mm-247 === Subject: Re: (n to the 5th power) mod 10> If p is prime, then for all integers a and b,> a^((p-1)b+1) = a (mod b).> More generally, if m is any squarefree integer, i.e., m is a product> of distinct primes, then> a^(phi(m)b+1) = a (mod m).> Here phi(m) is Euler's phi function. If m=p1*p2*...*pr, then> phi(m) = (p1-1)*(p2-1)*...*(pr-1).Ok, I was suspecting some such generalization. Now I see the essentialpremise is square free. Now phi p^k = p^k - p^(k-1) for prime p,phi p = p-1. To conclude for m that phi(m) = (p1-1)*(p2-1)*...*(pr-1).how is it that when n,m are coprime, phi mn = (phi m)(phi n) ?----== News==----http://www.newsfeed.com The #1 Newsgroup Service in the World! >100,000 Newsgroups---= 19 East/West-Coast Specialized Servers - Total Privacy via Encryption === =---Subject: 3x-1I was messing around with the 3x+1 (Collatz Conjecture) and flipped the signso that the algorithm was 3x-1. The results are similar, but not the same.I found for all integers less than 1,500,000 the set always resulted in apattern, I chose the following integers to end the sequence 1, 7, and 17 (aSpecial number, and two primes).Is 3x-1 the === same type of problem as 3x+1?Mike CurrySubject: 3x+3I again, took Collatz and changed the algorithm to 3x+3 and founbd that allof the numbers end with the integers of 12, 6, and 3.But the *interesting* thing is that any number that is a divisor of 8 === endsin a series like 3x+1 - 4, 2, 1.Subject: Re: 3x+3> I again, took Collatz and changed the algorithm to 3x+3 and founbd that all> of the numbers end with the integers of 12, 6, and 3.> But the *interesting* thing is that any number that is a divisor of 8 ends> in a series like 3x+1 - 4, 2, 1.See http://www.numbertheory.org/pdfs/survey.pdf where a discussion of the generalized Collatz function is made.Instead of {3x+1 if x is 1mod2, x/2 if x is 0mod2}I think the problem has been studied for {ax+f(a,p) for x being i mod p}, but I can't find a reference right now... I believe most other cases are easy to find nontrivial cycles. As for your 3x+3 case, one property you can see is if you start (or reach) a number, n, which can be divded by 3 exactly once (i.e. is a multiple of 3 but not a multiple of 9) then it always yields another number with that property. (since if n = 3k where k is NOT a multiple of 3, then 3n+3 = 9k+3 = 3(3k+1) which is 3 time some number not divisible by 3.) Now, convince yourself that starting with any number that is a multiple of 3 will always turn into (in one step!) a number which can be divided by 3 exactly once. Now, if you don't start with some multiple of 3 then it will turn into a multiple of 3 after 1 step and then into a number of 3-multiplicity of 1 on the next step. If you want to be doing exhaustive search on this, you could just search on the numbers with 3-multiplicity of 1: 3,6,12,15,21,24,30,33,39,42,48,51,57,60,66,69,75,78,84,87, etc... which, for every range of k to k+15, there are only 4 numbers to check so it saves (cuts out) 11/15 (over 70%) of the search space. There might be some other obvious properties that would lead to a complete solution. In short, the 3x+1 problem is the only one that will win you a million bucks === ;-)JSubject: Re: 3x+3> I again, took Collatz and changed the algorithm to 3x+3 and founbd that all> of the numbers end with the integers of 12, 6, and 3.> But the *interesting* thing is that any number that is a divisor of 8 ends> in a series like 3x+1 - 4, 2, 1.You can map the Collatz mapping to this mapping but not vice versa.Ie, if you have a Collatz sequence, m_0, m_1, ... map it to 3m_0,3m_1, ... with the mappings (3x+1)/2 and x/2 going to mapping (3x+3)/2and x/2.Then if the next element in the old sequence is (3x+1)/2 (priorelement being x and odd or even as appropriate), in the new mapping itis 3(3x+1)/2 = (3(3x)+3)/2 (where (3x) is the corresponding priorelement in the new sequence), and x/2 in the new mapping is (3x)/2.Hence, multiples of 3, ie, 3u in the 3x+3 scheme correspond exactlywith numbers u in the old 3x+1 scheme. As you note, 4, 2, and 1eventually end up in the {3,6} repeating sequence under the 3x+3scheme (or {3, 12, 6} if you don't automatically divide by 2).Karl === Hallowellkhallow@hotmail.comSubject: Re: a matrix problem related to eigevaluesinduction?I remember I once did similiar problem when I tooka matrix course, but I forgot what to do now.Try Matrix Analysis, there may have some math tools.Qianz> I have a matrix A(n),> where A(2)= [0 1/2> 1 1/2]> A(3)= [0 0 1/3> 0 1/2 1/3> 1 1/2 1/3]> A(4)= [0 0 0 1/4> 0 0 1/3 1/4> 0 1/2 1/3 1/4> 1 1/2 1/3 1/4]> And so on.> My purpose is to find the eigevalues. Through matlab simulation I> found they should be:> 1, -1/2, 1/3, -1/4,..., (-1)^n/n> But I cannot prove it.> === ClimberSubject: Re: Byron and Fuller> If you have access to Byron and Fuller's Mathematics of Classical and> Quantum Mechanics and would like to help me unravel some of their text> then please read on.> On page 222 (of volume 1) they have the following definition: Definition> 5.7. A set of orthonormal functions is said to be *closed* if no nonzero> function is orthogonal to every function in the set.> My question is, is this defintion correct, or is it rather a *complete*> set of functions which is defined in this way?Defnitions can't be correct or incorrect, but they may begenerally accepted or not generally accepted, alternatively,standard or nonstandard.I presume here that function effectively meanselement of a Hilbert (or simply an inner product) space.It is certainly not standard to call such a set closed.The word closed has a standard definition in general topologyand this is used in Hilbert space theory, as one often needsto deal with closed subspaces. I presume these authors avoid this,or otherwise have anothe nonstandard word for closed?> They then go on to discuss the difference between closed and complete as> applied to set of orthogonal functions, but before I tackle this I need> to know if their defintion is right.Hmmm. You just give the definition of complete as applied to orthogonalsets. I wonder (well, not really) what they mean by complete. On the otherhand complete is an overworked word: we also have complete metric spaces(Hilbert spaces are an example of course :-)) and that's a completely (:-))different concept!-- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlHis mind has been corrupted by === colours, sounds and shapes. The League of GentlemenSubject: Re: posting-host=gatekeeper.tripos.com; posting-account=35661; posting-date=1061456758X-URL: http://mygate.mailgate.org/mynews/sci/sci.math/ cc98765b0f194d9f385b091567b91abf.35661%40mygate.mailgate.org > Hmmm. You just give the definition of complete as applied to orthogonal> sets. I wonder (well, not really) not even a little bit? what they mean by complete. I've posted their definition of complete in reply to David Ullrich'sreply.complete when describing spaces. In this particular context the wordrefers to sets of functions, or === Mailgate.ORG Server - http://www.Mailgate.ORGSubject: Re: Byron and Fuller>> Anyway, now I'm very curious what their discussion of the>differences between closed and complete amounts to,>> because completeness and closedness (using their>> definition) are equivalent. Does their discussion of the>> differences end up by saying there are no differences? Theorem 5.2. A set of orthonormal functions in Hilbert space is complete> if and only if it is closed.So the answer is yes.And their use of closed is vile terminology :-(-- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlHis mind has been corrupted by colours, sounds and shapes. The League of === GentlemenSubject: Re: Calculus is irrational?Acid Pooh> You're reasoning about imaginary unicorns right there. Since> according to your definition of rationality, you cannot do this, and> you just did, your definition is obviously wrong.How does my defintion of rationality say that I cannot reason aboutunicorns?My defintion of rationality says: reasoning about something you'veobserved.It doesn't say that it's impossible to reason about something youhaven't observed. It just says that reasoning about something youhaven't observed isn't rational. It can still be reasonable. > Here's a pretty good definition of rationality: imagination bounded> by logic. If you can imagine something, you can reason about it.I don't disagree with the second sentence. But how about imaginationbounded by logic and observation as rationality? === Why resistence tothat suggestion?Subject: Re: Calculus is irrational?Hamish Reid > Most notions of God are amenable to rational analysis, yes. Do you > really think otherwise, or is it just blind faith that tells you > someone's god doesn't exist in the real world? Is three a rational > notion? Are you saying that your mind's model of the universe contains 3 asclearly defined as god?I'm saying rationality is limited by your ability to model theuniverse, and modeling requires observation which has it's ownlimitations.I can explain the cause of 3, but I explain the effects of it reallywell.> How about experiential? > Experimental is observation without the subsequent reason.> Did I say experimental?Ah, my bad. Still, experiential is anecdotal and not necessarilysomething I'd === like to equate to rationality.Subject: Re: Calculus is irrational?> Hamish Reid > Most notions of God are amenable to rational analysis, yes. Do you > really think otherwise, or is it just blind faith that tells you > someone's god doesn't exist in the real world? Is three a rational > notion? > Are you saying that your mind's model of the universe contains 3 as> clearly defined as god?What does that have to do with whether notions of God are amenable to rational analysis?> I'm saying rationality is limited by your ability to model the> universe, and modeling requires observation which has it's own> limitations.We know what you're *saying* -- and what you *think you're saying* (for most of the world, using standard definitions of rational, etc., they turn out to be different things. Never mind).> I can explain the cause of 3, but I explain the effects of it really> well.But can you explain what the sentence above means? A good translation into formal math or English would suffice.>How about experiential? > Experimental is observation without the subsequent reason.> Did I say experimental?> Ah, my bad. Still, experiential is anecdotal and not necessarily> something I'd like to equate to rationality.Well, I guess your grip on the standard usage and meaning of experiential is as good as it is with rational, so I'll just bite my tongue at the non sequitur experiential is anecdotal (this is like watching a slow-mo car accident).Look, this is getting boring very quickly, so this will be my last posting. What seems to have happened is that sometime in the recent past you stumbled across the fact that much of math (some would say all of it) is not directly rooted in real world human experience, and that despite its unreasonable effectiveness in the real world, calculus is actually pretty abstract and often based on stuff that just doesn't happen in the real world.Well, that's a pretty commonplace observation, and you probably wouldn't have had much argument one way or another here in sci.math (it's the sort of thing that starts drunken brawls in frat bars around Cal, but never mind...). But you didn't say anything like the above. Instead, you barged into a newsgroup that has a bunch of math-literate people in it (some of whom are actually real mathematicians...) and (in calculus assumes infinity to come to its answers (for example, the>limit of a function, we sum to infinity to find an answer) and because >infinity is irrational (infinity being defined by p/0 and rational >number defined by p/q where q <> 0) is it fair to say that any answer >given to us by calculus is by definition irrational as it assumes >irrationality in the solution?>I'm not saying that irrational is equal to bad or wrong or >useless because it is obviouslly none of those things, but am I right >in thinking that all solutions that require calculus are mathematically >defined as irrational?Well, what to say about a posting that has so much wrong with it? The first response is that the answer to the last question is obviously no. Mathematically irrational has a well-defined meaning, and it doesn't apply at all to calculus as a whole. This gave most readers here a strong clue that you didn't know much math or much about math. Strike one.You started with If calculus assumes infinity.... But -- as anyone with a decent knowledge of math would know -- modern calculus does not (have to) assume infinity. Strike two.You then continued and because inifinity is irrational (inifinity being defined by p/0 ...). Strike three. And so on. It was a mathematically- and logically-confused mish-mash, and most responses pointed this out, and tried to find out what you *really* meant.You responded by implicitly redefining rational to something it doesn't mean in standard or formal usage (or in usage by anyone I've ever read or met until now), and equivocating between the standard meaning(s) and your meaning(s). You then got irritated by our misunderstanding of what you meant when you said something like calculus is beyond human rationality. That phrase actually has quite well-defined formal and informal meanings in the real world, so it was hardly surprising that you copped a bit of flack for what was for most people a trivially false statement. It was as though you'd walked into an aviation newsgroup and used the word flaps when you meant elevators (maybe because elevators can actually be made to flap), and then got bent out of shape when someone called you on obviously-nonsensical phrases like and then I pushed the control yoke forward to force the flaps down....Anyway, back here, you then explictly redefined human rationality to mean something it doesn't currently mean (and your definition is much more restrictive than the standard one), and once again seemed surprised when no one else thought that that was a good idea. You resisted suggestions that perhaps you ought to use some standard existing phrase which covered the meaning you wanted, or coin your own phrase or word. You were unable to defend your non-standard usage beyond saying that it covered what you wanted it to cover, standard usage and definition be damned.Well, I'd be surpised if anything said here will change your mind. You *could* learn a little about communication, and probably say some interesting things (especially once you learn a little more about math), but until you do that, well, you'll be the only one who understands what you really mean -- and talking to yourself === gets old very quickly, no? HamishSubject: Circulant matrix of odd dimensionIs it true that if A is a n X n 0,1 circulant === matrix and n is odd thenthe rank of A is also odd ?Subject: Re: Circulant matrix of odd dimension> Is it true that if A is a n X n 0,1 circulant matrix and n is odd then> the rank of A is also odd ?Suppose n is oddThe eiegnvalues of a real circulant matrix with top rowa_0, a_1, ..., a_{n-1} are e(z) = sum_j a_j z^j wherez runs through the n-th roots of unity. If e(z) is nonzero then sois e(z*) where z* is the conjugate of z. So the parity of the numberof nonzero eigenvalues is even iff e(1) = 0. Hence the rank is oddiff e(1) =/= 0, that is iff the top row has nonzero sum.Hence if the entries are in {0,1} the rank is odd save for the trivialcase of the zero matrix.-- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlHis mind has been corrupted by colours, sounds and shapes. The League of === GentlemenSubject: Re: Cyclotomic polynomials with one negative coefficient>Is it true that if the n-th cyclotomic polynomial has exactly one>negative coefficient then n is of the form 2^i*3^j ?> It's true at least for n up to 2000. > If C_n is the n'th cyclotomic polynomial, it seems that > when n=2^i*3^j for i,j>0, C_n(t) = t^(n/3) - t^(n/6) + 1. Well, > that should be provable by induction, together with > C_n(t) = t^(n/2)+1 if n is a positive power of 2 and > C_n(t) = t^(2n/3) + t^(n/3) + 1 if n is a positive power of 3.> I don't know how to prove that these are the only cases where C_n has one > negative coefficient, though.> Of course since the zeros of C_n are symmetric under t -> 1/t,> the coefficients satisfy a_j = a_{d-j} where d is the degree,> and if you have one negative coefficient it can only be a_{d/2}.> Robert Israel israel@math.ubc.ca> Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia > Vancouver, BC, Canada V6T 1Z2Don Coppersmith now answered my question here :http://mathforum.org/discuss/sci.math/t/530160Again , I can't === find it in Google groups .TimSubject: Dark Matter PuzzleFeedback question from a reader with my answer:But if the 23% dark matter is exotic vacuum, wouldn'tthat behave as a uniform gravitating background which wouldhave cosmological consequences (Omega =~ 1) but, likelight axions, etc.) wouldn't clump sufficiently to explainflat galaxy rotational curves, galaxy velocities inclusters in excess of apparent escape velocity, or fitmodels of structure formation. How can vacuum clump?Do not think of /zpf(x) as a constant, but as a variable.The idea is that /zpf(x) is a variable local field. Also I should write/(x,L) at a scale L. Consider a small space-time volume elementat scale L if /zpf (x) > 0 in that volume element it acts as repulsivedark energy. If, alternatively /zpf(x) < 0 there it acts as attractive dark matter. So, for example, a galactic halo is simply roughly a spherical region of space where /zpf(x) < 0 with an appropriate variable field configuration to stabilize the galaxy. Do not think of /zpf as a constant - also it is scale dependent like a wavelet transform. On a large scale it approaches the Einsteincosmological constant in the FRW metric. But on smaller scalesclumps of exotic vacuum with /zpf(x) < 0 field configurations willappear like massive slow moving matter with apparent w ~ 0, even thoughin the interior of the clump the local w = pressure/energy density = -1with positive quantum pressure. There is no reason to assume that thisvariable local field always manifests as a uniform gravitational background. Quite the contrary. Indeed, I have shown that a dark matter exotic core on the scale of 1 fermi stabilizes the electron as a spatially extended object of size e^2/mc^2 ~ 1 fermi with a cloud of virtual electron positron pairs extending out to ~ h/mc ~ 10^-11 cm. because of the strong micro-curvature distortion. The exotic vacuum clumps precisely because it is represented by a local field /zpf(x,L) of residual zero point energy density at coarse-grained event x at scale L.Some typo corrections belowRe: Timothy Ferris on C-Span LA TIMES Book FairSeduction of ScienceFerris is a good writer, though not a physicist, he has a good understanding of the conventional wisdom in mainstream physics today.The Question is: What is The Question?Ferris describes the wrong question asked by all of the pundits in modern precision cosmology, which isthe dark matter?This is an important issue because, as Ferris was quick to point out, contrary to John Maddox's (NATURE) comment, inflationary cosmology is an excellent theory since 3D space is now measured to be flat on the large scale > 10 megaparsecs and the black body cosmic microwave background temperature variations ~ 10^-5 over the whole celestial sphere with scale invariant Gaussian fluctuations are now, since WMAP, measured with good accuracy and a precision ~ 2%.Heisenberg uncertainty principle that, for example, the product of the momentum and position uncertainties is greater than a quantity proportional to Planck's quantum of action. Similarly, for energy and time, for angular momentum and angle, for number and phase, electric and magnetic fields, generalized to any pair of incompatible non-commuting matrix observables in quantum BIT Hilbert space etc.interior Heisenberg uncertainty principle that the products of uncertainties in conflicting measurements is smaller than the There are two kinds of physical vacuum spacetime regions, exotic and non-exotic. The non-exotic vacuum does not gravitate and it is close to (including real photons forming radiation, and even giant quantum states of virtual photons forming near fields like magnetic fields in MHD plasmas, in pulsars etc.). The nonexotic vacuum on the large scale has zero cosmological constant / = 0.Gravity lensing, type 1a supernovae, WMAP et-al all show that the physical vacuum on a large scale is only about 4% in terms of the Omega of standard FRW cosmology from Einstein's theory of gravity. Our expanding 3D space is accelerating rather than slowing down. This means, from type 1a supernovae data, / > 0 that is an anti-gravity space warp field with an Omega that is ~ 73%. Flat 3D space then requires a dark matter component whose Omega is ~ 23%. The three main components of the stuff of our local universe then adds up to 100% with ordinary atoms and radiation in non-gravitating vacuum only ~ 4% of the universe. Our local universe is what we can see in our telescopes in all regions of the electromagnetic spectrum. That will include future gravity wave and neutrino detectors as well. That is, our local universe the past light cone in curved spacetime on the large scale. There are an infinity of parallel universes outside our past light cone. Indeed, one needs to use Cantor's theory of transfinite numbers with an infinity of infinities as Scientific American on Parallel Universes. These universes are geometrodynamical in the sense of Wheeler's IT not the same as David Deutsch's informational BIT multiverse (Fabric of Reality).Wheeler says IT FROM BIT which requires delving into the alternative meanings of quantum theory. I suggest Wheeler's idea is incomplete needing also BIT FROM IT to get his Oroborus of the universe as a self-excited circuit.http://www.metanexus.net/ultimate_reality/ agenda.htmEinstein's large-scale cosmological constant is really a limiting case of the local complex scalar field responsible for inflation. The Dirac electron vacuum dominates to a good approximation but it is unstable from the exchange of a virtual photon between a virtual electron-positron pair at the Fermi surface of zero energy with momentum 10^19Gev/c. The micro-quantum Dirac vacuum collapses to a macro-quantum coherent vacuum of lower entropy. The local complex scalar field is the vacuum polarization coherence. The coherence length is c/H(t) where the Hubble parameter is H(t) = R(t),t/R(t) in the large scale limit of the FRW metric including / =/= 0. This local complex numbered scalar field PSI(x) has a Goldstone phase and a Higgs amplitude. Einstein's geometrodynamic c-number field guv(x) of superfluid curved spacetime is simply emergent from the modulation of the Goldstone phase according to the IT FROM BIT Bohm pilot wave guidance equationguv(x) = nuv + (1/2)Lp^2 (Goldstone phase(x)){,u,v}, means ordinary partial derivative, { }means symmetrized anti-commutator and [ ] means commutator, nuv is the globally flat Minkowski metric tensorLp^2 = hG/c^3 consistent with black hole thermodynamics.Torsion fields correspond to the non-integrable anholonomy (Goldstone phase(x))[,u,v] =/= 0Susskind's world hologram corresponds to usingLp* = Lp^2/3(c/Ho)^1/3 ~ 1 fermi ~ 10^-13 cm rather than Lp ~ 10^-33 cmThe local exotic vacuum zero point energy fluctuation field is then/(x)zpf = Lp*^-2[1 - Lp*^3|Higgs Amplitude(x)|^2]where /(x) > 0 is anti-gravitating dark energy exotic vacuum and /(x) < 0 is gravitating dark matter exotic vacuum.The zero point energy induced electron rest mass ism~ (e^2/c^2)|/zpf|^1/2where |/zpf |~ (1 fermi)^-2 ~ Lp*^-2 on the scale of 1 fermi with /zpf < 0 dark matter exotic vacuum core to prevent the explosion of the self charge ~ e.The string theory alpha' corresponds to this same value for /zpf on the microscale.Dark matter ~ 23% of the stuff of our local universe is simply exotic vacuum with positive zero point energy pressure. In contrast, dark energy ~ 73% of our local universe is simply exotic vacuum with negative zero point energy pressure that is equivalent to Kip Thorne's exotic matter for traversable wormholes and Alcubierre timelike geodesic warp drive vacuum propulsion when we learn how to locally metric engineer the physical vacuum. Potential applications of this new physics are described in Chapter Nine of Sir Martin Rees's Our Final Hour.Jack SarfattiAmerican for background.Imagine a single Level 1 post inflationary bubble which (suppress 1 space dimension temporarily) is like an infinite flat rubber sheet Euclidean plane that is not only stretching isotropically, but is accelerating in its expansion rate. Our local universe is the interior of an expanding circle on the plane. The radius of the circle is c/H(t). The circumference of the circle is singular in the sense that the vacuum coherence PSI = 0 on the circumference where the micro incoherent --> macro quantum coherent vacuum inflationary phase transition happened. That is, the circumference is analogous to a vortex line in Type II superconductor with a lattice of vortex lines. Each vortex line is analogous to a local Level I parallel universe.The strong force hadronic resonances lie on a set of parallel lines of identical slope when spin J is plotted against square of mass m of the resonance peak. Each line corresponds to a fixed assignment of strong, weak and electric charge. The universality of the slope suggests strong short-range gravity induced by the zero point fluctuations of all the basic quantum fields.In my theory, the basic data of string theory for the strong sub-nuclear color force of SU(3) becomes simplyJ ~ |/zpf|^-1(mc/h)^2 = Lp*^2(mc/h)^2 = Lp^4/3(c/Ho)^2/3 = G*m^2/hc(+ intercept depending on the charges)where for the real electron (lepto-quark) on mass shell, for example, m ~ m(Planck)e^-1/z as in BCS theory. Ho is the large-scale cosmological Hubble parameter for our particular local universe in the sense of <0|e+e-|0> is the inflation complex scalar field.z is a dimensionless measure of the binding of virtual electron-positron pairs at the zero energy Fermi volume edge of the Dirac Sea vacuum with momentum 10^19 Gev/c in the micro -> macro quantum vacuum phase transition that lowers the nonclassical entropy kick-starting the arrow of time for emergent Einstein gravity with the unified dark energy/matter field./zpf = Lp*^-2[1 - Lp*^3|<0|e+e-|0>|^2] = -1/Lp*^2 in the dark matter exotic vacuum core of the electronThat is,1 - Lp*^3 |<0|e+e-|0>|^2 = -1 in that case.The Hubble horizon of radius c/H(t) of our local universe section on the large scale spatially flat expanding accelerating post-inflationary bubble is analogous to the core of the superconducting vortex singularity where the local macro-quantum coherent order parameter PSI vanishes. That is where the micro -> macro quantum vacuum phase transition happened after all in which the non-classical entropy (log of volume of phase space of the vacuum) is lowered to set the cosmic clock for the irreversible arrow of time in which the initial singularity big bang starts out with low entropy compared to the later universe. This Hubble horizon where PSI = 0 is a 2D surface out of which our 3D space of Einstein's curved 4D spacetime emerges like the projected holographic image in Lenny Susskind's sense. That's whyLp* = Lp^2/3(c/Ho)^1/3 ~ 1 fermi (10^-13 cm)where Lp ~ 10^-33 cm and c/Ho ~ 10^28 cmYou can think of the infinity of Tegmark's Level I parallel universes on a single post-inflationary bubble as a kind of type II superconductor lattice of vortices. Each vortex line maps to a Hubble horizon. Then there are, in the chaotic inflationary model of Tegmark's Level II, an infinity of such bubbles floating in hyperspace (in Kip Thorne's sense) like a glass of champagne (brewed by Rashi de Troyes 1000 years ago ;-)) These are not the QUBIT parallel worlds of David Deutsch's multiverse. The QUBIT multiverse is a landscape for the ensemble of IT system points of geometrodynamic post-inflationary bubbles to roll on like invs = (h/m)Grad(arg psi) in deBroglie-Bohm-Vigier theory in 3D, which for 4D geometrodynamics becomesdu = Lp*^2 argPSI,u,u is the ordinary partial derivative of the Goldstone phase argPSI, u = 0,1,2,3PSI = <0|e+e-|o> is the inflation complex scalar field|PSI| = Higgs field.du = Hagen Kleinert's world crystal lattice distortion field in which curvature and torsion are disclination and dislocation defect densities as shown for ordinary crystals by Bullough, Bilby et-al at Harwell AERE Theoretical Physics Division in studies of breakdown of nuclear reactor control rods. I was at Harwell in that division in the summer of 1966 after spending time at Heisenberg's Max Planck Institute in Munich at a Wheeler seminar.http://qedcorp.com/APS/Vigier4.pdfJack === SarfattiSubject: Re: Eigenvector problem>> Given non-negative matrix A with zero diagonal and rows sum to < 1,>> define B = A + A^2 + ... Problem: efficiently find all>> eigenvalues/eigenvectors of B (an algorithm is fine).>Since I + B = (I - A)^-1, the problem can be solved in general position>(ie. assuming A has a complete set of eigenvectors) by noting:> This doesn't require a complete set of eigenvectors.>x in Spec(B)> <==> 1/(x + 1) in Spec(I - A)> <==> 1 - 1/(x+1) in Spec(A)>and this correspondance in eigenvalues extends to a common basis>of eigenvectors for A and B.> Even without a complete set of eigenvectors, v is an eigenvector of B> for eigenvalue x iff v is an eigenvector of A for eigenvalue 1-1/(x+1).> In general, for any analytic function f in a neighbourhood of> Spec(B), Spec(f(B)) = f(Spec(B)), and if v is an eigenvector of B for > eigenvalue x then v is an eigenvector of f(B) for eigenvalue f(x). > Here f(B) is defined via the holomorphic functional calculus. For > rational functions, it's easy to prove directly. In this case, > f(z) = 1 - 1/(x+1) is one-to-one from C {-1} onto C {1}, its> inverse being also a rational function, so we have if and only if. > Numerically finding all the eigenvectors and eigenvalues of A (or B) > is not, I think, much different from finding the eigenvectors and > eigenvalues of a matrix in general. There are a number of standard > techniques. In practice, use Maple, Matlab or other software packages > which have good algorithms. for such properties of matrices, === eigenvectors, and so forth? -Shlomo-Subject: Re: Eigenvector problemin>> Given non-negative matrix A with zero diagonal and rows sum to < 1,>> define B = A + A^2 + ... Problem: efficiently find all>> eigenvalues/eigenvectors of B (an algorithm is fine).>Since I + B = (I - A)^-1, the problem can be solved in generalposition>(ie. assuming A has a complete set of eigenvectors) by noting:> This doesn't require a complete set of eigenvectors.>x in Spec(B)> <==> 1/(x + 1) in Spec(I - A)> <==> 1 - 1/(x+1) in Spec(A)>and this correspondance in eigenvalues extends to a common basis>of eigenvectors for A and B.> Even without a complete set of eigenvectors, v is an eigenvector of B> for eigenvalue x iff v is an eigenvector of A for eigenvalue 1-1/(x+1).> In general, for any analytic function f in a neighbourhood of> Spec(B), Spec(f(B)) = f(Spec(B)), and if v is an eigenvector of B for> eigenvalue x then v is an eigenvector of f(B) for eigenvalue f(x).> Here f(B) is defined via the holomorphic functional calculus. For> rational functions, it's easy to prove directly. In this case,> f(z) = 1 - 1/(x+1) is one-to-one from C {-1} onto C {1}, its> inverse being also a rational function, so we have if and only if.> Numerically finding all the eigenvectors and eigenvalues of A (or B)> is not, I think, much different from finding the eigenvectors and> eigenvalues of a matrix in general. There are a number of standard> techniques. In practice, use Maple, Matlab or other software packages> which have good algorithms.> for such properties of matrices, eigenvectors, and so forth?> -Shlomo-I'm not quite sure what sort of reference work you want, but I'll throwsome titles out.For a college level textbook you might pick up:Elementary Linear Algebra, by H. AntonFor a classic treatise on properties of matrices and eigenvectors:Matrix Theory (2 vols.), by F. R. GantmacherFor numerical methods with matrices and eigenvalues:Matrix Computations, by Golub === questionHi all,This must be a triviality to some of you, but not (yet) to me unfortunately. Here's the question:Suppose C[0,1] is our space, the space of continuous functions on the closed interval [0,1]. We have two norms on that space: the integral norm, N_1(f)=int_0^1 |f(x)|dx and the, well, let's call it moderated sup-norm, N_2(f)=||xf||_oo, i.e. sup{|x*f(x)|:x in [0,1]}. Now if we take f_n(x)=x^n we see that this is a sequence of functions converging to zero in the integral norm, but not in the moderated sup-norm, since N_2(f_n)=1 for all n, while N_1(f_n)=(n+1)^{-1}->0 if n->oo. This has led me to the strong belief that if a sequence goes to zero in the N_2-norm, that it also goes to zero in the N_1-norm. Can you confirm my intuition or am I way off here? If I'm correct, can you give a hint (so that I can still try === to complete it myself) how to prove this?MichaSubject: Re: Functional analysis question>Hi all,>This must be a triviality to some of you, but not (yet) to me >unfortunately. Here's the question:>Suppose C[0,1] is our space, the space of continuous functions on the >closed interval [0,1]. We have two norms on that space: the integral >norm, N_1(f)=int_0^1 |f(x)|dx and the, well, let's call it moderated >sup-norm, N_2(f)=||xf||_oo, i.e. sup{|x*f(x)|:x in [0,1]}. Now if we >take f_n(x)=x^n we see that this is a sequence of functions converging >to zero in the integral norm, but not in the moderated sup-norm, since >N_2(f_n)=1 for all n, while N_1(f_n)=(n+1)^{-1}->0 if n->oo. This has >led me to the strong belief that if a sequence goes to zero in the >N_2-norm, that it also goes to zero in the N_1-norm. I can't imagine why you think this is evidence of that...>Can you confirm my >intuition or am I way off here? You're way off here. Hint: the reason the counterexampleworks in one direction is that N_1(f_n) is much smallerthan N_2(f_n). To get a counterexample in the otherdirection look for functions g_n such that N_2(g_n) ismuch smaller than N_1(g_n).Another way to phrase the hint: If g(x) = 1/x thenN_2(g) = 1 while N_1(g) is infinite. Now, this isofficially nonsense, because g is not an elementof C[0,1]. But you can take g_n to be a sequenceof functions that approximates g in some sense...>If I'm correct, can you give a hint (so >that I can still try to complete it myself) how to prove === this?>Micha************************Subject: Re: Functional analysis question> Suppose C[0,1] is our space, the space of continuous functions on the > closed interval [0,1]. We have two norms on that space: the integral > norm, N_1(f)=int_0^1 |f(x)|dx and the, well, let's call it moderated > sup-norm, N_2(f)=||xf||_oo, i.e. sup{|x*f(x)|:x in [0,1]}. [...]This has > led me to the strong belief that if a sequence goes to zero in the > N_2-norm, that it also goes to zero in the N_1-norm. Can you confirm my > intuition or am I way off here? If I'm correct, can you give a hint (so > that I can still try to complete it myself) how to prove this?To find the answer, === consider the functions (1-a)/x^aas a -> 1Subject: Re: Functional analysis questionHi Micha,> Suppose C[0,1] is our space, the space of continuous functions on the > closed interval [0,1]. We have two norms on that space: the integral > norm, N_1(f)=int_0^1 |f(x)|dx and the, well, let's call it moderated > sup-norm, N_2(f)=||xf||_oo, i.e. sup{|x*f(x)|:x in [0,1]}. Now if we > take f_n(x)=x^n we see that this is a sequence of functions converging > to zero in the integral norm, but not in the moderated sup-norm, since > N_2(f_n)=1 for all n, while N_1(f_n)=(n+1)^{-1}->0 if n->oo. This has > led me to the strong belief that if a sequence goes to zero in the > N_2-norm, that it also goes to zero in the N_1-norm. Can you confirm > my intuition or am I way off here?You are off !Take, for each integer n>=1, f_n(x)= 1/(xLog(Log(n))), if 1/n<=x<=1, and f_n lineal between (0,0) and (1/n, f_n(1/n)).Then N_2(f_n)-> 0 and N_1(f_n)->+ inftySo these 2 norms give non comparable metrics on C[0,1].Is this O.K. for === you?Subject: Re: Functional analysis questionX-SessionID: bf540105Let f_n(x) = b_n/a_n for 0 <= x < a_n, b_n/x for a_n <= x <= 1, for somechoice of sequences {a_n}, {b_n} of positive numbers (with a_n in [0,1]). This is in C[0,1] and N_2(f_n) = b_n, while N_1(f_n) = b_n(1 - ln a_n). So any choice such that b_n -> 0 while b_n(1 - ln a_n) does not will give a counterexample to your claim, e.g. b_n = 1/n, a_n = === e^(1-n).JeffSubject: GRE Math Subject TestI want to do MS/MA in Math. Are there some standard text books that Ican refer === to? Besides, what is rescaled in the test?kjrSubject: Help with complex exponentiationHi all!Please help me with this (not homework):n^(p+qi) = a + bia = f(n,p,q)= ?b = g(n,p,q)= === ?N.Subject: Re: Help with complex exponentiation> Hi all!> Please help me with this (not homework):> n^(p+qi) = a + bi> a = f(n,p,q)= ?> b = g(n,p,q)= ?> N.n is positive? If so, then:n^(p+qi) = exp((p+qi) log n) = exp(p log n) exp(i q log n) = exp(p log n) cos(q log n) + i exp(p log n) sin(q log === n)Subject: Re: Help with complex exponentiation>Please help me with this (not homework):>n^(p+qi) = a + biStart with n integer, p zero, q one. Then try n real. Then try n = i. For that,see # 13 in:http://www.magisnetworks.com/pdf/cto_notes/ interviewQuestions01.pdfat least I remember 13. It is the problem, in electrical notation where j isused as i , of j to the j power, and to me, was illuminating.I don't have a ref for the proof that e to the ( pi times i ) equals negativeone, or that e to a complex equals the sum of a cosine term and a sine term butthat is fundamental. Can anyone here name that formula so N. can conduct asearch for it?Yours,Doug Goncz, Replikon Research, Seven Corners, VA The hormones work at different speeds. In a fight-or-flight scenario, glucocorticoids are the ones drawing up blueprints for new aircraft carriers;epinephrine is the one handing out === guns.Subject: Re: Help with complex exponentiation> Hi all!> Please help me with this (not homework):> n^(p+qi) = a + biMuch easier if n is in polar form.Let N = the magnitude of n, |n|, and lettheta = the phase of n. That is,n = N exp(i*theta).> a = f(n,p,q)= ?> b = g(n,p,q)= ?n^(p+qi) = [N*exp(i*theta)]^(p+qi) = [exp(log N) exp(i*theta)]^(p+qi) = exp(log N)^p exp(logN)^(qi) exp(i*theta)^p exp(i*theta)^(qi) = N^p exp(logN * qi) exp(i*p*theta) exp(-theta*q) = [N^p* exp(-theta*q)] * exp[i*(logN*q + theta*p)]So the result is a complex number with magnitudeN^p*exp(-theta*q), and phase (q log N + p*theta). a = magnitude*cos(phase) b = === magnitude*sin(phase) - RandySubject: Re: Help with complex exponentiation>Hi all!>Please help me with this (not homework):>n^(p+qi) = a + bi>a = f(n,p,q)= ?>b = g(n,p,q)= ?>N. Complete solution sent to the e-mail address you had the === integrityto provide.Subject: Re: how to make a (fired) bullet stay put> slightly above that. My reasoning is this, if an aircraft is flying> 200mph (feel free to interchange aircraft with car, truck, or bicycle)> and it fires a (400mph) bullet forward, the initial speed is 600mph> because the bullet is already traveling at 200mph. I> Handgun bullets are usually in the range of 800 fps to 1300 fps > while rifle bullets are usually in the range of 1700 fps to 3200 fps.There seem to be quite a few rifle shells that provide3300-3800 fps (2250-2591 mph) muzzle velocity (see e.g.http://www.chuckhawks.com/25mag.htm ). > The 400 mph bullet would be about 587 fps and I think that > even a 45 acp would beat that.I think you are saying it would be difficult to find a slow-enough bullet to get it to drop straight down when fired out the back of a plane flying 400 mph. However, a pellet gun like the Crossman 2200B (one third of the way throughhttp://www.gungarage.com/crossmanr.htm) would be close,at 595 feet per second or 405.7 miles per hour.Regarding the 45 acp, you probably mean .45, with 820 fps.I think 45's are more typically 1800-2525 fps, like the oneI saw on a === recent trip (http://pat7.com/jp/45caliber.jpg).-jiwSubject: Re: Is there a simple way to see if a matrix can be factorizedI found this indexing scheme in ACM. It uses factorization of a matrix. Perhapsthe author gives the factoring method.In this paper, we propose a novel document clustering method based on thenon-negative factorization of the term-document matrix of the given documentcorpus.These are large matrices. Are large matrices of interest?>Is there a simple way to see if a matrix can be factorized?It is in the:Annual ACM Conference on Research and Development in Information Retrieval Proceedings of the 26th annual international ACM SIGIR conference on Researchand development in informaion retrievalToronto, Canada SESSION: ClusteringPages: 267 - 273 ISBN:1-58113-646-3 Yours,Doug Goncz, Replikon Research, Seven Corners, VA The hormones work at different speeds. In a fight-or-flight scenario, glucocorticoids are the ones drawing up blueprints for new aircraft carriers;epinephrine is the one handing out === guns.Subject: Re: leech lattice generator matrixSorry I am still very confused. Could you please explain to me in moredetails.In the paper Amrani, O. et al, The Leech Lattice and the Golay Code: Bounded-Distance Decoding and Multilevel Constructions, IEEE Transactions on Information Theory, vol. 40, No. 4, pp. 1030-1043, Jul. 1994.on p1033 it first partition the checkboard lattice into 16 subsets and label each subset either Aijk or Bijk where i,j,k in GF(2).In fig3 it gives an example. The points of leech lattice may be represented by 2x6 arrays whoseentries are the points D2. A typical array is given by[Ai1j1k1, Ai3j3k3,....Ai11j11k11 Ai2j2k2, Ai4j4k4,....Ai12j12k12]The column is said to be even or odd according as the binary4-tuple(i1,j1,i2,j2) has even or odd weight. The index i1 is calledthe h-partiyof the column. The overall h-parity of the array is defined as themodulo-2 sumThe modulo-2 sum of k1+k2 is said to be the k-parity of the columns.The overal k-parity of the array is then the modulo-2 sum of the ksubscripts of the 12 points.The binary 4-tuple(i1,j1,i2,j2) is regarded as an interpretation ofsome charactor xinGF(4), while x is regarded the projections of theentire column.The leech lattice is defined asa) It is either A-type or B-typeb) It consists either of only even columns or only odd columnsc) The overall k-parity is even if the array is A-type, and oddotherwised) The overall h-parity is even if the array columns are even, and oddotherwisee) The projection of the array is a codeword of H6.This defintion seems very similiar to the MOG defintionon SPLAG p304, the defintion for Golay code is very similiar.p133 gives the generator matrixN=[8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 4 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 4 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 4 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 4 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0; 2 2 2 2 0 0 0 0 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0; 4 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0; 2 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 0 0 0 0 0 0 0 0; 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 0 0 0 0 0 0 0 0; 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 0 0 0 0 0 0 0 0; 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0; 2 0 2 0 2 0 0 2 2 2 0 0 0 0 0 0 2 2 0 0 0 0 0 0; 2 0 0 2 2 2 0 0 2 0 2 0 0 0 0 0 2 0 2 0 0 0 0 0; 2 2 0 0 2 0 2 0 2 0 0 2 0 0 0 0 2 0 0 2 0 0 0 0; 0 2 2 2 2 0 0 0 2 0 0 0 2 0 0 0 2 0 0 0 2 0 0 0; 0 0 0 0 0 0 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0; 0 0 0 0 0 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0; -3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1];So I assume the generator matrix on p133 is the generator matrix for this kind of leech lattice defintion.But I have no math tools to prove my assumption. That's my firstquestion.On paper Tables of sphere packings and spherical codesSloane, N.;Information Theory, IEEE Transactions on , Volume: 27 Issue: 3 , May1981Page(s): 327 -338It defines the lattice lattice as a(0 +2c + 4x)a(1 + 2c + 4y)a = 1/sqrt(8) c is any codword in the ginary extended Golay code oflenght 24, andx in Z^24, y in Z^24 saitisfysum x_i = 0 mod(2)sum y_i = 1 mod(2)In the paper it gives the generator matrix G = [4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 2 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 2 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 2 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 2 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 2 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0; 2 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0; 1 1 1 0 1 1 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0; 1 0 1 1 0 1 1 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0; 1 1 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0; 1 0 1 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0; 1 0 0 1 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0; 1 0 0 0 1 0 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0; 1 1 0 0 0 1 0 1 1 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0; 1 1 1 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0; 1 1 1 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0; 1 0 1 1 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0; 1 1 0 1 1 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0; -3/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/21/21/2 1/2 1/2 1/2 1/2 1/2 1/2];While in the paperFast decoding of the Leech latticeBe'ery, Y.; Shahar, B.; Snyders, J.;Selected Areas in Communications, IEEE Journal on , Volume: 7 Issue: 6,Aug. 1989Page(s): 959 -967 it saysThis assignment of coordinates agrees, within a scale factor,with that of SLOANE Tables of sphere packings and spherical codesBut I look at the two generator matrix, it seems to me not only thescalefactor, but there is some other difference.I am very curious which generator matrix to use if I follow the leech lattice defintion in Vardy's 1994 paper.The leech lattice is defined === asSubject: line - surface intersectionGiven line inside 3d surface,the line given by[x y z] = [x0 y0 z0] + [nx ny nz] * twhere [x0 y0 z0] , [nx ny nz] are knownI would like to find the intersection points between this line and the 3dsurface(in case the surface is convex there are supposed to be two such points)note that the 3d surface is a bunch of points - 3 x n === arraySubject: Re: LOGIC ^ ATHEISM> This is a sci.logic post === that may hold some interest here, No.-- G.C.Subject: Re: Math Trick> Here is a good Math Trick that I invented.> Check this web site with a very unique Math Game> http://w4u.sytes.net/MagicMath> You can send me your comments at gonzalep@.84scientist.com don't forget to> remove the .84 from the e-mail.It didn't work when I chose 2.-- Will === Twentymanemail: wtwentyman at copper dot netSubject: Re: Math TrickWhen clicking next it didn't always take me to the next step - it oftenskipped steps.> Here is a good Math Trick that I invented.> Check this web site with a very unique Math Game> http://w4u.sytes.net/MagicMath> You can send me your comments at gonzalep@.84scientist.com don't forget to> remove the .84 === from the e-mail.Subject: notion of a RingHi guys,My problem is why a Ring is defined as it is defined. What is the purpose ofsuch a definition. A ring has to me an association with === something infinite,no beginning no end.PeterSubject: Re: notion of a Ring> Hi guys,> My problem is why a Ring is defined as it is defined. What is the purpose> of such a definition.The existence of many examples having such properties. Such definitionsenable us to create a general theory of such algebraic structures.The term ring comes from the German Zahlring which was appliedby Dedekind to describe examples like Z[(1+sqrt(5))/2].> A ring has to me an association with something> infinite, no beginning no end.How strange.Pity mathematicians work with public definitons not private associations.-- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlHis mind has been corrupted by === colours, sounds and shapes. The League of GentlemenSubject: Re: notion of a RingBouwman P.> Hi guys,> My problem is why a Ring is defined as it is defined. What is the purposeof> such a definition. A ring has to me an association with somethinginfinite,> no beginning no end.> PeterAn object with those properties deserves _some_ name, because such objectsarise in quite a few different ways. But the name ring is prettyarbitrary, as names often are. I'm just guessing, but maybe Dedekind wasthinking of the annular rings that are seen in a tree trunk. Some rings innumber theory, such asZ[(-1+sqrt(-3))/2],have elements in concentric circles around 0, and if a,a' are in the samecircle, and b,b' are in the same circle, then ab and a'b' are in the === samecircle.LHSubject: number of treesDoes anyone know how to find :The total number of binary trees with N nodes and L leaves.Does this problem have a solution with general k-ary trees.The total number of binary trees(and k-ary) can be written using thecatalan number, but there is no constraint on === the number of leaves here.vijaySubject: Re: numeric matrix computation: how to reduce computation for this matrix computation involved in matrix transformation:>Y=A*X*A' where A is orthogonal... X is input, Y is output, if I only want>the upper triangular elements of the output matrix Y,... is there any good>way of computing it?I think we won't be able to take advantage of the orthogonalityof A. But some multiplications of A X A' can surely be saved.I'm appending some of my C codes made long time ago. I thinkit would be easy to translate even if you don't know C.This function is for a general A B A', where B must be squarein order for the expression to be defined but A can be non-square.So, A: m x n, B: n x n. You can modify the last loop to takeadvantage of the symmetry of (A B A'): m x m. This functionhas been well-tested according to the comments in my codes.And I hope it really is./* * The actual codes being used is of course an improved version * which does not repeat those multiplications which have been * done by allocating an extra n-vector to store the result of * A_i * B (where A_i is the i-th row of A) in the first places * because this row vector will be repeatedly used for computing * all elements of c_ij of the same row in C. B A' +---------+ +---------------------+ | ******* | | * | | ******* | | * | | ******* | | * | +---------+ +---------------------+ +---------+ +---------+ +---------------------+ | ******* | | ******* | | * | | A_i | | x=A_i B | | c_ij = x A_j | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | +---------+ +---------+ +---------------------+ A X = A B C = X A' = A B A' * * * Input A: m x n * Input B: n x n * Ouput C: m x m */REAL **MatrixMultABAt(REAL **A, int m, int n, REAL **B) /* A_ik B_kl A_jl */{ REAL **C, *x; int i, j, k, l; C = matrix(m, m); x = vector(n);/*Direct Multiplication: C = A B A' = c_ij = a_ik * b_kl * a_jl for (i=0; iOr please point me to some resources on the web?I think there must be some, but I don't think it wouldbe easy to locate one. :)Rudi-- Please remove the string u l t r a 5 1 to === mail me.Subject: Parallel Universes & Dark Energy/Matter Exotic VacuaPSJack earlier:Did you understand my answer to your physics question?Diligent reader replied:Yes, I understand it now.Good. You know I got that same question from an expert in cosmology at Texas APS who ran the session. Apparently all the cosmologists are really used to thinking of / as a constant, at least on spacelike surfaces because it is that in the FRW large scale metric. The idea that Einstein's cosmological constant may be really a limiting case of a variable local scalar field determined by the residual inflation field appears to take some getting used to. In terms of Occam's Razor it seems to be a natural idea. Dark energy as local exotic vacuum field with negative quantum pressure, dark matter as local exotic field with positive quantum pressure.Since in my two fluid vacuum picture/zpf(x) = Lp*^-2[1 - Lp*^3|PSI(x)|^2]exotic vacuum zero point field corresponding toTotal density = Superfluid macro-quantum density + Normal micro-quantum fluid density with Normal fluid density ~ /zpf(x)Dark energy is low density vacuum superfluidDark matter is high density vacuum superfluidwhere ordinary non-gravitating classical vacuum /zpf(x) = 0 corresponds to critical density|PSI(x)|^2 = Lp*^-3which is order of nuclear number density (1 fermi)^-3 if I use Susskind's world hologram formulaLp* = Lp^2/3(c/Ho)^1/3 ~ 1 fermi i.e. 1 Gev energy scale of hadronic physics.`Ho is the cosmological Hubble parameter for our universe as actually measured by astronomers as ~ 10^28 cm.Lp is the Planck quantum gravity scale ~ 10^-33 cmThere is no hierarchy problem - or so it appears from this POV.The singular real (not apparent) Hubble horizon of radius c/H(t) where t is an appropriate FRW universal cosmic time, t = 0 as Now in typical convention is that distance back in time along the past light cones of our EM/Gravity/Neutrino wave detectors to the initial singularity where the macro-quantum vacuum coherent order parameter PSI = 0 just like in the vortex core of a type II superconductor lattice of vortices. In fact, COBE and WMAP do not see back that far. They stop at ~ 300,000 years after the big bang when the neutral atoms form from the ionized plasma in ERA 3 as shown in http://CPEPweb.org The first stars and galaxies form at ~ 3 x 10^8 years. We are now at cosmic time ~ 14 x 10^9 years. Cal Tech/MIT/NSF/DOE/ARPA/ LIGO, LISA will see further back to the PSI = 0 Heart of Darkness - so will neutrino telescopes.Our Hubble sphere horizoned local universe is one of an infinity of parallel Level I geometrodynamic IT (Wheeler) universes in the Universes (not same as QBIT multiverse) on a single post-inflationary PSI =/= 0 bubble. Our bubble is spatially flat i.e. picture it in 2D (suppressing one space dimension) as an infinite Euclidean plane made from a rubber sheet that is approximately expanding the same in every direction from every point, indeed, even accelerating in the expansion at least on large scales L. There are, further, and infinity of such post-inflationary bubbles in the Level II chaotic inflation in hyperspace.Sir Martin Rees's Just Six Numbers allowing life in our little local universe is a contingent matter of Darwinian natural selection in the sense of the Weak Anthropic Principle as explained by Max Tegmark. God at all. However, that is not the end of the Greatest Story Ever Told as explained in http://qedcorp.com/APS/Ukraine.doc (Progress in role of advanced intelligence in the universes and their super-technology of metric engineering.Feedback question from a reader with my answer:But if the 23% dark matter is exotic vacuum, wouldn'tthat behave as a uniform gravitating background which wouldhave cosmological consequences (Omega =~ 1) but, likelight axions, etc.) wouldn't clump sufficiently to explainflat galaxy rotational curves, galaxy velocities inclusters in excess of apparent escape velocity, or fitmodels of structure formation. How can vacuum clump?Do not think of /zpf(x) as a constant, but as a variable.The idea is that /zpf(x) is a variable local field. Also I should write/(x,L) at a scale L. Consider a small space-time volume elementat scale L if /zpf (x) > 0 in that volume element it acts as repulsivedark energy. If, alternatively /zpf(x) < 0 there it acts as attractive dark matter. So, for example, a galactic halo is simply roughly a spherical region of space where /zpf(x) < 0 with an appropriate variable field configuration to stabilize the galaxy. Do not think of /zpf as a constant - also it is scale dependent like a wavelet transform. On a large scale it approaches the Einsteincosmological constant in the FRW metric. But on smaller scalesclumps of exotic vacuum with /zpf(x) < 0 field configurations willappear like massive slow moving matter with apparent w ~ 0, even thoughin the interior of the clump the local w = pressure/energy density = -1with positive quantum pressure. There is no reason to assume that thisvariable local field always manifests as a uniform gravitational background. Quite the contrary. Indeed, I have shown that a dark matter exotic core on the scale of 1 fermi stabilizes the electron as a spatially extended object of size e^2/mc^2 ~ 1 fermi with a cloud of virtual electron positron pairs extending out to ~ h/mc ~ 10^-11 cm. because of the strong micro-curvature distortion. The exotic vacuum clumps precisely because it is represented by a local field /zpf(x,L) of residual zero point energy density at coarse-grained event x at scale L.Additional typo corrections below:My non-analytic (as z -> 0) non-perturbative BCS Nambu-Jona Lasino inspired mass formulam = (e/c)^2|/zpf|^1/2 = (h/cLp)e^-1/zTherefore, using Susskind's world hologram formula(e^2/hc) (LpHo/c)^1/3 = e^-1/zSo that we can numerically compute zz ~ -1/loge[(e^2/hc) (LpHo/c)^1/3] ~ - 1/loge[10^-63]using data from both quantum gravity units and cosmology.The final theory should have some kind of polynomial in z whose roots determine the lepto-quark rest masses that go into QCD.One uses QCD to get the hadronic rest masses that are mostly kinetic in origin as in the bag model of QCD Lite (Frank Wilzcek, APS March & What Haisch, Rueda and Puthoff talk about should only be a small correction to this.Re: Timothy Ferris on C-Span LA TIMES Book FairSeduction of ScienceFerris is a good writer, though not a physicist, he has a good understanding of the conventional wisdom in mainstream physics today.The Question is: What is The Question?Ferris describes the wrong question asked by all of the pundits in modern precision cosmology, which isthe dark matter?This is an important issue because, as Ferris was quick to point out, contrary to John Maddox's (NATURE) comment, inflationary cosmology is an excellent theory since 3D space is now measured to be flat on the large scale > 10 megaparsecs and the black body cosmic microwave background temperature variations ~ 10^-5 over the whole celestial sphere with scale invariant Gaussian fluctuations are now, since WMAP, measured with good accuracy and a precision ~ 2%.Heisenberg uncertainty principle that, for example, the product of the momentum and position uncertainties is greater than a quantity proportional to Planck's quantum of action. Similarly, for energy and time, for angular momentum and angle, for number and phase, electric and magnetic fields, generalized to any pair of incompatible non-commuting matrix observables in quantum BIT Hilbert space etc.interior Heisenberg uncertainty principle that the products of uncertainties in conflicting measurements is smaller than the There are two kinds of physical vacuum spacetime regions, exotic and non-exotic. The non-exotic vacuum does not gravitate and it is close to (including real photons forming radiation, and even giant quantum states of virtual photons forming near fields like magnetic fields in MHD plasmas, in pulsars etc.). The nonexotic vacuum on the large scale has zero cosmological constant / = 0.Gravity lensing, type 1a supernovae, WMAP et-al all show that the physical vacuum on a large scale is only about 4% in terms of the Omega of standard FRW cosmology from Einstein's theory of gravity. Our expanding 3D space is accelerating rather than slowing down. This means, from type 1a supernovae data, / > 0 that is an anti-gravity space warp field with an Omega that is ~ 73%. Flat 3D space then requires a dark matter component whose Omega is ~ 23%. The three main components of the stuff of our local universe then adds up to 100% with ordinary atoms and radiation in non-gravitating vacuum only ~ 4% of the universe. Our local universe is what we can see in our telescopes in all regions of the electromagnetic spectrum. That will include future gravity wave and neutrino detectors as well. That is, our local universe the past light cone in curved spacetime on the large scale. There are an infinity of parallel universes outside our past light cone. Indeed, one needs to use Cantor's theory of transfinite numbers with an infinity of infinities as Scientific American on Parallel Universes. These universes are geometrodynamical in the sense of Wheeler's IT not the same as David Deutsch's informational BIT multiverse (Fabric of Reality).Wheeler says IT FROM BIT which requires delving into the alternative meanings of quantum theory. I suggest Wheeler's idea is incomplete needing also BIT FROM IT to get his Oroborus of the universe as a self-excited circuit.http://www.metanexus.net/ultimate_reality/ agenda.htmEinstein's large-scale cosmological constant is really a limiting case of the local complex scalar field responsible for inflation. The Dirac electron vacuum dominates to a good approximation but it is unstable from the exchange of a virtual photon between a virtual electron-positron pair at the Fermi surface of zero energy with momentum 10^19Gev/c. The micro-quantum Dirac vacuum collapses to a macro-quantum coherent vacuum of lower entropy. The local complex scalar field is the vacuum polarization coherence. The coherence length is c/H(t) where the Hubble parameter is H(t) = R(t),t/R(t) in the large scale limit of the FRW metric including / =/= 0. This local complex numbered scalar field PSI(x) has a Goldstone phase and a Higgs amplitude. Einstein's geometrodynamic c-number field guv(x) of superfluid curved spacetime is simply emergent from the modulation of the Goldstone phase according to the IT FROM BIT Bohm pilot wave guidance equationguv(x) = nuv + (1/2)Lp^2 (Goldstone phase(x)){,u,v}, means ordinary partial derivative, { }means symmetrized anti-commutator and [ ] means commutator, nuv is the globally flat Minkowski metric tensorLp^2 = hG/c^3 consistent with black hole thermodynamics.Torsion fields correspond to the non-integrable anholonomy (Goldstone phase(x))[,u,v] =/= 0Susskind's world hologram corresponds to usingLp* = Lp^2/3(c/Ho)^1/3 ~ 1 fermi ~ 10^-13 cm rather than Lp ~ 10^-33 cmThe local exotic vacuum zero point energy fluctuation field is then/(x)zpf = Lp*^-2[1 - Lp*^3|Higgs Amplitude(x)|^2]where /(x) > 0 is anti-gravitating dark energy exotic vacuum and /(x) < 0 is gravitating dark matter exotic vacuum.The zero point energy induced electron rest mass ism~ (e^2/c^2)|/zpf|^1/2where |/zpf |~ (1 fermi)^-2 ~ Lp*^-2 on the scale of 1 fermi with /zpf < 0 dark matter exotic vacuum core to prevent the explosion of the self charge ~ e.The string theory alpha' corresponds to this same value for /zpf on the microscale.Dark matter ~ 23% of the stuff of our local universe is simply exotic vacuum with positive zero point energy pressure. In contrast, dark energy ~ 73% of our local universe is simply exotic vacuum with negative zero point energy pressure that is equivalent to Kip Thorne's exotic matter for traversable wormholes and Alcubierre timelike geodesic warp drive vacuum propulsion when we learn how to locally metric engineer the physical vacuum. Potential applications of this new physics are described in Chapter Nine of Sir Martin Rees's Our Final Hour.Jack SarfattiAmerican for background.Imagine a single Level 1 post inflationary bubble which (suppress 1 space dimension temporarily) is like an infinite flat rubber sheet Euclidean plane that is not only stretching isotropically, but is accelerating in its expansion rate. Our local universe is the interior of an expanding circle on the plane. The radius of the circle is c/H(t). The circumference of the circle is singular in the sense that the vacuum coherence PSI = 0 on the circumference where the micro incoherent --> macro quantum coherent vacuum inflationary phase transition happened. That is, the circumference is analogous to a vortex line in Type II superconductor with a lattice of vortex lines. Each vortex line is analogous to a local Level I parallel universe.The strong force hadronic resonances lie on a set of parallel lines of identical slope when spin J is plotted against square of mass m of the resonance peak. Each line corresponds to a fixed assignment of strong, weak and electric charge. The universality of the slope suggests strong short-range gravity induced by the zero point fluctuations of all the basic quantum fields.In my theory, the basic data of string theory for the strong sub-nuclear color force of SU(3) becomes simplyJ ~ |/zpf|^-1(mc/h)^2 = Lp*^2(mc/h)^2 = Lp^4/3(c/Ho)^2/3 = G*m^2/hc(+ intercept depending on the charges)where for the real electron (lepto-quark) on mass shell, for example, m ~ m(Planck)e^-1/z as in BCS theory. Ho is the large-scale cosmological Hubble parameter for our particular local universe in the sense of <0|e+e-|0> is the inflation complex scalar field.z is a dimensionless measure of the binding of virtual electron-positron pairs at the zero energy Fermi volume edge of the Dirac Sea vacuum with momentum 10^19 Gev/c in the micro -> macro quantum vacuum phase transition that lowers the nonclassical entropy kick-starting the arrow of time for emergent Einstein gravity with the unified dark energy/matter field./zpf = Lp*^-2[1 - Lp*^3|<0|e+e-|0>|^2] = -1/Lp*^2 in the dark matter exotic vacuum core of the electronThat is,1 - Lp*^3 |<0|e+e-|0>|^2 = -1 in that case.The Hubble horizon of radius c/H(t) of our local universe section on the large scale spatially flat expanding accelerating post-inflationary bubble is analogous to the core of the superconducting vortex singularity where the local macro-quantum coherent order parameter PSI vanishes. That is where the micro -> macro quantum vacuum phase transition happened after all in which the non-classical entropy (log of volume of phase space of the vacuum) is lowered to set the cosmic clock for the irreversible arrow of time in which the initial singularity big bang starts out with low entropy compared to the later universe. This Hubble horizon where PSI = 0 is a 2D surface out of which our 3D space of Einstein's curved 4D spacetime emerges like the projected holographic image in Lenny Susskind's sense. That's whyLp* = Lp^2/3(c/Ho)^1/3 ~ 1 fermi (10^-13 cm)where Lp ~ 10^-33 cm and c/Ho ~ 10^28 cmYou can think of the infinity of Tegmark's Level I parallel universes on a single post-inflationary bubble as a kind of type II superconductor lattice of vortices. Each vortex line maps to a Hubble horizon. Then there are, in the chaotic inflationary model of Tegmark's Level II, an infinity of such bubbles floating in hyperspace (in Kip Thorne's sense) like a glass of champagne (brewed by Rashi de Troyes 1000 years ago ;-)) These are not the QUBIT parallel worlds of David Deutsch's multiverse. The QUBIT multiverse is a landscape for the ensemble of IT system points of geometrodynamic post-inflationary bubbles to roll on like invs = (h/m)Grad(arg psi) in deBroglie-Bohm-Vigier theory in 3D, which for 4D geometrodynamics becomesdu = Lp*^2 argPSI,u,u is the ordinary partial derivative of the Goldstone phase argPSI, u = 0,1,2,3PSI = <0|e+e-|o> is the inflation complex scalar field|PSI| = Higgs field.du = Hagen Kleinert's world crystal lattice distortion field in which curvature and torsion are disclination and dislocation defect densities as shown for ordinary crystals by Bullough, Bilby et-al at Harwell AERE Theoretical Physics Division in studies of breakdown of nuclear reactor control rods. I was at Harwell in that division in the summer of 1966 after spending time at Heisenberg's Max Planck Institute in Munich at a Wheeler seminar.http://qedcorp.com/APS/Vigier4.pdfJack === SarfattiSubject: Re: Primes and the Collatz conjecture!Christian Bau> For any Collatz sequence path find only the prime start numbers that> creates an odd prime for each of its odd integers except 1.> Here are the first few prime start numbers that meet these criteria> --- 3,5,7,11,13,17,19,29,37,53,59,67,89,101,131,149,157,179,181,197 , 241,269,277,349,397,739,853,1109,1237,1429,1621,1861,1877,2161, 2389,2531,2957,3413,3797,45> 49...> I believe I found them all up too prime 4549.> What is interesting also, only primes from the (above list) are> possible in any sequence where all the odd integers are prime except 1> and where this prime list above ------>oo.> If you do not believe this to be true, please show a counter example.> Also, the all odd integer = prime paths (except start # 3 and 5) all> seem to merge at this point 40,20,10,5,16,8,4,2,1> I think k = 22,369,621 is a prime, and 3k+1 = 2^26. The odd numbers on> the path are 22,369,621 -> 1 :-)Er, 22,369,621 = 8191*2731 (in hex, 0xAAB * 0x1FFF). But I get the idea.Anyhow, the Collatz conjecture is === not really about arithmetic :)LHSubject: Re: Publishing>Is this the cause of Aberg's animus against Serre?>Did Serre reject one of his papers?aberg's answer:> Serre, as editor, does not referee papers he does not want to see> published, but instead quickly returns a letter stating that he will> pretend that the already formally submitted paper has not been submitted> to him, and if the submitter insists on having the paper refereed, it will> not become published anyhow. I think there must be quite a few such Serre> letters around. (In the cases I know of, the papers where published> elsewhere and eventually judged to be of utmost quality, but Serre's> unusual editorial practises caused a long delay.)> This broke off from what the people I knew at the time thought were the> ethically correct procedures of scientific publishing, that papers> submitted to a scientific journal should be registered and properly> refereed. If the journal has some requirements that submitted papers> should be registered when submitted, then there is a formal clerical error> as well.> I think that if Serre says that he knows that a paper will not be> published even if refereed, prosecutors will say that it is circumstantial> evidence that he has a way to bypass proper refereeing. Such methods was> described to me in those days, but I have not made investigations as to> who is doing what (as I do not really care, not working in that circuit).or put === more simply: yes....Subject: Re: Publishing>Why would you appoint someone as editor in the first place if you didn't>want him to exercise his judgment about what should and should not be>published? >And once the editor has determined that something is not going to be>published, why should he waste the time of a referee by sending it out>anyway? It is idea that papers should be published based on scientific standards,within the declared scope of topics of the journal, and not on theindividuals whim, and the editor is set to ensure that those proceduresare honored and correct. You can compare with the procedures of say ajudge in a court, where the judge is supposed to listen to the evidencefirst, not tampering with it or the evidence gathering procedures, andthen make a judgment based on that evidence.The underlying principle has to do with the role diverse scientificknowledge is supposed to play in society. The principle is similar to thatof ensuring free and open competition in a market economy. There are ofcourse journals where the editors can do anything they please, but theycannot be called scientific.Your questions are rather chocking, because you say in effect that in thisparticular circuit there is no need to ensure scientific standards as inrest of science. Hans Aberg * Aberg * Home Page: === Subject: Re: set theory with atoms>In the first exercise on p.51 of Jech's book, The Axiom of Choice, he asks>the reader to prove the consistency of ZFC+there are infinitely many atoms.>He also gives a generous hint on how to construct a model. His chapter notes>don't attribute this construction to anyone. What I would like to know is a>reference to the publication in which this construction first appeared.AFAIK, this was first done in the 1920's by Fraenkel, asthe first step in the construction of Fraenkel-Mostowskimodels; there is some question whether Fraenkel had all thesteps correct. Fraenkel's paper showed the consistency ofthe negation of AC in an inner model of this model.If this is not the case, that paper should have a referenceto an earlier use.-- This address is for information only. I do not claim that these viewsare those of the Statistics Department or of Purdue University.Herman Rubin, Department of Statistics, Purdue Universityhrubin@stat.purdue.edu Phone: (765)494-6054 FAX: === (765)494-0558Subject: Re: solving algebraic equation> That is, what you are doing is a series of steps, some> of which involve if and only if transformations> (and are therefore reversible), and some of which involveimplies, i.e. only if transformations which are> not one way.> [maths elided]> Is that an also considered keeping> track of the sign of x (in the present case) as a remedy, but it is> merely another way of checking both solutions at the end. Right. However, it puts the cookbook recipe on a logicalfooting. The reason your transformed equation has solutionsthat are not solutions to the original is that the transformationis not reversible. However, it is important to note that thetransformation is valid in the sense that the solutionof the original is part of the solution set of thenew equation. You need that much of a guarantee.Stated more succinctly, you want x that solves A.You transform to B with the property that the solutionset of A, S(A) is a subset of S(B).> In the meantime I dug out a math textbook which states that for this> kind of algebraic equations one always needs to resubstitute the> solutions to tell the 'correct' one...Right, but that's just a method, not === an explanation. - RandySubject: Re: Synergetics your visions - concerning especially four dimensional explain my reply more detailed to your earlier>post before You can consider, or do You accept my arguments as such?>Of course - You are welcome to express counterarguments.> There is a model for a correspondence of xyz Cartesian and the xyz axes> of the equilateral triangle. The Cartesian axes could go through the> centers of the six square faces of the cuboctahedron. I can see the> ninety degree angles between the Cartesian axes and the one hundred and> twenty degree angles between the axes of the triangle at the same time.> I can't see ninety degree angles between the four dimensional Cartesian> xyzw axes or how they would correspond to the four axes of the regular> tetrahedron, so, Cartesian coordinates are an afterimage that just> gets in my way.>I consider synergetics coordinates is nothing but a new name for oldfacts>as a final goal to support somebody's effort to lift his profile as agreat>inventor.>(Please notice this message is no way against You and the honestinformation>You wanted to share with sci.math. community). My criticism was dedicated>against the original reference and against the inventor's ambitions.)>Tapio> I've been looking for web sites or books about something like> Synergetics coordinates for years.> Cliff Nelson>>Keywords: fourth dimension, simplex, tetrahedron, coordinate geometry,>>sixty degree coordinate system, sphere packing, linear algebra,>>operations research, linear programming, Newton's method, roots of>>unity, derivative, field, cyclotomic fields, B numbers, Bucky Numbers,>>Synergetics, R. Buckminster Fuller.>>I've read too much about most of the keywords from too many books. An>>afterimage, persistence of vision, time lag, stops me from seeing simple>>things from a new point of view for awhile.>> Cliff Nelson> Fyi: The nice property that was mentioned in the given link, is based> on the Viviani's theorem.>> http://mathworld.wolfram.com/VivianisTheorem.html>> Ternary phase diagrams have been used during the last century -> perhaps longer. See for example:>>http://www.sv.vt.edu/classes/MSE2094_NoteBook/ 96ClassProj/experimental/terna> ry2.html>> Btw: MS-Excel has radar plot property and, on the other hand,> xyz-cartesian plot, which can be viewed from any angle so that you> can result in synergetics view. (45-45 degrees rotation of the> point of view) Imho: I didn't know a certain angle of view in the xyz> cartesian coordination system is called synergetics coordinates.> ;-)>> Tapio>>What Eric W. Weisstein calls the x,y,z axes of the triangle are>>rotated one position to the right from mine in his description,>>but, it looks like at least some version of Synergetics coordinates>>might become respectable in academia now that they are mentioned>>at:>>http://mathworld.wolfram.com/ === SynergeticsCoordinates.html>>Cliff Nelson>>Subject: Re: Trying to understand Languages and Structures> Let L = Lab. Show that for every L-term t(x1, ..., xn) there are> integers k1, ..., kn such that for every abelian group A = (A, 0, -,> +) and every a1, ..., an belonging to An we have tA(a1, ..., an) => k1a1 + ... + knan.> You made no attempt to describe the context. What's Lab, tA ?L_ab is the Language of abelian groups. There are no relationalsymbols and the only function symbols are the constant symbol '0', theunary function symbol '-' and the binary function symbol '+'.t^A is the term applied to the structure A. So instead of t(x), wehave t^A(a) where a belongs to A.> Solution:> Let x = (x1, ..., xn), and a = a1, ..., an.> Case 1: t(x) is the variable xi (has arity 0)> Then tA(a) = ai by the definition of tA.> Define kj = 0 if j != i and ki = 1.> Then tA(a) = k1a1 + ... + knan.> Case 2: t(x) = - x (has arity 1)> Then tA(a) = - ai by the definition of tA.> Define kj = 0 if j != i and ki = -1.> Then tA(a) = k1a1 + ... + knan.> Case 3: t(x) = + xixj (has arity 2)> Then tA(a) = + aiaj by the definition of tA.> Define km = 0 if m != i and m != j and km = 1 if m = i or m = j.> Then tA(a) = k1a1 + ... + knan.> Case 4: t(x) = Ft1(x)t2(x)...tn(x) (has arity > 2)> Then tA(a) = FAt1A(a)t2A(a)...tnA(a) by the definition of tA.> Since each function of L is either nullary, unary, or binary> (definition of Lab), each ti will fall under case 1, case 2, or case> 3. Hence each ki will be either 0, 1, or -1.> Then tA(a) = k1a1 + ... + knan.> Since every term falls into either case 1, 2, 3, or 4, it follows that> the proposition is true.> QED.> I tried to put the proof in html, but I cant display html in these> groups. The actual proof with superscripts, subscripts and such is> listed at www.geocities.com/blakman211/lgprf1.htm> x sub n = x_n> x1, x2, ... alright if it's clear it's x_1, x_2, ...> x to the nth = x^n> F sub a super b = F_a^b> Tho it's best to present problems in body of post,> to be useful, present web addresses as> http://www.geocities.com/blakman211/lgprf1.htm> ----== News==----> http://www.newsfeed.com The #1 Newsgroup Service in the World! >100,000 Newsgroups> ---= 19 East/West-Coast Specialized Servers - Total Privacy via Encryption === =---Subject: Re: Trying to understand Languages and Structures>> Let L = Lab. Show that for every L-term t(x1, ..., xn) there are> integers k1, ..., kn such that for every abelian group A = (A, 0, -,> +) and every a1, ..., an belonging to An we have tA(a1, ..., an) => k1a1 + ... + knan.> You made no attempt to describe the context. What's Lab, tA ?>L_ab is the Language of abelian groups. There are no relational>symbols and the only function symbols are the constant symbol '0', the>unary function symbol '-' and the binary function symbol '+'.>t^A is the term applied to the structure A. So instead of t(x), we>have t^A(a) where a belongs to A.Ok, then it seems to me you'd prove this by induction on terms:If t is a term then either t = 0 or there exist terms t1, t2 of lowerarity such that t = t1 + t2 or t = -t1.> Solution:>> Let x = (x1, ..., xn), and a = a1, ..., an.> Case 1: t(x) is the variable xi (has arity 0)> Then tA(a) = ai by the definition of tA.> Define kj = 0 if j != i and ki = 1.> Then tA(a) = k1a1 + ... + knan.>> Case 2: t(x) = - x (has arity 1)> Then tA(a) = - ai by the definition of tA.> Define kj = 0 if j != i and ki = -1.> Then tA(a) = k1a1 + ... + knan.>> Case 3: t(x) = + xixj (has arity 2)> Then tA(a) = + aiaj by the definition of tA.> Define km = 0 if m != i and m != j and km = 1 if m = i or m = j.> Then tA(a) = k1a1 + ... + knan.>> Case 4: t(x) = Ft1(x)t2(x)...tn(x) (has arity > 2)> Then tA(a) = FAt1A(a)t2A(a)...tnA(a) by the definition of tA.> Since each function of L is either nullary, unary, or binary> (definition of Lab), each ti will fall under case 1, case 2, or case> 3. Hence each ki will be either 0, 1, or -1.> Then tA(a) = k1a1 + ... + knan.>> Since every term falls into either case 1, 2, 3, or 4, it follows that> the proposition is true.>> QED.>> I tried to put the proof in html, but I cant display html in these> groups. The actual proof with superscripts, subscripts and such is> listed at www.geocities.com/blakman211/lgprf1.htm> x sub n = x_n>> x1, x2, ... alright if it's clear it's x_1, x_2, ...> x to the nth = x^n>> F sub a super b = F_a^b> Tho it's best to present problems in body of post,>> to be useful, present web addresses as>> http://www.geocities.com/blakman211/lgprf1.htm>> ----== News==---->> http://www.newsfeed.com The #1 Newsgroup Service in the World! >100,000 Newsgroups>> ---= 19 East/West-Coast Specialized Servers - Total Privacy via Encryption === =---************************Subject: Re: Trying to understand Languages and Structures>> Let L = Lab. Show that for every L-term t(x1, ..., xn) there are>> integers k1, ..., kn such that for every abelian group A = (A, 0, -,>> +) and every a1, ..., an belonging to An we have tA(a1, ..., an) =>> k1a1 + ... + knan.>You made no attempt to describe the context. What's Lab, tA ?>>L_ab is the Language of abelian groups. There are no relational>>symbols and the only function symbols are the constant symbol '0', the>>unary function symbol '-' and the binary function symbol '+'.>>t^A is the term applied to the structure A. So instead of t(x), we>>have t^A(a) where a belongs to A.>Ok, then it seems to me you'd prove this by induction on terms:>If t is a term then either t = 0 or there exist terms t1, t2 of lower>arity such that t = t1 + t2 or t = -t1.Um, I should probably say that that's the idea of how you'd proveit; I'm probably not using the right notation and there are probablycases I omitted.>> Solution:>> Let x = (x1, ..., xn), and a = a1, ..., an.>> Case 1: t(x) is the variable xi (has arity 0)>> Then tA(a) = ai by the definition of tA.>> Define kj = 0 if j != i and ki = 1.>> Then tA(a) = k1a1 + ... + knan.>> Case 2: t(x) = - x (has arity 1)>> Then tA(a) = - ai by the definition of tA.>> Define kj = 0 if j != i and ki = -1.>> Then tA(a) = k1a1 + ... + knan.>> Case 3: t(x) = + xixj (has arity 2)>> Then tA(a) = + aiaj by the definition of tA.>> Define km = 0 if m != i and m != j and km = 1 if m = i or m = j.>> Then tA(a) = k1a1 + ... + knan.>> Case 4: t(x) = Ft1(x)t2(x)...tn(x) (has arity > 2)>> Then tA(a) = FAt1A(a)t2A(a)...tnA(a) by the definition of tA.>> Since each function of L is either nullary, unary, or binary>> (definition of Lab), each ti will fall under case 1, case 2, or case>> 3. Hence each ki will be either 0, 1, or -1.>> Then tA(a) = k1a1 + ... + knan.>> Since every term falls into either case 1, 2, 3, or 4, it follows that>> the proposition is true.>> QED.>> I tried to put the proof in html, but I cant display html in these>> groups. The actual proof with superscripts, subscripts and such is>> listed at www.geocities.com/blakman211/lgprf1.htm>x sub n = x_n>x1, x2, ... alright if it's clear it's x_1, x_2, ...>x to the nth = x^n>F sub a super b = F_a^b>Tho it's best to present problems in body of post,>to be useful, present web addresses as> via Newsfeed.Com - Unlimited-Uncensored-Secure Usenet News==---->http://www.newsfeed.com The #1 Newsgroup Service in the World! >100,000 Newsgroups>---= 19 East/West-Coast Specialized Servers - Total Privacy via Encryption === =---Subject: Re: Typing Monkeys Don't Write ShakespeareIndeed they don't. If documents (I won't say ideas in thisparaphrase of the negation of a good old saw) are responsible fortheir === readers, they write _Synergetics_.Lee RudolphSubject: Re: Who here believes maths is all there is?Which part of Love were my namesake's torture chambers made of?Still, we can each choose, to some extent, the lens through which we viewthe universe. I really have no problem with someone using a lens that makesrelationships central and I have no problem with another person that makesmathematics central. In fact, I use multiple lenses. But the person whothinks they have *the* panoptical === all-seeing lens is surely insane are theynot?--TorqueSubject: Re: Why not anonymous publication?>Well, why not type-A space?> Sure, but we'd run out of letters pretty soon, and it's usually easier> to remember names than arbitrary codes. I really don't understand> your problem with using names.> What to do when cultures meet up (what happens when>aliens come?). If you name things after people, you'll argue that you>want your culture's name, and it's a pointless battle.> We may well have pointless battles with aliens, but it won't be over> nomenclature. Obviously (except on Star Trek) the aliens don't speak> English.Just side-tracking...In anything but a scenario built in the current subculture of the audience,the best that you can do to capture the audience is to present someapproximation of cross-cultural things.Doug Hofstadter addresses this problem in exquisite detail in GEB as wellas MorotThe way that I can suspend disbelief in the English thing is that NOBODYspeaks English, and that Aliens ( i.e. Humanoids, usually) somehow cancommunicate with humans through some unspecified Universal translationMachine..Still a lot of logistic loopholes, But a Connecticut Yankee would have foundthe Language of King Arthur's Court unintelligible.There's just a time when this problem can't be addressed with its caveatsand loopholes.Goofy sounds and subtitles and five-tier chains of translators would cause a99% loss of viewers.(I recall Two people trying to communicate in ENGLISH which required thefollowing chainCreole English Dialect> Indian Language> Spanish> English)Same way about the 1970's social norms and speech idioms that all the crewseem to haveEven Star Wars' attempts at this result in a laughable m.8elange, withcharacters such as Jar Jar Binks ending up like === a Rasta Clown Act.Bob PeaseSubject: Re: Why not anonymous publication?>> However, if we carried this idea out for all definitions,>> think of how clumsy the notation would be. A Hilbert space>> would be a complete normed topological vector space with>> the norm given by an inner product, and these terms could>> be expanded.>In this particular example, couldn't we just use inproduct space?An inner product space does not have to be complete.> Would you like to use topological space for>> which all open coverings have countable refinements>> instead of Lindelof spaces.>countably-compact spaces?Countably compact has an entirely different meaning.>Or how about Stone-Cech>> compactification?>What is that? It could be that i actually know the construction,>but under another name...This is the compactification in which all boundedcontinuous real-valued functions can be extended.>> Or the Stone space of a Boolean algebra?>Canonic space, Prime filter space?Both of these are used elsewhere.-- This address is for information only. I do not claim that these viewsare those of the Statistics Department or of Purdue University.Herman Rubin, Department of Statistics, Purdue Universityhrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558