> 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 Christensen Easy 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 way Construct 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 Cantorâs > goof, or conversely it is just another mathematical goof about the > Cantorâs proof. > > A MATHEMATICAL PROOF IN THREE ACTS It 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 seconds I 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, and neither Google nor MathSciNet were helpful either. But I can tell you that Robin is not responsible for the Chapman-Kolmogorov equations. Robert Israel israel@math.ubc.ca Department 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. That would 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.ca Department 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 a where m(passive) = m(active) The approximation is two-fold A. Not near a space-time singularity, i.e. not falling behind the event horizon of a black hole. B. The scale is large enough so that quantum gravity metric fluctuations are ignorable. 2. To the same approximation the connection field g-force (which reduces to Newton's gravity force) is eliminated on a timelike geodesic. 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 equation d^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-like PZ: 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 tidal curvature tensor is NOT zero, then one can always locally, in principle, distinguish a true gravity field from a fake gravity field (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 any frame, whether you are in the presence of a permanent field, since permanent fields typically have non-vanishing tidal curvature, while inertial fields do not. JS: This is hardly stunning news. MTW never denied that. However, in ordinary conditions like spaceflight 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 gravitational JS: 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 acceleration imparted to this frame with respect to the source will then add calculable inertial contributions 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 & permanent gravitational fields, using locally measurable tidal effects, but we can in principle also determine 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 the g-force to the true gravitational field, and inertial effects to a separate fictitious inertial 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 gravitational field 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 connection field, then you cannot make such a distinction! PZ: Obviously. JS: Therefore, the problem here is only a semantic problem from wavering between the two different definitions of gravity field. PZ: No, it is not *merely* a semantical problem. It is a question of the proper definition of what in essence constitutes the *real gravitational field*. Of course, logically speaking, you can define the gravitational field to be a ham sandwich if that pleases you; but that doesn't mean that this is a physically appropriate 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 Einsteinian model, the gravitational field is reduced to the unified gravitational-inertial connection field, while the tidal curvature is left hanging. In the alternative model, the curvature field is integrated with the permanent component of the connection field as a fundamental aspect of the real gravitational field, while the inertial component of the connection field 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 is literally *annihilated*. In the alternative model, the unified connection field can vanish without annihilating the entire physical presence of the field. This difference in the two models has direct 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 of mass or a precession of a small spinning gyroscope of course. Also you can see deformations of shape of a geodesic droplet if the surface tension is small enough. PZ: Right. And this kind of effect is not *necessarily* scale-sensitive, notwithstanding the 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-parameter translation subgroup of the Poincare group, of course Einstein's early formulation of the equivalence principle was only an approximate rotational 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 curved space-time with the symmetric connection force field is simply the compensating gauge force field needed to restore the now local gauge symmetry which is equivalent to local general coordinate transformations. PZ: Whether this gauge symmetry has a physical basis is a *physical* question, and not just a 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 Tensor What 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's v(IT) = (h/m)Grad(Phase of Pilot BIT Wave) Note that Witten'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 - 1 Note 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 locally completely physically equivalent as Einstein supposed, then the gauge symmetry you refer 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 that his strict gravitational-inertial equivalence principle *explained* the Eotvos proportionality of the gravitational and inertial masses. It is very important to understand this aspect of Einstein's reasoning IMO. Otherwise, you cannot possibly understand his theory of general relativity. This is exactly what Einstein said to von Laue when responding to probing criticism of his theory in the early 1920s: ... what characterizes the existence of a gravitational field from the empirical standpoint 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 does not think in such intuitive [anschaulich] ways, one cannot grasp why something like curvature should have anything at all to do with gravitation. In any case, no rational person would have hit upon anything otherwise. The key to the understanding of the equality of gravitational mass and inertial mass would 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 now only be regarded as a heuristic tool of limited depth and applicability, then fine, I agree with that. Eddington, for one, took this position in 1921, and in fact this is what I have been 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.shtml I'm Costello BTW ;-) If, in addition, you locally gauge the 6-parameter Lorentz subgroup, then you get an additional torsion force field, i.e. an anti-symmetric piece of the connection field. This will obviously modify the predictions of GR for the tidal torques on extended test PZ: 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 gravity waves, comes from stringy (vortex core if one puts in a vacuum coherence O(1) ODLRO parameter) disclination defects in a Planck lattice in 4D. PZ: He shows that you can *get* tidal curvature from disclination defects in his 4D Planck lattice model. Of course that doesn't in itself prove that this is the true physical 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 was one in Nature, as Akimov & Shipov claim in Moscow, then it would correspond to dislocation lattice defects. The different 4D discrete world crystal symmetry groups of the tiled unit cell are different physical 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 principle now looks rather naive as a serious physical model. However, it was undeniably fruitful from a purely heuristic standpoint. JS: OK 7. There are still two more subgroups left in the 15 parameter Conformal Group. You will get new compensating gauge force field physics if you locally gauge the 1-parameter dilation subgroup and also locally gauge the 4-parameter special conformal transformation of constant acceleration boosts for hyperbolic motion in Special Relativity e.g. MTW has a chapter on this. PZ: This kind of formalistic groupology needs to be well-anchored in actual physics and 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 spin networks to pre-geometric spin foams to quantized Area operators and world holograms et-al I find Penrose saying: The algebra I have used to treat linear displacements and rotations together, or linear and angular momentum together, I call the algebra of twistors. I have used the term 'twistor' to denote a 'spinor' for the six-dimensional (++----) pseudo-orthogonal group O(2,4). 9. It is obvious to any School Boy that O(2,4) is simply Lorentz SR O(1,3) with string world sheet O(1,1) fiber. Therefore, string theory is inherent here. PZ: Wouldn't it be more correct to say that certain aspects of the *mathematical structure* 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 2004. BTW Second week March 14 to March 22 is looking better for me than the first week March 8 - 13. Penrose continues: The twistor group is the (++--) pseudo-unitary group SU(2,2) which is locally isomorphic with O(2,4). In turn O(2,4) is locally isomorphic with the fifteen-parameter (local) conformal group of space-time. Under a conformal transformation of space-time the twistors will transform linearly according to a representation of the group SU(2,2). p. 175 Combinatorial Space-Time Quantum Theory and Beyond (Cambridge, 1971). Suddenly the connections among 1. Quantum loop gravity of spin networks --> foams with quantized area etc. operators 2. Local gauge invariance 3. String theory 4. Twistors & Conformal Group are 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 the following triangle (not the Pascal triangle): 1 1 1 1 1 2 3 2 1 1 3 6 7 6 3 1 , where each element of row j is the sum of the three elements above it in row j-1 (null entries = 0). I want to show that the sum of each row is 3 ^ (i - 1), where i = 1 for the top row in the picture. I'm fairly new with induction and quite clumsy. It seems to me that I would want to show that each each element is the sum of three elements in the preceding row first. But, my attempts get sloppy quickly when trying to explain what a null entry is in this 'picture of numbers'. I'd like to avoid that route altogether if someone has a nicer idea to try. Any helpful ideas for this newby? Sincerely, Shane ==== >: 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, +31205924131 home: 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 k copies of the 2-point discrete space. Let X be the collection of all clopen 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 of finitely many S_t's is not contained in any other S_t. It follows that every subcollection of {S_t: t in T} generates a different topology on X. Thus there are 2^(2^k) different topologies on X. Since each homeomorphism class contains at most 2^k topologies, it follows that there are 2^(2^k) non-homeomorphic topologies on X. >>Is there a known combinatorial formula for finite k? >Beats me. See A001930 at > After I posted, I googled number of topologies. Apparently this > is a 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 click on 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).