mm-4269
===
Subject: Re: not a square ?
knowledge of representation theory is good but not complete.
and in a sence i am more intrested in the opposite
so instead of asking for representation of squares ,
which polynomials (multivariable of course) do representent all integers
apart from squares ??
i used to have a book , and i believe i have seen at least 3 solutions ,
but i cant remember them.
i suppose there are infinite ways ?
and what about cubic integers ?
i dont know the exact word for the opposite of representation.
i call it deniel or deny theory.
with operator deny
so
deny(nO) = ??
deny(nO) = ??
hope i learn something here , so i can fill this gaping hole in my
knowledge of number theory ( and maybe apply it elsewhere in number theory 
?
)
>Your post (above) inspired my ideas for the concept polynomially
>definable sets which I have described in a separate thread.
>Using the terminology I introduced in that thread, your question about
>non-squares can be rephrased as follows:
>If S is the set of squares of integers, and T=ZS, is T a pr-set?
Correction: T=NS.
Hmmm ... Well, if we leave it as ZS (which may have been your
>revised.
>It's clear that T is a nonempty recursively enumerable subset of the
>set of nonnegative integers, hence that means it's a diophantine set
>(I think). So that means there is a polynomial with integer
>coefficients whose range is T, when the inputs to the polynomial are
>restricted to nonnegative integers.
>It occurs to me (just now) that the restriction to nonnegative integer
>inputs can be removed by a simple trick, namely, replace every
>variable x by x1^2+x2^2+x3^2+x4^2 where now x1,x2,x3,x4 are arbitrary
>integers. If so, then the answer to your question is yes -- such a
>polynomial exists. Actually finding one is another matter.
But now I'm not so sure that the non-squares can be a polynomial
range. It's clearly a recursively enumerable set, but I realize now
that it's not automatic that a recursively enumerable set is the range
of a polynomial.
quasi
===
Subject: Re: not a square ?
my knowledge of representation theory is good but
> not complete.
>and in a sence i am more intrested in the
> opposite
>so instead of asking for representation of squares
> ,
>which polynomials (multivariable of course) do
> representent all integers apart from squares ??
>i used to have a book , and i believe i have seen
> at least 3 solutions , but i cant remember them.
>i suppose there are infinite ways ?
>and what about cubic integers ?
>i dont know the exact word for the opposite of
> representation.
>i call it deniel or deny theory.
>with operator deny
>so
> deny(n) = ??
> deny(no) = ??
>hope i learn something here , so i can fill this
> gaping hole in my knowledge of number theory ( and
> maybe apply it elsewhere in number theory ? )
>Your post (above) inspired my ideas for the concept
> polynomially
>definable sets which I have described in a
> separate thread.
>Using the terminology I introduced in that thread,
> your question about
>non-squares can be rephrased as follows:
>If S is the set of squares of integers, and T=ZS,
> is T a pr-set?
>Correction: T=NS.
>Hmmm ... Well, if we leave it as ZS (which may have
> been your
> would need to be
>revised.
>It's clear that T is a nonempty recursively
> enumerable subset of the
>set of nonnegative integers, hence that means it's
> a diophantine set
>(I think). So that means there is a polynomial with
> integer
>coefficients whose range is T, when the inputs to
> the polynomial are
>restricted to nonnegative integers.
>It occurs to me (just now) that the restriction to
> nonnegative integer
>inputs can be removed by a simple trick, namely,
> replace every
>variable x by x1^2+x2^2+x3^2+x4^2 where now
> x1,x2,x3,x4 are arbitrary
>integers. If so, then the answer to your question
> is yes -- such a
>polynomial exists. Actually finding one is another
> matter.
But now I'm not so sure that the non-squares can be a
> polynomial
> range. It's clearly a recursively enumerable set, but
> I realize now
> that it's not automatic that a recursively enumerable
> set is the range
> of a polynomial.
> quasi
could you give a counterexample then ?
it is logical to conclude that not all recursively enumerable sets are in
the range of a polynomial.
for instance the deniel of a*b become the primenumbers , which is no
polynomial ( at least not without the positive restriction of output).
however in my case i mainly consider one variable polynomials to deny
into polynomials ( in the deniel multivariable is allowed , in fact
neccesary )
luckily there are people here who claim to be infinitely smarter than me ,
so i guess we will see a solution soon to all these problems ( * rolleyes *
)
i have to admit i dont have a solution to my own question , so i forgive 
you
not having one yet :-)
welcome to the topic anyways.
you might not claim to be a genius , and you might not have a solution.
but unlike most others you did get the question ( what is already a lot 
here
:s )
you realised its difficulty and importance.
you have manners too.
and you were man enough to admit you cant solve it (yet).
how do you feel about my idea for rational function to represent primes ??
i hope my questions and posts here on sci.math gave you
more joy and interest
than pain and despair. (math is the ultimate opponent)
it is nice that you are intrested in my posts.
however you dont reply to all...
its a matter of taste i assume.
and i suppose your pet math is number theory.
greets
tommy1729
===
Subject: Re: not a square ?
> But now I'm not so sure that the non-squares can be a
> polynomial range. It's clearly a recursively enumerable
> set, but I realize now that it's not automatic that a recursively
> enumerable set is the range of a polynomial.
>
could you give a counterexample then ?
Well, I think the set of primes is the classic example.
There's no nonconstant polynomial which is always prime, hence the set
of primes can't be the range of a polynomial (with integer
coefficients, integer valued inputs).
However it's obvious that the set of primes is recursively enumerable.
Thus, not all recursively enumerable sets are the range of a
polynomial with integer coefficients (with a domain of all possible
integer inputs).
>it is logical to conclude that not all recursively enumerable sets are in
the range of a polynomial.
Yes, as noted above.
>for instance the deniel of a*b become the primenumbers , which is no
polynomial ( at least not without the positive restriction of output).
Right.
>how do you feel about my idea for rational function to represent primes ??
I haven't really thought about it, but yes, now that I consider it. it
could be a very interesting exploration, even in the univariate case.
>i hope my questions and posts here on sci.math gave you
more joy and interest
Yes, many of your problems are quite provoking. Some are too far out
for my taste, and others are too badly posed (from my point of view)
for me to deal with. But some are great -- often vague, but just the
right level of vague, while still revealing hints of mysterious
underlying truths.
You are clearly a kind of genius -- but what kind, I'm not sure!
quasi
===
Subject: Re: Size Theory
> So no doubt this theory is superior to the traditional ways
> of thinking.
> No, it is not no doubt since the notion of a theory being
superior
> to another theory is NOT merely that the one theory is richer than the
> other.
> MoeBlee
This theory is BOTH superior (i.e. has ZFC as a subtheory of it) and
> richer than the traditional ways of thinking.
YOU say i.e. that a theory is superior on account of the other
theory being a subtheory, as you simply missed my point that we are
NOT required to accept that i.e.
MoeBlee
===
Subject: Re: Size Theory
> So no doubt this theory is superior to the traditional ways
> of thinking.
> No, it is not no doubt since the notion of a theory being
superior
> to another theory is NOT merely that the one theory is richer than
the
> other.
> MoeBlee
> This theory is BOTH superior (i.e. has ZFC as a subtheory of it) and
> richer than the traditional ways of thinking.
YOU say i.e. that a theory is superior on account of the other
> theory being a subtheory, as you simply missed my point that we are
> NOT required to accept that i.e.
MoeBlee
I didn't say a theory is superior on account of the other theory being
a subtheory. There is no 'on account' here. you have a meaning of
superior to be something like better, while what I mean by superior IS
that the other theory is a subset of it. I have got lost in
translation here.
Let me omit the word superior since it caused all this confusion.
Let me rewrite it again: this theory has ZFC as subtheory of it
and it is richer than the cardinality approach of Cantor's.
and what I mean by richer is that it both have all the virtues of
Cantor's approach and on top of it it can define finer set sizes.
Zuhair
===
Subject: Re: Size Theory
> So no doubt this theory is superior to the traditional ways
> of thinking.
> No, it is not no doubt since the notion of a theory being
superior
> to another theory is NOT merely that the one theory is richer than
the
> other.
> MoeBlee
> This theory is BOTH superior (i.e. has ZFC as a subtheory of it) and
> richer than the traditional ways of thinking.
> YOU say i.e. that a theory is superior on account of the other
> theory being a subtheory, as you simply missed my point that we are
> NOT required to accept that i.e.
> MoeBlee
I didn't say a theory is superior on account of the other theory being
> a subtheory. There is no 'on account' here. you have a meaning of
> superior to be something like better, while what I mean by superior IS
> that the other theory is a subset of it. I have got lost in
> translation here.
Let me omit the word superior since it caused all this confusion.
Let me rewrite it again: this theory has ZFC as subtheory of it
> and it is richer than the cardinality approach of Cantor's.
> and what I mean by richer is that it both have all the virtues of
> Cantor's approach and on top of it it can define finer set sizes.
Okay, I'll accept that, if your theory has ZFC embedded in it and also
expresses other relations.
MoeBlee
===
Subject: Re: Axiom schema of Z-Hyperfinity.
> Is the set of all hereditary finite sets countably
> infinite?
> V_w is the union of the V_n's and each V_n is finite. Thus a
> denumerable union of finite sets, which, with countable choice,
gives
> us a countable set. But do we need choice? Do we need it for the
> particular case of the V_n's? Also, in general, do we need choice
to
> show a countable union of finite sets is countable?
> I too think you should take a look at some conventional textbook
> before houng on with your speculations; you'll spare a lot of time at
> the end.
> Textbooks are boring most of the time; but you can alternate them with
> some other books on philosophy of set theory, that seems to be your
> ultimate matter of concern.
> But, really, if you don't make an effort to learn the basics you'll
> find enormous difficulties to spell out your own thoughts.
> Best wishes
that when I will have the time.
No, won't. You've just been saying it for about two years now.
> I appreciate your advice.
Take it then.
MoeBlee
===
Subject: Re: Axiom schema of Z-Hyperfinity.
> Is the set of all hereditary finite sets countably
> infinite?
> V_w is the union of the V_n's and each V_n is finite. Thus a
> denumerable union of finite sets, which, with countable choice,
gives
> us a countable set. But do we need choice? Do we need it for the
> particular case of the V_n's? Also, in general, do we need choice
to
> show a countable union of finite sets is countable?
> I too think you should take a look at some conventional textbook
> before houng on with your speculations; you'll spare a lot of time at
> the end.
> Textbooks are boring most of the time; but you can alternate them
with
> some other books on philosophy of set theory, that seems to be your
> ultimate matter of concern.
> But, really, if you don't make an effort to learn the basics you'll
> find enormous difficulties to spell out your own thoughts.
> Best wishes
> that when I will have the time.
No, won't. You've just been saying it for about two years now.
> I appreciate your advice.
Take it then.
I will.
===
Subject: Re: Axiom schema of Z-Hyperfinity.
As well as a standard text, try as parallel reading Michael Potter,
Set Theory and its Philosophy, OUP, for a book with more than the
usual amount of discussion on the choice and motivation for set-
theoretic axioms.
logicmatters.blogspot.com
===
Subject: Re: Axiom schema of Z-Hyperfinity.
> As well as a standard text, try as parallel reading Michael Potter,
> Set Theory and its Philosophy, OUP, for a book with more than the
> usual amount of discussion on the choice and motivation for set-
> theoretic axioms.
logicmatters.blogspot.com
I know this book, it is present of Amazon.
Zuhair
===
Subject: Bug in Mathematica 6 - Integrate - 45 (Sqrt, freshman level
integral, regression bug, invalid value)
Our little demo continues... Hello again from the VM machine
which is still ignored by CAS manufacturers.
The following integral presents no difficulty for Maple 11
or a freshman...
Maple 11> int(sqrt(z-z^2), z= -I..I);
1/2*I*(1-I)^(1/2)+1/4*(1-I)^(1/2)-1/8*I*ln(5-2*(2*2^(1/2)+2)
^(1/2)+4*2^(1/2)-2*(2*2^(1/2)+2)^(1/2)*2^(1/2))+1/2*I*(1+I)^
(1/2)-1/4*(1+I)^(1/2)
Maple 11> evalf(%);
0. + 1.253281911*I
Maple 11> fnormal(evalf(Int(sqrt(z-z^2), z= -I..I)));
0. + 1.253281913*I
... but it presents a problem for Mathematica 6.
(* Yet another regression bug in Mathematica 6 *)
(* It worked correctly in Mathematica 5.2 *)
N[Integrate[Sqrt[z - z^2], {z, -I, I}]]
NIntegrate[Sqrt[z - z^2], {z, -I, I}]
0.785398 + 1.25328 I
0. + 1.25328 I
Best wishes,
Vladimir Bondarenko
VM and GEMM architect
Co-founder, CEO, Mathematical Director
http://www.cybertester.com/ Cyber Tester, LLC
http://maple.bug-list.org/ Maple Bugs Encyclopaedia
http://www.CAS-testing.org/ CAS Testing
===
Subject: Re: Bug in Mathematica 6 - Integrate - 45 (Sqrt, freshman level
integral, regression bug, invalid value)
> Our little demo continues... Hello again from the VM machine
> which is still ignored by CAS manufacturers.
The following integral presents no difficulty for Maple 11
> or a freshman...
Maple 11> int(sqrt(z-z^2), z= -I..I);
As Walter noted, integration in the complex plane is far from
freshman-level
mathematics. In general the answer depends on the path used, and it's not
at
all obvious which path should be taken. In this case I think Maple uses an
antiderivative which has its branch cut on the real line from -infinity to
0
and 1 to +infinity. Since the complement of the branch cut in the complex
plane is simply connected, the value of the integral doesn't depend on the
path, as long as the branch cut is avoided.
> 1/2*I*(1-I)^(1/2)+1/4*(1-I)^(1/2)-1/8*I*ln(5-2*(2*2^(1/2)+2)
> ^(1/2)+4*2^(1/2)-2*(2*2^(1/2)+2)^(1/2)*2^(1/2))+1/2*I*(1+I)^
> (1/2)-1/4*(1+I)^(1/2)
Maple 11> evalf(%);
0. + 1.253281911*I
Maple 11> fnormal(evalf(Int(sqrt(z-z^2), z= -I..I)));
0. + 1.253281913*I
I suppose evalf(Int(...)) uses a straight line between the endpoints.
In this case it hits a branch point, but that's harmless (the numerical
integrators might have some trouble with the lack of smoothness there, but
they seem to be able to deal with it in this case).
I also tried the integral on z = -1-I .. -1+I, where the straight-line path
does cross the branch cut. Both symbolic and numerical integration in
Maple
gave values of approximately 0.+1.04864837987469*I, which I think
corresponds
to the straight-line path (it is not the difference of the antiderivative
values, so presumably the symbolic version detects and compensates for the
discontinuity of the antiderivative across the branch cut).
--
Robert Israel israel@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada V6T 1Z2
===
Subject: Re: Bug in Mathematica 6 - Integrate - 45 (Sqrt, freshman level
integral, regression bug, invalid value)
Originator: roberson@ibd.nrc-cnrc.gc.ca (Walter Roberson)
>The following integral presents no difficulty for Maple 11
>or a freshman...
>Maple 11> int(sqrt(z-z^2), z= -I..I);
Hmmm, when I was a freshman, we never touched complex integration.
There was no complex analysis at all until third year, and
that was really a set theory course, no integration involved.
There might have been a fourth-year course that I didn't
look at.
Complex analysis might be a regular part of the freshman course stream
here these days, but I doubt it; they still dispair at teaching
introductory calculas to the freshmen.
--
There are some ideas so wrong that only a very intelligent person
could believe in them. -- George Orwell
===
Subject: Re: Re-inventing the Wheel: a new periodic table
On Jul 11, 9:56 pm, mensana...@aol.compost <mensana...@aol.com
> On Jul 11, 6:15 pm, mensana...@aol.compost <mensana...@aol.com
> On Jul 11, 1:47 am, mensana...@aol.compost <mensana...@aol.com > To call
something simply explosive is incredibly dense.
> Then why did you say it? Are you incredibly dense?
> No, but apparently you are.
That still doesn't explain why you said it.
I thought I was talking to someone with at least a grasp of high
school chemistry.
> TNT is
> explosive, therefore, TNT should be in the same column of the
> periodic table as hydrogen, according to you.
> TNT isn't an element, is it?
> No,
Says the Ph.D. chemist.
> but since you know so little about elements, I thought I'd use an
> example you might have heard of.
That's the definition of strawman. Go check with someone who
> has a Ph.D. in logic.
Maybe you ask a Ph.D. in logic to explaint to you ... very SLOWLY..
what the term Strawman actually means.
> Sodium reacts violently with water - hydrgoen doesn't react at all.
> Hydrogen react violently with oxygen - sodium reacts slowly (and
NON-
> violently) with oxygen.
> So there's no relationship of the violent reactions (whatever
> the circumstances are) to the single electron in the outer shell?
> That it's just a coincidence, like the violent reaction of TNT?
> TNT reacts violently only under certain conditions. C4, another
> explosie, can be be safely set on fire - it won't explode. Fire a
> bullet through TNT - it doesn;t explode. Mysterious, huh?
Not mysterious at all. TNT isn't lithium nor is it hydrogen.
The engineer's brilliant answer - the reason TNT explodes from a
blating cap, but not from a bullet is... IT ISN'T LITHIUM!
Why did the chicken cross the road?
Because it wasn't lithium!
Why did the engineer cross the road?
He was to stupid to just buy chicken in the store! OR:
Because he didn't take his lithium!
> The reason for storing alkali metals under oil is primarily to
prevent
> reaction with water vapor, not oxygen.
> Is pure hydrogen stored with pure oxygen or are they kept seperate?
> It is kept separate. Somehow, you can't tell the difference between
> oxygen and water.
I can tell the difference between the fuel and the ash of its
> oxidation. Water is the ash of hydrogen oxidation. Duh.
Nitwit you are, you don't even know your own ash from a whole in the
ground. So I guess the ash from the oxidation of carbohydrates is
CO2 and water.... Hint: next time you burn toast (and I doubt you have
enough wherewithal to actually MAKE toast), look at the ash, and
figure out how much oxygen it has in it.
> Here's a table
Poorly made. If only we had an ENGINEER.
> Element Reacts violentyly w/ water Reacts
> violently w/ oxygen
> Hydrogen NO YES
> Lithium YES
> NO
> Sodium YES NO
> Potassium YES NO
> Rubidiium YES NO
> In other words, this table shows that hydrogen is the same as the
> alkali metals, at least in your very confused mind.
_I_ never said it was the same as alkali metals. All _I've_ ever said
> was that it can have violent reactions because of the single electron
> in its outermost shell. You seem to think that I'm saying that since
> this property is also characteristic of alkali metals (which also
> have
> a single electron in their outermost shells) then all of hydrogen's
> properties must be the same as alkali metals. But I'm not saying that.
> Hmmm. Is TNT explosive?
> Hook it to a blasing cap - it's explosive.
> Shoot it with a bullet - it's NOT explosive.
> Does not compute! Does not compute! Warning! Danger Will
Robinson!
> This is what happens when chemists try to do math.
> No, when engineers try to do science.
It was your analogy, not mine.
> Just stick to plugging into
> formulae that someone else gives you and you don't understand, like
> most engineers.
I'm real good at that.
I'm real good at my job, too.
> It's not my job to know whether something
> is right, just that it's consistent - something that's beyond
> the understanding of scientists.
Consistency is a MINIMAL requirement for a scientist, not the end.
> If you can SEE the difference between lithium and hydrogen, you
need
> your eyes checked badly.
> I thought you said there WAS a visible difference? Are you
> changing your story again?
> No, typo, sorry - my bad. If you CAN'T see a difference between
> hydrogen and lithium, maybe your should increase your dosage of
> lithium. (Just joking :-)
> Yes, and your brilliant idea was to put helium in the alkali earth
> category,
> I never said that. I said helium was in the noble gas group.
> Why, if your point is to follow Quantum mechanics about which you
> know even less than you do about chemistry?
That's not my point. Do you have reading comprehension problems?
> Helium is 1s2. Why should it be in the same column as 2s2 2p6, 3s2
> 3p6, 4s2 4p6, etc... at least according to your quantum mechanics
I guess that answers my question.
> view. Shouldn't it go with beryllium 1s2 2s2, and the other alkali
> earths? Helium has almost as much in common with beryllium as hydrogen
> does with lithium.
Good thing I never said it belonged in the alkali earth group.
> with beryllium/magnesium/calcium, which is so ludicrous,
> Which is why I didn't say it.
> that you either never had a chemistry class,
> I'll have you know, sirrah, that I majored in chemistry at
> the University of Illinois.
> or you are 10 years old.
> 33 years ago. And I flunked out and became an electronics engineer.
> So why are so willing to argue about something that you don't know
> anything about?
I've never let that stop me. That's how you learn.
One can only hope.
> I know, because you are an engineer....
Right. Unlike chemists, my learning didn't stop when I left school.
Apparently, you knowledge of chemistry vanished when you left high
school.
> Which lead to my becoming an IT administrator.
> Which lead to my becoming a database administrator.
> Doing environmental remediation.
> So I've come full circle back to chemistry again. Spooky, eh?
> The environment is doomed.
Hate to disappoint you, but I don't DO the risk assessment.
> I manage the data that the scientists can't find their way around
> even with a white cane and a dog.
Wow, you are programmer. Big woop. Many scientists write their own
software, because it is much easier to teach programming to a chemist,
than chemistry to a programmer.
> Somewhat. If we are going to compare elements at vastly different
> pressure/temperatures/conditions, we wold find that almost
everything
> is the same
> Except how the elements interact, that's why we have a periodic
> table. Knowing that some things are gas and others solid doesn't
> help all that much.
> At least, hopefully even a computer programmer can recognize the
> difference betweeb a solid and a gas. In this case, no such luck.
I don't deal with reality, I deal with numbers.
Grabage in, Garbage out.
> While the halogens vary from gas to liquid to solids, ALL the alkali
> metals are solids, whereas hydrogen is a gas. Hmm.... why do you think
> that is?
Because it's the lightest? Just as the lightest halogen is a gas?
But ALL the alkali metals are solids.
> What do I win?
A brain transplant?
> Again, your eyesight is quite terrible:
> LiCl, NaCl, KCl, RbCl, CsCl, LiF, NaF, RbF, LiBr, NaBr, KBr, RbBr
are
> all ionic solids, with large lattice energies, usually white, with
> ionic bonding.
> One lithium atom bonds to one fluorine atom.
> One lithium atom bonds to one chlorine atom.
> One lithium atom bonds to one bromine atom.
> No wonder you flunked out. Did you even pass HIGH SCHOOL chemistry?
Yeah, and that's the reason I flunked out. I did so poorly in high
> school chemistry that I and one other student were invited to take a
> chemistry scholarship test. I should have known from that test that
> the reason I saw further than others was because I was surrounded
> by midgets.
Who passed the test when you didn't?
How long have you had these fellings of persecution?
How long will you blame other for your own (very obvious) failings?
> We write like NaCl just to show that thae balance of charges. Of
> course, there is NO SUCH THING. Sodium chloride as a solid has NO
> NaCl molecules. If you think so, you are both wrong and laughable.
> In the crystal, each sodium is surrounded by SIX chloride IONS, and
> each chloride ION is surrounded by SIX sodium ions. Simple crystal
> structures are taught in freshamn gen chem 1, so you didn't even last
> that long.
Ok, I admit the real reason I flunked out was because it was a
> liberal ARTS college. Couldn't hack that non-technical crap.
> I actually did ok in chemistry, but I thought that might bruise your
> ego.
What, did you take chem for non-majors - a hand-holding and watching
TV course?
> HF, HCl, HBr, HI are all gases, acids (all strong except for HF),
> which feature covalent bonding with partial ioic character.
> One hydrogen atom bonds to one fluorine atom.
> One hydrogen atom bonds to one chlorine atom.
> One hydrogen atom bonds to one bromine atom.
> Did you last in chemistry long enough to learn the difference between
> atoms and ions, and between ionic and covalent bonding?
Sure, at the time.
So, another example of how the engineer keeps on learning.
Funny, I never forget the control structures in C, or the differences
among data types.
> Isn't THAT why hydrogen is shown in Group IA (and also VIIA),
> and nothing to do with gas/solid, latice energies, color or
> ionic/covalent bonding?
> It is put there because they had to put it someplace
Oh, that explains everything.
> (hydrogen is kind
> of an important element to just leave off the table).
Duh.
> The problem is
> that its position is seized upon by the clueless as implying far more
> than its position should imply.
Far more? I never said hydrogen had to have the same properties
> as alkali metals. All I've ever said was that it has a single
> electron in its outer shell and reacts violently - same as alkali
> metals, but never far more than was warranted, that's your delusion.
> Take it from a Ph.D. in chemistry - hydrogen is NOTHING like the
> alkali metals, except that it has 1 s valence electron .
So it DOES belong in group IA.
It is duly noted that engineers who couldn't pass freshman chem vote
for hydrogen in group 1A (which is not what they call it anymore,
anyway, but of course, you couldn't know that, being as you keep up
learning so very well)
> As far as
> properties go, anytime you are tempted to think of hydrogen with the
> alkali metals -DON'T.
Don't fret, I never did.
I only fret about FRET. (A SCIENCE joke - way over your head)
> You'd be wrong 99% of the time to think of
> hydrogen as being like lithium.
In fact, you're wrong 99% of the time, anyway.
> Put KBr and NaCl in front of a chemist, he'll have to ddo some
tests
> to tell which is which. Put HCl and NaCl in front of you, but only
> you, and you would have apparently great difficuly telling a gas
from
> a solid.
> Why do you have a hair up your ass concerning gas vs. solid?
> Because telling the difference between a solid and a gas is SO EASY,
> an ENGINEER (or caveman) can do it.
But you still haven't explained why this is important. It's also easy
> to tell the difference between red and blue litmus paper (even a
> chemist
> can do that). But what does it MEAN?
The physical state of matter is a reflection of the inter- and
intramolecular forces. Solids have strong intermolecular forces,
liquids have weaker forces, and gases have very weak forces. The
strength of the forces depends on the electronic structure.
Differences between hydrogen (a gas) and the alkali metals (ALL
solids) demonstrates that there is a significant difference.
The halogens, on the other hand, progress gradually from gases to a
liquid to a solid, as the atomic number increases.
> Shouldn't such issues be of concern AFTER you've decided that
> hydrogen makes 1 to 1 bondings just like alkali metals and
> halogens?
> No, because the bonding is COMPLETELY different, and the compunds are
> COMPLETELY different. The whole point of periodic properties is
> similarities of the compounds.
But aren't you the one who said the properties aren't as black and
> white as the table implies? Changing your story again? Man, you're
> hard to pin down.
> NaCl, KCl, and RbCl are extremely similar - and completely different
> from HCl
Except that they all hurt like a son-of-a-bitch when you get one
> in a cut.
Those engineering paper cuts really test your metal - even if it's
lithium :-)
> Dissolve 1 mole of NaCl in L water, what is the pH?
> Dissolve 1 mole of HCl in 1L water, what is the pH?
> if I know, I flunked out of chemistry.
> Here's a hint - one of them will eat through your skin in less than 4
> minutes, and the other one won't. And who says chemistry isn't usful?
> But you should see the database hoops I have to jump through
> when doing pH-based soil remediation based on TACO
> (Tiered Approach to Corrective ...
> You should probably learn more chemistry,
Not necessary, I don't DO risk assessment.
> especially if you can't
> calculate the pH of 1 M solution of either HCl or NaCl.
It never comes up. I don't change the water, I merely observe it.
> Most
> reasonably intelligent high school students can do this in their
> heads...
Yeah, but I bet they can't do this:
Yes, a lot of high school students can do OOP. Programming in
assembler is considerably more challenging, but apparently, not as
challenging for you as simple chemistry.
SELECT tblSites.SiteName, tblLocations.IEPALocationNumber,
> tblSamples.TrueLocation, tblSamples.QAQCID,
> tblSampleEvent.SampleEventReportLabel,
> tblSamples.SampleNumber,
> tblSampleAnalyses.AnalysisID,
> tblRefParameterGroups.ParameterGroupCode,
> tblRefAnalyses.AnalysisTypeCode, tblResults.RunNumber,
> tblRefParameters.ParameterReportLabel, tblResults.Result,
> tblResults.Qualifier, tblResults.LabQualifier,
> tblRefUnitOfMeasure.Unit,
> tblRefParameters.Surrogate, tblSamples.SampleMatrixID,
> InStr([CollectorComment],-) AS hyph,
> tblSamples.SampleDepthTo, Master_Lookup.Tbl__B_ICIng,
> Master_Lookup.Tbl__B_ICInh, Master_Lookup.Tbl__B_CWIng,
> Master_Lookup.Tbl__B_CWInh, Master_Lookup.Tbl__B_ClassI,
> Master_Lookup.Tbl_pH_ClassI_Standard,
> IIf(Val(lowest_value([tbl__B_icing],[tbl__B_icinh]))>0,
> Val(lowest_value([tbl__B_icing],[tbl__B_icinh])),Null) AS
> IClow, lowest_value_who([tbl__B_icing],[tbl__B_icinh],G,H)
> AS
> ICwho, IIf(Val(lowest_value([tbl__B_cwing],[tbl__B_cwinh]))>0,
> Val(lowest_value([tbl__B_cwing],[tbl__B_cwinh])),Null) AS
> CWlow, lowest_value_who([tbl__B_cwing],[tbl__B_cwinh],G,H)
> AS
> CWwho, Val(Format$(IIf([unit]=ug/kg,[result]/1000,
> [result]),0.000)) AS
> the_ppm_result, IIf([the_ppm_result]>[iclow],!, ) AS
> icx, IIf([the_ppm_result]>[cwlow],!, ) AS
> cwx, [icx] & [cwx] AS
> xall, Master_Lookup.Tbl_pH_range, tblRefParameters.CASNumber,
> tblResults.SDG
> FROM tblSites
> INNER JOIN ((tblSampleEvent
> INNER JOIN (tblRefQAQCTypes
> INNER JOIN ((tblLocations
> INNER JOIN tblSamples
> ON tblLocations.LocationID = tblSamples.TrueLocation)
> INNER JOIN tblZones
> ON tblLocations.LocationID = tblZones.LocationID)
> ON tblRefQAQCTypes.QAQCID = tblSamples.QAQCID)
> ON tblSampleEvent.EventID = tblSamples.EventID)
> INNER JOIN ((tblRefAnalyses
> INNER JOIN tblSampleAnalyses
> ON tblRefAnalyses.AnalysisID = tblSampleAnalyses.AnalysisID)
> INNER JOIN (tblRefUnitOfMeasure
> INNER JOIN ((tblRefParameterGroups
> INNER JOIN (tblRefParameters
> LEFT JOIN Master_Lookup
> ON tblRefParameters.CASNumber = Master_Lookup.Tbl__A_cas)
> ON tblRefParameterGroups.ParameterGroupID =
> tblRefParameters.ParameterGroupID)
> INNER JOIN tblResults
> ON tblRefParameters.ParameterID = tblResults.ParameterID)
> ON tblRefUnitOfMeasure.UnitID = tblResults.UnitID)
> ON tblSampleAnalyses.SampleAnalysisID = tblResults.SampleAnalysisID)
> ON tblSamples.SampleID = tblSampleAnalyses.SampleID)
> ON tblSites.SiteID = tblLocations.SiteID
> WHERE (((tblSites.SiteName) Like *area 14*) AND
> ((tblSamples.QAQCID)=1 Or (tblSamples.QAQCID)=4) AND
> ((tblSampleAnalyses.AnalysisID)>1) AND
> ((tblRefParameters.Surrogate)=False) AND
> ((tblSamples.SampleMatrixID)=2 Or
> (tblSamples.SampleMatrixID)=4))
> ORDER BY tblSites.SiteName, tblLocations.IEPALocationNumber;
> read more
Oh dear, I wonder if he had more to say, but not being
> an ENGINEER, couldn't figure out how to use Google?
===
Subject: Re: Boolean spaces that are not extremally disconneceted
Originator: israel@math.ubc.ca (Robert Israel)
The place I'd look is Peter Johnstone's book Stone Spaces (Cambridge
University Press). I don't have it to hand, but he has various
examples to show that the various notions similar to total
disconnectedness really are different. My guess is that one of his
examples would answer your question.
I have heard people always saying .. there are boolean spaces which
> are not extremally disconnected. Where is it that I can find an
> example of this.. probably this is trivial, I have never given it a
> thought as it is not entirely in my field of research, nevertheless it
> interests me greatly.
[ Moderator's note on terminology:
> A boolean space is, I think, a totally disconnected compact Hausdorff
> space, also known as a Stone space.
> - RI ]
===
Subject: Re: Boolean spaces that are not extremally disconneceted
> The place I'd look is Peter Johnstone's book Stone Spaces (Cambridge
> University Press). I don't have it to hand, but he has various
> examples to show that the various notions similar to total
> disconnectedness really are different. My guess is that one of his
> examples would answer your question.
Yes, I've it at hand.. in fact even Wikipedia has an example (using
the characterization of complete Boolean algebras as mentioned by
Spaces have extensive literatures, Im glad about that :)
Jose Capco
===
Subject: sex
Cc: hamada
2sfm0iDHzc/LIMfd4cfjINPf0wo=
===
Subject: Re: Bug in Mathematica 6 - Integrate - 44 (Sin, Exp, regression
bug, false convergence)
> Our little demo continues... Hello again from the VM machine
> which is still ignored by CAS manufactureres.
> (* Yet another regression bug in Mathematica 6 *)
> N[Integrate[Sin[Exp[I z]], {z, 0, Infinity}]]
> NIntegrate[Sin[Exp[I z]], {z, 0, Infinity}]
> 0.946083 I (* I SinIntegral[1] *)
> 1.12689 10^244 + 1.33481 10^243 I
>Your title, together with the above, makes it seem as if you think the
>integral diverges to some infinity. It doesn't. _Both_ of the outputs
above
>are misleading.
>David
I can see why a numerical procedure would have
difficulty, as the expression is not absolutely
integrable, but as the expression is certainly real
on the real line, where does the I come from?
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: Bug in Mathematica 6 - Integrate - 44 (Sin, Exp, regression
bug, false convergence)
> Our little demo continues... Hello again from the VM machine
> which is still ignored by CAS manufactureres.
> (* Yet another regression bug in Mathematica 6 *)
> N[Integrate[Sin[Exp[I z]], {z, 0, Infinity}]]
> NIntegrate[Sin[Exp[I z]], {z, 0, Infinity}]
> 0.946083 I (* I SinIntegral[1] *)
> 1.12689 10^244 + 1.33481 10^243 I
>Your title, together with the above, makes it seem as if you think the
>integral diverges to some infinity. It doesn't. _Both_ of the outputs
>above are misleading.
>David
I can see why a numerical procedure would have
> difficulty, as the expression is not absolutely
> integrable, but as the expression is certainly real
> on the real line, where does the I come from?
Huh? Sin[Exp[I z]] is not real on the real line. For example,
In[5]:= Sin[Exp[I z]]/. z -> Pi/4
Out[5]= Sin[E^((I*Pi)/4)]
In[6]:= N[%]
Out[6]= 0.818927 + 0.583505*I
David
===
Subject: Re: Bug in Mathematica 6 - Integrate - 44 (Sin, Exp, regression
bug,
false convergence)
> Our little demo continues... Hello again from the VM machine
> which is still ignored by CAS manufactureres.
> (* Yet another regression bug in Mathematica 6 *)
> N[Integrate[Sin[Exp[I z]], {z, 0, Infinity}]]
> NIntegrate[Sin[Exp[I z]], {z, 0, Infinity}]
> 0.946083 I (* I SinIntegral[1] *)
> 1.12689 10^244 + 1.33481 10^243 I
> Your title, together with the above, makes it seem as if you think the
> integral diverges to some infinity. It doesn't. _Both_ of the outputs
> above are misleading.
> David
> I can see why a numerical procedure would have
> difficulty, as the expression is not absolutely
> integrable, but as the expression is certainly real
> on the real line, where does the I come from?
Huh? Sin[Exp[I z]] is not real on the real line. For example,
In[5]:= Sin[Exp[I z]]/. z -> Pi/4
Out[5]= Sin[E^((I*Pi)/4)]
In[6]:= N[%]
Out[6]= 0.818927 + 0.583505*I
David
Maple11 gives it as limit(-I*Si(exp(z*I))+Si(1)*I,z = infinity),
looking at Si on the circle shows why it is bounded, but does
not converge.
===
Subject: Re: Catenary at different elevations
>Can anyone help out with this one?
>We have two sprockets and a chain and need to find the sag. The support
>points of the chain are at different elevations (different diameter
sprockets).
>We are iterating through the following equation in Visual Basic to find a,
the
>y-coordinate of sag measured from the horizontal centerline of the two
sprockets.
2 * a * (sinh(L/(2 * a)))=(y1-y2)/sinh(arctanh((y1-y2)/S))
However, we keep getting a value above the highest support point. This
>equation should be valid for our scenario, so why is the solution so
>inaccurate?
Leigh
Did you ever get this resolved?
George
===
Subject: Re: New algorithm for the isomorphism of graphs
> Hello has all,
> Here a method to find the function of isomorphism between two
> isomorphous
> isomorphic
> graphs: For each vertex S of the graph G, one drawing a
> pseudo-tree in this manner.
> 1. in the first level there is the vertex S.
> 2. in bottom there are all the adjacent vertices at the
vertex S.
> 3. idem for this level (all adjacent vertices at the
vertices of the
> high level).
> 4. and one vertices when there N is more of the vertices.
> This is known as a Breadth-First-Search (BFS) tree of the graph G;
I
> will use this abbreviation below.
> In the second graph there is a vertex which has the same
pseudo-tree.
> You need to find this vertex and BFS tree. So what you might have
to
> do is to find all BFS trees for all vertices of G'; you may have
to
> permute the order of the vertices in the representation of G' to
do
> this, and this is a problem. (See below.)
> Since the construction of these pseudo-tree this fact in a time
> polynomial, then the problem of the isomorphism of graphs is
> polynomial.
> How do you know you will be constructing polynomially many BFS
trees?
> Sure, you may get lucky when you construct the second, but in the
> worst case, you may have to find them all, and there are graphs
with
> exponentially many BFS trees. This would ruin your polynomial
result.
> Prove:
> G and G' are isomorphous, then for a vertex S of G, there exists
in G'
> a vertex which has the same one dismantles and adjacent with a
group
> of the vertices (bottom grade) which has the same number is
> isomorphous with the group of the bottom grade for G.
> ??? I think what you're trying to say is: Suppose G and G' are
> isomorphic. We want to explicitly construct an isomorphism F from
G to
> G'. To do so, let S be a vertex of G.
> Thus to find F (S), F is related to isomorphism between the two
graphs
> G and G', one seeks in G' all the vertices which one dismantles
equal
> to dismantles S, then which is the level second isomorph on the
same
> level of S, and one finished when the correspondence is found.
> What this seems to say is Match up the neighbors of S with the
> neighbors of some candidate for F(S). Then match up the neighbors
of
> the neighbors of S with the neighbors of the neighbors of F(S).
And so
> on.
> The problem is that you have a non-polynomial number of
possibilities
> of matching up the neighbors of S with the neighbors of S' (a
> candidate for F(S)). If G [and G'] is a regular triangle-free
graph
> with degree n/2, then the number of ways to match up the neighbors
of
> S with the neighbors of S' is (n/2)!, which is exponential in n.
> Unless you have some way to eliminate a lot of these (and you
don't,
> since you're only looking at the first neighborhood of S and S'),
you
> have to look at them all.
> Any isomorphism-checking algorithm should also be able to deal
with
> the case where G and G' are not isomorphic.
> Or if
> one arrives towards the low level with several vertices then S
is
> isomorphous with one of its vertices.
> And to afflict for the syntax, spelling mistakes and grammar.
(A
> machine translation)
> An example of how your algorithm works would help. Let G be the
graph
> with vertex-set {1,2,3,4,5} and edges {12, 13, 15, 23, 34, 45},
and G'
> the graph with vertex-set {a,b,c,d,e} and edges {ab, ac, ae, bd,
be,
> cd}.
> f(2)=e.
> f(1)=a or f(1)=b.
> f(3)=b or f(3)=a.
> f(5)=c or f(5)=d.
> f(4)=d or f(4)=c.
> f(1,2,3,4,5)=(a,e,b,d,c).
> f(1,2,3,4,5)=(b,e,a,c,d).
> Okay. So what I said above really applies: In short, you have to
> consider the cases f(1)=a and f(1)=b and ahead of time, and then
> consider sub-cases from there.
> Since you have to choose between two options, and there's no
immediate
> reason for choosing one over the other, you (probably) won't have a
> polynomial-time algorithm. (Incidentally, more than one choice is
> why a lot of problems become NP-complete when some parameter becomes
2
> or 3. If you are 2-coloring a graph, once you color one vertex,
you're
> basically done, since the color of every other vertex is forced.
> However, if you are 3-coloring a graph, you have a choice between 2
> colors to color each of the neighbors of the first vertex. Since, as
> in your isomorphism algorith, you have to make a choice here, you
have
> to consider both options. My gut feeling, BTW, is that the graph
> isomorphism problem is NP-complete.)
> [ONE LAST THOUGHT:]
> A good test for your algorithm would be to pick two k-regular graphs
Actually, this should be k-vertex regular graphs with girth at least
> k/3.
> with girth at least (say) k/3. You can find a lot of regular graphs
> with a given number of vertices, and a given girth, at
>http://www.mathe2.uni-bayreuth.de/markus/reggraphs.html.
> proved that the two graphs are isomorphous.
This is true. However, the point of a simple example was to see if I
> understood your algorithm. The point of the more difficult graphs (k-
> vertex regular graphs with girth at least k/3) was to make sure that
> your algorithm runs in polynomial time, which is the relevant point
> here.
--- Christopher Heckman
> difficulty it is the translation.
> And thank you for the URL.
> -- mohamed mimouni- Masquer le texte des messages pr?c?dents -
- Afficher le texte des messages pr?c?dents -- Masquer le texte des
messages pr?c?dents -
- Afficher le texte des messages pr?c?dents -
I am sorry but I could not use the files of the URL (I do not have the
program to open these file). Then here a small example to make gone
the algorithm,
G=(12 ;24 ;34 ;36 ;45 ;56) and G'=(ab ;ae ;bc ;cf ;df ;ef)
One seeks for example f(e).
1) the pseudo-tree of e :
1) The vertix e.
2) Two vertices a and f.
3) Two vertices d (adjacent with vertix a and with vertix f) and c
(adjacent with vertix f).
4) The vertix b (adjacent with vertix c).
The vertix e has two degree, thus one compares with vertices 2, 3, 5
and 6. (I chose two only).
5) the pseudo-tree of 6 :
1) The vertix 6.
Two vertices 3 and 5. but they have both the same child (vertix 4),
what is not the case with vertices a and f second level of the pseudo-
tree of e, thus there' is a difference.
1) the pseudo-tree of 5 :
1) The vertix 5.
2) Two vertices 6 and 4. there' is an isomorph thus one contained.
3) Two vertices 3 (adjacent withx vertices 4 and 6) and 2 (adjacent
with vertix 2). there' is an isomorph thus one contained.
4) The vertix 1 (adjacent with vertix 2). Thus isomorphous total.
Thus according to the pseudo-tree of vertix 5, a solution will be :
f(1)=b ; f(2)=c ; f(3)=d ; f(4)=f ; f(6)=a and f(5)=e.
----for the demonstration if you to understand French it will be
good!---
--mohamed mimouni--
===
Subject: Re: New algorithm for the isomorphism of graphs
> Hello has all,
> Here a method to find the function of isomorphism between two
> isomorphous
> isomorphic
> graphs: For each vertex S of the graph G, one drawing a
> pseudo-tree in this manner.
> 1. in the first level there is the vertex S.
> 2. in bottom there are all the adjacent vertices at the
vertex S.
> 3. idem for this level (all adjacent vertices at the
vertices of the
> high level).
> 4. and one vertices when there N is more of the
vertices.
> This is known as a Breadth-First-Search (BFS) tree of the graph
G; I
> will use this abbreviation below.
> In the second graph there is a vertex which has the same
pseudo-tree.
> You need to find this vertex and BFS tree. So what you might
have to
> do is to find all BFS trees for all vertices of G'; you may have
to
> permute the order of the vertices in the representation of G' to
do
> this, and this is a problem. (See below.)
> Since the construction of these pseudo-tree this fact in a
time
> polynomial, then the problem of the isomorphism of graphs is
> polynomial.
> How do you know you will be constructing polynomially many BFS
trees?
> Sure, you may get lucky when you construct the second, but in
the
> worst case, you may have to find them all, and there are graphs
with
> exponentially many BFS trees. This would ruin your polynomial
result.
> Prove:
> G and G' are isomorphous, then for a vertex S of G, there
exists in G'
> a vertex which has the same one dismantles and adjacent with a
group
> of the vertices (bottom grade) which has the same number is
> isomorphous with the group of the bottom grade for G.
> ??? I think what you're trying to say is: Suppose G and G'
are
> isomorphic. We want to explicitly construct an isomorphism F
from G to
> G'. To do so, let S be a vertex of G.
> Thus to find F (S), F is related to isomorphism between the
two graphs
> G and G', one seeks in G' all the vertices which one
dismantles equal
> to dismantles S, then which is the level second isomorph on
the same
> level of S, and one finished when the correspondence is
found.
> What this seems to say is Match up the neighbors of S with
the
> neighbors of some candidate for F(S). Then match up the
neighbors of
> the neighbors of S with the neighbors of the neighbors of F(S).
And so
> on.
> The problem is that you have a non-polynomial number of
possibilities
> of matching up the neighbors of S with the neighbors of S' (a
> candidate for F(S)). If G [and G'] is a regular triangle-free
graph
> with degree n/2, then the number of ways to match up the
neighbors of
> S with the neighbors of S' is (n/2)!, which is exponential in
n.
> Unless you have some way to eliminate a lot of these (and you
don't,
> since you're only looking at the first neighborhood of S and
S'), you
> have to look at them all.
> Any isomorphism-checking algorithm should also be able to deal
with
> the case where G and G' are not isomorphic.
> Or if
> one arrives towards the low level with several vertices then S
is
> isomorphous with one of its vertices.
> And to afflict for the syntax, spelling mistakes and grammar.
(A
> machine translation)
> An example of how your algorithm works would help. Let G be the
graph
> with vertex-set {1,2,3,4,5} and edges {12, 13, 15, 23, 34, 45},
and G'
> the graph with vertex-set {a,b,c,d,e} and edges {ab, ac, ae, bd,
be,
> cd}.
> f(2)=e.
> f(1)=a or f(1)=b.
> f(3)=b or f(3)=a.
> f(5)=c or f(5)=d.
> f(4)=d or f(4)=c.
> f(1,2,3,4,5)=(a,e,b,d,c).
> f(1,2,3,4,5)=(b,e,a,c,d).
> Okay. So what I said above really applies: In short, you have to
> consider the cases f(1)=a and f(1)=b and ahead of time, and then
> consider sub-cases from there.
> Since you have to choose between two options, and there's no
immediate
> reason for choosing one over the other, you (probably) won't have a
> polynomial-time algorithm. (Incidentally, more than one choice
is
> why a lot of problems become NP-complete when some parameter becomes
2
> or 3. If you are 2-coloring a graph, once you color one vertex,
you're
> basically done, since the color of every other vertex is forced.
> However, if you are 3-coloring a graph, you have a choice between 2
> colors to color each of the neighbors of the first vertex. Since,
as
> in your isomorphism algorith, you have to make a choice here, you
have
> to consider both options. My gut feeling, BTW, is that the graph
> isomorphism problem is NP-complete.)
> [ONE LAST THOUGHT:]
> A good test for your algorithm would be to pick two k-regular
graphs
> Actually, this should be k-vertex regular graphs with girth at least
> k/3.
> with girth at least (say) k/3. You can find a lot of regular graphs
> with a given number of vertices, and a given girth, at
>http://www.mathe2.uni-bayreuth.de/markus/reggraphs.html.
> proved that the two graphs are isomorphous.
> This is true. However, the point of a simple example was to see if I
> understood your algorithm. The point of the more difficult graphs (k-
> vertex regular graphs with girth at least k/3) was to make sure that
> your algorithm runs in polynomial time, which is the relevant point
> here.
> difficulty it is the translation.
> And thank you for the URL.
> -- mohamed mimouni-
I am sorry but I could not use the files of the URL (I do not have the
> program to open these file).
The program that generated those files is GENREG. The compression used
for .scd files is described at
http://www.mathe2.uni-bayreuth.de/markus/manual/genreg.html
under the -s option. GENREG itself is available at
ftp://ftp.mathe2.uni-bayreuth.de/meringer/GENREG/ (which includes the
file readscd.c, which I assume is a program to convert scd files into
something readable).
> Then here a small example to make gone
> the algorithm,
> G=(12 ;24 ;34 ;36 ;45 ;56) and G'=(ab ;ae ;bc ;cf ;df ;ef)
> One seeks for example f(e).
> 1) the pseudo-tree of e :
> 1) The vertix e.
> 2) Two vertices a and f.
> [...]
> 1) the pseudo-tree of 5 :
> 1) The vertix 5.
> 2) Two vertices 6 and 4. there' is an isomorph thus one contained.
There is something called a partial isomorphism, which is a one-to-
one map from some vertices of G to some vertices of G', and that's
what you have at this point: phi(e)=5, phi(a)=6, and phi(f)=4.
But the real point is that at this point, you don't have one but TWO
partial isomorphisms, which I'll call phi1 and phi2:
phi1(e)=5, phi1(a)=6, and phi1(f)=4.
phi2(e)=5, phi2(a)=4, and phi2(f)=6.
The problem is that one of these partial isomorphisms might not be
part of an actual isomorphism, and you don't know which one you should
choose at this point. (You have arbitrarily chosen phi1 in your
example above, which fortunately works out for you. In analyzing the
worst possible case, you have to assume you don't get lucky like
this.)
If you start off with two graphs G and G', neither of which has a
triangle, and a vertex v0 in G and a vertex u0 in G' which have degree
d, then there are d! possible partial isomophisms you have to
consider. If d is n/3, where n is the number of vertices, then the
number of cases to check is already non-polynomial and, unless you get
lucky, you have to check them all.
> 3) Two vertices 3 (adjacent withx vertices 4 and 6) and 2 (adjacent
> with vertix 2). there' is an isomorph thus one contained.
> 4) The vertix 1 (adjacent with vertix 2). Thus isomorphous total.
Thus according to the pseudo-tree of vertix 5, a solution will be :
> f(1)=b ; f(2)=c ; f(3)=d ; f(4)=f ; f(6)=a and f(5)=e.
----for the demonstration if you to understand French it will be good!---
Unfortunately, I don't, and my English-French-German list of Graph
Theory terms is back at home. (I'm on vacation at the moment.) I got
this list from Claude Berge's book _Graphs_ (or maybe _Graphs and
Hypergraphs_).
--- Christopher Heckman
===
Subject: A green love song between two class 3 enviros
>
[wannabe]
> I majored in chemistry at the University of Illinois.
>
[enviro- aoler]
[wannabe]
> Dissolve 1 mole of NaCl in L water, what is the pH?
> Dissolve 1 mole of HCl in 1L water, what is the pH?
>
[enviro-aoler]
> *** if I know, I flunked out of chemistry. ****
> But you should see the database hoops I have to jump through
> when doing pH-based soil remediation based on TACO
> (Tiered Approach to Corrective Action Objectives).
>
[hanson]
... ahahaha... AHAHAHAHHHH... those great poster children
for environmentalist' causes... Here anthropogenically and
globally warming each other... Righteously...hahahaha....
Al Gore and his class 2 enviros do love you... sincerely!....
http://www.abc.net.au/tv/swindle/
ahahaha.... ahahahanson
===
Subject: Re: Dot product invariance under rotation
Originator: pouya@localhost
[EinsteinHelpMe <mandadopor-scimath@yahoo.com>]
[...]
>I'm stuck in going from eq 16 to eq 17, i.e. a_(ij)a_(ik) =
>delta_(jk), where the aij are the elements of the rotation matrix.
[...]
(a_ij a_ik) is the inner product of the j-th and k-th
columns of the rotation matrix. Since the rotation matrix
is orthonormal, this inner product vanishes unless j=k, in
which case it is equal to one.
--
Pouya D. Tafti
p dot d dot tafti at ieee dot org
===
Subject: Re: Dot product invariance under rotation
<1184264846_836@sicinfo3.epfl.ch [EinsteinHelpMe
<mandadopor-scim...@yahoo.com>]
> [...]>I'm stuck in going from eq 16 to eq 17, i.e. a_(ij)a_(ik) =
>delta_(jk), where the aij are the elements of the rotation matrix.
[...]
(a_ij a_ik) is the inner product of the j-th and k-th
> columns of the rotation matrix. Since the rotation matrix
> is orthonormal, this inner product vanishes unless j=k, in
> which case it is equal to one.
> --
> Pouya D. Tafti
> p dot d dot tafti at ieee dot org
clear rigth now.
Sergio
===
Subject: Re: Dot product invariance under rotation
> I've been asked to prove that having a rotation matrix R in SO(3), the
> transformation Rv (v been a vector) preserves dot and cross product.
> I've seen a proof in http://mathworld.wolfram.com/DotProduct.html
> (equations 14 to 19). They make use of einstein summation notation.
> I'm stuck in going from eq 16 to eq 17, i.e. a_(ij)a_(ik) =
> delta_(jk), where the aij are the elements of the rotation matrix.
Can anyone explain me?
The sum a_{ij}a_{ik}, with 1 <= i <= 3, is c_{jk}, where the matrix
(c_{jk})_{1 <= j,k <= 3} is the product A^t.A. This is the identity
matrix, since A is in SO(3). So, c_{jk} = 1 if j = k and c_{jk} = 0
otherwise.
Jose Carlos Santos
===
Subject: Differential Equation problem
I am wondering if there is easy method to solve a differential
equation with two variables such as the diffusivity equation:
dC/dt = d2C/dx2
the right size is the second order and C is a function of both x and t.
===
Subject: Re: Differential Equation problem
> I am wondering if there is easy method to solve a differential
> equation with two variables such as the diffusivity equation:
dC/dt = d2C/dx2
the right size is the second order and C is a function of both x and t.
These are called partial differential equations (PDE). This particular
one is also called the heat equation. Depending on the type of boundary
conditions, you might use separation of variables and Fourier series,
Laplace transform in one of the variables, Fourier transform in the x
variable, or several other techniques.
--
Robert Israel israel@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada V6T 1Z2
===
Subject: Re: Differential Equation problem
> I am wondering if there is easy method to solve a differential
> equation with two variables such as the diffusivity equation:
dC/dt = d2C/dx2
the right size is the second order and C is a function of both x and t.
(Yes, you corrected it to right side.)
agree that the methods described in textbooks on partial differential
equations of mathematical physics look very difficult. Because they
are, and we (definitely including me) have to live with it.
Try Google on key phrase diffusion equation - I got over
56 million hits. You can narrow it down by including another key phrase:
separation of variables - still over 80,000 hits.
There is a particular system of solutions: for every real c, the function
1/(2*sqrt(pi*t)) * exp(-(x-c)^2/(4*t))
satisfies the equation for all t>0 and all real x. In a sense, these
solutions are fundamental - find out what that means.
Good luck,
ZVK(Slavek).
===
Subject: Re: Differential Equation problem
>I am wondering if there is easy method to solve a differential
> equation with two variables such as the diffusivity equation:
dC/dt = d2C/dx2
the right size is the second order and C is a function of both x and t.
One of the easiest for this type of equation is separation of variables.
write C(x,t)= X(x)*T(t)
then you get 2 ODEs
dT/dt=ET and X''=EX for some constant E.
===
Subject: Re: Differential Equation problem
> I am wondering if there is easy method to solve a differential
> equation with two variables such as the diffusivity equation:
dC/dt = d2C/dx2
the right size is the second order and C is a function of both x and t.
Sorry I meant the right SIDE.
===
Subject: Re: An exact 1-D summation challenge - 14 - (go and surpass all
CASs)
Any progress?
> Hello computer algebra buffs,
None of the modern computer algebra systems can calculate this.
Is there a Summation Doc to invent the steps bringing the exact
> value of this sum
sum((-1)^n*(2*n)!*(Psi(1/2+n)+Psi(1+n)-Psi(5/4+n)-Psi(7/4+n))
?
Best wishes,
Vladimir Bondarenko
Considered harmful by Richard Fateman, UC Berkley
> VM and GEMM architect
> Co-founder, CEO, Mathematical Director
http://www.cybertester.com/ Cyber Tester, LLChttp://maple.bug-list.org/
Maple Bugs Encyclopaediahttp://www.CAS-testing.org/ CAS Testing
===
Subject: what is this called then ?
if you stand still, you will be shot with more and more bullets, but
if you move all of a sudden, the number of bullets reduces
significantly
That is one example, obvious!
Could you give me another one that is related to math please ?
===
Subject: Re: what is this called then ?
if you stand still, you will be shot with more and
> more bullets, but
> if you move all of a sudden, the number of bullets
> reduces
> significantly
That is one example, obvious!
Could you give me another one that is related to math
> please ?
>
the math on this forum is really amazing...
heres another one:
if i throw a lot of rotten tomato's with several at the same time , less
will hit you if you stand still.
tommy1729
===
Subject: Re: what is this called then ?
Could you give me another one that is related to math please ?
never eat more than you can lift.
===
Subject: Re: what is this called then ?
if you stand still, you will be shot with more and more bullets, but
> if you move all of a sudden, the number of bullets reduces
> significantly
That is one example, obvious!
Could you give me another one that is related to math please ?
Coincidence.
If point A and B are coincient, there are in the same place at the
same time.
- MO
===
Subject: Re: what is this called then ?
I forget to add in another explanation that, for example, there are
1000 gunmen pointing right toward a rectangular in which there is a
stickman, those people are not intentionally shooting the poor guy,
they just want to try out the rectangular. My original post's question
is why he is shot with a smaller number of bullets when he suddenly
jumps to another position in the rectangular than that when he stands
still to be shot ?
===
Subject: Re: what is this called then ?
I forget to add in another explanation that, for example, there are
> 1000 gunmen pointing right toward a rectangular in which there is a
> stickman, those people are not intentionally shooting the poor guy,
> they just want to try out the rectangular. My original post's question
> is why he is shot with a smaller number of bullets when he suddenly
> jumps to another position in the rectangular than that when he stands
> still to be shot ?
>
you are missing too much information to give a specific answer.
how big are the bullets?
how fast does he move?
when do they fire?
do the fire all at once?
do the fire at random, many times?
How big is the rectangle?
how thick are the lines of the stickman?
Assuming they fire randomly, and keep firing, it makes no difference if he
is moving or not.
===
Subject: Re: what is this called then ?
if you stand still, you will be shot with more and more bullets, but
> if you move all of a sudden, the number of bullets reduces
> significantly
That is one example, obvious!
Could you give me another one that is related to math please ?
>
If you stand out in the rain you get soaking wet.
If you go inside you dry off.
===
Subject: Bug in Mathematica 6 - Integrate - 46 (ArcTan, Sin, Catalan's
integral failure, 1989--2007--?)
Our little demo continues... Hello again from the VM machine
which is still ignored by CAS manufacturers.
Just for fun, there is a nice paper by Viktor Adamchik,
Thirty-Three Representations of Catalan's Constant
http://library.wolfram.com/infocenter/Demos/109/
Three Hundred and Thirty-Three Representations of Catalan's
Constant. A simple example from there.
The following integral presents no difficulty for Maple 11...
Maple 11> int(arctan(exp(-z)), z= 0..infinity);
1/8*Pi*ln(2+2^(1/2))-1/4*I*Pi*arctan(1/(2^(1/2)-1))+1/4*I*Pi*
arctan(1/(1+2^(1/2)))-I*dilog(-1/2*2^(1/2)-1/2*I*2^(1/2)+1)-I
*dilog(1/2*2^(1/2)+1/2*I*2^(1/2)+1)-1/8*Pi*ln(2)+5/96*I*Pi^2+
1/2*Catalan+1/8*Pi*ln(2-2^(1/2))
Maple 11> simplify(%);
Catalan
... but it presents a problem for ALL versions of Mathematica
during 1989-2007.
In[1] := Integrate[ArcTan[Exp[-z]], {z, 0, Infinity}]
Out[2] = Integrate[ArcTan[Exp[-z]], {z, 0, Infinity}]
Best wishes,
Vladimir Bondarenko
VM and GEMM architect
Co-founder, CEO, Mathematical Director
http://www.cybertester.com/ Cyber Tester, LLC
http://maple.bug-list.org/ Maple Bugs Encyclopaedia
http://www.CAS-testing.org/ CAS Testing
===
Subject: Hundreds and Hundreds of SM Available for Math, Science & Engr
If interested contact me at bwennny@hotmail.com
A Course in Algebraic Number Theory by Henry Cohen
A Course in Game Theory by Martin J. Osborne and Ariel Rubinstein
A First Course In Probability 7th Ed by Sheldon Ross
Adaptive Control 2nd Ed. by Karl.J.Astrom
Advanced Engineering Marhematics ERWIN KREYSZIG 9E
Advanced Engineering Mathematics 8Ed Erwin Kreyszig
Advanced Engineering Mathematics 8Ed Erwin Kreyszig in korean (all
even and odd no.)
Advanced Engineering Mathematics 2nd Ed. by Zill & Cullen
Advanced Engineering Mathematics 3RD ED. by ZILL & CULLEN
Advanced Macroeconomics by Jeffery Rohaly
Advanced Modern Engineering Mathematics 3rd Ed. by Glyn James
Analog Integrated Circuit Design by David Johns, Ken Martin
Analysis and Design of Analog Intergrated Circuits 4th Ed. by Paul R.
Gray
Analytical Dynamics by Barcuh (selected problems)
Analytical Mechanics 7th Ed. by Fowles
Antenna for all application 3rd Ed.by John D. Kraus (Instructor's
Manual)
Antenna Theory 2nd Ed. by balanis
Applied Linear Statistical Models 5th Ed. by MichaelH.Kutner
Applied Numerical Analysis 7th Ed. by Curtis F. Gerald, Patrick O.
Wheatley
Applied Numerical Methods with Matlab for Engineers and Scientists by
Chapra
Applied Statistics Probability for Engineers 3rd ed Montogomery
(selected problems)
Applied Strength of Materials 4th Ed. by Robert L. Mott
Automatic Control Systems 8th Ed. Kuo and Golnaraghi
Basic Engineering Circuit Analysis 7th Ed. by Irwin
Biochemistry 2nd Ed. by Philip W. Kuchel, Gregory B. Rakston
Bioprocess Engineering Principles by Doran, 1995 rare
C++ How to Program 3rd ed. Deitel, Deitel & Nieto (Instructor's
Manual)
Calculus 5th Ed. by James Stewart (Complete Instructors Solutions)
Calculus by Jerrold Marsden & Alan Weinstein Vol 1
Calculus Early Transcendentals 5th Ed. by James Stewart
Calculus of Variations with solutions by Russak 2002
Chemical And Engineering Thermodynamics 3Ed by Wiley
Chemical Engineering Solutions to the Problems in Volume 1 by J R
Backhurst
Classical Dynamics 5th Ed. by Thornton
Classical Dynamics by Marion
Classical Electrodynamics 3rd Ed. by J.D. Jackon
CMOS Analog Circuit Design 2nd Ed. by P.E. Allen
Combinatorics by Merris
Communication Systems 4th Ed. by Simon Haykin
Communication Systems Engineering 2nd Ed. by Proakis J.
Communication Systems: An Introduction to Signals and Noise in
Electrical Communication 4th. Ed. by Bruce Carlson
Computational Techniques for Fluid Dynamics by K. Srinivas and
C.A.J.Fletcher
Computer Networking: A Top-Down Approach Featuring the Internet 2nd
ed. by Jim Kurose and Keith Ross
Computer Networks 4th Ed. by Andrew.S.Tanenbaum
Computer Networks: A Systems Approach 3rd Ed. by larry L. Peterson and
Bruce S. Davie
Computer Organization And Design - The Hardware Software Interface 3rd
Ed.
Control Systems Engineering 3rd Ed. by Norman S. Nise
Control Systems Engineering by Nise, 4th
Design and Analysis of Experiments 6th Ed. by Montgomery
Design Of Analog Cmos Integrated Circuits by Razavi
Design with Operational Amplifiers and Analog Integrated Circuits 3rd
Ed. by Franco, Sergio
Device Electronics for Integrated Circuits 3rd Ed. by Muller and
Kamins
Differential Equations with Boundary-Value Problems 5th Ed. by Zill
and Cullen
Digital communication: Fundamentals and Applications 2nd Ed. by
Bernard Sklar
Digital communications 4th Ed. John Proakis with ebook
Digital Image Processing 2nd Ed. by Gonzalez (Instructor's Manual)
Digital Integrated Circuits by Rabaey 2nd Ed. (selected problems)
Digital signal processing A Computer-Based Approach 1st Ed. by Sanjit
K. Mitra
Digital Signal Processing by Proakis & Manolakis
Digital Signal Processing by Thomas J. Cavicchi
Discrete-Time Signal Processing 2nd Ed. by Oppenheim, R.W. Schafer and
J. R. Buck
Econometric Analysis 5th Ed. by William H. Greene
Econometric Analysis of Cross Section and Panel Data by Jeffrey M.
Wooldridge
Electric Circuits 7th Ed. by Nilsson
Electric Machinery 6th Ed. by Fitzgerald, Kingsley, Uman
Electric Machinery and Power System Fundamentals 1st Ed. by Stephen J.
Chapman (Instructor's Manual)
Electric Machinery Fundamentals 4th Ed. by Stephen J Chapma
(Instructor's_Manual)
Electrical Engineering Principles And Applications 4th Edition by
Giorgio Rizzoni
Electrical Engineering Principles And Applications by Hambley
Electronic Circuit Analysis and Design 2nd Ed. by Donald A. Neamen
Electronic Physics by Strabman
Electronics 2nd Ed. by Allan R. Hambley
Elementary Differential Equations and Boundary Value problems 7th Ed.
by Boyce, DiPrima (book and solutions)
Elementary Differential Equations and Boundary Value Problems 8th Ed.
Elementary Linear Algebra by K. R. Matthews (book and solution
manuals)
Elementary Mechanics and Thermodynamics by John W. Norbury
Elements of Chemical Reaction Engineering 3rd Ed. by Timothy Hubbard,
Jessica hamman and David Johnson
Elements of Information Theory by Thomas M. Cover
Engineering Biomechanics: Statics by Angela Matos, Eladio Pereira,
Juan Uribe and Elisandra Valentin
Engineering Economic Analysis 9E by Newman (selection problems)
Engineering Circuit Analysis 6th Ed. by Hayt
Engineering Electromagnetics 6th Ed. by William H. Hayt, Jr. and Hohn
A. Buck
Engineering Fluid Mechanics 7th Ed. by Crowe
Engineering Fluid Mechanics 8th Ed. by Crowe
Engineering Mathematics 4th Ed. by John Bird
Engineering Mechanics Dynamics 10th edition by Hibbeler
Engineering Mechanics Dynamics 11th edition by Hibbeler
Engineering Mechanics Dynamics 5th Ed. by Meriam
Engineering Mechanics Dynamics by Bedford and Fowler
Engineering Mechanics statics 10th edition by Hibbeler
Engineering Mechanics statics 11th edition by Hibbeler
Engineering Mechanics Statics 4th Ed. Bedford
Engineering Mechanics Statics by Meriam
Feedback Control of Dynamic Systems 4th Ed. by G. F. Franklin, J. D.
Powell, A. Emami
Foundations of Materials Science and Engineering, Ed. 3 by Smith and
Hashemi
Foundations of Materials Science and Engineering Ed. 4 by Smith and
Hashemi
Fractal Geometry: Mathematical Foundations and Approach by Kenneth
Falconer
Fluid Mechanics 5th Ed. by White
Fluid Mechanics by Shames
Fluid Mechanics and Thermodynamics of Turbomachinary, 4E & 5E by
Dixon
Foundations of Materials Science and Engineering 3E & 4E by Smith
Fundamentals of Machine Component Design 4th Ed. by Juvinall
Fundamentals of Aerodynamics 2nd Ed. by John D. Anderson
Fundamentals Of Aerodynamics 3rd Ed. by Anderson (Instructor's
Solutions Manual)
Fundamentals of Aerodyanmics 5 th Ed. by John D. Anderson (Instructor
Manual)
Fundamentals of Aircraft Structural Analysis by Howard Curtis
(Instructor's solutions)
Fundamentals of Applied Electromagenetics 5th Ed+
Fundamentals of Differential Equations by Nagle Saff Snider
Fundamentals of Digital Logic with Verilog Design 1st Ed. by S. Brown,
Z. Vranesic
Fundamentals of Digital Logic with VHDL Design 1st Ed. by S. Brown, Z.
Vranesic
Fundamentals of Electric Circuits 2nd Ed. by C.K.Alexander
M.N.O.Sadiku
Fundamentals of Engineering Thermodynamics 5th ed (M. J. Moran & H. N.
Shapiro)
Fundamentals of Financial Management With Infotrac Concise 4th Ed. by
Eugene Brigham
Fundamentals of Fluid Mechanics 3th Ed. by Munson (Student solution
manual)
Fundamentals of Fluid Mechanics 4th Ed. by Munson
Fundamentals of Fluid Mechanics 5th Ed. by Munson
Fundamentals of Heat and Mass Transfer 5th Ed. Incropera
Fundamentals of Logic Design 5th Ed. by Charles Roth
Fundamentals of Machine Component Design 3rd Ed. by Juvinall and
Marshek
Fundamentals of phsics 7th ed. by David Halliday, Robert Resnick, and
Jearl Walker (instructor's manual)
Fundamentals of Thermal-Fluid Sciences by Cengal
Fundamentals of Thermodynamics 6th Ed. Richard E. Sonntag, Claus
Borgnakke, Gordon J. Van Wylen
Gas Dynamics 2E by James John
Guide to Energy Management 5th Ed. by Klaus-Dieter E. Pawlik
Heat Transfer 2nd Ed. by Cengel
Hydraulics in Civil and Environmental Engineering 4th Ed. by Andrew
Chawick
Introduction To Algorithms 2nd Ed. by Philip Bille (selected
problems)
Introduction To Algorithms 2nd Ed. by Thomas H. Cormen, Charles
E.Leiserson, Ronald L. Rivest, Clifford Stein
Introduction to Antenna Theory by Palacios
Introduction to electric circuits 6th Ed. by Dorf-Svaboda
Introduction to electrodynamics 3rd Ed. by David Griffiths
Introduction to Fluid Mechanics 5ed. by Fox
Introduction to Fluid Mechanics 6th Ed. by Fox
Introduction to Linear Algebra 3th Ed. by Gilbert Strang
Introduction to Quantum Mechanics 2ed Griffiths, David J.
Linear Algebra and its Applications 3rd Ed. by Lay
Linear Algebra by Jim Hefferon
Linear circuit analysis by R. A. DeCarlo and P. Lin
Linear Systems and Signals by B P Lathi
Logic Computer Design Fundamentals 2th Ed. by Morris Mano (selected
problems)
Managerial Accounting 11th Ed.
Material Science and Engineering an Introduction 6th Ed. by Callister
Material Science and Engineering an Introduction 7th Ed. by Callister
Materials & Processing in Manufacturing 9th Ed. by Demargo
Mathematical Methods for Physics and Engineering 3rd Ed. by
K.F.RileyandM.P.Hobson
Mathematics for Economics by Simon and Blume
Mathematical Physics by Arfken
Matrix Analysis And Applied Linear Algebra by Carl D. Meyer
Mechanical Engineering Design 7th Ed. Shigley
Mechanics of Fluids 8th Ed. by Bernard Massey
Mechanics of materials Hibbeler 6th
Mechanics Of Materials(3rd Ed , By Beer, Johnston, & Dewolf)
Microeconomic Analysis 3rd Ed. by Hal R. Varian
Microeconomic Theory by Andreu Mas-Colell, Jerry R. Green, Michael
Dennis Whinston
Microelectronic Circuits 4th Ed. by Sedra, Smith
Microelectronic circuits 5th Ed. Sedra
Microelectronics Digital and Analog Circuits and Systems by Millman
Microwave and rf design of wireless systems by Pozar
Microwave Engineering 2nd Ed. David M Pozar
Microwave Engineering 3rd Ed. David M Pozar
Microwave Transistor Amplifiers Analysis and Design 2nd Ed. by
Guillermo Gonzalez
Modern Control Engineering 3rd Ed. by K.OGATA
Modern Control Systems 10E by Richard Dorff
Modern Digital and Analog Communications Systems by B P Lathi
Multi Variable Calculus 5th Ed.
Numerical Methods for Engineers by Steven C. Chapra and Raymond P.
Canale
Numerical Methods 3E by Faires
Operating System Concepts 6th Ed. by Abraham Silberschatz
Operating System Concepts 7th Ed. by Abraham Silberschatz
Operating Systems:Internals and Design Principles 4th Ed. by William
Stallings
Orbital Mechanics for Engineering Students by Curtis, 1st
Orbital Mechanics for Engineering Students Curtis
Organic Chemistry 5th Ed. by Atkins and Carey
Physical Chemistry 7th. Ed. by Peter Atkins & Julio de Paula
Physics for Scientists & Engineers by Tipler & Mosca 5th Edition
Physics For Scientists And Engineers 3rd Ed. by Douglas C. Giancoli
Physics for Scientists and Engineers: A Strategic Approach by Knight
Physics for Scientists and Engineers by KNIGHT
Physics for Scientists and Engineers 5th Extended Version by Serway
and Faughn
Physics for Scientists and Engineers 6th ed. Serway Jewett
Physics Volume 2 5th Ed. by Halliday, Resnick and Krane
Power System Analysis by John J. Grainger & William D. Stevenson, Jr
Principles of Communication 5th Ed. by R. E. Ziemer, William H.
Tranter
Principles of Physics 4th Ed. Calculus Based by Serway, Raymond A. and
John W. Jewett
Principles with Applications BY Giancoli 6TH Edition
Probability and Statistics for Engineers and Scientists (selected
problems)
Probability Random Variables and Stochastic Processes 4th Ed. by
Papoulis
Process Systems Analysis and Control by Donald R.coughanowr(selected
problems)
Quantum mechanics by Kyriakos Tamvakis
Recursive Macroeconomic Theory by Lars Ljungqvist and Thomas J.
Sargent
Recursive Methods in Economic Dynamics by Wright
RF circuit Design Theory and Application by Ludwig bretchko
Semiconductor Device Fundamentals 1st Ed. by Robert F. Pierret
Semiconductor Physics and Devices 3rd Ed. by Neamen
Signal Processing and Linear Systems by B P Lathi
Signals And Systems 2nd Ed. by Haykin
Signals And Systems 2nd Ed. by Oppenheim
Signals, Systems and Transforms 3rd Ed. by Oppenheim
Soil Mechanics: concepts and applications 2ed William Powrie
Solid State Electronic Device 5th Ed. by Ben Streetman (Instructor's
Manual)
Solid State Physics solutions
Solved and Unsolved Problems in Number Theory and Geometry by Daniel
Shanks
Strengths of Materials by Beer, Johnston, Dewolf
Structural Analysis by R.C. Hibbeler
System Dynamics by Ogata 3rd
The C Programming Language 2nd Ed. by Kernighan and Ritchie
The Econometrics of Financial Markets by Petr Adamek
The Science and Engineering of Materials 4th Ed. by Donald R. Askeland
Pradeep P. Phule (Instructor's Manual)
Theory and Applications of OFDM and CDMA Wideband Wireless
Communications (2005)
Thermodynamic 5th ed. Cengal
Thermodynamics: An Engineering Approach 5th Ed. by Cengel Boles
(Instructor's Solutions Manual)
Thermodyanamics of Turbomachinary 5E
Time-Harmonic Electromagnetic Fields by Harrington solutions, 2001
Thomas' Calculus Early transcendentals 10th Ed. by George B. Thomas
Vol 1
Thomas' Calculus Early transcendentals 10th Ed. by George B. Thomas
Vol 2
Transport Phenomena 2nd Ed. by Bird and Stewart(to class 1 and class 2
problems)
University Physics with Modern Physics (11th) by Hugh D. Young and
Roger A. Freedman.
Vector Calculus 3rd Ed. by Susan Jane Colley
Vector Mechanics for engineers dynamics by beer 6th edition
Vector Mechanics for engineers dynamics by beer 7th edition
Vector Mechanics for engineers statics by beer 6th edition
Vector Mechanics for engineers statics by beer 7th edition (nearly the
same with edtion 8)
Wireless Communications principles and practice 2nd Ed. by Theodore
Rappaport 1996
===
Subject: First time to write a paper, any tips?
This is the first time that I am trying to submit a paper to a
journal. So I hope some of you with experience could advise me. I
really really wanted to choose to submit a paper to an open-access
journal. So I tried the Nagoya Journal of Mathematics, according to
the editors my paper is too sophisticated and has to be submitted to
some algebra journal (unfortunately there is algebra journal that is
open-access, so I guess I have to choose a pay-journal). So I would
like to have some tips as to where else I could submit this paper I
have a third-party advise. Also, I'd like the paper to appear
somewhere as a preprint so people could read about it even before it
is published (which would take some time anyway). Could anyone advise
me on how I could publicize preprints? I could of course, post it in
my website but I do not know if that would attract interested readers
- my website is hardly visited.
Jose Capco
===
Subject: Re: First time to write a paper, any tips?
> ... So I would like to have some tips as to where else I
There is a list of open access journals maintained at the Directory
of Open Access Journals hosted at
http://www.doaj.org/
There are currently 2751 journals represented there --- all free and
full text. Of these, there are 104 mathematics journals:
http://www.doaj.org/doaj?func=subject&cpid=57
For a journal to be included in this list, that journal must have some
form of quality control involving peer review and/or an editor and/
or an editorial board.
http://www.doaj.org/doaj?func=loadTempl&templ=about#criteria
As already mentioned by Puppet_Sock, there is also Cornell's arXiv
server providing Open access to 427,314 e-prints in Physics,
Mathematics, Computer Science and Quantitative Biology at
http://www.arxiv.org/
ArXiv is the archive where Grigori Perelman posted his now famous
solution to the 100 year old Poincare Conjecture.
[Jose Capco]
> ...so I guess I have to choose a pay-journal ...
> Also, I'd like the paper to appear somewhere as a preprint ...
If you do find it necessary to publish in a journal where you must
forfeit your rights over your own work to a publisher, you may still
be able to legally post post-prints and/or preprints. To find out
which journals allow or don't allow such actions, you can consult
Journal Policies - Self-Archiving Policy By Journal at
http://romeo.eprints.org/
or Sherpa at
http://www.sherpa.ac.uk/
For example, here is the post-print/pre-print policies of a number of
journals dealing with algebra:
http://romeo.eprints.org/search.php?t=algebra
And as far as where you can post pre-prints/post-prints, again arXiv
would seem a good choice.
http://www.arxiv.org/
They maintain several categories which may be of interest to you:
Algebraic Geometry
Algebraic Topology
Commutative Algebra
Operator Algebras
Quantum Algebra
Rings and Algebras
I wish you all the best with finding a good way to share your results
^__^
Dan Greenhoe
On Jul 13, 4:59 am, Jose Capco <cliomse...@kriocoucke.mailexpire.com
> This is the first time that I am trying to submit a paper to a
> journal. So I hope some of you with experience could advise me. I
> really really wanted to choose to submit a paper to an open-access
> journal. So I tried the Nagoya Journal of Mathematics, according to
> the editors my paper is too sophisticated and has to be submitted to
> some algebra journal (unfortunately there is algebra journal that is
> open-access, so I guess I have to choose a pay-journal). So I would
> like to have some tips as to where else I could submit this paper I
> have a third-party advise. Also, I'd like the paper to appear
> somewhere as a preprint so people could read about it even before it
> is published (which would take some time anyway). Could anyone advise
> me on how I could publicize preprints? I could of course, post it in
> my website but I do not know if that would attract interested readers
> - my website is hardly visited.
Jose Capco
===
Subject: Re: First time to write a paper, any tips?
On Jul 12, 4:59 pm, Jose Capco
> So I would
> like to have some tips as to where else I could submit this paper I
If you are in grad school and want to publish a paper,
I'd say the very first person to ask is your prof.
Were I your prof and you didn't ask me that kind of
question, I might be a little annoyed.
Where you publish depends very strongly on the kind
of paper you are talking about. What you want to do
a journal that has the kind of paper you have written.
reason) representations of groups, you want to find
And also roughly the same length and complexity as
> Also, I'd like the paper to appear
> somewhere as a preprint
Have you considered the preprint arxiv?
http://www.arxiv.org/
First check it out and see if you can abide by there
(not very restrictive) rules. And see if their list
of topics matches your paper.
Socks
===
Subject: Re: Derive Software no longer available!?
<ysydnbA57aLz9QjbnZ2dnUVZ_u2mnZ2d@comcast.com>
<sO6dnUSCnpILmwvbnZ2dnUVZ_gCdnZ2d@rcn.net
> After that optimistic outlook by Julio Valella, now Derive is dead.
> Why would TI buy a product that had a very good reputation and kill
> it? I've never understood that. If it wasn't intentional, it had to
> be through incredible miss-management.
Frankly, I'm amazed at how many products that Borland had purchased that
> were sent to the scrap heap soon after.
I'm interested in seeing an example or two
of what you are thinking of, re: Borland.
===
Subject: Re: Derive Software no longer available!?
> Frankly, I'm amazed at how many products that Borland had purchased that
> were sent to the scrap heap soon after.
I'm interested in seeing an example or two
> of what you are thinking of, re: Borland.
They bought a CM package called Sorcerer's Apprentice. Soon it was
abandoned. Then they bought a popular programmer's editor called Brief from
Solutions Software. After buying Brief, Borland did no updates, and
eventually dropped the product. Then they bought the newer program editor
CodeWrite from Premia. That they actually did maintain for a while, but 
that
product was also allowed to lapse.
>
===
Subject: Re: chessalgebra
> On Sun, 08 Jul 2007 21:18:18 EDT, tommy1729
> <tommy1729@gmail.com
>let an infinite chessboard with one line on the real
> axis and the second on the imaginary axis define a
> set A.
>the lines separating the squares and the
> intersections of the lines (corners) define the
> complex elements in set A.
>let f be a multivariable function, how does one
> solve in general f(x,y,...)=0 for this set A ?
That's easy ...
Just plug f into your universal diophantine equation
> solver.
You mean you don't have such a machine?
Well, they are somewhat rare.
quasi
there is more to it than that.
note that it isnt a diophantine equation at all.
since the potential set of solutions has uncountable many elements.
and not countable as with diophantine.
you could say im vague
and ask for such a polymial , but im talking in general here...
and as for giving an example , i find it difficult at first sight to give a
polynomial that does not have a solution or even not an infinite set of
solutions,
therefore ive given none at all.
perhaps the construction of such a polynomials answers my question...
greets
tommy1729
===
Subject: Re: rivers of
> On Jul 11, 2:38 am, Vincenzo Librandi
> <vincenzo.librandw...@alice.it >there is a genius in scatology i swear
> we make philosophy?
> Vincenzo Librandi
In Mel Brooks' History of the World - Part 1,
> Brooks' Roman-era
> character was on an unemployment line, and said his
> occupation was
> Stand-up Philosopher.
And the unemployment clerk (played by Bea Arthur)
> responded, oh,
> you're a bull artist!.
So yes, philosophy = bullting.
In Mel We Trust! :-)
as long as it isnt mel gibson
tommy1729
===
Subject: Re: x^4+y^3=z^2
> For x,y,z integer>0
solve
x^4+y^3=z^2
after the solutions
(1,2,3) and (5,6,29)
Vincenzo Librandi
(6,9,45)
(7,15,76)
(9,27,162)
tommy1729
===
Subject: Re: x^4+y^3=z^2
> For x,y,z integer>0
solve
x^4+y^3=z^2
after the solutions
(1,2,3) and (5,6,29)
>(6,9,45)
>(7,15,76)
>(9,27,162)
Of course
(5,75,650)
(6,72,612)
(9,54,405)
(9,360,6831)
(63,5292,384993)
and so on
because
y^2=x^3+16
y^2=y^3+81
y^2=x^3+256
and so on
they do not have solutions?
Vincenzo Librandi
===
Subject: Re: x^4+y^3=z^2
For x,y,z integer>0
solve
x^4+y^3=z^2
after the solutions
(1,2,3) and (5,6,29)
>(6,9,45)
>(7,15,76)
>(9,27,162)
Of course
(5,75,650)
> (6,72,612)
> (9,54,405)
> (9,360,6831)
> (63,5292,384993)
> and so on
because
> y^2=x^3+16
> y^2=y^3+81
> y^2=x^3+256
> and so on
they do not have solutions?
whats that suppose to mean ??
i have given many solutions !!
and you added many too !!
what does not have a solution ???
its my turn now
8^4+y^3=z^2
tommy1729
Vincenzo Librandi
===
Subject: Re: x^4+y^3=z^2
>its my turn now
8^4+y^3=z^2
>
8^4+32^3=192^2
See Pell's equations.
Vincenzo Librandi
===
Subject: Re: Top Number Theory Journals??
Yes, but, this one is Swedish.
Do you know any good one from the US, and that doesn't require
a professor, I mean, I want one journal that do the analysis
the attention of those analyzers.
Thx
===
Subject: Book advice
Hi all,
I am about to buy some books about optimization theory and would
appreciate any advice.
I have a good math background but very limited knowledge about
optimization. I'll like to acquire the main techniques for linear and
non linear optimization.
I am currently considering:
Optimization theory and methods by Wenuy Sun and Ya-xiang Yuan
Introduction to linear and non linear programming by Luenberger
Numerical optimization by Nocedal
Non linear programming by Bertsekas
I'll probably buy 2 books. Which one should have my preference?
Any other book to recommend?
===
Subject: Re: Book advice
Originator: pouya@ioamac68.epfl.ch (Pouya D. Tafti)
> Hi all,
> I am about to buy some books about optimization theory and would
> appreciate any advice.
> I have a good math background but very limited knowledge about
> optimization. I'll like to acquire the main techniques for linear and
> non linear optimization.
Have you considered Boyd and Vandenberghe's Convex
Optimization? It has been published by Cambridge University
Press, but is also available online:
<http://www.stanford.edu/~boyd/cvxbook/>
--
Pouya D. Tafti
<p dot d dot tafti at i triple e dot org>
===
Subject: Re: set theory : the blunder
> On Jul 12, 12:56 pm, tommy1729 <tommy1...@gmail.com > On Wed, 11 Jul 2007
18:12:13 EDT, tommy1729
> <tommy1...@gmail.com > i said
> x = {x}
> Actually, there's no problem in set theory (say,
> a
> variant of ZFC
> where FUND is replaced by AFA) to have (some)
> such
> sets x.
> F.
> --
> E-mail: info<at>simple-line<dot>de
> then why did nobody agree on it ??
> and i mean all x.
> x=(x)
> btw i wonder , the original problem proposed to me
> , was about a set containing elements of its own
> subset.
> i said it was paradoxal or the 2 sets were equal
> (see the original plz since this description is
> incomplete)
> and the paradox was solved bye (x)=x
> but nobody else answered the question.
> that means they were stuck ?
> the cantorians cant answer the question defending
> cantor themselves ?
> how 'odd'
> tommy1729- Hide quoted text -
> - Show quoted text -
You say:
Ax x={x}
then:
N = {N}
and N has infinitely many members and just one
> member; wich is absurd.
yes what you say is absurd.
and has nothing to do with what i said.
the number of elements of a set x (or N ) is not even specified , so how do
you make a paradox ?
you confuse N with only one element, whereas N can be a set or a
number.(since undefined like x)
On the other hand, that a set contains all elements
> of all of its own
> subsets is trivially true. But perhaps you are
> requiring that it
> contain a subset of its own as element; that's quite
> another thing,
> but it's also possible; consider:
S = {1,2,{1}}
> I dare advice you to look at a good introduction to
> set theory.
i dare advice you thinking before you give a bad argument
tommy1729
===
Subject: Re: set theory : the blunder
On Thu, 12 Jul 2007 12:24:21 -0400, David Bernier
If x = {x}, then
> {x} = x, then
> {{x}} = {x} = x, then
> {{{x}}} = {{x}} = {x} (= x, of course...)
and in general
> {{{{{{{{.....{{{{{x}}}}.....}}}}} = x ,
> with n consecutive '{' symbols to the left of 'x' and n consecutive '}'
> symbols to the right of 'x',
> for any positive integer x.
Do you think that tells us something about what x is?
>
In SOME set theories it does.
Especially in Peter Aczel's variant of ZFC where the axiom FUND is
replaced with AFA:
[The axiom] AFA, is based on accessible pointed graphs (apg) and
states that two hypersets are equal if and only if they can be
pictured by the same apg. Within this framework, it can be shown that
the so-called Quine atom, formally defined by Q = {Q}, exists and is
unique.
http://en.wikipedia.org/wiki/Non-well-founded_set_theory
F.
--
E-mail: info<at>simple-line<dot>de
===
Subject: Re: set theory : the blunder
On Thu, 12 Jul 2007 06:56:36 EDT, tommy1729 <tommy1729@gmail.com
> i said
x = {x}
> Actually, there's no problem in set theory (say, a
> variant of ZFC where FUND is replaced by AFA) to
> have (some) such sets x.
> then why did nobody agree on it ??
>
Huh?! I already told you that there are (well established) theories
out there, where x = {x} for some sets x. Peter Aczel's variant of ZFC
for example.
[The axiom] AFA, is based on accessible pointed graphs (apg) and
states that two hypersets are equal if and only if they can be
pictured by the same apg. Within this framework, it can be shown that
the so-called Quine atom, formally defined by Q = {Q}, exists and is
unique.
Note though:
It is worth emphasizing that hyperset theory is an extension of
classical set theory rather than a replacement: the well-founded sets
within a hyperset domain conform to classical set theory.
http://en.wikipedia.org/wiki/Non-well-founded_set_theory
and i mean [for] all x:
x = (x)
>
Taking your (x) to mean {x}, I already told you the problem with
this _set_ theory: each and any set would have exactly one element.
That's not what _we_ want.
On the other hand, in mereology, you certainly can have x = [x] for
all x (depending on the definition of [.].) I personally once
devised a small theory of heaps where [a, b, c] denotes the heap
consisting of the constituents a, b, c. There for any heap x: x = [x].
Since a heap which consist of a heap is just a heap. In this heap
theory there wouldn't be an empty heap, since if there are no
constituents which make up a/the heap there is simply no heap (and not
some fictitious entity called the /the empty heap/). But this is not a
_necessary_ property of ALL mereological theories, it seems.
Have a look!
http://plato.stanford.edu/entries/mereology/
http://en.wikipedia.org/wiki/Mereology
F.
--
E-mail: info<at>simple-line<dot>de
===
Subject: Re: set theory : the blunder
On the other hand, in mereology, you certainly can have x = [x] for
> all x (depending on the definition of [.].) I personally once
> devised a small theory of heaps where [a, b, c] denotes the heap
> consisting of the constituents a, b, c. There for any heap x: x = [x].
> Since a heap which consist of a heap is just a heap. In this heap
> theory there wouldn't be an empty heap, since if there are no
> constituents which make up a/the heap there is simply no heap (and not
> some fictitious entity called the /the empty heap/). But this is not a
> _necessary_ property of ALL mereological theories, it seems.
>
Note that in a _set_ theory e and c necessarily coincide, as has
been shown by Kanamori in his paper The Empty Set, the Singleton, and
the Ordered Pair. I'll just quote the proof:
THEOREM. Aa(a = {a}) is equivalent to:
(*) AaAb(a c b <-> a e b).
PROOF. Suppose first that a = {a}. Then for any b, a c b if and only
if {a} c b. But {a} c b if and only if a e b, and hence we can
conclude that a c b if and only if a e b.
For the converse, first note that since a e {a}, (*) implies that a
c {a}. But also, since a c a, (*) implies that a e a, so that {a} c a.
Hence we can conclude that a = {a}. []
In other words, the identification of classes with their unit class
is equivalent to the very dissolution of the inclusion vs. membership
distinction [...].
Hence (after some additional considerations) I proposed for my heap
theory:
Let's stick with c (and forget about e) for heaps:
a c A
a is a constituent of the heap A
~~~~~~~~~~~~~~~~~~~~~~~~~
Another important point: If we use the symbol
[a, b, c]
to denote the heap consisting of the constituents a, b, c, this does
NOT mean that the heap necessarily consists of exactly 3 (atomic)
elements! Hence [a] does NOT necessarily denote a singleton here!
Example. Consider the heap C consisting of the 3 _atomic_ elements a,
b, c. Then we have C = [a, b, c]. But we also have C = [C]!
With other words, we cannot claim that [C] just consists of exactly
one (atomic) object.
~~~~~~~~~~~~~~~~~~~~~~~~~
Indeed, in contrast to a set there are no elements _in_ a heap. A heap
just consists of its constituents. As a consequence heaps merge:
Let d = [a,b], then [d,c] = [a,b,c].
For example, if Q would be the heap consisting of all rational
numbers, and I would be the heap of all irrational numbers, then
R := [Q,I] would be the heap of all real numbers.
With other words, the heap [Q,I] does not just have two elements Q and
I, but consists of all real numbers.
There's no hierarchy of heaps (like in set theory) - in this sense
heaps are flat.
F.
--
E-mail: info<at>simple-line<dot>de
===
Subject: Re: set theory : the blunder
> On the other hand, in mereology, you certainly can have x = [x] for
> all x (depending on the definition of [.].) I personally once
> devised a small theory of heaps where [a, b, c] denotes the heap
> consisting of the constituents a, b, c. There for any heap x: x = [x].
> Since a heap which consist of a heap is just a heap. In this heap
> theory there wouldn't be an empty heap, since if there are no
> constituents which make up a/the heap there is simply no heap (and not
> some fictitious entity called the /the empty heap/). But this is not a
> _necessary_ property of ALL mereological theories, it seems.
 Note that in a _set_ theory e and c necessarily coincide, as has
> been shown by Kanamori in his paper The Empty Set, the Singleton, and
> the Ordered Pair. I'll just quote the proof:
THEOREM. Aa(a = {a}) is equivalent to:
(*) AaAb(a c b <-> a e b).
PROOF. Suppose first that a = {a}. Then for any b, a c b if and only
> if {a} c b. But {a} c b if and only if a e b, and hence we can
> conclude that a c b if and only if a e b.
> For the converse, first note that since a e {a}, (*) implies that a
> c {a}. But also, since a c a, (*) implies that a e a, so that {a} c a.
> Hence we can conclude that a = {a}. []
In other words, the identification of classes with their unit class
> is equivalent to the very dissolution of the inclusion vs. membership
> distinction [...].
Hence (after some additional considerations) I proposed for my heap
> theory:
Let's stick with c (and forget about e) for heaps:
a c A
a is a constituent of the heap A
~~~~~~~~~~~~~~~~~~~~~~~~~
Another important point: If we use the symbol
[a, b, c]
to denote the heap consisting of the constituents a, b, c, this does
> NOT mean that the heap necessarily consists of exactly 3 (atomic)
> elements! Hence [a] does NOT necessarily denote a singleton here!
Example. Consider the heap C consisting of the 3 _atomic_ elements a,
> b, c. Then we have C = [a, b, c]. But we also have C = [C]!
With other words, we cannot claim that [C] just consists of exactly
> one (atomic) object.
~~~~~~~~~~~~~~~~~~~~~~~~~
Indeed, in contrast to a set there are no elements _in_ a heap. A heap
> just consists of its constituents. As a consequence heaps merge:
Let d = [a,b], then [d,c] = [a,b,c].
For example, if Q would be the heap consisting of all rational
> numbers, and I would be the heap of all irrational numbers, then
> R := [Q,I] would be the heap of all real numbers.
With other words, the heap [Q,I] does not just have two elements Q and
> I, but consists of all real numbers.
There's no hierarchy of heaps (like in set theory) - in this sense
> heaps are flat.
>
Now for my theory of heaps I rejected the idea of an empty heap based
on the following (philosophical) consideration:
Let's consider heaps of sand consisting of grains. Now if there
aren't any grains (say on some desk) then there simply (also) is no
heap of sand there (and not some sort of empty heap).
But now I think that this reasoning is not really sound (or forcing),
- I didn't' realize the significance of the conflation of c and e at
that time.
The problem was/is that I defined an /empty heap/ to be a heap such
that it does not have any constituent. But that definition is not
appropriate, as I see now.
So just lets consider an null heap 0. Since for any heap x we have x
c x, this also holds for 0: 0 c 0. (The null heap consists of the
null heap, though it doesn't consist of any other heap.) Moreover then
the following should hold:
Ax(0 e x).
The null heap is a constituent of any heap. Well, fine. :-)
And of course, we may also write 0 = [0], as Tommy proposed, since for
any heap x: x = [x] = [[x]], etc. (Again, we cannot interpret [x] to
mean that the heap [x] just consist of exactly one object.)
Just some thoughts.
F.
--
E-mail: info<at>simple-line<dot>de
===
Subject: Complete Solution Manuals for Various Textbooks in pdf. Get it
Quickly.
I have the comprehensive solution manual, solutions manual, solutions
manuals, in electronic format for the following textbooks. They
include complete solutions to all the problems in the text, except
where noted below in the listing. Payment is through Paypal, for the
amount of US $7-$15 per item (about 4 of my newest solution manuals
are a little higher than $15). Email me calgirl22[at]gmail.com but
please DO NOT POST HERE because I will not be able to help you, but
instead email and ask me for the solution that you need.
A Course in Game Theory by Osborne, Rubinstein
A Course in Algebraic Number Theory by Cohen
Adaptive Filter Theory, 4th Edition, by Haykin
Adaptive Control, 2nd. Ed., by Astrom, Wittenmark
Advanced Engineering Mathematics, 8th Editoin, by Erwin Kreyszig (even
solutions)
Advanced Engineering Mathematics, 9th Edition, by Erwin Kreyszig (even
solutions)
Advanced Macroeconomics, David Romer
Advanced Mathematical Concepts Precalculus With Applications by
Holliday [ISBN: 0028341759]
Advanced Modern Engineering Mathematics, 3rd Ed., by G. James
A First Course In Differential Equations, 7th Edition, by Zill, Cullen
Analog Integrated Circuit Design, 1st Ed., by Johns, Martin (text
ebook and solution manual)
Analysis and Design of Analog Integrated Circuits, 4th Ed., by Gray,
Hurst, Lewis, Meyer
Analytical Mechanics, 7th Edition, by Fowels, Cassiday
An Interactive Introduction to Mathematical Analysis, by Jonathan
Lewin
An Introduction to the Mathematics of Financial Derivatives, 2nd Ed.,
by Neftci [ISBN: 0125153929]
Antenna Theory, 2nd Ed., by Balanis
Antennas for all Applications, 3rd Edition, Kraus, Marhefka
Applied Linear Statistical Models, 5th Ed., by Neter (Selected Sol.)
Applied Numerical Analysis, 6th Edition, by Gerald, Wheatley
Applied Numerical Methods with MATLAB for Engineers and Scientists,
1st Ed,. by Chapra
Applied Statistics and Probability for Engineers, 3rd Ed., by
Montgomery, Runger (Selected Solutions)
Applied Strength of Materials, 4th Edition, by Mott
A Transition to Advanced Mathematics, 5th Edition, by Smith, Eggen,
Andre
Automatic Control Systems, 8th Edition, by Kuo, Golnaraghi
Basic Business Statistics: Concepts and Applications, 10th Ed., by
Berenson, Krehbiel, Levine (chap1-18)
Basic Engineering Circuit Analysis, 7th Ed., by J. David Irwin
Basic Engineering Circuit Analysis, 8th Ed., by J. David Irwin, Nelms
(Missing a chapter or 2)
Bioprocess Engineering Principles by Doran
Calculus Early Transcendental, 5th Ed., by James Stewart
Calculus - Early Transcendentals, 7th Ed., by Anton, Bivens, Davis
Calculus: Graphical, Numerical, Algebraic, 3rd Ed., Waits, Finney,
Demana, Kennedy
Calculus: Multivariable, 5th Edition, by James Stewart
Calculus: Single Variable, Early Transcendental, 5th Edition, by James
Stewart
Calculus, Single and Multivariable, 3rd Ed., by Hughes-Hallett,
McCallum
Calculus: Study and Solutions Guide, Vol. 1, 7th Ed., by Larson,
Hostetler, Edwards
Chemical and Engineering Thermodynamics, 3rd Ed., Stanley I. Sandler
Chemical Engineering Volume 1, Sixth Edition, by Richardson, Coulson,
Backhurst, Harker
Thornton
College Physics, Volume 1: 7th Edition, by Serway, Faugh
College Physics, Volume 2: 7th Edition, by Serway, Faughn
Communications Systems, 4th Ed., by Haykin
Communications Systems Engineering, 2nd Edition, by Proakis
Computational Techniques for Fluid Dynamics by Srinivas, Fletcher
Computer Networks, 4th Ed., by Andrew S. Tanenbaum
Computer Networks: A Systems Approach, 3rd Edition, by Davie
Control Systems Engineering, 4th Ed., by Norman Nise
Corporate Finance, 6th Edition, by Ross
C++ How to Program: Intro Object-Oriented Design with the UML, 3rd
Ed., by Deitel, Nieto
Data and Computer Communications, 8th Edition by Stallings
Database Management Systems, 3rd Ed., by Ramakrishnan, Gehrke (Sol.
for Chapters 2-21, odd only)
Design of Analog CMOS Integrated Circuits, 1st Edition, by Razavi
Design of Analysis of Experiments, 6th Edition, Montgomery (missing
chapter 6-8)
Design of Machinery, 3rd Ed by Robert L. Norton
Design With Operational Amplifiers and Analog Integrated Circuits, 2nd
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Design With Operational Amplifiers and Analog Integrated Circuits, 3rd
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Device Electronics for Integrated Circuits 3rd Edition by Muller
Differential Equations with Boundary Value Problems, 2nd Ed., by
Polking, Arnold
Digital And Analog Communication Systems 7th Ed., Leon
W. Couch
Digital Communications, 4th Edition, by Proakis
Digital Communications: Fundamentals and Applications, 2nd Ed, Skylar
Digital Design, 4th Edition, by Mano, Ciletti
Digital Image Processing, 2nd Edition, by Gonzalez, Woods
Digital Integrated Circuits, 2nd Ed., by Rabaey (Solutions ONLY for
Chapters 3, 5, 6, 10)
Digital Signal Processing: A Computer Based Approach, 1st Ed., by
Mitra
Digital Signal Processing: A Computer Based Approach, 2nd Ed., by S.
Mitra
Digital Signal Processing: A Computer Based Approach, 3rd Ed., by S.
Mitra
Digital Signal Processing: Principles, Algorithms and Applications,
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Discrete Time Signal Processing, 2nd Edition, Oppenheim
Dynamics of Mechanical Systems by C.T.F. Ross
Econometric Analysis, 5th Edition, by Greene
Wooldridge
Econometrics of Financial Markets, by Adamek, Cambell, Lo, MacKinlay,
Viceira
Electrical Properties of Materials, 7th Ed., by D. Walsh, L. Solymar
Electric Circuits 6th Ed. by Nilsson
Electric Circuits 7th Ed. by Nilsson
Electric Machinery, 6th Ed., Fitzgerald, Kingsley, Umans
Electric Machinery Fundamentals, 4th Ed by Chapman
Electromagnetic Fields and Waves by Iskander (...)
Electronic Circuit Analysis, 2nd Ed., by Donald Neamen
Electronics, 2nd Ed., by Allan R. Hambley
Elementary Differential Equations, 8th Edition, by Boyce, DiPrima
(some odd/even)
Elementary Principles of Chemical Processes, 3rd Ed., by Felder,
Rousseau
Elements of Chemical Reaction Engineering, 3rd Ed., by H. Scott Fogler
Engineering and Chemical Thermodynamics, by Koretsky [ISBN:
0471385867] (No sol. for chapt 6)
Engineering Circuit Analysis, 6th Edition, Hyat
Engineering Electromagnetics, 6th Ed W. Hayt, J. Buck
Engineering Electromagnetics, 7th Ed., Hayt, Buck
Engineering Fluids Mechanics 7th Edition by Crowe
Engineering Fluids Mechanics 8th Edition by Crowe
Engineering Mathematics, 4th Ed., by John Bird
Engineer Mechanics: Dynamics, 4th Ed., by Bedford
Engineering Mechanics: Dynamics, 10th Ed., by Russell C. Hibbeler
Engineering Mechanics: Dynamics 11th Ed. by Hibbeler
Engineering Mechanics: Dynamics 5th Ed. by Meriam, Kraige
Engineering Mechanics: Statics, 4th Edition - A. Bedford, Wallace
Fowler
Engineering Mechanics: Statics, 5th Ed., Meriam, Kraige
Engineering Mechanics: Statics, 6th Ed., Meriam, Kraige
Engineering Mechanics: Statics, 10th Ed., by Russell C. Hibbeler
Engineering Mechanics: Statics 11th Ed. by Hibbeler
Experiments with Economic Principles by Bergstrom, Miller
Feedback Control of Dynamic Systems, 4th Edition, by Powell, Emami-
Naeini
Financial Accounting, 4th Ed., by Libby, Short (Chap1-14)
Financial Accounting: An International Introduction, 2nd Ed., by
Alexander, Nobes
Finite Element Techniques in Structural Mechanics by Ross
Fluid Mechanics - 5th Edition by Frank M. White
Fluid Mechanics and Thermodynamics of Turbomachinery, 5th Ed., by S.
L. Dixon [ISBN: 0750678704]
Essentials of Fluid Mechanics: Fundamentals and Applications, 1st Ed.,
by Cengel & Cimbala
Fluid Mechanics with Engineering Applications, 10th Edition, by
Finnemore
Fundamentals of Aerodynamics, 3rd Edition, by J. D. Anderson, Jr.
Fundamentals of Applied Electromagnetics, 1st Ed., 2001 Media Edition,
by Ulaby
Fundamentals of Applied Electromagnetics, 5th Ed., 2008 Media Edition,
by Ulaby
Fundamentals of Digital Logic with Verilog Design, 1st Edition, by
Brown, Vranesic
Fundamentals of Digital Logic with VHDL Design by Stephen Brown,
Zvonko Vranesic
Fundamentals of Electric Circuits, 2nd Edition, by Alexander
Fundamentals of Electromagnetics with Engineering Appls by Wentworth
Fundamentals of Fluid Mechanics, 5th Ed. by Munson, Young..
Fundamentals of Heat and Mass Transfer, 4th Ed by Incropera...
Fundamentals of Heat and Mass Transfer, 5th Ed by Incropera...
Fundamentals of Heat and Mass Transfer, 6th Ed by Incropera...
Fundamentals of Logic Design, 5th Ed., by Roth Jr.
Fundamentals of Machine Component Design, 3rd Ed., by Juvinall
Fundamentals of Machine Component Design, 4th Ed., by Juvinall
Fundamentals of Machine Elements, 2nd Ed., Hamrock, Jacobson, Schmid
Fundamentals of Physics by Halliday, 7th Ed., Walker, Resnick
Fundamentals of Semiconductor Devices, 1st Edition by Anderson
Fundamentals of Structural Analysis, 2nd Ed., Chia-Ming Uang, Kenneth
Leet
Fundamentals of Thermal-Fluid Sciences, 2nd Ed. by Cengel
Fundamentals of Thermal-fluid Sciences, Int'l 2nd Ed. by Cengel
Fundamentals of Engineering Thermodynamics, 5th Ed. by Shapiro
Fundamentals of Thermodynamics, 5th Ed., by Sonntag, Borgnakke...
Fundamentals of Thermodynamics, 6th Ed., by Sonntag
Geometry, 04 Edition, by McGraw-Hill [ISBN: 0078296374]
Guide to Energy Management, 5th Edition, by Pawlik
Heat Transfer: A Practical Approach - 2nd Edition by Cengel
Hydraulics in Civil and Environmental Engineering, 4th Ed., by Andrew
Chadwick
Introduction to Algorithms, 2nd Ed by Cormen, Leiserson (Selected
Sol.)
Introduction To Chemical Engineering Thermodynamics, 7th Ed., by Van
Ness, Smith, Abbott
Introduction to Electric Circuits, 6th Ed., by Dorf, Svoboda
Introduction to Electric Circuits, 7th Ed., by Dorf, Svoboda
Introduction to Electrodynamics, 3rd Ed. by David Griffiths
Introduction to Fluid Mechanics - 5th Ed. by Fox..
Introduction to Fluid Mechanics - 6th Ed by Fox, McDonald...
Introduction to Linear Algebra, 3rd Ed., by Gilbert Strang
Introduction to Linear Algebra, 5th Ed., Arnold, Johnson, Riess
Introduction to Probability by Grinstead, Snell (odd solutions only,
not just answers but step by step solutions)
Introduction to Quantum Mechanics, 2nd Ed. by Griffiths
Introdution to Solid State Physics, 8th Edition by Kittel
Introduction to Thermal Systems Engineering: Thermodynamics, Fluid
Mechanics, and Heat Transfer by Moran, Shapiro, Munson, DeWitt
Introduction to Thermal Systems Engineering, by Moran, Shapiro
Linear Algebra, by J. Hefferon
Linear Algebra And Its Applications, 3rd Ed., by David C. Lay
Linear Algebra with Applications, 2nd Edition - by Otto Bretscher
Linear Algebra with Applications, 3rd Edition - by Otto Bretscher
Linear Circuit Analysis: Time Domain, Phasor and Laplace.., 2nd Ed,
Lin
Machine Design: An Integrated Approach, 2nd Ed., by Robert L. Norton
(same problems as third edition except for last 2-4 problems per
chapter that were added to the third edition)
Machine Design: An Integrated Approach, 3rd Ed., by Robert L. Norton
Managerial Accounting, 11th Ed., by Noreen, Brewer, Garrison
Materials Science and Engineering: An Introduction, 6th Ed. by
Callister
Matrix Analysis and Applied Linear Algebra by Carl Meyer
MC68HC11: An Introduction: Software/Hardware Interf, 2nd Ed, by Huang
Mechanical Engineering Design, 7th Ed. by Mischke, Shigley
Mechanical Vibrations, 3rd Edition, by S. S. Rao (99% same as 4th
Edition, No Solutions for Chapters 6, 9, and 12)
Mechanics of Fluids, 8th Ed., by Bernard Massey
Mechanics of Fluids, 4th Ed., Irving H. Shames
Mechanics of Fluids, 8th Ed., by Bernard Massey
Mechanics of Materials - 3rd Ed. by Beer, Johnston, Dewolf
Mechanics of Materials - 6th Ed. by Hibbeler
Mechanics of Materials, 6th Edition by James M. Gere (missing small
portion, section 8.5)
Mechanics of Materials, 6th Ed., by Sturges, Morris, Riley (part of
Chapt 2 is missing but only #1 thru #60)
Mechanics of Solids by C.T.F. Ross
Microeconomic Analysis, 3rd Ed., by H. Varian (Ans. to Exercises: Ch.
1- Ch.25)
Microeconomic Theory, by Mas-Colell, Whinston, Green
Microelctronic Circuits, 5th Ed. by Sedra and Smith
Microelectronic Circuit Design, 2nd Edition by Jaeger, Blalock
Microelectronic Circuit Design, 3rd Edition by Jaeger, Blalock
Microelectronics: Digital and Analog Circuits and Systems by Millman
Microwave and Rf Design of Wireless Systems, 1st Edition, by Pozar
Microwave Engineering, 3rd Ed., by David M. Pozar
Microwave Transistor Amplifiers: Analysis and Design, 2nd Ed., by
Guillermo Gonzalez
Miller & Freund's Probability and Statistics for Engineers, 7th
Edition, Johnson, Miller
Modern Compressible Flow, 3rd Edition, by Anderson
Modern Control Engineering, 3rd Edition, by Ogata
Modern Control Engineering, 4th Edition, by Ogata
Modern Digital and Analog Communication Systems, 3rd Ed., by Lathi
Modern Control Systems, 9th Ed., by Richard C. Dorf, Robert H Bishop
Modern Operating Systems,2nd Ed., by Andrew Tanenbaum
Modern Physics 4th Edition by Tipler
Monetary Theory and Policy, 2nd Edition, by Walsh
Multivariable Calculus, 5th Edition, by James Stewart
Numerical Methods, 3rd Ed., by J. Douglas Faires, Richard L. Burden
(Selected Solutions)
Operating Systems: Internals and Design Principles, 4th Edition, by
Stallings
Operating System Concepts, 7th Ed., Silberschatz, Galvin, Gagne
Options, Futures and Other Derivatives, 4th Ed., by John Hull
Options, Futures and Other Derivatives, 5th Ed., by John Hull
(Chapters 1 thru 18 ONLY)
Orbital Mechanics: For Engineering Students by Howard Curtis (includes
matlab scripts)
Organic Chemistry, 4th Ed., by Carey, Atkins (Student Study Guide and
Sol. Man.)
Partial Differential Equations with Fourier Series and Boundary Value
Problems, 2nd Ed., by Asmar (Student Solutions Manual)
Physical Chemistry - 7th Edition - by Julio de Paula, Peter Atkins
Physics, 6th Edition, by John Cutnell
Physics, 5th Edition, Vol 2 by Halliday, Resnick, Krane (Chap 25-52)
Physics for Scientist and Engineers by Knight (No Chapt 36-42)
Physics for Scientist and Engineers, 6th Ed., by Serway
Physics for Scientists and Engineers-Vol 1, 5th Edition, Serway,
Beichner (Chap. 1 - 22)
Physics for Scientists and Engineers-Vol 2, 5th Edition, Serway,
Beichner (Chap. 23 - 46)
Physics for Scientists and Engineers, 3rd Ed., by Douglas C. Giancoli
Physics for Scientist and Engineers, 5th Edition, by Tipler, Mosca
Physics: Principles with Applications, 6th Ed. by Giancoli
Power System Analysis and Design, 3rd Ed., by Glover, Sarma
Principles and Applications of Electrical Engineering 4th (Revised) Ed
by Rizzoni
Principles and Practices of Automatic Process Control, 3rd Edition by
Smith, Corripio [ISBN: 0471431907]
Principles of Communication: Systems, Modulation Noise, 5th Ed.,
Ziemer
Principles of Physics, 3rd Edition, by Serway
Principles of Physics, 4th Edition, by Serway
Principles of Statics, 10th Ed., by Russell C. Hibbeler [ISBN:
0131866745]
Probability and Statistics for Engineers and Scientists, 3rd Edition,
Hayter
Probability and Statistics for Engineering and the Sciences, 6th Ed.,
by Jay L. Devore
Probability Random Variables, and Stochastic Processes, 4th Ed., by
Papoulis, Pillai (..)
Quantum Mechanics: An Accessible Introduction, 1st Ed., by Robert
Scherrer
Recursive Macroeconomic Theory, 1st Ed., by Ljungqvist, Sargent
Recursive Methods in Economic Dynamics, (2002) by Irigoyen, Rossi-
Hansberg, Wright
RF Circuit Design: Theory & Applications, by Bretchko, Ludwig
Sears and Zemansky's University Physics 11th Edition by Young..
Semiconductor Device Fundamentals by Pierret
Semiconductor Devices: Physics and Technology, 2nd Ed., S.M. Sze
Semiconductor Physics And Devices -3rd Ed. by D. Neamen
Separation Process Principles, 2nd Ed., Seader, Henley
Signal Processing and Linear Systems by Lathi
Signals and Systems, 2nd Edition, by Haykin, Van Veen
Signals and Systems, 2nd Edition, Oppenheim, Willsky, Hamid, Nawab
Signals and Systems: Analysis Using Transform Methods and MATLAB, 1st
Ed., by M. J. Roberts
Signals, Systems, and Transforms: Charles L. Phillips, Eve A. Riskin,
John M. Parr
Shigley's Mechanical Engineering Design, 8th Ed. by Budynas, Nisbett
(No Sol. for Chapt 18 & 19)
Simply C#: An Application-Driven Tutorial Approach, by Deitel, Hoey...
(Chapters 1-32)
Soil Mechanics: Concepts and Applications, 2nd Ed., by Powrie
Solid State Electronic Devices - 5th Ed by Streetman
Solid State Electronic Devices - 6th Ed by Streetman
Statics and Mechanics of Materials: An Integrated Approach, 2nd Ed.,
by Riley, Sturges, Morris
Structural Analysis, 5th Edition, by Hibbeler
System Dynamics, 3rd Edition, by Ogata
Theory and Design for Mechanical Measurements, 4th Ed., Beasley,
Figliola
Thermal Physics, 2nd Edition, by Charles Kittel
Thermal Physics, by Ralph Baierlein
Thermodynamics: An Engineering Approach, 5th Ed., by Cengel, Boles
(Missing solutions #118-149 of Chapter 7)
Thermodynamics: An Engineering Approach, 6th Ed., by Cengel, Boles
The Science and Engineering of Materials, 4th Ed., by Donald R.
Askeland, Pradeep P. Phule
Thomas' Calculus, Early Trans., Part 1, 10th Ed. by Thomas, Weir,
Hass, Giordano
Thomas' Calculus: Part 2, 10th Ed. (Multivariable, chs. 11-16), by
Thomas, Weir, Hass, Giordano
Thomas' Calculus, Early Trans., Part 1, 11th Ed. by Thomas, Weir,
Hass, Giordano
Thomas' Calculus: Part 2, 11th Ed. (Multivariable, chs. 11-16), by
Thomas, Weir, Hass, Giordano
Transport Phenomena, 1st Edition, by R. Byron Bird
University Physics 11th Edition by Young..
Vector Mechanics: Statics 7th Edition by Beer
Vector Mechanics: Dynamics, 7th Ed., by Beer, Johnston, Staab, Clausen
Vibrations and Stability: Advanced Theory, Analysis, and Tools, 7th
Ed., by Thomsen
Wireless Communications: Principles and Practice, 2nd Ed, by Rappaport
===
Subject: yes
1) If you are looking for the truth to soothe and please your self,
just click on this link.
2) If you are looking for the secrt of the family happiness, just
click on this link
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click on this link.
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http://www.islam-guide.com/
===
Subject: Re: NASA exposes Apollo Hoax
<13971e5pqqbve4c@news.supernews.com>
<1398jmqelsvseb8@news.supernews.com>
<139aa0iahh49gde@news.supernews.com > A moon camera needs one heck of a UVa
cut-off filter.
They used regular Zeiss lenses which, like almost all lenses, do not
> transmit UV efficiently. To accomodate UV photography with film, one
> needs a special lens, usually with Quartz in the formula... and a UV
> sensitive film! They had neither.
However, filters would be no problem. Look at the material of the face
> shields.
> It also needs shielding from local gamma and hard-Xrays.
Not much at all was needed. First, the level of X-Rays is quite low.
> Gamma rays are not frequent enough for the time they were there to
> accumulate to a significant level.
> The highly electrostatic environment of that
> physically dark and dusty anticathode moon is simply asking too much
> of most any camera, or much less that of human DNA.
Once on the moon, they are the charge. No problem.
So much for your uninformed imagination.
Is that the very best Zion rusemaster of naysayism that you folks can
muster? How pathetic.
-
Brad Guth
===
Subject: Re: NASA exposes Apollo Hoax
<13971e5pqqbve4c@news.supernews.com>
<1398jmqelsvseb8@news.supernews.com>
<139aa0iahh49gde@news.supernews.com>
<P4bli.2946$iE5.56@fe3.news.blueyonder.co.uk
:
> : > A moon camera needs one heck of a UVa cut-off filter.
> :
> : They used regular Zeiss lenses which, like almost all lenses, do not
> : transmit UV efficiently. To accomodate UV photography with film, one
> : needs a special lens, usually with Quartz in the formula... and a UV
> : sensitive film! They had neither.
> :
> : However, filters would be no problem. Look at the material of the face
> : shields.
> :
> : > It also needs shielding from local gamma and hard-Xrays.
> :
> : Not much at all was needed. First, the level of X-Rays is quite low.
> : Gamma rays are not frequent enough for the time they were there to
> : accumulate to a significant level.
No, no, LOCAL gamma rays.Guththe Super Prospector knows
> how to find radioactive material right there in the Gruy?re holes.
> He learnt it from Johannes Wilhelm Geiger (1882 - 1945) but the old boy
> was 24 years dead in 1969. He spent the period 1906-12 in Manchester,
> England, working with Ernest Rutherford on radioactivity.
: > The highly electrostatic environment of that
> : > physically dark and dusty anticathode moon is simply asking too much
> : > of most any camera, or much less that of human DNA.
> :
> : Once on the moon, they are the charge. No problem.
> :
> : So much for your uninformed imagination.
Naked matter of most any kind that's outside of our badly failing
magnetosphere is anticathode worthy. Sorry about that. And, that
naked moon of ours is also representing the best ever cosmic morgue in
town. Terribly sorry about that too.
Solar and Cosmic radiation at Venus L2(VL2), similar to Earth L2
except less cosmic dosage.
Geoffrey A. Landis had a few kind words to share, although never any
specific numbers of what to expect at Venus L2 or even outside of
Earth's protective magnetosphere, such as residing at our moon's L1,
Earth's L1 or even that of Earth's L2. It's as though such science
pertaining to human space travel simply doesn't exist, other than in
various encrypted formats of any number of hocus-pocus conditional
physics, few of which seem to agree with one another.
SOHO Dosage
31 nifty pages of soft data on space radiation; meaning little if any
hard rad/rem data that pertains to human DNA that's having to reside
external to our magnetosphere.
http://spacecraft.ssl.umd.edu/academics/697S05/697S05L04.space env.pdf
Earth L2 Dosage
http://planetquest.jpl.nasa.gov/documents/JWST-RPT-radiation.pdf
Similar to the above JWST-RPT-radiation report, an official Raytheon/
TRW Space Data Report offered us a GSO satellite internal system
dosage pegged as 2e3 Sv/year while fully shielded by 5/16 aluminum (2
g/cm2), thus residing outside of our protective magnetosphere's Van
Allen belts as fully in the buff is likely capable of being similar to
that realm of dosage analogy, of there being at times 2e5 rem/year or
548 rem/day of local Xrays, Gamma and direct cosmic exposure that
you'll need sufficient spacecraft or habitat shielding in order to
long-term survive without incorporating a backup plan B of banked bone
marrow, especially on behalf of attinuating and/or somehow diverting
the moon's anticathode worth of hard Xrays and Gamma that's rather
unavoidable if your butt of frail DNA is situated within the moon's L1
for any extended period of time.
Fortunately for POOF City that's residing at the cool halo orbital
location or station-keeping realm of Venus L2(VL2) isn't nearly as
cosmic hot and nasty, whereas Venus and of its robust atmosphere
itself blocking and/or diverting the vast bulk of whatever nasty halo
CMEs our sun has to offer, as well as there being no such nearby moon
of secondary/recoil Xrays and Gamma to fret about.
Therefore, VL2 is not only a cool satellite environment in a solar IR
forced thermal sense of the word, but it's also offering a relatively
cool amount of humanly lethal radiation to deal with, as possibly
similar to or perhaps not much greater than what ISS/ESS has to
contend with. The primary habitat POOFs with their added shielding of
ice cold beer or whatever else you'd like to consume should manage to
protect our frail DNA rather nicely, as safely station-keeping us
within the VL2 halo orbit, that which could also host a new and
greatly improved set of ACE/SOHO/TRACE science instruments in addition
to whatever's intended for scoping out and/or probing Venus.
-
Brad Guth
===
Subject: Re: Solutions Manual print, on Cd and by email to low price
> My List of Solutions Manual
 (Available on print-original copy, Cd - for heavy files & by e-mail)
> Ship to you (print or Cds with supplementary material) by priority air
> mail, with tracking number to low cost.
> Delivery time : 3-5 days.
> contact me to : bergh...@yahoo.com
> If your solutions manual wanted ins't on this list, also can make the
> request. These are some only.
> Not interchange
 - Mechanics, MechanicalEngineering& AerospaceEngineering:
 Introduction to MechanicalEngineering(Rizza)
> MechanicalEngineeringPrinciples (Bird & Ross) + Ebook
> Mechanics of Fluids (8th Ed., Massey) + Ebook
> Fluid Mechanics (5th Ed., White) + Ebook
> Fluid Mechanics (6th Ed., White)
> Viscous Fluid Flow (3rd Ed., White) + Ebook
> Fundamentals of Thermal-Fluid Sciences (1st Ed.,Cengel) + Ebook
> Fundamentals of Thermal-Fluid Sciences (2nd Ed.,Cengel) + Ebook
> Fundamentals of Thermal-Fluid Sciences with Student Resource CD (3rd
> Ed.,Cengel& Turner)
>Thermodynamics: AnEngineeringApproach(5th Ed.,Cengel) + Ebook
>Thermodynamics: AnEngineeringApproach(6th Ed.,Cengel) + Ebook
> Essentials of Fluid Mechanics: Fundamentals and Applications (1st Ed.,
>Cengel) + Ebook
> Fluid Mechanics (1st Ed.,Cengel) + Ebook
> Heat Tranfer (2nd Ed.,Cengel) + Ebook
> Heat and Mass Transfer: A PracticalApproach(3rd. Ed.,Cengel) +
> Ebook
> Design and Simulation of Thermal Systems (Suryanarayana & Arici)
> Fluid Mechanics (5th Ed., Douglas)
> Fluid Mechanics (3rd Ed., Kundu)
> Fluid Mechanics withEngineeringApplications (Finnemore)
> Mechanics of Fluids (3rd Ed., Potter)
> Mechanics of Fluids (4th Ed., Shames)
>Thermodynamics: An Integrated Learning System (Schmidt, Ezekoye,
> Howell & Baker)
> Introduction to Thermal and FluidsEngineering(Kaminski & Jensen)
> Heating, Ventilating and Air Conditioning Analysis and Design (6th
> Ed., McQuiston)
> An Introduction to Fluid Dynamics: Principles of Analysis and Design
> (Middleman)
> Introduction to Mass and Heat Transfer: Principles of Analysis and
> Design (Middleman)
> Heat Transfer (2nd Ed., Mills)
> Convective Heat and Mass Transfer (4th Ed., Kays & Crawford)
> AdvancedEngineeringThermodynamics(3rd Ed., Bejan)
> Convection Heat Transfer (2nd Ed., Bejan)
> Convection Heat Transfer (3rd Ed., Bejan)
> Thermal Design and Optimization (Bejan)
> Shape and Structure, fromEngineeringto Nature (Bejan)
>Thermodynamics: Concepts and Applications (Stephen Turns)
> Thermal-Fluid Sciences: An IntegratedApproach(Stephen Turns)
> Principles of Heat Transfer (Kaviany)
> Heat Convection (Latif M. Jiji) + Ebook
> Heat Transfer (9th Ed., Holman)
> Fundamentals of Momentum, Heat and Mass Transfer (4th Ed., Welty)
> Momentum, Heat, and Mass Transfer Fundamentals (Kessler)
> Heat Tranfer (Rao)
> Heat Conduction (kakac)
> Heat Exchanges (Kakac)
> Heat Exchangers: Selection, Rating and Thermal Design (2nd Ed. Sadik
> Kakac & Hongtan Liu)
> Convective Heat Transfer (kakac)
> Fundamentals of Heat and Mass Transfer (5th Ed., Incropera, DeWitt)
> Fundamentals of Heat and Mass Transfer (6th Ed., Incropera, DeWitt)
> Introduction to Heat Transfer (4th Ed., Incropera, DeWitt)
> Introduction to Heat Transfer (5th Ed., Incropera, DeWitt)
> Radiation Detection and Measurement (3rd Ed., Glenn Knoll)
> Radiative Heat Transfer (2nd Ed., Michael Modest)
>EngineeringHeat Transfer (2nd Ed., Janna)
>EngineeringThermodynamics: Work and Heat Transfer (4th Ed., G.F.C.
> Rogers & Y.R. Mayhew)
> Elements of Heat Transfer (Yildiz Bayazitoglu and M. Necati Ozisik)
> Inverse Heat Transfer: Fundamentals and Applications (M.N. Ozisik &
> Helcio R.B. Orlande)
> Thermal Radiation Heat Transfer (4th Ed.,Robert Siegel & John R.
> Howell)
> Computational Heat Transfer (2nd Ed., Jaluria)
> Principles of Combustion (2nd Ed., Kenneth Kuan-yun Kuo)
> Incompressible Flow (3rd Ed., Panton)
> Non-Newtonian Flow : Fundamentals andEngineeringApplications (R P
> Chhabra & J F Richardson) + Ebook
> Computational Techniques for Fluid Dynamics (Srinivas, K., Fletcher,
> C.A.J.)
> Kinematic Chains and Machine Components Design (Dan B. Marghitu) +
> Ebook
> Kinematics and Dynamics of Machinery (3rd Ed., Wilson & Sadler)
> Kinematics, Dynamics, and Design of Machinery (2nd Ed., Waldron &
> Kinzel)
> Mechanism Design: Analysis and Synthesis-Volume 1 (4th Ed., Erdman &
> Sandor)
> Machines and Mechanisms: Applied Kinematic Analysis (3rd Ed.,
> Myszka)
> Mechanical Design: A ComponentsApproach(Peter Childs)
> Mechanical Design of Machine Elements and Machines: A Failure
> Prevention Perspective (Collins)
> Fundamentals of Machine Component Design (3rd Ed., Juvinall)
> Fundamentals of Machine Component Design (4th Ed., Juvinall)
> Design of Machine Elements (8th Ed., Spotts)
> Machine Design (Wentzell)
> Solutions to problems at the text : Problems on the Design of Machine
> Elements (Faires)
> Machine Elements in Mechanical Design (4th Ed., Mott)
> Mechanical Design: An IntegratedApproach(1st Ed., Ugural)
> Design of Machinery (3rd Ed., Norton)
> Machine Design (2nd Ed., Norton)
> Machine Design : An IntegratedApproach(3rd Ed., Norton)
> MechanicalEngineeringDesign (6th Ed., Shigley)
> MechanicalEngineeringDesign (7th Ed., Shigley)
> Shigley's MechanicalEngineeringDesign (8th Ed., Budynas)
> Fundamentals of Machine Elements (1st Ed., Hamrock)
> Fundamentals of Machine Elements (2nd Ed., Hamrock)
> Metal Fatigue inEngineering(2nd Ed., Stephens, Fatemi & Fuchs)
> Principles of Metal Manufacturing Processes (Beddoes & Bibby)
> Materials Selection in Mechanical Design (3rd Ed., Michael Ashby)
> Introduction to Manufacturing Processes (3rd Ed., Schey)
> Manufacturing,Engineering& Technology (4th Ed. Kalpakjian & Smith)
> Manufacturing,Engineering& Technology (5th Ed. Kalpakjian & Smith)
> Automation, Production Systems, and Computer-Integrated Manufacturing
> (2nd Ed., Groover)
> Introduction to Robotics: Mechanics and Control (3rd Ed, Craig)
> Applied Manufacturing Process Planning: With Emphasis on Metal Forming
> and Machining (Nelson, Schneider)
> Mechanics of Materials: A Modern Integration of Mechanics and
> Materials in Structural Design (Christopher Jenkins & Sanjeev Khanna)
> Mechanics of Materials (3th Ed., Beer)
> Mechanics of Materials (5th Ed., Gere)
> Mechanics of Materials (6th Ed., Gere)
> Vector Mechanics for Engineers: Statics (7th Ed., Ferdinand P. Beer)
> Vector Mechanics for Engineers: Statics (8th Ed., Ferdinand P. Beer)
> Vector Mechanics for Engineers: Dynamics (7th Ed., Ferdinand P. Beer)
> Vector Mechanics for Engineers: Dynamics (8th Ed., Ferdinand P. Beer)
> Statics: Analysis and Design of Systems in Equilibrium (Sheppard &
> Tongue)
> Statics and Mechanics of Materials: An IntegratedApproach(2nd Ed.,
> Riley, Sturges & Morris)
> Mechanics of Materials (6th Ed., Riley, Sturges & Morris)
> Deformable Bodies and Their Material Behavior (Haslach & Armstrong)
> Strength of Materials - Volume 1 : Elementary Theory and Problems
> (Timoshenko)
> Intermediate Mechanics of Materials, (1st Ed., Barber)
> Elasticity (2nd Ed., J.R. Barber) + Ebook
> Elasticity: Theory, Applications, and Numerics (Martin Sadd) + Ebook
> Elasticity inEngineeringMechanics (2nd Ed., Boresi)
> Advanced Mechanics of Materials (6th Ed., Boresi) + Ebook
>EngineeringMechanics: Dynamics (Boresi)
> Flight Performance of Fixed and Rotary Wing Aircraft (Antonio
> Filippone)
> Aircraft Structures forEngineeringStudents (4th Ed., T.H.G. Megson)
> Principles of Helicopter Aerodynamics (2nd Ed., Leishman)
> Fundamentals of Aerodynamics (3th Ed., Anderson)
> Fundamentals of Aerodynamics (4th Ed., Anderson)
> Modern Compressible Flow: With Historical Perspective (3rd Ed., John
> D. Anderson)
> Applied Fluid Mechanics (6th Ed., Mott)
> Applied Strength of Materials (4th Ed., Mott)
> Intermediate Dynamics for Engineers (Marcelo R.M & Crespo da Silva)
>EngineeringMechanics: Statics (2nd Ed., Pytel)
>EngineeringMechanics: Dynamics (2nd Ed., Pytel)
>EngineeringMechanics: Statics (2nd Ed., Shames)
>EngineeringMechanics: Statics (4th Ed., Shames)
>EngineeringMechanics: Dynamics (4th Ed., Shames)
> Introduction to Solid Mechanics (3rd Ed.., Shames)
> Elastic And Inelastic Stress Analysis (Shames)
> Advanced Dynamics (Greenwood) + Ebook
> AdvancedEngineeringDynamics (2nd Ed., Jerry Ginsberg) + Ebook
> Classical Dynamics (Jorge V. Jos.8e) + Ebook
> Impact Mechanics (W.J. Stronge)
> Statics and Strengths of Materials (6th Ed., Morrow & Kokernak)
>EngineeringMechanics : Statics (11th Ed., Hibbeler)-Not mathcad files
> converted to pdf, real instructor sol. manual
> Principles of Statics (10th Ed., Hibbeler)
>EngineeringMechanics : Dynamics (11th Ed., Hibbeler)-Not mathcad
> files converted to pdf, real instructor sol. manual
> Principles of Dynamics (10th Ed., Hibbeler)
> Mechanics of Materials (6th Ed, Hibbeler)
> Statics and Mechanics of Materials (2nd Ed., Hibbeler)
> Energy Principles and Variational Methods in Applied Mechanics (2nd
> Ed., Reddy)
> Theory of Vibrations with Applications (5th Ed., Thomson & Dahleh)
>EngineeringVibrations (2nd Ed., Inman)
>EngineeringVibrations (3rd Ed., Inman)
> Introduction to Finite Element Vibration Analysis (Maurice Petyt)
> Vibrations and Stability: Advanced Theory, Analysis, and Tools (2nd
> Ed., Jon J. Thomsen)
> Mechanical Vibrations (4th Ed., Rao)
> The Finite Element Method and Applications inEngineeringUsing ANSYS
> (Erdogan Madenci, Ibrahim Guven)
> Finite Element Analysis Theory and Application with ANSYS (3rd Ed.,
> Moaveni)
> Modeling and Analysis of Dynamic Systems (3rd Ed, Close, Frederick &
> Newell)
> System Dynamics: Modeling and Simulation of
...
read more E
can i get the thermodynamics 6e cengel
vhogan83@yahoo.com
===
Subject: similarity between two graphs
I was wondering if there are any standard metrics that can be used to
quantify the similarity between two undirected graphs. For example,
distribution of the node degrees, etc.
Roy
===
Subject: Re: similarity between two graphs
I was wondering if there are any standard metrics that can be used to
> quantify the similarity between two undirected graphs. For example,
> distribution of the node degrees, etc.
I'm not aware of anything is standard. Do you have a particular
application in mind?
You could, for instance, define d(G,H) to be the maximum number of
edges which are in exactly one of G and phi(H), where phi ranges over
all permutations of V(G).
--- Christopher Heckman
===
Subject: How to detect turning points in curves...
How to detect turning points in curves
Hi all,
If you take a look at the following plot,
http://img63.imageshack.us/img63/5050/gggyt1.jpg
You will agree with me that there are two turning points.
But is there a systematic way to let computer detect the turning points
automatically and programmatically?
Please be advised that the second turn isn't neccessarily turning down, it
can also possibly go up...
And in real-world applications, the turn can be more smooth and round, but
still, naked eyes should be able to find the turning points easily.
My program has to do a classification:
All the good curves should first go down, and then either make no turns; or
make a turn, and stay vertically flat and slightly up, going from the left
to the right.
There should be no second turn (up or down). If there is the second turn,
then that's the bad curves.
My program needs to decern the good curves from bad curves.
I cranked a few algorithms but they don't work well. Are there systematic
methods of handling this?
===
Subject: Re: How to detect turning points in curves...
> How to detect turning points in curves
Hi all,
If you take a look at the following plot,
http://img63.imageshack.us/img63/5050/gggyt1.jpg
You will agree with me that there are two turning points.
But is there a systematic way to let computer detect the turning points
> automatically and programmatically?
Please be advised that the second turn isn't neccessarily turning down,
it
> can also possibly go up...
And in real-world applications, the turn can be more smooth and round,
but
> still, naked eyes should be able to find the turning points easily.
My program has to do a classification:
All the good curves should first go down, and then either make no turns;
or
> make a turn, and stay vertically flat and slightly up, going from the
left
> to the right.
There should be no second turn (up or down). If there is the second turn,
> then that's the bad curves.
My program needs to decern the good curves from bad curves.
I cranked a few algorithms but they don't work well. Are there systematic
> methods of handling this?
>
My browser wouldn't show your image, but here's a
thought. If your points are sampled at equal intervals,
you can do a Fourier transform, use the analytic expression
of the derivative term by term and the Fourier coefficients
to calculate the derivative of the Fourier representation
of your function, and find the zeros of the derivative
and take those as the turning points. FWIW.
===
Subject: Re: How to detect turning points in curves...
> How to detect turning points in curves
> Hi all,
> If you take a look at the following plot,
> http://img63.imageshack.us/img63/5050/gggyt1.jpg
> You will agree with me that there are two turning points.
> But is there a systematic way to let computer detect the turning points
> automatically and programmatically?
> Please be advised that the second turn isn't neccessarily turning down,
> it
> can also possibly go up...
> And in real-world applications, the turn can be more smooth and round,
> but
> still, naked eyes should be able to find the turning points easily.
> My program has to do a classification:
> All the good curves should first go down, and then either make no turns;
> or
> make a turn, and stay vertically flat and slightly up, going from the
> left
> to the right.
> There should be no second turn (up or down). If there is the second
turn,
> then that's the bad curves.
> My program needs to decern the good curves from bad curves.
> I cranked a few algorithms but they don't work well. Are there
systematic
> methods of handling this?
 My browser wouldn't show your image, but here's a
> thought. If your points are sampled at equal intervals,
> you can do a Fourier transform, use the analytic expression
> of the derivative term by term and the Fourier coefficients
> to calculate the derivative of the Fourier representation
> of your function, and find the zeros of the derivative
> and take those as the turning points. FWIW.
>
How? Could you please elaborate?
===
Subject: Re: How to detect turning points in curves...
<f76pbs$mef$1@news.Stanford.EDU
> How to detect turning points in curves
> Hi all,
> If you take a look at the following plot,
>http://img63.imageshack.us/img63/5050/gggyt1.jpg
> You will agree with me that there are two turning points.
> But is there a systematic way to let computer detect the turning
points
> automatically and programmatically?
> Please be advised that the second turn isn't neccessarily turning
down,
> it
> can also possibly go up...
> And in real-world applications, the turn can be more smooth and round,
> but
> still, naked eyes should be able to find the turning points easily.
> My program has to do a classification:
> All the good curves should first go down, and then either make no
turns;
> or
> make a turn, and stay vertically flat and slightly up, going from the
> left
> to the right.
> There should be no second turn (up or down). If there is the second
turn,
> then that's the bad curves.
> My program needs to decern the good curves from bad curves.
> I cranked a few algorithms but they don't work well. Are there
systematic
> methods of handling this?
> My browser wouldn't show your image, but here's a
> thought. If your points are sampled at equal intervals,
> you can do a Fourier transform, use the analytic expression
> of the derivative term by term and the Fourier coefficients
> to calculate the derivative of the Fourier representation
> of your function, and find the zeros of the derivative
> and take those as the turning points. FWIW.
How? Could you please elaborate?- Hide quoted text -
- Show quoted text -
This is a brief summary. Hopefully it doesn't insult your
intelligence and isn't too sketchy.
Under certain conditions on F,
F(x) = a0 + a1 cos(kx) + a2 cos(2kx) + ... +
b1 sin(kx) + b2 sin(2kx) + ...
Using a discrete Fourier Transform (preferably using the Fast
Fourier Transform algorithm if you have very many data points),
you can calculate a discrete set of a's and b's the cause F to
agree with your function at the x values for which you have F
values. Then F' is given by
F'(x)= - ka1 sin(kx) - 2ka2 sin(2kx) - ... +
kb1 cos(kx) + 2kb2 cos(2kx) + ...
Set F'(x) = 0 and solve for x's that satisfy this equation.
These should approximate the turning point.
I don't know if it will work for your application, or how
practrical it will be.
===
Subject: Re: How to detect turning points in curves...
> How to detect turning points in curves
> Hi all,
> If you take a look at the following plot,
>http://img63.imageshack.us/img63/5050/gggyt1.jpg
> You will agree with me that there are two turning points.
> But is there a systematic way to let computer detect the turning
> points
> automatically and programmatically?
> Please be advised that the second turn isn't neccessarily turning
> down,
> it
> can also possibly go up...
> And in real-world applications, the turn can be more smooth and
round,
> but
> still, naked eyes should be able to find the turning points easily.
> My program has to do a classification:
> All the good curves should first go down, and then either make no
> turns;
> or
> make a turn, and stay vertically flat and slightly up, going from the
> left
> to the right.
> There should be no second turn (up or down). If there is the second
> turn,
> then that's the bad curves.
> My program needs to decern the good curves from bad curves.
> I cranked a few algorithms but they don't work well. Are there
> systematic
> methods of handling this?
> My browser wouldn't show your image, but here's a
> thought. If your points are sampled at equal intervals,
> you can do a Fourier transform, use the analytic expression
> of the derivative term by term and the Fourier coefficients
> to calculate the derivative of the Fourier representation
> of your function, and find the zeros of the derivative
> and take those as the turning points. FWIW.
> How? Could you please elaborate?- Hide quoted text -
> - Show quoted text -
This is a brief summary. Hopefully it doesn't insult your
> intelligence and isn't too sketchy.
Under certain conditions on F,
> F(x) = a0 + a1 cos(kx) + a2 cos(2kx) + ... +
> b1 sin(kx) + b2 sin(2kx) + ...
> Using a discrete Fourier Transform (preferably using the Fast
> Fourier Transform algorithm if you have very many data points),
> you can calculate a discrete set of a's and b's the cause F to
> agree with your function at the x values for which you have F
> values. Then F' is given by
> F'(x)= - ka1 sin(kx) - 2ka2 sin(2kx) - ... +
> kb1 cos(kx) + 2kb2 cos(2kx) + ...
> Set F'(x) = 0 and solve for x's that satisfy this equation.
> These should approximate the turning point.
I don't know if it will work for your application, or how
> practrical it will be.
>
Or better yet, use wavelets which is O(n)(rather than O(nlog n) of FFT) and
specifically used for detecting changes like this.
===
Subject: Re: How to detect turning points in curves...
<f76pbs$mef$1@news.Stanford.EDU>
<MfHli.6345$rL1.6014@newssvr19.news.prodigy.net
> How to detect turning points in curves
> Hi all,
> If you take a look at the following plot,
>http://img63.imageshack.us/img63/5050/gggyt1.jpg
> You will agree with me that there are two turning points.
> But is there a systematic way to let computer detect the turning
> points
> automatically and programmatically?
> Please be advised that the second turn isn't neccessarily turning
> down,
> it
> can also possibly go up...
> And in real-world applications, the turn can be more smooth and
round,
> but
> still, naked eyes should be able to find the turning points easily.
> My program has to do a classification:
> All the good curves should first go down, and then either make no
> turns;
> or
> make a turn, and stay vertically flat and slightly up, going from
the
> left
> to the right.
> There should be no second turn (up or down). If there is the second
> turn,
> then that's the bad curves.
> My program needs to decern the good curves from bad curves.
> I cranked a few algorithms but they don't work well. Are there
> systematic
> methods of handling this?
> My browser wouldn't show your image, but here's a
> thought. If your points are sampled at equal intervals,
> you can do a Fourier transform, use the analytic expression
> of the derivative term by term and the Fourier coefficients
> to calculate the derivative of the Fourier representation
> of your function, and find the zeros of the derivative
> and take those as the turning points. FWIW.
> How? Could you please elaborate?- Hide quoted text -
> - Show quoted text -
> This is a brief summary. Hopefully it doesn't insult your
> intelligence and isn't too sketchy.
> Under certain conditions on F,
> F(x) = a0 + a1 cos(kx) + a2 cos(2kx) + ... +
> b1 sin(kx) + b2 sin(2kx) + ...
> Using a discrete Fourier Transform (preferably using the Fast
> Fourier Transform algorithm if you have very many data points),
> you can calculate a discrete set of a's and b's the cause F to
> agree with your function at the x values for which you have F
> values. Then F' is given by
> F'(x)= - ka1 sin(kx) - 2ka2 sin(2kx) - ... +
> kb1 cos(kx) + 2kb2 cos(2kx) + ...
> Set F'(x) = 0 and solve for x's that satisfy this equation.
> These should approximate the turning point.
> I don't know if it will work for your application, or how
> practrical it will be.
Or better yet, use wavelets which is O(n)(rather than O(nlog n) of FFT)
and
> specifically used for detecting changes like this.- Hide quoted text
-
- Show quoted text -
Yes, wavelets are good. Kind of depends on whether there are
any reasons to prefer more global or local function behavior
to influence one's calculation of local turning points. There
didn't appear to be any given in the OP's post, so wavelets
might well be better there.
===
Subject: Re: How to detect turning points in curves...
> How to detect turning points in curves
> Hi all,
> If you take a look at the following plot,
>http://img63.imageshack.us/img63/5050/gggyt1.jpg
> You will agree with me that there are two turning points.
> But is there a systematic way to let computer detect the turning
> points
> automatically and programmatically?
> Please be advised that the second turn isn't neccessarily turning
> down,
> it
> can also possibly go up...
> And in real-world applications, the turn can be more smooth and
> round,
> but
> still, naked eyes should be able to find the turning points easily.
> My program has to do a classification:
> All the good curves should first go down, and then either make no
> turns;
> or
> make a turn, and stay vertically flat and slightly up, going from
the
> left
> to the right.
> There should be no second turn (up or down). If there is the second
> turn,
> then that's the bad curves.
> My program needs to decern the good curves from bad curves.
> I cranked a few algorithms but they don't work well. Are there
> systematic
> methods of handling this?
> My browser wouldn't show your image, but here's a
> thought. If your points are sampled at equal intervals,
> you can do a Fourier transform, use the analytic expression
> of the derivative term by term and the Fourier coefficients
> to calculate the derivative of the Fourier representation
> of your function, and find the zeros of the derivative
> and take those as the turning points. FWIW.
How? Could you please elaborate?- Hide quoted text -
- Show quoted text -
> This is a brief summary. Hopefully it doesn't insult your
> intelligence and isn't too sketchy.
> Under certain conditions on F,
> F(x) = a0 + a1 cos(kx) + a2 cos(2kx) + ... +
> b1 sin(kx) + b2 sin(2kx) + ...
> Using a discrete Fourier Transform (preferably using the Fast
> Fourier Transform algorithm if you have very many data points),
> you can calculate a discrete set of a's and b's the cause F to
> agree with your function at the x values for which you have F
> values. Then F' is given by
> F'(x)= - ka1 sin(kx) - 2ka2 sin(2kx) - ... +
> kb1 cos(kx) + 2kb2 cos(2kx) + ...
> Set F'(x) = 0 and solve for x's that satisfy this equation.
> These should approximate the turning point.
> I don't know if it will work for your application, or how
> practrical it will be.
 Or better yet, use wavelets which is O(n)(rather than O(nlog n) of FFT)
> and specifically used for detecting changes like this.
>
I admit that I've never heard about that... how to do that? Could you 
please
elaborate?
===
Subject: Re: How to detect turning points in curves...
<f76pbs$mef$1@news.Stanford.EDU>
<MfHli.6345$rL1.6014@newssvr19.news.prodigy.net>
<f77ojr$r32$1@news.Stanford.EDU
> How to detect turning points in curves
> Hi all,
> If you take a look at the following plot,
>http://img63.imageshack.us/img63/5050/gggyt1.jpg
> You will agree with me that there are two turning points.
> But is there a systematic way to let computer detect the turning
> points
> automatically and programmatically?
> Please be advised that the second turn isn't neccessarily turning
> down,
> it
> can also possibly go up...
> And in real-world applications, the turn can be more smooth and
> round,
> but
> still, naked eyes should be able to find the turning points
easily.
> My program has to do a classification:
> All the good curves should first go down, and then either make no
> turns;
> or
> make a turn, and stay vertically flat and slightly up, going from
the
> left
> to the right.
> There should be no second turn (up or down). If there is the
second
> turn,
> then that's the bad curves.
> My program needs to decern the good curves from bad curves.
> I cranked a few algorithms but they don't work well. Are there
> systematic
> methods of handling this?
> My browser wouldn't show your image, but here's a
> thought. If your points are sampled at equal intervals,
> you can do a Fourier transform, use the analytic expression
> of the derivative term by term and the Fourier coefficients
> to calculate the derivative of the Fourier representation
> of your function, and find the zeros of the derivative
> and take those as the turning points. FWIW.
> How? Could you please elaborate?- Hide quoted text -
> - Show quoted text -
> This is a brief summary. Hopefully it doesn't insult your
> intelligence and isn't too sketchy.
> Under certain conditions on F,
> F(x) = a0 + a1 cos(kx) + a2 cos(2kx) + ... +
> b1 sin(kx) + b2 sin(2kx) + ...
> Using a discrete Fourier Transform (preferably using the Fast
> Fourier Transform algorithm if you have very many data points),
> you can calculate a discrete set of a's and b's the cause F to
> agree with your function at the x values for which you have F
> values. Then F' is given by
> F'(x)= - ka1 sin(kx) - 2ka2 sin(2kx) - ... +
> kb1 cos(kx) + 2kb2 cos(2kx) + ...
> Set F'(x) = 0 and solve for x's that satisfy this equation.
> These should approximate the turning point.
> I don't know if it will work for your application, or how
> practrical it will be.
> Or better yet, use wavelets which is O(n)(rather than O(nlog n) of FFT)
> and specifically used for detecting changes like this.
I admit that I've never heard about that... how to do that? Could you
please
> elaborate?
> - Show quoted text -
Detecting turning points requires a model. Intervention Detection
methods compute the probability of observing what you observed BEFORE
YOU OBSERVED IT ... Thus if that probability ( for the last observed
poinr ) is small ...say less than some critical value ..one can
conclude (ceterus paribus) that an anomaly has been observed.
Going forward ...
1. download and install the freeware time series analysis and
forecasting ackage called FREEFORE from http://www.autobox.com/freef.exe
2. provide it with a history set of values (recordings) and examine
whether or not the LAST POINT is found to be anomolous.
3. step though time 1 period by period and continue the evaluation of
whether FREEFORE reports that an INTERVENTION has occurred at the most
recent point.
hth
dave reilly
automatic forecasting systems
http://www.autobox.com
215-675-0652
===
Subject: Re: How to detect turning points in curves...
Hi all,
Please be advised that my curves are not continuous -- they are discrete
data points. When I plotted them in Matlab, they are connected and looked
like a continous curve. So I am not sure if the second finite difference
will help..., and accurately...
> How to detect turning points in curves
Hi all,
If you take a look at the following plot,
http://img63.imageshack.us/img63/5050/gggyt1.jpg
You will agree with me that there are two turning points.
But is there a systematic way to let computer detect the turning points
> automatically and programmatically?
Please be advised that the second turn isn't neccessarily turning down, it
> can also possibly go up...
And in real-world applications, the turn can be more smooth and round, but
> still, naked eyes should be able to find the turning points easily.
My program has to do a classification:
All the good curves should first go down, and then either make no turns;
> or make a turn, and stay vertically flat and slightly up, going from the
> left to the right.
There should be no second turn (up or down). If there is the second turn,
> then that's the bad curves.
My program needs to decern the good curves from bad curves.
I cranked a few algorithms but they don't work well. Are there systematic
> methods of handling this?
>
===
Subject: Re: How to detect turning points in curves...
> Hi all,
Please be advised that my curves are not continuous -- they are discrete
> data points. When I plotted them in Matlab, they are connected and looked
> like a continous curve. So I am not sure if the second finite
difference
> will help..., and accurately...
>
My first thought would be to polynomial interpolation of 3-5 points at a
time, then use the second derivative of that polynomial.
I can't say anything about how efficient that would be, or how accurate,
though.
Peter Claussen
> How to detect turning points in curves
> Hi all,
> If you take a look at the following plot,
> http://img63.imageshack.us/img63/5050/gggyt1.jpg
> You will agree with me that there are two turning points.
> But is there a systematic way to let computer detect the turning points
> automatically and programmatically?
> Please be advised that the second turn isn't neccessarily turning down,
it
> can also possibly go up...
> And in real-world applications, the turn can be more smooth and round,
but
> still, naked eyes should be able to find the turning points easily.
> My program has to do a classification:
> All the good curves should first go down, and then either make no turns;
> or make a turn, and stay vertically flat and slightly up, going from the
> left to the right.
> There should be no second turn (up or down). If there is the second
turn,
> then that's the bad curves.
> My program needs to decern the good curves from bad curves.
> I cranked a few algorithms but they don't work well. Are there
systematic
> methods of handling this?
>
===
Subject: Re: How to detect turning points in curves...
<dakotajudo-0DB931.22201512072007@news.isp.giganews.com
> Hi all,
> Please be advised that my curves are not continuous -- they are
discrete
> data points. When I plotted them in Matlab, they are connected and
looked
> like a continous curve. So I am not sure if the second finite
difference
> will help..., and accurately...
My first thought would be to polynomial interpolation of 3-5 points at a
> time, then use the second derivative of that polynomial.
I can't say anything about how efficient that would be, or how accurate,
> though.
Peter Claussen
> How to detect turning points in curves
> Hi all,
> If you take a look at the following plot,
>http://img63.imageshack.us/img63/5050/gggyt1.jpg
> You will agree with me that there are two turning points.
> But is there a systematic way to let computer detect the turning
points
> automatically and programmatically?
> Please be advised that the second turn isn't neccessarily turning
down, it
> can also possibly go up...
> And in real-world applications, the turn can be more smooth and round,
but
> still, naked eyes should be able to find the turning points easily.
> My program has to do a classification:
> All the good curves should first go down, and then either make no
turns;
> or make a turn, and stay vertically flat and slightly up, going from
the
> left to the right.
> There should be no second turn (up or down). If there is the second
turn,
> then that's the bad curves.
> My program needs to decern the good curves from bad curves.
> I cranked a few algorithms but they don't work well. Are there
systematic
> methods of handling this?
>
I agree, we haven't been told how reliable or what the resolution of
the data is but a simple x(n)-2*x(n-1)+x(n-2)/(delta x)^2 will provide
the acceleration. One could use more points and fit that to a second
order polynomial if the points are noisy.
Peter Nachtwey
===
Subject: W. M.9fckenheim fails at Cantor bashing
W. Muekenheim argues in his paper at
http://arxiv.org/pdf/math.GM/0505648
that the Cantor proof that no set can surject onto its power set is
somehow flawed.
But it is WM's argument that is flawed, as his construction does not
deal with mappings from any set to its own power set but only to
smaller sets.
The Cantor argument goes:
Given an arbitrary set S, and its power set (set of all subset of S),
P(S), there exist functions from S to P(S). the mapping x --> {x}, for x
in S, is a trivial example.
Let f:S --> P(S) be any such function.
Then for THAT function, f, define set M_f = {x in S: not x in f(x)},
and it transpires that there cannot be any x in S such that f(x) is a
member of M_F.
Note that there is no constraint on S or on its members in that proof.
WM's argument is that for {1,a}, where a is not a number, there CAN be a
surjection from
{1,a} to P({1}).
But his argument is irrelevant to the issue, since P({1} ) =/= P({1,a}).
And one can verify, by ennumeratioon of cases if needed, that there is
no surjection from {1,a} to P({1,a}) = {{}, {1}, {a}, {1,a} }/.
It is only from a set to its OWN power set that Cantor said surjections
are impossible.
Re surjections from one set to the power set of a different set or
subset, Cantor has made no claims at all.
The thought of someone of such mathmatical ineptitude as WM has dispayed
here trying to teach anyone anything about mathematics gives one the
shudders.
===
Subject: Re: W. M?ckenheim fails at Cantor bashing
The thought of someone of such mathmatical ineptitude as WM has dispayed
> here trying to teach anyone anything about mathematics gives one the
> shudders.
===
Subject: Re: Equation of Locus
<469575a1$0$12834$5a62ac22@per-qv1-newsreader-01.iinet.net.au>,
> I can handle parts (i) and (ii). Any hints on part (iii) ?
The points P(2ap, ap^2) and Q(2aq, aq^2) lie on the parabola x^2 = 4ay.
The equation of the normal to the parabola at P is x + py = 2ap + ap^3
and
> the equation of the normal at Q is similarly given by x + qy = 2aq +
aq^3.
(i) Show that the normals at P and Q intersect at the point R whose
> coordinates are
> (-apq[p + q], a[p^2 + pq + q^2 + 2]).
(ii) The equation of the chord PQ is . y = (1/2)*(p+q)x - apq
> If the chord PQ passes through (0, a), show that pq = -1.
(iii) Find the equation of the locus of R if the chord PQ passes through
(0,
> a)....
Well, you know from (i) that the coordinates (x, y) of R are
x = -apq(p + q)
y = a(p^2 + pq + q^2 + 2)
and you also know from (ii) that pq = -1.
You then need to eliminate p and q between these three equations,
to get a single equation in x and y. I suggest that squaring the
equation for x should help.
Ken Pledger.
===
Subject: Re: Equation of Locus
> <469575a1$0$12834$5a62ac22@per-qv1-newsreader-01.iinet.net.au>,
> I can handle parts (i) and (ii). Any hints on part (iii) ?
> The points P(2ap, ap^2) and Q(2aq, aq^2) lie on the parabola x^2 = 4ay.
> The equation of the normal to the parabola at P is x + py = 2ap + ap^3
> and
> the equation of the normal at Q is similarly given by x + qy = 2aq +
> aq^3.
> (i) Show that the normals at P and Q intersect at the point R whose
> coordinates are
> (-apq[p + q], a[p^2 + pq + q^2 + 2]).
> (ii) The equation of the chord PQ is . y = (1/2)*(p+q)x - apq
> If the chord PQ passes through (0, a), show that pq = -1.
> (iii) Find the equation of the locus of R if the chord PQ passes through
> (0,
> a)....
> Well, you know from (i) that the coordinates (x, y) of R are
x = -apq(p + q)
y = a(p^2 + pq + q^2 + 2)
and you also know from (ii) that pq = -1.
You then need to eliminate p and q between these three equations,
> to get a single equation in x and y. I suggest that squaring the
> equation for x should help.
Ken Pledger.
===
Subject: Cardinality and set size:
Hi all,
Cardinality teaches that the existence of a bijection between a set x
and a set y is sufficient to determine equinumerousity between x and
y.
On the other hand Cardinality teaches that the existence of a strict
injection from a set x to a set y is a requirement for x to be
subnumerous to y but not sufficient to determine that x is subnumerous
to y.
The bias to bijection is clear! bijection is sufficient while strict
injection is just a requirement and not sufficient???
My concept of set size teaches that the existence of a bijection
between a set x and a set y is a requirement for x to be equinumerous
to y but not sufficient to determine that x is equinumerous to y.
SIMILARLY My concept of set size teaches that the existence of a
strict injection from a set x to a set y is a requirement for x to be
subnumerouse to y but not sufficient to determine that x is
subnumerous to y.
NO BIAS!!!
Furthermore my concept of set size defines the same situations under
which BOTH bijection and strict injection become sufficient to
determine equinumerousity and subnumerousity between sets
respecitively, and the same sitaution when they fail to be sufficient,
and that situation is z-homomorphisim, so
if two sets x and y are z-homomorphic then the mere existence of a
bijection between them is sufficient to determine their
equinumerousity, ALSO the mere existence of a strict injection from x
to y is sufficient to determine that x is subnumerous to y.
While when x and y are not z-homomorphic, then the existence of
bijection becomes insufficient to determine equinumerousity between
sets, but it continue to be a requirement, ALSO the existence of a
strict injection from x to y becomes insufficient to determine that x
is subnumerous to y but continues to be a requirement.
bijection to strict injection, therefore it is biased, on the other
hand set size is not biased.
x is z-homomorphic to y <-> ( Af((f:x->y, f is injective) ->
f is surjective) v ~Ef(f:x->y, f is bijective) )
Set Size 'S' :
A) if x is z-homomorphic to y then:
1) Ef ( f:x->y, f is bijective ) <-> x equinumerous_to y
2) Ef ( f:x->y, f is injective,~ f is surjective ) <->
x subnumerous_to y
This situation is similar to cardinality. No bias is involved here,
since bijection is biconditioned to equinuemrousity and so is
strict injection biconditioned to subnumerousity, so both have
biconditioning to the numerousity state they determine, so there is no
bias here.
B) if ~ x is z-homomorphic to y then:
1) x equinumerous_to y -> Ef ( f:x->y, f is bijective )
2) x subnumerous_to y -> Ef ( f:x->y, f is injective , ~ f is
surjective )
No Bias.
While with cardinality you have 2) but you have a biconditional in 1)
instead of a uni-conditional. Which make one see the clear bias to
bijection.
C) For any sets x and y
x equinumerous_to y <-> Sx = Sy
x subnumerous_to y <-> Sx<Sy
x supernumerous_to y <-> y subnumerous_to x <->
Sx>Sy <->Sy<Sx
In order to see details about set size, read the topic: Size Theory.
In this group.
many members a set has, it is rather an estimate of the bijectability
between two sets.
While set size is the measure of how many members a set has ( See Size
Theory ).
Zuhair
===
Subject: Re: Cardinality and set size:
>Hi all,
Cardinality teaches that the existence of a bijection between a set x
>and a set y is sufficient to determine equinumerousity between x and
>y.
It doesn't _teach_ that - this is just the _definition_ of
equinomerous.
>On the other hand Cardinality teaches that the existence of a strict
>injection from a set x to a set y is a requirement for x to be
>subnumerous to y but not sufficient to determine that x is subnumerous
>to y.
The bias to bijection is clear! bijection is sufficient while strict
>injection is just a requirement and not sufficient???
Bias is a curious word to use here. We're talking about the
statement There exists a bijection between X and Y if and only
if there exists a bijection between X and Y. Yes, I suppose that
there is some bias there.
Huh?
>My concept of set size teaches that the existence of a bijection
>between a set x and a set y is a requirement for x to be equinumerous
>to y but not sufficient to determine that x is equinumerous to y.
SIMILARLY My concept of set size teaches that the existence of a
>strict injection from a set x to a set y is a requirement for x to be
>subnumerouse to y but not sufficient to determine that x is
>subnumerous to y.
UNFORTUNATELY the standard notion of cardinality is very useful
in many places, while you haven't convinced anyone that any of
your revisions are of any interest at all.
>NO BIAS!!!
What if I were to say that _my_ definition of set size says that
any two sets are equinumerous? _Your_ definition has a clear bias
regarding the existence of functions of various sorts - in my
definition there is no such bias. In fact my definition has less
bias than any other possible definition of equinumerosity - no
bias at all, while any other definition has some bias in favor
of some pairs of sets being equinumerous and others not.
Would that make my definition better?
************************
David C. Ullrich
===
Subject: Re: Cardinality and set size:
<78se93phdjm2jhm845igj7uduig95nafdv@4ax.com >Hi all,
>Cardinality teaches that the existence of a bijection between a set x
>and a set y is sufficient to determine equinumerousity between x and
>y.
It doesn't _teach_ that - this is just the _definition_ of
> equinomerous.
>On the other hand Cardinality teaches that the existence of a strict
>injection from a set x to a set y is a requirement for x to be
>subnumerous to y but not sufficient to determine that x is subnumerous
>to y.
>The bias to bijection is clear! bijection is sufficient while strict
>injection is just a requirement and not sufficient???
Bias is a curious word to use here. We're talking about the
> statement There exists a bijection between X and Y if and only
> if there exists a bijection between X and Y. Yes, I suppose that
> there is some bias there.
Huh?
You understood nothing of what I am saying. What you said is nothing
but a tautology. I was not even mentioning this subject.
I am talking about Equinumerousity as an intutive concept
while you are talking about Equinumerousity as a defined concept
I am speaking about Equinumerousity in an informal sense
and I don't mean equinumerousity as its formally defined by
the existence of a bijection.
I am speaking in a predefinition status, while your speach is post
definition.
What I am trying to say is that I don't agree to the definition of
equinumerousity
as the existence of a bijection.
Equinumerousity between x and y informally means that x has as many
members
as y has, and this in my primitive opinion cannot be sufficiently
determined by the existence of
a bijection between two sets ( unless these two sets are Dedekindian
finite ).
To me the NON existence of a bijection between two sets is sufficient
to determine
their NON equinumerousity , While the EXISTENCE of a bijection between
two sets
is NOT sufficient to determine their equinumerousity (unless the two
sets
are Dedekindian finite ).
What I am trying to say here is that cardinality is not a measure of
how many
members are there in a set, it is only a measure of the injectability
of a set
to anther sets, that's it.
IF you tell me that Card x = Card y <-> Ef( f: x->y , f is bijective)
I will agree to this definition. Since the bias here is a primary bias
and in that sense every
definition is biased to the primary subject it defines.
But if you tell me that x equinumerous_to y <-> Ef( f: x->y, f is
bijective )
it is here were I object. since equinumerousity has an intutive
meaning in our minds
it means x has as many members as y has, and this cannot sufficiently
be determined
by the existence of a bijction alone, we need other information
besides the existence
of a bijection in order to determine weather two sets are equinumerous
or not.
>My concept of set size teaches that the existence of a bijection
>between a set x and a set y is a requirement for x to be equinumerous
>to y but not sufficient to determine that x is equinumerous to y.
>SIMILARLY My concept of set size teaches that the existence of a
>strict injection from a set x to a set y is a requirement for x to be
>subnumerouse to y but not sufficient to determine that x is
>subnumerous to y.
UNFORTUNATELY the standard notion of cardinality is very useful
> in many places, while you haven't convinced anyone that any of
> your revisions are of any interest at all.
Of course it is very useful. This is fortunately. I am not objecting
to cardinality, I am objecting to the informal interpretation of
cardinality.
Cardinality can be informally understood as a measure of injectability
and not as a measure of how many members there are in a set.
Actually perhaps regarding infinite sets injectability is a more
important
matter than how many members are there in a set, and that's why
cardinality is more useful than my set size definitions? unless
set size acheives better results ?
>NO BIAS!!!
What if I were to say that _my_ definition of set size says that
> any two sets are equinumerous? _Your_ definition has a clear bias
> regarding the existence of functions of various sorts - in my
> definition there is no such bias.
Yes your definition is not biased, your definition is not acceptable
at
all because it defies the simplist intuitive impression we have in our
mind about equinumerousity.
In fact my definition has less
> bias than any other possible definition of equinumerosity - no
> bias at all, while any other definition has some bias in favor
> of some pairs of sets being equinumerous and others not.
Would that make my definition better?
No, it doesn't have any meaning.
************************
David C. Ullrich
===
Subject: Re: Cardinality and set size:
<78se93phdjm2jhm845igj7uduig95nafdv@4ax.com [...] since equinumerousity has 
an
intutive meaning in
> our minds it means x has as many members as y has, and
> this cannot sufficiently be determined by the existence
> of a bijction alone, we need other information besides
> the existence of a bijection in order to determine weather
> two sets are equinumerous or not.
> I am not objecting to cardinality, I am objecting to the
> informal interpretation of cardinality.
Cardinality can be informally understood as a measure of
> injectability and not as a measure of how many members
> there are in a set.
What is an example of a set that you think cardinality
fails to capture the intuitive notion of the number of
members? The reason I ask is that I don't know what
intuitive notion you're talking about when you say
the number of members in the set, if it's not
cardinality.
Dave L. Renfro
===
Subject: ....a contradiction, ergo indeterminacy .
An event will occur and the outcome is indeterminate.
A coin will be flipped, the result will be either H or T.
In the following trial, the result is an H.
/ H (exists)
/
___mixed_state__________/
H,T, both exist partially
T (does not exist)
In this case the outcome H exists, the outcome T does not exist, and the
mixed state prior to the actual cointoss is a state of partial existence
of
both H and T. H and T both exist in this state with probability 50:50.
Interestingly, H and T are mutually exclusive. But they can coexist in this
mixed state of indeterminacy because they are existentially
indeterminate.
Yes, they are mutually exclusive, but their existence is indeternminate and
so they can coexist in this mixed state of mutual existential
indeterminacy.
H is fully existent only when T if completely nonexistent.
In the mixed state H and T are mutually exclusive yet coexist. A
contradiction. However, because they are partially existent this makes
tolerable. The existence of either H or T is indeterminate, and so these
mutually exclusive outcomes may coexist in the mixed state. As soon as the
uncertainty is destroyed and an outcome exists, then the other possible
outcomes become fully nonexistent.
The coexistence of mutually exclusives in the mixed state supplies the
neccesary contradiction to imply the indeterminacy whch we are attempting
to
prove.
????
http://sciphysicsopenmanuscript.blogspot.com/
===
Subject: Re: ....a contradiction, ergo indeterminacy .
On Jul 12, 11:47 pm, Tomoko Kanazawa
> An event will occur and the outcome is indeterminate.
A coin will be flipped, the result will be either H or T.
In the following trial, the result is an H.
/ H (exists)
> /
> ___mixed_state__________/
> H,T, both exist partially
 T (does not exist)
In this case the outcome H exists, the outcome T does not exist, and the
> mixed state prior to the actual cointoss is a state of partial existence
of
> both H and T. H and T both exist in this state with probability 50:50.
Interestingly, H and T are mutually exclusive. But they can coexist in
this
> mixed state of indeterminacy because they are existentially
indeterminate.
> Yes, they are mutually exclusive, but their existence is indeternminate
and
> so they can coexist in this mixed state of mutual existential
> indeterminacy.
H is fully existent only when T if completely nonexistent.
In the mixed state H and T are mutually exclusive yet coexist. A
> contradiction. However, because they are partially existent this makes
> tolerable. The existence of either H or T is indeterminate, and so these
> mutually exclusive outcomes may coexist in the mixed state. As soon as
the
> uncertainty is destroyed and an outcome exists, then the other possible
> outcomes become fully nonexistent.
The coexistence of mutually exclusives in the mixed state supplies the
> neccesary contradiction to imply the indeterminacy whch we are attempting
to
> prove.
????
http://sciphysicsopenmanuscript.blogspot.com/
A physical theory is determinate if it allows you to state what the
outcome of an experiment will be.
A physical theory is indeterminate if it does not allow you to state
what the outcome of some experiment will be. Usually indeterminate
theories provide statistical information about collections of
experiments.
The idea that an indeterminate physical theory allows one to say that
more than one state coexist in existential indeterminacy is
meaningless and is based in an attempt to salvage the idea that
reality can exist in some state of existential determinacy by
trying to assign more than one contradictory determinate states to the
same phenomenon and saying it exists in existential indeterminacy.
Don't think it works.
That notion is meaningless (in a sense) as can be learned from a
detailed study of phenomonology. The problem is that one is insisting
that an unexperienced state has properties even though by
hypothesis they are unexperienced. Note that this is not like the
back side of the moon which cannot in practice be experienced right
now but rather whether an experiment could be set up to do the
observation. We are talking about states that cannot be experienced on
principle - not just in practice.
Consider the theory of special relativity. One might be tempted to
call it an indeterminate theory. If I ask What is the distance from
the sun to the earth? then the theory does not allow one to determine
the answer. In fact it depends on the component of the velocity of the
observer in the direction of the sun to earth vector. However for
any experiment one might concieve, the theory of relativity does
provide a prediction of its outcome and so it is a determinate theory.
Still the notion that there exists an answer to the question What
is the distance between the sun and the earth? meaning the Sun
actualy is at some distance from the earth is not supported by the
theory. So the theory of relativity violates the notion that one can
imagine a reality that really is and from that image derive
all of the results of the theory for a give experiment. (Some may
argue that there are space-time realities that replace the spacial
realities and that's fair however there is no way to imagine
fourspace in the usual meaning of imagine, so in a sense the result
of the theory is a universe for which there does not correspond an
image - meaning a three dimensional image - that corresponds to the
way it is.)
Quantum mechanics is indeterminate because it will not for example
allow one to determine the point at which a photon will be detected
once it passes through a difraction slit. It also has an ambiguity of
answered. However the former is experimental and therefore in some
sense a geunuine indeterminacy while the latter is .an attempt to
impose a constraint on physical theories that they allow one to
imagine how the universe is at any point in time.
I believe a resolution of the problem occurs with a detailed
understanding of the meaning of objectivity and I would refer you to
Being and Nothingness by Sartre - especially the introduction. In it
one can see how being-for-itself, through its nihilating withdrawl
introduces nothingess into being. This is an interpretive tendency
that must be thrust aside in order to experience being as it is un-
interpreted, meaning as it reveals itself unconcealed in the sense
that Heidegar spoke of Being in Being and Time.
Once this happens one realizes that it is a misinterpretation of any
physical theory, even those that are completely determinate in all
senses e.g. clasical Newtonian mechanics, to infer that the theory
provides existential determinacy to things. No theory can do that
on principle and the concept itself is ultimately meaningless.
The only remaining reality to things is a kind of stability of
appearances that is a fact and is therefore and in that sense real. If
I go out of my hotel room my car will be there. I may think this is
true because my car exists and once something exists it would
require a cause to destroy it. That would be incorrect and
ultimately meaningless. There is actually no reason that my car
continues to exist from moment to moment. It simply does, or at least
it has so far in at least the vast majority of cases. That stability
is completely superfluous existentially and is determined soley by
Being.
We can see this in the fact that, although a Newtonian model of my car
would predict that it would be there, and in the fact that numerous
experiments and observations have failed to contradict the predictions
of this theory, still there exists the possibility that I will go out
and find it not there and consequently find that the physics was
incorrect. We truly do not know - for certain - that the sun will
rise. A predictive physical theory cannot be confirmed by experiment
it can only fail to be disproved over and over and over and over and
therin lies its usefulness. But therein also lies its failure to
provide existential determinacy in the sense that you mean it.
Once one realizes that all theories, those like classical physics that
allow an image of things which are in a certain state at a given
time and from then on evolve according to rules, those like relativity
which do not allow the existence of such an image but call such an
image relative to the observer and those like quantum mechanics that
do not allow one to determine the outcome of experiments, all of these
theories are the same in the fact that they do not confer existential
determinacy on things.
The problem lies deep in mathematics in the foundation of set theory
and the meaning of objects at its intersection with the philosophical
notions of being and cause. Once this problem is understood the
relation between religion (or at least meta-physics) and physics and
through physics chemistry and biology can be understood. What is
necessary is to understand the meaning of an element of a set from a
philosophical point of view of a metaphysics of being. The limitations
of this concept affects the ability of any mathematical theory to
provide existential determinacy at all. The notions of geometry
and its relation to imagination must be understood first.
So my response in a nutshell to the statement ...a contradiction,
ergo indeterminacy is ... contradiction or not there is always this
indeterminacy.
===
Subject: Re: ....a contradiction, ergo indeterminacy .
> On Jul 12, 11:47 pm, Tomoko Kanazawa
> An event will occur and the outcome is indeterminate.
> A coin will be flipped, the result will be either H or T.
> In the following trial, the result is an H.
> / H (exists)
> /
> ___mixed_state__________/
> H,T, both exist partially
 T (does not
exist)
> In this case the outcome H exists, the outcome T does not exist, and
the
> mixed state prior to the actual cointoss is a state of partial
existence of
> both H and T. H and T both exist in this state with probability 50:50.
> Interestingly, H and T are mutually exclusive. But they can coexist in
this
> mixed state of indeterminacy because they are existentially
indeterminate.
> Yes, they are mutually exclusive, but their existence is indeternminate
and
> so they can coexist in this mixed state of mutual existential
> indeterminacy.
> H is fully existent only when T if completely nonexistent.
> In the mixed state H and T are mutually exclusive yet coexist. A
> contradiction. However, because they are partially existent this makes
> tolerable. The existence of either H or T is indeterminate, and so
these
> mutually exclusive outcomes may coexist in the mixed state. As soon as
the
> uncertainty is destroyed and an outcome exists, then the other possible
> outcomes become fully nonexistent.
> The coexistence of mutually exclusives in the mixed state supplies the
> neccesary contradiction to imply the indeterminacy whch we are
attempting to
> prove.
> ????
> http://sciphysicsopenmanuscript.blogspot.com/
> A physical theory is determinate if it allows you to state what the
> outcome of an experiment will be.
A physical theory is indeterminate if it does not allow you to state
> what the outcome of some experiment will be. Usually indeterminate
> theories provide statistical information about collections of
> experiments.
The idea that an indeterminate physical theory allows one to say that
> more than one state coexist in existential indeterminacy is
> meaningless and is based in an attempt to salvage the idea that
> reality can exist in some state of existential determinacy by
> trying to assign more than one contradictory determinate states to the
> same phenomenon and saying it exists in existential
indeterminacy.
> Don't think it works.
A physical theory is determinate if it allows you to state what the
outcome of an experiment will be. Agreed.
A physical theory is indeterminate if it does _not_ allow you to state
what the outcome of some experiment will be. Agreed.
I am positing that existential indeterminacy allows the formulation of
physical theories which are probabilistic. Such a theory then predicts that
a particular outcome might, or might not exist.
A deterministic theory cannot produce predictions which are probabilistic,
because all of existence is assumed to be a giant Newtonian clockwork. This
precludes the possibility of probabilistic outcomes. Pure determinism
precludes the existence of randomness. The big paradox of science is that
we
have a universe where randomness occurs because the deterministic theory
said so.
> That notion is meaningless (in a sense) as can be learned from a
> detailed study of phenomonology. The problem is that one is insisting
> that an unexperienced state has properties even though by
> hypothesis they are unexperienced. Note that this is not like the
> back side of the moon which cannot in practice be experienced right
> now but rather whether an experiment could be set up to do the
> observation. We are talking about states that cannot be experienced on
> principle - not just in practice.
Actually, what is being suggested is that there is an state of uncertainty
which might or might not have properties because the outcome is
unexperienced.
By stating that a coin will be tossed, one is creating a situation
where
the coin toss clearly will occur, but has not yet occured. It will occur
nonetheless. This is quite different than considering arbitrary
hypothetical
coin tosses which may or may not even be tossed.
We have stated that there will be an outcome, and that the outcome will
be
unknown until some action occurs to destroy the uncertainty which is
created
by this determination of being of a yet unknown outcome.
Empirical evidence clearly shows that results from QM are in fact best
modelled probabilistically. Physics even claims that such processes are
truly, inherently indeterminate.
What I am proposing is that if you have outcome H and outcome T, and these
outcomes are mutually exclusive, that the only way to justify a state of
indeterminacy which allows them to coexist is by clarifying that
their
individual existential states is itself indeterminate in that we cannot say
if H exists or not, nor can we say that T exists or not. Clearly this is
not
a state of nonexistence, nor a state of existence, but a third state where
existence is simply unknown.
> Consider the theory of special relativity. One might be tempted to
> call it an indeterminate theory. If I ask What is the distance from
> the sun to the earth? then the theory does not allow one to determine
> the answer. In fact it depends on the component of the velocity of the
> observer in the direction of the sun to earth vector. However for
> any experiment one might concieve, the theory of relativity does
> provide a prediction of its outcome and so it is a determinate theory.
> Still the notion that there exists an answer to the question What
> is the distance between the sun and the earth? meaning the Sun
> actualy is at some distance from the earth is not supported by the
> theory. So the theory of relativity violates the notion that one can
> imagine a reality that really is and from that image
derive
> all of the results of the theory for a give experiment. (Some may
> argue that there are space-time realities that replace the spacial
> realities and that's fair however there is no way to imagine
> fourspace in the usual meaning of imagine, so in a sense the result
> of the theory is a universe for which there does not correspond an
> image - meaning a three dimensional image - that corresponds to the
> way it is.)
Quantum mechanics is indeterminate because it will not for example
> allow one to determine the point at which a photon will be detected
> once it passes through a difraction slit. It also has an ambiguity of
> answered. However the former is experimental and therefore in some
> sense a geunuine indeterminacy while the latter is .an attempt to
> impose a constraint on physical theories that they allow one to
> imagine how the universe is at any point in time.
I believe a resolution of the problem occurs with a detailed
> understanding of the meaning of objectivity and I would refer you to
> Being and Nothingness by Sartre - especially the introduction. In it
> one can see how being-for-itself, through its nihilating withdrawl
> introduces nothingess into being. This is an interpretive tendency
> that must be thrust aside in order to experience being as it is un-
> interpreted, meaning as it reveals itself unconcealed in the sense
> that Heidegar spoke of Being in Being and Time.
Once this happens one realizes that it is a misinterpretation of any
> physical theory, even those that are completely determinate in all
> senses e.g. clasical Newtonian mechanics, to infer that the theory
> provides existential determinacy to things. No theory can do that
> on principle and the concept itself is ultimately meaningless.
While it may be argued that a thing which is existentially indeterminate is
not really a thing, it is not so easy to discount the uncertainty which is
associated with such an object. The uncertainty itself is a thing in it's
own right, albeit uncertainly.
Again, this is different that simply not knowing. We have stated that
an
outcome will exist, but it does not yet exist. By stating that it will
exist, it is already invoked into existence subjectively by virtue of that
expectation.
> The only remaining reality to things is a kind of stability of
> appearances that is a fact and is therefore and in that sense real. If
> I go out of my hotel room my car will be there. I may think this is
> true because my car exists and once something exists it would
> require a cause to destroy it. That would be incorrect and
> ultimately meaningless. There is actually no reason that my car
> continues to exist from moment to moment. It simply does, or at least
> it has so far in at least the vast majority of cases. That stability
> is completely superfluous existentially and is determined soley by
> Being.
We can see this in the fact that, although a Newtonian model of my car
> would predict that it would be there, and in the fact that numerous
> experiments and observations have failed to contradict the predictions
> of this theory, still there exists the possibility that I will go out
> and find it not there and consequently find that the physics was
> incorrect. We truly do not know - for certain - that the sun will
> rise. A predictive physical theory cannot be confirmed by experiment
> it can only fail to be disproved over and over and over and over and
> therin lies its usefulness. But therein also lies its failure to
> provide existential determinacy in the sense that you mean it.
Once one realizes that all theories, those like classical physics that
> allow an image of things which are in a certain state at a given
> time and from then on evolve according to rules, those like relativity
> which do not allow the existence of such an image but call such an
> image relative to the observer and those like quantum mechanics that
> do not allow one to determine the outcome of experiments, all of these
> theories are the same in the fact that they do not confer existential
> determinacy on things.
The problem lies deep in mathematics in the foundation of set theory
> and the meaning of objects at its intersection with the philosophical
> notions of being and cause. Once this problem is understood the
> relation between religion (or at least meta-physics) and physics and
> through physics chemistry and biology can be understood. What is
> necessary is to understand the meaning of an element of a set from a
> philosophical point of view of a metaphysics of being. The limitations
> of this concept affects the ability of any mathematical theory to
> provide existential determinacy at all. The notions of geometry
> and its relation to imagination must be understood first.
So my response in a nutshell to the statement ...a contradiction,
> ergo indeterminacy is ... contradiction or not there is always this
> indeterminacy.
It would then also be true that randomness would be the result of some
deterministic process, which of course returns us to the same old paradox
which has been brewing for hundreds of years.
A complete disqualification of existential indeterminacy yields the
customary paradigm which is an existential dichotomy of that which exists
and that which does not exist. The result is a paradox which disallows
genuine randomness.
If indeterminacy is the result of some deterministic process, then it can
be
argued that indeterminacy is determined, clearly a paradoxical position.
Yet
unfortunately we find that this is also the current state of scientific
thinking. There really is only one solution, and that would be the
implementation of a third existential modality which is existential
indeterminacy, which simply means that there is a state of being where one
is not sure whether an object exists or not, regardless of whether that
object is objective or subjective, whether it exists or not is a matter of
uncertainty. It is a state of uncertainty which cannot be resolved by any
means other than the destruction of the uncertainty
If one takes the position that existence might be indeterminate or
probabilistic, then you have a universe where this property is simply out
there and might even be reflexive. So that you have a choice. You can
choose
to accept it or not.
It is no better or worse that being forced to acknowledge that
nonexistence
exists, within the paradigm of an existential dichotomy.
If you accept that existence can be uncertain, then reflexively this
property can disallow itself thereby resulting in the familiar dichotomy.
But the reflexivity of the position does not demand the dichotomy. This
too, would be uncertain.
The result of all this is a meaningful and sensible coexistence of order
and
disorder. Of randomness and determinism. A resolution of paradox,
regardless
of how strange it may appear.
It resolves the paradox that nonexistence exists, because it allows you
to
say that I dont know if nonexistence exists or not. It's indeterminate.
Existential quantifiers state that there exists an x, such that ...
XYZ.
What kind of quantifier can be used to quantify an uncertainty ? A cointoss
? You would have to write such a thing as It is uncertain whether there
exists an H or a T . Probability theory has avoided this approach, but
why
?
===
Subject: Re: subset of contractible
> Suppose X is a contractible topological space
> and S subset of X is a topological subspace of X.
> Is S contractible itself?
>
Of course not.
> How about the opposite? Suppose S is ANY topological space.
> Then S can be embedded in some contractible space X.
>
Hm. How about it?
The long line is embeddable in an contractible space?
===
Subject: Re: sets definable by polynomials
Based on replies so far, and some web searches, a few things are now
clearer to me. I'll summarize ...
Of the 3 concepts,
pr-sets (polynomial range sets)
pz-sets (polynomial zero sets)
pd-sets (polynomially definable sets)
it appears that the pz-sets are the same as the diophantine sets,
hence the concept of pz-sets is redundant, and can be dropped.
So forget pz-sets.
But the other two types of sets have survived (so far) as distinct
(and possibly useful) concepts.
For reference, here are the definitions.
First, pr-sets (polynomial range sets) ...
Let n be a positive integer. Call a subset S of Z^n a pr-set if
S=f(Z^m) for some polynomial map f:Z^m->Z^n with integer
coefficients.
Note: Every pr-set is a subset of Z^n for some positive integer n, but
different pr-sets need not live in the same Z^n.
Next, pd-sets (polynomially definable sets) ...
Here are the requirements:
(1) Z is a pd-set.
(2) If S is a pd-set, then so is f(S) for any polynomial map f.
(3) If S is a pd-set, then so is f^(-1)(S) for any polynomial map f.
(4) If S in Z^n and S is a pd-set, so is Z^nS.
Moreover, we require that every pd-set be obtainable by a finite
number of applications of (1), (2), (3), (4).
Note: In this context, polynomial map means a map from Z^m->Z^n
(where m,n are positive integers) such that the coordinate functions
are polynomials with integer coefficients.
Also, as before, every pd-set is a subset of Z^n for some positive
integer n, but different pd-sets need not live in the same Z^n.
We have the following relationships:
Every pr-set is diophantine. Not every diophantine set is a pr-set.
Every diophantine set is a pd-set. Not every pd-set is diophantine.
In other words, we have the proper inclusions
{pr-sets} subset {diophantine sets} subset {pd-sets}.
Since the diophantine sets are the same as the recursively enumerable
sets, and since not every recursively enumerable set is recursive,
this implies that there is no general algorithm for determining
whether a given diophantine set (as specified by a diophantine
equation or system of equations) has any elements.
I'm not sure whether the above fact suffices to yield answers to some
questions I've been thinking about. If so, maybe someone could explain
the tie? If not, are there other simple arguments (my knowledge of
logic is novice level) that can provide answers?
Here are the questions ...
(1) Is every pr-set recursive?
(2) Is there an algorithm to decide if a diophantine set is a pr-set?
(3) Is there an algorithm to decide if a pd-set is a diophantine set?
quasi
===
Subject: Re: sets definable by polynomials
<bk7e93hk1g8nf6h8dcofrcp7j6ahvndigp@4ax.com Based on replies so far, and 
some
web searches, a few things are now
> clearer to me. I'll summarize ...
Of the 3 concepts,
pr-sets (polynomial range sets)
pz-sets (polynomial zero sets)
pd-sets (polynomially definable sets)
it appears that the pz-sets are the same as the diophantine sets,
> hence the concept of pz-sets is redundant, and can be dropped.
So forget pz-sets.
But the other two types of sets have survived (so far) as distinct
> (and possibly useful) concepts.
For reference, here are the definitions.
First, pr-sets (polynomial range sets) ...
Let n be a positive integer. Call a subset S of Z^n a pr-set if
S=f(Z^m) for some polynomial map f:Z^m->Z^n with integer
> coefficients.
Note: Every pr-set is a subset of Z^n for some positive integer n, but
> different pr-sets need not live in the same Z^n.
Next, pd-sets (polynomially definable sets) ...
Here are the requirements:
(1) Z is a pd-set.
> (2) If S is a pd-set, then so is f(S) for any polynomial map f.
> (3) If S is a pd-set, then so is f^(-1)(S) for any polynomial map f.
> (4) If S in Z^n and S is a pd-set, so is Z^nS.
Moreover, we require that every pd-set be obtainable by a finite
> number of applications of (1), (2), (3), (4).
Note: In this context, polynomial map means a map from Z^m->Z^n
> (where m,n are positive integers) such that the coordinate functions
> are polynomials with integer coefficients.
Also, as before, every pd-set is a subset of Z^n for some positive
> integer n, but different pd-sets need not live in the same Z^n.
We have the following relationships:
Every pr-set is diophantine. Not every diophantine set is a pr-set.
Every diophantine set is a pd-set. Not every pd-set is diophantine.
In other words, we have the proper inclusions
{pr-sets} subset {diophantine sets} subset {pd-sets}.
Since the diophantine sets are the same as the recursively enumerable
> sets, and since not every recursively enumerable set is recursive,
> this implies that there is no general algorithm for determining
> whether a given diophantine set (as specified by a diophantine
> equation or system of equations) has any elements.
I'm not sure whether the above fact suffices to yield answers to some
> questions I've been thinking about. If so, maybe someone could explain
> the tie? If not, are there other simple arguments (my knowledge of
> logic is novice level) that can provide answers?
Here are the questions ...
(1) Is every pr-set recursive?
yes
since for any n
there are only a finite number of polynomial arguments
that evaluate <= n
we can always determine if n is in a _given_ pr-set or not
in a finite number of evaluations
we can get hard bounds on this finite set
by ensuring the leading term dominates
the absolute value sum of all other terms
> (2) Is there an algorithm to decide if a diophantine set is a pr-set?
if we can order it canonically as i showed in my other post
then if there is a finite number of delta applications to a constant
the collection is pr-
however
since one would need to consider infinite members in the general
case
we can not in general rule out a recursively enumerable set is pr-
in a finite time
that's handwaving but expository
the real proof would use a rigorous specification language
for recursively enumerable sets
and use the fact that being a polynomial range
is equivalent to being able to prove the specification implies
all potentially infinite elements
applied iterative delta mappings
become one constant
a diagonal argument gets applied
someone will pull rice's theorem out their @$$
and claim polynomality is a nontrivial property of partial functions
and recursive inseparability will be a result
or something like that.. :p
> (3) Is there an algorithm to decide if a pd-set is a diophantine set?
did i mention i don't like pd-sets?
they make me angry...
they are meaningless and possibly contradictory
so you can't ever trust them
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
===
Subject: Re: sets definable by polynomials
On Thu, 12 Jul 2007 23:36:40 -0700, galathaea <galathaea@gmail.com
> Here are the questions ...
> (1) Is every pr-set recursive?
yes
since for any n
> there are only a finite number of polynomial arguments
> that evaluate <= n
>we can always determine if n is in a _given_ pr-set or not
>in a finite number of evaluations
we can get hard bounds on this finite set
>by ensuring the leading term dominates
> the absolute value sum of all other terms
You seem to be stuck in one variable, but all these concepts are
multivariate.
If f is a multivariate polynomial, f^(-1) of a finite set need not be
finite.
> (2) Is there an algorithm to decide if a diophantine set is a pr-set?
if we can order it canonically as i showed in my other post
> then if there is a finite number of delta applications to a constant
> the collection is pr-
however
> since one would need to consider infinite members in the general
>case
>we can not in general rule out a recursively enumerable set is pr-
>in a finite time
that's handwaving but expository
the real proof would use a rigorous specification language
> for recursively enumerable sets
>and use the fact that being a polynomial range
>is equivalent to being able to prove the specification implies
> all potentially infinite elements
> applied iterative delta mappings
>become one constant
a diagonal argument gets applied
someone will pull rice's theorem out their @$$
> and claim polynomality is a nontrivial property of partial functions
>and recursive inseparability will be a result
or something like that.. :p
Well, in the above explanation, you lost me early on. I'm not even
sure whether your proposed answer is yes or no.
> (3) Is there an algorithm to decide if a pd-set is a diophantine set?
did i mention i don't like pd-sets?
they make me angry...
Hehe.
Sorry, but they don't get abandoned so easily. In fact, the concept of
what it means for a set to be polynomially definable is what this
thread is all about.
Let me try to rephrase the definition.
Start with Z.
Form the closure under finitely many applications of polynomial map
images, polynomial map inverse images and complements (where
complement is relative to an appropriate Z^n).
The new collection is the set of polynomially definable sets.
It's similar in spirit to the concept of group generated by a set.
>they are meaningless
I can try to amplify the definition further, if necessary, but there
is a concept here, and it's more general than diophantine sets.
Diophantine sets are the same as recursively enumerable sets, and
recursively enumerable sets are not closed under complementation.
The pd-sets include the diophantine sets, plus their complements, plus
whatever else is forced in by closure under polynomial map images,
polynomial map inverse images, and more complements. It's possible
that nothing else is forced in, I'm not sure. But as a minimum, the
complements of diophantine sets are pd-sets.
> and possibly contradictory
If they are contradictory (whatever that means), then the definition
gets revised, but hey -- innocent until proven guilty -- where's the
contradiction?
>so you can't ever trust them
Anything that proceeds from a known set to a new set by a finite
number of pre-specified admissible operations is very concrete, and in
my book, that inspires trust.
quasi
===
Subject: Re: sets definable by polynomials
Based on replies so far, and some web searches, a few things are now
clearer to me. I'll summarize ...
Of the 3 concepts,
pr-sets (polynomial range sets)
pz-sets (polynomial zero sets)
pd-sets (polynomially definable sets)
it appears that the pz-sets are the same as the diophantine sets,
hence the concept of pz-sets is redundant, and can be dropped.
So forget pz-sets.
But the other two types of sets have survived (so far) as distinct
(and possibly useful) concepts.
For reference, here are the definitions.
First, pr-sets (polynomial range sets) ...
Let n be a positive integer. Call a subset S of Z^n a pr-set if
S=f(Z^m) for some polynomial map f:Z^m->Z^n with integer
coefficients.
Note: Every pr-set is a subset of Z^n for some positive integer n, but
different pr-sets need not live in the same Z^n.
Next, pd-sets (polynomially definable sets) ...
Here are the requirements:
(1) Z is a pd-set.
(2) If S is a pd-set, then so is f(S) for any polynomial map f.
(3) If S is a pd-set, then so is f^(-1)(S) for any polynomial map f.
(4) If S in Z^n and S is a pd-set, so is Z^nS.
Moreover, we require that every pd-set be obtainable by a finite
number of applications of (1), (2), (3), (4).
Note: In this context, polynomial map means a map from Z^m->Z^n
(where m,n are positive integers) such that the coordinate functions
are polynomials with integer coefficients.
Also, as before, every pd-set is a subset of Z^n for some positive
integer n, but different pd-sets need not live in the same Z^n.
We have the following relationships:
Every pr-set is diophantine. Not every diophantine set is a pr-set.
Every diophantine set is a pd-set. Not every pd-set is diophantine.
In other words, we have the proper inclusions
{pr-sets} subset {diophantine sets} subset {pd-sets}.
Since the diophantine sets are the same as the recursively enumerable
sets, and since not every recursively enumerable set is recursive,
this implies that there is no general algorithm for determining
whether a given diophantine set (as specified by a diophantine
equation or system of equations) has any elements.
I'm not sure whether the above fact suffices to yield answers to some
questions I've been thinking about. If so, maybe someone could explain
the tie? If not, are there other simple arguments (my knowledge of
logic is novice level) that can provide answers?
Here are the questions ...
(1) Is every pr-set recursive?
(2) Is there an algorithm to decide if a diophantine set is a pr-set?
(3) Is there an algorithm to decide if a pd-set is a diophantine set?
quasi
===
Subject: Re: sets definable by polynomials
>Based on replies so far, and some web searches, a few things are now
>clearer to me. I'll summarize ...
suggesting the cross-post), but forgot, so I'll repost the prior
reply, this time both to sci.math and sci.logic.
Sorry for the duplicate post.
quasi
===
Subject: Re: sets definable by polynomials
> On Thu, 12 Jul 2007 00:31:01 -0700, galathaea
> <galathaea@gmail.com
> Once again, I acknowledge my debt to tommy1729 for
> asking unusual,
> insightful questions which in turn have lead me to
> questions of my
> own.
thank you quasi.
however you almost seem to be the only one.
are the others really to smart to consider my math ??
is it really way below their level.
they sure pretend too.
but im not buying it.
you cant fool tommy1729.
btw i hope you have not forgotten the other ideas eg factoring , which is
still miles away from totally solved.
> First, let's define pr-sets (polynomial range
> sets) ...
that is usually called a diophantine set or polynomial representation ...
but what the heck , pr-sets.
i assume you mean diophantine sets without any restrictions (like positive
output ) so ok pr-sets.
> Let n be a positive integer. Call a subset S of
> Z^n a pr-set if
> S=f(Z^m) for some polynomial map f:Z^m->Z^n
> with integer
> coefficients.
> Note: Every pr-set is a subset of Z^n for some
> positive integer n, but
> different pr-sets need not live in the same Z^n.
> Clearly every pr-set is recursively enumerable.
> Question (1): Is every nonempty recursively
> enumerable subset of Z a
> pr-set?
>yes
>every recursively enumerable subset of Z
>is diophantine over Z[x]
> and vice versa
I think you meant to say over Z[x_1,...,x_m] for
> some m.
this is a consequence of the famous
> robinson-putnam-davis theorem
Ok, I found some info in Wikipedia
<http://en.wikipedia.org/wiki/Hilbert's_tenth_problem
> which mentions the results you allude to.
Perhaps I should have spent a little more time
> checking for info
> before posting.
> I'm not sure exactly how this concept relates to
> the concept of
> diophantine sets. Letting N denote the set of
> positive integers, then
> as far as I understand it, a nonempty subset S of
> N is a diophantine
> set iff S = (T intersect N) where T is a subset of
> Z and T is a
> pr-set.
> Also, if I understand correctly, Matiyasevich
> proved that the
> recursively enumerable subsets of N are precisely
> the diophantine
> sets.
yes he did.
ill repeat :
Riemann , Matiyasevich and Wiles forever !
( and hope my list will extend some day )
>i am pretty certain matiyasevich's theorem
> does not restrict to positive naturals
Yes, I see that now. That restriction was what was
> bothering me, but I
> In fact they also
> mention the 4 squares trick which had occurred to me
> as well.
obvious trick
it is simply
>diophantine sets over Z <-> recursively enumerable
> sets over Z
i assume you mean primitive recursive.
dont recall partially recursive satisfying that.
( all in matheyasivich btw)
>i believe the restriction to positives
> is found in some of the examples like primes
>because of the methods of their construction
>
ive been wondering about rational functions instead of polynomials to
represent primes...
we would then remove the condition of positive with the condition of
integer.
it seems more logical since primes are mainly about divisibility, and we
wouldnt have to use tricks like sum of squares or anything.
perhaps also better to generate primes.
if it exists of course...
>i think these are related to exponential
> constructions
>and if i remmeber correctly
> matiyasevich gave some in his dissertation
yes he did.
> It's this restriction to positive range values
> that I find confusing,
> thus motivating for the question I asked above.
Ok, so based on what you are saying, and which seems
> to be confirmed
> subset of Z is the
> full range of a multivariate polynomial with integer
> coefficients when
> the inputs are allowed to vary over all integers. Is
> that correct?
But wait, that can't be right. There's no nonconstant
> polynomial with
> integer coefficients which only yields prime values
> over the set of
> all integer values for the input variables. But the
> set of primes is
> recursively enumerable. So what am I missing?
> Next, let's define pz-sets (polynomial zero sets)
> ...
> Let n be a positive integer. Call a subset S of
> Z^n a pz-set if
> S = { (x_1,...,x_n) in Z^n | f(x_1,...,x_m)=0
> for some polynomial
> f in Z[x_1,...,x_m], where m>=n }
> Note the allowed extra variables (m is potentially
> greater than n).
>these are normally called algebraic varieties
yep
>they are foundational constructions in algebraic
> geometry
>( later replaced by schemes )
Yes, certainly I intended definition I proposed was
> modeled on the
> concept of an algebraic set, although, the extra
> variables that I'm
> allowing (if m>n) might make the concepts different
> -- I'm not sure.
> Once again, every pz-zet is a subset of Z^n for
> some positive integer
> n, but different pz-sets need not live in the same
> Z^n.
> It's easy to show that every pr-set is a pz-set.
> Question (2): Is every nonempty pz-set a pr-set?
>yes
> because there is a finite method
> ( evaluation )
> to determine if a memeber is in the zero set
>and thus it is recursively enumerable
that.
> Finally, let's define pd-sets (polynomially
> definable sets) ...
> Here are the requirements:
> (1) Z is a pd-set.
> (2) If S is a pd-set, then so is f(S) for any
> polynomial map f.
> (3) If S is a pd-set, then so is f^(-1)(S) for
> any polynomial map f.
> (4) If S in Z^n and S is a pd-set, so is Z^nS.
> Moreover, we require that every pd-set be
> obtainable by a finite
> number of applications of (1), (2), (3), (4).
> Note: In this context, polynomial map means a
> map from Z^m->Z^n
> (where m,n are positive integers) such that the
> coordinate functions
> are polynomials with integer coefficients.
> Also, as before, every pd-zet is a subset of Z^n
> for some positive
> integer n, but different pd-sets need not live in
> the same Z^n.
> Clearly, every pz-set is a pd-set.
> Question (3): Is every pd-set a pz-set?
>this seems like yes again
>since each set is still diophantine
> by polynomial composition
Here I think the answer is no, since the pd-sets (in
> a given Z^n) are
> closed under complementation, whereas the recursively
> enumerable sets
> are not.
its getting late though
>so i'm probably missing something
Well, your reply was definitely helpful. For one
> thing, it makes clear
> that this ground has been well covered, and that the
> concept of
> pr-sets and pz-sets are (possibly) nothing more than
> recursively
> enumerable, hence diophantine. Thus, those 2 concepts
> appear to be
> subsumed by the concept of recursive enumerability,
> and can therefore
> safely be dropped.
On the other hand, based on the complementation issue
> as mentioned
> above, it appears that pd-sets are a strictly larger
> class of sets
> than the recursively enumerable sets. What to make of
> that, I'm not
> sure.
quasi
uhm all very well but
where are you going too with your definitions and conjectures ??
for instance suppose all of you conjectures were true , what would that
imply ?
as my friend timothy golden said recently
breaking things up , to the smallest parts of sets and definitions , does
not neccesarily mean better math or better understanding.
nor better communication.
what is your goal with these conjectures ?
i can see how i inspired you , but you seem to drift off ??
how does this help to find a polynomial that gives all integers except the
squares ?
maybe ive misted something.
but i dont think so...
dont use to much set theory plz :p
i dont like set theorems.
i prefer number theory and calculus
( the 'real' math :p )
but i suppose that was already clear bye now.
off topic : i was a bit surprised you did not reply to my question about
double series...
i expected you might be intrested in that.
im convinced you know what i meant there, so i assume your more intrested 
in
number theory than calculus ...
i am a constructivist
tommy1729
===
Subject: Re: sets definable by polynomials
On Thu, 12 Jul 2007 16:25:53 EDT, tommy1729 <tommy1729@gmail.com
> First, let's define pr-sets (polynomial range sets) ...
that is usually called a diophantine set or polynomial representation ...
but what the heck , pr-sets.
i assume you mean diophantine sets without any restrictions
>(like positive output ) so ok pr-sets.
A pr-set is a subset of Z^n which is the range of a polynomial map
from some Z^m to Z^n and where the (polynomial) coordinate functions
of the map have integer coefficients.
Every pr-set is diophantine, but not every diophantine set is a
pr-set. Diophantine sets are solution sets whereas pr-sets are
image sets. Not every solution set is an image set. Thus, a pr-set
is more of a representation.
>uhm all very well but
where are you going too with your definitions and conjectures ??
for instance suppose all of you conjectures were true , what would that
imply ?
Well for one thing, if we could characterize pr-sets, it might lead to
an instant answer to the question you asked about non-squares.
The set of pr-subsets of Z is a proper subset of the set of
diophantine subsets of Z. It would be nice to be able to determine
whether a given diophantine set is or is not a pr-set.
Is such a question algorithmically decidable? I'm not sure.
quasi
===
Subject: Re: sets definable by polynomials
<n8rb93tqhjj1qplr3lontbpc9o7k6b392o@4ax.com On Thu, 12 Jul 2007 00:31:01
-0700, galathaea <galath...@gmail.com
> Once again, I acknowledge my debt to tommy1729 for asking unusual,
> insightful questions which in turn have lead me to questions of my
> own.
> First, let's define pr-sets (polynomial range sets) ...
> Let n be a positive integer. Call a subset S of Z^n a pr-set if
> S=f(Z^m) for some polynomial map f:Z^m->Z^n with integer
> coefficients.
> Note: Every pr-set is a subset of Z^n for some positive integer n, but
> different pr-sets need not live in the same Z^n.
> Clearly every pr-set is recursively enumerable.
> Question (1): Is every nonempty recursively enumerable subset of Z a
> pr-set?
>yes
>every recursively enumerable subset of Z
>is diophantine over Z[x]
> and vice versa
I think you meant to say over Z[x_1,...,x_m] for some m.
well
there is a direct correspondence
due to recursive enumerability on the projections
but i see i have misinterpreted the maps
your maps are
X=(x1, x2, ..., xm) |-> (p1(X), p2(X), ..., pn(X))
whereas i mistook them as being simply
diophantine embeddings of R^m in R^n
ie.
solutions in yi of
p(x1, x2, ..., xm, y1, y2, ..., yn) = 0
for some polynomial p and given xi
i really need to read posts before responding!
[...]
> I'm not sure exactly how this concept relates to the concept of
> diophantine sets. Letting N denote the set of positive integers, then
> as far as I understand it, a nonempty subset S of N is a diophantine
> set iff S = (T intersect N) where T is a subset of Z and T is a
> pr-set.
> Also, if I understand correctly, Matiyasevich proved that the
> recursively enumerable subsets of N are precisely the diophantine
> sets.
>i am pretty certain matiyasevich's theorem
> does not restrict to positive naturals
Yes, I see that now. That restriction was what was bothering me, but I
> mention the 4 squares trick which had occurred to me as well.
yes
and other way of the equivalence is proved by
if x elementOf Z, thereExists x1, x2 elementsOf N
suchThat x = x1 - x2
>it is simply
>diophantine sets over Z <-> recursively enumerable sets over Z
>i believe the restriction to positives
> is found in some of the examples like primes
>because of the methods of their construction
>i think these are related to exponential constructions
>and if i remmeber correctly
> matiyasevich gave some in his dissertation
> It's this restriction to positive range values that I find confusing,
> thus motivating for the question I asked above.
Ok, so based on what you are saying, and which seems to be confirmed
> full range of a multivariate polynomial with integer coefficients when
> the inputs are allowed to vary over all integers. Is that correct?
no
that was my mistake
i should understand that when you say polynomials
you really mean polynomials
and not polynomial diophantine equations or something else
> But wait, that can't be right. There's no nonconstant polynomial with
> integer coefficients which only yields prime values over the set of
> all integer values for the input variables. But the set of primes is
> recursively enumerable. So what am I missing?
what am i missing?
patience?
probably
> Next, let's define pz-sets (polynomial zero sets) ...
> Let n be a positive integer. Call a subset S of Z^n a pz-set if
> S = { (x_1,...,x_n) in Z^n | f(x_1,...,x_m)=0 for some polynomial
> f in Z[x_1,...,x_m], where m>=n }
> Note the allowed extra variables (m is potentially greater than n).
>these are normally called algebraic varieties
>they are foundational constructions in algebraic geometry
>( later replaced by schemes )
Yes, certainly I intended definition I proposed was modeled on the
> concept of an algebraic set, although, the extra variables that I'm
> allowing (if m>n) might make the concepts different -- I'm not sure.
the projection of a variety is still a variety
[...]
> Finally, let's define pd-sets (polynomially definable sets) ...
> Here are the requirements:
> (1) Z is a pd-set.
> (2) If S is a pd-set, then so is f(S) for any polynomial map f.
> (3) If S is a pd-set, then so is f^(-1)(S) for any polynomial map f.
> (4) If S in Z^n and S is a pd-set, so is Z^nS.
> Moreover, we require that every pd-set be obtainable by a finite
> number of applications of (1), (2), (3), (4).
> Note: In this context, polynomial map means a map from Z^m->Z^n
> (where m,n are positive integers) such that the coordinate functions
> are polynomials with integer coefficients.
> Also, as before, every pd-zet is a subset of Z^n for some positive
> integer n, but different pd-sets need not live in the same Z^n.
> Clearly, every pz-set is a pd-set.
> Question (3): Is every pd-set a pz-set?
>this seems like yes again
>since each set is still diophantine
> by polynomial composition
Here I think the answer is no, since the pd-sets (in a given Z^n) are
> closed under complementation, whereas the recursively enumerable sets
> are not.
you are right here
>its getting late though
>so i'm probably missing something
Well, your reply was definitely helpful. For one thing, it makes clear
> that this ground has been well covered, and that the concept of
> pr-sets and pz-sets are (possibly) nothing more than recursively
> enumerable, hence diophantine. Thus, those 2 concepts appear to be
> subsumed by the concept of recursive enumerability, and can therefore
> safely be dropped.
On the other hand, based on the complementation issue as mentioned
> above, it appears that pd-sets are a strictly larger class of sets
> than the recursively enumerable sets. What to make of that, I'm not
> sure.
it seems
the only pd sets that are recursively enumerable
are the recursive sets
..
i also wanted to mention
for questions on recursive enumerability
you may want to post also to sci.logic
some of the regular posters there
are very well versed in all of this
and can offer very expert advise
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
===
Subject: Re: sets definable by polynomials
<n8rb93tqhjj1qplr3lontbpc9o7k6b392o@4ax.com>
[...]
i think we can give this a complexity-theoretic description
i hope you don't mind me starting with a clean slate
quasi
but i just want to reorganise the questions some
any polynomial of degree n is
completely determined by n+1 points
after that
the infinite other values are completely determined
which has infinite range or is constant
as you mentioned
so as you pointed out
no finite enumerable collection
with more than one member is possible
but the infinite collections
are determined by finite information
your polynomial ranges
are graded by this kolmogorov-type complexity
measured by degree
finite polynomials have finite complexity
but for any n+1 elements in a recursively enumerable collection
there is an entire collection of degree n
( indeed degree n-1 and lower are possible )
and therefore any
finite information recursively enumerable collection
can have finite intersection to any finite extent
with a polynomial collection
so to find nonpolynomial recursively enumerable sets
we must look to limit behavior
polynomials of complexity n
have constant delta^n
where deltas are from a canonical representation of the collection
ordered under the standard order of integers
note for integer polynomials of odd degree
thereExists an N suchThat
if x1, x2 are neighboring integers in this canonical
representation
|x1| > N, |x2| > N
then thereExists y1 elementOf Z
and thereExists p elementOf Z[x]
suchThat p(y1) = x1, p(y1+1) = x2
and for even degree you have two branches interleaved
so any recursively enumerable set
that doesn't have this property
of delta^n = C for some n
is not representable in your polynomial ranges
one can construct many
but one example is
{..., 1, 4, 6, 9, 11, 14, ...}
ie.
through finite information metadescriptions of delta patterns
any delta pattern that
does not become constant at finite iteration
is such a collection
..
this is very sloppy
i know
but i hope it seems a reasonable sketch
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
===
Subject: Re: sets definable by polynomials
On Thu, 12 Jul 2007 21:27:19 -0700, galathaea <galathaea@gmail.com[...]
i think we can give this a complexity-theoretic description
i hope you don't mind me starting with a clean slate
> quasi
>but i just want to reorganise the questions some
any polynomial of degree n is
> completely determined by n+1 points
after that
> the infinite other values are completely determined
> which has infinite range or is constant
>as you mentioned
so as you pointed out
> no finite enumerable collection
> with more than one member is possible
but the infinite collections
> are determined by finite information
your polynomial ranges
> are graded by this kolmogorov-type complexity
>measured by degree
finite polynomials have finite complexity
but for any n+1 elements in a recursively enumerable collection
>there is an entire collection of degree n
>( indeed degree n-1 and lower are possible )
and therefore any
> finite information recursively enumerable collection
>can have finite intersection to any finite extent
>with a polynomial collection
so to find nonpolynomial recursively enumerable sets
>we must look to limit behavior
polynomials of complexity n
> have constant delta^n
>where deltas are from a canonical representation of the collection
> ordered under the standard order of integers
note for integer polynomials of odd degree
> thereExists an N suchThat
> if x1, x2 are neighboring integers in this canonical
>representation
> |x1| > N, |x2| > N
> then thereExists y1 elementOf Z
> and thereExists p elementOf Z[x]
> suchThat p(y1) = x1, p(y1+1) = x2
>and for even degree you have two branches interleaved
so any recursively enumerable set
> that doesn't have this property
> of delta^n = C for some n
>is not representable in your polynomial ranges
one can construct many
> but one example is
>{..., 1, 4, 6, 9, 11, 14, ...}
ie.
> through finite information metadescriptions of delta patterns
any delta pattern that
>does not become constant at finite iteration
> is such a collection
..
this is very sloppy
> i know
>but i hope it seems a reasonable sketch
As your analysis demonstrates, sets represented by univariate
polynomials are quite restricted.
Thus, the analysis needs to be multivariate from the start.
A pr-subset of Z is allowed to be the range of a multivariate
polynomial.
A simple example is the polynomial f(x,y)=x^2+y^2.
Let S=f(Z^2).
Thus, S is a subset of Z, and by definition, S is a pr-set.
But S is not the range of a univariate polynomial -- at least 2
variables are needed to generate this set.
quasi
===
Subject: Re: sets definable by polynomials
<n8rb93tqhjj1qplr3lontbpc9o7k6b392o@4ax.com>
<b68e93pfs0nokclkp90eel9go9m7t0fa4j@4ax.com On Thu, 12 Jul 2007 21:27:19
-0700, galathaea <galath...@gmail.com >[...]
>i think we can give this a complexity-theoretic description
>i hope you don't mind me starting with a clean slate
> quasi
>but i just want to reorganise the questions some
>any polynomial of degree n is
> completely determined by n+1 points
>after that
> the infinite other values are completely determined
> which has infinite range or is constant
>as you mentioned
>so as you pointed out
> no finite enumerable collection
> with more than one member is possible
>but the infinite collections
> are determined by finite information
>your polynomial ranges
> are graded by this kolmogorov-type complexity
>measured by degree
>finite polynomials have finite complexity
>but for any n+1 elements in a recursively enumerable collection
>there is an entire collection of degree n
>( indeed degree n-1 and lower are possible )
>and therefore any
> finite information recursively enumerable collection
>can have finite intersection to any finite extent
>with a polynomial collection
>so to find nonpolynomial recursively enumerable sets
>we must look to limit behavior
>polynomials of complexity n
> have constant delta^n
>where deltas are from a canonical representation of the collection
> ordered under the standard order of integers
>note for integer polynomials of odd degree
> thereExists an N suchThat
> if x1, x2 are neighboring integers in this canonical
>representation
> |x1| > N, |x2| > N
> then thereExists y1 elementOf Z
> and thereExists p elementOf Z[x]
> suchThat p(y1) = x1, p(y1+1) = x2
>and for even degree you have two branches interleaved
>so any recursively enumerable set
> that doesn't have this property
> of delta^n = C for some n
>is not representable in your polynomial ranges
>one can construct many
> but one example is
>{..., 1, 4, 6, 9, 11, 14, ...}
>ie.
> through finite information metadescriptions of delta patterns
>any delta pattern that
>does not become constant at finite iteration
> is such a collection
>..
>this is very sloppy
> i know
>but i hope it seems a reasonable sketch
As your analysis demonstrates, sets represented by univariate
> polynomials are quite restricted.
Thus, the analysis needs to be multivariate from the start.
A pr-subset of Z is allowed to be the range of a multivariate
> polynomial.
A simple example is the polynomial f(x,y)=x^2+y^2.
Let S=f(Z^2).
Thus, S is a subset of Z, and by definition, S is a pr-set.
But S is not the range of a univariate polynomial -- at least 2
> variables are needed to generate this set.
yes but i wanna do the easy parts first
and drag my feet on the harder ones!!!
it's better if i make a fool of myself on easy things
because i will learn from it
if i jump into the harder ones
i will probably confuse myself about my inevitable errors
additive number theory is hard...
:l
well we have a similar tool like the delta
for the more multivariate case
called the shnirel'man density
to keep it simple for now
let me switch only to nonnegative sets
the density is defined as
sigma(X) = inf (#(X < n) / n)
n of N
given a collection X
call hX the collection of all numbers of the form
x1 + x2 + ... + xh (xi elementOf X)
X is called a basis of order h
if hX is all nonnegative numbers
ie. it is a finite basis if sigma(hX) = 1 for some h
i think it might be possible to prove
that any monic polynomial with 0 in its range is of finite order
by the waring problem
if thats true
then we can construct nonpolynomial collections
by choosing sufficiently sparse sets that are not of finite order
this is sufficient but i do not think it is necessary
one such nonpolynomial set is 2^n
which is not a basis of finite order
the number 111111111111111111111111111111111111111... in binary
shows we may need to add a bunch of them for some numbers
and there is no upper bound
..
okay
i'm done for tonight
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
===
Subject: Re: sets definable by polynomials
On Thu, 12 Jul 2007 11:55:48 -0700, galathaea <galathaea@gmail.com> 
Finally,
let's define pd-sets (polynomially definable sets) ...
> Here are the requirements:
> (1) Z is a pd-set.
> (2) If S is a pd-set, then so is f(S) for any polynomial map f.
> (3) If S is a pd-set, then so is f^(-1)(S) for any polynomial map f.
> (4) If S in Z^n and S is a pd-set, so is Z^nS.
> Moreover, we require that every pd-set be obtainable by a finite
> number of applications of (1), (2), (3), (4).
> Note: In this context, polynomial map means a map from Z^m->Z^n
> (where m,n are positive integers) such that the coordinate functions
> are polynomials with integer coefficients.
> Also, as before, every pd-zet is a subset of Z^n for some positive
> integer n, but different pd-sets need not live in the same Z^n.
>it seems
>the only pd sets that are recursively enumerable
> are the recursive sets
No, I don't think so.
It seems clear to me that every diophantine set is a pd-set.
Hence, since every recursively enumerable set is diophantine, it
follows that every recursively enumerable set is a pd-set.
However, as noted in my prior post, since the set of pd-sets is closed
under complementation (in a given Z^n), and since the same is not true
of the recursively enumerable sets, it follows that not every pd-set
is recursively enumerable.
quasi
===
Subject: Re: sets definable by polynomials
<n8rb93tqhjj1qplr3lontbpc9o7k6b392o@4ax.com>
<tsvd93p8glagemr87jvdch840m9aqm1a0n@4ax.com On Thu, 12 Jul 2007 11:55:48
-0700, galathaea <galath...@gmail.com
> Finally, let's define pd-sets (polynomially definable sets) ...
> Here are the requirements:
> (1) Z is a pd-set.
> (2) If S is a pd-set, then so is f(S) for any polynomial map f.
> (3) If S is a pd-set, then so is f^(-1)(S) for any polynomial map
f.
> (4) If S in Z^n and S is a pd-set, so is Z^nS.
> Moreover, we require that every pd-set be obtainable by a finite
> number of applications of (1), (2), (3), (4).
> Note: In this context, polynomial map means a map from Z^m->Z^n
> (where m,n are positive integers) such that the coordinate
functions
> are polynomials with integer coefficients.
> Also, as before, every pd-zet is a subset of Z^n for some positive
> integer n, but different pd-sets need not live in the same Z^n.
>it seems
>the only pd sets that are recursively enumerable
> are the recursive sets
No, I don't think so.
It seems clear to me that every diophantine set is a pd-set.
Hence, since every recursively enumerable set is diophantine, it
> follows that every recursively enumerable set is a pd-set.
However, as noted in my prior post, since the set of pd-sets is closed
> under complementation (in a given Z^n), and since the same is not true
> of the recursively enumerable sets, it follows that not every pd-set
> is recursively enumerable.
you know
this is why i've always hated impredicative definitions
they pretend to define things
that they haven't defined at all
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
===
Subject: Re: sets definable by polynomials
On Thu, 12 Jul 2007 12:41:21 -0400, David Bernier
> On Thu, 12 Jul 2007 11:26:43 -0400, David Bernier
>[...]
I've looked at the footnotes at:
> http://en.wikipedia.org/wiki/Diophantine_set#_note-1
> and it's still not clear to me that
> S diophantine ==> S a pr-set.
> In fact, now that I think about it, it's obvious that not every
> diophantine set is a pr-set.
> For example, let S be any finite subset of Z with more than one
> element. It's easy to see that S is diophantine. It's just as easy to
> see that S is not a pr-set since the range of a nonconstant polynomial
> is never finite, and the range of a constant polynomial is just a
> singleton set.
Right, that gives us a neat counterexample to the conjecture:
>If S is a diophantine set, then S is a pr-set.
(Eastern Time); but
>it's only 12:40 pm Eastern now...
I'm ahead of my time.
Actually, I think it's a bug in my newsreader. I am using an old
version of Agent. My system clock is correct.
quasi
===
Subject: #15 a working model of plasma physics turbulence and a good 
college
take home problem Re: Monograph-Book: FUSION BARRIER PRINCIPLE, Archimedes
Plutonium, Internet published 1997-2007
Last time I discussed FBP was in April. Today I made some progress and
insights into another working model
for the turbulence in a tokamak.
Let me first blurt out the problem for College physics students to
work on. I have a large 200 liter wheelbarrow
and I put inside it 120 liters of water. Now the trouble is that I
want to take the wheelbarrow of water a distance
on uneven ground and so the water sloshes back and forth in a wave of
disturbance. Now to minimize this
disturbance I place on top of the water four 12 liter plastic pails
that ride on top of the disturbance of water
and thus hopefully dampening the waves so no water sloshes out of the
wheelbarrow. Now the question I
have for College physics students as a problem for them to work out is
what configuration of those 4 pails
and how much water in those four pails to obtain maximum dampening of
water waves so I lose the
least amount of water?
Now here is the statement of the Fusion Barrier Principle:
Fusion Barrier Principle. Fission energy is the highest form of
energy that is able to be controlled and surpass breakeven. A Tokamak
such as JET or ITER can only reach 2/3 breakeven because 1/3 of the
input energy is forever lost in controlling the device.
Now the proof of the Fusion Barrier Principle follows from this logic:
The four Maxwell Equations are divided into two static and two dynamic
equations. The Coulomb force is a
static Maxwell Equation and the Faraday law is a dynamic Maxwell
Equation. Now the Coulomb law
is mathematically equivalent to a sphere and the Faraday Law is
equivalent to a cylinder. Now when you fit
a sphere into a cylinder the maximum surface area and maximum volume
is 2/3, which in mathematics
is called enclosing problems.
Now in physics, such as tokamak engineering fusion is attempted by
using the Coulomb force and to control
the plasma the tokamak uses Faraday's Law. So what happens is that
controlled fusion by any machine,
not just a tokamak, has to use either Faraday's Law or Ampere's Law to
control the fusion process and which
overcomes the Coulomb force. So essentially, all fusion machines
resolve into a case of enclosing a sphere
inside of a cylinder. Thus, all fusion machines that control fusion,
(the word control is essential), that
1/3 of all the input energy is irretrievably lost in the controlling
of the fusion events. Stated another way,
fusion machines can only reach a maximum of 2/3 breakeven.
In a fusion bomb, where the machine does not seek continuous control,
then breakeven is surpassed.
In the Sun and stars which are controlled machines, fusion reaches
only about 20% to 30% breakeven.
And alot of pupils of science have a tough time understanding that a
star fusion is so small compared
to the enormous inward energy of gravity as the controlling energy.
This means that the smallest fusion
machine that is controllable is the smallest fusion shining star. That
no machine ever built on Earth will
control and simultaneously reach breakeven. That all machines built on
Earth can only reach 2/3
breakeven and no more.
So what the working model above is an analogy to that of the FBP where
the water in the wheelbarrow
is the protons in a tokamak and the buckets or pails are controls over
the plasma protons. Think of the
turbulence in a tokamak.
So at what configuration of those buckets leads to the maximum
dampening of the water so that little
water is lost? Do we have the 2/3 and 1/3 ratio of buckets of water?
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
===
Subject: #16 tokamak fusion plasma guiding model problem Re: 
Monograph-Book:
FUSION BARRIER PRINCIPLE, Archimedes Plutonium, Internet published 
1997-2007
> Last time I discussed FBP was in April. Today I made some progress and
> insights into another working model
> for the turbulence in a tokamak.
Let me first blurt out the problem for College physics students to
> work on. I have a large 200 liter wheelbarrow
> and I put inside it 120 liters of water. Now the trouble is that I
> want to take the wheelbarrow of water a distance
> on uneven ground and so the water sloshes back and forth in a wave of
> disturbance. Now to minimize this
> disturbance I place on top of the water four 12 liter plastic pails
> that ride on top of the disturbance of water
> and thus hopefully dampening the waves so no water sloshes out of the
> wheelbarrow. Now the question I
> have for College physics students as a problem for them to work out is
> what configuration of those 4 pails
> and how much water in those four pails to obtain maximum dampening of
> water waves so I lose the
> least amount of water?
Now here is the statement of the Fusion Barrier Principle:
> Fusion Barrier Principle. Fission energy is the highest form of
> energy that is able to be controlled and surpass breakeven. A Tokamak
> such as JET or ITER can only reach 2/3 breakeven because 1/3 of the
> input energy is forever lost in controlling the device.
Now the proof of the Fusion Barrier Principle follows from this logic:
> The four Maxwell Equations are divided into two static and two dynamic
> equations. The Coulomb force is a
> static Maxwell Equation and the Faraday law is a dynamic Maxwell
> Equation. Now the Coulomb law
> is mathematically equivalent to a sphere and the Faraday Law is
> equivalent to a cylinder. Now when you fit
> a sphere into a cylinder the maximum surface area and maximum volume
> is 2/3, which in mathematics
> is called enclosing problems.
Now in physics, such as tokamak engineering fusion is attempted by
> using the Coulomb force and to control
> the plasma the tokamak uses Faraday's Law. So what happens is that
> controlled fusion by any machine,
> not just a tokamak, has to use either Faraday's Law or Ampere's Law to
> control the fusion process and which
> overcomes the Coulomb force. So essentially, all fusion machines
> resolve into a case of enclosing a sphere
> inside of a cylinder. Thus, all fusion machines that control fusion,
> (the word control is essential), that
> 1/3 of all the input energy is irretrievably lost in the controlling
> of the fusion events. Stated another way,
> fusion machines can only reach a maximum of 2/3 breakeven.
In a fusion bomb, where the machine does not seek continuous control,
> then breakeven is surpassed.
In the Sun and stars which are controlled machines, fusion reaches
> only about 20% to 30% breakeven.
> And alot of pupils of science have a tough time understanding that a
> star fusion is so small compared
> to the enormous inward energy of gravity as the controlling energy.
> This means that the smallest fusion
> machine that is controllable is the smallest fusion shining star. That
> no machine ever built on Earth will
> control and simultaneously reach breakeven. That all machines built on
> Earth can only reach 2/3
> breakeven and no more.
So what the working model above is an analogy to that of the FBP where
> the water in the wheelbarrow
> is the protons in a tokamak and the buckets or pails are controls over
> the plasma protons. Think of the
> turbulence in a tokamak.
So at what configuration of those buckets leads to the maximum
> dampening of the water so that little
> water is lost? Do we have the 2/3 and 1/3 ratio of buckets of water?
>
So let me clarify the generalized model. So as the body of water
inside the wheelbarrow
is perturbed by motion sets up a turbulence wave in the body of water.
If the disturbance is
large then the wave is large and some of the water is sloshed over the
sides and lost. What
the four buckets are there to place in a configuration so as to dampen
the wave in the water.
If the 4 buckets are resting on top of the water then the water wave
is little dampened.
But when the 4 buckets are somewhat submerged by filling them partly
with water, then
these buckets act as dampening of the water wave, partly because some
of the water
is now confined inside the walls of the buckets and partly because the
submerged buckets
dampen the standing wave of the wheelbarrow water.
The above is a model of turbulence in a tokamak and that in order to
achieve 100% breakeven
or to surpaass 100% is impossible because much of the input energy
goes to keeping
control. In the above model-analogy to reach 100% or more breakeven
would be to have a
lid built for the wheelbarrow where none of the water escapes.
Likewise in fusion to have
no perturbance of protons is impossible.
The answer to the above wheelbarrow model is that 1/3 of the water is
confined inside the
buckets and 2/3 is the wheelbarrow water. In that configuration the
perturbance of water
waves is kept to a minimum. So in a sense, 1/3 of the water is used to
dampen to a
minimum the remaining 2/3 of the water. So we again have the Coulomb
law
being controlled by the Faraday law.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
===
Subject: CEU Official Master on Computational Biotechnology (2007-2008)
San Pablo - CEU University organizes an Official Master on
Computational Biotechnology
for the academic course 2007/2008. For more information, please, visit
http://biolab.uspceu.com/CompBiotech
===
Subject: Re: How many handshakes.


> This post not CC'd by email
> On Wed, 11 Jul 2007 00:48:06 +1000, Peter Webb
>Actually, this handshake problem overlaps a whole lot of mathematical
>ideas.
G'day G'day Peter,
This is something I've been observing too. Most instances I was
> acquainted with were based on physical situations that generated the
> equation. With the handshake puzzle there wasn't this crutch to lean
> on. One had to apply mathematical strategies right from the start.
>Very early when you study calculus, you see another reason why it
works.
>(If you have, note that the nth derivative of x^n is n factorial, as is
>the
>last column of of the difference table for x^n).
Insightful.
I'm wondering how I can apply that snippet to other situations.
>Its a lot easier to work out the rules in the other direction. First note
>that the difference table for ax^n + bx^(n-1) +cx^(n-2) ... is the sum of
>the difference tables for each term separately (ie the difference table
>for
>ax^n plus the difference table for bx^(n-1) etc). This makes it easy to
>derive the expression for the whole table if you have the equation
>already.
>Another use for the handshake problem is you can simply observe the
>pattern,
>and prove the formula by induction, which you have probably already done.
>That's a common first demonstration of induction.
Tonight I typed up a couple of pages explaining Proof by induction
> using the handshake puzzle for a friend who has taken an interest in
> mathematics late in life.
Am I right in thinking proof by induction enables us to prove the
> validity of a formula we have already guessed or arrived at in some
> other way but doesn't help us derive the formula in the first place?
>
Yes.
>And its a nice introduction to permutations and combinations, with the
>rich
>theory they have.
That's a connection I haven't made yet. Could you give a simple
>
The number of handshakes is equal to the number of ways of selecting 2
people from n people if order isn't important, ie C(n,2) = n*(n-1)/2. The
same formula as you got by the other methods.
>Its great you like it - its a really interesting and simple problem, with
>lots of interesting answers.
I most certainly do. It's fascinating and fascination seems to be the
> introducing me to these possibilities.
>
I should also have mentioned the connection between the handshake problem
and graph theory.
http://en.wikipedia.org/wiki/Theorem_on_friends_and_strangers
might pique your interest even further. Just define a friend as somebody
you have shaken hands with and a stranger as somebody you haven't.
===
Subject: Re: proof of irrational numbers
CLAIM: Suppose you have two numbers, s and t, and you already know
> that one of them is irrational and, additionally, s^2 and t^2 are
> rational. Then s + t is irrational [if nonzero -wgd]
PROOF: Suppose that s is the irrational number and that s + t = q is
> rational. Then
> s^2 - t^2 = (s + t)(s - t) = q(s - t)
is rational. Therefore, since (s^2 - t^2) / q = s - t, we must have
> that s - t is rational. So, both s + t and s - t are rational. But
then (s + t) + (s - t) = 2s
is rational. So, 2s / 2 = s is rational. This is a contradiction.
CLEARER Field Q contains s^2-t^2 != 0, 2 != 0
--> s+t in Q <-> s-t in Q <-> s,t in Q
The proof simply shows that s+t is a primitive element for Q(s,t)
following the well-known proof of the PRIMITIVE ELEMENT THEOREM,
a viewpoint that I highlighted here in a prior post, namely
> If a and b are two prime numbers which are not equal
> show Q(/a, /b) = Q(/a + /b)
 suppose we start with Q and /a + /b. Cubing this quantity
> gives us (a+3b)/a + (3a+b)/b. Since a and b are unequal,
> it is easy to see that any field containing Q and /a+/b
> neccessarily contains /a and /b.
Yes, if field F has 2 independent F-linear combinations of /a, /b
you can solve for /a, /b in F. The Primitive Element Theorem
works that way, obtaining two such independent combinations by
Pigeonholing the infinite set F(/a + r /b), r in F, |F| = oo,
into the finitely many fields between F and F(/a, /b), e.g. see [1].
As I mentioned in another post, in this case it is simpler
to notice Q(/a + /b) contains the independent /a - /b since
a - b
/a - /b = ------- in Q(/a + /b)
/a + /b
--Bill Dubuque
[1] http://planetmath.org/encyclopedia/ProofOfPrimitiveElementTheorem2.html
http://planetmath.org/encyclopedia/PrimitiveElementTheorem.html
http://planetmath.org/encyclopedia/ProofOfPrimitiveElementTheorem.html
===
Subject: Re: proof of irrational numbers
On 13 Jul 2007 03:32:29 -0400, Bill Dubuque <wgd@nestle.csail.mit.edu> 
CLAIM:
Suppose you have two numbers, s and t, and you already know
> that one of them is irrational and, additionally, s^2 and t^2 are
> rational. Then s + t is irrational [if nonzero -wgd]
> PROOF: Suppose that s is the irrational number and that s + t = q is
> rational. Then
> s^2 - t^2 = (s + t)(s - t) = q(s - t)
> is rational. Therefore, since (s^2 - t^2) / q = s - t, we must have
> that s - t is rational. So, both s + t and s - t are rational. But
> then (s + t) + (s - t) = 2s
> is rational. So, 2s / 2 = s is rational. This is a contradiction.
CLEARER Field Q contains s^2-t^2 != 0, 2 != 0
> --> s+t in Q <-> s-t in Q <-> s,t in Q
The proof simply shows that s+t is a primitive element for Q(s,t)
>following the well-known proof of the PRIMITIVE ELEMENT THEOREM,
>a viewpoint that I highlighted here in a prior post, namely
> If a and b are two prime numbers which are not equal
> show Q(/a, /b) = Q(/a + /b)
> suppose we start with Q and /a + /b. Cubing this quantity
> gives us (a+3b)/a + (3a+b)/b. Since a and b are unequal,
> it is easy to see that any field containing Q and /a+/b
> neccessarily contains /a and /b.
Yes, if field F has 2 independent F-linear combinations of /a, /b
> you can solve for /a, /b in F. The Primitive Element Theorem
> works that way, obtaining two such independent combinations by
> Pigeonholing the infinite set F(/a + r /b), r in F, |F| = oo,
> into the finitely many fields between F and F(/a, /b), e.g. see [1].
As I mentioned in another post, in this case it is simpler
> to notice Q(/a + /b) contains the independent /a - /b since
a - b
> /a - /b = ------- in Q(/a + /b)
> /a + /b
>--Bill Dubuque
There are some high school students who are going to be competing in a
math competition this fall and this will be one of the problems.
I'll can almost bet that at least one of those students will figure
this out.
Some of those kids can really think if you give them the chance.
Brian
===
Subject: Re: Maplesoft Entertainment Inc. presents proudly the Worldwide
Maple 11 Roulette
---------------------------------------------------------------
$1895.00 only!
http://webstore.maplesoft.com/
---------------------------------------------------------------
# TEST CASE one of many 1000s
#
# Enjoy FOUR distinct outputs at random.
#
---------------------------------------------------------------
restart; evalf(int(ln(1+z^2)/(1+z^2+z^4), z= 0..infinity));
Error, (in assuming) when calling 'signum/... division by zero'
.3718816641+0.*I
.3718816648+0.*I
.3718816641-.737e-9*I
.3718816638+.5e-9*I
.3718816630+.329e-9*I
.3718816642-.37e-10*I
Error, (in assuming) when calling 'signum/... division by zero'
1.644934067
.3718816644+0.*I
.3718816634-.15e-8*I
1.644934067
Error, (in assuming) when calling 'signum/... division by zero'
.371881665+.4e-10*I
.3718816639-.18e-9*I
.3718816639+.11e-8*I
.3718816658-.397e-9*I
Error, (in assuming) when calling 'signum/... division by zero'
.3718816661+.3e-9*I
Error, (in assuming) when calling 'signum/... division by zero'
.3718816656-.137e-9*I
Error, (in assuming) when calling 'signum/... division by zero'
.3718816647-.5e-9*I
.3718816632+0.*I
.3718816652-.10e-8*I
.3718816643+0.*I
.3718816639+0.*I
Error, (in assuming) when calling 'signum/... division by zero'
Error, (in assuming) when calling 'signum/... division by zero'
Error, (in assuming) when calling 'signum/... division by zero'
Error, (in assuming) when calling 'signum/... division by zero'
.3718816637+0.*I
.3718816647+0.*I
.3718816649+0.*I
Error, (in assuming) when calling 'signum/... division by zero'
.3718816644+.5e-9*I
.3718816647-.5e-9*I
.3718816644-.5e-9*I
.3718816643+.234e-9*I
.3718816653-.46e-10*I
.371881663-.14e-9*I
.3718816644-.37e-10*I
.3718816650-.32e-9*I
Error, (in assuming) when calling 'signum/... division by zero'
.3718816641+.286e-9*I
.371881663-.23e-9*I
.371881663+.14e-9*I
Error, (in assuming) when calling 'signum/... division by zero'
.3718816653+0.*I
.3718816649+.26e-10*I
.3718816628+.1774e-8*I
.3718816644-.4e-9*I
Error, (in assuming) when calling 'signum/... division by zero'
.3718816651+0.*I
Error, (in assuming) when calling 'signum/... division by zero'
.3718816643+.1e-9*I
Error, (in assuming) when calling 'signum/... division by zero'
Error, (in signum) too many levels of recursion
Error, (in assuming) when calling 'signum/... division by zero'
.3718816640+.29e-10*I
Error, (in assuming) when calling 'signum/... division by zero'
.371881664-.24e-9*I
.3718816646-.537e-9*I
.3718816635-.6e-9*I
.3718816647-.1e-9*I
Error, (in assuming) when calling 'signum/... division by zero'
Error, (in assuming) when calling 'signum/... division by zero'
Error, (in assuming) when calling 'signum/... division by zero'
Error, (in assuming) when calling 'signum/... division by zero'
.3718816653+0.*I
.3718816650+0.*I
Error, (in assuming) when calling 'signum/... division by zero'
.3718816637+.71e-10*I
Error, (in assuming) when calling 'signum/... division by zero'
.3718816661+.4e-9*I
Error, (in signum) too many levels of recursion
.3718816642+.2e-10*I
.3718816633+0.*I
.3718816642+0.*I
.3718816641-.4e-9*I
.3718816633-.337e-9*I
Error, (in signum) too many levels of recursion
.3718816634+0.*I
.3718816643-.129e-9*I
Error, (in assuming) when calling 'signum/... division by zero'
.3718816635+.2e-9*I
.3718816632+0.*I
.371881663+.74e-9*I
.3718816654-.526e-9*I
.3718816641+0.*I
Error, (in assuming) when calling 'signum/... division by zero'
Error, (in assuming) when calling 'signum/... division by zero'
.3718816651+.1e-9*I
.3718816665+.12e-8*I
.3718816639-.1e-10*I
.3718816643+.26e-10*I
.3718816631+.329e-9*I
.3718816643+.1e-9*I
Error, (in assuming) when calling 'signum/... division by zero'
.3718816643-.1e-9*I
.3718816639+0.*I
---------------------------------------------------------------
$1895.00 only!
http://webstore.maplesoft.com/
---------------------------------------------------------------
Faites vos jeux!
Vladimir Bondarenko
VM and GEMM architect
Co-founder, CEO, Mathematical Director
http://www.cybertester.com/ Cyber Tester, LLC
http://maple.bug-list.org/ Maple Bugs Encyclopaedia
http://www.CAS-testing.org/ CAS Testing
===
Subject: Re: Maplesoft Entertainment Inc. presents proudly the Worldwide
Maple 11 Roulette
On Fri, 13 Jul 2007 00:55:09 -0700, Vladimir Bondarenko
>restart; evalf(int(ln(1+z^2)/(1+z^2+z^4), z= 0..infinity));
Error, (in assuming) when calling 'signum/... division by zero'
Even if you replace infinity by 1 it acts badly.
evalf(int(ln(1+z^2)/(1+z^2+z^4), z= 0..1));
Maple apparently ignores the evalf and gives a rather scary looking
symbolic result.
However, the workaround is easy. The following commands both give
correct answers:
evalf(Int(ln(1+z^2)/(1+z^2+z^4), z= 0..1));
evalf(Int(ln(1+z^2)/(1+z^2+z^4), z= 0..infinity));
quasi
===
Subject: Re: Maplesoft Entertainment Inc. presents proudly the Worldwide
Maple 11 Roulette
<4oge93p3mnd5aedn8laflsttom3hie417f@4ax.com>
Q> the workaround is easy.
I am afraid, this is not a workaround at all. I used evalf() to
just produce a compact output showing that there is a menacing
disease -- disease the long liver! -- of bad, invalid, false
randomness in the Maple engine.
Maple engine's output depends on the Moon's phase. We came
to Maplesoft NOT for numbers but for formulas.
Please let me add some clarifications.
Why millions persons use computer algebra systems?
Well, there are many reasons for it, but most probably it's
because, in contrast to piles of numbers showering from
THOUSANDS amateurish and professional numerical packages, a
computer algebra system can supply you with the highest math
aesthetics, one of the biggest math diamonds, with a - be in
awful fear, the numerical folks! - with Her Majesty FORMULA,
all hues of the rainbow over Her.
Not with a rank-and-file private, a pack of digits.
Thus, my message is about the full palsy in the symbolic mode
with maximize. Numeric calculation is trivial here, and it is
at the case neither here not there.
Let us imagine such a comparison.
You go to a shop (Maplesoft), and you see at the showcase a
yummy almond cake with custard and dusted with caster sugar,
the cat's whiskers (Maple, Help, ?maximize).
Then you enter the shop and ask the salesgirl the blonde job
about selling you this dainty bit, this very moment!
She agrees, and you hand her money (- Gotcha! - she adds
beneath her breath), and get the money payment receipt, and
a nice colorful parcel. Life *is* good! Purr-purr-purr...
Broiling with impatience, you order a cup of tea on the spot,
and are going to relish this wonder of higher cooking!!
You tear up the shrink-wrapper, and... and see... large dented
pieces of lump sugar fit to hammer nails in absence of a
hammer??!!
Strike me dead! - give you a cry - this is a piece of bamboozle!
Where on earth my pet yummy almond little cake with custard and
dusted with caster sugar?!
However, the belle gives you a chirp, Oh no, this sweet stuff
at our showcase is just a DEMO, this is why we put it into the
showcase (Maple Help, ?maximize). Actually, it does not work.
What we can offer you is the pack of lump sugar you already got.
- Okay, - say you hastily, - I want the refund.
Now the salesgirl sneers disdainfully.
Go to junior school, - she says between her teeth, - and learn
the ABC. Then, read our license agreement, carefully.
THERE IS NO REFUND. WE ARE NOT LIABLE FOR ANYTHING.
Take a load of this at last:
Maplesoft's guys are real programers.
1. Real programmers don't write specs.
2. Real programmers don't read manuals.
3. Reliance on a reference is a hallmark
of the novice and the coward.
4. Users should be grateful for whatever they get.
5. They are lucky to get any program at all.
----------------------------------------------------------------
He that hath ears to hear, let him hear.
Best wishes,
Vladimir Bondarenko
http://www.cybertester.com/
http://maple.bug-list.org/
http://www.CAS-testing.org/
P.S.
You ARE clever. You can beat this girl's trade show,
but this is another story to be told later.
> On Fri, 13 Jul 2007 00:55:09 -0700, Vladimir Bondarenko
>restart; evalf(int(ln(1+z^2)/(1+z^2+z^4), z= 0..infinity));
>Error, (in assuming) when calling 'signum/... division by zero'
Even if you replace infinity by 1 it acts badly.
evalf(int(ln(1+z^2)/(1+z^2+z^4), z= 0..1));
Maple apparently ignores the evalf and gives a rather scary looking
> symbolic result.
However, the workaround is easy. The following commands both give
> correct answers:
evalf(Int(ln(1+z^2)/(1+z^2+z^4), z= 0..1));
evalf(Int(ln(1+z^2)/(1+z^2+z^4), z= 0..infinity));
quasi
===
Subject: Re: Maplesoft Entertainment Inc. presents proudly the Worldwide
Maple 11 Roulette
<4oge93p3mnd5aedn8laflsttom3hie417f@4ax.com>
-------------------------------------------------
http://www.mapleprimes.com/forum/reproducibility
We do call these ordering problems. It can lead
to intermittent failure to produce any answer.
Sometimes it can lead to intermittently computing
an incorrect result.
Dave Linder
Mathematical Software, Maplesoft
-------------------------------------------------
$1895.00 only!
http://webstore.maplesoft.com/
-------------------------------------------------
# TEST CASE one of many 1000s
#
# Enjoy TWO distinct outputs at random.
#
restart: lcoeff(v*z - a*z);
-------------------------------------------------
# 1000 tests
-------------------------------------------------
OUTCOME TIMES
-------------------------------------------------
1 926
-1 74
-------------------------------------------------
$1895.00 only!
http://webstore.maplesoft.com/
-------------------------------------------------
Faites vos jeux!
Vladimir Bondarenko
VM and GEMM architect
Co-founder, CEO, Mathematical Director
http://www.cybertester.com/ Cyber Tester, LLC
http://maple.bug-list.org/ Maple Bugs Encyclopaedia
http://www.CAS-testing.org/ CAS Testing
===
Subject: Re: Maplesoft Entertainment Inc. presents proudly the Worldwide
Maple 11 Roulette
[Vladimir Bondarenko]
| restart: lcoeff(v*z - a*z);
|
| -------------------------------------------------
| # 1000 tests
| -------------------------------------------------
| OUTCOME TIMES
| -------------------------------------------------
| 1 926
| -1 74
So what? This is well documented in ?lcoeff:
- When x is not specified, the order of the indeterminates is given by
indets (more
specifically,
frontend(indets,[p],[`*`,`+`,`::`,constant,series,SDMPolynom,undefined]);).
In the multivariate case this ordering is session dependent.
SA
--
Stein Arild Strmme +47 55584825, +47 95801887
Universitetet i Bergen Fax: +47 55589672
Matematisk institutt, UiB http://math.uib.no/stromme/
Johs Brunsg 12, N-5008 BERGEN stromme@math.uib.no
===
Subject: Re: Maplesoft Entertainment Inc. presents proudly the Worldwide
Maple 11 Roulette
<4oge93p3mnd5aedn8laflsttom3hie417f@4ax.com>
<jxusl7sxx9v.fsf@math.uib.no>
SAS> So what? This is well documented in ?lcoeff:
Yes, it is well documented. But what Dave Linder,
Maplesoft, claims
http://www.mapleprimes.com/forum/reproducibility
We do call these ordering problems. It can lead
to intermittent failure to produce any answer.
Sometimes it can lead to intermittently computing
an incorrect result.
> [Vladimir Bondarenko]
| restart: lcoeff(v*z - a*z);
> |
> | -------------------------------------------------
> | # 1000 tests
> | -------------------------------------------------
> | OUTCOME TIMES
> | -------------------------------------------------
> | 1 926
> | -1 74
So what? This is well documented in ?lcoeff:
- When x is not specified, the order of the indeterminates is given by
indets (more
> specifically,
frontend(indets,[p],[`*`,`+`,`::`,constant,series,SDMPolynom,undefined]);).
> In the multivariate case this ordering is session dependent.
SA
> --
> Stein Arild Str?mme +47 55584825, +47 95801887
> Universitetet i Bergen Fax: +47 55589672
> Matematisk institutt, UiB http://math.uib.no/stromme/
> Johs Brunsg 12, N-5008 BERGEN stro...@math.uib.no
===
Subject: Re: Maplesoft Entertainment Inc. presents proudly the Worldwide
Maple 11 Roulette
[Vladimir Bondarenko]
|
| SAS> So what? This is well documented in ?lcoeff:
|
| Yes, it is well documented. But what Dave Linder,
| Maplesoft, claims
|
| http://www.mapleprimes.com/forum/reproducibility
|
| We do call these ordering problems. It can lead
| to intermittent failure to produce any answer.
| Sometimes it can lead to intermittently computing
| an incorrect result.
You are misquoting Linder, he did not end the sentence that way. Here
is the full paragraph :
These kind of effects discussed here can result in different
computation paths being taken, as has been noted in this thread by
a few people. We do call these ordering problems. It can lead to
intermittent failure to produce any answer. Sometimes it can lead
to intermittently computing an incorrect result, which is far more
serious. Ordering problems are not dismissed out of hand as being
unimportant.
The pages puts forward a rather good explanation of the problem, in my
view.
--
Stein Arild Strmme +47 55584825, +47 95801887
Universitetet i Bergen Fax: +47 55589672
Matematisk institutt, UiB http://math.uib.no/stromme/
Johs Brunsg 12, N-5008 BERGEN stromme@math.uib.no
===
Subject: Re: Maplesoft Entertainment Inc. presents proudly the Worldwide
Maple
11 Roulette
Besides that VB does not correctly cite Dave Lindner or does not
want to understand him, I find it really funny how VB handles
communication:
first he uses commands in obviously false manner and then talks
about something quite different ... like a trilling troll ...
===
Subject: Re: Maplesoft Entertainment Inc. presents proudly the Worldwide
Maple 11 Roulette
<4oge93p3mnd5aedn8laflsttom3hie417f@4ax.com>
<jxusl7sxx9v.fsf@math.uib.no>
<jxu3azsv2w5.fsf@math.uib.no>
<5fpqgaF3d9flkU1@mid.individual.net>
You see my way of quoting and comment it.
Why don't you see thousands Maple defects?
Why don't you comment this regression bug?
Maple 11> evalf(Int(1/(z^exp(I)+z^Pi), z=1..infinity));
-.5000000000
?
Best wishes,
Vladimir Bondarenko
http://www.cybertester.com/
http://maple.bug-list.org/
http://www.CAS-testing.org/
> Besides that VB does not correctly cite Dave Lindner or does not
> want to understand him, I find it really funny how VB handles
> communication:
first he uses commands in obviously false manner and then talks
> about something quite different ... like a trilling troll ...
===
Subject: Re: Maplesoft Entertainment Inc. presents proudly the Worldwide
Maple 11 Roulette
<4oge93p3mnd5aedn8laflsttom3hie417f@4ax.com>
<jxusl7sxx9v.fsf@math.uib.no>
<jxu3azsv2w5.fsf@math.uib.no>
SAS> The pages puts forward a rather good explanation
SAS> of the problem, in my view.
Yes, I agree with you. A nice discussion. In my view, especially
impressive comments belong to Prof. Jacques Carette who was for
long years a Waterloo Maple/Maplesoft top officer.
What's important, here Dave Linder admits in plain language that
there is a hereditary disease in ALL Maple versions (!) since at
least 1992 to 2007, that is during 25 years.
What Cyber Tester can add as QA engineers, over the last years
this false non-deterministic contagion propagates more and
more over Maple engine.
We are worried much as this could, being multiplied by other
unsolved development problems, gradually put the end to Maple
usefulness.
Best wishes,
Vladimir Bondarenko
VM and GEMM architect
Co-founder, CEO, Mathematical Director
http://www.cybertester.com/ Cyber Tester, LLC
http://maple.bug-list.org/ Maple Bugs Encyclopaedia
http://www.CAS-testing.org/ CAS Testing
> [Vladimir Bondarenko]
|
> | SAS> So what? This is well documented in ?lcoeff:
> |
> | Yes, it is well documented. But what Dave Linder,
> | Maplesoft, claims
> |
> |http://www.mapleprimes.com/forum/reproducibility
> |
> | We do call these ordering problems. It can lead
> | to intermittent failure to produce any answer.
> | Sometimes it can lead to intermittently computing
> | an incorrect result.
You are misquoting Linder, he did not end the sentence that way. Here
> is the full paragraph :
These kind of effects discussed here can result in different
> computation paths being taken, as has been noted in this thread by
> a few people. We do call these ordering problems. It can lead to
> intermittent failure to produce any answer. Sometimes it can lead
> to intermittently computing an incorrect result, which is far more
> serious. Ordering problems are not dismissed out of hand as being
> unimportant.
The pages puts forward a rather good explanation of the problem, in my
> view.
> --
> Stein Arild Str?mme +47 55584825, +47 95801887
> Universitetet i Bergen Fax: +47 55589672
> Matematisk institutt, UiB http://math.uib.no/stromme/
> Johs Brunsg 12, N-5008 BERGEN stro...@math.uib.no
===
Subject: Re: Maplesoft Entertainment Inc. presents proudly the Worldwide
Maple 11 Roulette
<4oge93p3mnd5aedn8laflsttom3hie417f@4ax.com>
<jxusl7sxx9v.fsf@math.uib.no>
<jxu3azsv2w5.fsf@math.uib.no>
....
What's important, here Dave Linder admits in plain language that
> there is a hereditary disease in ALL Maple versions (!) since at
> least 1992 to 2007, that is during 25 years.
What Cyber Tester can add as QA engineers, over the last years
> this false non-deterministic contagion propagates more and
> more over Maple engine.
>
has improved since I last visited it. It demonstrates rather
conclusively that people who know about Maple or want to know about
Maple can find more authoritative and informative data there. By
reading VB's posts here you might think the Maple technical people are
deaf; what you learn by visiting MaplePrimes is that they are quite
responsive. They just don't care to respond here, and who can blame
them, as long as VB tries to get in the record books for most content-
free irrelevant cross-postings.
2. Regarding the non-determinism of ordering. This has been an
issue discussed from the earliest days of the Maple design. Anyone
careful to SORT the terms before picking out (sight unseen) the 3rd
term of a result. There is an excellent discussion of the matter in
the MaplePrimes llink above. I would like to add only one point. I
have, several times, come up with algorithms for different tasks that
were substantially faster than the standard (in Mathematica, Maxima),
based on a novelty (for them) like hashcoding.
But Maple sometimes was faster yet. Why? Because it already used
hashing but did not have to sort the terms. Mathematica and Maxima
generally view a sum of 10,000 terms as something that must be
properly ordered to be considered simplified. Maple just blasts the
sum out using its internal order. Asking Maple to sort it properly
considerably slows down the process, so Maple is no longer fast. Of
course the original Maple sum IS not randomly ordered. It is just that
the ordering predicate is determined by a mechanism that is mostly
opaque to the user. Defenders of Maple have had to deal with this
issue since the original design, and seem resistant to changing it.
Mathematica also has what appears to be non-deterministic behavior,
apparently in some cases from caching results, and in other cases from
re-evaluating expressions because they were allocated space on the
same page of memory as others. It is unfortunate that in the
marketplace for software, speed tends to be the most prominent measure
of quality.
===
Subject: Re: Maplesoft Entertainment Inc. presents proudly the Worldwide
Maple 11 Roulette
<4oge93p3mnd5aedn8laflsttom3hie417f@4ax.com>
<jxusl7sxx9v.fsf@math.uib.no>
<jxu3azsv2w5.fsf@math.uib.noWhat's important, here Dave Linder admits in 
plain
language that
>there is a hereditary disease in ALL Maple versions (!) since at
>least 1992 to 2007, that is during 25 years.
Now who needs an accurate math package? :)
===
Subject: Re: Maplesoft Entertainment Inc. presents proudly the Worldwide
Maple 11 Roulette
<4oge93p3mnd5aedn8laflsttom3hie417f@4ax.com>
<jxusl7sxx9v.fsf@math.uib.no>
<jxu3azsv2w5.fsf@math.uib.no>
On Jul 13, 9:31 am, jmorr...@idirect.com <jmorr...@idirect.com>
JM> Now who needs an accurate math package? :)
A wildly bold hypothesis, The CAS customers??!!
On Jul 13, 9:31 am, jmorr...@idirect.com <jmorr...@idirect.com
>What's important, here Dave Linder admits in plain language that
>there is a hereditary disease in ALL Maple versions (!) since at
>least 1992 to 2007, that is during 25 years.
Now who needs an accurate math package? :)
===
Subject: Expected length of a random walk, counting only one end.
dimensional) random walk between 0 and n starting at k is k * (n - k)
steps.
Now I wonder what if I am not interested in the times when the random
walk ends at 0? That is what is the expected value of the length of
random walks that end at n?
Any answer, with our without proof would be very helpful!
Jonas Bjering
===
Subject: Re: Expected length of a random walk, counting only one end.
>dimensional) random walk between 0 and n starting at k is k * (n - k)
>steps.
[I started writing the text below some time ago. At the end I
found that much of it is silly - decided to post it anyway
to illustrate one way a person might try to answer the question.
I do think that the correct answer _is_ obtained below, but
there must be a much better way to get there...]
Hmm. Your statement is a little unclear - I can guess what the
missing details, let's see if I can prove the answer is correct,
in which case my guesses may be correct as well.
I'm guessing that we have a random walk staring at k, each
step is either plus or minus 1, each with probability 1/2,
and we're talking about the expected number of steps until
we reach 0 or n.
Let's say e(k) is this expected value. Then e(0) = e(n) = 0
and e(k) = [(1+e(k-1)) + (1+e(k+1)] / 2 for 0 < k < n. It's
not hard to see that those conditions determine e(k). So
the question is whether e(k) = k * (n - k) satisfies the
two conditions.
It's clear that e(0) = e(n) = 0. And
e(k-1) + e(k+2) = (k-1) * (n-k+1) + (k+1) * (n-k-1)
= 2k(n-k) - (n-k)+(k-1) + (n-k)-(k+1)
= 2 e(k) - 2.
Sure enough, that's right. So I imagine my guesses about
exactly what you meant by the problem statement are correct.
>Now I wonder what if I am not interested in the times when the random
>walk ends at 0? That is what is the expected value of the length of
>random walks that end at n?
Any answer, with our without proof would be very helpful!
Well let's see. First, we can simplify the notation by
solving an equivalent problem: We start at a point k >= 0
and ask for e(k), the expected number of steps until we
reach 0. (If e(k) is the answer to that question then
the answer to your question is e(|n-k|).)
There's a hidden assumption here, namely that we do at
some point reach 0. That's true with probability 1,
so in fact the number of steps is almost surely finite.
We still have e(0) = 0, and we still have the same
recurrence, which we can write as
(*) e(k+2) - 2e(k+1) + e(k) = -2.
To find the general solution to (*) we think about
standard methods for differential equations, and
start with the homogeneous equation
(**) e(k+2) - 2e(k+1) + e(k) = 0.
And we assume that a^k is a solution to (**). That
says
a^2 - 2a + 1 = 0,
which has a = 1 as a double root. Sure enough,
e_1(k) = k is a solution to (**). We need a second
linearly independent solution e_2 - too bad we got
a double root... Oh: e_2(k) = 1 is an obvious solution.
So the general solution to (**) is
e(k) = c + dk
for constants c and d. Which come to think of it
should have been obvious, sorry.
So what about (*)? The general solution would be
any solution plus the general solution to (**).
Again by analogy with DE it seems like (**) should
have a solution of the form Ak^2:
-2 = A(k+2)^2 - 2A(k+1)^2 + Ak^2
= 2A.
So the general solution to (**) is
e(k) = -k^2 + c + dk.
Since e(0) = 0 we have
e(k) = -k^2 + dk
for some constant d.
And no matter how we choose d this is negative for large k,
which is nonsense. I've looked for an error in the calculuations
without finding any. (The fact that k(n-k) _is_ a solution to
the recurrence shows that it's not just a sign error, it really
is -k^2, not k^2.) Could be there's an error I haven't found;
if not then there must be some hidden assumption that led
to the contradiction... Oh. While it's a fact that the number
of steps is almost surely finite that doesn't imply that it
has a finite expected value, which we've been assuming
throughout all this. If the expected value is infinite
then the steps above involving subtracting something from
both sides of an equation are invalid.
I bet, based on this roundabout reasoning, that in fact
the expected value is infinite. A more direct proof of this
would be better. (It seems like the fact that k(n-k)
tends to infinity as n -> infinity points in this direction,
but offhand I don't see how to use this fact to actually
prove that e(k) is infinite.)
>Jonas Bjering
************************
David C. Ullrich
===
Subject: ~ **DNA,RNA Related Bio Technology** ~
~ **DNA,RNA Related Bio Technology** ~
~ **DNA Related Biotechnology** ~
~ **DNA-protein interactions** ~
http://www.geocities.com/sanomexir
===
Subject: Using affine schemes to prove two rings are equal
Let A and B be two rings with A a subring of B, if the prime
spectrums of them are canonically homeomorphic,
i.e. Spec B --> Spec A , p |--> p / A is homeomorphic
and if their localizations with respect to the corresponding prime
ideals are also ring isomorphic
i.e. if p is in Spec B and q=p / A, then A/p --> B/q is an
isomorphism of domains
Im not if not sure if my logic is correct, but then Id like to say
that actually A and B are isomorphic as rings..
I thought that one could consider their affine schemes. The sheaf
spaces must be equal, no? The germs of the sheaves are equal (thats
what A/p = B/q is saying), their prime spectrum are equal.. one just
have to show that they have the same sheaf space,
For me a sheaf space is the triple (E,X, pi) , where X is a the prime
spectrum, E is the product of the germs (i.e.
prod_(Spec A ) A/p ) and pi is the local homeomorphism between E and
X (with some topology of E, making it so) bringing (e(p))_Spec A to p
problem is, I dont know if the topology of E coming from A and B are
the same ones... is it true, that if I just know the germs and the
prime spectrum of my affine scheme then I know the ring making the
affine scheme? how is this shown? Would it make any easier if I
consider A and B to have Krull dimension 0 (thus A/p is at once a
field)?
Jose Capco
===
Subject: Re: Using affine schemes to prove two rings are equal
On Jul 13, 12:43 pm, Jose Capco <cliomse...@kriocoucke.mailexpire.com Let A
and B be two rings with A a subring of B, if the prime
> spectrums of them are canonically homeomorphic,
i.e. Spec B --> Spec A , p |--> p / A is homeomorphic
and if their localizations with respect to the corresponding prime
> ideals are also ring isomorphic
i.e. if p is in Spec B and q=p / A, then A/p --> B/q is an
> isomorphism of domains
Im not if not sure if my logic is correct, but then Id like to say
> that actually A and B are isomorphic as rings..
I thought that one could consider their affine schemes. The sheaf
> spaces must be equal, no? The germs of the sheaves are equal (thats
> what A/p = B/q is saying), their prime spectrum are equal.. one just
> have to show that they have the same sheaf space,
For me a sheaf space is the triple (E,X, pi) , where X is a the prime
> spectrum, E is the product of the germs (i.e.
> prod_(Spec A ) A/p ) and pi is the local homeomorphism between E and
> X (with some topology of E, making it so) bringing (e(p))_Spec A to p
problem is, I dont know if the topology of E coming from A and B are
> the same ones... is it true, that if I just know the germs and the
> prime spectrum of my affine scheme then I know the ring making the
> affine scheme? how is this shown? Would it make any easier if I
> consider A and B to have Krull dimension 0 (thus A/p is at once a
> field)?
Jose Capco
Ok, I think I know how to show they are equal.. but its really messy
and informal I am hoping that there is a way to show this cleanly.. I
am like trying to say that sections of their schemes must be equal but
Im having a hard time formulating this as clean as possible.
===
Subject: Professor Haroon (Etymologist, Geologist & Author)
Professor Haroon Mustapha Leon
Etymologist, Geologist & Author
About the Author:
The Late Professor Haroon Mustapha Leon, M.A., Ph.D., LL.D., F.S.P.,
accepted Islam in 1882. He was a Fellow and Honorary Member of many
learned societies in Europe and America. He was an able Philologist,
Etymology of the Man's Language to the 'Isle of Man Examiner'. His
services to this important branch of science had frequently been
recognised by learned bodies. The Potomac University (U.S.A.)
conferred upon him the degree of M.A. Dr. Leon was also an earnest
geologist. He frequently lectured on scientific and literary subjects
before learned and other societies. He occupied the important position
of Secretaire-General of La Societe Internationale de Philologie,
Sciences et Beaux-Arts (founded 1875) and was the Editor of The
Philomathe a scientific magazine, published from London. Dr. Leon
received many decorations from Sultan Abdul Hamid Khan, the late Shah,
and the Emperor of Austria.
One of the glories of Islam is that it is founded upon reason, and
that it never demands from its followers an abnegation of that
important mental faculty. Unlike certain other faiths, which insist
upon their votaries implicitly accepting certain dogmas without
independent inquiry, but simply on the authority of The Church,
Islam courts inquiry and counsels its disciples to study, search and
investigate prior to acceptation. The Holy Prophet, of ever-blessed
memory, said:
Allah hath not created anything better than reason, the benefits
which Allah giveth are on its account, and understanding is begotten
of it.
On another occasion he said:
Verily, I tell you, a man may have performed prayers, fasts, charity,
pilgrimage and all other good works, but he will not be rewarded but
by the manner in which he hath used and applied his reason.
The parable of the 'Talents' narrated by Saiyiddena 'Issa', i.e. Jesus
(on whom be peace) is in strict accordance with Islamic doctrine, as
also is the maxim: 'Prove all things; hold fast to that which is
good.' The similitude of those who follow blindly and who neglect to
use the intelligence which the Divine Giver, of all good, hath
bestowed upon them, is declared in the imperishable pages of Al-Qur'an
( Sura 52: Al-Jumm'a - 'The Assembly') to be that of 'an ass laden
with books.'
The noble and learned Caliph, Hazrat Ali (on whom be peace) said:
The world is darkness; knowledge is light; but knowledge without
truth is a mere shadow.
Muslims believe that Islam is a term synonymous with truth, and that
under the glorious and ever-brilliant sun of Islam, by the light of
reason and knowledge, truth can be obtained but in order to obtain
that knowledge, and thus attain that truth, man must use his reasoning
faculties.
A most poignant pronouncement on this question was given by our Holy
Prophet only a few days prior to his decease.
There he lay, the last and greatest of the grand chain of mighty men
whom Allah, in His everlasting mercy and compassion, had sent to the
world as inspired messenger of truth and of righteousness, his saintly
head pillowed upon Hazrat 'Ayesha's loving knee.
The true believers of Medina, old and young, men and women -- nay,
even the children -- had gathered, in loving sympathy there around the
mat whereon lay Mustapha Al-amin, the chosen, the faithful, ar-Rasul-
Allah. Tears glistened in their eyes, and coursed down the cheeks of
even the most grizzled and valiant of the veteran warriors of Islam.
Their leader, their friend, their beloved pastor, and, above all,
their Prophet, he who had led them from the darkness of ignorance and
superstition into the radiant brightness of the truth, had brought
them into Islam, the habitation of peace, was about to pass from them.
No wonder, then, that their eyes became fountain of tears, and their
hearts were heavy and oppressed.
In the agony of distress, almost of despair, one exclaimed: O Prophet
of Allah, thou art ill, thou mayest die, then what is to become of
us?
You have Al-Qur'an said Allah's Messenger.
Oh, yes, Rasul-Allah, but even with that enlightening book and
unerring guide before us, we have had at times to ask from you advice,
counsel, and instruction, and if you are taken from us, O Prophet who
is there to be our guide? said the companions.
Do as I did and as I have said, was the reply.
But, O Prophet, after you have gone fresh circumstances may arise
which could not have arisen during thy blessed lifetime; what are we
to do then? And what are they to do who follow us?
The Prophet slowly raised his illustrious and saintly head, and with
the lurid light of prophecy and inspiration shining radiantly from his
noble eyes exclaimed: Allah hath given to every man as a personal
monitor, a conscience and as a guide, his reason; then, use them in
respect of all things and Allah's blessing will ever guide you
aright.
===
Subject: Apparel Magazine Article
Integrated IT Solutions for CMAI's Apparel Magazine, June 2007. More
Details click : http://www.reach-tech.com/news_apparelmagazine.htm
===
Subject: Re: Possible results from three variables
> On Thu, 12 Jul 2007 07:25:37 -0000,
> bakpao@gmail.com
Given three variables, say x, y, z with each
> variable being an integer
>from 1 to 10, how many possible values are there
> from the equation of
>x * y * z? The quickest and incorrect answer is 1000
> (from 10x10x10).
>This is wrong because some of the combinations
> actually the same, e.g.
>1*2*4 = 1*4*2 = 4*1*2 = 1*8*1. Doing 10C3 is also
> wrong.
>Is there any formula that we can use to find out the
> number of
>possibilities for the different integer range (e.g.
> 1 - 5 or 1 - 8)?
For positive integers n,k, let f(n,k) be the number
> of possible
> products of k integers from the range 1 to n
> inclusive.
It's obvious that f(1,k)=1 for all positive integers
> k.
It's also obvious that f(n,1)=n for all positive
> integers n,
For a fixed positive integer n, it appears that
> f(n,k) is always a
> polynomial in k.
However, for a fixed positive integer k, with the
> exception of k=1, it
> appears that f(n,k) is _not_ a polynomial in n. This
> seems somewhat
> mysterious.
Here are some of the formulas for f(n,k), for n from
> 1 to 10:
f(1,k) = 1
> f(2,k) = k + 1
> f(3,k) = ((k + 1)(k + 2))/2
> f(4,k) = (k + 1)^2
> f(5,k) = ((k + 1)(k + 2)(2 k + 3))/6
> f(6,k) = ((k + 1)^2 (k + 2))/2
> f(7,k) = ((k + 1)(k + 2)(k + 3)(3 k + 4))/24
> f(8,k) = ((k + 1)^2 (k + 2)(k + 3))/6
> f(9,k) = ((k + 1)^2 (k + 2)^2)/4
> f(10,k) = ((k + 1)^2 (k + 2) (2 k + 3))/6
Note that for the original example, with n=10, k=3,
> the formula gives
f(10,3) = ((4^2)(5)(9))/6 = 120
which agrees with the value obtained by a brute force
> count.
For n=1 and n=2, the formulas are easily proved, but
> in general, I
> don't have proofs.
It's clear from the data that as n increases, the
> degree of f(n,k) in
> k increases, but at slower rates. More data would be
> needed to
> estimate how the degree in k depends on n.
However, there's something else ...
I noticed the following fascinating subpattern:
f(2,k) = sum(i^0, i=1..k+1)
f(3,k) = sum(i^1, i=1..k+1)
f(5,k) = sum(i^2, i=1..k+1)
f(9,k) = sum(i^3, i=1..k+1)
I have no idea what to make of the above. Perhaps
> it's an accident,
> but it's definitely surprising.
quasi
i think there might be relationship with the euler numbers.
since they have combinatorical meaning in a similar way to the original
question here.
and also they are related to the polynomials you give.
and also some kind of Ehrhart polynomials (related), but ill stick with
euler at first sight.
tommy1729
===
Subject: Re: Possible results from three variables
<jhcc93lp4fnhgcl6dl20427smblh7du2a3@4ax.com On Thu, 12 Jul 2007 07:25:37
-0000, bak...@gmail.com
>Given three variables, say x, y, z with each variable being an integer
>from 1 to 10, how many possible values are there from the equation of
>x * y * z? The quickest and incorrect answer is 1000 (from 10x10x10).
>This is wrong because some of the combinations actually the same, e.g.
>1*2*4 = 1*4*2 = 4*1*2 = 1*8*1. Doing 10C3 is also wrong.
>Is there any formula that we can use to find out the number of
>possibilities for the different integer range (e.g. 1 - 5 or 1 - 8)?
For positive integers n,k, let f(n,k) be the number of possible
> products of k integers from the range 1 to n inclusive.
It's obvious that f(1,k)=1 for all positive integers k.
It's also obvious that f(n,1)=n for all positive integers n,
For a fixed positive integer n, it appears that f(n,k) is always a
> polynomial in k.
However, for a fixed positive integer k, with the exception of k=1, it
> appears that f(n,k) is _not_ a polynomial in n. This seems somewhat
> mysterious.
Here are some of the formulas for f(n,k), for n from 1 to 10:
f(1,k) = 1
> f(2,k) = k + 1
> f(3,k) = ((k + 1)(k + 2))/2
> f(4,k) = (k + 1)^2
> f(5,k) = ((k + 1)(k + 2)(2 k + 3))/6
> f(6,k) = ((k + 1)^2 (k + 2))/2
> f(7,k) = ((k + 1)(k + 2)(k + 3)(3 k + 4))/24
> f(8,k) = ((k + 1)^2 (k + 2)(k + 3))/6
> f(9,k) = ((k + 1)^2 (k + 2)^2)/4
> f(10,k) = ((k + 1)^2 (k + 2) (2 k + 3))/6
I was wondering how you are deriving your f(m,k).
In the case of f(5,k) ,say, would you be associating
1,2,3,4,5 with the polynomial ((1 +w+w^2) + x+y)^k
=(P+Q+R)^k and then using various relations
such as P^m =2m+1, Q^n = 1, etc.
to determine the number of distinct terms in the expansion
=(P+Q+R)^k = P^k + Q^k +R^k + P^(k-1)Q + P^(k-1)R + ?
===
Subject: Re: Possible results from three variables
On Fri, 13 Jul 2007 03:22:59 -0700, sttscitrans@tesco.net
> On Thu, 12 Jul 2007 07:25:37 -0000, bak...@gmail.com
>Given three variables, say x, y, z with each variable being an integer
>from 1 to 10, how many possible values are there from the equation of
>x * y * z? The quickest and incorrect answer is 1000 (from 10x10x10).
>This is wrong because some of the combinations actually the same, e.g.
>1*2*4 = 1*4*2 = 4*1*2 = 1*8*1. Doing 10C3 is also wrong.
>Is there any formula that we can use to find out the number of
>possibilities for the different integer range (e.g. 1 - 5 or 1 - 8)?
> For positive integers n,k, let f(n,k) be the number of possible
> products of k integers from the range 1 to n inclusive.
> It's obvious that f(1,k)=1 for all positive integers k.
> It's also obvious that f(n,1)=n for all positive integers n,
> For a fixed positive integer n, it appears that f(n,k) is always a
> polynomial in k.
> However, for a fixed positive integer k, with the exception of k=1, it
> appears that f(n,k) is _not_ a polynomial in n. This seems somewhat
> mysterious.
> Here are some of the formulas for f(n,k), for n from 1 to 10:
> f(1,k) = 1
> f(2,k) = k + 1
> f(3,k) = ((k + 1)(k + 2))/2
> f(4,k) = (k + 1)^2
> f(5,k) = ((k + 1)(k + 2)(2 k + 3))/6
> f(6,k) = ((k + 1)^2 (k + 2))/2
> f(7,k) = ((k + 1)(k + 2)(k + 3)(3 k + 4))/24
> f(8,k) = ((k + 1)^2 (k + 2)(k + 3))/6
> f(9,k) = ((k + 1)^2 (k + 2)^2)/4
> f(10,k) = ((k + 1)^2 (k + 2) (2 k + 3))/6
I was wondering how you are deriving your f(m,k).
Brute force count followed by interpolation.
>In the case of f(5,k) ,say, would you be associating
>1,2,3,4,5 with the polynomial ((1 +w+w^2) + x+y)^k
>=(P+Q+R)^k and then using various relations
>such as P^m =2m+1, Q^n = 1, etc.
>to determine the number of distinct terms in the expansion
>=(P+Q+R)^k = P^k + Q^k +R^k + P^(k-1)Q + P^(k-1)R + ?
I thought about this idea, but didn't try it.
quasi
===
Subject: Re: Possible results from three variables
On Thu, 12 Jul 2007 07:25:37 -0000, bakpao@gmail.com
>Given three variables, say x, y, z with each variable being an integer
>from 1 to 10, how many possible values are there from the equation of
>x * y * z? The quickest and incorrect answer is 1000 (from 10x10x10).
>This is wrong because some of the combinations actually the same, e.g.
>1*2*4 = 1*4*2 = 4*1*2 = 1*8*1. Doing 10C3 is also wrong.
Is there any formula that we can use to find out the number of
>possibilities for the different integer range (e.g. 1 - 5 or 1 - 8)?
Computational challenge:
How many possible values are there for the product of 100 positive
integers, each in the range 1 to 100 inclusive?
Note that the answer itself is not that big -- it's clearly less than
200 decimal digits. Thus, the problem is not presenting the number,
rather the problem is how to find it in a reasonable amount of time --
for example, within 100 years.
quasi
===
Subject: Re: Possible results from three variables
>On Thu, 12 Jul 2007 07:25:37 -0000, bakpao@gmail.com
>Given three variables, say x, y, z with each variable being an integer
>from 1 to 10, how many possible values are there from the equation of
>x * y * z? The quickest and incorrect answer is 1000 (from 10x10x10).
>This is wrong because some of the combinations actually the same, e.g.
>1*2*4 = 1*4*2 = 4*1*2 = 1*8*1. Doing 10C3 is also wrong.
>Is there any formula that we can use to find out the number of
>possibilities for the different integer range (e.g. 1 - 5 or 1 - 8)?
Computational challenge:
How many possible values are there for the product of 100 positive
>integers, each in the range 1 to 100 inclusive?
Note that the answer itself is not that big -- it's clearly less than
>200 decimal digits. Thus, the problem is not presenting the number,
>rather the problem is how to find it in a reasonable amount of time --
>for example, within 100 years.
My estimate for the answer is 43 decimal digits.
I doubt that anyone will ever be able to find it.
quasi
===
Subject: Re: Possible results from three variables
> On Thu, 12 Jul 2007 07:25:37 -0000, bakpao@gmail.com
Given three variables, say x, y, z with each variable being an integer
>from 1 to 10, how many possible values are there from the equation of
>x * y * z? The quickest and incorrect answer is 1000 (from 10x10x10).
>This is wrong because some of the combinations actually the same, e.g.
>1*2*4 = 1*4*2 = 4*1*2 = 1*8*1. Doing 10C3 is also wrong.
>Is there any formula that we can use to find out the number of
>possibilities for the different integer range (e.g. 1 - 5 or 1 - 8)?
For positive integers n,k, let f(n,k) be the number of possible
> products of k integers from the range 1 to n inclusive.
It's obvious that f(1,k)=1 for all positive integers k.
It's also obvious that f(n,1)=n for all positive integers n,
For a fixed positive integer n, it appears that f(n,k) is always a
> polynomial in k.
Very interesting...
You can think of it this way. Suppose p_1, ..., p_m are the primes
<= n. Each possible product can be represented uniquely in the form
y = p_1^y_1 ... p_m^y_m where y_1 ... y_m are nonnegative integers. If
a_{ix} is the power of p_i appearing in the representation of x and
y = 1^{x_1} 2^{x_2} ... n^{x_n} is one of the products we have
p_i = sum_{j=1}^n a_{ij} x_j with sum_{i=1}^n x_i = k. Thus f(n,k) is
the cardinality of T(K_k) where T is the linear map of R^n to R^m
corresponding to the m x n matrix (a_{ij}) and
K_k = {(x_1,...,x_n) in Z^n: all x_i >= 0, sum_i x_i = k}.
Essentially we're trying to count the lattice points in the dilations of a
fixed polytope.
A quick search turns up Ehrhart polynomials
<http://mathworld.wolfram.com/EhrhartPolynomial.html>...
--
Robert Israel israel@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada V6T 1Z2
===
Subject: Re: Possible results from three variables
On Thu, 12 Jul 2007 19:23:59 -0500, Robert Israel
> On Thu, 12 Jul 2007 07:25:37 -0000, bakpao@gmail.com
>Given three variables, say x, y, z with each variable being an integer
>from 1 to 10, how many possible values are there from the equation of
>x * y * z? The quickest and incorrect answer is 1000 (from 10x10x10).
>This is wrong because some of the combinations actually the same, e.g.
>1*2*4 = 1*4*2 = 4*1*2 = 1*8*1. Doing 10C3 is also wrong.
>Is there any formula that we can use to find out the number of
>possibilities for the different integer range (e.g. 1 - 5 or 1 - 8)?
> For positive integers n,k, let f(n,k) be the number of possible
> products of k integers from the range 1 to n inclusive.
> It's obvious that f(1,k)=1 for all positive integers k.
> It's also obvious that f(n,1)=n for all positive integers n,
> For a fixed positive integer n, it appears that f(n,k) is always a
> polynomial in k.
Very interesting...
You can think of it this way. Suppose p_1, ..., p_m are the primes
><= n. Each possible product can be represented uniquely in the form
>y = p_1^y_1 ... p_m^y_m where y_1 ... y_m are nonnegative integers. If
>a_{ix} is the power of p_i appearing in the representation of x and
>y = 1^{x_1} 2^{x_2} ... n^{x_n} is one of the products we have
>p_i = sum_{j=1}^n a_{ij} x_j with sum_{i=1}^n x_i = k. Thus f(n,k) is
>the cardinality of T(K_k) where T is the linear map of R^n to R^m
>corresponding to the m x n matrix (a_{ij}) and
>K_k = {(x_1,...,x_n) in Z^n: all x_i >= 0, sum_i x_i = k}.
>Essentially we're trying to count the lattice points in the dilations of a
>fixed polytope.
>A quick search turns up Ehrhart polynomials
><http://mathworld.wolfram.com/EhrhartPolynomial.html>...
Yes -- the interpretation as a count of lattice points makes sense.
The volume could probably be estimated by Monte Carlo methods and
that, together with some analysis of the eccentricity of the
boundary, might at least lead to good estimates.
Alternatively, the interpretation you describe can perhaps allow the
problem to be recast as an integer programming problem. Then again,
maybe not. We are not trying for feasibility or optimization. We want
to count the number of feasible solutions. Thus, integer programming
methods may not help -- I'm not sure.
quasi
===
Subject: Re: Possible results from three variables
>On Thu, 12 Jul 2007 07:25:37 -0000, bakpao@gmail.com
>Given three variables, say x, y, z with each variable being an integer
>from 1 to 10, how many possible values are there from the equation of
>x * y * z? The quickest and incorrect answer is 1000 (from 10x10x10).
>This is wrong because some of the combinations actually the same, e.g.
>1*2*4 = 1*4*2 = 4*1*2 = 1*8*1. Doing 10C3 is also wrong.
>Is there any formula that we can use to find out the number of
>possibilities for the different integer range (e.g. 1 - 5 or 1 - 8)?
For positive integers n,k, let f(n,k) be the number of possible
>products of k integers from the range 1 to n inclusive.
It's obvious that f(1,k)=1 for all positive integers k.
It's also obvious that f(n,1)=n for all positive integers n,
For a fixed positive integer n, it appears that f(n,k) is always a
>polynomial in k.
However, for a fixed positive integer k, with the exception of k=1, it
>appears that f(n,k) is _not_ a polynomial in n. This seems somewhat
>mysterious.
Here are some of the formulas for f(n,k), for n from 1 to 10:
f(1,k) = 1
> f(2,k) = k + 1
> f(3,k) = ((k + 1)(k + 2))/2
> f(4,k) = (k + 1)^2
> f(5,k) = ((k + 1)(k + 2)(2 k + 3))/6
> f(6,k) = ((k + 1)^2 (k + 2))/2
> f(7,k) = ((k + 1)(k + 2)(k + 3)(3 k + 4))/24
> f(8,k) = ((k + 1)^2 (k + 2)(k + 3))/6
> f(9,k) = ((k + 1)^2 (k + 2)^2)/4
> f(10,k) = ((k + 1)^2 (k + 2) (2 k + 3))/6
Note that for the original example, with n=10, k=3, the formula gives
f(10,3) = ((4^2)(5)(9))/6 = 120
which agrees with the value obtained by a brute force count.
For n=1 and n=2, the formulas are easily proved, but in general, I
>don't have proofs.
It's clear from the data that as n increases, the degree of f(n,k) in
>k increases, but at slower rates. More data would be needed to
>estimate how the degree in k depends on n.
However, there's something else ...
I noticed the following fascinating subpattern:
f(2,k) = sum(i^0, i=1..k+1)
f(3,k) = sum(i^1, i=1..k+1)
f(5,k) = sum(i^2, i=1..k+1)
f(9,k) = sum(i^3, i=1..k+1)
I have no idea what to make of the above. Perhaps it's an accident,
>but it's definitely surprising.
Ok, it appears that the power sum relations, as noted above, really
are some kind of weird accident. It never happens again -- in fact, I
believe I can prove that.
Based on my prior reply, I'll make a number of claims. The ones I
can't prove are labeled as conjectures ...
Proposition (1) [trivial]:
For each fixed positive integer k, f(n,k) is strictly increasing, as a
function of n.
For each fixed positive integer n>1, f(n,k) is strictly increasing, as
a function of k.
Proposition (2) [trivial]:
For all positive integers k, f(1,k)=1.
For all positive integers k, f(2,k)=k+1.
For all positive integers n, f(n,1)=n.
Conjecture (3) [I haven't tried this but I think it should not be too
hard to prove]:
For all positive integers k, f(3,k)=((k + 1)(k + 2))/2
Conjecture (4) [mysterious, possibly very hard]:
For each fixed positive integer n, f(n,k) is a polynomial in k.
Conjecture (5) [I think I can prove this]:
For each fixed positive integer k>1, f(n,k) is _not_ a polynomial in
n.
Conjecture (6):
For each fixed positive integer n>1, the polynomial f(n,k) has a
factor of k+1 in Q[k].
Conjecture (7):
For each fixed positive integer n>4, the polynomial f(n,k) has a
factor of (k+1)*(k+2) in Q[k].
Conjecture (8) [I think I can prove this]:
For all positive integers n, with the known exception of n=2 and the
possible exceptions of n=3,5,9, there does not exist a nonnegative
integer r such that, for all positive integers k, f(n,k)=sum(i^r,
i=1..k+1).
quasi
===
Subject: Re: Possible results from three variables
>On Thu, 12 Jul 2007 07:25:37 -0000, bakpao@gmail.com
>Given three variables, say x, y, z with each variable being an integer
>from 1 to 10, how many possible values are there from the equation of
>x * y * z? The quickest and incorrect answer is 1000 (from 10x10x10).
>This is wrong because some of the combinations actually the same, e.g.
>1*2*4 = 1*4*2 = 4*1*2 = 1*8*1. Doing 10C3 is also wrong.
Is there any formula that we can use to find out the number of
>possibilities for the different integer range (e.g. 1 - 5 or 1 - 8)?
>For positive integers n,k, let f(n,k) be the number of possible
>products of k integers from the range 1 to n inclusive.
Based on my prior reply, I'll make a number of claims. The ones I
>can't prove are labeled as conjectures ...
Proposition (1) [trivial]:
For each fixed positive integer k, f(n,k) is strictly increasing, as a
>function of n.
For each fixed positive integer n>1, f(n,k) is strictly increasing, as
>a function of k.
Proposition (2) [trivial]:
For all positive integers k, f(1,k)=1.
For all positive integers k, f(2,k)=k+1.
For all positive integers n, f(n,1)=n.
Conjecture (3) [I haven't tried this but I think it should not be too
>hard to prove]:
For all positive integers k, f(3,k)=((k + 1)(k + 2))/2
Conjecture (4) [mysterious, possibly very hard]:
For each fixed positive integer n, f(n,k) is a polynomial in k.
Conjecture (5) [I think I can prove this]:
For each fixed positive integer k>1, f(n,k) is _not_ a polynomial in
>n.
Conjecture (6):
For each fixed positive integer n>1, the polynomial f(n,k) has a
>factor of k+1 in Q[k].
Conjecture (7):
For each fixed positive integer n>4, the polynomial f(n,k) has a
>factor of (k+1)*(k+2) in Q[k].
Conjecture (8) [I think I can prove this]:
For all positive integers n, with the known exception of n=2 and the
>possible exceptions of n=3,5,9, there does not exist a nonnegative
>integer r such that, for all positive integers k, f(n,k)=sum(i^r,
>i=1..k+1).
Here's another property ...
Proposition (9) [easily proved]:
For any positive integer n>1, and any positive integer k,
f(n,k) <= 1 + sum (f(n-1,j), j=1..k)
with equality if and only if n is prime.
quasi
===
Subject: Re: Possible results from three variables
<bakpao@gmail.com> skrev i melding
> Given three variables, say x, y, z with each variable being an integer
> from 1 to 10, how many possible values are there from the equation of
> x * y * z? The quickest and incorrect answer is 1000 (from 10x10x10).
> This is wrong because some of the combinations actually the same, e.g.
> 1*2*4 = 1*4*2 = 4*1*2 = 1*8*1. Doing 10C3 is also wrong.
> Is there any formula that we can use to find out the number of
> possibilities for the different integer range (e.g. 1 - 5 or 1 - 8)?
In Maple:
a:= n -> nops({seq(seq(seq(x*y*z,x=1..n),y=1..n),z=1..n)});
> Hi.
(1*2*3*4*5*6*7*8*9*10) / (1*2*3*1*2*3*4*5*6*7) = 120
Generally: (n!) / (m! * (n-m)!)
It's a coincidence that (10 choose 3) is the right answer. For integers
1 to 11 the answer would be 173, not (11 choose 3) = 165.
See sequence A027425 in the On-Line Encyclopedia of Integer Sequences,
<http://www.research.att.com/~njas/sequences/A027425>. I don't think
there's a closed form for a(n) or a closed form generating function or
recurrence.
--
Robert Israel israel@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada V6T 1Z2
===
Subject: Re: Possible results from three variables
On Thu, 12 Jul 2007 17:23:43 -0500, Robert Israel
> <bakpao@gmail.com> skrev i melding
> Given three variables, say x, y, z with each variable being an integer
> from 1 to 10, how many possible values are there from the equation of
> x * y * z? The quickest and incorrect answer is 1000 (from 10x10x10).
> This is wrong because some of the combinations actually the same, e.g.
> 1*2*4 = 1*4*2 = 4*1*2 = 1*8*1. Doing 10C3 is also wrong.
> Is there any formula that we can use to find out the number of
> possibilities for the different integer range (e.g. 1 - 5 or 1 - 8)?
In Maple:
a:= n -> nops({seq(seq(seq(x*y*z,x=1..n),y=1..n),z=1..n)});
> Hi.
> (1*2*3*4*5*6*7*8*9*10) / (1*2*3*1*2*3*4*5*6*7) = 120
> Generally: (n!) / (m! * (n-m)!)
It's a coincidence that (10 choose 3) is the right answer. For integers
>1 to 11 the answer would be 173, not (11 choose 3) = 165.
See sequence A027425 in the On-Line Encyclopedia of Integer Sequences,
><http://www.research.att.com/~njas/sequences/A027425>. I don't think
>there's a closed form for a(n) or a closed form generating function or
>recurrence.
I'm late to this party and I couldn't go through all the replies, but
was there any counting method that started by ruling out any number <=
1000 whose smallest prime factor was greater than 10? I know that
still wouldn't be a formula, but I was wondering if it had been
pondered?
I know you could get the number of numbers relatively prime to 1000
using Euler's function, so that's why it crossed my mind.
Of course, this is just a starting point. Of course my thinking could
be pointless too.
Brian
===
Subject: Re: Possible results from three variables
Robert Israel <israel@math.MyUniversitysInitials.ca> skrev i melding
> <bakpao@gmail.com> skrev i melding
> Given three variables, say x, y, z with each variable being an integer
> from 1 to 10, how many possible values are there from the equation of
> x * y * z? The quickest and incorrect answer is 1000 (from 10x10x10).
> This is wrong because some of the combinations actually the same, e.g.
> 1*2*4 = 1*4*2 = 4*1*2 = 1*8*1. Doing 10C3 is also wrong.
> Is there any formula that we can use to find out the number of
> possibilities for the different integer range (e.g. 1 - 5 or 1 - 8)?
In Maple:
a:= n -> nops({seq(seq(seq(x*y*z,x=1..n),y=1..n),z=1..n)});
> Hi.
> (1*2*3*4*5*6*7*8*9*10) / (1*2*3*1*2*3*4*5*6*7) = 120
> Generally: (n!) / (m! * (n-m)!)
It's a coincidence that (10 choose 3) is the right answer. For integers
> 1 to 11 the answer would be 173, not (11 choose 3) = 165.
See sequence A027425 in the On-Line Encyclopedia of Integer Sequences,
> <http://www.research.att.com/~njas/sequences/A027425>. I don't think
> there's a closed form for a(n) or a closed form generating function or
> recurrence.
> --
> Robert Israel israel@math.MyUniversitysInitials.ca
> Department of Mathematics http://www.math.ubc.ca/~israel
> University of British Columbia Vancouver, BC, Canada V6T 1Z2
Hi.
I looked up this sequence earlier today (I have On-Line E... as a favourite
in IE). And I admit I was to fast on the trigger. It doesn't look like this
problem has an easy answer. I see they are using A027425 to generate 
A027426
and visa versa. Looking at quasi's replays there are some amazing things
going on. There are clear relations between some of the f(n,k) and for
others none.
A very fun problem.
Karl-Olav Nyberg
===
Subject: Re: Possible results from three variables
<139br4vmp9am060@corp.supernews.com>
<rbisrael.20070712220809$4d31@news.ks.uiuc.edu>
On Jul 13, 8:23 am, Robert Israel
> <bak...@gmail.com> skrev i melding
> Given three variables, say x, y, z with each variable being an
integer
> from 1 to 10, how many possible values are there from the equation of
> x * y * z? The quickest and incorrect answer is 1000 (from 10x10x10).
> This is wrong because some of the combinations actually the same,
e.g.
> 1*2*4 = 1*4*2 = 4*1*2 = 1*8*1. Doing 10C3 is also wrong.
> Is there any formula that we can use to find out the number of
> possibilities for the different integer range (e.g. 1 - 5 or 1 - 8)?
In Maple:
a:= n -> nops({seq(seq(seq(x*y*z,x=1..n),y=1..n),z=1..n)});
> Hi.
> (1*2*3*4*5*6*7*8*9*10) / (1*2*3*1*2*3*4*5*6*7) = 120
> Generally: (n!) / (m! * (n-m)!)
It's a coincidence that (10 choose 3) is the right answer. For integers
> 1 to 11 the answer would be 173, not (11 choose 3) = 165.
See sequence A027425 in the On-Line Encyclopedia of Integer Sequences,
> <http://www.research.att.com/~njas/sequences/A027425>. I don't think
> there's a closed form for a(n) or a closed form generating function or
> recurrence.
> --
> Robert Israel isr...@math.MyUniversitysInitials.ca
> Department of Mathematics http://www.math.ubc.ca/~israel
> University of British Columbia Vancouver, BC, Canada V6T 1Z2
Here are a few comments:
1. The numbers can be the same, there is no restriction.
2. quasi: that is a very interesting pattern you have there!
3. Stephen J. Herschkorn: that is also what I had in mind but it is
very slow. I was hoping for a simple formula to obtain it.
It seems there is no existing proven formula for this. But that's OK
since brute force seems to be sufficient to find out the answer.
Just for reference I have created a simple C# function that does brute
force search on it:
public void Solve()
{
Hashtable solution = new Hashtable();
int max = 10;
for (int x = 1; x <= max; x++)
{
for (int y = 1; y <= max; y++)
{
for (int z = 1; z <= max; z++)
{
solution[x * y * z] = 1;
}
}
}
MessageBox.Show(solution.Keys.Count.ToString());
}
===
Subject: Re: Possible results from three variables
<139br4vmp9am060@corp.supernews.com>
<rbisrael.20070712220809$4d31@news.ks.uiuc.edu On Jul 13, 8:23 am, Robert
Israel
> <bak...@gmail.com> skrev i melding
> Given three variables, say x, y, z with each variable being an
integer
> from 1 to 10, how many possible values are there from the equation
of
> x * y * z? The quickest and incorrect answer is 1000 (from
10x10x10).
> This is wrong because some of the combinations actually the same,
e.g.
> 1*2*4 = 1*4*2 = 4*1*2 = 1*8*1. Doing 10C3 is also wrong.
> Is there any formula that we can use to find out the number of
> possibilities for the different integer range (e.g. 1 - 5 or 1 -
8)?
> In Maple:
> a:= n -> nops({seq(seq(seq(x*y*z,x=1..n),y=1..n),z=1..n)});
> Hi.
> (1*2*3*4*5*6*7*8*9*10) / (1*2*3*1*2*3*4*5*6*7) = 120
> Generally: (n!) / (m! * (n-m)!)
> It's a coincidence that (10 choose 3) is the right answer. For
integers
> 1 to 11 the answer would be 173, not (11 choose 3) = 165.
> See sequence A027425 in the On-Line Encyclopedia of Integer Sequences,
> <http://www.research.att.com/~njas/sequences/A027425>. I don't think
> there's a closed form for a(n) or a closed form generating function or
> recurrence.
> --
> Robert Israel isr...@math.MyUniversitysInitials.ca
> Department of Mathematics http://www.math.ubc.ca/~israel
> University of British Columbia Vancouver, BC, Canada V6T 1Z2
> Here are a few comments:
> 1. The numbers can be the same, there is no restriction.
> 2. quasi: that is a very interesting pattern you have there!
> 3. Stephen J. Herschkorn: that is also what I had in mind but it is
> very slow. I was hoping for a simple formula to obtain it.
It seems there is no existing proven formula for this. But that's OK
> since brute force seems to be sufficient to find out the answer.
Just for reference I have created a simple C# function that does brute
> force search on it:
public void Solve()
> {
> Hashtable solution = new Hashtable();
int max = 10;
for (int x = 1; x <= max; x++)
> {
> for (int y = 1; y <= max; y++)
> {
> for (int z = 1; z <= max; z++)
> {
> solution[x * y * z] = 1;
> }
> }
> }
> MessageBox.Show(solution.Keys.Count.ToString());
You can brute force it using SQL.
TABLE [Table1]
--------------
1
2
3
4
5
6
7
8
9
10
QUERY [digit_product]
---------------------
SELECT [t1].[n]*[t2].[n]*[t3].[n] AS Product
FROM Table1 AS t1, Table1 AS t2, Table1 AS t3
GROUP BY [t1].[n]*[t2].[n]*[t3].[n];
Query returns 120 records:
Product
1
2
3
4
5
6
7
8
9
10
12
14
15
16
18
20
21
24
25
27
28
30
32
35
36
40
42
45
48
49
50
54
56
60
63
64
70
72
75
80
81
84
90
96
98
100
105
108
112
120
125
126
128
135
140
144
147
150
160
162
168
175
180
189
192
196
200
210
216
224
225
240
243
245
250
252
256
270
280
288
294
300
315
320
324
336
343
350
360
378
384
392
400
405
420
432
441
448
450
480
486
490
500
504
512
540
560
567
576
600
630
640
648
700
720
729
800
810
900
1000
===
Subject: Can we assign a unique number to a irreducible directional graph?
I am not a mathematician.
In order to solve a data networking problem, I need to know whether we
can assign a unique number to a irreducible directional graph? n-
tuplet? What if number of nodes is fixed?
I can't think of a way.
-Bhu
===
Subject: Re: Can we assign a unique number to a irreducible directional
graph?
On Fri, 13 Jul 2007 04:24:17 -0700, Bhu Joshipura
>I am not a mathematician.
>In order to solve a data networking problem, I need to know whether we
>can assign a unique number to a irreducible directional graph?
Well, presumably you're talking about finite graphs. Since there
are only countably many such graphs the answer is yes. But I
can't imagine how this is going to help with your networking
problem - surely you need the assignment to have some _property_
that you haven't told us about...
>n-
>tuplet? What if number of nodes is fixed?
>I can't think of a way.
>-Bhu
************************
David C. Ullrich
===
Subject: Projective space and fixed points
I have the following question related to the fixed points of
projective space mappings:
For which natural n is ti the case that every continuous mapping form
RP^n to RP^n has a fixed point.
The same question for CP^n.
I can prove that any such a mapping f, for even n, has a fixed point.
This follows form the fact that such f lifts to the
mapping f ' : S^n -> S^n of the covering space S^n over RP^n. But any
such f' has a point x such that f '(x) = x or -x,
because, otherwise, projecting f(x) onto the tangent space T_x S^n for
every x we would get a non-zero vector field
on S^n for even n, which contradicts the well known fact, that there
are no such vector fields.
Do you have any ideas for complex projective space.
===
Subject: Re: ** says: Definition: sum{i in N} i = 0
<virgil-86A403.11442707072007@comcast.dca.giganews.com>
<virgil-6B8CDD.15314507072007@comcast.dca.giganews.com>
<virgil-50315F.15565608072007@comcast.dca.giganews.com>
<virgil-CEDC73.13560809072007@comcast.dca.giganews.com>
<virgil-831C3A.14440210072007@comcast.dca.giganews.com>
<virgil-DC9FDC.21355011072007@comcast.dca.giganews.com
> Could you answer the following question, please:
> Why should anybody try to determine
> lim_{x->0} sinx/x by means of lim_{x->0} cosx/1
> if the function at x = 0 in fact (or by decision of Dr. hagman) was
> sin0/0 = 10^100?
Because in standard mathematics, the limit of a function at a point,
> when it exists, need not equal
...
That's the reason to determine the limit? Are you incapable of reading
or only of understanding? The question was: Why should anybody try to
determine lim_{x->0} sinx/x ...? The question was *not*: Why is it
useless to determine lim_{x->0} sinx/x ...?
===
Subject: Re: ** says: Definition: sum{i in N} i = 0
<virgil-4005B1.13562106072007@comcast.dca.giganews.com>
<virgil-86A403.11442707072007@comcast.dca.giganews.com>
<virgil-6B8CDD.15314507072007@comcast.dca.giganews.com>
<virgil-50315F.15565608072007@comcast.dca.giganews.com>
<virgil-CEDC73.13560809072007@comcast.dca.giganews.com>
<virgil-831C3A.14440210072007@comcast.dca.giganews.com>
<1afb1$46948483$82a1e228$10796@news1.tudelft.nl>
<virgil-68E911.17554711072007@comcast.dca.giganews.com
> To find the true value of sinx/x at x = 0 is a splendid example of
> this power.
It is the power to see what is not there while not seeing what is there,
> and WM relies on it to excess.
LOL! To see what's not there, like the existence of non existing real
numbers and uncountable infinities, is the prime private privilege of
matheologicians.
===
Subject: Re: ** says: Definition: sum{i in N} i = 0
<JKqJ9M.4Fw@cwi.nl>
<virgil-4005B1.13562106072007@comcast.dca.giganews.com>
<virgil-86A403.11442707072007@comcast.dca.giganews.com>
<virgil-6B8CDD.15314507072007@comcast.dca.giganews.com>
<virgil-50315F.15565608072007@comcast.dca.giganews.com>
<virgil-CEDC73.13560809072007@comcast.dca.giganews.com>
<virgil-831C3A.14440210072007@comcast.dca.giganews.com>
<1afb1$46948483$82a1e228$10796@news1.tudelft.nl>
<virgil-725752.17521011072007@comcast.dca.giganews.com
> The derivative of |x| is not
> continuous but -1 for negative x and +1 for positive x. In contrast
> sinx/x is continuous. Your example fails.
> The expression sin(x)/x is not even defined for x = 0, so that WM is
> claiming that it has a value when it does not have a value.
> Ah, My old friend the SINC-function:
> It IS defined for x = 0 and its value IS always 1 at x = 0 .
> Han de Bruijn
There are two similar functions, one of which is not even defined at x =
> 0 and another which is continuous, and even differentiable, there.-
And there are uncountably many others with f(x) = r in R. But only one
of these functions exists in useful mathematics. And f(x) for this one
can be determined by l'Hospital.
===
Subject: Re: Why the dark energy is small gr-qc/0602022
===
> Subject: RE: hcha gr-qc/0602022 (gr-qc/0602022) Metadata Edit
To verify abstract, usehttp://arxiv.org/abs/gr-qc/0602022
It would be interesting to take a look Jack's character:
http://en.wikipedia.org/wiki/Jack_Sarfatti
It seems to me that Jack Sarfatti has many times interesting
ideas although they are written with very technical and
difficult tensor mathematics.
I think that it is difficult to me many times to distinguish
in Jack's writings that what are real physical and acceptable
matters and what are just pointless bull cranking ???
Hannu
===
Subject: Re: Why the dark energy is small gr-qc/0602022
>
===
> Subject: RE: hcha gr-qc/0602022 (gr-qc/0602022) Metadata Edit
> To verify abstract, usehttp://arxiv.org/abs/gr-qc/0602022
It would be interesting to take a look Jack's character:
http://en.wikipedia.org/wiki/Jack_Sarfatti
It seems to me that Jack Sarfatti has many times interesting
> ideas although they are written with very technical and
> difficult tensor mathematics.
I think that it is difficult to me many times to distinguish
> in Jack's writings that what are real physical and acceptable
> matters and what are just pointless bull cranking ???
Hannu
I found one paper (which avoids difficult tensor
mathematics)
http://stardrive.org/ufos.shtml
in references above mentioned Wikipedia page
which enlights some Jack Sarfatti's ideas ???
I copy it (from the WORD-program copy) below.
Hannu
---COPY BELOW-------
UFOS AND THE NEW PHYSICS
By Kim Burrafato
in conversation with
Jack Sarfatti, Ph.D.
UFOs are a controversial and compelling enigma. We remain uncertain of
what these objects are, or whether they even exist. Conventional
wisdom is that most UFOs are likely erroneous observations, along with
a bit of mass hysteria and some outright charlatanism. Was there
really a UFO crash at Roswell in 1947? But what if we were to assume
-- purely as an exercise -- that UFOs are actually products of a
starfaring civilization?
The rational mind rebels at the idea. UFOs allegedly exhibit
outlandish behavior. Witnesses report incredible things like instant
accelerations, right angle turns at enormous velocities, and instant
stops. These objects are also reported to change their shapes and
colors, and simply disappear like phantoms instantly leaping from one
universe to another. UFOs violate the laws of physics. No terrestrial
spacecraft are capable of such magical feats of reality engineering.
It's no wonder that most scientists consider the subject of UFOs
patently absurd and not worthy of serious discussion. Can modern
physics ever hope explain the UFO phenomenon to fit within the current
address the question If UFOs are real, then how do they work?
Clear Intent, by Larry Fawcett and Barry Greenwood, presents evidence
of Air Force observations of these phenomena. Included in the book
were published Department of Defense documents, released under the
Freedom of Information Act, that report repeated UFO penetrations of
high security ICBM bases and nuclear weapons storage areas back in the
1970s. One major mission of the American intelligence community is to
anticipate technology that might disrupt our defense systems. Judging
from the documents Fawcett and Greenwood published, the Department of
Defense and Air Force are not on top of the situation, with regard to
UFOs buzzing some of our most sensitive defense areas.
Force is testing some new kind of black project aircraft, code named
Aurora. According to the news accounts, this ultra-secret, high
altitude hypersonic aircraft creates powerful vertical, earthquake-
like jolts, felt by thousands of Southern California residents, as it
passes high overhead at over four thousand miles per hour.
Anything that burns a propellant in a reaction chamber, and then
directs the exhaust products of that reaction out of a nozzle can be
described as a rocket. Aurora's propulsion system is obviously rocket-
like. It makes a lot of noise. It follows, then, that any rocket
capable of the tremendous speed and maneuvering attributed to UFOs
would be impossible not to notice.
But the majority of UFO witnesses report hearing nothing like the
continuous, low frequency rumbling sounds that would normally
accompany fuel combustion propulsion based rockets; nor do they report
seeing the kind of exhaust products one would expect to see with
rockets. If not reporting objects that seem to be entirely silent,
some witnesses have reported hearing loud, but brief bang-like sounds
as UFOs take off, as well as both low and high frequency humming
sounds.
We're forced to conclude, then, that UFOs don't operate on terrestrial
rocket science principles. In a single stroke, we can eliminate not
only all present solid and chemical fuel based rockets, but even
future exotic propulsion schemes like nuclear-electric ion rockets,
interstellar hydrogen fusion ramjets, and that favorite of science
fiction writers and scientists alike, antimatter-matter annihilation
rockets, as viable candidates for possible UFO propulsion.
The major problem with applying conventional rocket science to
interstellar travel is the Einstein barrier: the exhaust speed of the
propellant relative to a rocket is limited to three hundred thousand
kilometers per second, the speed of light in a vacuum. At present,
this barrier prevents practical interstellar travel because a
physically unreasonable quantity of conventional fuel is required to
get reasonably close to the speed of light in a rocket. Any starship
built on proven or even highly speculative principles of rocket
propulsion, such as the very efficient anti-matter drive popularized
in Star Trek, would still need to generate much more energy than is
generated on Earth every year. The manufacturing costs would be many
times the gross national products of America, all of Europe and Japan
combined. So, we're faced with both a physical and an economic
absurdity. The starship Enterprise still remains a playful notion at
best.
Cheap and practical interstellar flight is possible only if Einstein's
barrier can be overcome. If UFOs are indeed from other star systems,
then they have obviously broken that barrier.
Like conventional rockets and jets, UFOs seem to generate their own
internal propulsion. As far as we can tell, they are not pushed along
solar wind. They evidently operate on distinctly alien propulsion
principles than those we're accustomed to. Could we ever expect to
understand such principles?
Theoretical subtleties inherent within Einstein's general theory of
relativity (gravity) and quantum mechanics may hold the answer. If it
is possible to physically alter or deform four-dimensional space-time,
then a kind of space-time tunnel, or time travel gate, can be
constructed. In current physics, such theoretical tunnels are known as
traversable wormholes.
Cal Tech physicists Kip Thorne and Michael Morris, along with Igor
Novikov of Moscow State University and others, have published a number
of papers in the prestigious Physical Review on the physics of time
travel and traversable wormholes. A traversable wormhole is literally
a shortcut through both space and time.
An advanced extraterrestrial civilization , capable of cosmic-scale
engineering, could conceivably fabricate traversable wormhole time
travel gates, where their starships would enter a wormhole in one
region of space- time, and pop out in another extremely distant region
of space-time.
According to the equations of general relativity, one of the mouths of
the wormhole has to accelerate away from the other mouth to a very
high speed close to that of light and then return. If you step into
the mouth that accelerates and returns, you will step out the other
end into the relative past of when you entered. If you step into the
unaccelerated mouth and leave through the accelerated one, you will
jump into the relative future. That is, if you had an identical twin
who did not go through the wormhole, your twin would be older than
you.
How do you create a traversable wormhole time machine? Theoretical
physicist Jack Sarfatti, has also been working on that question. A
self-described rogue scholar, Sarfatti has had a long association
with fringe science. He and Michael Murphy organized and led the first
Physics of Consciousness Seminars at the Esalen Institute back in
1974, sponsored by former human potential movement superstar turned
pariah, Werner Erhard. Those Esalen seminars spawned an entirely new
genre of pop science literature, known as The New Physics -- of
which the recent best seller by Deepak Chopra, Quantum Healing, is but
one example.
Sarfatti's unorthodox theories on faster-than-light (superluminal)
communication continue to raise hackles among the more conservative
physics mainstream. When it comes to traversable wormhole time
machines, Sarfatti speculates that so-called dark matter, which may
comprise over 90% of the mass of the universe, could provide an
answer.
This elusive matter has so far remained unobservable directly by
conventional methods employing photons of electromagnetic radiation.
Yet, its gravitational effects on ordinary matter are observed
indirectly. If dark matter moves around in the imaginary time of
quantum gravity, then it would be difficult for it to emit any
photons, even if it wasn't electrically neutral. So, we wouldn't
expect to see it through optical telescopes, Sarfatti explains.
The distinction between imaginary time and real time is important to
clarify here. The late Nobel laureate, physicist Richard Feynman, was
attempting to unify the quantum theory with gravity -- for some time a
kind of Holy Grail to physicists. His intuitive genius led him to
reformulate the quantum theory as a sum over histories. According to
classical Newtonian and Einsteinian physics. Instead, it is supposed
to follow every possible path in spacetime.
There are two numbers that are associated with these histories, one
for the size of the wave, and the other for its position, or phase, in
particular point is then found by adding up the waves associated with
every possible history that passes through that point.
The problem is that when you actually perform this summing of all
possible histories, severe technical problems arise. To avoid the
technical difficulties with Feynman's sum over histories, says
physicist Stephen Hawking in A Brief History of Time,
one must use imaginary time... That is to say, for the purposes of
calculation, one must measure time using imaginary numbers, rather
than real ones. This has an interesting effect on spacetime: The
distinction between time and space disappears completely... we may
regard our use of Euclidean spacetime as merely a mathematical device
(or trick) to calculate answers about real spacetime.
Despite the fact that imaginary time is only a calculational device,
history has time and again that real physics can and does emerge from
such abstract mathematical constructs of the mind. Nevertheless, all
theories must still be verified by experiment for them to be
meaningful. That is the nature of physics and the scientific method.
Sarfatti employs this same trick of imaginary time in his
calculations on the possible nature of dark matter.
I've been investigating a new kind of causality violating quantum
field theory, which reveals that the projection of imaginary time
motion into real time can be both slower and faster than light. If
that's true, it means that the energy density of this hypothetical
neutral dark matter would get smaller as its speed increased, which
would make it the perfect stuff to construct traversable wormholes
with.
He believes it might also make a super-efficient starship fuel.
Remember the exhaust speed limitation of rocketry discussed earlier?
In theory, dark matter can produce a faster-than-light exhaust
velocity, which means it could be used to overcome the Einstein
barrier. Since its density would theoretically decrease as its exhaust
velocity increased, that means a small amount of dark matter fuel
could easily propel a massive payload to the stars,
says Sarfatti.
And, if dark matter is as uniformly distributed throughout the
universe, as it seems to be, you'd never run out of fuel. Even though
dark matter comprises 90% of the mass of the universe, its average
density is still too small for us to detect directly in an easy way.
But I think that situation will change in the near future.
Sarfatti admits that a number of important questions about dark matter
remain unanswered.
How do we find and use such matter if it only exists in imaginary
time? How does curvature in imaginary time influence curvature in real
time? Can it be both attractive and exotic? Can dark matter carry
electric charge? Does it have a magnetic moment? If it does, then it
could be electromagnetically manipulated. And finally, is it even
possible to scoop up and clump together enough dark matter with the
accuracy and precision required to fabricate a traversable wormhole or
power a starship?
Combine an inexhaustible fuel with a potential faster-than-light
exhaust speed, like the hypothesized dark matter, with a super-
technology capable of manipulating space-time geometry, and you have
all the physical requisites for a starfaring society. If UFOs are
interstellar in origin, then, they could be using traversable
wormholes, or, they have developed an efficient interstellar
propulsion system (like Sarfatti's proposed dark matter model), which
allows them to travel very close to the speed of light. Traversable
wormholes are preferable, since they would enable the
extraterrestrials to tunnel across vast interstellar distances, and
then tunnel back to where and when they came from.
exotic matter permits an effectively faster-than-light warp drive. The
ufo, or starship, freely floats along a gravitational shock without
time dilation. John Von Neumann, the mathematical genius and father of
the programmable computer, hypothesized that sufficiently advanced
extraterrestrial civilizations would be capable of constructing
artificially intelligent robot probes to scour the universe. These
intelligent Von Neumann probes would also be capable of universal
replication. They would be programmed to utilize whatever elements
they encountered in their travels to fabricate endless copies of
themselves. After a long enough period of time had elapsed, the
galaxies would be teeming with them. Von Neumann's hypothesis is often
used as an argument against the existence of starfaring
extraterrestrial societies, since we'd have likely already encountered
their intelligent probes if any such advanced civilizations were out
there.
If UFOs are not artificially intelligent Von Neumann robot probes, and
if they are not using traversable wormholes, then, they're probably
migrating as completely self-contained interstellar colonies. Since
they would be traveling extreme distances close to the speed of light,
they would find a much changed world from the one they left if they
eventually returned.
According to the equations of special relativity, the closer you get
to the speed of light, the more space-time contracts for you -- what's
known in relativistic physics as space-time dilation. The clocks
aboard your starship run much slower than your friends' clocks back
home. The result of all this is that your friends age and die much
faster than you. Assuming an extraterrestrial psychology and society
even vaguely similar to human society, with family and social ties as
important as they are in human society, then the lack of time travel
technology seems to necessitate the interstellar migratory colony mode
of travel. For intergalactic gypsies, home is where you are.
A strong case has been made that after over 30 years of searching for
extraterrestrial intelligence (SETI), we just have not observed the
kind of infra-red and electromagnetic signals that a number of
prominent scientists believe would be characteristic of an advanced
extraterrestrial civilization capable of cosmic scale engineering and
interstellar expansion. So, at this point in the evolution of the
universe, humanity may truly be alone. That fact weakens the case for
UFOs being purely extraterrestrial in origin from advanced alien
civilizations much older than our own. If UFOs are not
extraterrestrial, then what are they? Sarfatti suspects
They could be terrestrial time ships, originating from our own
future. In which case, their motivation might be to interfere in their
own past in order to assure their own coming into being in a globally
and logically self-consistent way.
British cosmologist Sir Fred Hoyle calls such acausal, circular
connections loops in time. A number of papers have been published in
respected physics journals that implicitly lend new credibility to
this scenario, e.g., Quantum mechanics near closed time-like lines,
Physical Review D, November 15, 1991 by Oxford's David Deutsch, and
Wormholes in space-time and their use for interstellar travel,
American Journal of Physics, May 1988, by Cal Tech's Michael Morris
and Kip Thorne.
NASA's Cosmic Background Explorer (COBE) satellite recently revealed
small ripples of one part in one hundred thousand in the angular
distribution of the cosmic background microwave thermal radiation left
over from the Big Bang. These observed ripples, possibly the remnants
of ultra-microscopic quantum fluctuations when the universe was even
smaller than a single electron, might provide the mechanism for galaxy
formation. This important new observation gives us a picture of the
universe as it was only about three hundred thousand years after the
initial moment of creation of real time out of imaginary time.
Sarfatti conjectures that the quantum ripples observed by the COBE
satellite may have been created from the far future, in a cosmic scale
loop in time by an ultra-advanced intelligence, quite possibly evolved
from us. A quantum optics experiment he had published a few years ago
in Physics Essays could actually test that idea in the laboratory.
If it works as the equations predict, it would be possible to build a
kind of quantum telescope that probes the future just as a classical
telescope probes the past. The power of prophecy would no longer be
the exclusive domain of mystics, psychics, and avatars. It would be
technology.
But how do we go from hypothetical extraterrestrial or future-
terrestrial super-technology, to present-day terrestrial engineering?
In order to even begin to answer that question, we have to embark on a
short trip through relativity theory and quantum mechanics.
Physics can be simplified by thinking in what's called inertial
frames. In an inertial frame, the gravitational force is canceled out,
Obviously, the surface of the earth is not an inertial frame, because
we feel gravity. In contrast, a freely floating non-rotating space
shuttle orbiting the earth is an inertial frame.
The intuitive basis of the general relativity theory of gravity is
something known as the equivalence principle. One version of this
equivalence principle says that a small non-inertial frame, like a
rocket or a UFO that is uniformly accelerating in empty, flat space-
time, far away from all large masses, is equivalent to a non-inertial
frame at rest, like an observer on earth. Whenever speaking of
physical equivalence, it is assumed that the observers agree not to
look outside their laboratory window or porthole of their spacecraft.
If they closed their eyes, or only looked at their gravitometers,
then, they could not decide which frame they were in. They would be in
locally equivalent frames.
Einstein showed that mass curves spacetime, and that those non-
spinning, local inertial frames, in which you are weightless, follow
special geodesic world lines. A geodesic is simply the straightest
path possible in a curved four-dimensional space-time. A skydiver in
freefall is riding along a geodesic world line. You age more on a
geodesic than on a non-geodesic. Conversely, one must apply non-
gravitational forces (like muscles and motors) to move along a world
line that is not a geodesic.
We now come to Jack Sarfatti's new key idea in understanding the
physics of hypothetical UFO maneuvers. The geodesic is highly curved
in three-dimensional space and corresponds to what a non-inertial
observer on the earth sees as accelerated motion. However, observers
living on the geodesic, like those in a UFO, experience a comfortable
free float weightless condition throughout their apparent high-speed,
high-acceleration curved-motion. That leads to a second version of the
equivalence principle: a small enough free floating or free falling
local inertial frame in geodesic motion near a large mass, is
equivalent to an inertial frame in flat space-time that is far away
from any large mass. That means, if we see a UFO make a sharp right
angle turn at very high speed, we can't jump to the conclusion that
the inhabitants are feeling enormous g forces, if they have technology
capable of controlling their local space-time curvature.
Imagine you are an astronaut in the space shuttle orbiting the earth.
Gravity is canceled out. You are weightless. Again, you are simply
following the straightest possible path, or geodesic, in curved four-
dimensional space-time. Your frame of reference is locally inertial.
As far as you can tell you are not accelerating. You feel no forces.
You are in free float. Give a short gentle push to an object floating
in the shuttle, and it moves at constant speed in a straight line.
In contrast, relative to a non-inertial, gravity bound observer down
at Mission Control, Houston, you are rapidly accelerating towards the
center of mass of the earth. It is only your conserved orbital angular
momentum that prevents you from crashing into the earth. That is
essentially the explanation given by Newtonian classical mechanics.
However, in Einstein's classical theory of general relativity, you are
moving locally on a geodesic, in the straightest path possible in
curved four-dimensional space-time. The curvature is primarily caused
by the mass of the earth.
We can now begin to see how what looks like a right angle turn at
enormous acceleration to a non-inertial observer on the surface of the
earth, might feel like nothing special to the freely floating inertial
passengers aboard the UFO. That's because the UFO isn't being
propelled in the familiar sense. It's simply freely floating in a
locally bent space-time geodesic in its immediate neighborhood.
If UFOs are really spacecraft, then somehow they're controlling their
local space-time curvature, maybe in angstrom thin boundary layers
fitting the outside surface of the UFO like a skin,
speculates Sarfatti.
And if that's true, they're probably doing it using loops in time.
It's just a hunch. Then he added, You know, when these things are
really pumping up their local curvature engines, let's say during a
high speed, right angle turn, standard Fourier signal analysis says
that they're going to be generating significant broad band
gravitational radiation. It happens that John Clauser at UC Berkeley
is building a new type of ultra-sensitive atomic beam gravitometer
that should be able detect just such radiation. Maybe his device could
be used as the first bona fide UFO detector.
A prerequisite for UFO propulsion may be the principle of global self-
consistency, or logical consistency, used by Kip Thorne and Igor
Novikov in their traversable wormhole-time travel papers. This same
key principle was independently discovered by Sarfatti and published
in Physics Essays, Vol. 4, No.3, Sept. 1991, in a paper provocatively
titled, Design for a Superluminal Communication Device. Sarfatti's
gedankenexperiment, which he no longer believes will work without a
significant coupling between mind and matter, attempted to use the
quantum spin correlation between special types of photon pairs, as a
communications channel. This spin correlation is non-local, that is,
faster-than-light.
Correlation, according to the equations of quantum mechanics, means
that each photon in every pair is permanently entangled with its
photon twin. Interfere with one photon, and its twin is always
affected. The correlation persists regardless of the distance between
the photon pair. This spooky, telepathic-like effect, known as the
Einstein-Podolsky-Rosen effect, has actually been observed in a number
of different atomic physics and quantum optics experiments.
Magazine, Henry Pierce Stapp, a physicist at Lawrence Berkeley
Laboratory, published a remarkable paper in the prestigious Physical
Review A (July, 1994, PP 18-24). Stapp cites an experiment by Helmut
Schmidt involving psychokinetic distortion of the statistical
prediction of orthodox quantum mechanics that works precognitively
backwards in time on radioactive nuclear decays! A theorem by Stapp's
Berkeley colleague, Philippe Eberhard, shows that Sarfatti's
prediction cannot work with an ordinary inanimate physical device that
obeys the statistical predictions of orthodox quantum mechanics. Nobel
Prize physicist, Brian Josephson has suggested that only living matter
can violate the statistical predictions of quantum mechanics and,
thereby, use the nonlocal quantum connection as a practical
communication channel the way Sarfatti claims to remote view.
Sarfatti now agrees with Josephson because of Stapp's July 94 paper.
What does all this quantum weirdness have to do with UFO propulsion? A
young Canadian physicist- engineer, Lyle Fuller, had an important
insight after reading Sarfatti's Physics Essays paper. In a series of
letters to Sarfatti, Fuller mathematically argued that any attempt to
create a time travel paradox with Sarfatti's proposed electro-optical
device, or any other type of superluminal signaling device, would
cohere the normally random and hidden, but infinite zero-point quantum
vacuum energy into coherent space-time curvature. The hypothetical
release of that energy would then change the geometry of space-time in
such a way that the possibility of paradox will always be prevented.
When we speak of the quantum vacuum, we don't mean a vacuum in the
ordinary sense of an empty void, but rather the opposite -- an
vacuum zero point energy exists, but at present there is no feasible
way of utilizing it. The basic idea of the connection between zero-
point energy fluctuations and gravity was first recognized by the
Russian Nobel physics laureate and political activist, the late Andre
Sakharov. Roger Penrose argues against this view in Shadows of the
Mind. On the other hand Hal Puthoff is working on Sakharov's theory.
Extrapolating from Fuller's argument, Sarfatti speculates that UFOs
could be employing some kind of paradox engine, or curvature
generator. By continually generating paradoxes, this engine would
force nature to avoid paradox by releasing inexhaustible quantities of
zero point vacuum energy, physically altering the geometry of
spacetime -- a starship fueled by paradox. Obviously, the
demonstration of such a device would dramatically prove that the
preservation of global, or logical consistency is a fundamental and
immutable law of nature. It would also probably fill our planet's
energy needs forever. In other words, energy may be mutable, but logic
is not. Nature abhors a paradox.
Sarfatti considers a paper by David Deutsch published in the November
25, 1991 Physical Review D to be a milestone.
Deutsch explicitly shows that influences from the future constrain
initial conditions on the classical level, and yield a globally self-
consistent pattern of probabilities at the quantum level,
he explained.
That's why the thermal radiation ripples observed by the COBE
satellite could be an intelligently created artifact from the future.
But here on the laboratory scale, any of our clever attempts to create
time travel paradoxes will likely cause one of the following three
things to happen: Either laboratory equipment or people will
malfunction with certainty -- in which case free will is an illusion,
isn't it? Or, zero point energy will be released, physically changing
space-time curvature in such a way as to always avoid paradox. Or, as
David Deutsch suggests, the universe will simply split.
Verification of any of these admittedly speculative hypotheses in the
laboratory would make the debate over UFOs moot -- we could make our
own starships.
The Starship Builders
Copyright (c) 1995 by Kim Burrafato and Jack Sarfatti. All rights
reserved.
http://stardrive.org/ufos.shtml
===
Subject: Re: Why the dark energy is small gr-qc/0602022
> A surprisingly simple holographic explanation for the low dark energy
> density is suggested. I derive the Einstein-Cartan disclination
> curvature tetrads and the physically independent dislocation torsion gap
> spin connections from an M-Matrix of non-closed Cartan 1-forms made
> from 8 Goldstone phase 0-forms of the vacuum ODLRO condensate inflation
> field in which the non-compact 10-parameter Poincare symmetry group is
> locally gauged for all invariant matter field actions. Quantum gravity
> zero point vacuum fluctuations should be renormalizable at the spin 1
> tetrad level where there is a natural scale-dependent holographic
> dimensionless coupling (hG/zpf/c^3)^1/3 ~ (Bekenstein BITS)^-1/3. The
> spacetime tetrad rotation coefficients play the same role as do the Lie
> algebra structure constants in internal symmetry spin 1 Yang-Mills local
> gauge theories. This suggests an intuitively pleasing natural
> organizing idea now missing in superstring theory. It is then clear
> why supersymmetry must break in order for our pocket universe to come
> into being with a small w = -1 negative pressure zero point exotic
> vacuum dark energy density. Just as the Michelson-Morley experiment gave
> a null result, this model predicts that the Large Hadron Collider will
> explain Omega(DM) ~ 0.23 as a matter of fundamental principle, neither
> will any other conceivable dark matter detector because dark matter
> forming galactic halos et-al is entirely virtual exotic vacuum w = - 1
> with positive irreducibly random quantum zero point pressure that mimics
> w = 0 CDM in its gravity lensing and all effects that we can observe
> from afar.
Given them the BS of GR, infinity can be reached.
The Dark Energy is the same as a negative mass density in vacuum. The
concept of a negative mass is so absurd beyond words, and yet the pile
is piling higher and higher. <shrug> Instead of re-examining the
very foundation of the hypothesis that leads to the silly conclusions
based on perhaps sound observations, the popular consensus is that the
hypothesis is more or less valid and just add more band-aides to allow
the modified hypothesis to fit the observations.
Some would say that I have no imagination to conceive the concept of a
negative mass. That is OK because I truly don't. Some would say I am
not gifted with special talent to cast a spell on someone. This is
OK, too, because I truly do not know how to cast a spell on someone.
Modern physics is so much like witchcraft. A physicist becomes merely
another name for a witch or a warlock. <shrug>
===
Subject: Re: Why the dark energy is small gr-qc/0602022
===
> Subject: RE: hcha gr-qc/0602022 (gr-qc/0602022) Metadata Edit
To verify abstract, usehttp://arxiv.org/abs/gr-qc/0602022
---------------------------------------------------------------------------
---
arXiv:gr-qc/0602022
> Title: Emergent Gravity and Torsion: String Theory Without String
> Theory, Why the Cosmic Dark Energy Is So Small
> Authors: Jack Sarfatti
> Categories: gr-qc
> Comments: This version is the second major revision addressing several
> unresolved fundamental empirical problems
A surprisingly simple holographic explanation for the low dark energy
> density is suggested. I derive the Einstein-Cartan disclination
> curvature tetrads and the physically independent dislocation torsion gap
> spin connections from an M-Matrix of non-closed Cartan 1-forms made
> from 8 Goldstone phase 0-forms of the vacuum ODLRO condensate inflation
> field in which the non-compact 10-parameter Poincare symmetry group is
> locally gauged for all invariant matter field actions. Quantum gravity
> zero point vacuum fluctuations should be renormalizable at the spin 1
> tetrad level where there is a natural scale-dependent holographic
> dimensionless coupling (hG/zpf/c^3)^1/3 ~ (Bekenstein BITS)^-1/3. The
> spacetime tetrad rotation coefficients play the same role as do the Lie
> algebra structure constants in internal symmetry spin 1 Yang-Mills local
> gauge theories. This suggests an intuitively pleasing natural
> organizing idea now missing in superstring theory. It is then clear
> why supersymmetry must break in order for our pocket universe to come
> into being with a small w = -1 negative pressure zero point exotic
> vacuum dark energy density. Just as the Michelson-Morley experiment gave
> a null result, this model predicts that the Large Hadron Collider will
> explain Omega(DM) ~ 0.23 as a matter of fundamental principle, neither
> will any other conceivable dark matter detector because dark matter
> forming galactic halos et-al is entirely virtual exotic vacuum w = - 1
> with positive irreducibly random quantum zero point pressure that mimics
> w = 0 CDM in its gravity lensing and all effects that we can observe
> from afar.
as always
jack..
very clever derivation!
i'm glad you pulled all of these pieces together
your ideas are always interesting
and just ahead the cutting edge!
(they're not hidden anymore!)
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
===
Subject: Re: Why the dark energy is small gr-qc/0602022
You are right, Jack, it _is_ surprisingly simple, almost obvious, in fact.
===
Living In The Present - Definition
(Version 3.0.1
on 13 July 2007)
(suitable for foreign
language students)
(View Summary
by skipping
indentation)
'
There is the fantastic relief of shedding all responsibility for
what is going on and for what has gone on and (by consequence) for
what will go on.
'
The fantastic relief
- of shedding (almost all) responsibility for seeing who is
who and what individuals have done,
and what these individuals - by consequence - will do
again;
- of shedding (almost all) responsibility for knowing what your
concerns and goals were, and are,
and what - by consequence - your concerns and goals will
be again, or indeed continue to be;
- of shedding (almost all) responsibility for knowing what your
abilities were and are,
and what - by consequence - continue to be your abilities,
even if dormant or taken away.
'
'
The relief is called - amongst other things - 'Death,'
by some. *(a)
However, many people at their death do not at all
drop their responsibility for life and for people,
but
they continue to nurture and protect life, your life
too, as much as they can and feel like doing, or are
invited by you to do, which they (try to) do as what
you commonly call angels. *(5)
Truly loving and responsible people will do
that, when they die
- till they find an opportunity to get born
again. *(5)(6)(7)
But you say, what about Heaven, then,
and what about the Judgment and what
about Hell?
Nobody ever told you the truth
about these things,
of course not, because it
was NOT THEIR INTENTION, to
tell you the truth,
so, as always, these things then
land on my shoulders - like on
'the broad shoulders of Plato,'
as the very faithful reader would
say.
(See also the upcoming
HRI 20070708 'What is True
about the Bible (and the
Koran and the Talmud)?')
Truth, as the faithful reader is
aware, can be defined - and is
defined indeed (in your soul) as
'What happened and who caused
it and with what intention.'
But this is about the living, meaning, with their soul in
their living body, so we go back to the living.
'
'
Others - knowing not at all what the present is - do call the shedding
of responsibility living in the present. Or being fully and only
in the present. *(b)
'
They omit, that the present is almost completely made up of the past,
and, that
the future is almost completely carrying on from and determined
by actions and by decisions made in the past - mostly even in a
very distant past.
THE MORE YOU ARE AWARE OF THAT PAST, THE MORE YOU ARE IN THE PRESENT.
'
And that gives, as promised, the actual Definition of 'Living in the
present':
The more you are aware of the past, the more you are in the
present, because
the present is almost completely made up of the past, and
the future is almost completely carrying on from and
determined by actions and decisions made in the past -
mostly even in a very distant past, and mostly by
others, too. *(5)(6)
'
'
I will explain it more, to your pleasure.
There is the fantastic relief of shedding all responsibility
for what is going on and for what has gone on, and what, by
consequence, will go on. ***
All the problems are still there, you see, but you are not
aware of these, you are only aware of the present
- meaning thus, to you, 'being only aware of where you are
now' and of 'with whom you are now,'
someone makes you feel very bad, or inflicts much
pain on you, and as only the present exists for
you,
the inflicted condition now is how you are,
you can NOT get out of the Energies inflicted,
because you have no reference point to the
past, nor to the future in which the pain
was not and will not be present:
All that exists, is the present, which
is the pain inflicted.
'
On top of that, you being in the present, are ONLY
aware of (you only consciously know) how they are NOW,
or how - in the case of Criminal Minds - they pretend to
be now:
you being in the present, are only aware of how these
PRETEND to be,
because you have 'switched off' your connection to
and awareness of the past, of your PAST EXPERIENCE
WITH THEM, with particular individuals,
who OF COURSE DO like you to be in the
present only,
so that you feel happy about them.
'
This gets of course much worse when you deny your friends,
when you are not connected to the past of you actual and
true friends either, but living in the present,
WHICH IS ACTUALLY, then, being immersed in
the Energies of Criminal Minds.
And so you see, that in those Religions and those
Philosophies, friendships become blotted out,
and the greatest Criminals suddenly become
friends, because everybody is the same,
though everybody is different.
(A paradox - an obvious conflict of
data - that you will find most in, and
that is typical of, all Energies used by
and inflicted on you by Criminal Minds.)
This creates tremendous Ugliness, Treachery, and
Destruction of Life,
which (the destruction of life) after all, IS the
- nicely worded, and well-hidden - goal of those
Religions and Philosophies that are planted on Earth
and that are maintained
to create on Earth a Paradise for Criminal
Minds. *(4)(3)(1)(2)
'
'
Thus there is the fantastic relief brought about by shedding
all responsibility for what is going on and for what has gone
on, and what, by consequence, will go on.
This means to you, then, that you are immersed in the illusion,
and you indeed experience the illusion - living in the
present -
that all is fresh and new, and that you don't know anything
about the people you meet, that you don't know anything about
your past, about who you really are.
'
While that not knowing is merely an illusion, (in fact
you do know but you only are temporarily disconnected from
it) and that provides you, with the experience, that
there is only existing what you see around you,
that you are 'living in the present'.
The ACTUAL Definition of 'Living in the Present,' is thus
formulated, and follows more precisely (in the next three
paragraphs of this Human Rights Issue) here below:
'
'
{Definition}
By regaining a viewpoint on the ACTUAL present - it consists of all
the past that shapes both the present and the future too - you
regain a viewpoint on yourself (and on others too) as you actually
are.
This, you can actually experience, with a great resurgence of all
that is dear and extremely valuable in you.
And THAT is true relief - resulting in correct actions and genuine
compassion that springs and flows from true identity.
'
Koos Nolst Trenite Cause Trinity
human rights philosopher and poet
'Solomon's wisdom was greater
than the wisdom of
all the men of the East,
and greater
than all the wisdom of Egypt.'
1 Kings 4:30
_________
Textnotes:
(a) Some use death (they go and die) in order to accomplish such a
relief
- usually with the idea and in the hope that they will get
reborn all fresh and new, or evolved and growing up in
better circumstances, or even, not reborn at all.
Whole religions or philosophies (all planted in and originating
from India) are based on and are trying to get people to achieve
this fantastic relief,
to not face life, not only to not face Criminal Minds, but
inevitably as a result, ALSO to deny one's friends,
to inevitably pervert and having to make love very
ugly indeed.
They have all manner of - very clever - methods to get you
to become irresponsible, and they (then have to) have - very
intelligent - lies about life, and about people, and (of
course also) about the past,
in order to make you or others want to strive for and
work towards this 'fantastic relief', *(2)
and they sell this as the ultimate goal in spiritual life, and
as spiritual Darwinism, (as spiritual Evolution) but
any even superficial investigation of any of the vast
amount of past life regression data does reveal most
easily and most thoroughly,
with data that they SHOULD be VERY interested in,
but, that quite understandably, they are not
at all interested in, because it reveals
most easily and fully, that
neither of these - neither the spiritual goal, nor their
spiritual Darwinism - do exist in actuality at all. *(1)
Quite and very much the contrary DOES exist, as the
faithful reader does know.
'
(b) They go and live only in the present - in order to experience
that, or at least as much as possible of that fantastic relief.
'
(c) This 'being unaware of responsibility,' is not at all necessarily
a voluntary action either *(8)
- most unawareness of the past has been (and is continuously
being) enforced on everyone, and kept up very heavily, *(8)
by those who have every reason to want you NOT aware of
the past, not only of your own past (your memories and
your abilities from the past) but especially they want you
NOT aware of THEIR past. *(9)
They work most violently, to reduce awareness in people,
in anybody, in you, and
they work most forcefully and persistently, to get false
ideas about the mind and about awareness accepted, and
applied (assumed and used by everyone).
(See under References also.)
'
_________
Footnotes:
(1) 'Penetrating and Exposing Hinduism and Buddhism'
{HRI 20060816-V5.0}
(16 August 2006 - Version 5.0 on 1 Nov 2006)
(2) 'The Nature of War'
{HRI 20051027-V1.8.1}
(27 Oct 2005 - Version 1.8.1 on 4 May 2007)
(3) 'About Aliens, Dimensions, Consciousness, Time, And Earth's Past
And Future'
(4) 'Your Role on Earth: A Very Simple Observation'
{HRI 20070629-II-V2.1-Q1-V1.3}
(2 July 2007 - Version 1.3 on 4 July 2007)
(5) 'If You Were Alive Now - in response to If Bach Lived Today...'
(6) 'Love On The Bridges Of Holland'
(18 Sept 2002 - Version 2.3 on 5 July 2007)
(7) 'A Poem Of Love And Adventure'
{HRI 20020717-V3.1.1}
(17 July 2002 - Version 3.1.1 on 4 June 2007)
(8) 'The First Law of Human Rights' (FLOHR)
{HRI 20060601-V3.1}
(1 June 2006 - Version 3.1 on 10 July 2007)
(9) ' Insane Defined By Criminal Minds As 'Ability To Perceive
Them' '
'
__________
'The Mafia Code Against Mankind'
{HRI 20021018-V2.0}
'Mechanics Of Awareness, Perception, Memory And Forgetfulness'
{HRI 20020913}
(13 September 2002)
'The Second Mozart-Effect - Connecting To Your Own Past'
{HRI 20020903}
(3 Sept 2002 - Version 1.1)
'Leonardo Da Vinci - About Aliens, Dimensions, Consciousness,
Time, And Earth's Past And Future'
'On Those Who Want Awareness Not Understood'
'
____________
Verification:
http://www.angelfire.com/space/platoworld
and poet
This is 'learnware' - it may not be altered, and it is free for
anyone who learns from it and (even if he can not learn from it)
who passes it on unaltered, and with this message included,
to others who might be able to learn from it.
None of my writings may be used, ever, to support any political
or religious or scientific agenda, but only to educate, and to
encourage people to judge un-dominated and for themselves,
about any organizations or individuals.
Send free-of-Envy and free-of-Hate, Beautiful e-mails to:
PlatoWorld at Lycos.com
===
Subject: B+ tree: How to obtain key from value
I am looking for solution (technique) that will allow me, while using
traditional B+ tree, to obtain key from value without creating
reversed table. Some overhead is acceptable but not reversed
table. Looping is also not an option.
===
Subject: Re: vote on cantor !!!
> Well, not really. Set theory is now almost universally accepted as a
> foundation for mathematics.
> Accepted as a *foundation* for mathematics? Or merely as a 'lingua
> universalis'?
> That is, as a language that allows mathematicians to easily communicate
> complex constructions and proofs across the borders of the various
> sub-disciplines and sub-cultures.
> My impression is that most practicing mathematicians avoid the subject
> of foundations like poop on the stoop. And the few who do care usually
> end up doing category theory or some variant of intuitionism. Almost as
> if practically -everyone- who thinks a bit deeper about it eventually
> concludes that set theory is sloppy, superficial, oversimplifying, in
> short: unconvincing.
Well, that's not the impression that I get. My impression is that most
> mathematicians are happy with ZFC as a foundation. Do you know of any
> empirical work that has been done on the matter?
'Are happy with' is exactly the right expression.
They're happy with it, and therefore don't think further (at least not
about foundations).
> Set-talk is widely used and accepted, yes. But the main reason of its
> popularity seems to be that it is an *anti-foundation*: with set-talk,
> mathematicians can do mathematics -without bothering further- about
> foundational issues. And there's nothing wrong with that.
> As long as we're honest about it, to ourselves, to our students, and to
> the rest of the world.
--
Herman Jurjus
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<rubrum-344939.19010608072007@newsclstr02.news.prodigy.com>
<nk6393p360vhcbuoje5e0tjmamu998cf5q@4ax.com
> The
> real numbers are represented by the paths of the decimal or binary
> tree. I think anybody can understand that.
> What the real numbers may or may not be represented by is
different
> from what the real numbers are defined to BE.
> Well, perhaps you will agree then to the following theorem: The set of
> all representations of all real numbers in the binary tree is
> equinumerous with the set of all the nodes of the tree.
What I would do is this:
Have us agree as to what logic, axioms, and definitions we're using in
> any given context.
You should know how a binary representation of a real number looks
like. That is enough to compare the numbers of paths and nodes in the
tree.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<11s29399vde550a5h7rh83t8lhpvth2gn9@4ax.com>
<rubrum-344939.19010608072007@newsclstr02.news.prodigy.com>
<nk6393p360vhcbuoje5e0tjmamu998cf5q@4ax.com
> Yes. It is wrong. Cantor's diagonal argument neglects the fact that
> for infinite series lim [n-->oo] 10^-n = 0. Hessenbergs argument
uses
> an impossible set.
> So, you claim it is _not_ a theorem of ZFC that the reals are
> uncountable? That is, you claim that there is _not_ a sequence of
> formulas that is a proof in first order logic from the axioms of Z,
> with the last entry in the sequence being the formula ~Ef f:N
->onto
> The sequence of formulas is without any value unless interpreted.
Interpretation is another matter. We can talk about that. But it would
> help to start if you would just say whether or not you recognize that
> the described sequence of formulas exists.
> This
> interpretation is wrong.
WHAT interpretation? We give interpretations through the method of
> structures for the language. What SPECIFICALLY and MATHEMATICALLY are
> you talking about that is a wrong interpretation?
> For Cantor's diagonal argument, which is an
> impossibility proof, we can easily find a counter example:
You can find using WHAT axioms and what logic?
That one which you are replying to below.
> The list
> 0.0
> 0.1
> 0.11
> 0.111
> ...
> with replacement rule 0 --> 1 yields a diagonal number which is
> already in the list.
First, I don't even know what diagonal you claim to be in the list.
> The diagonal you produced is 1.111111...
I think you are producing aleph_0 ones behind the point. And that
cannot be produced unless there is a line in the list with aleph_0 - 1
ones behind the point.
Second, more importantly, making a CERTAIN list and a CERTAIN diagonal
> on that list is NOT a refutation of the diagonal argument.
Sorry, you have not understood Cantor's argument. He does not talk
about any specific list of numbers but maintains to have proved that
the diagonal number is different from the entries of the list in
*every* list. Of course he can easily be proved wrong *if* he uses
real numbers (instead of infinite sequences) because of the double
representation of some rational numbers like 1.000... = 0.999.... But
this meaning of his proof cannot be proven, because he referred to it
in a somewhat ambiguous way. Compare the due discussion in and
subsequent to:
> The
> diagonal argument is that for ANY countable sequence f of denumerable
> binary (we'll use binary in this context) sequences, there is a
> denumerable binary sequence not in the range of the countable sequence
> f. To prove a counterexample is NOT to show a countable sequence f
> such that a CERTAIN diagonal is in the range of f, but rather to show
> a countable sequence f such that EVERY denumerable binary sequence is
> in the range of f.
My example shows that the diagonal argument is simply wrong.
Recognizing this any talking about uncountability becomes meaningless.
There is no clue in considering this notinon any further.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<virgil-391946.13594828062007@comcast.dca.giganews.com>
<46845cdd@news2.lightlink.com>
<spn8831u3hh23rp13c8t7cqhurop78bb36@4ax.com>
<46853961@news2.lightlink.com>
<46879459@news2.lightlink.com>
<468c0341@news2.lightlink.com>
<468fd687@news2.lightlink.com > If all real numbers exist, then they can be
written in that form in
> which they exist. This written expresses exactly the form in
which
> the real numbers in binary representation do exist.
> So you equate exists and can be written, and then you define
can
> be written as existing. Very unhelpful.
> What kind of existence do you have in mind?
None in particular, really.
Please let me know what your definition of to exist is.
> I was asking /you/ to clarify /your/
> statement If the reals exist, then... (some implication regarding
> 'writing'). Did you have something in mind when you said it, or were
> you just speaking automatically?
Contrary to you I have thought about the definition of existence. My
definition is, that if a number does exist, then it can be written (or
known). Or, the other way round, a number that cannot be written (or
known) does not exist.
> (Of course, can be
> written in mind, is also included into my definition.
Of course!
> But most people
> cannot write more the 40000 digits in mind.)
Well, what can you do? Most people aren't geniuses; they just get by
> as best they can. Truth be told, most people don't think about this
> stuff /at all/.
Yes, but at least mathematicians should switch now from belief to
proof.
> Bananas are not numbers. Not only that numbers do not taste, they
> cannot even exist without at least one representation.
I agree that I could hardly expect to /describe/ to you /which/ number
> I was speaking of, without /some/ form of representation; anymore than
> I could could describe to you the flavor of a banana, without some
> form of representation.
You could send me a banana.
However, that is surely instead a matter of the thorny philosophical
> problem of communicating my thoughts to you; and not a matter of the
> existence of the thing of which I would speak. Don't you agree?
No. You could not send me a number without having specified it. But
cannot specify more than countably many.
> I know from my experience that many people answer my questions as I
> expect it. Even most discutants here have understood the principle of
> the binary tree.
I certainly know what /I/ mean by it. However, it appears you mean
> some other thing than what I have in mind,
Try to understand how the tree would look if drawn down to level n.
Cut it there. Find that the set of cut edges is just the set of nodes
above the cut + 1. Repeat this in infinity. Then you have the result
for infinite paths if such exist.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<spn8831u3hh23rp13c8t7cqhurop78bb36@4ax.com>
<46853961@news2.lightlink.com>
<46879459@news2.lightlink.com>
<468c0341@news2.lightlink.com>
<468fd687@news2.lightlink.com
> If all real numbers exist, then they can be written in that form
in
> which they exist. This written expresses exactly the form in
which
> the real numbers in binary representation do exist.
> So you equate exists and can be written, and then you define
can
> be written as existing. Very unhelpful.
> What kind of existence do you have in mind?
> None in particular, really.
Please let me know what your definition of to exist is.
I don't speak for Chas, but for me, if I'm talking about a particular
mathematical theory (such as Z set theory), when I say x exists such
that P I mean that the theory proves the formula ExPx.
> I was asking /you/ to clarify /your/
> statement If the reals exist, then... (some implication regarding
> 'writing'). Did you have something in mind when you said it, or were
> you just speaking automatically?
Contrary to you I have thought about the definition of existence. My
> definition is, that if a number does exist, then it can be written (or
> known). Or, the other way round, a number that cannot be written (or
> known) does not exist.
If you were to put that in some mathematical language and add it as an
axiom of Z set theory, then I would guess that you would have an
inconistent theory. I don't think anyone disputes that your principle
is inconsistent with Z set theory. Okay, so if anything of any
mathematical interest is to come from you, then it would be for you to
state your own theory. Give us your inference rules, axioms, and
definitions and let's see what mathematics you can derive. If you can
derive all the mathematics needed for the various scientific purposes,
then I bet you will get some of the appreciation that you obviously
are not getting just by banging over and over on a drum that beats
nothing more really than that Z set theory does not match your
finitistic intuitions.
> Try to understand how the tree would look if drawn down to level n.
> Cut it there. Find that the set of cut edges is just the set of nodes
> above the cut + 1. Repeat this in infinity. Then you have the result
> for infinite paths if such exist.
Please state repeat this in infinity as a formula in the language of
set theory, or please state some mathematical language of your own in
which repeat this in infinity is a formula. Otherwise, you're
arguing as to you personal PHILOSOPHY about all this and are not
giving us a MATHEMATICAL argument.
MoeBlee
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
> If all real numbers exist, then they can be written in that form in
> which they exist. This written expresses exactly the form in
which
> the real numbers in binary representation do exist.
> So you equate exists and can be written, and then you define
can
> be written as existing. Very unhelpful.
> What kind of existence do you have in mind?
> None in particular, really.
Please let me know what your definition of to exist is.
> I was asking /you/ to clarify /your/
> statement If the reals exist, then... (some implication regarding
> 'writing'). Did you have something in mind when you said it, or were
> you just speaking automatically?
Contrary to you I have thought about the definition of existence. My
> definition is, that if a number does exist, then it can be written (or
> known). Or, the other way round, a number that cannot be written (or
> known) does not exist.
> (Of course, can be
> written in mind, is also included into my definition.
> Of course!
> But most people
> cannot write more the 40000 digits in mind.)
> Well, what can you do? Most people aren't geniuses; they just get by
> as best they can. Truth be told, most people don't think about this
> stuff /at all/.
Yes, but at least mathematicians should switch now from belief to
> proof.
> Bananas are not numbers. Not only that numbers do not taste, they
> cannot even exist without at least one representation.
> I agree that I could hardly expect to /describe/ to you /which/ number
> I was speaking of, without /some/ form of representation; anymore than
> I could could describe to you the flavor of a banana, without some
> form of representation.
You could send me a banana.
> However, that is surely instead a matter of the thorny philosophical
> problem of communicating my thoughts to you; and not a matter of the
> existence of the thing of which I would speak. Don't you agree?
No. You could not send me a number without having specified it. But
> cannot specify more than countably many.
I know from my experience that many people answer my questions as I
> expect it. Even most discutants here have understood the principle of
> the binary tree.
> I certainly know what /I/ mean by it. However, it appears you mean
> some other thing than what I have in mind,
Try to understand how the tree would look if drawn down to level n.
> Cut it there. Find that the set of cut edges is just the set of nodes
> above the cut + 1. Repeat this in infinity. Then you have the result
> for infinite paths if such exist.
You are using inductive proof, in the infinite case, are you not?
TY TO
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<virgil-391946.13594828062007@comcast.dca.giganews.com>
<46845cdd@news2.lightlink.com>
<spn8831u3hh23rp13c8t7cqhurop78bb36@4ax.com>
<46853961@news2.lightlink.com>
<46879459@news2.lightlink.com>
<468c0341@news2.lightlink.com>
<468fd687@news2.lightlink.com That is /obviously/ not equivalent to the
statement:
There exist a pair of distinct paths such that there does /not/ exist
> a node n where these paths separate.
The first statement is true; the second false.
Ok. Then use the reverse of this statement and conclude that for every
separation there is a node.
If, however, this seems to be too complicated, then cut the tree at
any given height x and find that the number of edges which you cut is
the same as the number of nodes above + 1. This should be sufficient
to convince you that for any given x the paths separated at that
height x are equinumerous with the nodes above this height.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<spn8831u3hh23rp13c8t7cqhurop78bb36@4ax.com>
<46853961@news2.lightlink.com>
<46879459@news2.lightlink.com>
<468c0341@news2.lightlink.com>
<468fd687@news2.lightlink.com
> That is /obviously/ not equivalent to the statement:
> There exist a pair of distinct paths such that there does /not/ exist
> a node n where these paths separate.
> The first statement is true; the second false.
Ok. Then use the reverse of this statement and conclude that for every
> separation there is a node.
If, however, this seems to be too complicated, then cut the tree at
> any given height x and find that the number of edges which you cut is
> the same as the number of nodes above + 1.
They're both aleph_0.
> This should be sufficient
> to convince you that for any given x the paths separated at that
> height x
What do you mean by 'the paths separated at x'? Do you mean some
particular subset of the set of infinite paths? What exact subset is
that? Perhaps you have defined that expression previously, but it
would help me if you'd restate your definition
MoeBlee
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<spn8831u3hh23rp13c8t7cqhurop78bb36@4ax.com>
<46853961@news2.lightlink.com>
<46879459@news2.lightlink.com>
<468c0341@news2.lightlink.com>
<468fd687@news2.lightlink.com>
<virgil-7EE547.15100210072007@comcast.dca.giganews.com>
<virgil-08A1D0.16410211072007@comcast.dca.giganews.com
> WM again conflates properties of members with properties of sets
having
> those members.
> Every natural number is a set of natural numbers. And from every such
> set you can say by what element it is distinguished from any smaller
> set. By what element is N distinguished from any smaller set?
That question implies that N should also be a natural number, i.e., be a
> member of itself, which it is not.
There is an element, by which n+1 is distinguished from any smaller
number n. What is the problem?
My question is only by what element N is distinguished from any
natural number? If such an element is not available, then I would
dispute that N is different from the collection of all natural
numbers.
> The tree also has an uncountably infinite
> number of infinite-length paths (which are isomorphic to the
> non-terminating rationals and irrationals in [0,1), but which
> cannot be mapped to the naturals).
> Because these paths do not consist of nodes?
> Because they DO consist of nodes. Each finite segment is a node.
> That is an obvious bijection. It can even be used to define finite
> segments. But what about the infinite segments?
If one has a set of such initial segments, say S, in which
> (1) the root node forms a segment in S and
> (2) each segment in S is a subset of all larger segments in the tree
> (3) no proper subset of S has properties (1) and (2)
> Then S is an infinite segment and defines a path in that tree.
Is there a node in the tree which belongs to S but does not belong to
any finite intial segments of S. If not, then S is nothing but the
union of all of its finite initial segments. And for all of them we
find P(x)/K(x) < 2.
> In OUR infinite binary trees, there is no path that can separate
itself
> from the rest by a single node. Just as with the infinite decimal
> expansions of all the reals in [0,1], there is no real which can be
> separated from all others by what happens only up to some fixed
digit
> position.
> Therefore there is no real number with non-periodical expansion which
> is only determined by its sequence of digits. (Or do you know of one
> such real number?)
I do. Here are infinitely many of them:
> f(b) = sum_{n in N} 1/(10^b)^(n!) , for every b in N{0}
I see a recipe for constructing a number. What I miss is the
construction of a number. (Compare the tiny intelligences of my toy
universe model 7 who claim that ... makes their universe infinite.
In truth the situation is quite different. They are only infinitely
stupid ...)
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<46879459@news2.lightlink.com>
<468c0341@news2.lightlink.com>
<468fd687@news2.lightlink.com>
<virgil-7EE547.15100210072007@comcast.dca.giganews.com>
<virgil-08A1D0.16410211072007@comcast.dca.giganews.com
> WM again conflates properties of members with properties of sets
having
> those members.
> Every natural number is a set of natural numbers. And from every such
> set you can say by what element it is distinguished from any smaller
> set. By what element is N distinguished from any smaller set?
> That question implies that N should also be a natural number, i.e., be
a
> member of itself, which it is not.
There is an element, by which n+1 is distinguished from any smaller
> number n. What is the problem?
My question is only by what element N is distinguished from any
> natural number? If such an element is not available, then I would
> dispute that N is different from the collection of all natural
> numbers.
Then you would dispute what is not disputed. We DON'T hold that N is
different from the set of all natural numbers. Indeed, by definition
of 'N' , we hold that N is the set of natural numbers. (Or maybe
you're using 'N' in some sense other than 'the set of natural
numbers.)
> The tree also has an uncountably infinite
> number of infinite-length paths (which are isomorphic to the
> non-terminating rationals and irrationals in [0,1), but which
> cannot be mapped to the naturals).
> Because these paths do not consist of nodes?
> Because they DO consist of nodes. Each finite segment is a
node.
> That is an obvious bijection. It can even be used to define finite
> segments. But what about the infinite segments?
> If one has a set of such initial segments, say S, in which
> (1) the root node forms a segment in S and
> (2) each segment in S is a subset of all larger segments in the tree
> (3) no proper subset of S has properties (1) and (2)
> Then S is an infinite segment and defines a path in that tree.
Is there a node in the tree which belongs to S but does not belong to
> any finite intial segments of S. If not, then S is nothing but the
> union of all of its finite initial segments. And for all of them we
> find P(x)/K(x) < 2.
Any denumerable sequence is the union of all its initial segments, yes
I agree. But please remind me, what is 'P' and 'K' here?
MoeBlee
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<468c0341@news2.lightlink.com>
<468fd687@news2.lightlink.com>
<11s29399vde550a5h7rh83t8lhpvth2gn9@4ax.com>
<lvd893tqg38gc8ib0p77lt0m8qohu8v0m6@4ax.com>
<lei8939v9u15og0iu41oeasj04vn9iqb3c@4ax.com>
<jqga93hki5fqpqkmooc79anl36rbj48lj6@4ax.com
> Now, if WM can prove, IN Z SET THEORY, that P is countable, then let
> him do so.
And the set of denumerable binary sequences is countable iff P is
> countable.
(This business of translating that into a question about a tree
> strikes me as unneeded, but WM is free to try whatever he can.)
And if WM claims that P is countable, but not necessarily proven in Z
> set theory, then I suggest he stipulate PRECISELY what axioms or
> inference rules he'd add to prove his claim.
Every path which is separated needs a node for this sake. Therefore
the number of nodes and paths is equal (well there is one more path,
because two paths spring off from the root node).
You can easily see this when you cut the tree at any height x: The
number of edges which you cut is at most the number of nodes above
your cut + 1. This is true for every height x and therefore is true
for all existing x.
If ZFC is missing this subtle result, then ZFC is without value to
support subtle thoughts.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<468c0341@news2.lightlink.com>
<468fd687@news2.lightlink.com>
<11s29399vde550a5h7rh83t8lhpvth2gn9@4ax.com>
<lvd893tqg38gc8ib0p77lt0m8qohu8v0m6@4ax.com>
<lei8939v9u15og0iu41oeasj04vn9iqb3c@4ax.com>
<jqga93hki5fqpqkmooc79anl36rbj48lj6@4ax.com
> Now, if WM can prove, IN Z SET THEORY, that P is countable, then let
> him do so.
> And the set of denumerable binary sequences is countable iff P is
> countable.
> (This business of translating that into a question about a tree
> strikes me as unneeded, but WM is free to try whatever he can.)
> And if WM claims that P is countable, but not necessarily proven in Z
> set theory, then I suggest he stipulate PRECISELY what axioms or
> inference rules he'd add to prove his claim.
Every path which is separated needs a node for this sake. Therefore
> the number of nodes and paths is equal (well there is one more path,
> because two paths spring off from the root node).
Every path which is separated needs a node for this sake. Unless you
tell me how that translates to a specific mathematical formula, I can
only surmise that what you're saying amounts to this: Given any two
different infinite paths, they differ upon the decision at at least
one node. Yes, I agree with that. But I have no idea by what logic you
think that entails that there is a bijection between the set of nodes
and the set of infinite paths.
(And there can't be just one more of one, since both sets are
infinite.)
Maybe we can start by just getting from you the very last line in your
proof: Suppose you say, There exists a bijection between the set of
nodes and the set of infinite paths.
What are the previous lines that you claim to have established and
that some principle of logic dictates that those exact previous lines
entail the aforementioned last line, and what is that EXACT principle
of logic (or, preferably, exaxt inference rule) that dictates that
your last line follows from those previous lines?
> You can easily see this when you cut the tree at any height x: The
> number of edges which you cut is at most the number of nodes above
> your cut + 1. This is true for every height x and therefore is true
> for all existing x.
Do you mean that for some NATURAL NUMBER x, the full binary tree with
ONLY x number of levels has deleted from the INFINITE full binary tree
exactly as many edges as it has deleted nodes +1? Well, since the set
of deleted edges and the set of deleted nodes are both denumerable, we
have that +1 has no affect, and the number is the same. Yes, the
number of deleted edges is the number of deleted nodes since that
number is aleph_0.
Anyway, I still don't know how you think this proves that there is a
bijection between the set of infinite paths and the set of nodes.
Please give me the principle of logic you use and the exact statements
that you apply that principle to in order to get that there exists a
bijection between the set of nodes and the set of infinite paths.
> If ZFC is missing this subtle result, then ZFC is without value to
> support subtle thoughts.
If ZFC is missing the result, then ZFC is consistent, since any theory
that fails to prove even a single result in its language is a
consistent theory. As to value in supporting subtle thoughts, I
don't think anyone has claimed that ZFC has or is supposed to have any
value in supporting your own particular subtle thoughts, especially
those that are based on an UNSTATED logic.
MoeBlee
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<11s29399vde550a5h7rh83t8lhpvth2gn9@4ax.com>
<rubrum-344939.19010608072007@newsclstr02.news.prodigy.com>
<nk6393p360vhcbuoje5e0tjmamu998cf5q@4ax.com>
<virgil-480E93.16121811072007@comcast.dca.giganews.com > The natural 
numbers
need not be defined at all.
Kronecker supposedly said,
> God gave us the natural numbers and all else is the work of man.
Kronecker went farther: Supposedly he said: God gave us the integers.
I don't believe that he gave us the negative numbers or zero.
Is WM worshipping at Kronecker's altar?
> They are there,
> notwithstanding of your or any one else's approval or axiomatics.
Where is there?
Every set of n elements is a member of the fundamental set of n. Every
such set is n. And if there is but one set with n elements in your
reach, then this set alone is the number n for you.
> The
> real numbers are represented by the paths of the decimal or binary
> tree. I think anybody can understand that.
The reals are also represented by Dedekind cuts of rationals and by
> equivalences of Cauchy sequences of rationals and by continued fractions.
Who told you how many Dedekind cuts and Cauchy equivalence classes do
exist? You know two or three candidates like sqrt(2), e , pi, log2.
But you don't know and cannot know more than countably many of them.
You believe and believe and believe ... that they exist somewhere.
Whereas I can and did tell you where my numbers exist.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<468c0341@news2.lightlink.com>
<468fd687@news2.lightlink.com>
<virgil-7EE547.15100210072007@comcast.dca.giganews.com
> Every natural number is a set of natural numbers. And from every
such
> set you can say by what element it is distinguished from any
smaller
> set. By what element is N distinguished from any smaller set?
> Given any natural number j there is a natural number k that is not a
> member of j. That distinguishes j from N, since every natural number
> is a member of N.
> Just use the axiom of extensionality, WM.
> Therefore I ask: By what element is N distinguished from any smaller
> set?
I TOLD you. Given any natural number j, we have that N is
> distinguished from j by N having as members all the natural numbers
> that are not members of j.
That is already satisfied for any k > j: k is different from j because
it contains all the natural numbers which are in k but not in j.
My question is not what distinguishes N from some given j, but what
distinguished N from the set of all j. Precisely: What distinguishes N
from the collection of all natural numbers? Or, in case of the binary
tree, what distinguishes an infinite path from the set of all finite
paths?
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
> Every natural number is a set of natural numbers. And from every such
> set you can say by what element it is distinguished from any smaller
> set. By what element is N distinguished from any smaller set?
> Given any natural number j there is a natural number k that is not a
> member of j. That distinguishes j from N, since every natural number
> is a member of N.
> Just use the axiom of extensionality, WM.
> Therefore I ask: By what element is N distinguished from any smaller
> set?
> I TOLD you. Given any natural number j, we have that N is
> distinguished from j by N having as members all the natural numbers
> that are not members of j.
That is already satisfied for any k > j: k is different from j because
> it contains all the natural numbers which are in k but not in j.
My question is not what distinguishes N from some given j, but what
> distinguished N from the set of all j. Precisely: What distinguishes N
> from the collection of all natural numbers? Or, in case of the binary
> tree, what distinguishes an infinite path from the set of all finite
> paths?
Always some finite node difference, since that's all there is.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<468c0341@news2.lightlink.com>
<468fd687@news2.lightlink.com>
<virgil-7EE547.15100210072007@comcast.dca.giganews.com
> Every natural number is a set of natural numbers. And from every
such
> set you can say by what element it is distinguished from any
smaller
> set. By what element is N distinguished from any smaller set?
> Given any natural number j there is a natural number k that is not
a
> member of j. That distinguishes j from N, since every natural
number
> is a member of N.
> Just use the axiom of extensionality, WM.
> Therefore I ask: By what element is N distinguished from any smaller
> set?
> I TOLD you. Given any natural number j, we have that N is
> distinguished from j by N having as members all the natural numbers
> that are not members of j.
That is already satisfied for any k > j: k is different from j because
> it contains all the natural numbers which are in k but not in j.
My question is not what distinguishes N from some given j, but what
> distinguished N from the set of all j. Precisely: What distinguishes N
> from the collection of all natural numbers?
What? N IS the set of all natural numbers. Or are using the letter 'N'
to stand for something else?
> Or, in case of the binary
> tree, what distinguishes an infinite path from the set of all finite
> paths?
I mentioned different senses in which one can take 'path' here:
sequence of alternating vertices and edges, sequences of vertices,
sequence of edges, or sequence of binary decisions. In any case,
unless you have some further argument that rests on the distinction,
any of those is sufficient for this discussion, and a decision
sequence seems to be the simplest.
Each finite path is a finite sequence. An infinite path is an infinite
sequence. No MEMBER of that infinite sequence is a finite sequence,
though SEGMENTS of that infinite path are finite sequences.
I don't know what answer you could be expecting beyond the utterly
obvious: An infinite sequence is a certin kind of set of ordered
pairs, and a finite sequence is a certain kind of set of ordered
pairs, and a set of ordered pairs (such as an infinite path) is not
here a set of sets of ordered pairs.
MoeBlee
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
> Every natural number is a set of natural numbers. And from every
such
> set you can say by what element it is distinguished from any smaller
> set. By what element is N distinguished from any smaller set?
> Given any natural number j there is a natural number k that is not a
> member of j. That distinguishes j from N, since every natural number
> is a member of N.
> Just use the axiom of extensionality, WM.
> Therefore I ask: By what element is N distinguished from any smaller
> set?
> I TOLD you. Given any natural number j, we have that N is
> distinguished from j by N having as members all the natural numbers
> that are not members of j.
> That is already satisfied for any k > j: k is different from j because
> it contains all the natural numbers which are in k but not in j.
> My question is not what distinguishes N from some given j, but what
> distinguished N from the set of all j. Precisely: What distinguishes N
> from the collection of all natural numbers?
What? N IS the set of all natural numbers. Or are using the letter 'N'
> to stand for something else?
> Or, in case of the binary
> tree, what distinguishes an infinite path from the set of all finite
> paths?
I mentioned different senses in which one can take 'path' here:
> sequence of alternating vertices and edges, sequences of vertices,
> sequence of edges, or sequence of binary decisions. In any case,
> unless you have some further argument that rests on the distinction,
> any of those is sufficient for this discussion, and a decision
> sequence seems to be the simplest.
Each finite path is a finite sequence. An infinite path is an infinite
> sequence. No MEMBER of that infinite sequence is a finite sequence,
> though SEGMENTS of that infinite path are finite sequences.
Define MEMBER of a sequence. Is that an element? Does not every
element of a sequence denote a finite sequence, provided the sequence is
countable?
I don't know what answer you could be expecting beyond the utterly
> obvious: An infinite sequence is a certin kind of set of ordered
> pairs, and a finite sequence is a certain kind of set of ordered
> pairs, and a set of ordered pairs (such as an infinite path) is not
> here a set of sets of ordered pairs.
MoeBlee
>
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<468c0341@news2.lightlink.com>
<468fd687@news2.lightlink.com>
<virgil-7EE547.15100210072007@comcast.dca.giganews.com
> By the way, would you please show that your definitions of
'countable'
> and 'uncountable' are equivalent with the ordinary definitions
without
> using choice, or say why you've elected to adopt defintions that
> require choice to prove equivalence with the ordinary definitions.
> The ordinary definitions are:
> A set is countable if it is finite or countably infinite.
> A set is countably infinite if there is a bijection with N.
And those aren't the definitions you gave earlier.
No? What did I misdefine or did you misunderstand?
By the way, just as a matter of style, I'd state it so that
> 'countable'/'countably' does not appear, respectively, on both sides
> of the first definition:
That is but an intermediate step. You can also state:
A set S is countable if S is finite or if there is a bijection of S
and N.
But I prefer to use statements as easily as possible to comprehend.
Therefore I chose the first version.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<468c0341@news2.lightlink.com>
<468fd687@news2.lightlink.com>
<virgil-7EE547.15100210072007@comcast.dca.giganews.com
> By the way, would you please show that your definitions of
'countable'
> and 'uncountable' are equivalent with the ordinary definitions
without
> using choice, or say why you've elected to adopt defintions that
> require choice to prove equivalence with the ordinary definitions.
> The ordinary definitions are:
> A set is countable if it is finite or countably infinite.
> A set is countably infinite if there is a bijection with N.
> And those aren't the definitions you gave earlier.
No? What did I misdefine or did you misunderstand?
You don't recall the definitions you posted?
MoeBlee
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
On Wed, 11 Jul 2007 15:14:26 -0700, MoeBlee <jazzmobe@hotmail.com> Starting
from node /a/ a path (or rather the edge the path goes
trough)
> either goes left or right. Hence we can represent paths by sequences of
> the form (<direction>, <direction>, <direction>, ...) where <direction> 
is
either /left/ or /right/. Let's abbreviate /left/ with 0 and /right/
> with 1. Then any path can be represented (or encoded) by a sequence
> (p_1, p_2, p_3, ...) with p_i e (0,1).
> Right, think it's called a 'decision path' or 'decision sequence' or
> something like that. It seems to me to be a good simple definition for
> the purpose of this discussion.
>
Yes. Even WM is aware of this way to represent paths.
So when talking about the set of paths we can as well talk about the
set P consisting of all sequences (p_1, p_2, p_3, ...) with p_i e
(0,1). (Since there exists a natural bijection between the set of
paths and P.)
Now, if WM can prove, IN Z SET THEORY, that P is countable, then let
> him do so. [...]
(This business of translating that into a question about a tree
> strikes me as unneeded, but WM is free to try whatever he can.)
>
Right.
And if WM claims that P is countable, but not necessarily proven in Z
> set theory, then I suggest he stipulate PRECISELY what axioms or
> inference rules he'd add to prove his claim. Because I'm sure no one
> disputes that WM can throw a bunch of terminology together and give
> his reasons why he believes reality shows that P is countable, but
> that is not mathematical proof.
>
Well, his arguments rest on some _intuitions_ concerning the number
of paths in an infinite binary tree. Granted, there _are_ some
counterintuitive states of affairs involved, imho. It's just that WM
does not understand that /counterintuitive/ does not imply /wrong/ in
math.
Learn some logic. Learn some mathematics. Or better yet,
give up mathematics and take up basket weaving.
(from sci.math, @M.9fckenheim)
F.
--
E-mail: info<at>simple-line<dot>de
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
> Infinite-case inductive proof doesn't contradict ZFC?
> Until we see a clear statement of what your Infinite-case Induction
> says, we have no way of judging, but I suspect that it will allow proof
> of both some proposition and also of its negation if added to ZFC.
Yes, I am sure it would, such as that there are many different sizes of
> countably infinite sets. I am certainly not trying to append it to ZFC.
> If it does, how do you judge which makes more sense?
> As a LOT more people have looked at ZFC and not found it
> self-contradictory, I trust it a lot more that I trust TO's
> mathematics.
It's not internally self contradictory as an axiomatic system, but
> plenty of people have issues with the results, and so maybe other axioms
> need to be explored as alternatives.
Lots of people explore NBG and NF and its offspring.
> Does fundamental truth mean anything?
What is your definition of fundamental truth?
> ToeKnee
It's a matter of priority. Fundamental truths are what you hold most
> inviolable.
Then they differ from person to person, as what WM holds inviolable I do
not, and vice versa.
> It seems to me, to preserve the fundamental character of
> what addition means, one can't sacrifice a+b=b+a or a>0 <-> a+b>b.
Then give me an unambiguous definition of what addition means that can
be applied outside of standard arithmetic. Within arithmetic, all the
additions that I am aware of arise from formal definitions followed by
deduction from those definitions of those properties which we expect
them to have.
> I
> think if an infinite number is greater than any finite, then a fact
> about all numbers greater than some finite should hold true for that
> infinite number. Those kinds of things. Basic facts.
How TO thinks about things does not make them work that way.
And what TO thinks about things are quite often not facts.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
Infinite-case inductive proof doesn't contradict ZFC?
Until we see a clear statement of what your Infinite-case Induction
> says, we have no way of judging, but I suspect that it will allow proof
> of both some proposition and also of its negation if added to ZFC.
> Yes, I am sure it would, such as that there are many different sizes of
> countably infinite sets. I am certainly not trying to append it to ZFC.
> If it does, how do you judge which makes more sense?
As a LOT more people have looked at ZFC and not found it
> self-contradictory, I trust it a lot more that I trust TO's
> mathematics.
> It's not internally self contradictory as an axiomatic system, but
> plenty of people have issues with the results, and so maybe other axioms
> need to be explored as alternatives.
Lots of people explore NBG and NF and its offspring.
Yes, and there are lots of muslims and neo-cons. So?
> Does fundamental truth mean anything?
> What is your definition of fundamental truth?
> ToeKnee
> It's a matter of priority. Fundamental truths are what you hold most
> inviolable.
Then they differ from person to person, as what WM holds inviolable I do
> not, and vice versa.
>
Perhaps.
> It seems to me, to preserve the fundamental character of
> what addition means, one can't sacrifice a+b=b+a or a>0 <-> a+b>b.
Then give me an unambiguous definition of what addition means that can
> be applied outside of standard arithmetic. Within arithmetic, all the
> additions that I am aware of arise from formal definitions followed by
> deduction from those definitions of those properties which we expect
> them to have.
>
Then please state them, and I'll tell you which rule I object to, or
show you how what I say follows.
> I
> think if an infinite number is greater than any finite, then a fact
> about all numbers greater than some finite should hold true for that
> infinite number. Those kinds of things. Basic facts.
How TO thinks about things does not make them work that way.
And what TO thinks about things are quite often not facts.
So, positive infinite numbers are not greater than all finites? Huh!
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
> Yes, trying to pin a size on a set with no bound gets tricky,
doesn't
> it?
> It only gets tricky when you refuse to define size. It has
been
> nearly
> three years now, and you still have not defined what you mean by
size.
> If N has a size, then its size is its largest element by obvious
> induction.
> Until you define size, who knows if N has a size? N may also
have
> a volume, and a girth and a favorite color.
> N clearly has a cardinality, but cardinality is actually defined.
> It has no largest element. Therefore it has no size.
> If that is how you define size, then that is true. But it
> is a rather uninteresting result. At least it is consistent.
> And according to Tony's definition, the size of the sets
> { ..., -3, -2, -1, 0}
> { -11, -7, -5, 3, -2, 0}
> is zero. Yeah, I know what Tony's response to this will be
> (I'm gifted in that regard), but it then falls upon Tony to be a
> little more precise about his obvious induction specification.
> Also, Tony must conclude, contrary to what he's stated many
> times in the past, that
> size( { 0, 2, 4, 6, ... } ) x 2 /= size( {0, 1, 2, 3, ...} )
> for the simple reason that neither set has a Tony-defined size.
> (What's twice an undefined amount?) So it's rather inconsistent
> of Tony say that there are twice as many naturals as even
> naturals. For Tony to make any sense of such a claim, he's
> going to have to (yes, hold on to your hat) actually commit to a
> definition of his elusive notion of set size.
> But Tony's heard all this before.
> Yes, and Tony's responded to it as well. N does not have a size.
> You still have not defined 'size'. (CIRCULAR definitions don't
> qualify.). However, your attempt seems to reduce to something along
> the lines of the the size of a set being the set's finite
cardinality.
> Okay, so N does not have a finite cardinality. Yeah, we knew that.
> It behaves along the same lines as finite counts, such that adding or
> subtracting a nonzero count changes it. That's basic and not to be
> sacrificed as a principle. The proper subset is smaller.
You STILL have not defined 'size'.
I just did in another post.
Not satisfactorily.
> Look, if you want to mention certain behviors or properties but not
> give a definition, then perhaps we should regard you as taking 'size'
> as primitive and giving axioms regarding it. That's fine as long as we
> (actually, starting with YOU) are clear about that.
I also just said that in the other post.
Not satisfactorily.
> If I say the smagglewaggle of a set behaves in such and such a way,
> then I have not defined 'the smagglewaggle of a set' but I have taken
> 'smagglewaggle' as primitive and have given my axioms to make it work
> out the way I want.
Right. I am treating size(S) as a primitive one-place operator, like
> 'e', and defining it as an extension of N, preserving certain
> fundamental rules which apply to finite n. I have been thinking of
> defining a set S of set sizes, but that perhaps gets tricky when you
> want to define a set of measures on a class of sets. Still, it seems to
> me that *N, the hyper-naturals, are the set of all set sizes.
It seems to me is mathematically unsatisfactory.
>
<snip Let me rephrase. Essentially, you claim to have a countably infinite 
set
> of nodes, and a countably infinite set of levels of nodes. On level n,
> there are 2^n nodes, which require n bits to address, corresponding to
> the n levels traversed, and whether path 0 or 1 was followed at that
> point. You thus have a power set relation between the countably infinite
> set of levels in the tree, and the countably infinite set of nodes in
> the tree. Can you have a power set relation which is also a bijection?
> No, you cannot. This is a contradiction within ZFC applied to the
> infinite binary tree.
TO implies that a relation that only is known to hold for finite levels
in a tree must also hold for infinite levels, which do not, in fact,
even exist!!!
This is of a piece with his confusion about infinite sets having to
contain infinite members.
Further, there is no possible TYPE of cardinality which you can place on
> the number of nodes (or levels, for that matter), which will yield a
> complete tree with a countably infinite number of paths. So, for you
> there can be no countably infinite set of paths in a tree? That sounds
> like a rather fishy result, don't you think?
There can be lots of countably infinite sets of paths, but the set of
ALL paths is not one of them.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
> Size is that measure of a set which increases with the addition of new
> elements, and decreases with the removal of existing elements.
> <snip
> ToeKnee
To quaify as a definition, you must specify explicitly how much of an
increase or decrease.
And among other things, you now need to prove that such a measure exists
for all, or at least for some classes, of sets.
You have already conceded that it does not hold for all unordered sets.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
> Size is that measure of a set which increases with the addition of new
> elements, and decreases with the removal of existing elements.
> <snip> ToeKnee
> To quaify as a definition, you must specify explicitly how much of an
> increase or decrease.
1 per element.
And among other things, you now need to prove that such a measure exists
> for all, or at least for some classes, of sets.
How so?
You have already conceded that it does not hold for all unordered sets.
For sets without order, it is impossible to determine whether the
bijection preserves order for both sets, and that is crucial to the
measure of infinite sets, which cannot be defined without a production
algorithm.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<46879971@news2.lightlink.com>
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<snip I reject it as absurd that changing the labels on the balls makes any
> difference as to how many there are in the vase, when the same sequence
> of insertions and removals is performed.
So do I, oddly enough. But then, no-one ever said that changing
labels on balls makes any difference. What does make a difference is
which balls you put in and which balls you take out. However, you've
surely been told this before, without effect...
> You're adding
> an infinite number of 9's and getting a zero sum.
No you are not. You are putting... oh, bother it, same as last
paragraph.
Never mind, Tony. Have you looked at Belgian lace-making?
Brian Chandler
http://imaginatorium.org
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<snip
> I reject it as absurd that changing the labels on the balls makes any
> difference as to how many there are in the vase, when the same sequence
> of insertions and removals is performed.
So do I, oddly enough. But then, no-one ever said that changing
> labels on balls makes any difference. What does make a difference is
> which balls you put in and which balls you take out. However, you've
> surely been told this before, without effect...
Oh. What happens if they have no labels? What if the labels aren't
revealed until after the experiment? It's silly, at best.
> You're adding
> an infinite number of 9's and getting a zero sum.
No you are not. You are putting... oh, bother it, same as last
> paragraph.
Never mind, Tony. Have you looked at Belgian lace-making?
Brian Chandler
> http://imaginatorium.org
>
How do you add ten, remove one, and not leave nine remaining? This has
all been explained. You are on the other side of the fence. Here, have a
purple berry.
Tony
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
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<4693b1d7@news2.lightlink.com>
<46964d3d@news2.lightlink.com>
Tony: just a followup to what I said...
<snip
> I reject it as absurd that changing the labels on the balls makes any
> difference as to how many there are in the vase, when the same sequence
> of insertions and removals is performed.
So do I, oddly enough. But then, no-one ever said that changing
> labels on balls makes any difference.
I now realise this isn't true. Your theory is the one where changing
labels really does make a difference.
Suppose you fill an urn with a ball for every natural. (Actually we
can pretend this might include your infinite values, chocolate
sundaes or whatever else you consider must inexorably be there as a
side effect of putting in one ball for each natural.) Your urn is sent
to planet X, where Annedroid goes through the balls, labelling them
(She sticks unremovable labels over whatever you used to identify
them). On planet X they write numbers in base 3 using two-ended
strings of symbols that look like Y, O, ^. So a few of them look like
this: 'O', '^', 'OY', 'OO', 'O^', '^Y', and so on. Annedroid simply
marks any balls with Tinfinite values or similar with a big cross
(leaving your identification intact). She passes the urn to Bobdroid,
who goes through them, ignoring ones marked with a cross, but removing
any whose label includes a 'HAt' sign '^', and dropping them in a
large pot labelled PRISONERS. The urn is then sent to Charles, an
agent on Earth who does this sort of thing. Charles's job is to
translate things from all over the galaxy into Earthspeak, and he's
overworked, but keen, so with a single glance he misremembers that
Planet X is one of the OY (pronounced 'roy' for obscure reasons)
binary planets, and he transscribes all of these 'O', 'Y', 'YO', 'YY',
'YOO' labels back to normal decimal numbers, leaving anything marked
with a big cross untouched. He then gives the urn back to you.
So you now have an urn containing one ball labeled with every (normal
pofnat) positive integer, plus possibly a pile of junk that has your
private meaning, but has been untouched. However, Bobdroid has this
pot full of PRISONERS. Where did they come from, then?
Brian Chandler
http://imaginatorium.org
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
> Tony: just a followup to what I said...
> <snip I reject it as absurd that changing the labels on the balls makes 
any
> difference as to how many there are in the vase, when the same sequence
> of insertions and removals is performed.
> So do I, oddly enough. But then, no-one ever said that changing
> labels on balls makes any difference.
I now realise this isn't true. Your theory is the one where changing
> labels really does make a difference.
Uh, no, it doesn't. You still have an unbounded sum of 9's, either way.
In any case, I find your humorous absurdification of my statements
annoying. Go play with your Annedroid.
Suppose you fill an urn with a ball for every natural. (Actually we
> can pretend this might include your infinite values, chocolate
> sundaes or whatever else you consider must inexorably be there as a
> side effect of putting in one ball for each natural.) Your urn is sent
> to planet X, where Annedroid goes through the balls, labelling them
> (She sticks unremovable labels over whatever you used to identify
> them). On planet X they write numbers in base 3 using two-ended
> strings of symbols that look like Y, O, ^. So a few of them look like
> this: 'O', '^', 'OY', 'OO', 'O^', '^Y', and so on. Annedroid simply
> marks any balls with Tinfinite values or similar with a big cross
> (leaving your identification intact). She passes the urn to Bobdroid,
> who goes through them, ignoring ones marked with a cross, but removing
> any whose label includes a 'HAt' sign '^', and dropping them in a
> large pot labelled PRISONERS. The urn is then sent to Charles, an
> agent on Earth who does this sort of thing. Charles's job is to
> translate things from all over the galaxy into Earthspeak, and he's
> overworked, but keen, so with a single glance he misremembers that
> Planet X is one of the OY (pronounced 'roy' for obscure reasons)
> binary planets, and he transscribes all of these 'O', 'Y', 'YO', 'YY',
> 'YOO' labels back to normal decimal numbers, leaving anything marked
> with a big cross untouched. He then gives the urn back to you.
So you now have an urn containing one ball labeled with every (normal
> pofnat) positive integer, plus possibly a pile of junk that has your
> private meaning, but has been untouched. However, Bobdroid has this
> pot full of PRISONERS. Where did they come from, then?
Brian Chandler
> http://imaginatorium.org
>
If you'd like me to follow your example, fill it with less crap and
stick to the point.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
> If N is a set of consecutive naturals starting at 1, then any size it
> has is also a largest element. The two are equal. If one exists, so does
> the other.
Which is why it makes more sense to start at 0, so that the size of the
set ending with n is always n+1 and is NOT a member of itself.
Note that in ZF and other set theories which prohibit a set being a
member of itself, TO's analysis is forbidden.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
> If N is a set of consecutive naturals starting at 1, then any size it
> has is also a largest element. The two are equal. If one exists, so does
> the other.
Which is why it makes more sense to start at 0, so that the size of the
> set ending with n is always n+1 and is NOT a member of itself.
Note that in ZF and other set theories which prohibit a set being a
> member of itself, TO's analysis is forbidden.
HAHAHAHAHA
Oh My.
Did I not point this out two years ago? Your littlest giant Omega is
based on a difference of 1, while you claim at the same time a
difference of 1 does not change the finite into the infinite. It's not a
valid argument, to claim there is some infinite number, just after the
finites, but with no predecessor. It's silly. My approach is forbidden
by those that wear their pants on their head.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
> It only gets tricky when you refuse to define size. It has been
nearly
> three years now, and you still have not defined what you mean by
size.
> A count of elements, a member of *N, such that adding an element
> increases the size by 1, and removing one decreases it by 1, and
x-1<x<x+1.
Then how does one compare sets neither of which is a subset of the
other?
Suppose, for example, that sets A and B are disjoint (no elements in
> common) and infinite, how do you compare their 'sizes'?
For example, your rule does not even allow one to compare the size of
> the set of odd naturals to the size of the set of even naturals.
Your notion is quite simple, and is valid enough for finite sets, but
> fails for infinite sets, at least if you want to be able to compare
> sizes of arbitrary infinite sets.
> We've been over this plenty before, and I know you don't think my
> statement above is meant to be some system for comparing all sets.
> However, it is a principle which should not be violated, even in the
> case of infinite sets.
Except that no one has yet produces a consistent size definition in
which bijectable infinite sets are ever of different sizes.
Where it is possible to compare two infinite sets quantitatively, each
> has its own quantitative order, and in both those orders there is a
> bijection expressed as a formulaic mapping. The quantitative formula
> expresses their relative size. That's IFR, remember?
As there is not IFR in standard mathematics, we are not bound by it.
And there are unordered (and without AoC unorderable) sets for which TO
theory flops miserably, but cardinality still works.
> If N has a size, then its size is its largest element by obvious
> induction.
> Until you define size, who knows if N has a size? N may also
have
> a volume, and a girth and a favorite color.
> See above.
> N clearly has a cardinality, but cardinality is actually defined.
> It has no largest element. Therefore it has no size.
Each point in a circle is the same size so there is no largest. What
size
> has the set of such points?
Huh? Now, you're talking about something else entirely. The points are
> not quantitatively measured or ordered, much less sequential, and there
> is no first element. We were talking about the identity function between
> element count and value which characterizes the naturals.
So TO's size does not measure comparative sizes for all sets, only a
chosen few, and is thus not a size measure at all, but only a measure
of orderings.
> I don't think numbers as points on the real line, or even infinitesimal
> intervals, is childish. I think it's foolish to rely entirely on
> symbolic arguments without some geometrical foundation. Oh well.
As geometry can be embedded in algebra, if it won't work in algebra, it
> won't in geometry either.
Okay. How about the other direction?
Can TO embed category theory in geometry?
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
> It only gets tricky when you refuse to define size. It has been
nearly
> three years now, and you still have not defined what you mean by
size.
> A count of elements, a member of *N, such that adding an element
> increases the size by 1, and removing one decreases it by 1, and
x-1<x<x+1.
> Then how does one compare sets neither of which is a subset of the
other?
Suppose, for example, that sets A and B are disjoint (no elements in
> common) and infinite, how do you compare their 'sizes'?
For example, your rule does not even allow one to compare the size of
> the set of odd naturals to the size of the set of even naturals.
Your notion is quite simple, and is valid enough for finite sets, but
> fails for infinite sets, at least if you want to be able to compare
> sizes of arbitrary infinite sets.
> We've been over this plenty before, and I know you don't think my
> statement above is meant to be some system for comparing all sets.
> However, it is a principle which should not be violated, even in the
> case of infinite sets.
Except that no one has yet produces a consistent size definition in
> which bijectable infinite sets are ever of different sizes.
Yes I have. It's called IFR.
> Where it is possible to compare two infinite sets quantitatively, each
> has its own quantitative order, and in both those orders there is a
> bijection expressed as a formulaic mapping. The quantitative formula
> expresses their relative size. That's IFR, remember?
As there is not IFR in standard mathematics, we are not bound by it.
You are not bound to anything. You can be as foolish as you choose.
And there are unordered (and without AoC unorderable) sets for which TO
> theory flops miserably, but cardinality still works.
Yeah, so? That doesn't mean cardinality actually means much in those cases.
> If N has a size, then its size is its largest element by obvious
> induction.
> Until you define size, who knows if N has a size? N may also
have
> a volume, and a girth and a favorite color.
> See above.
> N clearly has a cardinality, but cardinality is actually defined.
> It has no largest element. Therefore it has no size.
> Each point in a circle is the same size so there is no largest. What
size
> has the set of such points?
> Huh? Now, you're talking about something else entirely. The points are
> not quantitatively measured or ordered, much less sequential, and there
> is no first element. We were talking about the identity function between
> element count and value which characterizes the naturals.
So TO's size does not measure comparative sizes for all sets, only
a
> chosen few, and is thus not a size measure at all, but only a measure
> of orderings.
Look, in that case, you use a different method. If there are Big'un
points in (0,1], then there are 2*pi*Big'un points in the circumference
of the unit circle. That's pretty exact.
> I don't think numbers as points on the real line, or even infinitesimal
> intervals, is childish. I think it's foolish to rely entirely on
> symbolic arguments without some geometrical foundation. Oh well.
> As geometry can be embedded in algebra, if it won't work in algebra, it
> won't in geometry either.
> Okay. How about the other direction?
Can TO embed category theory in geometry?
Dunno. Maybe. I haven't thought about that.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
> I was just saying that the whole paradox here comes from
> imagining an end to a sequence that doesn't end, using the Zeno time
> machine. You cannot finish the finite naturals, and not touch infinite
> values.
Speak only for yourself, Tony.
Mathematicians have no problem doing just what TO has just declared
> impossible. They have, for example, a finished set, N, containing
> infinitely many finite values.
They claim it exists.
They axiomatize that it exists. And until someone has shown that all of
such axiomatizations are self-contradictory, they will continue to do so.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<snip > Yes, trying to pin a size on a set with no bound gets tricky, 
doesn't
> it?
> It only gets tricky when you refuse to define size. It has been
nearly
> three years now, and you still have not defined what you mean by
size.
> A count of elements, a member of *N, such that adding an element
> increases the size by 1, and removing one decreases it by 1, and
> x-1<x<x+1.
What is a 'count of elements'?
The sum one arrives at when incrementing a start value of 0 once for
> each element.
So show us your count of the power set of the naturals.
You are just using another undefined term.
And going on to define it...
Where?
What is *N?
N is the hyper-naturals, which include infinite values, such as the
> sizes of infinite sets.
What are the hyper-naturals in terms of something already defined?
Why does size appear in your definition of size?
Size is used in a statement which relates size to element membership
> and a basic axiom for addition which should be preserved.
That does not answer the question, but merely adds more confusion to
what was excessively confused before it.
> We start with {1}, of size 1, with max value 1.
Why not start with {} of size 0? If one starts with {1}, one does not
have any way of counting the members of {}.
> As we add each
> successive natural, the size and the max value simultaneous increment,
> and therefore remain equal with the addition of each and every natural
> to the set. The max value and count can never be unequal.
That is another reason to start with {} having count 0 and thereafter
adding 1 to the count of s to get the count of s / {s}.
Makes much more sense, but that would defeat TO's purpose.
That is true for sets of the form
> {1, 2, 3, ... n }
> for some n in N. But N itself is not of that form, neither are
> most sets.
If N has a size omega, then it is of the form {1,2,3,...,omega}
Not in standard, and no one follows TO's non-standard math, except TO.
What is the size of { 1, 2, e } ? And more importantly,
> how do you prove that is the size, using your actual definition of
size?
That set bijects with the initial segment of N ending in 3, with size 3,
> so that is its size. Simple bijection works for finite sets. It's more
> complicated with infinite sets, if it's to make sense.
But TO has not yet got his way complicated enough either to make sense
or to so befuddle those trying to understand it that they give up.
Seriously, can you provide a proof that the set { 1, 2, e} has a size,
> according to your definition of size?
> I think I just did.
> If N has a size, then its size is its largest element by obvious
> induction.
> Until you define size, who knows if N has a size? N may also
have
> a volume, and a girth and a favorite color.
> See above.
Your definition does not apply to N. As far as I can tell your
definition
> only applies to sets of the form { 1, 2, ... n}. Which is fine,
> if you want a definition of size that does not apply to all sets.
> If you want to apply a size to N, then you are going to have a maximum
> value, which causes a contradiction.
So we may roundfile TO's sizes of sets as self-contradictory.
<snip It is childish to be unable to think beyond the simple notion of
> numbers as points on a line. The number line is a tool used to
teach
> children about the real numbers. It is not the end all and be all of
> the concept of numbers. Anyway, what geometrical foundation
> are you relying on? Your definition of size above made no mention
> of geometry.
> When considering real numbers and subsets thereof, the real line is a
> perfect representation.
The real line presupposes a well understood set of real numbers as
well as any geometical notions.
> We can choose any point and call it 0. Using a
> compass, we can choose an arbitrary point 1, and then a next one, etc,
> and map out the naturals. We can subdivide any interval, and through
> such subdivisions, construct any digital fraction.
One can in theory, however tedious the practice might be, construct any
rational as well as any square root of a (positive) rational, as well as
a large variety of algebraic numbers as points on that line.
Think of each number as a vector, from 0 to whatever point indicates
> that number. If the number is to the right of 0, the vector points
> right. Any point you apply that vector to will indicate a point to the
> right of that point. Adding a positive quantity increases the quantity
> being added to. That's a pretty solid geometrical justification for x>0
> <-> x+y>y
Not until you establish more of a link between the geometry and the
number system which you have yet done.
You seem to have enough wisdom to recognize that Lester is full of
> nonsense.
> However you do not seem to have enough sense to not follow in his
> footsteps.
> Stephen
> I'm not following Lester anywhere.
> Tony
You seem to be following him down the road of crankdom. You both
> refuse to define your terms, and both post the same arguments over and
> over,
> and both insist on using peculiar private definitions of words. You
> are more civil, and sane, but if you keep at it, that may change.
Stephen
I used to be less civil. It depends how I'm addressed, but no, I don't
> just want to crank out the same old same old.
TO strives to krank our new same old instead.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<snip> Yes, trying to pin a size on a set with no bound gets tricky, 
doesn't
> it?
> It only gets tricky when you refuse to define size. It has been
nearly
> three years now, and you still have not defined what you mean by
size.
> A count of elements, a member of *N, such that adding an element
> increases the size by 1, and removing one decreases it by 1, and
> x-1<x<x+1.
> What is a 'count of elements'?
> The sum one arrives at when incrementing a start value of 0 once for
> each element.
 So show us your count of the power set of the naturals.
My count of power set of set size n is 2^n. Everyone knows that. You
claim 2^aleph_0=c? Prove it. c is beyond that. For every finite n, 2^n
is finite.
> You are just using another undefined term.
> And going on to define it...
Where?
Up there. Or, should I say, under there?
> What is *N?
> N is the hyper-naturals, which include infinite values, such as the
> sizes of infinite sets.
What are the hyper-naturals in terms of something already defined?
They are the sizes of all discrete sets, discrete meaning that every
object is either a member, or not.
> Why does size appear in your definition of size?
> Size is used in a statement which relates size to element membership
> and a basic axiom for addition which should be preserved.
That does not answer the question, but merely adds more confusion to
> what was excessively confused before it.
Your confusion is not my fault.
> We start with {1}, of size 1, with max value 1.
Why not start with {} of size 0? If one starts with {1}, one does not
> have any way of counting the members of {}.
>
I can start my inductive proof at whatever point I like, but you may
start at 0 if you like. No difference.
> As we add each
> successive natural, the size and the max value simultaneous increment,
> and therefore remain equal with the addition of each and every natural
> to the set. The max value and count can never be unequal.
That is another reason to start with {} having count 0 and thereafter
> adding 1 to the count of s to get the count of s / {s}.
Makes much more sense, but that would defeat TO's purpose.
Not really. {} has size 0. Add 1, have 1, its value is 1. So?
> That is true for sets of the form
> {1, 2, 3, ... n }
> for some n in N. But N itself is not of that form, neither are
> most sets.
> If N has a size omega, then it is of the form {1,2,3,...,omega}
Not in standard, and no one follows TO's non-standard math, except TO.
That's not true.
> What is the size of { 1, 2, e } ? And more importantly,
> how do you prove that is the size, using your actual definition of
size?
> That set bijects with the initial segment of N ending in 3, with size 3,
> so that is its size. Simple bijection works for finite sets. It's more
> complicated with infinite sets, if it's to make sense.
But TO has not yet got his way complicated enough either to make sense
> or to so befuddle those trying to understand it that they give up.
You don't give up. You never tried. Some do. You don't. So, you speak
for yourself.
> Seriously, can you provide a proof that the set { 1, 2, e} has a size,
> according to your definition of size?
> I think I just did.
> If N has a size, then its size is its largest element by obvious
> induction.
> Until you define size, who knows if N has a size? N may also
have
> a volume, and a girth and a favorite color.
> See above.
> Your definition does not apply to N. As far as I can tell your
definition
> only applies to sets of the form { 1, 2, ... n}. Which is fine,
> if you want a definition of size that does not apply to all sets.
> If you want to apply a size to N, then you are going to have a
maximum
> value, which causes a contradiction.
So we may roundfile TO's sizes of sets as self-contradictory.
>
What's self-contradictory about saying some set definitions don't lead
to a clear definition of size? N isn't a proper set in IST, for that
very reason.
<snip It is childish to be unable to think beyond the simple notion of
> numbers as points on a line. The number line is a tool used to
teach
> children about the real numbers. It is not the end all and be all of
> the concept of numbers. Anyway, what geometrical foundation
> are you relying on? Your definition of size above made no mention
> of geometry.
> When considering real numbers and subsets thereof, the real line is a
> perfect representation.
The real line presupposes a well understood set of real numbers as
> well as any geometical notions.
>
Oh. And I thought is was some childish teaching device, for those with
no deep understanding. Funny, that.
> We can choose any point and call it 0. Using a
> compass, we can choose an arbitrary point 1, and then a next one, etc,
> and map out the naturals. We can subdivide any interval, and through
> such subdivisions, construct any digital fraction.
One can in theory, however tedious the practice might be, construct any
> rational as well as any square root of a (positive) rational, as well as
> a large variety of algebraic numbers as points on that line.
Yes, so there is a geometrical foundation to the algebraic operations we
entertain, on the finite level. And?
> Think of each number as a vector, from 0 to whatever point indicates
> that number. If the number is to the right of 0, the vector points
> right. Any point you apply that vector to will indicate a point to the
> right of that point. Adding a positive quantity increases the quantity
> being added to. That's a pretty solid geometrical justification for x>0
> <-> x+y>y
Not until you establish more of a link between the geometry and the
> number system which you have yet done.
A number is a point on the infinite line. A greater number is a point to
the right.
> You seem to have enough wisdom to recognize that Lester is full of
> nonsense.
> However you do not seem to have enough sense to not follow in his
> footsteps.
> Stephen
> I'm not following Lester anywhere.
> Tony
> You seem to be following him down the road of crankdom. You both
> refuse to define your terms, and both post the same arguments over and
> over,
> and both insist on using peculiar private definitions of words. You
> are more civil, and sane, but if you keep at it, that may change.
Stephen
> I used to be less civil. It depends how I'm addressed, but no, I don't
> just want to crank out the same old same old.
 TO strives to krank our new same old instead.
At least it's new.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
 Well, we can always take solace in the fact: no matter what, there's
> always fluglemops. Delicious, crunchy fluglemops! Enjoy some every
> day!
> I'm preparing mine right now. Mmmmmmmm.
> Fluglemops sound a good deal better than anything WM has come up with
so
> far.
Tasty, crunchy, and completly abstract! Now with 0% trans-fats!
>
And low in calories to boot!
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<468fd687@news2.lightlink.com>
<11s29399vde550a5h7rh83t8lhpvth2gn9@4ax.com>
<rubrum-344939.19010608072007@newsclstr02.news.prodigy.com>
<nk6393p360vhcbuoje5e0tjmamu998cf5q@4ax.com
> Yes. It is wrong. Cantor's diagonal argument neglects the fact that
> for infinite series lim [n-->oo] 10^-n = 0. Hessenbergs argument uses
> an impossible set.
> So, you claim it is _not_ a theorem of ZFC that the reals are
> uncountable? That is, you claim that there is _not_ a sequence of
> formulas that is a proof in first order logic from the axioms of Z,
> with the last entry in the sequence being the formula ~Ef f:N ->onto
The sequence of formulas is without any value unless interpreted. This
> interpretation is wrong.
Again, I note that you are afraid of giving a straight answer. Is
there or is there not such a sequence of formulas?
> For Cantor's diagonal argument, which is an impossibility proof, we can
easily find a counter example:
The list
0.0
> 0.1
> 0.11
> 0.111
> ...
with replacement rule 0 --> 1 yields a diagonal number which is
> already in the list.
Who claimed that diagonalisation produces a non-listed number for all
lists and for all replacement rules? Do you really thinks this is a
valid objection? Are you really a mathematics teacher?
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<11s29399vde550a5h7rh83t8lhpvth2gn9@4ax.com>
<rubrum-344939.19010608072007@newsclstr02.news.prodigy.com>
<nk6393p360vhcbuoje5e0tjmamu998cf5q@4ax.com
> Yes. It is wrong. Cantor's diagonal argument neglects the fact that
> for infinite series lim [n-->oo] 10^-n = 0. Hessenbergs argument
uses
> an impossible set.
> So, you claim it is _not_ a theorem of ZFC that the reals are
> uncountable? That is, you claim that there is _not_ a sequence of
> formulas that is a proof in first order logic from the axioms of Z,
> with the last entry in the sequence being the formula ~Ef f:N
->onto
> The sequence of formulas is without any value unless interpreted. This
> interpretation is wrong.
Again, I note that you are afraid of giving a straight answer. Is
> there or is there not such a sequence of formulas?
> For Cantor's diagonal argument, which is an impossibility proof, we can
easily find a counter example:
> The list
> 0.0
> 0.1
> 0.11
> 0.111
> ...
> with replacement rule 0 --> 1 yields a diagonal number which is
> already in the list.
Who claimed that diagonalisation produces a non-listed number for all
> lists and for all replacement rules?
Cantor and each of his followers who understood the principle of an
impossibility proof.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<rubrum-344939.19010608072007@newsclstr02.news.prodigy.com>
<nk6393p360vhcbuoje5e0tjmamu998cf5q@4ax.com
> Yes. It is wrong. Cantor's diagonal argument neglects the fact
that
> for infinite series lim [n-->oo] 10^-n = 0. Hessenbergs argument
uses
> an impossible set.
> So, you claim it is _not_ a theorem of ZFC that the reals are
> uncountable? That is, you claim that there is _not_ a sequence of
> formulas that is a proof in first order logic from the axioms of Z,
> with the last entry in the sequence being the formula ~Ef f:N
->onto
> The sequence of formulas is without any value unless interpreted.
This
> interpretation is wrong.
> Again, I note that you are afraid of giving a straight answer. Is
> there or is there not such a sequence of formulas?
> For Cantor's diagonal argument, which is an impossibility proof, we
can easily find a counter example:
> The list
> 0.0
> 0.1
> 0.11
> 0.111
> ...
> with replacement rule 0 --> 1 yields a diagonal number which is
> already in the list.
> Who claimed that diagonalisation produces a non-listed number for all
> lists and for all replacement rules?
Cantor and each of his followers who understood the principle of an
> impossibility proof.
WRONG. YOU don't understand the diagonal argument. In a previous post,
I mentioned the details with regard to this particular example of
yours.
MoeBlee
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<rubrum-344939.19010608072007@newsclstr02.news.prodigy.com>
<nk6393p360vhcbuoje5e0tjmamu998cf5q@4ax.com
> Yes. It is wrong. Cantor's diagonal argument neglects the fact
that
> for infinite series lim [n-->oo] 10^-n = 0. Hessenbergs argument
uses
> an impossible set.
> So, you claim it is _not_ a theorem of ZFC that the reals are
> uncountable? That is, you claim that there is _not_ a sequence of
> formulas that is a proof in first order logic from the axioms of Z,
> with the last entry in the sequence being the formula ~Ef f:N
->onto
> The sequence of formulas is without any value unless interpreted.
This
> interpretation is wrong.
> Again, I note that you are afraid of giving a straight answer. Is
> there or is there not such a sequence of formulas?
Still no answer. Do you even know?
> For Cantor's diagonal argument, which is an impossibility proof, we
can easily find a counter example:
> The list
> 0.0
> 0.1
> 0.11
> 0.111
> ...
> with replacement rule 0 --> 1 yields a diagonal number which is
> already in the list.
> Who claimed that diagonalisation produces a non-listed number for all
> lists and for all replacement rules?
Cantor and each of his followers who understood the principle of an
> impossibility proof.
Wrong.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<rubrum-344939.19010608072007@newsclstr02.news.prodigy.com>
<nk6393p360vhcbuoje5e0tjmamu998cf5q@4ax.com > For Cantor's diagonal 
argument,
which is an impossibility proof, we
can easily find a counter example:
> The list
> 0.0
> 0.1
> 0.11
> 0.111
> ...
> with replacement rule 0 --> 1 yields a diagonal number which is
> already in the list.
> Who claimed that diagonalisation produces a non-listed number for all
> lists and for all replacement rules?
> Cantor and each of his followers who understood the principle of an
> impossibility proof.
Wrong.
Betrachten wir alsdann das Element
E0 = (b1, b2, b3, ...)
von M, so sieht man ohne weiteres, da? die Gleichung
E0 = Em
f?r keinen positiven ganzzahligen Wert von m erf?llt sein kann, da
sonst f?r das betreffende m und f?r alle ganzzahligen Werte von n
bn = amn
also auch im besonderen
bm = amm
w?re, was durch Definition von bn ausgeschlossen ist.
This statement is wrong for real numbers. But for more than 3 years no
mathematician was clever enough to note that. Therefore I have
reasonable hope that after five years or so some mathematicians will
turn out clever enough understand my arguments.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<11s29399vde550a5h7rh83t8lhpvth2gn9@4ax.com>
<rubrum-344939.19010608072007@newsclstr02.news.prodigy.com>
<nk6393p360vhcbuoje5e0tjmamu998cf5q@4ax.com>
[Muckenheim]
> For Cantor's diagonal argument, which is an impossibility proof, we can
easily find a counter example:
> The list
> 0.0
> 0.1
> 0.11
> 0.111
> ...
> with replacement rule 0 --> 1 yields a diagonal number which is
> already in the list.
Who claimed that diagonalisation produces a non-listed number for all
> lists and for all replacement rules?
Nobody, but even if they had it still isn't an objection. Every number
explicitly shown has two ends, and the ... should mean this is true
of every number in the list. But the diagonal has only one end, as
indicated by the _different_ ... in 0.111... So the diagonal is
not in the list; but obviously this is beyond Muckenheim, and you knew
already really, so why do we bother?
> Are you really a mathematics teacher?
Ah, yes, a second mystery.
Brian Chandler
http://imaginatorium.org
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<rubrum-344939.19010608072007@newsclstr02.news.prodigy.com>
<nk6393p360vhcbuoje5e0tjmamu998cf5q@4ax.com
> [Muckenheim]
> For Cantor's diagonal argument, which is an impossibility proof, we
can easily find a counter example:
> The list
> 0.0
> 0.1
> 0.11
> 0.111
> ...
> with replacement rule 0 --> 1 yields a diagonal number which is
> already in the list.
> Who claimed that diagonalisation produces a non-listed number for all
> lists and for all replacement rules?
Nobody,
You are wrong. Cantor's proof is an impossibility proof. It is not
given for special numbers, but in the general case for any list of
numbers.
> but even if they had it still isn't an objection. Every number
> explicitly shown has two ends, and the ... should mean this is true
> of every number in the list. But the diagonal has only one end, as
> indicated by the _different_ ... in 0.111... So the diagonal is
> not in the list;
That is but one side of the medal. The other side, usually overlooked
by fiery Cantorians, is that there must be a number in the list which
has as many digits as the diagonal. Otherwise the diagonal simply
cannot exist. (Well, one digit less. But infinity minus one is
infinity.) Therefore your arguing is invalid.
Or would you like to propose that the diagonal can exist where no
lines are, in the nirvana? Strong belief required!
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<rubrum-344939.19010608072007@newsclstr02.news.prodigy.com>
<nk6393p360vhcbuoje5e0tjmamu998cf5q@4ax.com
> [Muckenheim]
> For Cantor's diagonal argument, which is an impossibility proof, we
can easily find a counter example:
> The list
> 0.0
> 0.1
> 0.11
> 0.111
> ...
> with replacement rule 0 --> 1 yields a diagonal number which is
> already in the list.
> Who claimed that diagonalisation produces a non-listed number for all
> lists and for all replacement rules?
> Nobody,
You are wrong. Cantor's proof is an impossibility proof. It is not
> given for special numbers, but in the general case for any list of
> numbers.
It is given for any list of DENUMERABLE (say, binary, or whatever
base) SEQUENCES.
> but even if they had it still isn't an objection. Every number
> explicitly shown has two ends, and the ... should mean this is true
> of every number in the list. But the diagonal has only one end, as
> indicated by the _different_ ... in 0.111... So the diagonal is
> not in the list;
That is but one side of the medal. The other side, usually overlooked
> by fiery Cantorians, is that there must be a number in the list which
> has as many digits as the diagonal. Otherwise the diagonal simply
> cannot exist. (Well, one digit less. But infinity minus one is
> infinity.) Therefore your arguing is invalid.
PLEASE answer this questions: Do you understand what it means for a
logic to be monotonic, and do you understand that first order logic is
monotonic?
The list is a list of denumerable binary sequences. The diagonal is a
denumerable binary sequence. And the anti-diagonal is a denumerable
binary sequence. And the anti-diagonal is not on the list. Therefore,
the list is not a list of ALL denumerable binray sequences. I don't
know why you don't understand this.
MoeBlee
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
[Muckenheim]
> For Cantor's diagonal argument, which is an impossibility proof, we
can
> easily find a counter example:
> The list
> 0.0
> 0.1
> 0.11
> 0.111
> ...
> with replacement rule 0 --> 1 yields a diagonal number which is
> already in the list.
> Who claimed that diagonalisation produces a non-listed number for all
> lists and for all replacement rules?
Nobody, but even if they had it still isn't an objection. Every number
> explicitly shown has two ends, and the ... should mean this is true
> of every number in the list. But the diagonal has only one end, as
> indicated by the _different_ ... in 0.111... So the diagonal is
> not in the list; but obviously this is beyond Muckenheim, and you knew
> already really, so why do we bother?
Are you really a mathematics teacher?
Ah, yes, a second mystery.
AS I understand it, WM does not teach actual mathematics, but only
history about mathematics, or some such.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
On Thu, 12 Jul 2007 21:11:41 -0700, Brian Chandler
> [Muckenheim]
> For Cantor's diagonal argument, which is an impossibility proof,
> we can easily find a counter example:
> The list
> 0.0
> 0.1
> 0.11
> 0.111
> ...
> with replacement rule 0 --> 1 yields a diagonal number which is
> already in the list.
>
Strange claim. Maybe he meant with the rule 1 --> 0?
Though it's clear that this rule does not necessarily lead to a number
not in the list, since the rule doesn't apply to the first entry in
the list; hence it allows for the diagonal number to be identical
with the first entry in the list.
And right, this is exactly what happens in this case! :-)
Given the list:
0.0
0.1
0.11
0.111
...
The diagonal number (got applying the rule from above) might be:
0.000... = 0.0. (Though we usually only take into consideration the
decimals after the ., but well...)
> Are you really a mathematics teacher?
 Ah, yes, a second mystery.
>
Indeed.
F.
--
E-mail: info<at>simple-line<dot>de
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<46879971@news2.lightlink.com>
<4687f9e7@news2.lightlink.com>
<468938a4@news2.lightlink.com>
<468a73ee@news2.lightlink.com>
<468c05dc@news2.lightlink.com>
<468c5ec8@news2.lightlink.com>
<468fdc7c@news2.lightlink.com>
<4693b1d7@news2.lightlink.com>
<46964d3d@news2.lightlink.com > Ugh. Back to basics.
> Let's call the following Experiment A.
> At time -1/n, for perfectly ordinary finite naturals n, insert a ball
> labelled n into the vase. What is in the vase at time 0?
> My answer : For each pofnat, a ball labelled with that pofnat. This
> collection of balls is countably infinite.
> Your answer : ...?
> More experiments forthcoming, if you can answer this one without
> descending into blathering incoherence.
> You would seem to have all of N in there, but there is still a paradox
> in the timeline. At every moment before 0, there are balls not yet in
> the vase, and at time 0, no balls are inserted, so there is no moment
> when you are done. The Zeno time machine is the culprit here.
> Does there have to be a moment when you are done?
> It seems to me that you reject the entire thought experiment as
> pointless because it is not possible, in the real world, to perform
> it. If so then I am utterly bemused at the effort you've spent arguing
> about it.
I reject it as absurd that changing the labels on the balls makes any
> difference as to how many there are in the vase, when the same sequence
> of insertions and removals is performed. That's nonsense. You're adding
> an infinite number of 9's and getting a zero sum. Please! There's
> something wrong with that picture.
Stick to experiment A. There is _no_ point in discussing the original
experiment if we disagree about a much more basic one.
> The fact that the experiment involves
> time but that there is no point in that time when your supposed event
> could occur is a pretty strong indication that it doesn't.
In experiment A, the earliest time at which the vase contains all of N
is time 0. No balls are inserted at time 0. You claim this is a
paradox in the timeline. Why?
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
> Ugh. Back to basics.
> Let's call the following Experiment A.
> At time -1/n, for perfectly ordinary finite naturals n, insert a ball
> labelled n into the vase. What is in the vase at time 0?
> My answer : For each pofnat, a ball labelled with that pofnat. This
> collection of balls is countably infinite.
> Your answer : ...?
> More experiments forthcoming, if you can answer this one without
> descending into blathering incoherence.
> You would seem to have all of N in there, but there is still a paradox
> in the timeline. At every moment before 0, there are balls not yet in
> the vase, and at time 0, no balls are inserted, so there is no moment
> when you are done. The Zeno time machine is the culprit here.
> Does there have to be a moment when you are done?
> It seems to me that you reject the entire thought experiment as
> pointless because it is not possible, in the real world, to perform
> it. If so then I am utterly bemused at the effort you've spent arguing
> about it.
> I reject it as absurd that changing the labels on the balls makes any
> difference as to how many there are in the vase, when the same sequence
> of insertions and removals is performed. That's nonsense. You're adding
> an infinite number of 9's and getting a zero sum. Please! There's
> something wrong with that picture.
Stick to experiment A. There is _no_ point in discussing the original
> experiment if we disagree about a much more basic one.
> The fact that the experiment involves
> time but that there is no point in that time when your supposed
event
> could occur is a pretty strong indication that it doesn't.
In experiment A, the earliest time at which the vase contains all of N
> is time 0. No balls are inserted at time 0. You claim this is a
> paradox in the timeline. Why?
>
Because there is no time at which the vase contains all of N, without
containing all of *N.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<virgil-080907.10424602072007@comcast.dca.giganews.com>
<4689a90d@news2.lightlink.com>
<virgil-6646A9.23363202072007@comcast.dca.giganews.com>
<468a6db0@news2.lightlink.com>
<virgil-43A784.11005803072007@comcast.dca.giganews.com>
<468bf027@news2.lightlink.com>
<virgil-4C26F4.14273604072007@comcast.dca.giganews.com>
<468c5876@news2.lightlink.com>
<468fd777$1@news2.lightlink.com>
<4693b06f@news2.lightlink.com>
<46964c81@news2.lightlink.com > Yes, trying to pin a size on a set with no
bound gets tricky,
doesn't
> it? If N has a size, then its size is its largest element by obvious
> induction.
> Nope.
> Yep. {1} has 1 element, and a max value of 1. Every subsequent natural
> added to the set increases the set size by one, and the max value by
1,
> so they can never be unequal. x=y <-> x+1=y+1.
> So, by induction, every finite initial segment of N has a 'size' equal
> to its largest element.
> This doesn't imply anything in particular about the 'size' of N.
If N is a set of consecutive naturals starting at 1, then any size it
> has is also a largest element. The two are equal. If one exists, so does
> the other.
Nope.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
> Yes, trying to pin a size on a set with no bound gets tricky,
doesn't
> it? If N has a size, then its size is its largest element by obvious
> induction.
> Nope.
> Yep. {1} has 1 element, and a max value of 1. Every subsequent natural
> added to the set increases the set size by one, and the max value by
1,
> so they can never be unequal. x=y <-> x+1=y+1.
> So, by induction, every finite initial segment of N has a 'size' equal
> to its largest element.
> This doesn't imply anything in particular about the 'size' of N.
> If N is a set of consecutive naturals starting at 1, then any size it
> has is also a largest element. The two are equal. If one exists, so does
> the other.
Nope.
>
What does it mean for two things to be equal, Mike?
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
What is *N?
> N is the hyper-naturals, which include infinite values, such as the
> sizes of infinite sets.
And you proved the existence of such a set how?
E *N
tada!
And since your definition of 'hypernatural' is not the ordinary one,
> why don't you just use a word such as 'T-hypernatural' or something so
> that one could read your comments without confusion.
MoeBlee
They're essentially the same.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<virgil-6646A9.23363202072007@comcast.dca.giganews.com>
<468a6db0@news2.lightlink.com>
<virgil-43A784.11005803072007@comcast.dca.giganews.com>
<468bf027@news2.lightlink.com>
<virgil-4C26F4.14273604072007@comcast.dca.giganews.com>
<468c5876@news2.lightlink.com>
<468fd777$1@news2.lightlink.com>
<f6u95n$cie$1@news.msu.edu>
<4693b3da@news2.lightlink.com>
<f70f9r$7oc$1@news.msu.edu>
<469642b9@news2.lightlink.com>
<469664e4@news2.lightlink.com
> What is *N?
> N is the hyper-naturals, which include infinite values, such as the
> sizes of infinite sets.
> And you proved the existence of such a set how?
E *N
tada!
Too bad you don't have any system whatsoever in which to be making
such existence axioms.
> They're essentially the same.
Not in ordinary treatments. If you have your own definition, then you
should use 'T-hyperreal' to avoid confusion.
MoeBlee
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
> Until you count to omega, you have not counted omega naturals.
NeN
> Which version of set theory is TO trying to sell in which any set
is
> allowed to a member of itself.
> sum(n=1->x: 1)=x.
> How does than justify NeN?
> Okay, excuse me. I was kinda quoting Ross in that. I suppose it's
more
> properly |N|eN. :)
> Since by definition, for all n in N, n+1 in N; it follows that |N|+1
> in N; and so on. So N contains strictly more elements than |N|
by
> your unusual measure of more?
> Yes, trying to pin a size on a set with no bound gets tricky, doesn't
> it?
> It only gets tricky when you refuse to define size. It has been
nearly
> three years now, and you still have not defined what you mean by size.
> A count of elements, a member of *N, such that adding an element
> increases the size by 1, and removing one decreases it by 1, and
x-1<x<x+1.
> What is a 'count of elements'?
> The sum one arrives at when incrementing a start value of 0 once for
> each element.
> You are just using another undefined term.
> And going on to define it...
> What is *N?
> N is the hyper-naturals, which include infinite values, such as the
> sizes of infinite sets.
> Why does size appear in your definition of size?
> Size is used in a statement which relates size to element membership
> and a basic axiom for addition which should be preserved.
> We start with {1}, of size 1, with max value 1. As we add each
> successive natural, the size and the max value simultaneous increment,
> and therefore remain equal with the addition of each and every natural
> to the set. The max value and count can never be unequal.
> That is true for sets of the form
> {1, 2, 3, ... n }
> for some n in N. But N itself is not of that form, neither are
> most sets.
> If N has a size omega, then it is of the form {1,2,3,...,omega}
Perhaps, but you still have not defined size, so it remains to
be seen if that is actually true or not.
> What is the size of { 1, 2, e } ? And more importantly,
> how do you prove that is the size, using your actual definition of
size?
> That set bijects with the initial segment of N ending in 3, with size 3,
> so that is its size. Simple bijection works for finite sets. It's more
> complicated with infinite sets, if it's to make sense.
> Seriously, can you provide a proof that the set { 1, 2, e} has a size,
> according to your definition of size?
> I think I just did.
So your definition of size is based on bijection for finite sets.
How does your definition of size work for infinite sets? Apparently
some sets, such as N, do not even have a size. Can you give
an example of an infinite set that does have a size, and a proof
that shows what the size is? Does the set { x | 0 <= x <= 1 } have
a size? And if so, what is it, and how do you prove it?
> If N has a size, then its size is its largest element by obvious
> induction.
> Until you define size, who knows if N has a size? N may also have
> a volume, and a girth and a favorite color.
> See above.
> Your definition does not apply to N. As far as I can tell your
definition
> only applies to sets of the form { 1, 2, ... n}. Which is fine,
> if you want a definition of size that does not apply to all sets.
> If you want to apply a size to N, then you are going to have a maximum
> value, which causes a contradiction.
This is your definition of size we are talking about. It is clear that
according to your definition, N does not have a size. Which as I said,
is fine, if you do not mind that your definition of size does not apply
to all sets.
> N clearly has a cardinality, but cardinality is actually defined.
> It has no largest element. Therefore it has no size.
> If that is how you define size, then that is true. But it
> is a rather uninteresting result. At least it is consistent.
> See above.
> Aleph_0
> is a phantom, a number only in the sense that that is what it does to
> one's mind, numbs it.
> More stupid comments. aleph_0 is perfectly well defined. It may
> not match your childish notion of numbers as points on a line, but one
can
> actually provide a non-circular definition of aleph_0, unlike your
> arbitray Big'Uns.
> I don't think numbers as points on the real line, or even infinitesimal
> intervals, is childish. I think it's foolish to rely entirely on
> symbolic arguments without some geometrical foundation. Oh well.
> It is childish to be unable to think beyond the simple notion of
> numbers as points on a line. The number line is a tool used to
teach
> children about the real numbers. It is not the end all and be all of
> the concept of numbers. Anyway, what geometrical foundation
> are you relying on? Your definition of size above made no mention
> of geometry.
> When considering real numbers and subsets thereof, the real line is a
> perfect representation.
Yes, but there are other numbers than real numbers.
> We can choose any point and call it 0. Using a
> compass, we can choose an arbitrary point 1, and then a next one, etc,
> and map out the naturals. We can subdivide any interval, and through
> such subdivisions, construct any digital fraction.
> Think of each number as a vector, from 0 to whatever point indicates
> that number.
No, because my concept of number is much richer than that. There
are numbers other than real numbers.
> If the number is to the right of 0, the vector points
> right. Any point you apply that vector to will indicate a point to the
> right of that point. Adding a positive quantity increases the quantity
> being added to. That's a pretty solid geometrical justification for x>0
> <-> x+y>y
Only for certain geometries. Which geometry are you using?
Stephen
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
> Until you count to omega, you have not counted omega naturals.
NeN
> Which version of set theory is TO trying to sell in which any
set is
> allowed to a member of itself.
> sum(n=1->x: 1)=x.
> How does than justify NeN?
> Okay, excuse me. I was kinda quoting Ross in that. I suppose it's
more
> properly |N|eN. :)
> Since by definition, for all n in N, n+1 in N; it follows that
|N|+1
> in N; and so on. So N contains strictly more elements than |N|
by
> your unusual measure of more?
> Yes, trying to pin a size on a set with no bound gets tricky, doesn't
> it?
> It only gets tricky when you refuse to define size. It has been
nearly
> three years now, and you still have not defined what you mean by
size.
> A count of elements, a member of *N, such that adding an element
> increases the size by 1, and removing one decreases it by 1, and
x-1<x<x+1.
> What is a 'count of elements'?
> The sum one arrives at when incrementing a start value of 0 once for
> each element.
> You are just using another undefined term.
> And going on to define it...
What is *N?
> N is the hyper-naturals, which include infinite values, such as the
> sizes of infinite sets.
> Why does size appear in your definition of size?
> Size is used in a statement which relates size to element membership
> and a basic axiom for addition which should be preserved.
>
That made a little sense, eh?
> We start with {1}, of size 1, with max value 1. As we add each
> successive natural, the size and the max value simultaneous increment,
> and therefore remain equal with the addition of each and every natural
> to the set. The max value and count can never be unequal.
> That is true for sets of the form
> {1, 2, 3, ... n }
> for some n in N. But N itself is not of that form, neither are
> most sets.
> If N has a size omega, then it is of the form {1,2,3,...,omega}
Perhaps, but you still have not defined size, so it remains to
> be seen if that is actually true or not.
>
How would you like size defined? Simply in terms of set membership and
bijection? With infinite sets there are additional considerations before
you reach a sensible conclusion.
> What is the size of { 1, 2, e } ? And more importantly,
> how do you prove that is the size, using your actual definition of
size?
> That set bijects with the initial segment of N ending in 3, with size 3,
> so that is its size. Simple bijection works for finite sets. It's more
> complicated with infinite sets, if it's to make sense.
See?
Seriously, can you provide a proof that the set { 1, 2, e} has a size,
> according to your definition of size?
> I think I just did.
So your definition of size is based on bijection for finite sets.
No, my system of determining relative size is based on the system for
finite sets, with a different kind of extension for infinite sets.
Infinite sets must be parametrically compared over some variable range
without bound. That requires infinite-case induction.
> How does your definition of size work for infinite sets? Apparently
> some sets, such as N, do not even have a size.
That is correct. N has no exact size. However, when you compare sets
such as the evens with the wholes, naturals and hyper-naturals alike,
you can only compare unending sets over a particular range variable,
infinite as that might be, using a formulaic relationship. If you want
to restrict your set to the countably infinite, that applies even more.
If you want to address an uncountable set, it's very likely that it
indeed does have an end, but no middle, and some artificial count, like
Big'un, can be applied to a metric space, based on the units of measure.
Can you give
> an example of an infinite set that does have a size, and a proof
> that shows what the size is?
Yes. I say there exists 0eR. Then I declare the existence of 1eR. Then I
use the axiom that, between any two different reals, exists another
halfway between them, as per bisection of the line segment, an easy
geometrical operation. It follows logically that any finite number of
subdivisions produces subintervals of finite length, which can be
further subdivided. So, no finite number of intermediate points can fill
the segment, and the segment contains some infinite number of points, or
reals. Then, we axiomatically state that, in a one dimensional segment
of length 1, we have Big'Un points or reals. That ties measure to count,
for the infinite case.
|{xeR ^ x>0 ^ ~x>1}|=Big'Un.
Does the set { x | 0 <= x <= 1 } have
> a size? And if so, what is it, and how do you prove it?
It i9s Big'Un+1, since you have added one extra point/real to the
half-open interval (0,1]. See? I'm not crazy.
> If N has a size, then its size is its largest element by obvious
> induction.
> Until you define size, who knows if N has a size? N may also
have
> a volume, and a girth and a favorite color.
> See above.
> Your definition does not apply to N. As far as I can tell your
definition
> only applies to sets of the form { 1, 2, ... n}. Which is fine,
> if you want a definition of size that does not apply to all sets.
> If you want to apply a size to N, then you are going to have a
maximum
> value, which causes a contradiction.
This is your definition of size we are talking about. It is clear
that
> according to your definition, N does not have a size. Which as I
said,
> is fine, if you do not mind that your definition of size does not
apply to all sets.
>
I am comfortable with the fact that not all set definitions lead to an
identifiable size? Does that put ants in your pants? Not every
collection of objects is a set. Does that bother you too?
> N clearly has a cardinality, but cardinality is actually defined.
> It has no largest element. Therefore it has no size.
> If that is how you define size, then that is true. But it
> is a rather uninteresting result. At least it is consistent.
> See above.
> Aleph_0
> is a phantom, a number only in the sense that that is what it does to
> one's mind, numbs it.
> More stupid comments. aleph_0 is perfectly well defined. It may
> not match your childish notion of numbers as points on a line, but one
can
> actually provide a non-circular definition of aleph_0, unlike your
> arbitray Big'Uns.
> I don't think numbers as points on the real line, or even infinitesimal
> intervals, is childish. I think it's foolish to rely entirely on
> symbolic arguments without some geometrical foundation. Oh well.
> It is childish to be unable to think beyond the simple notion of
> numbers as points on a line. The number line is a tool used to
teach
> children about the real numbers. It is not the end all and be all of
> the concept of numbers. Anyway, what geometrical foundation
> are you relying on? Your definition of size above made no mention
> of geometry.
> When considering real numbers and subsets thereof, the real line is a
> perfect representation.
Yes, but there are other numbers than real numbers.
Yes, and they are compounded upon reals, largely by adding extra real
lines, or replacing such lines with angular measures.
> We can choose any point and call it 0. Using a
> compass, we can choose an arbitrary point 1, and then a next one, etc,
> and map out the naturals. We can subdivide any interval, and through
> such subdivisions, construct any digital fraction.
> Think of each number as a vector, from 0 to whatever point indicates
> that number.
No, because my concept of number is much richer than that. There
> are numbers other than real numbers.
>
There are numbers built upon the reals, but I ask you to consider the
root node first.
> If the number is to the right of 0, the vector points
> right. Any point you apply that vector to will indicate a point to the
> right of that point. Adding a positive quantity increases the quantity
> being added to. That's a pretty solid geometrical justification for x>0
> <-> x+y>y
Only for certain geometries. Which geometry are you using?
Stephen
For starters, a linear geometry of the reals.
Tony
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<snip
> CBrown:
> Feel free to supply a formal description of what you mean.
Tony:
> Here, as below, I am using 'E' as the existential quantifier
> E e_0
> E e_n -> E e_2n+1
> E e_n -> E e_2n+2
CBrown:
> I.e., it is logically equivalent to:
> E e_0
> E e_n -> E e_(n+1)
> You appear to /want/ to define something like what is called an L-
> system (or Lindenmayer-system); but this again requires a lot more
> than what you have stated above.
> The difference between the Peano-like definition you gave is that it
> doesn't describe a tree.
Neither does the set of rules of implications for existence that you
> listed, without further definitions.
> If the logical implications stated express the
> parent child edge, then my three rules describe an infinite binary tree.
Yes indeedy - /if/ (some additional stuff) then you're right. Just as,
> in my system, /if/ we consider a parent child edge as the relationship
> between the existing nodes e_n and e_(2n+1) and the existing nodes e_n
> and e_(2n+2), then it /also/ describes a tree. Which is to say, also
> in my system, /if/ (some additional stuff), then it describes a tree.
But (that additional stuff) is not the /mathematical meaning/ of:
E e_n -> E e_2n+1
If you /meant/ something else, then /say/ something else.
No, I meant exactly that. What do you think the edges on the tree
represent, if not logical implication and instantiation? The information
regarding the edges is captured in those statements.
> :) I've always marveled at all sorts of intricate structures in nature,
> certainly including plants. So, it might not change that much. I don't
> see ugly weeds.
Well, you /do/ own a hippo, after all.
>
Yeah, I actually don't have much weeds left. Hungry hippo.
Actually I have tons of weeds. Comes with the territory.
> But anyway, even an L-system based on rules you seem to allude to
> above does not correspond to the infinite binary tree. It's the usual
> problem: you cannot acheive an infinite set simply by adding 1 element
> again and again without further axioms or rules.
> Um, I don't see why. Don't the Peano axioms define an infinite set? I'll
> see what an L-system is, anyway...
Short answer: No, Peano's system does not assert the /existence of/ an
> infinite set. More on this difficult idea at some other time. I'm sure
> it will come up again. And again. And again. :).
<snip
> In any case,
> nodes cannot be defined to be equal to ordered pairs of nodes in a
> definition which is not circular.
> It's that very self-reference that makes the infinite structure, isn't
it?
Uh, no; it's that very self-reference that makes mathematical
> gibberish.
Ordered pairs of nodes are not nodes. We first define nodes; then we
> define ordered pairs of nodes. The fact that we have an infinite
> structure in our case is because the number of nodes is not finite.
<snip
> Once something exists, it simply exists. The method of the proof of
> its existence does not, by itself, add additonal structure to that
> object.
> The above is equivalent to saying: Exists an element e_0. If an
> element e_n exists for some natural n, then the element e_(n+1)
> exists.
> <snip> Except that that produces a sequence, not a tree. Think of the -> 
as
> being, not just logical implication, but a production operator from
> parent to child.
/Make up your mind/ for God's sake: which is it? Are you defining - as 
logical
implication, or as a function?
Why can't it be both? What does an edge represent?
Every car has four tires. But if you choose to start calling a car
> by the name a tire, then the sentence Every tire has four tires
no
> longer makes any kind of sense at all. Am I supposed to think that you
> mean Every car has four cars? Or Every tire has four cars? Why
> must I guess?
I said you should think of it as both.
Whenever you /define/ a symbol like -> to mean something in /
> particular/, this is at the expense that it can no longer mean
> anything you choose it to; it now means /exactly/ what you defined
> it to be. You can't have it both ways /at the same time/ and expect to
> be understood, just as you can't use the word tire to mean both a
> car and a tire and expect to be understood.
Okay, but the fact that the existence of one object implies the
existence of another constitutes an instantiation derived from the
existence of some root object. The use of logical implication and the
existential quantifier acts as a production function. You're smart
enough to see that.
<snip
> Of course, given this definition of the tree, most
> subsets of N will give us disconnected subtrees rather than a single
> one. {1,2,3} looks like:
> 1 2
> /
> 3
> I'm sure it looks like that /to you/. Mathematics is a way of
> formalizing what looks like means, so we can agree that we actually
> see the same thing.
> That what my 2n+1 and 2n+2 were.
As long as you are /explicit/ and /consistent/ with your definitions,
> it's not a problem. But if you don't say I define an additional
> relation between edge-nodes called the parent-child relationship,
> defined as pairs of the form (m, 2m+1) and (m, 2m+2), then it's just
> an idea in your head. There's no way I can see that idea unless you
> explicitly /state/ that idea out loud.
When you /state/ your idea, then I can say well, I call those buggers
> 'edges' and I call edge-nodes nodes, but I define them /exactly
> the same way/, just using different /words/ as /placeholders/. So when
> you are 'counting edge-nodes', I am 'counting nodes'. Right, then!
There's only a difference of one between them, anyway, and no difference
at all, with a sub-root node. :)
<snip
> Instead of a set of node pairs to describe the tree, I'm doing it with a
> start node and a production formula. That's all.
Well, there's still problems with your approach in the infinite case,
> which I won't comment on right now as I noted above. But actually, it
> works perfectly fine for finite trees, once you made /explicit/ your
> production formula.
I think the problems in the infinite case are minimal.
In usual mathematical terms, your production rule is represented as
> a relation, which is a set of ordered pairs. (a,b) is in this set if,
> and only if, it is true that b is produced by a.
Yes, it can certainly be viewed as a set of pairs. I don't deny that.
So we're all happy; or at least /I'm/ happy, because I also know what
> a relation is (what properties it must have), and so I can understand
> what you are saying by production rule.
>
Well, good!
> Please, feel free to show that set in your system.
> S = {{n,p}: neN ^ (p=2n+1 v p=2n+2)}
<snip
> And that meets the definition of the set usually called edges in
> what is usually called a tree. Congratulations! The ordered pairs
of
> sets (N, S) meets the usual definition of a tree.
No need to get snippy! Before it was just an idea in your head, and an
> idea in my head; now it's an idea we can share.
We now /know/ we are talking about /the same thing/; despite the fact
> that what I call the edge from a to b, you call the production of b
> from a, via the production rule.
Just remember: when I say the edges, I /mean/ a production rule
> which satisfies: the production rule produces b from a if, and only
> if, (a,b) is an edge.
>
Yes. You know, I DID study trees and graphs, and I DO know the normal
definitions and approaches. I am just suggesting a formulaic logical
approach, which did lead to an interesting idea, after all....
> <snip
> An interesting thought just occurred to me. Say we start at the root,
> node 0, and try to work backwards to find a predecessor, using the two
> production operators I defined.
<snip
> Now that we agree on what we're talking about, I can understand your
> interest. Consider these questions:
Instead of starting with E e_0, start with for all n in Z, E e_z.
> Apply your production rules to this set of existing elements. If we
> apply your production rule repeatedly a finite number of times
> starting with a particular element e_j, and e_k is in the set of
> elements produced this way, we will say e_j is an ancestor of e_k.
(i) Which nodes are self-producing; i.e., for which e_n does e_n
> produce e_n?
(ii) Does there exist a node e_x which is not produced by any other
> node?
(iii) What is the /smallest set/ of elements such that for /every/
> integer j, an element of this set is an ancestor of e_j?
To me, this looks like two identical binary trees, each with a little
> loopy thing on top like a bubble blowing wand. But that's just a
> picture, of course.
Translation into edge/node language:
Let the set of nodes be the integers Z. Let the set of edges be
> {(n,m) : n in Z, and m = 2n+1 or m = 2n+2}. If there is a finite chain
> of edges (j, n_1), (n_1, n_2), ..., (n_z, k), then we will say j is
> an ancestor of k.
(i) Which nodes have self-edges; i.e., which edges are of the form
> (n,n)?
(ii) Does there exist a node x such that there is no node of the form
> (j, x)?
(iii) What is the /smallest set/ of integers such that for /every/
> integer j, an element of this set is an ancestor of j?
Sidebar: I didn't previously state this explicitly, but the usual
> definitions distinguish between two beasties: graphs and simple
> graphs.
In the most generous definition of graph, you can have self-
> productions like -1 produces -1, and still say this meets the
> definition of a graph, but we also define a simple graph as a graph
> where that never happens: a simple graph is a graph which has no self-
> to-self edges.
So, your graph produced by e_(-1) does /not/ meet the definition of
> a /simple/ graph; and since trees are defined to be a certain type of /
> simple/ graph, it doesn't meet the /usual/ definition of a tree.
Simple graphs are so much more common and usual than graphs that
> people tend to say graph when they really mean simple graph.
Well,
> what can you do? Now you know: trees are defined to be particular
> kinds of /simple/ graphs.
<snip>
I knew you'd object on that technical ground, but I didn't expect you to
be so verbose. :) Anyway, it's true that the loopy thing on the top
violates the standard meaning of the tree, but I think you're being a
little unimaginative to protest on those grounds. Think of what happened
here. We take a production mechanism that assigns naturals to the nodes
arithmetically, and then try to follow it backwards, and find it has
this one self-referential sub-root node. In the process, we have now
made edges and nodes exactly equal. We had 2^(n+1)-1 nodes up to level
n, and now we have 2^(n+1), a nice coincidental simplification. It's not
the standard tree, but it's not much different, and closed under
parent(x). Hmmm...
> Except that parent-child relationship can be captured arithmetically
to
> describe the edges.
> It cetainly /can/ be; as you have done above. But it's never the case
> that we can simply equate the former set with the set of relationships
> (e.g., equate nodes with edges, so they have the same number).
> That's where your statements became incoherent.
> See my epiphany above.
It /still/ doesn't imply that things and the relationships between
> those things are identical. A thing is not the production rule that
> generated that thing. n is simply /not/ the same thing as (n,n),
> no matter how you cut it; unless your actual /goal/ is to be
> misunderstood.
>
I don't know what you're talking about. It's like complaining I call
your dog a terrier because its great grandmother was part beagle.
> <snip
> My claim had more to do with trying to formulate the relations between
> objects in the tree.
> But that's /already been done/, ya maroon!
> Incorrectly, except by WM.
Spare me! He's pointing at ascii art and claiming that it defines
> something, if we just look and use our imagination and so on.
> How is anyone supposed to know /exactly/ what the hell he means by
> such phrases?
>
I was arguing a similar case a long time ago. Every path produced from
the root on up does so in conjunction with the production of two nodes.
Therefore, there are half as many paths as nodes. See?
Hey! This even applies now to my root path, which requires the
production of the pre-root node and the root node. Man, I'm good! :)
> E .
> E x -> E x0
> E x -> E x1
<snip
> For example, the above rules are equivalent to:
> The empty string exists.
> The string 0 exists
> The string 1 exists
> If strings x and y exist, then the string x + y exists.
> Not if the statements of implication are taken as the production
> operators represented as edges in the tree.
But then they /cease/ to be logical axioms. Pick /one/, for God's
> sake! Use two /different/ symbols if you mean two /different/ things,
> like car, and tire for a car, and a tire.
/Why/ is that so very hard to do for a person of your obvious
> intelligence?
Sometimes I feel like I'm talking to my little nephew Ben. Use your
> words, Ben! Use your words!
>
The use of the existential quantifier in the implication of a logical
statement implies production of some sort, does it not?
> Your definition...
Not my /definition/, my /set of axioms/. They're different things.
> ... simply says we can
> combine any strings of 0's and 1's.
Then you misundertsand me. That is /not/ my intended /exact/ meaning;
> and /not/ what those symbols /mean/ in general mathematical discourse.
> The distinction is a /very/ important one; however trivial you may
> think I'm being.
E x and E y -> E xy instead says:
For /some unknown reason beyond the scope of our discussion/, whenever
> we find that x exists and y exists, it /turns out/ that is always
> true that xy /also/ exists, /without exception/.
The existence of xy is not thereby caused by the existence of
x
> and y or by the combining of x and y.
Instead, the axiom asserts a true observation: the existence of
xy
> is /correlated/ with the existence of x and y in just that way.
> But correlation is not causation.
Of course. But, what do you think the edges ARE?
The axiom above is not a /neccessary/ condition. xy may exist, even
> when x does not exist. We would simply lack a /proof/ of this fact
> in that case in my system.
For example, under my axioms (and yours), we cannot /prove/ that the
> string 01022 does /not/ exist; nor can we /prove/ that it /does/
> exist.
If it /does/ exist, we cannot prove that therefore 2 exists; nor can
> we prove that 2 does /not/ exist.
The existence (or non-existence) of a thing, and a /proof/ of the
> existence (or non-existence) of a thing, are /two/ /different/ /
> things/. Not the same. /Different/.
When I say my system is equivalent to your system, I mean : I can /
> prove/ the existence of a string from my axioms if, and only if, I
> can /prove/ the existence of that same string from your axioms; and
> the same follows for /proving/ the non-existence of a string.
If that is /not/ the kind of thing you /mean/ by E a -> E b, then I
> would recommend not abusing such fundamental notation as E a -> E b.
> It would be like using a - b in the naturals to mean a when
> multiplied by b, as well as a, when b is subtracted.
Use some /other/ notation that actually captures what you /do/ mean.
> Like an L-system! :)
> It doesn't say two follow from each
> one, so it doesn't describe a tree.
In my system, we have:
E (axiom)
We also have:
> E 0 (axiom)
> E x and E y -> E xy (axiom)
> E x and E 0 -> E x0 (substitution)
Conclusion: (by the inference: ( ((A and B) -> C) and B ) implies (A -
> C) )
E x -> E x0 (theorem)
Similarly, we can prove in my system:
E x -> E x1 (theorem)
I don't see a statement that E 1. Isn't that required?
So all your axioms are either axioms or /theorems/ in my system; which
> is to say that they are /exactly/ as true as statements in my system
> as they are in your system (no more, and no less).
Do you see that the above is correct? Thus my confusion: you claim
> that your system that:
E
> E x -> E x0.
> E x -> E x1.
are true statements and describes a tree, but that my system does /
> not/ describe a tree, even though each of those statements is also
> true in my system.
>
You'd have to include E 1, and your E x and E y -> E xy
allows
for the production of non-child relationships, if the edges have any
relation to the logical implication defining the tree. Aren't the edges
logical relationships?
> A string exists in your system if, and only if, there exists a
> corresponding string in the latter system prepended with the character
> ..
> Yeah, sure, butt hat doesn't describe the edges that connected them.
(heh hehe. You said butt hat. heh hehe.)
Yeah yeah! Heh! butt hat!!
Statements such as E a -> E b, as they are generally understood,
> don't describe anything /except/ the fact of the existence of b, /
> given/ the fact of existence of a.
>
Yes. Now picture that graphically. Does it remind you of a graph?
> It
> doesn't describe the bitwise production of the set of strings.
Oh, butt hat, it /does/ indeed describe (i.e., the two existence
> rules you describe as being true by your system are described as being
> true by mine as well). See the above theorems.
But your system (with E 1 included) also describes additional means
of production, namely, the appending of more than one character at a
time. Therefore, it does not describe a tree.
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<46812bda@news2.lightlink.com>
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<4681496d@news2.lightlink.com>
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<46853961@news2.lightlink.com>
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<468fd687@news2.lightlink.com>
<4693a9da@news2.lightlink.com>
<4696623a@news2.lightlink.com No, I meant exactly that. What do you think 
the
edges on the tree
> represent, if not logical implication and instantiation? The information
> regarding the edges is captured in those statements.
Do you know what a graph edge IS?
> Why can't it be both? What does an edge represent?
Why don't you find out what a graph edge IS?
> Yes. You know, I DID study trees and graphs, and I DO know the normal
> definitions
And so you pile on heaps and heaps of other UNDEFINED terminology on
top of the ordinary defintions. And that is supposed to be some kind
of mathematical contribution of yours...
MoeBlee
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
> No, I meant exactly that. What do you think the edges on the tree
> represent, if not logical implication and instantiation? The information
> regarding the edges is captured in those statements.
Do you know what a graph edge IS?
I know that standard definition, as a pair of atomic nodes. I am asking
what you think it MEANS?
> Why can't it be both? What does an edge represent?
Why don't you find out what a graph edge IS?
A pair of nodes. How...set-theoretical.
An edge indicates a production or relation. Close your eyes for a minute
and think.
> Yes. You know, I DID study trees and graphs, and I DO know the normal
> definitions
And so you pile on heaps and heaps of other UNDEFINED terminology on
> top of the ordinary defintions. And that is supposed to be some kind
> of mathematical contribution of yours...
MoeBlee
>
No, I offer extensions to the standard debate.
ToeKnee
===
Subject: Re: PARADISE LOST: Debunking Cantor's theory
<virgil-080907.10424602072007@comcast.dca.giganews.com>
<4689a90d@news2.lightlink.com>
<virgil-6646A9.23363202072007@comcast.dca.giganews.com>
<468a6db0@news2.lightlink.com>
<virgil-43A784.11005803072007@comcast.dca.giganews.com>
<468bf027@news2.lightlink.com>
<virgil-4C26F4.14273604072007@comcast.dca.giganews.com>
<468c5876@news2.lightlink.com>
<468fd777$1@news2.lightlink.com>
<f6u95n$cie$1@news.msu.edu>
<4693b3da@news2.lightlink.com>
<46955bed@news2.lightlink.com>
<46965154@news2.lightlink.com > It has been nearly
> three years now, and you still have not defined what you mean by
size.
> A count of elements, a member of *N, such that adding an element
> increases the size by 1, and removing one decreases it by 1, and
x-1<x<x+1.
> CIRCULAR, you dork!
> I didn't call you a dork, but you seem to like to bring it back there.
> If I recall, it was Brian you called a dork.
Yes, he was being obnoxious. Why do you feel it necessary to jump to his
> defense? He's a big boy. He can take care of himself.
No, I was incorrect. It was David R. Tribble you called a dork. He
was being blunt saying straight out what should be said.
And I didn't jump to his defense; I just called you what you called
him as my doing that highlights that YOU are the one who deserves the
apellation. Mr. Tribble is not the one giving circular definitions,
and giving circular definitions is definitely dorky! And even if I had
defended him at that time, that wouldn't imply that I don't think he
can't defend himself.
> I defined how size behaves.
> That's not a definition of 'size'.
No?
No, you dork.
> That is how one defines a measure such as
> size, in relation to set membership.
> No, it's not. You have no concept of mathematical definition.
> If an element is added, size is
> incremented, and if an element is removed, size is decremented, by 1.
> Size starts at 0, and every pure set size is a counting number,
> whether finite or infinite. You can certainly define a countable
subset,
> say, if the adics.
> Let us know when you have a definition of 'size'.
> You just used 'size' to try to define 'size'!
> I referred to size in describing the rules applicable to size.
> What's your DEFINITION of 'size'?
Weren't you the one ranting at me long ago about how definitions don't
> matter? I'm giving you axiomatic statements about a one-place primitive
> function size(S).
I spent a lot of effort explaining both in detail and in general terms
about this. You missed understanding ANY of it apparently. If you use
a term, then its primitive or defined. OF COURSE definitions matter in
the sense that you have to GIVE them for terms that are not primitive.
That doesn't contradict my remarks and explanations for you as to the
fact that with correct definitions we get only conservative extensions
of theories, which is to say that adding defined terms with correct
defintions does not affect the substance of the theory, but that IF
you do add a term and do NOT want to affect the substance of the
theory by taking the term as primitive THEN you do have to give a
correct (ESPECIALLY non-circular) definition.
I told you where to read a an excellent and quite down to earth
explanation of more about this, but you prefer to remain an ignoramus.
> Size is that measure of a set which increases with the addition of new
> elements, and decreases with the removal of existing elements.
Undefined now: 'measure', 'increases'.
Why don't you read a mathematics book for a change?
MoeBlee
===
Subject: Parametrical solutions of y^(2*x)=x^(3*y)
Good evening,
when putting y = x*t , we obtain :
x = t^(2/(3*t-2)) , y = t^(3*t/(3*t-2))
And for example t = 2 , w've got x =sqrt(2) ,y =2*sqrt(2)
Starting from the value of x in the given equation :
y^(2*sqrt(2) = sqrt(2)^(3*y) , will give other solutions
than y =2*sqrt(2) ;
So, what about parametrization?
Alain
===
Subject: Funny feature of serial correlation coefficient...
Hallo,
let {x_0,x_1,x_2,...} a sequence of outcomes of the random variable X
which can take only two values 0 or 1 (the sequence can be considered
as a string of bit). Does anybody knows where I can find the proof
(books, weblinks, papers...) that the serial correlation coefficient
(SCC) with lag 1, corresponds to the sum of transition probabilities
between an outcome and the following one:
SCC(x_i,x_(i+1)) = p(1->1) + p(0->0) - p(0->1) - p(1->0)
David Tischler
===
Subject: Re: Using C's rand() to generate a random number
> another between 0 and 24 and yet another between 0 and 25.
I am using this:
int getRandomNumber(int maxn)
> {
> return (unsigned int)(rand() % maxn);
}
The computations I am doing are extremely sensitive and I believe this
> random number generate to be biased. Since rand() returns a number
> between 0 and 32767, and not all 23, 24 and 25 can possibly divide
> 32767, some are more likely to be chosen than others.
In this program, I am eventually generating 500,000,000 random
> numbers, so this bias will likely to be seen.
I am not a programmer by trade so I don't really know a lot, but is
> there an easy way to get a unbiased random number?
>
I would recommend Mersenne Twister, which is good for trillions of
random 32 bit numbers. I use it for any application where I need more
than a few random numbers.
===
Subject: Collision probability (extension)
Hi guys,
I'd be grateful if someone can show me how to work out this:
What is the probabiliy of C different collisions after inserting N values
(chosen randomly from set of [1..D]) ?
differrent collisions mean something like if i've inserted 1..10, and 3
and 5, then i have 2 collision. But if i then insert 3 again, i still have 
2
collisions.
This looks like an extension of the Birthday paradox, but i can't work out
how.
===
Subject: The most minimal injective function
Hi all,
If x and y are subsets of w .
I want to define a relation f from x and y in the following manner:
we arrange x and y in an ascending manner after the cardinality of
their members.
The first member z1 in f would be the orderd pair that has the first
member of x denoted as r1 and the member s(r1) in y that is nearest
in value to r1 in x.
what I mean by nearest id the following:
s(r1) is the nearest to r1 means that from all ordered pairs
<r1,s> were sey , s(r1) is the s in y that has the smallest interval
value |r-s| , and if two such members exists then s(r1) is the
smaller of the two.
Example: let x = { 4,5,6,7,...}, y={ 1,3,5,....}
Now the first member in f:x->y would be z1=<4,3>
so although <4,5> has the same interval value
as <4,3> but since 3<5, then we select <4,3>
as the minmal ordered pair in f.
Now the second ordered pair will be z2=<r2 s(r2)> were
s(r2) would be the nearst value in y{s(r1)} to r2 in x.
were r2 is the second member in x.
In a similar manner the i-th ordered pair in f would be
zi= <ri , s(ri) > were ri is the i-th member of x
and s(ri) is the member in y{ s(r1),s(r2),...,s(r(i-1)) } that has
the nearest value ri.
Now f according to this definition would be an injection from x to y.
f is called the minimal injective function from x to y.
Now for any two sets that are Dedekindian infinite and
has bijection between them, then we will have two such minimal
injective functions , one in each direction.
Now to decide which of these two minimal injective functions is the
most minimal injective function, we order both functions
as below:
f1:x->y , f1={<r1,s(r1)> , < r2,s(r2) > , ............}
f2:y->x , f2={<k1,s(k1)> , < k2,s(k2) > , ............}
Now we see which pair is the first pair to have a bigger interval
betwen its paired member, then the function having this ordered pair
will not be the most minimal injective function.
Example:
x={ 0,4,8,12,.....}
y= { 1,3,5,7,9,11,13,......}
Now: f1:x->y is {<0,1>,<4,3>,<8,7>,<12,11>,.......}
and: f2:y->x is {<1,0>,<3,4>,<5,8>,...................}
Now f2 will be excluded from being the most minimal injective
function
between x and y , becasue <5,8> has interval of |5-8|=3
while <8,7> in f1 has |8-7|=1 interval.
So f1 is the most minimal injective function.
Now we define comparisons of Sx and Sy as:
1) Sx=Sy <-> the most minimal injective function between x and y is
bijective.
2) Sx<Sy<-> the most minimal injective function between x and y
is from x to y and is injective and not surjective.
3) Sx>Sy <-> Sy<Sx
Zuhair
===
Subject: Re: ATLAS
Well, I bought the ATLAS (withdrawn from a library,
> only $80!) What a great book. I read the first 8
> chapters right away, comprehending a fair amount,
> with much being over my head, of course. My question
> today is fairly
> simple, could someone give me a simple definition
> of the
> Schur Multiplier? I have a rough idea, but am a
> little
> hung up on Central Extension. I have looked at
> Wikipedia
> (requires homology theory). Mathworld has nothing.
> I'll answer your first question!
The Schur Multiplier of a group G is the second
> homolgy group H_2(G,Z)
> (Z = integers).
OK, that probably wasn't very helpful, so here is the
> equivalent group-
> theoretical definition.
A central extension of G is defined to be a group
> E with a
> subgroup N <= Z(E) such that E/N ~= G,
or equivalently an exact sequence 1 -> N -> E -> G - 1 with image(N)
> <= Z(G).
Call it a stem extension if, in addition, N <=
> [E,E], the commutator
> subgroup of E.
We can order such extensions and say a central
> extension (N',E') is
> larger than (N,E) if (N,E) is a proper epimorphic
> imageof (N',E') in
> the sense that there is a commutative diagram
1 -> N' -> E' -> G -> 1
| | | 1
> v v v
1 -> N -> E -> G -> 1
with all of the vertical maps epimorphic.
It can be proved that maximal stem extensions exist
> for any group G.
> Let (N,E) be one such. Then it turns out that N is
> uniquely determined
> up to isomorphism by G, and N is the Schur Multiplier
> of G.
PH asks:
does Z(E) and Z(G) mean? I am close to getting this,
> In general, E is not uniquely determined by G. For
> example, If G is a
> Klein 4-group, then N is cyclic of order 2, but E can
> be the dihedral
> group D8 or the quaternion group Q8.
PH asks:
So N is 2, and E could be Dihedral(4 elements)?
> However, if G is perfect (and in particular if G is
> simple), then E is
> uniqely determined by G.
For example, if G = A5, then E = SL(2,5).
PH asks:
I see. What would N be in this case?
PGH
===
Subject: Welcome into the Maple crisis continuing and extending! A
quadrature's ridiculous result
Maple bugs ARE ubiquitous. Our little demo continues...
Here is yet another salute to symbolic community and all
honest mathematicians from the VM machine which is still
ignored by CAS manufacturers.
So before you yet another sample of MULTIPLE regression bug
in Maple. Once it worked (1992-1996), then was broken (1997-
1999), then repaired (2000-2006), and now it's broken AGAIN.
...
Maple 11> evalf(Int(1/(z^exp(I)+z^Pi), z= 1..infinity));
-.5000000000 # where on earth is my non-zero imaginary part?!
# for adorers of workarounds
Maple 11> evalf(Int(1/(z^exp(I)+z^Pi), z=1..infinity, _Dexp));
.3384941890-.1978429309e-1*I
All other Maple versions are OK save Maple V 5 of 1997 which
returns
Error, (in evalf/int) unable to handle singularity
.................................................................
Man+Machine Review Of Maple Crisis
The beta 0.1 of the first world's document written by
a human in a close cooperation with a successor of the
GEMM machine, our unique VM automated testing expert
system which is a failure prediction oracle
http://maple.bug-list.org/maple-crisis.php
.................................................................
Best wishes,
Vladimir Bondarenko
VM and GEMM architect
Co-founder, CEO, Mathematical Director
http://www.cybertester.com/ Cyber Tester, LLC
http://maple.bug-list.org/ Maple Bugs Encyclopaedia
http://www.CAS-testing.org/ CAS Testing
===
Subject: This Week's Finds in Mathematical Physics (Week 254)
Also available as http://math.ucr.edu/home/baez/week254.html
July 13, 2007
This Week's Finds in Mathematical Physics (Week 254)
John Baez
This week I'd like to talk about exceptional Lie algebras and the
Standard Model, Witten's new paper on the Monster group and black
holes in 3d gravity, and Connes and Marcolli's new book! Then
I want to continue the Tale of Groupoidification.
However, I don't have the energy to do this all now. And even
if I did, you wouldn't have the energy to read it.
So, I'll just point you towards Connes and Marcolli's new book,
which you can download for free:
1) Alain Connes and Mathilde Marcolli, Noncommutative Geometry,
Quantum Fields and Motives, available at
http://www.alainconnes.org/downloads.html
I hope to discuss it sometime, especially since it tackles a
question I've been mulling lately: is there a good explanation
For now, I'll start by discussing Witten's latest paper:
2) Edward Witten, Three-dimensional gravity revisited, available
as arXiv:0706.3359.
This is a bold piece of work, which seeks to relate the entropy of
black holes in 3d quantum gravity to representations of the
Monster group - the largest sporadic finite simple group, with
about 10^{54} elements.
If the main idea is right, this gives a whole new view of
Monstrous Moonshine - the bizarre connection between the Monster
and fundamental concepts in complex analysis like the j-function.
(See week66 for a quick intro to Monstrous Moonshine.)
As the title hints, Witten had already tackled quantum gravity in
3 spacetime dimensions. In this earlier work, he argued it was an
exactly soluble problem: a topological field theory called
Chern-Simons theory. However, this theory is really an
*extension* of gravity to the case of degenerate metrics:
roughly speaking, geometries of spacetime where certain regions
get squashed down to zero size. Degenerate metrics are weird.
So, what happens if we try to quantize 3d gravity while insisting
that the metric be nondegenerate?
It's hard to say. So, Witten takes a few clues and cleverly fits
them together to make a surprising guess. He considers 3d general
relativity with negative cosmological constant. This has 3d
anti-DeSitter space as a solution. Anti-DeSitter space has a
boundary at infinity: a 2d cylinder with a conformal structure.
The AdS-CFT idea, also known as holography, suggests that in
this sort of situation, 3d quantum gravity should be completely
described by a field theory living on this boundary at infinity -
a field theory theory with all conformal transformations as
symmetries.
Which conformal field theory should correspond to 3d quantum
gravity with negative cosmological constant? It depends on the
value of the cosmological constant! Some topological arguments
suggest that the Chern-Simons description of 3d quantum gravity
is only gauge-invariant when the cosmological constant Lambda
takes certain special values, namely
Lambda = -1/(16 k)^2
where k is an integer, known as the level in Chern-Simons theory.
By the way: I'm working in Planck units here, and I'm assuming
our Chern-Simons theory is left-right symmetric, just to keep
things simple. I may also be making some small numerical errors.
This quantization of the cosmological constant must seem strange
if you've never seen it before, but it's not really so weird.
What's weird is that Witten is using Chern-Simons theory to
determine the allowed values of the cosmological constant even
though he wants to study what happens if gravity is *not*
described by Chern-Simons theory!
Witten knows this is weird: later he says we used the gauge
theory approach to get some hints about the right values of the
cosmological constant simply because it was the only tool
available.
Indeed, the whole paper seems designed to refute the notion that
make at each stage the most optimistic possible assumption.
Perhaps he has some evidence for his guesses that he's not
revealing yet. Or perhaps he's decided it takes courage verging
on recklessness to track the Monster to its lair.
Anyway: next Witten relates the level k to something called the
central charge of the conformal field theory living at the
boundary at infinity.
What's the central charge? This is a standard concept in
conformal field theory. Perhaps the simplest explanation is that
in a conformal field theory, the total energy of the vacuum state
is -c/24, where c is the central charge. So, naively you'd expect
c = 0, but quantum effects make nonzero values of the vacuum energy
possible, and even typical. A closely related cool fact is that
the partition function of a conformal field theory is only a
well-defined number up to multiples of
exp(2 pi i c / 24)
This means the partition function is a well-defined number when
c is a multiple of 24. This happens in certain especially nice
conformal field theories which are said to have holomorphic
factorization.
The appearance of the magic number 24 here is the first taste of
Monstrous Moonshine! For more on the importance of this number
in string theory, see week124, week125 and week126.
As you can see, there are lots of subtleties here, which I really
don't want to get into, but feel guilty about glossing over.
Here's another. There are really *two* conformal field theories
in this game: one that describes ripples of the gravitational
field moving clockwise around the boundary at infinity, and
another for ripples moving counterclockwise. Our simplifying
assumption about left-right symmetry lets us describe these
right-movers and left-movers with the same theory. So, both
have the same central charge.
In this case, the relation between central charge and level
is simple:
c = 24 k
Next, Witten considers the situation where k takes its smallest
interesting value: k = 1, so c = 24. It just so happens that
c = 24 conformal field theories with holomorphic factorization
have been classified, at least modulo a certain conjecture:
2) A. N. Schellekens, Meromorphic c = 24 conformal field theories,
Comm. Math. Phys. 153 (1993) 159-196. Also available as
hep-th/9205072.
It's believed there are 71 of them. Which one could describe
3d quantum gravity?
Of these 71, ALL BUT ONE have gauge symmetries! Now, Witten
is assuming 3d quantum gravity is *not* described by Chern-Simons
theory, which is a gauge theory. So, he guesses that the one
exceptional theory is the right one!
And this is a very famous conformal field theory. It's a theory
of a bosonic string wiggling around in a 26-dimensional spacetime
curled up in clever way with the help of a 24-dimensional lattice
called the Leech lattice. This theory is famous because its
symmetry group is the Monster group! It is, in fact, the simplest
thing we know that has the Monster group as symmetries.
For more details, try these - in rough order of increasing
thoroughness:
3) Terry Gannon, Postcards from the edge, or Snapshots
of the theory of generalised Moonshine, available as
arXiv:math/0109067.
Terry Gannon, Monstrous Moonshine: the first twenty-five years,
available as arXiv:math/0402345.
4) Richard Borcherds, online papers, available at
http://math.berkeley.edu/~reb/papers/
5) Igor Frenkel, James Lepowsky, Arne Meurman, Vertex operator
algebras and the Monster, Academic Press, New York, 1988.
Now, if this monstrous conformal field theory turned out to be
3d quantum gravity in disguise - viewed from infinity, so to
speak - it might someday give a much better understanding of
Monstrous Moonshine. However, Witten gives no explanation as to
*why* this theory should be 3d gravity, except for the indirect
argument I just sketched. The precise relation between 3d
quantum gravity and the bosonic string wiggling around in
26 dimensions remains obscure.
However, while Witten leaves this mysterious, he does offer a
tantalizing extra tidbit of evidence that the relation is real!
The partition function of the monstrous conformal field theory
I just mentioned is the j-function, or more precisely:
J(q) = q^{-1} + 196884 q + 21493760 q^2 + ...
As I mentioned, this function shows up naturally in complex
analysis. More precisely, it parametrizes the moduli space of
elliptic curves (see week125). But, its bizarre coefficients
turn out to be dimensions of interesting representations of the
Monster group. For example, the smallest nontrivial representation
of the Monster has dimension 196883; adding the trivial
representation, we get 196884. This was one of several strange
clues leading to the discovery of Monstrous Moonshine.
What Witten does is assume that the monstrous conformal field
theory describes 3d quantum gravity for k = 1, and then use
properties of the j-function to compute the entropy of black
holes!
I won't attempt to explain the calculation. Suffice it to say
that the lightest possible black hole turns out to have
196883 quantum states - its space of states is the smallest
nontrivial representation of the Monster group. So, its entropy
is:
ln(196883) ~ 12.19
On the other hand, Hawking's semiclassical calculation gives
4 pi ~ 12.57
The match is not perfect - but it doesn't need to be, since we
expect quantum corrections to Hawking's formula for small black
holes.
What's more impressive is that Witten can guess the entropy of
the lightest possible black hole for other values of k - meaning,
other values of the cosmological constant. The space of states
of these black holes are always representations of the Monster
group, so we get logarithms of weird-looking integers. For
example, for k = 4 the entropy is
ln(81026609426) ~ 25.12
while Hawking's formula gives
8 pi ~ 25.13
Much better! And, using a formula of Petersson and Rademacher
for asymptotics of coefficients of the j-function, together with
some facts about Hecke operators, he shows that as k -> infinity,
the agreement becomes perfect!
In short, there are some fascinating hints of a relation between
the Monster group and black hole entropies in 3d gravity, but the
details of Witten's hoped-for AdS-CFT correspondence between 3d
gravity and the monstrous conformal field theory remain obscure.
Indeed, there are lots of problems with Witten's proposal:
6) Jacques Distler, Witten on 2+1 gravity,
http://golem.ph.utexas.edu/~distler/blog/archives/001335.html
But, time will tell. In fact, if history is any guide, we can
expect to see armies of string theorists marching into this
territory any day now. So, I'll just pose one question.
There's a well-known route from 2d rational conformal field
theories (or RCFTs) to 3d topological quantum field theories
(or TQFTs), which passes through modular tensor categories.
For example, an RCFT called the Wess-Zumino-Witten model gives
the TQFT called Chern-Simons theory.
But, now Witten is saying 3d quantum isn't Chern-Simons theory;
instead, it's something related to the monstrous CFT.
So: is the monstrous conformal field theory known to be an RCFT?
If so, what 3d TQFT does it give? Could this TQFT be the 3d
quantum gravity theory Witten is seeking?
Even though Witten is now claiming 3d quantum gravity *can't be*
a TQFT, I think this is an interesting question. At the very
least, I'd like to know more about this Monster TQFT - if it
exists.
physics...
Last week I described some mathematical relations between the
theories, and some exceptional structures in mathematics: the
exceptional Lie group E6, the complexified octonionic projective
plane, and the exceptional Jordan algebra.
This week I want to go a bit further, and talk about the work
of Kac, Larsson and others on the exceptional Lie superalgebras
E(3|6), E(3|8) and E(5|10).
As before, my goal is to point out some curious relations between
structures we find in mathematics. By this, I mean structures
that show up when you classify algebraic gadgets, but don't fit
into nice systematic infinite families. Right now the Monster
is the king of all exceptional structures, the biggest of the
26 sporadic finite simple groups. But, there are lots of other
such structures, and they all seem to be related.
As mentioned back in week66, Edward Witten once suggested that
the correct theory of our universe could be an exceptional
structure of some sort. There's even a fun hand-wavy argument
for this idea. It goes like this: the theory of our universe
*must* be incredibly special, since out of all the theories we can
write down, only one describes the universe that actually exists!
In particular, lots of very simple theories do *not* describe our
universe. So there must be some principle besides simplicity that
picks out the theory of our universe.
Unfortunately, when we try to think about these issues seriously,
we're quickly led into very deep waters. In practice, people
quickly muddy these waters and create a quagmire. It's very hard
to discuss this stuff without uttering nonsense. If you want
to see my try, look at week146.
But right now, I prefer to act like a sober, serious mathematical
physicist. So, I'll tell you a bit about exceptional Lie
superalgebras and how they could be related to the Standard Model.
First, some history. In 1887, Wilhelm Killing sent a letter to
Friedrich Engel saying he'd classified the simple Lie algebras.
Besides the classical ones - namely the infinite series sl(n,C),
so(n,C) and sp(n,C) - he found 6 exceptions: a 14-dimensional one,
two 52-dimensional ones, a 78-dimensional one, a 133-dimensional
one and a 248-dimensional one.
In 1894, Eli Cartan finished a doctoral thesis in which he cleaned
up Engel's work. In the process, he noticed that Engel's two
52-dimensional Lie algebras were actually the same. Whoops!
So, we now have just 5 exceptional simple Lie algebras. In
order of increasing size, they're called G2, F4, E6, E7 and E8.
In 1914, Cartan realized that the smallest exceptional Lie algebra,
G2, comes from the symmetry group of the octonions! Later it was
realized that all 5 are connected to the octonions. I've written
a lot about this in previous Weeks, but most of that material can
be found here:
6) John Baez, Exceptional Lie algebras,
http://math.ucr.edu/home/baez/octonions/node13.html
Now, whenever mathematicians do something fun, they want to keep
doing it, which means *generalizing* it.
One way to generalize Cartan's work is to study symmetric spaces,
which I defined last week. Briefly, a symmetric space is a
manifold equipped with a geometrical structure that's very
symmetrical: so much so that every point is just like every other,
and the view in any direction is the same as the view in the
opposite direction.
In fact, it was Cartan himself who invented the concept of
symmetric space, and after he classified the simple Lie algebras
he went ahead and classified these.
More precisely, I think he classified the compact Riemannian
symmetric spaces. Every simple Lie algebra gives one of these,
namely a compact simple group. But, there are others too. So,
compact Riemannian symmetric spaces are a nice generalization
of simple Lie algebras - and I believe Cartan succeeded in
classifying them all.
Again, there are some infinite series, but also some exceptions
coming from the octonions. I talked about one of these last week,
namely EIII, the complexified octonionic projective plane. You
can see a list here:
7) Wikipedia, Riemannian symmetric space,
http://en.wikipedia.org/wiki/Riemannian_symmetric_space
For a quick intro to the classification of simple Lie algebras and
compact Riemannian symmetric spaces, try this great book:
For a slower, more thorough introduction, try the book by Helgason
mentioned last Week.
A second way to generalize Cartan's work is to consider simple
Lie *superalgebras*.
Lie superalgebras are just like Lie algebras, except they're split
into an even or bosonic and odd or fermionic part. The idea
is that we stick minus signs in the usual Lie algebra formulas
whenever we switch two odd elements.
This is very natural from a physics viewpoint, since whenever
you switch two identical fermions, the wavefunction of the
universe gets multiplied by -1. (Take my word for it - I've seen
it happen!)
It's also very natural from a math viewpoint, since super vector
spaces form a symmetric monoidal category with almost all the
nice properties of plain old vector spaces. This lets crazed
mathematicians and physicists systematically generalize pretty
much all of linear algebra to the super world. So, why not Lie
algebras?
The simple Lie superalgebras were classified by Victor Kac in
1977:
9) Victor Kac, Lie superalgebras, Adv. Math. 26 (1977), 8-96.
Not counting the ordinary simple Lie algebras, there are 8
series of simple Lie superalgebras and a few exceptional ones.
At least some of these exceptions come from the octonions:
10) Anthony Sudbery, Octonionic description of exceptional
Lie superalgebras, Jour. Math. Phys. 24 (1983), 1986-1988.
Do they all? I don't know! Someone please tell me!
A third way to generalize Cartan's work is to classify
*infinite-dimensional* simple Lie algebras - or for that matter,
Lie superalgebras.
So far I've implicitly assumed all our algebraic gadgets are
finite-dimensional, but we can lift that restriction. If you
try to classify infinite-dimensional gadgets without *any*
restrictions, it can get really hairy. It turns out the nice
thing is to classify linearly compact infinite-dimensional
simple Lie algebras. I won't define the quoted phrase, since
it's technical and it's explained near the beginning
of this paper:
11) Victor Kac, Classification of infinite-dimensional simple
linearly compact Lie superalgebras, Erwin Schroedinger Institut
preprint, 1998.
Available at http://www.esi.ac.at/Preprint-shadows/esi605.html
Anyway, back in 1880 Lie himself made a guess about
infinite-dimensional Lie algebras that would solve the problem
I'm talking about now, though he didn't phrase it in the modern
way. And, Cartan proved Lie's guess in 1909! Actually, there
was a hole in Cartan's proof, which was only noticed much later.
It was filled by Guillemin, Quillen and Sternberg in 1966.
So, here's the answer: there are 4 families of linearly compact
infinite-dimensional simple Lie algebras, and no exceptions.
Ignoring an important nuance I'll explain later, these are:
A) The Lie algebra of all complex vector fields on C^n.
B) The Lie algebra of all complex vector fields v on C^n
that are divergence-free:
div v = 0
C) The Lie algebra of all complex vector fields v on C^{2n}
that are symplectic:
L_v omega = 0
where omega is the usual symplectic structure on C^{2n},
and L means Lie derivative
D) The Lie algebra of all complex vector fields v on C^{2n+1}
that are contact:
L_v alpha = f alpha
for some function f depending on v, where alpha is the usual
contact structure on C^{2n+1}.
If you don't know about symplectic structures or contact
structures, don't worry - we won't need them now. The main point
is that they're differential forms that show up throughout
classical mechanics. So, this classification theorem is
surprisingly nice.
Notice: no exceptions! That's a kind of exception in its own
right.
In 1998, Victor Kac proved the super version of this result.
In other words, he classified linearly compact infinite-dimensional
Lie superalgebras! This result is Theorem 6.3 of his paper above.
There turn out to be 10 families and 6 exceptions, which are
called E(1|6), E(2|2), E(3|6), E(3|8), E(4|4) and E(5|10).
Many of the families are straightforward super generalizations
of the 4 families I just showed you. Some are stranger. Most
important for us today are the exceptions discovered by Irina
Shchepochkina in 1983:
12) Irena Shchepochkina, New exceptional simple Lie superalgebras,
C. R. Bul. Sci. 36 (1983), 313-314.
The easiest to explain is E(5|10). And, you'll soon see that the
number 5 here is related to the math of the SU(5) grand unified
theory, which I explained last Week!
The even part of E(5|10) is the Lie algebra of divergence-free complex
vector fields on C^5.
The odd part of E(5|10) consists of closed complex 2-forms on C^5.
The bracket of two even guys is the usual Lie bracket of vector
fields.
The bracket of an even guy and an odd guy is the usual Lie
derivative of a differential form with respect to a vector field.
The only tricky bit is the bracket of two odd guys! So, suppose
mu and nu are closed complex 2-forms on C^5. Their wedge product
is a 4-form mu ^ nu. But, we can identify this with a vector
field v by demanding:
i_v vol = mu ^ nu
Here vol is the volume form:
vol = dx_1 ^ dx_2 ^ dx_3 ^ dx_4 ^ dx_5
and i_v vol is the interior product, which feeds v into vol and
leaves us with a 4-form. You can check that this vector field v
is divergence-free. So, we define the bracket of mu and nu to be
v.
sl(5,C) sits inside the even part of E(5|10) in a nice way, as
the divergence-free vector fields whose coefficients are *linear*
functions on C^5. So, since su(5) sits inside sl(5,C), we get a
tempting relation to SU(5).
(Now I'll come clean now and explain the important nuance I
ignored earlier. For the classification theorems I mentioned
earlier, we must use vector fields and differential forms with
*formal power series* as coefficients. But for the purposes
of mathematical physics, we should keep a more flexible attitude.)
Next, what about E(3|6)? This is contained in E(5|10). To define
it, we give E(5|10) a clever grading where x_1, x_2, x_3 are
treated differently from the other two variables. Then we take
the subalgebra of degree-zero guys. The details are explained in
the above papers - or more simply, here:
13) Victor Kac, Classification of infinite-dimensional simple
groups of supersymmetries and quantum field theory, available as
math.QA/9912235.
All this is reminiscent of how SU(5) contains the gauge group of
the Standard Model, namely S(U(3) x U(2)). In particular, the
even part of E(3|6) contains the Lie algebra
sl(3,C) + sl(2,C) + gl(1,C)
in a canonical way. So, any representation of E(3|6) automatically
gives a representation of the Standard Model Lie algebra
su(3) + su(2) + u(1)
And in the above paper Kac goes even further! He defines a fairly
natural class of representations of E(3|6), and proves something
remarkable: these restrict to representations of
su(3) + su(2) + u(1)
that correspond precisely to the gluon, the photon and the W and
Z bosons, and the quarks and leptons in one generation...
boson, but instead acts like a gluon with electric charge +-1.
Darn.
One nice thing is how these Lie superalgebras get both bosons and
fermions into the game in a natural way without forcing the
existence of a bunch of unseen superpartners. One unfortunate
thing is that the above result gives no hint as to why there
should be three generations of quarks and leptons. However, Kac
and Rudakov develop some mathematics to address that question
here:
14) Victor Kac and Alexi Rudakov, Representations of the
exceptional Lie superalgebra E(3,6): I. Degeneracy conditions.
Available as math-ph/0012049.
Representations of the exceptional Lie superalgebra E(3,6): II.
Four series of degenerate modules. Available as math-ph/0012050.
Representations of the exceptional Lie superalgebra E(3,6)
III: Classification of singular vectors. Available as
math-ph/0310045.
15) Victor Kac, Classification of supersymmetries, Proceedings
of the ICM, Beijing, 2002, vol. 1, 319-344. Available as
math-ph/0302016.
the stable region [whatever that means], but the situation with
quarks is more complicated: this model predicts a complete fourth
generation of quarks and an incomplete fifth generation (with
missing down type triplets).
So, while I don't understand this model, it seems tantalizingly
close to capturing the algebraic patterns in the Standard Model...
without quite doing so.
Some more nice explanations and references can be found here:
16) Irina Shchepochkina, The five exceptional simple Lie
superalgebras of vector fields. Available as hep-th/9702.3121.
17) Pavel Grozman, Dimitry Leites and Irina Shchepochkina,
Defining relations for the exceptional Lie superalgebras of
vector fields pertaining to The Standard Model, available as
math-ph/0202025.
18) Pavel Grozman, Dimitry Leites and Irina Shchepochkina,
Invariant operators on supermanifolds and Standard Models,
available as math.RT/0202193.
Thomas Larsson has been working on similar ideas, mainly
using E(3|8) instead. This also contains
sl(3,C) + sl(2,C) + gl(1,C)
in a canonical way.
19) Thomas A. Larsson, Symmetries of everything, available as
math-ph/0103013.
Exceptional Lie superalgebras, invariant morphisms, and a
second-gauged Standard Model, available as math-ph/020202.
Thomas A. Larsson, Maximal depth implies su(3)+su(2)+u(1),
available as hep-th/0208185.
Alas, E(3|8) gets the hypercharges of some fermions wrong.
Larsson seems to say this problem also occurs for E(3|6),
which would appear to contradict what Kac claims - but I
could be misunderstanding.
I'll end with few questions. First, is there any relation
between the exceptional Lie superalgebras E(5|10), E(3|6) or
E(5|10) and the exceptional Lie algebra e6? Last week I
explained some relations between e6 and the Standard Model;
are those secretly connected to what I'm discussing this week?
Second, has anyone tried to unify all three generalizations of
Cartan's classification of simple Lie algebras? Starting from
simple Lie algebras, we've seen three ways to generalize:
A) go to symmetric spaces,
B) go the super version,
C) go to the infinite-dimensional case.
So: has anyone tried to classify infinite-dimensional super
versions of symmetric spaces? Or even finite-dimensional ones?
(Maybe the super version of a symmetric space should be
called a supersymmetric space, just for the sake of a nice pun.)
Next, the Tale of Groupoidification! I'll keep this week's
episode short, since you're probably exhausted already.
I want to work my way to the concept of Hecke operator
through a series of examples. The examples I'll use are a bit
trickier than the concept I'm really interested in, since the
examples involve integrals, where the Hecke operators I ultimately
want to discuss involve sums. But, the examples are nice if
you like to visualize stuff...
In these examples we'll always have a relation between two sets
X and Y. We'll use this to get an operator that turns functions
on X into functions on Y - a Hecke operator.
A) The Radon transform in 2 dimensions
Suppose you're trying to do a CAT scan. You want to obtain
a 3d image of someone's innards. Unfortunately, all you do is
take lots of 2d X-ray photos of them. How can you assemble all
this information into the picture you want?
Who better to help you out than a guy named after a
radioactive gas: Radon!
In 1917, the Viennese mathematician Johann Radon tackled a related
problem one dimension down. You could call it a CAT scan for
flatlanders.
Suppose you want to obtain a complete image of the insides of a
2-dimensional person, but all you can do is shine beams of X-rays
through them and see how much each beam is attenuated.
So, mathematically: you have a real-valued function on the plane -
roughly speaking, the density of your flatlander. You're trying
to recover this function from its integrals along all possible
lines. Someone hands you this function on the space of *lines*,
and you're trying to figure out the original function on the space
of *points*.
(Points lying on lines! If you've been following the Tale of
Groupoidification, you'll know this incidence relation is
connected to Klein's approach to geometry, and ultimately to
spans of groupoids. But pretend you don't notice, yet.)
Now, it's premature to worry about this tricky inverse problem
before we ponder what it's the inverse of: the Radon transform.
This takes our original function on the space of *points* and
gives a function on the space of *lines*.
Let's call the Radon transform T. It takes a function f on the
space of points and gives a function Tf on the space of lines,
as follows. Given a line y, (Tf)(y) is the integral of f(x) over
the set of all points x lying on y.
What Radon did is figure out a nice formula for the inverse of
this transform. But that's not what I'm mainly interested in now.
It's the Radon transform itself that's a kind of Hecke operator!
Next, look at another example.
B) The X-ray transform in n dimensions
This is an obvious generalization to higher dimensions of what
I just described. Before we had a space
X = {points in the plane}
and a space
Y = {lines in the plane}
and an incidence relation
S = {(x,y): x is a point lying on the line y}
If we go to n dimensions, we can replace all this with
X = {points in R^n}
Y = {lines in R^n}
S = {(x,y): x is a point lying on the line y}
Again, the X-ray transform takes a function f on the space of
points and gives a function Tf on the space of lines. Given a
line y, (Tf)(y) is the integral of f(x) over the set of all x
with (x,y) in S.
Next, yet another example!
C) The Radon transform in n dimensions
Radon actually considered a different generalization of the
2d Radon transform, using hyperplanes instead of lines. Using
hyperplanes is nicer, because it gives a very simple relationship
between the Radon transform and the Fourier transform. But never
mind - that's not the point here! The point is how similar
everything is. Now we take:
X = {points in R^n}
Y = {hyperplanes in R^n}
S = {(x,y): x is a point lying on the hyperplane y}
And again, the Radon transform takes a function f on X
and gives a function Tf on Y. Given y in Y, (Tf)(y) is the
integral of f(x) over the set of all x with (x,y) in S.
We're always doing the same thing here. Now I'll describe
the general pattern a bit more abstractly.
We've always got three spaces, and maps that look like this:
S
/
/
P/ Q
/
v v
X Y
In our examples so far these maps are given by
P(x,y) = x
Q(x,y) = y
But, they don't need to be.
Now, how do we get a linear operator in this situation?
Easy! We start with a real-valued function on our space X:
f: X -> R
Then we take f and pull it back along P to get a function on S.
Pulling back along P is just impressive jargon for composing
with P:
f o P: S -> R
Next, we take f o P and push it forwards along Q to get a
function on Y. The result is our final answer, some function
Tf: Y -> R
Pushing forwards along Q is just impressive jargon for
integrating: Tf(y) is the integral over all s in S with Q(s) = y.
For this we need a suitable measure, and we need the integral
to converge.
This is the basic idea: we define an operator T by pulling
back and then pushing forward functions along a span, meaning
a diagram shaped like a bridge:
S
/
/
P/ Q
/
v v
X Y
But, the reason this operator counts as a Hecke operator
is that it gets along with a symmetry group G that's acting
on everything in sight. In the examples so far, this is
the group of Euclidean symmetries of R^n. But, it could be
anything.
This group G acts on all 3 spaces: X, Y, and S. This makes the
space of functions on X into a representation of G! And, ditto
for the space of function on Y and S.
Furthermore, the maps P and Q are equivariant, meaning
P(gs) = gP(s)
and
Q(gs) = gQ(s)
This makes pulling back along P into an intertwining operator
between representations of G. Pushing forwards along Q will
also be an intertwining operator if the measure we use is
G-invariant in a suitable sense. In this case, our transform T
becomes an intertwining operator between group representations!
Let's call an intertwining operator constructed this way a Hecke
operator.
If you're a nitpicky person, e.g. a mathematician, you may wonder
what I mean by a suitable sense. Well, each fiber Q^{-1}(y)
of the map
Q: S -> Y
needs a measure on it, so we can take a function on S and
integrate it over these fibers to get a function on Y. We need
this family of measures to be invariant under the action of G,
for pushing forwards along Q be an intertwining operator.
This stuff about invariant families of measures is mildly
annoying, and so is the analysis involved in making precise
*which* class of functions on X we can pull back to S and then
push forward to Y - we need to make sure the integrals converge,
and so on. When I really get rolling on this Hecke operator
business, I'll often focus on cases where X, Y, and S are
*finite* sets... and then these issues go away!
Hmm. I'm getting tired, but I can't quit until I say one more
thing. If you try to read about Hecke operators, you *won't*
see anything about the examples I just mentioned! You're most
likely to see examples where X and Y are spaces of lattices in
the complex plane. This is the classic example, which we're
trying to generalize. But, this example is more sophisticated
than the ones I've mentioned, in that the functions on X and
Y become sections of vector bundles over X and Y. The same
sort of twist happens when we go from the Radon transform to the
more general Penrose transform.
Anyway, next time I'll talk about some really easy examples,
where X, Y, and S are finite sets and G is a finite group.
These give certain algebras of Hecke operators, called Hecke
algebras.
In the meantime, see if you can find *any* reference in the
literature which admits that Hecke algebras are related
to Hecke operators. It ain't easy!
It's a great example of a mathematical cover-up - one we're
gonna bust wide open.
-----------------------------------------------------------------------
Addendum: for more discussion, go to the n-Category Cafe:
http://golem.ph.utexas.edu/category/2007/07/this_weeks_finds_in_mathematic_1

5.html
-----------------------------------------------------------------------
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twfcontents.html
A simple jumping-off point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
===
Subject: Re: This Week's Finds in Mathematical Physics (Week 254)
>July 13, 2007
>This Week's Finds in Mathematical Physics (Week 254)
>John Baez
This week I'd like to talk about exceptional Lie algebras and the
>Standard Model, Witten's new paper on the Monster group and black
>holes in 3d gravity, and Connes and Marcolli's new book! Then
>I want to continue the Tale of Groupoidification.
Bunghole Baez strikes again! ahahaha... Physics has taken another
giant leap backwards as Bunghole Baez pontificates about yet more
irrelevant math puzzles. ahahaha... In the meantime, real physics gets
buried in a sea of useless complexity while Bunghole Baez calls home
and says, Look, ma. I am so smart. ahahaha... Sometimes I wonder if
Bunghole Baez is a friend of that little con artist in the wheelchair,
over in England. ahahaha... AHAHAHA... ahahaha...
Louis Savain
Why Software Is Bad and What We Can Do to Fix It:
http://www.rebelscience.org/Cosas/Reliability.htm
===
Subject: Re: (non)complete graphs of polytopes and (non)unique
decomposition
as a convex combination of vertices
<slrnf8rub0.rg9.unsenet@localhost.localdomain * Erik Quaeghebeur
<equaeghe@nospammail.net>:
> I think (but am not sure) that when one has a polytope with a complete
> graph as adjacency graph (of its vertices), any point of the polytope
> can be written as a unique convex combination of these vertices _and_
> that the converse also holds: a polytope with a noncomplete graph as
> adjacency > graph does not have this property,
Yes, you are correct on both counts. Radon's theorem says that
> the vertex set of a polytope of dimension d with more than d+1
> points can be partitioned into two blocks whose convex hulls
> intersect, implying your second claim above.
before). The one thing that strictly speaking, I still have to solve, is
going from the (number of edges is the) adjacency graph to the dimension
of the corresponding polytope (usually much smaller than the space it's
embedded in, in my case).
Any suggestions welcome,
Erik
===
Subject: Re: Ultimate debunking of Cantor's Theory
<4694dd53$0$13845$afc38c87@news.optusnet.com.au
> THE ONLY SETS WHICH EXIST ARE FINITE SETS.
> THE INFINITE IS ONLY A PRODUCT OF THE IMAGINATION.
> Then what is the largest natural number?
> I can't see that restricting set theory to finite sets
> means there has to be a largest natural number.
The original post doesn't restrict set theory to finite
> sets. It says that only sets which are finite exist.
> Obviously natural numbers exist, and can be used for
> example to count the words in this post. Since they
> exist, and there are supposedly only a finite number
> of them, then there must be a largest one.
That is a naive misunderstanding. There is a limited number of bits.
numbers is merely limited by the ingeniousity to introduce suitable
abbreviations.
===
Subject: Re: Ultimate debunking of Cantor's Theory
> THE ONLY SETS WHICH EXIST ARE FINITE SETS. THE INFINITE IS ONLY A
> PRODUCT OF THE IMAGINATION.
What is everyone's obsession with debunking Cantor? This is insane, I
> think there are three forums devoted to this topic. I would have
> thought that only students new to set theory would have a problem with
> aleph_0. sigh
Once upon a time only students new to electrodynamics would have a
problem with ether. What is simultaneity? There are some other simple
questions which seem to be silly, but require advanced wisdom to
answer. Why is it dark at night? Why is the sky blue? Why is Cantor's
proof wrong?
===
Subject: Re: Ultimate debunking of Cantor's Theory
<46959e73$0$13845$afc38c87@news.optusnet.com.au On Jul 11, 11:24 pm, Peter 
Webb
> But set theory minus the Axiom of infinity is a perfectly valid set
theory.
> The advantage for the poster is that Cantor's diagonal construction of
the
> Reals doesn't exist, or indeed any form of the diagonal argument applied
to
> infinite sets. ...
I assume that by 'Cantor's diagonal construction' you
> mean considering a hypothetical countable list of all of
> the decimal expansions of the real numbers between 0 and
> 1, then going down the diagonal, changing each digit to
> any digit other than the one at each diagonal position,
> and then noticing that the real number so constructed by
> using the changed diagonal elements cannot be in the
> original list. Thus the existence of a countable list
> of decimal expansions of the reals between 0 and 1 is
> disproven.
A variation of that which I subjectively like is making
> it a list of binary expansions instead of decimal. Then
> it is only necessary to 'flip' the diagonal, changing
> all ones to zeros and all zeros to ones.
No, this doesn't work. In fact, it fails spectacularly:
a(1) = .011111...
a(2) = .011111...
a(3) = .011111...
...
Your new binary decimal turns out to be .100000..., which is equal
to .01111...; so you don't get a new number after all!
> Are there other noteworthy forms of the diagonal argument?
The Halting problem?
...
This is as good of a place to any to present my motivation for this
thread: (1) I had read somewhere about the Infinity Axiom and the
fact that you can have a set theory without (actual) infinite sets,
and I was wondering if it actually works; and (2) I was sick and tired
of anti-Cantor threads in sci.math.
--- Christopher Heckman
===
Subject: Re: Ultimate debunking of Cantor's Theory
<46959e73$0$13845$afc38c87@news.optusnet.com.au > A variation of that which 
I
subjectively like is making
> it a list of binary expansions instead of decimal. Then
> it is only necessary to 'flip' the diagonal, changing
> all ones to zeros and all zeros to ones.
No, this doesn't work. In fact, it fails spectacularly:
a(1) = .011111...
> a(2) = .011111...
> a(3) = .011111...
> ...
Your new binary decimal turns out to be .100000..., which is equal
> to .01111...; so you don't get a new number after all!
I have no idea what you are talking about.
Your a(1), a(2), and a(3) above do not suggest
a hypothetical list of the binary representations
of the real numbers between 0 and 1, which is what
I was talking about.
I'll take a wild guess at what you mean. Maybe
you are saying that there could be a hypothetical
countable list of the reals between 0 and 1 such
that for the nth element of the list (n>1), the binary
expansion is .0 followed by all 1's out to the
diagonal position, and whatever else beyond the
diagonal position.
But you can't make the list that way, because many
(infinitely many, actually) of the reals between
0 and 1 would be missing from such a list.
Similarly you couldn't make a list of the rational
numbers between 0 and 1 that way. It's absurd.
===
Subject: Re: Ultimate debunking of Cantor's Theory


> A variation of that which I subjectively like is making
> it a list of binary expansions instead of decimal. Then
> it is only necessary to 'flip' the diagonal, changing
> all ones to zeros and all zeros to ones.
> No, this doesn't work. In fact, it fails spectacularly:
> a(1) = .011111...
> a(2) = .011111...
> a(3) = .011111...
> ...
> Your new binary decimal turns out to be .100000..., which is equal
> to .01111...; so you don't get a new number after all!
I have no idea what you are talking about.
> Your a(1), a(2), and a(3) above do not suggest
> a hypothetical list of the binary representations
> of the real numbers between 0 and 1, which is what
> I was talking about.
I'll take a wild guess at what you mean. Maybe
> you are saying that there could be a hypothetical
> countable list of the reals between 0 and 1 such
> that for the nth element of the list (n>1), the binary
> expansion is .0 followed by all 1's out to the
> diagonal position, and whatever else beyond the
> diagonal position.
But you can't make the list that way, because many
> (infinitely many, actually) of the reals between
> 0 and 1 would be missing from such a list.
Similarly you couldn't make a list of the rational
> numbers between 0 and 1 that way. It's absurd.
>
What Prog said is quite true. He just picked an example which (whilst
correct) is probably a little bit too clever.
He hasn't claimed that you can write down a list of all Reals in base 2. He
claims that the Cantor diagonal argument cannot be used in base 2 to prove
this.
Here is a somewhat clearer example.
Imagine the list is:
a(1) = 0.011111 ...
a(2) = 0.01
a(3) = 0.001
a(4) = 0.0001
.
.
Form the diagonal by flipping bits. You get
cantor diagonal = 0.1000000...
But 0.0111... is the same number 0.1 (unless you also believe that 0.999...
<> 1), and so the Cantor diagonal does appear on the list.
Now we all know that the list above doesn't contain every Real, but the
Cantor diagonal construction doesn't itself prove this to be true.
(His example was a bit too clever because he set a(2) = a(3) = a(4)
....,
which is quite valid but obscures his central argument).
===
Subject: Re: Ultimate debunking of Cantor's Theory
<46959e73$0$13845$afc38c87@news.optusnet.com.au>
<469715aa$0$10708$afc38c87@news.optusnet.com.au>
On Jul 13, 2:03 am, Peter Webb
> What Prog said is quite true. He just picked an example which (whilst
> correct) is probably a little bit too clever.
He hasn't claimed that you can write down a list of all Reals in base 2.
He
> claims that the Cantor diagonal argument cannot be used in base 2 to
prove
> this.
Here is a somewhat clearer example.
Imagine the list is:
a(1) = 0.011111 ...
> a(2) = 0.01
> a(3) = 0.001
> a(4) = 0.0001
> .
> .
Form the diagonal by flipping bits. You get
cantor diagonal = 0.1000000...
But 0.0111... is the same number 0.1 (unless you also believe that
0.999...
> <> 1), and so the Cantor diagonal does appear on the list.
Now we all know that the list above doesn't contain every Real, but the
> Cantor diagonal construction doesn't itself prove this to be true.
(His example was a bit too clever because he set a(2) = a(3) = a(4)
....,
> which is quite valid but obscures his central argument).
My problem with what he did is not a failure to
understand that 0.0111... is the same number as 0.1;
I accepted that immediately.
My problem with it is that he produced a CONTRIVED list,
flipped the diagonal and said that the new number
thus produced is indeed already in the list. So
what?
That's not the sort of thing Cantor's argument is.
He takes a traditional reductio ad absurdum approach
and 'assumes' that what he wants to prove is not true.
If it were not true, then a countable list of decimal
(or binary) expansions of the reals between 0 and 1
could exist. He didn't say how such a list might be
generated. How could he? Why would he? He wants
to show that the existence of such a complete list
is impossible.
To make a simple analogy, if you are proving that
there is no rational number the square of which
is 2, you assume the contrary and posit integers p
and q in lowest terms, q not 0, for which the square
of p/q is 2. You don't claim to know what p and q
would have to be. That would be outrageously absurd,
since your motive is to prove that they don't exist
as posited.
I could be very much in sympathy with a complaint
against Cantor's diagonal argument that said it is
not acceptable to operate on the assumed impossible
list to the extent that he does in order to 'show'
that such a countable list cannot be complete.
I've always been uneasy about that. But the sort
of approach that Proginoskes uses to 'refute' Cantor
in the case of a list of binary forms of the reals
is an insult to everyone's intelligence, it seems
to me.
===
Subject: Re: Ultimate debunking of Cantor's Theory


> On Jul 13, 2:03 am, Peter Webb
> What Prog said is quite true. He just picked an example which (whilst
> correct) is probably a little bit too clever.
> He hasn't claimed that you can write down a list of all Reals in base 2.
> He
> claims that the Cantor diagonal argument cannot be used in base 2 to
> prove
> this.
> Here is a somewhat clearer example.
> Imagine the list is:
> a(1) = 0.011111 ...
> a(2) = 0.01
> a(3) = 0.001
> a(4) = 0.0001
> .
> .
> Form the diagonal by flipping bits. You get
> cantor diagonal = 0.1000000...
> But 0.0111... is the same number 0.1 (unless you also believe that
> 0.999...
> <> 1), and so the Cantor diagonal does appear on the list.
> Now we all know that the list above doesn't contain every Real, but the
> Cantor diagonal construction doesn't itself prove this to be true.
> (His example was a bit too clever because he set a(2) = a(3) = a(4)
> ....,
> which is quite valid but obscures his central argument).
My problem with what he did is not a failure to
> understand that 0.0111... is the same number as 0.1;
> I accepted that immediately.
My problem with it is that he produced a CONTRIVED list,
> flipped the diagonal and said that the new number
> thus produced is indeed already in the list. So
> what?
That's not the sort of thing Cantor's argument is.
> He takes a traditional reductio ad absurdum approach
> and 'assumes' that what he wants to prove is not true.
> If it were not true, then a countable list of decimal
> (or binary) expansions of the reals between 0 and 1
> could exist. He didn't say how such a list might be
> generated. How could he? Why would he? He wants
> to show that the existence of such a complete list
> is impossible.
To make a simple analogy, if you are proving that
> there is no rational number the square of which
> is 2, you assume the contrary and posit integers p
> and q in lowest terms, q not 0, for which the square
> of p/q is 2. You don't claim to know what p and q
> would have to be. That would be outrageously absurd,
> since your motive is to prove that they don't exist
> as posited.
I could be very much in sympathy with a complaint
> against Cantor's diagonal argument that said it is
> not acceptable to operate on the assumed impossible
> list to the extent that he does in order to 'show'
> that such a countable list cannot be complete.
> I've always been uneasy about that. But the sort
> of approach that Proginoskes uses to 'refute' Cantor
> in the case of a list of binary forms of the reals
> is an insult to everyone's intelligence, it seems
> to me.
>
I said this last time, I will say it again now.
He is not arguing that you can construct a list of all Reals in base 2. He
was correcting somebody who said the Cantor diagonal proof was easy in base
2, which it isn't, which is interesting because base 2 is the only base
where the Cantor construction can't be used (in its simplest form).
The problem is that if you just flip bits, you could end up with a number
which is already on the list, because (in the example given) the procedure
creates 0.100.. when 0.0111.. is already on the list.
Now, you only need to show one example where the Cantor construction fails
to produce a number not on the list, and you can no longer claim that the
number produced by the Cantor construction is always a number not on the
list. So it is possible it is already on the list.
Its easy to get around this problem that (for example) 0.5 = 0.4999... in
other base other than base 2, and if you have a look at the Cantor
construction the problem doesn't exist in base 10 or any other base (except
2).
The standard workaround in base 2 is to consider pairs of digits, which is
equivalent to using base 4. Not that this is needed; if the Reals aren't
equinumerous with the integers in base 10, then they aren't in any base.
===
Subject: Re: Ultimate debunking of Cantor's Theory
<46959e73$0$13845$afc38c87@news.optusnet.com.au>
<469715aa$0$10708$afc38c87@news.optusnet.com.au>
<469790d2$0$26579$afc38c87@news.optusnet.com.au I said this last time, I 
will
say it again now.
> He is not arguing that you can construct a list
> of all Reals in base 2.
I was being sloppy in my desperation, when I suggested
earlier that he was arguing that.
> He was correcting somebody who said the Cantor
> diagonal proof was easy in base 2,
That would be me.
> which it isn't, which is interesting because base 2
> is the only base where the Cantor construction can't
> be used (in its simplest form).
Yes, that is interesting, and I'm not doubting you.
What I'm still doubting is the original poster's
contrived list.
> The problem is that if you just flip bits, you could
> end up with a number which is already on the list,
Fine, and I would like to see a demonstration of that,
not that I doubt it in principle.
> because (in the example given) the procedure
> creates 0.100.. when 0.0111.. is already on the list.
That's where you lose me. Yes, 0.100.. is the same
as 0.0111.. which is already on the *contrived* list.
> Now, you only need to show one example where
> the Cantor construction fails
> to produce a number not on the list, and you can
> no longer claim that the number produced by the
> Cantor construction is always a number not on the
> list. So it is possible it is already on the list.
Yes, I understand that, but I still want to understand
that any such example is valid.
> Its easy to get around this problem that (for example)
> 0.5 = 0.4999... in
> other base other than base 2, and if you have a look
> at the Cantor construction the problem doesn't exist
> in base 10 or any other base (except 2).
The standard workaround in base 2 is to consider pairs
> of digits, which is equivalent to using base 4. Not
> that this is needed; if the Reals aren't equinumerous
> with the integers in base 10, then they aren't in any
> base.
I have no problem believing any of that, and again I
clearly understand that 0.0111... is the same as
0.111...
I wont ask you to repeat anything yet again. I'll just
meditate on what looks to me like an obviously bogus
example given by the original poster, which you defend.
I don't know whether the original poster is an idiot
or not, but you obviously are not, and you defend his
example, so I must be missing something. Let me just
think about it for a while, and if the light ever dawns
I will say so.
===
Subject: Re: Ultimate debunking of Cantor's Theory
<46959e73$0$13845$afc38c87@news.optusnet.com.au>
<469715aa$0$10708$afc38c87@news.optusnet.com.au>
<469790d2$0$26579$afc38c87@news.optusnet.com.au ... again I
> clearly understand that 0.0111... is the same as
> 0.111...
Another unfortunate typo.
0.0111... is the same as 0.1
===
Subject: Re: Ultimate debunking of Cantor's Theory


> ... again I
> clearly understand that 0.0111... is the same as
> 0.111...
Another unfortunate typo.
0.0111... is the same as 0.1
>
As you have been very polite, and spent some time on this, I will give it
another go.
When you do the Cantor trick in base 10, you can prove to yourself that it
always produces a number not on the list. Even if you have 0.500.. 
somewhere
on the list, you can be certain that you will never get the same number in 
a
different form, such as 0.49999.. as a result of the construction.
If you could find a single example where the Cantor construction failed to
produce a different number - if for example it generated 0.4999.. when 0.5
was on the list - then you can no longer claim that the Cantor construction
ALWAYS produces a new number. You need this for it to be a number not on 
the
list already. Now you can see for yourself that this isn't a problem in the
standard base 10 construction.
It is, however, a problem in base 2. The examples posted are of lists where
the number that is constructed is in fact already on the list in a 
different
form. WE only need one example of a list where flipping the bits doesn't
produce a number not already on the list to break the central premise of
the construction, which is that the constructed number is NEVER on the 
list.
Sometimes, obviously, in base 2 it is already on the list.
===
Subject: Re: Ultimate debunking of Cantor's Theory
<46959e73$0$13845$afc38c87@news.optusnet.com.au>
<469715aa$0$10708$afc38c87@news.optusnet.com.au>
<469790d2$0$26579$afc38c87@news.optusnet.com.au>
<4697a382$0$14825$afc38c87@news.optusnet.com.au
> ... again I
> clearly understand that 0.0111... is the same as
> 0.111...
> Another unfortunate typo.
> 0.0111... is the same as 0.1
As you have been very polite, and spent some time on this, I will give it
> another go.
When you do the Cantor trick in base 10, you can prove to yourself that
it
> always produces a number not on the list. Even if you have 0.500..
somewhere
> on the list, you can be certain that you will never get the same number 
in
a
> different form, such as 0.49999.. as a result of the construction.
If you could find a single example where the Cantor construction failed
to
> produce a different number - if for example it generated 0.4999.. when
0.5
> was on the list - then you can no longer claim that the Cantor
construction
> ALWAYS produces a new number.
Such an example is easy to find. Consider the list
0.0
0.1
0.11
0.111
...
and switch 0 to 1 on the diagonal. Then you have at the diagonal the
number 0.111..., but only if this number (with one digit less) is also
in the list.
===
Subject: Re: Ultimate debunking of Cantor's Theory
<46959e73$0$13845$afc38c87@news.optusnet.com.au>
<469715aa$0$10708$afc38c87@news.optusnet.com.au>
<469790d2$0$26579$afc38c87@news.optusnet.com.au>
<4697a382$0$14825$afc38c87@news.optusnet.com.au>
On Jul 13, 12:08 pm, Peter Webb
> As you have been very polite, and spent
> some time on this, I will give it
> another go.
> ...
This finally dawned on me, while you were making your
new attempt to make me understand:
A countably infinite list of binary expansions of
reals between 0 and 1 can't be shown to fail to have
a diagonal-flip-produced number on the list.
The OP's example, among many others, can be used to
prove this proposition by illustration.
I understand now.
I still quibble with the OP's example, because it is
a list of the same number over and over again. But
that could easily be fixed in a variety of ways. At
first I didn't assume that he meant the same number
over and over again, but now I think he did.
===
Subject: Re: Ultimate debunking of Cantor's Theory


> On Jul 13, 12:08 pm, Peter Webb
> As you have been very polite, and spent
> some time on this, I will give it
> another go.
> ...
This finally dawned on me, while you were making your
> new attempt to make me understand:
A countably infinite list of binary expansions of
> reals between 0 and 1 can't be shown to fail to have
> a diagonal-flip-produced number on the list.
The OP's example, among many others, can be used to
> prove this proposition by illustration.
I understand now.
I still quibble with the OP's example, because it is
> a list of the same number over and over again. But
> that could easily be fixed in a variety of ways. At
> first I didn't assume that he meant the same number
> over and over again, but now I think he did.
>
If you really understand, you will see that producing even one such list
disproves the requirement that the construction always produces the same
number. The example he gave trivially had that property. He produced a list
where the diagonal expansion failed to provide a number not on the list, in
fact he only had two different numbers on the list and the diagonal number
happened to be one of the two. My example was very similar, except I picked
a list with an infinite number of different reals. His worked exactly the
same, but only needed two different numbers in the list to prove that it
isn't a different number; I had an infinite number.
Hence my statement that maybe it was too clever.
===
Subject: Re: Ultimate debunking of Cantor's Theory
<46959e73$0$13845$afc38c87@news.optusnet.com.au>
<469715aa$0$10708$afc38c87@news.optusnet.com.au>
<469790d2$0$26579$afc38c87@news.optusnet.com.au>
<4697a382$0$14825$afc38c87@news.optusnet.com.au>
<4697afad$0$14986$afc38c87@news.optusnet.com.au>
On Jul 13, 1:00 pm, Peter Webb
> On Jul 13, 12:08 pm, Peter Webb
> As you have been very polite, and spent
> some time on this, I will give it
> another go.
> ...
> This finally dawned on me, while you were making your
> new attempt to make me understand:
> A countably infinite list of binary expansions of
> reals between 0 and 1 can't be shown to fail to have
> a diagonal-flip-produced number on the list.
> The OP's example, among many others, can be used to
> prove this proposition by illustration.
> I understand now.
> I still quibble with the OP's example, because it is
> a list of the same number over and over again. But
> that could easily be fixed in a variety of ways. At
> first I didn't assume that he meant the same number
> over and over again, but now I think he did.
If you really understand, you will see that producing even one such list
> disproves the requirement that the construction always produces the same
> number. The example he gave trivially had that property. He produced a
list
> where the diagonal expansion failed to provide a number not on the list,
in
> fact he only had two different numbers on the list and the diagonal
number
> happened to be one of the two. My example was very similar, except I
picked
> a list with an infinite number of different reals. His worked exactly the
> same, but only needed two different numbers in the list to prove that it
> isn't a different number; I had an infinite number.
Hence my statement that maybe it was too clever.
I just looked back at his list, and it is all of the
same number. I understand his example now, but it's
just a little bothersome that we are willing to talk
about infinite lists of the same number over and over
again. But that example does prove his point, I have
to admit.
===
Subject: Re: Ultimate debunking of Cantor's Theory
<46959e73$0$13845$afc38c87@news.optusnet.com.au>
<469715aa$0$10708$afc38c87@news.optusnet.com.au>
<469790d2$0$26579$afc38c87@news.optusnet.com.au>
<4697a382$0$14825$afc38c87@news.optusnet.com.au>
<4697afad$0$14986$afc38c87@news.optusnet.com.au ... it's
> just a little bothersome that we are willing to talk
> about infinite lists of the same number over and over
> again. ...
Going off on this tangent, not as part of the main
discussion, does standard set theory allow a set
such as {1,1,1,...} where there are a countably
infinite number of the single element, 1? If so, then
that means there is a 1 in the set for every natural
number.
What then prevents there being a set containing a
1 for every real number? And if that is allowed,
how do you distinguish between the two sets, one
of which is countable and the other not?
===
Subject: Re: Ultimate debunking of Cantor's Theory
<46959e73$0$13845$afc38c87@news.optusnet.com.au Indeed, if you consider 
that
the existence of more than one
> type of infinity to be absurd, you can use this to show that the Axiom
of
> infinity must be false through a reductio-ad-absurdum argument. I suspect
> this is what the OP believes.
I suspect not. Familiarity with Christopher Heckman's posts suggests
that this thread was either a forgery or a joke, and my limited
ability to understand headers makes me think it was the latter.
===
Subject: Re: Ultimate debunking of Cantor's Theory
<virgil-40A2C0.23470210072007@comcast.dca.giganews.com>
<4695a163$0$10708$afc38c87@news.optusnet.com.au>
<virgil-122F12.22080311072007@comcast.dca.giganews.com>
<vYkli.1$Fa7.162@wagner.videotron.net I think x=y needs to be expanded
further, since = is a defined
> predicate, not a primitive one.
I expect you know more about this that I do, but I was under the
impression that it *is* a primitive one, and hence the name axiom of
extensionality rather than definition of extensionality. I expect
that one needs to use the axioms of first order logic with equality
when constructing formal proofs in ZF, so it would make sense to keep
it as a primitive predicate and have those axioms implicitly there.
===
Subject: Re: Ultimate debunking of Cantor's Theory
> On Jul 11, 11:24 pm, Peter Webb
> But set theory minus the Axiom of infinity is a perfectly valid set
theory.
> The advantage for the poster is that Cantor's diagonal construction of
the
> Reals doesn't exist, or indeed any form of the diagonal argument applied
to
> infinite sets. ...
I assume that by 'Cantor's diagonal construction' you
> mean considering a hypothetical countable list of all of
> the decimal expansions of the real numbers between 0 and
> 1, then going down the diagonal, changing each digit to
> any digit other than the one at each diagonal position,
> and then noticing that the real number so constructed by
> using the changed diagonal elements cannot be in the
> original list. Thus the existence of a countable list
> of decimal expansions of the reals between 0 and 1 is
> disproven.
A variation of that which I subjectively like is making
> it a list of binary expansions instead of decimal. Then
> it is only necessary to 'flip' the diagonal, changing
> all ones to zeros and all zeros to ones.
Are there other noteworthy forms of the diagonal argument?
A related theorem and proof is the theorem that for any set S, there is
no surjective function from it to its power set, P(S) (the set of all
subsets of S).
Suppose f is a function from some S to its power set P(S), f: S --> P(S)
consider the subset of S of all members which do not map into their
images under f, i.e., M_f = {s in S: not s in f(s)}.
Given any f:S -->P(S), such a set is always well defined, but a moment's
thought shows that one cannot have f(x) in M_f for any x in S.
===
Subject: Re: Ultimate debunking of Cantor's Theory
<46959e73$0$13845$afc38c87@news.optusnet.com.au>
<virgil-F6B09D.12554712072007@comcast.dca.giganews.com > Are there other
noteworthy forms of the diagonal argument?
A related theorem and proof is the theorem that for any set S, there is
> no surjective function from it to its power set, P(S) (the set of all
> subsets of S).
Suppose f is a function from some S to its power set P(S), f: S --> P(S)
> consider the subset of S of all members which do not map into their
> images under f, i.e., M_f = {s in S: not s in f(s)}.
Given any f:S -->P(S), such a set is always well defined, but a moment's
> thought shows that one cannot have f(x) in M_f for any x in S.
You're presenting half of the continuum hypothesis, right?
You're showing that the power aleph1 is greater than
the power aleph0, but you're not really showing that
it is the same as the power of the reals (called c),
the continuum hypothesis being that aleph1 = c.
My limited understanding is that the continuum
hypothesis has been shown to be neither provable nor
disprovable.
===
Subject: Re: Ultimate debunking of Cantor's Theory


> Are there other noteworthy forms of the diagonal argument?
>
There are many. It is the underlying tool used for many of the most
important results in set theory and logic. In addition to the two results
already described, it is the basis of:
* Goedel's first incompleteness thereom;
* Turing's proof of the unsolvability of the halting problem
* Church's proof that the set of theorems in ZF is not recursively
enumerable
Asking if there are any other noteworthy forms of the diagonal construction
in set theory is a bit light asking if there are any other noteworthy uses
of limits in calculus.
> A related theorem and proof is the theorem that for any set S, there is
> no surjective function from it to its power set, P(S) (the set of all
> subsets of S).
> Suppose f is a function from some S to its power set P(S), f: S --> P(S)
> consider the subset of S of all members which do not map into their
> images under f, i.e., M_f = {s in S: not s in f(s)}.
> Given any f:S -->P(S), such a set is always well defined, but a moment's
> thought shows that one cannot have f(x) in M_f for any x in S.
You're presenting half of the continuum hypothesis, right?
>
No, he is showing another use of the diagonal argument, which shows that 
you
are given a set of any cardinality, you can construct a set with strictly
larger cardinality.
> You're showing that the power aleph1 is greater than
> the power aleph0, but you're not really showing that
> it is the same as the power of the reals (called c),
> the continuum hypothesis being that aleph1 = c.
No. Aleph1 > Aleph0, by definition.
The diagonal argument shows c > Aleph0
It doesn't prove c=Aleph1, and indeed that cannot be proven in ZFC. You are
however free to add it as an axiom if you which, in which case it obviously
is provable.
> My limited understanding is that the continuum
> hypothesis has been shown to be neither provable nor
> disprovable.
>
That is correct.
===
Subject: Re: Ultimate debunking of Cantor's Theory
> Are there other noteworthy forms of the diagonal argument?
> A related theorem and proof is the theorem that for any set S, there is
> no surjective function from it to its power set, P(S) (the set of all
> subsets of S).
> Suppose f is a function from some S to its power set P(S), f: S --> P(S)
> consider the subset of S of all members which do not map into their
> images under f, i.e., M_f = {s in S: not s in f(s)}.
> Given any f:S -->P(S), such a set is always well defined, but a moment's
> thought shows that one cannot have f(x) in M_f for any x in S.
> You're presenting half of the continuum hypothesis, right?
No, nothing he said has anything to do with the continuum hypothesis, or
with aleph_1.
> You're showing that the power aleph1 is greater than
> the power aleph0, but you're not really showing that
> it is the same as the power of the reals (called c),
> the continuum hypothesis being that aleph1 = c.
The cardinality of P(N) is, in fact, 2^aleph_0 = c. It's not equal to
aleph_1 unless we assume CH, but nobody said anything about assuming
that.
To see that 2^aleph_0 is the right cardinality, consider that P(N), the
set of all subsets of N, has a natural bijection with 2^N, the set of all
characteristic functions defined on N. But the cardinality of the latter
set is exactly 2^|N| = 2^aleph_0.
> My limited understanding is that the continuum
> hypothesis has been shown to be neither provable nor
> disprovable.
Within ZFC, at least.
--
Dave Seaman
Oral Arguments in Mumia Abu-Jamal Case heard May 17
U.S. Court of Appeals, Third Circuit
<http://www.abu-jamal-news.com/>
===
Subject: Re: Ultimate debunking of Cantor's Theory
<46959e73$0$13845$afc38c87@news.optusnet.com.au>
<virgil-F6B09D.12554712072007@comcast.dca.giganews.com>
<f76kqd$lp5$1@mailhub227.itcs.purdue.edu > Are there other noteworthy 
forms
of the diagonal argument?
> A related theorem and proof is the theorem that for any set S, there
is
> no surjective function from it to its power set, P(S) (the set of all
> subsets of S).
> Suppose f is a function from some S to its power set P(S), f: S -->
P(S)
> consider the subset of S of all members which do not map into their
> images under f, i.e., M_f = {s in S: not s in f(s)}.
> Given any f:S -->P(S), such a set is always well defined, but a
moment's
> thought shows that one cannot have f(x) in M_f for any x in S.
> You're presenting half of the continuum hypothesis, right?
No, nothing he said has anything to do with the continuum hypothesis, or
> with aleph_1.
> You're showing that the power aleph1 is greater than
> the power aleph0, but you're not really showing that
> it is the same as the power of the reals (called c),
> the continuum hypothesis being that aleph1 = c.
The cardinality of P(N) is, in fact, 2^aleph_0 = c. It's not equal to
> aleph_1 unless we assume CH, but nobody said anything about assuming
> that.
To see that 2^aleph_0 is the right cardinality, consider that P(N), the
> set of all subsets of N, has a natural bijection with 2^N, the set of all
> characteristic functions defined on N. But the cardinality of the latter
> set is exactly 2^|N| = 2^aleph_0.
I know he didn't say anything about the continuum
hypothesis, but I thought he was pointing in that
direction for anyone who wanted to go there.
If aleph1 is not the power of the set of all
subsets of a set with power aleph0, then what
is it? I thought each aleph number was the
power of the set of all subsets of a set with
the power of the previous aleph number, there
always being a next aleph number, defined this
way.
But now you're saying, I believe, that the power
of the continuum is the same as the power of the
set of all subsets of the set of natural numbers,
so the definition of aleph0 has to be something
else for the continuum hypothesis ever to have
held any interest for anyone.
> My limited understanding is that the continuum
> hypothesis has been shown to be neither provable nor
> disprovable.
Within ZFC, at least.
Isn't ZFC the set theory that we are assuming whenever
we are discussing the real numbers and countable and
uncountable sets such as the natural numbers, the
rationals, and the reals?
===
Subject: Re: Ultimate debunking of Cantor's Theory
<46959e73$0$13845$afc38c87@news.optusnet.com.au>
<virgil-F6B09D.12554712072007@comcast.dca.giganews.com>
<f76kqd$lp5$1@mailhub227.itcs.purdue.edu If aleph1 is not the power of the 
set
of all
> subsets of a set with power aleph0, then what
> is it? I thought each aleph number was the
> power of the set of all subsets of a set with
> the power of the previous aleph number, there
> always being a next aleph number, defined this
> way.
aleph_1 was originally defined (and usually still is
defined) to be the cardinality of the set of all
countable ordinal numbers. One can then *prove*
(and Cantor did so, I believe) that if b is any
uncountable cardinal number, then b >= aleph_1.
Thus, aleph_1 can be *proved* to be the next larger
cardinal after aleph_0. [It can actually be proved
that there is always a next larger cardinal number
for any specified cardinal number, although one
doesn't have to prove this more general statement
to prove that aleph_1 is the next larger cardinal
number after aleph_0.]
For a long time it was not known whether the
cardinality of the set of real numbers was aleph_1.
It was certainly known that the cardinality of the
set of real numbers is uncountable, but what wasn't
known is whether the cardinality was actually aleph_1,
or whether the cardinality was some other cardinal
greater than aleph_0. It is now known that the ZFC
axioms for set theory allow both yes and no, much
like the axioms for a group allow both yes and no
to the statement of whether one always has xy = yx
in a group.
Dave L. Renfro
===
Subject: Re: Ultimate debunking of Cantor's Theory
<46959e73$0$13845$afc38c87@news.optusnet.com.au>
<virgil-F6B09D.12554712072007@comcast.dca.giganews.com>
<f76kqd$lp5$1@mailhub227.itcs.purdue.edu For a long time it was not known
whether the
> cardinality of the set of real numbers was aleph_1.
> It was certainly known that the cardinality of the
> set of real numbers is uncountable, but what wasn't
> known is whether the cardinality was actually aleph_1,
> or whether the cardinality was some other cardinal
> greater than aleph_0.
In fact, there was quite a bit of discussion in the
mathematical literature during the 1890s to at least
1920 as to whether the real numbers even had a cardinality.
we would accept the existence of a well-ordering for the
set of real numbers, but I think there were other things
not completely-settled/understood/correctly-formulated
during this time, besides well-ordering issues, that led
some people to have these doubts.
Dave L. Renfro
===
Subject: Re: Ultimate debunking of Cantor's Theory
> Are there other noteworthy forms of the diagonal argument?
> A related theorem and proof is the theorem that for any set S, there
is
> no surjective function from it to its power set, P(S) (the set of all
> subsets of S).
> Suppose f is a function from some S to its power set P(S), f: S -->
P(S)
> consider the subset of S of all members which do not map into their
> images under f, i.e., M_f = {s in S: not s in f(s)}.
> Given any f:S -->P(S), such a set is always well defined, but a
moment's
> thought shows that one cannot have f(x) in M_f for any x in S.
> You're presenting half of the continuum hypothesis, right?
> No, nothing he said has anything to do with the continuum hypothesis, or
> with aleph_1.
> You're showing that the power aleph1 is greater than
> the power aleph0, but you're not really showing that
> it is the same as the power of the reals (called c),
> the continuum hypothesis being that aleph1 = c.
> The cardinality of P(N) is, in fact, 2^aleph_0 = c. It's not equal to
> aleph_1 unless we assume CH, but nobody said anything about assuming
> that.
> To see that 2^aleph_0 is the right cardinality, consider that P(N), the
> set of all subsets of N, has a natural bijection with 2^N, the set of
all
> characteristic functions defined on N. But the cardinality of the
latter
> set is exactly 2^|N| = 2^aleph_0.
> I know he didn't say anything about the continuum
> hypothesis, but I thought he was pointing in that
> direction for anyone who wanted to go there.
> If aleph1 is not the power of the set of all
> subsets of a set with power aleph0, then what
> is it? I thought each aleph number was the
> power of the set of all subsets of a set with
> the power of the previous aleph number, there
> always being a next aleph number, defined this
> way.
Aleph_1 is the cardinality of omega_1, the set of all countable ordinals.
This makes omega_1 the smallest uncountable ordinal, and aleph_1 the
smallest uncountable cardinal.
> But now you're saying, I believe, that the power
> of the continuum is the same as the power of the
> set of all subsets of the set of natural numbers,
> so the definition of aleph0 has to be something
> else for the continuum hypothesis ever to have
> held any interest for anyone.
> My limited understanding is that the continuum
> hypothesis has been shown to be neither provable nor
> disprovable.
> Within ZFC, at least.
> Isn't ZFC the set theory that we are assuming whenever
> we are discussing the real numbers and countable and
> uncountable sets such as the natural numbers, the
> rationals, and the reals?
It's braoder than that; ZFC is generally considered to be the foundation
of nearly all of mathematics, in the sense that virtually all
mathematical concepts are defined in terms of sets.
--
Dave Seaman
Oral Arguments in Mumia Abu-Jamal Case heard May 17
U.S. Court of Appeals, Third Circuit
<http://www.abu-jamal-news.com/>
===
Subject: Re: Ultimate debunking of Cantor's Theory


> Are there other noteworthy forms of the diagonal argument?
> A related theorem and proof is the theorem that for any set S, there
> is
> no surjective function from it to its power set, P(S) (the set of all
> subsets of S).
> Suppose f is a function from some S to its power set P(S), f: S --> P(S)
> consider the subset of S of all members which do not map into their
> images under f, i.e., M_f = {s in S: not s in f(s)}.
> Given any f:S -->P(S), such a set is always well defined, but a
> moment's
> thought shows that one cannot have f(x) in M_f for any x in S.
> You're presenting half of the continuum hypothesis, right?
> No, nothing he said has anything to do with the continuum hypothesis, or
> with aleph_1.
> You're showing that the power aleph1 is greater than
> the power aleph0, but you're not really showing that
> it is the same as the power of the reals (called c),
> the continuum hypothesis being that aleph1 = c.
> The cardinality of P(N) is, in fact, 2^aleph_0 = c. It's not equal to
> aleph_1 unless we assume CH, but nobody said anything about assuming
> that.
> To see that 2^aleph_0 is the right cardinality, consider that P(N), the
> set of all subsets of N, has a natural bijection with 2^N, the set of
all
> characteristic functions defined on N. But the cardinality of the
latter
> set is exactly 2^|N| = 2^aleph_0.
I know he didn't say anything about the continuum
> hypothesis, but I thought he was pointing in that
> direction for anyone who wanted to go there.
If aleph1 is not the power of the set of all
> subsets of a set with power aleph0, then what
> is it?
A very good question.
> I thought each aleph number was the
> power of the set of all subsets of a set with
> the power of the previous aleph number, there
> always being a next aleph number, defined this
> way.
>
No.
Aleph1 is simply defined as the smallest cardinal strictly larger than
Aleph0. We can prove such an animal exists. There is no reason to think 
this
is the same as c. Cantor knew this well; AFAIR, he actually suspected that
c=Aleph2. We now know that c can be pretty well any Aleph, with some
occasional exceptions.
> But now you're saying, I believe, that the power
> of the continuum is the same as the power of the
> set of all subsets of the set of natural numbers,
This is very easy to prove.
> so the definition of aleph0 has to be something
> else for the continuum hypothesis ever to have
> held any interest for anyone.
>
Yes, Aleph1 has a different definition to c.
> My limited understanding is that the continuum
> hypothesis has been shown to be neither provable nor
> disprovable.
> Within ZFC, at least.
Isn't ZFC the set theory that we are assuming whenever
> we are discussing the real numbers and countable and
> uncountable sets such as the natural numbers, the
> rationals, and the reals?
>
Yes, ZFC is sufficient for those purposes.
However, ZFC cannot prove or disprove that c=Aleph1. Its simply not 
powerful
enough. ZFC works the same whether this statement is true or false. You 
need
to add something to ZFC to be able to address this question. You can simply
define it as an axiom if you want, and then set theory will continue to
work - ZFC + CH is exactly the same as ZFC, except that c=Aleph1. Or you
can add an axiom which says that (for example) c=Aleph2, and ZFC will
continue to work properly.
There have been many, many extensions proposed to ZF to allow it to address
a wider range of questions. The Axiom of Choice and the Continuum 
hypothesis
are just two of these. The Axiom of Infinity could also in this category if
you wish (the point I was originally making); ZF would work perfectly well
for many purposes without the Axiom of Infinity; it is included because
without the Axiom of Infinity, we can't talk about infinite sets at all, 
and
that is so boring that it was bundled into the core axioms of ZF.
A similar process has occurred with the Axiom of Choice. This cannot be
proved in ZF. It is however such a useful axiom that it is just often
bundled in to form ZFC, just as the Axiom of Infinity was bundled into ZF,
to form a richer and more powerful system than ZF alone.
The same probably won't happen with CH. ZF + ~CH is not so spectacularly
weaker than ZF+CH as to make it uninteresting in its own right, so there is
no huge advantage in adding CH in as a core axiom of set theory. Its only
added when actually needed.
===
Subject: Re: Ultimate debunking of Cantor's Theory
On Fri, 13 Jul 2007 13:23:46 +1000, Peter Webb
ZF would work perfectly well for many purposes without the Axiom
> of Infinity; it is included because without the Axiom of Infinity,
> we can't talk about infinite sets at all, and that is so boring
> that it was bundled into the core axioms of ZF.
>
Moreover a set theory could hardly serve as a (the) mathematical
foundation, if we could not even derive the Peano Axioms from them.
The other way round: we can derive virtually all mathematics (or at
least a good deal of it) from ZFC.
F.
--
E-mail: info<at>simple-line<dot>de
===
Subject: Re: Ultimate debunking of Cantor's Theory


> On Fri, 13 Jul 2007 13:23:46 +1000, Peter Webb
> ZF would work perfectly well for many purposes without the Axiom
> of Infinity; it is included because without the Axiom of Infinity,
> we can't talk about infinite sets at all, and that is so boring
> that it was bundled into the core axioms of ZF.
> Moreover a set theory could hardly serve as a (the) mathematical
> foundation, if we could not even derive the Peano Axioms from them.
> The other way round: we can derive virtually all mathematics (or at
> least a good deal of it) from ZFC.
>
Whilst I agree with you, it is my suspicion that you don't need the Axiom 
of
Infinity for the Peano axioms. The only place you come close is in the
induction axiom, which can (I suspect) be re-written to avoid the concept 
of
all numbers.
You probably know if this is true or not - you don't need the Axiom of
Infinity for PA, just as we don't need the AxC for PA either?
The Axiom of Infinity seems to me to be a lot like the Axiom of Choice -
very useful in set theory, but not completely self evident, and not
necessary for the construction of numbers.
===
Subject: Re: Ultimate debunking of Cantor's Theory
> On Fri, 13 Jul 2007 13:23:46 +1000, Peter Webb
 ZF would work perfectly well for many purposes without the Axiom
> of Infinity; it is included because without the Axiom of Infinity,
> we can't talk about infinite sets at all, and that is so boring
> that it was bundled into the core axioms of ZF.
> Moreover a set theory could hardly serve as a (the) mathematical
> foundation, if we could not even derive the Peano Axioms from them.
> The other way round: we can derive virtually all mathematics (or at
> least a good deal of it) from ZFC.
 Whilst I agree with you, it is my suspicion that you don't need the
> Axiom of Infinity for the Peano axioms. The only place you come close
> is in the induction axiom, which can (I suspect) be re-written to
> avoid the concept of all numbers.
You probably know if this is true or not - you don't need the Axiom of
> Infinity for PA, just as we don't need the AxC for PA either?
This may be true. After all, in ZF, we can still carry out induction
through the ordinals. A natural number can be defined without Infinity,
and then we can make statements like For all x, if x is a natural
number, then... In fact, can't we just use transfinite induction to
prove such statements?
However, it seems that without Infinity, we wouldn't be able to
establish the existence of negative numbers, let alone rationals and
reals. We could still define them and make statements about them, I
think, but we would have no guarantee that the statements are not vacuous.
But I am no expert. Experts care to chime in?
The Axiom of Infinity seems to me to be a lot like the Axiom of Choice
> - very useful in set theory, but not completely self evident, and not
> necessary for the construction of numbers.
--
Stephen J. Herschkorn sjherschko@netscape.net
Math Tutor on the Internet and in Central New Jersey and Manhattan
===
Subject: Re: Ultimate debunking of Cantor's Theory
On Thu, 12 Jul 2007 18:54:01 -0700, Calvin <crice5@windstream.net> My 
limited
understanding is that the continuum
> hypothesis has been shown to be neither provable nor
> disprovable.
> Within ZFC, at least.
> Isn't ZFC the set theory that we are assuming whenever
> we are discussing the real numbers and countable and
> uncountable sets such as the natural numbers, the
> rationals, and the reals?
>
Well, in most cases this may indeed be the case (since ZFC is some
sort of standard set theory), though not necessarily. On the other
hand, your claim is also true for other popular set theories, NBG
for example.
F.
--
E-mail: info<at>simple-line<dot>de
===
Subject: Re: Ultimate debunking of Cantor's Theory
<46959e73$0$13845$afc38c87@news.optusnet.com.au>
<virgil-F6B09D.12554712072007@comcast.dca.giganews.com>
<f76kqd$lp5$1@mailhub227.itcs.purdue.edu>
(in my previous post)
> But now you're saying, I believe, that the power
> of the continuum is the same as the power of the
> set of all subsets of the set of natural numbers,
> so the definition of aleph0 has to be something
> else for the continuum hypothesis ever to have
> held any interest for anyone.
BAD TYPO ABOVE
aleph0 should read aleph1
===
Subject: Re: x^4+y^3=z^2
Use standard techniques for solving Pell's equation to get solutions
to x^2 = 3m^2 + 3m + 1. Then let y = m^2 + m and z = m^3 + a^2
and you'll have a solution to x^4 + y^3 = z^2.
===
Subject: Re: A Review of Mathematica
For a *tiny* starter, try this
Vladimir Bondarenko
http://www.cybertester.com/
http://maple.bug-list.org/
http://www.CAS-testing.org/
> A Review of Mathematica by Richard J. Fateman, UC Berkeley
>http://www.cs.berkeley.edu/~fateman/papers/mma.review.pdf
> The Mathematica computer system is reviewed from the perspective
> of its contributions to symbolic and algebraic computation, as
> well as its stated goals.
> Design and implementation issues are discussed.
Damn it! You should have warned us. This review is 15-16 years old!
How about some recent reviews.
I'm not interested in a review of Mathematica 1.2
===
Subject: Re: A Review of Mathematica


> On Thu, 12 Jul 2007 08:22:13 -0700, Mike M
A Review of Mathematica by Richard J. Fateman, UC Berkeley
http://www.cs.berkeley.edu/~fateman/papers/mma.review.pdf
The Mathematica computer system is reviewed from the perspective
> of its contributions to symbolic and algebraic computation, as
> well as its stated goals.
Design and implementation issues are discussed.
>As far as I am concerned, Mathematica is the greatest invention since
>sliced
>bread!
>If you know of a better multi-purpose software system, please name it.
> I like Maple better.
Maple has a simpler syntax, closer to math in look and feel, whereas
> Mathematica has more of a computer science look and feel.
quasi
http://amath.colorado.edu/computing/mmm/brief.html
I believe that it is most probably the case that most major US (and
European) universities have Mathematica, Maple and Matlab on their computer
networks.
http://www.bitwisemag.com/copy/reviews/software/maths/maple10_mathematica52.

html
Mike M
===
Subject: Re: A Review of Mathematica
<zq6dneIXsIW12gvbnZ2dnUVZ_uiknZ2d@comcast.com>
<ghoc93hm2eofq3ut57g4cv4pmsq7qrr0nc@4ax.com>
<b_Wdnd9ebZs08ArbnZ2dnUVZ_sOknZ2d@comcast.com
> On Thu, 12 Jul 2007 08:22:13 -0700, Mike M
>A Review of Mathematica by Richard J. Fateman, UC Berkeley
>http://www.cs.berkeley.edu/~fateman/papers/mma.review.pdf
> The Mathematica computer system is reviewed from the perspective
> of its contributions to symbolic and algebraic computation, as
> well as its stated goals.
> Design and implementation issues are discussed.
>As far as I am concerned, Mathematica is the greatest invention since
>sliced
>bread!
>If you know of a better multi-purpose software system, please name it.
> I like Maple better.
> Maple has a simpler syntax, closer to math in look and feel, whereas
> Mathematica has more of a computer science look and feel.
> quasi
http://amath.colorado.edu/computing/mmm/brief.html
> I believe that it is most probably the case that most major US (and
> European) universities have Mathematica, Maple and Matlab on their
computer
> networks.
http://www.bitwisemag.com/copy/reviews/software/maths/maple10_mathema...
Mike M
1. Regarding the review above: Neither of the programs being compared
is the current version. It is mma 5.2 vs maple 10.
2. The reviewer admits Before Maple 10, I would have said that
Mathematica would be the preferred choice: Maple didn't have the same
level of graphical user interface and text formatting capability.
This like an automobile reviewer who says he is comparing cars on the
basis of the cupholders. Not that cupholders are unimportant; it would
be bad to hold a coffee cup between your knees while driving. But...
===
Subject: Re: A Review of Mathematica
<zq6dneIXsIW12gvbnZ2dnUVZ_uiknZ2d@comcast.com>
<ghoc93hm2eofq3ut57g4cv4pmsq7qrr0nc@4ax.com>
I agree with Herman Rubin on the importance of notation, but I think
we disagree on what that notation should be.
Here is an example. (better set your news reader to fixed-width
font...)
A formula that is part of Filon quadrature, written in infix.
How clear is it compared to the 3 choices written in prefix (lisp) ,
below?
Which is equivalent?
beta= 2*((1+cos(theta)^2)/theta^2 -sin(2*theta)/theta^3),
;; is this a difference of two items? [a]
(setf beta (- (* 2 (/ (+ 1 (expt (cos theta) 2)) (expt theta 2)))
(/ (sin (* 2 theta)) (expt theta 3))))
;; or is it a product of 2 times the difference of two items? [b]
(setf beta (* 2 (- (/ (+ 1 (expt (cos theta) 2)) (expt theta 2))
(/ (sin (* 2 theta)) (expt theta 3)))))
;; or is it a quotient of a difference divided by theta^3? [c]
(setf beta (/ (- (* 2 (/ (+ 1 (expt (cos theta) 2)) (expt theta
2)))
(/ (sin (* 2 theta))))
(expt theta 3)))
;; or something else?
;; answer is [b].
;; I contend that [b] is clearer than the infix version. And the
indentation is
;; done free, by the editor.
You might argue that we should be using different brackets, and in
this case it would help. But once the size of the expression grows
beyond a line or two, even a whole zoo of bracket alternatives may not
be enough to save you.
RJF
ps, if you really want a review of Mathematica 6.0, and you are
watching this on some other newsgroup than sci.math.symbolic, go look
at sci.math.symbolic.
===
Subject: Re: A Review of Mathematica
<zq6dneIXsIW12gvbnZ2dnUVZ_uiknZ2d@comcast.com>
<ghoc93hm2eofq3ut57g4cv4pmsq7qrr0nc@4ax.com I agree with Herman Rubin on 
the
importance of notation, but I think
> we disagree on what that notation should be.
Here is an example. (better set your news reader to fixed-width
> font...)
A formula that is part of Filon quadrature, written in infix.
> How clear is it compared to the 3 choices written in prefix (lisp) ,
> below?
> Which is equivalent?
beta= 2*((1+cos(theta)^2)/theta^2 -sin(2*theta)/theta^3),
;; is this a difference of two items? [a]
(setf beta (- (* 2 (/ (+ 1 (expt (cos theta) 2)) (expt theta 2)))
> (/ (sin (* 2 theta)) (expt theta 3))))
;; or is it a product of 2 times the difference of two items? [b]
(setf beta (* 2 (- (/ (+ 1 (expt (cos theta) 2)) (expt theta 2))
> (/ (sin (* 2 theta)) (expt theta 3)))))
> ;; or is it a quotient of a difference divided by theta^3? [c]
(setf beta (/ (- (* 2 (/ (+ 1 (expt (cos theta) 2)) (expt theta
> 2)))
> (/ (sin (* 2 theta))))
> (expt theta 3)))
;; or something else?
;; answer is [b].
> ;; I contend that [b] is clearer than the infix version. And the
> indentation is
> ;; done free, by the editor.
If I were entering the infix as maple code in emacs, maplev-mode
will gladly do
beta := 2*( (1+cos(theta)^2) / theta^2
- sin(2*theta)/theta^3
);
which seems every bit as clear as [b].
--
Joe Riel
===
Subject: Re: A Review of Mathematica
>A Review of Mathematica by Richard J. Fateman, UC Berkeley
> http://www.cs.berkeley.edu/~fateman/papers/mma.review.pdf
> The Mathematica computer system is reviewed from the perspective
> of its contributions to symbolic and algebraic computation, as
> well as its stated goals.
> Design and implementation issues are discussed.
>As far as I am concerned, Mathematica is the greatest invention since
sliced
>bread!
>If you know of a better multi-purpose software system, please name it.
>Mike M
Was sliced bread such a great invention? It is much
harder to get good bread since it came out.
A multi-purpose system should have a simple flexible
format, much as the early FORTRAN did, with all its
faults, which should have been corrected, rather than
clumsier systems invented.
A major problem is NOTATION; if one has to type 100
characters instead of 10, errors get propagated.
There needs to be an easy way of indicating bracketing,
with many alternatives, instead of the overly rigid and
unusual type used by Mathematica, which is one of
its greatest drawbacks.
I think the importance of a compact notation is
underestimated. Being able to write a+b instead
of plus(a,b) already makes a difference in the
intelligibility of the expression.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: A Review of Mathematica
As far as I am concerned, Mathematica is the greatest invention since
sliced
>bread!
>If you know of a better multi-purpose software system, please name it.
>Mike M
Was sliced bread such a great invention? It is much
> harder to get good bread since it came out.
I eventually decided that the expression was first coined as irony;
to mark hyperbole in claims for a novelty.
--
Michael Press
===
Subject: Re: Some Thoughts on the Four Color Theorem
> 1. The minimal counter example (MCE) to the 4CT
> The vertex-coloring version of the 4CT.
> is a simple
> loopless triangulation such that every vertex has
degree
> equal 5 or greater.
> 2. The dual of the MCE is a triangle-free cubic planar
graph.
> There are no triangular _faces_ in a triangulation with
> minimum degree>= 5, but there might be triangles in a
> planar graph which are not planar.
> What are you trying to say?
> You might have a triangle which has a vertex inside of it and a
vertex
> outside of it, sometimes called a separating triangle.
Seehttp://math.asu.edu/~checkman/septri.jpgforanexample.
>However, this is not true for a MCE. (It takes a little bit of
> work, though, which I'll let bill do.)
> Exactly what is not true for a MCE?
> The statement G has not facial triangles and a separating
triangle.
> I am working on the problem! But right now, I think that a cubic
with
> a separating triangle must also have a bridge?. If, this is true,
then
> the dual cannot be a simple loopless triangulation!
> Yes. Last night after posting I realized that this is true, and the
> proof's not that bad.
> If uvw is a separating triangle, and u', v', w' are the neighbors of
> u, v, and w not in the triangle, then we may assume that u' is inside
> the triangle and v' and w' are outside of it (using symmetry and the
> fact that a separating triangle is not facial). Then uu' is a
cut-edge
> (bridge).
> 3 The MCE must be the dual of a triangle-free cubic
planar graph!
> This is the same thing you said in (2.)
> 4 Every triangle-less cubic planar graph is 3 edge
colorable.
> You can eliminate triangle-less here, but you need to put
in bridge-
> free (or 2-connected) here. If every triangle-free
2-connected cubic
> planar graph is 3-edge colorable, then every 2-connected cubic
planar
> graph is 3-edge colorable. Again, I'll leave this as an
exercise for
> bill, hinting to do it by induction on the number of triangles
in the
> graph.
> 5 The 4CT is true if statement 4 above is true!
> This is a statement of the form (False => something), so is
vacuuously
> true. After adding the 2-connected condition, it becomes
equivalent.
> FWIW. The dual of the MCE has no 3-cycles nor
> 4-cycles; only 5-cycles and 6-cycles. No cycles
> larger than 6?
What if you had a vertex of degree 7 in your MCE? If you can't
> definitely rule it out, it must be possible.
> If the dual has a 3-cycle, then
> he MCE would have a vertex of degree 3; if the
> dual has a 4-cycle, then the MCE would have a degree 4 vertex.
Not necessarily. The proof of no separating 3-cycles above doesn't go
> through: If you have a separating 4-cycle C = uvwx, and you let u',
> v', w', x' be the neighbors of u,v,w,x not in C, then you can have u'
> and w' inside of C and v' and x' outside of C, and not have a bridge.
(As of about 2000, Robin Thomas hadn't heard about a proof that a MCE
> doesn't have a separating 4-cycle and would like to have found one,
> since it simplifies the new proof of the 4CT.)
--- Christopher Heckman
===
Subject: Euler Angle Order convertion
Hi
I'm trying to acheive a particular orientation of an object in a piece of
software. I have the rotation angles to get it in the correct orientation 
by
rotating about Y, then Z, then X. However, due to the structure of the
program (which is prohibitively difficult to rewrite), I have to go about 
Z,
then Y, then X.
If anyone knows the process for finding equivalent Euler Angles in a
different order of application and would be willing to post here, I would 
be
most appreciative! I haven't been able to find any good resources online.
-Pat
===
Subject: Re: Bug in Mathematica 6 - Integrate - 43 (simple real-valued
integral, MathKernel instability)
<593656.1184348088653.JavaMail.jakarta@nitrogen.mathforum.org>
BB> This is not a bug.
Do not say this way within your company, especially
to Arnoud Buzing, your superior. You will be fired.
Best wishes,
Vladimir Bondarenko
VM and GEMM architect
Co-founder, CEO, Mathematical Director
http://www.cybertester.com/ Cyber Tester, LLC
http://maple.bug-list.org/ Maple Bugs Encyclopaedia
http://www.CAS-testing.org/ CAS Testing
> This is not a bug. You should use arbitrary precision, e.g. N[#,20]&.
Bhuvanesh,
> Wolfram Research
===
Subject: Re: Bug in Mathematica 6 - Integrate - 43 (simple real-valued
integral, Mat
<593656.1184348088653.JavaMail.jakarta@nitrogen.mathforum.org>
Oh no, this worked OK in Mathematica 5.2 and should
work OK in Mathematica 6.
Best wishes,
Vladimir Bondarenko
http://www.cybertester.com/
http://maple.bug-list.org/
http://www.CAS-testing.org/
> This is not a bug. You should use arbitrary precision, e.g. N[#,20]&.
Bhuvanesh,
> Wolfram Research