mm-321 

===
Subject:
: Re: Sphinx Tiling Explanation?>That's not the
aperiodic tiling. Take the template used to make the>larger
16-sphinx shape from the unit sphinx, and use it to produce>a
256-sphinx (256 = 16^2) shape. Next, use the same template to
produce>a 4096-sphinx shape out of 16 of those, next a
65536-sphinx shape,>a 1048576-sphinx shape, and so forth. As I
see it, this ultimately>tiles only the upper half plane, so a
mirror image (or 180 degree>rotation) would be needed as well,
to fill in the entire plane.Surely you don't need this
reflection if you just perform the recursioncentred about the
uppermost upright (magenta) sphinx, rather than any ofthe four
on the bottom row? -- Erick===
===
Subject:
: Re: Sphinx Tiling
Explanation?True, you can extend the sphinx in any direction
at any time, but thenyou can't guarantee that you have an
infinite line of unit sphinxes, andthen if you don't have such
a line, it's much harder (if not impossible)to guarantee that
it's aperiodic.Justin: Surely you don't need this reflection
if you just perform the recursion: centred about the uppermost
upright (magenta) sphinx, rather than any of: the four on the
bottom row?: -- Erick===
===
Subject:
: Re: Turing machine question>
A Turing machine question:> Can you build a Turing machine if
- instead of a pen and an> eraser to mark on the tape you have
a *finite* number of paper> clips, which can be clipped on and
off the tape?You could certainly build such a machine, but its
power, as,say, a language recognizer, would be severly limi.
Therewould certainly be useful things such a machine could
do--unary addition is just one example. > You could not use
binary representations - you would have> to use large numbers
to store things - stored in the form:> 1: .X> 2: ..X> ...> 50:
..................................................X> ...etc.Of
course you could use binary representation: 0: .X 1: .XX 2:
..X 3: .XXX 4: .X.X ... 50: ..XX..X > If the answer is no,
then why not?If you star out with a fixed number C of
clips,your machine would not even be capable of recognizingall
regular languages, since many regular languagescontain strings
of length greater than C, so youwouldn't even be able to
represent some inputs.If, on the other hand, the number of
clips you coulduse was O(n) for inputs of length n a quick
guessmight be that your proposed machine might be ableto
recognize all context-sensitive languages, thoughI really
haven't given this much thought.Rick===
===
Subject:
: Re: Turing
machine question
3QLpj-NoP*NzsIC,boYU]bQ]H'<P%JD/,.J>y<#4ga3$21:> A Turing
machine question:> Can you build a Turing machine if - instead
of a pen and an> eraser to mark on the tape you have a *finite*
number of paper> clips, which can be clipped on and off the
tape?>> You could certainly build such a machine, but its
power, as,> say, a language recognizer, would be severly limi.
There> would certainly be useful things such a machine could
do--> unary addition is just one example.The power would not
be severely limi unless you mean merely that it would be
slower. Two-counter machines are capable of simulating
arbitrary Turing machines. If you can distinguish some
position on the tape (say the start of the input) then two
paper clips are enough to simulate two counters, by their
distance from the distinguished position; otherwise you could
use three paper clips.-- David Eppstein
http://www.ics.uci.edu/~eppstein/Univ. of California, Irvine,
School of Information & Computer Science===
===
Subject:
: Re: Turing
machine question>A Turing machine question:>>Can you build
a Turing machine if - instead of a pen and an>eraser to mark
on the tape you have a *finite* number of paper>clips, which
can be clipped on and off the tape?>>You could certainly
build such a machine, but its power, as,>>say, a language
recognizer, would be severly limi. There>>would certainly be
useful things such a machine could do-->>unary addition is
just one example.> The power would not be severely limi
unless you mean merely that it > would be slower. Two-counter
machines are capable of simulating > arbitrary Turing
machines. If you can distinguish some position on the > tape
(say the start of the input) then two paper clips are enough
to > simulate two counters, by their distance from the
distinguished > position; otherwise you could use three paper
clips.Ho ho. Of course you're right. Looks like I made the
samesort of mistake I warn my students about--assuming that*a*
mental model is *the only* one.Rick ===
===
Subject:
: Re: Turing
machine question>> A Turing machine question: Can you build a
Turing machine if - instead of a pen and an>> eraser to mark
on the tape you have a *finite* number of paper>> clips, which
can be clipped on and off the tape?>> You could certainly build
such a machine, but its power, as,>> say, a language
recognizer, would be severly limi. There>> would certainly be
useful things such a machine could do-->> unary addition is
just one example.> The power would not be severely limi unless
you mean merely that it > would be slower. Two-counter machines
are capable of simulating > arbitrary Turing machines.
[...]Thanks - this and Minsky machines proved to be useful
search terms.-- __________ |im |yler http://timtyler.org/
tim@tt1lock.org Remove lock to reply.===
===
Subject:
: Re: Image of
a straight line in a convex cylindrical mirror> In 3D, y+b=0
is a plane. Right? Are you looking for its reflection, or>
reflection of a line?> What is the reflection in a cylindrical
mirror x^2+y^2 = a^2 of a> straight line y+b = 0 as seen from a
viewpoint (0,-c,h) ( a < b << c)> ? Is it a catenary ? If not,
what is it?> It's probably safe to assume, since this is a
line on the floor of the> mall, that the equation is y+b = 0
and z = 0. Now that it's been a few> days, can I give a hint?
The symmetry of the problem -- what> coordinate system should
you be using?Looking for the reflection of line y+b=0,z=0, an
omit typo. Shouldbe using polar/cylindrical (r-th-z)
coordinates for convenience ofsymmetry.===
===
Subject:
: Re: Image
of a straight line in a convex cylindrical mirror> In 3D,
y+b=0 is a plane. Right? Are you looking for its reflection,
or> reflection of a line?Looking for the reflection of line
y+b=0,z=0, omit typo.===
===
Subject:
: conjugate subgroupsWhy can't
a conjugate of a subgroup be a proper subset of it?===
===
Subject:
:
Re: conjugate subgroups> Why can't a conjugate of a subgroup be
a proper subset of it?Because if H is a subgroup of G, h in H
and g in G with gHg' asubset of H, then g'Hg is also a subset
of H. What does thatimply about the equation
gkg'=h?===
===
Subject:
: Re: conjugate subgroups> Why can't a
conjugate of a subgroup be a proper subset of it?> Because if
H is a subgroup of G, h in H and g in G with gHg' a> subset of
H, then g'Hg is also a subset of H. You sure? I would have
thought a superset, not a subset.> What does that> imply about
the equation gkg'=h?===
===
Subject:
: Re: conjugate subgroups>> Why
can't a conjugate of a subgroup be a proper subset of it?>>
Because if H is a subgroup of G, h in H and g in G with gHg'
a>> subset of H, then g'Hg is also a subset of H. > You sure?
I would have thought a superset, not a subset.Well, I may have
been sure, but as several posters have poin out, I was also
wrong. >> What does that>> imply about the equation
gkg'=h?===
===
Subject:
: Re: conjugate subgroups>> Why can't a
conjugate of a subgroup be a proper subset of it?It
can!>Because if H is a subgroup of G, h in H and g in G with
gHg' a>subset of H, then g'Hg is also a subset of H.That's not
true. Take H cyclic subgroup of GL(2,Q) genera byh = ( 1 1 ), (
0 1 )and let g = ( 2 0 ) ( 0 1 )ghg^-1 = h^2 but g^-1hg = ( 1
1/2 ) ( 0 1 )Derek Holt.> What does that>imply about the
equation gkg'=h?===
===
Subject:
: Re: conjugate subgroups> Why can't
a conjugate of a subgroup be a proper subset of it?Surely it
can (consider the subgroup genera by the matrix (Mathematica
notation) {{1,1},{0,1}} inside GL_2(Q)). Why do you ask?Jyrki
Lahtonen, Turku, Finland===
===
Subject:
: Re: conjugate subgroups>
Why can't a conjugate of a subgroup be a proper subset of it?>
Surely it can (consider the subgroup genera by the matrix >
(Mathematica notation) {{1,1},{0,1}} inside GL_2(Q)). Why do
you ask?> Jyrki Lahtonen, Turku, FinlandYes, I realized the
result was false shortly after posting. This meansthere is an
error in Spanier's Algebraic Topology, p. 74. There heclaims
that two subgroups of a group are equal because they
areconjugate and one is a subset of the other, but actually
they areequal for a different reason.===
===
Subject:
: Re: conjugate
subgroupsWhy can't a conjugate of a subgroup be a proper subset
of it?> Yes, I realized the result was false shortly after
posting. This means> there is an error in Spanier's Algebraic
Topology, p. 74. There he> claims that two subgroups of a
group are equal because they are> conjugate and one is a
subset of the other, but actually they are> equal for a
different reason.Sorry, but which chapter, section, etc is
that? I only have the Tata-McGraw hillpaperback edition
available, and there p.74 is at the beginning of section 4of
chapter 2 (section titled The Lifting Problem). I couldn't
immediatelylocate the claim you mentioned on that page or any
other page near it.Jyrki===
===
Subject:
: Re: conjugate subgroupsWhy
can't a conjugate of a subgroup be a proper subset of it?>
Yes, I realized the result was false shortly after posting.
This means> there is an error in Spanier's Algebraic Topology,
p. 74. There he> claims that two subgroups of a group are equal
because they are> conjugate and one is a subset of the other,
but actually they are> equal for a different reason.> Sorry,
but which chapter, section, etc is that? I only have the
Tata-McGraw hill> paperback edition available, and there p.74
is at the beginning of section 4> of chapter 2 (section titled
The Lifting Problem). I couldn't immediately> locate the claim
you mentioned on that page or any other page near it.>
JyrkiChap. 2, Sec. 3, Theorem 11===
===
Subject:
: Re: Basic
factorization ideasIn sci.physics,
<jstevh@msn.com><3c65f87.0311120854.6eb55db1@
posting.google.com>:>> In sci.physics, >> <jstevh@msn.com>
<3c65f87.0311111302.799cf8e3@posting.google.com>:>> If you saw
(c_1 x + 7)(c_2 x + 7)( c_3 x + 1) = > 49(x^3 + 5x^2 + 3x +
1) with the c's algebraic integers, I think few of you would
have a>> problem realizing that only two of the c's have 7 as
a factor.>> Oh, look, he's changed polys on us! But OK, let's
check>> this one.>> Recall that, if P(x) = 49 * (x - x_1) * (x
- x_2) * (x - x_3),>> then P(x) = (c_1 * x + 7) * (c_2 * x + 7)
* ( c_3 * x + 1)>> implies that c_1 = -7 / x_1, c_2 = -7 / x_2,
c_3 = -1 / x_3,>> for some permutation of the x_i. Again, c_1 *
c_2 * c_3 = 49,>> as required, since x_1 * x_2 * x_3 = -1.>>
Therefore, c_1 and c_2 both satisfy the equation>> c^3 * P(-7
/ c)/49 = c^3 - 21*c^2 + 245*c - 343>> and c_3 of course
satisfies>> c^3 * P(-1 / c)/49 = c^3 - 3*c^2 + 5*c - 1>> so it
turns out that this time, James, you got it more>> or less
right. (And c_3 is even a unit, to boot.>> Come to think of
it, so are the x_i, which is probably>> one big reason why
this particular case actually works.)>> Also, it is obvious
that c_1 and c_2 have factors of 7,>> as well, in the ring of
algebraic integers. But, of course, you're looking at
*functions* of x, as you have f_1(x) = c_1 x, f_2(x) = c_2 x,
and f_3(x) = c_3 x, so I could also write it as >> (f_1(x) +
7)(f_2(x) + 7)( f_3(x) + 1) = 49(x^3 + 5 x^2 + 3x + 1). Notice
that dividing both sides by 49 gives (f_1(x)/7 + 1)(f_2(x)/7 +
1)( f_3(x) + 1) = x^3 + 5 x^2 + 3x + 1 as long as you're in a
ring where 7 is not a factor of 22.>> OK, stupid question.
Where did the 22 come from?> Oh, I've been used to arguing
with people about> (5 a_1(x) + 7)(5 a_2(x)+ 7)(5 b_3(x) + 22)
=> 49(300125 x^3 - 18375 x^2 - 360 x + 22)> where the a's are
roots of> a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x)>
and b_3(x) = a_3(x) - 3, which is a substitution made because
a_3(0) => 3, so that I can isolate constant terms.> My point
is that certain rules apply as even with my example above, I>
just have rather basic functions of x.Great. Try your logic on
this polynomial.(c_1 x + 7)(c_2 x + 7)( c_3 x + 8) = 49 * (x^3
+ 8).If you don't like that one, try this one:(c_1 x + 7)(c_2
x + 7)( c_3 x + 2) = 49 * (x^3 + 2).>> But yes, for this
particular example, your odd math actually does>> work, as you
are dealing with x_i which are in fact units.> So you believe
that math is quirky?Your math sure is.[rest snipped]-- #191,
ewill3@earthlink.netIt's still legal to go
.sigless.===
===
Subject:
: Faustus of Witten-burgMusic from Gounod's
Fausthttp://www.classicalarchives.com/m/023/act_1_du.midplay
the above in background as you read like the piano in a silent
movieat the Nickelodeon 100 + years ago on the New York City
Bowery.Z as MephistophelesMe as Dr. FaustusPaul, all my books
are packed ready to move so I cannot get the precise formulaI
remember. Basically though you can split the torsion-free
Levi-Civitaconnection for parallel transport of tensors (Lie
dragging along vectorfields) along world lines in curved base
spacetime of the tangent bundlein terms of a bona-fide 3rd
rank Diff(4) group tensor + the inhomogeneousnon-tensor part
that causes all the trouble in the gravity vacuum issue ofa
local stress-energy density tensor.All the above is from the
local gauging of the energy-momentum Pugenerators of the
4-parameter translation sub-group of the 15-parameterConformal
Group.Locally gauge the 6-parameter O(1,3) subgroup of the
Conformal Groupin the tangent space to get the torsion fields
which appear as anew Diff(4) 3rd rank tensor addition to the
connection that isanti-symmetric in two of the
indices.According to Tony Smith, Segal locally gauged the
remaining 5 parametersub-group and I think that should be
unified exotic vacuum dark energy/matter.That is only a wild
guess at this stage. I have not worked that out yet.Fast
ForwardM-theory may be the final proof of the bankruptcy and
heuristicexhaustion of theformal-empirical methods of modern
physics. On the other hand, underthe rightinterpretation, it
might open up new vistas. We'll see.No argument from me on
that one! What a double standard eh? -- abouthype in science I
mean.People are always telling me to connect my theories with
experimentsand observations. Wellthat's what I have done more
so that anything on NOVA's ElegantUniverse although I
doconnect my math with their stringy alpha' and indeed I think
what Ihave done inhttp://qedcorp.com/APS/EmergentGravity.pdf is
good for Witten & Co Ihave connec theirvaporware more closely
to observations of important stuff like thecosmological
constant.Z: Well, of course there has to be an empirical
bottom line -- cash value.Let me suggest that the biggest
payoff -- and the best potential sales point for yourprogram
-- would be the prediction of *novel phenomena*, which only a
newfundamental theory of gravitation of the kind you are
offering is likely to be capableof.Dark Energy/Matter IS that!
It's as novel as they come. Mike Turner and Ed Wittenadmit
readily in print they really have no clue at all! They are
honest men. Untilshown otherwise I think I have the correct
explanation. Also UFOs are novelphenomena, but try getting
that into Nature!Don't forget Regge trajectories, stability of
electron (evaded by string theorists)and point structure of
lepto-quarks in deep inelastic scattering. I got plenty
ofphenomena.Of course it is the string tension that stabilizes
the self-electric chargebut what is the origin of that tension?
It is dark matter /zpf < 0 exotic vacuumnegative zero point
energy density with positive pressure that doesthe trick
providing the mechanism for the string tension or
micro-geonquasi Kerr Newman black hole where the extra
space-dimensions providethe strong and electro-weak charges.
The micro-black holes have hairin hyperspace.Also the open end
strings can connect different branes not just the same brane.I
think the branes come from O(N) symmetries of the
MACRO-QUANTUMorder parameter beyond my approximate O(2) model
that makes quantized string topological defects in 3Dspace
from imposing single-valuedness on the local vacuum coherence
field PSI.Giant Star Gates stabilized by exotic vacuum dark
energy /zpf > 0also connect different brane parallel worlds
floating in hyperspace.This is the UFO phenomenon.Z: But the
extravagant resort to these multi-dimensional hyperspaceslooks
verysuspicious to me -- regardless of how neatly the Standard
Model fitsinto theformalism.Pythagorism run amuck?In any case,
Puthoff is not claiming that his PV theory is any kind
offundamentaltheory of gravitation -- just a highly computable
working model thatmight beat least pragmatically justified by
some future deeper theory.Not so, I think he claims he has a
viable alternative to Einstein.Z: I never read that from him.
He issued repea disclaimers that PV was not atthe present
stage of the game to be regarded as anything more than
anengineering model.We must have read different things.
Puthoff, for example, claims there are no black holes, no
event horizons.Z: Although he did say that *if* the Yilmaz
theory or some other field theory thatrecovers his exponential
metric expansion is eventually vindica, then that*could*
provide a deeper justification for his model.But even then PV
would not reach down as far as quantum gravity, obviously.That
is unless an empirically valida Yilmaz-type field can be and is
successfullyquantized.I think Hal's approach violates
Einstein's Rule:Make physics as simple as possible, but not
simpler than is possible.Ditto for Evan Harris Walker's.Z:
Your original strategy, as I understand it, was to piggyback
your BEC modelby way of correspondence (emergence) on
canonical GR as expounded in MTW,and on the authority of their
big black bible.NO! My approach comes from the following
sources:1. Andrei Sakharov's metric elasticity2. P. W.
Anderson's More is different3. Hagen Kleinert's world crystal
lattice 4D elastic-plastic formulationof Einstein's gravity
allowing torsion in addition to curvature.4. Bohm's
pilot-wave/hidden variable interpretation.5. Wheeler's IT FROM
BIT and his early Geometrodynamics ofMass without mass, Charge
without charge, Spin without spinWheeler's idea here came from
Einstein's Vision6. Feynman's inability to solve electron
self-energy/self-charge problemapart from the shell game (his
term) of renormalization.7. Unrenormalizability of Einstein's
classical gravity made into a localquantum field theory by
standard methods.8. Regge data in hadrons.9. Point structure
of lepto-quarks in deep inelastic scattering.10. Dark
energy/matter as 95% of the Universe.Z: If I am right and MTW
are the Enron of gravitational physics, then I would saythis
is a serious mistake.You are not right.Z: If I am right, then
your best strategy would be to consistently and explicitly
breakwith the naive Einstein interpretation of the EEP and
look for a radically differentcorrespondence model vis a vis a
consistently re-interpre GR.I have already done it. Do you not
understand what I am saying?Z: As I see it, that would entail
a meta-theoretic mapping of your Goldstone phase to aseparate
physical metric that represents the phenomenological g-field.I
do not understand what your words mean? I have the equations
written down inhttp://qedcorp.com/APS/EmergentGravity.docDo
you have different equations to replace them?Without math I
simply do not understand what you are saying here. What
difference does it make?As far as I am concerned the local
classical vacuum stress-energy density Diff(4) tensor is
simplytuv(vac) = (c^4/8piG)GuvEinstein's local field equation
in non-exotic vacuum is thentuv(classical vac) +
Tuv(matter-radiation) = 0Adding exotic vacuum dark
energy/matter istuv(classical vac) + tuv(quantum zero point) +
Tuv(matter-radiation) = 0tuv(quantum zero point) =
(c^4/8piG)/zpfguv/zpf = (alpha')^-1[(alpha')^3/2|Vacuum
Coherence|^2 - 1]alpha' = Witten's string
parameterD^uDu(Vacuum Coherence) + dU(Vacuum
Coherence)/d(Vacuum Coherence*) = 0 is Landau-Ginzburg eq. BIT
FROM ITdu(x) = alpha'(Goldstone phase of Vacuum Coherence),u is
IT FROM BITguv(x) = Minkowski + du,v(x) + dv,u(x)BIT is Vacuum
Coherence = The Wordguv and /zpf are IT = The Fleshif you want
to decode The Cipher of Genesis or The Da Vinci Code - the real
one not the fake one.
http://qedcorp.com/APS/StarGate1.mov(typo correc yesterday
on dark matter slide)Z: You would then of course also have to
recover the inertial metric, but you wouldfuse this with the
gravitational metric at the correspondence level, and not on
themost fundamental theoretic level.Ditto. I do not understand
what you mean.Z: Your theory could then provide a physical
basis -- based on your virtual BECcoherence -- for the *limi*
physical analogy between gravitational and inertial fields.Not
could but doesZ: While you seem to be on the right track with
your emergence thesis, I don'tthink you are as yet entirely
consistent in your approach, for the fundamentalreasons I have
sta.I do not understand your reasons and I never will as long
as you rely totally onvague ambiguous ordinary language
without mathematics and/or precise
operationalgedankenexperiments like in the Einstein-Bohr
Dialogues.Z: As for elegance -- as Boltzmann once said, when
in the process of lookingfor deeper models of physical
reality, elegance is for tailors.I honestly don't understand
why you think you are in directcompetition with Hal.Because he
got all the money people in the fringe BPP people to thinkhe is
on theright path - not that there is a lot of money there. But
because of hispast top securityclearances he has influence and
access inside USG Black Ops and in factthere is afactional
fight inside USG right now about all this - more than that
Icannot say. So theissue is one of whistle blowing on wasting
tax payer money on thingsthat have nochance of working whilst
ignoring things that do.Z: OK -- now that makes sense.I'm not
sure we're on the same page here. But I *am* talking
aboutspinningEinstein equivalence-- within the framework of
canonical GR.space-time. It's complica butstandard in MTW
et-al.Z: And does he mention any empirically testable
scale-independent multipoleeffects?If so I'll have a look at
that.Don't know off hand.If ALSO there is TORSION then
additional coupling of spinning testZ: OK, but that's another
ball game.Is there any convincing empirical evidence for
non-vanishing torsion?Good question. Akimov in Moscow says so.
Looking at Kleinert it wouldbe strange if it were not there.
How big ? is the issue.Akimov in Moscow says that is a big
effect with weapons applications.One guy here who came to ISSO
2000 also said that and he was working onareport. Most
researchers pooh pooh that. I do not knowfirst-hand.Z: I said
convincing. Apply same standard to Brian Greene's Elegant
Universe.String theory is like 2D field theory because the
string sweeps out aworld sheet in ordinary space-time rather
than the 1D world line. Thisis a great thing because it
essentially does away with the infinitiesCosmological Constant
problem, which worries Witten and rightly so.problems.NO!
String theory give bad prediction /zpg = 10^66 cm^-2
whenexperiment says/zpf = 10^-56 cm^-2 because Witten's
strings are only 10^-33 cm asBrian Greenerepeats many times in
the NOVA vaporware.String theory is no better than local field
theory on this issue! Wellonly a little better, butstill 122
Powers of Ten off-target!Witten understands that!They lack my
giant vacuum coherent field!Z: No? So I am wrong to think that
string theory avoids all the renormalizationheadaches of QFT at
least partly because it posits extended entities as modelsYou
have garbled renormalization with the cosmological
constant.String theory does avoid the infinite renormalization
constants offinite numbers. The string theory prediction for
the Einsteincosmological constant is/ ~ 1/Lp^2 ~ alpha'^-1 =
10^66 cm^-2In contrast observation is/ ~ (Ho/c)^2 = 10^-56
cm^-2That's why Ed Witten the smartest man in physics today
according to Brian Greene on NOVA Elegant Universe so so
worried about it!My solution to that problem is simply/ =
alpha'[alpha'^3/2|Vacuum Coherence|^2 - 1]Where alpha' is
Witten's favorite thing (string tension)^-1 = c^4/G*What none
of the Pundits have from Witten to Puthoff (that's many powers
of ten in mathematical depth ) is VACUUM
COHERENCE!IMHO.http://qedcorp.com/APS/EmergentGravity.docZ:
OK, I'll read it.After all - you get an infinite mass density
and infinite electricalforces insidegravitationalforce is
infinite at zero separation of two mass points.The classical
electron was a hopeless conundrum even in Lorentz'stheory.THIS
IS NOT THE PROBLEM.Z: I know you are talking about exploding
perturbation expansions, massrenormalization, that kind of
thing. If these go away under a string model,I imagine the
extended nature of the model might be relevant to the
problem.But then, what do I know?imprecise and ambiguous -
goodfor poetry not for physics.Z:Your simple dichotomy between
ordinary words and math is a falseone IMO.There is ordinary
language, there is *technical* and *scientific*language,
andthere is the highly refined and abstract language of
naturalosophy.Then there is mathematics.If the formal
mathematics is to have a well-defined physical meaning-- a
physicalinterpretation -- then it must be arrived at and
expressed inextra-mathematical technicaland osophical
language.Even Einstein had an equivalence principle --
expressed in what youcall ordinarylanguage -- before he had
the Einstein-Hilbert field equations andbefore he even hada
metric tensor representation of the g-field.Yet we still don't
know what these field equations mean exactly,except in
purelyformal-empirical terms, since Einstein's equivalence
principle isproblematic.I do not mean that ordinary language
is not useful or important. Imean't that it is not enough.Z:
And of course in that case I completely agree with
you.Einstein had math to back up his essential points
inordinary language.Z: Of course. But the math is for the most
part standard and almost completely workedout. We have the
canonical theory, with the EEP, and a raft of bi-metric
theories.The problem on this level, as I see it, is for the
most part finding the correct physicalinterpretation of the
math that is already available.Which, incidentally, and
perhaps ironically, is precisely how Einstein arrived
atspecial relativity. After Lorentz and Poincare, he took the
last *interpretive* steps.After all, they are called the
*Lorentz* transformations -- and not the
Einsteintransformations.What I have to do is show that when
you back out the non-tensor inertial contributionto the vacuum
gravitational stress-energy density, you get something that
transformslocally like a world-tensor (under Diff(4)) .My
solution is simplytuv(classical geometry) = (c^4/8piG)GuvZ:
That could begin to make sense of the theory of gravitational
waves, which is currentlya puzzle within a puzzle. Anyone who
claims to understand it obviously doesn't.I have to start with
the standard weak-field (linearized) model and go from
there.When I have finally done this I'll write it up and then
you can read all about it.I am not going to write anything
until I have the math completely worked out.I think Kip Thorne
correctly worked all that out in the LIGO/LISA project.Do not
reinvent the wheel. I doubt you can do it better than Kip
Thorne & Co.So far, I am not aware of even ONE math formula to
back up any of yourkey points that you have expressed so far
only in ordinary informallanguage.Am I mistaken here? Please
correct me if I missed something. We havebeen goingback and
forth on this for almost 2 years now without any relevant
mathfrom your corner.Actually, the basic mathematical approach
is easy. You just locally separate the inertialand
gravitational contributions to the metric gradients:g_uv, w =
g_uv, w |G + g_uv, w |I, for all u, v, and w.(Here G =
Gravitational, I = Inertial)Then the gammas automatically
decompose in an analogous manner.I do not see what those terms
are.I think you are wrong here.guv is a tensor.The place to
start is the connection like I did above.Apart from my
formulaguv = Minkowski + World Crystal Strain Tensorthere is
nothing new here.Z: I am looking for a theorem in differential
geometry that locally splits the deformationaland
transformational contributions of the metric gradients on an
abstract curved4-dim Riemannian manifold.That's eqs (1.5) &
(II.2) & (II.3) inhttp://qedcorp.com/APS/EmergentGravity.docZ:
I've just been busy with other things, but I'll get down to it
eventually.It would be different if I were getting money to do
it.I have already done it.Z: The physical stress energy (of the
permanent field) can then be construcout of gammas from which
the g_uv, w |I have been removed.No, because the gammas are
zero in the LIF.Z: The only thing that makes the vacuum
stress-energy a pseudo-tensor, as far asI can see, is the
inertial contribution to the gammas.You mean the non-tensor
term in the gamma. They mix with the tensor term when you
makea quadratic form in the the gammas. I fail to see how that
gets one anywhere.tuv(geometry) = (c^4/8piG)Guvsolves the
problem trivially because it is a trivial problem IMHO.The
Question is: What is the The Question?Z: At this point the
math is pretty straightforward. What is important is what it
*means*in the context of GR.Then write it down.Z: What it
*means* is that without Einstein equivalence there is no such
thing as generalrelativity. It's a mirage. That also raises
serious questions about *special relativity*.And I am NOT a
crank....Uh Oh!Einstein's equivalence principle is classical
and the boundary of itsdomain of validity isfuzzy and
problematical, which is part of its greatness like great
Artand Poetry.Z: Yes, and of course I agree. I'm not in the
business of trashing dear old Albertfor the sake of it.Like
some we know.Z: But I suspect you are here talking about the
EEP. I say Einstein equivalenceis in fact invalid
*everywhere*. But if you are only talking about EEP, then I
agree.In any case, as to Einstein equivalence, that is what I
mean when I use the termheuristic. This is of course all very
important in the context of discovery,even if it does
eventually turn out to be a red herring.But I am working in
the context of evaluationFor example the classical Einstein
equivalence principle and thequantum Heisenberguncertainty
principle work hand in hand to produce the
randommicro-quantumDiff(4) covariant additional term
/zpf(x)guv(x) in Einstein's localfield equation.See John
Peacock's Cosmological Physics pp 25-6 for that derivation.Z:
My guess is that this is a formalistic grafting of some of QM
(which no oneunderstands) onto GR (which almost no one
understands).It leads to the very nice eq. (III.1)
inhttp://qedcorp.com/APS/EmergentGravity.docEinstein's
equivalence principle has a nice mathematical expression inthe
tetradtransformationFLAT SPACE-TIME = TETRAD (CURVED
SPACE-TIME)and inverselye.g.nab = ea^u(P)eb^v(P)guv(P)Z: As an
expression of the bare EEP, yes.Yes yes yes.Will you take yes
for an answer?That's all I ever need and use.Z: But this is
still no more than a mathematical re-packaging of the EEP. It
adds no newphysical content.It sure does and it's all
inhttp://qedcorp.com/APS/EmergentGravity.docplain as day.Z:
And it still does not in and of itself fully capture the
content of Einstein equivalence,which reaches much deeper than
the formal-empirical level of the EEP.As far as I can see the
math of the point you are making is thatTORSION-FREE
CONNECTION = HOMOGENEOUS 3rd RANK TENSOR +
INHOMOGENEOUSPIECESo your inertial field is the second piece
on RHS.Z: Yes, exactly. If you interpret this inhomogeneous
piece as a kinematical field.Fine.Z: The idea is that if you
pull this out of the connection field, what you have left is
much morelike a classical field, although it is represen by
tensor quantities and integrates rubberrod and clock metric
effects with forcelike dynamical effects.*That* is what a
deeper theory should recover based squarely on a physical
model.I do that in
http://qedcorp.com/APS/EmergentGravity.docZ: And it will, if I
am right, also have a fully localizable vacuum stress-energy
which transformsmuch more like the EM stress-energy under
Diff(4) in both weak- and strong-field domains.Again that's
simply in the Wittenesque symbology of the real Da Vinci Code
[What is the alpha in Egyptian hieroglyphs and Babylonian
cuniform? Is alpha the Aleph? No it's clearly the Bayt
(bait)]tuv(vac) = (c^4/G*)[Guv(classical Einstein) + /zpfguv]=
(alpha')^-1 [Guv(classical Einstein) + /zpfguv] = (String
Tension)xCurvature(MACRO-QUANTUM + Micro-Quantum Zero
Point)What more is there to be said here? It's in the standard
Einsteintheory.Z: But again, that is *not* Einstein's
theory.Let's not quibble. None of this would be in discussion
if Einstein never exis.Z: With all due respect Jack, you do
not seem to know what Einstein's theory actuallywas!What *you*
mean by general relativity is not what *Einstein* meant by
general relativity.With different punctuation, what *you* mean
by general relativity (at least on Mondays,Wednesdays, and
Fridays) is *not general relativity*.I have already given you
fully adequate documentation on this.Power-down reset?But in
any case, I actually agree with *your* current definition,
since Einstein's generalrelativity doesn't exist.As long as
you adhere to it consistently.You are splitting very fine
hairs here. No mainstream physicist doing perfectly good work
inprecision cosmology would understand you here or even care
to.One can quibble about the informal language attached to
this formula,but I think it is a quibble.Z: You are saying
that refutation of Einstein's fundamental argument for the
general relativity ofall motion -- which he himself protes was
the very cornerstone of his theory of gravitation-- is a
quibble?Jack, pull the other one.No, I did not say that. You
did.The main content of Einstein's theory, the key algorithm,
isGuv = alpha'Tuvwhat he may or may not have said in heuristic
informal language at some early point in the emergence of the
theory is not that important.EEP here also means that locally
at event P one can find timelikegeodesic frames in which
theTORSION-FREE CONNECTION = 0 at P but not in neighborhood of
P whentensor curvature =/= 0 at P.Z: OK.What gyroscopes do if
there is a torsion field was not considered inEinstein's
original theory.Z: He ruled out any empirically measurable
effects that would locally distinguish gravitationaland
inertial fields. In this sense, he did consider them as
belonging to a class of possible effectsthat were prohibi by
his equivalence principle, which was supposed to hold -- and
indeedhad to hold -- universally and without restriction.No he
did not later on in his unified field theory later period.No
principle in physics holds universally without restriction.
What Einstein said in 1915pre 1925 on this should not be taken
literally. See Bohm on this BTW.But that is a straight forward
extension.Einstein's theory guv(P) is from locally gauging the
LINEARTRANSLATIONS inCONFORMAL = DILATION x NONLINEAR SPECIAL
TRANSLATIONS x LINEARTRANSLATIONS x SPACE-TIME
ROTATIONSTORSION is local compensating gauge field from
SPACE-TIME ROTATIONS.We should LOCALLY GAUGE EVERYTHING! Why
stop at a subgroup?Z: More Pythagorean groupology!It works.Z:
Where's the beef?Yum, yum. Niman not Wendy's.The torsion-free
connection can probably be split into the homogeneoustensor
part + the inhomogeneous part.Indeed I remember that and can
quickly look it up. So if that is allyou mean, then that is
standard Einstein.What then is the point here?That's not it.
The decomposition I am talking about is orthogonal tothis.I
have no idea of what you mean here if you cannot produce a
mathformula togive the informal sentence meaning.See above.I
have explained the point many times. We are separating the
inertialand gravitationalcontributions to the connection
field.You have explained nothing to my mind unless there is
math to give thewords meaning and/ora precise operational
gedankenexperiment in sense of theBohr-Einstein debates.See
above.Also, showing how your new insight predicts some new
physics and/orexplains animportant mystery in the old physics
is also desired.It's not new, and it's not mine.(1) The
equivalence principle is a mere conjecture that is not to be
taken too seriously;(2) The GR inertial field is associa with
a stress-energy density that is a non-tensor.That was
*1922*.Einstein had a slugfest with von Laue in the early 20s
on this, in which von Laue arguedfor the purely heuristic
character of Einstein's principle.gradients g_uv, w asmetric
potentials of the [permanent] gravitational field.Bergmann
also referred to a pure inertial stress-energy.Feynman more or
less trashed Einstein equivalence in the 60s.Precise quotes
please.Z In the 90s Weinberg announced in his Gravitation and
Cosmology that he isa heretic.Ditto.Z: I am just trying to
push this skeptical argument to its logical conclusions.I
gather deformational means Ruvwl curvature tensor =/= 0 at P
?It means the contribution to the connection field (i.e.
metricgradients) that resultsfrom the deformation of the
spacetime manifold, where Ruvwl =/= 0.Transformational is when
Ruvwl = 0?It's what you have left when you set Ruvwl = 0. It is
obviously stillpresent even when Ruvwl =/= 0 in almost all
frames.Fine.OK.I think that is what I said above. It's already
there in Einstein'soriginal theory. Nothing new here IMHO.Z: Of
course this is not new. It is the (logical) conclusions that I
am drawing from it thatare new.Yet I wonder: Why does Weinberg
feel compelled to call himself a heretic when heis writing
about this in the mid-1990s?If it's all so standard and
obvious??Feynman and Weinberg quotes would help here.Z: This
is really a matter of pure differential geometry, since
itholdsgenerally in theabstract, without any specific
reference to spacetime manifolds.I do not understandwhat you
mean by inertial compensation above.Z: Just think of how
inertial and electrical forces interact and howthisis
describedmathematically in *standard GR*. The EM field is
notgeometrodynamic,while thepure inertial field (no
gravitational sources) is.EM field is geometrodynamical in
Kaluza-Klein with one extra spacedimension.Z: That's not
*standard GR*, JackZ: Are you telling me that Kaluza-Klein is
now canonical? Or are youtalking aboutsuperstring theory?Yes,
in sense there is well-defined math meaning to their
concepts,which is lacking inmost of your thesis here unless I
have missed something specific?Z: The math is for the most
part completely standard.It is the interpretive conclusions
that are new.I think the fictitious or inertial g-force i.e.
the metrictorsion-free connection for parallel transport in
the original1915theory is LOCALLY equivalent to agravity
force.Z: Einstein's principle was that a uniform gravitational
field is*fundamentallyindistinguishable* from an inertial
field. That is all in thehistorical record.Depends what one
means by fundamentally.Z: It is quite clear what Einstein
meant. He was quite forthcoming onthis. I think it's important
to recognize that this is the cornerstone ofEinstein'sconcept
of general relativity. The meaning of Einstein's principle
isquiteunambiguous when placed in context.Fine, but nothing
wrong with what Einstein said above except thatphysically we
never havea global uniform gravity field.Z: There is nothing
wrong with it except that it is almost certainlyfalse as a
matterof empirical fact.Yes,Z: Stop right there!Yes.So you
actually agree with me.Einstein was playing with a useful
idealization, theorists do that all the time. It led to the
correct conception. It's like the principle of virtual
work.Remember without electromagnetism no light hence no
gravity!First:Let there be lightGravity came on the Second
Day.Decoding The Cipher of Genesis.but one could imagine a
universe like that in terms of theparameters of the theory.Z:
One could.I'm interes in this universe.Although I agree that
Einstein was at least consistent -- which I cannot say for
MTW.Einstein had to pretend that while Riemann curvature
distinguishesalmostall permanent gravitational fields
(Eddington, Tolman, Bergmann)from all inertialfields, it has
no direct physical meaning, at least in the
immediateneighborhoodof a spacetime point.No, again I think
you are mis-interpreting here.If the local curvature is not
zero it is not zero in any frame at P.Z: Yes...However,
it'stidal effect at P is ignorable in a small enough
neighborhood of Pprovided1. Not near a space-time
singularity2. Not inside the effective Planck area Lp*^2Z:
Isn't that like arguing that since we cannot measure Gaussian
curvature at ageometric point, the Earth is really flat
there?No, we can measure it at a coarse-grained point
determined by band width ofthe detectors.Z: So is that a
flat-earth argument?No.Z: Or pseudo-positivistic
gobbledygook?No, that's physics not shmizziks.Z: The Planck
area is another question -- although this has to be
consideredat some point I agree.I have been doing that in
http://qedcorp.com/APS/EmergentGravity.docNote Planck Area =
Witten's alpha' (h = c = 1 convention) = 1/(String
Tension)gravity becausespacetime is infinitely rigid but
unstable.End of story. It's an approximate statement or a
correspondenceprinciple.Z: Exactly. And that is *all* the
so-called EEP is.But that is not Einstein equivalence --
unless you add an additional layer ofphysical
interpretation.Einstein was not thinking about Lp^2 = hG/c^3
in 1915, although I thinkhis thinguntil 1925!Z: Immaterial to
my general argument.Which is historical and not important in
solving real problems of the day IMHO.Z: As far as I can see
Einstein was simply (although understandably)confused aboutthe
distinctions between empirical correspondence,
formalcorrespondence, formalanalogy, and the literal identity
of inertial and gravitationalphenomena.Quite possibly, but so
what? It's no big deal today unless you can useyour new
insightZ: Not new.to explain why cosmological constant is
small, and the true nature ofboth dark energy anddark matter
and also the string structure of lepto-quarks and gaugeforce
bosons etc.Z: I'll leave the details to you.God is in the
details. In this caseVacuum Coherence = Mind of GodZ:
Remember, I don't claim to be a physicist.As a minor
historical point maybe you are correct.Z: I think I am
correct, but it is a major historico-critical point that has
been thesubject of much controversy over the years.The issue
is in fact taken quite seriously and I believe it has profound
implicationsfor future gravitational physics.I am concerned now
with Physics Today.I don't know withoutstudying theclassic
writings more deeply.Z: As I said, it turns out that there is
in fact no such thing as general relativity.As far as I can
see this approach is a hangover from the
fundamentallywrong-headed logical empiricism of the 1920s.How
is the MTW argument regarding localized vacuum stress-energy
basedon theEEP any different from arguing that since the speed
of an objectcannot be measuredat an instant, the speed of the
object has no physical meaning at aninstant?Hence, as Zeno
might argue, motion is impossible, since at eachinstant the
objectcannot be said to be moving?We enlightened moderns have
the differential calculus. We are supposedto
knowbetter.Eddington poin out in 1922 that there is no
rational foundation forinsistence on sucha sweeping universal
assumption (Einstein equivalence) -- that it ismerely a
conjecturethat is subject to empirical test and that it is a
mistake to assumethat the hypothesis*must* be true.And whether
it is or is not is as far as *formal-empirical* GR isconcerned
a purelyempirical matter.In other words, it was merely
Einstein's guess. A sweepingextrapolation. A
boldhypothesis.Based on what we know now, as a general law of
nature, quite possiblya huge REDHERRING.This was later reitera
by Feynman in his Lectures on Gravitation,where he actuallymade
some gentle fun of Einstein (or at least Einstein's
morefanatical followers).Clearly, we know there is a
formal-empirical *analogy* betweengravitational and
inertialfields, but only a partial one (Eddington 1922). But a
partial analogy-- short of identification-- is a perfectly
adequate basis for a weaker physical interpretationof the
so-called EEP astechnically formula by Wheeler et al. that
does not imply theidentity of gravitationaland inertial
phenomena.That's only an idealization.The EEP is an
idealization. But the EEP does not capture Einsteinequivalence
withoutan additional layer of interpretation.IMO EEP amounts to
an *ad hoc* stratagem whereby, in order to appearto save the
Einsteinequivalence hobbyhorse, the fundamental distinction
betweenequivalence as identity, andequivalence as mere local
correspondence with flat-spacetime SR,
issystematicallyobfusca, and a minimal agnostic epsilon-delta
type technicalformulation is misleadinglyrepresen as somehow
capturing the content of Einstein's principle(hence the
misnomerEEP).It doesn't. It is *consistent* with it, but it
does not capture thefull content of Einstein equivalence.That
is why I call the MTW development of canonical GR a shell
game.You have to keepyour eyes on that pea.What they do in
Gravitation in certain important contexts is totacitly attach
the additional layerof Einsteinian interpretation in their
supposed proof of thenon-locality of vacuum stress-energy.The
physical field is supposed to be elimina (i.e. annihila)at a
point in a free-fall frame.But that is Einstein equivalence,
and not merely EEP.If Einstein equivalence is empirically
false, then it can hardly beinvoked as the theoretic basis
forthe official interpretation of the GR formalism, of the
EEP, or asthe basis for an argumentagainst localized vacuum
stress-energy, as MTW would have it.However, under a suitable
reinterpretation, the entiregeneral-covariant formal machinery
ofcanonical GR still works fine. But at the same time, the
reinterpretheory may point in verydifferent directions as
regards development of deeper quantum theoriesof gravitation
and thecorrespondence relation between the two theories.That
this very bold Einstein concept of
gravitational-inertialequivalence historically led Einstein
tothe GR tensor field equations in the manner of a *heuristic
guide*, orthat Einstein himselfmay never have explicitly
abandoned his bold hypothesis, are neitherhere nor there as
far as thecontemporary objective critical discussion of the
subject isconcerned. IMHO.However, this was all poin out by
Eddington and von Laue in theearly 1920s.Too bad M, T, and W
apparently ignored them.Tidal differences in the g-force
between P and P + dP meanscurvaturei.e. real gravity as
opposed to simula gravity.Yes, exactly. That is what is meant
by a permanent or intrinsicgravitational field.Fine, so what's
the problem?The problem is the GR definition of vacuum
stress-energy, the questionof its covariantformal
representation, and the question of its localizability -- as
Ithink you know.Again your point here is only interesting if
you say there should bemore terms inGuv + /zpfguv =
-alpha'Tuvalpha' = Ed Witten's inverse string tension (h = c =
1 convention).i.e. if you say there is some kind of power
series in alpha'^n with nfactors of Guv?Z: That would require
a specific underlying model, which I am not specifying.That's
your job.The power series so far is not needed.Z: I am
proposing a correspondence template for a more fundamental
theoryfrom which phenomenological gravitation, as you say,
emerges.I am arguing that there is no rational objection to
conceiving of gravitation asa local physical disturbance of
the physical vacuum that *causes* the motionof bodies to
deviate from their normal geodetic paths.geodesics.EM force
pulls them off timelike geodesic in 4D if they carry electric
charge.Z Such a disturbance would propagate physically through
the new aether,conceived as a non-material (or proto-material)
medium of physical propagation.Old Hat. So what?Z: In other
words, HYPOTHESES FINGO. I am throwing out all the old
pseudo-positivistic taboos that have cast such a hideous spell
over theoretical physics for wellover a century, if not
longer.This influence is not propaga through a void, but
through a physical mediumthat lies in the twilight zone
somewhere between a material space-filling substance anda
literal void.You seem to have forgotten about QED vacuum
polarization plasma of virtual electron-positron pairs and
virtual photons.Z: I strongly believe this was Maxwell's
inchoate view, for which he was lampooned by
therelativists.100 + years out of date.Z: In a deeper theory,
under this basic model, the physical process through which
gravitationmanifests itself will be shown to be closely
analogous to the physical process throughwhich the phenomenon
of inertia manifests itself. This *limi physical analogy* --
andnot curved spacetime and gravitational-inertial equivalence
-- will then fundamentallyexplain the so-called EEP together
with gravitational attraction and mechanical inertia.I have
done this all already
inhttp://qedcorp.com/APS/EmergentGravity.docZ: If the EEP can
be rela on a formal-empirical level to a gauge symmetry of
someunderlying quantum field, fine. That is all quite
consistent with the abstract templateI am proposing.Then you
have something interesting. Otherwise your point is trivialZ:
The immediate corollary of this would be that Einstein's
general relativity thesis was trivial.As I said, pull the
other one.becauseLOCAL CLASSICAL GEOMETRIC VACUUM
STRESS-ENERGY DENSITY TENSOR =alpha'^-1Guv = Tuv(Geometry)Z:
Is this in linearized weak-field GR? Or are you talking about
the exact solutions?The latter! This is strong field
non-perturbative. I am talking the MACRO-QUANTUM local field
equation.LOCAL MICRO-QUANTUM RANDOM ZERO POINT VACUUM
STRESS-ENERGY DENSITYTENSOR = alpha'^-1/zpfguvZ: The effect of
t_uv (vac) in the weak-field model on the non-linear
corrections to the vacuum stressenergy density is obviously
closely bound up with the non-linear character of the field
equations.You missed my point here.Z: A deeper model would be
expec to unpack this -- e.g. the non-Abelian character of the
spin-1fields in the Deser-Feynman approach, which physically
explains the self-gravitating nature of theEinstein g-field at
a deeper level as another indelible characteristic of the
permanent gravitational field.Is that the kind of thing you
are getting at here?NO! If you read
http://qedcorp.corp/APS/EmergentGravity.doc you would not ask
that.Therefore Einstein's local geometrodynamic field equation
is:Sum off all local stress-energy density tensors from
classical geometry+ random virtual zero point quantum
fluctuations + matter + coherentvirtual near field + radiation
= 0Z: But what, in your model, is the *physical origin* of the
local stress-energy density tensors fromclassical
geometry?Read my paper. Eq. (I.5) is the answer.Z: Not curved
spacetime, surely.Exotic vacuum /zpf =/= 0 is coupled to guv.
The former is from modulating the Higgs intensity ofthe vacuum
coherence, the latter is from modulating the Goldstone phase of
the vacuum coherence.When no torsion and no hyperspace
violations of metricity then we havea simple situation from
the Bianchi identities that the 4-covariantdivergence of the
classical geometry stress-energy density currentslocally
vanish if the vacuum is NOT exotic, but NOT OTHERWISE!Z: OK,
this looks interesting.You bet it is.Z: Again, you are
reverting to your straw man here according to which Iam
supposed tobe finding a technical problem with Wheeler's
minimal agnostictechnical statement of the so-called EEP.I'm
not -- except that it is only valid at a spacetime point.As
every School Boy knows I suppose -- at least at Cal Tech.Z:
Because they are reminded of this every school day by Dick
Feynman -- orDick Feynman's ghost...Walks the parapets when
the Night Wind Howls.Z: But if you want to glimpse part of
what I'm getting at, just exchangethe epsilon and thedelta in
Wheeler's formulation of EEP. For any given finiteneighborhood
of aspacetime point, we can in (GR) principle always construct
agravitometer that is sufficientlysmall and sensitive to
detect the effects of non-vanishing Riemanncurvature around
thatpoint, regardless of the state of motion of the object.All
inside Einstein's famous elevator.This turns EEP' on its
head.You lost me.Z: I guess so.No I was just getting tired of
reading and my eyes glazed over. I was likeAlcibiades with
Socrates after a night of heavy drinking and ofawzing.Yes, I
understand what you are saying. I have said it myself, but
your attitude about it is odd.This is simply a matter of
bandwidth/resolving power of the curvature meter untilone
reaches the Planck area. The larger the bandwidth the smaller
the size of theLIF. The key number is (L/r)^2 where L is size
of LIF and r is scale of localcurvature. Approach to
singularity is r --> 0 so you cannot obeyEEP which requires
(L/r)^2 L << 1. This is elementary my dear Z.Z: It turns out
(and I didn't know this when I first got onto it) thatthe
position Iam advancing here is essentially Eddington's as set
out in 1924. Atthat time heviewed Einstein equivalence as
heuristic and tentative in character,and not asa fundamental
principle.I say he was right on the money.This is the view
that was later echoed by Bergmann, Synge, andothers.EEP as
technically formula by Wheeler is fine -- but it isconsistent
with eitherinterpretation (Einstein's or Eddington's).So it
seems your point is one of the history of relativity on
howEinstein's ideas evolved.Z: You don't seem to see that this
entails a profound shift in thephysical meaning ofthe GR field
equations, and that this means that *we do not haveactual
generalrelativity*, regardless of the impressive empirical
accuracy ofstandard GR.You lost me again.Z: The empirical
accuracy of GR is neutral with respect to the
re-interpretationof the EEP that I am proposing.If we do not
have actual general relativity, then we have not in
fact*explained* theequality of gravitational and inertial
mass, as Einstein thought hehad. We havesimply *assumed*
it.Again you are too subtle for me here.Z: Then so is
Einstein, Jack.This is *his* reasoning.There is no question
about this. I have all the quotes and citations.The crux here
is simply thatlight rays move on null geodesics, and nothing
moves on spacelikegeodesicsin torsion-free space-time.Z: The
crux here is, what, physically speaking, is a geodesic?The
straightest path in a curved spacetime that obeys the geodesic
equation forIt is the longest proper time for a bundle of
neighboring world lines withsame starting and end points in
sense of Calculus of Variations.Z: *Why* do material objects
and light rays move in straight lines? *Why* dothey deviate
from these straight geodetic paths under certain
conditions?Why?Scientifically meaningless. It's the Ansatz of
the Action Principle.Physics is not asking why it is asking
how.That is, if there is in physical reality no such thing as
curved spacetime, exceptin the sense of an abstract
mathematical model?This is precisely the problem as Feynman
presen it.'Cite Feynman here. Look you can say this about any
theory of physics.This is not interesting. This is Wigner's
The unreasonable effectivenessof mathematics in physics. is
metaphysics not physics.You cannot have light, you cannot have
anything move on a timelikenon-geodesicnull geodesicsphysically
if there were no electric charges. This already shows
thatgravity must emegefrom electromagnetism. No charges no
weight!Z: I suspect this is a fallacy. Let me think about
it.It is not a fallacy. You cannot have light without electric
charges. Indeed, in Wheelerand Feynman they only have electric
charges NO LIGHT! Without lightno light cones as rocklike IT
geometrodynamic realities as distinct from thoughtlike BIT
formal possibilities.Therefore, electricity is more
fundamental than gravity. This is whyelectricity is
micro-quantum renormalizable and gravity ain't. This was
Sakharov's keyidea I bet! Gravity is an emergent MACRO-QUANTUM
collective propertyof the unstable micro-quantum
electrodynamical vacuum.The Question is: What is The
Question?Z: The term general relativity, like EEP, is thus in
that case amisnomer. In the MTW development of GR it amounts
to an elaboratebait-and-switch routine.It also opens up a
prospect for the resurrection of a suitably
refinedneo-Lorentzianconcept of a non-material aether
(physical yet non-material mediumof propagationacross the
vacuum) to be sharply distinguished from spacetime as avoid.
Under thismodel, it is the *physical disturbance* of this
aether by thepresence of matter thatcauses deviations from
straight-line motion.Less words more math would help clarity
here.Z: The words are carefully chosen, and every word I use
here means something.Math is objective. Meaning here is too
subjective.Z: If we don't actually have *general* relativity,
then we don't haverelativity, period -- sincethat implies and
demands the physical relativity of *all* motion
(A.Einstein).Relativity is the stillness in the motion, i.e.
invariants of groups transformations.That's all it is. There
are different groups like the 15 parameter ConformalGroup for
spacetime structure broken down to 10-parameter Poincare
Groupto get invariant rest mass from dark matter /zpf < 0
exotic vacuum coresi.e. Brian Greene's vibrating strings of
energy. Brian should have saidvibrating strings of negative
micro-quantum zero point energy densitywith positive pressure
with vibrations in the extra space dimensions.Z: No Einstein
equivalence, no general relativity. This *according
toEinstein's ownclassic arguments*.This is more a legal
argument then a physics argument.Z: Natural osophy -- which I
believe is about to make a major comeback.Once the dead hand
of pseudo-positivism is finally lif.Hawking proudly declares
he is a positivist BTW.Z: This is IMO profoundly iconoclastic
and could have deep implicationsfor the furtherdevelopment of
gravitational theory. But the logic is inescapable IMO.Depends
what you mean by logic.Z: At least I know what I mean.How do I
know you know? Note I said How not Why.Is renormalization
theory forZ: Good question.Logic never stopped great physics
discoveries before and why should itnow?Look at Niels Bohr as
a good example.Z: Of course logic can be misapplied. That
doesn't mean it should simply beignored. Syntax and semantics
must always work hand-in-hand.Logic is like geometry - many
different kinds, e.g. quantum logic is not classical logic.I
do not see however how this will lead to new physics today
unlessyoumean thataddition torsion local gauge field and
possibly local gauge fieldsfromALL of theconformal group will
modify EEP which is only based upon the4-parametertranslation
group of the 15 parameter Conformal Group for
Minkowskispace-time.Z: I am not trying to modify EEP. I am
simply proposing an alternativeinterpretation to thosebased on
Einstein equivalence.The alternative interpretation points to a
very differentcorrespondence model betweenstandard GR and a
deeper theory of gravitation. It may hold someimportant clues
to howthe analogical relationship between physical gravitation
andmechanical inertia, and the basisfor weak equivalence
(strict proportionality of gravitational andinertial mass)
will beexplained in the deeper theory -- which will presumably
be some kindof quantum fieldtheory of the gravitational
vacuum.I don't see how without specific toy models of
gedankenexperimentsand/or mathematics.Also I have already
explicitly solved all this with equations!Electricity is more
fundamental than gravity.Quantum electricity is micro-quantum
unstable. Gravity emerges out ofthis instability (inflation in
large scale) along with exoticvacuum dark energy/matter as
emergent MACRO-QUANTUMmore stable smooth curved spacetime
maybe with torsion.That is the more fundamental theory! It
gives Einstein's equations!Gravity is like the Meissner effect
- a MACRO-QUANTUM effect.It's not that there is no general
relativity, there is NO CLASSICAL PHYSICS!Its ALL
MACRO-QUANTUM PHYSICS.Z: You should take my arguments
seriously, Jack, even without the math. I reallythink my
proposals could help you with your program in the longer
term.I see no evidence for that. Your proposal is like asking
a caterpillar to think how it walks.I have all the essential
empirical mysteries accoun for in a semi-quantitative
algebraic fashionfree from any kind of passion at a distance.I
refer to the paper by Utiyama I think in mid-60's also
Kibble'slaterpaper inwhich the general relativity principle is
rederived as a kind of localYang Millsgauge principle in the
curved space-time base space of the fiberbundlewhere thefiber
is the U(1)SU(2)SU(3) or its GUT internal
symmetrygeneralization. Roughly, the gauge forces arein the
principle bundle with unstable globally flat space-time as
acoset space mod the internal symmetrysubgroup of the total
group and the lepto-quarks are the associavector bundle. One
thendoes the same trick again inside the coset space whose
symmetry groupis the Conformal Group.Fine. This was also
Feynman's approach. He also alluded to gaugesymmetries asthe
basis for EEP *by way of formal correspondence* with standard
GR.Yes, that is how everyone in theoretical physics today
thinks about it.Z: OK. This does not require Einstein
equivalence.WRONG! Einstein's equivalence is simply a
consequence of locally gauging the 4 (energy-momentum)
generatorsof the translation subgroup of the 15-parameter
Conformal Group.This gives the locally flat tangent bundle of
GR and you can locally gauge using tetrads which is the formal
contentof EEP! This local gauge compensating field that
replaces the global translation group with the local Diff(4)
group is the distortion field of the worldcrystal lattice
where in the infrared limit >> Planck area, curvature =
disclination topological string defect density and torsion =
dislocation topologicalstring defect density (see Hagen
Kleinert's math here).That is EEP is simply an almost trivial
application of the Local Gauge Principle to a very simple
group -- thetranslation group.Z: So why does Steve Weinberg --
who knows something about QFT -- call himself aheretic when he
discusses such matters in Gravitation and Cosmology?Does this
gauge symmetry of the underlying Yang-Mills field imply
thatgravitational andinertial fields are, physically speaking,
literally one and the samephenomenon?I do not understand your
question unless you give the math.Z: This is not a
mathematical question. The math is neutral to the question.I
do not understand what you mean by an inertial field?I mean by
inertial field the symmetric torsion-free Levi-Civita parallel
transport connection field = inertial field forThe g-force is
the inertial force.In the LIF the g-force is made to be zero
to a good approximation for all Earth and NASA near space
physics.This means that locally the tensor + non tensor part
of the connection cancel each other.The tidal curvature
differential is still there, but small in Earth and near space
physics.Connection = Tensor + non-TensorZ: OK.Real gravity is
from tidal gradients of Tensor part of ConnectionZ: Permanent
gravitational fields are mathematically described by the
tensorcomponent of the connection field.So where's the beef? I
see no there there.Connection = 0 in any LIF.Z: *Net combined
gravitational-inertial* connection = 0 at some point in any
LIF.g-FORCE = ConnectionZ: But Jack, mathematically speaking,
we get a non-vanishing connection fieldeven in a purely
classical 2- or 3-space in *any* non-linear coordinate
system.Formally yes. But that formalism will not describe a
real physical situation unlessother conditions are met like a
Tuv source for example.Given any atlas of overlapping local
coordinate patches one must computeGuv from them. If there is
no suitable Tuv there that obeysGuv(Geometry) = -(String
Tension)^-1 Tuv(Source Stress-Energy Density)OrMarble Geometry
LOCAL Stress-Energy Density + Wood Source LOCAL Stress-Energy
Density = 0that atlas in not physically realized nor is any
Diff(4) equivalent bundle of local gauge transformations of
it.Remember this is manifold theory with overlapping patches
sewn together by Diff(4) transformations at fixedevents P.
Part of your problem is that you think globally not locally.
Euclid's plane geometry and globalCartesian geometry breakdown
in the micro-quantum --> MACRO QUANTUM vacuum instabilityphase
transition in which total phase space volume collapses to
allow emergence of new ordersetting the direction of the
irreversible arrow of time in the early post-inflationary
universe.Z: We get metric distortions in purely classical
(i.e. Newtonian) accelera frames,and not just under Lorentz
transformations in Minkowski space.That doesn't necessarily
mean there is a *physical effect*.So this is about *physical
interpretation* of the math. The math is the same orat least
very similar in either case. Just because you have a
*mathematical* connection field doesn't mean you havea
*physical field*. You may have a *purely* kinematical field.
Sorry if I'm re-stating the painfully obvious.You are not
asking the right question.Given any atlas you compute Tuv and
then look to see if that Tuv is there.If it's not then that
atlas is not physically realized.I mean an equivalence class
of atlases mod Diff(4) for zero torsion case.I also mean a
Diff(4) equivalence class of Tuv representations of the same
actual source distribution./zpfguv also acts like the Tuv
source if there is also exotic vacuumrandom virtual zero point
dark energy/matter.Note that MHD fields in say pulsars are
MACRO-QUANTUM coherentnonrandom states of virtual photons not
same as the random zero pointvirtual quanta of the exotic
vacuum phases of anti-gravitating dark energyand gravitating
dark matter.If your atlas has a curvature tensor field over
the patches, it is not a kinematical field.If the
corresponding Tuv is not there that atlas does notdescribe the
actual curved spacetime region of interest.You have a space of
solutions, or in this case a space ofatlases covering a
manifold with a metric. Not every point in thatspace of
atlases is the correct solution for that metricmanifold.In
LIFTensor g-force + non-Tensor g-force = 0Z:P OK.So that's
compensation at g-force level.Z: OK -- if it is interpre that
way.That was not Einstein's interpretation.Sure it was! The
painter falling off the ladder felt no weight.This was
Einstein's motivation in his own words.Tidal force is the
derivative of g-Force!i.e. f(P) = 0 but df(P)/dx^u =/= 0 in
LIFZ: OK.If not, then such theories are inconsistent with
Einstein equivalence-- although perhapsnot with the so-called
EEP in its most agnostic formulation.You lost me.Z: I guess
so.Any quantum field theory of gravitation -- or any other
fundamental theory ofgravitation -- that entails locally
measurable physical distinctions between theinertial and
permanent gravitation fields is *inconsistent with
Einsteinequivalence*. That should be completely clear at this
point.First of all there is no local micro-quantum field
theory of gravitation that isconsistent and useful because
that is a category error asking an ill-posedquestion of
Nature. No one claims a locally measurable physical
distinctions between theinertial and permanent gravitation
fields that is your Red Herring.You can ask. Is there a
curvature tensor at P that does not vanish?Am I in an LIF at P
or in a LNIF at P?The curvature tensor not vanishing is so in
any local frame LIF or LNIF.Those are the only meaningful
questions.You got any others? Apart from issues of torsion and
exotic micro-quantum zero point vacuum effects?Random /zpf is
to smooth guv as normal fluid is to superfluid./zpf = 0
gravitationally neutral non-exotic vacuum is 100%
superfluid.Omega(/zpf = 0) seems to be not bigger than 0.05
where total Omega = 1.wasEinstein's. It isinconsistent with
Eddington's as sta in Mathematical Theory ofRelativity.MTW
seem to use both, which strikes me as an example of
Bohriancomplementarity-- AKA speaking with forked tongue -- in
the context of spacetimephysics.That is a very bold hypothesis,
as Feynman poin out on manyoccasions.But it is at the core of
Einstein's concept of generalrelativity,as Einsteinhimself
consistently emphasized (specifically to von Laue).Of course,
this was later formula as a local cancellation of
thegravitationalforce field in a free-fall frame, together
with our ability torendertidal effectsnegligible by going to a
sufficiently small neighborhood. Thismakesit lookon the surface
like the Einstein principle is still valid --
hencethemisleadingacronym EEP.Forget what Einstein allegedly
thought in his early days.Jack, this was integral and
indispensable to Einstein's idea ofgeneral relativity.So, if
you forget about this -- and also Pauli's entire
1921exposition of GR --you then have general relativity
*without actual generalrelativity*, as I said earlyon.I mean,
this is a historical point about the initial
immatureconception of relativity that gotfurther refined into
the modern ideas launched in MTW.Z: No, it means that there is
in fact no such thing as generalrelativity. That isa very
profound shift of interpretation IMO.All we really have now is
*general covariance* and a tensorrepresentation of
thegravitational field, and EEP for empirical correspondence
with SR.That's good enough.Z: OK. So you actually seem to
agree with me.That was not obvious a year or so ago.Even
Einstein eventually admit that general covariance in and
ofitself is emptyof physical content (see Autobiographical
Notes).Common knowledge.Z: But the implications of this in
conjunction with certain other observationsare most definitely
not common knowledge. To the contrary, they are considered to
be heresy.Examples?Looks like I was right.But if you are
willing to drop this and hew to a consistentnon-Einsteinian
interpretationof EEP, then fine.But then, the MTW argument
against localized vacuum gravitationalstress-energyfalls apart
(though of course that does not in itself mean there areno
other valid objections).The only way you can attack MTE there
is to say that their initialassumption that the
localizedvacuum tensor is quadratic in the connection is a
Straw Man, i.e. notcomplete.Z: Here you are ignoring, or not
understanding, my proposal for acanonical localdecomposition
of the connection field, which I contend avoids allthese
difficultieseven within the existing GR formalism with its
unified abstract metric.like I did above.it goes.You don't
seem to remember, but I gave you as much as this more than a
yearago in an e-mailYou are right I do not remember that.That
there isanother form based only on the nonlinear products of
the curvaturetensor but that wouldchange the field equation
from (neglecting zero point /zpfguv)Guv = -(8piG/c^4)TuvtoGuv
+ Nonlinear products of 4th rank curvature tensor =
-(8piG/c^4)TuvThe extra terms are a power series in Lp^2 =
hG/c^3they may be there, but are too small on the
macro-scale.Is that what you mean?Z: I don't think so. I am
talking about a local decomposition of thestandard
Christoffelconnection.Like I showed above - standard stuff. So
what?Z: So what? So what?THERE IS NO SUCH THING AS GENERAL
RELATIVITY.I cannot understand really what you mean until I
see formal statementsin math that pin down the ambiguities of
ordinary language.OK. Of course this has to be done.I think I
did it above. This should not take years to do.Z: That much I
did years ago. The mathematical development goes further than
that.Your point is elusive and ineffable. You seem to be stuck
on some early informal remarks that Einstein may have made as
scaffolding. Now hisstatue is complete, the Ceiling of the
Sistine Chapel is there with the scaffolding removed. You have
confused the temporary scaffold with thepermanent work of
art!Z: I gave you a formal expression some time ago but you
apparentlyignored it.I forgot it.Z: I guess.I recrea it
above.Z: See above.This is principally a matter of proper
interpretation of the metricgradients -- even inthe abstract
context of pure differential geometry.which breaks down when
scale ~ Lp* and which emerges from local giantvacuum coherent
wave.Z: On the physical level, it is tied to a close analysis
of exactly whatit means to mathematicallymeld physical time
into a Riemannian spacetime manifold, and preciselyhow to
properlyinterpret the local transformational properties of the
connectionfield on a curved (i.e.deformed) Riemannian
manifoldYou lost me without the math.That *is* math. You mean
with out the symbolic expression?I could call the Riemannian
manifold M and the the coordinate transformationsT and the
stretchless deformation (of M) D if it helps.ThenT(M) -->
g_uv, w | IandD(M) --> g_uv, w | Gandg_uv, w| Einstein = g_uv,
w | I + g_uv, w | Gand then, locally,Is that what you mean?I
don't know. Your notation is not immediately obvious to me.I
would writeguv(locally curved) = nuv(globally flat) + (world
crystal strain tensor)uvDiff(4) is then local Goldstone phase
transformations.This is NOT perturbative linear approximation,
this is non-perturbativeThe coupling strength of second term on
RHS is alpha' = (string tension)^-1infinite string tension
means NO GRAVITY.Z: Einstein may simply have gone wrong there
-- a victim of his ownwishful thinking --regardless of his
impressive heuristic success in arriving at the GRfield
equations.I crave heuristic success!Music from Gounod's
Fausthttp://www.classicalarchives.com/m/023/act_1_du.midAs you
should, since you are working in the context of
discovery.Z.ZThis is why I feel uneasy about Bohr's reliance
on ordinary languageinthe quantum reality problem.This is a
complex issue that I don't really have the time to get
intohere. But I do agree thatBohr's insistence on the
irreducible use of ordinary language andcommon sense
conceptsin describing the results of experiments is very
dubious. At the sametime I don't think thismeans that the only
satisfactory language for discussion and analysisof the
problems ofQM is formal mathematics. I would say that this
approach simply avoidsthe entire problemof the proper
interpretation of the QM formalism.I believe this was Bohm's
view.Which is what I originally argued: the bare EEP does not
establishthat thegravitational energy density must vanish at a
point, or that itscomponents dependupon the frame of reference
in a non-covariant manner. It onlyrequires that thenet
measurable combined gravitational and inertial forces vanish
at apoint in somecoordinate system.To get the true
gravitational stress-energy, we simply ignore theinertial
contribution-- that's all.Eddington himself explicitly states
that in GR the inertial field hasa stress-energythat is
frame-dependent and represen by a non-tensor. But this
isa*fictitious*stress-energy, unless you believe in Einstein
equivalence *asoriginally sta* --which it now appears that you
don't.He wasperhaps a bit confused whilst the theory was
immature.The point is that it's coherent conceptually today in
the form MTWhave.I cannot agree. See above. This is a *real
interpretive dichotomy* atthe most fundamentallevel.Now you
could say somethinglike this about Schroedinger, who
reallythought he had awave theory in the classical sense --
but later had to give this upfor a number of reasons.Einstein
never explicitly abandoned equivalence and true
generalrelativity. Not even inhis Autobiographical Notes,
which is just about his last word onthesubject.You cannot
render tidal effects negligible in the approach to
asingularity until effective Planck scale is reached.That
scale may not be 10^19 Gev after all. It may be 1 Gev?That may
well be the case, but this is not part of canonical GR. Nodoubt
a deepertheory, if it ever finally comes, will answer such
questions.Ed Witten thinks that's M-theory. It's now on NOVA!
I am beginning toread Witten. I am getting a little
uneasyabout all this duality stuff in string theory because it
is beginningto remind me of mydebate with Hal Puthoff on his
use of isotropic r vs. curvature rin the SSS model.This debate
is described in my Dec 2002 book Space-Time and
BeyondII.M-theory may be the final proof of the bankruptcy and
heuristicexhaustion of theformal-empirical methods of modern
physics. On the other hand, underthe rightinterpretation, it
might open up new vistas. We'll see.No argument from me on
that one! What a double standard eh? -- abouthype in science I
mean.People are always telling me to connect my theories with
experimentsand observations. Wellthat's what I have done more
so that anything on NOVA's ElegantUniverse although I
doconnect my math with their stringy alpha' and indeed I think
what Ihave done inhttp://qedcorp.com/APS/EmergentGravity.pdf is
good for Witten & Co Ihave connec theirvaporware more closely
to observations of important stuff like thecosmological
constant.But the extravagant resort to these multi-dimensional
hyperspaceslooks verysuspicious to me -- regardless of how
neatly the Standard Model fitsinto theformalism.Pythagorism
run amuck?My debate with Puthoff was that he was
misinterpreting the physics notrealizingthat the expansion of
space inside a critical radius was an illusionofnot
realizingwhat the coordinate patch was in the curved
differential manifold.OK.MattVissermakes this clear in
Lorentzian Wormholes in contest of Kruskalcoordinates.There is
a kind of duality there withr' = r*^2/rbut r' and r are both
OUTSIDE the horizon. Puthoff and Ibison makethe conceptual
error that r' is INSIDE the horizon!Couldn't this simply be
the result of a mathematical ambiguity?No, good books like
Matt Visser's Lorentzian Wormholes are good onthis.I don't
think Puthoff and Ibison ever really read that book or
ponderedits meaningif they did.In any case, Puthoff is not
claiming that his PV theory is any kind offundamentaltheory of
gravitation -- just a highly computable working model thatmight
beat least pragmatically justified by some future deeper
theory.Not so, I think he claims he has a viable alternative
to Einstein.I honestly don't understand why you think you are
in directcompetition with Hal.Because he got all the money
people in the fringe BPP people to thinkhe is on theright path
- not that there is a lot of money there. But because of
hispast top securityclearances he has influence and access
inside USG Black Ops and in factthere is afactional fight
inside USG right now about all this - more than that Icannot
say. So theissue is one of whistle blowing on wasting tax
payer money on thingsthat have nochance of working whilst
ignoring things that do.I am wondering if Witten, Greene & Co
are making a similar conceptualerror in that really they
ignore the physics INSIDELp*^2 which G.E. Volovik in Universe
in a Helium Drop says isessential for solving the smallness of
the CosmologicalConstant problem. It's like in condensed matter
physics not goinginside the unit cell of the crystal lattice.
If you do that youwill get a zero point energy for phonons
that is much too large! Thisis Volovik's insight.String theory
is like 2D field theory because the string sweeps out aworld
sheet in ordinary space-time rather than the 1D world line.
Thisis a great thing because it essentially does away with the
infinitiesCosmological Constant problem, which worries Witten
and rightly so.problems.NO! String theory give bad prediction
/zpg = 10^66 cm^-2 whenexperiment says/zpf = 10^-56 cm^-2
because Witten's strings are only 10^-33 cm asBrian
Greenerepeats many times in the NOVA vaporware.String theory
is no better than local field theory on this issue!Witten
understands that!They lack my giant vacuum coherent
field!http://qedcorp.com/APS/EmergentGravity.docAfter all -
you get an infinite mass density and infinite electricalforces
insidegravitationalforce is infinite at zero separation of two
mass points.The classical electron was a hopeless conundrum
even in Lorentz'stheory.THIS IS NOT THE PROBLEM.There are
analogies of string theory with lattice spin Ising modelswhich
have a duality relating a T theory to a 1/T theory in a
duallattice.Witten introduces the famous alpha' as basically a
new stringfundamental constant. He does not seem to connect it
to the Salamgravity I worked on in 1973.Yet more knobs and
dials.Is any of this really physics? Can this kind of theory
really be saidto predictanything? Other than in the *ad hoc*
formal manner of abstractmathematicaldescription?I'll see if I
can comment on the stuff below after I have looked
atWitten.Using his h = c = 1 units. Witten's alpha' is
essentially aneffectivefield Planck area. If we believe
Susskind's world hologram we havealpha' = Lp*^2 =
Lp^4/3L^1/3at scale LIf we choose L = c/Ho in a kind of Mach
Principle of large scaleinfluence on small scaleHo = HubbleLp*
~ 1 fermiThe problem with this is the time dependence in H(t)
where Ho is thepresent epoch value.Note, in Witten's
conventions, energy is 1/LengthThereforealpha' is the
universal Regge slope, which for hadronic resonance hasalpha'
~ (1 fermi)^2Quantized spin of resonance = alpha'(Energy of
resonant peak)^2 +interceptalpha' is reciprocal string tension
since Energy/Length ~ 1/Area intheh = c = 1 convention used by
Witten & Co.In my theory/zpf = (alpha')^-1[(alpha')3/2|Vacuum
Coherence|^2 - 1}Einstein's gravity emerges fromguv =
(Minkowski)uv + (alpha'){Goldstone Phase of
VacuumCoherence)(,u,v)}So this seems to map directly over to
some bi-metric version of GR.No I do not see that at all. The
SUM on RHS is not bimetric anymore than2 = 1 + 1 is in reality
a pair of two 1's with independent parallelexistences.Youhave a
flat Minkowski background with a Lorentz metric, with
inertialfieldsconceived as artifacts of non-linear spacetime
coordinatetransformations,PLUS a *physical metric* describing
gravitational phenomena that isdirectlyrela to your BEC
Goldstone phase.Am I right?Yes, but it is only the sum that is
physical. I do not see twocoexisting independent metrics apart
fromlocally flat tangent bundle idea of standard GR with
tetrad map betweenLIF geodesic fiber and non-geodesic LNIF
base local framesIf so, then this is NOT Einstein's gravity.
It is *Eddington's*gravity.This was also Feynman's general
approach in his Lectures onGravitation.It is BOTH! I haveGuv +
/zpfguv = alpha'TuvAs I said, I think your task is going to be
a lot easier, as far ascorrespondencewith GR is concerned,
under my proposed alternative interpretation ofthe EEP.You
cannot simply sweep all these problems under the
carpetemergence IMO.anholonomic torsion field potential =
(alpha'){Goldstone Phase ofVacuum Coherence)[,u,v]The O(2)
internal symmetry of this complex numbered scalar
VacuumCoherence local field implies 1D string topological
defects in 4Dspacetime asa rigorous theorem in the text
books.OK.More generalized models of O(N) hypercomplex numbered
MACRO-QUANTUMVACUUM More is different (P.W. Anderson) coherence
order parameterswill give the brane world topological defect
generalizations ofthesestrings that should also have torsion
on the microscale.OK.The most from the least.BTW, how does
your theory explain mechanical inertia?Exactly as in Wheeler's
book Geometrodynamics with Mass withoutmass except now I have
G* = 10^40 G(Newton) at the fermi scale.OK. I'll have a look.I
star working on Wheeler's book Geometrodynamics in 1966.
Wheelerspoke at UCSD frequentlythen.The characteristic rest
mass of the spatially extended lepto quark ofCharge without
charge rotating with Spin without spin ism = e^2/zpf^1/2 =
e^2/Lp* = e^2(alpha')^-1/2because Vacuum Coherence -> inside
the Type II superconductorquantized flux vortex vibrating
string exotic vacuum dark mattercore with ends pinned to the
3Dim brane world we are stuck on like EdAbbott's Flatlanders
if the basic Witten & Co ideas are on righttrack.So in your
theory, inertial effects result from the physicalinteraction
of matter,conceived as branes, with the quantum vacuum,
conceived as a virtualBEC?BRANES emerge FROM virtual
BEC.strings emerge from virtual O(2) BEC which is only low
energylong-range limiting case of shorter-scale O(N) BEC in
some kind ofO(N') or G(Mystery) group hypercomplex number
generalization ofEinstein's real space-time continuum.Say bye
bye to Mach's principle.Not ifLp*^2 = Lp^4/3(c/Ho)^2/3which
has a problem however.These 1 Dim strings vibrate in the extra
boson space dimensions andtheextra fermion supersymmetry matrix
space-dimensions forming theCalabi-Yau space generalization of
the Kaluza-Klein hyperspace of the1920's.OK.But it only takes
one physical effect that is not locallynegligiblein this
senseto blow out Einstein equivalence.Einstein equivalence is
only a classical weak field approximation oryour
correspondence principle in the informal sense for the
zerotorsion limit.I'm not sure exactly what you mean here.
Einstein thought it heldwithoutrestriction.Looks like you
don't agree.I am not sure without doing a careful look. In any
case, of coursethere have beensubtle modifications or
refinements of Einstein's early thinking.Theseare
smallcorrections so to speak. Only minor clarifications of his
greatinsights.This is simply not the case, Jack. What I am
talking about is aprofound counter-revolutionary paradigmatic
shift back to some kind of an aether,conceived as
anon-material yet physical medium of propagation across the
physicalvacuum.So it is precisely Einstein's original insight
-- that bothgravitation and inertiacould be jointly reduced to
a distortion of an abstract metric in anabsolute void,without
reference to any physical medium of propagation -- that is
inquestion here.Look, Einstein was a wonderful man and a
creative genius, but businessis business.Music from 'The God
Father plays in background.We need to move forward. Let's not
be too sentimental about this.There isnothing revolutionary in
the distinctions you propose here that I amable tounderstand at
least.Then perhaps -- and I don't say this lightly -- you don't
reallyunderstand Einstein.You don't seem to realize that you
have in fact abandoned his theoryof generalrelativity.Quite
rightly, IMO, since THERE IS NO GENERAL RELATIVITY.There you
go again sounding like Brian Greene on NOVA! That is a lot
more than a mere detail of history. That can hardly
bedescribed as a slightadjustment or refinement to Einstein's
original insight.You lost me. I am only a humble post-Quantum
non-mechanic.It is in fact a wholesale *abandonment* of
Einstein's originalinsight, which is nowdemo to the status of
a mere heuristic tool -- mere scaffolding tobe discarded
oncethe cathedral is built, in favor of more solid and durable
buttressingthat is more inkeeping with the final architecture.I
am not sure what happens when there is torsion-spincoupling. It
can break down for a variety of reasons.Exactly -- you're not
sure. The fact is it's anybody's guess and isultimatelyan
empirical question.Not quite - there is a lot of work in the
published literature - it'sonly my personal lack of knowledge
here I allude to.I'm not sure we're on the same page here. But
I *am* talking aboutspinningEinstein equivalence-- within the
framework of canonical GR.space-time. It's complica
butstandard in MTW et-al.If ALSO there is TORSION then
additional coupling of spinning testAkimov in Moscow says that
is a big effect with weapons applications.One guy here who came
to ISSO 2000 also said that and he was working onareport. Most
researcher pooh pooh that. I do not knowfirst-hand.Yet
Einstein thought he was completely sure, and said so. That's
whyFeynmanmade fun of him in Lectures on Gravitation.Z.Again
this is very minor compared to what is happening now and what
Iam talking about above in relation toWitten & Co.See above.
It changes the entire meta-theoretic relationship betweenGR
and thedeeper field theories of gravitation. It profoundly
changes ourunderstanding of thevacuum and its relationship to
gravitational and inertial phenomena.Show me how in detail.I
do not have time to absorb all your points below.
Later.OK.===
===
Subject:
: Need advice on letters of recommendation
I am a senior and I will be needing to fill out applications
for graduateschool. My problem is that I am on the quarter
system and the quarter is aboutto end. We will not resume
until the beginning of January. I'm pretty shy andhave not
asked any of my professors for letters. I'm just not sure how
to askthem and now that I basically have no time left, I was
just wondering how Ishould go up to them and ask. I have had
the same professors for a lot ofclasses in some cases but I'm
not sure whether or not lower level classes wouldgive a
professor enough to say anything about me. Is there any
standard way togo about this? Most schools seem to want three
and for me that is a lot.Thanks for any advice.===
===
Subject:
: Re:
Need advice on letters of recommendation> I am a senior and I
will be needing to fill out applications for> graduate school.
My problem is that I am on the quarter system and> the quarter
is about to end.> We will not resume until the beginning of
January. I'm pretty shy> and have not asked any of my
professors for letters. I'm just not sure> how to ask them and
now that I basically have no time left, I was just> wondering
how I should go up to them and ask. I have had the same>
professors for a lot of classes in some cases but I'm not sure
whether> or not lower level classes would give a professor
enough to say> anything about me. Is there any standard way to
go about this?> Most schools seem to want three and for me that
is a lot.> Thanks for any advice.letters of recommendation are
about the most ludicrous mechanism inplace in academia - it's
a shame. institutional admissions exams wouldbe far
superior.anyways, since letters are inevitable, you need to
get them asap fromprofessors who you know will give only
good/excellent reports andremarks. if possible, get the
professors to issue them to you, thatway you do not have to
bother them repealy, plus that will ensurethat all your
materials arrive timely. full contact information andtitle
should remove any doubt about legitimacy...===
===
Subject:
: Re:
Need advice on letters of recommendation> letters of
recommendation are about the most ludicrous mechanism in>
place in academia - I don't know about most ludicrous, but
there are some kinksin the system. We talk about how lazy and
irresponsible studentsare, but the percentage of slacking
professors is also a bithigh. My opinions about the writing of
recommedations:1. When a student (or anyone) asks for a letter,
it is implicit that they are asking Will you write me a
_good_recommendation. And if you agree to write a
recommendation,then you are implicitly agreeing to write a
_good_ one.And by this I mean, you write a letter about
whatever goodqualities the student might have, however few
that mightbe. If that damns him with faint praise, one has at
leastac with honor.2. The writer must write honestly and
should do his homework. Actually look up the grades, and
determine ifthe student really was in the top 5% of students
or the best student I've ever had. Check out any factsfrom the
resume that are referenced in the letter.3. The purpose of the
letter is not to be just another transcriptor another academic
exam score. Something else is wan here.Of course the student
has passed classes and exams. Has thestudent changed since he
entered college as a freshman?The graduate school is going to
admit a pile of students, and several of them are going to
drop out, taking their stipendswith them, which is not an
effient use of their financial aid.They want to know if this
student is a good risk of a TA-ship.This is the heart of the
letter, I think. And everyone has toget at it in their own
way. My style is that I try to thinkof at least one concrete
anecdote that demonstrates the student's perserverence,
longevity, stability and adulthood.(Which shows that I know
first hand what I'm talking about.)Bart===
===
Subject:
: Re: Need
advice on letters of recommendationThanks for the replies. I
talked to one of my professors yesterday that I havethe most
experience with. He sugges the other 2 professors that
shouldwrite my recommendations. He told me not to ask one
Professor that I hadpretty well with, so he must have known
something I don't.===
===
Subject:
: Re: Need advice on letters of
recommendation>letters of recommendation are about the most
ludicrous mechanism in>place in academia - it's a shame.
institutional admissions exams would>be far superior.It's
doubtful that, even if examinations were superior, they'd
be*far* superior. And saying something's true doesn't make it
so.===
===
Subject:
: Re: Need advice on letters of
recommendation>letters of recommendation are about the most
ludicrous mechanism in>place in academia - it's a shame.
institutional admissions exams would>be far superior.Letters
of recommendation may well be ludicrous. I have no reasonto
believe, and much reason to disbelieve, that institutional
admissions exams would be any better at deciding who would bea
good graduate student. Go into detail on how you'd make suchan
exam, if you don't mind.I, by the way, am sorely lacking in
the experience of administeringadmissions evaluations of any
sort whatever (but I do have all toomuch experience at setting
exams, hence my dubiety that they wouldbe any damned good for
this purpose). While you're describing your proposed exam, you
might just run over your own experience in administering
admissions.Lee Rudolph===
===
Subject:
: Re: Need advice on letters
of recommendation> letters of recommendation are about the
most ludicrous mechanism in> place in academia - it's a shame.
institutional admissions exams> would be far superior.> Letters
of recommendation may well be ludicrous. I have no reason> to
believe, and much reason to disbelieve, that institutional>
admissions exams would be any better at deciding who would be>
a good graduate student.perhaps you do not believe that a solid
undergraduatepreparation is key component in graduate studies
success.is that it?> Go into detail on how you'd make such an
exam, if you don't mind.i do not need to go into any detail to
ascertain that aresounding reason for the lack of institutional
exams in theuni states is correla with the economic and
logisticsrequired to put such systems in place.> I, by the
way, am sorely lacking in the experience of administering>
admissions evaluations of any sort whatever (but I do have all
too> much experience at setting exams, hence my dubiety that
they would> be any damned good for this purpose). While you're
describing your > proposed exam, you might just run over your
own experience in > administering admissions.>ps posting
through mathforum.org bites===
===
Subject:
: Re: Need advice on
letters of recommendation permission for an emailed response.
The GNU project will probably not be Posix conformant, Tom
said noncommittally> I, by the way, am sorely lacking in the
experience of administering> admissions evaluations of any
sort whatever (but I do have all too> much experience at
setting exams, hence my dubiety that they would> be any damned
good for this purpose). While you're describing your > proposed
exam, you might just run over your own experience in >
administering admissions.One year when I worked there I was an
admissions reader for MIT. Weread undergraduate admissions. The
system as it exis then was toocomplex to go into now, but the
recommendations were in factimportant. One question MIT asked
was how has this student dealtwith failure in the past, and
the most illuminating answer I got fromthat was a teacher who
said Johnny has never experienced significantfailure in his
academic work--apparently thinking this was
good.Thomas===
===
Subject:
: Re: Need advice on letters of
recommendation>I am a senior and I will be needing to fill out
applications for>graduate school. My problem is that I am on
the quarter system and>the quarter is about to end. We will
not resume until the beginning>of January. I'm pretty shy and
have not asked any of my professors>for letters.Everyone else
has given you fine ideas about letters. But I'd like topoint
out something about this comment. It's OK to be shy, and many
math majors are, but college and grad schoolare really good
times to grow past this. We all know the jokes abouthow you
can distinguish the extrover mathematician (looks at YOURshoes
when talking to you) but the reality is that mathematics
isreally a group activity, and in order to thrive in
mathematics you needto learn to step forward and talk to
people you don't know. You'regoing to have to ask the faculty
a lot of questions in grad school, andI don't just mean about
your classes: you're going to have to get in thehabit of
knocking on a door and saying, Prof. X, I'm one of the
gradstudents here and I've been reading this book about [your
specialty] andit makes some references to results in [X's
specialty]; could I meet withyou some time and ask you a few
questions about this?. I still do thisquite often (except I
don't call myself a grad student). Get used to it!Practice it
and you'll have an icebreaker for the departmental tea.The
nice part about this is that after years of teaching
calculus,mathematicians are usually starved for an opportunity
to talk aboutsomething in their own area, or some bit of
mathematics (or whatever!)that they feel qualified to talk
about. If you walk in with someclear questions and the ability
to show that you've been making areasonable effort to
understand things on your own, your only realchallenge will be
to get Prof X to shut up.(Just take a look at this newsgroup;
some of us seem to suffer fromlogorrhea!)dave===
===
Subject:
: Re:
Need advice on letters of recommendation> Is there any
standard way to go about this? I've had students walk into my
office with the letter written for me and asking me to sign
it. Let's callthat not standard (and not acceptable.)As others
have said, professors do this all the time, andoften for
students they barely know, except as a list ofnumbers in a
grade book. If you can help make the connectiona bit, that is
a good thing. Some students walk in andask for recommendations
and then walk out. Later I getsome addresses. OK, I can deal
with it, and they get thebest letter I can put together.But
other students hand me a little resume of their careers,which
helps me immensely. It reminds me that they were inthat
production of Othello I enjoyed so much and that theyare
always lying next to me when I'm giving blood at the blood
drive and it helps me find their records amongst allmy
spreadsheets when they've lis the classes they took.Oh, yeah,
she DID win that freshman math contest.It helps keep the
letter from looking generic.Plus the fact that the student had
it together enough tohand me the resume says something in
itself. Grad school also takes perserverence, and the letter
ought to addressthe student's character if it can. So if I can
say somethingpositive that indicates that the student not only
has somemathematical talent, but also exhibits the academic
maturityfor sticking things out, then I will. The brief
inteview you get when asking for the letter is a quick, but
importantchance to demonstrate those characteristics.So, my
advice is that while you're asking for the recommendation,you
connect personally with the professor a bit. It's
anopportunity to ask him for advice on the choice of grad
schoolsor field of study or tips for getting through and it
willhelp him write the letter if he knows you a bit
better.Bart===
===
Subject:
: Re: Need advice on letters of
recommendation permission for an emailed
response.X-Zippy-Says: I selec E5... but I didn't hear ``Sam
the Sham and the Pharaohs''!> I've had students walk into my
office with the letter > written for me and asking me to sign
it. Let's call> that not standard (and not acceptable.)Eek!
The thing to do of course is to agree to write a letter
anyway,insist that it be confidential, and then mention the
details of thetransaction in the letter.Thomas===
===
Subject:
: Re:
Need advice on letters of recommendation>> I've had students
walk into my office with the letter >> written for me and
asking me to sign it. Let's call>> that not standard (and not
acceptable.)> Eek! The thing to do of course is to agree to
write a letter anyway,> insist that it be confidential, and
then mention the details of the> transaction in the
letter.Well, I should have said student, not students. Andin
this case, I trea it as a special situation. First,it was a
(very) foreign student. Second, the letter wasonly for a
summer job, not an internship or grad school.Third, the
situation came up suddenly, where few opportunitiesare
available for (very) foreign students and it was a rush job
(due that day) and the student thought she was doing me a
favor AND was quite polite about it. Fourth, the letterwas
accurate and well-written, thanks to a collaborationof some
friends. So I had no trouble signing my name toit. I would
have written a more glowing letter for her,since she was a
stellar student.Then we had a discussion about proper
procedure. No harm done. (She _was_ jerking her boyfriend
around at the time,so if that character trait manifests itself
again in a moreserious situation, then she _will_ get nuked.
But it won'tbe because someone wasn't frank with her at least
once.)Bart===
===
Subject:
: Re: Need advice on letters of
recommendation permission for an emailed
response.X-Zippy-Says: Yow! Am I in Milwaukee?> Well, I should
have said student, not students. And> in this case, I trea it
as a special situation. First,> it was a (very) foreign
student. Second, the letter was> only for a summer job, not an
internship or grad school.> Third, the situation came up
suddenly, where few opportunities> are available for (very)
foreign students and it was a > rush job (due that day) and
the student thought she was doing > me a favor AND was quite
polite about it. Fourth, the letter> was accurate and
well-written, thanks to a collaboration> of some friends. So I
had no trouble signing my name to> it. I would have written a
more glowing letter for her,> since she was a stellar
student.Ah yes, that is a particular singular situation
indeed.===
===
Subject:
: Re: Need advice on letters of
recommendation Adjunct Assistant Professor at the University
of Montana.>> I've had students walk into my office with the
letter >> written for me and asking me to sign it. Let's
call>> that not standard (and not acceptable.)>Eek! The thing
to do of course is to agree to write a letter anyway,>insist
that it be confidential, and then mention the details of
the>transaction in the letter.I disagree. There's a chance, at
least, that the individual issincerely trying to be helpful. I
had, on occasion, had professors Iasked for letters of
recommendation tell me to just write it and I'llsign it.
Needless to say, I decided to go ask other people forletters
when that was the case, and I frown thoroughly on
thatpractice. But perhaps the student had encountered a
similarlyill-disposed professor, and thought he would save
time by bringing aletter ready? Absent other experience, he
may well think that it->is<- standard.I would certainly tell
the student that the practice is not standard,and should not
be acceptable. I would also suggest that any professorof his
who has sugges this method to him is NOT a good
referencewriter, and he should seek to replace him. And I will
usually try torefuse to write letters of recommendation for
admission to graduateschool for people that I will not
recommend (i.e., people I would notsay good things in
balance). That does not mean that my letters do notmention any
problems, just that if I have a generally bad opinion
ofsomeone, then I recommend to the student to find someone
else to writeit. (It's happened once or twice; once, I asked a
student who had notdone very well in my class why he was asking
me. He explained that hewas seeking to be admit to a specific,
applied math program, and hewas hoping that by getting a
letter from an algebraist, he would showhis breadth. I sugges
he just get letters from applied math and usehis transcripts
to show breadth, and he acquiesced. I usually try tosuggest
other courses of actions rather than flatly refuse, and
oncethe student insis and I said that I would not be able to
writeanything glowing, and would point out that on the balance
his work had===
===
Subject:
: Re: Need advice on letters of
recommendation> I've had students walk into my office with
the letter >> written for me and asking me to sign it. Let's
call>> that not standard (and not acceptable.)>>Eek! The thing
to do of course is to agree to write a letter anyway,>insist
that it be confidential, and then mention the details of
the>transaction in the letter.> I disagree. There's a chance,
at least, that the individual is> sincerely trying to be
helpful. I had, on occasion, had professors I> asked for
letters of recommendation tell me to just write it and I'll>
sign it. Needless to say, I decided to go ask other people
for> letters when that was the case, and I frown thoroughly on
that> practice. But perhaps the student had encountered a
similarly> ill-disposed professor, and thought he would save
time by bringing a> letter ready? Absent other experience, he
may well think that it> ->is<- standard.For the most part I
disagree with this, but there are certainsituations as you
mention where I think this is acceptable. One ofthem is
happening to me at present. I am undergraduate, graduating
inspring. I am applying for a program which is completely
unrela tomathematics, yet two recommendations have to come
from professors ofmine. So what's a person to do? I have to
get letters ofrecommendation from math professors, but none of
them really can saymuch about the skills which should be
mentioned in the letter (inparticular, ability to teach,
leadership qualities, speaking ability,ability to work under
pressure, work ethic, not cracking under thepressure of living
in a foreign country for the first time). So Iapproached one
professor with whom I have had about 3 advancedclasses, and he
thought it would be good to just make stuff up (orrather,
stretch the truth a little bit and take qualities he
knowsabout me and extend them to qualities he doesn't), but
anotherprofessor told me to just write the letter for him and
he'd rewrite itin his own voice. His complaint, perfectly
valid, is that he had noidea what to say about me for these
kinds of qualities. It turned outthe class I had had him for
was an upper division class taught usingthe Moore method, so I
mentioned lots of stuff in my letter about howI was very eager
to go the board all the time, was able to make
keenobservations and was one of the leaders of the classroom,
goodspeaker when it was my turn at the board, etc. All very
honest, buthe simply wasn't used to writing letters of this
sort.===
===
Subject:
: Re: Need advice on letters of recommendation>
I disagree. There's a chance, at least, that the individual is>
sincerely trying to be helpful. I had, on occasion, had
professors I> asked for letters of recommendation tell me to
just write it and I'll> sign it. Needless to say, I decided to
go ask other people for> letters when that was the case, and I
frown thoroughly on that> practice. But perhaps the student
had encountered a similarly> ill-disposed professor, and
thought he would save time by bringing a> letter ready? Absent
other experience, he may well think that it> ->is<- standard.If
the student did well in your class but you don't know very
muchabout her you can always have a chat with her to get to
know herbetter. And you can ask the student for a copy of her
resume, too. If you really can't say much specific about the
student, even if shedid well in your class, perhaps you
shouldn't write the reference.===
===
Subject:
: Re: Need advice on
letters of recommendation>>I've had students walk into
my office with the letter >written for me and asking me to sign
it. Let's call>that not standard (and not acceptable.)>Eek!
The thing to do of course is to agree to write a letter
anyway,>>insist that it be confidential, and then mention the
details of the>>transaction in the letter.>>I disagree.
There's a chance, at least, that the individual is>sincerely
trying to be helpful. I had, on occasion, had professors
I>asked for letters of recommendation tell me to just write it
and I'll>sign it. Needless to say, I decided to go ask other
people for>letters when that was the case, and I frown
thoroughly on that>practice. But perhaps the student had
encountered a similarly>ill-disposed professor, and thought he
would save time by bringing a>letter ready? Absent other
experience, he may well think that it>->is<- standard.>I would
certainly tell the student that the practice is not
standard,>and should not be acceptable. I would also suggest
that any professor>of his who has sugges this method to him is
NOT a good reference>writer, and he should seek to replace
him.Perhaps (well, more than perhaps) I am just lazy, but I
disagree. It sometimes happens that a student has very few
choices amongst referees. This happened to me (as the referee)
once. It was thie middle of fall semester, a first-year grad
student needed a reference for some scholarship fast, and I
had only had him that semester. I looked at the form and what
it asked and I knew that I could not knowledgeably answer the
questions. I sugges he write a recommendation himself and give
it to me. Then I could modify it to what I thought was
appropriate. I would certainly not just sign it carte
blanche.However, in this case, I also sugges as a better
procedure that the student contact professors at his
undergraduate department; this is what he did.-- ===
===
Subject:
:
Re: Need advice on letters of recommendation Adjunct Assistant
Professor at the University of Montana.>I am a senior and I
will be needing to fill out applications for graduate>school.
My problem is that I am on the quarter system and the quarter
is about>to end. We will not resume until the beginning of
January. I'm pretty shy and>have not asked any of my
professors for letters. I'm just not sure how to ask>them and
now that I basically have no time left, I was just wondering
how I>should go up to them and ask. I have had the same
professors for a lot of>classes in some cases but I'm not sure
whether or not lower level classes would>give a professor
enough to say anything about me. Is there any standard way
to>go about this? Most schools seem to want three and for me
that is a lot.Asking for letters is about the most
uncomfortable thing I've had todo, both when applying for grad
school, and now for jobs. The goodnews is that most people are
very nice about it.First: do it, and do it SOON. The letters
have to be in, and youshould give the professors at least a
couple of weeks to do it; as thesemester winds down, a
professor's time supply is short. Ask for theletters this week
or next, and don't delay.Pick upper division classes in which
you did well. They don't all haveto be math classes, but at
least one (preferably two) should be.Go to the professor,
remind him who you are and how he knows you, andsay simply
that you are applying for grad school, and since you didwell
in his/her class, you would very much appreciate it if
he/shecould could write you a letter of recommendation for
yourapplication. Take the forms with you, and be ready with
informationabout deadlines and so on. Probably the best is to
catch them duringtheir regularly scheduled office hours. If
the professor only knows you from lower division courses, it
isprobably not a good bet for a letter, but if you had him for
bothupper and lower divisions, then feel free to go for it.And,
keep this in mind: most professors understand that this is
partof their job. If you did well in the class, they will be
happy tohelp. ===
===
Subject:
: Re: Need advice on letters of
recommendationKatie, Your professors are used to writing
letters, so don't be afraid tosimply go talk to them and ask.
Keep in mind that your professors aremathematicians because
they really like math, so they'll generally bevery happy that
you're going to continue your studies of theirfavorite
subject. The best letters come from professors from whom
youhave either taken an advanced course or from whom you have
takenmultiple courses.Your professors will know what procedure
they like to follow. Mostgrad schools have a separate form that
they ask the professor to fillout, plus they ask for a letter.
So you'll need to give your professorthe forms from the
schools to which you're applying. Your professormay also ask
that you to provide an addressed (and stamped) envelopefor
them to mail the letter to the grad school. OTOH, some
gradschools will have the professor seal his/her letter in an
envelope andgive it back to you to include with your
application. So you'll needto read the instructions for each
application.If you want to make your professors' job a little
easier, and alsohelp them write the strongest possible letter
for you, then you couldgive them a copy of the essay that
you'll be including with your gradschool application. In that
essay you'll be explaining, in one form oranother, why you
want to go to grad school. [It isn't _necessary_ togive this
to your professors, but as someone who has written 100+
suchletters over the past decade or so, I can tell you that I
find it veryhelpful. BTW, the same advice applies when you
graduate with yourPh.D. and are applying for jobs; the more
information you can give toyour letter writers, the
better.]Finally, if you won't have time to get these materials
to yourprofessors before the quarter ends, then you should
definitely go talkto them _now_, and tell them that you'll
mail them the materials inDecember. Then assemble the
necessary material at home and send it tothem.Good luck with
your applications.JoeS===
===
Subject:
: Re: Need advice on letters
of recommendation> I am a senior and I will be needing to fill
out applications for graduate> school. My problem is that I am
on the quarter system and the quarter is > about> to end. We
will not resume until the beginning of January. I'm pretty shy
> and> have not asked any of my professors for letters. I'm
just not sure how to > ask> them and now that I basically have
no time left, I was just wondering how I> should go up to them
and ask. I have had the same professors for a lot of> classes
in some cases but I'm not sure whether or not lower level
classes > would> give a professor enough to say anything about
me. Is there any standard way > to> go about this? Most schools
seem to want three and for me that is a lot.What worked for me
many years ago was to ask the professors of the upper-division
classes where I had done well. I agree it can feel a little
embarrassing to feel like you have to hustle like this, but
the profs are quite accustomed to it, it's part of their jobs.
Just pick the three profs who have seen you do your best work,
and go to them during office hours and simply ask for a
letter. They've done this before and they'll know what to
do.===
===
Subject:
: Re: Need advice on letters of
recommendationThanks. I have another (probably stupid)
question. If you're applying tomultiple schools and they all
want a letter of recommendation, does theprofessor have to
print out a lot and sign them all or can I xerox
them?Thanks===
===
Subject:
: Re: Need advice on letters of
recommendation>Thanks. I have another (probably stupid)
question. If you're applying to>multiple schools and they all
want a letter of recommendation, does the>professor have to
print out a lot and sign them all or can I xerox them?Usually
the department office can handle that; I know that when I was
interviewing for professor positions, our graduate secretary
handled thedistribution of letters of rec. I just asked for
letters, and then toldwhere I wan them to go (I had to label
the envelopes, too).===
===
Subject:
: Re: Need advice on letters of
recommendation permission for an emailed
response.X-Tom-Swiftie: You've put too much peanut butter on
the sandwich, Tom said thickly> Thanks. I have another
(probably stupid) question. If you're applying to> multiple
schools and they all want a letter of recommendation, does
the> professor have to print out a lot and sign them all or
can I xerox them?When I applied to grad school (not in math,
but the procedure isbasically the same regardless), each
school had a separaterecommendation form.I provided each
professor with a labelled envelope, the form for eachschool to
which I was applying. I had filled out the applicantportion of
each form and as much of the recommender portion as madesense,
and then gave each professor their stack.Most of the schools
wan the recommenders to mail theirrecommendations directly to
the school; for those, the labelledenvelope was completely
addressed and stamped. Other schools wanthe recommenders to
give sealed signed-across-the-flap envelopes tothe applicant,
and for that purpose I gave the professor a largeenvelope
addressed to me to put them all in and drop in the mail tome.I
made it my responsibility to do as much as I could to minimize
thehassle on the part of the recommenders, while being careful
topreserve the confidentiality of the recommending process.
Thomas===
===
Subject:
: Re: Need advice on letters of
recommendation> I provided each professor with a labelled
envelope, tYeah, that's about the extent of it. You want to
make their job as easyas possible, but there is tto much
variation. But I write my letters inWord (heresy!) anyway, so
with cut and paste that part is not too muchtrouble. You might
offer to send them the addresses over email so thatthey can
cut/paste that if they write the letters electronically.V.--
email: lastname at cs utk eduhomepage: cs utk edu tilde
lastname===
===
Subject:
: Re: Need advice on letters of
recommendation> Thanks. I have another (probably stupid)
question. If you're applying to> multiple schools and they all
want a letter of recommendation, does the> professor have to
print out a lot and sign them all or can I xerox them?Ask
someone in the math department office, for example the office
manager. As I recall, they didn't give me my letters, in fact
I never saw them. You just tell the prof what schools and the
rest gets handled somehow.===
===
Subject:
: Cantor Enumeration for
ZZ^3Dear Readers,does anybody know an explicit bijective map
IN -> ZZ^3 (for computation in MAGMA)?Your help would be much
apprecia.Guido===
===
Subject:
: Re: Cantor Enumeration for ZZ^3>Dear
Readers,>does anybody know an explicit bijective map IN -> ZZ^3
>(for computation in MAGMA)?> Start withan explicit bijection f
between N and Z (you can code the sequence 0,1,-1,2,-2,3,-3,
... using (-1)^n judiciously)an explicit bijection g between N
and N^2, e.g., n -> (g1(n),g2(n)), where n = 2^(g1(n)) *
(2*g2(n)+1)then n -> (g1(n),g2(n)) ->
(g1(n),g1(g2(n)),g2(g2(n))) ->
(f(g1(n)),f(g1(g2(n))),f(g2(g2(n))))shows how to compose these
into the desired bijection.Note: g1(n)=floor(2log(n)) and
g2(n)=(n*2^(-g1(n))-1)/2KP-- E-MAIL: K.P.Hart@EWI,TUDelft,NL
PAPER: Faculty EWIPHONE: +31-15-2784572 TU DelftFAX:
+31-15-2786178 Postbus 5031URL: http://aw.twi.tudelft.nl/~hart
2600 GA Delft the Netherlands===
===
Subject:
: OPE BookI am looking
for supplemental text to aid in my learning of ODE andPDE next
semester. Currently my undergraduate text is
ElememntaryDifferential Equations with Boundry Value Problems
by Derrick andGrossman. Ideally, I would appreciate an text
which emphsizestheoretical foundation over applied examples.
Any suggestions aregreatly apprecia.Thanks.Owen===
===
Subject:
: Re:
OPE Book> I am looking for supplemental text to aid in my
learning of ODE and> PDE next semester. Currently my
undergraduate text is Elememntary> Differential Equations with
Boundry Value Problems by Derrick and> Grossman. Ideally, I
would appreciate an text which emphsizes> theoretical
foundation over applied examples. Any suggestions are> greatly
apprecia.how about two books?1. _ordinary differential
equations_ by wolfgang waltera well-balanced emphasis on both
theory and applications2. _partial differential equations_ by
fritz johnpleasant development of the theoryor2'. _partial
differential equations: an introduction_ by walter strauss>
Thanks.> Owen===
===
Subject:
: Re: OPE Book>I am looking for
supplemental text to aid in my learning of ODE and>PDE next
semester. Currently my undergraduate text is
Elememntary>Differential Equations with Boundry Value Problems
by Derrick and>Grossman. Ideally, I would appreciate an text
which emphsizes>theoretical foundation over applied examples.
Any suggestions are>greatly apprecia.Wow, that's a refreshing
change: most students these days are lookingfor lots of
examples and no theory.You might look at M. Braun,
Differential Equations and Their Applications, Applied
Mathematical Sciences 15, Springer-Verlag 1975.Or a somewhat
more challenging one: G. Birkhoff and G.-C. Rota, Ordinary
Differential Equations, Ginn 1962.Robert Israel
israel@math.ubc.caDepartment of Mathematics
http://www.math.ubc.ca/~israel University of British Columbia
Vancouver, BC, Canada V6T 1Z2===
===
Subject:
: Re: Question about a
C^infty functionThe police and so forth only exist insofar as
they can demonstratetheir authority. They say they're here to
preserve order, but in factthey'd go absolutely mad if all the
criminals of the world went onstrike for only a month. They'd
be on their knees waiting for acrime. That's the only
existence they have.William S. Burroughs (American writer)
Guardian, 1966huffy===
===
Subject:
: Re: Question about a C^infty
functionThe police and so forth only exist insofar as they can
demonstratetheir authority. They say they're here to preserve
order, but in factthey'd go absolutely mad if all the
criminals of the world went onstrike for only a month. They'd
be on their knees waiting for acrime. That's the only
existence they have.William S. Burroughs (American writer)
Guardian, 1966huffy===
===
Subject:
: Re: Question about a C^infty
functionThe police and so forth only exist insofar as they can
demonstratetheir authority. They say they're here to preserve
order, but in factthey'd go absolutely mad if all the
criminals of the world went onstrike for only a month. They'd
be on their knees waiting for acrime. That's the only
existence they have.William S. Burroughs (American writer)
Guardian, 1966huffy===
===
Subject:
: Re: Question about a C^infty
function> The police and so forth only exist insofar as they
can demonstrate> their authority. They say they're here to
preserve order, but in fact> they'd go absolutely mad if all
the criminals of the world went on> strike for only a month.
They'd be on their knees waiting for a> crime. That's the only
existence they have.> William S. Burroughs (American writer)>
Guardian, 1966(off-topic)When I heard that Bill Burroughs had
died, I immediately askedhow could they tell?.(apologies to
Dorothy Parker)-- ===
===
Subject:
: Re: Banach limit and almost
convergenceThe police and so forth only exist insofar as they
can demonstratetheir authority. They say they're here to
preserve order, but in factthey'd go absolutely mad if all the
criminals of the world went onstrike for only a month. They'd
be on their knees waiting for acrime. That's the only
existence they have.William S. Burroughs (American writer)
Guardian, 1966huffy===
===
Subject:
: Re: Banach limit and almost
convergence>Hi!>It was no several times in Google Groups (by
Ronald Bruck), then>uniqueness of Banach limit of a sequence
is equivalent to almost>convergence of this sequence. Can you
give me a hint where to find a>proof of this result. (Web
resources are prefered, but also book or>paper in a journal
would help.)> Have you tried searching the _web_ using Google?
If you type> Banach limit almost convergence> into the search
box you get a few hits, one of which says this> result was
proved by Lorentz in A contribution to the theory> of
divergent sequences, Acta Math 80(1948).>Thanks in
advance!>Martin Sleziak>The police and so forth only exist
insofar as they can demonstratetheir authority. They say
they're here to preserve order, but in factthey'd go
absolutely mad if all the criminals of the world went onstrike
for only a month. They'd be on their knees waiting for acrime.
That's the only existence they have.William S. Burroughs
(American writer) Guardian, 1966huffy===
===
Subject:
: Re: Banach
limit and almost convergenceContent-transfer-encoding:
8bit>Hi!>It was no several times in Google Groups (by Ronald
Bruck), then>uniqueness of Banach limit of a sequence is
equivalent to almost>convergence of this sequence. Can you
give me a hint where to find a>proof of this result. (Web
resources are prefered, but also book or>paper in a journal
would help.)> Have you tried searching the _web_ using Google?
If you type> Banach limit almost convergence> into the search
box you get a few hits, one of which says this> result was
proved by Lorentz in A contribution to the theory> of
divergent sequences, Acta Math 80(1948).Thank you, David. But
with this big a hint, one ought to be able toprove it oneself.
As is common, it's discovering the STATEMENT TO BEPROVED which
is the original contribution and the hard part. I supposethat
Fermat's Last Theorem is an exception to the rule--Ron
Bruck===
===
Subject:
: Re: There are no other solutions... how to
prove it?> how can I prove that the ONLY solutions of the ODE>
y'' = -y are the ones given by A cos(x) + B sin(x) ?Easy:
Wronskian W(cos,sin) = cos^2 + sin^2 = 1 != 0, soVariation of
Parameters (2) specialized to _homogeneous_ case shows that
any other solution is a linear combination of cos, sin with
_constant_ coef's. Below are easy proofs,which generalizes to
higher-order LODEs (and recurrences).The proof is slicker in
matrix form, e.g. Lemma 4.1 in (3).THEOREM If f,g,h are
solutions on an interval I of the LODE y'' = p y' + q y, p,q
continuous on Iand gh'-g'h != 0 for all x in I, then there
exist constants c,dsuch that f = c g + d h on IPROOF: The
below equations [0],[1] have a unique solution (c,d)since det
= W(g,h) = gh'- g'h != 0 on I.[0] f = c g + d h[1] f' = c g' +
d h'[2] qf+pf' = f = c g + d h via q[0]+p[1], g=qg+pg',
h=qh+ph'[3] 0 = c'g + d'h via [0]'-[1][4] 0 = c'g' + d'h' via
[1]'-[2] (1b)The above equations [3],[4] have unique solution
(c',d') = (0,0)since det = gh'-g'h = W(g,h) != 0 on I. Thus
c,d are constants.-------THEOREM If f,g,h are solutions of the
recurrence y'' = p y' + q y, where y'(n) := y(n+1)with
Wronskian W = gh'-g'h != 0 then there exist constants c,dsuch
that f = c g + d hPROOF: [0],[1] below have unique solution
(c,d) via det = W != 0[0] f = c g + d h[1] f' = c g' + d h'
Now q[0] + p[1] yields:[2] qf+pf' = f = c g + d h via qg+pg' =
g, qh+ph' = h[3] 0 = (c'-c)g' + (d'-d)h' via [0]'-[1][4] 0 =
(c'-c)g + (d'-d)h via [1]'-[2]The above equations [3],[4] have
solution (c'-c,d'-d) = (0,0),unique via det = W' = g'h-gh' !=
0. So c,d are constants,since c' = c means c(n+1) = c(n).-Bill
Dubuque(1) L. E. Pursell: A simple uniqueness theory for
ordinary linearhomogeneous differential equations, Amer. Math.
Monthly, 74, 1967, 47-50
http://links.jstor.org/sici?sici=0002-9890(196701)74:1%3C47%3E
(2) Variation of
Parametershttp://planetmath.org/encyclopedia/
VariationOfParameters.htmlhttp://ltcconline.net/greenl/courses
/204/appsHigherOrder/variationHigher.htm(3) Marius van der
Put: Symbolic analysis of differential
equationshttp://msri.org/activities/programs/9899/focm/soggy/
MSRIintro/van_der_Put2.psd:Symbolic analysis of differential
equations -- Put.ps.gz===
===
Subject:
: Re: if x has normally
distribu digits, does 1/x?>> If so, how do you prove its
normality?>He can't. That's why he said it was his guess, not
his theorem. Well at least it would be nice to get an idea why
such a reciprocalwould be normal.>No one knows how to prove pi
is normal, but the smart money >is all riding on that guess.
Same for the sugges number.Smart money? Did I miss the JSH
lecture of marketing in mathematicsagain?===
===
Subject:
: Re: if x
has normally distribu digits, does 1/x?> If so, how do you
prove its normality?>>He can't. That's why he said it was his
guess, not his theorem. > Well at least it would be nice to
get an idea why such a reciprocal> would be normal.>No one
knows how to prove pi is normal, but the smart money >is all
riding on that guess. Same for the sugges number.> Smart
money? Did I miss the JSH lecture of marketing in mathematics>
again?The set of non-normal numbers has Lebesgue measure zero.
If you pick a real uniformly at random from the interval [0,
1] then with probability 1 you have picked a normal number.
This leads many to take the position that if some number pops
up somewhere & there's no good reason to think it's not normal
- some number like pi, or like the reciprocal of the number you
get when you replace every other digit in the decimal expansion
of pi with a zero - then it's safe to assume that the number is
normal. Safe to assume doesn't mean we know how to prove, and I
don't know what JSH has to do with it as I stopped reading his
threads years ago.-- ===
===
Subject:
: Axiom of Foundation
(absymally stupid question)Hi folks,Okay, this is going to be
some very stupid question to most of you.The axiom of
foundation says that: ForAll B ~= emptyset, ThereExists y in B
such that (y intersect B)=emptyset.Now suppose B = {lime,
orange, lemon}. What is such y among this setof citruses?
Suppose y = lime. How can lime be a set? (myunderstanding is
that only sets can intersect).I told you, it is a stupid
question. But I want to know.===
===
Subject:
: Re: Axiom of
Foundation (absymally stupid question)The whole problem is
just that you're misquoting the axiom. You say: Everynonempty
B contains a y with (B intersect y) = empty. I say: Every
nonempty Bcontains a y for which there is no z with z in y and
z in B. My axiom sayssubstantially the same thing, but it
allows for the possibility that there existcitrus fruits that
are not sets. Alternatively, you could banish such
citrusfruits from your theory. Citrus fruits that have no
elements, but aren't theempty set, are technically called
individuals. Whether you want them to existor not is basically
a matter of taste, which I suppose would have to be fairlysour.
HTH| Hi folks,|| Okay, this is going to be some very stupid
question to most of you.|| The axiom of foundation says
that:|| ForAll B ~= emptyset, ThereExists y in B such that (y
intersect B)=| emptyset.|| Now suppose B = {lime, orange,
lemon}. What is such y among this set| of citruses? Suppose y
= lime. How can lime be a set? (my| understanding is that only
sets can intersect).|| I told you, it is a stupid question. But
I want to know.===
===
Subject:
: Re: Axiom of Foundation (absymally
stupid question)>Citrus fruits that have no elements, but
aren't the>empty set, are technically called individuals.
>Whether you want them to exist>or not is basically a matter
of taste, >which I suppose would have to be fairly>sour. HTHIs
that a consequence of Zorn's Lemon?Lee Rudolph===
===
Subject:
: Re:
Axiom of Foundation (absymally stupid question)> Hi folks,>
Okay, this is going to be some very stupid question to most of
you.> The axiom of foundation says that: > ForAll B ~=
emptyset, ThereExists y in B such that (y intersect B)=>
emptyset.> Now suppose B = {lime, orange, lemon}. What is such
y among this set> of citruses? Suppose y = lime. How can lime
be a set? (my> understanding is that only sets can
intersect).> I told you, it is a stupid question. But I want
to know.It's not stupid, just based on a bit of a confusion
between formal and intuitive set theory.The thing about formal
set theory (or at least ZFC formal set theory - I think there
are versions where this isn't true) is that *everything* is a
set. For example 0 = empty set, 1 = {0}, 2 = {0, 1}, etc. You
define the natural numbers in terms of sets, and then build up
from there. When things get complica enough things stop
*looking* like sets - for example writing down the real number
1 as a set would be highly non-trivial (and very dependent on
how you construc the real numbers), but the point remains that
at some fundamental level they are still sets.So you couldn't
form a set like B = {lime, orange, lemon}, because limes,
oranges and lemons aren't sets (well, you could give the name
'lime' to a set, but it would still be a set).Hope that
helps,David===
===
Subject:
: Re: Axiom of Foundation (absymally
stupid question)Jared> The axiom of foundation says that:>
ForAll B ~= emptyset, ThereExists y in B such that (y
intersect B)=> emptyset.> Now suppose B = {lime, orange,
lemon}. What is such y among this set> of citruses? Suppose y
= lime. How can lime be a set? (my> understanding is that only
sets can intersect).lime intersect B = emptysetMaybe the
confusion is due tolime <> {lime}and indeed{lime} intersect B
<> emptyset.LH===
===
Subject:
: Re: Usenet Posting Guide?> (snip)>
Not to get into a flame war, but coming from sci.math> I have
to object to what seems to me to be some false> theorems in
your reasoning, namely that> politically liberal => PETA>
Sierra Club => environmental terrorist> environmentalist =>
every wacko animal rights sentiment> I would describe myself
as a political liberal and an> environmentalist (to the point
of bicyling across> adelphia to work on occasion). But I
don't> identify with anything on the right hand side.> I
refuse to get into an argument about my positions,> or yours.
I just want to point out that you have> dismissed the other
side of the political spectrum> as not being a spectrum.> -
Randy> Randy, dear boy, if I might jump in here, it's rather
quite axiomatic> that liberals never identify themselves as
being liberal in their> politicsReally? Liberals never call
themselves liberal? Areyou sure? How do you read the sentence
beginningI would describe myself...? Just curious, if
you'regoing to throw words like axiomatic around.> the
presumption being that liberals by definition (their>
definition, BTW) are open to all views they see as reasonable
(again,> their definition of what's reasonable).So you would
say all liberals consider themselves opento, say, Ann Coulter
or Rush Limbaugh? Where are yougetting this from?> As for your
above equations, they ignore the larger reality that most> PETA
freaks are overwhelmingly liberal in their politics.No they
don't. Again, I'm not going to get into a politicalwar, just
commenting on the error in logic. Your statementis that A =>
B, where A is person X is a PETA freakand B is person Y is a
liberal.That does not in any way mean that B => A.> So, too,
is> the white wine and brie crowd who by and large make up the
Sierra Club> constituency. As for the environmentalists, as a
group, if anything> they're even worse than liberals in their
politics, the vast majority> of them being
Bolsheviks.Bolsheviks? Are you a fan of Tom Potter? He's the
onlyother person I know to use that term to describe
peoplepost-1920 or so.> No, dear man, these people can be
called a> lot of things, but conservative or republican aren't
numbered among> them.Nor did I imply that it should be.Consider
the phrase political spectrum. The politicalspectrum is
actually a multivariate space, and there's lotsof room in the
quadrant called liberal to separateenvironmentalist from
eco-terrorist or PETA freak.Again you seem to be making the
same logical error(s)that cause you to be unable to recognize
that people,even liberal people, have multiple attributes
thatdistinguish their political views.By the way, I'll also
point out that democrat/republicanis not equivalent to
liberal/conservative, thoughthere is certainly a high degree
of correlation. Up tilla couple of years ago, I lived in a
liberal congressionaldistrict in a liberal state with a
well-loved Republicancongressional representative.In fact,
prior to the early 80s, I identified myself asa liberal
Republican. - Randy===
===
Subject:
: Re: Usenet Posting Guide?> So,
too, is> the white wine and brie crowd who by and large make
up the Sierra Club> constituency. As for the
environmentalists, as a group, if anything> they're even worse
than liberals in their politics, the vast majority> of them
being Bolsheviks.> Bolsheviks? Are you a fan of Tom Potter?
He's the only> other person I know to use that term to
describe people> post-1920 or so.It is interesting to see that
this poster thinks that the Bolsheviks just vanished into
nothingness.The fact of the matter is that the Bolsheviks
instiga the class wars of the 1900's for power and wealth, and
after their class wars were discredi, and the Native Russians
regained controlled of their government, millions of the
Bolsheviks who had lived high and mighty in Russia, migra to
Israel and New York, from where they are instigating the
religious wars of the 2000's to get back into the chips as the
loot from their class wars is almost gone.I suggest that
intelligent, rational, moral folks, reject the media
brainwashing, open their eyes, look around and see who instiga
the class wars, who is instigating the religious wars, and who
profits from both, while others, such as Blacks, Rednecks and
Latinos, sacrifice their lives, limbs, liberties and fortunes
to fight folks they would otherwise get along with just
fine.As can be seen by studying history, and by observing
current events, the Bolsheviks have a long history of for
power and wealth.Instigating conflict and war is the stock in
trade of the Bolsheviks, much as fortune telling is the stock
in trade of Gyspies.--Tom Potter
http://tompotter.us===
===
Subject:
: Re: Usenet Posting Guide?>
So, too, is> the white wine and brie crowd who by and large
make up the Sierra Club> constituency. As for the
environmentalists, as a group, if anything> they're even worse
than liberals in their politics, the vast majority> of them
being Bolsheviks.> Bolsheviks? Are you a fan of Tom Potter?
He's the only> other person I know to use that term to
describe people> post-1920 or so.> It is interesting to see
that this poster thinks > that the Bolsheviks just vanished
into nothingness.> The fact of the matter is that the
Bolsheviks instiga the > class wars of the 1900's for power
and wealth, > and after their class wars were discredi, > and
the Native Russians regained controlled > of their government,
millions of the Bolsheviks > who had lived high and mighty in
Russia, > migra to Israel and New York, from where > they are
instigating the religious wars of the 2000's > to get back
into the chips as the loot from their class wars > is almost
gone.> I suggest that intelligent, rational, moral folks, >
reject the media brainwashing, open their eyes, > look around
and see who instiga the class wars, > who is instigating the
religious wars, > and who profits from both, > while others,
such as Blacks, Rednecks and Latinos, > sacrifice their lives,
limbs, liberties and fortunes > to fight folks they would
otherwise get along with just fine.> As can be seen by
studying history, > and by observing current events, > the
Bolsheviks have a long history of > for power and wealth.>
Instigating conflict and war is the stock in trade > of the
Bolsheviks, much as fortune telling > is the stock in trade of
Gyspies.Tom, I believe you and I are also on the same page here
as it reflectsa proper understanding of Bolshevism. These
ill-tutored twits who seethemselves as sophistica and schooled
in world politics generallydon't know their gluteous maximus
from a hole in the ground. Theseclueless suckers love to
snicker at things they know absolutelynothing about. This
whole system is crumbling right before their veryeyes, and
it's bread and circuses as usual for these politicalgeniuses
while race is set against race, class is set against
class,gender is set against gender and we all get sold down
the river for 30pieces of silver. We're in meltdown and these
fools are asleep at thecontrol panel.I liked the OP's comments
about the environmental white wine and briecrowd, for they are
truly the useful idiots in this whole equation whodo the most
damage.===
===
Subject:
: Re: Usenet Posting Guide?>> So, too, is>>
the white wine and brie crowd who by and large make up the
Sierra Club>> constituency. As for the environmentalists, as a
group, if anything>> they're even worse than liberals in their
politics, the vast majority>> of them being Bolsheviks.>>
Bolsheviks? Are you a fan of Tom Potter? He's the only>> other
person I know to use that term to describe people>> post-1920
or so.<Potteriana snipped>Randy, you're right!-- ===
===
Subject:
:
Re: Usenet Posting Guide?>> So, too, is>> the white wine and
brie crowd who by and large make up the Sierra Club>>
constituency. As for the environmentalists, as a group, if
anything>> they're even worse than liberals in their politics,
the vast majority>> of them being Bolsheviks. Bolsheviks? Are
you a fan of Tom Potter? He's the only>> other person I know
to use that term to describe people>> post-1920 or so.>
<Potteriana snipped>> Randy, you're right!Just curious
Robin.You seem to be a pretty bright boy, what compeled you to
make this comment?Do you, or did you, have some kind of
relationship with the people who instiga the class wars of the
1900's?I assume that you understand correlations, and have
thought a little about the existence of things, including
objects and non-objects.Have you ever correla what people, or
what groups of people have been central to most conflict and
war, particularly the class wars of the 1900's and the
religious wars of the 2000's?How do you explain the extinction
of an object or a non object, such as an idea?As you may know,
the Bolsheviks were not only central to the class wars of the
1900's, they held most of the power positions in Russia up to
the present time.After their class wars were discredi, and the
loyal Russians began to retake control of the government (A
battle which is still going on, as can be seen from recent
news reports.), hundreds of thousands of the sons and
daughters of the former Russian leadership migra to Israel and
New York, where they are up to their old tricks. Do you think
that the ideas and tactics of the Bolsheviks simply vanished
into nothingness?I assert that:1. Populations exist.2.
Populations are quantum entities.3. Contiguous populations
form an environment.4. Populations increase exponentially in a
nurturing environment.5. Populations decrease exponentially in
a non-nurturing environment.6. An event is a population
change.Do you know what happens to the objects and the fission
or fusion non-objects when events occur?If you think objects
and ideas can simply vanish without a trace, please explain
how it happens, and when one knows that the final quantum has
vanished.--Tom Potter http://tompotter.us===
===
Subject:
: Re:
Usenet Posting Guide?>> So, too, is> the white wine and brie
crowd who by and large make up the Sierra> Club constituency.
As for the environmentalists, as a group, if> anything they're
even worse than liberals in their politics, the> vast majority
of them being Bolsheviks.Bolsheviks? Are you a fan of Tom
Potter? He's the only> other person I know to use that term to
describe people> post-1920 or so.>> <Potteriana snipped>
Randy, you're right!<even more off-topic P_o_t_t_e_r_i_a_n_a
snipped (thanx mensanator)>Randy, you're right squared!!--
===
===
Subject:
: Re: Usenet Posting Guide?>Subject: Re: Usenet
Posting Guide?>Message-id: <bp2aqi$v76ro$1@athena.ex.ac.uk>>
So, too, is> the white wine and brie crowd who by and large
make up the Sierra Club> constituency. As for the
environmentalists, as a group, if anything> they're even worse
than liberals in their politics, the vast majority> of them
being Bolsheviks.Bolsheviks? Are you a fan of Tom Potter? He's
the only> other person I know to use that term to describe
people> post-1920 or so.><Potteriana snipped>Did you notice
how Potter but into this thread with his boiler-platebull? He
mentioned once that he doesn't sit around _reading_ sci.math
butthat he regularly does searches on his own name to see if
anyone is talkingabout him behind his back.If you want to keep
his odius presence out of the discussion, it would be bestto
take an example from that other Potter and dare not speak his
name. Isuggest referring to him in a way to frustrate his
search engine, such asT_o_m_P_o_t_t_e_rOf course, he'll
probably start looking for _this_ pattern, so pick some
othervariation. If he shows up, then he's caught on to that
pattern, so switch toanother.>Randy, you're right!>--
>>Needless to say, I had the last laugh.> Alan Partridge,
_Bouncing Back_ (14 times)--===
===
Subject:
: I'm reviewing
Calc3-need integral help.I'm having trouble setting up these 3
integrals. Let S be the integralsymbol.1) Evaluate the integral
SS (over R) dA/(1 + X^2 + Y^2)^2 over the regionenclosed by one
loop of the lemniscate (X^2 + Y^2)^2 - (X^2 -Y^2) =0.2) Find
the surface area of the portion of the sphere X^2 + Y^2 + Z^2
= 16between the planes Z = 1 and Z = 2.3)Find the volume cut
from the solid sphere X^2 + Y^2 + Z^2 = a^2 by thecylinder r =
a * sin (theta).Thanks.===
===
Subject:
: Re: I'm reviewing Calc3-need
integral help.> I'm having trouble setting up these 3
integrals. Let S be the integral> symbol.> 1) Evaluate the
integral SS (over R) dA/(1 + X^2 + Y^2)^2 over the region>
enclosed by one loop of the lemniscate (X^2 + Y^2)^2 - (X^2
-Y^2) =0.Use plane polars (x = r cos(phi), y = r sin(phi), dA
= r dr d(phi) )[Hint: substitute u = r^2, take the limits of
the theta integral as-(pi)/4, pi/4]> 2) Find the surface area
of the portion of the sphere X^2 + Y^2 + Z^2 = 16> between the
planes Z = 1 and Z = 2.Use cylindrical polars (x = r cos(phi),
y = r sin(phi), z = z,dA = r d(phi) dz )> 3)Find the volume
cut from the solid sphere X^2 + Y^2 + Z^2 = a^2 by the>
cylinder r = a * sin (theta).Use spherical polars ( x = r
cos(phi) sin(theta),y = r sin(phi) sin(theta), z = r
cos(theta),dx dy dz = r^2 sin(theta) dr d(theta) d(phi) )(I'm
not sure r = a *sin(theta) is a cylinder; I think the cylinder
isgiven by a = r *sin(theta) [ x^2 + y^2 = a^2, -oo < z < oo ]
)-- ===
===
Subject:
: Re: Fastest factorial algorithm> What is the
fastest algorithm for computing factorial, for very large >
numbers (e.g. 10000!)?> Normally this would take n-2
multiplications, by multiplying out each > term
n(n-1)(n-2)(n-3)...3.2. Is a better way known?Just do the
multiplications carefully in the right order. Multiplying
k-digit numbers takes O(k log k) time using the best known
algorithms, and obviously requires at least O (k) time. If you
start with 1, multiply by 2, multiply by 3, and so on, the
numbers involved get larger and larger, so the time will be O
(n^2 log^2 n). Instead you should use a binary tree for the
multiplications: To find the product of k numbers for k > 2,
divide the numbers into two sets of equal size, calculate the
product of the numbers in the first set, the product of the
numbers in the second set, and multiply the products. That
way, you should be able to calculate n! in O (n log^3 n).
(Actually, I wouldn't count 10000 as a very large
number)===
===
Subject:
: Re: Fastest factorial algorithmX-ID:
Tbl9NaZ1we++y9QKom6kuOERHmzG5xwqLk5ZS+wfxmx9HlTWqmzpYu> What
is the fastest algorithm for computing factorial, for very
large> numbers (e.g. 10000!)?>(...)> (Actually, I wouldn't
count 10000 as a very large number)Yes, but what about the
OP's 10000!  Gottfried Helms===
===
Subject:
: Re: Fastest
factorial algorithm> What is the fastest algorithm for
computing factorial, for very large> numbers (e.g.
10000!)?(...)> (Actually, I wouldn't count 10000 as a very
large number)> Yes, but what about the OP's 10000!  I
believe what Christian meant was: I wouldn't count 10000 as a
very large number [to compute the factorial for].Computing
10000! using the straightforward algorithm in bc takes only
about 13.8 seconds on a 600MHz Pentium III machine I have
lying around. And of course, bc isn't what I'd call a speed
demon.-M-- http://www.dartmouth.edu/~sting/ | Dartmouth
College, Hanover, NH, USA===
===
Subject:
: Re: Fastest factorial
algorithm E29yc_kQC&^> What is the fastest algorithm for
computing factorial, for very large> numbers (e.g.
10000!)?(...)> (Actually, I wouldn't count 10000 as a very
large number)> Yes, but what about the OP's 10000!  That's
large itself, but it doesn't take long tocompute. computing
10000!! isn't feasible whatever algorithmyou use.-- J.97n
Fairbairn Jon.Fairbairn@cl.cam.ac.uk===
===
Subject:
: Re: Fastest
factorial algorithm>> Yes, but what about the OP's 10000!
>That's large itself, but it doesn't take long to>compute.
computing 10000!! isn't feasible whatever algorithm>you use.
True [at least for (10000!)!, as opposed to the
Mathematicameaning]. OTOH, its last 10000!/4 or so digits are
easy to calculate,at least to base 10, which gets you part of
the way; and so are thefirst few digits. It's just those in
the middle that are hard ....-- Andy Walker, School of
MathSci., Univ. of Nott'm, UK.anw@maths.nott.ac.uk===
===
Subject:
:
Re: Fastest factorial algorithm E29yc_kQC&^>> Yes, but what
about the OP's 10000! >That's large itself, but it doesn't
take long to>compute. computing 10000!! isn't feasible
whatever algorithm>you use.> True [at least for (10000!)!, as
opposed to the Mathematica> meaning]. OTOH, its last 10000!/4
or so digits are easy to calculate,> at least to base 10,
which gets you part of the way; and so are the> first few
digits. It's just those in the middle that are hard ....I
wasn't thinking so much of the difficulty, as where to
storethem when you're done.-- J.97n Fairbairn
Jon.Fairbairn@cl.cam.ac.uk===
===
Subject:
: Re: Fastest factorial
algorithm>> computing 10000!! isn't feasible whatever
algorithm>>you use.>> True [at least for (10000!)!, as opposed
to the Mathematica>> meaning]. OTOH, its last 10000!/4 or so
digits are easy to calculate,>> at least to base 10, which
gets you part of the way; and so are the>> first few digits.
It's just those in the middle that are hard ....>I wasn't
thinking so much of the difficulty, as where to store>them
when you're done. Well, I've saved you a few, 'cos you don't
really need tostore the insignificant zeros. The number of
digits is only aroundthe 3/2 power of the number of different
games of chess, so oncechess is completely solved, this could
be the next project ... justneeds a bigger bit-bucket. Or you
could think laterally: in base10000!!, the answer is just 10.
[But there is no base in which itis 42, sadly.]-- Andy Walker,
School of MathSci., Univ. of Nott'm,
UK.anw@maths.nott.ac.uk===
===
Subject:
: Re: Fastest factorial
algorithm What is the fastest algorithm for computing
factorial, for very large>> numbers (e.g. 10000!)?>> (...)
(Actually, I wouldn't count 10000 as a very large number)>>
Yes, but what about the OP's 10000!  > That's large itself,
but it doesn't take long to> compute. computing 10000!! isn't
feasible whatever algorithm> you use.Actually, 10000!! is
easy, using Mathematica. The !! notationrepresents the double
factorial function, such that n!! =n*(n-2)*(n-4)*....I think
you meant (10000!)!, which is a bit larger.-- Dave SeamanJudge
Yohn's mistakes revealed in Mumia Abu-Jamal
ruling.<http://www.commoncouragepress.com/index.cfm?action=
book&bookid=228>===
===
Subject:
: Re: Fastest factorial algorithm
E29yc_kQC&^> Actually, 10000!! is easy, using Mathematica. The
!! notation> represents the double factorial function, such
that n!! => n*(n-2)*(n-4)*....I wasn't aware of that usage.> I
think you meant (10000!)!,I did indeed.> which is a bit
larger.Several bits.-- J.97n Fairbairn
Jon.Fairbairn@cl.cam.ac.uk
===
Subject:
: Re: Fastest factorial
algorithm <3FB37A98.1E6E3DCF@uni-kassel.de>
<wfk76477u4.fsf@pictogramme.al.cl.cam.ac.uk>
<bp00r2$qjh$1@mozo.cc.purdue.edu>
<wfwua4tfac.fsf@calligramme.cl.cam.ac.uk>===> Actually,
10000!! is easy, using Mathematica. The !! notation>
represents the double factorial function, such that n!! =>
n*(n-2)*(n-4)*....> I wasn't aware of that usage.I think it
bogus and that 10000!! means just what you sayit means and for
those who insist it means 10000(!!), letthem make it known
they're omiting those '(' and ')'!!> I think you meant
(10000!)!,> I did indeed.===
===
Subject:
: Re: Fastest factorial
algorithmActually, 10000!! is easy, using Mathematica. The !!
notation> represents the double factorial function, such that
n!! => n*(n-2)*(n-4)*....I wasn't aware of that usage.I think
it bogus and that 10000!! means just what you say> it means
and for those who insist it means 10000(!!), let> them make it
known they're omiting those '(' and ')'!!The double factorial
is a standard (and, in my experience, useful)notation. It is
hardly bogus.David Cantrell> I think you meant (10000!)!,I did
indeed.===
===
Subject:
: Re: Fastest factorial algorithm>Actually,
10000!! is easy, using Mathematica. The !! notation>
represents the double factorial function, such that n!! =>
n*(n-2)*(n-4)*....I wasn't aware of that usage.I think it
bogus and that 10000!! means just what you say> it means and
for those who insist it means 10000(!!), let> them make it
known they're omiting those '(' and ')'!!> The double
factorial is a standard (and, in my experience, useful)>
notation. It is hardly bogus.Usefulness of the
double-factorial function aside, the notation is, in my
not-so-humble opinion, an unintuitive abomination that should
have been stamped out before it took hold.Does anyone know who
first coined that notation?-- Unpatched IE vulnerability: WMP
local file bounceDescription: Switching security zone,
arbitrary command execution, automatic email-borne command
executionReference:
http://www.ntbugtraq.com/default.asp?pid=36&sid=1&A2=ind0307&L
=ntbugtraq&F=P&S=&P=6783Exploit:
http://www.malware.com/once.again!.html===
===
Subject:
: Re: Fastest
factorial algorithm>> Actually, 10000!! is easy, using
Mathematica. The !! notation>> represents the double factorial
function, such that n!! =>> n*(n-2)*(n-4)*....> I wasn't aware
of that usage.>> I think it bogus and that 10000!! means just
what you say>> it means and for those who insist it means
10000(!!), let>> them make it known they're omiting those '('
and ')'!! > The double factorial is a standard (and, in my
experience, useful)> notation. It is hardly bogus.I'll agree
that the double factorial can be quite useful, but I'llnot go
so far as to say that the _notation_ is useful. When one
firstencounters something such as 10000!!, everyone no doubt
assumes it tomean (10000!)! instead of the double factorial.
Perhaps if everyoneput their minds to it, someone could invent
a more intuitivenotation (or at the very least, not negatively
intuitive). Perhaps ?will work? Or maybe
&!?-----------------------------------------------------------
---------- | Good and evil both increase at compoundBen
Hocking, Grad Student | interest. That is why the
littlehocking@cs.virginia.edu | decisions you and I make every
day are of | such infinite importance. - C. S.
Lewis---------------------------------------------------------
------------===
===
Subject:
: Re: Fastest factorial algorithm>>
Actually, 10000!! is easy, using Mathematica. The !!
notation>> represents the double factorial function, such that
n!! =>> n*(n-2)*(n-4)*....> I wasn't aware of that usage.>> I
think it bogus and that 10000!! means just what you say>> it
means and for those who insist it means 10000(!!), let>> them
make it known they're omiting those '(' and ')'!!> The double
factorial is a standard (and, in my experience, useful)>
notation. It is hardly bogus.> I'll agree that the double
factorial can be quite useful, but I'll> not go so far as to
say that the _notation_ is useful.I grant that the notation is
an unfortunate one. But there are several ofthose in
mathematics.On the board:dy--dxHey, prof, why don't you reduce
the fraction to y/x by cancelling the d's?> When one first>
encounters something such as 10000!!, everyone no doubt
assumes it to> mean (10000!)! instead of the double factorial.
Perhaps if everyone> put their minds to it, someone could
invent a more intuitive> notation (or at the very least, not
negatively intuitive). Perhaps ?> will work? Or maybe
&!?Here's one idea. For the double factorial, place a
subscript of 2 on thefactorial sign. For example, 10000!_2.
For the triple factorial, use asubscript of 3; etc.But even if
such a notation were to become popular, the previous
notation,now well established, would most likely persist. Then
we'd have _two_notations, causing even more confusion
perhaps...David===
===
Subject:
: Re: Fastest factorial
algorithmDistribution: inet
<wfk76477u4.fsf@pictogramme.al.cl.cam.ac.uk>
<bp00r2$qjh$1@mozo.cc.purdue.edu> Actually, 10000!! is easy,
using Mathematica. The !! notation>> represents the double
factorial function, such that n!! =>> n*(n-2)*(n-4)*....>> I
think it bogus and that 10000!! means just what you say>> it
means and for those who insist it means 10000(!!), let>> them
make it known they're omiting those '(' and ')'!!> The double
factorial is a standard (and, in my experience, useful)>
notation. It is hardly bogus.> I'll agree that the double
factorial can be quite useful, but I'll> not go so far as to
say that the _notation_ is useful. When one first> encounters
something such as 10000!!, everyone no doubt assumes it to>
mean (10000!)! instead of the double factorial. Perhaps if
everyone> put their minds to it, someone could invent a more
intuitive> notation (or at the very least, not negatively
intuitive). Perhaps ?> will work? Or maybe &!? I think
10000(!!) works quite well in ASCII, and FWIW I agreethat
10000!! looks like it should mean take 10000, apply the
!operator, then apply it again, not take 10000, apply the
!operator, then divide out all the odd-numbered factors.
Soregardless of what Mathematica may think, this math student
isgoing to use those parentheses [next time I have a reason
towant the double factorial of anything ;] . I believe 10000?
is common notation for the 10000th triangularnumber, sum(i=1
to 10000)(i). Do I remember correctly? 10000&! just looks
silly. :)-Arthur===
===
Subject:
: Re: Fastest factorial algorithm>
Actually, 10000!! is easy, using Mathematica. The !! notation>
represents the double factorial function, such that n!! =>
n*(n-2)*(n-4)*....Just curious... what's the use of such a
function? - Risto -===
===
Subject:
: Re: Fastest factorial
algorithm>> Actually, 10000!! is easy, using Mathematica.
The !! notation>> represents the double factorial function,
such that n!! =>> n*(n-2)*(n-4)*....> Just curious... what's
the use of such a function?A Google search turns up
<http://mathworld.wolfram.com/DoubleFactorial.html>,among
others.-- Dave SeamanJudge Yohn's mistakes revealed in Mumia
Abu-Jamal
ruling.<http://www.commoncouragepress.com/index.cfm?action=
book&bookid=228>===
===
Subject:
: Errata for Spivak's Calculus on
ManifoldsDoes anyone know if an errata exists for this text? I
wan to checkbefore going through the drudgery of contacting the
publisher. As faras I can tell there is no errata available on
the web. Thanks inadvance.===
===
Subject:
: definition of
UniformityHiI'm working in the study of some crystals with
vacancies. Crystal arestudied like an orderen arrangement of
atoms that forms a net. In one dimension this can be represen
like a sequence of points at theseme distance:(i would
represent the unit cell, this part of the secuence isrepetead
infynit times.). . . . . . . . . . . . . . Now, if we have put
some vacancies (faults of atoms) The secuence canbe. . . . . .
. . . . . I want to know if there is a definition of
uniformity in the folowingform:The most uniform distributions
of the vacances in such a sequence iswhen the vacancies are as
far as we can from each other. (There is asmal trick that the
secuence is repetead so, the end and begining ofthe secuence
are conec)TIAZunbeltz IzaolaUniversity of the Basque Country--
Remove XXX from email: zunbeltz@wm.lc.ehu.esXXX===
===
Subject:
: Re:
Combinatorial proof> Show with combinatorial reasoning that>
C(3n,3) = 3*C(n,3)+6n*C(n,2)+n^3, where C(n,k) is n choose k.>
Am I supposed to show that both sides of the equation counts
the thing? The text where you found this exercise should
probably explain what they mean by combinatorial reasoning.
You should attempt to use their definition. It also might help
to try a simpler one:C(2n, 2) = 2*C(n, 2) + n^2> Does it
qualify as combinatorial reasoning if I use combinatorial >
identities or prove it by induction?In my definition of
combinatorial reasoning or proof, sometimes identities or
arithmetic or induction can be used, but the -main- intent is
to interpret the symbols as statements about combinatorial
objects (sets, subsets, tuples, trees, etc) and the manner in
which they are coun. To show an equality (as above) you take
one mathematical situation (most likely simply described by
one side of the equation), and interpret it another way
(corresponding to the other side of the equation).So I would
say -first- use combinatorial ideas, and only then for small
lemmas might you bother with
identities/induction/arithmetic.Of course, the above identity
is -very- trivial with arithmetic, just replace C(n,k) with
the appropriate factorials and simplify. (No, I can't do that
in my head, but I can do the combinatorics very easily in my
head.) Trivial means mindlessly, just applying the rules you
have without thought.> Any good hints, by the way?I will
refrain from commments about the other hint ;).My hints are
these:+ corresponds to disjoint union of sets.* corresponds to
...what? ... you can think of it as repea summation, or as
first doing one thing, then doing another (i.e. 3*C(n,3) means
do something in three possible 3 ways first, then do something
taking C(n,3) ways).n choose k is not only the -number- of
subsets of size k out of n.^ corresponds to ... what?... repea
multiplication or tuples or functions from x to y (which is
counterintuitively coun by y^x)Combinatorial reasoning just
assigns meaning to the parts of the formula.MitchPS Hi
273!===
===
Subject:
: Re: Sex and Math Discussion, linux)> recently
came up with the excellent idea of using modern> advertising
techniques to get his theories through to the public. I> have
decided that this is an excellent way for me to promote my>
anti-Cantorian doctrine to the public. Now, it is well known
that sex> sells. Therefor, without further ado (please
maximize your window):Brilliant. You've convinced me that
you're right, although I've beentemporarily distrac from what
the question was.Also, an 11-year-old shouldn't have access to
such explicit images. Ihope someone tells your momma.-- I've
been thinking about my problems with getting any kind
ofadmission that my math arguments showing the core error in
mathematicsare correct, so I've gone to marketing books. --
James S. Harris, on when mathematics isn't enough