a, d].
> (You do of course need to fill in the details there).
> Once we've proved the integrability of f, the second part is
kinda
> easy, all we have to do is observe that |Integral (a to c)
f(x) dx| <=
> (c-a) W(f), W(f) = oscillation of on [a,b]. Therefore, when
c ->a the
> second member gets as small as desired. OK?
> With a little bit more work the proof I sketched also gives
you that the
> integral over [a, d] is the limit of the integral over [c,
d] as c->a,
> but it's probably easier to do that second part like you did
it anyway.
> (Or in a similar manner at any rate)
Thank you, David. Actually, it's true the solution you suggest
gives a
better insight than the one based on Lebesgue's criterion.
Amanda
Subject: Re: A TRUE love story
: The Pauling observation is straighforward:
: 1) Do not do it if you are prone to oxalate kidney stones.
Well, I'm out...
--
--
William Dave Thweatt
Robert E. Welch Postdoctoral Fellow
Chemistry Department
Rice University
Houston, TX
thweatt@ruf.rice.edu
dave.thweatt@us.army.mil
Subject: Method of proof help
We have a regular n-gone (polygon with N sides) whose tops are
numbered
0,
1..., n-1, where top I follows the top i-1 in the
anti-clockwise order. For
entire positive T < = N, an anti-clockwise rotation by an
angle of 360t/n
degrees sends top 0 towards the top T.
Show that the smallest strictly positive number of
anti-clockwise rotations
by 360t/n degrees which sends to the top 0 to itself is
n/pgcd(t, n).
Deduct that N rotations anti-clockwise by 360t/n degrees are
necessary to
send to the top 0 to itself if and only if PGCD(t, n)=1.
Subject: Brian Josephson on Strings and ESP
Here is the announcement from a Nobel Laureate in Physics from
Cambridge
University UK :
physics/0312012 [abs, html] :
Title: String Theory, Universal Mind, and the Paranormal
Authors: Brian D. Josephson
Comments: 20KB HTML file. To appear in the Proceedings of the
Second
string theory, anthropic principle, thought bubble, universal
mind,
mental state
Subj-class: General Physics
A model consistent with string theory is proposed for so-called
paranormal phenomena such as extra-sensory perception (ESP).
Our
mathematical skills are assumed to derive from a special
'mental vacuum
state', whose origin is explained on the basis of anthropic and
biological arguments, taking into the need for the
informational
processes associated with such a state to be of a
life-supporting
character. ESP is then explained in terms of shared 'thought
bubbles'
generated by the participants out of the mental vacuum state.
The paper
concludes with a critique of arguments sometimes made claiming
to 'rule
out' the possible existence of paranormal phenomena.
...........................................
Now you know where Carl Sagan got those words billions and
billions
in Cosmos. :-)
The same thing thing goes with the String Theory sales pitch
to the
public.
Give me more money and I will show you the way to paradise....
Yes, Brian Greene is a great salesman! He reminds me of Werner
Erhard
with a PhD in physics. :-)
Joy Christian has pointed out serious philosophical problems
with
string/M-theory. See the book Physics Meets Philosophy at the
Planck
Scale - Penrose and Christian, Baez, Bohmian quantum gravity,
etc...
Latest paper by Christian:
Passage of Time in a Planck Scale Rooted Local Inertial
Structure
http://www.arxiv.org/abs/gr-qc/0308028
Authors: Joy Christian (Oxford)
Comments: 17 pages, 2 figures (included), RevTeX4
It is argued that the `problem of time' in quantum gravity
necessitates a
refinement of the local inertial structure of the world,
demanding a
replacement of the usual Minkowski line element by a 4+2n
dimensional
pseudo-Euclidean line element, with the extra 2n being the
number of
internal phase space dimensions of the observed system. In the
refined
structure, the inverse of the Planck time takes over the role
of
observer-independent conversion factor usually played by the
speed of
light,
which now emerges as an invariant but derivative quantity. In
the
relativistic theory based on the refined structure, energies
and momenta
turn out to be invariantly bounded from above, and lengths and
durations
similarly bounded from below, by their respective Planck scale
values.
Along
the external timelike world-lines, the theory naturally
captures the `flow
of time' as a genuinely structural attribute of the world. The
theory also
predicts expected deviations--suppressed quadratically by the
Planck
energy--from the dispersion relations for free fields in the
vacuum. The
deviations from the special relativistic Doppler shifts
predicted by the
theory are also suppressed quadratically by the Planck energy.
Nonetheless,
in order to estimate the precision required to distinguish the
theory from
special relativity, an experiment with a binary pulsar
emitting TeV range
gamma-rays is considered in the context of the predicted
deviations from
the
second-order shifts.
Looks interesting. Thanks.
ALSO:
Why the Quantum Must Yield to Gravity
Author: Joy Christian (Oxford University)
http://www.arxiv.org/pdf/gr-qc/9810078
----- Original Message -----
Subject: Re: Kleinert/Zaanen World Nematic Crystal
Jack,
Thanks for information. Paper is really interesting. The
appearance
of stringy sources is important, as well as
relation to Ginzburg-Landau (Higgs) model. I intend
to consider such sources caused by the Kerr ring excitations.
Alexander
----- Original Message -----
Subject: Re: Kleinert/Zaanen World Nematic Crystal
Thanks. Interesting that Kleinert comes up with a model that
forbids
torsion fields.
reply to sarfatti@well.com not mindspring if you want me to
see your
message mindspring is
local out server at the Caffe.
Dear Alfred, Jack and Tony :
Undoubtedly UFO, paranormal and consciousness research is a
fundamental
human endeavor that cannot be dismissed as crackpot stuff , as
Weinberg would say.
Fantasy, imagination, dreams, etc... are essential for the
advancement
of knowledge besides being indispensable in the world of human
emotions. Take Jules Verne as example.
I am open about UFO , SETI, paranormal research etc..... It is
plausible that UFOs harness their energy from the vacuum
fluctuations.
I do not know how they do it. Some believe they use volcanic
and
geomagnetic activity, this is why Mexico is such a hot UFO
spot.
I do not think the saucers need volcanos or geomagnetism. I
suspect
(just a hunch based on what Colonel Phil Corso reported) that
their
metric engineering technology is at micro to nanometer scale
all inside
the thin fuselage.
There is a proverb which says : The human mind is like a
parachute, it
only works if it is open .
Dear Brian ( and friends ) :
Despite the initital problems you had in posting the paper in
the archives,
it did appear in the physics section. I guess the moderators
at the
archives realized who they were dealing with !
However, notice that there is no PDF nor PS files available.
Only html ? Usually when you sned a Latex or Tex file it is
automatically converted to PDF and PS files.
Here is the announcement from a Nobel Laureate in Physics from
Cambridge
University UK :
physics/0312012 [abs, html] :
Title: String Theory, Universal Mind, and the Paranormal
Authors: Brian D. Josephson
Comments: 20KB HTML file. To appear in the Proceedings of the
Second
string theory, anthropic principle, thought bubble, universal
mind,
mental state
Subj-class: General Physics
A model consistent with string theory is proposed for so-called
paranormal phenomena such as extra-sensory perception (ESP).
Our
mathematical skills are assumed to derive from a special
'mental vacuum
state', whose origin is explained on the basis of anthropic and
biological arguments, taking into the need for the
informational
processes associated with such a state to be of a
life-supporting
character. ESP is then explained in terms of shared 'thought
bubbles'
generated by the participants out of the mental vacuum state.
The paper
concludes with a critique of arguments sometimes made claiming
to 'rule
out' the possible existence of paranormal phenomena.
...........................................
Subject: Phil Corso, Ed Teller & SDI
PAOLA: A popular misconception has been that somehow UFO
Magazine
Publisher Bill Birnes -- Phil Corso's co-author and editor --
somehow
altered Corso's manuscript and made the aliens appear hostile.
To All,
I asked Bill Birnes about the above and this is his reply.
Don Ecker
**********************************************
Don:
I believe that Corso said the belief was or something to that
effect
that the laser weapons would hold the aliens at bay or at
least pose
something of a weapon that could inflict some damage. Whether
true in
fact or not, Corso is reporting on an underlying belief that
this is
what the weapons could do. That was the premise of that
statement.
Bill
I have heard from a confidential reliable source at NSC level
during
Reagan Era that Ed Teller used that scenario as one reason, of
several,
to go ahead with SDI so that Corso should not be faulted on
that one. It
was an alleged Mission Objective not that we had such weapons
in place
ready to go.
My http://qedcorp.com/APS/EmergentGravity.pdf
certainly points to the possibility that metric engineering by
advanced
ET's is not all that implausible.
Subject: Re: Topology/metric space question...hints, please
>> For a metric space, countable (or sequential) compactness
is equivalent
>> to ordinary compactness.
> Eh? I thought countable compactness was when every open
cover reduces
> to a countable covering...
No: that's to be a Lindelof space.
Best regards,
Jose Carlos Santos
Subject: Re: Topology/metric space question...hints, please
Oh...then is M or any subset of M Lindelof?
>> For a metric space, countable (or sequential) compactness is
equivalent
>> to ordinary compactness.
>Eh? I thought countable compactness was when every open cover
reduces
>to a countable covering...
> No. That's Lindelof. Countable compactness is when every
countable cover
> has a finite subcover.
> --Dan Grubb
Subject: Nobel Prize Physicist on ESP & Physics
http://www.arxiv.org/html/physics/0312012
String Theory, Universal Mind, and the Paranormal *
Brian D. Josephson
Department of Physics, University of Cambridge
Cavendish Laboratory, Madingley Rd, Cambridge CB3 0HE, U.K.
http://www.tcm.phy.cam.ac.uk/~bdj10
ABSTRACT
A model consistent with string theory is proposed for so-called
paranormal phenomena such as extra-sensory perception (ESP).
Our
mathematical skills are assumed to derive from a special
mental vacuum
state, whose origin is explained on the basis of anthropic
and
biological arguments, taking into the need for the
informational
processes associated with such a state to be of a
life-supporting
character. ESP is then explained in terms of shared thought
bubbles
generated by the participants out of the mental vacuum state.
The paper
concludes with a critique of arguments sometimes made claiming
to
rule
out the possible existence of paranormal phenomena.
Keywords: ESP, string theory, anthropic principle, thought
bubble,
universal mind, mental state
* To appear in the Proceedings of the 2nd. European Samueli
Symposium,
1. Introduction
Critics of claims of the paranormal, e.g. Deutsch (2001), have
declared
extrasensory perception (ESP) or other paranormal phenomena to
be
nonsense . Such absolutist positions give
little weight to the
experimental evidence (Radin 1997) in support of the reality
of such
processes, and seem naive given the range of imaginative
proposals
concerning the nature of reality currently being put forward
for serious
consideration by conventional physicists. One important
advance has
been the superseding of the so-called Standard Model as a
fundamental
theory of nature by string theory
(http://superstringtheory.com), where
the Standard Model features merely as a subset of the set of
permitted
alternative Randall-Sundrum picture) has suggested, such a
change in
perspective opens up new possibilities in science, including
the
possibility of accommodating paranormal phenomena within
physics. In
the following a number of concepts are combined, each in
essence
consistent with accepted ideas, resulting in a qualitative
explanation
for ESP, with the promise of an eventual clear cut basis for
understanding paranormal phenomena in general.
2. A separate mental reality
A key assumption we make is one which, while it has no clear
connections
with experimental physics, does make contact with a position
that was
advocated by mathematicians such as G.9adel (Davis and Hersh
1981, Penrose
1994). This is the idea that some aspects of mentality involve
a realm
of reality largely, but not completely, disconnected from the
phenomena
manifested in conventional physics. The idea of a disconnected
realm
does have precedents, for example in the way two of the
fundamental
forces (the strong and weak forces) play no role in large
areas of
physics and chemistry, whilst in other contexts they have a
very
important part to play. Next note that string theory,
involving as it
does spaces having more dimensions than the usual three, and
also a
large number of such states), is consistent with there being
such a
separate realm, in a way that the Standard
Model, with its unique
vacuum state contained within a limited number of spatial
dimensions,
did not.
The point in regard to mathematical thinking, which motivates
our model,
is the following. Consider first of all what the brain does in
visual
perception. Here the primary information from the visual
receptors goes
through various levels of processing until it ends up as a
high-level
representation of the content of the visual field. It is not
unreasonable to identify mathematics as a similar process,
except that
higher levels of abstraction are involved in this case. With
the visual
case, the mechanics are straightforward: the visual field
typically
contains for example edges, for which abstraction a dedicated
neural
system has evolved, related to our ability to perceive edges.
It is
hard to see why we should have such ready access to higher
mathematical
abstractions having little connection with experience (Penrose
1994).
One resolution of the problem would be for mathematical
concepts to be
in some way in the physics, rather than being
emergent properties of
brains. In case it is felt that such a drastic solution is not
necessary to explain our ready access to mathematical ideas,
and that
neural networks can provide an adequate explanation, a
stronger argument
for the existence of some kind of Platonic realm can be made
on the
basis of the aesthetic aspect of music (Josephson and
Carpenter 1996).
So far, in shifting the locus of mathematical thinking (and
music?) to
another realm, we have only replaced one mystery by another.
But why
should such a realm exist at all? The explanation we provide
is of a
biological character, taking account of the fact that
information
processing is an essential component of biological
functioning, but with
only very specific informational processes having a
life-supportive
character. While it is commonly taken that the informational
processes
involved are mediated by ordinary physical means, it is not a
logical
necessity that this should be the case. Some informational
processes in
an organism are specialised to the nature and circumstances of
the
organism concerned, but some have a more abstract and universal
character, and so could be mediated by a quite different
system with
which individual organisms would interact.
Next we observe that a form of proto-life, defined as
fluctuation
patterns surviving longer than typical patterns do, can be
hypothesised
as occurring at the Planck scale, evolution of such life being
expected
to involve evolution of the accompanying informational systems
also. We
get to the proposed model by supposing that the ordinary
physical
component and the informational component can evolve
separately. and
that the informational component can even survive the creation
and
destruction of individual universes, remaining as an
ever-present
background with which new universes, Planck scale fluctuations
and more
developed life forms can all beneficially interact. Assuming an
indefinitely extended time scale, the most persistent part of
the
informational background can evolve indefinitely, so that its
dynamics
might come to include features corresponding to mathematical
concepts
and operations as well.
This idea can be fruitfully connected with anthropic ideas,
particularly
our universe seems to be mysteriously fine tuned to develop in
such a
way that life is possible in terms of it being only one of a
vast number
of coexisting universes, a small proportion of which have such
a
property, one of which we find ourselves occupying. Our
speculations
can be seen as the application of a similar idea to the
informational
aspect of life.
While Susskind treats life as a passive occupant of whatever
universe
can permit it to develop, our proposals see life in a more
general
light, able to shape its environment in a partnership with it,
in a
manner analogous to the proposals of Lovelock (1995) (the Gaia
hypothesis, for which there is now considerable supportive
evidence), to
the effect that life may be able to interact cooperatively
with its
environment, discovering how to operate upon it to its best
advantage.
3. A model for ESP
We need to add another piece of detail to our model. In order
that it
can model individual thought, we suppose that individual life
forms can
perturb the background state so as to create a localised
thought
bubble, tied to the individual concerned. This suggests
that the
vacuum state involved is close to a phase transition, so that
an
appropriate perturbation can create a domain with a different
kind of
order to that of the vacuum.
Assuming the validity of the scenario that has been described,
the
picture proposed can be adapted to account for the phenomena
we set out
to explain, namely telepathy or ESP. In the first, the grounds
for the
existence of such a process can be taken to be the advantages
that might
be conferred in certain situations if two life forms could in
some way
share their mental states (there could also be accompanying
disadvantages, the significance of which will become clear
later). It
is natural to postulate, in this case, that a shared mental
bubble,
whose contents are available to both life-forms, is involved.
We
assume, as would need to be assumed generally in the model,
that the
state of this bubble plays the role of information that is
meaningful in
the context and, by virtue of this, usable by the connected
systems.
The physics involved in the sharing that has
to be assumed in the
above can be clarified by means of an analogy based on the
M.9assbauer
effect, which is a phenomenon involving the decay of
radioactive nuclei
embedded in a crystal (M.9assbauer 1961). In a certain
fraction of cases,
depending on parameters such as the decay energy and the
temperature,
the recoil from such a decaying nucleus is in effect
transmitted to the
crystal as a whole rather than generating activity in the
vicinity of
the decay. These no local recoil processes
involve a certain subset
of all possible final states of the system, for which, as
quantum
mechanics allows, the state of the lattice vibrational system
(phonons)
is unchanged by the decay. This somewhat esoteric possibility
suggests
a mechanism, dependent on analogous constraints upon the
possible states
of the thought bubble, that could fit our requirement of a
system state
being shared by two individuals as in the ESP situation.
4. Countering the critics
The problem any such analysis has to face is that of
explaining how it
is that, if such a mechanism for ESP or other paranormal
processes
exists, these processes manifest themselves only in very
specific ways,
and in ways that are not readily controllable. This should not
be seen
as an insuperable objection, since other phenomena (e.g. those
involving
the weather), have similar features. The point to bear in mind
is that
in the biological realm the phenomena that manifest are
governed not
only by what is physically possible, but also by which of those
physically permitted possibilities are likely to be of overall
benefit
to the organism concerned. In the ESP context, an
undifferentiated
sensitivity to the thoughts of all other people, as would
result from
the uncontrolled sharing of thought bubbles, would tend to be
disadvantageous rather than of benefit, leading to the
individual being
overwhelmed by thoughts of others. Most of these would be
merely
distracting, and interfere with constructive activity. The
right way to
think about ESP is therefore to see it as a slowly developing
phenomenon
for a given individual, and one which may not develop at all if
conditions are unfavourable. We see from this analysis that the
frequently made counter-argument to the existence of ESP, that
if it
were possible it would have such a survival value that we
would all
evolve to be very good at it, is based on a misleading concept
of what
would be involved.
A related problem is the one raised by Weinberg (1993), who
asks what
possible physical signal could move distant objects and yet
have no
effect on scientific instruments? Such a question ignores the
possibility that there might be a threshold for psychokinetic
effects.
A similar argument would lead one to be equally sceptical of
claims that
the heat of the sun can induce chemical reactions (i.e.
burning) in a
piece of paper, analogously something that happens only under
special
circumstances (e.g. using a magnifying glass to focus the
suns rays on
to a spot on the paper), the amount of burning under normal
conditions
being negligible.
The moral to draw would seem to be that one should not be too
ready to
dismiss paranormal phenomena on the basis of apparently
plausible
arguments; as t Hooft (2001) has said in a slightly
different context,
plausible arguments come with their own small
print, viz. assertions
to the effect that assumptions that seem reasonable to their
authors may
be violated in the real world.
5. Concluding comments
This work was motivated primarily by the perception that the
arguments
commonly made against the possible existence of paranormal
phenomena are
not well-founded, suggesting a need to discover how they might
be
accommodated within conventional science. Proposals with this
aim have
been made previously, based upon Bohms causal interpretation
of quantum
mechanics (Josephson and Pallikari-Viras 1991, Valentini
1991), but the
fact that the causal interpretation of quantum mechanics has
not
developed in ways relevant to current scientific concerns
suggests it
may be more fruitful to look elsewhere for ideas. The present
paper is
the outcome of such an investigation. Clearly, it is at best a
sketch of
a theory, since the arguments are of a very qualitative
character, but
this qualitative sketch brings to light a number of specific
issues
whose resolution may provide the basis for a more complete
account of
the phenomena.
6. Supplementary remarks (added after submission of paper for
Proceedings)
Susskind's arguments suggest that reality may be much more
complex than
has normally been assumed. Further changes in fundamental
science
(which may include consideration of the influence of life) may
be
required to address this complexity. Since our proposals (such
as
thought bubbles emerging from some kind of background) do not
involve
the precise details of string theory, they may survive any
such changes
that fundamental science may undergo.
7. Acknowledgements
The author is indebted to Dr. Fotini Pallikari for many
illuminating
discussions concerning the nature and mechanisms of ESP. No
funding
from counter-innovative sources was involved with the
preparation of
this paper.
References
B. Carr (2001), Can physics be extended to accommodate
psi?,
Proceedings of the 22nd Annual International Meeting of the
Alternative
Natural Philosophy Association, ed. Arleta Griffor, ANPA.
P. J. Davis and R. Hersh (1981), The Mathematical Experience,
Brighton:
Harvester Press.
D. Deutsch (2001). quoted in Robin McKie, Royal
Mails Nobel guru in
telepathy row, The Observer, September 30, 2001,
http://observer.guardian.co.uk/uk_news/story/0,6903,560604,00.
html
G t Hooft (2001), How Does God Play Dice?
(Pre-)Determinism at the
Planck Scale, arxiv:hep-th/0104219
B.D. Josephson and T. Carpenter (1996), What can Music tell
us about
the Nature of the Mind? A Platonic Model, in Toward a
Science of
Consciousness, ed. S.R. Hameroff, A.W. Kaszniak and A.C.
Scott, 691-694,
B.D. Josephson and F. Pallikari-Viras, Found. Phys., Vol. 21,
pp.
197-207, 1991,
http://www.tcm.phy.cam.ac.uk/~bdj10/papers/bell.html
J. Lovelock (1995), The Ages of Gaia : a biography of our
living earth
(2nd ed.) Oxford: Oxford University Press.
R.L. M.9assbauer (1961), Recoilless Nuclear Resonance
Absorption of Gamma
Radiation, Nobel Lecture,
http://www.nobel.se/physics/laureates/1961/
mossbauer-lecture.pdf
R. Penrose (1994), Shadows of the Mind, Oxford: Oxford
University Press.
D. I. Radin (1997), The conscious universe : the scientific
truth of
psychic phenomena, New York: HarperEdge.
S. Weinberg (1993), Dreams of a Final Theory, London:
Hutchinson Radius.
arxiv:hep-th/0302219.
A. Valentini (1991), Physics Letters A158, 1-8 (abstract at
http://www.fourmilab.ch/rpkp/valentini.html)
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Subject: Re: Nobel Prize Physicist on ESP & Physics
i have sample data that proves ESP if you'd only reply you
would probably
have the capablity to assess it.
>crazy moon lovers!
>Herc
///////////
all you moon-huggers!
-----------------
Greg Neill
scribe2b
Roundtable
John L
--------------------------------------------------------------
--------------
----
Examine what is written in each post, because these are all
chronological
twist in each message. As I have attempted to explain, within
my
environment
reality is described for me.
The 4 names follow the post, 1 of them is the real author, in
this example,
since the post contains a number of scribes ////// the answer
is scribe2b.
To make the test much smaller, where possible only the authors
comments
from the post are shown.
--------------------------------------------------------------
--------------
----
>this is another version :
>I can guess peoples names by analysing what they reply to me
>in newsgroups providing their reply is not contrived
>Doesn't matter what I say, 20 emails to ozskeptics saying
>half a dozen times I can guess names, 5 emails to Randi
>saying I can guess names, 10 emails to ukskeptics I can
>guess names, 20 posts to sci.skeptics I can guess names.
>so can I, not a testable claim, doesn't say how, doesn't
>say why, doesn't say where.
>I give a possible test scenerio to all of you, you all say its
>up to us to make the test, look at the claim, according to
>that you can pass this, oh no, now you've changed your
>claim disqualified. Its a claim which I CAN prove, its
>not the proof mechanism, its a CLAIM. I can do it *somehow*
>not every which way but loose.
>I come here and say I can do paranormal, look a match question
>baker makes and you all refute me 50 different ways, I
>answer every one. after the fact you say, not 1 in 6. ok so
>I set up a test and guess what, get 1 in 6 again and this time
>its not an assertion after the fact, 1 in 36, oh now the judge
>is paranormal, what next?
>Your so entrenched with scientific method you can't say, what
>if we allow the skeptics clause even if we don't understand
it.
>Science is made going against the grain.
>Herc
So if I get one of my kids to post under my name- because my
name is
obvious (and only one kid, always the same one), you will be
able to
guess *his* or *her* name, right? not the surname, but the
first and
middle names?
-----------------
patty-anne-lea
Wally Anglesea
Matt Giwer
J.y.n.x
--------------------------------------------------------------
--------------
----
The bumblebee has, like a helicopter, insufficient wing area
to fly.
My parrot can fly, but then he has wheels.
The wrens visiting my bird feedeing station don't have wheels
so I'm only fantasizing that they can fly, I suppose.
Just thinking out loud ...
Wood stumps warm me thrice,once while splitting them, once
while they are on
the
fire, and again when I read their posts to usenet. (Henry
David Thoreau).
-----------------
NormDePloom
Edward Caruthers
malcolm burton
Lawrence & Bobbie
--------------------------------------------------------------
--------------
----
The problem is that everyone has the answers in advance.
That's not a test, that's a demonstration. In a test,
have access to the answers until *after* the test.
Nothing you described so far possesses this property.
-----------------
Chas
Xcott Craver
Roundtable
Odysseus
--------------------------------------------------------------
--------------
----
Gee Zedenk, that's real interesting. A couple of questions
though. Where
did the continuum come from? Who or what lit the fuse that
caused the
spark? What set in motion the totality that we call the vacuum
of space
and
LOL! At the end of it all, physics and science are no more
than Bro'
Rabbit, and the mystery of God is Tar Baby :) No matter how
much a man
rambles and babbles, he cannot escape the reality of a First
Cause.
Think on this Zedenk. Take the simplest of things, the
Hydrogen atom. One
proton, and one electron orbiting it. Now apply the trinity
theory to it.
The physical existence of the proton and the electron are the
Son. The
perpetual motion of the electron flying in it's orbit is the
Holy
Spirit
or will of God. The force that holds it forever in place and
does not let
it collapse into the center or fly off into space is the
Father or mind
or
Law of God.
Now set this in your mind Zedenk. If even for a fraction of a
second, any
of these 3 realities ceased to work, the universe would
explode! Your
universe Zedenk, has nothing in it's existence to do with
physical
properties or time or this dimension, it exists and continues
to exist
because something unexplainable has not withdrawn the Divine
Power that
glues it all together! Chaos is not a reasonable explanation
for
never-ending Law!
C YA,
-----------------
Greg Neill
Wanda
sertec
Mitch Dickson
--------------------------------------------------------------
--------------
----
> I read that if you put something large and highly visible in
your
> yard, say a bright tarp, you can release your birds and
> they can find their way home. Is this worth experimenting on?
I would have to say a big NO. A bird is gonna get confused and
scared and
panic and just go! Not to mention preditors and such. Now, if
you're
talking homeing pigeon, then maybe, but again, predators must
be taken into
account.
-----------------
Someone
Mercury481
Lawrence & Bobbie
Tim Kozusko
--------------------------------------------------------------
--------------
----
Go for it.
-----------------
Lawrence & Bobbie
Ralph Hertle
J.y.n.x
The Pervert
--------------------------------------------------------------
--------------
----
:^) if any of you had a half open
Not me, mine is all the way open, and yours ?
You must not have seen my post advising that
typos should be ignored - not trolled.
Open your mind - if you can -
:^) Absolutely not. Intolerance is !
-----------------
Hold, I'll think of it in just a nanosecond or two
Ralph Hertle
John L
TheKid
--------------------------------------------------------------
--------------
----
>He said email him so maybe wont get your reply. The theory
seemed
>alright, and even though SETI purposes you are right that
doesn't
>discount the figure of only 100 planets in Milky Way that we
could
>reach escape velocity? Anyway planets are becoming
increasingly
>common even nearby so I doubt 100 billion stars would be near
>empty at all. Me, still working on my age halting serum, few
centuries
>to enjoy the lower G on Mars then maybe retire for a few
millenia
>enjoying the view from one of Saturns larger moons.
>Herc
The theory seemed alright but only assuming the assumptions he
(we) makes.
The
assumptions of the scientist at the SETI institute are most
likely more
plausible that those by an unknown person in as obscure
newsgroup who
frequently spams multiple groups with off the wall crackpot
ideas. But
hey,
what do I know.
-----------------
First Name
Hold, I'll think of it in just a nanosecond or two
Mercury481
Greg Evans
--------------------------------------------------------------
--------------
----
> Sound is a mechanical compression transmitted through a
material.
> The speed with which a mechanical compression is transmitted
through a
> material is called the speed of sound.
> For copper this is about 3560 m/sec at 20 C.
> but if you push a 3560 m rod of copper a foot, the other end
> will push out one foot in much less than a second. Sound is
the
> transmission of vibration, not the compression of the
structure.
> If you hit the rod with a sledge hammer, at the other end it
would
> move and one second later you would hear the noise. Actually
> one second minus a very small amount for the structural
compression.
> Herc
Thank you for better describing my thought
-----------------
Chris
raven1
sertec
Chas
--------------------------------------------------------------
--------------
----
///////////
all you moon-huggers!
-----------------
Greg Neill
scribe2b
Roundtable
John L
--------------------------------------------------------------
--------------
----
I've spent basically my whole life at 28 degrees lat. and have
gotten quite
good at telling the time of day by the sun. In the past couple
years I've
been to England three times and was baffled by the sun there -
even knowing
to expect it. It was wild there in February actually being
able to feel
the
day getting longer. That was different.
Also different was the snow we got today. The Space Center had
enough that
it piled up on cars.
-----------------
CNote
Mitch Dickson
Tim Kozusko
J.y.n.x
--------------------------------------------------------------
--------------
----
The evidence based on metallurgical analysis of fractured
surfaces
(produced by Geller) indicates that a paranormal
influence must have been operative in the
formation of the fractures.
Dr Wilbur Franklin (Physics Department,
Kent State University - U.S.A.)
We have observed certain phenomena with
...
-----------------
G=EMC^2 Glazier
CNote
malcolm burton
Greg Neill
--------------------------------------------------------------
--------------
----
> hey I'm the one solving dillemmas round here, contribute
> something useful yourself. its been well discussed b4 that
> odd binaries are allowed here.
No.
> formal rules such as attachments
> only in binaries named groups are easily programmed into
> news servers yet they dont actually exist do they?
> now which part of too bad didn't you get?
> Herc
Welcome to the brackish depths of my killfile, Sparky.
*plonk* ...problem solved.
-----------------
Rich Shewmaker
Ian
Greg Neill
scribe2b
--------------------------------------------------------------
--------------
----
thanks for the suggestions
the plastic was held by the magnet with a small metal
superficial insert
the bottom of the tear drop was the same proportions as the
sphere, and in
all was much heavier
I was considering the possibility of the teardrop wabbling and
causing
turbulence on its first moments of fall, but wouldn't that
have showed up
as
a cubic term on the plot?
thanks
> some ideas :
> how is a plastic sphere being held by a magnet?
> the magnetic field will remain after the voltage is cut
> and different shapes and materials will drop sooner
> the weight (also size) will influence the fall, a teardrop
> might be faster than a sphere in general but lighter
> material will fall slower in atmosphere in general aswell
> maybe your teardrop doesn't maintain vertical at low speeds
> Herc
> Someone HAS to know what's going on.
> If it helps, it's not anything wrong with our particular
apparatus
because
> similar phenomenon happened with other groups on different
workstations.
> I'm doing a first year university physics lab on free fall..
the
apparatus
> is simple. An electromagnet positioned above two, moveable,
photogates. When
> the voltage is cut, the object falls and the time interval
between the
two
> gates is measured on a timer accurate to 0.1 ms. We used two
different
> objects to examine their free fall, we used a plastic sphere
and
a
> streamlined (i.e. looks like a teardrop) steel object.
> We hypothesized that the plastic sphere would feel greater
drag during
its
> fall, and would thus, take longer to fall.
> However, when we performed the lab, we found that the
plastic sphere
was
> consistently falling approximately 5-15 ms faster than the
steel (for
> heights of 50-200 cm).
> When we fitted the two sets of data our confusion grew as
the plot for
the
> plastic sphere had a cubic term, representing the 1/3 the
drag
coefficient,
> while the steel did not. (verified by chi-squared results).
 Can anyone explain this phenomenon?
> Thanks
-----------------
Shanx
Wanda
TheKid
raven1
--------------------------------------------------------------
--------------
----
|-|erc:
The picture looks great.
Interestingly, the ill-logic
of the basic design conguration
is more visible in the sketch
than in photos. The problems
would be that, numerous strong
points in the fuselage are needed
to support the heavy components
that are exterior to the main tank
and to resist and distribute high
local stresses that are due to the
high forces that are applied to the
structural components.
A vertical tower provides for the
inline stacking of components. The
symmetrical design of the USSR rocket
system would probably have far less
structural problems.
Neat sketch, however.
The unit spacing worked just fine.
Thanks,
_________________________________
>>|-|erc:
>>Please advise us regarding the font setting that
>>you used in making the picture. We can then select
>>the same font in order to see the picture as you
>>intended.
> this is the original ascii art, I altered it for my default
outlook font,
> but if you are adjusting font this one is better (any
monospaced) :
> /
> /
> /
> /
> /
> ^ /__________ ^
> / | | /
> /___| / |/___
> | || / || |
> | || /____ || |
> | || // || |
> | || //______ || |
> | || | || | || |
> |___|| | || | ||___|
> | || | || | || |
> | || | || | || |
> | || | || | || |
> | || | || | || |
> | ||/| || ||| |
> | |/ | || | | |
> | / | || | |
> | / | || | |
> | / |___||___| |
> |/ | | |
> / |___||___|
> (______|____||____|______)
> /___ /_/||/_ /___
> // //
> //// ////
> //// // //// //
> /// // // // //// //// /
> / / ///// / //// // / //
> / /// // // /// /// /// / //
> / // / /// //// / //// // //
> /// /// //// // // ///////
> / // / ///// ///// / /// // / /
> /// /// / /// / // //// // // / /
> // // ///// /// // //// ///// // /
> ////// //// // // /// // ////// //
> /// / ///// / /// // // ///// /// /
> -Jas
-----------------
Rust
Ralph Hertle
Mercury481
PlanetaryMatrix
--------------------------------------------------------------
--------------
----
> thats really good, if you have a spare 30 secs download it ,
hollywood
> doesn't do much better.
I will humbly accept the compliment even though I am never
humble.
http:www.sworld.org/artiv/fs.mpg is the next scene.
I was very surprised with how good it looked with almost no
effort. Of
course
comments on what is unrealistic are appreciated as I want to
improve it.
-----------------
Tim Kozusko
Matt Giwer
Chris
Apostate
--------------------------------------------------------------
--------------
----
Yep, I took the data from both sources for 1 bar
-----------------
Someone
Lawrence & Bobbie
Odysseus
malcolm burton
--------------------------------------------------------------
--------------
----
> First of all I need a volunteer from the audience,
> give me the name of any rec newsgroup!
> Dont be shy, say you there, any rec newsgroup, step
> right up and see real magic!
> Herc
You're doing it wrong.
Please immediately purchase a copy of Jim Cellini's DVD and
view it five
times
in a row. (Potty breaks are allowed; it's a long video.)
It'll be a good start for you. The rest of us are praying,
burning white
candles, chanting, thinking positive thoughts, and/or invoking
any number
of
Chopra Quantum Whatevers to help bring about the release of
volume two
sometime
real soon now.
-----------------
Rust
John L
Greg Evans
Ben Sauvin
--------------------------------------------------------------
--------------
----
Randi will test you when you properly apply to be tested. Sign
up here:
http://www.randi.org/research/challenge.html
-----------------
Rich Shewmaker
CNote
Wanda
Rust
--------------------------------------------------------------
--------------
----
It really all depends on the situation.
-----------------
Shanx
See You In Hell My Friend.
Someone
Greg Neill
--------------------------------------------------------------
--------------
----
If ever I actually found myself in that situation, I'd hold it
upright,
with the intent of attacking my assailant's knife hand.
-----------------
cliff86
Rust
Shanx
NormDePloom
--------------------------------------------------------------
--------------
----
If you 'found yourself' in a knife fight, work with the grip
that
comes fastest to you in an ambush situation. I prefer the
reversed
grip for obvious reasons.
If you have more time, or are the aggressor, you probably want
it
blade forward, edge out.
-----------------
Ian
Mercury481
Rich Shewmaker
Chas
--------------------------------------------------------------
--------------
----
This is fascinating stuff.. hopefully in a few decades I will
be a Guru
like
you guys :)
-----------------
Saad Malik
Chris
Matt Giwer
John L
--------------------------------------------------------------
--------------
----
> Yeah I know, I only posted here on a bet.
You lost, eh?
-----------------
cliff86
Greg Evans
beavith
scribe2b
--------------------------------------------------------------
--------------
----
Should and do are often very different things.
If you've been hard of hearing for a long time, you'll
probably understand when I claim that odd noises can be
distracting.
I'm very comfortable driving around with my natural hearing,
being
unable to hear all the various tickings and bangings and
screechings
of my own car, or those around me.
I tried driving while wearing a hearing aid, once, and was
almost startled out of my skin when a truck right next to my
good ear
let loose a blast of air as its brakes actuated.
In controverse argument, I offer the legions of people who
insist on driving while running their mouths on their stupid
cell
phones.
I wear glasses. I do not know the numbers to describe my
visual deficit, but while I can operate comfortably without my
glasses
when I'm not behind the wheel, there are very few
circumstances under
which I would ever be caught behind a steering wheel without
the
glasses.
-----------------
Ben Sauvin
First Name
Tim Kozusko
G=EMC^2 Glazier
--------------------------------------------------------------
--------------
----
> Will a float in an aquarium go down a bit when the aquarium
is put in
> a fast elevator that starts moving up? Why is this or isn't
this?
> Thanks in anticipation.
> Mv
> As the elevator starts to move up the float will rise a
little.
> Then it will sink a little when the lift stops.
> This is because the apparent gravity rises shortly as the
elevator
> accelerates upwards then returns to normal and it goes down
a bit when
the
> elevator is decelerating.
> As the gravity rises the net upward force of the water on
the float
rises
> by the same amount as the increased force down from the
weight of
> the float, cancelling out.
> Herc
Yes I agree with the theory. Still, I have actually seen what
I described.
(At about 2g) but know I am doubting if I remember correctly.
Maybe what I
saw whas the other way round.
-----------------
Terry Wilder
Edward Caruthers
(Hildo) news multikabel
Odysseus
--------------------------------------------------------------
--------------
----
its those added SRB's. ugh! i don't think its man-rated like
the
saturn 5 was.
-----------------
See You In Hell My Friend.
sertec
beavith
Edward Caruthers
--------------------------------------------------------------
--------------
----
[expletives deleted] I did at first ... :(
Thanks for catching that ... I guess. ;)
OK, that should be between 81.6 and 93.3 million straight-line
trips,
and OTOH the original statement implies a round trip to be
about 1.03
million km. So while there's not as much of a discrepancy as my
regrettable error made me think, it's still large enough not to
require high-precision comparisons -- just a little care with
exponents. ;)
-----------------
Saad Malik
PlanetaryMatrix
CNote
Odysseus
--------------------------------------------------------------
--------------
----
At least we know why you are using a computer to
communicate...you're not
allowed any sharp pointy things, such as pens, right?
-----------------
Wally Anglesea
beavith
raven1
malcolm burton
--------------------------------------------------------------
--------------
----
Well it is possible that the shape of a rain drop is not the
best way to
enter the atmosphere.The reason is the front of the rain drop
would
always be getting the friction(heat) Nature creates that shape
when
drops of water are moving through the air at a top speed 0f
200 mph
Lets think about this idea of mine. Back to a round ball,and a
round
ball inside this ball. The outer ball made to rotate,by
electro-magnetisum,and the inner ball(cabin) not rotating.
This is done
very easy. Outer ball changing its surface by rotating is much
like
passing your finger through a flame. There is a difference of
high
temperature and the build up of heat. Bert
-----------------
Kurst
G=EMC^2 Glazier
Odysseus
Chas
--------------------------------------------------------------
--------------
----
If you are not trying to argue anything, why do you keep
posting to me? I
never said you were a liar, so stop trying to convince me that
you're not.
I
really don't care if The Truman Show is based on your life, if
you
predicted
the upcoming war, or if your dick falls off, so please leave
me out of any
further discussion.
-----------------
Xcott Craver
J.y.n.x
Saad Malik
Wanda
--------------------------------------------------------------
--------------
----
<PLONK
-----------------
scribe2b
Greg Evans
Mitch Dickson
Lee S. Billings
--------------------------------------------------------------
--------------
----
It's also much farher away from the Sun than the moon is, so
Titan
doesn't have as much of solar blasting that stripped closer
planets.
-----------------
Impmon
gecko
scribe2b
Someone
--------------------------------------------------------------
--------------
----
Couldn't time be defined as the passage of an instance, no
matter how small
that passing was? Wouldn't the passing of, let's say, one
billion billion
billion billion billion billionth of a second taking a
trillion years to
complete (because time itself was slowed down to such an
extreme), still be
called time since an instant did pass? Just thinking... but
the observer
would never know time was slowed to this extreme, since the
observer most
certainly would also be slowed down, and time would seem
normal to him.
Then you could really never know the age of the universe to
any accuracy,
since the fluctuation of times' length could not be calculated
and added to
the perceived time passage the observer experiences.
-----------------
G=EMC^2 Glazier
PlanetaryMatrix
Odysseus
gecko
--------------------------------------------------------------
--------------
----
Hmm which mint made the coin ?
> If I use my hand to push a coin across a table, can you
pinpoint the
cause of the
> coins movement?
> I think you would say the fundamental cause is the decision
to move the
coin, or
> would it? Could you ever pinpoint the cause?
> the decision happens 1 to 1.5 seconds before the movement,
and can
> be detected, i.e. predicted.
> There are two classes of theories : your person is an
idependant being
that
> made this decision, probably some meta scientific training
that made you
> think of it and inspired by an unresolved thirst for mothers
milk.
> Alternatively your interactions within the universe are
entwined within it
and
> the cause was distributed beyond a single comprehension,
perhaps the
future
> outcomes of the coins movement has a greater effect, such as
getting me
to
> type upon it.
> Herc
-----------------
David H. Lipman
Kurst
Lee S. Billings
Mitch Dickson
--------------------------------------------------------------
--------------
----
what?
-----------------
malcolm burton
gecko
Edward Caruthers
Ben Sauvin
--------------------------------------------------------------
--------------
----
--Agreed. But there are lots of strange things going on due to
--network congestion caused by the worm.
--- You must think this, look you, that the worm will do his
kind.
:-)
-- I refuse to write LOL since that always makes me think of
someone
-- lolling round on a sofa, one arm trailing on the ground,
eyes turned
-- upwards in their sockets, his tongue lolling out of his
mouth, making
-- lollll...lolllll...lollll noises like a nutcase.
--
-- But I laughed out loud.
--
-- Without lolling.
- for years I thought the internet was a friendly place, all
these
people
- sending me lots of love!
-
- Herc
Ahhhhhh.... ha ha ha ha ha ha ha!
That really made me laugh out loud.
Big hug, big kiss - after all, you deserve it.
-----------------
Roundtable
sertec
(Hildo) news multikabel
gecko
--------------------------------------------------------------
--------------
----
OOC: Eat a fig, figgy.
-----------------
cliff86
Saad Malik
Terry Wilder
Rust
--------------------------------------------------------------
--------------
----
Oh, so you're qualified/annointed/authorized to speak for
people as a
whole now? Are you further saying that the opinions you
expressed are
not
what you really believe, but are merely immature bull assumed
(for what
reason I'm not entirely sure) to be the opinion of the people
as a
whole?
> Its easy to sit in the top economic section of society and
say everything
is dandy.
I wouldn't know about the top economic section of society, but
it's also
easy to look at reality and say Life is good. If it isn't for
you,
well,
that's a shame. And if you have to blame women or feminists
for your
inadequacies, that, too, is a shame. (I lied about that last
part.)
-----------------
TheKid
Mercury481
The Pervert
(Hildo) news multikabel
--------------------------------------------------------------
--------------
----
<snip schizophrenic ramble
Someone's off their meds, I see.
-----------------
The Pervert
raven1
Lee S. Billings
scribe2b
--------------------------------------------------------------
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----
Could be too much medication. Like Chief Wiggam said to
Ralphie, if
your nose starts bleeding it means you're picking it too
much... or
not enough.
-----------------
Gary Rockley
PlanetaryMatrix
gecko
NormDePloom
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--------------
----
Only under the right circumsatnces.
-----------------
Lee S. Billings
Lawrence & Bobbie
NormDePloom
PlanetaryMatrix
--------------------------------------------------------------
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----
I was just thinking that it would be cool to put a pin hole
camera in the
bow of a fast boat and drive it using VR glasses.
-----------------
Edward Caruthers
Mercury481
Ralph Hertle
First Name
--------------------------------------------------------------
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----
You may know a lot about the cosmos but you can't spell to
save your soul.
The word is prodigy not prodogy.
-----------------
NormDePloom
TheKid
John L
First Name
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----
Basically...
Your Off Topic posts are; unwarranted, annoyining and idiotic.
This is a discussion group on the use of Mr. David Harris'
Pegasus mail and
utilities.
If you can't focus on that topic...GET LOST ! Your posts are
more akin to
spam.
Please go find a Psychiatry news forum, they may be able to
help you with
your
problem.
Are you taking your Prozac ? You should stay on the
regiment...It'll help
you
with your problem.
-----------------
Hold, I'll think of it in just a nanosecond or two
Mercury481
David H. Lipman
Greg Evans
--------------------------------------------------------------
--------------
----
 Ah yes, the things we learn when we least expect it. Feels
good to
take
> control, doesn't it? *back rub hugs*
> back rub hugs, my favourite
> do 10 years of weight training and having strong arms means
> YOU do the massage!!
> Herc
Haa! Have been in the Air Force for 12 years, Have Rank, Guess
who gives
the
orders.
-----------------
(Hildo) news multikabel
Apostate
patty-anne-lea
See You In Hell My Friend.
--------------------------------------------------------------
--------------
----
This is a good point. We may not be further evolving, but that
doesn't
mean
selection isn't going on.
In the last forty years, the welfare state has had some very
powerful
positive side effects. Women have on the whole been able to
shag who
they like, without need for a man's money. (The latest no fault
divorce laws allow women men's money basically regardless.)
I believe the meteoric rise in height is partly due to women's
freedom
to shag about. Also the majority of people were poor in the
19th
century, and sleeping with rich men failed to work, as they
could just
deny parentage.
One other thing you never hear these days is Older men are more
attractive
Obviously they never were, but with money coming from the
state, women
no longer have to butter up the wealthy.
-----------------
Chris
Lee S. Billings
Ian
sertec
--------------------------------------------------------------
--------------
----
That would be nice if I knew where the top and bottom were..or
even where
the left to right borders were..then I could do that. Not sure
if there's
an easy way to find that out though with a triangle.
-----------------
scribe2b
Matt Giwer
cliff86
Chris
--------------------------------------------------------------
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----
How about if we just called it the empty set?
-----------------
Rich Shewmaker
The Pervert
Terry Wilder
gecko
--------------------------------------------------------------
--------------
----
>> I merely asked how, IYO, mathematics relates to belief
>> or nonbelief in supernatual beings.
>because in mathematics you only believe what you KNOW is true.
And so it goes, possibly as long as the cosmos itself.
OK, Mr. Only Agnostics Know Which Side Is Up, what belief is it
that you are trying to inject into every atheist's veins, so
you can
claim
that we/they all believe things we don't know?
Are you prepared to believe that I know that I don't believe
in any
gawds?
Are you able to get a faint glimpse of the possibility that I
could hold
no
belief in any gawds, while not making any rash claims I can't
demonstrate
the
truth of (much less ask anyone else to believe as much or as
little as I
do)? Is
my not being so smug as to troll atheist groups chiding the
posters for
their
na.95ve faith in a strawman religious stance I posit for them,
enough
grounds for
you to count me among the question-begging 'atheist
fundamentalists'?
Or have you even heard of agnostic atheists?
Does it make you feel all warm and secure, just before you
fall off to
sleep, to
know for sure that you haven't angered any gawds by scoffing
at them?
Just in case, you know, you die before you wake?
-----------------
David H. Lipman
Apostate
malcolm burton
Chris
--------------------------------------------------------------
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Answers Below.
1 Wally Anglesea see angel
2 Edward Caruthers car
3 Xcott Craver no cot
4 Mitch Dickson first cause
5 Lawrence & Bobbie fence bird
6 J.y.n.x tempt, act, silence!
7 Hold, I'll think of it in just a nanosecond or two open mind
8 Mercury481 boiling atmosphere
9 sertec sir technical
10 Scribe2b scribes
11 Tim Kozusko mount koziosko
12 CNote see note
13 Greg Neill nil
14 Shanx show then thanks
15 Ralph Hertle alpha display hurtle, hurtle into space
16 Matt Giwer give away
17 Someone one bar
18 John L loo
19 Rich Shewmaker rich showmaker
20 See You In Hell My Friend. depends
21 Rust attacks metal
22 Chas chase
23 Saad Malik well said
24 Greg Evans even
25 Ben Sauvin save
26 (Hildo) news multikabel hill gravity
27 beavith animal
28 Odysseus odysey
29 malcolm burton button
30 G=EMC^2 Glazier glacier
31 Wanda wand dissapears
32 Lee S. Billings leave
33 Impmon impossible for man
34 PlanetaryMatrix space dimension
35 Kurst cursed
36 gecko primitive
37 Roundtable round
38 cliff86 jump off a cliff
39 The Pervert inadequate
40 raven1 raving
41 Gary Rockley rock
42 NormDePloom normal circumstances
43 First Name first person perspective
44 TheKid child
45 David H. Lipman tell off
46 patty-anne-lea pat
47 Ian I Agree
48 Chris nice
49 Terry Wilder empty will do
50 Apostate agnostic atheist
--
www.StealthHostiing.com You rule Truman.
http://tinyurl.com/iky4
Hey Trueman...love the show. YOU ARE the Truman I heard him.
Very
spooky!
>Is the truman living in Townsville? I've been hearing stuff,
yeah.
Webmasters help the TRUEman by joining www.theBanner.net
Current:1
Goal:1000
--------------------------------------------------------------
--------------
------
> String Theory, Universal Mind, and the Paranormal *
> Brian D. Josephson
> Department of Physics, University of Cambridge
> Cavendish Laboratory, Madingley Rd, Cambridge CB3 0HE, U.K.
> http://www.tcm.phy.cam.ac.uk/~bdj10
> ABSTRACT
> A model consistent with string theory is proposed for
so-called
> paranormal phenomena such as extra-sensory perception (ESP).
Our
> mathematical skills are assumed to derive from a special
mental vacuum
> state, whose origin is explained on the basis of anthropic
and
> biological arguments, taking into the need for the
informational
> processes associated with such a state to be of a
life-supporting
> character. ESP is then explained in terms of shared thought
bubbles
> generated by the participants out of the mental vacuum
state. The paper
> concludes with a critique of arguments sometimes made
claiming to
rule
> out the possible existence of paranormal phenomena.
> Keywords: ESP, string theory, anthropic principle, thought
bubble,
> universal mind, mental state
> * To appear in the Proceedings of the 2nd. European Samueli
Symposium,
> 1. Introduction
> Critics of claims of the paranormal, e.g. Deutsch (2001),
have declared
> extrasensory perception (ESP) or other paranormal phenomena
to be
> nonsense . Such absolutist positions give little
weight to the
> experimental evidence (Radin 1997) in support of the reality
of such
> processes, and seem naive given the range of imaginative
proposals
> concerning the nature of reality currently being put forward
for serious
> consideration by conventional physicists. One important
advance has
> been the superseding of the so-called Standard Model as a
fundamental
> theory of nature by string theory
(http://superstringtheory.com), where
> the Standard Model features merely as a subset of the set of
permitted
> alternative Randall-Sundrum picture) has suggested, such a
change in
> perspective opens up new possibilities in science, including
the
> possibility of accommodating paranormal phenomena within
physics. In
> the following a number of concepts are combined, each in
essence
> consistent with accepted ideas, resulting in a qualitative
explanation
> for ESP, with the promise of an eventual clear cut basis for
> understanding paranormal phenomena in general.
> 2. A separate mental reality
> A key assumption we make is one which, while it has no clear
connections
> with experimental physics, does make contact with a position
that was
> advocated by mathematicians such as G.9adel (Davis and Hersh
1981,
Penrose
> 1994). This is the idea that some aspects of mentality
involve a realm
> of reality largely, but not completely, disconnected from
the phenomena
> manifested in conventional physics. The idea of a
disconnected realm
> does have precedents, for example in the way two of the
fundamental
> forces (the strong and weak forces) play no role in large
areas of
> physics and chemistry, whilst in other contexts they have a
very
> important part to play. Next note that string theory,
involving as it
> does spaces having more dimensions than the usual three, and
also a
> large number of such states), is consistent with there being
such a
> separate realm, in a way that the Standard Model,
with its unique
> vacuum state contained within a limited number of spatial
dimensions,
> did not.
> The point in regard to mathematical thinking, which
motivates our model,
> is the following. Consider first of all what the brain does
in visual
> perception. Here the primary information from the visual
receptors goes
> through various levels of processing until it ends up as a
high-level
> representation of the content of the visual field. It is not
> unreasonable to identify mathematics as a similar process,
except that
> higher levels of abstraction are involved in this case. With
the visual
> case, the mechanics are straightforward: the visual field
typically
> contains for example edges, for which abstraction a
dedicated neural
> system has evolved, related to our ability to perceive
edges. It is
> hard to see why we should have such ready access to higher
mathematical
> abstractions having little connection with experience
(Penrose 1994).
> One resolution of the problem would be for mathematical
concepts to be
> in some way in the physics, rather than being
emergent properties of
> brains. In case it is felt that such a drastic solution is
not
> necessary to explain our ready access to mathematical ideas,
and that
> neural networks can provide an adequate explanation, a
stronger argument
> for the existence of some kind of Platonic realm can be made
on the
> basis of the aesthetic aspect of music (Josephson and
Carpenter 1996).
> So far, in shifting the locus of mathematical thinking (and
music?) to
> another realm, we have only replaced one mystery by another.
But why
> should such a realm exist at all? The explanation we provide
is of a
> biological character, taking account of the fact that
information
> processing is an essential component of biological
functioning, but with
> only very specific informational processes having a
life-supportive
> character. While it is commonly taken that the informational
processes
> involved are mediated by ordinary physical means, it is not
a logical
> necessity that this should be the case. Some informational
processes in
> an organism are specialised to the nature and circumstances
of the
> organism concerned, but some have a more abstract and
universal
> character, and so could be mediated by a quite different
system with
> which individual organisms would interact.
> Next we observe that a form of proto-life, defined as
fluctuation
> patterns surviving longer than typical patterns do, can be
hypothesised
> as occurring at the Planck scale, evolution of such life
being expected
> to involve evolution of the accompanying informational
systems also. We
> get to the proposed model by supposing that the ordinary
physical
> component and the informational component can evolve
separately. and
> that the informational component can even survive the
creation and
> destruction of individual universes, remaining as an
ever-present
> background with which new universes, Planck scale
fluctuations and more
> developed life forms can all beneficially interact. Assuming
an
> indefinitely extended time scale, the most persistent part
of the
> informational background can evolve indefinitely, so that
its dynamics
> might come to include features corresponding to mathematical
concepts
> and operations as well.
> This idea can be fruitfully connected with anthropic ideas,
particularly
> our universe seems to be mysteriously fine tuned to develop
in such a
> way that life is possible in terms of it being only one of a
vast number
> of coexisting universes, a small proportion of which have
such a
> property, one of which we find ourselves occupying. Our
speculations
> can be seen as the application of a similar idea to the
informational
> aspect of life.
> While Susskind treats life as a passive occupant of whatever
universe
> can permit it to develop, our proposals see life in a more
general
> light, able to shape its environment in a partnership with
it, in a
> manner analogous to the proposals of Lovelock (1995) (the
Gaia
> hypothesis, for which there is now considerable supportive
evidence), to
> the effect that life may be able to interact cooperatively
with its
> environment, discovering how to operate upon it to its best
advantage.
> 3. A model for ESP
> We need to add another piece of detail to our model. In
order that it
> can model individual thought, we suppose that individual
life forms can
> perturb the background state so as to create a localised
thought
> bubble, tied to the individual concerned. This suggests that
the
> vacuum state involved is close to a phase transition, so
that an
> appropriate perturbation can create a domain with a
different kind of
> order to that of the vacuum.
> Assuming the validity of the scenario that has been
described, the
> picture proposed can be adapted to account for the phenomena
we set out
> to explain, namely telepathy or ESP. In the first, the
grounds for the
> existence of such a process can be taken to be the
advantages that might
> be conferred in certain situations if two life forms could
in some way
> share their mental states (there could also be accompanying
> disadvantages, the significance of which will become clear
later). It
> is natural to postulate, in this case, that a shared mental
bubble,
> whose contents are available to both life-forms, is
involved. We
> assume, as would need to be assumed generally in the model,
that the
> state of this bubble plays the role of information that is
meaningful in
> the context and, by virtue of this, usable by the connected
systems.
> The physics involved in the sharing that has to be
assumed in the
> above can be clarified by means of an analogy based on the
M.9assbauer
> effect, which is a phenomenon involving the decay of
radioactive nuclei
> embedded in a crystal (M.9assbauer 1961). In a certain
fraction of
cases,
> depending on parameters such as the decay energy and the
temperature,
> the recoil from such a decaying nucleus is in effect
transmitted to the
> crystal as a whole rather than generating activity in the
vicinity of
> the decay. These no local recoil processes involve a
certain subset
> of all possible final states of the system, for which, as
quantum
> mechanics allows, the state of the lattice vibrational
system (phonons)
> is unchanged by the decay. This somewhat esoteric
possibility suggests
> a mechanism, dependent on analogous constraints upon the
possible states
> of the thought bubble, that could fit our requirement of a
system state
> being shared by two individuals as in the ESP situation.
> 4. Countering the critics
> The problem any such analysis has to face is that of
explaining how it
> is that, if such a mechanism for ESP or other paranormal
processes
> exists, these processes manifest themselves only in very
specific ways,
> and in ways that are not readily controllable. This should
not be seen
> as an insuperable objection, since other phenomena (e.g.
those involving
> the weather), have similar features. The point to bear in
mind is that
> in the biological realm the phenomena that manifest are
governed not
> only by what is physically possible, but also by which of
those
> physically permitted possibilities are likely to be of
overall benefit
> to the organism concerned. In the ESP context, an
undifferentiated
> sensitivity to the thoughts of all other people, as would
result from
> the uncontrolled sharing of thought bubbles, would tend to be
> disadvantageous rather than of benefit, leading to the
individual being
> overwhelmed by thoughts of others. Most of these would be
merely
> distracting, and interfere with constructive activity. The
right way to
> think about ESP is therefore to see it as a slowly
developing phenomenon
> for a given individual, and one which may not develop at all
if
> conditions are unfavourable. We see from this analysis that
the
> frequently made counter-argument to the existence of ESP,
that if it
> were possible it would have such a survival value that we
would all
> evolve to be very good at it, is based on a misleading
concept of what
> would be involved.
> A related problem is the one raised by Weinberg (1993), who
asks what
> possible physical signal could move distant objects and yet
have no
> effect on scientific instruments? Such a question ignores the
> possibility that there might be a threshold for
psychokinetic effects.
> A similar argument would lead one to be equally sceptical of
claims that
> the heat of the sun can induce chemical reactions (i.e.
burning) in a
> piece of paper, analogously something that happens only
under special
> circumstances (e.g. using a magnifying glass to focus the
suns rays
on
> to a spot on the paper), the amount of burning under normal
conditions
> being negligible.
> The moral to draw would seem to be that one should not be
too ready to
> dismiss paranormal phenomena on the basis of apparently
plausible
> arguments; as t Hooft (2001) has said in a slightly different
context,
> plausible arguments come with their own small print,
viz. assertions
> to the effect that assumptions that seem reasonable to their
authors may
> be violated in the real world.
> 5. Concluding comments
> This work was motivated primarily by the perception that the
arguments
> commonly made against the possible existence of paranormal
phenomena are
> not well-founded, suggesting a need to discover how they
might be
> accommodated within conventional science. Proposals with
this aim have
> been made previously, based upon Bohms causal interpretation
of
quantum
> mechanics (Josephson and Pallikari-Viras 1991, Valentini
1991), but the
> fact that the causal interpretation of quantum mechanics has
not
> developed in ways relevant to current scientific concerns
suggests it
> may be more fruitful to look elsewhere for ideas. The
present paper is
> the outcome of such an investigation. Clearly, it is at best
a sketch of
> a theory, since the arguments are of a very qualitative
character, but
> this qualitative sketch brings to light a number of specific
issues
> whose resolution may provide the basis for a more complete
account of
> the phenomena.
> 6. Supplementary remarks (added after submission of paper for
Proceedings)
> Susskind's arguments suggest that reality may be much more
complex than
> has normally been assumed. Further changes in fundamental
science
> (which may include consideration of the influence of life)
may be
> required to address this complexity. Since our proposals
(such as
> thought bubbles emerging from some kind of background) do
not involve
> the precise details of string theory, they may survive any
such changes
> that fundamental science may undergo.
> 7. Acknowledgements
> The author is indebted to Dr. Fotini Pallikari for many
illuminating
> discussions concerning the nature and mechanisms of ESP. No
funding
> from counter-innovative sources was involved with the
preparation of
> this paper.
> References
> B. Carr (2001), Can physics be extended to accommodate
psi?,
> Proceedings of the 22nd Annual International Meeting of the
Alternative
> Natural Philosophy Association, ed. Arleta Griffor, ANPA.
> P. J. Davis and R. Hersh (1981), The Mathematical
Experience, Brighton:
> Harvester Press.
> D. Deutsch (2001). quoted in Robin McKie, Royal Mails
Nobel guru in
> telepathy row, The Observer, September 30, 2001,
>
http://observer.guardian.co.uk/uk_news/story/0,6903,560604,00.
html
> G t Hooft (2001), How Does God Play Dice?
(Pre-)Determinism at the
> Planck Scale, arxiv:hep-th/0104219
> B.D. Josephson and T. Carpenter (1996), What can Music tell
us
about
> the Nature of the Mind? A Platonic Model, in Toward a Science
of
> Consciousness, ed. S.R. Hameroff, A.W. Kaszniak and A.C.
Scott, 691-694,
> B.D. Josephson and F. Pallikari-Viras, Found. Phys., Vol.
21, pp.
> 197-207, 1991,
http://www.tcm.phy.cam.ac.uk/~bdj10/papers/bell.html
> J. Lovelock (1995), The Ages of Gaia : a biography of our
living earth
> (2nd ed.) Oxford: Oxford University Press.
> R.L. M.9assbauer (1961), Recoilless Nuclear Resonance
Absorption of
Gamma
> Radiation, Nobel Lecture,
>
http://www.nobel.se/physics/laureates/1961/
mossbauer-lecture.pdf
> R. Penrose (1994), Shadows of the Mind, Oxford: Oxford
University Press.
> D. I. Radin (1997), The conscious universe : the scientific
truth of
> psychic phenomena, New York: HarperEdge.
> S. Weinberg (1993), Dreams of a Final Theory, London:
Hutchinson Radius.
> arxiv:hep-th/0302219.
> A. Valentini (1991), Physics Letters A158, 1-8 (abstract at
> http://www.fourmilab.ch/rpkp/valentini.html)
> Home |Horowitz Bio/Articles |Today's Articles |Columnists
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Subject: Re: Almost an Integer - e^?
In sci.math, Dan
<30pack@sbcglobal.net
<fefe1afc.0312021353.ed3ed83@posting.google.com>:
>> I remember there is an explanation for these numbers.
>> Anyone knows where to find the reference (books better)?
>> E.g. J.Silverman, Advanced Topics in the Arithmetic of
Elliptic
>> Curves or something like that (Springer GTM series) has the
>> explanation involving class fields and their relations to
the
>> j-invariant. There may be other explanations.
> Cheers,
> Jyrki Lahtonen
> On the much lighter side ---
> Another interesting fact about e^(pi * sqrt(163)) is ---
> If you add this large composite reciprocal to 163 which is a
(29)
> digit long integer ---
> e^(pi * sqrt(163 + (1/43072298941682041177938098750))) =
>
262537412648768743.99999999999999999999999999999999999999999821
9574092..
> Gives (41) 9's in its decimal expansion.
> More interesting is --
> e^(1/43072298941682041177938098750) =
> 1.000000000000000000000000000023216777942...
> Where if you -1 from the above --- =
> 1/43072298941682041177938098750 ;-)
You do realize that
e^x = 1 + x + x^2/2! + x^3/3! + ...
so
e^(1/43072298941682041177938098750) = 1 +
1/43072298941682041177938098750
+ (1/43072298941682041177938098750)^2/2! + ...
so this isn't really all that remarkable. :-)
> Probably the largest integer reciprocal that could be added
to 163
> giving the same floor (value) of the original (e^(pi *
sqrt(163))) and
> producing a limit of (41) 9s in the decimal expansion
compared to
> (12) 9's in the original.
> Dan
--
#191, ewill3@earthlink.net
It's still legal to go .sigless.
Subject: Re: Almost an Integer - e^?
> Can anyone recall the transcendental number which, if crudely
approximated,
> appears as an integer? It's something like e^(sqrt159) or
e^(pi*sqrt159).
pi sqrt(163)
e = 262537412640768743.99999999999925...
The fascinating explanation of why this is so close to an
integer
involves deep results from algebraic number theory (class
fields,
complex multiplication, modular functions, the j-function,
Kronecker's
Jugendtraum, etc). For much further information see my prior
post:
http://google.com/groups?selm=y8zohirmj49.fsf%
40martigny.ai.mit.edu
-Bill Dubuque
Subject: Re: Almost an Integer - e^?
>> Also: (Pi*Sqrt(163))^Exp(1)= 22806.9992386
>Do you have any reason to believe that is more than
>a coincidence?
Subject: Re: Numerical curiosity!
> I have seen discussions of the special nature of sqrt(163) as
> it relates to Pi. The reversed version is so inelegant, I
> hope its just a mathematical coincidence.
> I discovered for myself the lesser known (and less
impressive)
> fact that (Pi*Sqrt(163))^Exp(1)= 22806.9992386 by getting
> my keying sequence confused (damn reversed Polish
calculators)
>when I was verifying the former (more well known expression).
So it has been found by pure coincidence. Yes: it is a very
minor
conundrum at the level of probability = 10^(-3), but being
close to an
INTEGER, it is quite possible that somebody will find an
analytical
explanation for it in the future (perhaps related to
Ramanujan's
famous hint-guess).
[end Garcia quote, start another quote re Ramanujan's guess]
Subject: Re: pi Gets Fixed
newsgroups: talk.origins
Ramanujan knew that e^(pi*sqrt(163)) was a
near-integer, and he knew why. In fact (with far more detail
than is
appropriate for t.o), class field theory tells us:
If n is a positive integer such that the field Q(sqrt(-n)) has
class number one, then the value j((1+sqrt(-n))/2) is an
integer,
where j(z) is the elliptic modular function. [I'm fudging; see
the
note below.] Now j(z) has a power series, not in z but in
q = e^(2*pi*i*z), which starts:
j(z) = 1/q + 744 + 196884*q + 21493760*q^2 + ... [***]
Let z = (1 + sqrt(-163))/2. Then q = e^(2*pi*i*z) =
e^(2*pi*i*(1 + sqrt(-163))/2) = e^(pi*i*(1 + sqrt(-163)) =
e^(pi*i) * e^(pi*i*sqrt(-163)) = -e^pi*i*i*sqrt(163) =
-e^(-pi*sqrt(163)) = -1/e^(pi*sqrt(163))
Now if you look at the series [***] you see that all but the
first
two
terms are really small. So
some integer = -e^(pi*sqrt(163)) + 744 + something very small,
so e^(pi*sqrt(163)) is very close to an integer.
Actually, something even cooler is true: j((1+sqrt(-163))/2)
is not
only an
integer, but the cube of an integer. So in fact,
(e^(pi*sqrt(163) - 744)^(1/3) =
640319.9999999999999999999999993903175...
(end second quote)
John Bailey
http://home.rochester.rr.com/jbxroads/mailto.html
Subject: Another Complex Question
Thanks in advance to those who help me. (My Final is on
friday.)
Show that
Arg( (z-1)/(z+1)) = -1/2(pi) if Im z< 0
= 1/2(pi) if Im z > 0
Where |z| = 1.
Aside from rearranging the function to get Arg( (z-1)/(z+1)) =
Arg(z-1) -
Arg(z+1) or rewriting (z-1)/(z+1) as -i*(cos(q) - 1) / sin(q)
, I'm not
sure
what else to do.
I keep asking, why does the imaginary part of z matter? Which
leads me to
believe I should break z up into it's real & imaginary
components.
(And hence, if z = x + iy, x = cos(q) and y = sin(q).)
So, with the results I've got (unless I'm wrong) I'm asked to
show that for
any angle q, (cos(q) - 1)/sin(q) = Pi/2.
I can't see how to get that answer.
GREG
Subject: Re: Another Complex Question
> Show that
> Arg( (z-1)/(z+1)) = -1/2(pi) if Im z< 0
> = 1/2(pi) if Im z > 0
> Where |z| = 1.
Multiply and divide by the conjugate of the denominator to get
2iy/(something positive), whose argument is obvious.
Subject: Re: Another Complex Question
> Thanks in advance to those who help me. (My Final is on
friday.)
> Show that
> Arg( (z-1)/(z+1)) = -1/2(pi) if Im z< 0
> = 1/2(pi) if Im z > 0
> Where |z| = 1.
Are you familiar with the idea of a mobius transformation? I
presume
not, but if you are then it will give a much better proof of
this
result; consider where the circle |z|=1 and the line Im(z)=0
are mapped.
> Aside from rearranging the function to get Arg( (z-1)/(z+1))
= Arg(z-1)
-
> Arg(z+1)
You need to be a little careful when you do that. The problem
is that
argument is well defined - you need to choose the range you
want it to
lie in. e.g. (-Pi, Pi). It doesn't matter too much, but you
may need to
throw in a correction term after that. (It's not very useful
here
anyway, so never mind).
> or rewriting (z-1)/(z+1) as -i*(cos(q) - 1) / sin(q) , I'm
not sure
> what else to do.
> I keep asking, why does the imaginary part of z matter?
Which leads me to
> believe I should break z up into it's real & imaginary
components.
> (And hence, if z = x + iy, x = cos(q) and y = sin(q).)
> So, with the results I've got (unless I'm wrong) I'm asked
to show that
for
> any angle q, (cos(q) - 1)/sin(q) = Pi/2.
No, that isn't right. You're trying to find the argument of i
( 1 -
cos(q) ) / sin(q). The argument isn't the same as the
imaginary part -
it's an angle (usually the angle the line between 0 and the
number makes
with the line).
Consider where the number is. It's pure imaginary, so it lies
on the y
axis in your complex plane. So the angle it makes with the x
axis will
only depend on whether it's above or below the plane. Try and
think
about when each of those two cases occur depending on q.
(Notice that (1
- cos (q) ) > 0, so all that matters here is the sign of
sin(q) ).
Hope this helps,
David
(E-mail address spam-blocked in the obvious way)
Subject: Re: Another Complex Question
> Thanks in advance to those who help me. (My Final is on
friday.)
> Show that
> Arg( (z-1)/(z+1)) = -1/2(pi) if Im z< 0
> = 1/2(pi) if Im z > 0
> Where |z| = 1.
> Aside from rearranging the function to get Arg( (z-1)/(z+1))
= Arg(z-1)
-
> Arg(z+1) or rewriting (z-1)/(z+1) as -i*(cos(q) - 1) /
sin(q) , I'm not
sure
> what else to do.
It's a geometry question in disguise:
(Re[z], Im[z]) lies on a circle, centre (0,0) of radius 1.
Arg(z - 1) is the angle given by measuring anti-clockwise from
the line
{(x,0) : x > 1} to the line between (1,0) and (Re[z], Im[z]);
Arg(z + 1) is defined by replacing 1 with -1 in the above.
Drawing a diagram would be exceedingly helpful for this
problem.
If Im[z] > 0, elementary geometry should tell you that the
triangle with
corners (1,0) (-1,0) and (Re[z], Im[z]) has angles pi - Arg(z
- 1),
Arg(z + 1), and pi/2 respectively. Therefore
Arg(z + 1) + (pi - Arg(z-1)) + pi/2 = pi
=> Arg(z + 1) - Arg(z - 1) = -pi/2
=> Arg(z - 1) - Arg(z + 1) = Arg((z - 1)/(z + 1)) = pi/2
In order that some sense of mystery should still prevail, the
case
Im[z] < 0 is left as an exercise.
--
P.A.C. Smith
The vast majority of Iraqis want to live in a peaceful, free
world.
And we will find these people and we will bring them to
justice.
Subject: Re: What does .sig mean?
Thanks everyone! Now, I won't irk anyone, at least with a .sig.
Lurch
> I have been told to get rid of .sig in my responses. What
does that
mean?
> Lurch
Subject: Re: What does .sig mean?
>I have been told to get rid of .sig in my responses. What
does that mean?
It means not to quote the signature[*] of the posts you're
following
up. Under *nix the the signature is usually contained in
~/.signature,
hence '.sig'.
To me, get rid of .sig in your responses is just an
application of
the general principle: quote correctly, i.e. only the relevant
material you're answering to.
[*] BTW: if you use a smart client, generally it will do it
for you!
Michele
--
> Comments should say _why_ something is being done.
Oh? My comments always say what _really_ should have happened.
:)
- Tore Aursand on comp.lang.perl.misc
Subject: Re: What does .sig mean?
Adjunct Assistant Professor at the University of Montana.
>I have been told to get rid of .sig in my responses. What
does that mean?
A signature. Usually a number of lines (from 1, the name, to
several) which include identification and/or phrases and/or
links, but
do not really form part of the body of the message you are
replying
to.
In my case, for example, anything from the first line of ==='s
until
the end.
Removing text that is not addressed or germain in your replies
is
considered polite when you post.
==============================================================
========
It's not denial. I'm just very selective about
what I accept as reality.
--- Calvin (Calvin and Hobbes)
==============================================================
========
Arturo Magidin
magidin@math.berkeley.edu
Subject: Re: What does .sig mean?
Discussion, linux)
>>I have been told to get rid of .sig in my responses. What
does that
mean?
> A signature. Usually a number of lines (from 1, the name, to
> several) which include identification and/or phrases and/or
links, but
> do not really form part of the body of the message you are
replying
> to.
> In my case, for example, anything from the first line of
==='s until
> the end.
To be sure, this delimiter (===...) is non-standard. Signatures
which are automatically appended to a post should be prefaced
by a
line consisting only of -- (note the space) according to some
RFC
somewhere which I can look up if required.
I assume that you don't add every signature by hand.
That same RFC allows for no more than four lines in each
signature,
but this advice is much more commonly ignored than the
delimiter. (It
also accounts for the sometimes horrible editing one sees in
my own
.sigs.)
--
Jesse Hughes
Like the ski resort full of girls hunting for husbands
and husbands hunting for girls, the situation is not as
symmetrical as it might seem. -- Alan MacKay
Subject: Re: What does .sig mean?
Adjunct Assistant Professor at the University of Montana.
>I assume that you don't add every signature by hand.
automatically created by Pnews when I ask to reply to a post
through
trn. Can't remember anymore, but there was some reason why
there were
problems if I asked Pnews to append it at the end. Then again,
the
reason might have been I keep adding it by hand, so my posts
ends
up with two copies of my sig...
--
==============================================================
========
It's not denial. I'm just very selective about
what I accept as reality.
--- Calvin (Calvin and Hobbes)
==============================================================
========
Arturo Magidin
magidin@math.berkeley.edu
Subject: Re: What does .sig mean?
>A signature. Usually a number of lines (from 1, the name, to
>several) which include identification and/or phrases and/or
links, but
>do not really form part of the body of the message you are
replying
>to.
>In my case, for example, anything from the first line of
==='s until
>the end.
BTW: you surely know that the correct .sig separator is n-- n;
as
I hinted in my other post, some clients automatically remove
anything
following (and including) the standard separator when
replying...
Michele
--
> Comments should say _why_ something is being done.
Oh? My comments always say what _really_ should have happened.
:)
- Tore Aursand on comp.lang.perl.misc
Subject: Re: What does .sig mean?
Adjunct Assistant Professor at the University of Montana.
>>In my case, for example, anything from the first line of
==='s until
>>the end.
>BTW: you surely know that the correct .sig separator is n--
n; as
>I hinted in my other post, some clients automatically remove
anything
>following (and including) the standard separator when
replying...
Yeah; I seem to recall having some trouble with that at some
early
stage (I've been using this format since 1993), so I stopped
trying. Don't know if it is still a problem. Maybe I should
try it
again.
--
==============================================================
========
It's not denial. I'm just very selective about
what I accept as reality.
--- Calvin (Calvin and Hobbes)
==============================================================
========
Arturo Magidin
magidin@math.berkeley.edu
Subject: Re: What does .sig mean?
> I have been told to get rid of .sig in my responses. What
does that
mean?
Some people have an automatic signature under their posts.
Check out the
two lines after -- in this message. If you reply to such a
message,
please quote only the part you are replying to, and in
particular not
the signature.
The .sig refers to the fact that on unix systems your
signature is in
a file called .signature.
V.
--
email: lastname at cs utk edu
homepage: cs utk edu tilde lastname
Subject: degrees of certain zeros
Keywords: roots of unity minimal polynomial
Let P be an integer coefficient polynomial and P(a) = 0
such that a/|a| is a root of unity.
Can one say anything interesting about an upper bound on the
degree of a/|a| in terms of degree(P)?
If not, what about the case where a/|a| is a root of unity for
all zeros a of P?
Please reply to:
bruce@cs.jcu.edu.au
Subject: Re: degrees of certain zeros
> Let P be an integer coefficient polynomial and P(a) = 0
> such that a/|a| is a root of unity.
> Can one say anything interesting about an upper bound on the
> degree of a/|a| in terms of degree(P)?
> If not, what about the case where a/|a| is a root of unity
for
> all zeros a of P?
Trivially, the degree of a/|a| is bounded above by the degree
of P, but I can't see doing any better than this. After all,
a (and all its conjugates) could be real, in which case a/|a|
is plus-or-minus one and thus a root of unity.
Posted & emailed.
--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
Subject: Troubles getting MD5 digest to work.
X-AuthenticatedUsername: NoAuthUser
Does any one have a functioning MD5 program that they can give
me
intermediate
results from. I've made two different programs and neither one
is giving
me
the proper results even with an empty file.
Sincerely,
Gregory D. MELLOTT
for Kenton W. MELLOTT
Subject: Re: Troubles getting MD5 digest to work.
>Does any one have a functioning MD5 program that they can
give me
>intermediate
>results from. I've made two different programs and neither
one is giving
me
>the proper results even with an empty file.
>Sincerely,
>Gregory D. MELLOTT
>for Kenton W. MELLOTT
MD5 is described in rfc1321:
http://www.faqs.org/rfcs/rfc1321.html
There are test vectors at the bottom of the page.
If you want public-domain code that works, search on google
for the
version that says written by Colin Plumb in 1993, I know it
works,
like the one here http://www.fourmilab.ch/md5/
Subject: Mathematics software for Linux
Where can I find free mathematics software for Linux? I love
using
Sketchpad
and Maple under Windows, but I need something that can compare
to those two
for Linux. Any suggestions?
Subject: Re: Mathematics software for Linux
> Where can I find free mathematics software for Linux? I love
using
Sketchpad
> and Maple under Windows, but I need something that can
compare to those
two
> for Linux. Any suggestions?
Scilab (www.scilab.org) might be useful.
Subject: Re: Mathematics software for Linux
> Where can I find free mathematics software for Linux? I love
using
Sketchpad
> and Maple under Windows, but I need something that can
compare to those
two
> for Linux. Any suggestions?
Try mupad...
http://www.sciface.com/
Subject: Re: Chained arrow notation: 2->3->2->2->2 sketch?
>Regarding Graham?s number,here is a diagram that I?ve seen on
the web
> / 1) 3^^^^3
> |
> | 2) 3^^...1)...^^3 [where there are 1)= 3^^^^3 up-arrows]
> |
> | 3) 3^^...2)...^^3 [where there are 2) up-arrows]
> | .
>64 levels < .
> | .
> | .
> |63) 3^^...62)...^^3
> |
> 64) 3^^...63)...^^3 <---- Graham's #
>(the process of so and so many arrows is far from being
mathemat-
>cally precise and needs to be reworked.However, it does lend
itself
>to being spacially compact) :-)
>Labelling sections and bays as below?
> one section
________________^________ |
> / |
> 2 bays |
> _____________^___________ |
> / |
> bay 1 bay 2 |
> ____^___ ______^________ |
> / / > one section
> |
> / 1) 3^^^^3 |
> | 2) 3^..1)..^3 |
> | 3) 3^..2)..^3 |
> 64 levels< . |
> | . |
> | . |
> 64) 3^..63)..^3 /
>where a bay (OL) includes all levels and a section is all the
bays
>and their levels.
>Using this mechanism, the Conway-Guy chained arrow expression
> a -> b -> ...x -> y -> z
>can be defined in terms of Knuth up-arrows,e.g.?
> *2->3->3->4*
> 8 bays
>
__________________________________^___________________________
____
>/ bay 1 bay 2 bay 3 4-7 bay 8
> __^__ _________^_________ _________^_________ __^_ ____^_____
>/ / / / /
> / 1) 2^3 = 8 /1) 8 / /1) 8
> | 2) 2^..1)..^3 |2) 2^..1)..^3 | |2)2^..1)..^3
> 8 | 3) 2^..2)..^3 | . | | .
>levels< . | . | | .
> | . | . | 4 | .
> 8) 2^..7)..^3 levels< . |bays | .
> =2->3->8->2 | . | here| .
> =2->3->2->3 ..2^.......^3 levels<.....< .
> = 2->3->3->3 | | .
> ..2^......^3
> = 2->3->8->3 bays
> =2->3->2->4
>
___________________________________________________________^__
____
>/ / 1) 8 /1) 8 / / 1)8
> 8 | . | . | | .
>levels< . | . | | .
> | . | . | | .
> 8) 2^..7)..^3 levels< . | | .
> = 2->3->2->3 | . | | .
> ..2^.......^3 levels<.....< .
> = 2->3->3->3 | | .
> ..2^......^3
> = 2->3->3->4
>-all the above can be called 2 section levels.
>crunch the diagram down to this-by dropping bays #1 and 3 --
> 7 bays
> __________^______________
> /
> 1) 8 / /1) 8
> . | | .
> . | | .
> 8)2^..7)..^3<..< .
> levels| | .
> ..2^.....^3 = 2->3->2->4
> bays
> ___________________^_____
> /1) 8 / /1) 8
> . | | .
> . | | .
> 8)2^..7)..^3<..< .
> levels| | .
> ..2^.....^3 = 2->3->3->4
>Following and extending the concept---
>the following sketch is 2 -> 3 -> 2 -> 2 -> 2 -a five number
chain
>Reads from top left to bottom right...
> /
> |
> |
>[ the big brackets < and ____^____ are critical parts of the
> | / diagram]
> |

>
7 super-bays
>
______________________________________________________________
_______________
______________________________________________________________
_______________
______________________________________________________________
_______________
______________________________________________________________
__^____________
______________________________________________________________
_______________
____________
______________________________________________________________
______________
______________________________________________________________
_______________
_________________________
| | | | |
>/
| | | | |
>
7 bays of
levels of bays of section bays
| | | | |
>
______________________________________________________________
_______________
______________________________________________________________
_______^_______
______________________________________________________________
_______________
__________________________________________________
| | | | |
>/ 7
bays of section bays
| | | | |
>
______________________________________________________________
_________^_____
___________________________________________________________
| | | | |
>/ 7 section bays
| | | | |
>
____________________________________^_________________________
__
| | | | |
> / / 7 bays
| | | | |
> | _____________^____________
| | | | |
> |/1) 2^3 = 8 / /1) 8
| | | | |
> |2)2^..1)..^3| |2)2^..1)..^3
| | | | |
> |3)2^..2)..^3| |3)2^..2)..^3
| | | | |
> | . | | .
| | | | |
> | . | | .
| | | | |
> |8)2^..7)..^3<..< .
| | | | |
> | levels | | .
| | | | |
> | ..2^......^3
| | | | |
> |[=2->3->2->4] --> bays
| | | | |
> | ___________________^______
| | | | |
> |/
| | | | |
> |1) 8 / /1) 8
| | | | |
> |2)2^..1)..^3| |2)2^..1)..^3
| | | | |
> | . | | .
| | | | |
> | . | | .
| | | | |
> |8)2^..7)..^3<..< .
| | | | |
> / 7 < levels | | .
| | | | |
> |sect| ..2^......^3
| | | | |
> |lvls|[=2->3->3->4] --> bays
| | | | |
> | | ___________________^______
| | | | |
> | |/ .
| | | | |
> | | .
| | | | |
> | | [=2->3->7->4 bays].
| | | | |
> | | ___________________^______
| | | | |
> | |/
| | | | |
> | |1) 8 / /1) 8
| | | | |
> | |2)2^..1)..^3| |2)2^..1)..^3
| | | | |
> | | . | | .
| | | | |
> | | . | | .
| | | | |
> | |8)2^..7)..^3<..< .
| | | | |
> | | levels | | .
| | | | |
> | | | | . / / 7 bays
| | | | |
> | | | | . | |
____________^______________
| | | | |
> | | | | . | |/1) 8 / /1) 8
| | | | |
> | | | | . | | 2)2^..1)..^3| |2)2^..1)..^3
if anyone is interested,the rest of the sketch is at--
http//www.mathforum.org/discuss/sci.math/m/493049/552788
with,unfortunately,the topmost long horizontal bracket
displaced to
the right.
It should be directly under the super bay
-cs
Subject: An Easy Complex Quesition
Hei Greg,
You got stuck may be cause You got irritated by the third root
of
unit.
One or two steps back and a new start (to jump higher):
Take a constant vector and multiply by itself, and again and
again,
draw it and You'll see the length grows (or shrinks) in a
geometric
row, whereas the angle grows arithmetically - l, l^2,l^3,..
versus
@,2*@,3*@,..
Now opposite, You look for a vector, which - powered by four -
gives
a known
constant vector. So take the forth root from the length (as
You did)
and a quarter of the angle (it was 120 degrees , so it's 30
degrees).
And You can add 1,2 ,3 ,..full turns to the constant vector
and You'll
get more solutions:
the length again taken to the forth root and the angle
(360+120)/4,(2*360+120)/4,..
Your comment : i = 1, 2 shows You need some practice.
Plane graph is a good tool, You find a link via
http://i-is-no-longer-imaginary.gmxhome.de
Subject: Re: Few Diagonal Determinants
says...
> Given a determinant with only three major diagonals nonzero
it is easy
> to construct a three term recursion relation to evaluate the
> determinant. Is there any similar result for a determinant
with only
> five major diagonals nonzero?
> Are you referring to tridiagonal and pentadiagonal matrices,
resp.?
> That is, are the diagonals located consecutively about the
main
diagonal?
> Do you require a linear recurrance, or will a nonlinear
recurrance do as
> well?
> regards, chip
The diagonals are centered on the main diagonal. A linear
recursion
would be nice but for numerical work a nonlinear one would be
fine.
Subject: two questions on the dirac delta function
(1) Let f(x_0)=0, f'(x_0) = 0. I'm trying to prove the property
delta(f(x)) = (1/|f'(x_0)| ) delta (x-x_0)
by observing
INT(-oo,oo) delta(f) df = INT(-oo,oo) delta(x-x_0) dx.
The left-hand side contributes to the integral only when
f(x)=0, i.e., when
x=x_0, and the right-hand side also only contributes when
x=x_0. But how
can
we then take everything out of the integral and assert
delta(f(x)) = (1/|f'(x_0)| ) delta (x-x_0) ?
(2) How do you show that INT(0,oo) f(x)delta(x) dx = (1/2)
f(0) ?
I know that INT(-oo,oo) f(x) delta(x) dx = f(0), so I'd like
to show that
INT(-oo,0) f(x) delta(x) dx = INT(0,oo) f(x) delta(x) dx.
But, the best I can do as of yet is to obtain
INT(-oo,0) f(x) delta(x) dx = - INT(oo,0) f(-x) delta(-x) dx =
INT(0,oo)
f(-x) delta(-x) dx = INT(0,oo) f(-x) delta(x) dx,
using first a change of variable and second that delta(x) is
an even
function. Unless I messed up on the change of variable,
though, I don't see
how to take this the next step and assert that it = INT(0,oo)
f(x) delta(x)
dx without some special condition of f (i.e. f is even).
Suggestions please! Thx.
Subject: pursuit curve, slow turning pursuer
hello, i wondered if anyone could direct me to some online
information on a
variation of the pursuit curve as described here:
http://www.2dcurves.com/power/powerp.html
Assume its a man chasing a dog. The variation is as follows:
the man can
run at a reasonable pace but cant turn easily so it leads to
overshoots and
corrections and for my money more interesting curves.
Here is an attempt at showing what sort of curve i mean:
< <
< <
< <
< <
< <
< < < < < < < < < < < < < < < < < < < < < < < < < < < < < < <
< < < fox
<
<
man
Im interested in how to mofidy the equation y=
(x**(1+a))/(1+a)) -
(x**(1-a))/(1-a)), given on the link mentioned above, to
accomodate the
reduced turning capability of the chaser.
Thanks in advance for any help given.
Subject: Re: pursuit curve, slow turning pursuer
> hello, i wondered if anyone could direct me to some online
information on
a
> variation of the pursuit curve as described here:
> http://www.2dcurves.com/power/powerp.html
> Assume its a man chasing a dog. The variation is as follows:
the man can
> run at a reasonable pace but cant turn easily so it leads to
overshoots
and
> corrections and for my money more interesting curves.
> Here is an attempt at showing what sort of curve i mean:
> < <
> < <
> < <
> < <
> < <
> < < < < < < < < < < < < < < < < < < < < < < < < < < < < < <
< < < < fox
<snip the bottom half of the diagram and anything thereafter
In keeping with the working ethos of this NG, I'll be the
first to
point out that (fox != dog). While they're both canids and
occupy
similar ecological niches, they're certainly not the same
thing,
which you'd know if you ever had a den of foxes living under
your
front porch.
That nit picked, I'll admit that it does sound like an
interesting
problem, to which I can provide no assistance whatsoever.
HTH <grins
Rick
Subject: Trying to design a simple function
I'm trying to design a function with certain properties, and
it's just
driving me crazy... it really should be simple, but I just
don't succeed.
A rough scetch can be found on
http://www.wernhoff.com/div/graf1.gif.
I'm searching a function f(x) that behaves like the scetched
functions
on the niterval [0,1]. As you see I sketched three functions;
the
function should be dependent on some constant ( i e A) that
changes the
graph as in the sketch. For example, A could be 5, 2 and -3
from top to
bottom. For some value A (in this example A = 0) you should
get y=x, as
you see.
f(0) = 0, f(1) = 1 of course, it's also easy to see/understand
that
f'(0,5) = 1.
As an example, this is exactly what you would use when
compressing or
expanding an audio signal, if x is input as the absolute of the
amplitude and y is output. Max-volume is still max-volume,
f(1) = 1, and
silence is still silence, f(0) = 0, but between that you want
to expand
or compress the signal.
Any suggestions?
Thanks in advance!
Subject: Re: Trying to design a simple function
> I'm trying to design a function with certain properties, and
it's just
> driving me crazy... it really should be simple, but I just
don't succeed.
> A rough scetch can be found on
http://www.wernhoff.com/div/graf1.gif.
Consideration of so-called super-quadric functions led me to
the family
of
functions shown at http://www.vcnet.com/~simonp/function.png
--Peter
Subject: Re: Trying to design a simple function
Distribution: inet
If you rotate by 90 degrees x^n+y^n=1 n>0 is close.
(1-x)^n+y^n=1 accounts for the rotation I think. When
n=1 you get x=y.
Does that help ?
G
Subject: Re: Trying to design a simple function
> I'm trying to design a function with certain properties, and
it's just
> driving me crazy... it really should be simple, but I just
don't succeed.
> A rough scetch can be found on
http://www.wernhoff.com/div/graf1.gif.
This family of curves resembles a Receiver Operating Curve,
which can be
viewed as the area under two Gaussian PDFs as a threshold is
varied.
Good luck,
OUP
Subject: Re: Trying to design a simple function
> I'm trying to design a function with certain properties, and
it's just
> driving me crazy... it really should be simple, but I just
don't succeed.
> A rough scetch can be found on
http://www.wernhoff.com/div/graf1.gif.
> I'm searching a function f(x) that behaves like the scetched
functions
> on the niterval [0,1]. As you see I sketched three
functions; the
> function should be dependent on some constant ( i e A) that
changes the
> graph as in the sketch. For example, A could be 5, 2 and -3
from top to
> bottom. For some value A (in this example A = 0) you should
get y=x, as
> you see.
> f(0) = 0, f(1) = 1 of course, it's also easy to
see/understand that
> f'(0,5) = 1.
> As an example, this is exactly what you would use when
compressing or
> expanding an audio signal, if x is input as the absolute of
the
> amplitude and y is output. Max-volume is still max-volume,
f(1) = 1, and
> silence is still silence, f(0) = 0, but between that you
want to expand
> or compress the signal.
> Any suggestions?
f(x) = x^A for positive real A
David
Subject: Re: Trying to design a simple function
<<
f(x) = x^A for positive real A
>
It works only if the simmetry along the (0,1 - 1,0) line is
not required
http://lzkiss.netfirms.com/cgi-bin/igperl/igp.pl?dir=calculus&
name=test
Subject: Re: Trying to design a simple function
> <<
> f(x) = x^A for positive real A
 It works only if the simmetry along the (0,1 - 1,0) line is
not required
Agreed. I apologize for having responded too hastily, not
noticing the
symmetry indicated by his graph.
But now, looking back, I'm also confused by the statement it's
also easy
to see/understand that f'(0,5) = 1. That certainly does not
seem to be
true for the uppermost of the curves shown in his graph.
David
Subject: Re: Prime numbers, my find, and discovery
> I should be a rather happy guy. After all, over 18 months
ago I found
> this partial difference equation I call dS(x,y), and the sum
of dS
> from dS(x,2) to dS(x,sqrt(x)) is the count of primes up to
and
> including x.
Hey James. You know what? I actually have faith in you. I'm
pretty
sure that if you got some professional help, took your
medicine, and then
signed up for some math courses, you could actually do some
interesting math. I think you're smart enough.
Subject: Re: Prime numbers, my find, and discovery
> I should be a rather happy guy. After all, over 18 months
ago I found
> this partial difference equation I call dS(x,y), and the sum
of dS
> from dS(x,2) to dS(x,sqrt(x)) is the count of primes up to
and
> including x.
> Hey James. You know what? I actually have faith in you. I'm
pretty
> sure that if you got some professional help, took your
medicine, and then
> signed up for some math courses, you could actually do some
> interesting math. I think you're smart enough.
Well, I'm diagnosed with 'mental illness' and I've been
reading a book by,
psychologists Chadwick et al, about 'Cognitive Therapy For
Delusions,
Voices
and Paranoia'. They propose that the psychiatric concept
concerning
delusions is flawed and propose a psychological model for
understanding and
modifying 'delusions.'
The psychiatric concept is roughly ( check the DSM and ICD
manuals) that
delusions are convictions held by the person that are seen as
irrational by
other people in the same culture and that are not open to
modification or
reaccessment, despite overwhelming evidence to the contray
that the
conviction is false.
They are regarded by most psychiatrists as not being beliefs
and in a
separate category to beliefs and are classed as one symptom in
some forms
of
mental illness. However, to be diagnosed, you generally have
to have a
cluster of symptoms, of degree to a greater or lesser extent,
that
correspond to a particular syndrome such as schizophrenia,
etc. Generally,
psychiatrists( especially biological psychiatrists-biological
psychiatry is
the dominant paradigm in psychiatry at the moment) regard
these syndromes
as
discrete illnesses each indicating a brain pathology with an
underlying
physical cause/s, which are not amenable to psychotherapies.
Chadwick et al have empirical evidence that 'delusions' can be
weakened or
modified. They conceptualise 'delusions' as being at the
extreme end of a
continuum with normal beliefs held by so called normal people
in the
population, and thus can be understood within the discourse of
psychology.
They say there is evidence that 'delusions' relate to the
persons
underlying
psychological vulnerabilities that relate to their view of
themselves and
their world.
Their technique, to briefly describe it, I hope not
simplistically, is
called collaborative empiricism. They create a non threatening
relationship
with the person and with the person develop empirical tests to
test the
evidence for the 'delusion.'
But, firstly they proceed through stages to gain the
confidence of the
person in a non threatening manner. One technique mentioned is
called
'Reaction to Hypothetical Contradiction.' This is where they
agree with the
person a hypothetical test which would contradict the
'delusion' to see how
they react to contradiction, rather than confronting them
directly which
could be counter productive.
Subject: Re: Prime numbers, my find, and discovery
> I should be a rather happy guy. After all, over 18 months
ago I
found
> this partial difference equation I call dS(x,y), and the sum
of dS
> from dS(x,2) to dS(x,sqrt(x)) is the count of primes up to
and
> including x.
> Hey James. You know what? I actually have faith in you. I'm
pretty
> sure that if you got some professional help, took your
medicine, and
then
> signed up for some math courses, you could actually do some
> interesting math. I think you're smart enough.
> Well, I'm diagnosed with 'mental illness' and I've been
reading a book
by,
> psychologists Chadwick et al, about 'Cognitive Therapy For
Delusions,
Voices
> and Paranoia'. They propose that the psychiatric concept
concerning
> delusions is flawed and propose a psychological model for
understanding
and
> modifying 'delusions.'
> The psychiatric concept is roughly ( check the DSM and ICD
manuals) that
> delusions are convictions held by the person that are seen
as irrational
by
> other people in the same culture and that are not open to
modification or
> reaccessment, despite overwhelming evidence to the contray
that the
> conviction is false.
> They are regarded by most psychiatrists as not being beliefs
and in a
> separate category to beliefs and are classed as one symptom
in some forms
of
> mental illness. However, to be diagnosed, you generally have
to have a
> cluster of symptoms, of degree to a greater or lesser
extent, that
> correspond to a particular syndrome such as schizophrenia,
etc.
Generally,
> psychiatrists( especially biological
psychiatrists-biological psychiatry
is
> the dominant paradigm in psychiatry at the moment) regard
these syndromes
as
> discrete illnesses each indicating a brain pathology with an
underlying
> physical cause/s, which are not amenable to psychotherapies.
> Chadwick et al have empirical evidence that 'delusions' can
be weakened
or
> modified. They conceptualise 'delusions' as being at the
extreme end of a
> continuum with normal beliefs held by so called normal
people in the
> population, and thus can be understood within the discourse
of
psychology.
> They say there is evidence that 'delusions' relate to the
persons
underlying
> psychological vulnerabilities that relate to their view of
themselves and
> their world.
> Their technique, to briefly describe it, I hope not
simplistically, is
> called collaborative empiricism. They create a non
threatening
relationship
> with the person and with the person develop empirical tests
to test the
> evidence for the 'delusion.'
edit: * not would *could*
> But, firstly they proceed through stages to gain the
confidence of the
> person in a non threatening manner. One technique mentioned
is called
> 'Reaction to Hypothetical Contradiction.' This is where they
agree with
the
> person a hypothetical test which *could* contradict the
'delusion' to see
how
> they react to contradiction, rather than confronting them
directly which
> could be counter productive.
Subject: Re: Prime numbers, my find, and discovery
> If Gauss were alive today, would he cheer me?
> Gauss's motto was Few, but ripe. He would probably call your
> contributions, Many, and rotten.
I disagree that a great man and mind like Gauss would do that.
The
insults to open minded creative people (whether they are right
or
wrong) are rather an attribute of closed minded envious
persons.
Carlos L
Subject: Re: Prime numbers, my find, and discovery
> If Gauss were alive today, would he cheer me?
 Gauss's motto was Few, but ripe. He would probably call your
> contributions, Many, and rotten.
> I disagree that a great man and mind like Gauss would do
that. The
> insults to open minded creative people (whether they are
right or
> wrong) are rather an attribute of closed minded envious
persons.
James Harris is not creative, and definitely not open minded.
Subject: Re: Prime numbers, my find, and discovery
> If Gauss were alive today, would he cheer me?
 Gauss's motto was Few, but ripe. He would probably call your
> contributions, Many, and rotten.
> I disagree that a great man and mind like Gauss would do
that. The
> insults to open minded creative people (whether they are
right or
> wrong) are rather an attribute of closed minded envious
persons.
James's mind is so open his brain fell out and was carried off
by an ant.
James is creative all right, he creates garbage.
Subject: Re: Prime numbers, my find, and discovery
>I should be a rather happy guy. After all, over 18 months ago
I found
>this partial difference equation I call dS(x,y), and the sum
of dS
>from dS(x,2) to dS(x,sqrt(x)) is the count of primes up to and
>including x.
>Afer talking with mathematicians all over the world by email
and
>Usenet, and searching math references, both bought and on the
>Internet, I know that I have a first-find.
>Somehow, I am the first human being in recorded human history
to find
>a partial difference equation that sums to give the count of
prime
>numbers.
> Not true. Won't become true through repetition. See
> http://mathworld.wolfram.com/LegendresFormula.html
Are you saying David Ullrich that what's shown at the link you
provide
is a partial difference equation that sums to give the count
of prime
numbers?
>This post is about some of the significance of that beyond
>it being a first-find.
> Really? Curious that it's such a long post, then.
>Prime numbers have fascinated people for some time, and
mathematicians
>especially. The great mathematician Karl Gauss is credited
with
>making an important hypothesis in the field of prime numbers,
as he'd
>noticed something.
>Gauss noticed that the count of primes numbers could be
approximated
>by x/ln x, for instance, the count of primes up to 1000 is
168, and
>1000/ln 1000 approximately is 144.76. The count of primes up
to 10000
>is 1229, and 10000/ln 10000 is approximately 1085.73, which
is a
>closeness that continues as you go higher.
>Gauss wondered what the discrete count of prime numbers could
have to
>do with continuous functions like x/ln x, and while
mathematicians
>made progress in finding relations that gave limits, like
Chebyshev's
>use of the zeta function discovered by Euler, they never
found a
>reason why.
> Not true. A reason why (that is, a proof of the Prime Number
Theorem)
> was found long ago, I think in the 1890's. More or less
simultaneously
> by two people, who I think are the people whose names I
think are
> spelled something like Hadamard and de-Vallee Poisson.
That is false. Can someone help David Ullrich out by *giving*
the
Prime Number Theorem? It's a boundary condition, and doesn't
tell
why.
My discovery is a direct connection between the discrete and
the
continuous because the partial difference equation I found has
a
partial differential equation analog.
For you physicists, remember that in calculus integration is
usually
discussed by considering *discrete* sums as approximations to a
solution. Then you shrink your delta and consider the limit as
it
goes to 0.
What I have is a first approximation, which shows that the
*count of
primes numbers* is a first approximation in the integration of
a
continuous function!!!
That has NEVER been shown before in recorded human history, and
offers, for the first time, a reason for *why* the prime
distribution
is related to a continuous function like x/ln x.
Remember, Gauss noticed that the count of primes numbers could
be
approximated by x/ln x, for instance, the count of primes up
to 1000
is 168, and 1000/ln 1000 approximately is 144.76. The count of
primes
up to 10000 is 1229, and 10000/ln 10000 is approximately
1085.73,
which is a closeness that continues as you go higher.
You don't have to just trust me--some guy posting on
Usenet--check the
literature on partial difference equation, the Prime Number
Theorem, and integration.
I'm putting in quotes phrases that needed to be put in quotes
for
those who wish to do Google searches for maximum efficiency.
The Internet is important to me as it offers an independent
source
that readers can quickly check.
James Harris
My math discoveries, found for profit
http://mathforprofit.blogspot.com/
Subject: Re: Prime numbers, my find, and discovery
> <ullrich@math.okstate.edu
>I should be a rather happy guy. After all, over 18 months
ago I found
>>this partial difference equation I call dS(x,y), and the sum
of dS
>>from dS(x,2) to dS(x,sqrt(x)) is the count of primes up to
and
>>including x.
>>Afer talking with mathematicians all over the world by email
and
>>Usenet, and searching math references, both bought and on the
>>Internet, I know that I have a first-find.
>>Somehow, I am the first human being in recorded human
history to find
>>a partial difference equation that sums to give the count of
prime
>>numbers.
> Not true. Won't become true through repetition. See
> http://mathworld.wolfram.com/LegendresFormula.html
 Are you saying David Ullrich that what's shown at the link
you provide
> is a partial difference equation that sums to give the count
of prime
> numbers?
if he isn't i will. did you read the damn page? it does note
that it is an
inefficient way to compute pi(n).
<snip
Subject: Re: Prime numbers, my find, and discovery
>I should be a rather happy guy. After all, over 18 months
ago I found
>>this partial difference equation I call dS(x,y), and the sum
of dS
>>from dS(x,2) to dS(x,sqrt(x)) is the count of primes up to
and
>>including x.
>>Afer talking with mathematicians all over the world by email
and
>>Usenet, and searching math references, both bought and on the
>>Internet, I know that I have a first-find.
>>Somehow, I am the first human being in recorded human
history to find
>>a partial difference equation that sums to give the count of
prime
>>numbers.
> Not true. Won't become true through repetition. See
> http://mathworld.wolfram.com/LegendresFormula.html
Are you saying David Ullrich that what's shown at the link
you provide
>is a partial difference equation that sums to give the count
of prime
>numbers?
Uh, yes.
>>This post is about some of the significance of that beyond
>>it being a first-find.
> Really? Curious that it's such a long post, then.
>Prime numbers have fascinated people for some time, and
mathematicians
>>especially. The great mathematician Karl Gauss is credited
with
>>making an important hypothesis in the field of prime
numbers, as he'd
>>noticed something.
>>Gauss noticed that the count of primes numbers could be
approximated
>>by x/ln x, for instance, the count of primes up to 1000 is
168, and
>>1000/ln 1000 approximately is 144.76. The count of primes up
to 10000
>>is 1229, and 10000/ln 10000 is approximately 1085.73, which
is a
>>closeness that continues as you go higher.
>>Gauss wondered what the discrete count of prime numbers
could have to
>>do with continuous functions like x/ln x, and while
mathematicians
>>made progress in finding relations that gave limits, like
Chebyshev's
>>use of the zeta function discovered by Euler, they never
found a
>>reason why.
> Not true. A reason why (that is, a proof of the Prime
Number Theorem)
>> was found long ago, I think in the 1890's. More or less
simultaneously
>> by two people, who I think are the people whose names I
think are
>> spelled something like Hadamard and de-Vallee Poisson.
>That is false. Can someone help David Ullrich out by *giving*
the
>Prime Number Theorem?
Everyone but you _knows_ the PNT.
>It's a boundary condition,
Huh? Exactly how is the statement that pi(x) is asymptotic to
x/log(x) a boundary condition?
> and doesn't tell
>why.
No it doesn't - I didn't say it did. The _proof_ of the PNT is
what
explains why it's true.
Duh.
>My discovery is a direct connection between the discrete and
the
>continuous because the partial difference equation I found
has a
>partial differential equation analog.
Except that you've never shown that the solution to what you
insist on incorrectly calling that pde has anything whatever to
do with pi(x).
>For you physicists, remember that in calculus integration is
usually
>discussed by considering *discrete* sums as approximations to
a
>solution. Then you shrink your delta and consider the limit
as it
>goes to 0.
>What I have is a first approximation, which shows that the
*count of
>primes numbers* is a first approximation in the integration
of a
>continuous function!!!
>That has NEVER been shown before in recorded human history,
and
>offers, for the first time, a reason for *why* the prime
distribution
>is related to a continuous function like x/ln x.
Uh, no, the reason why was given over a century ago.
Otoh _you_ have never given any explanation for the connection.
Just _saying_ that your difference equation has an analogous
pde does not prove that pi(x)/(x/log(x)) tends to 1 as x tends
to
infinity.
Unless maybe I missed it. How does your proof of that fact
go again?
>Remember, Gauss noticed that the count of primes numbers
could be
>approximated by x/ln x, for instance, the count of primes up
to 1000
>is 168, and 1000/ln 1000 approximately is 144.76. The count
of primes
>up to 10000 is 1229, and 10000/ln 10000 is approximately
1085.73,
>which is a closeness that continues as you go higher.
>You don't have to just trust me--some guy posting on
Usenet--check the
>literature on partial difference equation, the Prime Number
>Theorem, and integration.
>I'm putting in quotes phrases that needed to be put in quotes
for
>those who wish to do Google searches for maximum efficiency.
You know, when you equate Google with the mathematical
literature you sound like an idiot. Just a hint.
>The Internet is important to me as it offers an independent
source
>that readers can quickly check.
>James Harris
>My math discoveries, found for profit
>http://mathforprofit.blogspot.com/
************************
Subject: Re: Prime numbers, my find, and discovery
> I should be a rather happy guy. After all, over 18 months
ago I found
> this partial difference equation I call dS(x,y), and the sum
of dS
> from dS(x,2) to dS(x,sqrt(x)) is the count of primes up to
and
> including x.
Which partial difference equation? You haven't told us what it
is.
> Afer talking with mathematicians all over the world by email
and
> Usenet, and searching math references, both bought and on the
> Internet, I know that I have a first-find.
Excellent. Have you searched MathSciNet for papers which may
have
preempted your work? Which mathematicians have you discussed
it with?
Please name them so I can verify this with them.
> Somehow, I am the first human being in recorded human
history to find
> a partial difference equation that sums to give the count of
prime
> numbers. This post is about some of the significance of that
beyond
> it being a first-find.
I'm not convinced. I have seen such things before.
Even if this were true, it is not the significant part of any
claim you
are making. It is only significant if what you have proved
using this
technique is useful. I could just as easily claim that I am
the first
person to use a difference equation to evaluate coefficients
of modular
equations. However if my technique does not have some
theoretical
significance or establish some new result, then it is
worthless. More
must come out than goes in.
> Prime numbers have fascinated people for some time, and
mathematicians
> especially. The great mathematician Karl Gauss is credited
with
> making an important hypothesis in the field of prime
numbers, as he'd
> noticed something.
I prefer conjecture, not hypothesis in mathematical contexts.
The field
would be number theory, not prime numbers. The prime numbers
are a set of
numbers, and certainly not a field.
> Gauss noticed that the count of primes numbers could be
approximated
> by x/ln x, for instance, the count of primes up to 1000 is
168, and
> 1000/ln 1000 approximately is 144.76. The count of primes up
to 10000
> is 1229, and 10000/ln 10000 is approximately 1085.73, which
is a
> closeness that continues as you go higher.
That's not a correct account. Legendre noted in 1798 that
pi(x) is
approximately x/(log x - 1.08366). Gauss made the conjecture
in a letter
to Encke in 1849 that pi(x) is approximately Li(x) (the
principle value
of integral of 1/log u from u=0 to u=x).
> Gauss wondered what the discrete count of prime numbers
could have to
> do with continuous functions like x/ln x, and while
mathematicians
> made progress in finding relations that gave limits, like
Chebyshev's
> use of the zeta function discovered by Euler, they never
found a
> reason why.
That's incorrect. The reason was discovered in 1896 when the
prime number
theorem was completely proved by Hadamard and de la Vall.8ee
Poisson.
Also,
the link to primes is built into Euler's definition of the
zeta function.
It is not mysterious in the least.
> I may have found that reason.
Please be specific. What is it? Given the ignorance you have
displayed
above on the historical matters, I now doubt this to be true.
> Not surprisingly, a first-find in the area of prime numbers
*should*
> be a big deal, but despite the ease with which I link my
discovery to
> some of the biggest names and biggest issues in mathematics,
there is
> the value to society of the discoverer.
A first find is not a big deal unless it proves to be a
significant find.
Specifically, it's mathematical significance will be measured
by how
useful it is to other mathematicians.
As soon as you investigate primes you are ostensibly linking
your work to
just about any of the great number theorists. They all studied
primes.
But in reality this adds nothing to your work. Significance is
only
measured by how many people use and acknowledge your work, not
by how
many people's work it is `related to' or relies upon.
> Since when has modern society decided that discoverers
should be
> attacked instead of cheered?
What are we cheering? What is your discovery and how is it
useful?
> Now you may have seen a LOT of postings from people trying
to attack
> the worth of my find, which can be a healthy process--if
they stick
> with the facts.
Actually people are inclined to ignore what has no worth. It
is a healthy
process to submit your research for peer review so it can be
checked and
critiqued. You seem to have confounded rejection of your work
with a
critique of it and the peer review process with usenet
posting. These are
not the same thing in general.
> Unfortunately posters have shown a dismaying tendency to
lie, but
> that's minor to the problem I've faced where mainstream
mathematicians
> have tried to ignore or downplay my result.
Above you said that mainstream mathematicians had looked at
your work and
said it was new. Now you say that they have been worse than
those who
have rejected you and your work in this forum. Who has lied?
Where is
this documented. Please be specific.
> I have a first-find in the area of prime numbers, and my not
being a
> mathematician does not mean that mathematicians can just
deny the
> reality if it suits them. While they may feel they have many
reasons
> to attack the value of an important find from a
non-mathematician,
> those reasons cannot be in the best interests of society.
That would be the field of number theory, again. You are free
to submit
your work to a peer reviewed journal. If you are really
worried you can
exclude referees who you believe would be prejudiced against
your work. I
would even suppose that in extreme cases you could ask to be
anonymous.
Where is your peer reviewed publication? Do you have a
publication list.
> If Gauss were alive today, would he cheer me?
I doubt it. Gauss would have already learned of the proof of
the prime
number theorem over 100 years ago and would doubtlessly think
that your
remarks about the reasons behind these phenomena not being
well known
showed ridiculous credibility. I also doubt he would be
impressed that
you had misrepresented his own work which de la Vall.8ee
Poussin showed
was
far more precise than Legendre's estimate, regardless of the
constant
that is used in the latter.
> I like to think he would, as he was someone interested in
asking
> questions *and* in getting answers. First and foremost I
think he
> would have been driven to find out just where my discovery
led, and if
> it was the answer to the question that intrigued him.
A mathematician will not in general be interested in where a
particular
random technique might lead. They will usually devote their
time to
techniques which have proved their worth already, or about
which they
hold deep theoretical hunches. Your work fails the first test,
and thus
you are the most likely person to satisfy the second
requirement, if you
had any theoretical knowledge to base your hunch on. Gauss
would almost
certainly not be interested unless he could see something in
your
technique which you clearly have not.
> As I've found a partial difference equation, it leads to a
partial
> differential equation. That partial differential equation
may answer
> many questions.
The first sentence here is tautological and pointless. Please
be specific
about the second statement. What answers does it lead to? I am
not
convinced it leads anywhere useful. You have not given me any
reasons
which indicate that it should. What are those reasons? Please
support
your assertions with evidence.
> Or more importantly, it should raise many more.
What are they?
> You should not allow mathematicians to continue to pervert a
process
> that has helped humanity for so many thousands of years. You
must not
> show a loss of faith in the future of humanity, as if
discoverers are
> no longer needed.
You have this the wrong way around. Mathematicians have
provided the
discoveries which have helped people for many thousands of
years. There
is no `process' which is capable of making discoveries, only
mathematicians (which are by definition people who know and
understand
mathematics and make a careful study of it).
> Academic institutions can no more constrain who can make a
major
> discovery, than they could limit who will be a great painter,
> composer, or architect.
Although they seem to try. However the mathematicians push on
at those
institutions regardless. And mathematicians peer review each
other's work
and then after publication they subsequently review that work
to see what
impact it has had. This process continues day and night all
around the
world despite the best efforts of the institutions they work
for.
> Maybe that's part of the problem as we know that architects
require a
> lot of schooling beyond just art, as they need to know
physics, like
> materials science, and engineering, among many other things.
Mathematicians these days also need to know a lot in general
to make
worthwhile discoveries. There are exceptions but they are
extremely rare.
Agrawal, Kayal and Saxena were recent examples of this. They
were
essentially computer scientists. I also believe that Praeda
Mihalescu was
essentially a computer scientist. In both cases the discoveies
suprised
the mathematical world since the mathematics they relied upon
was
elementary (but not simple let me assure you). Their work was
accepted
however.
At any rate, your chance of success is much higher if you have
a
postgraduate degree in mathematics. This is borne out by the
thousands of
mathematical papers published each year by PhD mathematicians
as opposed
to the extremely rare exceptions which come from amateurs or
those
qualified in other fields.
> So it's easy to assume that a great building can only come
from
> someone heavily trained in academia who can manage a huge
structure.
The architects of old did not have university degrees any more
than the
mathematicians of old did.
> However, sometimes something a little smaller in terms of
physical
> size can be huge in terms of social value, and the person
who built
> it, might be someone from just around the corner, outside of
academia.
Sure, this happens in mathematics. There are people who
publish problems
for amateurs to solve. Take Gardner, Dudeny and the like.
There is
nothing wrong with having mutual appreciation societies which
have social
value but whose academic value is rather low.
> Maybe I'm pushing the analogy, but I hope that you'll agree
that at
> the end of the day, what's important is the *information*
and petty
> squabbles and personal attacks are irrelevant, and often
forgotten
> over history anyway.
You'll only be remembered by history if your discovery was
important and
useful. Yes, the facts are more important, but they have to be
correct.
So far almost nothing you have said has been.
> It's the knowledge that remains--pure.
> dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) -
p(y-1,
> sqrt(y-1))],
> S(x,1) = 0, p(x, y) = floor(x) - S(x, y) - 1,
> and S(x,y) is the sum of dS from dS(x,2) to dS(x,y).
Finally we get some mathematical symbols. But what do they
mean. Where
are your definitions? Which is the statement of your theorem,
and where
is it's proof. By the way, is this it? Three lines of
mathematics. What
is it useful for? Perhaps you have a preprint and this is just
an
announcement of a result you have obtained. Where is that
preprint?
> And p(x,sqrt(x)) is the count of prime numbers up to and
including x.
Is this a definition or your result?
> That's pure knowledge. Information discovered by me, and
hey, it
> wasn't like it just jumped in my lap you know. There's a
value to
> cheering on discovery, and not attacking it.
What information did you discover. Please be specific. In what
way was
this not known before? Why is it useful.
> The value is hope for the future. Hope that there may be
answers out
> there from unlikely sources. Hope that every person can be
valuable.
Mathematics does not progress by the existence of hope. We all
hope that
answers may come to our questions. We don't care if they are
from likely
or unlikely sources. What answers have you provided?
Whether or not every person has value has nothing to do with
mathematics.
That is a philosophical or religious question. You are yet to
demonstrate
that your usefulness lies in your ability to do mathematics.
How has it
been useful to anyone? Please give a list of citations of your
work or
examples of where it has influenced the world at large.
> Maybe mathematicians want a reality that has them ordained
as the only
> route for new mathematical knowledge. Possibly they wish
control over
> the creative process, and total dominion over mathematical
discovery.
Where do I sign up? That would be nice. Then everyone would
have to ask
me when they have mathematical questions.
Unfortunately that is unrealistic. Mathematicians these days
can only
learn the mathematics in a very small subfield of mathematics
if they are
lucky. They are reliant on other mathematicians, logicians,
computer
scientists, computers, physicists and even occasional amateurs
to make
even the most humble progress. This is the major reason why
conferences
and journals exist. Mathematicians must share ideas in order
to make
progress. But what would be the worth of the proceedings and
publications
if they were full of errors and full of irrelevant and
insignificant
facts. Why if that were the case..... well we'd have progress
roughly at
the current rate!
> But hey, they're only human.
Aint it the truth.
> James Harris
My math discoveries, found for profit
> http://mathforprofit.blogspot.com/
And you have made how much profit so far? Who has profited?
Oh, that's it! I thought you said you had made some great
discovery.
Where is it?
Bill Hart.
Subject: Re: Prime numbers, my find, and discovery
> I should be a rather happy guy. After all, over 18 months
ago I found
> this partial difference equation I call dS(x,y), and the
sum of dS
> from dS(x,2) to dS(x,sqrt(x)) is the count of primes up to
and
> including x.
>>Which partial difference equation? You haven't told us what
it is.
> Afer talking with mathematicians all over the world by
email and
> Usenet, and searching math references, both bought and on
the
> Internet, I know that I have a first-find.
>>Excellent. Have you searched MathSciNet for papers which may
have
>>preempted your work? Which mathematicians have you discussed
it with?
>>Please name them so I can verify this with them.
>Lagarias, Odlyzko, Granville, are just a few.
Good, these seem like reasonable people to have spoken with on
this topic
(although I'll say nothing of the appropriateness of
discussing three lines
of mathematics with people who have produced hundredss of
pages of it).
Sadly I do not know any of them personally. However I will do
you a deal.
If
you write up your discovery in the form of a set of
definitions, a theorem
and a proof of that theorem, I will contact Lagarias and ask
him what he
said about your research. I'm not going to do that until I
know what it is
that I am discussing with him. Please also post this to the
group, even if
you email me directly as well.
>I can give others, but why don't you start with them.
>You have an interesting take on Gauss's hypothesis. However,
I know that
li
>(x) is a mystery to a lot of people while x/ln x is more
accessible.
Actually, Gauss did start off with x/ln x as his estimate. I
perhaps should
have mentioned that. It was a bit unfair to criticize you for
this exactly.
But it seems odd to me to talk about Gauss' work with x/ln x
and not his
Li(x) stuff which is more precise. It seems that people all
too often
ignore
stuff they can't understand and then crow about how their own
work is
better
than what has come before, as if the problem hasn't moved much
further on
since then. OK, so perhaps you thought your work was somehow
specifically
related to the prime number theorem itself. But you seem to
say that it is
not later on. What exactly are you saying? It doesn't seem to
make a whole
lot of sense. How is your work related to the work of these
people?
>Also I know that the Prime Number Theorem does NOT tell the
'why' of the
>connection between prime numbers and continuous functions
like x/ln x, or
li(x),
>as it's just a boundary theorem.
That is irrelevant. It is the proof of the theorem which gave
the insight.
OK, so sometimes proofs don't always give much insight, they
just say `
'tis
so', but in the case of the Prime Number Theorem, if you can't
understand
the old proof (more than likely, it doesn't give much insight
to most
people, even some decent mathematicians I perceive), there is
an elementary
proof (still very hard) by Erdos and Selberg (or both). I
believe there is
an elementary version reprinted in Hardy and Wright (which you
should own).
I believe that this proof will give you more insight if you
think about it
hard enough.
But perhaps there is a better explanation for the phenomena.
You seem to be
saying that you have it. Please do tell. What is that better
explanation.
You have not offered it yet.
>You typed a lot of stuff in reply to me, which I see as
typical from math
society.
You have typed a lot of stuff into the usenet! That is the pot
calling the
>You're used to bull-dozing people and getting away with it.
>But you see, I know your tactics.
You on the other hand ignore it when everyone says you are
wrong and push
on
regardless, even when people point out errors you have made.
That sounds
more like bulldozing to me. But I am happy to accept you are
right if you
can demonstrate that you are. As I say, please outline your
discovery and
explain why it has the significance that you say it has. Then
if you are
right, I will accept it. Before then you are crowing about
something which
isn't even written in a way that people can follow and which
has no
demonstrated benefit over anything that came before, if indeed
it is even
new.
>Why don't you check with those mathematicians like you
*claimed* you
would,
>unless you're a liar.
> I dare you.
>James Harris
I have to admit, I am very reluctant to speak to these people
about your
work. They are extremely busy people and I think I can predict
their
response. However I will do so if you think they will back up
your story.
But as I say, the cost to you is that you need to write up
your stuff so I
can follow it and so I know what I am discussing with them.
Actually I only
need to check with one of them.
By the way, just what exactly am I going to discover that they
had to say
about you? I'm certainly not keen to expose myself by talking
to eminent
people about the work of someone who they feel is a crank. I'm
not going to
waste their time any further. So I hope that you think that
they will tell
me that you have made a discovery of some (limited)
theoretical interest at
the very least and that it is new in their opinion. Please
tell me this
before I write to them and embarrass myself.
Bill Hart.
Subject: Re: Prime numbers, my find, and discovery
...
> Afer talking with mathematicians all over the world by
email and
> Usenet, and searching math references, both bought and on
the
> Internet, I know that I have a first-find.
>>Excellent. Have you searched MathSciNet for papers which may
have
>>preempted your work? Which mathematicians have you discussed
it with?
>>Please name them so I can verify this with them.
>Lagarias, Odlyzko, Granville, are just a few.
> Good, these seem like reasonable people to have spoken with
on this
topic
> (although I'll say nothing of the appropriateness of
discussing three
lines
> of mathematics with people who have produced hundredss of
pages of it).
I have met Odlyzko when he was at the CWI to discuss some
stuff about the
Riemann conjecture (actually it was Mertens' conjecture, but
that is
closely related). We had done some quite huge calculations on
it. He
is a reasonable enough person.
However, as far as I remember from earlier posts by James, he
just mailed
interesting and that he had gone on to other fields than prime
counting.
So much for a discussion.
> Sadly I do not know any of them personally. However I will
do you a deal.
If
> you write up your discovery in the form of a set of
definitions, a
theorem
> and a proof of that theorem, I will contact Lagarias and ask
him what he
> said about your research. I'm not going to do that until I
know what it
is
> that I am discussing with him. Please also post this to the
group, even
if
> you email me directly as well.
I do not think Lagarias will even remember it.
> Also I know that the Prime Number Theorem does NOT tell the
'why' of
the
> connection between prime numbers and continuous functions
like x/ln x,
or
> li(x), as it's just a boundary theorem.
> That is irrelevant. It is the proof of the theorem which
gave the
insight.
> OK, so sometimes proofs don't always give much insight, they
just say `
'tis
> so', but in the case of the Prime Number Theorem, if you
can't
understand
> the old proof (more than likely, it doesn't give much
insight to most
> people, even some decent mathematicians I perceive), there
is an
elementary
> proof (still very hard) by Erdos and Selberg (or both). I
believe there
is
> an elementary version reprinted in Hardy and Wright (which
you should
own).
> I believe that this proof will give you more insight if you
think about
it
> hard enough.
That there is indeed deep insight in the connection is shown
by the
existence
of Skewes' number (and the proof of the existence by
Littlewood in 1912).
Also the connection between pi(x), Li(x) and the Riemann
hypothesis is
pretty well understood. See for instance:
<http://mathworld.wolfram.com/SkewesNumber.html
Note that it was well known that for calculated values pi(x)
was always
smaller than Li(x). Littlewood has shown that that was not
always true
and there was a number above which that would be false. Skewes
actually
calculated the upper bound where the condition pi(x) < Li(x)
would
fail (and also proved that the functions pi(x) and Li(x) would
cross
each other infinitely often). He calculated the upper bound at
exp(exp(exp(79))) when the Riemann hypothesis is true and at
10^(10^(10^(10^3))) when the hypothesis is true. The first
upperbound
has subsequently been lowered to exp(exp(27/4)) ~ 10^371.
Still well
beyond computing capabilities.
So in a way Li(x) is weaving around pi(x).
When I was much younger I have tried to prove that an
algebraically
complete field of characteristic 2 did not exist. (If such a
field
does not exist, the Axiom of Choice is false.) Of course, I
just
doodled around and was surprised (happily so) to see in John
Conway's
On Numbers and Games an algebraically complete field of
characteristic
2. Much done for nothing, but it gave me insight. I did learn
from it.
--
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: Prime numbers, my find, and discovery
Summary: nX-No-Archive: yes
>I should be a rather happy guy. After all, over 18 months ago
I found
>this partial difference equation I call dS(x,y), and the sum
of dS
>from dS(x,2) to dS(x,sqrt(x)) is the count of primes up to and
>including x.
Golly, we have another Einstein and Maxwell among us, folks!
Somebody
alert the Nobel Society.
---
Yes, George W. Bush is an unelected baby killing fascist
dictator.
Those who are _against_ freedom call another's fight to be free
terrorism.
Subject: Re: Prime numbers, my find, and discovery
> I should be a rather happy guy. After all, over 18 months
ago I found
> this partial difference equation I call dS(x,y), and the sum
of dS
> from dS(x,2) to dS(x,sqrt(x)) is the count of primes up to
and
> including x.
Crank Information
http://www.crank.net/harris.html
http://www.crank.net/usenet.html
http://www.google.com/search?q=harris+site%3Awww.crank.net
Subject: Re: Prime numbers, my find, and discovery
> I should be a rather happy guy.
Shouldn't that last 'h' be an 's'?
Subject: Re: Why knowledge is NOT power
Regardless of whether James Harris made a worthwhile discovery
about
prime numbers, your following criticism is off the mark by a
wide
margin.
> 1) An explanation of why it is important--for example, does
it enable
> faster computation of primes? (no); does it make for faster
> factorizations? (no). Did you know there is a diophantine
equation
> whose roots are exactly the set of primes? I could tweak that
> equation in a million ways and produce a previously unknown
> formula for primes, but that's hardly very interesting.
The existence of such a diophantine equation is an interesting
mathematical fact. The existence of an exponential diophantine
equation for any recursively enumerable set is a more
interesting
fact.
Indeed, if these are not interesting mathematical facts, I can
hardly
think of any mathematical fact worth deeming valuable.
Let us leave the value of a mathematical proposition to
mathematicians
in proper and free them of philosophical obscurity about what
is
interesting or valuable. Such arguments can achieve no better
than
clouding their vision.
Encoding computation in diophantine equations will be
enormously
interesting to a theoretical computer scientist and a number
theorist,
and a proof of Poincare conjecture will be of interest to a
topologist. I believe the specialists have a much better idea
of what
is of interest to them!
Sincerely,
--
Eray Ozkural
Subject: Re: An Easy Complex Quesition
> This should be an easy question, but I can't seem to get it.
> Q: Solve the following equations in polar form and locate
the roots in
the
> complex plane:
> c) z^4 = -1 + sqrt(-3)
> This is taken from Bak, Newman, Complex Analysis. Pg 18.
> They don't give the answer for the roots in the book, but
clearly,
> (z^4)/2 = (-1 + sqrt(-3))/2 is the third root of unity.
> Let w = (-1 + sqrt(-3))/2
> Then when z = (2^{1/4}) w, z^4 = 2w. Done.
> However, I'm stumped as to how to find the polar coordinates.
> Can I just say
> z = 2^{1/4} e^{ (2 (PI) i )/ 3}
> Then r = 2^{1/4} and arg z = 2 (PI) i / 3, i = 1, 2
> However, this isn't the answer in the book.
> So, any help is appreciated.
> Thanks,
> GREG
Thre is an easy solution, and it is not clear to me that I
should be
giving it to you, but I can't help myself.
z^4 = -1 + isqrt(3).
It is extremely easy to take roots of things in polar
coordinates, and
you almost had it. What are the polar coordinates of -1 +
isqrt(3)?
Well, it is not too hard to find r, since r^2 = (-1)^2 +
sqrt(3)^2 = 4
and thus r = 2. How about theta? Since you oberved that 1/2
the left
side is a cube root of unity and since theta is not changed by
multiplication by a real number, clearly theta must be 2*pi/3
or
4*pi/3. Using the fact that exp(i*theta) = cos(theta) +
i*sin(theta),
root of 4 and divide theta by 4. There is a little bit of a
wrinkle.
In some sense theta is only determined mod 2*pi, and when we
divide by
4, we get extra values. I am expressing this terribly, but
this is
the reason that we get four different roots. The possible
values for
theta in the fourth root are thus (2*pi/3)/4, ((2*pi/3) +
2*pi)/4,
((2*pi/3) + 4*pi)/4, and
((2*pi/3) + 6*pi)/4. Thus your answer will have an r of
2^(1/4) (real
version) and a theta from among the above four values, all of
which
should be simplified, of course.
Hope this helps,
Achava
Subject: Re: An Easy Complex Quesition
> This should be an easy question, but I can't seem to get it.
> Q: Solve the following equations in polar form and locate
the roots in
the
> complex plane:
> c) z^4 = -1 + sqrt(-3)
What does sqrt(-3) mean? Is that i*sqrt(3) or does it
refer to +-i*sqrt(3)? I'll assume it means i*sqrt(3).
Then the magnitude of z^4 is 2 and the phase is 2*pi/3, i.e.
z^4 = 2*exp(i*2*pi/3)
> This is taken from Bak, Newman, Complex Analysis. Pg 18.
> They don't give the answer for the roots in the book, but
clearly,
> (z^4)/2 = (-1 + sqrt(-3))/2 is the third root of unity.
> Let w = (-1 + sqrt(-3))/2
> Then when z = (2^{1/4}) w, z^4 = 2w. Done.
> However, I'm stumped as to how to find the polar coordinates.
> Can I just say
> z = 2^{1/4} e^{ (2 (PI) i )/ 3}
> Then r = 2^{1/4} and arg z = 2 (PI) i / 3, i = 1, 2
No. Review how you find the three units of unity:
1 = exp(i*(0 + 2*pi*k)), k = 0, 1, 2, ...
That is, you can add arbitrary multiples of 2*pi to the phase.
So 1^(1/3) = exp(i*(0+2*pi*k)/3)
k = 0 -> 1^(1/3) = 1
k = 1 -> 1^(1/3) = exp(i*2*pi/3)
k = 2 -> 1^(1/3) = exp(i*4*pi/3) = exp(-i*2*pi/3).
Now apply the same technique to the fourth root
of the above expression.
z^4 = 2*exp(i*2*pi/3)
= 2*exp(i*[(2*pi/3) + 2*pi*k]), k = 0, 1, 2, ...
z = 2^(1/4)*exp(i*[(pi/6) + k*pi/2]), k = 0, 1, 2, ...
- Randy
Subject: Re: An Easy Complex Quesition
> This should be an easy question, but I can't seem to get it.
> Q: Solve the following equations in polar form and locate
the roots in
the
> complex plane:
> c) z^4 = -1 + sqrt(-3)
> This is taken from Bak, Newman, Complex Analysis. Pg 18.
> They don't give the answer for the roots in the book, but
clearly,
> (z^4)/2 = (-1 + sqrt(-3))/2 is the third root of unity.
> Let w = (-1 + sqrt(-3))/2
> Then when z = (2^{1/4}) w, z^4 = 2w. Done.
> However, I'm stumped as to how to find the polar coordinates.
> Can I just say
> z = 2^{1/4} e^{ (2 (PI) i )/ 3}
> Then r = 2^{1/4} and arg z = 2 (PI) i / 3, i = 1, 2
> However, this isn't the answer in the book.
> So, any help is appreciated.
> Thanks,
> GREG
Try using DeMoivre's theorem
Subject: Re: An Easy Complex Quesition
The easiest way to do this is to represent -1+sqrt(-3) in polar
coordinates. -1+sqrt(-3) has an absolute value of 2, and as you
noted, is a multiple of a third root of unity, which implies
that its
arg is 2/3 Pi radians. -1+sqrt(-3) = (2, 2Pi/3).
The solutions to your equation are the fourth roots of
-1+sqrt(-3).
DeMoivre's theorem tells us that all the fourth roots of
(2,2Pi/3)
will have an absolute value of the real fourth root of 2. One
of the
roots will have an argument of (2Pi/3) divided by 4, and the
others
will have an argument that differs by an integer multiple of
2Pi/4 =
Pi/2.
So the solutions are:
(2^(1/4), Pi/6)
(2^(1/4), Pi/6+Pi/2)
(2^(1/4), Pi/6+2Pi/2)
(2^(1/4), Pi/6+3Pi/2)
All of the 2^(1/4) in the solutions above refer to the real
fourth
root of 2.
~ Chris
> This should be an easy question, but I can't seem to get it.
> Q: Solve the following equations in polar form and locate
the roots in
the
> complex plane:
> c) z^4 = -1 + sqrt(-3)
> This is taken from Bak, Newman, Complex Analysis. Pg 18.
> They don't give the answer for the roots in the book, but
clearly,
> (z^4)/2 = (-1 + sqrt(-3))/2 is the third root of unity.
> Let w = (-1 + sqrt(-3))/2
> Then when z = (2^{1/4}) w, z^4 = 2w. Done.
> However, I'm stumped as to how to find the polar coordinates.
> Can I just say
> z = 2^{1/4} e^{ (2 (PI) i )/ 3}
> Then r = 2^{1/4} and arg z = 2 (PI) i / 3, i = 1, 2
> However, this isn't the answer in the book.
> So, any help is appreciated.
> Thanks,
> GREG
Subject: Re: Riemann Surfaces in Analysis
<o0m6svg71cl6qgc1mejd0a2p5nlvsisvn9@4ax.com
<8083d02b.0311252242.6067a0af@posting.google.com
<1ob9sv87acg95ij245tvhan9vg9r34dau7@4ax.com
<8083d02b.0311262322.73600bbf@posting.google.com
X-Cise: tanbanso@iinet.net.au
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X-Tinguish: Mark Griffith <markgriffith@rocketmail.com
X-Treme: C&C,DWS
at 11:22 PM, conesetter@btopenworld.com (conesetter) said:
> A mathematical entity produced by this or any process to be a
>Riemann surface must have an equivalent specification using
cuts
No. The equivalence can by defined via homotopy, independent
of cuts.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT
Unsolicited bulk E-mail will be subject to legal action. I
reserve
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Reply to domain Patriot dot net user shmuel+news to contact
me. Do
not reply to spamtrap@library.lspace.org
Subject: Re: Riemann Surfaces in Analysis
<o0m6svg71cl6qgc1mejd0a2p5nlvsisvn9@4ax.com
<8083d02b.0311252242.6067a0af@posting.google.com
X-Cise: tanbanso@iinet.net.au
X-CompuServe-Customer: Yes
X-Coriate: admin@interspeed.co.nz
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X-Tinguish: Mark Griffith <markgriffith@rocketmail.com
X-Treme: C&C,DWS
at 10:42 PM, conesetter@btopenworld.com (conesetter) said:
> This is where we disagree. Analytic continuation does not
deliver
>uniqueness. What you get depends on the path.
No, only on the homotopy class of the path.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT
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Subject: Calculating a Flux
ath: meganewsservers.com
Hi. My calculus 3 textbook has a math problem with the
following question:
Calculate the flux across the triangle (a,0,0), (0,a,0),
(0,0,a), a > 0. v
=
xi + yj + zk
I'm looking at a solution to it but it still does not make
sense. The
answer
in the back of the book says (1/2)a^3, but another solution
says (1/2)(sqrt
3)(a^3).
I calculatted with a double integral and got (1/2)a^3.
I have attached an image with a book solution.
Thank you.
Subject: Re: Calculating a Flux
ath: meganewsservers.com
Sorry, found out I can't upload an binaries to this newsgroup.
So I
couldn't
put in the image.
Subject: Groups, algorithms, logic?
I presume this is an easy question for experts in some
field of mathematics (logic? algorithms?).
Suppose we have an identity, for example,
aaa=a (*)
Then (*), inverting the sides of the equality,
implies a=a^3 and hence, multiplying side by side,
(a^3)a=a(a^3).
I can keep playing with identity (*) deriving many identities
more.
Of course (*) also implies b=b^3 and hence (a^3)b=a(b^3).
I say that an identity u=v (where u and v are words on some
alphabet)
has type (n,m) if in the word u appear n (equal or different)
letters
and in the word w appear m letters. Therefore (*) is of type
(3,1)
and the identity (a^3)a=a(a^3) is of type (4,4).
(a^3)b=a(b^3) is also of type (4,4).
My question is the following: How can I design an algorithm
that can
be given to a computer so that, for any given (n,m), it
generates all
identities
of type (n,m) that can be derived from the original
identity (or set of identities)? (For the example above, how
can
I design an algorithm that generates all identities of type
(5,9) that
can
be derived from aaa=a?).
I thank any help.
Subject: Re: Sets vs. categories as a foundation
<bpqhpm$fvv0$1@netnews.upenn.edu
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<xes65gzyxra.fsf@greeneg-cs.cs.unc.edu
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Please rebut statements that I actually made, instead of
statements
that nobody made.
>ANYthing that you take as fundamental, purely BY VIRTUE of
the fact
>that you have taken it as fundamental, becomes MORE THAN JUST
a tool
>for other branches of mathematics.
What does that have to do with what the whole point of
categories is?
Category Theory was around before people decided to take it as
fundamental. Why? The answer, of course, is that it was a
unifier.
>This thread was started by
Irrelevant to the issue of what the point of Category Theory
is?
>No, it isn't. There are a lot of things you CAN'T DO with
groups.
Irrelevant. Category started out as a way of unifying several
different concepts. The justification for that was the same as
the
justification for unifying different concepts into group
theory. I did
not claim that Group Theory was equivalent to Category Theory.
>Because the vicious circle principle is a matter of basic
>intellectual hygiene, that's why. Also because circular
definitions
>risk sheer irrelevance and meaninglessness.
Your argument is viciously circular. Why do you consider a set
theory
without foundation to be irrelevant and meaningless? Why do you
consider self-referential sets in a consistent theory to be
poor
intellectual hygiene?
>My point is: a) if sets are foundational then the category of
all
>sets is just irrelevantly circular.
That's flat wrong. GBN handles it quite easily be saying that
it is a
proper class, with no circularity.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT
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Subject: Re: Sets vs. categories as a foundation
Let me add a couple points here. Before that, let me say that
my
appeared there. It was solicited by the then and current editor
Chandler Davis who practically guaranteed publication about a
dozen
years ago. Two years after that I asked him and he said a
report
would be arriving shortly. I am still waiting. Ok, he didn't
like
it, but he could have had the courtesy to let me know.
That off my chest, let me say a few words on the subject. Sets
(and
membership) can be used perfectly well as a foundation for
mathematics. A set is a well-founded epsilon tree. (I am aware
of
non-well-founded set theory, but for my purposes, let us stick
to ZF
or ZFC.) So the ordinal omega (= natural numbers) is a tree
with one
empty node, one node that goes down one level to an empty
node, a node
that that itself has two nodes, one empty and one that goes
down two
levels to an empty node, a node that had three nodes.... Does
all
this structure ever matter? Do you ever think of it as you
talk about
N? That is what I don't like about using sets as the
foundation.
A category has two sorts, objects and arrows. (In a pinch, you
can do
without the objects.) It has operations of domain and codomain
and a
partial operation of composition of arrows, whose domain is
definable
in terms of the domain and codomain operations. Among the
categories
we single out some very special ones called toposes. They have
a
terminal object, pullbacks, and power objects. Add that epis
split,
an object for recursion, and two-valued-ness (the terminal
object has
but two subobjects) and you have a set theory. Clean and neat.
And
no irrelevant structure. You prefer ZFC? Fine, I prefer this.
Subject: Re: Sets vs. categories as a foundation
: >Perhaps. I came into this thread secondhand. Why don't you
tell us
: >what you mean when you write, all categories are in a sense
: >imitations of the category of sets, the objects being
imitations of
: >sets and the morphisms being imitations of functions. It's
very
: >plausible that I don't know what the heck you mean.
: Seems pretty clear to me.
That's your problem.
: The fact that set theory and category theory
: can be used as foundations for mathematics is irrelevant to
james
dolan's
: point,
No, it isn't.
Suppose you started out as a committed categorist,
believing that using categories as a foundation was good
and using sets was bad. Then, suppose a fellow categorist
had reminded you, all categories are in a sense imitations
of the category of sets. That INVITES you to react, Hmmm --
WHY is the category of SETS so SPECIAL -- WHAT is it ABOUT
THAT category that makes it beat this relationship to ALL
other categories? Why is it so central? Could it be -- gasp --
adequately FOUNDATIONAL??
: and is being dragged in uninvited.
No, really, it isn't. The person alleging an unusually
anybody else.
: Suppose one had said instead,
: All homology and cohomology theories are in a sense
imitations of
: simplicial (co)homology. Presumably nobody is tempted to use
simplicial
: homology as a foundation for mathematics, and hence nobody
is tempted
: into failing to understand this statement.
The analogy fails. I don't personally know WHETHER
other homology and cohomology theories imitate
simplicial cohomology. But the question of whether other
categories imitate the category of sets is a lot more
attackable, even on the basis of very limited general concerns
like the ones I have been raising.
: The fact that there are other ways of thinking about
categories
Other THAN WHAT??
Nobody ever suggested that any particular way was more or less
relevant to the question of whether other categories imitate
the category of sets! If the imitation is real then one would
certainly expect it to remain equally real under ALL such ways!
Subject: Re: Sets vs. categories as a foundation
> |I think I agree with George here. One can take the
set-theoretic
> |intuitions too far. What about posets as categories? The
arrows
> |aren't imitations of functions, are they?
> sure they are; specifically, of inclusion functions between
subsets
> (is one way to think of it).
> A category can be thought of in many ways; for instance,
that it's
> nothing more than a typed monoid, or that it's an automaton
whose
> states are its objects and state transitions its arrows.
> Trying to force the idea in one mould completely misses the
point that
> it's all these things, and therefore none of them at all.
> Besides, everyone knows that categories are actually
automata and
> automata are categories.
Everyone doesn't know that. Since Category theory
is like most things related to gibberish set theory.
If you ask a category theorist where these
vacuous objects are referred to to in set theory,
he will obviously say ask a physicist. Since all I
know is that category theory is full-funded by the
French Government, hence it MUST be TRUE.
Subject: Re: Printing Math Symbols
Just to mention. Microsoft Word as an Equation Editor. It's
not the
easiest to use, but maybe an option for small stuff.
Go to <Insert> <Object> and look for something similar to
Microsoft
Equation 3.0.
HTH
--
Dana
= = = = = = = = = = = = = = = = =
> I just learned I will need to turn in math homework and so I
thought I
> may as well type it. What software would simplify this chore?
> Tom Adams
Subject: Re: Printing Math Symbols
> I just learned I will need to turn in math homework and so I
thought I
> may as well type it. What software would simplify this chore?
LaTeX, if your courage is limitless. In order to know what it
can do,
take a look at
http://www.tug.org/texshowcase/
Best regards,
Jose Carlos Santos