mm-131 Make no mistake, IOm in this for the money.If you really believe that someone is going to pay you for discovering a newmathematical fact, IOd like to help you invest that money so that it will dome, er, I mean _both_ of us some good. ThereOs this bridge that connectsManhattan to New Jersey. No, no, not _that_ one, the other one! Got it? Good.Now I have it on excellent authority that the bridge is for sale to aquali'ed loony, er, I mean party, because the Port of New York Authorityneeds cash. The deal involves a few million up front plus a percentage of thetolls for the next three centuries. IOm sure that with your enormousarithmetical skill, you will see instantly that purchasing this bridge willmake you very, very rich - far richer than mere mathematical discoveries willever make you. For pointing you in the direction of this investment, andbrokering the deal on your behalf, IOm asking a piddling 30% of net pro'ts.blind.owl@third.tree.from.the.corner.com, and we can work out a deal. Be sureto use RSA encryption - otherwise some damn unethical mathematician will beable to sneak in and take advantage of this opportunity before we can do so.DonOt delay! This is a once in an evening opportunity. Oh, and you need notsend me the key - IOve found a method of decrypting RSA, which I will sharewith the world as soon as it recognises my superior genius and provides meloadsadough and excellent babes - a genius superior even to yours, IOm sad tosay, but themOs the breaks of the genetic lottery.-- Wolf Kirchmeir, Blind River ON CanadaNature does not deal in rewards or punishments, but only in consequences.(Robert Ingersoll) > Well I claim that my prime formula is a great discovery, while others> keep posting that itOs not important at all!> > Yes, that would be because itOs not important at all.> > -jcrReally? Then clearly if you *know* that so that you can be socertain, you can explain how it works, right?Now then, why donOt you try and explain how it works.For your reference, here it is again:dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1,sqrt(y-1))],S(x,1) = 0.And p(x, y) = §oor(x) - S(x, y) - 1, and you get S as the sum of dSfrom dS(x,2) to dS(x,y).The count of primes is given by p(x, sqrt(x)), now then John C.Randolph, why donOt you explain why.Can you? Or are you just another Usenet loser trying to blow offsteam at my expense?James HarrisMy math discoveries, found for pro'thttp://mathforpro't.blogspot.com/ > Well I claim that my prime formula is a great discovery, while others> keep posting that itOs not important at all!>> Yes, that would be because itOs not important at all.>> -jcr>> Really? Then clearly if you *know* that so that you can be so> certain, you can explain how it works, right?>> Now then, why donOt you try and explain how it works.Why donOt *you* explain how it works? If your discovery is as important as you claim it is, the propercourse of action is to write a paper describing its operation in great detail. Either publish the paper ina reputable journal or publish it yourself. But why in the world are you asking *other* people to explainhow it works?You have succeeded in elevating idiocy to unprecedented heights, Wacky. Go back to your playpen.--There are two things you must never attempt to prove: the unprovable -- and the obvious.--Democracy: The triumph of popularity over principle.--http://www.crbond.com >> Well I claim that my prime formula is a great discovery, while others> keep posting that itOs not important at all!>> Yes, that would be because itOs not important at all.>> -jcr>> Really? Then clearly if you *know* that so that you can be so> certain, you can explain how it works, right?>> Now then, why donOt you try and explain how it works.> > Why donOt *you* explain how it works? If your discovery is as important as you claim it is, the proper> course of action is to write a paper describing its operation in great detail. Either publish the paper in> a reputable journal or publish it yourself. But why in the world are you asking *other* people to explain> how it works?IOve explained in the past but noticed that other posters would justuse information I provided to try and confuse others. Yup, theyOdpervert the process.Here, by not explaining 'rst in this thread, IOm showing readers thatall these people trying to convince them that my prime area discoveryis in fact unimportant, are in fact, lying.If theyOre not lying then they have the expertise to answer thequestion of how my partial difference equation works.> You have succeeded in elevating idiocy to unprecedented heights, Wacky. Go back to your playpen.> The essential point is that IOm someone who made a nice discovery inthe area of prime numbers, but rather than at a minimum acknowledge mydiscovery, mathematicians chose to ignore or downplay it.ItOs clearly a political decision at odds with claims ofmathematicians about the importance to them of pure math or beauty,or math for mathOs sake.ItOs also a high-risk strategy which clearly could back-'re withstupendous consequences, so why would they do it?I want you to think about that question without me trying to explainit to you, except to remind you of others in a high stakes areaengaging in seemingly strange behavior. Like consider the currentpresident of the United States.James HarrisMy math discoveries, found for pro'thttp://mathforpro't.blogspot.com/ *you* explain how it works? If your discovery is as important as you claim it is, the proper> course of action is to write a paper describing its operation in great detail. Either publish the paper in> a reputable journal or publish it yourself. But why in the world are you asking *other* people to explain> how it works?>> IOve explained in the past but noticed that other posters would just> use information I provided to try and confuse others. Yup, theyOd> pervert the process.What process? Nothing prevents you from publishing a complete, detailed exposition of your discovery withunambiguous, step-by-step examples of its operation and explanation of the operating theory. Any decentresearcher would do so as a matter of course.> Here, by not explaining 'rst in this thread, IOm showing readers that> all these people trying to convince them that my prime area discovery> is in fact unimportant, are in fact, lying.Unimportant? To whom? Each reader is entitled to judge its importance for themselves. It is decidedlyunimportant to me and IOm entitled to say so. So far, it appears that your discovery is only important to*you*. You are entitled to make that judgment. Others are equally entitled to make theirs.> If theyOre not lying then they have the expertise to answer the> question of how my partial difference equation works.If they are not lying, then they are sincere in their judgment that your discovery is unimportant. If it isunimportant (to them) they will have little interest or motivation in doing your work for you. It is *your* jobto explain how it works. It is each readerOs job to determine whether your discovery is of any use to him.> You have succeeded in elevating idiocy to unprecedented heights, Wacky. Go back to your playpen. The essential point is that IOm someone who made a nice discovery in> the area of prime numbers, but rather than at a minimum acknowledge my> discovery, mathematicians chose to ignore or downplay it.Tough luck. No one is under any obligation to value your work. Each reader has a right to ignore your work orto downplay or even condemn it.> ItOs clearly a political decision at odds with claims of> mathematicians about the importance to them of pure math or beauty,> or math for mathOs sake.Political? How does politics enter into this? The issues of pure math, beauty, or math for mathOs sake arephilosophical matters, not political ones. The choice to ignore or downplay your work is a judgment call thateach reader is entitled to make -- by right. And it stretches credulity to imagine that political motives haveany role in that choice.> ItOs also a high-risk strategy which clearly could back-'re with> stupendous consequences, so why would they do it?Why high-risk strategy? ThereOs no strategy at all in evidence, much less any risk. You offered a solution toa prime counting problem, others are unimpressed. ThatOs their prerogative.> I want you to think about that question without me trying to explain> it to you, except to remind you of others in a high stakes area> engaging in seemingly strange behavior. Like consider the current> president of the United States.What high stakes. What has the president got to do with your discovery? Are you mad???..Or are you just alegend in your own mind?--O wad some PowOr the giftie gie us, to see oursels as others see us! (from: To A Louse, by Robert Burns.)--Democracy: The triumph of popularity over principle.--http://www.crbond.com =Already got one reply, and itOs not worth discussing much. Seemscomputer scientists defer to mathematicians. Oh well, not really abig surprise. __JSH> Well, IOve started, and like in the past, I may post responses from> editors, as they can be so amusing. And yes, for those of you who> donOt know, IOve got a nice collection of interesting responses from> journal editors!> > ___JSH> > Why not submit your code to a numerical analysis or computer> journal or maybe even take it to the Computer Science> Department at Vanderbilt and see what they think?> IOll tell you why not. Because we both know that> it is not at the level of original research. ItOs so child-like> to maintain the fantasy of being a misunderstood genius.> Your childhood glory days are over, James. Now youOre a> grown-up troll and/or crank. If you canOt face it hoist another> > Wait though, it seems to me that the need of the poster Uncle Al to> question the correctness of my work shows I think the popular feeling> that a valid result in the area of prime numbers *should* be worth> noting, especially by mathematicians!!!> > The math formula is> > dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1,> sqrt(y-1))],> > S(x,1) = 0.> > And p(x, y) = §oor(x) - S(x, y) - 1, where you get S(x,y) as the sum> of dS from dS(x,2) to dS(x,y).> > And, amazingly enough, p(x,sqrt(x)) gives the count of prime numbers> up to and including x, like p(100,10) = 25, the count of primes up to> and including 100.> > IOll also include a straightforward Java implementation, which also> prints out prime numbers found by my formula along the way! ___JSH> ThatOs a red herring as the speed issue is secondary. I admit that a> straightforward implementation directly from the math is slow, but my> point is uniqueness, beauty and purity, as in pure math.You have written various versions, some simpler, some more complicated, some faster, some slower. I have posted code that is simpler than your simplest version, uses less memory, and runs faster. I have also published code that runs about 1000 times faster than your fastest version. And as far as beauty is concerned, any experienced Java programmer having a look at your code will just throw up. However, I have a suspicion that the people whoOve been boldest in> claiming that my work isnOt important are ones who canOt explain how> it works!!!On my webpage www.cbau.freeserve.co.ukwhich contains my source code for a prime counting algorithm that beats yours by a factor of about 1000 for large values, there is also a .pdf 'le MeisselLehmerAlgorithm.pdf which on the 'rst one and a half pages describes the Legendre algorithm which is equivalent to your algorithm, and describes in two sentences the basics of MeisselOs algorithm and LehmerOs algorithm which are far superior. It is quite sad that you never bothered to improve your algorithm by at least using MeisselOs method which is really very easy. If you had done that, your algorithm wouldnOt choke on memory consumption for smallish results like pi (10^16). genuinely _new_ stuff, unlike yours! Well the data are coming from a population that follows a> normal distribution, in my case is just data from a disease,> but in these data part of them are coming from another > different disease. The fact is the second disease values > are always much bigger than the mean of the previous one> and the problem is they are not so numerous to separate then> with a mixture of gaussian. Because of they are a small number > my algorithm does not works. And I need to separate them> because they will be noise in my results. Looking to the> histogram they can consider outliers(because they are> far away from the peak) so stimating the variance I can> take the 90 % of the data that belongs to the 'rst disease> and do my results with themWell, from this description it seems clear that the observations are described well by a mixture model,with one bump for each disease. Making a hard assigment of each observation to one bumpor the other is an approximation to the right solution, which is to count each observation in proportion to howwell it belongs to each bump. If there is a lot of overlap,partial assignment yields substantially different resultsthan all-or-nothing assignment.ItOs not very dif'cult to work with partial assignments,so I donOt see that thereOs much to gain by thinking up various hacks. Another consideration is that disease #2may be of greater importance in some slightly differentcontext; why not get in the habit of working carefully,so that you can say something interesting about both diseases.For what itOs worth,Robert Dodier--... much of what is called rational seems more like rationalization, to me; a ruse intended to makesomething desired appear necessary. -- Jeff Inman IOve always understood the ideal gas law PV=nRT to refer to an ideal> gas contained in a container with a boundary, the boundary playing some> role in the de'nition of pressure. On the other hand, suppose you> consider an ideal gas in a compact manifold without boundary, e.g.> the 3-sphere. How would one formulate the ideal gas law in that> context or in the full generality of manifolds? Surely someone> must have worked that out, and references to relevant literature> would be helpful.> > Would it simply be used to *de'ne* pressure in the case of a compact> manifold without boundary?You can turn the gas law from global to local by dividing through byvolume: P = nRTwhere P is the pressure, n is the density in moles/unit volume, and Tis the temperature. Pressure can be measured by inserting a small,evacuated box into the gas and measuring the force inward on itswalls.Dale =ItOs well-known that compact (real) 2-manifolds have little variety,they are just spheres with zero or more toruses and cross-caps graftedon. If you look at the orientable manifold of genus 1, the torus, itis homogeneous in that there is a homeomorphism that takes any pointto any other point. And it seems to be isotropic, in that there is ahomeomorphism that rotates the neighborhood of any point in any wayyou want.If you look at manifolds with an ordinary metric structure(differentiable manifolds with metric?), this breaks down. Forinstance, the torus made by identifying the opposite sides of a square(R^2 mod Z^2) is homogeneous, but it isnOt isotropic, because the fourshortest non-trivial loops from a point to itself are aligned on theaxes of R^2.Is there a torus as a manifold with a metric structure that is bothhomogeneous and isotropic?What about higher dimensions? Is there a general classi'cationtheory of manifolds with metrics?Dale ItOs well-known that compact (real) 2-manifolds have little variety,> they are just spheres with zero or more toruses and cross-caps grafted> on. If you look at the orientable manifold of genus 1, the torus, it> is homogeneous in that there is a homeomorphism that takes any point> to any other point. And it seems to be isotropic, in that there is a> homeomorphism that rotates the neighborhood of any point in any way> you want.> As you probably know, every smooth manifold admits a self-diffeomorphismthat exchanges any two of its points (and more generally, that achievesan arbitrary permutation of any of its 'nite pointsets [that diffeo-morphism will of course depend on what subset youOre looking at, andwhat permutation you have in mind]).However, IOm sure I donOt understand your notion of isotropic, sincefor any manifold M, if you give a point x and a suf'ciently smallneighborhood U of x in M, there is self-diffeomorphism of M that rotatesthat neighborhood any way you want (i.e., given an element R of SO(n),there is a map f: M > M, that 'xes x, and maps U to itself via Mas follows: f U -> U | | | | | | V V D^n --> D^n Rwhere the vertical maps are the (same) coordinate map to a disc in R^n).That map f can also be 'xed as the identity outside an epsilonneighborhood of U.So, you clearly mean much more than this.On the other hand, as far as I know, there is no effective circleaction that 'xes any point of T^2 (i.e., an action by S^1 = SO(2),for which every non-identity element actually moves some point). SinceS^1 is connected, any circle action will consist of maps homotopic tothe identity, sending the obvious generators of H_1 to themselves. Thisseems to be a version of the problem you identify below [viz, the fourshortest non-trivial loops]:> If you look at manifolds with an ordinary metric structure> (differentiable manifolds with metric?), this breaks down. For> instance, the torus made by identifying the opposite sides of a square> (R^2 mod Z^2) is homogeneous, but it isnOt isotropic, because the four> shortest non-trivial loops from a point to itself are aligned on the> axes of R^2.> > Is there a torus as a manifold with a metric structure that is both> homogeneous and isotropic?> > What about higher dimensions? Is there a general classi'cation> theory of manifolds with metrics?> Classi'cation of maps among manifolds, or of maps from a manifold toitself, has been an active research area: the buzz word is mappingclass group meaning the group of smooth isotopy classes (i.e., homotopythrough self-diffeomorphisms) of self-diffeomorphisms. It is related tostudy of the homotopy theory of the space of self-diffeomorphisms. Sincehomotopy reduces continuous questions to discrete ones, perhaps itOs notwhat you care to look at.There are also the areas of homogeneous spaces (spaces with a transitiveaction by a Lie group), and group actions on manifolds. Somewhere in mydistant past I recall hearing people discuss (as a measure of symmetryof a manifold M) the largest dimension of a Lie group that had an effective action on M; the upshot of the discussion was the result thatamong the various exotic differential structures on spheres [I donOt recall whether this was speci'cally S^7, or N in general], only thestandard, round sphere could achieve the value of N(N-1)/2, the dimension of SO(N+1).Along these lines, I came across this paper by Volker Puppe: Do manifolds have little symmetry? www.inf.uni-konstanz.de/Schriften/ papers/2002/preprint-181.pdf> DaleDale. ItOs well-known that compact (real) 2-manifolds have little variety,> they are just spheres with zero or more toruses and cross-caps grafted> on. If you look at the orientable manifold of genus 1, the torus, it> is homogeneous in that there is a homeomorphism that takes any point> to any other point. And it seems to be isotropic, in that there is a> homeomorphism that rotates the neighborhood of any point in any way> you want.> > If you look at manifolds with an ordinary metric structure> (differentiable manifolds with metric?), this breaks down. For> instance, the torus made by identifying the opposite sides of a square> (R^2 mod Z^2) is homogeneous, but it isnOt isotropic, because the four ^^^^> shortest non-trivial loops from a point to itself are aligned on the ^^^^^^^^ How did you get four? I only see two...> axes of R^2.> > Is there a torus as a manifold with a metric structure that is both> homogeneous and isotropic?> > What about higher dimensions? Is there a general classi'cation> theory of manifolds with metrics?Maybe you are looking for *Zoll manifolds*, Riemannian manifolds whose geodesics are all simple closed curves of equal length. If yes, you will 'nd a lot of interesting results in A. L. Besse, Manifolds All of Whose Geodesics Are Closed (Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge. A Series of Modern Surveys in Mathematics Bd. 93)By a result of Bott and Samelson, every Zoll manifold is topologically a CROSS (compact rank-one symmetric space), and so must be Sn , RPn, CPn, HPn, or the Cayley plane. > Tired light.IMO tired light is the true cause of thered-shifying.[snip in agrrement ]>> Tired light. OccamOs razor.>> John>Tired light doesnOt work. IOve repeated ad nauseam what OwillO work, and>why nothing else will. To wit, relative motion of source and observer,>or a source frequency shift over time. Any other proposals result in a>loss of information. Frequency is by de'nition cycles of some system>per unit time. A cycle is an absolute event and the information>associated with that cycle is just the energy-change propagating away>from the source of the OchangeO. Since observed frequency is just the>result of interception of the outgoing information, then for bodies at>rest wrt each other you have simply:>>f * delta t = fO * delta tO >>Here the time intervals are just those required to OexperienceO a given>absolute portion of the information stream.>>If fO < f, then >>delta tO > delta t>>Thus if the clocks measuring these intervals are at rest wrt each other,>then they must be ticking at different rates. If OTOH it is noted that>delta t was the time interval wrt the source at the era of emission of>the information stream, and that this was millions of years ago in our>present discussion, then it is not necessary that the clocks be>simultaneously ticking at different rates, only that the rate was>different at that time than it is now. Tired light theory assumes no>change in ticking rates of the clocks, which latter may very well be the>very radiation source and detector respectively. Neither does it assume>a simultaneous difference in ticking rates between the clocks, and thus>it cannot account for the doppler shift without requiring a loss of>objective events (cycles) wrt one of the frames.>>In the case that the clocks are not at rest wrt each other, then changes>in ticking rates are unnecessary in order to account for the red shift,>the relative motion is suf'cient to produce ordinary doppler. Special>relativity OTOH assumes both, i.e. that clockOs rates shift OandO there>is relative motion between source and detector. Tired light in this>context still involves the loss of information, and thus is incorrect,>no question.>Richard Perry(yes question:-)I like Richards approach but disagree withhis conclusion. I do agree with Richard, itis a fact that INFO = frequency * duration = invariant.where Richard calls INFO,. A cycle is an absolute event and the information...But the duration is expressed by (N = cycles)duration = N * wavelength, then INFO = N*c, (c=freq*wavelength). Because the wavelength increases there isno loss of information carried by the photon.Think about how information transfer rateincreases proportional to frequency, andthink about how no information is lost whena photon is re§ected from a receding mirror.A receding mirror will red-shift the photon onre§ection (doppler effect) but no INFO is lost. As a photon propagates across the empty voidsof inter-galatic space, it is being de§ected by allthe matter that gravitationally in§uences that location, and gravitational de§ection sucks photonmomentum, and frequency, but INFO remainsinvariant.Ken S. Tucker >> Tired light. IMO tired light is the true cause of the> red-shifying.> [snip in agrrement ]> >> Tired light. OccamOs razor.>> JohnTired light doesnOt work. IOve repeated ad nauseam what OwillO work, and>why nothing else will. To wit, relative motion of source and observer,>or a source frequency shift over time. Any other proposals result in a>loss of information. Frequency is by de'nition cycles of some system>per unit time. A cycle is an absolute event and the information>associated with that cycle is just the energy-change propagating away>from the source of the OchangeO. Since observed frequency is just the>result of interception of the outgoing information, then for bodies at>rest wrt each other you have simply:>>f * delta t = fO * delta tO>>Here the time intervals are just those required to OexperienceO a given>absolute portion of the information stream.>>If fO < f, then>>delta tO > delta t>>Thus if the clocks measuring these intervals are at rest wrt each other,>then they must be ticking at different rates. If OTOH it is noted that>delta t was the time interval wrt the source at the era of emission of>the information stream, and that this was millions of years ago in our>present discussion, then it is not necessary that the clocks be>simultaneously ticking at different rates, only that the rate was>different at that time than it is now. Tired light theory assumes no>change in ticking rates of the clocks, which latter may very well be the>very radiation source and detector respectively. Neither does it assume>a simultaneous difference in ticking rates between the clocks, and thus>it cannot account for the doppler shift without requiring a loss of>objective events (cycles) wrt one of the frames.>>In the case that the clocks are not at rest wrt each other, then changes>in ticking rates are unnecessary in order to account for the red shift,>the relative motion is suf'cient to produce ordinary doppler. Special>relativity OTOH assumes both, i.e. that clockOs rates shift OandO there>is relative motion between source and detector. Tired light in this>context still involves the loss of information, and thus is incorrect,>no question.>Richard Perry> > (yes question:-) I like Richards approach but disagree with> his conclusion. I do agree with Richard, it> is a fact that> > INFO = frequency * duration = invariant.> > where Richard calls INFO,. A cycle is an> absolute event and the information... But the duration is expressed by (N = cycles)> > duration = N * wavelength, thenWrongtime interval source = N / frequencyfrequency = c/wavelengthThus time interval source = N * wavelength/ c> > INFO = N*c, (c=freq*wavelength).INFO = N> > Because the wavelength increases there is> no loss of information carried by the photon.True, but how are you going to shift the wavelength? Each end of thewave represents the information of the beginning of two separateobjective events, e.g. cycles, both of these information ObitsO ispropagating at the same speed wrt the observer, and thus theirinstantaneous displacement in space wrt each other is 'xed wrt theobserver from the time of its emission to its absorption, i.e.throughout its entire trip. The only way to change a wavelength in§ight is to change the speed of the wave, OTOH, this will not change> > Think about how information transfer rate> increases proportional to frequency, and> think about how no information is lost when> a photon is re§ected from a receding mirror.> A receding mirror will red-shift the photon on> re§ection (doppler effect) but no INFO is> lost.> > As a photon propagates across the empty voids> of inter-galatic space, it is being de§ected by all> the matter that gravitationally in§uences that> location, and gravitational de§ection sucks photon> momentum, and frequency, but INFO remains> invariant.The frequency never changes wrt a given inertial frame.> > Ken S. TuckerRichard Perry Tired light.>> Besides, I donOt think universal expansion is possible, do you?Who are you to say whatOs possible?> If IOm at some arbitrary center, and IOm expanding, you are also> expanding, but you must also be accelerating away from me, and> someone behind you has to accelerate away even faster, etc, etc, to> where things really far behind must be moving at incredible speeds> away from us.>> Tired light. OccamOs razor.One of Fred HoyleOs theories was that the universe isnOt expanding,the atoms inside are shrinking. That would produce an apparent redshift since the light would have been emitted by larger atoms emittinglonger wavelengths.-- http://hertzlinger.blogspot.com Within a week or so, I will be releasing a free beta version download for my> new proof checking software, DC Proof 1.0.> > In the mean time, here is a sampler from the User Guide:> > http://members.allstream.net/~dchris/DCProofT.chm> > It contains a tutorial that illustrates many of the main features of DC> Proof. Readers may be interested in both theoretical and a pedagogical> aspects of this application. Example 3, is a resolution of RussellOs Paradox> without the usual prohibition on self-reference.> > Enjoy.> > Dan Christensen> Toronto, CanadaYour system appears to be a great aid to logicians writing proofs, andwill also hopefully lend more insight into the exact nature of proofs. I will be happy to obtain a copy.While a great bookkeeping aid, your system doesnOt seem to do anythingthat isnOt being done by hand already. Indeed, the user explicitlyenters the proof itself. This is a helpful tool, but I donOt see howyou have resolved the Russell Paradox. You have only computerizedthe same proof that is written out by hand. You correctly (IMHO)conclude that there is no Russell Set (the set of sets that donOtcontain themselves), which is the common conclusion one reaches fromseeing the contradiction.However, the question remains, what do we do about it? How do wede'ne sets to include the sets that mathematicians use, but excludethe Russell Set? Are you simply saying donOt allow it? But what dowe allow? Everything else?ZF and various other axiomitizations of set theory attempt to de'nesets in a way that that meets these two needs (completeness withoutcontradiction.) Do you really have a solution to that problem?Charlie VolkstorfCambridge, MAhttp://www.mathpreprints.com/math/Preprint/CharlieVolkstorf /20021008.1/1http://www.arxiv.org/html/cs.lo/0003071 = Within a week or so, I will be releasing a free beta version download for mynew proof checking software, DC Proof 1.0.In the mean time, here is a sampler from the User Guide:http://members.allstream.net/~dchris/DCProofT.chmIt contains a tutorial that illustrates many of the main features of DCProof. Readers may be interested in both theoretical and a pedagogicalaspects of this application. Example 3, is a resolution of RussellOs Paradoxwithout the usual prohibition on self-reference.Enjoy.Dan ChristensenToronto, Canada Are there any number-theoretic statemtents which are known to be > undecidable in ZFC? If so, some examples?> > This is a follow-up to some comments in the 2^pi thread, where JIP, I > believe, asked about unknown results in number theory, and GM responded > with something about decidability.> > This may depend, of course, on what you mean by number-theoretic.> With a slightly liberal interpretation, the following is an admissible> solution, I guess: If a family of sets of natural numbers cannot be> put into one-to-one correspondence with the natural numbers, can it> then be put into one-to-one correspondence with the family of all sets> of natural numbers?> > I imagine, however, that this isnOt what you had in mind. A better> solution, but one IOm not entirely sure about, may a form of> MatijasevicOs theorem: there is (perhaps?) a family of diophantine> equations (namely, a single equation with one of the variables treated> as a parameter) such that the set of values of the parameter for which> the equation has a solution cannot be determined within ZFC. Once> again, IOm not quite sure this statement is correct - IOll be glad to> know if it is.reviewed in the quote from the Book review:MatiyasevichOs most important contribution since solving the problem has to be his introduction of new exponential Diophantine coding techniques. With such, he improved the initial Matiyasevich-Robinson small-number-of-variables result fromthe algorithmic unsolvability of the general 13-variable Diophantine problem tothe algorithmic unsolvability of the 9-variable problem.If ZFC could decide every 9-variable problem, then I think the 9-variable problem would be algorithmically solvable.So I think some 9-variable problem is undecidable in ZFC.HereOs a link to the book review:http://www.ams.org/journals/bull/pre-1996-data/199501/ 199501014.htmlDavid Bernier If ZFC could decide every 9-variable problem, then I > think the 9-variable problem would be algorithmically solvable.??? why ??? I think perhaps the question is really: Are there any independently> interesting number theoretic statements which are undecidable in ZFC?Do a search for Paris-Harrington & see what you think.-- > I think perhaps the question is really: Are there any independently> interesting number theoretic statements which are undecidable in ZFC?> > Do a search for Paris-Harrington & see what you think.IsnOt this an example of independence within PA, but not ZFC? I knowthat GoodsteinOs theorem is of this kind (unprovable in PA, true inZFC, actually in PA+some simple ordinals) and I thought thatParis-Harrington was, too; perhaps I got it wrong. > I think perhaps the question is really: Are there any independently> interesting number theoretic statements which are undecidable in ZFC?> > Do a search for Paris-Harrington & see what you think.> > IsnOt this an example of independence within PA, but not ZFC? I know> that GoodsteinOs theorem is of this kind (unprovable in PA, true in> ZFC, actually in PA+some simple ordinals) and I thought that> Paris-Harrington was, too; perhaps I got it wrong.YouOre probably right, most likely I was confused.-- > I think perhaps the question is really: Are there any independently> interesting number theoretic statements which are undecidable in ZFC?> > Do a search for Paris-Harrington & see what you think.Paris-Harrington is provable in an extension by re§ection ofPA. =This question is related to the planning of electric systems. I havea sequence of functions (c_n), de'ned on [0, inf) and with values inthis same set. These functions, usually called marginal costs, arestrictly increasing but have discontinuities. They all have a same setD of discontinuities on [0, inf), which is 'nite. In addition, I knowthat, for every x>=0, the sequence (c_n(x)) is strictly increasing andconverges to a c(x), but IOm not sure if the convergence c_n -> c isuniform. All I can assure is that itOs pointwise.What I want is to 'nd a good approximation of the total cost for eachx>0. So, I can integrate, at least numerically, the function c_n over[0, x], getting a function C_n that, for each x, gives the total costof supplying the load x. But, are the given condition suf'cient toguarantee this provides an accurate aproximation? If I could guaranteethe set of discontinuities of c is again the set D, then I could applyDinis theorem to guarantee the convergence c_n -> c was piecewiseuniform on [0, M] for some M. Therefore, the convergence of C_n to Cwould be piecewise uniform and IOd get a good approximation for thetotal cost.Does any one have a clue if, from the given conditions, there issomething interesting we can conclude?Artur > This question is related to the planning of electric systems. I have> a sequence of functions (c_n), de'ned on [0, inf) and with values in> this same set. These functions, usually called marginal costs, are> strictly increasing but have discontinuities. They all have a same set> D of discontinuities on [0, inf), which is 'nite. In addition, I know> that, for every x>=0, the sequence (c_n(x)) is strictly increasing and> converges to a c(x), but IOm not sure if the convergence c_n -> c is> uniform. All I can assure is that itOs pointwise.> > What I want is to 'nd a good approximation of the total cost for each> x>0. So, I can integrate, at least numerically, the function c_n over> [0, x], getting a function C_n that, for each x, gives the total cost> of supplying the load x. But, are the given condition suf'cient to> guarantee this provides an accurate aproximation? If I could guarantee> the set of discontinuities of c is again the set D, then I could apply> Dinis theorem to guarantee the convergence c_n -> c was piecewise> uniform on [0, M] for some M. Therefore, the convergence of C_n to C> would be piecewise uniform and IOd get a good approximation for the> total cost.> > Does any one have a clue if, from the given conditions, there is> something interesting we can conclude?For each b > 0, Cn(b) -> C(b) by the monotone convergence theorem. This implies Cn -> C uniformly on each [0,b]: For any x in [0,b], 0 <= C(x) - Cn(x) = int_[0,x] (c - cn) <= int_[0,b] (c - cn) = C(b) - Cn(b);we used the nonnegativity of c - cn to get the second inequality. I have an unsolved problem. ItOs about a Diophantine Equation.>The problem is ... Find, with proof, all positive integers a and b such that>a^4+(a^2-1)^2=b^2. .The only rational numbers, c/b, close enough to sqrt(2) to suf'ce inequation [3] are from the continued fraction expansion of sqrt(2); i.e. 1 3 7 17 41 - , - , - , -- , -- , ... 1 2 5 12 29Where both the numerators and denominators follow the recursive rule x = 2 x + x [4] n n-1 n-2However, we can only use every other fraction, since equation [3] needsan underestimate; i.e. 1 7 41 - , - , -- , ... 1 5 29Thus, if there is another pair (a,b) that satis'es [1], b needs to begreater than 10^7656. This seems to indicate that there may not be anypairs other than (1,1) and (2,5). However, I donOt have a proof of thisyet.Rob Johnson take out the trash before replying > To : Jose Carlos Santos> No. It Os not help anything. I well understand in pytagorus triples.> a^2 + b^2 = c^2> it can be prove that : (a,b,c) = (st, (s^-t^2)/2, (s^2 + t^2)/2)> for s, t is odd and s > t >= 1 and gcd(s,t) = 1> > or (a, b,c) = (2uv, u^ - v^, u^2 + v^2)> for u - v = odd, u > v >=1, and gcd(u, v) = 1> > I try and try ..... to solve but itOs not so children. You should to> solve and you will 'nd that itOs very very dif'cult.> > Now. I believe that there Os only 2 solutions. (a, b) = (1,1), (2,5)> But. I canOt prove.> > No one can solve it. ??? !!!!!!!!!!There are several regulars here who IOm sure could solve it with enougheffort, including possibly me (although IOm somewhat hampered by beingcriminally careless, as you noticed ;-(Why not try the same type of approach I used in case 2 of my OsolutionO?If I get time IOll look at the problem again next week.-- -John R Ramsden (jr@adslate.com) Eternity is a long time, especially towards the end. Woody Allen =Larry Hammick> phongthong> I have an unsolved problem. ItOs about a Diophantine Equation.> The problem is ... Find, with proof, all positive integers a and b such> that> a^4+(a^2-1)^2=b^2. .> aa = mm - nn> aa - 1 = 2mn> or> aa - 1 = mm - nn> aa = 2mn.> Therefore> (m-n)^2 - 2n^2 = 1> or> (m-n)^2 - 2n^2 = -1.> These are Pell equations, and we can write down all the solutions. Let Abe> the square matrix> 1 1> 2 1> and consider the powers A^n. The upper row (x,y) of A^n satis'es> xx - 2yy = -1 if n is odd> = 1 if n is even> and these are the only solutions of the two Pell equations for m-n and n.> But we also need> mm - nn to be a square (a^2). Still looking...As you thought, the only solutions for (a,b) are (1,1) and (2,5). I managedto prove it using the lemma below.Lemma: The equationx^4 - 2y^2 =1has no solutions except y=0 and x=+-1.Proof: In the ring Z[i] we have this factorization:x^4-1 = (x-1)(x+1)(x-i)(x+i)and if we have(x-1)(x+1)(x-i)(x+i) = 2yyand y>0 then the number 1+i (which is prime in Z[i]) divides the left sidean odd number of times (see why?), and the right side an even number oftimes, which cannot be true, since factorization in Z[i] is unique. Thusy=0, as claimed.The solution a=1 corresponds to a solution of the Pythagorean triple witha=mm-nn, and the other solution a=2 is from a=mn with m=2 and n=1.Larry Larry Hammick>> phongthong>> I have an unsolved problem. ItOs about a Diophantine Equation.>> The problem is ... Find, with proof, all positive integers a and b such> that>> a^4+(a^2-1)^2=b^2. (1,1) and (2,5). I managed>to prove it using the lemma below.>>Lemma: The equation>x^4 - 2y^2 =1>has no solutions except y=0 and x=+-1.>Proof: In the ring Z[i] we have this factorization:>x^4-1 = (x-1)(x+1)(x-i)(x+i)>and if we have>(x-1)(x+1)(x-i)(x+i) = 2yy>and y>0 then the number 1+i (which is prime in Z[i]) divides the left side>an odd number of times (see why?), and the right side an even number of>times, which cannot be true, since factorization in Z[i] is unique. Thus>y=0, as claimed.Suppose n is a real integer and that n is divisible by 1+i in Z[i]. Thenn/(1+i) = n(1-i)/2 is in Z[i]. This means that n must be even andn/2 = in/(1+i)^2 is also a real integer. Thus, n is divisible by 1+itwice. So any real integer is divisible by 1+i an even number of timesin Z[i].Rob Johnson take out the trash before replying =Soit E lOensemble {1,2,3}. Trouvez des relations R et S sur E quisont transitives mais telles que R o S nOest pas transitiveEst ce possible? Soit E lOensemble {1,2,3}. Trouvez des relations R et S sur E qui> sont transitives mais telles que R o S nOest pas transitive>> Est ce possible?Yes, for example: R = { (1,2), (2,3), (1,3) } S = { (1,1), (2,2) } RoS = { (1,2), (2,3) }Dirk Vdm Soit E lOensemble {1,2,3}. Trouvez des relations R et S sur E qui>sont transitives mais telles que R o S nOest pas transitive>>Est ce possible?(Sorry, my french is lousy)R o S = {(a,b) : (a,c) in R, (c,b) in S, c in {1,2,3} }R = {(1,1), (2,3)} est transitiveS = {(1,2), (3,3))} est transitiveR o S = {(1,2), (2,3)} nOest pas transitive. =ItOs not denial. IOm just very selective about what I accept as reality. Calvin (Calvin and Hobbes) IOm doing a report on Grice for an undergraduate philosophy class IOm> taking, and I came across this post:>> http://philosophy.wisc.edu/920/_disc2/00000015.htm>> which suggests that GriceOs theory might fall apart because it forces> an in'nite regress, the only way out of which is to say GriceOs> theory is wrong. IOve heard a few arguments against GriceOs theory,> but never one like this...has anyone written on this in'nite regress> in GriceOs work? If someone could provide any other references to> explore Grice from this point of view, I would very much appreciate> it...9 Speaker MeaningGriceOs (1956) initial account of speaker meaning appealed to aself-referential intention.Speaker Meaning:A meantNN something by x is roughly equivalent to A uttered x with theintention of inducing a belief by means of the recognition of this intention (384).Grice added, This seems to involve a re§exive paradox, but it does notreally do so. Later, as various complications were noted, Grice (1969) andSchiffer (1972) replaced the self-referential analysis with ones involving aseries of intentions, with later intentions about the earlier intentions.This let to issues about the existence of the potentially in'nite regressof intentions required, issues that could have been avoided by staying withself-referential formulations.http://www.princeton.edu/~harman/Papers/ Adler.html[the reason there are so many ways to say the same thing is that meaningis simply a (convergence) upon similar effects of particular perceptions,irrelevent of truth or falsehood being relayed between people?] [he shouldhave also noted ealier intentions about a range of future intentions to goalong with the future intentions about past intensions] [it appears to be anextension in space AND time, though a series, there be no need to discardthe self-reference or iterations since the series makes the iteration §oat]GriceOs concept of speakerOs meaning was an ingenious re'nement of thecrude idea that communication is a matter of intentionally affecting anotherpersonOs psychological states. He discovered that there is a distinctive,rational means by which the effect is achieved: by way of getting oneOsaudience to recognize oneOs intention to achieve it. The intention includes,as part of its content, that the audience recognize this very intention bytaking into account the fact that they are intended to recognize it. Acommunicative intention is thus a self-referential, or re§exive, intention.It does not involve a series of nested intentions --the speaker does nothave an intention to convey something and a further intention that the 'rstbe recognized, for then this further intention would require a still furtherintention that it be recognized, and so on ad in'nitum. Confusing re§exivewith iterated intentions, to which even Grice himself was prone, led to anextensive literature replete with counterexamples to ever more elaboratecharacterizations of the intentions required for genuine communication (see,e.g., Strawson 1964 and Schiffer 1972), and to the spurious objection thatit involves an in'nite regress (see Sperber and Wilson 1986, whose ownRELEVANCE theory neglects the re§exivity of communicative intentions).Although the idea of re§exive intentions raises subtle issues (see theexchange between Recanati 1986 and Bach 1987), it clearly accounts for theessentially overt character of communicative intentions, namely, that theirful'llment consists their recognition (by the intended audience). This ideaforms the core of a Gricean approach to the theory of speech acts, includingnonliteral and indirect speech acts (Bach and Harnish 1979). Different typesof speech acts (statements, requests, apologies, etc.) may be distinguishedby the type of propositional attitude (belief, desire, regret etc.) beingexpressed by the speaker.http://libra.sfsu.edu/~kbach/grice.htm[Iterative and recursive events are circular not straight lines of proofseach depending upon the other before it. Iteration that feedback onto itselfwhat went before can alter its course by internal reference to points alongthe circular chain.]--A theory of meaning within a linguistic system is another goal of the NewHumanism, because, in pragmatic everyday practice and in literature, we domean things by making utterances in a language. Thus, in the spirit of theabove proof of the existential signi'cance of Self, meaning exists, aswell. Structuralist and post-structuralist criticism seems to forget thetrue purpose of language: to communicate. The importance of meaning as afunction of communication brings us to the need for a theory ofcommunication, if the meaning of meaning (or structurality of structure?) isto be made explicit. What is it then for an utterer to mean something in alanguage? Our model for determining this will be a Grice-Schiffer hybrid(see Paul GriceOs Studies in the Way of Words and Stephen SchifferOsMeaning). This is an intentionalist approach in that it focuses on thesubject (also, the self) as the source of the utterance and meaning as anemergent property that occurs between the utter u and audience A. In themodi'ed, logical Gricean terms, the conception can be formed simplisticallyas such:an utterance p means x if utterer u intended to produce some effect E, inaudience A, and the E is produced by AOs recognition of uOs intention toutter p to mean x (Grice 88).The Schiffer side of the hybrid incorporates the notion of mutual knowledge*in order to avoid the possibility of an in'nite regress of intentions andrecognitions (e.g. u intends that A recognize that u intends that Arecognize that u intends that A recognizes that. . .). Mutual knowledge* isthe knowledge that A knows that p, and B knows that p, and A knows that Bknows that p, and B knows that A knows that p. In language and a de'nitionof meaning, it is the recognition on behalf of both the speaker, S, and theaudience, A, that there has been a precedence set in which x has taken on aproperty in relation to a circumstance or fact that is the object ofexpression p, such that when S wishes to express or mean p, S will utter x(Schiffer 30-31). The * following knowledge is to make clear that themutuality of the knowledge is dependent upon the context of the environmentin which meaning is engendered. Thus, we have provisional account of what itis to mean something pragmatically in a semiotic system.http://www.janushead.org/JHSpg99/orr.cfm-- inaryO language. Grice is also subject to criticism from the twopsychologists - for example, his idea of mutual knowledge is implausible inpsychological terms - for to be manifest is weaker than to know or assumesomething, so is less implausible. Instead Sperber and Wilson posit a theoryof mutual manifestness, which does indeed sidestep the in'nite regress ofGrice, and appeal to ostensive acts which alter the cognitive environment ofboth speaker and audience.GriceOs notion of mutually accepted assumptions can also be criticised - inorder for both agents in a discourse to know they share mutual knowledge,these assumptions need to be drawn by them both, and they both need to knowthis fact as well, and so on ad in'nitum. Since a cognitive environmentdoes not get caught in an in'nite number of assumptions, Sperber and Wilsonstate that it can explain situations where information is exploited in atheory, unlike the Gricean perspective.They also claim that communication in general has the aim of increasing themutuality of the cognitive environment...rather than guarantee...strictduplication of thoughts..., which of course is GriceOs assertion of whatimplicatures achieve. By claiming that the speaker only wants to alter hisown cognitive environment, not the thoughts of the hearer, Sperber andWilson escape this.Sperber and Wilson, remember, posit two models - code and inferential, bothof which are essential to explain human discourse. GriceOs maxims seem touse just the inferential model, by inferring a set of conclusions from a setof premises. Moreover, their principle of Relevance suggest the existence ofheuristics, some innate, some acquired via experience, and it implies thatthere is a degree of relevance in all communication. This can be worked outfrom contextual effect and processing effort. Grice merely appeals to us toObe relevantO in his maxim of relation; further, that norms are acquired andneed to be known for proper communication. In contrast, Sperber and Wilsonargue that communicators do not follow their principle of relevance: indeed,they could not violate it even if they so desired. The principle ofrelevance is always commun ...http://tinyurl.com/x8uj[The assumption above, Since a cognitive environment does not get caught inan in'nite number of assumptions is very problematical and stands as nocondemnation of relationships in series be they circular or recursive]--Good luck gotta go..hereOs the page I left off on:http://tinyurl.com/x8v0 IOm doing a report on Grice for an undergraduate philosophy class IOm> taking, and I came across this post:> > http://philosophy.wisc.edu/920/_disc2/00000015.htm> > which suggests that GriceOs theory might fall apart because it forces> an in'nite regress, the only way out of which is to say GriceOs> theory is wrong. IOve heard a few arguments against GriceOs theory,> but never one like this...has anyone written on this in'nite regress> in GriceOs work? If someone could provide any other references to> explore Grice from this point of view, I would very much appreciate> it...Just about every philosophical theory IOve seen goes into an in'niteregress if pushed too hard. (This is why I no longer studyphilosophy--itOs much too easy to be destructive instead ofconstructive)Ocid =HereOs the post I was talking about:A note (or hypothesis) on Griceregress, but it makes Grice wrong.IOm 'nding it dif'cult to clearly explain how the regress arises.Here goes. Grice says that the hearer gets a strong conditional via aconversational implicature on a material conditional. That is, allthat is actually *said* or *uttered* is a material conditional, andthe rest has to be inferred from conversational context. This works byinference to the best explanation, or abduction. The hearer thinks,That utterance is strange in that it violates some ConversationalMaxim(s), and I have every reason to suspect that the speaker wouldfollow these Maxims under normal circumstances. In order to preservethe speakerOs following the Maxims, I must take him to be trying toconvey something not identical with what was, strictly speaking,uttered. Example: Professor B reads a letter of recommendation fromProfessor A regarding grad student C. The letter reads: C showed upto every class, and has an excellent command of English. There is aclear conversational implicature that C is not a good philosopher, andA does not recommend C to B. OK, so you need abduction in order togenerate an implicature. The problem is that you must have, in yourbrain, a representation of strong conditionality in order to make anabduction. This is because explanations operate on strongconditionals. The thing doing the explaining (explanans) is related tothe thing being explained (explanandum) in the same way as theantecedent is related to the consequent in a strong conditional. Youmust already be able to understand and manipulate strong conditionalsin order to perform an abduction. So hereOs the regress. The speakerutters material conditional M. In connection with context C, thehearer gets strong conditional S by implicature (by getting S as thebest explanation of why M was uttered in C). This means the hearermust be able to represent a strong conditional E to himself (or useit) in order to generate the abduction. But where did E come from? I't came from a separate implicature, then we have a further abductionwhich needed a strong conditional E*, and so on. The way to stop theregress is to say that E is not created by implicature, that wealready have a mental symbol for strong if . But if this is thecase, then why canOt we say that sometimes if means strong if,that if is ambiguous? That is, if E has to be primitive (in thesense that it was not generated by implicature), isnOt that immediateevidence that thereOs more than one sense of ifO? Grice seems keen onusing the Razor to keep senses to a minimum. But here we seem forcedinto admitting one. And why do we need to keep senses to a minimum? Itis not as though there are SENSES §oating in platonic space, and weneed to keep the world of Forms as small as we can. ItOs not asthough, in saying that ifO is ambiguous, weOre postulating some*thing*.Does this regress idea make any sense to anyone? The complaint isessentially that strong conditionality cannot depend for itsgeneration on our already having command of strong conditionality. Forthen we have another strong conditional to reduce, and it is reallyimplausible to suppose that it is reducible in the same way.More generally, suppose our account of reduction looks like this: Forany strong conditional S, S is reducible to a corresponding materialconditional M plus some extra considerations XYZ. What I want to sayabout such an account is this: our extraction (or construction, orwhat have you) of S from M and XYZ must not require us touse/have/represent-to-ourselves another strong conditional T. For thenwhat does T reduce to? Some M* plus XYZ? The way out of the loomingregress is to take T as already in our possession independently of theS-XYZ story. But this leaves T unaccounted for, and seems to indicatethat strong conditionals are not cashable in terms of materialconditionals, or at least that not all strong conditionals are socashable.> IOm doing a report on Grice for an undergraduate philosophy class IOm> taking, and I came across this post:> > http://philosophy.wisc.edu/920/_disc2/00000015.htm> > which suggests that GriceOs theory might fall apart because it forces> an in'nite regress, the only way out of which is to say GriceOs> theory is wrong. IOve heard a few arguments against GriceOs theory,> but never one like this...has anyone written on this in'nite regress> in GriceOs work? If someone could provide any other references to> explore Grice from this point of view, I would very much appreciate> it...Any thoughts??? ... So hereOs the regress. The speaker> utters material conditional M. In connection with context C, the> hearer gets strong conditional S by implicature (by getting S as the> best explanation of why M was uttered in C). This means the hearer> must be able to represent a strong conditional E to himself (or use> it) in order to generate the abduction. But where did E come from? If> it came from a separate implicature, then we have a further abduction> which needed a strong make any sense to anyone? The complaint is> essentially that strong conditionality cannot depend for its> generation on our already having command of strong conditionality. For> then we have another strong conditional to reduce, and it is really> implausible to suppose that it is reducible in the CarrollOs paradox. (Textsavailable via Google.)Perhaps i understand either of them incorrectly, though.Herman Jurjus > Advanced, and Challenge. Please visit us at>> http://math.smsu.edu/~les/POTW.html>> [I will be tackling the baclog of old problems over the Christmas break.]seems to be interested.Skip =HereOs a little something else about Venus, regarding their rigidairships and/or of what we could do if push comes down to shove, as wecould somewhat narrow the rigid airship gap, possibly even creating ahybrid shuttle/airship of which hopefully they donOt have just yet.Also a little more pertaining to the utilization of good old basaltthat a few too many Earthly folks donOt seem to have a clue about.http://guthvenus.tripod.com/airship-01.htmhttp:// guthvenus.tripod.com/gv-basalt.htmLunar basalt composite applications, besides the LSE-CM/ISS tether;http://guthvenus.tripod.com/gv-lm-1.htm =-- www.StealthHostiing.com You rule Truman. http://tinyurl.com/iky4 Hey Trueman...love the show. YOU ARE the Truman I heard him. Very spooky! >Is the truman living in Townsville? IOve been hearing stuff, yeah.Webmasters help the TRUEman by joining www.theBanner.net Current:1 Goal:1000- CanOt remember the rest. introduction music] Roger Ramjet and his Eagles 'ghting for our freeeeedom, Fly through in- and outer space, not to join Oem but to beat Oem. Roger Ramjet heOs our man, hero of our nation. For his adventures just be sure and stay tuned to this station. So come and join us all you kids for lots of fun and laughter As Roger Ramjet and his men get all the crooks theyOre after. Roger Ramjet heOs our man, hero of our nation. For his adventures just be sure and stay tuned to this station. VOICEOVER: As this encroaching episode starts to begin, we 'nd Roger Ramjet and The American Eagles about to join thousands of other Americans for a fun weekend: a LetOs build something out of snow. DOODLE EAGLE: What Roger? egomaniac was about to get started a strange phenomenon took place. DOODLE: Hey, whatOs going on? YANK EAGLE: The snow, DAN EAGLE: ThatOs avalanche, but whoOs counting. VOICEOVER: Yes, all over America it was happening. The snow was being stolen! But how? And why? SOLENOID ROBOT MEREDITH: Soon we will be ready. Soon our interplanetary snow swiping machine will have stolen all the snow in the world and piled it here . We will be the only ones with any snow! SOLENOID ROBOT STANLEY: Yes. Then all the skiers and skaters will have to come to us to play in the snow and we will charge them a lot of money . MEREDITH: We will make a fortune before taxes . VOICEOVER: The Solenoid Robots! So they were the plots behind this rat, were you!?! DOODLE: early thaw... YANK: No, Roger. The snow didnOt melt, it just slid away. It .. it .. disappeared. DAN: I think IOve got the stole the snow. And that means theyOll have to pile it YANK: LetOs get to our Oplanes and scout around. We ought to be able to 'nd it from the air. VOICEOVER: And as the scrambling American Eagles take to the air... MEREDITH: That just about does it Stanley . We have stolen all of the snow in the world . Get the signs up . STANLEY: Right you are Meredith . VOICEOVER: And as the money-mad robots waited for their 'rst customers, the American Eagles were hot on cockpit looking up> Where? I donOt see any? YANK: Below us piled in a gigantic hill. LetOs land and investigate. MEREDITH: Oh look Stanley, our 'rst customers! STANLEY: Our 'rst customers are troublemakers. ThatOs Roger you are the ones who took all the snow, and now youOre charging people to play in it. ThatOs not fair! STANLEY: Quickly, put on your skis. WeOll escape to the bottom of the hill and get in our spaceship. VOICEOVER: And, no sooner said than done, the two crafty robots quickly Put on those skis and get after them . YANK: Roger lookout, what? Yiippes!!! I wonder how I did ... Oooff! ... that. MEREDITH: Quickly Stanley, letOs go down this ski-jump. That should get those Eagles off our trail. Woooaah!!! [Robots now standing thoughtfully at bottom of ski-jump, looking up...] MEREDITH: Look up in the sky ... STANLEY: ItOs a bird ... MEREDITH: ItOs a Oplane ... STANLEY: ItOs ... [Ramjet and tree land your snow-stealing is over. You must return all this snow immediately. The people in our world should be able to play in the snow without paying all kinds of money for it. ThatOs the American way. MEREDITH: Have you been to Squaw Valley lately? VOICEOVER: And so once aain, cunning, daring and blind luck has paid off for Roger Ramjet, and heOs made the world a better one in which to ski! Who threw that? [cheesy organ 'nale music as credits roll] When Ramjet takes a proton pill the crooks begin to worry. They canOt escape their awful fate from protonOs mighty fury. Roger Ramjet heOs our man, hero of our nation. For his adventures just be sure and stay tuned to this station. So come and join us all you kids for lots of fun and laughter As Roger Ramjet and his men get all the crooks theyOre after. Roger Ramjet heOs our man, hero of our nation. For his adventures just be sure and stay tuned to this station. Topology is totally irrelevant to, e.g., projective>spaces over 'nite 'elds.It is? Surely the cohomology (or cohomologies) of projective spaces of'nite 'elds is central to their study?-- Tim Chow tchow-at-alum-dot-mit-dot-eduThe range of our projectileseven ... the artilleryhowever great, willnever exceed four of those miles of which as many thousand separate us fromthe center of the earth. Galileo, Dialogues Concerning Two New Sciences >Topology is totally irrelevant to, e.g., projective>>spaces over 'nite 'elds.>>It is? Surely the cohomology (or cohomologies) of projective spaces of>'nite 'elds is central to their study?Of course, the non-visionary sticks-in-the-mud among us (making noreference to anyone or -ones in particular) would say, that ainOttopology, thatOs just more gol-derned algebray, tricked out in adisguise!Lee RudolphX-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft = at 01:46 PM, tb+usenet@becket.net (Thomas Bushnell, BSG) said:>For example, you canOt prove that> If G = {G} and H = {H}, then G=H.Yes you can. That comes from Extesionality - you donOt needFoundation.>Your set X (which is X={X})He did not de'ned such a set; he gave an axiom asserting that oneexists; X={X} is *NOT* a de'nition of X, only a property. He didnot give an axiom asserting that only one such set exists.>might make the axiom of in'nity unnecessary,No.-- Shmuel (Seymour J.) Metz, <3fcbbc14$17$fuzhry+tra$mr2ice@news.patriot.net> at 01:46 PM, tb+usenet@becket.net (Thomas Bushnell, BSG) said:For example, you canOt prove that If G = {G} and H = {H}, then G=H.>> Yes you can. That comes from Extesionality - you donOt need> Foundation.No, it doesnOt. LetOs try to prove it together. WeOll aim for usingExtensionality, as you suggest.That means that we need to show that, for all x, x in G iff x in H.Now, x in G iff x = G and x in H iff x = H. Thus, if we can showthat, for all x, x = G iff x = H, then we may conclude (byExtensionality) that G = H.You got any shortcuts for showing that claim? I havenOt.Thomas is right. Extensionality is insuf'cient.-- Jesse HughesSuch behaviour is exclusively con'ned to functions invented bymathematicians for the sake of causing trouble. -Albert EagleOs _A Practical Treatise on FourierOs Theorem_X-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft = at 03:35 PM, particularily interesting work been done on this sort of>thing? That depends on what you consider interesting. Certainly Quinepublished a set theory without Foundation.-- Shmuel (Seymour J.) Metz, SysProg and JOATnot reply to > >>For no especially good reason I recently became curious as to what>>axiomatic set theory would look like without the axiom of>>foundation.> > > The general opinion is that there isnOt anything interesting that> comes about from ~Foundation.> > >> More interestingly what it would look like with some sort>>of strong negation of foundation. IOm not sure exactly what that>>should mean, but IOm sure it can be made precise in some sense. > > > Immediately you have a problem, which is to decide whether the shape> of a set suf'ces for uniqueness.> > For example, you canOt prove that> > If G = {G} and H = {H}, then G=H.> > Which means that you might have two sets, both which are:> > {{{{{{...}}}}}} {{{{{{...}}}}}}> > and which would not be equal.> > With foundation, of course, you can prove that if two sets have the> same shape, then the must be equal. (Not by anything deep:> foundation simply excludes the cases where the axiom of extensionality> is not suf'cient to answer the question.) I know IOm being vague> here about what I mean by shape, but I hope my example and> statements make the intuitive concept clear enough.That is quite interesting. I donOt suppose thereOs an easy example of models to show independence? It seems to me that if you have a model of them without equality then you can do some clever thing with taking equivalences over OshapeO to get a model with equality of such sets (I canOt immediately see how to make this precise, but it looks plausible). I donOt immediately see how to get a model of existence of such sets without equality though.> Before foundation was as popular as it is today, set theorists and> logicians worried about these cases. If you drop foundation, then you> need to decide what to do. If you donOt mind unequal sets with the> same shape, then thereOs no problem. But if you donOt want that,> then when you drop foundation you need to add something to the axiom> of extensionality to cover these cases.> > As for adding something in place of foundation, that is, an axiom> asserting some non-well-founded set, that might go. > > Your set X (which is X={X}) might make the axiom of in'nity> unnecessary, but I havenOt thought much about it to be sure. (Just> based on the intuition that there is something in'nite about it.) I see where youOre coming from, but canOt you use a set of 'nite graphs to provide a model of the theory without in'nity? (Using Ox extends yO for y is an element of xO). IOm not entirely sure about that particular construction, but it seems likely that you can; I seem to recall you can construct a model of ZFC without In'nity using well-founded graphs, so if you relax the well-foundedness condition to allow loops youOre probably going to get a model of this theory without in'nity.Maybe, kindof, sortof. I donOt know. :) IOm committing my usual crime of posting to sci.math while half-asleep (which I should really learn not to do).of some of the references you supplied shortly.David >>For no especially good reason I recently became curious as to whataxiomatic set theory would look like without the axiom offoundation.>> The general opinion is that there isnOt anything interesting that>> comes about from ~Foundation. More interestingly what it would look like with some sortof strong negation of foundation. IOm not sure exactly what thatshould mean, but IOm sure it can be made precise in some sense. >> Immediately you have a problem, which is to decide whether the shape>> of a set suf'ces for uniqueness.>> For example, you canOt prove that>> If G = {G} and H = {H}, then G=H.>> Which means that you might have two sets, both which are:>> {{{{{{...}}}}}} {{{{{{...}}}}}}>> and which would not be equal.>> With foundation, of course, you can prove that if two sets have the>> same shape, then the must be equal. (Not by anything deep:>> foundation simply excludes the cases where the axiom of extensionality>> is not suf'cient to answer the question.) I know IOm being vague>> here about what I mean by shape, but I hope my example and>> statements make the intuitive concept clear enough.>That is quite interesting. I donOt suppose thereOs an easy example of >models to show independence?Take any model of NBG set theory whatever (ZF will alsowork), and use any well-de'ned permutation F of theelements. De'ne a new element relation E by x E Y iffF(x) in Y. There is no problem whatever in showing thatthis is a model of set theory; each of the axioms goesthrough. It can look very odd.-- 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 University >> |> no one will have to read them.> |> |what, exactly, gives you the authority to do such thing?>> i said he could.> > and, who or what exactly, gives you the authority to do such thing?> > I said he could.and who made you the authority??? later this evening> |> so that no one will have to read them.> |> |what, exactly, gives you the authority to do such thing?>> i said he could.>> and, who or what exactly, gives you the authority to do > such thing?>> I said he could.> > and who made you the authority???Sorry, but I donOt have the authority to tell you that.Fortunately, I do have the authority to tell you I donOt havethe authority to tell you that.Have a nice day.Jim Burns =Given a determinant with only three major diagonals nonzero it is easy to construct a three term recursion relation to evaluate thedeterminant. Is there any similar result for a determinant with only 've major diagonals nonzero? Given a determinant with only three major diagonals nonzero it is easy> to construct a three term recursion relation to evaluate the> determinant. Is there any similar result for a determinant with only> 've major diagonals nonzero?Are you referring to tridiagonal and pentadiagonal matrices, resp.?That is, are the diagonals located consecutively about the main diagonal?Do you require a linear recurrance, or will a nonlinear recurrance do aswell? =can some one please break this down for me, im not all together clearon this con cept as a whole =I donOt know the notation for a subscript character without the propercharacter set (e.g. 2^2 = two squared (superscript), but there is no downarrow to represent subscript characters). So IOm going to use |x for thisexample.If z|1 = 2 + 3i, z|2 = -1 + 2i and z|3 = -3 - 4i, express 2z|1 + 3z|2 in theform a + ib.I have more examples but I need this one to start me off.TIA. I donOt know the notation for a subscript character without the proper> character set (e.g. 2^2 = two squared (superscript), but there is no down> arrow to represent subscript characters)The underscore character is used for this. x_1 = x_2 + x_3> 2z|1 + 3z|2 in theSee? This is how I read the above:Two times z times the absolute value of (1 + 3 times z) times2.Carlos-- I donOt know the notation for a subscript character without the proper> character set (e.g. 2^2 = two squared (superscript), but there is no down> arrow to represent subscript characters). So IOm going to use |x for this> example.> > If z|1 = 2 + 3i, z|2 = -1 + 2i and z|3 = -3 - 4i, express 2z|1 + 3z|2 in the> form a + ib.> > I have more examples but I need this one to start me off.> > TIA.> > > 0. If youOre learning complex numbers in a class (or from a book), you should have a reference that spells all the mathematics in steps 2-3 out explicitly. That reference may be your instructor.1. Subscripts are frequently written using underscore, as in z_1 = 2 + 3i z_2 = -1 + 2i z_3 = -3 - 4iJust in case you are trying to pick all this up by osmosis, or 'ndyour text impenetrable:2. To form the combination 2 z_1 + 3 z_2 you need to know these things: (a) how to multiply a complex number by a real number (b) how to add two complex numbers For (a), a complex number is a sum of two quantities: the real part, and i times the imaginary part. If you treat the i as an algebraic symbol (like x), multiplied by its coef'cient the imaginary part, then multiplying the complex number by a real number is no different from multiplying a polynomial by a number: multiply each term by the new factor. Then combine terms. For instance, 5*(-2 + i) = 5*(-2) + 5*i = -10 + 5i 7*(7 - 6i) = 7*7 + 7*(-6i) = 49 - 42i. For (b), addition of complex numbers is again similar to addition of polynomials. The real parts are added together, and the imaginary parts are added together. The sum of real parts is the real part of the result, and the sum of imaginary parts is the imaginary part of the result. For instance: (1 + i) + (2 - 3i) = (1 + 2) + (1 - 3)i = 3 - 2i (7 - 3i) + (12 - 7i) = (7 + 12) + (-3 - 7)i = 19 - 10iThis information should be more than enough to get you through yourproblem. Good luck.Dale. I donOt know the notation for a subscript character without the proper> character set (e.g. 2^2 = two squared (superscript), but there is no down> arrow to represent subscript characters). So IOm going to use |x forthis> example.>> If z|1 = 2 + 3i, z|2 = -1 + 2i and z|3 = -3 - 4i, express 2z|1 + 3z|2 inthe> form a + ib.>> I have more examples but I need this one to start me off.>> TIA.2(2+3i)+3(-1+2i)4+6i-3+6i1+12iDavid Moran <6vigsvsqpjpbbptor2nt81ft88pam673it@4ax.com>X-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft Im saying that both INFINITY and ZERO, represent a discontinuity.What do you mean by discontinuity? Why doesnOt 1 represent adiscontinuity? 666? 3.14159? If youOre going to use a privatelanguage, de'ne your terms, and choose words with different spellingfrom those the rest of us use.>we no longer carePerhaps you meant to post to sci.psychology? We no longer care isnot a Mathematical concept.>When a cardinal number gets to that grand/small a scope,What do you mean by a cardinal number?>more practically relevant and interesting,Again, that has nothing to do either with Mathematics or withIn'nity.>it becomes necessary to plot them on cartesian co-ordinates Do you know what Cartesian coordinates are?>if that were not the case then we could hardly claim to have found a>discontinuity,We donOt claim that - you do. >and polar coordinatesPlease donOt use termss you donOt understand.-- Shmuel (Seymour J.) Metz, SysProg and tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft tylerneylon@yahoo.com (Tyler Neylon) said:>However, having said that, I believe Stepan is>actually trying to express some rather valid>mathematical intuition along the lines of>nonstandard analysis. such an hypothesis to be tenable.>We can express an>extended set of numbers as vectors a+b*d,>which we could write in the shorthand (a,b).>Hence (0,b) and (1,y) can be thought of as>in'nitely far apart regardless of b and y.No. But there are pairs of nonstandard reals that we can think of thatway.>IOm not sure exactly how well imaginary numbers 't in here. They donOt.-- Shmuel (Seymour J.) Metz, <0ksgsvkktgi0fpfj2517qh6a7d9jug8fkn@4ax.com> tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft My personal idea of in'nity goes like:>In'nity (not a number) is a property of a partially ordered set,>that says: for every element thereOs a great element (wrt to that>order).So if x and y are real numbers, (x,y) is in'nite and [x,y] is not?-- Shmuel (Seymour J.) Metz, SysProg and JOATnot reply to tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft A nice profound explanation of ZERO and INFINITY:~nice, ~profound, ~explanation.>You are standing at a point on the ground marked ZERO,A number is not a point on the ground.>EXTREMELY far away Extremely far away is not in'nity.>The measured distance between the two of you can be written>mathematically in polar coordinate or as an imagenary number>s=25-i50That would not be a distance.>ZERO is an arbitrary frame of reference on the near side of a>discontinuity. Do you imagine that your statement has any meaning?>If we decide exactly where this INFINITY point is>supposedly located, then we can start to say things like In'nity>plus 3 etc.Or we could label it eggnog and start saying things like eggnogplus 3. That would make just as much sense.-- Shmuel (Seymour J.) Metz, SysProg and JOATnot reply to So you may say, 'ne, IOll DEFINE 1/0 for ya. ItOs in'nity!...or alternately de'ne 1/in'nity as well.>Then you have to invent your own in'nity arithmetic. For you cannot>have in'nity follow the same rules that we all agree regular numbers>follow. For example, 1 / 0 = in'nity, 2 / 0 = in'nity, so by>de'nition of division 0 * in'nity has all the values of the rainbow.>So in'nity * 0 is not unique. Incidentally, 0 / 0 is therefore not>unique either, since the answer is the number which multiplied by 0>gives 0 but this is true for all numbers. So you can replace the>with and and have yourself a nice little world with your in'nity>and zero arithmetic....so now you have two different sets of rules. Does this mean thatyou now have two different ways/methode for solving a real worldproblem which can be put to paper? If the answer is yes, then can weconvert from from one system of arithmetic to the other? Does thenotion of 0 x INFINITY = ? somehow make this connection impossible?Intuitively it seems to me that the number ZERO which is 'niteunder the conventional set of rules, would be in'nately small ifveiewed in the context of the alternate set of rules. Conversely, thenumber INFINITY under the alternate rules, should be 'nite and behavemuch like OUR number 0 does. What I tried to explore with the original example that started thiswhole thread, is whether you can have two such systems inco-existence. In order for them to co-exist, you have to keep thenumbers from one system, separate from the numbers of the other,because the rules are in fact different. I tried to achieve thatseparation by using the REAL numbers to represent quantities whichshould be manipulated by the conventional rules of arithmetic, and atthe same time using the IMAGINARY numbers to represent quantitieswhich should be manupulated using the unconventional rules.>No one says you canOt do that. It just has to be consistent and>logical. You can invent your own math. If itOs interesting and no one>has done it before, you can even publish it. :-)I 'nd it deeply disturbing that no one has done this. Does anyone outthere have any idea why that is? Is it because there is no real worldapplication for the answer to what is meant by 0 x INFINITY ?When you think about it, the number 0 as a real number (in themathematical sense) is not that meningfull for measurement, becausewhichever fractional quantity you are measuring eventually becomesunmeasarable at the point where it becomes so small that it getsobfuscated by noise or measurement accuracy. At this point where themeasured quantity is undetectable, it becomes unde'ned, zero, orin'nately small. At this point there is no need to make a distinctionbetween 0 and Unmeasurably-Small, so for the purpose of solving realworld problems we can de'ne the small quantity as either 0 oralternately and equally correctly as x/in'nity where x is anyarbitrary 'nite positive value. The exact same argument holds true atthe opposite end of the spectrum, ie INFINITY versus 2*INFINITY etc.In this case, the number INFINITY is any number grand enough in scope,that increasing the measured value beyond that quantity is irellevantto the real world problem we are trying to solve. In other words, ifINFINITY is out of scope, then the product of INFINITY and anycardinal number is also out of scope and therefore, for all practicalpurposes relevant to the real world problem we are trying to solve, wecan say that 6 x INFINITY = INFINITY. For example, look straight upinto the sky on a clear night. How far can you see? Can you see 20miles? Yes. Can you see 2 million miles? You see all the stars so Yes.Can you see to 37.6 X 10 ^ 387 miles out? The answer at this point isthat for practical purposes of solving any real world problems, itdoesnOt matter whether you can see that far. If it doesnOt matterwhether we can see that far, then we will not bother assigning anumeric value to that distance, but we will instead call that distanceINFINITY. Now it should be apparent that there is no need to make adistinction between INFINITY and 3xINFINITY etc....Stepan...Stepan >So you may say, 'ne, IOll DEFINE 1/0 for ya. ItOs in'nity!>> ...or alternately de'ne 1/in'nity as well.>>Then you have to invent your own in'nity arithmetic.ThatOs not necessary. ItOs already been invented.>For you cannot>have in'nity follow the same rules that we all agree regular numbers>follow. For example, 1 / 0 = in'nity, 2 / 0 = in'nity, so by>de'nition of division 0 * in'nity has all the values of the rainbow.>So in'nity * 0 is not unique.In the system to which I refer, the answer is unique, but it is a set ofnumbers, rather than a single number. Speci'cally, the answer is the setof extended reals, [-oo, +oo]. And, as you say below, the situation is thesame for 0/0.>Incidentally, 0 / 0 is therefore not>unique either, since the answer is the number which multiplied by 0>gives 0 but this is true for all numbers. So you can replace the>with and and have yourself a nice little world with your canOt do that. It just has to be consistent and>logical. You can invent your own math. If itOs interesting and no one>has done it before, you can even publish it. :-)ItOs been done before, as I said. It is consistent, logical, and quiteuseful in the context of interval arithmetic (or, if you wish to be moreabstract, in the context of algebraic structures known as wheels). One o'ts primary exponents is Bill (G. William) Walster, an engineer at Sun.are available on the web.> I 'nd it deeply disturbing that no one has done this. Does anyone out> there have any idea why that is? Is it because there is no real world> application for the answer to what is meant by 0 x INFINITY ?Ah, then you need be disturbed no more!David Cantrell ItOs been done before, as I said. It is consistent, logical, and quite>useful in the context of interval arithmetic (or, if you wish to be more>abstract, in the context of algebraic structures known as wheels). One of>its primary exponents is Bill (G. William) Walster, an engineer at Sun.>are available on the web.>Where on the web >ItOs been done before, as I said. It is consistent, logical, and quite>useful in the context of interval arithmetic (or, if you wish to be more>abstract, in the context of algebraic structures known as wheels). One>of its primary exponents is Bill (G. William) Walster, an engineer at>system, are available on the web.>> Where on the web is this stuff g. william walster interval arithmeticYou should 'nd plenty. Choose his most recent papers to look at.David g. william walster interval arithmeticSince +oo is the 'rst member of the SET OF NUMBERS to the right ofthe SET OF REALs, and since -oo is the last member of the SET OFNUMBERS to the left of the SET OF REALs, it follows that including +ooand/or -oo in the SET OF REALS, is a formal way of declaring one orboth of the following two points:1) We are UNABLE to de'ne an upper (or lower) bound in the context ofthe problem we are trying to solve. In this case we should expect anapproximate result. This is the real world problem we have when we tryto quantify(write down as a number) analog measurements in thepresence of uncertainty, noise, and lack of time to acquire a propermeasurement.2) Or we know that there is no upper (or lower) bound in the contextof the problem we are trying to solve. In this case we should expectan accurate result. This appears to be the case in calculus....Stepan In the system to which I refer, the answer is unique, but it is a set of>numbers, rather than a single number. Speci'cally, the answer is the set>of extended reals, [-oo, +oo]. And, as you say below, the situation is the>same for 0/0.>What you say makes perfect sense if the following is observed:The answer you refer to above should be an INFINATE number of UNIQUEand ORDERED sets, in other words, the theory should be placingmarkers(de'ning sets) at regular intervals in the progression from 0to INFINITY (AND beyond). In this case, INFINITY is NOT locatedarbitrarily. It is a boundary value which de'nes the scope ofrelevant numeric values(the working set) for any given problem.Speci'cally, this boundary is de'ned to exist at the very 'rstunnatainable value in a progression. In other words, INFINITY pointsto the FIRST element of the NEXT set, where both the sets and also thevalues within the sets are ordered.Then it all makes perfect sense in the world of applied mathematics,and I can see the (possibly weak) analogy to wheels. Such a groupingwould allow us to perform mathematical operations upon an in'natenumber of UNIQUE sets, and on paper using the proper notation, itwould be a single mathematical operation. It would probably feel likevector arithmetic, but instead of being 2-Ddimensional (real andimaginary numbers), or three dimensional, it would bein'nate-Dimensional.However beware:The answer to which you are referring, can not be just ONE SINGLE SETof numbers. That would not be a usefull extension....Stepan I am of the beleif that math in and of itself is completely>meaningless, just a set of symbols (numbers) and relationships between>those symbols (operators) To make math usefull as opposed to Just A>Form Of Art (applied mathematics) you need to declare a relationship>between the numbers and some real-world>(measurable/countable/observable) properties, so that the operators>can help you solve a problem.You still need rigorous mathematical de'nitions as a basis for yourapplied mathematics. While you can de'ne in'nity as an arbitraryframe of reference on the far side of a discontinuity just as well asEuclid could de'ne a point as that which has no dimension, neitheris very helpful in developing useful mathematical frameworks that arerequired in real life problems like, say, predicting the asymptoticbehavior of a system described by ODEs.>You are talking about INFINITY as a form of art, while I am talking>about INFINITY as a practical entity. They are two very different>worlds and I can neither agree nor disagree with you.I donOt see how your de'nition is any more practical than any of theother quasi-mathematical de'nitions of in'nity IOve seen passedaround by philosophists of various schools. IOm making a huge gear-shift from the original topic. > so part of whatOs going on here is this: 'rst, all categories are in>> a sense imitations of the category of sets, the objects being>> imitations of sets and the morphisms being imitations of functions (or>> maps or mappings or whatever you call them). >> This is a very set-centric view. I thought the whole purpose of> bothering with category theory in the 'rst place was to escape this> view.I think I agree with George here. One can take the set-theoreticintuitions too far. What about posets as categories? The arrowsarenOt imitations of functions, are they? What about a category inwhich the objects are formulas and the arrows are proofs? [...]> How exactly one would one even DEFINE the category of sets if one> were NOT STARTING with ZFC or some other rich set theory as a> foundation? My point is simply that if you have ZFC, what do you> need categories for? Sets are already adequate as a foundation; you> can do EVERYthing, INCLUDING categories, AS sets. The category of> sets starts to get viciously circular. But if you donOt have a set> theory, if you are using categories as a foundation instead, then> the category of sets is simply nowhere in evidence: how do you> even DEFINE set?And here, I think George goes too far (or I donOt get his point).Some folks want category theory as an alternate foundation, itOs true.Others just like it for its unifying qualities and ability to makeapparently disparate phenomena particular instances of a commonstructure. I donOt see why thereOs any particular issue for talkingabout the category of sets as a particular category for *either*group.That said, the foundations folk may still have a good argument againsttaking set theory as the real foundation. Namely, the aims ofstructuralism seem to be much more easily attained via categoricalfoundations than set theoretic foundations. But here, IOm talking abit out my ass and the interested reader should 'nd McClartyOsNumbers can be just what they have to, an eminently accessiblenumbers could not be. -- Destiny is a funny thing. Once I thought I was destined to become Emperor of Greenland, sole monarch over its 52,000 inhabitants. Then I thought I was destined to build a Polynesian longship in my garage. I was wrong then, but IOve got it now. -- The Tick =||> IOm making a huge gear-shift from the original topic. |>|>|>> so part of whatOs going on here is this: 'rst, all categories are in|>> a sense imitations of the category of sets, the objects being|>> imitations of sets and the morphisms being imitations of functions (or|>> maps or mappings or whatever you call them). |>|> This is a very set-centric view. I thought the whole purpose of|> bothering with category theory in the 'rst place was to escape this|> view.||I think I agree with George here. One can take the set-theoretic|intuitions too far. What about posets as categories? The arrows|arenOt imitations of functions, are they?sure they are; speci'cally, of inclusion functions between subsets(is one way to think of it). (thereOs things to say about why this isa special degenerate case but i donOt feel like saying any of thembecause i think this whole thread is misguided.)|What about a category in which the objects are formulas and the|arrows are proofs?in such circumstances itOs often crucial to think of such an arrow asa function from proofs of the antecedent formula to proofs of theconsequent formula.again, though, i probably shouldnOt be replying to this thread otherthan to point out that yOall are reading way too much that isnOt thereinto <87smk4gm61.fsf@phiwumbda.org> |> IOm making a huge gear-shift from the original topic. > |>> |>> |>> so part of whatOs going on here is this: 'rst, all categories are in> |>> a sense imitations of the category of sets, the objects being> |>> imitations of sets and the morphisms being imitations of functions (or> |>> maps or mappings or whatever you call them). > |>> |> This is a very set-centric view. I thought the whole purpose of> |> bothering with category theory in the 'rst place was to escape this> |> view.> |> |I think I agree with George here. One can take the set-theoretic> |intuitions too far. What about posets as categories? The arrows> |arenOt imitations of functions, are they?>> sure they are; speci'cally, of inclusion functions between subsets> (is one way to think of it). (thereOs things to say about why this is> a special degenerate case but i donOt feel like saying any of them> because i think this whole thread is misguided.)I think that the whole categories are just sets and set functions ismissing the point of category theory. There is nothing aboutcollections and elementhood which is fundamentally more basic thanobjects and arrows between them. To believe that *every* poset isjust essentially an abstraction from sets and inclusions seems torather miss the point IOd think. ThereOs something to be said for theview that orderings on collections is epistemically prior to ZF.> |What about a category in which the objects are formulas and the> |arrows are proofs?>> in such circumstances itOs often crucial to think of such an arrow as> a function from proofs of the antecedent formula to proofs of the> consequent formula.Even if we do that, your point isnOt made. A formula is not a set ofproofs of that formula, is it? If not, then the arrows are notfunctions from the set of proofs of the antecedent to the set ofproofs of the consequent.Maybe things can be viewed that way, but thatOs not particularlynatural.> again, though, i probably shouldnOt be replying to this thread other> than to point out that yOall are reading way too much that isnOt there> into imitation.Perhaps. I came into this thread secondhand. Why donOt you tell uswhat you mean when you write, all categories are in a senseimitations of the category of sets, the objects being imitations ofsets and the morphisms being imitations of functions. ItOs veryplausible that I donOt know what the heck you mean.You might mean merely that category theory can be interpreted in settheory. But thatOs obvious. You might mean something else. IhavenOt a clue.-- My proof has been checked very thoroughly, both by me and others.Those others apparently decided that they would not believe the proofwas correct, but cannot support that position using mathematics. Buthey, theyOre just human beings. --JSH, prover of FermatOs Last Thm tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose: george_cox@btinternet.comX-Punge: Micro$oft = at 12:50 PM, greeneg@cs.unc.edu (George Greene) said:>This is a very set-centric view.>I thought the whole purpose of bothering with>category theory in the 'rst place was to escape this view.No. Category theory is a tool for other branches of Mathematics,whether you take categories as fundamental or take sets asfundamental.>My point is simply that if you have ZFC, what do>you need categories for? Because you can prove things once in Category Theory and then applythem to various branches of Mathematics. Your question is like askingwhat you need groups for.>The category of sets starts to get viciously circular. Why is that an issue? You can do Set Theory without the Axiom ofFoundation.-- Shmuel (Seymour J.) Metz, SysProg =|IOm making a huge gear-shift from the original topic. |||> so part of whatOs going on here is this: 'rst, all categories are in|> a sense imitations of the category of sets, the objects being|> imitations of sets and the morphisms being imitations of functions (or|> maps or mappings or whatever you call them). ||This is a very set-centric view.|I thought the whole purpose of bothering with|category theory in the 'rst place was to escape this view.well, i donOt want to get involved much in a side discussion aboutthis; i know that you indicated that itOs a shift of topic, but ithink itOs even much more off the topic than that. anyway, i justdonOt agree that the whole purpose of bothering with category theoryis anything remotely like what you suggest it is.-- = : >> so part of whatOs going on here is this: 'rst, all categories are in : >> a sense imitations of the category of sets, the objects being : >> imitations of sets and the morphisms being imitations of functions (or : >> maps or mappings or whatever you call them). : > This is a very set-centric view. : > I thought the whole purpose of bothering with : > category theory in the 'rst place was to escape this view. : That is a bit exaggerated. : My impression is, that categorists have no problem with sets as such, : but would like to regard function as the primitive concept instead : of elementhood, because this is closer to mathematical practice.Well, maybe Marc and James should conduct competing pollsof the category of all categorists; I mean, I doubt itcan simultaneously be the case that both all categoriesare imitations of the category of sets AND that function,as opposed to element, should be the primitive concept. : the Math Intelligencer (?). It should still be available via : : > You donOt have to presume a set of objects upon which to : > found categories.... : > How exactly one would one even DEFINE the category of sets : > if one were NOT STARTING with ZFC or some other rich set theory : > as a foundation? ... : > if you are using categories as a foundation instead, then the category of : > sets is simply nowhere in evidence: how do you even DEFINE set? : You would de'ne set like categories.Just how hard is that? DonOt you need to add, to the basic axioms,some de'nitions describing which categories are categories-of-categories, and then some more de'nitions, describingwhich of THOSE categories have sets-as-their-objects?ThatOs a lot of layers up, for something that was allegedlysupposed to be the foundational template. : First you can add the topos axioms, then you can add more axioms : to narrow down the candidates. This is already sketched in the appendix : of the 2nd edition of MacLanes CWM;Well, sure, topoi are categories with sets in them.But how does J.Random Topos relate to the category ofall sets? There are a big new bunch of topos axioms (aboveand beyond the handful de'ning a category), and isnOt it simplyABSURD to allege that all categories are imitations of thecategory of all sets, when a great many of those categoriesneed FAR FEWER additional axioms than are needed by topoi?I mean, topoi are complicated. The category of all setsis, in at least SOME peopleOs opinion, simple -- simple enoughto be analogous to a lot of other simple small categories. : see also the new book by Lawvere and Rosebrugh Sets for Mathematics.-- ItOs dif'cult ... you need to be united to have any strength, but internal issues have to be addressed. E. Ray Lewis, on liberalism in America Does the trace>essentially contain every other sort of information that we could get>from the complete set of latent roots of the representative matrix?Others have pointed out that if your only tool is the calculationof traces, you can still say everything you might want to about asingle similarity class of matrices -- as long as you are willing toapply that tool to other (similarity classes of) matrices too, andas long as you didnOt want any information which couldnOt beobtained from the eigenvalues themselves. (Some information aboutthe similarity class IS lost that way, e.g. the question of whetherthe matrices are diagonalizable. But non-diagonalizable matricesdonOt show up in the context described below anyway.)Your subject line is a little different, though: when you are lookingat a character of a ('nite) group, youOre looking at a _function_de'ned on the group, namely chi(g) = trace( R(g) ) where R(g) isa representation matrix. That is, you havenOt got just one matrixbut lots of them. In fact, one is not usually so interested in _a_character of a group but rather the whole character table, which shedslight on the structure of the group. These pieces of information makea difference in your question: itOs not just linear algebra now butrather group theory. For example, we have some theorems:1. If R_1 and R_2 are two complex representations of a 'nite groupwhose characters are equal, then the representations are equivalent(that is, there is a single invertible P with P R_1(g) = R_2(g) P forevery P; you might say the R1Os and R2Os are uniformly similar.)In other words: the trace is all you need to distinguish two representations anyway.2. Any complex-valued function on G which is constant on conjugacy classes is a linear combination of characters. In other words, thetraces of all the characters already give you all the kinds of functionsyou could make out of the similarity classes anyway.Taken together, (1) and (2) sort of tell us that traces are the onlysimilarity invariant we need in classical representation theory.dave > Besides, my textbook says nothing about the traces of the exterior> powers of the linear transformation either. Do they ever really enter> group character theory?> > Yes, they give another way of constructing new representations> from a given one> (as well as adding or multiplying representations).You mean the exterior powers of the representing lineartransformations are themselves representations? I rekhon you meandirect sum and direct product in the second line. I just gotstarted...anything smartly stated beats me.> The book (Ledermann, Intro. to group> characters) characterizes groups by traces completely. As far as I> know thatOs what the tradition is too.> > You do take the trace in this case too.> If the transformation T acts on the vector space V,> then there is a corresponding transformation T^(r)> on the exterior product V^(r),> and you take the trace of T^(r).If this is what Chapman and Dolan are talking about, corresponding means itrespects the exterior product like f(y ^ v) = f(y) ^ f(v)? =|> |> ||> |> Similarity transformations preserve much more than the tracethe|> |> characteristic equation itself, |> ||> |The coef'cients of the characteristic polynomial are all traces:|> |the traces of the exterior powers of the linear transformation.|> |> the traces of the exterior powers of the linear transformation by|> which a group element acts arenOt among the traces directly mentioned|> in the character of a group representation, though, so wouldnOt it be|> more relevant to point out that the traces of the _ordinary_ powers of|> a linear transformation contain a lot of information about the|> similarity class of the transformation? (assuming i didnOt|> misunderstand the original question.)||I think you didnOt. But I didnOt quite get what you said (as before).|The eigenvalues of the ordinary powers of a linear transformation are|the ordinary powers of the eigenvalues, arenOt they (if Av = av, then|A^2v = Aav = aAv = a^2v and induction)?yes.|Their traces would be the sum|of all the same powered eigenvalues, right (sum_i a_i^2, sum_i|a_i^3)? yes.|Am I missing something?i donOt know. do you agree that the traces of all the ordinary powersof a linear transformation gives a lot more information than just thetrace of the linear transformation itself? thatOs the only real pointi was trying to make.(in the present context i suspect the traces of the ordinary powerscompletely determine the linear transformation up to similarity, butoffhand i forget some of the details of how that should work.)-- =I said *data*, as I donOt agree with much of Dr.> ArpOs cosmological theories. Having followed the topic for some years, I have some questions andobservations; In§ation theory proposes that space is §at/Omega=1, that expansionand gravitational contraction are balanced. Observations show this tobe true. If the collapsing other out, where is the additional expansion for theuniverse as a whole to expand? My 'rst thought in reading of this prediction was that a convectiveprocess made more sense that the apparent coincidence that BBTsuggests. Big Bang theory is essentially based on the assumption that lightfrequencies do not deteriorate, therefore the redshift can only beexplained by recession. The 'rst problem with this is that we are willing to accept thatspace is not an absolute and is gravitationally malleable. Now gravityeffectively collapses our measure of space, BUT it is constantlyradiating the energy of which this matter consisted! So it would seemextremely logical to assume that radiation has the opposite effect andexpands the measure of space. This would result in a very basic and understandable convectivecycle, as matter collapses and energy expands. Say that empty space has a very low threshold for holding stableradiation, such as 2.7k. At that phase transition point it starts tocondense out as hydrogen. This propels the system, as there is moreradiation than space to hold it. This vital explanation for thesmoothness of background radiation would be more logical than as theremnant of a 13 billion year old event. Assuming the universe is in'nite and already full, local space hasnowhere to expand to. The only place for the pressure to express isonto the gravitationally collapsing vortex of galaxies. This wouldprovide a very neat explanation for the excess spin attributed to darkmatter. When the light of distant sources passes through intermediategravitational 'elds, it is magni'ed by the well known lensingeffect. Suppose this compresses the light waves, blueshifting. Giventhe distribution of galaxies, light from distant sources is going topass through the residual gravity 'eld of a number of intermediatesources. This blueshifting will reduce the overall redshift, so thatthe average redshift of closer sources is greater, which would explainthe phenomena for which dark energy is proposed. As it is, Big Bang theory proposes a universe in which ninety-sevenpercent of the matter and energy is invisible to everything but themath. All because we are completely convinced that we know theproperties of light over distances we can never positively quantify,through a medium we can never test. NINETY_SEVEN PERCENT UNKNOWN!!??ALL BECAUSE WE ARE SURE LIGHT IS INFLEXIBLE, EVEN THOUGH WE ACCEPT THESPACE ITOS CROSSING ISNOT!!! Not to mention what we are willing toaccept with in§ation theory! Another point, the greater the distance, the greater the radius ofthe volume of space being considered, but according to BBT, thegreater the distance, the smaller space is!!!!! IOm not an expert and when I 'rst considered the topic I assumed theexperts were right, but I just donOt see it. > 1) Measure redshift of object.>> 2) Measure distance to object.>>We canOt measure distances directly beyond parallax range (about 300>parsecs). The OmeasurementsO that we use beyond the nearest few dozen>galaxies (based mostly on cepheids) are all based squarely on assumingthe>big bang.>> Describe those methods and tell me where this assumption comes into> the method, or stop repeating this ridiculous bit of nonsense.You can easily show me my error by describing (i.e. not just naming) onemethod that does NOT use the big-bang assumption -- either directly orindirectly (i.e. for calibration).I simpy wonOt bother answering an open-ended question, from someone whosnips and ignores all prior evidence in the thread. No matter how manymethods I describe that do use the BB directly or for calibration, you canalways complain that I missed one.SO much easier for you (and educational for everyone) to simply show me acase where IOm wrong.--greywolf42ubi dubium ibi libertas > 1) Measure redshift of object.>> 2) Measure distance to object.>>We canOt measure distances directly beyond parallax range (about 300>parsecs). The OmeasurementsO that we use beyond the nearest few dozen>galaxies (based mostly on cepheids) are all based squarely on assuming> the>big bang.>> Describe those methods and tell me where this assumption comes into> the method, or stop repeating this ridiculous bit of nonsense.> > You can easily show me my error by describing (i.e. not just naming) one> method that does NOT use the big-bang assumption -- either directly or> indirectly (i.e. for calibration).ThatOs a very rational request for Randy Poe. Can he give a clear anddirect answer to it? > I simpy wonOt bother answering an open-ended question, from someone who> snips and ignores all prior evidence in the thread. No matter how many> methods I describe that do use the BB directly or for calibration, you can> always complain that I missed one.Yeah, itOs odd, but thatOs Randy Poe. He creatively deletes to createfalse implication, and then keeps posting repeatedly until he drivesaway the person heOs arguing with by replying, and replying, andreplying.ItOs very odd behavior to me as it involves a bit of energy,persistence, and continual scanning of threads.Randy Poe must do all of those things, but why?> SO much easier for you (and educational for everyone) to simply show me a> case where IOm wrong.That makes sense. LetOs see if Randy Poe has the energy andpersistence to reply again now--with a straight answer.James Harris > Describe those methods and tell me where this assumption comes into>> the method, or stop repeating this ridiculous bit of nonsense.>>You can easily show me my error by describing (i.e. not just naming) one>method that does NOT use the big-bang assumption -- either directly or>indirectly (i.e. for calibration).I posted another message asking you to explain what that laststatement means, since I expect youOll use it uses the Big Bang forcalibration as a catch all for everything, whether it has anything todo with the Big Bang or redshifts at all. YouOve already illustrated awillingness to grossly misread with your circular argument post.Meanwhile, IOll just note that HubbleOs red shift data was publishedin 1929 (with distances measured by parallax), but the calibrationcurve for Cepheid variables was published by Henrietta Leavitt in1912. Big Bang theory in its present form is mostly credited to Gamowin the 1940s with a successful prediction of the 3-degree background,though Lemaitre in 1927 did propose an explosive-origin theory.Explain to me how Leavitt managed to use Big Bang theory for hercalibration in 1912, and what use Big Bang for calibration means,and IOll explain both LeavittOs calibration and the Cepheid variablemethod. - Randy >> Describe those methods and tell me where this assumption comes into>> the method, or stop repeating this ridiculous bit of nonsense.>>You can easily show me my error by describing (i.e. not just naming) one>method that does NOT use the big-bang assumption -- either directly or>indirectly (i.e. for calibration).>> I posted another message asking you to explain what that last> statement means, since I expect youOll use it uses the Big Bang for> calibration as a catch all for everything, whether it has anything to> do with the Big Bang or redshifts at all. YouOve already illustrated a> willingness to grossly misread with your circular argument post.Yes, you avoided the question in the parallel post, too.> Meanwhile, IOll just note that HubbleOs red shift data was published> in 1929 (with distances measured by parallax), but the calibration> curve for Cepheid variables was published by Henrietta Leavitt in> 1912.Did you have a point to make?> Big Bang theory in its present form is mostly credited to Gamow> in the 1940s with a successful prediction of the 3-degree background,Which was a false claim, as the lowest temperature predicted by Gamow, priorto Penzias and Wilson was 50 degrees (a factor of 10,000 error in energydensity -- which was the basis for his estimate).> though Lemaitre in 1927 did propose an explosive-origin theory.Yes. Carl Wirtz 'rst published an empirical redshift-distance relation in1924 (pre Cepheid variable identi'cation). LemaitreOs publication of theOexpanding universeO theory came in 1927, and was based partly on WirtzOempirical work. HubbleOs version of the redshift relation was not publisheduntil 1929 (after Cephied variable identi'cation made WirtzO relationshipmore certain).There have been at least 've major revisions of the explosive origintheory that is now called the Obig bang.O Which one are you defending?> Explain to me how Leavitt managed to use Big Bang theory for her> calibration in 1912,I never claimed that the cepheid period-luminosity relationship was based onthe big bang theory. What I noted was that HubbleOs law was based on thecepheid curve.> and what use Big Bang for calibration means,> and IOll explain both LeavittOs calibration and the Cepheid variable> method.Not necessary. All IOve (repeatedly) asked you to do was simply describeone, modern distance estimatation method -- applicable beyond the range ofcepheid variable resolution -- that does not depend upon the hubbleconstant, and/or is not calibrated by same.--greywolf42ubi dubium ibi libertas > Meanwhile, IOll just note that HubbleOs red shift data was published>> in 1929 (with distances measured by parallax), but the calibration>> curve for Cepheid variables was published by Henrietta Leavitt in>> 1912.>>Did you have a point to make?Cepheid calibration 1912.Big Bang Theory post-1940s.Cepheid calibration canOt be based on Big Bang Theory.Only understand short sentences?>> and what use Big Bang for calibration means,>> and IOll explain both LeavittOs calibration and the Cepheid variable>> method.>>Not necessary. All IOve (repeatedly) asked you to do was simply describe>one, modern distance estimatation method -- applicable beyond the range of>cepheid variable resolution As I asked in your other post, what would beyond the range of cepheidvariable resolution be, since those are as far as I know the mostdistant sources used for Hubble Law tests? - Randy Explain to me how Leavitt managed to use Big Bang theory for her>calibration in 1912, and what use Big Bang for calibration means,>and IOll explain both LeavittOs calibration and the Cepheid variable>method.In FowlesO book on optics I read about MichelsonOs stellar interferometer, which measures stellar diameters. And there was an intensity method thatOs supposed to have improved precision, but I donOt really understand it. Fowles didnOt give any numbers, but it seemed to me that when parallax fails, you can keep going if you can measure diameter, and combine that with brightness and temperature. Assuming diameter measurements can be made farther out than parallax measurements.-- And donOt skimp on the mayonnaise! >Explain to me how Leavitt managed to use Big Bang theory for her>calibration in 1912, and what use Big Bang for calibration means,>and IOll explain both LeavittOs calibration and the Cepheid variable>method.> > In FowlesO book on optics I read about MichelsonOs stellar interferometer, > which measures stellar diameters. And there was an intensity method > thatOs supposed to have improved precision, but I donOt really understand > it. Fowles didnOt give any numbers, but it seemed to me that when > parallax fails, you can keep going if you can measure diameter, > and combine that with brightness and temperature. Assuming diameter > measurements can be made farther out than parallax measurements.What I found in reading pages on astronomical distanceestimation is that many authors seem to use a techniquecalled main sequence 'tting. Google on that term andyouOll learn more than you ever wanted to.The original idea is to use the brightness: based on otherdata about the source, how bright should it be, and thenhow bright does it actually appear to be. The differencetells you distance in a more or less obvious way.This idea has been modernized. Now rather than brightnessitOs some sort of multi-spectral measure that gives muchmore accurate determinations. But itOs still the samebasic idea: if you know how bright something is, and yousee how bright it appears to be, then you know how faraway it is. No Big Bang assumptions. The big unknownalways is how bright is this object really? and thatOswhere main sequence 'tting comes in.IOm probably mangling this a little. I need to read thosepages in more detail before responding directly togreywolf. - Randy > >>Explain to me how Leavitt managed to use Big Bang theory for her>>calibration in 1912, and what use Big Bang for calibration means,>>and IOll explain both LeavittOs calibration and the Cepheid variable>>method.>> >> In FowlesO book on optics I read about MichelsonOs stellar interferometer, >> which measures stellar diameters. And there was an intensity method >> thatOs supposed to have improved precision, but I donOt really understand >> it. Fowles didnOt give any numbers, but it seemed to me that when >> parallax fails, you can keep going if you can measure diameter, >> and combine that with brightness and temperature. Assuming diameter >> measurements can be made farther out than parallax measurements.>>What I found in reading pages on astronomical distance>estimation is that many authors seem to use a technique>called main sequence 'tting. Google on that term and>youOll learn more than you ever wanted to.>>The original idea is to use the brightness: based on other>data about the source, how bright should it be, and then>how bright does it actually appear to be. The difference>tells you distance in a more or less obvious way.>>This idea has been modernized. Now rather than brightness>itOs some sort of multi-spectral measure that gives much>more accurate determinations. But itOs still the same>basic idea: if you know how bright something is, and you>see how bright it appears to be, then you know how far>away it is. No Big Bang assumptions. The big unknown>always is how bright is this object really? and thatOs>where main sequence 'tting comes in.>>IOm probably mangling this a little. I need to read those>pages in more detail before responding directly to>greywolf.I vaguely recall main sequence charts. What could be considered a problem for distance determination is that some things are big and red, big and white, small and red, small and white So thereOs some more theory involved in turning brightness and temperature into a distance. I thought having a diameter measurement must be more direct.-- When the fool walks through the street, in his lack of understanding he calls everything foolish. -- Ecclesiastes 10:3, New American Bible >> 1) Measure redshift of object. 2) Measure distance to object.>We canOt measure distances directly beyond parallax range (about 300>>parsecs). The OmeasurementsO that we use beyond the nearest few dozen>>galaxies (based mostly on cepheids) are all based squarely on assuming>the>>big bang.> Describe those methods and tell me where this assumption comes into>> the method, or stop repeating this ridiculous bit of nonsense.>>You can easily show me my error by describing (i.e. not just naming) one>method that does NOT use the big-bang assumption -- either directly or>indirectly (i.e. for calibration).Noted, for the record, that you made a blanket statement without anyknowledge. Now you want me to provide the actual information so youcan try to 'nd a place to claim your silliness applies.>I simpy wonOt bother answering an open-ended question, from someone who>snips and ignores all prior evidence in the thread. No matter how many>methods I describe that do use the BB directly or for calibration, you can>always complain that I missed one.and IOll describe distance measurement via Cepheid variables. Foundseveral good links. - Randy 1) Measure redshift of object. 2) Measure distance to object.>We canOt measure distances directly beyond parallax range (about 300>>parsecs). The OmeasurementsO that we use beyond the nearest few dozen>>galaxies (based mostly on cepheids) are all based squarely on assuming>the>>big bang.> Describe those methods and tell me where this assumption comes into>> the method, or stop repeating this ridiculous bit of nonsense.>>You can easily show me my error by describing (i.e. not just naming) one>method that does NOT use the big-bang assumption -- either directly or>indirectly (i.e. for calibration).>> Noted, for the record, that you made a blanket statement without any> knowledge.I have knowledge of several different methods of estimating distance. Butthey are all based directly or indirectly on assuming the hubble constant.Obviously, you think you know one. All it takes is one, to show me (and therest of the newsgroup) my error. Go on! DonOt you want to provideeducation?> Now you want me to provide the actual information so you> can try to 'nd a place to claim your silliness applies.I 'gured you couldnOt provide any.>I simpy wonOt bother answering an open-ended question, from someone who>snips and ignores all prior evidence in the thread. No matter how many>methods I describe that do use the BB directly or for calibration, you>can always complain that I missed one.>> What does use the Big Bang for calibration even mean?Every standard candle distance method requires a calibration step. Thereare methods used for distance estimation that are used beyond the range ofcepheids. Look up the section entitled secondary distance indicators inthe book The Cosmological Distance Ladder. These are de'ned as ...indicators which depend for their calibration on our knowing the distance tosome representative nearby galaxies through primary distance indicators.Taking the 'rst method in that section -- for no other reason than itOs'rst in the book -- we have the HII regions method. It is based on theassumption that one can estimate the dimensions of core and halo diameterswithin the HII regions of a galaxy (and that these surround new O and Bstars). A OcorrelationO was found between the HII region diameter and thegalaxy luminosity class. There are several problems with this method(including the fact that the relationship was not the one that was 'rstidentied, but was OforcedO as a secondary method when the 'rst was found tobe nearly useless), which are listed in the book. The primary one beingthat the method of core and halo diameters on the plates are subjective and