mm-25
Is there an easy way to trisect a line segment, i.e. split
it into> three equal portions, with just straightedge and
compass?> If not an easy way, is there any way at all?>
Jonathan ChristensenEasy way.Set the compass to approximately
L/3.Mark off 3 compass increments on L.If three L/3 is too
long or too short adjust the compass accordingly. Keep doing
this until L/3 is just right.Hard wayConstruct an equilateral
triangle with L as the altitude.Draw a second altitude. This
will divide L into L/3 and 2L/3.(the hard part is constructing
the triangle)
> I would like to know if this is a
mathematical proof of the Cantors > goof, or conversely it is
just another mathematical goof about the > Cantors proof.> A
MATHEMATICAL PROOF IN THREE ACTSIt is imaginative, and in three
acts, but it is neither mathematical nor a proof.Nor did Cantor
make a goof in either his statement or his proof that there is
no surjection from N to Q.The goofs are all those of the
people who assert that Cantor goofed.
>It took about 22
seconds to print out 1,000,000 primes. How much faster is>it
able to be?> also the program produces 10,000,000 primes
in 22.27 secondsI think you will find that print out is what
takes the time. > The program stores all the primes in a bit
array and can be accessed to see > if it is prime or
not.What about Chapman inequality?Is it named after
Robin Chapman?I don't recall seeing anything called Chapman
inequality, andneither Google nor MathSciNet were helpful
either. But I can tell you that Robin is not responsible for
the Chapman-Kolmogorovequations.Robert Israel
israel@math.ubc.caDepartment of Mathematics
http://www.math.ubc.ca/~israel University of British Columbia
Vancouver, BC, Canada V6T 1Z2
>>What about Chapman
inequality?Is it named after Robin Chapman?>I don't recall
seeing anything called Chapman inequality, and>neither Google
nor MathSciNet were helpful either. Oh, maybe you mean the
Chapman-Robbins (or Chapman-Robbins-Kiefer, or
Hammersley-Chapman-Robbins) inequality from statistics.
Thatwould come from Chapman, D.G. and Robbins, H. (1951).
Minimum variance estimation without regularity assumptions.
Ann. Math. Statist. 22, 581-586. A bit before Robin's time, I
think.Robert Israel israel@math.ubc.caDepartment of
Mathematics http://www.math.ubc.ca/~israel University of
British Columbia Vancouver, BC, Canada V6T 1Z2
JS: The
following statements are true:1. To a good approximation the
non-tensor connection field g-force on awhere m(passive) =
m(active)The approximation is two-foldA. Not near a space-time
singularity, i.e. not falling behind the eventhorizon of a
black hole.B. The scale is large enough so that quantum
gravity metricfluctuations are ignorable.2. To the same
approximation the connection field g-force (whichreduces to
Newton's gravity force) is eliminated on a
timelikegeodesic.PZ: Newton's gravity force did NOT include
fictitious inertial forces.JS: Irrelevant my statement 2 is
true obviously since, e.g.goo = (1 - 2V(Newton)/c^2) = grr^-1
is the warp factor in the Schwarzschild solution.The Einstein
geodesic equation in weak field limit becomes Newton's gravity
force equationd^2r/dt^2 = - GradV(Newton)Inertial forces in
Newton's theory are also independent of the mass m of The
connection field is non-zero only on time-likePZ: Right. But
starting from zero gravity everywhere, the field can be made
non-zero by(1) going to an accelerated frame; and/or (2)
switching on gravitational sources.JS: Obvious, so what? Also
if you want to use tidal stretch-squeeze curvature as the
criterion for a real gravity field rather than the GR
connection field geometrization of Newton's gravity force,
then it is not enough that the curvature vanish at isolated
zeros, it must vanish in a finite region of space-time.3. The
curvature tensor is also a local observable.PZ: Yes. And local
measurements of tidal curvature are not *necessarily*
scale-sensitive.JS: Red Herring. Depends what you mean by
scale-sensitive? Any classical procedure (liquid drops without
surface tension, precessing spinning tops ...) breaks down at
quantum gravity level and also in approach to a singularity -
assuming dark energy exotic vacuum zero point energy does not
smooth out the classical singularity.IF you define a true
gravity field as one in which the tidalcurvature tensor is NOT
zero, then one can always locally, inprinciple, distinguish a
true gravity field from a fake gravityfield (where the tidal
tensor vanishes in all frames geodesic LIF &non-geodesic LNIF)
by definition.PZ: It's really quite simple: you can for all
practical purposes always locally tell, in anyframe, whether
you are in the presence of a permanent field, since permanent
fieldstypically have non-vanishing tidal curvature, while
inertial fields do not.JS: This is hardly stunning news. MTW
never denied that. However, in ordinary conditions
likespaceflight in solar system local tidal effects on scale
of spaceships is really tiny.PZ: This does not in itself allow
you to locally distinguish between the gravitationalJS: That's
not significant. I suppose, in principle, if you did
simultaneous measurements of both g-force and local tidal
curvature you could make a computer program to integrate the
tidal information and subtract it from total g-force. However,
Bohr's generalized complementarity may forbid, since you need
to measure tidal curvature in LIF, but you need to measure
g-force in LNIF. This is tricky.You have the astronauts in
orbit. How would they do the measurement?PZ: we know that in a
frame that is unaccelerated with respect to a static source,we
can determine the strength of the true gravitational force.JS:
How? We can easily determine the TOTAL g-force in that LNIF.
How do you extract the true piece of it? I suppose that means,
how do you measure components of Ruvwl in the LNIF frame since
most gedankenexperiments to measure it assume a LIF? In
principle it is possible of course because a tensor is a
tensor in any local frame LIF or LNIF.PZ: Any
accelerationimparted to this frame with respect to the source
will then add calculable inertialcontributions to the observed
g-forces.JS: But your rest frame is already an LNIF.PZ: So we
can not only distinguish between pure inertial fields and
mixed inertial & permanentgravitational fields, using locally
measurable tidal effects, but we can in principle
alsodetermine the true strength of the permanent part of the
observed g-forces.JS: This is useless. It's too vague. Show
explicitly in detail in a gedankenexperiment how to do it.
Show a real constructive procedure.PZ: So we could equally
attribute both the tidal curvature and the permanent part of
theg-force to the true gravitational field, and inertial
effects to a separate fictitiousinertial field -- as I am
proposing.JS: You have no math for this and words are
inadequate. I do not know what it really means. Even if you
can do it mathematically I do not see how it adds any new
physics?PZ: I see nothing in the basic formal-empirical
framework of GR that prevents this.This seems like a perfectly
feasible alternative definition of the real gravitationalfield
which corresponds very naturally with Newtonian theory.JS:
Until you can show this explicitly it's idle speculation.
Maybe Ruffini et-al have a clue in their discussion of
experiments?On the other hand, if you define a gravity field
as the connectionfield, then you cannot make such a
distinction!PZ: Obviously.JS: Therefore, the problem here is
only a semantic problem from waveringbetween the two different
definitions of gravity field.PZ: No, it is not *merely* a
semantical problem. It is a question of the proper
definitionof what in essence constitutes the *real
gravitational field*.Of course, logically speaking, you can
define the gravitational field to be aham sandwich if that
pleases you; but that doesn't mean that this is a
physicallyappropriate definition.JS: Not a fair comparison
since using the connection field definition IS the proper
generalization of how Newton thought of the gravity force
field. The motivation is seamless integration of the larger
covering theory with the smaller theory.PZ: The bottom line
here is that there are two alternative models. In the
Einsteinianmodel, the gravitational field is reduced to the
unified gravitational-inertial connectionfield, while the
tidal curvature is left hanging. In the alternative model, the
curvature fieldis integrated with the permanent component of
the connection field as a fundamentalaspect of the real
gravitational field, while the inertial component of the
connectionfield is left hanging.JS: You need operational
procedures here that are lacking.PZ: So this is a matter of
two alternative physical models of the gravitational field,
and not*merely* a matter of definitions or semantics.In the
Einstein model, when the unified connection field vanishes,
the gravitational field isliterally *annihilated*. In the
alternative model, the unified connection field can vanish
withoutannihilating the entire physical presence of the field.
This difference in the two models hasdirect consequences for
the definition of gravitational vacuum stress-energy.JS: Show
exactly what you mean here with mathematics, otherwise your
point is too vague.principle locally see tidal effects as a
torque around the center ofmass or a precession of a small
spinning gyroscope of course. Also youcan see deformations of
shape of a geodesic droplet if the surfacetension is small
enough.PZ: Right. And this kind of effect is not *necessarily*
scale-sensitive, notwithstandingthe so-called EEP.JS: I do not
understand your use of scale-sensitive in your context.5.
Since Einstein's GR comes from locally gauging ONLY the
4-parametertranslation subgroup of the Poincare group, of
course Einstein's earlyformulation of the equivalence
principle was only an approximaterotational degrees of
freedom.PZ: Exactly.JS: Again hardly news. The Ruffini book
looks interesting BTW and I will read it cover to
cover.Einstein's guv field of curvedspace-time with the
symmetric connection force field is simply thecompensating
gauge force field needed to restore the now local
gaugesymmetry which is equivalent to local general
coordinatetransformations.PZ: Whether this gauge symmetry has
a physical basis is a *physical* question, and not justa
formal mathematical one.JS: It is physical. It is clearly
physical in my vacuum coherence model that builds upon Hagen
Kleinert's metric elasticity model.guv(Einstein Locally Curved
Space-Time) = Special Relativity Metric (Globally Flat
Space-Time) + Elasticity Strain TensorWhat is strained? It is
the World Crystal Lattice. When it gets over-strained it
fractures into fault lines of curvature disclination defects
and also maybe fault lines of torsion dislocation defects on
scales large compared to the unit cell.The physical distortion
field of the stressed World Crystal is my Bohm Pilot Wave
Constraint:Distortion Field = (Loop Gravity Quantum of
Area)(Partial Derivative of Goldstone Phase of Vacuum
Coherence)This equation does for the elastic-plastic World
Crystal Lattice what Bohm'sv(IT) = (h/m)Grad(Phase of Pilot
BIT Wave)Note thatWitten's alpha' = hc/(String Tension) = Loop
Gravity Quantum of Area from Spin Networks of combinatorial
pre-geometry (Penrose).Quantum Gravity Area = (Loop Quantum of
Area)(QED electron-photon dimensionless coupling)What stresses
the the World Crystal causing it to distort, warp and finally
buckle and then fracture?Exotic vacuum zero point pressure
causes it!All ordinary matter are simply micro-geons of
STRONGLY ATTRACTIVE positive zero point pressure string vortex
cores with quantized trapped gauge force fluxes where/zpf =
(Loop Gravity Quantum of Area)^-1[(Loop Gravity Quantum of
Area)^3/2|Vacuum Coherence|^2 - 1Note that the Vacuum
Coherence drops to zero inside the stringy vortex core whose
diameter is the coherence length. The gauge force fluxes vary
on the scale of the Meissner effect penetration depth like in
a Type II superconductor.* This essentially qualitatively
conceptually solves the 100 + year old Abraham-Becker-Lorentz
extended electron problem that Johnny Glogower and I struggled
with without success in Walter Breen's group in 1954. It took
me 50 years to solve this problem. It was the first serious
physics problem I worked on at age 14. Johnny was 13, but he
had already mastered Synge & Schild's Tensor Calculus a book
with a rough orange hard cover.PZ: If the gravitational and
inertial fields are not even locallycompletely physically
equivalent as Einstein supposed, then the gauge symmetry
yourefer to may not have any deep physical meaning.JS: You are
very wrong there IMHO. Way off the mark. Ruffini does not say
that? Does he? I would be surprised. The local gauge principle
is perhaps the most useful principle in modern theoretical
physics.PZ: The point is Einstein was convinced that it did
have a deep meaning; he believed thathis strict
gravitational-inertial equivalence principle *explained* the
Eotvos proportionalityof the gravitational and inertial
masses. It is very important to understand this aspect
ofEinstein's reasoning IMO.Otherwise, you cannot possibly
understand his theory of general relativity. This isexactly
what Einstein said to von Laue when responding to probing
criticism of histheory in the early 1920s:... what
characterizes the existence of a gravitational field from the
empiricalstandpoint is the non-vanishing of the [components of
the affine connection],not the non-vanishing of the [components
of the Riemann tensor].JS: There you have it! That's a point I
was making to you. I was not aware of Einstein's remark you
cite.PZ: If one doesnot think in such intuitive [anschaulich]
ways, one cannot grasp why somethinglike curvature should have
anything at all to do with gravitation. In any case, norational
person would have hit upon anything otherwise.The key to the
understanding of the equality of gravitational mass and
inertial masswould have been missing.-- A. Einstein, quoted in
How Einstein Discovered General Relativity: A With Some
Contemporary Morals, J.J. Stachel (1986)).JS: Ruffini calls
Einstein a Sleep Walker in Arthur Koestler's sense. BTW I met
Koestler at the Uri Geller Bohm Birkbeck tests. Went to his
home on Montpelier Square for drinks ... See Martin Gardner's
Magic and Paraphysics.PZ: If you want to say that this very
strong Einsteinian hypothesis of equivalence can nowonly be
regarded as a heuristic tool of limited depth and
applicability, then fine, I agreewith that. Eddington, for
one, took this position in 1921, and in fact this is what I
havebeen saying all along.JS: So that's what you have been
saying? Why didn't you say so? That's basically what I have
been saying!Who's on
first?http://www.baseball-almanac.com/humor4.shtmlI'm Costello
BTW ;-)If, in addition, you locally gauge the
6-parameterLorentz subgroup, then you get an additional
torsion force field, i.e.an anti-symmetric piece of the
connection field. This will obviouslymodify the predictions of
GR for the tidal torques on extended testPZ: Yes, this is a
good point. Now, are such effects *necessarily*
scale-sensitive?JS: Define scale-sensitive.6. Hagen Kleinet
shows that tidal curvature, the basis for gravitywaves, comes
from stringy (vortex core if one puts in a vacuumcoherence
O(1) ODLRO parameter) disclination defects in a Plancklattice
in 4D.PZ: He shows that you can *get* tidal curvature from
disclination defects in his 4DPlanck lattice model. Of course
that doesn't in itself prove that this is the truephysical
origin of tidal curvature, or even that such a lattice
actually exists.JS: If it walks like a Duck, and talks like a
Duck ...There is no torsion field in 1915 GR, but if there
wasone in Nature, as Akimov & Shipov claim in Moscow, then it
wouldcorrespond to dislocation lattice defects. The different
4D discreteworld crystal symmetry groups of the tiled unit
cell are differentphysical vacuum structures quite obviously.
This is 4D (and maybe 11D)crystallography with additional
supersymmetry matrix dimensions).PZ: Right. Physics has moved
forward since 1915-16. Einstein's equivalence principlenow
looks rather naive as a serious physical model. However, it
was undeniablyfruitful from a purely heuristic standpoint.JS:
OK7. There are still two more subgroups left in the 15
parameterConformal Group. You will get new compensating gauge
force fieldphysics if you locally gauge the 1-parameter
dilation subgroup and alsolocally gauge the 4-parameter
special conformal transformation ofconstant acceleration
boosts for hyperbolic motion in SpecialRelativity e.g. MTW has
a chapter on this.PZ: This kind of formalistic groupology needs
to be well-anchored in actual physicsand actual physical
reasoning IMHO.But still, this looks interesting.JS: Read any
book on QCD's SU(3) and the electro-weak U(1)xSU(2) with
elegant fashion show of weaves of the fabric of reality from
spinnetworks to pre-geometric spin foams to quantized Area
operators andworld holograms et-al I find Penrose saying:The
algebra I have used to treat linear displacements and
rotationstogether, or linear and angular momentum together, I
call the algebraof twistors. I have used the term 'twistor' to
denote a 'spinor' forthe six-dimensional (++----)
pseudo-orthogonal group O(2,4).9. It is obvious to any School
Boy that O(2,4) is simply Lorentz SRO(1,3) with string world
sheet O(1,1) fiber. Therefore, string theoryis inherent
here.PZ: Wouldn't it be more correct to say that certain
aspects of the *mathematicalstructure* of string theory is
inherent?JS: OK. I am still seeing this Grand Synthesis
Through The Glass Darkly on a Foggy Day in London Town March
for me than the first week March 8 - 13.Penrose continues:The
twistor group is the (++--) pseudo-unitary group SU(2,2) which
islocally isomorphic with O(2,4). In turn O(2,4) is locally
isomorphicwith the fifteen-parameter (local) conformal group
of space-time. Undera conformal transformation of space-time
the twistors will transformlinearly according to a
representation of the group SU(2,2). p. 175Combinatorial
Space-Time Quantum Theory and Beyond (Cambridge,1971).Suddenly
the connections among1. Quantum loop gravity of spin networks
--> foams with quantized areaetc. operators2. Local gauge
invariance3. String theory4. Twistors & Conformal Groupare
becoming clearer.PZ: If so, then at least something is
becoming clearer. :-)JS:
http://www.classicalmidiconnection.com/cgibin/x.cgi?midi/n3/
zsunrise.mid
> I think we have to center the tread,
because at this moment it looks >> like the chats in the
auditorium before the concert.> POINTS>> Point 1:>> The
surjection f: N -> R is not possible because the definition of
>> naturals and reals do not allow it.>>This statement reqires
proof, as it is not self evident.> See latest version.
Point 2:>> Cantor clearly stated that we can say that an
infinite set K is >> countable if and only if it is possible
to establish a one-to-one >> correspondence between K and N,
i.e. f: N <-> K.>>More precisely, a set K is countable if
there is a surjective map from N >to K and countably infinite
(or some say denumerable) if there is a >bijective map.>
Good mathematician! >> Point 3:>> As the one-to-one
correspondence N <-> R is not possible (Point 1), >> the
criterion settled by Cantor (Point 2) is not valid to resolve
>> whether R is countable or not. In other words, we cannot
use this >> criterion with R. (See note below)>>You assert it
to be impossible, but you provide no proof. In >mathematics,
no assertion need be accepted without proof. Since you
>provide no proof, we may assume that you do not know whether
the reals >are countable.> See latest version. Point
4: >> If a proof (any) comes to the conclusion that R is not
countable >> because it is not possible to accomplish the
criterion established in >> Point 2, then the proof is useless
because that criterion is not >> valid for that purpose (Point
3). Note that the proof could be >> correct and the conclusion
too.>>There were, at last count, well over 100 proofs of the
Pythagorean >theorem. Which ones are invalid? > If all of
them are correct, all of them are valid. Why?Give an example,
outside of Cantor's various proofs, of a proof you would
categorize as useless.>> DEFINITION>> Useless proof: A
correct proof that proves nothing.>> Obviously you never have
done an useless work. Otherwise you would > know what a
useless proof is.Useless work and useless proof are not
synonymous. No proof, if actually a proof of something, is
worthless. Each of those many different proofs of the
Pythagorean theorem is worthwhile. If nothing else, the
crerator of the proof learned something.>This is an empty
definition, since it can never be instantiated. A >correct
proof, by its own definition, proves something. You are
defining >the members of the empty set.>This Futz is nearly
as wacky as JSH.> This statement requires proof, as it is
not self evident.To anyone who has followed this thread and
the many JSH postings, it IS self evident.
I'm having some
problems trying to come up with an induction proof for
thefollowing triangle (not the Pascal triangle): 1 1 1 1 1 2 3
2 11 3 6 7 6 3 1, where each element of row j is the sum of the
three elements above it inrow j-1 (null entries = 0). I want to
show that the sum of each row is 3 ^ (i - 1), where i = 1 for
thetop row in the picture. I'm fairly new with induction and
quite clumsy. It seems to me that Iwould want to show that
each each element is the sum of three elements inthe preceding
row first. But, my attempts get sloppy quickly when trying
toexplain what a null entry is in this 'picture of numbers'.
I'd like to avoid that route altogether if someone has a nicer
>: Assume I have a TM that always finds a natural number larger
than >: any natural number on any given finite tape. >: This
same TM will try to find the largest natural on an infinite
tape. >: The TM has no way of knowing the tape is infinitely
long. >: This TM will always say there is a natural number
larger than >: any natural it has found on the tape. >What
is the output of the following program if it never reaches
>the end of input? > I am assuming the computer can perform
an infinite number of > operations in finite time.I thought
you were talking about a TM.-- dik t. winter, cwi, kruislaan
413, 1098 sj amsterdam, nederland, +31205924131home: bovenover
215, 1025 jn amsterdam, nederland;
http://www.cwi.nl/~dik/
>>You probably mean 2^(2^k), seeing
as (2^k)^k = 2^k. Anyway, that is>the answer to your question:
there are 2^(2^k) non-homeomorphic>topologies on a set of
cardinality k. There are 2^(2^k) topologies on>a set of
(infinite) cardinality k, and each homeomorphism
class>contains at most 2^k of those topologies.>> Yes, sorry -
I meant 2^(2^k), the cardinality of P(P(k)). Is the> proof
relatively easy? If so, hints? If not, reference?Let k be an
infinite cardinal. Let T be the Tychonoff product of kcopies
of the 2-point discrete space. Let X be the collection of
allclopen subsets of T. Note that |X| = k and |T| = 2^k. For
each t in T,let S_t = {x in X: x contains t}. Note that the
intersection offinitely many S_t's is not contained in any
other S_t. It follows thatevery subcollection of {S_t: t in T}
generates a different topology onX. Thus there are 2^(2^k)
different topologies on X. Since eachhomeomorphism class
contains at most 2^k topologies, it follows thatthere are
2^(2^k) non-homeomorphic topologies on X.>>Is there a known
combinatorial formula for finite k?>Beats me. See A001930
at><http://www.research.att.com/~njas/sequences/> After I
studied, hard combinatorial problem. Even for small k, this>
gets hard to do by hand. For k = 3, I get 9. Confirmation?I
think that's right, but why take my word for it, why don't you
clickon that link I gave you?> Next question: How many
non-homeomorphic connected subspaces of R^n> (under the usual
topology) are there? For n = 1, the answer is 3;> for n = 2,
the answer is at least omega.For n = 2 the answer is
2^(2^omega).