mm-464 === Subject: : Re: Isotopic knot diagrams.>Sorry about the confusion I manged to produce. I did mean diagrams in>R^2.. typo.. sorry. The terminology I've learned so far is the>following: 2 diagrams>D and D' in R^2 are called isotopic if there is a sequence of>Reidemeister moves( and their inverses) and planar isotopies s.t>performing this sequence on D gives the diagram D'. If we can get from>D to D' without using R1 (and its inverse) then D and D' are regularly>isotopic.Okay, I can see that that's a reasonable terminology, and if it's(now) standard I'll agree to it. (I would have called D and D'regularly homotopic in that case, and said that the *framed* knotsin R^3 determined by the diagrams were regularly isotopic, or isotopic respecting framing, or something.)>Since last I have seen an exercise in a book, where diagrams are drawn>on S^2 rather than in R^2. And that exercise is actually to prove that>the answer to my question is no if diagrams are drawn on S^2... So>I'll have to think about whether there is any essential difference>between diagrams in R^2 and on S^2.Yes, you will have to think about that.=== Subject: : Understanding the definition of k-clique and k-tree math folks,I have read the definition of k-tree, and it appears to be grown from ak-clique, which would seem to just be a complete graph of k vertices. I amnot understanding the naming convention here. A tree has no cycles, but itseems that if it is based on a k-clique, it has cycles.advance.Definition of k-tree. The k-trees are the graphs that arise from ak-clique by 0 or more iterations of adding a new vertex joined to a k-cliquein the old graph.=====================================================God made the integers, all else is the work of man.L. Kronecker, Jahresber. DMV 2, S. 19.=== Subject: : Re: Understanding the definition of k-clique and k-treeIt sounds as if a k-tree starts from a complete graph with k vertices, andthen has n-k little spikes or leaves, each attached to the initial completegraph. What I am wondering is if these legs can grow longer than one inlength?> math folks,> I have read the definition of k-tree, and it appears to be grown from a> k-clique, which would seem to just be a complete graph of k vertices. I am> not understanding the naming convention here. A tree has no cycles, but it> seems that if it is based on a k-clique, it has cycles.> advance.> Definition of k-tree. The k-trees are the graphs that arise from a> k-clique by 0 or more iterations of adding a new vertex joined to ak-clique> in the old graph.> -- > =====================================================God made the integers, all else is the work of man.> L. Kronecker, Jahresber. DMV 2, S. 19.=== Subject: : Roots of Primitive Polynomials windows-nt)Let's say p(x) is a primitive polynomial overGF(2) of degree 4 (just to use for an example). 1. We know this polynomial has 4 roots in some extension field E. Is it necessarily true that all roots are primitive in E? If not, are there ever two or more such roots that are primitive in E? If so, how do you distinguish one from the other? If there is only ever one such primitive element over E, then how do you know that this is the root that is used to construct GF(2^4)? 2. Let's say the primitive root of p(x) is a. Is there a way to identify in a concrete manner the field GF(2)(a), i.e., the field of GF(2) adjoined by a?These are elementary, and I think I know the answers tosome of them, but I'm looking for clarification/firmingup. % Randy Yates % How's life on earth? %% Fuquay-Varina, NC % ... What is it worth? %%% 919-577-9882 % 'Mission (A World Record)', %%%% % *A New World Record*, ELOhttp://home.earthlink.net/~yatescr=== Subject: : Re: Roots of Primitive Polynomials> Let's say p(x) is a primitive polynomial over> GF(2) of degree 4 (just to use for an example).> 1. We know this polynomial has 4 roots in some> extension field E. Is it necessarily true that all> roots are primitive in E? If not, are there ever> two or more such roots that are primitive in E?> If so, how do you distinguish one from the other?I take it a primitive polynomial of degree d is an irreducible polynomialone of whose roots is a primitive element of GF(p^d).In that case all the roots are primitive elements.If one root is a, the others are a^p, a^{p^2}, ....If a is a primitive element of GF(p^d)then so is a^r if gcd(r,p^d-1) = 1.(Alternatively, if one root a has order r then p(x) | x^r - 1and so any other root has order | r.)> 2. Let's say the primitive root of p(x) is a.> Is there a way to identify in a concrete manner> the field GF(2)(a), i.e., the field of GF(2) adjoined> by a?I'm not sure I follow this question.If p(x) is of degree d then GF(2)[a] = GF(2^d).Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ietel: +353-86-2336090, +353-1-2842366s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland=== Subject: : Re: (xpost bait) Axioms> Deny that 1+1=2, and a mathematician won't play with you.> (Redefining 1,2,+,= or counts as cheating :-)> But does Joe Blow have axioms in this sense .... Even if that message was a prank, I'd like to respond. It's not just a matter of axioms, but of the whole self-authenticating nature of mathematics. Students of it need *never* accept something (other than a definition) just because the teacher says so. They can always demand proof. Also any teacher who makes a mistake should always back down. Mathematical reasoning has a strength and authority within itself which puts it beyond mere opinions in a way you don't find in most other areas of human thought. That's one of the things that makes it so special. (Lyrical mode off. :-) .=== Subject: : Re: (xpost bait) Axioms> Even if that message was a prank, I'd like to respond.Oh, to the contrary, it's deadly (pun intended) serious. > It's not just a matter of axioms, but of the whole > self-authenticating nature of mathematics. Students of it need *never* > accept something (other than a definition) just because the teacher says > so. They can always demand proof. Also any teacher who makes a mistake > should always back down. Mathematical reasoning has a strength and > authority within itself which puts it beyond mere opinions in a way you > don't find in most other areas of human thought. That's one of the > things that makes it so special.Yes, but it is still an if-then business: IF you accept Peano and standard meaning of symbols, THEN 1+1=2and that's provable in this system.IF God xyz exists, THEN you have to slay the infidels for Hishigher glory and that's provable in this system. IF the premisse can be demolished...found an area which is intimately connected with religioussensations, so God may get hacked, so to say, b: just for fun, religion of terror, and sqrt(2)...nobody expects the SpanishInquisition!)Do I sound confused? As an absolute skeptic, I have to :-)Hauke Reddmann <:-EX8 fc3a501@uni-hamburg.deals man ankam wollte man werden, die geschichte schreiben,die doofen sollen sterben, der plan als man damals nach hamburg kam(Kettcar)=== Subject: : Re: (xpost bait) Axioms> Even if that message was a prank, I'd like to respond.>Oh, to the contrary, it's deadly (pun intended) serious. found an area which is intimately connected with religious>sensations, Well, of course. The whole point of the exercise is to learn the biofeedback to get the brain into that statewhen the believer is in Church. I always thought it wasself-induced rush of serotinin (I may have my chemicalsmixed up.) Deny that 1+1=2, and a mathematician won't play with you.> (Redefining 1,2,+,= or counts as cheating :-)> But does Joe Blow have axioms in this sense ....> Even if that message was a prank, I'd like to respond.> It's not just a matter of axioms, but of the whole >self-authenticating nature of mathematics. Students of it need *never* >accept something (other than a definition) just because the teacher says >so. They can always demand proof. Also any teacher who makes a mistake >should always back down. What? > ...Mathematical reasoning has a strength and >authority within itself which puts it beyond mere opinions in a way you >don't find in most other areas of human thought. That's one of the >things that makes it so special.Good grief. Where in your world did math become confrontational?The fun of math is that it can't be changed because some humanhas a difference of opinion. When the world is completelyinsane and nobody makes sense, I sit down and do some mathproblems to soothe me. 1. It proves there still exists logicand 2. I demonstrate to myself I can still think well./BAHSubtract a hundred and four for e-mail.=== Subject: : Re: L^p functions> If f is a L^p function, with p>1, on a open subset A of R^n or on R^n , can> I say that f is a L^1 function locally, i.e. on the compact subset of A?Sure, L^p for p > 1 is always contained in L^1 if the underlying measure is finite - which Lebesgue measure is on any compact set. You can prove this easily with Holder, or even more basic, consider the sets where |f| < 1 and |f| >= 1.=== Subject: : Re: L^p functions> If f is a L^p function, with p>1, on a open subset A of R^n or on R^n , can> I say that f is a L^1 function locally, i.e. on the compact subset of A?Indeed. Use Hoelder's inequality on f and the characteristic function gof somecompact set C. Choose q such that 1/q + 1/p = 1. Then g is a L^qfunction on C, because C is a) closed and thus measureable and b) bounded and thereforeof finite (Lebegue) measure. By Hoelder's inequality f = fg is a L^1function on C.My favorite reference is Royden, H. L : Real Analysis, 3rd Edition1988.Carl Christian K. MikkelsenSoon I run the rapids of lifeA man's life, like the frothing watersRising high, falling lowAll vanity!- Death poem of Masatsune, Lord of Suruga (L&C, vol 17)=== Subject: : universal enveloping algebraAs I understand it, the universal enveloping algebra U of a Liealgebra L can be constructed as the tensor algebra of L modulo theideal I generated by all xy-yx-[x,y]. I've seen it argued (Humphreys)that U contains an isomorphic copy of the scalars, because I obviouslycontains no degree-0 elements. Humphreys also claims that it's not atall obvious that U contains an isomorphic copy of L, but isn't it? Iknow this fact follows from the PBW theorem, but isn't it obvious thatI doesn't contain any elements of degree 1. I guess I'm missingDave=== Subject: : Re: universal enveloping algebra> As I understand it, the universal enveloping algebra U of a Lie> algebra L can be constructed as the tensor algebra of L modulo the> ideal I generated by all xy-yx-[x,y]. I've seen it argued (Humphreys)> that U contains an isomorphic copy of the scalars, because I obviously> contains no degree-0 elements. Humphreys also claims that it's not at> all obvious that U contains an isomorphic copy of L, but isn't it? I> know this fact follows from the PBW theorem, but isn't it obvious that> I doesn't contain any elements of degree 1. I guess I'm missingWhy is it so obvious to you that no linear combination of elements ofthe form u.(xy - yx - [x,y]).v (with u and v in the tensor algebra of Land x and y in L) can have degree 1? For me it isn't. But if you thinkthat it is simple, please provide a proof.=== Subject: : Re: universal enveloping algebra> Why is it so obvious to you that no linear combination of elements of> the form u.(xy - yx - [x,y]).v (with u and v in the tensor algebra of L> and x and y in L) can have degree 1? For me it isn't. But if you think> that it is simple, please provide a proof.> Jose Carlos Santosonly slightly more complicated than for the scalars, but I see nowthat's a mistake. I guess I should have given it more thought beforeposting my question, but at least your response encouraged me to bemore thorough. I agree now: it isn't obvious.=== Subject: : Re: Hey guys, isolated singularities?> What Wade actually meant to say was Look up the > definition of pole and essential singularity.Right, yep, I know that poles and essential singularities are isolatedby definition. (Also the removable singularities.)I was asking about singularities in general mainly because I didn'tsee references to them in the literature, yet they seem to appear hereand there (e.g. the natural boundary).I agree what you said in the other post -- the function elements arecontinued analytically. (What's the rigorous definition of this, btw?A function element is a function from a Riemann surface into C whichis restricted to a particular subset of the complex plane, orsomething like that? I haven't seen the definition of this either.)I guess this function element thing boils down to multivaluedfunctions and the concept Riemann surface (which I think of as thedomain of all possible analytic continuations of a particular functionelement). Should I think of it instead as some kind of complexmanifold? Is that the same thing?Anyway, I have only one important question about this whole business:If I continue a function analytically around a branch point and comeback to the same place I may or may not get a different value. (Idon't think you can ever get a Laurent series around a branch point.)But if I continue around a pole or essential singularity I get thesame function value, correct? After all, I always can get a Laurentseries around those (some people define them in fact in terms of theLaurent series).So the monodromy theorem for complex analysis says that you only getdifferent values if you run around branch points, but not essentialsingularities or poles?-Greg=== Subject: : Re: Hey guys, isolated singularities?> What Wade actually meant to say was Look up the > definition of pole and essential singularity.>Right, yep, I know that poles and essential singularities are isolated>by definition.You know that now. When Wade said what he said you'd justsaid you _think_ that poles have to be isolated and asked whetheressential singularities had to be isolated, suggesting sin(1/z) asa possible counterexample...> (Also the removable singularities.)>I was asking about singularities in general mainly because I didn't>see references to them in the literature, yet they seem to appear here>and there (e.g. the natural boundary).>I agree what you said in the other post -- the function elements are>continued analytically. (What's the rigorous definition of this, btw?>A function element is a function from a Riemann surface into C which>is restricted to a particular subset of the complex plane, or>something like that? I haven't seen the definition of this either.)How can you possibly agree with what I said about functionelements if you don't know what a function elelement is???Discussions of concepts one has only a fuzzy understandingof can be useful. But I gather you're a graduate student inmathematics - at some point (very soon!) you need to realizethat if you don't know the definition of a term you need tolearn the definition before trying to say anything meaningfulor ask sensible questions.If you're trying to learn about complex analysis you need abook on the subject. If you have a book that gives a seriousduscussion of analytic continuation (at the first-year-gradstudent level, in a moderately abstract context, ie not justcommenting on continuing log and sqrt) that does _not_contain a definition of function element I'll be verysurprised.>I guess this function element thing boils down to multivalued>functions and the concept Riemann surface (which I think of as the>domain of all possible analytic continuations of a particular function>element). Should I think of it instead as some kind of complex>manifold? Is that the same thing?>Anyway, I have only one important question about this whole business:>If I continue a function analytically around a branch point and come>back to the same place I may or may not get a different value. (I>don't think you can ever get a Laurent series around a branch point.)>But if I continue around a pole or essential singularity I get the>same function value, correct? After all, I always can get a Laurent>series around those (some people define them in fact in terms of the>Laurent series).>So the monodromy theorem for complex analysis says that you only get>different values if you run around branch points, but not essential>singularities or poles?I explained yesterday why this question doesn't quite make sense(or to be fair, why it doesn't quite make sense to me). I couldrepeat what I said yesterday - I don't see much point to that.I gather you'd prefer a simple yes or no. You can't have one -the question doesn't make sense - instead of asking for ayes or no you should try to understand why not.>-Greg=== Subject: : Re: Fourier analysis on groups question windows-nt)Hi Moshe,I would say a resounding Yes!.I am not an expert, just a student in coding theory, but there issomething in coding theory that comes very close to the notion you arequerying about: Reed-Solomon codes.A Reed-Solomon code is a type of cyclic linear block code. In asuch a code you take a vector of input symbols u(x) (the symbols arecoefficients of a polynomial) and map them into a vector of codedsymbols c(x) by applying a generator matrix G mod x^n - 1. Youtransmit this vector through the channel and receive a possiblycorrupted code vector y(x). The parity-check matrix matrix H is amatrix you can use to check the received vector to see if it haserrors. If y(x)H(x) mod g(x) = 0, then the vector was uncorrupted.Now in a RS code, H(x) takes the form of a Vandermonde matrix, so thatthe operation y(x)H(x) looks a lot like a DFT over a finite field (didI say these code symbols are over finite fields?). So the Reed-Solomoncodes are codewords which have 0 DFT in the first 2t frequencies, where t is known as the designed distance of the code.Anyway, I think there's a connection here somewhere, and you have a few keywords to search on. I hope I haven't messed anything up too badly.--Randy> Are there any practical applications for Harmonic/Fourier analysis on> groups other than R or R/Z? Is Harmonic analysis ever really used on groups> other than R or R/Z? (I suppose anyone could study Harmonic analysis on> groups other than R or R/Z, but is it useful in any way other than the> advancement of Math?)> Moshe% Randy Yates % And all that I can do%% Fuquay-Varina, NC % is say I'm sorry, %%% 919-577-9882 % that's the way it goes...%%%% % Getting To The Point', *Balance of Power*, ELOhttp://home.earthlink.net/~yatescr=== Subject: : Re: Fourier analysis on groups question> Are there any practical applications for Harmonic/Fourier analysis on> groups other than R or R/Z? Is Harmonic analysis ever really used on groups> other than R or R/Z? (I suppose anyone could study Harmonic analysis on> groups other than R or R/Z, but is it useful in any way other than the> advancement of Math?)> MosheHarmonic analysis on locally compact groups has importance in modernnumber theory (p-adic fields and adele rings) and probably thereforein such practical applications as secure communications.=== Subject: : Computer based proofsHi all,What is the consensus here in sci.math on the use of computers in proofs?Should they be allowed, or not?Lurch=== Subject: : Re: Computer based proofs> Hi all,> What is the consensus here in sci.math on the use of computers in proofs?> Should they be allowed, or not?They should be allowed though a shorter proof that a human can handlein a reasonable time is far better. The real problem here IMHO is thelack of useful, *standard*, proof checking tools. It also would helpwith a lot of papers (particularly preprints) to be able to run aprogram which verifies that the logical arguments of the paper arecorrect. The theorems/proofs might be misleading or uninteresting, butat least you should be able to certify correctness for a broadcategory of mathematics. A bonus is that the proof checker can be madevery reliable (eg, checked by hand and gobs of empirical examples tobe to high probability mathematically correct).The biggest drawback I see to such a tool is the need to standardizethe rigor of virtually all mathematics at a pretty high level. Inparticular, some fields (eg, low dimensional topology, some algebraicgeometry) often slink by :-) with a lower level of rigor due to theintuitive nature of the mathematical concepts involved. Others (eg,String Theory) have fundamental concepts (eg, Feynman and Stringintegral actions) with no rigorous mathematical backing (perhaps thereare rigorous constructions, but they aren't used).A proof checker isn't great for every field of endevour here. But itdoes provide a stronger proof checking capability than the currentregime. I doubt anyone who has published half a dozen or more seriousproofs doesn't have a paper out there without some typo or logicerror. Usually, these errors don't change the outcome of the proof,but it does mean more work for the reader while the error is workedout and potential embarrassments for the author.Karl Hallowellkhallow@hotmail.com=== Subject: : Re: Computer based proofs> Hi all,> What is the consensus here in sci.math on the use of computers in proofs?> Should they be allowed, or not?It's interesting that you bring that up now. Recently the Annals ofMathematics has decided that Hales' computer-based proof of the KeplerConjecture should be published in two parts, in two different journals. The theoretical part will be published in the Annals and thecomputational part will be published in Discrete and ComputationalGeometry.osition=There are numerous inaccuracies (as usual), but it gives the gist. Berkeley mathematician) to be some kind of crank, even going so far asquoting Frank Quinn as saying Hsiang has a bad track record. I don'tunderstand why they're quoting Quinn to begin with; as far as I know,he's not an expert in sphere-packing at all. And I especially don'tunderstand why Quinn is badmouthing Hsiang, when as far as i know,Hsiang has a very good track record and pretty famous guy in his ownfield. I hope the journalist either misquoted or took things reallyout of context or whatever.As for your question, Lurch, I'll probably side with the mainstreamview that computers are ok and fairly reliable (as long as you test ondifferent architectures, etc, etc) in some situations, but it's oftennot as elegant or informative as a traditional proof. As for allowing computer proofs, I don't really see anything I coulddo to stop them, even if I wanted to. People are publishing them moreand more.I personally don't think computers have really changed the waymathematicians do pure mathematics. It's certainly nice to use someprogram to check that such-and-such is true or not true, but the samegame of conjecturing and forming connections continues on. In somecases, computers give us a way to visualize that we weren't able tobefore, but then they just become another tool that we use to do thesame kinds of things. Ultimately what mathematicians seek is humanunderstanding of mathematics. Computers haven't changed that, butmaybe they've made it clearer what we humans are trying to do.=== Subject: : Re: Computer based proofs> Berkeley mathematician) to be some kind of crank, even going so far as> quoting Frank Quinn as saying Hsiang has a bad track record. I don't> understand why they're quoting Quinn to begin with; as far as I know,> he's not an expert in sphere-packing at all. And I especially don't> understand why Quinn is badmouthing Hsiang, when as far as i know,> Hsiang has a very good track record and pretty famous guy in his own> field. I hope the journalist either misquoted or took things really> out of context or whatever.Well, I heard with my own ears an expert on the 4-color problem,Gerhard Ringel, give his approval of this proof. (His words were,There's no other way to do it.) So phooey on Quinn!=== Subject: : Re: Computer based proofs> What is the consensus here in sci.math on the use of computers in proofs?> Should they be allowed, or not?(snip)>It's interesting that you bring that up now. Recently the Annals of>Mathematics has decided that Hales' computer-based proof of the Kepler>Conjecture should be published in two parts, in two different journals. >As for your question, Lurch, I'll probably side with the mainstream>view that computers are ok and fairly reliable (as long as you test on>different architectures, etc, etc) in some situations, but it's often>not as elegant or informative as a traditional proof. Performing any computation that provides a definite output istantamount to proving that the observed output is one of the possibleresults of the given computation. This was the case classically too,but in the absence of (quantum computing) interference effects it isalways possible to keep a record of the steps of the computation, andthereby produce (and check the correctness of) a proof that satisfiesthe classical definition - as a sequence of propositionseach of which is either an axiom or follows from earlier propositionsin the sequence by the given rules of inference. Now we are forced toleave that definition behind. Henceforward, a proof must be regardedas a process |the computation itself | for we must accept that infuture, quantum computers will prove theorems by methods that neithera human brain nor any other arbiter will ever be able to checkstep-by-step, since if the `sequence of propositions' corresponding tosuch a proof were printed out, the paper would fill the observableuniverse many times over. (The above quoted from: Machines, Logic andQuantum Physics by David Deutsch, Artur Ekert, and RossellaLupacchini, http://xyz.lanl.gov/abs/math.HO/9911150)The abstract of this paper has some stimulating remarks too:Though the truths of logic and pure mathematics are objective andindependent of any contingent facts or laws of nature, our knowledgeof these truths depends entirely on our knowledge of the laws ofphysics. Recent progress in the quantum theory of computation hasprovided practical instances of this, and forces us to abandon theclassical view that computation, and hence mathematical proof, arepurely logical notions independent of that of computation as aphysical process. Henceforward, a proof must be regarded not as anabstract object or process but as a physical process, a species ofcomputation, whose scope and reliability depend on our knowledge ofthe physics of the computer concerned. (end quote 2)John Baileyhttp://home.rochester.rr.com/jbxroads/mailto.html=== === Subject: : Re: Computer based proofsIf something's *proven*, then it's irrelevant how the proof was obtained.Just because the four-colour theorem was proven by computer, that doesn'tmean that it's not true.Or are you talking about something else? Hopefully you're talking aboutsomething else, because the question as originally posed is silly.Doug=== Subject: : Re: Computer based proofs* Doug Norris> If something's *proven*, then it's irrelevant how the proof was obtained.> Just because the four-colour theorem was proven by computer, that doesn't> mean that it's not true.> Or are you talking about something else? Hopefully you're talking about> something else, because the question as originally posed is silly.I do agree with you that a proof is a proof. But I do not agree withyou that the question is silly. I seem to remember that the fourcolor theorem was critisized because there was no way to check if itwas valid (and perhaps that it did not bring forth any new _knowledge_of mathematics).However, I think computer based proof will have a great impact on thefuture. A simple example can illustrate this. A few years ago Iasked for the best strategy to solve the bookshelf problem, (usagesort). Given a bookshelf with 6 books numbered from 1-6 but thatare in random order. Consider yourself lazy and you don't want tosort the books, but you will each time you use a book, insert it onthe position that the orderness of the books increases. The questionis what your best strategy is so that the expected number of insertionis minimized. (The next book to use is random.)I am not sure but I think that a very good strategy is to put the bookat the place where the number of books well placed in relation to thisbook is maximized. However, someone did a computer search and found abetter strategy. The shock was that this strategy was notunderstandable in any easy terms.I have thus made the conclusion that there is a vast mass ofmathematics that we shall have problems to understand in our typicalclean abstract manner. Perhaps we can make computers to generate suchmathematics, but who knows?Jon Haugsand Dept. of Informatics, Univ. of Oslo, Norway, mailto:jonhaug@ifi.uio.no http://www.ifi.uio.no/~jonhaug/, Phone: +47 22 85 24 92=== Subject: : Re: Computer based proofs> * Doug Norris> If something's *proven*, then it's irrelevant how the proof was obtained.> Just because the four-colour theorem was proven by computer, that doesn't> mean that it's not true.> Or are you talking about something else? Hopefully you're talking about> something else, because the question as originally posed is silly.> I do agree with you that a proof is a proof. But I do not agree with> you that the question is silly. I seem to remember that the four> color theorem was critisized because there was no way to check if it> was valid (and perhaps that it did not bring forth any new _knowledge_> of mathematics).I think the real objection was that the proof didn't provide muchinsight, ie. it did not really find anything interesting or new. Thisis the real purpose of proofs, discovering the answer to the questionis secondary. In this proof was enumberated a large number of cases,so large they could not be checked by hand. Even if the number ofcases were low enough for hand-checking it would not have been a greatproof. In fact, if it were hand-checkable it would have been a lesscelebrated result: the computer involvement was the contribution tomathematics.> However, I think computer based proof will have a great impact on the> future. A simple example can illustrate this. A few years ago I> asked for the best strategy to solve the bookshelf problem, (usage> sort). Given a bookshelf with 6 books numbered from 1-6 but that> are in random order. Consider yourself lazy and you don't want to> sort the books, but you will each time you use a book, insert it on> the position that the orderness of the books increases. The question> is what your best strategy is so that the expected number of insertion> is minimized. (The next book to use is random.)> I am not sure but I think that a very good strategy is to put the book> at the place where the number of books well placed in relation to this> book is maximized. However, someone did a computer search and found a> better strategy. The shock was that this strategy was notunderstandable in any easy terms.Very interesting! Last time I looked at computer-generated proofsthey hadn't gotten anywhere. They couldn't even check an existingproof.Of course, you cannot know whether a simplification will be discoveredsomeday.> I have thus made the conclusion that there is a vast mass of> mathematics that we shall have problems to understand in our typical> clean abstract manner. Very likely so. > Perhaps we can make computers to generate such> mathematics, but who knows?Aren't you telling us that this has already happened once?=== Subject: : Re: Computer based proofs> * Doug Norris> If something's *proven*, then it's irrelevant how the proof was > obtained.> Just because the four-colour theorem was proven by computer, that > doesn't> mean that it's not true.> Or are you talking about something else? Hopefully you're talking > about> something else, because the question as originally posed is silly. > I do agree with you that a proof is a proof. But I do not agree with> you that the question is silly. I seem to remember that the four> color theorem was critisized because there was no way to check if it> was valid (and perhaps that it did not bring forth any new _knowledge_> of mathematics).> However, I think computer based proof will have a great impact on the> future. As I understand it, both the four color theorem and a problem aboutstacking spheres were solved with special purpose programs. Thereare also a number of programs for general purpose mathematics. Irefer to ACL2, Nuprl, and HOL, for example. Here's a paper documentingsome of the math discovered by one of these programs: Hi all,> What is the consensus here in sci.math on the use of computers in proofs?> Should they be allowed, or not?As I recall, the four color theorem was the first such . . . and unless I'm mistaken, there is still no non-computer proof. When you ask if they should be allowed, what do you mean? The four-color theorem was proved way back in 1977. If anyone was going to disallow this proof, it would have happened by now.=== Subject: : Re: Computer based proofs> Hi all,> What is the consensus here in sci.math on the use of computers in proofs?> Should they be allowed, or not?> As I recall, the four color theorem was the first such . . . and unless > I'm mistaken, there is still no non-computer proof. When you ask if they > should be allowed, what do you mean? The four-color theorem was proved > way back in 1977. If anyone was going to disallow this proof, it would > have happened by now.people were not happy with the proof at all - not with the fact that most of the work was done by computer, but the bits that were done by hand were so complicated that they have never been completely verified. Since then there has been a new version of the proof, much simplified, still requiring the use of a computer, but apparently the complete proof can and has now be verified by verifying the correctness of some computer programs and by running these programs.=== Subject: : Re: Computer based proofs> As I recall, the four color theorem was the first such . . . and unless > I'm mistaken, there is still no non-computer proof. I don't understand this. There can not not be non-computer proof;there is nothing that a computer can do that can not be done withouta computer (unless the thing to do includes timing requirements, andunless we're talking about something specific to the computer -- like,sure, things like execute an assembler instruction that puts electricalsignals on a bus and causes a hard disk to write some data).Can you elaborate on this there is no non-computer proof?Carlos=== Subject: : Re: Computer based proofs> As I recall, the four color theorem was the first such . . . and unless > I'm mistaken, there is still no non-computer proof. > I don't understand this. There can not not be non-computer proof;> there is nothing that a computer can do that can not be done without> a computer (unless the thing to do includes timing requirements, and> unless we're talking about something specific to the computer -- like,> sure, things like execute an assembler instruction that puts electrical> signals on a bus and causes a hard disk to write some data).> Can you elaborate on this there is no non-computer proof?If the phrase computer proof means anything at all, then it's clear what a non-computer proof is. If you say that any computer-based proof could ultimately be reduced to a human being carrying out the same steps by hand -- IN THEORY -- then you deny the meaning of the phrase computer proof in the first place.=== Subject: : Re: Computer based proofs> As I recall, the four color theorem was the first such . . . and unless > I'm mistaken, there is still no non-computer proof. > I don't understand this. There can not not be non-computer proof;> there is nothing that a computer can do that can not be done without> a computer (unless the thing to do includes timing requirements, and> unless we're talking about something specific to the computer -- like,> sure, things like execute an assembler instruction that puts electrical> signals on a bus and causes a hard disk to write some data).> Can you elaborate on this there is no non-computer proof?I sense a trap, but someone has to spring it. I believe here fishfrymeans a proof construction that didn't use a computer to construct theproof, and has been checked without using a computer. In theory, onecan construct a proof of the four-color theorem and check that theoremwithout resorting to the computer (except perhaps as a place to storythe proof and make it look pretty). To my knowledge, that hasn't beendone. That is what is meant by no non-computer proof in my humbleopinion.Karl Hallowellkhallow@hotmail.com=== Subject: : Re: Computer based proofs> As I recall, the four color theorem was the first such . . . and > unless I'm mistaken, there is still no non-computer proof. > I don't understand this. There can not not be non-computer proof;> there is nothing that a computer can do that can not be done without> a computer (unless the thing to do includes timing requirements, and> unless we're talking about something specific to the computer -- like,> sure, things like execute an assembler instruction that puts electrical> signals on a bus and causes a hard disk to write some data).> Can you elaborate on this there is no non-computer proof?The only known proof breaks the problem down into many, many cases. It is virtually impossible to cover or check these all by hand. The proof uses a computer to cover all the cases. Researchers needed to verify the logic of the program and check that it worked on different platforms.=== Subject: : Re: Computer based proofs> What is the consensus here in sci.math on the use of computers in proofs?> Should they be allowed, or not?Students shouldn't be allowed to use them until and after they know themath and understand the concepts.Kids who can't divide 10,000 by 100 without a calculator, shouldn't beallowed to use them until the can do arithmetic and understand numbers.=== Subject: : Re: Computer based proofs> What is the consensus here in sci.math on the use of computers in proofs?> Should they be allowed, or not?>Students shouldn't be allowed to use them until and after they know the>math and understand the concepts.I'll bite. How does one do proofs with a computer withoutunderstanding the concepts?I'm not interested in mathematics that might have anythingto do with reality. -- Russell Easterly, in sci.math=== Subject: : Re: Computer based proofs> What is the consensus here in sci.math on the use of computers in proofs?> Should they be allowed, or not?>Students shouldn't be allowed to use them until and after they know the>math and understand the concepts.> I'll bite. How does one do proofs with a computer without> understanding the concepts?Some students use a graphic computer as a guessing machine.Like today's corrupt congress critters asking for revotes upon ameasure until they get a result they like.=== Subject: : Re: Computer based proofs>Students shouldn't be allowed to use them until and after they know the>math and understand the concepts.> I'll bite. How does one do proofs with a computer without> understanding the concepts?>Some students use a graphic computer as a guessing machine.Many theories began as simply educated guesses. If computers helppeople make those guesses, then all the better. But you still haven'texplained how computers allow one to write proofs withoutunderstanding the mathematics involved. Apart from posting yourhomework in sci.math.>Like today's corrupt congress critters asking for revotes upon a>measure until they get a result they like.How is this relevant?I'm not interested in mathematics that might have anythingto do with reality. -- Russell Easterly, in sci.math=== Subject: : Re: Computer based proofs>Students shouldn't be allowed to use them until and after they know the>math and understand the concepts.> I'll bite. How does one do proofs with a computer without> understanding the concepts?>Some students use a graphic computer as a guessing machine.> Many theories began as simply educated guesses. If computers help> people make those guesses, then all the better. But you still haven't> explained how computers allow one to write proofs without> understanding the mathematics involved. Apart from posting your> homework in sci.math.Student's who use calculators as guessing machine aren't making educatedguesses.>Like today's corrupt congress critters asking for revotes upon a>measure until they get a result they like.> How is this relevant?Wrong numbers don't compute, only good ones need apply.=== Subject: : Re: Computer based proofs>Students shouldn't be allowed to use them until and after they know the>math and understand the concepts.> I'll bite. How does one do proofs with a computer without> understanding the concepts?>Some students use a graphic computer as a guessing machine.> Many theories began as simply educated guesses. If computers help> people make those guesses, then all the better. But you still haven't> explained how computers allow one to write proofs without> understanding the mathematics involved. Apart from posting your> homework in sci.math.>Student's who use calculators as guessing machine aren't making educated>guesses.Nope, you still don't make any sense. What does students usingcalculators have to do with using computers to do proofs?>Wrong numbers don't compute, only good ones need apply.Can I have some of that you're smoking?I'm not interested in mathematics that might have anythingto do with reality. -- Russell Easterly, in sci.math=== Subject: : Re: Computer based proofs> today's corrupt congress critters Isn't this, like, a triple redundancy?Carlos.=== Subject: : Re: Computer based proofs> today's corrupt congress critters > Isn't this, like, a triple redundancy?You don't think they had corrupt congress critters in other eras?Dave SeamanJudge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. Students shouldn't be allowed to use them until and after they know the> math and understand the concepts.> Kids who can't divide 10,000 by 100 without a calculator, shouldn't be> allowed to use them until the can do arithmetic and understand numbers.http://john.regehr.org/reading_list/power.htmlCarlos== = === Subject: : Re: Computer based proofs> Hi all,> What is the consensus here in sci.math on the use of computers in proofs?> Should they be allowed, or not?> LurchAllowed where, precisely? On sci.math?=== Subject: : Re: Can all/everything be in a set, or in a mathematical group (which are two different things in math)?> I define all/everything (or all as a unit) by saying> Is there anything outside all/everything? No.>My father grew up in a small town. They only had one barber. The>barber shaved everybody and only everybody who did not shave themselves.>That does not make sense, fool, if he or she shaved everybody >he or she did not shave only everybody who did not shave themselves.>If you say everybody you can't qualify it with an only part/subset.>You are posing an ill-posed, psychotic question. My definition holds.It would be sort of like saying everything in the galaxy and onlyeverything in the solar system which is why I stopped you (brokeinto the question) where I did before going any further.DRD=== Subject: : Re: Can all/everything be in a set, or in a mathematical group (which are two different things in math)?>This sounds like you mean The set of all observable things. Ie, the >set of all matter/energy in the universe.No, everything does not imply necessarily observable. I defineit by Is there anything outside all/everything? No. Thusif there is something outside the universe (space and its contents)such as the Christian God, then obviously the universe isnot everything and in that case both the universe and the Christian God would be subsets of everything. However I thinkthe Christian God definition of all powerful ultimate creator mapsto the exact or nearest match someone to all/everything whichhowever is not perfect, but perfection and perfect goodness are not mentioned in the Nicene creed.Also your definition of existence as tied to matter and energyseems quite naive. If I make a definition, that definitionexists but a match to it may or may not. === Subject: : Re: Can all/everything be in a set, or in a mathematical group (which are two different things in math)?>This sounds like you mean The set of all observable things. Ie, the >set of all matter/energy in the universe.> No, everything does not imply necessarily observable. I define> it by Is there anything outside all/everything? No. This statement is equivalent to saying U' = {} where U is the universal set, {} is the empty set. It does nothing for computing U, it is simply a property of U, regardless of the contents. In fact, it is possible that U={} and this would still be correct. > Thus> if there is something outside the universe (space and its contents)> such as the Christian God, then obviously the universe is> not everything and in that case both the universe and the > Christian God would be subsets of everything. However I think> the Christian God definition of all powerful ultimate creator maps> to the exact or nearest match someone to all/everything which> however is not perfect, but perfection and perfect goodness > are not mentioned in the Nicene creed.> Also your definition of existence as tied to matter and energy> seems quite naive. If I make a definition, that definition> exists but a match to it may or may not. First of all, I am a Christian, so would consider God part of everything, though obviously not directly observable or measurable.Second, it seems like you are trying to mix math and philosophy/religion, which can be a disastrous mixture. It looks like you're trying to find a mathematical model of reality, without understanding the mathematics you are attempting to model it with. I recommend getting a book that will introduce you to set theory before you try to use it to model anything. I can appreciate the desire to formalize discussions about reality, but with so many different takes on what reality is, you are likely to find that mathematics is an ineffective tool.In your original post, you cross-posted to a.p.taoism, t.r.pantheism, t.r.misc. You have additionally referenced Christianity. This is where the problem comes in: to have a set, it must be well-defined. A pantheist will not include God as described by Christians as part of everything. Taoists may not include observable reality in everything. If your description does not lead to a common understanding of what you mean to include in your set, it is not well-defined. Until you can formulate what you have in mind, you will make no progress.email: wtwentyman at copper dot net=== Subject: : Re: Antidiagonal, InfinityI'm interested in a simple counterexample that is a function inequational form that is not necessarily expressed as a composition oftwo functions. You should be able to figure that out.Did you say a G-rail? Arthur found the Grail and sailed to Avalon.Please explain to me how the naturals are a compact set or similar toa compact set.I'm going to belabor the leading zeros argument for a moment. Youmight say your rule would be that if the zero was a leading zero wherehaving it stay the same value would still allow a different number,but you could leave that or any other element of the expansionunchanged only if you changed some other element. So, you just startbeyond the leading zeros. Yet, where that is so, some of the numbersmight have less leading zeros. Anyways I add leading zeros and sayantidiagonalize them and the antidiagonal is a number that is not anelement of the range, codomain, or coimage. So now please doaddress the leading zeros, another way to change the representationwithout changing the value.What, you can't figure out Borel and Mr. Combinatorics? I just sayhalf.The antidiagonal argument is one of about two results, both by GeorgCantor, about mapping bijectively the (natural) integers to the realnumbers, and contradictions that ensue from that. Please provide aproof besides the antidiagonal or nested intervals argument thatsupports that same conclusion.RossRoss FinlaysonFinlayson Consulting=== Subject: : Re: Antidiagonal, Infinity> I'm interested in a simple counterexample that is a function in> equational form that is not necessarily expressed as a composition of> two functions. You should be able to figure that out.A counterexample to what?> Please explain to me how the naturals are a compact set or similar to> a compact set.Do some research for yourself.> I'm going to belabor the leading zeros argument for a moment. You> might say your rule would be that if the zero was a leading zero where> having it stay the same value would still allow a different number,> but you could leave that or any other element of the expansion> unchanged only if you changed some other element.I would never say anything using so many words to convey so little sense. So, you just start> beyond the leading zeros. Yet, where that is so, some of the numbers> might have less leading zeros. Anyways I add leading zerosTo what are you adding leading zeros. What are they supposed to lead? and sayantidiagonalize them and the antidiagonal is a number that is not an> element of the range, codomain, or coimage. So now please do> address the leading zeros, another way to change the representation> without changing the value.As I have no idea what your leading zeros are all about, I am in no case to address them.> What, you can't figure out Borel and Mr. Combinatorics? I just say> half.> The antidiagonal argument is one of about two results, both by Georg> Cantor, about mapping bijectively the (natural) integers to the real> numbers, and contradictions that ensue from that. Please provide a> proof besides the antidiagonal or nested intervals argument that> supports that same conclusion.If two valid arguments are not enough to penetrate your ignorance, why should I believe that even two thousand will be?> --> Ross Finlayson> Finlayson ConsultingI can imagine Ross consulting others, as he does here, but why would anyone who sees Ross' performance here ever want to consult him, I wonder?=== Subject: : Re: Antidiagonal, InfinityThe infinitesimal iota doesn't have all properties of definite reals,it is among a form of what may be called indefinite reals, realnumbers to be sure but those not representable as a rational as afraction of integers or an irrational as a convergnt sequence.My opinion is that real numbers of the form x+ni, real x plus somenon-zero integer multiple n of iotas, are nearer to x than to anyother real number y for any finite value n, yet are different than x.About the leading zeros, it's like saying: consider a function fromthe integers to [0.0, 0.1]. Make a list and construct theantidiagonal..000.......001...010......011...Construct the antidiagonal:.1xxx...The antidiagonal never except in the case of .011... = .10...represents an element of the range. The antidiagonal always differsfrom each of the elements because each of the elements has a zero inthe first integer modulus after the radix where the antidiagonal doesnot, and the antidiagonal is not an element of [0.0, 0.1).Again in the case of all reals, an antdiagonal could be constructedfor the list of all reals by at index one constructing the element ofthe antidiagonal to the left of the radix, at index two to the right,and so on and so forth as the possible values of the expansion extendinfinitely far, yet finitely, to the left and infinitely to the right. Please consider whether some consider the hyperintegers to containinfinite integers.Explain why that is irrelevant or not so or unbecoming amathematician. I might not be concerned. I'm still trying to figureout dual representation of infinity and zero for the powerset issue. The set of all sets would be its own powerset, and have both a lesserand greater cardinal number than itself. Model only ordinals beingsets.Here's one way I consider iota: dividing it in half is aninfinitesimal again, what with infinitesimals everywhere within thecontinuous real numbers. This is perhaps a strange rationalization,but iota is a limit point of sorts, for example how omega is a limitordinal, iota is a limit ordinalet. Twice iota is some sum ofiota.Why iota? It's little i, infinity is big I, the imaginary root i isnot iota but rather i. Why iota, indeed.Anyways: we discuss here the antidiagonal in terms of infinity. Explain why the antidiagonal is always an element of the range, ordon't.It's a means of support. I'm a passingly decent computer programmer. Is digit summation congruence a worthwhile step in computerizednumerical factorization of integers? What fraction of all integersare multiples of 3, 5, 7, 11, 13, and other factors of small Mersennenumbers, and why? What's the most efficient way to enumerate thek-subsets of a set electronically? What's the best way to translatebetween error function return values and exceptions?Assume the Equivalency Functiion is as I have stated, and more andless and so on and so forth. May you derive any results from it? Doso and they are yours.Ross FinlaysonFinlayson Consulting / Apex Internet Software=== Subject: : Re: Antidiagonal, Infinity> About the infinitesimals vis-a-vis the standard reals, this scalar> infinitesimal iota is within a model of the real numbers where iota is> defined as the least positive real number and through deduction about> the known properties of the real numbers, and as well the understood> limitations of such a definition within a model of the reals, iota can> be determined to fulfill or satisfy certain properties ascribed to an> immediate neighbor of zero in this model of the reals, one of several> within the Finlayson numerical model, contiguous reals. > Run-on sentence notwithstanding, there is no smallest positive real in > the standard model of the reals. You can prove that by noting that for > any e>0, > 0 < e/2 < e.I say that for any _definite_ real e that e/2 exists. Theinfinitesimal iota is atomic, for any definite real number, forexample 0, 1, e, pi, etcetera, infinitesimal differences are declaredto be in increments of iota.This has something along the lines of iota/2 being undefined, but2iota/2 is iota. 3iota/2 is undefined, 4iota/2 is 2iota. Another wayo consider it is iota/2 or iota/3 or iota/n for finite n is iota, andr>iota>0 for definite real r.Also I say the sum from zero to infinity of iota is one. The integralfrom zero to one of 1dx is one.The Equivalency Function doesn't have to use iota.Ross=== Subject: : Re: Antidiagonal, Infinity>About the infinitesimals vis-a-vis the standard reals, this scalar>infinitesimal iota is within a model of the real numbers where iota is>defined as the least positive real number and through deduction about>the known properties of the real numbers, and as well the understood>limitations of such a definition within a model of the reals, iota can>be determined to fulfill or satisfy certain properties ascribed to an>immediate neighbor of zero in this model of the reals, one of several>within the Finlayson numerical model, contiguous reals. > Run-on sentence notwithstanding, there is no smallest positive real in > the standard model of the reals. You can prove that by noting that for > any e>0, > 0 < e/2 < e.> I say that for any _definite_ real e that e/2 exists. The> infinitesimal iota is atomic, for any definite real number, for> example 0, 1, e, pi, etcetera, infinitesimal differences are declared> to be in increments of iota.> This has something along the lines of iota/2 being undefined, but> 2iota/2 is iota. 3iota/2 is undefined, 4iota/2 is 2iota. Another way> o consider it is iota/2 or iota/3 or iota/n for finite n is iota, and> r>iota>0 for definite real r.> Also I say the sum from zero to infinity of iota is one. The integral> from zero to one of 1dx is one.> yson> The Equivalency Function doesn't have to use iota.I used to think you were a deliberate troll. Now I realize you're a nut.=== Subject: : Re: Antidiagonal, Infinity>About the infinitesimals vis-a-vis the standard reals, this scalar>infinitesimal iota is within a model of the real numbers where iota is>defined as the least positive real number and through deduction about>the known properties of the real numbers, and as well the understood>limitations of such a definition within a model of the reals, iota can>be determined to fulfill or satisfy certain properties ascribed to an>immediate neighbor of zero in this model of the reals, one of several>within the Finlayson numerical model, contiguous reals. > Run-on sentence notwithstanding, there is no smallest positive real in > the standard model of the reals. You can prove that by noting that for > any e>0, > 0 < e/2 < e.> I say that for any _definite_ real e that e/2 exists. The> infinitesimal iota is atomic, Careful that you don't concatenate a critical mass of those atomic thingies.for any definite real number, for> example 0, 1, e, pi, etcetera, infinitesimal differences are declared> to be in increments of iota.Nonsense.> This has something along the lines of iota/2 being undefined, but> 2iota/2 is iota. 3iota/2 is undefined, 4iota/2 is 2iota. Another way> o consider it is iota/2 or iota/3 or iota/n for finite n is iota, and> r>iota>0 for definite real r.In other words, more nonsense. If iota is to be an element of any ring containing the rationals then iota/2 must be defined.> Also I say the sum from zero to infinity of iota is one. Your saying it does not make it so. You first have to prove that such iota object is consistent with the known proporties of the reals, which it clearly is not, before you can insist that such an object exists.=== Subject: : Herbert Frohlich's CoherenceBack in San Francisco:sending me his summary of Herbert Frohlich's work upon which I base the following remarks on the physics of matter, mind and consciousness.once matter has been understood, penetrated by mind, the matter itself is transformed H. FrohlichClassical physics is IT.IT = matterQuantum physics isIT piloted by BITBIT = mindIn quantum physics IT receives its marching orders from BIT but not vice versa.Quantum physics is the sound of one hand clapping.IT FROM BIT is monologue one-way relation, action without direct reactionThe IT FROM BIT quantum potential is fragile with signal locality, no cloning a quantumPost-quantum physics isIT FROM BIT + BIT FROM ITThe two way relation, dialogue with signal nonlocality hence presponse, animal telepathy and consciousness.coherence is a powerful, inherently nonlocal property, in terms of which diverse equilibrium and non-equilibrium macroscopic quantum phenomena find a unified description. HylandMy theory of emergent Einstein gravity with dark energy/matter residual zero point energy density fields comes from vacuum coherence.Vacuum coherence is a special case of Frohlich's more general coherence which is also closely related to P. W. Anderson's More is different and spontaneous breaking of symmetry for the ground state of real on-mass-shell processes like a superconductor and the vacuum state for virtual off-mass-shell processes like Einstein's gravity as the bending of smooth space-time.The creative tension between the themata of nonlocal and local should be noted.Coherence has nonlocal long-range motional phase stability or rigidity with a local order parameter (AKA giant or macro quantum IT/BIT wave).Coherence being of a rather general nature was enthusiastically espoused by Frohlich following Yang's introduction of the associated concept of 'off-diagonal-long-rang-order' (AKA ODLRO) - a concept which Frohlich preferred to re-express in terms of macroscopic wave-functions exhibiting long-range phase correlations. G.J.H.The living conscious mind is such a macroscopic ground state wave.Einstein's gravity field and the dark energy and dark matter emerge from a macroscopic vacuum wave.into coherent macro-quantum waves and a residual random zero point/normal fluid.N = 1 in superfluid HeliumN = 2 in the real superconductor and also in emergent Einstein gravity from a virtual electron-positron vacuum condensate giant quantum wave.Andrei Sakharov's metric elasticity is essentially an effect of the long-range phase correlations of the giant quantum vacuum wave that coheres the great bulk of the random zero point vacuum fluctuations of all the substrate quantum fields of all spins.Coherence is oil calming the turbulent waters of the void.This is the decoding of the Cipher of Genesis (Carlo Suares, Paris, 1973).In next installment some comments on how Frohlich applied some of these same general ideas to living matter.Frohlich's paper is not readable by non-specialists.He does show how open dissipative systems far from thermal equilibrium above a critical threshold of energy flowcan develop a giant quantum wave analogous to a Bose-Einstein condensate in quantum equilibrium statistical mechanics.This wave can develop as an electric polarization in a living membrane for example - or in the microtubules I suspect.In my theory the BIT component of this giant electric polarization field is the physical mind field with a two-way relation between it's IT and BIT components, which make it conscious with signal nonlocality - a necessary condition for Dick Beirman's presponse data, for Rupert Sheldrake's data and for Puthoff-Targ SRI RV data. Signal nonlocality of course strongly violates micro-quantum theory, which in my view only works for non-living matter in thermal equilibrium or close to thermal equilibrium obeying the linear response fluctuation-dissipation theorem and the Kramers-Kronig retarded causality relations, both violated above a critical threshold of emergent complexity in open systems. This is the threshold between the nonliving and the living.Micro-quantum theory with signal locality is to macro-quantum theory with signal nonlocality as special relativity is to general relativity.Consciousness emerges in the ground state phase transition from micro to macro-quantum theory in very much the same formal waythat gravity (curvature) emerges in the micro-quantum to macro-quantum vacuum phase transition of inflationary cosmology.Hesse) mathematics of category theory (functors, morphisms) there is a formal parallelism between consciousness and gravity though not the way Roger Penrose means it.Frohlich gives all sorts of enticing tidbits about non-thermal effects of very low intensity micro and millimeter waves on biological complex open systems including cancer, in which the pulsing profile as well as the specific sharp frequencies are important - for effects not seen with continuous irradiation at same mean intensity.Frohlich's model is organic in a very similar way to Bohm's quantum potential but with significant dynamical differences of course.