mm-609 === >Now I face a situation where I claim to have important mathematics >which other people claim is false. > >Um, speaking as one of those other people, I have provided >simple arithmetic that refutes your conclusion in the paper >youre so hot under the collar about. I have shown it in >who so unceremoniously yanked your paper. >Sigh. Now hes bragging about it! >He is telling it because you _still_ have not addressed the >counterproof apart from complaining about the consequences of its >existence. > Thats not true. Sez you. > What is happening here is just a basic case of one group of people > saying one thing--without regard to the facts--versus one person > backed up by multiple facts that are ignored. The fact is that the conclusion of the so-called Primary Argument in your ill-fated paper is simply incorrect, as elementary arithmetic demonstrates. I have provided that elementary arithmetic for all to see, and repeated it below. You, on the other hand, have persisted in your campaign of name-calling and baby threats. > After all, Im not just some Usenet poster, as I actually followed the > rules, sent my paper to a peer reviewed math journal (well they claim > they peer review and theyre listed) which after nine months told me > they liked my paper, said reviewers liked my paper, and they put it > up. > Those are facts. Yes, those are facts. You were treated unfairly by the editors at > Some sci.mathers heard about it, got all hot and bothered, and in a > thread where they ripped on the journal and its editors, some came up > with the idea of emailing the journal claiming my paper is wrong, > using objections that I shot down long ago. This is possibly what you believe, but it does not resemble the truth. When you claimed that my argument made some assumption regarding the existence of some ring containing the algebraic integers, I asked you to point out *which statement*, out of a short list that outline my argument, was incorrect. You had the statements, you had the arithmetic that I maintain is correct, and you had the opportunity to rebut whatever I claimed. You still have that opportunity, yet you avoid the issue. > The journal editor *immediately* withdrew my paper, responding to the > emails, which have specious arguments that do not in fact show it to > be wrong: > See http://rattler.cameron.edu/swjpam/vol2-03.html Again with the false claim. I have a single argument that demonstrates the conclusion of your argument to be false. Show that it is specious, as you have just stated, and Ill accept that. Heres the argument, just so you dont have to go looking: To refresh everyones memory, the error that I have already noted several times, is this concluding statement from your paper: Therefore, with the factorization 65x^3 .89 12x+ 1 = (a_1x + 1)(a_2x + 1)(a_3x+ 1) one of the a.89s is coprime to 5, which shows where some of the algebraic integer factors distribute despite the factors being irrational. What I have proven is that, in fact, *each* of the as has a factor in the algebraic integers, in common with 5. This is at least as strong as the condition that each coefcient a *not* be coprime to 5. I have shown you these factors. For any of your as in that section, the number (as shown in step 2 below): r(-a) = 8 a^2 + 4a - 45 is an algebraic integer that divides a, and also divides 5. You dont have to believe me. All you have to do is follow some elementary arithmetic. Why wont you do elementary arithmetic, if you think youre correct? OK, now for the repeat of my too-often posted argument: In this outline, a refers to one of the coefcients that you claim must be coprime to 5. P(x) refers to the polynomial x^3 - 12 x^2 + 65, and I note that P(-a) = 0, for each of your values a. 1. The following formulas are true: q(x)r(x) = (64 x + 128)P(x) + 5 r(x)s(x) = (32 x + 72)P(x) + x, where q,r,s are dened as follows: q(x) = 8 x^2 - 76 x - 185 r(x) = 8 x^2 - 4 x - 45 s(x) = 4 x^2 - 37 x - 104 2. Since -a is a root of P(x), we have the following factorizations: q(-a)r(-a) = 5 r(-a)s(-a) = -a This shows that this number: r(-a) = 8 (-a)^2 - 4 (-a) - 45 = 8 a^2 + 4a - 45 is a divisor of -a (i.e., the coefcient a that you claim to be coprime to 5), as well as a divisor of 5. 3. The minimal polynomial of r(-a) is given as: MP_r = x^3 - 969 x^2 + 315 x + 5 which is irreducible over Q. This shows that r(-a) is an algebraic integer, but not a unit in that ring. shared by a and 5 is not a unit in the ring of algebraic integers. 5. a and 5 are not coprime in the ring of algebraic integers, since they share a non-unit factor in common. In any commutative ring, if two elements share a non-unit factor, they cannot be coprime. I suppose the result holds for arbitrary rings, with some restrictions, but thats irrelevant for this case. It is well known that the algebraic integers form a commutative ring. Therefore, a and 5 are not coprime in the ring of algebraic integers. > I got an email from Mathematical Reviews checking with me to set up my > listing about my published paper and called them up to tell them the > story. The person I talked to had never heard of such a thing, but > wished me luck! > You can now at least see the paper at > http://www.ne-plus-ultra.net/index.php?option=content&task= view&id=46&Itemid= 26 > but sci.mathers successfully got it censored out of a peer reviewed > math journal and now you can see one of them *bragging* about it!!! Again claiming that for me to stand up for the truth, and informing everyone of what Ive done, constitutes bragging. Now, you claim Im and you know it. > These things are unheard of in math circles but its happening now, > and posters try to act like its normal because, well, theyre knee > deep in it. Perhaps you think its OK for a person to stand idly by while another makes false statements and publishes them. I dont care whether you are published, or where, but I will not ignore falsehoods masquerading as truth, and will not knowingly allow an editor to be kept in the dark about this. > I have a cabal of posters who now know that they are freaking SCREWED > if the real story comes out and people understand their role. Do you imagine that the real story is not out? Why not take out a big ad in a national paper, write to the editors of all journals you know about, and tell your story? Is it so difcult? > So they keep lying. Tell me a lie that Ive written. > Im wondering now what it will take before some of you wake up to the > real story here as the big deal about that paper is it outlines > techniques that disprove a central idea in Galois Theory. You have a greater chance of overturning all of physics than of debunking Galois Theory. Good luck. > Basically the paper is revolutionary in a way not ever before seen in > the math world as it shows an error in thinking that has persisted for > more than a hundred years to this day. No, it is not. It is fundamentally awed, and to claim that for you to cling to your errors is in any way revolutionary is the height of conceit. Clinging to ones own errors is utterly conventional. > Teachers are actually still teaching it, to this day. > There is nothing like this story in human history, let alone > mathematics. Yeah, you bet. Know-nothing makes outlandish claims about revolutionary mathematical discovery. No one has ever done that before, not in human history. Got it. > James Harris Dale. === Subject: Re: What would it take? > 1. The following formulas are true: > RQ = U = (4x+8)16P + 5 > RS = V = (4x+9) 8P + x > where P,Q,R,S are dened as follows: > P = x^3 - 12 xx + 65 > Q = 8 x^2 - 76 x - 185 > R = 8 x^2 - 4 x - 45 > S = 4 x^2 - 37 x - 104 Simpler: To verify that R is a common divisor of U and V in Z[x] one need only verify that theyre both 0 mod R, i.e. (mod R Z[x]). This can be done in a couple minutes of mental arithmetic, viz: 16P = 16 ( x^2 (x-12) + 65) = 2 (4x+45) (x-12) + 16*65 via 8x^2 = 4x+45, i.e. R = 0 = 8 x^2 - 6x - 40 via 16*65-24*45 = 8*5(2*13-3*9) = -40 = -2x + 5 via 8x^2 = 4x+45 So U = (4x+8)16P+5 = (4x+8)(5-2x)+5 = -R = 0 (mod R Z[x]) V = (4x+9) 8P+x = (4x+9)(5-2x)/2+x = -R/2 = 0 (mod R Q[x]) Note that the prior equation only yields V = RS for S in Q[x], but S must be in Z[x] by Gauss Lemma [1] because R is primitive. Note also that since R is in (5,x) it is not only a common divisor but is further also a gcd of 5,x in Z[x]/P (and any superring). --Bill Dubuque === Subject: Re: What would it take? |Ive heard a similar story, but without quite so many dramatic |ourishes (especially the graduation thing). | |I wonder if this story has any relation to actual events and, if so, |what the events were. I would also like to know whether this is based on a true story. Ive heard several similar stories, and I doubt that theyre all correct. In one version its a professor whose lifes work is demolished by showing that the kind of structure hed studied does not exist. Sometimes the person in the story was studying functions f that are Holder continuous with exponent e > 1. That means there exists a C>0 such that |f(a)-f(b)| <= C|a-b|^e. When e>1, its easy to show that only constant functions have this property: applying the inequality to a, a+(b-a)/n, a+2(b-a)/n, ..., b, we get |f(a+k(b-a)/n)-f(a+(k+1)(b-a)/n)| <= C |(b-a)/n|^e, and using the triangle inequality n-1 times this implies |f(a)-f(b)|<=C|b-a|^e * n^(1-e). As n goes to innity, the right hand side goes to 0, so |f(a)-f(b)|=0 or f(a)=f(b). A lot of the stories seem like urban legends to me. On the other hand, that doesnt mean there isnt some original basis in fact. Keith Ramsay === Subject: Re: What would it take? >|Ive heard a similar story, but without quite so many dramatic >|ourishes (especially the graduation thing). >|I wonder if this story has any relation to actual events and, if so, >|what the events were. >I would also like to know whether this is based on a true story. >Ive heard several similar stories, and I doubt that theyre >all correct. In one version its a professor whose lifes work is >demolished by showing that the kind of structure hed studied >does not exist. >Sometimes the person in the story was studying functions f that are >Holder continuous with exponent e > 1. That means there exists a C>0 >such that |f(a)-f(b)| <= C|a-b|^e. When e>1, its easy to show that >only constant functions have this property: And hence the class of all such functions is naturally isomorphic to C (or to R, whatever). Which seems to me makes it a very intersting class of functions... >applying the inequality >to a, a+(b-a)/n, a+2(b-a)/n, ..., b, we get >|f(a+k(b-a)/n)-f(a+(k+1)(b-a)/n)| <= C |(b-a)/n|^e, and using the >triangle inequality n-1 times this implies >|f(a)-f(b)|<=C|b-a|^e * n^(1-e). As n goes to innity, the right >hand side goes to 0, so |f(a)-f(b)|=0 or f(a)=f(b). >A lot of the stories seem like urban legends to me. On the other >hand, that doesnt mean there isnt some original basis in fact. >Keith Ramsay ************************ David C. Ullrich === Subject: Re: What would it take? : Im wondering now what it will take before some of you wake up to the : real story here as the big deal about that paper is it outlines : techniques that disprove a central idea in Galois Theory. : Basically the paper is revolutionary in a way not ever before seen in : the math world as it shows an error in thinking that has persisted for : more than a hundred years to this day. : Teachers are actually still teaching it, to this day. Im still not clear on what the alleged hundred-year-old error is. Could you nd a citation of the alleged erroneous statement in a textbook so that we can all see clearly the central idea youre supposedly disproving? This would also be a great opener for your paper: The following so-called theorem, which appears in textbooks around the world, is false: ... then cite the theorem. Then you could provide your alleged counterexample and show that it satises all the hypothesis of the so-called theorem but not the conclusion. Even just for sci.math puproses, citing a theorem you believe to be wrong would provide some common ground for the discussion and avoid a lot of the endless dialogue about what background conditions, denitions, and so on you are using. It is also possible that you will discover you misunderstood the conventional mathematical wisdom and are not overturning it after all. Just a suggestion. Mike === Subject: Re: What would it take? X-RFC2646: Original [David Kastrup] > ... > like that happens only with math. In no other discipline can > years of diligent and consistent hard work turn into a heap of rubbish > in a minute. Indeed, if Mr. Harris did produce a result undermining Galois theory, hes dead wrong that it would meet extreme resistance. To the contrary, I expect most mathematicians would nd it intellectually delightful -- not to mention that textbook publishers would rejoice at nding a compelling reason to release new editions. The best example I know about: after the second volume of Freges Basic Laws of Arithmetic (Grundgesetze der arithmetik) went to the publisher, Russell communicated his now-famous paradox to Frege, which happened to destroy one of the foundations of Freges conception of set theory. Did Frege whine, pout, obfuscate, threaten, accuse Russell of incompetence or lying? Nope. Within a week he agreed that Russell had demolished his work, and hastily added an appendix to the volume. A translation I found on the web says it starts: http://intranet.woodvillehs.sa.edu.au/pages/resources/maths/ History/Frg.htm Hardly anything more unwelcome can befall a scientic writer than that one of the foundations of his edice be shaken after the work is nished. I have been placed in this position by a letter of Mr Bertrand Russell just as the printing of the second volume was nearing completion... Thats class. Indeed, too much class to be a set . === Subject: Re: What would it take? > Thats class. Indeed, too much class to be a set . Believe it or not, that class was almost literally beaten into the European intellectuals of the time. Seems kind-of exotic nowadays........... === Subject: Re: What would it take? > He is telling it because you _still_ have not addressed the > counterproof apart from complaining about the consequences of its > existence. > Thats not true. [57 lines that still fail to address the counterexamples that have been posted deleted] Still waiting. At least four posters have given different counterexamples now that demonstrate your theorems are not true. Until you can address these, youve got *nothing*. Perhaps if you tried to actually *prove* your results, instead of just throwing them out and asserting that they are true, you wouldnt make these *elementary* mistakes. -- --Tim Smith === Subject: Re: What would it take? Wahhhhhhhhhhh!!! Wahhhhhhhhhhh!!! Quick! Someone call the Wahhbulance! > Now I face a situation where I claim to have important mathematics > which other people claim is false. > Im just one person, there are many of them, and they refuse to just > talk out the math logically and objectively. > years over the same things, and they cheat. > And before you say Im being paranoid consider what happened with a > paper of mine, which a group of sci.math posters successfully managed > to censor out of publication in a peer reviewed math journal by gang > emailing the editors: > See http://rattler.cameron.edu/swjpam/vol2-03.html > I mean, come on, Im just one guy, but if a group of sci.mathers are > willing to use those kind of tactics, what can I do? > The chief editor yanked my paper basically immediately upon getting > *claims* that it was wrong, without there being time enough for him to > have actually bothered to check, when the journal had my paper for > over NINE MONTHS before! > You can see this group operating on these newsgroups now as they make > sure to try and reply to my posts when I post content, mathematical or > otherwise. > If theres math, then usually itll be Nora Baron or Dik Winter > replying to me, so that Arturo Magidin can reply to them since he says > that he wont reply to me now. > If theres a lot of commentary then itll be Jesse Hughes, David > Ullrich, or Gib Bogle, among others. > Thats just 6 of the standard crew, and you can see some of the others > that pop in and out. > Im ONE GUY. I need to know what it would take to convince a sizeable > number of you that theres a chance I could be right and that all of > this dedicated activity is actually--oddly and bizarrely enough--meant > to hide the truth. > Now Ive demonstrated my willingness to talk about the details of my > work from prime counting to advanced polynomial factorization but the > people who reply to me are part of the cabal. > Are there any others willing to talk out the math? > Is there anyone who has a solution? > And, consider, I tried publication. The sci.mathers gang emailed the > editors of the Southwest Journal of Pure and Applied Mathematics. > Other editors are saying to me that I should go to other journals!!! > Its a massive passing of the buck on a huge academic scale. > So what do I do? What would you do? > James Harris === Subject: Re: What would it take? otherwise. > If theres math, then usually itll be Nora Baron or Dik Winter > replying to me, so that Arturo Magidin can reply to them since he > says that he wont reply to me now. > You ASKED Arturo not to reply to you. As I remember it, it was actually in the form of an order involving such considerate terms as (these arent actual quotes, just my recollection of the language used) just f* off, cant you understand? and yes, I want you to never reply to me again, just f* off and leave me alone I seem to remember also that this outburst occured after Arturo had spent a very considerable amount of time explaining, very politely, where he thought JSH was making some mistakes. I suggest that saying JSH ASKED Arturo not to reply to [JSH] is putting it far too mildly for the violent outburst that actually occurred. the well known post in which JSH was less than polite to David Ulrich - another post gone forever :( Ivan. === Subject: Re: What would it take? > otherwise. > > If theres math, then usually itll be Nora Baron or Dik Winter > replying to me, so that Arturo Magidin can reply to them since he > says that he wont reply to me now. > > > You ASKED Arturo not to reply to you. > > As I remember it, it was actually in the form of an order involving such > considerate terms as (these arent actual quotes, just my recollection of > the language used) just f* off, cant you understand? and yes, I want > you to never reply to me again, just f* off and leave me alone > I seem to remember also that this outburst occured after Arturo had spent a > very considerable amount of time explaining, very politely, where he > thought JSH was making some mistakes. > I suggest that saying JSH ASKED Arturo not to reply to [JSH] is putting it > far too mildly for the violent outburst that actually occurred. > the well known post in which JSH was less than polite to David Ulrich - > another post gone forever :( > Ivan. Your recollection agrees with mine. Arturo several times said essentially, If you want me to quit replying to your posts, just say so and I will stop. Finally he said this when Harris was at the peak of one of his ts of rage. Harris very obscenely said yes. Arturo has not posted a direct reply to Harris since that time. I believe Harris would like for all of his opponents to do the same. That is the closest he can come to proving himself right. Yes, Harris has been deleting posts of which he is apparently ashamed. He deleted one yesterday which contained a simple algebraic mistake. It doesnt do him much good because, I believe, all of his posts are still out there on mathforum. Plus some people save some of his choicer contributions for future reference. But its curious that he has tried to change parts of his own history that he doesnt want people to know about. Nora B. === Subject: Re: What would it take? Discussion, linux) > otherwise. > > If theres math, then usually itll be Nora Baron or Dik Winter > replying to me, so that Arturo Magidin can reply to them since he > says that he wont reply to me now. > > You ASKED Arturo not to reply to you. > As I remember it, it was actually in the form of an order involving such > considerate terms as (these arent actual quotes, just my recollection of > the language used) just f* off, cant you understand? and yes, I want > you to never reply to me again, just f* off and leave me alone > I seem to remember also that this outburst occured after Arturo had spent a > very considerable amount of time explaining, very politely, where he > thought JSH was making some mistakes. > I suggest that saying JSH ASKED Arturo not to reply to [JSH] is putting it > far too mildly for the violent outburst that actually occurred. > the well known post in which JSH was less than polite to David Ulrich - > another post gone forever :( Yep, its gone, but Arturos reply is still there, quoting Jamess original. (missing-in-action) and Arturos reply is . Shame I dont have that post. For a while now, Ive been collecting a few JSH gems, but only for entertainments sake. Now that hes enjoying revisionism, I wish I had gotten a few of his controversial ones. But I dont know which post about Ullrich you were searching for. The lapdog post is still out there with Message-ID <7s8n87$adn$1@nntp5.atl.mindspring.net>. -- Jesse F. Hughes Well, if I can get [my proof of FLT accepted], then I hopefully get a book deal down the road, and maybe I get to go on Oprah. James Harris, on the rewards of mathematical endeavours. === Subject: Re: What would it take? >[...] >the well known post in which JSH was less than polite to David Ulrich - Uh, theres more than one post that meets that description. >another post gone forever :( Yup. Hes been deleting a lot of crap. Luckily people have hard drives... >Ivan. ************************ David C. Ullrich === Subject: extending ZFC I present here a proposal for a family of set theories. I believe these can be a good framework for working with set theory, providing an overall organization to various alternative approaches. In general I like ZFC, but I think the usual formulation leaves some subtle philosophical loose ends. I am trying to have a basis for handling these better. My approach below departs somewhat from the mainstream set theoretic opinion. The usual view of what should be reasonable canonical extensions of ZFC would be agnostic about V=L or else to believe V ~= L. I have been led to eventually come around to accept V=L in as a part of the basis. I am aware that this is unusual, and of the usual reasons for accepting V ~= L, (in fact I used to believe these myself), but I have my own reasons for coming back to V=L. I am aware that Godel, he rst dened L, did not even believe V=L. In fact my settling on V=L does not have the full ramications it may rst seem, because I can recover some of the non-L alternatives in an interpreted way. There are several distinct philosphical difculties I had with simple ZFC, and there are accordingly distinct aspects of my systems to handle these. So it is something of a hodge-podge, throwing in everything but the kitchen sink. So there are several parallel motivating ideas co-existing. Just as I recognize including V=L as controversial as noted above, I also even recognize AC as controversial. I do think about such aspects, but have so far come back to the AC side. Another big issue in looking for alternative set theories is to consider weakening ZFC. I have considered that too, but I think in the end it does not solve the philosphical problems it was intended to solve. In fact I strengthen ZFC. Strengthening to beyond ZFC is the usual approach among set theoriests. So against the background of sci.math my strengthening may seem controversial. Compared to usual set theorists though my version will look different by its relative weakness. By including V=L I am holding back from the higher levels. I have philosphical reasons for that. But along the lines of my comments about non-L above, I do manage to recover something of these levels less directly. For this rst post I am just going to give a bare bones description of the systems along with the opening comments above. For now I wont go into detail about the motivations leading me to such systems. So on to the systems. There will be a base system. Other systems from the same family can be produced by adding more axioms and possibly extending the language. The base system will have language rst order with = and primitive binary relation epsilon, two individual constants u and U, and two binary function symbols, which I will write in forms u^beta_alpha and U^beta_alpha, ie us or Us with superscripts and subscripts. So basically I have an indexed family of us with superscripts and subscripts, and an indexed family of Us with superscripts and subscripts, but also I have two very important special cases that are written without superscripts or subscripts, they will act like the correponding letters adorned to terms, but being the most important I make them easy to write as u and U. The axioms will included the corresponding ZFC axioms for the expanded language, ie we allow separation and replacement axioms for formulas with the extra terms. Now to get to the V=L aspect of the systems. This is one of the features of the system which makes the banishment of non-L and high consistency strength beyond L incomplete, that those aspects are still around in the background. This was the outcome of my struggles with the difculties Godels incompleteness theorem makes for producing a foundational system that could be judged as ultimate. So in the end I am interest in V=L universes. We want a system to generate theorems about that. Going to high universes gives us more proof power to know about L, but that extra power also puts more complexity into the universe and geenrates new difcult questions. So I break the cycle. The real universe is L, but as a guide to what is true in L, I can look out and see what higher universes tell me about L. I think of them as a formal tool for generating an r.e. list of theorems about L. By Craigs theorem, any r.e. axiomatized theory can be equivalently recursively axiomized (different axioms but same theorems). So I cna consider that I have given a system telling about L even when I step outside and then cut the answers back to L. So though the language is rst order language, I am giving an unusual proof system for it. I am not literally adding to the extended ZFC axiomatization above the axiom V=L. Instead I will dene the nal theorems of my system to be those with L-relativization provable in an underlying ZFC theory. In this way V=L gets into the the nal system. But it allows me to sneak power beyond L in the background, the underlying theory generating those L theorems might use large cardinals incomatible with L. abandoning non-L, or large cardinal strength beyond L. This is a non-standard notion of proof system. It is my nal resolution of the struggles trying to reconcile Godels theorem with system building. So so far we have ZFC in the expanded language, and this indirect working of V=L into the system. So more axioms now, for te base system, about those new constants. We axiomatize that U is an inaccessible L-rank, ie L_kappa for some strongly inaccessible cardinal. This is inaccessiblity in the background full universe, not just its L. We axiomatize that for all alpha countle^L ordinals and beta countable^L ordinals, with alpha < beta, that U^beta_alpha is an L rank for an ordinal inaccessible in L. This is weaker than the corresponding U axiom, because we axiomatized that Us rank was real inaccessible, ie in the full background universe, but for this axiom we are only claiming inaccessible in L. (Note that in discussing the axioms we are using the language literally. It is only the nal reinterpretation step that automatically realtizes back to L. The axioms and proofs stay in the base set theory language for a while and only as a last step do we check which are the theorems produced that realtized to L.) We axiomatize that the u and U functions behind the terms have domain as just mentioned: L-countable ordinals superscript beta subscript alpha with alpha < beta. We axiomatize with universal quantiers that for suitable alpha1 < alpha2 and beta U^beta_alpha1 member U^beta_alpha2 and for suutable alpha1, alpha2, beta1, beta2 with beta1 < beta2 U^beta1_alpha1 member U^beta2_alpha2. Ie each beta makes an incrreasing sequence as alpha increases, and the next beta sequence starts strictly above the entire previous beta sequence. Suitable above means all displayed terms are well-dened with the alpha-beta pair being in the U domain. We axiomatize that every U^beta_alpha term as above (ie proper beta, alpha to be in the dimain) is an elementary substructure of the full L of the universe, elementary in the language of =, epsilon. This is a schematum over every formula in the laguage. (We leave off the U^beta_alphas from the language since those may not be in the small model). This schematum is a schematum over choice of formula in expanding the denition of elementary substructure. But the betas and alphas abpve are expressed with full universal quantiers, ie for each formula we have an axiom saying for all suitable beta,alpha, actual universal quantier in the axiom. Next axioms. We axiomatize as a schematum over formulas, with beta and alpha universally quantied in the axiom, that for each L-countable ordinal beta, the sequence ^ , ie the seuquence with U followed by all the U^beta_alpha with xed beta and varying alpha, is a sequence of strong order indiscerables in the full universes L. Ie, this means for any formula with parameters in L of rank < both starting members of two increading n-tuples from the sequence, we have the indescernabilty axiom: the 2 formulas in those respective tuples are equivalent. We axiomatize that L satises separation and replacement for formulas with the new symbols: u and U, both the constants and the function symbols. Now a suble point, for the next axioms. Above we axiomatized L-inaccessiblilty and L-indiscernability for the Us, with full universal quantiers. I want corresponding axioms for the full background universe. But I must do these in a weaker form. Otherwise some of the intended strengthenings of the base system would be inconsistent. For inaccessiblity, I axiomatize a schematum, over all terms t1 and t2 in the language, terms with parameters, and the full language with all the new symbols, saying that if t1 denoted an L-countable ordinal and t2 denotes an L-countable ordinal, and t1 < t2, then U^t2_t1 has rank strongly inaccessible in the background universe. Similarly, I axiomatize the indiscernability axiom, but not for L-formulas, but for the full background universe, but instead of quantifying over all suitable beta, alpha as in the L version above, I just do each axiom in the schematum for a specic tuple of terms making the betas and alphas for sequences of indiscernables. I also axiomatize, in similar schematum fashion, that the U^beta_alphas have correponding full ranks (ie take the V rank in the background universe with same height as U^beta_alphas height) which are elementary substructures in =,epsilon to the full background universe. The previous axiom making the L levels elementary substuctures of L may seem reundant after the last. But the L version quantied over all the U^beta_alphas and the full version is only a schematum so we retain the L version too. That covers U and U^beta_alpha. Now u and u^beta_alpha, which I have not previously discussed. We axiomatize using universal quantiers that the u function has same domain as the U function, ie L-countable ordinals both with alpha < beta. We axiomatize that u and all suitable u^beta_alpha (using universal quantiers on alpha,beta) are countable L ranks. And we axiomatize that for any formula in =,epsilon with parameters exactly U^beta_alpha1 U^beta_alpha2 ... U^beta_alphan with these all suitable (ie superscripts and subscripts in the domain of the U function) such a formula holding in is equivalent to having satisfying the same formula with each U^beta_alphai replaced by the corresponding u^beta_alphai. Ie: we have U and the various U^beta_alphas sitting as inccessible ranks in L and the universe. u and u^beta_alpha are a miniature corresponding version, of small L-countable ranks that look like the uncountable hierarchy sitting above regarding internal formulas. This last axiomatization is done as a single axiom. We quantify over all tuples of ordinals, of all arities (ie n) in the quantication, and we quantify over all formulas, since we are only talking about their truth in set sized structures we can dene that truth and so formalize all that in one axiom instead of a schematum. Thats it, for the axiomatization. The base system is the axiomatization just given, with the understanding that the nal list of the theorems generated will be those with L relativization provable in the base axiomatization given above. I started off writing that I was giving a family of related systems. So we branch out into the family by adding more axioms to the above, axioms about the underlying universe, and possibly get new theorems about its L, hence new theorems about the nal system. We can also extend the language. For example, if we anted to add new constant symbols we would typically axiomatize them to denote a member of U. And we would probably make the real version of them in upper case in U, and a lower case version in u, and extend the axiom reecting the Us to us in the obvious way. So even though L is in the foreground, we can add large cardinal strength beyond L in the background, to the base axioms which dont assume outright that V=L. We can also get some strength from non-AC. If you have a strong ~AC prinicple you would like to get into the background, in the framework above you could axiomatize that there is an uncountable standard model of the other principle. This will often be good enough to import things into L. You can work in that model to get things about its L. Having things hold in an uncounble L rank is as good for many purposes as having it in all of L. This is actually a theorem if you are already beyond 0#. Sometimes we want to treat the universe of set theory as the enveloping context. Formalize this in the theiry above by just forgetting all the extra machinery and working in the ZFC or ZFC + V=L part. Sometimes we want to talk as if the set theoretic universe could itself be a bounded object, having global properties to discuss. For this use U. Sometimes we want to consider the universe as one of a population of many possible universes. For this use u, and consider all the standard countable ZF models to be the entire population. If you just want to be working in ZFC or even ZFC + V=L, just use those and interpret them straightforwardly in the axiomatization above. (Any consequence of V=L is autimatically generated by the nal theorems list). If you want to be in GB or Morse-Kelley, interpret sets as members of U and classes as subsets (in the original sense) of U. If you want to be in ZFC + V ~= L or ZF + ~AC, interpret your universe as one of the countable structures along with the u s. -- David Libert ah170@FreeNet.Carleton.CA === Subject: Re: extending ZFC > I present here a proposal for a family of set theories. I believe these > can be a good framework for working with set theory, providing an overall > organization to various alternative approaches. > In general I like ZFC, but I think the usual formulation leaves some > subtle philosophical loose ends. I am trying to have a basis for handling > these better. > ... > This last axiomatization is done as a single axiom. We quantify over all > tuples of ordinals, of all arities (ie n) in the quantication, and we > quantify over all formulas, since we are only talking about their truth in > set sized structures we can dene that truth and so formalize all that in > one axiom instead of a schematum. > Thats it, for the axiomatization. > ... Hi David, Yeah, V=L! Thats an excellent post. It deserves a cogent and structured response from set theory. I disagree on principle, but it would take me a while to determine why. Anyways, Im just happy that you think V=L. What are V and L? I ask for the benet of others, you can read what I say about V and L. Ross F. === Subject: Re: bush tax cut and small businesses > Mr. Weatherby is referring to > > Paul M. Romer, Endogeneous Technical Change, Journal of Political > Economy. V. 98, N. 5, p. S71-S101. > > The rst numbered equation in that paper is a production function. > The arguments of this function include heterogeneous capital goods. > These arguments are measured in numeraire units. I pointed out some > time ago that, because of price Wicksell effects, this equation (and > the remainder of the paper) is incorrect. > > Mr. Weatherbys belief that this objection depends on whether > these units are dollars or some other numeraire reveals he doesnt > understand the objection. > It is funny you still do not what that numeraire is. Your comment was > that because capital is measured in dollars it is subject to Price > Wicksell effects. I say above that this objection doesnt depend on whether these units are dollars or some other numeraire. So what is Mr. Weatherbys excuse for his falsehood? > You still do not what capital is measured in. You > speak against things before you have fully read and understand them. > Your beliefs will not allow you to look at model and interpert it. > Instead you catch one word, misinterpert it and then declare it is wrong. Mr. Weatherby, of course, is making up. Notice Mr. Weatherbys comments, inasmuch as they have any cognitive content at all, merely demonstrate that Mr. Weatherbys belief that this objection depends on whether these units are dollars or some other numeraire reveals he doesnt understand the objection. And wasnt Mr. Weatherby just telling us that Cohen and Harcourt clearly explain what Wicksell effects are? In short, Mr. Weatherby has no idea of what he is talking about and is not addressing the post to which he is pretending to reply. By the way, Romer (1990) is mistaken. [ > Finally your application of capital switching models to labor is ] [ > solely your application. ] [ > Two posts ago, I said that I think the usage of the phrase ] [ > capital switching or capital reswitching is indicative of ] [ > somebody that does not understand the conclusions agreed to ] [ > by both sides on the Cambridge Capital Controversy. Does Mr. ] [ > Weatherby explain why he thinks capital switching is an ] [ > appropriate term for switches of technique? No, he just repeats ] [ > his perhaps inappropriate terminology. This does not seem ] [ > like honest behavior. ] And Mr. Weatherbys explanation of what he means by capital switching is? [ > In the post to which Mr. Weatherby is pretending to respond, I ] [ > cited the following URL as an application of my favorite sort ] [ > of argument to labor markets: ] > > > Mr. Weatherby deleted that URL without comment, but makes the > above already-refuted statement. This does not seem like honest > behavior. > No it is the decision to not to read undecipherable babble. The decision > is due to your inability to write. See, in particular, the fth footnote and surrounding text in the paper at the above URL. Does the paper at the above URL argue that the Cambridge Capital Controversy has implications for the labor market? So what is Mr. Weatherbys excuse for not having retracted his false statement that my application of capital switching models to labor is solely [my] application? And what is Mr. Weatherbys excuse for pretending my writing abilities are relevant to the clarity of the paper at that URL? > If you would > provide one referred journal reference showing the model I will happy to > read it. Since this claim is immediately followed by a quote from a referred journal showing a model (but shortly, as is typical of papers), Mr. Weatherby is clearly making a false statement about what he is willing to do. What is his excuse for his falsehoods? Furthermore in Footnote 11 of Sraffa3.pdf, I reference ve textbooks justifying my favorite model. Any one of these textbooks would do. Sraffa3.pdf is available off a link at the URL in my sig. The link is labeled A Critique of Disaggregated Neoclassical Theory. > One cannot match a proof like that of Equation (5) by nding a > valid proof for the stationary state conjecture > > d(C/L)/di <= 0. (6) > > Why not? Because, as the next section will illustrate with numerical > examples, such a conjecture is simply not true! > -- Paul Samuelson, A Modern Post-Mortem on Bohms Capital > Theory: Its Vital Normative Flaw Shared By Pre-Srafan > Mainstream Capital Theory. Journal of the History of > Economic Though. V. 23, N. 3. 2001. > Ill explain Samuelsons notation. i is the interest rate. In > the simple model under discussion, a lower interest rate is > associated with a higher real wage. C/L is consumption per > person-year. Samuelson is saying a lower interest rate is not > necessarily associated with higher consumption per worker, that > is, a less labor-intensive (L/C) technique. > > In other words, a higher wage may be associated with a switch > to a more labor-intensive technique. In other words, as I > have proven, cost-minimizing rms may want to hire more > workers at a higher wage, given the available technology, > perfect competition, and the level of output. > No this does not say that. Where are the techinique used. You are > skipping a lot of steps here. Saying a lower interest rate is not > necessiarly associated with higher consumption does not automatically > lead to highering more workers when wages rises. You have a lot of steps > to go through. You have to do that coherently. This is just another > example of your complete inability to write. Does Display 5 and Theorem 1 in Samuelsons paper say that a lower interest rate is associated with a higher real wage? Does Samuelson imply in the above quote that a lower interest rate is not necessarily associated with a less labor-intensive (L/C) technique? Consider the point (S sub BC) in Figure 1 in Samuelsons paper. Around this point is a higher wage associated with the adoption of a more labor intensive (higher L/C) technique? How does this not imply that cost-minimizing rms will hire more labor around this point at a higher wage, given the technology, perfect competition, and the level (C) of output? > Im not going to learn how to right properly from Mr. Weatherby. >and explain models rather than write down several >on >your website in a Post-Keynesian journal. I would be interested in >knowing if the Post-Keynesians other than you would agree with your >analysis. Assuming that you could present it in a coherent way. > Notice Mr. Weatherby does not say anything at all about where he > nds anything confused, unclear, or incoherent about the following: > > > > That URL was in the post to which Mr. Weatherby is pretending to > respond. His behavior does not seem honest. > I am not going to read this babble again. I can not respond when it is > completely incoherent and pieces are missing. Notice Mr. Weatherby does not say anything at all about where he nds anything confused, unclear, or incoherent about the paper at the above URL. > [ Babble I dont feel like comment on - deleted. ] >I can post working papers on the web as well. It means little. The true >test is if colleagues in your eld agree that the analysis is correct >and interesting. This is only shown by publishing in the journals of >your eld. > No. Neither authority nor popularity determines the validity of > an argument. > This is not authority or popularity. It shows conscensus that the > techniques are correct and properly applied. Whatever. Conscensus does not determine the validity of an argument. > To be published you have to > explain things better. If you dont believe send one of these PDFs to a > professor at one of the schools that recognizes Post-Keynesian thoughts. > See their comments. They will say the same. Nope. I dont think I have ever had any interaction with a heterodox professor who was not encouraging. Some use some of my materials for teaching. Not everybody who has used my materials for teaching has even told me about it. > You substance may or may not > be correct. I can not comment because it is too poorly explained and > poorly written. You throw down a bunch of equations with little > explanation of what you are doing or why it is proper then jump to > how to write. Mr. Weatherby can easily see that I have already read some journal not? A remark: how is it possible for professional academic economists to be as dishonest as weve seen on sci.econ? -- Mostly economics: r c v s a Whether strength of body or of mind, or wisdom, or i m p virtue, are found in proportion to the power or wealth e a e of a man is a question t perhaps to be discussed by n e . slaves in the hearing of their masters, but highly @ r c m unbecoming to reasonable and free men in search of d o the truth. -- Rousseau === Subject: Re: bush tax cut and small businesses > Mr. Weatherby can easily see that I have already read some journal > not? > A remark: how is it possible for professional academic economists > to be as dishonest as weve seen on sci.econ? How about dishonesty here. Here are the class notes you speak of. # Frequently Asked Questions about the Labor Theory of Value * Materials for Reviewing Marx, Dan Ryans History of Sociological Thought Class Notes. (My FAQ appears in other class notes too.) * Socialist Movement links, from Washington State Universitys Program in American Studies * Google directory links for Labor Economics * Serebella Directory for Labor Economics (There are many other directories with a link to my FAQ.) * List of links from Vision? Nary!, a magazine of satire. * Satoshi Miyamuras resources for economics * Catallarchy Web Log discussion * D-Squared Crooked Timber Web Log discussion * Discussion of LTV several years ago on Agent Causation. * Discussion Several years ago on another mail list, this one described as General nonsense, fun for everyone. * Random Balderdash Your class notes a heterodox economist have used deals with the labor theory of value not the model you present. In fact it is one paragraph I think this shows your political motivation. It shows I am correct. You cling so heavily to one argument with little proof for it because you are a Marxist and this is the only thing can you nd that might put a chink in the armor of economic theory. You want to replace economics with Marx. That is why you do not have an alternative theory just complaints about the current theory. and Harcourts argument and cite it then call it a fumble. Please dont put me on your fumble list again. Did you really think I would not nd your homepage Rob? === Subject: Re: bush tax cut and small businesses [ > To be published you have to ] [ > explain things better. If you dont believe send one of these ] [ > PDFs to a professor at one of the schools that recognizes ] [ > Post-Keynesian thoughts. See their comments. They will say ] [ > the same. ] [ > Nope. I dont think I have ever had any interaction with a ] [ > heterodox professor who was not encouraging. Some use some of ] [ > my materials for teaching. Not everybody who has used my ] [ > materials for teaching has even told me about it. ] > A remark: how is it possible for professional academic economists > to be as dishonest as weve seen on sci.econ? > How about dishonesty here. Here are the class notes you speak of. > [ Mr. Weatherbys clip of text off my home page. The text describes ] > [ my FAQ on the Labor Theory of Value and some links to it. ] > Your class notes a heterodox economist have used deals with the labor > theory of value not the model you present. In fact it is one paragraph I took Mr. Weatherbys whine to be referring to my writing in general, even though he does say PDF. So I dont see how the fact that the links I provide at that point are to my LTV FAQ, and not much CCC commentary, is relevant. Furthermore, I dont know what the one paragraph above refers to. Furthermore, I dont know why Mr. Weatherby thinks professors using my stuff for teaching are conned to my LTV FAQ. I know of one who has told me that he has used my FAQ in teaching, but did not give me a page linking to it. (They probably just gave out a URL in class.) There are many more pages attempting to link to my FAQ than I list. If I were to attempt to ensure all these pages had updated links, it would be a lot of work. How does Mr. Weatherby know that teachers dont use in their lectures, for example, my spreadsheet on the choice of technique? I once asked a heterodx economist to take a look at it. His response was encouraging. Perhaps he uses it in a lecture for all I know. > I think this shows your political motivation. It shows I am correct. You > cling so heavily to one argument with little proof for it because you > are a Marxist and this is the only thing can you nd that might put a > chink in the armor of economic theory. The above is mistaken ad hominem. There are many more objections to neoclassical theory I could raise. My supposed motivations can have nothing to do with the validity of the following claim: In this view, the central mistake in neoclassical theory concerns the scarcity explanation of distribution which must lead into difculties whenever capital, being produced in the form of heterogeneous capital goods, changes its price in a process of substitution, hence its amount as capital. Mr. Weatherby continues his fumbling blunders: > You want to replace economics > with Marx. That is why you do not have an alternative theory just > complaints about the current theory. Being able to say intelligent things about the Labor Theory of Value does not establish one is a Marxist or that one wants to replace (neoclassical?) economics with Marx. It is odd to say one wants to replace (neoclassical?) economics with Marx and simultaneously say one does not have an alternative theory. I have no idea what Mr. Weatherby means by being a Marxist. Last time he brought up neo-Marxist he had no coherent explanation of what he meant. Neither did poor Chris Auld. And I have pointed out before that many Marxists attack Srafans. > and Harcourts argument and cite it then call it a fumble. This is a fumble: If you read Cohen and Harcourt you would that is exactly why General Equilibrium analysis started. Did you read the whole arose after the failure of the parables. It was motivated by Samuelsons quest, in his surrogate production function to provide some rationalization for the validity of the simple you previously agreed with? > Please dont put me on your fumble list again. I may put Mr. Weatherby on my fumbles list again. But I am being kind to him. Im not sure if he has ever posted anything that doesnt contain a fumble. For example, this question, given my sig, is extremely odd: > Did you really think I would not nd your homepage Rob? -- Mostly economics: r c v s a Whether strength of body or of mind, or wisdom, or i m p virtue, are found in proportion to the power or wealth e a e of a man is a question t perhaps to be discussed by n e . slaves in the hearing of their masters, but highly @ r c m unbecoming to reasonable and free men in search of d o the truth. -- Rousseau === Subject: Re: bush tax cut and small businesses > Notice Mr. Weatherbys comments, inasmuch as they have any cognitive > content at all, merely demonstrate that Mr. Weatherbys belief that > this objection depends on whether these units are dollars or some > other numeraire reveals he doesnt understand the objection. That was your statement not mine. I had to put out to you that capital goods were measured in units of the consumpion good not dollars. I had to point out to you that money never enters a growth model. >If you would >provide one referred journal reference showing the model I will happy to >read it. > Since this claim is immediately followed by a quote from a referred > journal showing a model (but shortly, as is typical of papers), Mr. > Weatherby is clearly making a false statement about what he is > willing to do. What is his excuse for his falsehoods? The paper you cite actually deals with labor markets being out of equilibrium due to rigdities. It does not deal with the possibility that labor demand can slope upward. In fact in the introduction they cite Mankiw and Romer 1992 that shows rigidities can keep labor demand from equaling labor supply. This has nothing to do with your arguement. No where does the paper say a rm can demand more labor when wages rise which is your arguement. The refer to the fact that the neo-classical assumption of exible prices and wages maybe wrong. It says that policies that lower the real wage may not induce employment. This is possible if real rigidities keep the labor market from clearing. There is no place in this paper where the authors support your analysis that labor demand can be increasing in the wage. The converse does not follow from the argument. Real rigidities can mean that labor demand and labor supply will not equate with a falling real wage. The same sort of issue is dealt with in Real Business cycle models. In this case it deals with labor hording where employers keep the same level of employment when wages rise. This does not say nor is it a reasonable conclusion from the paper that labor demand curves can slope upward. You are really stretching to support your claims here. > Furthermore in Footnote 11 of Sraffa3.pdf, I reference ve textbooks > justifying my favorite model. Any one of these textbooks would do. > Sraffa3.pdf is available off a link at the URL in my sig. The link is > labeled A Critique of Disaggregated Neoclassical Theory. Do you they make the same applications to wages. You have in the past made far reaching statements such as unskilled labor increases when your model only had one type of labor in it. >and explain models rather than write down several >on >your website in a Post-Keynesian journal. I would be interested in >knowing if the Post-Keynesians other than you would agree with your >analysis. Assuming that you could present it in a coherent way. >Notice Mr. Weatherby does not say anything at all about where he >nds anything confused, unclear, or incoherent about the following: > >That URL was in the post to which Mr. Weatherby is pretending to >respond. His behavior does not seem honest. >I am not going to read this babble again. I can not respond when it is >completely incoherent and pieces are missing. > Notice Mr. Weatherby does not say anything at all about where he nds > anything confused, unclear, or incoherent about the paper at the above > URL. It is hard to do when so much is missing. I like other professional will drop the paper and move on to something else long before we try to decipher what you meant. > Mr. Weatherby can easily see that I have already read some journal > not? That is impossible for me to see. You do not post your CV. Only a bunch of URLs. > A remark: how is it possible for professional academic economists > to be as dishonest as weve seen on sci.econ? Dishonesty is posting a URL to a paper that has absolutely nothing to do with your ideas and saying it supports your model. === Subject: Re: bush tax cut and small businesses > Notice Mr. Weatherbys comments, inasmuch as they have any cognitive > content at all, merely demonstrate that Mr. Weatherbys belief that > this objection depends on whether these units are dollars or some > other numeraire reveals he doesnt understand the objection. > That was your statement not mine. I had to put out to you that capital > goods were measured in units of the consumpion good not dollars. I had > to point out to you that money never enters a growth model. Mr. Weatherby is making up. Consider: Paul M. Romer, Endogeneous Technical Change, Journal of Political Economy. V. 98, N. 5, p. S71-S101. The rst numbered equation in that paper is a production function. The arguments of this function include heterogeneous capital goods. These arguments are measured in numeraire units. Because of price Wicksell effects, this equation (and the remainder of the paper) is incorrect. This objection doesnt depend on whether these units are dollars or some other numeraire. And wasnt Mr. Weatherby just telling us that Cohen and Harcourt clearly explain what Wicksell effects are? What does Mr. Weatherby understand price Wicksell effects to be? And Mr. Weatherbys explanation of what he means by capital switching is? Consider: See, in particular, the fth footnote and surrounding text in the paper at the above URL. Does the paper at the above URL argue that the Cambridge Capital Controversy has implications for the labor market? So what is Mr. Weatherbys excuse for not having retracted his false statement that my application of capital switching models to labor is solely [my] application? And what is Mr. Weatherbys excuse for pretending my writing abilities are relevant to the clarity of the paper at that URL? The Graham White paper whose URL is given above does NOT assert that the labor market would approach equilibrium more quicky if the labor market were more exible. It fact it asserts such a belief is unfounded. Does Mr. Weatherby see that Graham White draws on the Cambridge Capital Controversy analysis of competitive (exible) markets to support his arguments? >If you would >provide one referred journal reference showing the model I will happy >to read it. > Since this claim is immediately followed by a quote from a referred > journal showing a model (but shortly, as is typical of papers), Mr. > Weatherby is clearly making a false statement about what he is > willing to do. What is his excuse for his falsehoods? > The paper you cite actually deals with labor markets being out of > equilibrium due to rigdities. Notice the cited refereed paper by Samuelson below. It has nothing to do with rigdities. Furthermore in Footnote 11 of Sraffa3.pdf, I reference ve textbooks justifying my favorite model. Any one of these textbooks would do. Sraffa3.pdf is available off a link at the URL in my sig. The link is labeled A Critique of Disaggregated Neoclassical Theory. One cannot match a proof like that of Equation (5) by nding a valid proof for the stationary state conjecture d(C/L)/di <= 0. (6) Why not? Because, as the next section will illustrate with numerical examples, such a conjecture is simply not true! -- Paul Samuelson, A Modern Post-Mortem on Bohms Capital Theory: Its Vital Normative Flaw Shared By Pre-Srafan Mainstream Capital Theory. Journal of the History of Economic Though. V. 23, N. 3. 2001. Does Display 5 and Theorem 1 in Samuelsons paper say that a lower interest rate is associated with a higher real wage? Does Samuelson imply in the above quote that a lower interest rate is not necessarily associated with a less labor-intensive (L/C) technique? Consider the point (S sub BC) in Figure 1 in Samuelsons paper. Around this point, is a higher wage associated with the adoption of a more labor intensive (higher L/C) technique? How does this not imply that cost-minimizing rms will hire more labor around this point at a higher wage, given the technology, perfect competition, and the level (C) of output? Consider: Notice Mr. Weatherby does not say anything at all about where he nds anything confused, unclear, or incoherent about the paper at the above URL. > Mr. Weatherby can easily see that I have already read some journal > not? > That is impossible for me to see. You do not post your CV. Only a bunch > of URLs. The references in LaborDemand.pdf, Sraffa3.pdf, etc. suggest I have Weatherby is pretending to respond. > Dishonesty is posting a URL to a paper that has absolutely nothing to > do with your ideas and saying it supports your model. I have no idea what Mr. Weatherby imagines he is talking about. How is it possible for professional academic economists to be as dishonest as weve seen on sci.econ? -- Mostly economics: r c v s a Whether strength of body or of mind, or wisdom, or i m p virtue, are found in proportion to the power or wealth e a e of a man is a question t perhaps to be discussed by n e . slaves in the hearing of their masters, but highly @ r c m unbecoming to reasonable and free men in search of d o the truth. -- Rousseau === Subject: Expectation(X), characteristic function Hi Let f(x) be a prob density function. Is there any signicance to the value f(E(x)), where E(.) is expectation operator ? Why is the Fourier transform of f(x) ( known as characteristic function ) signicant ? Note that Fourier transform indicates the frequency content of f(x). A link explaining practical application of these concepts in engineering / science will be helpful. shankar === Subject: Re: Expectation(X), characteristic function X-RFC2646: Original > Hi > Let f(x) be a prob density function. > Is there any signicance to the value f(E(x)), where > E(.) is expectation operator ? > Why is the Fourier transform of f(x) ( known as characteristic function ) > signicant ? Note that Fourier transform indicates the frequency > content > of f(x). > A link explaining practical application of these concepts in > engineering / science will be helpful. A very simple example from the real world of engineering: We have a resistor bridge circuit for remote control of a motor-driven or solenoid-driven rotary switch. At the controlled end is the switch and one pole of the switch rotates over a series string of resistors of equal value. At the controlling end is a manual switch and the pole of the switch also rotates over a series string of resistors of equal value. The strings of resistors have the same voltage applied. The poles of the controlled and controlling switches are to be compared with a difference amplier. When the voltages are the same the drive to the controlled switch is turned off. When the voltages are different the drive to the controlled switch is energized. The controlled switch is on a portable device. The controller needs to work with any portable device thats connected. The resistors on both sides were selected from a random selection of resistors of nominally the same value and the distribution of actual resistor values is known or can be approximated - likely cases and worst cases. What is the Expected value of the proportional voltage on one side of the bridge for each setting of the switch? What is the probability distribution for the proportional voltage on one side of the bridge for each setting of the switch? Is there any situation in the worst case where the voltages will appear to be different even though the switches are in the same relative position? ... such that the drive to the controlled switch never goes off? What is the optimum threshold for the voltage difference for same position and different position. Hints: The probability distribution of two resistors in series is the convolution of the individual probability distributions. This can be calculated by multiplying the Fourier Transforms of each and inverse transforming the result. Or, one can convolve the two distributions. So, with a square distribution for one resistor, you would get a triangular distribution, twice as wide, for two resistors in series, etc. Its also a rather neat demonstration of the central limit theorem ... just keep increasing the number of resistors in series until you reach something very much like a truncated Gaussian distribution - which Im sure has a fancier name... Fred === Subject: Re: Expectation(X), characteristic function ... > Its also a rather neat demonstration of the central limit theorem ... just > keep increasing the number of resistors in series until you reach something > very much like a truncated Gaussian distribution - which Im sure has a > fancier name... Binomial distribution? Where the densities are the binomial coefcients (a row across Pascals triangle) of order N? Jerry -- Engineering is the art of making what you want from things you can get. [OS lash] [OSl ash] [OSl ash] [OSl ash] === Subject: Re: Expectation(X), characteristic function X-RFC2646: Original > ... > Its also a rather neat demonstration of the central limit theorem ... > just > keep increasing the number of resistors in series until you reach > something > very much like a truncated Gaussian distribution - which Im sure has a > fancier name... > Binomial distribution? Where the densities are the binomial coefcients > (a row across Pascals triangle) of order N? In discrete terms - yep. What about continuous? === Subject: Re: Expectation(X), characteristic function > Hi > Let f(x) be a prob density function. > Is there any signicance to the value f(E(x)), where > E(.) is expectation operator ? Not that I can think of. The mean of a random variable need not correspond to anything signicant, such as the peak (if any) of the pdf. > Why is the Fourier transform of f(x) ( known as characteristic function ) > signicant ? Note that Fourier transform indicates the frequency content > of f(x). Its closely related to the moment-generating function, which uniquely species all the moments, and the pdf itself. Its often easier to do calculations and proofs in terms of the MGF or characteristic function than in terms of the pdf. > A link explaining practical application of these concepts in > engineering / science will be helpful. I dont know about scientic applications. Its a powerful mathematical tool for doing difcult probability calculations. Here are a couple of links on characteristic functions: http://www2.sjsu.edu/faculty/watkins/charact.htm http://planetmath.org/encyclopedia/CharacteristicFunction2. html - Randy === Subject: Re: Expectation(X), characteristic function > Hi > Let f(x) be a prob density function. > Is there any signicance to the value f(E(x)), where > E(.) is expectation operator ? Im not sure, but I *think* you might nd an interesting property of the probability E(f(x)) P(x < E(f(x))) = integral f(y) dy. -inf I dont think the integral as above can be evaluated analytically in general, but you might be able to do so for particular fs and E(f(x))es. Rune === Subject: Re: Expectation(X), characteristic function > Hi > Let f(x) be a prob density function. > Is there any signicance to the value f(E(x)), where > E(.) is expectation operator ? > Why is the Fourier transform of f(x) ( known as characteristic function ) > signicant ? Note that Fourier transform indicates the frequency content > of f(x). > A link explaining practical application of these concepts in > engineering / science will be helpful. > shankar > Is there any signicance to the value f(E(x)), where > E(.) is expectation operator ? Let the lowercase Greek letter mu denote E(X) (which I will write as u in this post) (assuming you meant E(X) where X denotes the random variable under examination). I dont see any special signicance to the density value at u unless X possesses some extra special properties. For example, if X is a unimodal and (perfectly) symmetric distribution such as a Gaussian distribution, then (u, f(u)) will be where the density function peaks. > Why is the Fourier transform of f(x) ( known as characteristic function ) > signicant ? Note that Fourier transform indicates the frequency content > of f(x). As I recall, the density of the sum of two independent random variables is the convolution of their densities (assuming their densities exist). Furthermore the characteristic function of a sum of two independent random variables equals the product of their characteristic functions. These mathematical properties allow one to rephrase the Central Limit Theorem (CLT) in the following manner: The product of the characteristic functions of a nite number of density functions approaches the characteristic function of the Gaussian density as the number of density functions approaches innity (under the usual conditions stated for the central limit to hold such as independent and identically distributed random variables from the same population, etc, etc). Furthermore, as the characteristic function of a random variable is essentially a Fourier transform of its density, one can cast the CLT in the language of Fourier transforms. Shedar === Subject: Re: Expectation(X), characteristic function > Hi > Let f(x) be a prob density function. > Is there any signicance to the value f(E(x)), where > E(.) is expectation operator ? > Why is the Fourier transform of f(x) ( known as characteristic function ) > signicant ? Note that Fourier transform indicates the frequency > content > of f(x). > A link explaining practical application of these concepts in > engineering / science will be helpful. > shankar > Is there any signicance to the value f(E(x)), where > E(.) is expectation operator ? > Let the lowercase Greek letter mu denote E(X) (which I will write as u > in this post) (assuming you meant E(X) where X denotes the random variable > under examination). I dont see any special signicance to the density > value at u unless X possesses some extra special properties. For example, > if X is a unimodal and (perfectly) symmetric distribution such as a Gaussian > distribution, then (u, f(u)) will be where the density function peaks. BTW, from probability measure theory, Jensens inequality relates the behavior of f(E(X)) to that of E(f(X))depending on whether f is a convex map or a concave map. For example, see the following links for more information: http://planetmath.org/encyclopedia/JensensInequality.html http://www.probability.net/WEBjensen.pdf#jensen:inequality http://www.probability.net http://www.engineering.usu.edu/classes/ece/7680/lecture2/ node5.html Shedar === Subject: Re: Expectation(X), characteristic function >[...] >Let the lowercase Greek letter mu denote E(X) (which I will write as u >in this post) Priceless. ************************ David C. Ullrich === Subject: Re: Expectation(X), characteristic function > Hi > Let f(x) be a prob density function. > Is there any signicance to the value f(E(x)), where > E(.) is expectation operator ? Does not make much sense to me. Becasue for a random variable x, E(x) is a number. So, f(E(x)) is another number if you consider f(x) as a function of x. > Why is the Fourier transform of f(x) ( known as characteristic function ) > signicant ? Note that Fourier transform indicates the frequency content > of f(x). Because it is called the moment generating function. Looking at the fourier formula as a mathematical transformation than something that gives you the frequency content is much more helpful. Intutively, the higher order moments in someway tells you about the higher order harmonics of the signal in hand. But remember that frequencies as we speak of it, does not make much sense for random signals. Mathematically, the characteristic function of a random variable X is just phi(X) = E(exp(sX)) or E(exp(iwX)), and this function is helpful in calculating the moments of X, like, E(X) = d(phi(X))/dx, E(X^2) = d^2(phi(X))/dx^2 (not precise formulas, but can be derived easily). For literature, see Proakis - Digital Communications, Papoulis: Random variables and stochastic process, or any descent book on probability theory. I recommend Wozencraft and Jacobs, Principles of communications engineering, but it is out of print. === Subject: Re: Expectation(X), characteristic function > Let f(x) be a prob density function. >Is there any signicance to the value f(E(x)), where >E(.) is expectation operator ? I assume youre talking about E[X], the expected value of the random variable of which f is the density, rather than E[x] where x stands for a number: E[x] = x. The answer is no, in general. In fact, no particular value of f is signicant (a density is only dened up to sets of measure 0 anyway; change its value at a single point and its still just as good as a density for the same random variable). >Why is the Fourier transform of f(x) ( known as characteristic function ) >signicant ? Note that Fourier transform indicates the frequency content >of f(x). Because a lot of analysis can be done using Fourier transforms. One example: the characteristic function of a sum of independent random variables is the product of their characteristic functions. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Carom billiards as a stochastic process > Ive been working on masse and jump shots. Im better with the jump > shot, lifting the ball basically 12-16 inches away from the cue spot > and proceeding in a direct line. The masse (maa-SAY) shot is not the > push shot, basically shooting down into the ball to give it so much > side spin that it rolls in a curve instead of a straight line, where > English has more effect only on the rebound of the ball from object or > wall. Im not good with either of those shots. We are a little bit off track here... What I am trying to do requires very little knowledge of the game. I consider that with the balls in a given position, the shot has some known probability to be successful. Also if it is successful I assume I know the probability for the balls to end up in any position. Of course I know the probability to score from this nal position and the probability that the balls end up in any position, etc. What Id like to nd is the probability to score n points in a row. The fact that this is drawn from billiards does not matter this much. It is in fact a Markovian problem, but I need to simplify it because I do not know the probability to get an arbitraty nal position given an arbitrary initial position. Mathieu === Subject: Re: Carom billiards as a stochastic process ... > What I am trying to do requires very little knowledge of the game. I > consider that with the balls in a given position, the shot has some > known probability to be successful. Perhaps, although I doubt it. But even *if* there are probabilities involved, the probability depends on the player. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Carom billiards as a stochastic process > ... > What I am trying to do requires very little knowledge of the game. I > consider that with the balls in a given position, the shot has some > known probability to be successful. > Perhaps, although I doubt it. But even *if* there are probabilities > involved, the probability depends on the player. Of course it depends on the player. I am looking for a general formalism applicable to any player. Only parameters would depend on the player. For instance in the simple case of a Bernoulli process, the probability to score at least n points is p^n for anybody, what changes for different players is the value of p not the analytical form of the expression p^n. Mathieu === Subject: Re: Carom billiards as a stochastic process ... > Of course it depends on the player. I am looking for a general > formalism applicable to any player. Only parameters would depend on > the player. For instance in the simple case of a Bernoulli process, > the probability to score at least n points is p^n for anybody, what > changes for different players is the value of p not the analytical > form of the expression p^n. And I seriously doubt that. The probability of making a shot is highly dependent on the ability to make just such a shot. There is no overall ability to make all kinds of shots. Just these last few days I have seen some television coverage on the world-championship three cushion carom. When you look at that the most you can think is that misses are due to miscalculations. Giving the cue ball too much speed, giving a bit too much english, and whatever. But even then. Say you start with a game of libre. Experienced players have a high probability to make the aquit (for most that will be 1). What is the probability on their next shot? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Minimum Distance at Complex root of non-intersecting Curves At the point of {intersection/minimum distance}, {intersecting/stand-off} curves produce {real/complex} roots. What is the minimum distance dmin [dmin^2=(x2-x1)^2 + (y2-y1)^2] between stand-off curves f1(x,y)=0 and f2(x,y)=0 when its complex roots at minimum distance are ( a +/- b i)?. It is a function of b. For a simple example, the minimum distance between parabola y = x^2/4 + 4 and x-axis ( y = 0) at complex roots (x1,x2)=( 0 +/- 4 i) is 4 units. === Subject: Re: Minimum Distance at Complex root of non-intersecting Curves >At the point of {intersection/minimum distance}, >{intersecting/stand-off} curves produce {real/complex} roots. >What is the minimum distance dmin [dmin^2=(x2-x1)^2 + (y2-y1)^2] >between stand-off curves f1(x,y)=0 and f2(x,y)=0 when its complex >roots at minimum distance are ( a +/- b i)?. It is a function of b. >For a simple example, the minimum distance between parabola y = x^2/4 >+ 4 and x-axis ( y = 0) at complex roots (x1,x2)=( 0 +/- 4 i) is 4 >units. Interesting that you just happened to use those coefcients... In a case like this, where one curve is the x axis and the other y = f(x), and the roots youre talking about are the roots of f(x), what happens if you multiply f by some positive constant c? You multiply the distance by c, but you dont change the roots. So the distance is _not_ just a function of b. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Minimum Distance at Complex root of non-intersecting Curves >At the point of {intersection/minimum distance}, >{intersecting/stand-off} curves produce {real/complex} roots. >What is the minimum distance dmin [dmin^2=(x2-x1)^2 + (y2-y1)^2] >between stand-off curves f1(x,y)=0 and f2(x,y)=0 when its complex >roots at minimum distance are ( a +/- b i)?. It is a function of b. >For a simple example, the minimum distance between parabola y = x^2/4 >+ 4 and x-axis ( y = 0) at complex roots (x1,x2)=( 0 +/- 4 i) is 4 >units. > Interesting that you just happened to use those coefcients... > In a case like this, where one curve is the x axis and the other y = f(x), > and the roots youre talking about are the roots of f(x), what happens if you > multiply f by some positive constant c? You multiply the distance by c, but > you dont change the roots. So the distance is _not_ just a function > of b. Minimum distance d is not a function of b only !! I mentioned b,the imaginary part, as it inuences d more strongly and because such dependence is not fully intuitive. The imaginary part of a root of a quadratic equation y= a x^2 + b x + c when b^2-4ac < 0 increases as the separation/stand off between the x-axis and parabola sqrt(c - b^2/4a )/2a increases. Also, the curves touch for b=0 as a repeated root. In the general case, it is of course expected that d depends on b, a, f1, f2 and partial derivatives etc.,the result sought. I hope someone would identify not only d, but also points on each curve for which the minimimum distance occurs. Solution is obtained from variational calculus, wanted to know if a well known/ready result is available. === Subject: Re: Derivative of exponential function > Robin, > How about this - dene a function f(x) to be the same as its > derivative. Assume you can do a series expansion of f(x). > You dont have to assume this --- it follows. > Well, there is another function which satises the condition; its series > expansion is Sigma a_ix^i where all a_i are 0. :) Yes, and that equals its Macalurin serie too :-) -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Derivative of exponential function > Im in the process of home-schooling my son in calculus. Were about to > nish up on limits, so that we can start differentiation. Ive relearned > enough that Ive been able to derive, from the denition of derivative, > the derivatives of c, x, cx, f(x)+g(x), f(x)*g(x), f(g(x)), sin(x), > (f(x))^n, (f(x))^(-n), tan(x). A pretty solid beginning. But theres > one big hole: I cant prove that the derivative of e^x is e^x. > > I can see that if Lim(h->0) ((e^h-1)/h) == 1, then Im set. Numerically, > I can crank out values for ((e^h-1)/h) with h getting very small, and > see that they get really close to 1. But, thats not proof. > > I was going to base a proof on the Taylor series for e^x, but that > depends upon already knowing the nth derivatives of e^x, so that was > out. Then I was going to try using my knowledge of the behavior of > e^x, but upon examination, that was all based on knowing things about > its derivative. > > Is there some simple (or subtle) trick that Im overlooking? Is the > proof of this limit actually incredibly hard? > > I cant look in the book, because its with him (he lives with his > mother). Ive even tried typing ((e^h-1)/h) into Google, but that > just turned up a bunch of PDF les. Any help, or am I going to need > to do some serious hand-waving? >How about this - dene a function f(x) to be the same as its >derivative. > With the condition f(0) = 1 that is indeed a way to dene > the exponential, and its fairly elegant. But to make it an > actual valid denition you rst need to know something > about existence and uniqueness for differential equations. I think I was suggesting a slightly simpler approach. Use a power series expanision for f(x) (which remember we are dening is equal to its derivative). Then we can differentiate f(x) simply by knowing how to differentiate powers of x. Then we simply set the two sides equal, and if we insist that the constant term in the expansion is equal to 1, we get the power series expansion for f(x). We can then use this power series (which of course is equal to exp(x)) to show all the usual properties of exp(x). The way I am proposing doing this is not by explicitly solving the ordinary differential equation (at least not in the conventional way by using an integrating factor). >Assume you can do a series expansion of f(x). Explicitly >do the series expansion for f(x) and d/dx (f(x)) and set them equal to >each other (for all x). This should dene the coefcients in the >series expansion. Now call this function f(x) the exponential >function, exp(x). >I have seen this done before and thought it was quite neat. I dont >know if it can mathematically justied, as I am only a physicist, but >it looks quite neat to me, and you can avoid all the handwaving. >Ian Taylor > ************************ > David C. Ullrich === Subject: Re: Derivative of exponential function > Im in the process of home-schooling my son in calculus. Were about to > nish up on limits, so that we can start differentiation. Ive relearned > enough that Ive been able to derive, from the denition of derivative, > the derivatives of c, x, cx, f(x)+g(x), f(x)*g(x), f(g(x)), sin(x), > (f(x))^n, (f(x))^(-n), tan(x). A pretty solid beginning. But theres > one big hole: I cant prove that the derivative of e^x is e^x. > > I can see that if Lim(h->0) ((e^h-1)/h) == 1, then Im set. Numerically, > I can crank out values for ((e^h-1)/h) with h getting very small, and > see that they get really close to 1. But, thats not proof. > > I was going to base a proof on the Taylor series for e^x, but that > depends upon already knowing the nth derivatives of e^x, so that was > out. Then I was going to try using my knowledge of the behavior of > e^x, but upon examination, that was all based on knowing things about > its derivative. > > Is there some simple (or subtle) trick that Im overlooking? Is the > proof of this limit actually incredibly hard? > > I cant look in the book, because its with him (he lives with his > mother). Ive even tried typing ((e^h-1)/h) into Google, but that > just turned up a bunch of PDF les. Any help, or am I going to need > to do some serious hand-waving? > >How about this - dene a function f(x) to be the same as its >derivative. > With the condition f(0) = 1 that is indeed a way to dene > the exponential, and its fairly elegant. But to make it an > actual valid denition you rst need to know something > about existence and uniqueness for differential equations. >I think I was suggesting a slightly simpler approach. Use a power >series expanision for f(x) (which remember we are dening is equal to >its derivative). Then we can differentiate f(x) simply by knowing how >to differentiate powers of x. Then we simply set the two sides equal, >and if we insist that the constant term in the expansion is equal to >1, we get the power series expansion for f(x). We can then use this >power series (which of course is equal to exp(x)) to show all the >usual properties of exp(x). Well ok, you can do that - this amounts to just starting with the power series, another standard approach. >The way I am proposing doing this is not by explicitly solving the >ordinary differential equation (at least not in the conventional way >by using an integrating factor). >Assume you can do a series expansion of f(x). Explicitly >do the series expansion for f(x) and d/dx (f(x)) and set them equal to >each other (for all x). This should dene the coefcients in the >series expansion. Now call this function f(x) the exponential >function, exp(x). > >I have seen this done before and thought it was quite neat. I dont >know if it can mathematically justied, as I am only a physicist, but >it looks quite neat to me, and you can avoid all the handwaving. > >Ian Taylor > ************************ > David C. Ullrich ************************ David C. Ullrich === Subject: Re: Derivative of exponential function by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i98HLmM29405; >A solution to your problem given two standard denitions of e^x have >already been given. >Here is a further possibility: if e^x is dened to be >lim (1+x/n)^n, >as it sometimes is in school (in connection with cumulative interest, >for example), then you can argue as follows: the derivative of each >term in the limit is >(1+x/n)^{n-1}, >and >lim (1+x/n)^{n-1} = lim 1/(1+x/n) * lim (1+x/n)^n = 1 * e^x. >Since the convergence of the derivatives is, in fact, (locally) >uniform in x, the original function is differentiable and the >derivative of the limit is the limit of the derivatives. Of course you >wouldnt really try to say the latter to a high school student, but >this might give a reasonable justication. >Hope this helps, >Lasse >--- >(lasse@remove.for.spam.maths.warwick.ac.uk) This denition is useful in other ways, too -- see the intuitive explanation given by Conway and Guy in The Book of Numbers on why exp(pi*i) = -1. Similar arguments show exp(it) = cos(t) + i*sin(t). Ruminations similar to that explanation yield a geometric way of visualizing e: consider the system of lines in the plane which pass through the origin O (radial lines), and starting from any point P away from O, draw a curve through P which meets all the radial lines in a 45 degree angle. (This curve is a logarithmic spiral.) Mark the point Q where the angle POQ is one radian. Then the distances d(O, P) and d(O, Q) differ by a factor of e. Perhaps this example is best contemplated by examining conch shells. Todd Trimble === Subject: Baseball skill vs. raw chance Suppose after this season is over you had each of the Major League Baseball clubs choose a representative. Or appoint your own unofcial ones. Each real game would be duplicated as to who played who how many times. But each game would be decided by random number gereration, cutting cards, or the like rather than actual baseball or anything else involving skill. And be absolutely honest and involving only random chance. How much difference would you expect between the two seasons? And if you prefer have it cricket teams or soccer teams or footie teams or whathaveyou:) But have 30 teams play a 162 game season where each plays each as often as they would in the MLB season. === Subject: Re: Baseball skill vs. raw chance > Suppose after this season is over you had each of the Major League Baseball > clubs choose a representative. Or appoint your own unofcial ones. Each real > game would be duplicated as to who played who how many times. But each game > would be decided by random number gereration, cutting cards, or the like rather > than actual baseball or anything else involving skill. And be absolutely honest > and involving only random chance. How much difference would you expect between > the two seasons? And if you prefer have it cricket teams or soccer teams or > footie teams or whathaveyou:) But have 30 teams play a 162 game season > where each plays each as often as they would in the MLB season. The other 4 are simulated based on coin toss outcomes. The actual best and worst stand out against the simulations, but the whole distribution of the actual is spread out more. The actual is denitley distinct from chance outcomes, but the difference is not really so dramatic. 68 94 66 95 51 111 67 95 65 96 71 90 67 95 58 104 71 91 71 91 72 90 69 92 63 99 72 89 72 90 73 89 69 93 67 94 72 90 73 89 74 88 70 92 67 94 72 90 74 88 75 87 73 89 67 95 74 88 75 87 75 87 74 88 68 94 75 86 77 85 76 86 76 85 70 91 76 86 77 85 76 86 76 86 71 91 76 86 78 83 77 85 79 82 72 89 77 84 78 84 78 84 79 83 72 90 77 85 79 82 78 84 80 82 76 86 79 83 79 83 79 83 80 82 78 84 79 83 79 83 80 82 82 80 80 82 79 83 79 83 80 82 82 80 83 79 80 82 80 82 80 82 82 80 83 79 81 80 81 81 81 81 83 79 86 76 81 81 81 81 81 81 84 78 87 75 83 79 82 80 82 80 84 78 89 73 83 79 83 79 83 79 84 78 89 73 85 77 83 79 84 78 85 77 91 71 85 77 83 79 86 75 86 76 91 71 85 77 85 77 87 75 86 76 92 70 86 76 85 77 87 75 87 75 92 70 88 74 85 77 88 74 87 75 92 70 88 74 86 76 89 73 88 74 93 69 89 73 87 75 89 73 90 72 96 66 91 71 89 72 89 73 91 71 98 64 91 71 90 72 93 68 93 69 101 61 92 70 94 68 97 64 96 66 105 57 94 68 98 64 === Subject: Re: Baseball skill vs. raw chance Heres a comparison where the teams have a weighted likelihood of winning from 0.44 through 0.56, in even steps. I opted for computational convenience over theoretical endorsement. I didnt actually do games, per se. I ran each teams record as statistically independent, but discarded nonzero-sum tables. This takes on the order of 100 retries, but gives a w/l table which satisies the obvious checks, including the number of teams playing 161 and 162 games. In this case its a lot harder to pick out the actual table from the simulations. This is really an ANOVA problem of course, but the exact model is a little tricky to dene because of the nonindependence of each teams record. 51 111 59 103 55 107 59 102 54 108 58 104 61 101 59 103 60 101 61 101 63 99 62 99 60 102 61 101 62 100 67 94 66 96 63 99 62 100 63 99 67 94 68 94 66 95 65 97 64 98 67 95 69 92 66 95 67 95 69 92 68 94 69 93 66 96 67 95 70 92 70 91 70 92 69 93 68 94 71 91 71 91 70 92 70 92 70 92 75 87 72 89 72 90 73 89 73 89 76 86 72 90 74 88 74 88 74 88 78 84 76 86 74 88 77 85 76 86 79 83 78 84 75 87 77 85 76 86 79 83 80 82 79 83 77 85 77 85 80 81 83 79 79 83 80 82 77 85 80 82 83 79 80 82 82 80 78 84 81 80 86 76 84 78 83 78 78 84 82 80 87 75 85 77 84 78 85 77 83 79 89 73 86 76 87 75 86 75 85 77 89 73 86 76 88 74 86 76 87 75 91 71 91 71 88 74 91 71 87 75 91 71 92 70 89 72 91 71 87 75 92 70 93 69 90 72 93 69 88 74 92 70 94 67 97 65 94 68 92 70 92 70 94 68 98 64 99 62 92 70 93 69 95 67 100 62 99 63 94 68 96 66 97 65 101 61 101 61 95 67 98 64 100 61 102 60 103 59 102 60 101 61 100 62 103 59 105 57 103 58 105 57 104 58 104 58 107 55 109 53 === Subject: Re: Baseball skill vs. raw chance >Suppose after this season is over you had each of the Major League Baseball >clubs choose a representative. Or appoint your own unofcial ones. Each real >game would be duplicated as to who played who how many times. But each game >would be decided by random number gereration, cutting cards, or the like rather >than actual baseball or anything else involving skill. And be absolutely honest >and involving only random chance. How much difference would you expect between >the two seasons? And if you prefer have it cricket teams or soccer teams or >footie teams or whathaveyou:) But have 30 teams play a 162 game season >where each plays each as often as they would in the MLB season. If every game were random, each team would have an 0.73% chance of winning 65 games or less, and equal 0.73% chance of winning 97 games or more (losing 65 games or less). The other 98.54% would be in the middle. So if there are consistently 1 or more teams which do this (not necessarily the same ones every year), it would indicate that baseball is a game of skill, or at least that games are not independent random events. 60 games or less --> 0.06% 56 games or less --> 0.005% --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Re: Baseball skill vs. raw chance === >Subject: Re: Baseball skill vs. raw chance >Message-id: >Suppose after this season is over you had each of the Major League Baseball >clubs choose a representative. Or appoint your own unofcial ones. Each >real >game would be duplicated as to who played who how many times. But each game >would be decided by random number gereration, cutting cards, or the like >rather >than actual baseball or anything else involving skill. And be absolutely >honest >and involving only random chance. How much difference would you expect >between >the two seasons? And if you prefer have it cricket teams or soccer teams >footie teams or whathaveyou:) But have 30 teams play a 162 game season >where each plays each as often as they would in the MLB season. >If every game were random, each team would have an 0.73% chance of winning >65 games or less, and equal 0.73% chance of winning 97 games or more (losing >65 games or less). The other 98.54% would be in the middle. The regular season just ended three teams won over 97 games and three teams won less than 65 games. Nobody won exactly 97 or 65. I hope you can acess this URL http://aolsvc.cnnsi.sports.aol.com/baseball/mlb/standings/ So if there >are consistently 1 or more teams which do this (not necessarily the same >ones every year), it would indicate that baseball is a game of skill, or at >least that games are not independent random events. >60 games or less --> 0.06% >56 games or less --> 0.005% >--Keith Lewis klewis {at} mitre.org >The above may not (yet) represent the opinions of my employer. === Subject: Re: Baseball skill vs. raw chance >The regular season just ended three teams won over 97 games and three teams >won less than 65 games. Nobody won exactly 97 or 65. > I hope you can acess this URL >http://aolsvc.cnnsi.sports.aol.com/baseball/mlb/standings/ Thats pretty solid evidence that the results were more than just chance. --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Re: Baseball skill vs. raw chance > Suppose after this season is over you had each of the Major League Baseball > clubs choose a representative. Or appoint your own unofcial ones. Each real > game would be duplicated as to who played who how many times. But each game > would be decided by random number gereration, cutting cards, or the like rather > than actual baseball or anything else involving skill. And be absolutely honest > and involving only random chance. How much difference would you expect between > the two seasons? And if you prefer have it cricket teams or soccer teams or > footie teams or whathaveyou:) But have 30 teams play a 162 game season > where each plays each as often as they would in the MLB season. (1) Are you a baseball coach? (2) Is a tie allowed between two teams in a game? Shedar === Subject: Re: Baseball skill vs. raw chance X-RFC2646: Original > Suppose after this season is over you had each of the Major League > Baseball > clubs choose a representative. Or appoint your own unofcial ones. Each > real > game would be duplicated as to who played who how many times. But each > game > would be decided by random number gereration, cutting cards, or the like > rather > than actual baseball or anything else involving skill. And be absolutely > honest > and involving only random chance. How much difference would you expect > between > the two seasons? And if you prefer have it cricket teams or soccer > teams or > footie teams or whathaveyou:) But have 30 teams play a 162 game > season > where each plays each as often as they would in the MLB season. > (2) Is a tie allowed between two teams in a game? Only at the all-star break === Subject: Re: Baseball skill vs. raw chance > Suppose after this season is over you had each of the Major League > Baseball > clubs choose a representative. Or appoint your own unofcial ones. Each > real > game would be duplicated as to who played who how many times. But each > game > would be decided by random number gereration, cutting cards, or the like > rather > than actual baseball or anything else involving skill. And be absolutely > honest > and involving only random chance. How much difference would you expect > between > the two seasons? And if you prefer have it cricket teams or soccer > teams or > footie teams or whathaveyou:) But have 30 teams play a 162 game > season > where each plays each as often as they would in the MLB season. > > (2) Is a tie allowed between two teams in a game? > Only at the all-star break Pardon my ignorance. I know next-to-nothing about baseball. 1. How many games does each team play in a season of 162 games? 2. Does a pair of opposing teams meet exactly once or more than once in a given season? Shedar === Subject: Re: Baseball skill vs. raw chance === >Subject: Re: Baseball skill vs. raw chance >Message-id: <2Yqad.709290$gE.608120@pd7tw3no> > Suppose after this season is over you had each of the Major League >Baseball > clubs choose a representative. Or appoint your own unofcial ones. Each >real > game would be duplicated as to who played who how many times. But each >game > would be decided by random number gereration, cutting cards, or the like >rather > than actual baseball or anything else involving skill. And be absolutely >honest > and involving only random chance. How much difference would you expect >between > the two seasons? And if you prefer have it cricket teams or soccer >teams or > footie teams or whathaveyou:) But have 30 teams play a 162 game >season > where each plays each as often as they would in the MLB season. >(1) Are you a baseball coach? No, I am just curious. >(2) Is a tie allowed between two teams in a game? Only on extemely rare occasions for unusual events and it always leads to ill feelings (the spectators feel extremely cheated, etc.) and controversy over whether the occurence was unusual enough >Shedar === Subject: Representations of afne type A Hello. Consider the cyclic quiver Q of type tilde{A_{n-1}} with orientation i->i-1 for all vertices i. Let k be a eld. Let l be a natural number, and let i be a vertex of Q. Denote by k[i] the simple representation of Q at the vertex i. It seems to be a standard fact that there is (up to isomorphism) a unique indecomposable representation of Q of length l with top k[i]. Unfortunately I have not been able to nd a proof anywhere ... any hints? -- Michael Knudsen === Subject: Re: Representations of afne type A > Hello. > Consider the cyclic quiver Q of type tilde{A_{n-1}} with orientation > i->i-1 for all vertices i. > Let k be a eld. Let l be a natural number, and let i be a vertex > of Q. Denote by k[i] the simple representation of Q at the vertex i. It > seems to be a standard fact that there is (up to isomorphism) a unique > indecomposable representation of Q of length l with top k[i]. Youll have to remind me what length is. If it is what I think it is, this goes wrong (badly) when n = 2 :-) -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Representations of afne type A > Youll have to remind me what length is. If it is what I think it is, > this goes wrong (badly) when n = 2 :-) It is the length of a composition series with simple factors. Do you know about quiver representations? If this is so, I can elaborate on how the indecomposable modules are constructed. Perhaps this will help. -- Michael Knudsen === Subject: Re: Representations of afne type A > Youll have to remind me what length is. If it is what I think it is, > this goes wrong (badly) when n = 2 :-) > It is the length of a composition series with simple factors. When n = 2, and you are talking about A_1, then the path algebra is the ground eld, and its only indecomposable representation is the 1-dimensional one. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Representations of afne type A > When n = 2, and you are talking about A_1, then the path algebra > is the ground eld, and its only indecomposable representation > is the 1-dimensional one. the cyclic graph with n vertices. -- Michael Knudsen === Subject: Re: Representations of afne type A > When n = 2, and you are talking about A_1, then the path algebra > is the ground eld, and its only indecomposable representation > is the 1-dimensional one. > the cyclic graph with n vertices. Aha. That makes more sense. Now what about the edge-orientations? Were they cyclic? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Representations of afne type A > Aha. That makes more sense. Now what about the edge-orientations? > Were they cyclic? If you would like to see a place where you can see that it is well known, you can as an example check out page 4 of http://www.mathematik.uni-bielefeld.de/~fdlist/PAPERS/ dengdu.ps -- Michael Knudsen === Subject: Re: Representations of afne type A > Aha. That makes more sense. Now what about the edge-orientations? > Were they cyclic? Right. Here is a length (and dimension) l indecompasable. Its spanned by vectors e_1, ..., e_l with e_j being at vertex j modulo n. (So for say l = n+1 the space at vertex 1 is spanned by e_1 and e_{n+1} while for at the other vertex j it is spanned by e_j). Let the arrows take e_j to e_{j+1}, except e_l goes to zero. This has top the simple at vertex 1. As it is generated by any vector not in its radical it is indecomposable. Why are these the only ones? I cant quite see this right now, but the obvious attack is as follows. Take an indecomposable with top S_1. Let v be a vector in V_1 but not in the radical. Show that the cyclic module it generates is a direct summand (I hope this works :-)). -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Representations of afne type A > As it is generated by any vector not in its radical > it is indecomposable. Is that easy to see? I am not that familiar with the notion of the radical of a module... -- Michael Knudsen === Subject: Re: Representations of afne type A > As it is generated by any vector not in its radical > it is indecomposable. > Is that easy to see? I am not that familiar with the notion of the > radical of a module... I guess that the argument is as follows(?): If M = direct sum of M_1 and M_2, then M_1 or M_2 must contain something which is not in rad(M). Assume that x is in M_1 and not in rad(M). Since M_1 is a submodule it must contain the submodule of M generated by x, hence M_1=M. Thus, M_2=0, so M is indecomposable. Can this be said in a nicer way? -- Michael Knudsen === Subject: Re: Representations of afne type A > As it is generated by any vector not in its radical > it is indecomposable. > Is that easy to see? I am not that familiar with the notion of the > radical of a module... Erm. the radical of M is the largest submodule with M/rad(M) semisimple (M/rad(M) is the top of M). -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Representations of afne type A > Erm. the radical of M is the largest submodule with M/rad(M) > semisimple (M/rad(M) is the top of M). Yes, I know the denition of the radical, but I still cannot see what it has to do with *indecomposable* modules. -- Michael Knudsen === Subject: Re: Representations of afne type A > Right. Here is a length (and dimension) l indecompasable. Its > spanned by vectors e_1, ..., e_l with e_j being at vertex j modulo n. > (So for say l = n+1 the space at vertex 1 is spanned by e_1 and e_{n+1} > while for at the other vertex j it is spanned by e_j). Let the arrows > take e_j to e_{j+1}, except e_l goes to zero. This has top the simple > at vertex 1. As it is generated by any vector not in its radical > it is indecomposable. > Why are these the only ones? I cant quite see this right now, but the > obvious attack is as follows. Take an indecomposable with top S_1. > Let v be a vector in V_1 but not in the radical. Show that the cyclic > module it generates is a direct summand (I hope this works :-)). -- Michael Knudsen === Subject: Re: Representations of afne type A > Aha. That makes more sense. Now what about the edge-orientations? > Were they cyclic? > If you would like to see a place where you can see that it is well > known, you can as an example check out page 4 of > http://www.mathematik.uni-bielefeld.de/~fdlist/PAPERS/ dengdu.ps The authors make a stipulation which you omitted; that these representations be nilpotent, so that the endomorphism induced by a cycles worth of composition be nilpotent. (I had noticed that there were plent of isomorphism classes of indecomposable quivers with a 1-dimensional space at each vertex.) -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Representations of afne type A > The authors make a stipulation which you omitted; that these > representations be nilpotent, so that the endomorphism induced > by a cycles worth of composition be nilpotent. Sorry! I forgot that when typing my question. Do you nd it trivial now when you know that the representations must be nilpotent (I dont)? -- Michael === Subject: Re: Representations of afne type A > Sorry! I forgot that when typing my question. Do you nd it trivial now > when you know that the representations must be nilpotent (I dont)? Nevermind. I had not seen your latest post. -- Michael Knudsen === Subject: Re: Representations of afne type A > Aha. That makes more sense. Now what about the edge-orientations? > Were they cyclic? Yes. -- Michael Knudsen === Subject: Context-free grammar question Lately Ive been thinking more about an old problem: How do I write a context-free grammar that accepts any string of a and b so that there are as many as as bs? I came up with this solution: S ::= (empty) | BaS | SaB | AbS | SbA A ::= SaS B ::= SbS The idea is that I keep track of the balance between as and bs. Each S means theyre balanced, each A means theres one a less, and each B means theres one b less. Would this work? -- /-- Joona Palaste (palaste@cc.helsinki.) ------------- Finland -------- -------------------------------------------------------- rules! --------/ To doo bee doo bee doo. - Frank Sinatra === Subject: Re: Context-free grammar question * Joona I. Palaste > Lately Ive been thinking more about an old problem: How do I write a > context-free grammar that accepts any string of a and b so that > there are as many as as bs? > I came up with this solution: > S ::= (empty) | BaS | SaB | AbS | SbA > A ::= SaS > B ::= SbS > The idea is that I keep track of the balance between as and bs. Each > S means theyre balanced, each A means theres one a less, and each B > means theres one b less. Would this work? Looks right and should be easy to prove. But isnt the following simpler? S ::= empty | S a S b S | S b S a S -- Jon Haugsand Dept. of Informatics, Univ. of Oslo, Norway, mailto:jonhaug@i.uio.no http://www.i.uio.no/~jonhaug/, Phone: +47 22 85 24 92 === Subject: Re: Context-free grammar question > * Joona I. Palaste >Lately Ive been thinking more about an old problem: How do I write a >context-free grammar that accepts any string of a and b so that >there are as many as as bs? >I came up with this solution: >S ::= (empty) | BaS | SaB | AbS | SbA >A ::= SaS >B ::= SbS >The idea is that I keep track of the balance between as and bs. Each >S means theyre balanced, each A means theres one a less, and each B >means theres one b less. Would this work? > Looks right and should be easy to prove. But isnt the following > simpler? > S ::= empty | S a S b S | S b S a S Or even simpler... S ::= empty | a S b S | b S a S What about an unambiguous grammar? -- Mitch Harris (remove q to reply) === Subject: Re: Context-free grammar question * Mitch Harris > Or even simpler... > S ::= empty | a S b S | b S a S Ah yes. > What about an unambiguous grammar? Long time since I studied this. But it looks impossible. -- Jon Haugsand Dept. of Informatics, Univ. of Oslo, Norway, mailto:jonhaug@i.uio.no http://www.i.uio.no/~jonhaug/, Phone: +47 22 85 24 92 === New job listings at http://jobs.phds.org - Jobs for PhDs List your job at no cost! http://jobs.phds.org/jobs/post * Entry-Level Quantitative Analyst: Discovery Europe, London, UK. Our client, a major global investment bank has an excellent opportunity in their derivatives research group for a quantitative analyst. The role will involve the research and development of models for option trading, using state of the art mathematical and... * Sr. Front Ofce Quantitative Developer: The Execu|Search Group, New York, NY. We are building a new team that will focus on delivering Excel-based pricing tools/models to the trading desk quickly. One of our main clients will be a brand new trading desk that will focus on some new strategies. 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Areas of expertise could include molecular dynamics, simulations of bimolecular systems and protein structures, binding free energies and... === Subject: Clique Graph at mathworld X-RFC2646: Original Hi I was wondering about this explanation of a clique graph at mathworld. http://mathworld.wolfram.com/CliqueGraph.html It shows that the K4 is the clique graph of G. I cannot see why. As I read the description of a clique at mathworld then it is a complete subgraph of a graph. I can see 4 complete subgraphs of G (all the triangles). Then mathworld says the clique graph is the intersection of all these clique graphs, but that would yield a graph of three nodes (the triangle in the center of G) and not of 4 nodes? Can anyone explain why K4 is the clique graph of G as shown on the above mathworld link? Mark === Subject: Re: Clique Graph at mathworld > Hi > I was wondering about this explanation of a clique graph at mathworld. > http://mathworld.wolfram.com/CliqueGraph.html If you follow the link of graph intersection, you will nd the answer to your question: the clique graph of $G$ is the graph whose vertices are the (maximal) cliques of $G$, two of which are connected if and only if they intersect. Hope this helps, Lasse --- (lasse@remove.for.spam.maths.warwick.ac.uk) === Subject: Re: Clique Graph at mathworld > Hi > I was wondering about this explanation of a clique graph at mathworld. > http://mathworld.wolfram.com/CliqueGraph.html > It shows that the K4 is the clique graph of G. I cannot see why. As I read > the description of a clique at mathworld then it is a complete subgraph of a > graph. I can see 4 complete subgraphs of G (all the triangles). Then > mathworld says the clique graph is the intersection of all these clique > graphs, No, it says the clique graph is the graph intersection of the family of cliques of G. Per http://mathworld.wolfram.com/GraphIntersection.html the graph intersection of a family F will be a graph with as many nodes as there are members of F. There are 4 cliques in G, etc. > but that would yield a graph of three nodes (the triangle in the > center of G) and not of 4 nodes? Can anyone explain why K4 is the clique > graph of G as shown on the above mathworld link? -jiw === Subject: Re: Clique Graph at mathworld X-RFC2646: Original > Hi > I was wondering about this explanation of a clique graph at mathworld. > http://mathworld.wolfram.com/CliqueGraph.html > It shows that the K4 is the clique graph of G. I cannot see why. As I > read > the description of a clique at mathworld then it is a complete subgraph > of a > graph. I can see 4 complete subgraphs of G (all the triangles). Then > mathworld says the clique graph is the intersection of all these clique > graphs, > No, it says the clique graph is the graph intersection of the family of > cliques of G. Per http://mathworld.wolfram.com/GraphIntersection.html > the graph intersection of a family F will be a graph with as many nodes > as there are members of F. There are 4 cliques in G, etc. > but that would yield a graph of three nodes (the triangle in the > center of G) and not of 4 nodes? Can anyone explain why K4 is the clique > graph of G as shown on the above mathworld link? > -jiw Hi James Mark === Subject: A simple question on algebraic extensions Let K be a base eld and let L be its extension formed by adding an element alpha not in K. Let p(alpha) = 0 where p(x) is a polynomial. The eld L is of the form L = {f(alpha)/g(alpha), f,g in K[x], g(alpha) neq 0 } This extension eld L forms a vector space over K, with { 1, alpha, alpha^2, ..., alpha^(n-1) } as the basis. The question is how can any rational function f(x)/g(x) be written as a polynomial in x with coefcients from K just because there is a polynomial p(x) = 0? Also, nothing is said about p(x), it need not be prime, any polynomial would do. When i do some examples, like taking p(x) = x^2 - 2, and forming Q(sqrt(2)) which is an extension of Q, i have 1/sqrt(2) = (1/2 )*sqrt(2), and 1/(1 + sqrt(2) ) = ( 1 - sqrt(2) ) / (1 - sqrt(2))(1 + sqrt(2)) = sqrt(2) - 1 So, the above statement is validated. But how to validate the above statement in general. Prasanna. === Subject: Re: A simple question on algebraic extensions > Let K be a base eld and let L be its extension formed by adding an > element > alpha not in K. Let p(alpha) = 0 where p(x) is a polynomial. The > eld L is of the form > L = {f(alpha)/g(alpha), f,g in K[x], g(alpha) neq 0 } > This extension eld L forms a vector space over K, with > { 1, alpha, alpha^2, ..., alpha^(n-1) } as the basis. The question > is how can any rational function f(x)/g(x) be written as a polynomial > in x with coefcients from K just because there is a polynomial p(x) > = 0? Also, nothing is said about p(x), it need not be prime, any > polynomial would do. No, not any polynomial would do; youre assuming that p(alpha) = 0. Now, take p = minimal polynomial of alpha. Then not only p(alpha) = 0, but, furthermore, p is irreducible. Take some element of L of the form f(alpha)/g(alpha), where f and g are in K[x] and g(alpha) is not 0. Then there are polynomials a(x) and b(x) in K[x] such that a(x).g(x) + b(x).p(x) is a gcd of g(x) and p(x). But p(x) is irreducible and g(x) cannot be a multiple of p(x), since g(alpha) is not 0. Therefore, you can take a(x) and b(x) such that a(x).g(x) + b(x).p(x) = 1; so, a(alpha).g(alpha) = 1. And it follows that f(alpha)/g(alpha) = f(alpha).a(alpha), which is a linear combination of 1, alpha, alpha^2, ... , alpha^{n - 1}, where n = degree of p. Jose Carlos Santos === Subject: factoring GCF Given something like, determine the GCF of each pair of terms: x(x + 3) and 2(x + 3) Why is the greatest common factor x + 3? Why is it a single factor? Why are two mutually exclusive terms considered a single factor? x(x + 3) is x^2 + 3x and 2(x + 3) is 2x + 6 from the former, the greatest common factor is x and in the latter, the GCF is 2? Other examples: x(x - 2) and x - 2 GCF: x - 2 Another: 2(x + y) and 3x(x + y) And guess what the GCF is? x + y! No thought whatsoever. Im just recognizing the pattern and Im not satisied with this or the lack of explanation offered by my book. They just seem to present it and for you to recognize the pattern and use that to solve the problem. But I am not satisied with this. I am a pedant. And I want to know the specics behind why the GCF from any distributive form is always two terms considered as a single factor. If I see: 6x + 18 then I know the GCF of each term is 3. and that I factor out the GCF with the distributive property: 6 * x + 6 * 3 = 6(x + 3) But with something like this: x(5x - 2) + 7(5x - 2) the GCF is (5x - 2). However, watch this: x(5x - 2) = 5x^2 - 2x. GCF = x and 7(5x - 2) = 35x - 14. GCF = 7 Hm. Odd. Two different GCFs, x and 7. so how do they gure (5x - 2) is the GCF? === Subject: Re: factoring GCF > determine the GCF of each pair of terms: > x(x + 3) and 2(x + 3) > Why is the greatest common factor x + 3? Because x + 3 divides both of them and anything else that divides both of them will divide x + 3. > x(x + 3) is x^2 + 3x and 2(x + 3) is 2x + 6 > from the former, the greatest common factor > is x and in the latter, the GCF is 2? The gcf is dened for two elements and not for just one. However gcf of x(x + 3) and x(x + 3) is x(x + 3). > If I see: 6x + 18 then I know the GCF > of each term is 3. and that I factor out > the GCF with the distributive property: > 6 * x + 6 * 3 = 6(x + 3) There no such thing as a gcf for a single element. However the gcf of something and the same something is the same old something. > But with something like this: > x(5x - 2) + 7(5x - 2) the GCF is > (5x - 2). However, watch this: > x(5x - 2) = 5x^2 - 2x. GCF = x > and 7(5x - 2) = 35x - 14. GCF = 7 7 doesnt divide (x + 7)(5x + 2) so it cant be a factor. > Hm. Odd. Two different GCFs, x and 7. No, the oddness is in the mind of the beholder. > so how do they gure (5x - 2) is the GCF? The gcf of (x + 7)(5x + 2) and (x + 7)(5x + 2) is (x + 7)(5x + 2). === Subject: Re: factoring GCF > determine the GCF of each pair of terms: > x(x + 3) and 2(x + 3) > Why is the greatest common factor x + 3? > Because x + 3 divides both of them and anything else that divides both of > them will divide x + 3. This appears to be a rather badly constructed problem. If x=-2, the GCF is 2 while x+3 is 1. If x=2, the GCF is 10 while x+3 is 5. x+3 is not always the GCF. === Subject: Re: factoring GCF > determine the GCF of each pair of terms: > x(x + 3) and 2(x + 3) > > Why is the greatest common factor x + 3? > Because x + 3 divides both of them and anything else that divides both of > them will divide x + 3. > This appears to be a rather badly constructed problem. > If x=-2, the GCF is 2 while x+3 is 1. > If x=2, the GCF is 10 while x+3 is 5. > x+3 is not always the GCF. Were not talking about gcfs of integers, were talking about gcfs of polynomials. Evaluations of polynomials have little to do to determine the gcd of polynomials. The gcfs of integers and polynomials are computed with the Euclidean algorythm, which is a sequence of division algorythms for all a and d (dividing a/d) some q and some r with 0 <= r < d, a = qd + r for integers in Z and for polynomials in Z[x] for all a(x) and d(x) (dividing a(x)/d(x)) some q(x) and some r(x) with deg r(x) < deg d(x) or r = 0 and a(x) = q(x) d(x) + r(x) === Subject: Natural Densities are Probabilities Related threads: < Re: How to do magic with innity < Re: Zenkins paper on Cantor In 1933, Kolgomorov gave the rst widely accepted axiomatization of Probability Theory. An axiomatization which has the following strange consequence, when embedded in mainstream mathematics. By intuition, we expect that the probability that a randomly chosen integer is even equals 1/2, but then we have the following theorem. There exists no probability law on N such that P(aN) = 1/a (a in N) where aN := { n in N : n = 0 mod a } With other words, quite contrary to intuition: The probability that a natural is divisible by a is NOT 1/a . A proof of this can be found here (I am not the author): Because Probability Theory - as it has been embedded in mainstream mathematics - gives this unwanted result, the following denition has been widely accepted instead (skipping densities in general): The natural density dA of a sequence A of naturals is: dA = lim (x->innity) 1/x # { n <= x : n in A) Therefore the natural density of the even numbers is, exactly as you would expect: 1/2 . I have major objections against this new denition, simply because its NOT NEW at all. For example, if you ip a coin, it will come up either heads or tails. Suppose we perform x of these tosses. Dene the set A as all tosses where the coin comes up heads. The natural density of the heads is then: dA = lim (x->innity) [ 1/x # { n <= x : n in A) ] Everybody knows that the part between square brackets is simply called a CHANCE and that these chances converge to a Probability in the limit. Thus natural densities are simply a re-formulation of Probabilities. And there is no need, at all, for a new concept. Consequently, there must be a aw in the abovementioned proof too. Because, otherwise, it would be impossible to even ip a coin. Han de Bruijn === Subject: Re: Natural Densities are Probabilities It is interesting to come up with two sets A,B (subsets of N={1,2,3,4,...} ) such that both A and B have natural density, but their intersection does not. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: Natural Densities are Probabilities > It is interesting to come up with two sets A,B (subsets of > N={1,2,3,4,...} ) such that both A and B have natural density, but > their intersection does not. Let C be a subset of N without natural density. Let A = {2n: n in N} and B = {2n: n in C} u {2n-1: n notin C}. Then A and B have density 1/2, yet A intersect B is trying its best to have denisty half of that of C :-) -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Natural Densities are Probabilities on porbabilistic number theory. They are valuable, unlike the remainder of your posting. Have you sought Dr Steudings permission to host them on your website? Incidentally Dr Steudings own website is http://www.math.uni-frankfurt.de/~steuding/steuding.shtml -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Natural Densities are Probabilities > In 1933, Kolgomorov gave the rst widely accepted axiomatization of > Probability Theory. An axiomatization which has the following strange > consequence, when embedded in mainstream mathematics. > By intuition, we expect that the probability that a randomly chosen > integer is even equals 1/2, but then we have the following theorem. intuition? You need to say what it means for an integer to be randomly chosen. > There exists no probability law on N such that > P(aN) = 1/a (a in N) > where aN := { n in N : n = 0 mod a } > With other words, quite contrary to intuition: > The probability that a natural is divisible by a is NOT 1/a . There are many useful probability distributions on N, e.g. the Poisson distributions, not just one. > A proof of this can be found here (I am not the author): No scare quotes necessary: this is a valid proof that there is no probability measure on N with the property that P(aN) = 1/a for all a. > Because Probability Theory - as it has been embedded in mainstream > mathematics - gives this unwanted result, Unwanted? Unwanted by you maybe, but your wants are irrelevant to mathematics. > the following denition > has been widely accepted instead (skipping densities in general): > The natural density dA of a sequence A of naturals is: > dA = lim (x->innity) 1/x # { n <= x : n in A) Make A a subset of N (not a sequence) and add the proviso that this limit exists (not automatic). > Therefore the natural density of the even numbers is, exactly as you > would expect: 1/2 . Yes. > I I! I! I! > have major objections against this new denition, Its a perfectly straightforward denition. Its only you who is calling it new. > simply because > its NOT NEW at all. Thats a stupid objection. > For example, if you ip a coin, it will come up > either heads or tails. Suppose we perform x of these tosses. Dene > the set A as all tosses where the coin comes up heads. The natural > density of the heads is then: > dA = lim (x->innity) [ 1/x # { n <= x : n in A) ] If you toss it innitely many times. Why dont you go and do so, and report to us when youve nished? > Thus natural densities are simply a re-formulation of Probabilities. > And there is no need, at all, for a new concept. What the are you talking about. The denition of density is a simple mathematical denition, and is useful (see the text you copied on your website (with the authors permission?)). What would be the sense in abandining itfor the sole reason you dont like it? > Consequently, there must be a aw in the abovementioned proof too. > Because, otherwise, it would be impossible to even ip a coin. What proof? The proof you cited says nowt about tossing coins; it was a result on the theory of probability measures. All it says is that there is no map f from N to the nonnegative reals with the property that sum_{j=1}^innity f(an) = 1/a for all a in N. Your nal sentence is a nonsequitur. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Integration of elliptic rhumb distance If LT_a = LT_1 = Latitude_1, LT_b = LT_inf = Latitude_2 and LN_a = LN_1 = Longitude_1, LN_b = LN_inf = Longitude_2, the rhumb distance on a sphere is simply cos(LT)_m = (cos(LT_1) + cos(LT_2) ... + cos(LT_inf))/inf A = r * (LT_b - LT_a) B = r * cos(LT)_m * (LN_b - LN_a) C = sqrt( sq(A) + sq(B)) What about a spheroid? If M_n, N_n are the surface radiuses for LT_n, is it ++++++++++++M_m = (M_1 + M_2 ... + M_inf)/inf (N * cos(LT))_m = ((N_1 * cos(LT_1)) + (N_2 * cos(LT_2)) ... + ++++++++++++++++++ (N_inf * cos(LT_inf)))/inf A = M_m * (LT_b - LT_a) B = (N * cos(LT))_m * (LN_b - LN_a) C = sqrt( sq(A) + sq(B)) or is it sq(N * cos(LT))_m = (sq(N_1 * cos(LT_1)) + sq(N_2 * cos(LT_2)) ... + ++++++++++++++++++ sq(N_inf * cos(LT_inf)))/inf ++++++A = M_m * (LT_b - LT_a) sq(B)_m = sq(N * cos(LT))_m * sq(LN_b - LN_a) ++++++C = sqrt( sq(A) + sq(B)_m) ????? -------------- .b3[CapitalYAc ute] .bd[P aragraph] KORNET ------------- === Subject: Integration of elliptic rhumb distance If LT_a = LT_1 = Latitude_1, LT_b = LT_inf = Latitude_2 and LN_a = LN_1 = Longitude_1, LN_b = LN_inf = Longitude_2, the rhumb distance on a sphere is simply cos(LT)_m = (cos(LT_1) + cos(LT_2) ... + cos(LT_inf))/inf A = r * (LT_b - LT_a) B = r * cos(LT)_m * (LN_b - LN_a) C = sqrt( sq(A) + sq(B)) What about a spheroid? If M_n, N_n are the surface radiuses for LT_n, is it ++++++++++++M_m = (M_1 + M_2 ... + M_inf)/inf (N * cos(LT))_m = ((N_1 * cos(LT_1)) + (N_2 * cos(LT_2)) ... + ++++++++++++++++++ (N_inf * cos(LT_inf)))/inf A = M_m * (LT_b - LT_a) B = (N * cos(LT))_m * (LN_b - LN_a) C = sqrt( sq(A) + sq(B)) or is it sq(N * cos(LT))_m = (sq(N_1 * cos(LT_1)) + sq(N_2 * cos(LT_2)) ... + ++++++++++++++++++ sq(N_inf * cos(LT_inf)))/inf ++++++A = M_m * (LT_b - LT_a) sq(B)_m = sq(N * cos(LT))_m * sq(LN_b - LN_a) ++++++C = sqrt( sq(A) + sq(B)_m) ????? -------------- .b3[CapitalYAc ute] .bd[P aragraph] KORNET ------------- === Subject: Integration of elliptic rhumb distance 
If LT_a = LT_1 = Latitude_1, LT_b = LT_inf = Latitude_2 and
LN_a = LN_1 = Longitude_1, LN_b = LN_inf = Longitude_2,
the rhumb distance on a sphere is simply
cos(LT)_m = (cos(LT_1) + cos(LT_2) ... + cos(LT_inf))/inf
A = r * (LT_b - LT_a)
B = r * cos(LT)_m * (LN_b - LN_a)
C = sqrt( sq(A) + sq(B))
What about a spheroid?
If M_n, N_n are the surface radiuses for LT_n, is it
++++++++++++M_m = (M_1 + M_2 ... + M_inf)/inf
(N * cos(LT))_m = ((N_1 * cos(LT_1)) + (N_2 * cos(LT_2)) ... +
++++++++++++++++++ (N_inf * cos(LT_inf)))/inf
A = M_m * (LT_b - LT_a)
B = (N * cos(LT))_m * (LN_b - LN_a)
C = sqrt( sq(A) + sq(B))
or is it
sq(N * cos(LT))_m = (sq(N_1 * cos(LT_1)) + sq(N_2 *
cos(LT_2)) ... +
++++++++++++++++++ sq(N_inf * cos(LT_inf)))/inf
++++++A = M_m * (LT_b - LT_a)
sq(B)_m = sq(N * cos(LT))_m * sq(LN_b - LN_a)
++++++C = sqrt( sq(A) + sq(B)_m)
?????
--------------
.b3[CapitalYAc
ute]
.bd[P
aragraph] KORNET -------------
===
Subject: Integration of elliptic rhumb distance
If LT_a = LT_1 = Latitude_1, LT_b = LT_inf = Latitude_2 and
LN_a = LN_1 = Longitude_1, LN_b = LN_inf = Longitude_2,
the rhumb distance on a sphere is simply
cos(LT)_m = (cos(LT_1) + cos(LT_2) ... + cos(LT_inf))/inf
A = r * (LT_b - LT_a)
B = r * cos(LT)_m * (LN_b - LN_a)
C = sqrt( sq(A) + sq(B))
What about a spheroid?
If M_n, N_n are the surface radiuses for LT_n, is it
++++++++++++M_m = (M_1 + M_2 ... + M_inf)/inf
(N * cos(LT))_m = ((N_1 * cos(LT_1)) + (N_2 * cos(LT_2)) ... +
++++++++++++++++++ (N_inf * cos(LT_inf)))/inf
A = M_m * (LT_b - LT_a)
B = (N * cos(LT))_m * (LN_b - LN_a)
C = sqrt( sq(A) + sq(B))
or is it
sq(N * cos(LT))_m = (sq(N_1 * cos(LT_1)) + sq(N_2 *
cos(LT_2)) ... +
++++++++++++++++++ sq(N_inf * cos(LT_inf)))/inf
++++++A = M_m * (LT_b - LT_a)
sq(B)_m = sq(N * cos(LT))_m * sq(LN_b - LN_a)
++++++C = sqrt( sq(A) + sq(B)_m)
?????
--------------
.b3[CapitalYAc
ute]
.bd[P
aragraph] KORNET -------------
===
Subject: Integration of elliptic rhumb distance

If LT_a = LT_1 = Latitude_1, LT_b = LT_inf = Latitude_2 and
LN_a = LN_1 = Longitude_1, LN_b = LN_inf = Longitude_2,
the rhumb distance on a sphere is simply
cos(LT)_m = (cos(LT_1) + cos(LT_2) ... + cos(LT_inf))/inf
A = r * (LT_b - LT_a)
B = r * cos(LT)_m * (LN_b - LN_a)
C = sqrt( sq(A) + sq(B))
What about a spheroid?
If M_n, N_n are the surface radiuses for LT_n, is it
++++++++++++M_m = (M_1 + M_2 ... + M_inf)/inf
(N * cos(LT))_m = ((N_1 * cos(LT_1)) + (N_2 * cos(LT_2)) ... +
++++++++++++++++++ (N_inf * cos(LT_inf)))/inf
A = M_m * (LT_b - LT_a)
B = (N * cos(LT))_m * (LN_b - LN_a)
C = sqrt( sq(A) + sq(B))
or is it
sq(N * cos(LT))_m = (sq(N_1 * cos(LT_1)) + sq(N_2 *
cos(LT_2)) ... +
++++++++++++++++++ sq(N_inf * cos(LT_inf)))/inf
++++++A = M_m * (LT_b - LT_a)
sq(B)_m = sq(N * cos(LT))_m * sq(LN_b - LN_a)
++++++C = sqrt( sq(A) + sq(B)_m)
?????
--------------
.b3[CapitalYAc
ute]
.bd[P
aragraph] KORNET -------------
===
Subject: Integration of elliptic rhumb distance

If LT_a = LT_1 = Latitude_1, LT_b = LT_inf = Latitude_2 and
LN_a = LN_1 = Longitude_1, LN_b = LN_inf = Longitude_2,
the rhumb distance on a sphere is simply
cos(LT)_m = (cos(LT_1) + cos(LT_2) ... + cos(LT_inf))/inf
A = r * (LT_b - LT_a)
B = r * cos(LT)_m * (LN_b - LN_a)
C = sqrt( sq(A) + sq(B))
What about a spheroid?
If M_n, N_n are the surface radiuses for LT_n, is it
[Capi
talEHat][CapitalEHat
]M_m = (M_1 + M_2 ... + M_inf)/inf
(N * cos(LT))_m = ((N_1 * cos(LT_1)) + (N_2 * cos(LT_2)) ... +
++++++++++++++++++ (N_inf * cos(LT_inf)))/inf
A = M_m * (LT_b - LT_a)
B = (N * cos(LT))_m * (LN_b - LN_a)
C = sqrt( sq(A) + sq(B))
or is it
sq(N * cos(LT))_m = (sq(N_1 * cos(LT_1)) + sq(N_2 *
cos(LT_2)) ... +
++++++++++++++++++ sq(N_inf * cos(LT_inf)))/inf
++++++A = M_m * (LT_b - LT_a)
sq(B)_m = sq(N * cos(LT))_m * sq(LN_b - LN_a)
++++++C = sqrt( sq(A) + sq(B)_m)
?????
--------------
.81[NonBreakingSpac
e] .bc.a6 KORNET
-------------
===
Subject: Integration of elliptic rhumb distance

If LT_a = LT_1 = Latitude_1, LT_b = LT_inf = Latitude_2 and
LN_a = LN_1 = Longitude_1, LN_b = LN_inf = Longitude_2,
the rhumb distance on a sphere is simply
cos(LT)_m = (cos(LT_1) + cos(LT_2) ... + cos(LT_inf))/inf
A = r * (LT_b - LT_a)
B = r * cos(LT)_m * (LN_b - LN_a)
C = sqrt( sq(A) + sq(B))
What about a spheroid?
If M_n, N_n are the surface radiuses for LT_n, is it
[Capi
talEHat][CapitalEHat
]M_m = (M_1 + M_2 ... + M_inf)/inf
(N * cos(LT))_m = ((N_1 * cos(LT_1)) + (N_2 * cos(LT_2)) ... +
++++++++++++++++++ (N_inf * cos(LT_inf)))/inf
A = M_m * (LT_b - LT_a)
B = (N * cos(LT))_m * (LN_b - LN_a)
C = sqrt( sq(A) + sq(B))
or is it
sq(N * cos(LT))_m = (sq(N_1 * cos(LT_1)) + sq(N_2 *
cos(LT_2)) ... +
++++++++++++++++++ sq(N_inf * cos(LT_inf)))/inf
++++++A = M_m * (LT_b - LT_a)
sq(B)_m = sq(N * cos(LT))_m * sq(LN_b - LN_a)
++++++C = sqrt( sq(A) + sq(B)_m)
?????
--------------
.81[NonBreakingSpac
e] .bc.a6 KORNET
-------------
===
Subject: Integration of elliptic rhumb distance

If LT_a = LT_1 = Latitude_1, LT_b = LT_inf = Latitude_2 and
LN_a = LN_1 = Longitude_1, LN_b = LN_inf = Longitude_2,
the rhumb distance on a sphere is simply
cos(LT)_m = (cos(LT_1) + cos(LT_2) ... + cos(LT_inf))/inf
A = r * (LT_b - LT_a)
B = r * cos(LT)_m * (LN_b - LN_a)
C = sqrt( sq(A) + sq(B))
What about a spheroid?
If M_n, N_n are the surface radiuses for LT_n, is it
[Capi
talEHat][CapitalEHat
]M_m = (M_1 + M_2 ... + M_inf)/inf
(N * cos(LT))_m = ((N_1 * cos(LT_1)) + (N_2 * cos(LT_2)) ... +
[Capi
talEHat][CapitalEHat
][Capi
talEHat](N_inf * cos(LT_inf)))/inf
A = M_m * (LT_b - LT_a)
B = (N * cos(LT))_m * (LN_b - LN_a)
C = sqrt( sq(A) + sq(B))
or is it
sq(N * cos(LT))_m = (sq(N_1 * cos(LT_1)) + sq(N_2 *
cos(LT_2)) ... +
++++++++++++++++++ sq(N_inf * cos(LT_inf)))/inf
++++++A = M_m * (LT_b - LT_a)
sq(B)_m = sq(N * cos(LT))_m * sq(LN_b - LN_a)
++++++C = sqrt( sq(A) + sq(B)_m)
?????
--------------
.81[NonBreakingSpac
e] .bc.a6 KORNET
-------------
===
Subject: Integration of elliptic rhumb distance

If LT_a = LT_1 = Latitude_1, LT_b = LT_inf = Latitude_2 and
LN_a = LN_1 = Longitude_1, LN_b = LN_inf = Longitude_2,
the rhumb distance on a sphere is simply
cos(LT)_m = (cos(LT_1) + cos(LT_2) ... + cos(LT_inf))/inf
A = r * (LT_b - LT_a)
B = r * cos(LT)_m * (LN_b - LN_a)
C = sqrt( sq(A) + sq(B))
What about a spheroid?
If M_n, N_n are the surface radiuses for LT_n, is it
[Capi
talEHat][CapitalEHat
]M_m = (M_1 + M_2 ... + M_inf)/inf
(N * cos(LT))_m = ((N_1 * cos(LT_1)) + (N_2 * cos(LT_2)) ... +
[Capi
talEHat][CapitalEHat
][Capi
talEHat](N_inf * cos(LT_inf)))/inf
A = M_m * (LT_b - LT_a)
B = (N * cos(LT))_m * (LN_b - LN_a)
C = sqrt( sq(A) + sq(B))
or is it
sq(N * cos(LT))_m = (sq(N_1 * cos(LT_1)) + sq(N_2 *
cos(LT_2)) ... +
[Capi
talEHat][CapitalEHat
][Capi
talEHat]sq(N_inf *
cos(LT_inf)))/inf
[Capi
talEHat]A = M_m * (LT_b - LT_a)
sq(B)_m = sq(N * cos(LT))_m * sq(LN_b - LN_a)
[Capi
talEHat]C = sqrt( sq(A) + sq(B)_m)
?????
--------------
.81[NonBreakingSpac
e] .bc.a6 KORNET
-------------
===
Subject: Integration of elliptic rhumb distance

If LT_a = LT_1 = Latitude_1, LT_b = LT_inf = Latitude_2 and
LN_a = LN_1 = Longitude_1, LN_b = LN_inf = Longitude_2,
the rhumb distance on a sphere is simply
cos(LT)_m = (cos(LT_1) + cos(LT_2) ... + cos(LT_inf))/inf
A = r * (LT_b - LT_a)
B = r * cos(LT)_m * (LN_b - LN_a)
C = sqrt( sq(A) + sq(B))
What about a spheroid?
If M_n, N_n are the surface radiuses for LT_n, is it
[Capi
talEHat][CapitalEHat
]M_m = (M_1 + M_2 ... + M_inf)/inf
(N * cos(LT))_m = ((N_1 * cos(LT_1)) + (N_2 * cos(LT_2)) ... +
[Capi
talEHat][CapitalEHat
][Capi
talEHat](N_inf * cos(LT_inf)))/inf
A = M_m * (LT_b - LT_a)
B = (N * cos(LT))_m * (LN_b - LN_a)
C = sqrt( sq(A) + sq(B))
or is it
sq(N * cos(LT))_m = (sq(N_1 * cos(LT_1)) + sq(N_2 *
cos(LT_2)) ... +
[Capi
talEHat][CapitalEHat
][Capi
talEHat]sq(N_inf *
cos(LT_inf)))/inf
[Capi
talEHat]A = M_m * (LT_b - LT_a)
sq(B)_m = sq(N * cos(LT))_m * sq(LN_b - LN_a)
[Capi
talEHat]C = sqrt( sq(A) + sq(B)_m)
?????
--------------
.81[NonBreakingSpac
e] .bc.a6 KORNET
-------------
===
Subject: Sorry :(
Sorry about the multiple copies (I thought I was just
previewing!)! :(
--------------
.b3[CapitalYAc
ute]
.bd[P
aragraph] KORNET -------------
===
Subject: Integration of elliptic rhumb distance
If LT_a = LT_1 = Latitude_1, LT_b = LT_inf = Latitude_2 and
LN_a = LN_1 = Longitude_1, LN_b = LN_inf = Longitude_2,
the rhumb distance on a sphere is simply
cos(LT)_m = (cos(LT_1) + cos(LT_2) ... + cos(LT_inf))/inf
A = r * (LT_b - LT_a)
B = r * cos(LT)_m * (LN_b - LN_a)
C = sqrt( sq(A) + sq(B))
What about a spheroid?
If M_n, N_n are the surface radiuses for LT_n, is it
[Capi
talEHat][CapitalEHat
]M_m = (M_1 + M_2 ... + M_inf)/inf
(N * cos(LT))_m = ((N_1 * cos(LT_1)) + (N_2 * cos(LT_2)) ... +
[Capi
talEHat][CapitalEHat
][Capi
talEHat](N_inf * cos(LT_inf)))/inf
A = M_m * (LT_b - LT_a)
B = (N * cos(LT))_m * (LN_b - LN_a)
C = sqrt( sq(A) + sq(B))
or is it
sq(N * cos(LT))_m = (sq(N_1 * cos(LT_1)) + sq(N_2 *
cos(LT_2)) ... +
[Capi
talEHat][CapitalEHat
][Capi
talEHat]sq(N_inf *
cos(LT_inf)))/inf
[Capi
talEHat]A = M_m * (LT_b - LT_a)
sq(B)_m = sq(N * cos(LT))_m * sq(LN_b - LN_a)
[Capi
talEHat]C = sqrt( sq(A) + sq(B)_m)
?????
--------------
.81[NonBreakingSpac
e] .bc.a6 KORNET
-------------
===
Subject: Re: Integration of elliptic rhumb distance
> If LT_a = LT_1 = Latitude_1, LT_b = LT_inf = Latitude_2 and
> LN_a = LN_1 = Longitude_1, LN_b = LN_inf = Longitude_2,
> the rhumb distance on a sphere is simply
> cos(LT)_m = (cos(LT_1) + cos(LT_2) ... + cos(LT_inf))/inf
sin{LT_d} - sin{LT_b}
Even better, cos{LT}_m = ---------------------
LT_d - LT_b
> A = r * (LT_b - LT_a)
> B = r * cos(LT)_m * (LN_b - LN_a)
> C = sqrt( sq(A) + sq(B))
> What about a spheroid?
> If M_n, N_n are the surface radiuses for LT_n, is it
> M_m = (M_1 + M_2 ... + M_inf)/inf
> (N * cos(LT))_m = ((N_1 * cos(LT_1)) + (N_2 * cos(LT_2))
... +
> (N_inf * cos(LT_inf)))/inf
> A = M_m * (LT_b - LT_a)
> B = (N * cos(LT))_m * (LN_b - LN_a)
> C = sqrt( sq(A) + sq(B))
> or is it
> sq(N * cos(LT))_m = (sq(N_1 * cos(LT_1)) + sq(N_2 *
cos(LT_2)) ... +
> sq(N_inf * cos(LT_inf)))/inf
> A = M_m * (LT_b - LT_a)
> sq(B)_m = sq(N * cos(LT))_m * sq(LN_b - LN_a)
> C = sqrt( sq(A) + sq(B)_m)
Given typical differentiation and integration order and
protocol, since it
involves squaring, Id say the latter:
(B^2)_m = ([N * cos(LT)]^2)_m * [LN_b - LN_a]^2
~Kaimbridge~
-----
Wanted?Kaimbridge (w/mugshot!):
http://www.angelre.com/ma2/digitology/Wanted_KMGC.html
----------
Digitology?The Grand Theory Of The Universe:
http://www.angelre.com/ma2/digitology/index.html
***** Void Where Permitted; Limit 0 Per Customer. *****
===
Subject: Integration of elliptic rhumb distance
If LT_a = LT_1 = Latitude_1, LT_b = LT_inf = Latitude_2 and
LN_a = LN_1 = Longitude_1, LN_b = LN_inf = Longitude_2,
the rhumb distance on a sphere is simply
cos(LT)_m = (cos(LT_1) + cos(LT_2) ... + cos(LT_inf))/inf
A = r * (LT_b - LT_a)
B = r * cos(LT)_m * (LN_b - LN_a)
C = sqrt( sq(A) + sq(B))
What about a spheroid?
If M_n, N_n are the surface radiuses for LT_n, is it
[Capi
talEHat][CapitalEHat
]M_m = (M_1 + M_2 ... + M_inf)/inf
(N * cos(LT))_m = ((N_1 * cos(LT_1)) + (N_2 * cos(LT_2)) ... +
[Capi
talEHat][CapitalEHat
][Capi
talEHat](N_inf * cos(LT_inf)))/inf
A = M_m * (LT_b - LT_a)
B = (N * cos(LT))_m * (LN_b - LN_a)
C = sqrt( sq(A) + sq(B))
or is it
sq(N * cos(LT))_m = (sq(N_1 * cos(LT_1)) + sq(N_2 *
cos(LT_2)) ... +
[Capi
talEHat][CapitalEHat
][Capi
talEHat]sq(N_inf *
cos(LT_inf)))/inf
[Capi
talEHat]A = M_m * (LT_b - LT_a)
sq(B)_m = sq(N * cos(LT))_m * sq(LN_b - LN_a)
[Capi
talEHat]C = sqrt( sq(A) + sq(B)_m)
?????
--------------
.81[NonBreakingSpac
e] .bc.a6 KORNET
-------------
===
Subject: Re: Zenkins paper on Cantor
> Eray Ozkural says...
> Well, constructivism was not invented at Cantors time,
but there
> is nothing nonconstructive about Cantors proof.
>Abstraction of actual innity seems to stand in the way,
Daryl.
> Where does the concept of actual innity come into play in
> Cantors proof?
> Intuitionists have a different interpretation of Cantors
> result than do classical mathematicians, but they accept its
> correctness.
I had thought like that.
> For a classical mathematician, if a set is countable then
any subset is
> also countable (or nite). So countability is a measure of
size. For an
> intuitionist, a subset of a countable set may be
uncountable. For
example,
> the set T of integers n such that n codes a Turing machine
program that
> computes a total function from N to N. T is a subset of N,
and N is
countable.
I see.
> However, T is not countable (intuitionistically) because
there is no
> computable bijection between T and N.
Can you explain a little more? Im not sure I understood this
correctly.
> Because of this problem, intuitionists dont view
cardinality as a
measure
> of size.
What do they use instead to measure size? I nd this
interesting.
> But they still agree that there is no bijection between the
reals
> and the naturals.
OK. Thats fair.
--
Eray Ozkural
===
Subject: Re: Zenkins paper on Cantor
> Eray Ozkural says...
> Well, constructivism was not invented at Cantors time,
but there
> is nothing nonconstructive about Cantors proof.
>Abstraction of actual innity seems to stand in the way,
Daryl.
> Where does the concept of actual innity come into play in
> Cantors proof?
> Intuitionists have a different interpretation of Cantors
> result than do classical mathematicians, but they accept its
> correctness.
I had thought like that.
> For a classical mathematician, if a set is countable then
any subset is
> also countable (or nite). So countability is a measure of
size. For an
> intuitionist, a subset of a countable set may be
uncountable. For
example,
> the set T of integers n such that n codes a Turing machine
program that
> computes a total function from N to N. T is a subset of N,
and N is
countable.
I see.
> However, T is not countable (intuitionistically) because
there is no
> computable bijection between T and N.
Can you explain a little more? Im not sure I understood this
correctly.
> Because of this problem, intuitionists dont view
cardinality as a
measure
> of size.
What do they use instead to measure size? I nd this
interesting.
> But they still agree that there is no bijection between the
reals
> and the naturals.
OK. Thats fair.
--
Eray Ozkural
===
Subject: Re: Zenkins paper on Cantor
> prepositions, shall we?
> As in sworn to you?
I thought I made it awfully clear that by from an intellectual
vacuum, I meant the subjects relation to physics, biology,
and
complexity sciences in general.
I now conceive that the very mistake might be to view
Chaitins
results from a Platonist perspective.
--
Eray Ozkural
===
Subject: Re: Zenkins paper on Cantor
> prepositions, shall we?
> As in sworn to you?
> I thought I made it awfully clear that by from an
intellectual
> vacuum, I meant the subjects relation to physics, biology,
and
> complexity sciences in general.
Its still just abuse.
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square
root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Re: Metacyclic groups and cyclic Sylow-p-subgroups
> I was trying to prove myself that all groups of squarefree
order are
> solvable.
I think the most instructive way to prove this result is to
use Burnsides
Transfer Theorem, which says that if a group G has a Sylow
p-subgroups P
such that P is in the centre of the normalizer in G of P,
then G has a
normal
subgroup N which complements P - that is, G = PN with P
intersect N
trivial.
For a group os squarefree order, let p be the smallest prime
dividing G,
and
let P be a Sylow p-subgroup of G. Since Aut(P) has order p-1,
the elements
in
the normalizer in G of P must induce the trivial automorphism
on P, and so
P
is in the centre of its normalizer, and we can use Burnsides
Transfer
Theorem to infer the existence of a normal subgroup N
complementing P. You
can now prove the theorem by applying induction to N.
You may not know Burnsides Transfer Theorem, but it is not
too difcult,
and is worth learning. This does not prove the stronger
result that G is
metacyclic, but you were not asked to prove that.
And by the way, your denition of metacyclic below is
non-standard. The
standard denition is G is metacyclic if it has a normal
subgroup N such
that N and G/N are both cyclic.
Derek Holt.
> I got these intermediate results but am missing the nal
> part:
> Lemma:
> Let |G| be squarefree. Then all Sylow-p-subgroups of G are
cyclic.
> Proof:
> We can write |G| = p_1^(k_1) p_2^(k_2) ... p_n^(k_n) by the
> Fundamental Theorem of Arithmetic, where all p_i are
different
> primes. Since |G| is squarefree, we have that k_i <= 1 for
all i,
> thus |G| = p_1 p_2 ... p_j. Then, by Sylows First Theorem,
the
> Sylow-p_i-subgroups of G have prime order and are cyclic.
> Denition:
> Group G is said to be metacyclic if G is cyclic and G/G
is cyclic.
> Lemma:
> Metacyclic groups are solvable.
> Proof:
> G is normal in G and G/G is abelian. Thus (G, G, {1}) is
a
> solvable series for G, since G/{1} = G is abelian.
> Now all thats missing is to show that a group is
metacyclic if all
> Sylow-p-subgroups are cyclic and the result follows by
chaining these
> statements. How does one show that?
===
Subject: odd perfect
Just wondering if anyone out there is currently undertaking
research on the
odd perfect number problem?
Tim
t.roberts@cqu.edu.au
===
Subject: Re: odd perfect
> Just wondering if anyone out there is currently undertaking
research on
the
> odd perfect number problem?
Apparently there is. See the page on MathWorld about odd
perfect numbers.
number must have at least 47 prime factors.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam,
nederland,
+31205924131
home: bovenover 215, 1025 jn amsterdam, nederland;
http://www.cwi.nl/~dik/
===
Subject: Re: odd perfect
> Just wondering if anyone out there is currently
undertaking research on
the
> odd perfect number problem?
> Apparently there is. See the page on MathWorld about odd
perfect
numbers.
> number must have at least 47 prime factors.
See, also, http://www.mersenneforum.org/showthread.php?t=3101
and the
recent NMBRTHRY archives.
Paul
--
Hanging on in quiet desperation is the English way.
The time is gone, the song is over.
Thought Id something more to say.
===
Subject: Re: odd perfect
> Just wondering if anyone out there is currently
undertaking research
on the
> odd perfect number problem?
>
> Apparently there is. See the page on MathWorld about odd
perfect
numbers.
> number must have at least 47 prime factors.
> See, also,
http://www.mersenneforum.org/showthread.php?t=3101 and the
> recent NMBRTHRY archives.
Indeed. And then grab an ECMNET client, and GMPECM, and then
point
the client to 62.236.152.54 port 8192, and help crack the
last of
the three numbers!
Phil
--
They no longer do my traditional winks tournament lunch -
liver and bacon.
Its just what you need during a winks tournament lunchtime
to replace lost
===
Subject: Groups of order 30, 105 and pqr
I am working some problems in Milnes excellent new version
Prob 64. Prove or give counter-example:
(a) Every group of order 30 has a normal subgroup of order 15.
(b) Every group of order 30 is nilpotent.
--------------
The Sylow thms --> either n_5 or n_3 = 1, i.e., either the
subgroup
N such that |N| = 5 or H such that |H| = 3 is normal.
This works if |G| = 105 = 3.5.7, and I think for |G| = pqr,
product of 3
primes.
If n_5 = 1, and N = , |G/N| = 6 --> there is a y in G =
G/N such that
|y| = 3,
and if H = , H is normal in G.
Then there is y in G such that y = yN = y , so there is
z = yx in G, and |y| = 3, |x| = 5, so |z| = 15.
I want to say that M =  is normal in G, which seems to be
true,
but its not clear to me how to prove it.
Can anyone help? Please be patient with me, I am just learning
this, and have mental blocks at places that appear obvious to
those who are trained in math.
Van
===
Subject: Re: Groups of order 30, 105 and pqr
> I am working some problems in Milnes excellent new version
> Prob 64. Prove or give counter-example:
> (a) Every group of order 30 has a normal subgroup of order
15.
> (b) Every group of order 30 is nilpotent.
> --------------
> The Sylow thms --> either n_5 or n_3 = 1, i.e., either the
subgroup
> N such that |N| = 5 or H such that |H| = 3 is normal.
> This works if |G| = 105 = 3.5.7, and I think for |G| = pqr,
product of 3
> primes.
Would it work if |G| = 465 = 3.5.31?
> If n_5 = 1, and N = , |G/N| = 6 --> there is a y in G
= G/N such
that
> |y| = 3,
> and if H = , H is normal in G.
OK
> Then there is y in G such that y = yN = y , so there is
> z = yx in G, and |y| = 3, |x| = 5, so |z| = 15.
> I want to say that M =  is normal in G, which seems to
be true,
> but its not clear to me how to prove it.
It is true that yN is nite, but all you need is that yN is
a subgroup. It is so since its the inverse image of H
under the projection from G to G/N.
What if n_5 = 6?
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square
root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Re: Groups of order 30, 105 and pqr
> I am working some problems in Milnes excellent new version
> Prob 64. Prove or give counter-example:
> (a) Every group of order 30 has a normal subgroup of order
15.
> (b) Every group of order 30 is nilpotent.
> --------------
> The Sylow thms --> either n_5 or n_3 = 1, i.e., either the
subgroup
> N such that |N| = 5 or H such that |H| = 3 is normal.
> This works if |G| = 105 = 3.5.7, and I think for |G| =
pqr, product of 3
> primes.
> Would it work if |G| = 465 = 3.5.31?
> If n_5 = 1, and N = , |G/N| = 6 --> there is a y in G
= G/N such
that
> |y| = 3,
> and if H = , H is normal in G.
> OK
> Then there is y in G such that y = yN = y , so there is
> z = yx in G, and |y| = 3, |x| = 5, so |z| = 15.
> I want to say that M =  is normal in G, which seems to
be true,
> but its not clear to me how to prove it.
> It is true that yN is nite, but all you need is that yN is
> a subgroup. It is so since its the inverse image of H
> under the projection from G to G/N.
> What if n_5 = 6?
-----------
I dont understand this statement:
>It is true that yN is nite, but all you need is that yN is
> a subgroup.
Of course it is nite. Do you mean that N =  is a
subgroup?
---------
If n_5 = 6, n_3 = 1 mod 3 = 1 or 10. If n_3 = 10, there are
20 elements g with |g| = 3. But there are 6.4 = 24 elements x
with |x| = 5, so 20 + 24 > 30, AND n_3 = 1.
Let N =  with |x| = |N| = 3. y in G = G/N has |y| = 5.
Let H = , so |H| = 5.
Here is the part I have trouble with. How does this imply that
there is H < G such that p:H-->H < G = G/N ?
p(h) = h = hN = h , where |x| = 3 and |h| = 5.
For some reason, I get confused here. Perhaps I just havent
thought it through carefully.
--------
Would it work if |G| = 465 = 3.5.31?
I still have to do the case of n = 105 = 3.5.7.
n_7 = 1 mod 7 = 1 or 15. 6.15 = 90 elements if n_7 = 15.
n_5 = 1 mod 5 = 1 or 21 ==> either n_7 or n_5 = 1, same as
for |G| = 30.
If |N| = 7 is normal, G = G/N =~ Z_15 = Z_3 x Z_5.
So there is a normal subgroup of order 35, and if |H| = 35,
G = G/H = Z_3.
n_3 = 1 or 7. n_3 = 1 gives an Abelian group, n_3 = 7 gives
the
non-Ableian group.
I think these are the only 2 groups of order 105.
(I could be wrong).
If |x| = 35 and |y| = 3.
If yxy^(-1) = x^r, we nd r = 16, which is the non-Ableian
group.
Van
===
Subject: Re: Groups of order 30, 105 and pqr
> I am working some problems in Milnes excellent new version
>
> Prob 64. Prove or give counter-example:
> (a) Every group of order 30 has a normal subgroup of order
15.
> (b) Every group of order 30 is nilpotent.
> --------------
>
> The Sylow thms --> either n_5 or n_3 = 1, i.e., either the
subgroup
> N such that |N| = 5 or H such that |H| = 3 is normal.
>
> This works if |G| = 105 = 3.5.7, and I think for |G| = pqr,
product of
3
> primes.
> Would it work if |G| = 465 = 3.5.31?
> If n_5 = 1, and N = , |G/N| = 6 --> there is a y in G
= G/N such
that
> |y| = 3,
>
> and if H = , H is normal in G.
> OK
> Then there is y in G such that y = yN = y , so there is
>
> z = yx in G, and |y| = 3, |x| = 5, so |z| = 15.
>
> I want to say that M =  is normal in G, which seems to
be true,
> but its not clear to me how to prove it.
> It is true that yN is nite, but all you need is that yN is
> a subgroup. It is so since its the inverse image of H
> under the projection from G to G/N.
> What if n_5 = 6?
> -----------
> I dont understand this statement:
> It is true that yN is nite, but all you need is that yN is
> a subgroup.
> Of course it is nite. Do you mean that N =  is a
subgroup?
> ---------
> If n_5 = 6, n_3 = 1 mod 3 = 1 or 10. If n_3 = 10, there are
> 20 elements g with |g| = 3. But there are 6.4 = 24 elements
x
> with |x| = 5, so 20 + 24 > 30, AND n_3 = 1.
> Let N =  with |x| = |N| = 3. y in G = G/N has |y| = 5.
> Let H = , so |H| = 5.
> Here is the part I have trouble with. How does this imply
that
> there is H < G such that p:H-->H < G = G/N ?
> p(h) = h = hN = h , where |x| = 3 and |h| = 5.
> For some reason, I get confused here. Perhaps I just havent
> thought it through carefully.
> --------
> Would it work if |G| = 465 = 3.5.31?
> I still have to do the case of n = 105 = 3.5.7.
> n_7 = 1 mod 7 = 1 or 15. 6.15 = 90 elements if n_7 = 15.
> n_5 = 1 mod 5 = 1 or 21 ==> either n_7 or n_5 = 1, same as
for |G| = 30.
> If |N| = 7 is normal, G = G/N =~ Z_15 = Z_3 x Z_5.
> So there is a normal subgroup of order 35, and if |H| = 35,
> G = G/H = Z_3.
> n_3 = 1 or 7. n_3 = 1 gives an Abelian group, n_3 = 7 gives
the
> non-Ableian group.
> I think these are the only 2 groups of order 105.
> (I could be wrong).
> If |x| = 35 and |y| = 3.
> If yxy^(-1) = x^r, we nd r = 16, which is the non-Ableian
group.
> Van
If |G| = 465 = 3.5.31, n_31 = 1 --> always and x of order 31,
and
 = N is normal.
n_5 = 31 is allowed, as far as I can see, so it does not hold
for
|G| = pqr in general.
As for one of the isomorphism thms., I know it is true that
if H < G = G/N and H is normal in G, then H = the inverse
image of
H is normal in G, where p: G --> G and p(H) = H.
I think this follows since p is a homomorphism, since
if h = p(h) and x = p(x), xh(x)^(-1) = p(x) p(h) p(x^-1)
= p(xhx^-1) = p(h*) = (h*) = (h)* is in H iff h* = xhx^-1
is in H,
so H normal in G iff H normal in G.
But H must contain N. I still dont have this clear in my
mind.
Van
===
Subject: Re: Groups of order 30, 105 and pqr
> If |G| = 465 = 3.5.31, n_31 = 1 --> always and x of order
31, and
>  = N is normal.
> n_5 = 31 is allowed, as far as I can see, so it does not
hold for
> |G| = pqr in general.
There is a G with n_3 = n_5 = 31.
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square
root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Re: Groups of order 30, 105 and pqr
> If |G| = 465 = 3.5.31, n_31 = 1 --> always and x of order
31, and
>  = N is normal.
> n_5 = 31 is allowed, as far as I can see, so it does not
hold for
> |G| = pqr in general.
> There is a G with n_3 = n_5 = 31.
Interesting. I will look into this.
One nds groups with unusual properties in unexpected places.
This is one of the things I like about nite groups.
structure--Im not sure how to put it, but anyone who has
worked
with them, even at a basic level, like me, knows what I mean.
Van
===
Subject: Re: Groups of order 30, 105 and pqr
> It is true that yN is nite, but all you need is that yN is
> a subgroup. It is so since its the inverse image of H
> under the projection from G to G/N.
> What if n_5 = 6?
> -----------
> I dont understand this statement:
>It is true that yN is nite, but all you need is that yN is
Doh! nite -> cyclic :-(
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square
root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Re: Suggestions for grad school?

X-CompuServe-Customer: Yes
X-Coriate: interspeed.co.nz
X-Ecrate: tanandtanlawyers.com
X-Pose: George Cox
X-Punge: Micro$oft
X-Sanguinate: The MVS Guy
X-Terminate: SPA(GIS)
X-Tinguish: Mark Grifth
X-Treme: C&C,DWS
at 03:57 PM, hrubin@odds.stat.purdue.edu (Herman Rubin) said:
>Frankly, I believe one could, and should, teach a good real
analysis
>course, with little computation, before calculus.
Yes, and Im also wondering whether it might not make sense
to include
exterior algebra and differential forms in that.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT

Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply
to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spamtrap@library.lspace.org
===
Subject: Re: Suggestions for grad school?
X-RFC2646:  Original
> at 03:57 PM, hrubin@odds.stat.purdue.edu (Herman Rubin)
said:
>Frankly, I believe one could, and should, teach a good real
analysis
>course, with little computation, before calculus.
> Yes, and Im also wondering whether it might not make sense
to include
> exterior algebra and differential forms in that.
> --
> Shmuel (Seymour J.) Metz, SysProg and JOAT

> Unsolicited bulk E-mail subject to legal action. I reserve
the
> right to publicly post or ridicule any abusive E-mail.
Reply to
> domain Patriot dot net user shmuel+news to contact me. Do
not
> reply to spamtrap@library.lspace.org
Are there any book titles that use this order of teaching?
Brett
===
Subject: Re: Suggestions for grad school?
Check learn.wisconsin.edu for self-study courses in math, at
tuition
of less that $200 a credit hour. You will have to nd
appropriate
proctors on your own, and area schools will perhaps be
reluctant to
cooperate in your avoidance of their tuition costs. On the
other
hand, if you are doing a couple of courses, so as to take two
midterms
or two nals at a time, then an occasional trip to Madison
might
agree with your plans. Ask whether other students have used
their
courses for credit or placemenr elsewhere.
David Ames
> Thats around 2 years of taking only undergrad courses,
since Ill
probably
> limit myself to 3 hours per semester, beings I want to be
sure Im
grasping
> everything in the current course before moving forward.
Also, I live in
DC
> and nearly all schools (that its feasible for me to
attend) charge around
> $950/credit. I need to try and decrease the total number of
undergrad
> courses to the lowest possible.
> Self study and any type of advancement (placement) exams
are where I need
to
> try for these undergrad courses.
> Brett
> Try the next course in line Calculus III, then
Differential Equations.
> After that you can take Fourier Analysis, which is a
great review all
> previous math courses (Im nding this out now). Linear
Algebra is
mostly
> new stuff, mostly based on matrices. Abstract Algebra is
mostly new
stuff
> dealing with group theory, I dont remember much calculus
at all...maybe
a
> few derivatives thats about it. Real Analysis at least
so far is a
course
> in proving all the ideas that we used in lower
mathematics, of course
you
> need a good handle on this ideas in the rst place before
you begin to
> prove them. I also completed a 300 sequence in
probability and stats.
This
> takes me to the end of my math major. You may also want
to consider
> complex analysis, and topology as prep for grad school.
Ive applied to
> two so far with my current status, well see what happens.
> Good Luck!
> Id like recommendations on suggested mathematics
courses for
preperation
> of mathematics grad school. Im not sure yet which
school but Id like
a
> general scenario.
>
> Ive taken up to calculus II and Maxtrix Theory. That
was about seven
> years ago. What is the best way to review my previous
courses without
> taking time to go through those courses again? Id like
to use that
time
> for attending higher level courses such as abstract
algebra and real
> analysis.
>
> Brett
>
===
Subject: Re: Prime numbers and the RSA algorithm
Robin Chapman:
> There *are* more efcient ways of checking primality.
Much more
> effcient if you allow the slight possibility that a
composite is
> falsely stated as prime, e.g., the Miller-Rabin test.
Thats already the answer to the OPs question.
> And if you do not allow that possibility you can always do
a full
> primality test on numbers that are probably prime by
Miller-Rabin.
> BTW, this is how primality provers work. First trial
division by
> small primes. Next Miller-Rabin, or suchlike. Next a full
proof.
In the scenario of the OPs question, Miller-Rabin could be
sufcient,
and a full proof not necessary. When constructing the two
primes, one can
do it in a way that one knows the factorisation of p-1 for
the prime p. If
then, for each prime factor d of p-1 there is an x such that
x^((p-1)/d)
is not 1 but x^(p-1) is 1 (both mod p), then p is proven
prime. This is
quickly tested.
Helmut Richter
===
Subject: Re: Prime numbers and the RSA algorithm
> I have been reading about this and I have a question. It
is probably
> naive or simplistic, but I would like to understand.
>
> The question is not about the RSA algorithm itself but
the nature of
> prime numbers and factorization.
> You want Professor Caldwells Prime Pages:
> http://primepages.org/
> In particular the section on proving primality.
> Phil
The distributed computing NFSNET project doesnt seem
to get much publicity ... Their goal is to factor
some large numbers.
The NFSNET homepage is at: http://www.nfsnet.org/
Also of interest is the News section:
http://www.nfsnet.org/status.html
David Bernier
===
Subject: Confused on convex functions/sets
X-RFC2646:  Original
I see the following line in my text : If f : [a,b] ---> R is
convex, then it
follows from Proposition 3.2 that f(x) <= max {f(a), f(b)}.
Proposition 3.2 says : A function f : [a,b] --> R is convex
iff the set A =
{(x,y) : a <= x <= b and f(x) <= y} is convex.
First of all, I dont think I understand this set A. If f(x)
= x^2 on
[0,1], then is A the set of ALL (x,y) s.t. a<=x<=b and x^2 <=
y? That is, A
is the innite strip above x^2 on [0,1]? (i.e. points like
(0,100), (.5,
50000) are in A?)
Secondly, why does it follow from Proposition 3.2 that for
any convex f,
f(x) <= max{f(a),f(b)}?
Isaac
===
Subject: Re: Confused on convex functions/sets
* Isharu@yahoo.com
> I see the following line in my text : If f : [a,b] ---> R
is convex, then
it
> follows from Proposition 3.2 that f(x) <= max {f(a), f(b)}.
> Proposition 3.2 says : A function f : [a,b] --> R is convex
iff the set A
=
> {(x,y) : a <= x <= b and f(x) <= y} is convex.
> First of all, I dont think I understand this set A. If
f(x) = x^2 on
> [0,1], then is A the set of ALL (x,y) s.t. a<=x<=b and x^2
<= y? That is,
A
> is the innite strip above x^2 on [0,1]? (i.e. points like
(0,100), (.5,
> 50000) are in A?)
Yes, that is so.
> Secondly, why does it follow from Proposition 3.2 that for
any convex f,
> f(x) <= max{f(a),f(b)}?
Suppose f(x) > max(f(a),f(b)). Then the line l from f(a) to
f(b)
would contain a point (x,y) such that y:
>That questions trivial to answer.
>Yes. It can be done in O(1) time. Just throw away all but the
>lowest |o| bits of a.
I was afraid it was that simple.. 
Still, the answer is much appreciated and very helpful.
>However, with your notational inconsistencies and
ambiguities,
>Im still not entirely sure thats the question you wanted
to ask.
Hm. Perhaps it would be best if I corrected it - at least in
my mind:
> d = ( 1 + o ( -o^{e-2} mod e ) ) / e,
>By mod I assume you mean the remainder function?
I wonder if a mod m is not standard notation? (Or was this
only
confusing in light of the rest of my post?)
> a (mod 2^|o|)
>Now youre using mod in a way reminiscent of modular
arithmetic
>rather than using the remainder function. Gramattically
mod cant
>be a binary function in the above.
This was a bad choice on my part - I think I was trying to
emphasize
mod 2^|o|.
> d = ( ( 1 + o ( -o^{e-2} mod e ) ) * e^{-1} ) mod 2^{|o|},
> where e^{-1} indicates the modular inverse of e.
>What do you mean by the modular inverse of e?
>The inverse of e modulo n makes sense, but modular inverses
only
>make sense if a base is specied. You have not specied such
a base.
Here I was just using some shorthand, though not very well
apparently.
The mod 2^{|o|} was supposed to be distributed and in that
context I
was hoping the modular inverse would be assumed in base
2^{|o|}.
In any case, I do apologize and again thank you for the quick
(albeit
sadly trivial) answer!
--
Jay Miller
PGP Fingerprint | 5f7adbbe bfc60727 96dca94c 616d5080 09e3e846
===
Subject: Re: (mod 2^4096) on a 4096 bit MAA?
> d = ( 1 + o ( -o^{e-2} mod e ) ) / e,
>By mod I assume you mean the remainder function?
> I wonder if a mod m is not standard notation? (Or was
this only
> confusing in light of the rest of my post?)
Its used in several different ways in different contexts.
a mod b to a programmer in some languages is the remainder
when
a is divided by b. i.e. the application of a function with two
parameters. Some programmers in other languages would call
this
same value a%b, % being the C operator for remainders. Note
- different languages treat negative values differently --
dont
assume anything, or make your audience assume anything, if
youre
dealing with negative values.
a mod b to a mathematician would often be recognised as a
sloppier
way of writing a (mod b). a (mod b) is a single entity, like a
number, and does not represent the application of a function
to two
parameters.
Due to the ambiguity, if you mean the latter, its best to
always put
the brackets in -- a (mod b), which _cannot_ be interpreted
as a
binary function.
Assignment and comparison of values modulo a xed base is
another area
where syntax can become ambiguous. The modulo comparison
operator in
properly typeset maths is the three line equals, which has
no ASCII
equivalent. However, its a de-facto standard to use == as
a stand-in.
So if you want to assert the modular equivalence of two
values, youd
write a == n (mod b), pronounced a is equivalent to n modulo
b.
Be prepared to see others write a = n (mod c) for the same
thing.
> a (mod 2^|o|)
>Now youre using mod in a way reminiscent of modular
arithmetic
>rather than using the remainder function. Gramattically
mod cant
>be a binary function in the above.
> This was a bad choice on my part - I think I was trying to
emphasize
> mod 2^|o|.
> d = ( ( 1 + o ( -o^{e-2} mod e ) ) * e^{-1} ) mod
2^{|o|},
>
> where e^{-1} indicates the modular inverse of e.
>What do you mean by the modular inverse of e?
>The inverse of e modulo n makes sense, but modular
inverses only
>make sense if a base is specied. You have not specied
such a base.
> Here I was just using some shorthand, though not very well
apparently.
> The mod 2^{|o|} was supposed to be distributed and in
that context I
> was hoping the modular inverse would be assumed in base
2^{|o|}.
If everything in an expression is to be performed in modular
arithmetic
to a single base, then you only need one (mod b) in the whole
line,
and at the end -- so yes it does distribute over the whole
expression.
However, remember to keep the brackets in.
The problem was that you also had -o^{e-2} mod e embedded in
that
expression. It was therefore non-obvious which of the two
meanings
for mod was being used.
> In any case, I do apologize and again thank you for the
quick (albeit
> sadly trivial) answer!
No harm done.
Phil
--
They no longer do my traditional winks tournament lunch -
liver and bacon.
Its just what you need during a winks tournament lunchtime
to replace lost
===
Subject: Re: (mod 2^4096) on a 4096 bit MAA?
Words by Phil Carmody :
>Its used in several different ways in different contexts.
> ...
Excellent info - thank you again!
--
Jay Miller
PGP Fingerprint | 5f7adbbe bfc60727 96dca94c 616d5080 09e3e846
===
Subject: Re: (mod 2^4096) on a 4096 bit MAA?
Words by Peter Webb :
>Isnt the mod 4096 of a binary number just the last 11 bits
of the number?
>Cant you just mask off all the higher bits?
Doh! Thats an embarrassingly simple answer.
--
Jay Miller
PGP Fingerprint | 5f7adbbe bfc60727 96dca94c 616d5080 09e3e846
===
Subject: Re: factoring GCF
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9BE82C02901;
>Given something like,
>determine the GCF of each pair of terms:
>x(x + 3) and 2(x + 3)
>Why is the greatest common factor x + 3?
>Why is it a single factor? Why are two
>mutually exclusive terms considered a single
>factor?
May I suggest that you start by learning the denition?
The GCD (or GCF as you call it) of A and B is the largest
integer
that divides both A and B.
(x+3) is the GCF of x(x+3) and 2(x+3) because it is the
largest integer that divides both. Why is this a mystery?
What do you mean by single factor? A factor is a factor is a
factor...
What do you mean by mutually exclusive terms? To what two
mutually exclusive terms do you refer?
There is a language difculty here. You are using terminology
that is vague, ambiguous, and devoid of mathematical meaning.
>x(x + 3) is x^2 + 3x and 2(x + 3) is 2x + 6
>from the former, the greatest common factor
>is x and in the latter, the GCF is 2?
Huh? GCF refers to the greatest common factor of two
integers. When you say from the former, I assume you mean
x(x+3). This is a single integer. It does not have a GCF
all by itself. 2x+6 is also a single integer. It too does not
have a GCF. However, the two integers x(x+3) and 2x+6 can have
and do have a GCF. Why is this a mystery?
>Other examples: x(x - 2) and x - 2
>GCF: x - 2
>Another:
>2(x + y) and 3x(x + y)
>And guess what the GCF is?
>x + y! No thought whatsoever.
>Im just recognizing the pattern
I suggest that you forget about pattern and learn the
denition. If you believe that you can rely on pattern,
please explain how to nd the GCF of x^3-5x^2+11x-6 and
x^2 - 5x+ 6 by nding the pattern?
===
Subject: Re: factoring GCF
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9BE82h02888;
>Given something like,
>determine the GCF of each pair of terms:
>x(x + 3) and 2(x + 3)
>Why is the greatest common factor x + 3?
This is only true iff x is odd. Otherwise the
GCD is equal to 2(x+3).
>Why is it a single factor? Why are two
>mutually exclusive terms considered a single
>factor?
>x(x + 3) is x^2 + 3x and 2(x + 3) is 2x + 6
>from the former, the greatest common factor
>is x and in the latter, the GCF is 2?
>Other examples: x(x - 2) and x - 2
>GCF: x - 2
>Another:
>2(x + y) and 3x(x + y)
>And guess what the GCF is?
>x + y! No thought whatsoever.
>Im just recognizing the pattern and
>Im not satisied with this or the lack
>of explanation offered by my book. They
>just seem to present it and for you to recognize
>the pattern and use that to solve the problem.
>But I am not satisied with this. I am a pedant.
>And I want to know the specics behind why the GCF
>from any distributive form is always two terms
>considered as a single factor.
>If I see: 6x + 18 then I know the GCF
>of each term is 3. and that I factor out
>the GCF with the distributive property:
>6 * x + 6 * 3 = 6(x + 3)
>But with something like this:
>x(5x - 2) + 7(5x - 2) the GCF is
>(5x - 2). However, watch this:
>x(5x - 2) = 5x^2 - 2x. GCF = x
>and 7(5x - 2) = 35x - 14. GCF = 7
>Hm. Odd. Two different GCFs, x and 7.
>so how do they gure (5x - 2) is the GCF?
I think youll nd what you are looking for.
Here is an introduction:
http://www.cut-the-knot.org/blue/Euclid.shtml
Geert
===
Subject: Re: Comma category
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9BE84302974;
>If X is any category, then a representing object for
> a |--> X^{(-)->(0)<-(+)} (Diag(a), x -f-> z <-g- y)
>is just the pullback of f and g. However, F|G is not the
>pullback of F and G. It *is* however a pullback of the
>diagram
> F dom cod G
> C ---> E <--- E^2 ---> E <--- D
>and while were at it we might as well cast this in the
>language of weighted limits: take V = Cat (where V-categories
>are now *2-categories*), let J be the category (-) -> (0) <-
(+)
>(considered as a 2-category by viewing the hom-sets as
discrete
>categories), and let F: J --> V be the functor (or 2-functor)
>into Cat which sends (+) and (-) to the terminal category 1,
>(0) to the category 2 = (0 -> 1), the arrow (-) -> (0) to the
>inclusion i_0: 1 --> 2 valued at 0, and the arrow (+) -> (0)
>to the inclusion i_1: 1 --> 2 valued at 1.
>Then given a 2-category X, and a 2-functor G: J --> X (viz.,
>a diagram of 1-cells C -F-> E <-G- D in X), a limit of G
>w.r.t. the weight F dened above is called a *comma object*
>F|G in X. I believe it is also called a *lax pullback* of F
>and G, although I tend not to like such terminology, as lax
>is overused, sometimes confusingly so.
>Perhaps this is the type of statement you were aiming for?
>Todd Trimble
Bleah, bad choice of notation; both F and G used twice. Call
the diagram of 1-cells C -f-> E <-g- D. Then a limit of G
w.r.t. the weight F dened above is called a *comma object*
f|g in X, etc.
===
Subject: Re: Natural Densities are Probabilities
Cant resist ...
In message  Robin Chapman:
> By intuition, we expect that the probability that a
randomly chosen
> integer is even equals 1/2, but then we have the following
theorem.
> intuition? You need to say what it means for an integer
> to be randomly chosen.
Take a dice and throw it. That is randomly chosen from
{1,2,3,4,5,6}
As a professional mathematician (: Ive looked it up), you
should have
no problems with generalizing the simple concept of throwing
a dice.
Have you ?
> The probability that a natural is divisible by a is NOT
1/a .
> What is The probability ?
The same as The natural density. Read my drivel (as you call
it).
> A proof of this can be found here (I am not the author):
> No scare quotes necessary: this is a valid proof [ ... ]
This is NOT a valid proof. Valid proofs cannot lead to a
false results.
> Unwanted? Unwanted by you maybe, but your wants are
irrelevant to
> mathematics.
Time will learn if my wants are irrelevant to mathematics.
> The natural density dA of a sequence A of naturals is:
> dA = lim (x->innity) 1/x # { n <= x : n in A)
> Make A a subset of N (not a sequence) and add the proviso
that
> this limit exists (not automatic).
Overkill. So called mathematical precision. Yaaawn ...
> For example, if you ip a coin [ ... ]
> If you toss it innitely many times. Why dont you go and
do so,
> and report to us when youve nished?
If you make a bijection between all even and all natural
numbers.
Why dont you go and do so, and report to us when youve
nished?
> What the are you talking about. The denition of density
> is a simple mathematical denition, and is useful [ ... ]
Its useful because its simply a re-denition of something
that
we know for centuries: how chances converge to probabilities.
> [ ... ] for the sole reason you dont like it ?
Oh no ! Id like to have another chance ;-)
> Consequently, there must be a aw in the abovementioned
proof
> Because, otherwise, it would be impossible to even ip a
coin.
> Your nal sentence is a nonsequitur.
My nal sequence is a sequitur. Throwing a dice is described
quite
accurately by a natural density. Of course it is !
> http://www.math.uni-frankfurt.de/~steuding/steuding.shtml
Han de Bruijn
===
Subject: Re: Natural Densities are Probabilities
> Cant resist ...
> In message  Robin Chapman:
> By intuition, we expect that the probability that a
randomly chosen
> integer is even equals 1/2, but then we have the
following theorem.
> intuition? You need to say what it means for an integer
> to be randomly chosen.
> Take a dice and throw it. That is randomly chosen from
{1,2,3,4,5,6}
And obviously anything that works on nite sets must
work in innite sets, right?
Dene the procedure for generating an random integer which
could be chosen from anywhere in N.
> As a professional mathematician (: Ive looked it up), you
should have
> no problems with generalizing the simple concept of
throwing a dice.
> Have you ?
The simple concept of dice generalizes very nicely to
uniformly
choosing from among n possibilities, for any *nite* value
of n. You can also consider the continuous uniform
distribution,
uniformly distributed on [a,b] to be a generalization of the
dice throw. But that doesnt extend to innite intervals
either.
The question was to you.
> intuition? You need to say what it means for an integer
> to be randomly chosen
- Randy
===
Subject: Re: Natural Densities are Probabilities
> Cant resist ...
> In message  Robin Chapman:
> By intuition, we expect that the probability that a
randomly chosen
> integer is even equals 1/2, but then we have the
following theorem.
> intuition? You need to say what it means for an integer
> to be randomly chosen.
> Take a dice and throw it. That is randomly chosen from
{1,2,3,4,5,6}
A die (dice is singular).
So then, I presume, you are choosing 1 with probability 1/6,
2 with probability 1/6, 3 with probability 1/6, 4 with
probability 1/6,
5 with probability 1/6 and 6 with probability 1/6. So
the probability that you choose an even number is 1/6 + 1/6 +
1/6 =
1/2,
the probability that you choose a multiple of three is 1/6 +
1/6 = 1/3,
the probability that you choose a multiple of four is 1/6,
the probability that you choose a multiple of ve is 1/6,
the probability that you choose a multiple of six is 1/6
the probability that you choose a multiple of seven is zero
etc.
> As a professional mathematician (: Ive looked it up), you
should have
> no problems with generalizing the simple concept of
throwing a dice.
> Have you ?
Throwing a die is a physical, not a mathematical process.
> The probability that a natural is divisible by a is NOT
1/a .
> What is The probability ?
> The same as The natural density. Read my drivel (as you
call it).
I did, and if you admit that what you write is drivel, then
please cease
posting it.
And also you admit that natural density is a valid concept.
> A proof of this can be found here (I am not the author):
> No scare quotes necessary: this is a valid proof [ ... ]
> This is NOT a valid proof. Valid proofs cannot lead to a
false results.
It is a valid proof. It leads to no false results.
Are you too stupid to understand it?
> Unwanted? Unwanted by you maybe, but your wants are
irrelevant to
> mathematics.
> Time will learn if my wants are irrelevant to mathematics.
No time needed. No ones wants are relevant to mathematics.
> The natural density dA of a sequence A of naturals is:
>
> dA = lim (x->innity) 1/x # { n <= x : n in A)
> Make A a subset of N (not a sequence) and add the proviso
that
> this limit exists (not automatic).
> Overkill. So called mathematical precision. Yaaawn ...
Is that your usual response when confronted with mathematical
rigour?
A bit inadequate of you, isnt it?
If you are not interested in mathematics, why post here?
> For example, if you ip a coin [ ... ]
> If you toss it innitely many times. Why dont you go and
do so,
> and report to us when youve nished?
Done it yet?
> If you make a bijection between all even and all natural
numbers.
> Why dont you go and do so, and report to us when youve
nished?
f(n) = n/2. Done!
> What the are you talking about. The denition of density
> is a simple mathematical denition, and is useful [ ... ]
> Its useful because its simply a re-denition of something
that
> we know for centuries: how chances converge to
probabilities.
?
Anyway, what are chances save for a term in your private
language?
> [ ... ] for the sole reason you dont like it ?
> Oh no ! Id like to have another chance ;-)
Another chance at posting drivel :-(
> Consequently, there must be a aw in the abovementioned
proof
> Because, otherwise, it would be impossible to even ip a
coin.
> Your nal sentence is a nonsequitur.
> My nal sequence is a sequitur. Throwing a dice is
described quite
> accurately by a natural density. Of course it is !
No, it is a non sequitur. The nonexistence of a probability
distribution with certain properties has nothing to do
with physical actions such as throwing a die. Hold on, I
already
explained this! Why am I doing it again? If an idiot is too
thick to
understand this the rst time, hes not going to understand
it again.
> http://www.math.uni-frankfurt.de/~steuding/steuding.shtml
Did you seek Dr Steudings permission to pirate his MS onto
your website?
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square
root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Re: Average Length of a line
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9BEVNN05394;
>Hi;
> Suppose we have an isosceles triangle, with the base angles
equaling 30
degrees and a height of 1. Naturally the apex angle is equal
to 120 degrees.
If I draw a random line from the apex of the triangle to the
base what is the
average length of that line? Now obviously the minimum length
is equal to 1
and the maximum length is one of the sides of the triangle
which is 2. I get
an average value of the line of about 1.25. I do not know how
to get the
exact analytical answer. Can someone help?
>Angela
First you need to know the probability density function of
angles or
of termination points on the baseline.
phil
===
Subject: Re: Average Length of a line
>Hi;
> Suppose we have an isosceles triangle, with the base
angles equaling 30
degrees and a height of 1. Naturally the apex angle is equal
to 120 degrees.
If I draw a random line from the apex of the triangle to the
base what is the
average length of that line? Now obviously the minimum length
is equal to 1
and the maximum length is one of the sides of the triangle
which is 2. I get
an average value of the line of about 1.25. I do not know how
to get the
exact analytical answer. Can someone help?
>Angela
> First you need to know the probability density function of
angles or
> of termination points on the baseline.
The length of that line would be the secant of the angle from
the
vertical.*
Then, the area under the curve of the length function from 0
to PI/3
(60 degrees) would be
int(sec(t),t=0..(PI/3)) = 1.316957...
Then the average height would be that gure divided by the
angle
change, PI/3, or
1.257602...
--
john
* picture half the isosceles triangle as an angle; then the
vertical
line from the apex to the base, of length 1, is the adjacent
side of the
angle, and the unknown length is hypotenuse. hyp/adj =
secant, and the
denominator is 1, so the unknown length is just the secant of
an angle
going from 0 to 60 degrees.
===
Subject: Re: Triples correspond to sequences
>For every triple of positive integers (a,b,c) there is
associated a
sequence {
>(a^n + b^n) mod c }.
>Is this a one-to-one correspondence?
>With the condition a < b < c < (a+b), is this one-to-one?
>With the addtional condition gcd(a,b,c), how about now?
The triples ( x(x+y)z+1, (x+2y)(x+y)z+1, 2z(x+y)^2 )
and ( x(2x+y)z+1, (x+y)(2x+y)z+1, z(2x+y)^2 )
all give the sequence 2, 2, 2 ...
This is the only innite set of triples I can nd which yield
the
same sequence.
You can get lots of nite sets by taking sequences of period
2, say
2, k, 2, k ... Then a+b = c+k and c | 2a^2 - 2ak + k^2 - 2 and
c | 3ka^2 - 3k^2a + k^3 - 1. Hence c | k(k^2-4).
Choose c > k and c | k(k^2-4). Now choose a so that k < a <
(c+k)/2
and c | 2a^2 - 2ak + k^2 - 2. Then the triple (a, c+k-a, c)
will give
the sequence 2, k, 2, k ...
For example, 2, 3, 2, 3 ... yields (4, 14, 15) only, but
2, 4, 2, 4 yields (5, 7, 8), (5, 11, 12), (5, 23, 24) and
(11, 17, 24).
Mike Guy
===
Subject: Does this have a name?
Is there a name for the series of numbers a[n] such that:
a[1] = 1
a[n] = lcm(a[n-1], n) = lcm(n!, n)
(lcm - lowest common multiple)
i.e. a[n] is the smallest number that has 1, 2, ...., n as
factors.
Up to n = 40 it seems to be (very) approximately equal to
exp(n).
Just curious.
Jon
===
Subject: Re: Does this have a name?
> Is there a name for the series of numbers a[n] such that:
> a[1] = 1
> a[n] = lcm(a[n-1], n) = lcm(n!, n)
> (lcm - lowest common multiple)
> i.e. a[n] is the smallest number that has 1, 2, ...., n as
factors.
The On-Line Encyclopedia of Integer Sequences doesnt seem to
have a
specic name for it:
http://www.research.att.com/projects/OEIS?Anum=A003418
--
Daniel W. Johnson
panoptes@iquest.net
http://members.iquest.net/~panoptes/
039 53 36 N / 086 11 55 W
===
Subject: Re: Does this have a name?
> Is there a name for the series of numbers a[n] such that:
> a[1] = 1
> a[n] = lcm(a[n-1], n)
[...]
> (lcm - lowest common multiple)
> i.e. a[n] is the smallest number that has 1, 2, ...., n as
factors.
> Up to n = 40 it seems to be (very) approximately equal to
exp(n).
> Just curious.
> Jon
Yes, it is called U(x), Hardy & Wright p. 340, but more
common is its
logarithm, psi(x) = log(U(x)). Used in H&Ws proof of the
Prime
Number Theorem.
For any a > 1, U(x) < exp(ax) for large x;
for any a < 1, U(x) > exp(ax) for large x.
--
G. A. Edgar
http://www.math.ohio-state.edu/~edgar/
===
Subject: Re: i dont like irrationals
>So. I accept only Natural numbers. Even i had
Should a base-10 positional number system also be prohibited,
or
should we use IIIIIIIIIIIIIIIIIIII for 20?
-
http://mysite.verizon.net/vze8adrh/news.html (prole)
--Tim923 My email is
valid.
===
Subject: Re: i dont like irrationals
Is that really a rational position to take?
--irascible since 1957
===
Subject: Re: i dont like irrationals




Discussion, linux)
>The matererial i have been composed burned about 6 billion
years ago
>in supernova. It is the only place in this universum where
heavier
>atoms are made of from less heavier.
> Think again.
Again? Thats pretty presumptuous.
--
Jesse F. Hughes
Now pure math makes sense as well as clearly its a peacock
game,
where some of you see it as a way to show you as being highly
intelligent and thus more desirable to women. -- James S.
Harris
===
Subject: Re: i dont like irrationals
>The matererial i have been composed burned about 6 billion
years ago
>in supernova. It is the only place in this universum where
heavier
>atoms are made of from less heavier.
> Think again.
>Again? Thats pretty presumptuous.
<
--
Stan Brown, Oak Road Systems, Tompkins County, New York, USA
http://OakRoadSystems.com
Fortunately, I live in the United States of America, where we
are
gradually coming to understand that nothing we do is ever our
fault, especially if it is really stupid. --Dave Barry
===
Subject: Re: i dont like irrationals
>The matererial i have been composed burned about 6 billion
years ago
>in supernova. It is the only place in this universum where
heavier
>atoms are made of from less heavier.
> Think again.
> The source of heat in our sun is the creation of helium
atoms
> (atomic mass 4) from hydrogen atoms (atomic mass 1). There
are
> heavier atoms also being made, but I believe that process
is much
> less common in our star.
> The process of fusion is also the source of the energy of a
> hydrogen bomb, which is essentially a little bit of a star
not
> supernova) shining briey.
> If you had said supernovas were the only place in the
universe where
> elements heavier than _iron_ were made from less heavy
elements,
> then I believe youd be right. But ordinary stars make the
elements
> up through iron quite well.
Not quite. Ordinary, low-mass stars like the sun only get up
through
oxygen. High-mass stars do get up through iron from ordinary
fusion,
before going supernova. And then everything heavier than iron
comes
from the supernova itself.
Keckman was largely correct in saying that heavier elements
making up
Earth come from supernovae; although low-mass stars get up
through
oxygen, it gets locked up in white dwarfs. Only supernovae
DISTRIBUTE
the heavy elements throughout space.
But youre right: the claim [A supernova] is the only place
in this
universum where heavier atoms are made of from less heavier is
big bang nucleosynthesis.
But why count your spiritual birth from when your atoms were
formed?
rst 10 microseconds.
===
Subject: Re: i dont like irrationals
> That sure makes it easy but ancient. Calculus and
Engineering and
> Physics couldnt exist.
Ratios exist and they are wery usefull
but they are not numbers. They are results.
What is number? In a deepest i think that there does not exist
numbers either. Only thing that exist is logic.
Like in computers we use on/off logic. There is no numbers
before human come and make a denation that those bits queues
represent something we call numbers.
--
amount and biggest
1+1+1+...= innite and nite in math today
Petri Keckman
===
Subject: Re: i dont like irrationals
>Ratios exist and they are wery usefull
>but they are not numbers. They are results.
So 1/2 is not a number?
I cant gure out whether youre actually trying to say
something
sensible but just saying it REALLY badly, or youre
deliberately
trolling, or youre severely challenged intellectually.
--
Stan Brown, Oak Road Systems, Tompkins County, New York, USA
http://OakRoadSystems.com
Fortunately, I live in the United States of America, where we
are
gradually coming to understand that nothing we do is ever our
fault, especially if it is really stupid. --Dave Barry
===
Subject: Re: i dont like irrationals
>Ratios exist and they are wery usefull
>but they are not numbers. They are results.
> So 1/2 is not a number?
> I cant gure out whether youre actually trying to say
something
> sensible but just saying it REALLY badly, or youre
deliberately
> trolling, or youre severely challenged intellectually.
The two last statement hit the mark. Keckman has been
trolling at the Finnish math newsgroup as well, and he
isnt making any more sense, when hes using his native
language. Take my word for that:)
Just ignore him and all the other trolls that dont have
the honest desire to learn and understand. Initially a lot
of people feel like helping someone having trouble at some
point. After their responses to a couple of such explanations
their true color shows.
Jyrki Lahtonen, Turku, Finland
===
Subject: Re: i dont like irrationals
> [...]
Indeed, sci.math is no place for irrationals
(and other stupid posters).
Wlod
===
Subject: Rigid congurations of unit circles in squares
> I must say that that packing
Of 24 circles rigidly in a square at

Its been many months since Id looked at the site. Im
pleased to see new
rigid congurations of 13, 14, 15 and 23 circles by Craig
Clapp, replacing
previous unimaginative congurations of mine.
> is, in my (not exactly humble) opinion,
> extraordinarily beautiful!
> Very interesting.
Here are a few comments which might be interesting:
1. This type of problem is of course very different from that
of an
ordinary packing; in the former, we wish to maximize the
enclosing gure,
while in the latter we wish to minimize it. Indeed, I wonder
if these
things which Erich calls rigid packings are properly packings
at all. I
think that rigid congurations might be better. (Does
somebody have a
still better name? Or are they truly packings?)
2. I must suppose that, for most of the rigid congurations
shown at
Erichs site, rigidity has not been proven. Indeed, its
conceivable that
some of them might not actually be rigid. Note that, by just
looking at a
conguration, one can be easily deceived into thinking that
its rigid
when in fact its not. As an example, looking at Clapps rigid
conguration of 13 circles, remove the three circles in the
lower left
corner, leaving a conguration of 10 circles. Does it look
rigid?
3. Erichs site doesnt show alternatives. As an example, if
you look at
my rigid conguration of 11 circles, theres a spot where
another circle
can be snuggly inserted. Doing so gives an alternative
conguration of 12
circles.
4. You may have noticed from the above that the side length
of the squares
is not strictly increasing with the number of circles. But
its worst
than that. Side length sometimes _decreases_, something which
never happens
with ordinary packings of course. For the simplest example of
that, look at
Erichs rigid congurations of 5 and 6 circles.
5. Can computers help us to nd rigid congurations of
circles? Programs
like that at Dave Bolls

revolutionized ordinary packings of circles. Maybe something
similar could
be done for rigid congurations.
6. Is there an existence problem? (Of course there can be no
such problem
for ordinary packings.) For a given number n of circles, can
we be sure
that a rigid conguration exists? So that people understand
what Im
talking about, it might be good to look at some ordinary
packings of
circles in squares at, say,

noting the loose
circles (shown in magenta) called rattlers. Maybe there is a
simple
argument that rigid congurations must exist for all n, but
such an
argument hasnt occurred to me yet. And if it happens that
there is no
rigid conguration of some n unit circles in a square, what
is the
smallest such n?
> At rst it looked to me like that one was the rst
> one with some symmetry (since the simple cases) but then I
noticed that
> they are all symmetric around the diagonal. Reason? It sort
of makes
> sense that a number like 24 with lots of divisors would
have more
> symmetry, but then why 17?
My question about 17 is Why the heck didnt I nd that
conguration
earlier myself?! I believe that I probably did consider it,
but
mistakenly
discarded it, thinking that it wasnt rigid. Anyway,
concerning symmetry,
theyre not all symmetric about a diagonal; note that Clapps
conguration
of 14 circles has no symmetry of any sort.
Finally, let me encourage readers to try their own hands at
lling in gaps
or improving existing packings, coverings, rigid
congurations, etc. at
Erichs site.
David Cantrell
===
Subject: Re: Rigid congurations of unit circles in squares
|2. I must suppose that, for most of the rigid congurations
shown at
|Erichs site, rigidity has not been proven. Indeed, its
conceivable that
|some of them might not actually be rigid. Note that, by just
looking at a
|conguration, one can be easily deceived into thinking that
its rigid
|when in fact its not.
Im guessing from this that its not sufcient that its not
possible
to move any one disk by itself, that there are nonrigid cases
where
the only motion possible involves more than one circle. True?
Keith Ramsay
===
Subject: Re: Rigid congurations of unit circles in squares
> |2. I must suppose that, for most of the rigid
congurations shown at
> |Erichs site, rigidity has not been proven. Indeed, its
conceivable
> |that some of them might not actually be rigid. Note that,
by just
> |looking at a conguration, one can be easily deceived into
thinking
> |that its rigid when in fact its not.
> Im guessing from this that its not sufcient that its
not possible
> to move any one disk by itself, that there are nonrigid
cases where
> the only motion possible involves more than one circle.
True?
True. Thats the notion of rigidity used on Erichs site.
But there is a notion of rigidity which is different from
Erichs. That
alternative type of rigidity is far easier to establish. It
requires merely
that no single disk cant be moved if all the others are held
xed.
David
===
Subject: Re: Rigid congurations of unit circles in squares
Heres a response from Erich, which he said I could post here.
David
-------------------------------------------
> 1. This type of problem is of course very different from
that of an
> ordinary packing; in the former, we wish to maximize the
enclosing
> gure, while in the latter we wish to minimize it. Indeed,
I wonder
> if these things which Erich calls rigid packings are
properly
> packings at all. I think that rigid congurations might be
better.
> (Does somebody have a still better name? Or are they truly
packings?)
packing means tting inside without overlap, so they ARE
packings.
> 2. I must suppose that, for most of the rigid congurations
shown at
> Erichs site, rigidity has not been proven.
some have, some havent.
> 5. Can computers help us to nd rigid congurations of
circles?
> Programs like that at Dave Bolls

> revolutionized ordinary packings of circles.
the same sort of approach works. squish the circles until you
get
stuck. but this time you hope to get a LARGE value, not a
small one.
and computers can be used to check rigidity as well, by
checking all
the innitesimal perturbations.
> 6. Is there an existence problem?
it seems incredibly unlikely to me for circles. for other
shapes, i
can see this happening. but there IS an unsolved problem
there.
> Finally, let me encourage readers to try their own hands at
lling in
> gaps or improving existing packings, coverings, rigid
congurations,
> etc. at Erichs site.
most denitely!
erich
===
Subject: Re: Rigid congurations of unit circles in squares
>Heres a response from Erich, which he said I could post
here.
> 5. Can computers help us to nd rigid congurations of
circles?
> Programs like that at Dave Bolls

> revolutionized ordinary packings of circles.
>the same sort of approach works. squish the circles until
you get
>stuck. but this time you hope to get a LARGE value, not a
small one.
>and computers can be used to check rigidity as well, by
checking all
>the innitesimal perturbations.
I thought about that a little bit today.
Suppose you tried to handle the one-dimensional variant, with
just
two pieces. That is, you want to know the largest box
(=interval)
into which two 1-balls (=intervals) of radius 1 will t
rigidly.
What is there to pick here? If the enclosing box is the
interval
[0, L], for example, then you need to specify the centers of
the
balls, say x and y, each of which will be a number in [0, L].
There are three constraints:
(1) The balls dont stick out on the left: x-1 >= 0, y-1 >= 0.
(2) The balls dont stick out on the right: x+1 <= L, y+1 <=
L.
(3) The balls dont overlap: |x-y| >= 2 .
Well, there are lots of ways to put the 2 balls in the box, at
least if L is large enough. Each such conguration corresponds
to a point (x,y) in [0,L]^2, that is, we can visualize each
such conguration as a point inside a square in the rst
quadrant
of R^2. The three constraints tell us, respectively, that we
have to
stay away from the left and bottom edges; we have to stay
away from
the top and right edges; and we have to stay away from the
diagonal.
Thus the conguration space consists of two components: a
little
triangle near the top-left corner of the square, and another
triangle
near the bottom-right corner of the square. The fact that
there are
two components just reects the obvious fact that you cant
continuously move the left ball to the right of the right ball
while preserving the three constraints.
Now, as L decreases, the positions and sizes of these two
triangles
will change, until nally when L=4 we nd that they consist
only
of isolated points at (x,y) = (1,3) and (3,1). The fact that
the points
are isolated means there is no way to move the two balls at
all now,
consistent with the three constraints. (Were actually
relying here
on the fact that the constraints describe a space which is
locally
path connected!)
If you like, you can stack up the images of the regions (one
for each
value of L ) into a 3D shape whose horizontal cross-sections
are
the regions just described. The fact that we have two
triangles for
every L until L=4 just means that the complete 3D shape is a
union of two triangular cones resting with their vertices on
the
plane L=4. You can imagine lling the shape with water and
then
letting the pool evaporate; the last wet spots are the rigid
congurations.
You might want to play the same game with three intervals; the
conguration space corresponds to a cube with three fat cuts
made
along the planes x=y, x=z, and y=z. What remains is a set of
six
tetrahedra which, when L=6, reduce to six isolated points
(corresponding to the 3! permutations of the three intervals).
You can play the water-draining game if you can visualize the
assembly of these regions in R^4 ...
In exactly the same way, every placement of N ordinary disks
into
a box B in the plane corresponds to a point in B^N (which in
turn
is a subset of R^(2n) ). The constraints are nearly identical:
we must have each coordinate of each point p_i=(x_i,y_i) be
more
than 1 and less than L-1, and for any two points we must have
dist( p_i, p_j ) >= 2. This means that the set of possible
positions
of the N disks is an open subset of R^(2n) bounded by 4N
linear
hyperplanes and N(N-1)/2 quadratic hypersurfaces. When L is
sufciently large, this open set is (pathwise) connected --
clearly we
can slide the disks around from any non-overlapping
conguration to
any other. As L starts to shrink, these open sets also
shrink, and
what we are looking for is the values of L for which there are
isolated points. All you need to do, if you like, is to stack
the
level curves together to get a landscape in R^(2n+1); as the
water
drains out you get more and more isolated puddles, and we
just want
to nd the last places within each puddle to dry out. (Note
that
the landscape is not necessarily very regular: some of the
pits and
valleys will be deeper than others, so there can be rigid
congurations
which are packed into larger boxes than other rigid
congurations.)
Now, unless one of the puddle-bottoms happens to be at a place
where a quadratic surface has a rounded pit, we would expect
that
each quadratic constraint behaves locally like a linear one.
In
that case, as the puddle dries up, it would assume a small
polygonal
shape which generically speaking we would expect to have
(2N+1) sides.
That is, the boundary of the puddle ought, in general, to be
formed
by some set of (2N+1) of the hypersurfaces. When the puddle
nally
dries out, we would nd the point where these 2N+1 surfaces
come together.
So this would be a mechanism to nd rigid congurations: pick
any
set of 2N+1 constraints (which would typically include some
which
refer to L ) and see when they have a common solution. You
wouldnt
expect, in general, that you can satisfy more equations than
the
number of variables, but you can if L is just right. More
precisely,
you can eliminate L from the set of 2N+1 equations in 2N+1
unknowns
(including L) and get a single equation in L whose roots are
exactly the magic values where the 2N+1 equations do have a
common
solution in B^N. These Ls are the sizes of the boxes in the
rigid congurations. (The corresponding values of the other
variables
tell us where the N disks are located.)
For example, when N=2, there is a rigid conguration
corresponding to
the following 5 equations:
x1=1, y1=1, x2=L-1, y2=L-1, (x1-x2)^2+(y1-y2)^2=2^2
These equations in (x1,y1,x2,y2) are only consistent if L = 2
+- sqrt(2)
(and taking L = 2 - sqrt(2) leads to values of x2, y2 which
violate
one of the constraints not mentioned here). So there is an
N=2 rigid
conguration with a box of width 2 + sqrt(2).
Algebraically, then, all you need to do is to look at the
4N+N(N-1)/2 equations, consider each possible set of 2N+1 of
them,
and solve in each case for the 2N+1 variables; the winner is
the
combination which has the highest value of the L coordinate.
For example, there is no conguration with N=25 shown, but we
can nd one: we simply consider all subsets of 51 equations
from
among the 400 inequalities to be satised, in each case
solving
the 51 equations for the all values of the 51 unknowns...
(There are about 10^65 such cases to consider!)
Well, this is not only very hand-waving-ish, but I think its
actually
false, or at least hides some difculties beyond the obvious
ones.
For example, this reasoning suggests that in an optimal
conguration,
we ought to have 2N+1 of the inequalities turn in to
equations.
That is, there ought to be 2N+1 points of contact. Well,
thats not
too far off. I think I see
4*, 5, 7, 12*, 11, 16*, 15, 17, ...
points of contact in the pictures shown on
http://www.stetson.edu/~efriedma/rigid/
Thats indeed 2N+1 except in the starred cases, where there
are extra
points of contact.
So, can a computer really help? Well, yes and no. I think
whats true
is that if you have a gut sense of how the congurations
ought to
look, then you can use this method to nd the precise
coordinates
(they will all be algebraic) in an optimal conguration; you
dont
have to start drawing triangles and whatnot. I guess you can
even
prove rigidity in nondegenerate cases by showing that the
gradient
of each of the bounding constraints points up (that is, by
proving
that all the water nearby would drain into the puddle).
As far as _nding_ those rigid congurations, and especially
nding
the _optimal_ one, I am not optimistic. Clearly the number of
cases I
suggested above is a gross overestimate (we can, after all,
rely on
symmetry to reduce the number of cases by a factor of roughly
N!),
and no doubt one can intelligently sort through some
combinations of
constraints which are clearly inferior. But I would not
expect someone
to develop a standalone program which could regularly spit
out the
optimal congurations for the next two dozen cases.
Im not sure whether the cited web page is claiming that the
precise
values for L are known when N=9 or N=10. These could probably
be
computed precisely by this technique, and rigidity could be
proven.
I havent actually tried it, though...
dave
===
Subject: Re: Rigid congurations of unit circles in squares
Dave,
> So, can a computer really help? Well, yes and no. I think
whats true
> is that if you have a gut sense of how the congurations
ought to
> look, then you can use this method to nd the precise
coordinates
> (they will all be algebraic) in an optimal conguration;
you dont
> have to start drawing triangles and whatnot.
Sure. I used Mathematica for all the dirty algebraic work.
The key is to
know how the congurations ought to look rst, and thats were
nickels
on the kitchen table come in very handy.
> I guess you can even
> prove rigidity in nondegenerate cases by showing that the
gradient
> of each of the bounding constraints points up (that is, by
proving
> that all the water nearby would drain into the puddle).
Right. One could, at least in theory.
> As far as _nding_ those rigid congurations, and
especially nding
> the _optimal_ one, I am not optimistic. Clearly the number
of cases I
> suggested above is a gross overestimate (we can, after all,
rely on
> symmetry to reduce the number of cases by a factor of
roughly N!),
> and no doubt one can intelligently sort through some
combinations of
> constraints which are clearly inferior. But I would not
expect someone
> to develop a standalone program which could regularly spit
out the
> optimal congurations for the next two dozen cases.
I dont know about that. I wouldnt be extremely surprised if
someone
devised such a program.
> Im not sure whether the cited web page is claiming that
the precise
> values for L are known when N=9 or N=10.
Right. Theres no indication there, but thats only because
the precise
expression for side length s is too big to t conveniently.
Since N=10
is my conguration, I just looked back at the Mathematica
notebook which
I sent to Erich, and found s to be precisely
(7*(20 + Sqrt[2] + Sqrt[6]) + 2*Sqrt[98 + 98*Sqrt[2] +
77*Sqrt[3]
+ 42*Sqrt[6]])/28.
For N=9, I dont have the precise value at my ngertips
because that
wasnt my conguration. Erich might know it right away. In
any event, it
shouldnt be difcult to calculate.
> These could probably be
> computed precisely by this technique, and rigidity could be
proven.
> I havent actually tried it, though...
Id love to see a proof of rigidity for, say, Clapps
asymmetric N=14.
I suspect it would be horribly messy.
David
===
Subject: HELP on network stability
hi all,
I am working hard on a model and I need to know whether the
following
is true or false:
PROP:
- consider an undirected unweighted graph g; if g is stable
under
the payoff functions Y_i(g) (i index of nodes) then any of its
components must be complete -
DEF: a graph is stable under the payoff functions Y_i iff:
i) link ij does not belong to graph g implies
[Y_i(g+ij)>Y_i(g)
implies that Y_j(g+ij)If a triangle has three sides of length zero, can it still
be reasonably
>described as a triangle?
No. For one thing, it violates the triangle inequality which
states that
the sum of any two sides is greater than the remaining side.
Another problem you would have is that the angles would be
indeterminate.
You could argue that triangle AAA is an equilateral and a
right triangle at
the same time.
--Keith Lewis klewis {at} mitre.org
The above may not (yet) represent the opinions of my employer.
===
Subject: Re: Triangles with zero-length sides
>If a triangle has three sides of length zero, can it still
be reasonably
>described as a triangle?
> No. For one thing, it violates the triangle inequality
which states that
> the sum of any two sides is greater than the remaining side.
Doesnt the triangle inequality say greater than or equal to?
===
Subject: Re: Triangles with zero-length sides

<416B164F.DD78F6A3@spambtinternet.com.invalid>
>If a triangle has three sides of length zero, can it
still be
reasonably
>described as a triangle?
> No. For one thing, it violates the triangle inequality
which states
that
> the sum of any two sides is greater than the remaining
side.
>Doesnt the triangle inequality say greater than or equal to?
Youre right. The inequality Im talking about must be called
something
else.
--Keith Lewis klewis {at} mitre.org
The above may not (yet) represent the opinions of my employer.
===
Subject: Re: Triangles with zero-length sides
> Perhaps a strange question...
No perhaps about it.
> If a triangle has three sides of length zero, can it still
be reasonably
> described as a triangle?
That depends on your denition of reasonably. ;-)
Indeed, a triangle with just one side of length zero would
normally be
thought of as a digon, instead of as a triangle.
David Cantrell
===
Subject: Re: Triangles with zero-length sides
> Perhaps a strange question...
>No perhaps about it.
> If a triangle has three sides of length zero, can it still
be reasonably
> described as a triangle?
>That depends on your denition of reasonably. ;-)
>Indeed, a triangle with just one side of length zero would
normally be
But, for instance, it is very reasonable to consider (say)
the set,
or space, of unordered triples of points in R^2 (or R^n) and
not
entirely unreasonable (depending on the application in view)
to
call a point of that set, or space, a triangle in R^2 (or
R^n).
Then the triples with 2 or 3 coincidences would be degenerate
triangles, of course.
Lee Rudolph
===
Subject: Re: Triangles with zero-length sides
> Perhaps a strange question...
>No perhaps about it.
> If a triangle has three sides of length zero, can it
still be
> reasonably described as a triangle?
>That depends on your denition of reasonably. ;-)
>Indeed, a triangle with just one side of length zero would
normally be
> But, for instance, it is very reasonable to consider (say)
the set,
> or space, of unordered triples of points in R^2 (or R^n)
and not
> entirely unreasonable (depending on the application in
view) to
> call a point of that set, or space, a triangle in R^2 (or
R^n).
> Then the triples with 2 or 3 coincidences would be
degenerate
> triangles, of course.
I agree.
David
===
Subject: Re: THE THREE LAWS OF THOUGHT
I am really greatful for your kind response.
> The basis for mathematics was never decided. [...]
Right. If I may quote another poster I tremendously respect
and value,
it is turtles all the way down (btw, I do not even know what
it
really means).
Hopefully, you really can forgive me not quoting the learned
discourse
you kindly included in your correspondence. I must admit that
reading
your posts always gives me a thrill of excitement, I think to
myself
incredible.
*NO*, I WILL *NOT* DARE contradict your rejecting the
necessity for
rst reecting on the idea of existence while working on the
principles. Please, kindly assume, that is not my intention
*WHATSOEVER*.
Would you be kind enough as to comment on the importance of
the ve
laws of thought (existence, philosophy, science, you name it)
for
mathematics, and I quote from Heidegger again for convenience:
[1] The principle of ground
[2] The principle of difference
[3] The principle of identity
[4] The principle of contradiction
[5] The principle of excluded middle
Firstly, I would *NEVER* *EVER* believe these are the basic
laws of
thought unless I had read them in Heideggers work (which I
had) or
had been told by Some Posters. :-)
Secondly, it makes me very sad that authors of books on the
philosophy
of mathematics seem to assume the extensive knowledge of these
principles on the part of the reader, and make only very few
and
(please, forgive the word) casual remarks as to their
principial
importance, source and TRUE meaning. That, to my deepest
sadness,
pertains to Russells PM.
principles of philosophy (The principle of the ground, Die
Frage nach
dem Ding). Please, kindly note he (at least formally) was a
mathie
(too).
Fourthly, Frege uses the the [4]th principle as means for
dening the
You) he later uses for creating his system of arithmetic.
Lastly, principle no. [1] speaks of the Aristotles Logos, the
absolute source of truth, Freges Pure Thought, Heideggers
Dasein (to
mention the thinkers whose credibility is, imho, absolute).
Again, please read this post not as an attempt to contradict
you. My
knowledge will NEVER be as broad and detailed as yours, and
all i
can do is ask in case I am not sure. Shall I abandon a
detailed study
in this direction (and not read MHs Die Frage nach dem Ding,
An
introduction to metaphysics etc.)?
Tom
P.S. My delayed response is due to the assumption of there
being no
further responses to my previous posts. I am sorry.
===
Subject: Re: THE THREE LAWS OF THOUGHT
> I am really greatful for your kind response.
> The basis for mathematics was never decided. [...]
> Right. If I may quote another poster I tremendously respect
and value,
> it is turtles all the way down (btw, I do not even know
what it
> really means).
Google does.
--
Dave Seaman
Judge Yohns mistakes revealed in Mumia Abu-Jamal ruling.

===
Subject: Re: THE THREE LAWS OF THOUGHT
> I am really greatful for your kind response.
> The basis for mathematics was never decided. [...]
> Right. If I may quote another poster I tremendously
respect and value,
> it is turtles all the way down (btw, I do not even know
what it
> really means).
> Google does.
Stephen Hawking in A Brief History Of Time starts with the
anecdote. A
well-known
scientist (some say it was Bertrand Russell) once gave a
public lecture on
astronomy. He described how the earth orbits around the sun
and how the sun,
in
turn, orbits around the centre of a vast collection of stars
called our
galaxy.
At the end of the lecture, a little old lady at the back of
the room got up
and
said: What you have told us is rubbish. The world is really a
at plate
supported on the back of a giant tortoise.
The scientist gave a superior smile before replying, What is
the tortoise
standing on?
Youre very clever, young man, very clever, said the old
lady. But
its
turtles all the way down.
Theres a hole in the middle of the sea
Theres a hole in the middle of the sea
Theres a hole, theres a hole
Theres a hole in the middle of the sea
Theres a log in the hole in the middle of the sea
Theres a log in the hole in the middle of the sea
Theres a log, theres a log
Theres a log in the hole in the middle of the sea
Theres a bump on the log in the hole
In the middle of the sea
Theres a bump on the log in the hole
In the middle of the sea
Theres a bump, theres a bump
Theres a bump on the log in the hole
In the middle of the sea
Theres a frog on the bump on the log
In the hole in the middle of the sea
Theres a frog on the bump on the log
In the hole in the middle of the sea
Theres a frog, theres a frog
Theres a frog on the bump on the log
In the hole in the middle of the sea
Theres a y on the frog on the bump on the log
In the hole in the middle of the sea
Theres a y on the frog on the bump on the log
In the hole in the middle of the sea
Theres a y, theres a y
Theres a y on the frog on the bump on the log
In the hole in the middle of the sea
Theres a wing on the y on the frog
On the bump on the log in the hole
In the middle of the sea
Theres a wing on the y on the frog
On the bump on the log in the hole
In the middle of the sea
Theres a wing, theres a wing
Theres a wing on the y on the frog
On the bump on the log in the hole
In the middle of the sea
Theres a ea on the wing on the y
On the frog on the bump on the log
In the hole in the middle of the sea
Theres a ea on the wing on the y
On the frog on the bump on the log
In the hole in the middle of the sea
Theres a ea, theres a ea
Theres a ea on the wing on the y
On the frog on the bump on the log
In the hole in the middle of the sea?
> --
> Dave Seaman
> Judge Yohns mistakes revealed in Mumia Abu-Jamal ruling.
>

===
Subject: Re: THE THREE LAWS OF THOUGHT
> Youre very clever, young man, very clever, said the old
lady. But
> its turtles all the way down.
Perhaps to the innity?
That old lady is like math today.
Math have come to a conclusion, that from n=n+1 in Peanos
axiom follow
that
N is innity allthough that n=n+1 grows up as much bigness as
amount of
numbers.
Math has put to Natural numbers something that does not
comes from
Peanos axiom.
Because everything is relative you can allways multiple the
amount by 2.
The more far away you are the more you have to go.
Of course, im not saying that N is nite in a way that there
is biggest
number.
Just N->oo but not N=oo.
--
amount and bigness
1+1+1+...= innite and nite in math today
Petri Keckman
===
Subject: Re: THE THREE LAWS OF THOUGHT
> Youre very clever, young man, very clever, said the old
lady.
But
> its turtles all the way down.
> Perhaps to the innity?
> That old lady is like math today.
> Math have come to a conclusion, that from n=n+1 in Peanos
axiom follow
> that
> N is innity allthough that n=n+1 grows up as much bigness
as amount of
> numbers.
> Math has put to Natural numbers something that does not
comes from
> Peanos axiom.
> Because everything is relative you can allways multiple the
amount by 2.
> The more far away you are the more you have to go.
You might have a hard time showing that it is deductive that
You can
always
multiply the amount by two because that is an inductive
theory based upon
a
probability which you have not shown. You might or might not
be able to
always
multiply the remainder by 2 but it is not that case that it
is necessarily
the
case by denition that you can always continue multiplying or
not.
An antinomy produces a self-contradiction by accepted ways of
reasoning.
It
establishes that some tacit and trusted pattern of reasoning
must be made
Quine, in The Ways of Paradox (1966), p.7.
Antinomies are contradictions that Kant believed follow
necessarily from
our
attempts to conceive the nature of transcendent reality. Kant
thought the
Antinomies cannot be resolved and that attempts to conceive
the transcendent
will
always produce irresolvable contradictions. This does not
mean that there is
no
transcendent or that attempts to conceive the transcendent
are meaningless.
They
are, just as Kant said, necessitated by reason itself. It
does mean,
however,
that the transcendent defeats rational representation.
antinomies (conict of laws) which are usually described as
paradox or
contradiction. An example of one Kant sought to deal with
is whether the
universe has a beginning (rst cause) or whether it has
always existed.
The contradiction arises because valid arguments
can be made in favour of both views. If
unresolved this antimony could lead to the
euthanasia of pure reason (skepticism).
Thus Kant believed antinomies must be reconciled.
http://www.faithnet.org.uk/Philosophy/kant.htm
> Of course, im not saying that N is nite in a way that
there is biggest
> number.
> Just N->oo but not N=oo.
By denition that is a function. True. It is analytic in the
sense that
the
function is necessary by denition. But the ability to
continue this
function is
synthetic in the sense the what is in the predicate is not
necessarily
dened
even covertly by the subject and hence you must go outside
the subject to
determine the predicates value. For the function describes
one operation and
you
are adding to it by saying how many times this can happen and
without
evidence
loc.
> --
> amount and bigness
> 1+1+1+...= innite and nite in math today
> Petri Keckman
===
Subject: Re: BROUWER in dimension=2 + sequences
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9BGx3b18768;
>hi all,
>here is a rather interesting problem:
>consider a complex function f from the unit closed ball of
the
>complex plane into itself. Then we can dene the following
sequence:
>U(n+1)=U(n)+(1/n)*( f(U(n))-U(n) ) with arbitrary U(1) in
the unit ball.
>How could we prove that this sequence converges to a xed
point of f.
>I nd this result really amazing. What do you think about it
?
>any idea,
interesting, really. Though not true I think.
Your reccurence can be rewritten as: U[0] arbitrary,
U[n] = 1/n Sum[f[U[k]], k=0..n-1].
Thus its perfectly true for contractions - evident.
But I think that for special continuous maps (I suppose that
you
accidentally omitted the word continuous in your post),
something like
local anti-contractions (like |f(x)-f(y)| > k |x-y|, k>1) it
is not true,
although I failed to rigorously disprove it so far.
But do an experiment if you wish - try the map
/ 2*z*e^i for Abs[z]<=1/2
f(z) =
z/Abs[z]*e^i for Abs[z]>=1/2
this continuous map has only one xed point-zero, but your
sequence
doesnt seem to converge. (I used mma to compute some values
up to 10000).
If I will gure out the disproof, Ill add it in here.
highegg
===
Subject: Re: BROUWER in dimension=2 + sequences
>consider a complex function f from the unit closed ball of
the
>complex plane into itself. Then we can dene the following
sequence:
>U(n+1)=U(n)+(1/n)*( f(U(n))-U(n) ) with arbitrary U(1) in
the unit ball.
>How could we prove that this sequence converges to a xed
point of f.
>I nd this result really amazing. What do you think about
it ?
>interesting, really. Though not true I think.
>Your reccurence can be rewritten as: U[0] arbitrary,
>U[n] = 1/n Sum[f[U[k]], k=0..n-1].
>Thus its perfectly true for contractions - evident.
>But I think that for special continuous maps (I suppose that
you
>accidentally omitted the word continuous in your post),
something like
>local anti-contractions (like |f(x)-f(y)| > k |x-y|, k>1) it
is not
>true, although I failed to rigorously disprove it so far.
>But do an experiment if you wish - try the map
> / 2*z*e^i for Abs[z]<=1/2
>f(z) =
> z/Abs[z]*e^i for Abs[z]>=1/2
>this continuous map has only one xed point-zero, but your
sequence
>doesnt seem to converge. (I used mma to compute some values
up to
>10000). If I will gure out the disproof, Ill add it in
here.
Suppose 0 < |U[n]| <= 1/2. Then U[n+1] = (1 + (2 e^i-1)/n)
U[n].
Since Re(2 e^i - 1) > 0, |U[n+1]| > |U[n]|. So its impossible
for U[n] to converge to 0. Since thats the only xed point,
youre done.
I suspect, however, that the OP may have meant f to be
analytic
in the open unit disk. Then I suspect the result is true.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
===
Subject: Question on Rosser/Godel Theorems
The goal being the formalization of Rossers 1936 theorem
(extending
Godels theorem to apply to any consistent system) and its
reduction
to as simple a form as possible, consider the following two
programs A
and B (A1, B1 and B2 signify points in the programs):
A:
FOR every Proof X (coded as natural numbers)
IF X proves that A Loops THEN Halt [A1]
B:
FOR every Proof X (coded as natural numbers)
IF X proves that B Loops THEN Halt [B1]
IF X proves that B Halts THEN LOOP [B2]
Propositional variables A1, B1 and B2 state that the
corresponding
points in the programs are reached. CONS states that our
proof system
in consistent (provable and refutable are disjoint.) HALT(z)
and
LOOP(z) are the propositions that z halts and z loops,
respectively,
where z=A or B.
Now consider the following properties of A and B:
1. A1 = HALT(A)
2. A1 => |- LOOP(A)
3. HALT(B) => |- LOOP(B)
Explanation:
1: We reach point A1 in program A iff program A halts. That is
because we do halt if we reach point A1 and there is no other
point in
program A where we halt.
2: We reach point A1 only if we can prove that program A
loops. That
is because we must pass the IF X proves that A Loops test to
reach
point A1.
3: Program B halts only if we reach point B1, which is
reached only if
X proves that B Loops for some proof X.
Question: What other properties of A and B may we conclude?
C-B
(Note: A actually proves Smullyans Dual Form theorem, not
Godels 1st
Theorem based on soundness, despite what popular texts say.)
===
Subject: Optimization of integral
I am interested in proving the following conjecture. It seems
like it
should have a neat proof, as opposed to, say, using calculus
of
variations...
*************************************************************
**********
If G(x) is a given continuous function on x in [0,1] satisying
int_0^1 G(x) dx = 1
and
G(x) ne 0 on x in [0,1]
and Q(x) is allowed to be any continuous function on x in
[0,1]
satisying
int_0^1 Q(x) dx = 1
Then,
int_0^1 ( Q(x) / G(x) ) ^2 dx ge 1
with equality iff Q(x) = G(x) for all x in [0,1]
*************************************************************
Any ideas?
-- Mark
===
Subject: Re: Optimization of integral
> I am interested in proving the following conjecture. It
seems like it
> should have a neat proof, as opposed to, say, using
calculus of
> variations...
[stuff re the mistaken notion that int_0^1 G(x) dx = 1,
G(x) ne 0 on x in [0,1], int_0^1 Q(x) dx = 1, with
G and Q continuous, implies int_0^1 ( Q(x) / G(x) ) ^2 dx ge
1 ]
...
You posted about the same thing last week, but seem to have
counterexample, G(x) = x+(1/2); Q(x) = (7/4)x+(1/8), is
simpler than
mine (which Ive repeated below with minor changes) but my
example
shows that int_0^1 {(Q(x)/G(x))^2} can be arbitrarily close
to zero.
Let 1/12 > e > d > 0. Dene Q = 1/e - d/e for x <= e;
0 for x >= 4e/3; and continuous between (e, 1/e - d/e) and
(4e/3, 0)
with int_e^{4e/3} Q = d < e. Dene G = 1/(3e) for x <= 4e/3,
else
continuous with area 5/9 for x >= 4e/3. (Ie, int_{4e/3}^1 G =
5/9.)
So int_0^1 Q = int_0^1 G = 1 and G and Q are continuous on
[0,1].
Now Q/G <= 3 on [0,4e/3] and Q/G = 0 for x >4e/3, so
int_0^1 Q^2/G^2 <= 9*4e/3 = 12e < 1.
-jiw
===
Subject: Re: Optimization of integral
> I am interested in proving the following conjecture. It
seems like it
> should have a neat proof, as opposed to, say, using
calculus of
> variations...
>
*************************************************************
**********
> If G(x) is a given continuous function on x in [0,1]
satisying
> int_0^1 G(x) dx = 1
> and
> G(x) ne 0 on x in [0,1]
> and Q(x) is allowed to be any continuous function on x in
[0,1]
> satisying
> int_0^1 Q(x) dx = 1
> Then,
> int_0^1 ( Q(x) / G(x) ) ^2 dx ge 1
> with equality iff Q(x) = G(x) for all x in [0,1]
>
*************************************************************
> Any ideas?
> -- Mark
counterexample...
G(x) = x+(1/2); Q(x) = (7/4)x+(1/8);
--
G. A. Edgar
http://www.math.ohio-state.edu/~edgar/
===
Subject: Re: Methods that count primes without counting
primes or referring
to them...
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9BIPI426725;
Would it be safe to say, then, that if someone found a way to
express pi(x)
in a way that didnt refer at all to primes or lower values
of pi(x) or to
complex numbers or to calculus or even to irrational or
transcendental
numbers in any way, it would be a noteworthy mathematical
achievement, as
well as a highly improbable one?
The Searcher of the Erstwhile Random
>Yes, any exact calculation of pi(x) that I know of uses
methods
>like this. However, you dont ever have to have a list of
primes
>up to x, or even up to sqrt(x) (such as youd get from an
Erastosthenes
>sieve) in order to calculate pi(x). I think the best known
version
>of the recursive sum youre talking about only needs to go up
>to the cube root of x. For large x, the difference between
sqrt(x)
>and cub(x) is enormous. For x = 10^24, its the difference
between
>10^12 (storing that many 24-digit integers would require a
>trillion 80-bit values, or 10 terabytes) and 10^8 (which
would
>require a factor of 10000 less, or a gigabyte). So those kind
>of savings are the difference between doable and not-doable.
>We know pi(10^22) now (its tabulated at the Prime Pages) but
>we dont know pi(10^23).
> - Randy
===
Subject: Re: Methods that count primes without counting
primes or referring
to them...
> Would it be safe to say, then, that if someone found a way
to express
pi(x)
> in a way that didnt refer at all to primes or lower values
of pi(x) or to
> complex numbers or to calculus or even to irrational or
transcendental
> numbers in any way, it would be a noteworthy mathematical
achievement, as
> well as a highly improbable one?
No.
Heres a formula from p. 19 of (the highly recommended book)
Crandall
and Pomerance, Prime Numbers:
pi(n) = sum from 2 to n of [ ((j - 1)! + 1) / j ] - [ (j -
1)! / j ],
where [x] is the greatest integer not exceeding x.
I think this formula meets your criteria, it is not a
noteworthy
mathematical achievement, and it has probability 1.
More examples are given in the books exercises.
--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
===
Subject: Re: Math books online
>Hi everybody !
>Here is a lot of usefull mathematical books, lectures
notes,... for
>free and online :
>http://www.sciencedaily.com/directory/Science/Math/
Publications/Online_Text
s
There dont appear to be any online books (unless a link to
amazon
counts as an online book). The short news items are quite
interesting
though.
--
Jeremy Boden
===
Subject: Re: Math books online
>Hi everybody !
>Here is a lot of usefull mathematical books, lectures
notes,... for
>free and online :
>
>http://www.sciencedaily.com/directory/Science/Math/
Publications/Online_Texts
> There dont appear to be any online books (unless a link to
amazon
> counts as an online book). The short news items are quite
interesting
> though.
Try clicking on Links. (And those news items are
advertisements.)
cid ooh
===
Subject: Re: Measuring the randomness of a signal...
>I need an algorithm that can measure the randomness of a
signal (variable
>length set of real numbers).
>Idealy it would return a number between 0 and 1, where 0
represents a
>straight line and 1 represents white noise.
>I currently use a clumsy (and rather processor intensive)
method whereby I
>take a variety of exponential moving averages with different
smoothing
>factors and look at the intersections between them. Im sure
there must be
a
>far more well developed and formal way of measuring such a
thing.
The thing about randomness is you can never empirically prove
you have it,
only that you dont. So the best thing to do is run a bunch
of different
tests.
Heres one: Correllate the signal with itself, with a small
time shift.
If
you get +1 or -1 you have a perfectly repeating signal (and
your time shift
is related to the period of repetition); 0 indicates it could
be random.
--Keith Lewis klewis {at} mitre.org
The above may not (yet) represent the opinions of my employer.
===
Subject: Re: Measuring the randomness of a signal...
[...]
> Correllate the signal with itself, with a small time
> shift. If you get +1 or -1 you have a perfectly repeating
signal (and
your time
> shift is related to the period of repetition); 0 indicates
it could be
> random.
It is called autocorrelation test, but it is just a test (and
it can be
said weak). A stronger answer can be gotten using a good test
suite.
Cristiano
===
Subject: Re: Who thinks Goldbachs Conjecture is unprovable?
>I want to take an informal survey to nd out what people
believe:
>1) Goldbachs Conjecture is possible to prove.
>2) Golbachs Conjecture is impossible to prove but is
nevertheless
>true.
>3) Goldbachs Conjecture is impossible to prove, because
there is a
>counterexample.
>I am interested to hear any reasons for your beliefs.
>Craig
> Im inclined for the 2nd belief. I support the thesis that
the
> Goldbach
> Conjecture is a probabilistic assertion disguised in an
analytic
> language.
> Simply, its very improbable that suming all the pairs of
primes
> lesser than
> half an even number, then results that this number do not
appear as a
> sum.
> Empirically, if S is the even number, then the times it
will appear as
> a sum of two primes,is aproximately = S/(Log(S)*(Log(S)-1)).
> If that number is the mean of a Binomial Distribution, then
its very
> improbable that it can attain the frequency zero, (but not
> impossible).
> If we postulate that the Eratosthenes Sieve is equivalent
to a chaotic
> process, then its easy to show that the Goldbach Conjectute
is
> undemostrable.
But the Sieve of Eratosthenes has a wealth of relations and
is far from
chaotic.
I. Boxen
===
Subject: Federal Judge Rules in Favor of Exponential Function
( from the New Hampster: www.javaspider.com/tnh )
Federal judge Phil Orouti ruled in favor of the exponential
function [ticker EXP] in a closely watched antitrust case
Wednesday.
The court ruling did not deny charges that the exponential
had a
monopoly in functional behavior, but also did not move to
break up the
function into separate domains.
A spokesman for the leading competitor Bessel function group
[JV],
Arfken, was dissappointed in the ruling. The exponential
behavior not dependent on the exponential. We hope the appeal
process
can stop further predation in complex domains. Arfken also
complained the the law was unprepared for modern functional
behavior
ownership litigation, citing the relation:
[..]
Under the stewardship of Leonard Euler in the mid 1700s, the
exponential was able to execute a hostile takeover over of the
circular functions, including functional behavior giants sine
and
cosine, by exercising its power in the complex domain.
Leveraging of
its complex assets allowed the exponential to gain control of
frequency space transforms, including the well known brand
Fourier
[FFT]. Such manipulations with imaginary assets have since
become a
staple of modern nance.
Functional advocacy groups have differed in their criticism of
industry giant EXP. Some complain that use of the exponential
in the
complex plane is buggy, and users are unprepared for
treacherous
behaviour around so called branch cuts used to patch
discontinuities. Others complain that the exponentials
growth leads
to unsustainable behavior and eventual crashes. Some even
fear a
return to the mythical ination age, when the domination of
the
exponential affected the very fabric of space and time. None
the
less, most agree that the amazing popularity of the function
is due to
the expandability and diversity, enabling countless
innovations.
Spokesman for the exponential Lyapunov remained aloof and
condent in his comments after the court decision. df/dt = f,
he
said while walking toward his car. No further comment. His
security team then told us to take a drunken walk off a
natural log.
===
Subject: Re: Federal Judge Rules in Favor of Exponential
Function
> ( from the New Hampster: www.javaspider.com/tnh )
> Federal judge Phil Orouti ruled in favor of the exponential
> function [ticker EXP] in a closely watched antitrust case
Wednesday.
> The court ruling did not deny charges that the exponential
had a
> monopoly in functional behavior, but also did not move to
break up the
> function into separate domains.
> A spokesman for the leading competitor Bessel function
group [JV],
> Arfken, was dissappointed in the ruling. The exponential
> behavior not dependent on the exponential. We hope the
appeal process
> can stop further predation in complex domains. Arfken also
> complained the the law was unprepared for modern functional
behavior
> ownership litigation, citing the relation:
> [..]
> Under the stewardship of Leonard Euler in the mid 1700s, the
> exponential was able to execute a hostile takeover over of
the
> circular functions, including functional behavior giants
sine and
> cosine, by exercising its power in the complex domain.
Leveraging of
> its complex assets allowed the exponential to gain control
of
> frequency space transforms, including the well known brand
Fourier
> [FFT]. Such manipulations with imaginary assets have since
become a
> staple of modern nance.
> Functional advocacy groups have differed in their criticism
of
> industry giant EXP. Some complain that use of the
exponential in the
> complex plane is buggy, and users are unprepared for
treacherous
> behaviour around so called branch cuts used to patch
> discontinuities. Others complain that the exponentials
growth leads
> to unsustainable behavior and eventual crashes. Some even
fear a
> return to the mythical ination age, when the domination of
the
> exponential affected the very fabric of space and time.
None the
> less, most agree that the amazing popularity of the
function is due to
> the expandability and diversity, enabling countless
innovations.
> Spokesman for the exponential Lyapunov remained aloof and
> condent in his comments after the court decision. df/dt =
f, he
> said while walking toward his car. No further comment. His
> security team then told us to take a drunken walk off a
natural log.
Anyone brave enough to try this at the Improv? :-)
Shedar
===
Subject: Re: odd perfect
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9BJX5E32284;
> See, also, http:
//www.mersen
neforum.org/showthread.php?t=3101 and the
> recent NMBRTHRY archives.

>Indeed. And then grab an ECMNET client, and GMPECM, and then
point
>the client to 62.236.152.54 port 8192, and help crack the
last of
>the three numbers!
Why? I intend no disrespect, but all this will do is extend
the lower bound a little bit by purely computational means.
The
method for doing so is well established. Pushing the lower
bound
by extensive computation (which at this point is unlikely to
succeed given the number of ECM trials already expended) does
little
toward furthering a proof.
BTW, I am not the only one to say this. Richard Brent (who
established the
10^300 bound by computation means) said essentially
the same thing to me when I asked about re-running his code
given
recent factoring results. His opinion was that such pursuits
werent
really worth pursuing.
===
Subject: Yet ANOTHER lame math joke!! Argh!
Q: Why are the Japanese and Chinese so good at real analysis?
A: Their alphabets have a transnitude of symbols to use for
numbers!
Theologian: You mathematicians are blind. Dont you know man
is more
than just numbers?
Mathematician: Youre right! ...(prolonged pause)... man is
sets!
===
Subject: Re: Yet ANOTHER lame math joke!! Argh!
> Q: Why are the Japanese and Chinese so good at real
analysis?
> A: Their alphabets have a transnitude of symbols to use for
> numbers!
> Theologian: You mathematicians are blind. Dont you know
man is more
> than just numbers?
> Mathematician: Youre right! ...(prolonged pause)... man is
sets!
I believe the appropiate word here is groan.
^_^
-paul
===
Subject: Re: Yet ANOTHER lame math joke!! Argh!
> Q: Why are the Japanese and Chinese so good at real
analysis?
> A: Their alphabets have a transnitude of symbols to use for
> numbers!
> Theologian: You mathematicians are blind. Dont you know
man is more
> than just numbers?
> Mathematician: Youre right! ...(prolonged pause)... man is
sets!
Please, make the hurting stop.
Okay, there arent very many good math jokes out there. And
there still
arent. I do appreciate the effort, though.
- Tim
--
Timothy M. Brauch
NSF Fellow
Department of Mathematics
University of Louisville
email is:
news (dot) post (at) tbrauch (dot) com
===
Subject: Re: Question about induction argument
> The proposition:
> Let X be a nite collection of open intervals that cover a
closed
> interval I = [a0, b0]. Show that there exists a nite
subcollection
> Y = {(a_i, b_i): i = 1...m} of X such that Y covers I , a1
< a0
> < b1, a_m < b0 < b_m, and a_(k+1) < b_k < b_(k+1) for 1 <=
k <= m-1.
> Outline of the induction step: Assume true for any interval
closed by a
> cover with n intervals and that |X| = n+1. Let a be the
least
> value of lower bounds of intervals in X and let (a1, b1) be
any
> interval in X where a1 = a. Then X {(a1,b1)} covers [b1,
b0].
say b1 < a0.
Is it then possible for X (a1,b1) to not cover [b1, b0]?
Since a0
must belong to at least one (a_k , b_k) should we start with
a1 = a_k
===
Subject: Re: Question about induction argument
>Consider interval [a_0, b_0]. There is a collection {(a_j,
b_j):
>1<=j<=n} of (open) intervals such that
>
>Union (j=1 -> n) (a_j, b_j) contains [a_0, b_0].
>It is intuitively clear that there exists a subset of {(a_j,
b_j)},
>{(a_k, b_k): 1<= k <= m <= n} such that :
>a_1 < a_0 < b_1, a_m < b_0 < b_m and
>for m>1, a_(k+1) < b_k < b_(k+1), 1 <= k <= m-1
>How does one construct a formal induction argument to prove
this?
>The following is a rough proof ignoring the induction idea.
You may
>need to ll in a few details to make it rigorous. You know
that
>[a_0,b_0] is a subset of Union (j=1,n) of (a_j,b_j). There
are two
>possibilities:
>1) [a_0,b_0] is a subset of one of the (a_j,b_j). This
trivially
>satises the conditions.
>2) [a_0,b_0] is not a subset of any of the (a_j,b_j). In
this case,
>consider a minimal collection (with renumbering as
necessary) of
>(a_k,b_k) where k=1 to m, a_1(a_k,b_k) contains [a_0,b_0]. Now, to be a minimal
collection,
>b_1(a_(k+1),b_(k+1)). Now, suppose a_1a_0 < a_1, or b_1violating the fact that the union is a cover. In the second
case, the
>(a_k,b_k) where k=2 to m must be a cover, violating the
minimality.
>Similarly, a_m < b_0 < b_m. Now, suppose there is a k,
1<=k<=m-1,
>where a_(k+1) < b_k < b_(k+1) is not true. We know from
above that
>b_knot cover [b_k,a_(k+1)]. Since it does cover [a_0,b_0],
this must mean
>that the intersection of [b_k,a_(k+1)] and [a_0,b_0] is
empty. So,
>either b_1 b_1or b_0 b_0 < a_m, a contradiction with
the above.
>QED
>I think the only detail that could be strengthened is the
existence of
>the minimal covering of [a_0,b_0], but since there is a
nite cover of
>open sets to begin with, getting a minimal cover by
throwing out the
>excess sets shouldnt be too hard.
> more stringent than that of the original problem: In the
minimal
> collection, as you indicate, both a_js and b_js are
ordered,
> whereas in the original only the b_js are ordered. I guess
one would
> still require some kind of induction to establish its
existence. That
> is, one can somehow use induction to perform the throwing
out and
My own inclination would be to do another proof by
contradiction. Then
again, my inclination frequently includes proof by
contradiction. I
would try to show if you dont have the sequence with the
a_js and
b_js ordered that you have one of the (a_j,b_j) contained in
another,
violating the minimality.
--
Will Twentyman
email: wtwentyman at copper dot net
===
Subject: Introductory books on point processes?
Can anyone suggest an introductory book on point processes?
Im
especially interested in spatial and space-time point
processes.
-- Matti
===
Subject: Re: Introductory books on point processes?
I know a book by Cox/Isham: Point processes. It is an
introductory text
without too much mathematicel formalities, but not much about
the
special topics you mention.
Ciao
Karl
> Can anyone suggest an introductory book on point processes?
Im
> especially interested in spatial and space-time point
processes.
> -- Matti
===
Subject: Re: Skolems Paradox and why is math the way it is?
> J.E. says...
>The proof just demonstrates the lack of a bijection.
> And thats what it means for a set to be uncountable: that
there
> is no bijection between that set and the set of naturals.
>You may, in reality, be correct, but your statement wasnt
very
>convincing (or detailed or complete). Can you prove (in
ZF) that the
>sequence exists?
> Who cares? Why is it important (for applications to
physics, for
> example) that a sequence be denable in ZF?
You dened the real numbers thusly: It is the smallest set
containing the rational numbers which is closed, in the sense
that
every Cauchy sequence converges. Then you claimed that that
wasnt
true of some other things. But every cauchy sequence with a
proof
that the sequence exists has a real number that is its limit
that has
a proof that the real number exists. So whats wrong with the
collection of real numbers with a proof that they exist? The
only
problem I can see is if you FIRST bring in a cauchy sequence
that you
didnt prove exists, THEN it might not have a limit point.
But get
real, if your going to honestly allow yourself to do that,
why not
just sneak in a real that doesnt have a proof that it
exists. Its
the same thing. Apparantly concerning yourself with things
that have
proofs that they exist makes you a mathematical pariah? When
did this
happen?
> To work exclusively with constructible, or denable
reals is much
> more difcult, with no actual benet for physics or
science in
> general. You say that you are motivated by physics and
science, but
> I certainly dont see any motivation from *physics* to
restrict our
> attention to constructible reals, or to assume that all
reals are
> denable.
>How is it more difcult?
> Try it and see.
If I use the countable model, it seems easy. Nothing terrible
happens
to me or anything I prove theorems about or any calculations
I do.
When you say that its hard and I can tell it isnt, then I
get
nervous, and then I try to ask sci.math questions that come
out like
statements, and then everyone gets mad at me. Its not hard,
its not
even really different. You comments are the only evidence I
have that
you dont do it to. Your proofs (when you actually give them)
dont
demonstrate it, your theorems dont either.
>How can you even work with things you do
>NOT dene? THAT seems hard.
> Well, its not.
Ive seen you talk about every Cauchy sequence, but you only
prove
theorems about cauchy sequences that are SETS, and you havent
demonstrated the existance of any cauchy sequence that is NOT
represented in the countable model. It doesnt count as
working with
things you didnt dene, if you only pretend to work with
them. I
havent ever seen an example of you working with things you
didnt
dene, but you said it was easy. Ive seen some nonscientists
do
that, but they were just babbling, Im very interested in
seeing how
YOU do this. How can I tell when you are doing this easy
thing? Have
you done it before?
===
Subject: Re: Skolems Paradox and why is math the way it is?
> You dened the real numbers thusly: It is the smallest set
> containing the rational numbers which is closed, in the
sense that
> every Cauchy sequence converges. Then you claimed that that
wasnt
> true of some other things. But every cauchy sequence with a
proof
> that the sequence exists has a real number that is its
limit that has
> a proof that the real number exists.
You are coming from a point of view that many of us do not
understand.
You said, above: But every cauchy sequence with a proof that
the
sequence exists ...
You seem to attach meaning to the idea that there is a thing,
whether it
be a number or a Cauchy sequence; and that there is also a
proof that
the thing exists. As if there is the number 5, and also proof
that the
number 5 exists.
It is true that if you believe in the number 4, then its
easy to show
that 5 exists. (via the standard Von Neumann construction of
the
ordinals, where 5 = 4 union {4}. )
But what do you mean by every Cauchy seq with a proof that
the sequence
exists ...
I can not conceive of a Cauchy sequence (of reals, say) that
does not
have a proof that it exists. I think this is the crux of your
confusion.
You are referencing some meaning in your mind to this notion
of a
sequence along with a proof that the sequence exists, and you
are
ascribing great depth and meaning to that concept, whereas
the rest of
us dont know what you mean.
===
Subject: Re: Skolems Paradox and why is math the way it is?
The denition of the real number system may be expressed by:
the set of
rationals together with the axiom that every Cauchy sequence
converges
(this
means converges TO A NUMBER, i.e., a member of the system;
the number can
be
thought of as a new member of the older system.
An example is root 2, which did not exist in the old
sysytem, since it is
not
rational.
There are equivalent denitions, such as every sequence which
is
bounded
above has a least upper bound .These all serve to dene a set
which is
the
completion of the rationals. The axioms used can all be based
on
Zermelo-Frankel set theory (plus a system of logic which is
rst- order
predicate calculus with equality) and all arithmetic can be
built from
these
axioms. You mst not question the truth of these axioms, as
they are
simply
put down to form a basis for the theory. If you want to
change them you
may
get a different system. The construction of the whole
numbers, which as
you
mention, can be built up via the Peano axioms, are all part
of the axioms
for
the real numbers (when I taught this, I found I could do with
10 axioms,
the
tenth of which is the axiom of completeness, namely the one
discussed at
the
start of this argument. The complex niumbers can be built up
fromn the
reals,
but you need more than the axioms given for the reals. (for
instance the
order
condition among the reals is false here - there being no
linear order
among
the complex numbersI)
My book on this subject, Real Numbers is no longer in
print, but if you
can
get it youll nd more detailsl on all this. It is indeed a
fascinating
subject, forming, as it does, the fondation for alll modern
mathematics,
including calculus.
===
Subject: Re: Skolems Paradox and why is math the way it is?
The real numbers form a system which can be expressed by the
set of
rationals
plus an axiom of completeness which can be expressed by
every Cauchy
sequence converges. By converges, we mean to a number of the
system, a
new
numnber if you like. For instance, root 2 is not a member of
the
rationals,
but it is indeed the limit of a sequene of rationals, so that
iindeed there
is
a
cauchy sequence which convergers to it. There are many forms
of this axiom
of
completeness, one of which is a set which is bounded above
has a least
upper
bound:.
===
Subject: Re: Skolems Paradox and why is math the way it is?
> [...]
> |The axioms dont put anything into a line, the whole point
is that the
> |axioms dont generate ALL the points on the line, only a
countable
> |number of them. Consistently one can add more axioms to
have more
> |numbers on the line, thats what the diagonal arguement
says, but if
> |one doesnt add more axioms, then you only have a
countable number of
> |points that the original ZF(C) axioms talk about directly.
And sadly,
> |even after adding more axioms you still only have a
countable number.
> |> Are second order logic and countable models of the
uncountable
not
> |> slippery slopes? (They are.)
> |I dont see how it is a slippery slope to be clear what
you are
> |tlaking about and what you are not.
> Then avoid such unclarities as set that the ZFC axioms talk
> about. If you actually plan to go anywhere with this, itll
be
> really important not to use phrases like that without
dening
> what you mean by it.
> Its not a phrase that is ordinarily used. I can come up
with
> various things that you *might* mean by it; I just dont
know
> which you have in mind, if you actually have a specic one
> in mind after all.
> Keith Ramsay
I was trying to repeat other peoples words back at them
since they
were the ones that (I thought) claimed these unprovable
subsets and
real numbers exist. I think I failed at repeating it
correctly, and I
now regret trying. If you make a countable model of ZF inside
ZF and
then take a bijection B (not in the model) from the naturals
(in the
model) to the reals (in the model) then there is a real
number that is
different than all the ones you could prove exist in ZF. The
base
three expansion of the real number has a 0 whenever the ith
digit in
the base three expansion of the real (in the model)
corresponding
(with the bijection B) to the ith integer (in the model) is a
1 or a 2
and the ith digit in the base three expansion of the real
number has a
1 whenever the ith digit in the base three expansion of the
real (in
the model) corresponding (with the bijection B) to the ith
integer (in
the model) is a 0. There is clearly no proof that this real
number
exists in ZF. So does it belong to R? Isnt R supposed to
have all
reals? Isnt that the standard interpretation other people get
frustrated with me for not adopting as my own? There is no
proof it
exists, but I (prehaps mistakenly) thought that people have
claimed on
this group that all the real numbers exist in ZF. I was
trying to be
nice when (I thought) people claimed that that number existed
by
saying that ZFC doesnt talk about it, which helps maintain
their
ction that it is there. Is this still unclear?
===
Subject: Re: Skolems Paradox and why is math the way it is?

X-CompuServe-Customer: Yes
X-Coriate: interspeed.co.nz
X-Ecrate: tanandtanlawyers.com
X-Pose: George Cox
X-Punge: Micro$oft
X-Sanguinate: The MVS Guy
X-Terminate: SPA(GIS)
X-Tinguish: Mark Grifth
X-Treme: C&C,DWS
>I am listening. Thats why I respond to posts that say
things I
>dont understand, Im asking for clarication because I am
TRYING to
>understand what they are saying.
One thing that they are saying is that you cant simply sling
words
around and expect to be taken seriously; you need to either
use their
standard meanings in the context or to clearly, explicitly and
unambiguously dene what you mean. If youre going to use
such terms
as exists and larger in an idiomatic fashion then you can
expect to be
dismissed as a kook. When you insist that others are
pretending, you
can expect to be dismissed as a liar.
>For instance you just said that people are did so.
They told you that the existence of a bijection is the
denition of
two sets having the same cardinality. You insist on using it
with some
other meaning, but to date have refused to say what that
other meaning
is.
>What was the error if there was one?
The error is that you are attempting to discuss Mathematics
without
understanding the vocabulary and without dening the
vocabulary that
you prefer to use.
>How does ZF meet my expectations if it really did?
Nobody cares; you havent formulated realistic expectations.
>ZF doesnt distinguish between the set of subsets that could
exist
>consistentl with ZF
There you go again. What does that mean? Sets are constructs
of a
theory, and it is only meaningful to speak of sets existing
in a
particular theory. If the only theory that youre discussing
is ZF,
then the only sets that exist are the sets of ZF. If youre
discussing
some other theory then you need to specify what that other
theory is
and you need to be explicit about which theory youre
discussing when.
>It just plan doesnt,
It doesnt count leprechauns either.
>I can dene relations
What do you mean by dene? What do you mean by relations?
>that should be sets,
What do you mean by should be sets?
These are questions that you keep dodging.
>It seems that ZF is incomplete in a bad way, that
specically not
>all well-dened subcollections are subSETS.
No. If P(s) is a proposition in ZF and S is a set in ZF then
{s S:P(s)} is a set in ZF.
>Are these relations sets in the second order axiom system?
What second order axiom system? And, again, what do you mean
by
relations?
>No one tells me.
Because you havent asked a coherent question. Dene your
terms, then
use the terms only in accordance with the denitions you
give. If you
cant dene them coherently then stick to the standard
denitions.
But what youve been doing is hand waving, substituting
nonstandard
denitions and expecting it not change the truth of sentences
using
those terms.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT

Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply
to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spamtrap@library.lspace.org
===
Subject: Re: Skolems Paradox and why is math the way it is?
>I am listening. Thats why I respond to posts that say
things I
>dont understand, Im asking for clarication because I am
TRYING to
>understand what they are saying.
> One thing that they are saying is that you cant simply
sling words
> around and expect to be taken seriously; you need to either
use their
> standard meanings in the context or to clearly, explicitly
and
> unambiguously dene what you mean. If youre going to use
such terms
> as exists and larger in an idiomatic fashion then you can
expect to be
> dismissed as a kook. When you insist that others are
pretending, you
> can expect to be dismissed as a liar.
First off I would like to thank you for being clear when
explaining
why others (possibly yourself included) didnt understand
what I was
saying. I appreciate the time you spent to do that, and I
hope it
will be helpful to all parties. The problem Im facing is
that we
have some popular theorems that have a usual interpretation,
and I
disagree with the interpretation, but not the theorem. I do
not know
how to talk about interpretations without using nonstandard
denitions, so I will try active phrases instead of discussing
interpretations at all.
>For instance you just said that people are did so.
> They told you that the existence of a bijection is the
denition of
> two sets having the same cardinality. You insist on using
it with some
> other meaning, but to date have refused to say what that
other meaning
> is.
Im ne with using that meaning. Claim 1: The power set Y of
a set X
has a different cardinality than the set X in ZF. (Proof, we
agree
this has been proved before). Claim 2: This is about a lack
of a
bijection. (Follows from the denition) Claim 3: One can add a
bijection B to ZF without breaking consistency. (Proof: Im
assuming
you understand that claim from previous posts, I can prove it
again if
you want, Im not trying to wave hands). Claim 4: If you do
that,
then the power set of X changes from Y to Y, which is not
the image
of the bijection B. So in this new system Y (the power set
of X) is
uncountable. (Proof: diagonal arguement).
Expected rebuttal: that the set Y was already the same as Y.
If you
believe that, submit a proof. Claim 5: No proof exists that Y
contains an element that Y does not. N.B. remember that Y is
a set
in ZF and Y is a set in a new axiom system. (Proof: Should
be clear
from earlier posts, will post against if you want, not trying
to wave
hands, Im not sure I understand rst order logic and have
seen
different forumlations of the axiom of specication and am
unclear as
to how to present it to use to your satisfaction. If you
supply
standards, I will try to supply proof.)
>What was the error if there was one?
> The error is that you are attempting to discuss Mathematics
without
> understanding the vocabulary and without dening the
vocabulary that
> you prefer to use.
I thought things were more clear than they apparantly were.
That may
be an error in my post, it does not imply that there was an
error in
my idea, which I am rewriting in this post to be more clear.
>How does ZF meet my expectations if it really did?
> Nobody cares; you havent formulated realistic expectations.
I want clear distinctions between what exists and what can be
proven
to exist. Is that unrealistic?
>ZF doesnt distinguish between the set of subsets that
could exist
>consistentl with ZF
> There you go again. What does that mean? Sets are
constructs of a
> theory, and it is only meaningful to speak of sets existing
in a
> particular theory. If the only theory that youre
discussing is ZF,
> then the only sets that exist are the sets of ZF. If youre
discussing
> some other theory then you need to specify what that other
theory is
> and you need to be explicit about which theory youre
discussing when.
The power set axiom talks about the set of all subsets of a
set A.
But applying the axiom (schema) of specication to the set A
over and
over again produces a collection of subsets. Is the
collection of
the power set of the set A? Can I prove that they are or
arent the
same? To me the ZF axioms are vague on that front. Is that an
explicit enough question in the ZF theory? If not, please
tell me
what is wrong with it.
>It just plan doesnt,
> It doesnt count leprechauns either.
>I can dene relations
> What do you mean by dene? What do you mean by relations?
>that should be sets,
> What do you mean by should be sets?
> These are questions that you keep dodging.
Hopefully the questions will be unnecissary if you read the
previous
parts of this post. If they are, jsut ask again, Im not
trying to
hide anything.
>It seems that ZF is incomplete in a bad way, that
specically not
>all well-dened subcollections are subSETS.
> No. If P(s) is a proposition in ZF and S is a set in ZF then
> {s S:P(s)} is a set in ZF.
Do you have a set of propostions? If so, is the set countable?
>Are these relations sets in the second order axiom system?
> What second order axiom system? And, again, what do you
mean by
> relations?
There are axioms I can consistantly add, that increase the
set of
bijections to contain more members, that also require that new
elements of the power set be provable to exist. Let ZF2 be
the second
order ZF theory (second order ZF axioms, with a second order
axiom of
specication with no inaccessible ordinals). Are these new
bijections that I created with an extended rst order theory
already
in ZF2?
>No one tells me.
> Because you havent asked a coherent question. Dene your
terms, then
> use the terms only in accordance with the denitions you
give. If you
> cant dene them coherently then stick to the standard
denitions.
> But what youve been doing is hand waving, substituting
nonstandard
> denitions and expecting it not change the truth of
sentences using
> those terms.
Did I do ANY better this time? Any improvement at all?
===
Subject: Re: Skolems Paradox and why is math the way it is?
> Im ne with using that meaning. Claim 1: The power set Y
of a set X
> has a different cardinality than the set X in ZF. (Proof,
we agree
> this has been proved before).
Check.
Claim 2: This is about a lack of a
> bijection. (Follows from the denition)
Yes, but I can already see youre looking for trouble. There
IS no
bijection, because assuming there is a bijection leads to a
contradiction so obvious that the proof can be shown to a
high school
student.
Claim 3: One can add a
> bijection B to ZF without breaking consistency.
Oh no, that is most assuredly false. There is no bijection.
By the way I am the same person who was chatting with you
this afternoon
on that other forum. I no longer use the shfry handle on
that board.
===
Subject: Re: Skolems Paradox and why is math the way it is?

<414e25b3$17$fuzhry+tra$mr2ice@news.patriot.net>

<4152642f$0$7207$8fcfb975@news.wanadoo.fr>
<4156ed70$62$fuzhry+tra$mr2ice@news.patriot.net>
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X-Treme: C&C,DWS
>If Ive being clear in asking why people like the ZF axioms,
then why
>arent people answering my question?
I did; you didnt like my answer. Maybe others gured that it
was a
waste of time. Maybe they dont like being lied about.
>If Im being clear about why I
>think they fail to do what we expect, then why dont people
correct
>my expectations or show me how the axioms do meet my
expectations?
How many times?
--
Shmuel (Seymour J.) Metz, SysProg and JOAT

Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply
to
domain Patriot dot net user shmuel+news to contact me. Do not
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===
Subject: Re: Skolems Paradox and why is math the way it is?
>If Ive being clear in asking why people like the ZF
axioms, then why
>arent people answering my question?
> I did; you didnt like my answer. Maybe others gured that
it was a
> waste of time. Maybe they dont like being lied about.
Im not trying to lie, Im trying to interpret vague and
confusing
things that people say to me. Obviously if it wasnt vague and
confusing to the original author, then Ive interpreting it
differently. That doesnt make my attempt dishonest in
anyway. How
am I supposed to be more clear about how Im interpreting
what people
say so that they dont interpret it as me lying about them?
No one
tells me how I should have phrased things differently, so how
am I
supposed to learn? I can tell that people are unhappy, but
the golden
rule and empathy only works so well. They dont create
knowledge
about how not to upset people. Ive read posts between other
people
on this group and I see a lot of sarcasm and name calling and
such.
Maybe there are specic people whose posts I should look at?
>If Im being clear about why I
>think they fail to do what we expect, then why dont
people correct
>my expectations or show me how the axioms do meet my
expectations?
> How many times?
When people say that theyve said something before without
saying what
it was that they say before, then its vague. For instance
you didnt
even tell me if you thought people said I was wrong (and
people have
pointed out errors that I admitted were errors) or if you were
claiming that people told me that the axioms do meet my
expectations.
I cant tell what you are saying. I can tell you are
frustrated. If
the purpose of this forum is to discuss your feelings I could
do that.
I thought the forum was for talking about math related topics.
Is there a way to distinguish between the collection of reals
that we
can prove exist with the axioms of ZF and the set of reals
the axioms
of ZF prove exists? (Yes or no)
If yes, what is the difference?
If no, are they the same? (Yes or no)
---If yes, what is the proof?
---If no, what is the proof?
And for the meta-question, has this already been answered? If
so,
where.
===
Subject: Re: Skolems Paradox and why is math the way it is?
<4150a021$4$fuzhry+tra$mr2ice@news.patriot.net>
X-CompuServe-Customer: Yes
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X-Ecrate: tanandtanlawyers.com
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X-Tinguish: Mark Grifth
X-Treme: C&C,DWS
at 12:25 PM, troubled6man@yahoo.com (J.E.) said:
>Your comments are very vague. People are introducing
unobservable
>phenomena to make theories where observations are consistent
with the
>matter we do see? Why? What is the point?
The point is that they believe that they can devise better
theories by
doing so.
>As for not doing better than GR, Ive seen claims of better
results
>from people whose work I dont understand.
Well, if NASA doesnt kill it theres some instrumentation
that might
get us closer to an answer. I know that B-D didnt fair too
well when
tested, but that could change with more accurate
measurements. IAC,
the real problem is Quantum Gravity, and Im not aware of any
current
approach that comes remotely close to your criteria.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT

Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply
to
domain Patriot dot net user shmuel+news to contact me. Do not
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===
Subject: Re: Skolems Paradox and why is math the way it is?
> -----BEGIN PGP SIGNED MESSAGE-----
> Hash: SHA1
>WHY cant I use a countable model while doing physics?
> Why do you need to use any model at all?
>To me we you language, and the axioms of set theory I
introduce as
>denitions of my terms.
> But isnt that where your mistaken thinking begins?
> Surely you should be starting with terms that come from
physics.
> terms.
> The standard mathematical conception of the real numbers
really
> arises as an idealization of the measurements used by
physicists.
> The axioms of set theory are a late comer to the scene.
> Mathematicians were using real numbers long before they
studied set
> theory.
> For the mathematician, the role of the axioms is clear. It
provides
> a consistent logical framework for discussing mathematical
objects,
> and proving theorems about them. There relevance to physics
is less
> obvious.
> Mathematical foundations gives the impression that the real
numbers
> are constructed from set theory. But this is mostly a clever
> pretense. Mathematicians went to a lot of trouble to make
sure that
> what they could appear to construct from set theory, was
really the
> old familiar real numbers that arose out of the needs of
physics.
> If you jump to a non-standard model (such as countable
model of set
> theory) you lose all of that. Maybe you will feel more
comfortable
> with the countable model. But you will be dealing with a
model whose
> real numbers have at most a dubious connection with physics.
> Take an example. Suppose that you settle on the
constructable
> reals. Then you know that every mathematical operation you
could do
> would nish up with a constructable number.
> But now lets suppose that a physical event is observed.
You want to
> say that the event occurred at time t. Is reality such that
the time
> t is constrained to be a constructable real? If so, how
would you
> prove that?
> Or are you going to say that if the time t is not
constructible, then
> the event didnt actually occur, since t is not in your
model of the
> reals?
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In reality the accuracy of our time measurements are limited
by the
energy available to us to do experiments. Rationals are good
enough
for time measurements in the lab. I agree that ZF(C) is good
enough
for experimentalists that just take measurements and use the
existing
theories that are already made in ZF(C). It is theorists who
must
make better theories that predict more phenomina where we
need a
powerful math engine that contains enough stuff and not too
much
stuff. Because a real world symmetry can use innite precision
cancellation to generate real world effects, and so theory
needs more
than the nite precision needed in the lab to take
measurements. We
need a mathematical language that allows the appropriate real
world
symmetries to exist and to have the right operations. Do you
understand anything I am saying? ZFC appears to be good
through an
eloborate social illusion that it contains ALL subsets, when
there is
no truth behind the claims that it does, as the countable
model
EXPOSES to mathematical observation.
J.E.
===
Subject: Re: Skolems Paradox and why is math the way it is?
> -----BEGIN PGP SIGNED MESSAGE-----
> Hash: SHA1
> J.E. says...
>The proof is that the bijection doesnt exist, not that
the range
>of the non-existant bijection is different than the
power set.
> I cant make any sense of that statement. The claim is
that there
> is no bijection between the reals and the naturals.
Thats what
> it *means* to say that the reals are uncountable.
>If I look at the set containing every real in the
countable model,
>lets call it R-, then how can you prove that that set is
NOT the same
>as R?
> Cantors proof shows that.
> How can you prove that Plancks constant is in R-? And if
you cannot
> prove that, how is the countable model useful for
physicists?
> Clearly the set R- is UNcountable only because a certain
>countable set has been EXCLUDED by ZF, not because it has
many
>members.
> Thats a bit silly. Its your choice of a model (rather
than ZF)
> that did the excluding.
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Cantors proof is by contradiction. He FIRST assumes that a
set B
exists and then nds a contraction. This PROVES that the set
B does
not exist, not ANYTHING else. In fact THAT is THE proof that
the set
was excluded from YOUR model. Claiming that it proves
something about
R- and/or R and/or R+ is WRONG, all it proves is that B
doesnt exist,
BECAUSE the axioms cant tell the difference between R- and
R+. You
are missing my point entirely and this conversation is goind
in
circles. My model makes the limitations of the ZF axioms
OBVIOUS,
yours just hide them, they dont x them. There are relations
that
should have sets correspondaning to them if you think R is
R+, but
there are no proofs that the sets exist, in fact they cant
be proven
to exist with the ZFC axioms.
For instance, look at R-, you cant prove that it exists. You
can
prove that R exists, but you cant prove that R- is the same
as R.
But why should that subclass of R (the class determined by
R-) NOT be
a set since ZFC is supposed to have ALL subsets of R? THAT is
the
question you have NOT answered.
===
Subject: Re: Skolems Paradox and why is math the way it is?
-----BEGIN PGP SIGNED MESSAGE-----
Hash: SHA1
>If I look at the set containing every real in the
countable model,
>lets call it R-, then how can you prove that that set is
NOT the same
>as R?
> Cantors proof shows that.
> Clearly the set R- is UNcountable only because a certain
>countable set has been EXCLUDED by ZF, not because it has
many
>members.
> Thats a bit silly. Its your choice of a model (rather
than ZF)
> that did the excluding.
>Cantors proof is by contradiction. He FIRST assumes that a
set B
>exists and then nds a contraction. This PROVES that the set
B does
>not exist, not ANYTHING else.
Thats a bit simplistic.
Keep in mind that Cantor was not working in ZF. His proof is
older
than ZF.
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--
vote for regime change in Washington, Nov 02.
===
Subject: Re: Skolems Paradox and why is math the way it is?
|I HAVE written out denitions of countable models of ZF(C),
and they
|are well-founded and pure, AND obviously incomplete in that
there are
|things that should be sets that arent in it. They are not
|complicated, the only problem with using them in practise is
the
|OBVIOUSNESS of the INcompleteness of ZF(C).
When have you written out such denitions? I havent seen
any such denition on the internet. I dont know of any
specic
countable model of ZFC whose denition I would call simple,
at least by comparison with the usually intended (proper
class)
model. In Cohens book describing his proof of the
independence
of the axiom of choice he denes a (relatively natural) one,
but it
takes him a fair bit of preliminary work to build up to it.
The proofs
of Goedels completeness theorem or of Lowenheim-Skolem
suggest various other possibility, that again require pages
worth
of preliminary denitions, conventions, and so on.
If there are ones that arent complicated, perhaps you could
post
one of them.
Keith Ramsay
===
Subject: Re: Skolems Paradox and why is math the way it is?
> |I HAVE written out denitions of countable models of
ZF(C), and they
> |are well-founded and pure, AND obviously incomplete in
that there are
> |things that should be sets that arent in it. They are not
> |complicated, the only problem with using them in practise
is the
> |OBVIOUSNESS of the INcompleteness of ZF(C).
> When have you written out such denitions? I havent seen
> any such denition on the internet. I dont know of any
specic
> countable model of ZFC whose denition I would call simple,
> at least by comparison with the usually intended (proper
class)
> model. In Cohens book describing his proof of the
independence
> of the axiom of choice he denes a (relatively natural)
one, but it
> takes him a fair bit of preliminary work to build up to it.
The proofs
> of Goedels completeness theorem or of Lowenheim-Skolem
> suggest various other possibility, that again require pages
worth
> of preliminary denitions, conventions, and so on.
> If there are ones that arent complicated, perhaps you
could post
> one of them.
> Keith Ramsay
rst order logic has required a crash course in rst order
logic
that Im still going through. I assumed that if I could do it
quickly, that someone else had already done it earlier. Its
really
hard for me to nd other peoples wheels, I usually end up
making
them up myself because its just faster, but again maybe I
just took
too many classes with the Moore method. I remember after I
nished
my Moore topology class someone asked me how many blackboards
I used
I dont know how much space the words I spoke would have
taken, but
good notation makes proofs and denitions as easy as is
necissary.
My favorite proof of the generalized stokes theorem (not my
own) is
just one line long. But since Im terrible about nding other
peoples work I dont want to publish my own for fear of
having
someone think Im taking credit for it. Im not interested in
the
credit, I just want to talk about the ideas.
Is there really no simple model? You just formalize a proof
that the
specic set exists and say that the formalized proof
reresents the
set itself. Whats hard about that, especially since it is so
similar
to really old results and techniques? Now a sub collection of
all
formalized proofs is a countable model. If you think no one
has
seriously done that before then Ill publish after Im done
with my
crash course, but only if you tell me no one has done it
before.
===
Subject: Re: Skolems Paradox and why is math the way it is?
> |I HAVE written out denitions of countable models of
ZF(C), and they
> |are well-founded and pure, AND obviously incomplete in
that there are
> |things that should be sets that arent in it. They are not
> |complicated, the only problem with using them in practise
is the
> |OBVIOUSNESS of the INcompleteness of ZF(C).
> When have you written out such denitions? I havent seen
> any such denition on the internet. I dont know of any
specic
> countable model of ZFC whose denition I would call simple,
> at least by comparison with the usually intended (proper
class)
> model. In Cohens book describing his proof of the
independence
> of the axiom of choice he denes a (relatively natural)
one, but it
> takes him a fair bit of preliminary work to build up to it.
The proofs
> of Goedels completeness theorem or of Lowenheim-Skolem
> suggest various other possibility, that again require pages
worth
> of preliminary denitions, conventions, and so on.
> If there are ones that arent complicated, perhaps you
could post
> one of them.
There is at least one model, that is not complicated, and
easily
formulated with computability in the limit. I tentatively
titled such
a draft Algorithmic Set Theory, but its pointless to expect
any use
from such a work if it were to be completed, except to show
that
mathematics has an empirical nature.
--
Eray Ozkural
===
Subject: Re: Skolems Paradox and why is math the way it is?
> If there are ones that arent complicated, perhaps you
could post
> one of them.
> There is at least one model, that is not complicated, and
easily
> formulated with computability in the limit.
Are you going to show us?
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square
root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Re: Skolems Paradox and why is math the way it is?
>
> If there are ones that arent complicated, perhaps you
could post
> one of them.
>
> There is at least one model, that is not complicated, and
easily
> formulated with computability in the limit.
> Are you going to show us?
If I feel that its correct, I will. Its really simple, and
uses
LISP-like symbolic expressions to represent potentially
innite sets.
Its nothing big, its not important at all as you will see
if I post
it, but I argue that we can use information theory to reason
about
sets, e.g. how slow a generation procedure is with respect to
another
using asymptotic ratios. I rst pick a machine, e.g. a
universal
discrete computer, and show what kinds of sets can be dened
using
this, then I try to show which axioms of ZFC hold in this
formulation.
Naturally, I examine questions such as whether Russells
paradox or
Galileos paradox occurs in this system. Then I dene a
super-turing
computation model and show how this can be used to dene the
continuum in the same way. Nothing seems to contradict with
classical
set theory. Additional arguments are made with respect to
physical
plausibility in mathematics, the whole effort is to make it
clear that
we can choose the physics to set the rules straight as in
employing
the assumptions and limitations of digital physics, quantum
physics or
a continuous model (e.g. general relativity). An easy
philosophical
consequence of this is that mathematical is partly solipsist
and
partly empirical.
--
Eray Ozkural
===
Subject: Re: Skolems Paradox and why is math the way it is?
> Eray are there integers with an innite number of digits?
Ozkural
>
> If there are ones that arent complicated, perhaps you
could post
> one of them.
>
> There is at least one model, that is not complicated,
and easily
> formulated with computability in the limit.
> Are you going to show us?
> If I feel that its correct, I will.
The answer is NO then :-)
> Its really simple, and uses
> LISP-like symbolic expressions to represent potentially
innite sets.
What does it use to represent actually innite sets?
> Additional arguments are made with respect to physical
> plausibility in mathematics,
Drifting into irrelevance.
> An easy philosophical
> consequence of this is that mathematical is partly
solipsist and
> partly empirical.
Youve already provided conclusive evidence that your approach
to mathematics is totally solipsitic.
I remain sceptical.
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square
root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Re: Skolems Paradox and why is math the way it is?
...
> An easy philosophical
> consequence of this is that mathematical is partly
solipsist and
> partly empirical.
>Youve already provided conclusive evidence that your
approach
>to mathematics is totally solipsitic.
Solipsistic is essentially singular. Bollocks, as Skeat
pointed out, is a rare remnant in modern English of an Anglo-
Saxon dual (and quite right, too).
Thus, 1=2.
Lee Rudolph
===
Subject: Re: Skolems Paradox and why is math the way it is?

Discussion, linux)
> Eray are there integers with an innite number of digits?
> There is at least one model, that is not complicated,
and easily
> formulated with computability in the limit.
> Are you going to show us?
> If I feel that its correct, I will.
Since you havent shown us, I guess you dont (yet) feel its
correct.
So perhaps you should have written, There is at least one
model,
that is not complicated and *might even be correct* (but
might not),
and easily formulated with computability in the limit.
A bit less punch, perhaps, but youll sleep better at night.
--
Jesse F. Hughes
You see 300 of something, anything, and you go `[Man], thats
a lot of
stuff. -- Jim Bigler, quoted in the Pittsburgh Post-Gazette.
===
Subject: Re: Skolems Paradox and why is math the way it is?
> The countable model isnt the intended model
because there
are some
> collections of its elements of rank smaller
than some xed
ordinal
> which dont get counted as sets. Why would
you want to x
the axioms?
> Theres nothing wrong with them.
>
> I was interested in xing the axioms because I
dont see the
point of
> numbers that we cant talk about. If we had a
bijection that
said,
> these are the sets and the other axioms dont
apply to this
particular
> bijection, then wed know what the model was
conned too, no
> mysterious other elements whose properties are
based on what
order I
> same as your model, how can you tell from the
inside that it
is
> defective, and what makes it defective in your
opinion.
>
> The fact that its missing a bijection.
>
> Why do you keep saying that *I* am missing a
bijection, when I
say
> that *you* are missing a logical correspondance
that satises
the
> same logical properties that the bijections of your
model do. You
are
> the one that refuses to formalize a true statement
with the
properties
> of a bijection just so you can make some type
theory on your class
of
> sets, that has NO application that Ive ever seen.
>
>
> The fact that you say your model is countable means
you must admit
> there is a bijection not in the model. This is
presumably what you
are
> calling a logical correspondence. I freely admit that
the
bijection
> exists, and I say this shows your model is
incomplete. Im asking
you
> to consider a model that has *all* of the bijections
in it.
>
> I think I understand what you are saying, and its
really CLOSE to
> answering my questions. Let me try to ask it again more
carefully.
> Apparantly the rst order ZF axioms dont adequately
describe the
> universe of sets because they allow countable models.
You tell me to
> x this by considering a different model. Wouldnt it
be better
to
> change the axioms so that the objectional model was no
longer a
model?
> It seems better than assuming that things exist that
are independant
> of the axioms. And it sounds like you are saying that
you need
second
> order axioms to avoid the countable model. Since
IF-logic is rather
> new, is it possible that there are IF axioms that are
sufcient to
> remove substandard models? Is it just going to be the
case that all
> axioms of any order are substandard?
>
>
> Second-order axioms can x the model up to isomorphism. I
dont know
> whether IF axioms will do the job.
> Dont second order axioms assume the existance of all the
functions as
> a basis? That means you basically already have to have the
reals to
> use it, dont you?
Well, we require that models for second-order axioms be
models in a
second-order sense, where the second-order quantiers are
regarded as
ranging over *all* possible subsets of the domain of
discourse.
To prove that a given set of second-order axioms has a model
you have
to use a theory of sets like ZFC.
===
Subject: Re: Skolems Paradox and why is math the way it is?
> The countable model isnt the intended
model because there
are some
> collections of its elements of rank smaller
than some
xed ordinal
> which dont get counted as sets. Why would
you want to x
the axioms?
> Theres nothing wrong with them.
>
> I was interested in xing the axioms because
I dont see
the point of
> numbers that we cant talk about. If we had a
bijection
that said,
> these are the sets and the other axioms dont
apply to this
particular
> bijection, then wed know what the model was
conned too,
no
> mysterious other elements whose properties
are based on what
order I
> same as your model, how can you tell from the
inside that it
is
> defective, and what makes it defective in
your opinion.
>
> The fact that its missing a bijection.
>
> Why do you keep saying that *I* am missing a
bijection, when I
say
> that *you* are missing a logical correspondance
that satises
the
> same logical properties that the bijections of
your model do.
You are
> the one that refuses to formalize a true
statement with the
properties
> of a bijection just so you can make some type
theory on your
class of
> sets, that has NO application that Ive ever seen.
>
>
> The fact that you say your model is countable means
you must
admit
> there is a bijection not in the model. This is
presumably what you
are
> calling a logical correspondence. I freely admit
that the
bijection
> exists, and I say this shows your model is
incomplete. Im asking
you
> to consider a model that has *all* of the
bijections in it.
>
> I think I understand what you are saying, and its
really CLOSE to
> answering my questions. Let me try to ask it again
more carefully.
> Apparantly the rst order ZF axioms dont adequately
describe the
> universe of sets because they allow countable models.
You tell me
to
> x this by considering a different model. Wouldnt it
be
better to
> change the axioms so that the objectional model was
no longer a
model?
> It seems better than assuming that things exist that
are
independant
> of the axioms. And it sounds like you are saying that
you need
second
> order axioms to avoid the countable model. Since
IF-logic is
rather
> new, is it possible that there are IF axioms that are
sufcient to
> remove substandard models? Is it just going to be the
case that
all
> axioms of any order are substandard?
>
>
> Second-order axioms can x the model up to isomorphism.
I dont know
> whether IF axioms will do the job.
>
> Dont second order axioms assume the existance of all the
functions as
> a basis? That means you basically already have to have
the reals to
> use it, dont you?
> Well, we require that models for second-order axioms be
models in a
> second-order sense, where the second-order quantiers are
regarded as
> ranging over *all* possible subsets of the domain of
discourse.
> To prove that a given set of second-order axioms has a
model you have
> to use a theory of sets like ZFC.
I mean this in a way that isnt trying to accuse anyone of
anything
bad, but that is very confusing to me. And I dont think you
described it badly, its just the ZFC axioms system doesnt
do a good
job of distinguishing between the set P- of subsets that can
be
proven to exist in ZFC and the set P+ of all subsets. It does
SUCH
a terrible job that I dont even know which word to remove
the shudder
quotes from. In fact, most weak axioms (fewest assumptions,
not
inferior in deductive ability) have the power axiom assert the
existance of a random unknown set that merely has P- as a
subset (if
one assumes consistency of ZFC). But many people talk as if
ZFC
proves that P+ is a set. So which sets (the P- kind or the P+
kind)
does a second order quanitier range over? If you have to use
ZFC to
range over, then I cant tell which you are doing becuase
common
(historical) mathematical practise is just plain sloppy about
distinguishing the two. I dont want to sweep this important
distinction under the rug.
===
Subject: Re: Skolems Paradox and why is math the way it is?
> The countable model isnt the intended
model because
there are some
> collections of its elements of rank
smaller than some
xed ordinal
> which dont get counted as sets. Why
would you want to
x the axioms?
> Theres nothing wrong with them.
>
> I was interested in xing the axioms
because I dont see
the point of
> numbers that we cant talk about. If we had
a bijection
that said,
> these are the sets and the other axioms
dont apply to
this particular
> bijection, then wed know what the model
was conned too,
no
> mysterious other elements whose properties
are based on
what order I
> same as your model, how can you tell from
the inside that
it is
> defective, and what makes it defective in
your
opinion.
>
> The fact that its missing a bijection.
>
> Why do you keep saying that *I* am missing a
bijection, when I
say
> that *you* are missing a logical correspondance
that satises
the
> same logical properties that the bijections of
your model do.
You are
> the one that refuses to formalize a true
statement with the
properties
> of a bijection just so you can make some type
theory on your
class of
> sets, that has NO application that Ive ever
seen.
>
>
> The fact that you say your model is countable
means you must
admit
> there is a bijection not in the model. This is
presumably what
you are
> calling a logical correspondence. I freely admit
that the
bijection
> exists, and I say this shows your model is
incomplete. Im
asking you
> to consider a model that has *all* of the
bijections in it.
>
> I think I understand what you are saying, and its
really CLOSE
to
> answering my questions. Let me try to ask it again
more
carefully.
> Apparantly the rst order ZF axioms dont
adequately describe
the
> universe of sets because they allow countable
models. You tell me
to
> x this by considering a different model. Wouldnt
it be
better to
> change the axioms so that the objectional model was
no longer a
model?
> It seems better than assuming that things exist
that are
independant
> of the axioms. And it sounds like you are saying
that you need
second
> order axioms to avoid the countable model. Since
IF-logic is
rather
> new, is it possible that there are IF axioms that
are sufcient
to
> remove substandard models? Is it just going to be
the case that
all
> axioms of any order are substandard?
>
>
> Second-order axioms can x the model up to
isomorphism. I dont
know
> whether IF axioms will do the job.
>
> Dont second order axioms assume the existance of all
the functions
as
> a basis? That means you basically already have to have
the reals to
> use it, dont you?
>
> Well, we require that models for second-order axioms be
models in a
> second-order sense, where the second-order quantiers are
regarded as
> ranging over *all* possible subsets of the domain of
discourse.
>
> To prove that a given set of second-order axioms has a
model you have
> to use a theory of sets like ZFC.
> I mean this in a way that isnt trying to accuse anyone of
anything
> bad, but that is very confusing to me. And I dont think you
> described it badly, its just the ZFC axioms system doesnt
do a good
> job of distinguishing between the set P- of subsets that
can be
> proven to exist in ZFC and the set P+ of all subsets. It
does SUCH
> a terrible job that I dont even know which word to remove
the shudder
> quotes from. In fact, most weak axioms (fewest assumptions,
not
> inferior in deductive ability) have the power axiom assert
the
> existance of a random unknown set that merely has P- as a
subset (if
> one assumes consistency of ZFC). But many people talk as if
ZFC
> proves that P+ is a set. So which sets (the P- kind or the
P+ kind)
> does a second order quanitier range over? If you have to
use ZFC to
> range over, then I cant tell which you are doing becuase
common
> (historical) mathematical practise is just plain sloppy
about
> distinguishing the two. I dont want to sweep this important
> distinction under the rug.
What youre objecting to is that the language of the axioms
can be
given a different semantics. This deciency is irremediable.
It is
*always* open to you to adopt a different semantics. We just
have to
agree that we are talking about the standard semantics.
===
Subject: Re: Skolems Paradox and why is math the way it is?
> The countable model isnt the intended
model because
there are some
> collections of its elements of rank
smaller than some
xed ordinal
> which dont get counted as sets. Why
would you want to
x the axioms?
> Theres nothing wrong with them.
>
> I was interested in xing the axioms
because I dont
see the point of
> numbers that we cant talk about. If we
had a bijection
that said,
> these are the sets and the other axioms
dont apply to
this particular
> bijection, then wed know what the model
was conned
too, no
> mysterious other elements whose
properties are based on
what order I
> same as your model, how can you tell from
the inside
that it is
> defective, and what makes it defective in
your
opinion.
>
> The fact that its missing a bijection.
>
> Why do you keep saying that *I* am missing a
bijection, when
I say
> that *you* are missing a logical
correspondance that
satises the
> same logical properties that the bijections
of your model
do. You are
> the one that refuses to formalize a true
statement with the
properties
> of a bijection just so you can make some type
theory on your
class of
> sets, that has NO application that Ive ever
seen.
>
>
> The fact that you say your model is countable
means you must
admit
> there is a bijection not in the model. This is
presumably what
you are
> calling a logical correspondence. I freely
admit that the
bijection
> exists, and I say this shows your model is
incomplete. Im
asking you
> to consider a model that has *all* of the
bijections in it.
>
> I think I understand what you are saying, and
its really CLOSE
to
> answering my questions. Let me try to ask it
again more
carefully.
> Apparantly the rst order ZF axioms dont
adequately describe
the
> universe of sets because they allow countable
models. You tell
me to
> x this by considering a different model.
Wouldnt it be
better to
> change the axioms so that the objectional model
was no longer a
model?
> It seems better than assuming that things exist
that are
independant
> of the axioms. And it sounds like you are saying
that you need
second
> order axioms to avoid the countable model. Since
IF-logic is
rather
> new, is it possible that there are IF axioms that
are sufcient
to
> remove substandard models? Is it just going to be
the case that
all
> axioms of any order are substandard?
>
>
> Second-order axioms can x the model up to
isomorphism. I dont
know
> whether IF axioms will do the job.
>
> Dont second order axioms assume the existance of all
the functions
as
> a basis? That means you basically already have to
have the reals
to
> use it, dont you?
>
> Well, we require that models for second-order axioms be
models in a
> second-order sense, where the second-order quantiers
are regarded
as
> ranging over *all* possible subsets of the domain of
discourse.
>
> To prove that a given set of second-order axioms has a
model you have
> to use a theory of sets like ZFC.
>
> I mean this in a way that isnt trying to accuse anyone
of anything
> bad, but that is very confusing to me. And I dont think
you
> described it badly, its just the ZFC axioms system
doesnt do a good
> job of distinguishing between the set P- of subsets that
can be
> proven to exist in ZFC and the set P+ of all subsets. It
does SUCH
> a terrible job that I dont even know which word to
remove the shudder
> quotes from. In fact, most weak axioms (fewest
assumptions, not
> inferior in deductive ability) have the power axiom
assert the
> existance of a random unknown set that merely has P- as a
subset (if
> one assumes consistency of ZFC). But many people talk as
if ZFC
> proves that P+ is a set. So which sets (the P- kind or
the P+ kind)
> does a second order quanitier range over? If you have to
use ZFC to
> range over, then I cant tell which you are doing becuase
common
> (historical) mathematical practise is just plain sloppy
about
> distinguishing the two. I dont want to sweep this
important
> distinction under the rug.
> What youre objecting to is that the language of the axioms
can be
> given a different semantics. This deciency is
irremediable. It is
> *always* open to you to adopt a different semantics. We
just have to
> agree that we are talking about the standard semantics.
I dont think this is entirely about semantics. Different
semantics
reveal different things about the underlying theory. The
underlying
theory has sets whose existance are independant of the
axioms. The
existance of these sets imply the existance of real numbers
that
cannot be proven to previously exist in ZF, so there are
missing real
numbers too. Changin semantics to make it hard to see,
doesnt change
the fact that one can consistently add an axiom that says
that a
SPECIFIC real number does NOT exist. I agree that maybe there
will
always be incompleteness and maybe there is always some thing
that
should exist, but an axiom can consistently be added to say it
doesnt. I think that if we cant have our cake and eat it
too, then
I *AT LEAST* want to have the real numbers (or something else
that
works as well for physics) and if we eat some higher sets out
in
homotopy theory where no one cares about incompleteness,
thats ne
with me.
WHY can we not x the axioms to AT LEAST produce all the
reals, even
if there is something else still missing?
===
Subject: Re: Skolems Paradox and why is math the way it is?
> The countable model isnt the
intended model because
there are some
> collections of its elements of rank
smaller than
some xed ordinal
> which dont get counted as sets. Why
would you want
to x the axioms?
> Theres nothing wrong with them.
>
> I was interested in xing the axioms
because I dont
see the point of
> numbers that we cant talk about. If we
had a
bijection that said,
> these are the sets and the other axioms
dont apply to
this particular
> bijection, then wed know what the
model was conned
too, no
> mysterious other elements whose
properties are based
on what order I
> same as your model, how can you tell
from the inside
that it is
> defective, and what makes it defective
in your
opinion.
>
> The fact that its missing a bijection.
>
> Why do you keep saying that *I* am missing
a bijection,
when I say
> that *you* are missing a logical
correspondance that
satises the
> same logical properties that the bijections
of your model
do. You are
> the one that refuses to formalize a true
statement with
the properties
> of a bijection just so you can make some
type theory on
your class of
> sets, that has NO application that Ive
ever seen.
>
>
> The fact that you say your model is countable
means you must
admit
> there is a bijection not in the model. This
is presumably
what you are
> calling a logical correspondence. I freely
admit that
the bijection
> exists, and I say this shows your model is
incomplete. Im
asking you
> to consider a model that has *all* of the
bijections in it.
>
> I think I understand what you are saying, and
its really
CLOSE to
> answering my questions. Let me try to ask it
again more
carefully.
> Apparantly the rst order ZF axioms dont
adequately describe
the
> universe of sets because they allow countable
models. You
tell me to
> x this by considering a different model.
Wouldnt it be
better to
> change the axioms so that the objectional model
was no longer
a model?
> It seems better than assuming that things exist
that are
independant
> of the axioms. And it sounds like you are
saying that you
need second
> order axioms to avoid the countable model.
Since IF-logic is
rather
> new, is it possible that there are IF axioms
that are
sufcient to
> remove substandard models? Is it just going to
be the case
that all
> axioms of any order are substandard?
>
>
> Second-order axioms can x the model up to
isomorphism. I dont
know
> whether IF axioms will do the job.
>
> Dont second order axioms assume the existance of
all the
functions as
> a basis? That means you basically already have to
have the reals
to
> use it, dont you?
>
> Well, we require that models for second-order axioms
be models in a
> second-order sense, where the second-order quantiers
are regarded
as
> ranging over *all* possible subsets of the domain of
discourse.
>
> To prove that a given set of second-order axioms has
a model you
have
> to use a theory of sets like ZFC.
>
> I mean this in a way that isnt trying to accuse anyone
of anything
> bad, but that is very confusing to me. And I dont
think you
> described it badly, its just the ZFC axioms system
doesnt do a good
> job of distinguishing between the set P- of subsets
that can be
> proven to exist in ZFC and the set P+ of all subsets.
It does
SUCH
> a terrible job that I dont even know which word to
remove the
shudder
> quotes from. In fact, most weak axioms (fewest
assumptions, not
> inferior in deductive ability) have the power axiom
assert the
> existance of a random unknown set that merely has P- as
a subset (if
> one assumes consistency of ZFC). But many people talk
as if ZFC
> proves that P+ is a set. So which sets (the P- kind or
the P+ kind)
> does a second order quanitier range over? If you have
to use ZFC
to
> range over, then I cant tell which you are doing
becuase common
> (historical) mathematical practise is just plain sloppy
about
> distinguishing the two. I dont want to sweep this
important
> distinction under the rug.
>
> What youre objecting to is that the language of the
axioms can be
> given a different semantics. This deciency is
irremediable. It is
> *always* open to you to adopt a different semantics. We
just have to
> agree that we are talking about the standard semantics.
> I dont think this is entirely about semantics. Different
semantics
> reveal different things about the underlying theory. The
underlying
> theory has sets whose existance are independant of the
axioms. The
> existance of these sets imply the existance of real numbers
that
> cannot be proven to previously exist in ZF, so there are
missing real
> numbers too. Changin semantics to make it hard to see,
doesnt change
> the fact that one can consistently add an axiom that says
that a
> SPECIFIC real number does NOT exist.
Actually, this isnt obvious to me. All the *denable* real
numbers
(the ones that you can refer to in axioms) exist in all the
models of
the theory.
> I agree that maybe there will
> always be incompleteness and maybe there is always some
thing that
> should exist, but an axiom can consistently be added to say
it
> doesnt. I think that if we cant have our cake and eat it
too, then
> I *AT LEAST* want to have the real numbers (or something
else that
> works as well for physics) and if we eat some higher sets
out in
> homotopy theory where no one cares about incompleteness,
thats ne
> with me.
> WHY can we not x the axioms to AT LEAST produce all the
reals, even
> if there is something else still missing?
Every consistent theory in a countable language has a
countable model.
===
Subject: Re: Skolems Paradox and why is math the way it is?
> The countable model isnt the intended
model because there
are some
> collections of its elements of rank smaller
than some
xed ordinal
> which dont get counted as sets. Why would
you want to x
the axioms?
> Theres nothing wrong with them.
>
> I was interested in xing the axioms because
I dont see
the point of
> numbers that we cant talk about. If we had a
bijection
that said,
> these are the sets and the other axioms dont
apply to this
particular
> bijection, then wed know what the model was
conned too,
no
> mysterious other elements whose properties
are based on what
order I
> same as your model, how can you tell from the
inside that it
is
> defective, and what makes it defective in
your opinion.
>
> The fact that its missing a bijection.
>
> Why do you keep saying that *I* am missing a
bijection, when I
say
> that *you* are missing a logical correspondance
that satises
the
> same logical properties that the bijections of
your model do.
You are
> the one that refuses to formalize a true
statement with the
properties
> of a bijection just so you can make some type
theory on your
class of
> sets, that has NO application that Ive ever seen.
>
>
> The fact that you say your model is countable means
you must
admit
> there is a bijection not in the model. This is
presumably what you
are
> calling a logical correspondence. I freely admit
that the
bijection
> exists, and I say this shows your model is
incomplete. Im asking
you
> to consider a model that has *all* of the
bijections in it.
>
> I think I understand what you are saying, and its
really CLOSE to
> answering my questions. Let me try to ask it again
more carefully.
> Apparantly the rst order ZF axioms dont adequately
describe the
> universe of sets because they allow countable models.
You tell me
to
> x this by considering a different model. Wouldnt it
be
better to
> change the axioms so that the objectional model was
no longer a
model?
> It seems better than assuming that things exist that
are
independant
> of the axioms. And it sounds like you are saying that
you need
second
> order axioms to avoid the countable model. Since
IF-logic is
rather
> new, is it possible that there are IF axioms that are
sufcient to
> remove substandard models? Is it just going to be the
case that
all
> axioms of any order are substandard?
>
>
> Second-order axioms can x the model up to isomorphism.
I dont know
> whether IF axioms will do the job.
>
> Dont second order axioms assume the existance of all the
functions as
> a basis? That means you basically already have to have
the reals to
> use it, dont you?
> Well, we require that models for second-order axioms be
models in a
> second-order sense, where the second-order quantiers are
regarded as
> ranging over *all* possible subsets of the domain of
discourse.
> To prove that a given set of second-order axioms has a
model you have
> to use a theory of sets like ZFC.
I mean this in a way that isnt trying to accuse anyone of
anything
bad, but that is very confusing to me. And I dont think you
described it badly, its just the ZFC axioms system doesnt
do a good
job of distinguishing between the set P- of subsets that can
be
proven to exist in ZFC and the set P+ of all subsets. It does
SUCH
a terrible job that I dont even know which word to remove
the shudder
quotes from. In fact, most weak axioms (fewest assumptions,
not
inferior in deductive ability) have the power axiom assert the
existance of a random unknown set that merely has P- as a
subset (if
one assumes consistency of ZFC). But many people talk as if
ZFC
proves that P+ is a set. So which sets (the P- kind or the P+
kind)
does a second order quanitier range over? If you have to use
ZFC to
range over, then I cant tell which you are doing becuase
common
(historical) mathematical practise is just plain sloppy about
distinguishing the two. I dont want to sweep this important
distinction under the rug.
===
Subject: Re: Skolems Paradox and why is math the way it is?
>
> Read this:
>
>
> Does Goedel submarine higher-order logic?
> When I follow that link I get a comment about a footnote,
and all it
> says is how did Godel know results that were proved by
other people
> years later?. Did you perhaps give the wrong link?
>
> Why do we care about uncountably many real
numbers when in
reality
> there are only a countable number that we can
prove theorems
about?
>
> Actually, thats not true. Although there can be
only a
countable
> number of theorems, they can still address an
uncountable number
of
> reals. As a simple example, there are an
uncountable number of
reals
> in the Cantor set, and a simple procedure by
which to determine
given
> any real if it is a member (namely, can its
decimal expansion,
when
> convereted to base 3, be written without using
the digit 1).
>
> Jonathan Hoyle
>
> question more clearly. There are many sets, we
agree on that.
Some
> sets have bijections with the set of naturals,
others dont, we
agree
> on that.
>
> I dont. You can apply a canonical ordering operator
onto any set
that is
> not ordering-sensitive.
>
> Which statement were you disagreeing with? Are you
saying that all
> sets have bijections with the set of naturals, because
all I said was
> that some do and others dont?
>
>
> No, only innite sets are equivalent, in my theory. There
are also
> alternatives where that is not so.
>
> This has to do with dening, or rather, acknowledging,
that P(X) = X +
1.
> I reference for that would be nice, it seems like some
formal symbols
> to me, and I dont understand what the intended
interpretation is
> supposed to be.
> ... But each time you use an axiom to prove a
theorem, the axiom
> produces, at most, a nite number of sets, and you
can use each
axiom
> only a nite number of times in a single proof.
Doesnt it seem
that
> each proof generates only a nite number of sets,
even though
some of
> these sets might be considered large themselves?
And if the
number
> of theorems is somewhat countable, then doesnt it
seem like
there
> is only a somewhat countable number of sets that we
can prove
> theorems about?
>
>
> I eschew axioms, but if you call innity an axiom
then it
generates
> innitely many sets and all the proofs about those
sets.
>
> The axiom of innite I know asserts the existance of
ONE set (that
> either is, or at least contains omega). And then the
axiom of
> specication creates a countable number of subsets, and
one can
> invoke the other axioms a nite number of times, and
there is still
> just an countable number of sets youve made.
>
>
> I just say there is at least nothing and then that
through excluded
middle there
> is everything else, and that the axiom is actually a
theorem.
>
> In the above reference, there is discussion about Goedel
essentially
agreeing
> with Skolem and Loewnheim. Also, the reals are the reals
are the
reals.
> I didnt see that in the reference.
> You, for instance, cited a theorem about ONE set
(the cantor set).
So
> there is at least one set in the universe. But you
want be to
believe
> that you just proved a theorem about many sets.
However, all you
> proved is that there is a certain subset of another
set, namely
the
> cantor set is a subset of the reals. That is a
theorem about ONE
set.
> How am I supposed to tell if there are as MANY sets
in the
universe
>
> Easily, put them in a line.
>
> The axioms dont put anything into a line, the whole
point is that
the
> axioms dont generate ALL the points on the line, only
a countable
> number of them. Consistently one can add more axioms to
have more
> numbers on the line, thats what the diagonal arguement
says, but if
> one doesnt add more axioms, then you only have a
countable number of
> points that the original ZF(C) axioms talk about
directly. And
sadly,
> even after adding more axioms you still only have a
countable number.
>
>
> Thats not sad. There are just countably many, innite
sets are
equivalent.
> I think I believe that too, thats why I dont like axioms
like ZF
> that make it appear otherwise.
> Are second order logic and countable models of the
uncountable not
> slippery slopes? (They are.)
>
> I dont see how it is a slippery slope to be clear what
you are
> tlaking about and what you are not.
>
>
> The issue I hope to make clear to you is that pushing off
the resolution
into
> high-order logics, in an admitted way that there is no
resolution in
resort to
> some metalogic, is handwaving that does not offer a
resolution, and that
if a
> resolution is to exist it exists in rst order logic, and
that all
high-order
> and meta logic is enframed or emposed within rst order
logic.
> I dont know what you mean by emframed and emposed. And
what about
> something between rst order and second order logic?
> Its a theory about one set: all of them.
>
> I lost you here. There is no set of all sets.
>
> Why not?
> We need some way of saying what is and is not a set, what
is your
> dention and/or what axioms do you use to assert the
existance of the
> sets you claim exist?
Hi Dr. J.,
I dont know the answers to all of your questions. Theyre not
insoluble.
About that link, actually it is the correct link. Also, it
does.
Even Goedels right sometimes.
About P(X) = X+1, that basically says that P(X) equals X plus
one.
About emposed, that is about questions posed. Abut enframed,
that is
about there only being rst order logic.
I dont claim to understand your logic-and-a-half, or IF
logic.
I say: nothing, and not nothing, and not that, ad innitum,
those
are each sets.
For a given set X, and a given universal set U which may well
be the
actual universal set, I say that U X is the context of X.
In set theory, anything is a set. Thats the only reason set
theory
can be said to be exible enough to be the foundation of
mathematics,
it has to be all-inclusive. To meet that goal, it is
necessary that
it resolve its issues instead of restating them to dizzying
heights of
stacked plates, because such pablum is meaningless and less
than
worthless. What that means is that if you have to resort to
second
order logic, some higher logic, than logic, to utilize logic,
then
there is a problem with that logic.
So, even the unicorn and Haephaestus, mythical creatures and
beings,
are sets, the set of all sets is a set, the empty set is a
set, and
anything and everything that can possibly be dened is an
object.
Why not a set? Its that, too.
Plainly, as far as I know, the uncountable is meaningless in
physics
and for the most parts in theories used to explain physics,
it is
ephemera that entails from that most educated people in
mathematics
have been exposed to the word and many use it. That mostly
means
thats one of the rst words they use to describe the real
numbers,
because they cant count them. For most of them, trying to
justify
mapping the naturals to the reals doesnt enter into their
equations
because they have seen their tool of the integral calculus
justied
in terms of nite approximations. Newton, Leibniz, Robinson,
and
Conway believe in innitesimals.
Is not the frequency of light innitely variable? It is,
because it
spins around in a circle, and pi is irrational.
What do you think about the observed forces of gravity being
constant
repulsion from all sides, blocked by nearby masses in the
direction
that mass? Most people assume and rightly that the Earths
gravity
draws mass towards it, because its mass is larger than
anything else
on Earth. How about the opposite?
I hope you think I have a decent mindset for the pursuit of
the study
of physics.
In terms of Skolems paradox, and why math is the way it is,
Skolem
basically tells us that innite sets are equivalent.
Assume it is your task to describe a function bijecting the
naturals
and reals. Whats the shortest answer you can use here on
sci.math
that would be mutually understood?
Are you often a computer programmer, Dr. J.?
Ross F.
===
Subject: Re: Skolems Paradox and why is math the way it is?
>
> Read this:
>
>
> Does Goedel submarine higher-order logic?
>
> When I follow that link I get a comment about a footnote,
and all it
> says is how did Godel know results that were proved by
other people
> years later?. Did you perhaps give the wrong link?
>
>
> Why do we care about uncountably many real
numbers when
in reality
> there are only a countable number that we can
prove
theorems about?
>
> Actually, thats not true. Although there can
be only a
countable
> number of theorems, they can still address an
uncountable
number of
> reals. As a simple example, there are an
uncountable number
of reals
> in the Cantor set, and a simple procedure by
which to
determine given
> any real if it is a member (namely, can its
decimal expansion,
when
> convereted to base 3, be written without using
the digit 1).
>
> Jonathan Hoyle
>
> question more clearly. There are many sets, we
agree on that.
Some
> sets have bijections with the set of naturals,
others dont, we
agree
> on that.
>
> I dont. You can apply a canonical ordering
operator onto any set
that is
> not ordering-sensitive.
>
> Which statement were you disagreeing with? Are you
saying that all
> sets have bijections with the set of naturals,
because all I said
was
> that some do and others dont?
>
>
> No, only innite sets are equivalent, in my theory.
There are also
> alternatives where that is not so.
>
> This has to do with dening, or rather, acknowledging,
that P(X) = X
+ 1.
>
> I reference for that would be nice, it seems like some
formal symbols
> to me, and I dont understand what the intended
interpretation is
> supposed to be.
>
> ... But each time you use an axiom to prove a
theorem, the
axiom
> produces, at most, a nite number of sets, and
you can use each
axiom
> only a nite number of times in a single proof.
Doesnt it
seem that
> each proof generates only a nite number of sets,
even though
some of
> these sets might be considered large themselves?
And if the
number
> of theorems is somewhat countable, then doesnt
it seem like
there
> is only a somewhat countable number of sets that
we can
prove
> theorems about?
>
>
> I eschew axioms, but if you call innity an axiom
then it
generates
> innitely many sets and all the proofs about those
sets.
>
> The axiom of innite I know asserts the existance of
ONE set (that
> either is, or at least contains omega). And then the
axiom of
> specication creates a countable number of subsets,
and one can
> invoke the other axioms a nite number of times, and
there is
still
> just an countable number of sets youve made.
>
>
> I just say there is at least nothing and then that
through excluded
middle there
> is everything else, and that the axiom is actually a
theorem.
>
> In the above reference, there is discussion about
Goedel essentially
agreeing
> with Skolem and Loewnheim. Also, the reals are the
reals are the
reals.
>
> I didnt see that in the reference.
>
> You, for instance, cited a theorem about ONE set
(the cantor
set). So
> there is at least one set in the universe. But
you want be to
believe
> that you just proved a theorem about many sets.
However, all
you
> proved is that there is a certain subset of
another set, namely
the
> cantor set is a subset of the reals. That is a
theorem about
ONE set.
> How am I supposed to tell if there are as MANY
sets in the
universe
>
> Easily, put them in a line.
>
> The axioms dont put anything into a line, the whole
point is that
the
> axioms dont generate ALL the points on the line,
only a countable
> number of them. Consistently one can add more axioms
to have more
> numbers on the line, thats what the diagonal
arguement says, but
if
> one doesnt add more axioms, then you only have a
countable number
of
> points that the original ZF(C) axioms talk about
directly. And
sadly,
> even after adding more axioms you still only have a
countable
number.
>
>
> Thats not sad. There are just countably many, innite
sets are
equivalent.
>
> I think I believe that too, thats why I dont like
axioms like ZF
> that make it appear otherwise.
>
> Are second order logic and countable models of the
uncountable not
> slippery slopes? (They are.)
>
> I dont see how it is a slippery slope to be clear
what you are
> tlaking about and what you are not.
>
>
> The issue I hope to make clear to you is that pushing
off the
resolution into
> high-order logics, in an admitted way that there is no
resolution in
resort to
> some metalogic, is handwaving that does not offer a
resolution, and
that if a
> resolution is to exist it exists in rst order logic,
and that all
high-order
> and meta logic is enframed or emposed within rst order
logic.
>
> I dont know what you mean by emframed and emposed. And
what about
> something between rst order and second order logic?
>
> Its a theory about one set: all of them.
>
> I lost you here. There is no set of all sets.
>
> Why not?
>
> We need some way of saying what is and is not a set, what
is your
> dention and/or what axioms do you use to assert the
existance of the
> sets you claim exist?
> Hi Dr. J.,
> I dont know the answers to all of your questions. Theyre
not
> insoluble.
> About that link, actually it is the correct link. Also, it
does.
> Even Goedels right sometimes.
How is predicting or anticipating results supposed to mean
anything to
me. Maybe you are expecting me to have certain views about the
results. It seemed like Godel thought higher orders save the
day, and
I thought you didnt think so, so what was I supposed to get
from the
> About P(X) = X+1, that basically says that P(X) equals X
plus one.
> About emposed, that is about questions posed. Abut
enframed, that is
> about there only being rst order logic.
> I dont claim to understand your logic-and-a-half, or IF
logic.
I think youd nd it interesting. The book I read is called
The
Principles of Mathematics Revisited by Jaakko Hintikka. There
are
some errors, I thought they were obvious, he repeats himself
enough
that when he says it wrong one time you know what he meant,
but you
can always pick up a critical review to make sure you know
what you
can trust.
> I say: nothing, and not nothing, and not that, ad innitum,
those
> are each sets.
> For a given set X, and a given universal set U which may
well be the
> actual universal set, I say that U X is the context of X.
> In set theory, anything is a set. Thats the only reason
set theory
> can be said to be exible enough to be the foundation of
mathematics,
> it has to be all-inclusive. To meet that goal, it is
necessary that
> it resolve its issues instead of restating them to dizzying
heights of
> stacked plates, because such pablum is meaningless and less
than
> worthless. What that means is that if you have to resort to
second
> order logic, some higher logic, than logic, to utilize
logic, then
> there is a problem with that logic.
As I understand it, you start with the empty set, then the
universe,
then the universe missing somthing, then that something, then
the
universe with some different set, then that different set,
and you
have some method to keep going. Did I get that right. Dont
you have
to be very very careful to have the property that the
powerset of a
set is just after the set itself? Obviously if you make your
construction general enough, then youd include anything you
can
describe, but then Id just wonder what description
corresponds to
something similar to the real numbers? My whole point is that
I
havent seen them described.
> So, even the unicorn and Haephaestus, mythical creatures
and beings,
> are sets, the set of all sets is a set, the empty set is a
set, and
> anything and everything that can possibly be dened is an
object.
> Why not a set? Its that, too.
> Plainly, as far as I know, the uncountable is meaningless
in physics
> and for the most parts in theories used to explain physics,
it is
> ephemera that entails from that most educated people in
mathematics
> have been exposed to the word and many use it. That mostly
means
> thats one of the rst words they use to describe the real
numbers,
> because they cant count them. For most of them, trying to
justify
> mapping the naturals to the reals doesnt enter into their
equations
> because they have seen their tool of the integral calculus
justied
> in terms of nite approximations. Newton, Leibniz,
Robinson, and
> Conway believe in innitesimals.
But we try to solve equations, and sometime mathematicians do
weird
things when trying to solve equations, how do we know it
doesnt
matter?
> Is not the frequency of light innitely variable? It is,
because it
> spins around in a circle, and pi is irrational.
A plane wave has a frequency, but plane waves dont exist,
they are
idealizations. Real elds dont have well-dened frequencies.
And
honestly if the energy involved is limited then there is
going to be a
maximum frequency possible, and even if your equipment/budget
are
really fancy, a high enough frequency of light would have an
event
horizon form around it, which would obscure it from you pretty
effectively, then there is the complication that high
frequency light
light and not anything else seriously breaks down. But Ill
assume
you mean innitely tunable within a bounded range. And I
disagree
with that too (but its basically the nonexistance of plane
waves
again in fancy/detailed talk). The light originally came from
someplace and went someplace, otherwise its unobservable and
I cut it
from my model, (cause if you postulate strictly unobservable
things
that dont interaction with what you do observe then you can
make up
anything), and otherwise the creation and destruction
endpoints
restrict what frequencies were possible. So you have limited
tunability of a laser.
> What do you think about the observed forces of gravity
being constant
> repulsion from all sides, blocked by nearby masses in the
direction
> that mass? Most people assume and rightly that the Earths
gravity
> draws mass towards it, because its mass is larger than
anything else
> on Earth. How about the opposite?
> I hope you think I have a decent mindset for the pursuit of
the study
> of physics.
The most obvious aw is about mass. Gravity is based on
energy. A
hot glasss of N moles water weighs more than the same glass
with N
moles of coller water, a compressed spring wieghs more than
the same.
Self-contained nuclear power plants heating themselves up do
NOT
reduce the mass of the earth system. SO on and so forth. So
if your
argument depends on MATTERs allegedly unique ability to block
repulsion, then it is clearly false. As for distinguishing
the two
cases (repulsion, versus attraction), the little tiny person
pulls the
earth towards themself with a force EQUAL in magnitude to the
force
that the earth pulls the little tiny person. How does your
theory
explain that result? How about a denser planet than the earth
with
the same size, why does it pull stronger? Maybe your theory
explains
all this, but it is vague as described.
> In terms of Skolems paradox, and why math is the way it
is, Skolem
> basically tells us that innite sets are equivalent.
He shows as that in truth they are equivalent, but he doesnt
x the
axioms to reect that truth. And if we have to step out of the
axioms, then why have them.
> Assume it is your task to describe a function bijecting the
naturals
> and reals. Whats the shortest answer you can use here on
sci.math
> that would be mutually understood?
I dont understand WHICH set is supposed to be the reals, so
its hard
to biject onto a moving target. You can formalize the proofs
of
existance of every specic real that has a proof and
correspond the
positive integer that corresponds to the formalization to the
real
about which it proves. There are multiple proofs for some
reals, so a
subset of those naturals is in one-to-one bijection, that
subset can
be put into one-to-one bijection with the entire naturals. I
dont
know if thats a bijection between the reals and the
naturals. I
cant prove that there is another real number in ZF, since in
ZF I
cant prove that that bijection I just described exists. So
Im left
with an undecidable proposition about whether its a
bijection between
the reals and naturals. It denately is NOT in the sense that
it
isnt a set at all. But to me that is a worse problem than
anything
else.
> Are you often a computer programmer, Dr. J.?
When Im not teaching, yes. And I thought I told you earlier
that Im
not a doctor, when you rst suggested calling me Dr. J. If I
failed,
or it didnt post. Im not a doctor. I havent defended my
thesis
(on an entirely unrelated subject), and Im not sure its
worth the
bother or expense, so I might not.
===
Subject: Re: Skolems Paradox and why is math the way it is?
> Why not?
>
> We need some way of saying what is and is not a set,
what is your
> dention and/or what axioms do you use to assert the
existance of
the
> sets you claim exist?
>
> Hi Dr. J.,
>
> I dont know the answers to all of your questions.
Theyre not
> insoluble.
>
> About that link, actually it is the correct link. Also,
it does.
> Even Goedels right sometimes.
> How is predicting or anticipating results supposed to mean
anything to
> me. Maybe you are expecting me to have certain views about
the
> results. It seemed like Godel thought higher orders save
the day, and
> I thought you didnt think so, so what was I supposed to
get from the
> About P(X) = X+1, that basically says that P(X) equals X
plus one.
>
> About emposed, that is about questions posed. Abut
enframed, that is
> about there only being rst order logic.
>
> I dont claim to understand your logic-and-a-half, or IF
logic.
> I think youd nd it interesting. The book I read is called
The
> Principles of Mathematics Revisited by Jaakko Hintikka.
There are
> some errors, I thought they were obvious, he repeats
himself enough
> that when he says it wrong one time you know what he meant,
but you
> can always pick up a critical review to make sure you know
what you
> can trust.
> I say: nothing, and not nothing, and not that, ad
innitum, those
> are each sets.
>
> For a given set X, and a given universal set U which may
well be the
> actual universal set, I say that U X is the context of X.
>
> In set theory, anything is a set. Thats the only reason
set theory
> can be said to be exible enough to be the foundation of
mathematics,
> it has to be all-inclusive. To meet that goal, it is
necessary that
> it resolve its issues instead of restating them to
dizzying heights of
> stacked plates, because such pablum is meaningless and
less than
> worthless. What that means is that if you have to resort
to second
> order logic, some higher logic, than logic, to utilize
logic,
then
> there is a problem with that logic.
> As I understand it, you start with the empty set, then the
universe,
> then the universe missing somthing, then that something,
then the
> universe with some different set, then that different set,
and you
> have some method to keep going. Did I get that right. Dont
you have
> to be very very careful to have the property that the
powerset of a
> set is just after the set itself? Obviously if you make your
> construction general enough, then youd include anything
you can
> describe, but then Id just wonder what description
corresponds to
> something similar to the real numbers? My whole point is
that I
> havent seen them described.
> So, even the unicorn and Haephaestus, mythical creatures
and beings,
> are sets, the set of all sets is a set, the empty set is
a set, and
> anything and everything that can possibly be dened is an
object.
> Why not a set? Its that, too.
>
> Plainly, as far as I know, the uncountable is meaningless
in
physics
> and for the most parts in theories used to explain
physics, it is
> ephemera that entails from that most educated people in
mathematics
> have been exposed to the word and many use it. That
mostly means
> thats one of the rst words they use to describe the
real numbers,
> because they cant count them. For most of them, trying
to justify
> mapping the naturals to the reals doesnt enter into
their equations
> because they have seen their tool of the integral
calculus justied
> in terms of nite approximations. Newton, Leibniz,
Robinson, and
> Conway believe in innitesimals.
> But we try to solve equations, and sometime mathematicians
do weird
> things when trying to solve equations, how do we know it
doesnt
> matter?
> Is not the frequency of light innitely variable? It is,
because it
> spins around in a circle, and pi is irrational.
> A plane wave has a frequency, but plane waves dont exist,
they are
> idealizations. Real elds dont have well-dened
frequencies. And
> honestly if the energy involved is limited then there is
going to be a
> maximum frequency possible, and even if your
equipment/budget are
> really fancy, a high enough frequency of light would have
an event
> horizon form around it, which would obscure it from you
pretty
> effectively, then there is the complication that high
frequency light
> light and not anything else seriously breaks down. But Ill
assume
> you mean innitely tunable within a bounded range. And I
disagree
> with that too (but its basically the nonexistance of plane
waves
> again in fancy/detailed talk). The light originally came
from
> someplace and went someplace, otherwise its unobservable
and I cut it
> from my model, (cause if you postulate strictly
unobservable things
> that dont interaction with what you do observe then you
can make up
> anything), and otherwise the creation and destruction
endpoints
> restrict what frequencies were possible. So you have limited
> tunability of a laser.
> What do you think about the observed forces of gravity
being constant
> repulsion from all sides, blocked by nearby masses in the
direction
> that mass? Most people assume and rightly that the
Earths gravity
> draws mass towards it, because its mass is larger than
anything else
> on Earth. How about the opposite?
>
> I hope you think I have a decent mindset for the pursuit
of the study
> of physics.
> The most obvious aw is about mass. Gravity is based on
energy. A
> hot glasss of N moles water weighs more than the same glass
with N
> moles of coller water, a compressed spring wieghs more than
the same.
> Self-contained nuclear power plants heating themselves up
do NOT
> reduce the mass of the earth system. SO on and so forth. So
if your
> argument depends on MATTERs allegedly unique ability to
block
> repulsion, then it is clearly false. As for distinguishing
the two
> cases (repulsion, versus attraction), the little tiny
person pulls the
> earth towards themself with a force EQUAL in magnitude to
the force
> that the earth pulls the little tiny person. How does your
theory
> explain that result? How about a denser planet than the
earth with
> the same size, why does it pull stronger? Maybe your theory
explains
> all this, but it is vague as described.
> In terms of Skolems paradox, and why math is the way it
is, Skolem
> basically tells us that innite sets are equivalent.
> He shows as that in truth they are equivalent, but he
doesnt x the
> axioms to reect that truth. And if we have to step out of
the
> axioms, then why have them.
> Assume it is your task to describe a function bijecting
the naturals
> and reals. Whats the shortest answer you can use here on
sci.math
> that would be mutually understood?
> I dont understand WHICH set is supposed to be the reals,
so its hard
> to biject onto a moving target. You can formalize the
proofs of
> existance of every specic real that has a proof and
correspond the
> positive integer that corresponds to the formalization to
the real
> about which it proves. There are multiple proofs for some
reals, so a
> subset of those naturals is in one-to-one bijection, that
subset can
> be put into one-to-one bijection with the entire naturals.
I dont
> know if thats a bijection between the reals and the
naturals. I
> cant prove that there is another real number in ZF, since
in ZF I
> cant prove that that bijection I just described exists. So
Im left
> with an undecidable proposition about whether its a
bijection between
> the reals and naturals. It denately is NOT in the sense
that it
> isnt a set at all. But to me that is a worse problem than
anything
> else.
> Are you often a computer programmer, Dr. J.?
> When Im not teaching, yes. And I thought I told you
earlier that Im
> not a doctor, when you rst suggested calling me Dr. J. If
I failed,
> or it didnt post. Im not a doctor. I havent defended my
thesis
> (on an entirely unrelated subject), and Im not sure its
worth the
> bother or expense, so I might not.
Doctor is just a term of affection, in an academic way. Its a
nickname. Are you often an assembly coder? (Hi Steve.)
Wassup, doc.
Doctor, doctor, doctor, ....
About the gravity thing, I am thinking more about the hot
water
weighing more than the cool water and the compressed spring
more than
the free spring. So, could a ship carry for its ballast
springs and
compress them for ballast, or is that a mass system? Is the
force
only applied in the direction of the springs compression? Is
the
warmer water having higher gas pressure throughout, affecting
and
effecting force in all directions? Do you have any inertial
capacitors?
The gravity conjecture there is really just that, and
uninformed to
boot.
Youre telling me that you cant innitely tune a laser, but
laser is
coherent light of one wavelength and frequency, uncoherent
light is
not coherent. My thoughts on the matter are not particularly
coherent.
I would like some example of something in nature that is
innitely
variable. I think the universe itself is innite, in
space-time.
About the nothing, not nothing, not not nothing, etcetera, I
hadnt
thought to consider that the rst step after nothing was U,
but
rather that it was 1.
You ask for a construction of the real numbers. In a sense,
the real
numbers are the closure of the hyperintegers to division,
except that
the hyperintegers (from NSA) are just integers and the
closure of the
integers to division is the rational numbers. Irrational
numbers
exist, that is well-known.
The real numbers exhibit all qualities of being continuous.
Draw a
line from point A to point B without lifting the pencil.
Pi is an irrational number, its not a rational number, being
a ratio
of two integers, or algebraic irrational number, being a root
of a
polynomial (not power series) with integer coefcients. The
Earth
orbits the Sun each year, sweeping through each of 2Pi
radians. Pi is
also absolutely normal, probably, and as well, you can
calculate any
digit of the expansion of Pi without calculating intervening
digits.
Is each irrational absolutely normal? (No.) Is any rational
absolutely normal? (No.)
Lets focus on Loewenheim-Skolem. Loewenheim-Skolem appears
to posit
that innite sets are equivalent. Is that so? Is the notion of
uncountability meaningless or useless?
About Goedels footnote, it may be along the lines that
higher-order
logic is just an escapist fantasy from rst-order logic.
Theorize. What is your thesis? What does physics need from
mathematics?
Ross F.
===
Subject: Re: Skolems Paradox and why is math the way it is?
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at 09:29 PM, troubled6man@yahoo.com (J.E.) said:
>No apparatus measures it perfectly, but any hamiltonian that
causes
>correlations between the wavefunction part corresponding to
the
>and the wavefunction part corresponding to that cross section
Would you care to put that into English? You seem to be
missing parts
of the sentence. You might also explain what you mean by a
wavefunction part.
>Any Bohmian model should describe this to you in
>whatever language you like, since Im unlike to happen to
pick word
>you already understand.
In the sense that every interpretation that Ive read of has
had holes
in it.
>Well Ive been asking why people like ZF axioms, and you
havent
>been disparaging them in favor of GBN axioms.
So? That just means that the differences arent relevant in
this
context.
>Can you describe how this is anything other than an
ambiguous result
>to have?
It follows from the axioms. It doesnt much matter which set
theory I
use; its true in any powerful enough to be useful. There is
no
ambiguity.
>The ZF axioms fail to describe an uncountable number of
things,
Wrong; you just disapprove of the denition and of the
description.
>because they are incapable of proving that that many
distinct things
>exist.
Wrong; they prove[1] that they exist; they dont enumerate
them.
>If I new how to do that rst, I would. It seems hard when
physics
>uses math that already has elements that are independant of
proof to
>exist.
Youre confusing constructibility with existence.
>Incompleteness is a problem,
Its a problem that youre stuck with, as shown by Kurt
G.9adel. Any
theory powerful enough to be useful will have the problem.
>Descriptive completeness isnt beautiful or elegant to
>mathematicians?
It isnt possible.
>It seems vague,
Lots of things seem vague, without detracting from their
usefulness.
>like if someone claimed there was some deep symmetry
>on the oor of a room that was mostly covered with a bland
rug.
You mean like quarks and gluons?
>If the deep symmetries are forever unobservable, then what
is the
>point.
forever unobservable?
>So why consider them?
Because Mathematics would be crippled without them.
>Why not stick to denite results about things
>we have observed and statistical claims about the things we
havent?
Because Mathematics is not Physics, and because even
Physicists dont
do that. Mathematics is about what we can prove, not about
what we can
observe.
>A theory can only describe things that exist in all its
models,
>right?
Correct.
>So if consistency is independant of the theory, then it
cant be
>described in the theory.
What do you mean by independent? It is true that a consistent
theory
encompassing the integers cannot prove its own consistency.
>If truth is independent of the theory,
What is truth, and what does it have to do with Mathematics?
>I dont know what your denition of construct is, Im
talking about
>sets that you can prove exist from the axioms.
I can prove from the axioms that uncountably many sets exist.
The fact
that you dont like such proofs is irrelevant.
>The fact that no bijection exists from
>the two classes, is an inadequacy of the thoery
No, its an inadequacy of your perception.
>it says NOTHING about the alleged larger class of subsets
that COULD
>have been proved with MORE axioms.
With more axioms it would be a different theory. You could
add axioms
asserting the existence of sets not derivable from ZF, but
you could
equally easily add axioms asserting their non existence. As
long as
each addition is consistent with ZF, they are equally valid.
>Then a theory that is clear about the incompleteness being
in the
>parts we dont use should be good enough then.
There is no such thing.
>That statement is not a theorem of ZF.
Irrelevant.
>How can we tell that the incomplete parts of the theory of
ZF dont
>interest with the physically interesting parts of our
physical models
>based on ZF.
By running into them. Should that happen, you can extend ZF,
but there
is no guaranty that the incompleteness of the extended ZF
wont also
hit you.
>A single point is less than a drop in the bucket.
Its but one of many, but its a showstopper.
>A single point is less than a drop in the bucket. Besides
doesnt
>the countable model have a mean value for every function
that can be
>proven to exist?
No, ZF has an MVT, and that translates to a statement about
the model.
>Then surely you can see how the class of sets you prove
theorems
>about is not different in kind than the class of natural
numbers.
Irrelevant.
>There are not MORE sets you prove theorems about than
numbers you can
>write down.
What do you mean by are? What do you mean by more? I know what
those terms mean in Mathematics, and am not interested in
what they
mean in, e.g., Philosophy when sci.math is concerned with
Mathematics.
>In that case there, the existance of some sets, that if they
existed
>would be elements of the power set depends on the OTHER
axioms.
What do you mean by existence? It is a theorem of the theory
that
they exist; no other axioms are needed to prove it.
>I dont need inaccessible cardinals, there are missing sets
in the
>power set of the naturals.
What do you mean by are? What do you mean by missing? This
isnt
rec.metaphysics, and the relevant denitions are the
mathematical
ones.
>Why isnt this a problem to anyone else?
It may be a problem to the constructivists. Why should it be
a problem
to anyone else?
>I dont publish proofs of this because it seems obvious and
trivial.
Proofs of what? That not every set in ZF is constructible in
ZF?
>Show me this theorem. I have only seen theorem about lack of
>bijections not about existances of many sets.
Youre talking about some vague mystical concept of existence
and Im
talking Mathematics. Youre talking some metaphysical concept
of
number and Im talking Mathematics.
>The two concepts are different, as I HOPE the skolem
paradoxs
>resolution has already made clear to you.
What the Skolem paradox has made clear to me is that there is
a
countable model. Only this, and nothing more.
>There arent proofs that all things that COULD consistently
be added
>to the theory ARE already in the theory.
There arent proofs of the Tooth Fairy either. The fact that
any
usable theory is incomplete is more than half a century old.
What does
that have to do with the price of tea in China? Youre not
going to
nd a complete theory; deal with it.
>I can describe things that should be sets
What do you mean by should be? There is no should and should
not; there is only theorem and not a theorem.
>Im sorry that we will have to disagree on this (I would have
>preferred for you to understand my arguement and either
adopt it
>yourself or convince me of a better one,
This entire group hasnt been able to convince you to see the
obvious;
either you will eventually see or you wont. I wont lose any
sleep
over it.
>The theory is about the lab results and the calculation
results,
That hasnt been true for a century.
>but you dont seem to either not want to or not be able to
>understand it).
Thats an interesting theory. Therre are, alas, no data to
substantiate it.
>Those are DESCRIPTIONS of the theory.
In your world, perhaps, not in the Physics literature.
>Each logical problem from math
Your lack of agreement or comprehension does not constitute a
logical
problem.
>I already said Closed timelike curves,
statement as written made no sense.
>do you consider that to be merely a constraint that he is
>interested in,
Do you have a problem with reading comprehension? Or are you
deliberately asking about things that you know arent there?
Since
youve already claimed multiple times that I dont understand
the
physical vocabulary, Im sure that you wont embarrass
yourself by
asking for the title of one of the relevant papers.
>Im sure hed eventually admit that he wanted to be told,
That depends on what hes doing; perhaps *YOU* dont
understand it
well enough to accurately describe it, and have left out some
unimportant qualiers.
>How about a decriptively complete model?
Sure, if its nite. Not if it includes the integers.
>ZF seems to have its undecidable statements buried in
innite
>sets, so it seems like a theory with smaller sets could
remove the
>undecidable sets.
Sure. That would mean giving up, e.g., induction. Lots of
luck using
it for Physics.
>There are things that should be sets,
Should be sets is not a term in Mathematics. Nor do I
understand
what you mean by are.
>Axioms havent been well ordered. Adding an axiom that makes
the
>system inconsistent doesnt seem very helpful, and what if an
>existing axiom is inconsistent with the axiom we really
should add,
What if the one that you removed isnt the one that you
should add?
Doesnt it make more sense to not tinker with the axioms
until you
know what youre doing?
>Isnt the prudent thing to do, to start out with as few as
possible
No, because either that would leave you with a theory
incapable of
proving some things of interest or with an equivalent theory
in which
the axioms were more terse.
>If not, whats wrong with what I said?
Your assumption that you know whats needed.
>Observations are local, the wavefunction is not. Clearly
only a
>portion is applicable to the verifyable parts of any
particular
>experiment.
Thats not clear to me. Do you understand, e.g., the
difference
between position space and momentum space?
>The other parts can be lled in with anything that makes the
>computation easier and can be thrown away when you are done.
No.
>Some physicists might start by counting their toes and
drinking
>coffee, it doesnt mean that affects me. What matters is the
>initial conditions and the potentials and the evolution
equations.
What kind of equations? If youre talking about equations
involving
operators, then it does affect you.
>You ask such details questions sometimes that I forget you
may not
Well, somebody doesnt understand. When the likes of Ryder
disagree
with you, I can draw the obvious conclusion.
>You keep assuming that I do my physics badly just because I
dont
>do it like you.
Or like any of the QFT books that Ive seen.
>Im trying to get your perspective and I treat you with
respect
Respect? Attributing things to me that I didnt write is not
respect.
Nor is telling me that what you are writing is above my head.
>You seem to assume that Im dishonest just because you arent
>familiar with my work.
I assume that youre dishonest because youre been dishonest
in this
thread.
>I have trouble believing you.
Then perhaps you shoudl look up sarcasm.
>What equipment are you using to get these results?.
It doesnt matter, because the commutation relations make it
impossible to measure position and momentum concurrently,
regardless
of the apparatus. You can increase the precision of one only
at the
expense of the other.
>There are observable consequences to how much funding you
have?
Thats an example of how you convinced me that youre
dishonest.
>You subject your nger to a hamiltonian from your brain that
>correlates the brain state with the position of your nger
such
>that the position of you nger is correlated with the parts
of the
>experimental apparratus that you decide to observe.
That doesnt measure the wave function.
>If its gonna be approximate anyway, why not nd a better
basis
>then?
Why not x the real problems instead of chasing after
imaginary
problems?
>For bounded energy bound states there is.
So your theory is no good for free electrons?
>Models tell you things about your axioms. In this case they
pointed
>out the fact that lack of bijections and size are not
intrinsically
>related.
No. You still dont get it. This isnt metaphysics and isnt
mysticism; the statements in ZF use the language of ZF.
Cardinality is
dened in terms of the existence of bijections.
>But these lack of satisifactions only occur under the rug
where we
>cant observe them anyway with our nite precision.
We most certainly can observe them; h isnt *that* small. Nor
is a
model the same as a calculation regime.
[1] More precisely, you can derive that result from the
axioms.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT

Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply
to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spamtrap@library.lspace.org
===
Subject: Re: Skolems Paradox and why is math the way it is?
> at 09:29 PM, troubled6man@yahoo.com (J.E.) said:
>No apparatus measures it perfectly, but any hamiltonian
that causes
>correlations between the wavefunction part corresponding
to the
>and the wavefunction part corresponding to that cross
section
> Would you care to put that into English? You seem to be
missing parts
> of the sentence. You might also explain what you mean by a
> wavefunction part.
>Any Bohmian model should describe this to you in
>whatever language you like, since Im unlike to happen to
pick word
>you already understand.
> In the sense that every interpretation that Ive read of
has had holes
> in it.
Read about the Bohmiam interpretation. The only problem with
it is
only thing wrong with that is that it is unnecissary because
the same
experimental set up causes the wavefunction of the pen to
move and
that causes the wavefunction of the ink to move over the lab
book.
Thats all that matters. Look up the SSC theorem, anything
other than
a wavefunction can be cut away.
>Well Ive been asking why people like ZF axioms, and you
havent
>been disparaging them in favor of GBN axioms.
> So? That just means that the differences arent relevant in
this
> context.
Youve decided that I wont like GBN even though I dont like
ZF. You
can prejudge anything you want, I dont know why you object
to me
inferring those judgements on you.
>Can you describe how this is anything other than an
ambiguous result
>to have?
> It follows from the axioms. It doesnt much matter which
set theory I
> use; its true in any powerful enough to be useful. There
is no
> ambiguity.
There are collections that clearly should be sets. Why arent
they
sets. The ONLY purpose ANYONE has explained to me is that the
lack of
sethood to these well-dened collections is necissary to
maintain an
eloborate ILLUSION that the set of reals is of a different
size than
the set of naturals by constructing a series of missing sets
that
maintain the ction of a theory of caridinality as if that
theory is
REALLY about something OTHER than the formal limitations of
ZF.
>The ZF axioms fail to describe an uncountable number of
things,
> Wrong; you just disapprove of the denition and of the
description.
You can DEFINE a SET to be uncountable, but that doesnt MEAN
that
there are more real numbers in them than in the naturals, all
it MEANS
is that you have a limited number of one-to-one
correspondances that
you are considering. It doesnt MEAN a single thing about the
images
of the disallowed one-to-one correspondances.
>because they are incapable of proving that that many
distinct things
>exist.
> Wrong; they prove[1] that they exist; they dont enumerate
them.
Show me how it follows from the axioms. You agree that you
cant list
them. Just show me that P-(N) the collection of collections S
of
natural numbers that that follow from ZF to be sets, is
different (or
the same as) P+(N) the set of all subsets of naturals
numbers. If
P-(N) is not a set, then it is neither countable nor
uncountable
(thats why I HATE that word), but if P-(N) is a set then it
clearly
SHOULD be countable, even if it is not in fact countable. IF
P-(N) is
equal to P+(N) (I hope you agree with me that it follows from
ZF that
P+(N) is a set) then clearly P-(N) isnt countable. The axiom
of
equality says they are the same iff they have the same
members. So
prove whether or not they have the same members. If you can
prove
that there exists an element of P+(N) that is not in P-(N),
then Ill
feel like an idiot, but Ill be happy. I dont know how you
are going
to prove it though, its independant of the ZF axioms. How
about that
alleged real number the diagnomal arguement fallaciously
claims
exists. Is it in P+(N)?
>If I new how to do that rst, I would. It seems hard when
physics
>uses math that already has elements that are independant
of proof to
>exist.
> Youre confusing constructibility with existence.
It is proof theory that attempts to conate the two and fails.
Because things are either provable or they arent, but some
alleged
sets neither exist nor dont exist. Thats a very very bad
situation.
Is that xed by second order logic?
>Incompleteness is a problem,
> Its a problem that youre stuck with, as shown by Kurt
G.9adel. Any
> theory powerful enough to be useful will have the problem.
G.9adel assumed a rst order theory. IF-logic is just a tad
bigger
than rst order, so maybe it escapes. And maybe second order
is
better than rst order if it doesnt sweep everything under
the rug,
which it seems to do when people base it on ZF.
>Descriptive completeness isnt beautiful or elegant to
>mathematicians?
> It isnt possible.
What about IF-logic. Have you even seen Hintikka
generalization of
the so-called axiom of choice? And if its not possible, what
is the
basis for your claim. Are you reteating back to rst order
nitely
axiomizable theories again? Just because you do that doesnt
make
other people who dont all of a sudden be doing impossible
things.
>It seems vague,
> Lots of things seem vague, without detracting from their
usefulness.
Vagueness and incompleteness dont sit well on the same shelf.
>like if someone claimed there was some deep symmetry
>on the oor of a room that was mostly covered with a bland
rug.
> You mean like quarks and gluons?
We can peek under the rug whenever we want to. Why do you
assume
insane and terrible things about all physicists and physics
and then
project it onto other people.
>If the deep symmetries are forever unobservable, then what
is the
>point.
> forever unobservable?
I dont think there is a point. Id call the theory a waste
of time,
or maybe even dishonest depending of the details of the
theory.
>So why consider them?
> Because Mathematics would be crippled without them.
How is it crippling to face reality. In physics if something
is
proven wrong by reality, we try again. This is science.
>Why not stick to denite results about things
>we have observed and statistical claims about the things
we havent?
> Because Mathematics is not Physics, and because even
Physicists dont
> do that. Mathematics is about what we can prove, not about
what we can
> observe.
You cant prove that P-(N) and P+(N) are the same or
different, so why
do you talk about them being different?
>A theory can only describe things that exist in all its
models,
>right?
> Correct.
An uncountable universe doesnt exist in all models. The
universe
isnt a set, so it is neither countable nor uncountable, but
there is
no basis to say the universe is large, since the countable
model shows
that it can be quite small.
>So if consistency is independant of the theory, then it
cant be
>described in the theory.
> What do you mean by independent? It is true that a
consistent theory
> encompassing the integers cannot prove its own consistency.
Independent means that it can be neither proven nor disproven
in the
theory. Maybe the word undecidable is more familiar to you.
Im not
familiar with the denintion of that, but its likely the
same.
Eculids parrell postulate is independant of the theory
composed of
the rst four axioms, as an example. And yes I know that
Eculids
axioms as self-stated are not sufcient to make a consisten
theory,
but there are consistent theories called Euclidean and
Non-Euclidean
geometry that axiomation such that there is a subcolection of
common
axioms, and that is was Im expecting you to be familiar
with. And I
apologize if you arent/werent.
>If truth is independent of the theory,
> What is truth, and what does it have to do with Mathematics?
Tarski showed its independant of ZF. If you dont want to
worry
about, you dont have to, since I think you said you spend
your time
living in worlds that are independent of truth.
>I dont know what your denition of construct is, Im
talking about
>sets that you can prove exist from the axioms.
> I can prove from the axioms that uncountably many sets
exist. The fact
> that you dont like such proofs is irrelevant.
I never said I didnt LIKE the proofs. I disagree about WHAT
is
proved by your proof. You can prove that ONE set lacks a
bijection.
But if you want to prove whether than makes the set of
bijecions small
or the image set large, you need ANOTHER proof to distinguish
the too
cases. Its not that I dont LIKE the second proof required,
its
that people are HIDING the proof under a rug, IF they have
one at all.
YOU havent shown one for instance.
>The fact that no bijection exists from
>the two classes, is an inadequacy of the thoery
> No, its an inadequacy of your perception.
Imagine a person C makes a theory like Euclidean geometry but
without
any kind of parrallel axiom. Now imagine C said the theory
described
ALL lines. I could ask if there exists a line M though a
point P that
never intersects a line L that P is not on. Person A would be
forced
to say that they didnt know it it did or not. Then what if
someone
created a model and called it TCM, where there were no such
lines.
But then two other people, Z, and F, create models where
there ARE
such lines. Person C then CLAIMS that they like like models Z
and F
and claim that they are better models than TCM (because they
are
non-compact likely), but Z has innitely many different lines
through
point P that dont intersect line L, while F has only one
such line
for each P not on a line L. So the noncompact theories are
NOT the
same.
Well in ZFC we have a similar situation, you dont like The
Countable
Model because it is countable (just like C didnt like TCM
because it
was compact), but its JUST as good a model as the other
models. And
they are different than each other, so its not like there is
a
preferred model of the theory. The faithful models are
faithful. If
C kept babbling ON AND ON about geometry not being compact,
then he
should add an axiom to make it so. And lucky for C, he can.
The
problem is that YOU are babbling about proving things you
cant prove
and you dont HAVE an axiom to add that REALLY xes the
problem, but
you still insist that the universe is uncountable, when that
is
indendant of the axioms.
>it says NOTHING about the alleged larger class of subsets
that COULD
>have been proved with MORE axioms.
> With more axioms it would be a different theory. You could
add axioms
> asserting the existence of sets not derivable from ZF, but
you could
> equally easily add axioms asserting their non existence. As
long as
> each addition is consistent with ZF, they are equally valid.
If you assert that they dont exist, then it is clear that
P+(N) isnt
the set of all subsets that could exist. I think the problem
here is
that you want to consider the set of all sets that could
consistently
exist, but that you cant because depedning on which axioms
you add,
different sets will be existant or not. This is an idicator
that some
axioms could be more or less interesting or useful, and that
we should
make more. If the incompleteness of ZF is big enough to hide
these
things, then make the rug we should remove is the one that
makes
cardinal theory possible. Its the most radical adjustment I
can come
up with, and to decide how to do it, I wanted to know why we
like it.
How does lacking bijections HELP anyone. This is INDEPENDANT
of your
axioms so it WILL NOT disprove any of your theorems, so why
are you
resistant to contemplating it. I understand not xing things
that
arent broken, but ZF is broken.
>Then a theory that is clear about the incompleteness being
in the
>parts we dont use should be good enough then.
> There is no such thing.
Proof?
>That statement is not a theorem of ZF.
> Irrelevant.
Relevant.
>How can we tell that the incomplete parts of the theory of
ZF dont
>interest with the physically interesting parts of our
physical models
>based on ZF.
> By running into them. Should that happen, you can extend
ZF, but there
> is no guaranty that the incompleteness of the extended ZF
wont also
> hit you.
I agree that we wont get a garantee. I wasnt planning on
putting
Godel patches all over everyeher transnitely. My plan is to
break
cardinality, not patch around it. I want to know what will
REALLY
break if I break cardinality, not what will just require
different
proofs. So what REALLY depends on this lack of bijections?
>A single point is less than a drop in the bucket.
> Its but one of many, but its a showstopper.
Not when you assumed the point existed by making a false
assumption.
Then its an audience putter to sleep. Except for the
memebers that
like MAGIC tricks, and think that something else is going on.
>A single point is less than a drop in the bucket. Besides
doesnt
>the countable model have a mean value for every function
that can be
>proven to exist?
> No, ZF has an MVT, and that translates to a statement about
the model.
Its a FAITHFUL model! You cant exhibit a contradiction in
it withou
diproving the consistency of ZF! If you believed you could do
that
you wouldnt be talking to me on sci.math. youd be
publishing your
paper right now!
>Then surely you can see how the class of sets you prove
theorems
>about is not different in kind than the class of natural
numbers.
> Irrelevant.
See my above question about what REALLY breaks if you break
cardinality. Thats why my question is relevant.
>There are not MORE sets you prove theorems about than
numbers you can
>write down.
> What do you mean by are? What do you mean by more? I know
what
> those terms mean in Mathematics, and am not interested in
what they
> mean in, e.g., Philosophy when sci.math is concerned with
Mathematics.
I want to know what theorems DEPEND on theses bijections being
missing. I dont think there really are any, and you havent
exhibiting any yet. You MVT didnt hold water (the drops
turned out
not to be real, only falsly assumed to exist).
>In that case there, the existance of some sets, that if
they existed
>would be elements of the power set depends on the OTHER
axioms.
> What do you mean by existence? It is a theorem of the
theory that
> they exist; no other axioms are needed to prove it.
Prove that R contains an element that is not in R- (the set
of reals x
such that the exists a rst order statement P(x) such that
P(x) is
satisied (is true) for only one x). You havent done that.
>I dont need inaccessible cardinals, there are missing
sets in the
>power set of the naturals.
> What do you mean by are? What do you mean by missing? This
isnt
> rec.metaphysics, and the relevant denitions are the
mathematical
> ones.
I can dene a well-dened set A such that I can introduce an
axiom
to ZF(C) to make an new axiom system ZF(C)+ where the set A
is a
subset of the naturals and I can prove that ZF(C)+ is
consistent iff
ZF(C) is consistent. Thats what I mean. That isnt rec.
anything.
>Why isnt this a problem to anyone else?
> It may be a problem to the constructivists. Why should it
be a problem
> to anyone else?
Why do mathematicians make a whole arthimetic about their
blind spot?
This is vague an imprecise. The diagonal arguement proves
that a
bijection doesnt exist (the bijection to all aubsets
provable in
ZF(C)), when it should exist since the power set is SUPPOSED
to
contain ALL subsets. The reason it doesnt is because ZF(C)
accidentally scrubed the real number and
>I dont publish proofs of this because it seems obvious
and trivial.
> Proofs of what? That not every set in ZF is constructible
in ZF?
I dont know what you mean by constructable, but do you
really claim
that more sets exist in ZF than we can prove exist. If so
what is
your basis for that claim? All I see PROVED is that ZF is so
SMALL
that it doesnt include the set corresponding to all proofs,
which is
just plain silly for something that is supposed to have ALL
sets.
>Show me this theorem. I have only seen theorem about lack
of
>bijections not about existances of many sets.
> Youre talking about some vague mystical concept of
existence and Im
> talking Mathematics. Youre talking some metaphysical
concept of
> number and Im talking Mathematics.
Im trying to talk about mathematics and bending over
backwards to try
to be clear about something that ZF is obsfucating and you
are stating
the theorem you CLAIM exists. The theorems you do claim dont
prove
what you claim they do, and yet you keep mentioning more
theorems.
>The two concepts are different, as I HOPE the skolem
paradoxs
>resolution has already made clear to you.
> What the Skolem paradox has made clear to me is that there
is a
> countable model. Only this, and nothing more.
The skolem paradox explains that a countable model can lack
bijections
to count countable sets, thus articially making them appear
the be
uncountable and making them LITERALLY uncountable on the
inside of the
model, while showing that the set need not have many many
members.
>There arent proofs that all things that COULD
consistently be added
>to the theory ARE already in the theory.
> There arent proofs of the Tooth Fairy either. The fact
that any
> usable theory is incomplete is more than half a century
old. What does
> that have to do with the price of tea in China? Youre not
going to
> nd a complete theory; deal with it.
You are assuming that because any rst theory I made would
ALSO be
incomplete that it follows that cardinality theory is USEFUL.
I can
deal with incompleteness. I just dont want to make a theory
(cardinality theory) OUT of an incompleteness and pretend
that it is
something different. You could take the word uncountable and
REPLACE
it with INCOMPLETE, and then Id probably be happy because
then it is
clear what it means to not have a bijection in the model, it
means the
model incompletely DESCRIBES the set. As I believe the ZF
axioms do
not adequately describe the real numbers. Isnt there
somekind of
model theory that says that a complete ordered eld is not
nitely
axiomizable? I dont know for sure because I its hard to
remember
the interpretation of theorems that have a word that is used
differently in different context (model completeness,
algebraically
closed completeness, limit completeness, etc.). But the point
is that
I wouldnt AT ALL be surprised to nd out that ZF just cant
model
the reals well enough to do a full thoery of the reals.
>I can describe things that should be sets
> What do you mean by should be? There is no should and should
> not; there is only theorem and not a theorem.
In addition to theorem and not theorem, there is CAN which is
the
same as the intersection between theorem and not theorem. And
should
comes from a CAN that when introduced appears to have already
been
there. For instance is already a well-dened subset of a
previously
dened and previously proven to exist set. If you want to
call P(S)
the power set of S instead of calling it the set of all
subsets, then
Id be happy to drop the should. But that too close to
admitting that
your theory is incomplete in a way that matters, so I doubt
youll do
that.
>Im sorry that we will have to disagree on this (I would
have
>preferred for you to understand my arguement and either
adopt it
>yourself or convince me of a better one,
> This entire group hasnt been able to convince you to see
the obvious;
> either you will eventually see or you wont. I wont lose
any sleep
> over it.
Ive seen very different views from people in the group, and
frankly I
dont think most people listen to what I say. And its sad
that so
many people come here and say similar things to what Im
saying,
because then the group responds to what others have said
previously
instead of guring out what Im saying right now. Shouldnt
there be
some more evidence about there being more numbers in sets
that are
uncountable? After all these years, wouldnt you expect some
large
set to be able to do something that a small set cant.
Something that
you could PROVE it does instead of hypothetical cases
involving doing
things only AFTER you make false assumptions. I mean, come
on, any
set can do anything after you make a false assumption.
>The theory is about the lab results and the calculation
results,
> That hasnt been true for a century.
Are you talking Bohr, all he cared about was lab results. Are
you
calling string theorists and mathematical physicists as
physicists. I
dont know what you are saying at all, but aparantly Ive
found an
e-mail communication with someone that is alive 100 years
after Ive
died, or you dont think Im a physicsist. Im tired of
explaining
how REAL physics is done to mathematicians that dont do it
themselves
read or the next physicist they meet. This shouldnt have any
bearing
on the question about what mathematical results TRULY depend
on the
lack of bijections between sets that contain elements you can
prove
exist. You can take my criteria as given and if you want to
pretend
that I just want to make physics models for revisionist
historical
novels, thats ne with me, I dont care about convincing you
about
how this relates to the world anymore. I just want you to
answer my
question. What mathematical results TRULY depend on the lack
of
bijections from sets that contain elements that you can prove
exist.
Or I want you to explain how it is necissary and/or helpful
to assume
that sets MUST contain elements that you cannot prove exist.
>but you dont seem to either not want to or not be able to
>understand it).
> Thats an interesting theory. Therre are, alas, no data to
> substantiate it.
You lost me completely. Im asking why there is a set of all
subsets
of a set but no set of all subsets that you can prove exist?
Im
asking why excluding sets makes a better theory than
including them.
And nally Im asking why cardinal theory and/or
incompleteness seem
like valid reasons to IGNORE these questions. I dont think
these
questions are HARD to understand, so I do not know WHY no one
is
TRYING. Answer so other question you heard long ago just
because you
remmeber the answer doesnt answer MY question(s).
>Those are DESCRIPTIONS of the theory.
> In your world, perhaps, not in the Physics literature.
The physics literature is a large area, I think its safe to
say that
neither of us has read it all, and I nd many many people
that agree
with me. I know of many many more that disagree with me, but
that
when I talk about experimentally veriable results and the
outcomes
of experiments THEN they listen to me. Similarly I tend to
only
listen to those people when they discuss exprimentally
veriable
results and teh outcomes of experiments. Its enough to get
along.
>Each logical problem from math
> Your lack of agreement or comprehension does not constitute
a logical
> problem.
You havent explained the differences Ive asked. Why is
there no set
of subsets S of the naturals such that there exists a proof
in ZF that
S exists? No one has explained that. I could add an axiom to
assert
that it does. It wouldnt introduce any problems. So why
hasnt
anyone done it earlier? No one has given me a reason, except
maybe
the people who mentioned second order systems. And they
havent
actually said that such a set exists in the second order
system, but
at least there answer was vague enough that I couldnt tell
right away
that it was wrong, like I can with yours.
>I already said Closed timelike curves,
> statement as written made no sense.
I dont know what was confusing about it to you. Does it
really
matter to you? Do you know the answers to my questions but I
have to
earn them by proving that it matters to me? Isnt it easier
for
everyone if you just tell me if you do know and admit that
you dont
know if you dont know?
>do you consider that to be merely a constraint that he is
>interested in,
> Do you have a problem with reading comprehension? Or are you
> deliberately asking about things that you know arent
there? Since
> youve already claimed multiple times that I dont
understand the
> physical vocabulary, Im sure that you wont embarrass
yourself by
> asking for the title of one of the relevant papers.
I dont know what you mean about asking about things that
arent
there. Now you are discussing being embarrassed. Are you
asking for
a physics paper title
from me? Are you saying that Ive embarrassed myself by not
understanding technical math terms and that I should ask you
for a
mathematical paper title? What are you saying, is it
something else?
You said something vague, I told you it was vague, and I was
specic
about what was vague, and now you respond by saying that I
cant read,
or so it seems. And then you follow up with something vaguer
than the
rst thing. All to ask about physics? Why cant you just
answer my
math questions?
>Im sure hed eventually admit that he wanted to be told,
> That depends on what hes doing; perhaps *YOU* dont
understand it
> well enough to accurately describe it, and have left out
some
> unimportant qualiers.
Thats why you could tell me the result to me and I could
tell him,
without telling him you gave me the result. Then if Im an
idiot, it
doesnt look bad on you. Normally I wouldnt care, but they
sound
like references that I should really know about anyway. I
mean they
sound both interesting and important, and if they are old
results to
other people, thats still good if they are new results to me.
>How about a decriptively complete model?
> Sure, if its nite. Not if it includes the integers.
Have you studied IF-logic? Im not trying to claim it has
super
powers, but I want to know where you are coming from. Just
because
rst order is weak and secdon order introduces its own
quantiers.
Besides Hinttikka had the best generalization fo the axiom of
choice
Ive ever seen, its shame not to put it to use and see what
it can
clean up.
>ZF seems to have its undecidable statements buried in
innite
>sets, so it seems like a theory with smaller sets could
remove the
>undecidable sets.
> Sure. That would mean giving up, e.g., induction. Lots of
luck using
> it for Physics.
We can add as many axioms as we need for physics. Its not
the same
goals as pure math, where you want a small number of axioms to
generate as much as possible. If we want to generate a small
number
of theorems with a large number of weaker (in their deductive
powers)
axioms, thats ne.
>There are things that should be sets,
> Should be sets is not a term in Mathematics. Nor do I
understand
> what you mean by are.
I thought youd read Godel? I can consitently add sets to the
universe of ZF with new axioms that maintain consistency. And
these
arent weird inaccessible cardinals, they are subsets of
naturals. To
me its puzzling why mathematicians left them out in the rst
place.
I can understand if it was an accident, but surely thousands
of people
have known for decades, so instead of publishing that I found
a new
axiom, Im asking sci.math WHY the sets werent added. There
should
be either a technical or an obvious reason WHY.
>Axioms havent been well ordered. Adding an axiom that
makes the
>system inconsistent doesnt seem very helpful, and what if
an
>existing axiom is inconsistent with the axiom we really
should add,
> What if the one that you removed isnt the one that you
should add?
> Doesnt it make more sense to not tinker with the axioms
until you
> know what youre doing?
Yes it makes sense to nd out what they are doing. Thats
WHAT Im
trying to ask, but YOU seem to think that people here have
ALREADY
told me. They havent. They pointed at theorems I already
knew that
said things I already said, theorems that I used in MY
proofs. Or
theyve sited names of mathematicians that Ive looked up and
cant
nd any work that relates to this topic. Im sure it was
obvious to
them in their speciality, but I cant nd it. And other
people have
mentioned work in totally different axiom systems than ZF,
and others
mentioned second order theories without telling me if it
actually
xes the problem of missing sets, or if it just does
something weird
to make the uncountable model go away without really changing
anything.
>Isnt the prudent thing to do, to start out with as few as
possible
> No, because either that would leave you with a theory
incapable of
> proving some things of interest or with an equivalent
theory in which
> the axioms were more terse.
Terse axioms are BAD? Why? And if you had a weak theory,
different
people could make different axioms to solve them, and you
could
compare to see which ones lead to more fruitful results
and/or more
fruitful proofs of the same results. Why is THAT bad?
>If not, whats wrong with what I said?
> Your assumption that you know whats needed.
As I said I wanted to physics. You claim that mathematicians
dont
care about physics, so its up to someone to represent the
needs of
physicists. If I knew someone else doing that, then Id go
help them
out instead of wading through here.
>Observations are local, the wavefunction is not. Clearly
only a
>portion is applicable to the verifyable parts of any
particular
>experiment.
> Thats not clear to me. Do you understand, e.g., the
difference
> between position space and momentum space?
I understand that momentum space is crutch that people use
when making
approximations of experiments, to compute say, a scattering
matrix.
There arent any momentum eigenstates because the states
arent
renormalizable. Yes you can work with a rigged hilbert space,
but now
things are getting complicated and whats the point? Bohmiam
people
prefer a full theory in conguration space including having
the
the wavefunction. And they dont wave their hands about
approximation
expect when they are REALLY approximating. Yes there are
people who
try to quantize general symplectic manifolds, but that is a
tedious
theory anyway, since most times once has to put a causality
lter on
the interactions and we believe that it physically happens in
the full
conguration space anyway, so the fancy symmetries of some
theories
is wasted space. Momentum space is not really an equal
partner to
anti-symmetrization) occurs in position space.
>The other parts can be lled in with anything that makes
the
>computation easier and can be thrown away when you are
done.
> No.
Yeah right. Like you the wavefunction far away affects
anything. Im
happy to put a one-point compactication of the vector space
and say
the wave function goes to zero at innity. In reality thats
no
constraint at all since it has to go to zero pretty fast
anyway. I
think you dont understand how I do physics and you dont
really care.
>Some physicists might start by counting their toes and
drinking
>coffee, it doesnt mean that affects me. What matters is
the
>initial conditions and the potentials and the evolution
equations.
> What kind of equations? If youre talking about equations
involving
> operators, then it does affect you.
You keep citing these general theories where in reality I
only need
some parts of it, and then only approximately. I read the
theories,
but you dont actually think people have hyper-computation
machines
chugging away to compute things in the lab, so you? Its all
nite
discrete computers.
>You ask such details questions sometimes that I forget you
may not
> Well, somebody doesnt understand. When the likes of Ryder
disagree
> with you, I can draw the obvious conclusion.
I have a book by Ryder, does he have more than one? Do you
know where
is disagrees with me. Big names dont scare me, but at least
hell
write more than a no and Ill have a chance to understand
what hes
really saying.
>You keep assuming that I do my physics badly just because
I dont
>do it like you.
> Or like any of the QFT books that Ive seen.
You know if you wanted to talk about QFT, you could have said
so at
the beginning. I myself use innite bases there because I
havent
been doing it long enough to trust myself to always be right
doing it
any other way than what I learned in school. I thought you
were
use QFT for ne corrections or scattering matrices, and in
both cases
there, they choose an order of approximation and do a few
integrals
to get a few coefcients of a matrix, that would in theory
have an
innite number of elements. But no one does it all, or even
tries.
Maybe you know people that do cooler stuff then anyone Ive
seen. Im
neither famous not that well traveled, being so interested in
teaching
as I am.
>Im trying to get your perspective and I treat you with
respect
> Respect? Attributing things to me that I didnt write is
not respect.
> Nor is telling me that what you are writing is above my
head.
Im trying my best. Most of what you write is brusque or
vague, and
usually both. And it seems like the questions come out of
nowhere for
no purpose that I can see and I try my best to respond to
what you
say. I really do. I can tell that you dont like my posts. I
dont
know what to do differently. So have you studied IF-logic,
are you
familiar with the SSC theorem, have you seen Bohmian models
of spin
measurements, double slit experiments? Im not TRYING to
assume what
you know. Your statements come across as accusing me of bad
science
of of not getting the results I do, so had been assuming you
didnt
know the things I knew. I meant no offense. Im sure you are
WAY
smarter than me, and know more than me about everything. You
can post
faster than I can, and probably make fewer errors too, and you
probably get paid more than me too, so its a reasonable
expectation
that you really do know more than me. But just because you
know a
bunch doesnt mean you know how to explain it so that others
know what
you are talking about. If you know all these things, then I
dont
know why you are asking the questions you do. I really dont
know why
you think Im wrong, and thats why I wasnt sure if you even
knew
what I was doing. Im sorry for doubting any skill you have.
I just
want help understanding why these sets are excluded from ZF,
and if
you can help me Im read papers you write and show them to
others, or
whatever it takes to demonstrate that I really do respect
you. Right
now I dont think youve answered my question, and if you
have I
didnt catch it.
>You seem to assume that Im dishonest just because you
arent
>familiar with my work.
> I assume that youre dishonest because youre been
dishonest in this
> thread.
If I made a mistake Id be happy to own up. I dont know what
you
think is dishonest. I agree that enough people have reacted
like I
said something that I didnt say that the reasonable
assumption is
that Im dishonest or that I was unclear about my actual
statements
and/or questions. My own introsepction tells me I was honest,
so Im
concluding that I was unclear. Previous on this group when I
said I
was being unclear I was told something about it being
arrogant to say
that, so I was trying not to mention it again. Im not trying
to
upset anyone, this is my rst thread on sci.math. it takes
awhile to
learn the culture. So far the culture seems to be one that
would be
very rude in the world I live in, so I dont know how people
here
expect me to act.
>I have trouble believing you.
> Then perhaps you shoudl look up sarcasm.
Oh, I can recognize sarcasm in RL, but Ill miss it everytime
in
e-mail. Does that mean you dont know any of the things Ive
been
asking all along? Is there anything you were being serious
about. As
I said I cant tell on e-mail.
>What equipment are you using to get these results?.
> It doesnt matter, because the commutation relations make it
> impossible to measure position and momentum concurrently,
regardless
> of the apparatus. You can increase the precision of one
only at the
> expense of the other.
Ive never measured momentum directly, even experiment ends
up being a
position measurement in the end, before we look at it. Ive
seen some
photoelectric effect experiments where the position of the
knob on the
tunable laser tuner has an effect on current that is fairly
related to
the momentum of the light, but its still positions I observe
and
record. Yeah I put some dust over the laser before and
afterwards to
verify the different colors, but I use the position
measurements of an
oscilloscpe to back up that human observation. I want hard
data, and
that ends up being the position of something.
>There are observable consequences to how much funding you
have?
> Thats an example of how you convinced me that youre
dishonest.
Whats dishonest. People with different funding have different
equipment and that requires a different degree of
approximation to
accurate predict and compare results. Whats dishonest about
that?
>You subject your nger to a hamiltonian from your brain
that
>correlates the brain state with the position of your nger
such
>that the position of you nger is correlated with the
parts of the
>experimental apparratus that you decide to observe.
> That doesnt measure the wave function.
The wavefunction IS the common cause that is behind the
correlations
between matter. You are never going to see the whole thing.
And to
see a cross section you need to have a body of correlations
to see it.
If thats not what you do with the wavefunction, then we are
just
doing different things.
>If its gonna be approximate anyway, why not nd a better
basis
>then?
> Why not x the real problems instead of chasing after
imaginary
> problems?
I agree that if the problems were imaginary then youd have a
point.
Why attempt to use bad tools to x a bad machine that was
designed
with those very same bad tools?
>For bounded energy bound states there is.
> So your theory is no good for free electrons?
Youve seen a free electron? You cant observe it if it were
truly
free!
>Models tell you things about your axioms. In this case
they pointed
>out the fact that lack of bijections and size are not
intrinsically
>related.
> No. You still dont get it. This isnt metaphysics and isnt
> mysticism; the statements in ZF use the language of ZF.
Cardinality is
> dened in terms of the existence of bijections.
But WHY, what actually depends on those denitions to work.
If no
one made those denitions would all the theorems that dont
use those
denitions STILL be true. So what REALLY depends on a lack of
bijections besides that ONE denition that looks really dumb
from the
perspective of the countable model? Thats a question Im
asking, and
you are making fun of me INSTEAD of answering it.
>But these lack of satisifactions only occur under the rug
where we
>cant observe them anyway with our nite precision.
> We most certainly can observe them; h isnt *that* small.
Nor is a
> model the same as a calculation regime.
Im saying that if you show me an experiment DESIGNED to
measure h
(and Ive done them before), then when push comes to shove I
eventually measure the position of something. It always comes
down to
that. Fancy non-operational symmetries dont negate that
paractical
fact. The imprecisions of an approximation DESIGNED
CONSISTENTLY WITH
THE APPARATUS will be UNobservable provided one does the
approximation
correctly. Which I do, when I do physics, and as others do
when thet
do physics. We pick a signal out of the noise, its pretty
common
even. We can do that and measure h at the same time. If you
want to
create another thread on a physics forum, Id be happy to do
that (you
start the thread with a topic or suggest a topic to me), how
about on
this forum we talk about what the consequences are and are
not of the
common and uncommon denitions in ZF and the axioms of ZF and
the
axioms that can be consistently added.
===
Subject: Re: Skolems Paradox and why is math the way it is?
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>ZF doesnt distinguish between the set of subsets that could
exist
>consistentl with ZF and the set of subsets that are provable
to exist
>in ZF.
Why do you see this as a problem?
Numbers such as pi are provable to exist. But, as far as I
can tell,
there couldnt be a proof in ZFC, that Plancks constant
exists.
Thats because it is a physical constant, rather than a
mathematical
constant.
You could, of course, say that
h_1 nu < E < h_2 nu
and give explicit values for h_1 and h_2 . In that case you
will be
giving constructible h_1 and h_2, so you can manage without
assuming
that h exists as a real number. But it is much simpler to
assert
E = h nu
The standard reals are constructed so that we can be sure
there
is nothing missing. There isnt any problem with Plancks
constant
existing. But once you try to go to constructible reals, or to
some other countable model, then you lose that assurance.
And, of course, it is not just Plancks constant. There are
other
physical constants which raise the same kind of issue.
> It seems that ZF is incomplete in a bad way, that
specically
>not all well-dened subcollections are subSETS.
ZF is incomplete, but I dont see that this is in a bad way.
Rather, I see that as a limitation of our formal methods. ZF,
or any
axiom system, has to be a nite specication. That is, it can
be
written down in a nite number of words. I suggest that what
you
are seeing is a result of the limitation of nite
specication.
The stand real numbers dont come from ZF. They come from
what is
needed to make physics work.
> This makes it VERY
>confusing (to me at least) about what the power set is
supposed to
>contain. Are these relations sets in the second order axiom
system?
>No one tells me.
I dont count myself as a mathematical logician, so I dont
try to
answer that sort of question. But I would suggest that ZF
isnt
something that need concern physicists. Most mathematicians
dont
much concern themselves with it either.
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--
vote for regime change in Washington, Nov 02.
===
Subject: Re: Skolems Paradox and why is math the way it is?
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> Hash: SHA1
>ZF doesnt distinguish between the set of subsets that
could exist
>consistentl with ZF and the set of subsets that are
provable to exist
>in ZF.
> Why do you see this as a problem?
> Numbers such as pi are provable to exist. But, as far as I
can tell,
> there couldnt be a proof in ZFC, that Plancks constant
exists.
> Thats because it is a physical constant, rather than a
mathematical
> constant.
> You could, of course, say that
> h_1 nu < E < h_2 nu
> and give explicit values for h_1 and h_2 . In that case you
will be
> giving constructible h_1 and h_2, so you can manage without
assuming
> that h exists as a real number. But it is much simpler to
assert
> E = h nu
> The standard reals are constructed so that we can be sure
there
> is nothing missing. There isnt any problem with Plancks
constant
> existing. But once you try to go to constructible reals, or
to
> some other countable model, then you lose that assurance.
> And, of course, it is not just Plancks constant. There are
other
> physical constants which raise the same kind of issue.
Planks constant, h, is either a rational number (and h-bar is
then
derived from pi and h) or it constructed from OTHER measure
constants
which are then rational numbers again. If you are talking
about a
THEORY that PREDICTS planks constant, then we care VERY MUCH
if h can
be proven to exist in ZFC, because if it cant then we are
going to
have to ADD more axioms to ZFC to make our model that
PREDICTS the
value of h. And currently there are real numbers that we can
describe
(using countable models and cantors arguement) but that we
cannot
prove exist using ZFC, so HOW do we know that this
predictable h
(instead of the measurable one) is in R- (provable) instead
of just in
R+ (consistent with ZF).
> It seems that ZF is incomplete in a bad way, that
specically
>not all well-dened subcollections are subSETS.
> ZF is incomplete, but I dont see that this is in a bad way.
> Rather, I see that as a limitation of our formal methods.
ZF, or any
> axiom system, has to be a nite specication. That is, it
can be
> written down in a nite number of words. I suggest that
what you
> are seeing is a result of the limitation of nite
specication.
> The stand real numbers dont come from ZF. They come from
what is
> needed to make physics work.
But there is no evidence that the ZF model of the real
numbers is a
correct physical model. We cant even tell how many points
there are
in R.
> This makes it VERY
>confusing (to me at least) about what the power set is
supposed to
>contain. Are these relations sets in the second order
axiom system?
>No one tells me.
> I dont count myself as a mathematical logician, so I dont
try to
> answer that sort of question. But I would suggest that ZF
isnt
> something that need concern physicists. Most mathematicians
dont
> much concern themselves with it either.
> -----BEGIN PGP SIGNATURE-----
> Version: GnuPG v1.2.6 (SunOS)
>
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> DzwBJEP3ThYlix58UvKqlj4=
> =s5IH
> -----END PGP SIGNATURE-----
How do we know the incomplete axioms of ZFC are strong enough
for
physics? Thats basically what you are suggesting that I
assume.
J.E.
===
Subject: Re: Skolems Paradox and why is math the way it is?
-----BEGIN PGP SIGNED MESSAGE-----
Hash: SHA1
> And, of course, it is not just Plancks constant. There
are other
> physical constants which raise the same kind of issue.
>Planks constant, h, is either a rational number (and h-bar
is then
>derived from pi and h) or it constructed from OTHER measure
constants
>which are then rational numbers again.
I will give you the benet of the doubt, and assumed that you
misunderstood the point.
Physicists carry out empirical measurements to estimate h.
Their estimate is a rational value.
Lets say that they rst estimate it as the rational h_1.
Later, they rene their measuring instruments, and get a more
precise value h_2 (say, at least one more digit of precision).
We imagine that there are continued improvements, leading to
a sequence
of ever more precise values h_1, h_2, h_3, ...
This is a cauchy sequence of rationals.
In the standard reals, this converges to the value h, which
need not
be rational.
I am asking for some basis to conclude that the sequence
converges in
your preferred countable model.
> The stand real numbers dont come from ZF. They come from
what is
> needed to make physics work.
>But there is no evidence that the ZF model of the real
numbers is a
>correct physical model. We cant even tell how many points
there are
>in R.
Why do we need to tell how many points there are? All we
need, is
that whenever we need a point to be there, it is there.
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kcmSs5YwIDjIpGRC6B1Ec3M=
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--
vote for regime change in Washington, Nov 02.
===
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===
Subject: Dice array combinations
Given that p=sum of points on a roll of dice, s=number of
faces on each
die,
and n=number of dice rolled, I know that
k=oor((p-n)/s)
sum(0,k) (-1)^k n!/((n-k)! k!) (p-sk-1)! /((p-sk-n)! (n-1)!)
will give the number of combinations that will yield a
particular sum.
If you had an array of such dice, and summed the rows and
columns, how
would
you calculate the total number of combinations that could
yield that set of
sums? For example, if the dice rolled and the sums were:
3 3 3 | 9
3 3 3 | 9
3 3 3 | 9
_____
9 9 9
How would you calculate how many other combinations would
also yield
(9,9,9)
and (9,9,9)?
-Michael VanDeMar
===
Subject: Re: Dice array combinations
> Given that p=sum of points on a roll of dice, s=number of
faces on each
die,
> and n=number of dice rolled, I know that
> k=oor((p-n)/s)
> sum(0,k) (-1)^k n!/((n-k)! k!) (p-sk-1)! /((p-sk-n)! (n-1)!)
> will give the number of combinations that will yield a
particular sum.
> If you had an array of such dice, and summed the rows and
columns, how
would
> you calculate the total number of combinations that could
yield that set
of
> sums? For example, if the dice rolled and the sums were:
> 3 3 3 | 9
> 3 3 3 | 9
> 3 3 3 | 9
> _____
> 9 9 9
> How would you calculate how many other combinations would
also yield
(9,9,9)
> and (9,9,9)?
> -Michael VanDeMar
Write a program with 9 loops and get
Row &
Col Sums Possibilities
3 1
4 6
5 21
6 55
7 120
8 231
9 355 <-- your example
10 432
11 432
12 355
13 231
14 120
15 55
16 21
17 6
18 1
Example: The 6 arrangements yielding rowsums=colsums=4 are
1 1 2
1 2 1
2 1 1
1 1 2
2 1 1
1 2 1
1 2 1
1 1 2
2 1 1
1 2 1
2 1 1
1 1 2
2 1 1
1 1 2
1 2 1
2 1 1
1 2 1
1 1 2
Hugo Pfoertner
===
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boundary=B_613229C9.0E6D.0
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and Staff,
All desktop computers are brand-new packed in their original
boxes,
and come with a full manufacturers warranty plus
a 100% satisfaction guarantee.
These professional grade Desktops are fully equipped with 2005
next generation technology, making these the best performing
computers money can buy.
Avtech Direct is offering these feature rich, top performing
Desktops with the latest technology at an amazing price
to all who call:
The fast and powerful AT-3200 series Desktop features:
* IBM Processor for amazing speed and performance
* 20 GB UDMA Hard Drive, -- Upgradeable to 80 GB
* 52X CD-Rom Drive, -- Upgradeable to DVD/CDRW
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* Soft Touch Keyboard and scroll mouse
* Internet Ready
* Network Ready
* 1 Year parts and labor warranty
* Priority customer service and tech support
MSRP $499 ........................................ Your Cost
$227
How to qualify:
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2. All desktop computers will be available on a
rst come rst serve basis.
and we will hold the desktops you request on will call.
4. You are not obligated in any way.
5. 100% Satisfaction Guaranteed.
6. Ask for Department C.
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Avtech Direct
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Woodland Hills, CA 91364
--B 613229C9.0E6D.0--
===
Subject: Find a parameterization of solution space of
det(I+XY)>0
Hi all,
The problem is to parameterize the solution space (X,Y) of
det(I+XY)>0.
For now, we can assume X and Y are square matrices.
Eventually, we wish to
consider the case where X and Y are retangular matrices of
appropriate
dimension.
I think I have a way to parameterize the solution in the
square matrices
case, but it is long and ugly (glad to discuss my solution
with you, but
its too lengthy for this forum). Can you think of an easy
way to generate
all much solutions (X,Y)? Or can you refer me to a related
known problem?
Akai
===
Subject: Re: The beast, again.
>We all know that Rev. 13:18 says 666 is the number of the
Beast
Actually there are TWO numbers of the beast: 666 and 616 from
Latin
sources, which are the total letter values of Nero Caesar in
Hebrew
and Latin. Case closed IMO.
Or maybe it is the CD burning software, Nero.
-
http://mysite.verizon.net/vze8adrh/news.html (prole)
--Tim923 My email is
valid.
===
Subject: Re: The beast, again.
> We all know that Rev. 13:18 says 666 is the number of the
Beast,
> but did you know that...
4973 - the beasth prime
16661, 26669, 46663, 56663,
76667, 96661, 96667 - Primes with The Beast Inside.
Hugo
===
Subject: Re: The beast, again.
> We all know that Rev. 13:18 says 666 is the number of the
Beast,
> but did you know that...
666^666 = 1 Beastol (B666)
B666-83 and B666+4163 the next prime neighbors of the Beastol.
Hugo
===
Subject: Re: The beast, again.
> The bestial primes:
> 61, 661, 6661, 6666666661, 666666666666666661,
6666666666666661
> 6666666666666666666661, 6666666666666666666666666661
> As the Bestial Primes are of the form : 2*(10^n -1)/3 -5
> The diabolical exponents n are:
2,3,4,10,18,21,22,28,43,66,121,133,178
...
> The Conjecture of the Beast : There are innitely many
Bestial Primes
> and are more numerous than the Mersenne Primes.
And bast not beast: 1 5 8 10 19 22 40 62 ... the number of
6s in 666667-
like primes. (Why not start with 0? OK, ne, since 7 is
prime.)
This sequence is (still) not in the OEIS. Incredible.
http://www.alpertron.com.ar/ECM.HTM
Rainer Rosenthal
r.rosenthal@web.de
===
Subject: Re: The beast, again.
> The bestial primes:
> 61, 661, 6661, 6666666661, 666666666666666661,
6666666666666661
> 6666666666666666666661, 6666666666666666666666666661
> As the Bestial Primes are of the form : 2*(10^n -1)/3 -5
> The diabolical exponents n are:
> 2,3,4,10,18,21,22,28,43,66,121,133,178 ...
> The Conjecture of the Beast : There are innitely many
Bestial
> Primes
> and are more numerous than the Mersenne Primes.
> a beastly magic square...
> 3 107 5 131 109 311
> 7 331 193 11 83 41
> 103 53 71 89 151 199
> 113 61 97 197 167 31
> 367 13 173 59 17 37
> 73 101 127 179 139 47
> all entries are prime numbers, and each row, column, and
> diagonal adds up to 666.
Congratulations!
Now, here are the only Hyper-bestial primes:
HBP(1) = 6^2 + 7 ..... = 43
HBP(2) = 66^2 + 7 .... = 4363
HBP(3) = 666^2 + 7 ... = 44563
HBP(5) = 66666^2 + 7 = 4444355563
HBP(7) = 6666666^2 + 7 = 44444435555563
HBP(8) = 66666666^2 + 7 = 4444444355555563
HBP(n) = 4*(10^n - 1)^2/9 + 7
===
Subject: Re: Deep Thoughts # 17: Liar Paradox is a Formal
Metamathematical
Theorem
>ny total tutorial in predicate logic was reading
>_The Laws of Form_ through to the middle
>of the chapter with second-order equations;
>it purports to be the unmentioned arithmetic
>of boolean/predicate logic. I agree.
Since I (and presumably most others) have not read that book,
if you want to
have a serious conversation then you need to state the
relevant ideas here.
> of course, Ive come across examples, before & since.
A few well-chosen examples that illustrates whatever your
point is would be
helpful too.
>my point was taht the time dimension is simply ignored,
>although it is naturally required, even to ennunciate the
paradox;
So you say. As I said, it sure doesnt look that way. Please
PROVIDE some
of the allegedly deeper analysis, rather than just asserting
that there is
one.
>the same applies to Lord Bertys Village Barber conundrum;
>its just a conundrum!
So where do you disagree with my analysis?
[unsnip]
>And again, where is the explicit or implicit reference to
time
>in the Barber paradox?
>And the Liar seems much more intractable than the
>Barber: the simple solution to the Barber paradox is that
there can not be
>any such barber (just as the usual solution to the
Russell-set paradox is
>that there can not be any such set). But applying that to
the Liar would
be
>saying that there can not be any such statement, but in
actual fact there
it
>is right before your eyes.
> It sure doesnt look that way: there is no explicit
reference to time in
> This statement is false. By what deeper analysis to do you
impute
some
> time reference?
>--ils duces dEnron!
>http://larouchepub.com
(BTW, that isnt a useful link to include in a
sci.logic/sci.math post,
unless you WANT to be thought of as a kook.)
--
---------------------------
| BBB b Barbara at LivingHistory stop co stop uk
| B B aa rrr b |
| BBB a a r bbb | Quidquid latine dictum sit,
| B B a a r b b | altum viditur.
| BBB aa a r bbb |
-----------------------------
===
Subject: Re: Deep Thoughts # 17: Liar Paradox is a Formal
Metamathematical
Theorem
> If, as it appears, you think provability is a
property of
sentences, you
> obviously dont have any idea what you are talking
about.
>
> Of course it is. Whats the problem?
>
> OK, is (x)(y)(fx = fy -> x=y) provable? Yes or no.
>
> Its a function of your axioms and rules (and
denitions.)
>
> C-B
> Which is exactly why provability isnt a property of
*sentences.*
>
> cid ooh
> for approval. He had only 2 or 3 questions. Then I noticed
that all
> of his questions had to do with material from pages 1 and
2. (I dont
> know if its incompetence or laziness that makes people pass
judgment
> without having really examined an issue.)
Youre accusing someone on incompetence because they asked 2
or 3
questions about your preliminary material. A few points: 2 or
3
questions are not a large enough sample to be statistically
relevant.
Perhaps it was all a coincidence. Maybe your presentation of
the
basics was unclear but his understanding of the subject was
sufcient
for him to understand the rest. Since you didnt realize
this, it
seems fair, if ironic, to apply your parenthetical remark to
your
unjustied accusation.
> So your comment is that provability isnt just a function
of a
> sentence, but also of the axioms and rules used? Well,
yeah, I think
> we all know that.
You obviously didnt demonstrate that, since you had to be
corrected.
> So howd you like my two Metamathematical theorems?
> C-B
What theorems?
cid ooh
===
Subject: Re: Deep Thoughts # 17: Liar Paradox is a Formal
Metamathematical
Theorem
> If, as it appears, you think provability is a
property of
sentences, you
> obviously dont have any idea what you are talking
about.
>
> Of course it is. Whats the problem?
>
> OK, is (x)(y)(fx = fy -> x=y) provable? Yes or no.
>
> Its a function of your axioms and rules (and
denitions.)
>
> In other words, provability is not a property of
sentences. Congrats.
>
> Typically the axioms and rules are xed.
> Right, x axioms and rules and you get a notion of
provability. In
> other words, provability is not a property of sentences.
Congrats.
> Chris Menzel
Once you x the axioms and rules it is - but that is just
quibbling
over English terminology.
So howd you like my two metamathematical theorems?
C-B
===
Subject: Polynomials, Fourier Series, and function
decomposition
As I understand it, any (continuously differentiable)
function can be
written as an innite polynomial, that is:
f(x) = SUM(n=0...inf) a(n)*x^n
Where SUM is a lame attempt at sigma summation notation, and
a(n) are
constants.
Likewise, if we have a multidimensional function, we can also
represent
it as a polynomial, including all cross terms. For the two
dimensional
case:
f(x,y) = SUM(n=0...inf) SUM(m=0...inf) a(n,m) * x^n * y^m
Alternatively, if a function is periodic, it can be
represented by a
Fourier series:
f(x) = SUM(n=0...inf) a(n) * cos(nx) + SUM(m=0...inf) a(m) *
sin(mx)
or equivalently:
f(x) = SUM(n=-inf...inf) F(x) * e^(i*n*x)
It isnt mentioned in very many places that I could nd (or
understand), but I think the multidimensional form of the
Fourier series
is as follows:
f(x,y) = SUM(n=-inf...inf) SUM(m=-inf...inf) F(x,y) *
e^(i*n*x + i*m*y)
My question is if it is possible (or wise) to combine the two
forms when
you have a multidimensional function which is known to be
periodic in
some parameters, but aperiodic in others. Naively I would
expect:
f(w,x,y,z) = SUM(n=0...inf) SUM(m=0...inf) SUM(p=-inf...inf)
SUM(q=-inf...inf) F(w,x,y,z) * w^n * x^m * e^(i*p*y + i*q*z)
Am I correct here? Or am I barking up the wrong tree?
***
The reason I ask is that I would like to nd extrema for a
multidimensional function where the full analytical form is
unknown to
me. Im able to sample the function at discrete points,
though, and
some of the parameters are angles, so Im know they are
periodic with a
period of 2*pi.
What Im hoping to do is sample the function at a set of
points, and
then use linear algebra to nd coefcients for a truncated
form of the
innite series. I can then nd the min-max values of the
truncated
form analytically, and use that to know where to sample on
the next
round. Repeat until I get sufcient convergence of the new
points, and
untill the parameter range is sampled sufciently.
Is my scheme reasonable? Is there a better way to approach
this?
-Rocco
===
Subject: Re: Polynomials, Fourier Series, and function
decomposition
>As I understand it, any (continuously differentiable)
function can be
>written as an innite polynomial, that is:
>f(x) = SUM(n=0...inf) a(n)*x^n
>Where SUM is a lame attempt at sigma summation notation, and
a(n) are
>constants.
As others have noted, this is incorrect. You want to look up
analytic function.
>Likewise, if we have a multidimensional function, we can
also represent
>it as a polynomial, including all cross terms. For the two
dimensional
case:
>f(x,y) = SUM(n=0...inf) SUM(m=0...inf) a(n,m) * x^n * y^m
Again, this is OK for an analytic function.
>Alternatively, if a function is periodic, it can be
represented by a
>Fourier series:
>f(x) = SUM(n=0...inf) a(n) * cos(nx) + SUM(m=0...inf) a(m) *
sin(mx)
You want
sum_{n=0}^innity a(n) cos(nx) + sum_{m=1}^innity b(m)
sin(mx)
if the function has period 2 pi. Some conditions have to be
satised,
depending on what sense of represented you are using: for
example,
if f is continuous the series exists but might diverge at
some points
(although it does converge almost everywhere).
>or equivalently:
>f(x) = SUM(n=-inf...inf) F(x) * e^(i*n*x)
sum_{n=-innity}^innity c(n) e^(inx)
>It isnt mentioned in very many places that I could nd (or
>understand), but I think the multidimensional form of the
Fourier series
>is as follows:
>f(x,y) = SUM(n=-inf...inf) SUM(m=-inf...inf) F(x,y) *
e^(i*n*x + i*m*y)
Again, the coefcient is c(n,m), not F(x,y).
>My question is if it is possible (or wise) to combine the
two forms when
>you have a multidimensional function which is known to be
periodic in
>some parameters, but aperiodic in others. Naively I would
expect:
>f(w,x,y,z) = SUM(n=0...inf) SUM(m=0...inf) SUM(p=-inf...inf)
>SUM(q=-inf...inf) F(w,x,y,z) * w^n * x^m * e^(i*p*y + i*q*z)
Again, its OK under appropriate assumptions, and the
coefcient
is a function of n,m,p,q, not w,x,y,z.
>The reason I ask is that I would like to nd extrema for a
>multidimensional function where the full analytical form is
unknown to
>me. Im able to sample the function at discrete points,
though, and
>some of the parameters are angles, so Im know they are
periodic with a
>period of 2*pi.
>What Im hoping to do is sample the function at a set of
points, and
>then use linear algebra to nd coefcients for a truncated
form of the
>innite series. I can then nd the min-max values of the
truncated
>form analytically, and use that to know where to sample on
the next
>round. Repeat until I get sufcient convergence of the new
points, and
>untill the parameter range is sampled sufciently.
Trying to t a polynomial to a function sampled at nitely
many points
tends to be numerically unstable. Your polynomial is likely
to have
extrema at unexpected places. A better idea is to use
something more
local, such as splines.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
===
Subject: Re: Polynomials, Fourier Series, and function
decomposition
>Alternatively, if a function is periodic, it can be
represented by a
>Fourier series:
>f(x) = SUM(n=0...inf) a(n) * cos(nx) + SUM(m=0...inf) a(m)
* sin(mx)
> You want
> sum_{n=0}^innity a(n) cos(nx) + sum_{m=1}^innity b(m)
sin(mx)
> if the function has period 2 pi. Some conditions have to be
satised,
> depending on what sense of represented you are using: for
example,
> if f is continuous the series exists but might diverge at
some points
> (although it does converge almost everywhere).
For example, if f is continuously differentiable on R, then
the Fourier
series of f will converge uniformly to f.
===
Subject: Re: Polynomials, Fourier Series, and function
decomposition
X-RFC2646:  Response

> The reason I ask is that I would like to nd extrema for a
> multidimensional function where the full analytical form is
unknown to me.
> Im able to sample the function at discrete points, though,
and some of
> the parameters are angles, so Im know they are periodic
with a period of
> 2*pi.
> What Im hoping to do is sample the function at a set of
points, and then
> use linear algebra to nd coefcients for a truncated form
of the
> innite series. I can then nd the min-max values of the
truncated form
> analytically, and use that to know where to sample on the
next round.
> Repeat until I get sufcient convergence of the new points,
and untill
> the parameter range is sampled sufciently.
> Is my scheme reasonable? Is there a better way to approach
this?
> -Rocco
well, your method will not work in general. When you goto the
power series,
it will possibly be missing a lot of information, and
sometimes, not matter
how many terms it has, it cannot get that information.
Basicaly, the problem is, when you are sampling, between any
to points there
can be a max or min and you your power series for that will
represent
that... it will just t those points smoothly.... though, if
you know your
functions are smooth enough, then it can be true. If not,
then you can never
truely know if it has a local extrema.
as you can see, if above some nth der. of f(x), they are all
virtually zero,
then the power series for f(x) will be represented acurately
using n terms
or so, so you can then use that power series to approximate
the extrema.
Though, Im sure there are much better numerical methods than
that... since
your method involves several methods that are all inaccurate,
so its just
not that good IMO.
Its better just to directly compute the partials at each
sample point and
use those to try and determine if there is a local extrema(if
the sign
changes, then you have an extrema between those points... but
the problem
here is, that again, you need smooth enough functions... the
less smooth,
the denser your parition must be.. while it doesnt tell you
your extrema,
it does tell you the intervals they are contained in, then
you can zoom
in
on it.
I might be wrong on a few points, since Im not much into
numerical
analysis... but I know that you need to worry about a lot of
stuff if you
want a good algorithm, and Im sure there are plenty out
there.
===
Subject: Re: Polynomials, Fourier Series, and function
decomposition
Supersedes: 
!3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(
5eZ41to5f%E@ELIi
$t^
VcLWP@J5p^rst0+(>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw
Cancel-Key: sha1:AkYX+YQO1lCQ3AXAK5FSu9rStdY=
> As I understand it, any (continuously differentiable)
function can be
> written as an innite polynomial, that is:
> f(x) = SUM(n=0...inf) a(n)*x^n
> Where SUM is a lame attempt at sigma summation notation,
and a(n) are
> constants.
So what is the polynomial for exp(-1/x^2) (with an explicit
value of 0
at 0)? This function is continously differentiable in the
reals to an
arbitrary degree. And, by the way, all derivatives at x=0 are
0. So
its Taylor approximation at x=0 is identical to that of
f(x)=0.
--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
===
Subject: Re: Polynomials, Fourier Series, and function
decomposition
> As I understand it, any (continuously differentiable)
function can be
> written as an innite polynomial, that is:
> f(x) = SUM(n=0...inf) a(n)*x^n
No. The function f(x) = cube of absolute value of x is
differentiable
with continuous derivative but there is no power series in x
that
converges to it in a neighborhood of zero.
--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
===
Subject: Have you tried MITs openCourseware?
X-RFC2646:  Original
Looking MITs opencourseware (OCW) for Analysis I (Fall
2002), it doesnt
seem very helpful. There is only a listing of chapers you
should cover in
some time frame. Assingments are also listed. Where exactly
are the
explanations besides the book? I could buy the book and learn
on my own.
What is the advantage of using MIT OCW?
Brett
===
Subject: Re: Intermediate values of Chebyshev polynomials T(r
,x)
>
> Chebyshev polynomials are very well known.
> Here x in(-1,1)
> T2=2x^2-1, T3=4*x^3-3*x, T4=8*x^4-8*x^3+1 ...
> Since T2=cos(2*cos^[-1](x)) ,I believe we may compute Tr
for all real
> values of r.
> That the reason why I use the word INTERMEDIATE,
>
> Id like to, but: What is the question?
>Of course, the functions cos(r*arccos(x)) make sense for
all real r and
>for all x between -1 and 1. (Extensions are also
possible.) They may not
>be polynomials. For example,
cos((1/2)*arccos(x))=sqrt((1+x)/2) is
>manifestly not a polynomial.
> Alain.
> You are of course right ,intermediate values are not
polynomial.
> I am ,in fact interested in iterated functions ;
> Look the example you gave
cos((1/2)*arccos(x))=sqrt((1+x)/2) allows
> us to compute any iterate of sqrt((1+x)/2)
;sqrt((1+x)/2)^[r] r real.
> ( sqrt((1+x)/2)seems just an inverse of T2=2*x^2-1 )
> What for cos(sqrt(2)*arccos(x)) ?
There is an old book that deals, among others, with iterated
functions:
Marek Kuczma: Functional Equations in a Single Variable
Monograe Matematyczne 46, Warsaw 1968
The same author, perhaps with others, has written newer books
on
iterations.
What you touched upon is called conjugation: you can make the
study
of iterations of a function f easier if you invent a suitable
function
g with its inverse function G, so that the new function
F(x) = G(f(g(x)))
has iterates
F_{n times}(x) = G(f_{n times}(g(x)))
or backwards
f_{n times}(y) = g(F_{n times}(G(y)))
A good key expression for a search is Abel functional
equation.
===
Subject: Re: Division by zero. Go ahead and laugh.
>No, I do not divide by zero. But it did occur to me to
mention it
here
> as
>an illustration of mathematical culture.
>What the material below reveals is that you have no
understanding of
>mathematical culture or mentality. Im curious, how far
have you gotten
>in your math training? A lot of the things youve been
talking about
>reveal gaping holes in your mathematical knowledge.
> Apologize if I offended you Will, and maybe Im just trying
to ll that
> ditch. Nothing personal, if there is something I need to
understand that
Im
> open to learn whatever I need to.
Start with the denition of paradox and inconsistency. Part
of the
frustration you are observing is your persistent misuse of
words. If
you want to have a philosophical argument, there are plenty
of places to
take it. As it stands, those words have a dened meaning that
doesnt
change to meet your desires.
> Yet, I was told by someone who really is a huge math guru
that
> mathematicians are indeed all just plain LAZY, (not my
words), and I even
> knew one who I would never accuse of being lazy however he
never wore
shoes
> even in winter. Are you personally lazy ? Maybe, maybe not.
In some ways I am lazy, in others not. The trick is to know
*when* you
can afford to be lazy.
> To answer your question about how far I got, I did pretty
well on
undergrad
> coursework and completed a BS, studied some higher stuff
but never had
time
> or opportunity to pursue graduate level material as I would
have liked to.
I
> did better in analysis than anything else but my math days
are pretty
much
> over - Im doing other things. I am not a professional
mathamatician.
Ok, fair enough.
>If 1/0 is undened, what else are you going to do besides
avoid it? If
>gricklesmak is undened, are you going to avoid using it in
>conversation? Are you going to avoid it like the plague?
Please
stop
>using emotionally charged language.
>There are no paradoxes or inconsistencies in mathematics,
unless you
> remove
>all the ad hoc stipulations and band aids which hold it all
together.
>You have it backwards. There are no paradoxes except as
historical
>artifacts. There are inconsistencies, but they arise from
systems that
>are not explored except for illustrative purposes. The ad
hoc
>stipulations and band aids are a result of *preventing*
inconsistencies
>while allowing things to work roughly as we would like them
to.
> Im going to email you a sense of humor because you dont
seem to have one
> yet.
The doctors tried to transplant one into me at birth, but my
body
rejected it. Ive been forced to live without one or risk my
health.
> All Im saying is that arithmetic falls apart if you allow
1/0 = innity.
A
> well known fact.
I dont have a problem with that, its your explanations of
*why* that I
take issue with.
> To this I add only one thing, that this phenomena may be
caused by forces
> which are external to the framework of the general
mathematical model.
This presumes there *is* something external that can affect
it. The
mathematical model is outside everything else.
> The
> fact that the human mind cannot construct an arithmetic
where division by
> zero would be logical, might be due to the topology of
space/time which
> allows paradoxes to inuence logic in our minds, or perhaps
our logic
has
> no inconsistencies but the math which exists in thie
universe as we know
it,
> as we can understand it, must contain these wierd
singularities.
Or perhaps math has nothing to do with reality and attempts
to make the
two mesh is pointless.
--
Will Twentyman
email: wtwentyman at copper dot net
===
Subject: Re: Division by zero. Go ahead and laugh.
> Physicists just cant live without a terra rma.
> The science of physics is less real than physical reality.
Physics is
> our cleverest way of making good guesses about how the
world works.
> Bob Kolker
Agreed. The models which are used to describe the natural
world are mostly
algebraic and abstract. The natural world, however, is
non-abstract. These
two worlds do not mesh precisely.
Yet, in the physical universe, there must be a thing known as
truth.
And
this truth is (by denition) as precise and exact as
mathematical
precision
in the abstract world. Mathematical exactness is manifest in
the
physical
universe by virtue of truth.
I think that using the naturals to describe physical objects
introduces
error, because no two objects are identical. This implies a
number system
for the physical universe, one which is precise and exact. I
really do not
like it, but it seems logical. If I had my way, the naturals
would describe
things perfectly. But if I said that I had 5 marbles, then by
denition of
what 5 implies, these marbles must be exactly identical to
each other. I
cant believe that this is possible.
The amazing thing is that even if there is an error
introduced into the
model of physics because we allow all naturals, all the basic
operations
still seem valid. Im perplexed by this, but I still see this
as a aw in
the current model of physics.
The naturals cannot be exact, precise representations of
physical objects.
===
Subject: Re: Methods that count primes without counting
primes or referring
to them...
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9C1NLd29544;
Here is an automatic way of nding all odd primes:
generate the numbers given by f(a,b)=4ab+6a+6b+9=
(2a+3)(2b+3) for all
nonnegative integers a less or equal to b and *eliminate* the
results from
the sequence of all odd integers. What you get is precisely
the set of odd
prime numbers.
It looks super at rst glance but ordering to *order* these
(composite odd
numbers) generated by the formula above is very costly.
Youll get lots of
repetitions, of course, when computing f(a,b) and this adds
to the problem
further.
===
Subject: Re: Average Length of a line
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9C1NLG29564;
>Hi;
> Suppose we have an isosceles triangle, with the base angles
equaling 30
degrees and a height of 1. Naturally the apex angle is equal
to 120 degrees.
If I draw a random line from the apex of the triangle to the
base what is the
average length of that line? Now obviously the minimum length
is equal to 1
and the maximum length is one of the sides of the triangle
which is 2. I get
an average value of the line of about 1.25. I do not know how
to get the
exact analytical answer. Can someone help?
>Angela
> First you need to know the probability density function of
angles or
> of termination points on the baseline.
>The length of that line would be the secant of the angle
from the
vertical.*
> Then, the area under the curve of the length function from
0 to PI/3
>(60 degrees) would be
>int(sec(t),t=0..(PI/3)) = 1.316957...
>Then the average height would be that gure divided by the
angle
>change, PI/3, or
>1.257602...
>john
>* picture half the isosceles triangle as an angle; then the
vertical
>line from the apex to the base, of length 1, is the adjacent
side of the
>angle, and the unknown length is hypotenuse. hyp/adj =
secant, and the
>denominator is 1, so the unknown length is just the secant
of an angle
>going from 0 to 60 degrees.
You have assumed that all angles from the vertical are equally
likely.
===
Subject: Re: Average Length of a line
>Hi;
>
>Suppose we have an isosceles triangle, with the base angles
equaling 30
degrees and a height of 1. Naturally the apex angle is equal
to 120 degrees.
If I draw a random line from the apex of the triangle to the
base what is the
average length of that line? Now obviously the minimum length
is equal to 1
and the maximum length is one of the sides of the triangle
which is 2. I get
an average value of the line of about 1.25. I do not know how
to get the
exact analytical answer. Can someone help?
>
>Angela
>First you need to know the probability density function of
angles or
>of termination points on the baseline.
>The length of that line would be the secant of the angle
from the
vertical.*
> Then, the area under the curve of the length function from
0 to PI/3
>(60 degrees) would be
>int(sec(t),t=0..(PI/3)) = 1.316957...
>Then the average height would be that gure divided by the
angle
>change, PI/3, or
>1.257602...
>--
>john
>* picture half the isosceles triangle as an angle; then the
vertical
>line from the apex to the base, of length 1, is the
adjacent side of the
>angle, and the unknown length is hypotenuse. hyp/adj =
secant, and the
>denominator is 1, so the unknown length is just the secant
of an angle
>going from 0 to 60 degrees.
> You have assumed that all angles from the vertical are
equally
> likely.
Yes, there is that assumption. The calculation would be wrong
if
anything differentiates one angle difference from another.
--
john
===
Subject: Re: Average Length of a line
Tiangle can be taken as (0,1), (-root3, 0) (root3,0). (easily
checked).
Length
of line from (0,1) to (x,0) is [root(xsquared) +1]. Thus
average length is
1/
root3 times integral of root(xsquared +1) from 0 to root3.
(Think of
approximating by little steps of x from 0 to root3.)
Substitute
x = tan a, say, then dx ={sec squared) a} da, giving integral
of sec cubed
a
from 0 to
pi/3. This is a well-known integral (use integration by
parts, writing
(sec
cubed) a as sec a d/da (tana) and then (inthe next integral)
tan squared a =
[(secsquared a) - 1] The answer is 2 - [(reciprocal of root
3) times log
((base e) of 2 + root 3), which gives approx. 1.25. Sorry
about the
awkward
symbols, but if you write it out you will nd it quite
simple. Godfrey.
===
Subject: Re: New paper, algebraic integers, Galois Theory
Fleshed out a bit, some editing. ___JSH
-------------------------------------------------------------
---------
I. First section
The following are in a commutative ring.
Start with
P(m) = f^2 ((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1 + mf^2) xu^2
+ u^3 f)
with the factorization
P(m) = (a_1 x + uf)(a_2 x + uf)(a_3 x + uf)
and note that at
m=0, P(0) = u^2 f^2(3x + uf),
which gives you terms that do not vary as m varies.
So what about (a_1 x + uf), (a_2 x + uf), and (a_3 x + uf)?
(a_1 x + uf)(a_2 x + uf)(a_3 x + uf) = u^2 f^2 (3x + uf)
which shows that at least two of the as have to equal 0 at
m=0, while
one equals 3.
Since, at m=0, two of the as must equal 0, its convenient
to just
arbitrarily select a_1 and a_2 as those two.
Then you have uf for the rst, uf for the second and 3x + uf
for the
third as terms that do not vary when m varies.
Now then, if m=1, what are the *constant* terms?
They are uf, for the rst, uf for the second, and 3x + uf for
the
third.
Thats logical because they do not vary with m, so if
m=1003909273,
what are the constant terms?
They are uf, for the rst, uf for the second, and 3x + uf for
the
third.
Now divide f^2 from both sides, which gives
P(m)/f^2 = (m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1 + mf^2) xu^2
+ u^3 f
P(m)/f^2 = (a_1 x + uf)(a_2 x + uf)(a_3 x + uf)/f^2
and you note that P(0)/f^2 = u^2(3x + uf), which means that
now your
constant terms are u for the rst, u for the second and 3x +
uf for
the third.
Now then, if m=1, what are the constant terms now?
They are u for the rst, u for the second, and 3x + uf for
the third.
If m = 2938479378, what are the constant terms now?
They are u for the rst, u for the second, and 3x + uf for
the third.
How can the constant terms of the rst two go from uf to u?
They must be divided by f.
Now, the constant term of a_1 x + uf, is uf, but when f^2 is
divided
from P(m), it is u; therefore, a_1 x + uf is divided by f,
and you
have
a_1 x/f + u
and the constant term of a_2 x + uf is uf, but when f^2 is
divided
from P(m), it is u; therefore, a_2 x + uf is divided by f,
and you
have
a_2 x/f + u
while the constant term of a_3 x + uf is 3x + uf, and after
f^2 is
divided off, it is 3x + uf, so you have
a_3 x + uf
so, dividing P(m) by f^2 gives
P(m)/f^2 = (a_1 x/f + u)(a_2 x/f + u)(a_3 x + uf).
II. Second section
Now take
P(m)/f^2 = (a_1 x/f + u)(a_2 x/f + u)(a_3 x + uf)
and multiply inside the parentheses by f^2/(a_1 a_2 a_3), and
outside
by f^2(a_1 a_2 a_3) and you have
P(m)/f^2 = ((a_1 a_2 a_3)/f^2)(x + uf/a_1)(x + uf/a_2)(x +
uf/a_3)
and since a_1 a_2 a_3 = f^2(m^3 f^4 - 3m^2 f^2 + 3m), that is
P(m)/f^2 =
(m^3 f^4 - 3m^2 f^2 + 3m)(x + uf/a_1)(x + uf/a_2)(x + uf/a_3).
Now consider the case that m, f, and u are algebraic
integers, then I
have the ratios of algebraic integers:
uf/a_1, uf/a_2, and uf/a_3,
and now let
v_1/w_1 = uf/a_1, v_2/w_2 = uf/a_2, and v_3/w_3 = uf/a_2
where the vs and ws are algebraic integers in each case
coprime to
each other.
Making the substitutions I have
P(m)/f^2 =
(m^3 f^4 - 3m^2 f^2 + 3m)(x + v_1/w_1)(x + v_2/w_2)(x +
v_3/w_3).
And I have from before that
P(m)/f^2 = (m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1 + mf^2) xu^2
+ u^3 f
so
(m^3 f^4 - 3m^2 f^2 + 3m)(v_1 v_2 v_3)/(w_1 w_2 w_3) = f
as that is the last coefcient from the last term u^3 f,
which proves
that
(m^3 f^4 - 3m^2 f^2 + 3m) has w_1, w_2 and w_3 as factors, so
let
(m^3 f^4 - 3m^2 f^2 + 3m) = w_1 w_2 w_3
then I have
P(m)/f^2 = (w_1 x + v_1)(w_2 x + v_2)(w_3 x + v_3)
but I still have that
P(m)/f^2 = (a_1 x/f + u)(a_2 x/f + u)(a_3 x + uf).
III. Third section
Now in the ring of algebraic integers consider the
possibility that
a_1/f is not an algebraic integer to see if that leads to a
contradiction.
First, if a_1/f is not an algebraic integer and w_1 is, they
obviously
cannot be equal.
But I have
P(m)/f^2 = (w_1 x + v_1)(w_2 x + v_2)(w_3 x + v_3)
and
P(m)/f^2 = (a_1 x/f + u)(a_2 x/f + u)(a_3 x + uf)
so far simultaneously true without contradiction, so there
must exist
z_1, z_2, and z_3 such that
w_1 = (a_1 x z_1)/f, w_2 = (a_2 x z_2)/f and w_3 = a_3 x z_3
and z_1 z_2 z_3 = 1,
where algebraically the zs are units, but z_1, z_2 and z_3
are not
units in the ring of algebraic integers.
James Harris
===
Subject: Re: New paper, algebraic integers, Galois Theory
Commented on Galois Theory at end. ___JSH
-------------------------------------------------------------
---------
I. First section
The following are in a commutative ring.
Start with
P(m) = f^2 ((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1 + mf^2) xu^2
+ u^3 f)
with the factorization
P(m) = (a_1 x + uf)(a_2 x + uf)(a_3 x + uf)
and note that at
m=0, P(0) = u^2 f^2(3x + uf),
which gives you terms that do not vary as m varies.
So what about (a_1 x + uf), (a_2 x + uf), and (a_3 x + uf)?
(a_1 x + uf)(a_2 x + uf)(a_3 x + uf) = u^2 f^2 (3x + uf)
which shows that at least two of the as have to equal 0 at
m=0, while
one equals 3.
Since, at m=0, two of the as must equal 0, its convenient
to just
arbitrarily select a_1 and a_2 as those two.
Then you have uf for the rst, uf for the second and 3x + uf
for the
third as terms that do not vary when m varies.
Now then, if m=1, what are the *constant* terms?
They are uf, for the rst, uf for the second, and 3x + uf for
the
third.
Thats logical because they do not vary with m, so if
m=1003909273,
what are the constant terms?
They are uf, for the rst, uf for the second, and 3x + uf for
the
third.
Now divide f^2 from both sides, which gives
P(m)/f^2 = (m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1 + mf^2) xu^2
+ u^3 f
P(m)/f^2 = (a_1 x + uf)(a_2 x + uf)(a_3 x + uf)/f^2
and you note that P(0)/f^2 = u^2(3x + uf), which means that
now your
constant terms are u for the rst, u for the second and 3x +
uf for
the third.
Now then, if m=1, what are the constant terms now?
They are u for the rst, u for the second, and 3x + uf for
the third.
If m = 2938479378, what are the constant terms now?
They are u for the rst, u for the second, and 3x + uf for
the third.
How can the constant terms of the rst two go from uf to u?
They must be divided by f.
Now, the constant term of a_1 x + uf, is uf, but when f^2 is
divided
from P(m), it is u; therefore, a_1 x + uf is divided by f,
and you
have
a_1 x/f + u
and the constant term of a_2 x + uf is uf, but when f^2 is
divided
from P(m), it is u; therefore, a_2 x + uf is divided by f,
and you
have
a_2 x/f + u
while the constant term of a_3 x + uf is 3x + uf, and after
f^2 is
divided off, it is 3x + uf, so you have
a_3 x + uf
so, dividing P(m) by f^2 gives
P(m)/f^2 = (a_1 x/f + u)(a_2 x/f + u)(a_3 x + uf).
II. Second section
Now take
P(m)/f^2 = (a_1 x/f + u)(a_2 x/f + u)(a_3 x + uf)
and multiply inside the parentheses by f^2/(a_1 a_2 a_3), and
outside
by f^2(a_1 a_2 a_3) and you have
P(m)/f^2 = ((a_1 a_2 a_3)/f^2)(x + uf/a_1)(x + uf/a_2)(x +
uf/a_3)
and since a_1 a_2 a_3 = f^2(m^3 f^4 - 3m^2 f^2 + 3m), that is
P(m)/f^2 =
(m^3 f^4 - 3m^2 f^2 + 3m)(x + uf/a_1)(x + uf/a_2)(x + uf/a_3).
Now consider the case that m, f, and u are algebraic
integers, then I
have the ratios of algebraic integers:
uf/a_1, uf/a_2, and uf/a_3,
and now let
v_1/w_1 = uf/a_1, v_2/w_2 = uf/a_2, and v_3/w_3 = uf/a_2
where the vs and ws are algebraic integers in each case
coprime to
each other.
Making the substitutions I have
P(m)/f^2 =
(m^3 f^4 - 3m^2 f^2 + 3m)(x + v_1/w_1)(x + v_2/w_2)(x +
v_3/w_3).
And I have from before that
P(m)/f^2 = (m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1 + mf^2) xu^2
+ u^3 f
so
(m^3 f^4 - 3m^2 f^2 + 3m)(v_1 v_2 v_3)/(w_1 w_2 w_3) = f
as that is the last coefcient from the last term u^3 f,
which proves
that
(m^3 f^4 - 3m^2 f^2 + 3m) has w_1, w_2 and w_3 as factors, so
let
(m^3 f^4 - 3m^2 f^2 + 3m) = w_1 w_2 w_3
then I have
P(m)/f^2 = (w_1 x + v_1)(w_2 x + v_2)(w_3 x + v_3)
but I still have that
P(m)/f^2 = (a_1 x/f + u)(a_2 x/f + u)(a_3 x + uf).
III. Third section
So, even if a_1/f is not an algebraic integer, you can nd
w_1 an
algebraic integer.
But if a_1/f is an algebraic integer and w_1 is not, they
cannot be
equal.
But I have
P(m)/f^2 = (w_1 x + v_1)(w_2 x + v_2)(w_3 x + v_3)
and
P(m)/f^2 = (a_1 x/f + u)(a_2 x/f + u)(a_3 x + uf)
so how do you reconcile a case where a_1 x/f is not an
algebraic
integer?
There must exist z_1, z_2, and z_3 such that
w_1 = (a_1 x z_1)/f, w_2 = (a_2 x z_2)/f and w_3 = a_3 x z_3
and z_1 z_2 z_3 = 1,
so algebraically the zs are units, but z_1, z_2 and z_3 are
not units
in the ring of algebraic integers, if a_1/f is not.
Ive often faced arguments over the result from Section 1,
and at
times Ive dealt with people claiming that Galois Theory
proves
something about the factors of roots of monic polynomials
with integer
coefcients.
The basic claim is that *each* of the roots of a monic
polynomial with
integer coefcients that is irreducible over rationals must
share
non-unit factors with ALL of the prime factors of the last
coefcient.
For instance, with P(x) = x^2 + x + 6, the claim would be
that the two
roots:
(-1 + sqrt(32))/2 and (-1 - sqrt(32))/2
would, supposedly, each have to share non-unit factors with 2
and 3.
My works shows that its possible that actually neither does
and you
have to check using advanced polynomial factorization
techniques.
Faced with the algebra, certain people simply claimed that
Galois
Theory *forces* that result, when in fact, it does not.
Thats kind of obvious as consider
P(x) = x^2 + 5x + 6 = (x+2)(x+3)
and if Galois Theory forced the previous on irrationals, why
wouldnt
it force it on rationals as well?
It doesnt force anything on either. They were just wrong.
James Harris
===
Subject: Re: New paper, algebraic integers, Galois Theory
Getting to the last part:
> For instance, with P(x) = x^2 + x + 6, the claim would be
that the two
> roots:
> (-1 + sqrt(32))/2 and (-1 - sqrt(32))/2
Should be (-1 +- sqrt(-23))/2.
> would, supposedly, each have to share non-unit factors with
2 and 3.
Dene:
r1 = (-1 + sqrt(-23))/2 and r2 = (-1 - sqrt(-23))/2 (the
roots),
p1 = (3 + sqrt(-23))/2 and p2 = (3 - sqrt(-23))/2,
q1 = (-2 - sqrt(-23)) and q2 = (-2 + sqrt(-23)).
You can verify the following:
p1 * q1 = r1^3
p2 * q2 = r2^3
p1 * p2 = 8
q1 * q2 = 27
So the common factor between r1 and 2 is p1^(1/3), and
between r1 and 3
it is q1^(1/3).
What surprises me is that each time you come with claims that
something
might not be the case while it is easily shown that it *is*
the case.
This one took me only a few minutes.
> My works shows that its possible that actually neither
does and you
> have to check using advanced polynomial factorization
techniques.
The checking was done using simple arithmetic. Finding p1 and
q1
requires some knowledge, but that can be found in the
literature
(and not in your paper).
> Faced with the algebra, certain people simply claimed that
Galois
> Theory *forces* that result, when in fact, it does not.
Oh, yes, it does. It states something about the elds you get
when you adjoin roots of polynomials to the eld you are and
so
also states things about the ring of integers in that eld.
Strange enough, quite some time ago you already stated that
you had
looked at Galois theory, and it appears that only recently
you have
read the rst page on that theory. To give you a shortcut:
Galois states that if you have an irreducible polynomial over
a
certain eld you can not algebraically distinguish the roots
of
that polynomial.
This implies quite a bit. Being an integer in the eld can be
shown algebraically (you can construct a monic polynomial with
coefcients that are integer in the eld), so either all
roots are
integer in the eld or are not. And, you have a ring of
integers in
the eld.
Also co-primeness of integers in the ring of integers in the
eld can
be shown algebraically. Two integers in the eld are co-prime
whenever
you can nd other integers in the eld such that some
algebraic relation
holds.
> Thats kind of obvious as consider
> P(x) = x^2 + 5x + 6 = (x+2)(x+3)
> and if Galois Theory forced the previous on irrationals,
why wouldnt
> it force it on rationals as well?
You seem not to get the distinction between reducible and
irreducible
polynomials. Galois theory states something about irreducible
polynomials.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam,
nederland,
+31205924131
home: bovenover 215, 1025 jn amsterdam, nederland;
http://www.cwi.nl/~dik/
===
Subject: Re: New paper, algebraic integers, Galois Theory
> Getting to the last part:
> For instance, with P(x) = x^2 + x + 6, the claim would be
that the two
> roots:
>
> (-1 + sqrt(32))/2 and (-1 - sqrt(32))/2
> Should be (-1 +- sqrt(-23))/2.
Oh, yeah, thats right. I was wrong there.
Lets see what else you have.
> would, supposedly, each have to share non-unit factors
with 2 and 3.
> Dene:
> r1 = (-1 + sqrt(-23))/2 and r2 = (-1 - sqrt(-23))/2 (the
roots),
> p1 = (3 + sqrt(-23))/2 and p2 = (3 - sqrt(-23))/2,
> q1 = (-2 - sqrt(-23)) and q2 = (-2 + sqrt(-23)).
> You can verify the following:
> p1 * q1 = r1^3
> p2 * q2 = r2^3
> p1 * p2 = 8
> q1 * q2 = 27
> So the common factor between r1 and 2 is p1^(1/3), and
between r1 and 3
> it is q1^(1/3).
Yeah, but how do you know that q2 doesnt have 27 itself as a
factor?
In fact, let that be a test for you Dik Winter.
I say youre an asshole as you copied from me to your own
webpage
without my permission violating international copyright laws
and
common decency.
Basically youre a thief, and besides being a thief, you add
insult to
injury by continuing to annoy me on Usenet.
Now Im calling you out.
What makes you think that your q2 cant have 27 itself as a
factor?
If you answer it cant in the ring of algebraic integers, my
answer
is, so what?
What *exactly* does that mean?
> What surprises me is that each time you come with claims
that something
> might not be the case while it is easily shown that it *is*
the case.
> This one took me only a few minutes.
What surprises me is how stupid you are.
Now you replied before to a question in this area, right?
And seeing a mistake, how could you not reply again now?
Answer my question about about your q2, and Ill explain
again, how
youre wrong, and also let people see how badly you deal with
the
truth.
LOL.
James Harris
===
Subject: Re: New paper, algebraic integers, Galois Theory
Discussion, linux)
> I say youre an asshole as you copied from me to your own
webpage
> without my permission violating international copyright
laws and
> common decency.
What are international copyright laws?
In any case, one has the fair use right to excerpt anothers
work for
purposes of criticism (in the U.S., that is --- I dont know
Dutch
copyright law).
--
Jesse F. Hughes
What does soap kill? Germs or Germans?
-- Quincy P. Hughes (age 3 1/2) asks for clarication
===
Subject: Re: New paper, algebraic integers, Galois Theory
...
> would, supposedly, each have to share non-unit factors
with 2 and
3.
>
> Dene:
> r1 = (-1 + sqrt(-23))/2 and r2 = (-1 - sqrt(-23))/2 (the
roots),
> p1 = (3 + sqrt(-23))/2 and p2 = (3 - sqrt(-23))/2,
> q1 = (-2 - sqrt(-23)) and q2 = (-2 + sqrt(-23)).
>
> You can verify the following:
> p1 * q1 = r1^3
> p2 * q2 = r2^3
> p1 * p2 = 8
> q1 * q2 = 27
> So the common factor between r1 and 2 is p1^(1/3), and
between r1 and
3
> it is q1^(1/3).
> Yeah, but how do you know that q2 doesnt have 27 itself as
a factor?
Perhaps because q2 is a root of x^2 + 4x + 27, and so q2^2 +
4.q2 + 27 = 0?
> In fact, let that be a test for you Dik Winter.
Passed.
> I say youre an asshole as you copied from me to your own
webpage
> without my permission violating international copyright
laws and
> common decency.
If it is a violation of international copyright laws (I think
you
mean the Berne convention to which the US subscribed fairly
late),
I do not see it. Please start a suit and I will see you in
court,
be aware however about the fair-use clausule.
> Basically youre a thief, and besides being a thief, you
add insult to
> injury by continuing to annoy me on Usenet.
Well, as long as you are making wrong claims I see no reason
to refrain
from what I have been doing. Moreover, you have been
insulting me quite
a few times...
> Now Im calling you out.
> What makes you think that your q2 cant have 27 itself as a
factor?
See above.
> If you answer it cant in the ring of algebraic integers,
my answer
> is, so what?
> What *exactly* does that mean?
Yes, I have no idea what you mean with your remark. You were
talking
about co-primeness; that is only possible if you start with a
ring.
It is obvious to most that the factorisation I gave indeed
works in
the ring of algebraic integers. So it is up to you to provide
some
contrary information.
> What surprises me is that each time you come with claims
that
something
> might not be the case while it is easily shown that it
*is* the case.
> This one took me only a few minutes.
> What surprises me is how stupid you are.
> Now you replied before to a question in this area, right?
> And seeing a mistake, how could you not reply again now?
> Answer my question about about your q2, and Ill explain
again, how
> youre wrong, and also let people see how badly you deal
with the
> truth.
Good. Show, how I am wrong.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam,
nederland,
+31205924131
home: bovenover 215, 1025 jn amsterdam, nederland;
http://www.cwi.nl/~dik/
===
Subject: Re: New paper, algebraic integers, Galois Theory
...
Note this one:
> and since a_1 a_2 a_3 = f^2(m^3 f^4 - 3m^2 f^2 + 3m),
And these substitutions:
> v_1/w_1 = uf/a_1, v_2/w_2 = uf/a_2, and v_3/w_3 = uf/a_2
Multiply these three together and you get:
(v1.v2.v3)/(w1.w2.w3) = u^3.f^3/(a1.a2.a3)
Using your formula above, for (a1.a2.a3), we nd:
(m^3 f^4 - 3m^2 f^2 + 3m)(v1.v2.v3)/(w1.w2.w3) = u^3.f
and not what you nd:
> (m^3 f^4 - 3m^2 f^2 + 3m)(v_1 v_2 v_3)/(w_1 w_2 w_3) = f
So you made an error somewhere.
But lets see:
> Making the substitutions I have
> P(m)/f^2 =
> (m^3 f^4 - 3m^2 f^2 + 3m)(x + v_1/w_1)(x + v_2/w_2)(x +
v_3/w_3).
Seems right.
> P(m)/f^2 = (m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1 + mf^2)
xu^2 + u^3 f
Yup.
> so
> (m^3 f^4 - 3m^2 f^2 + 3m)(v_1 v_2 v_3)/(w_1 w_2 w_3) = f
No. u^3.f you mean, check the math for x = 0.
> (m^3 f^4 - 3m^2 f^2 + 3m) has w_1, w_2 and w_3 as factors,
Not necessarily. Only if w1 coprime to v2 and v3, etc.
> so let
> (m^3 f^4 - 3m^2 f^2 + 3m) = w_1 w_2 w_3
You are making an assumption you can not make. You can not
out of the
blue equalise the rst to the second.
> P(m)/f^2 = (w_1 x + v_1)(w_2 x + v_2)(w_3 x + v_3)
No you get at most:
(v1.v2.v3).P(m)/f^2 =
u^3.f.(w1.w2.w3)(x + v1/w1)(x + v2/w2)(x + v3/w3) =
u^3.f.(w1 x + v1)(w1 x + v2)(w3 x + v3)
Only with your assumption that (v1.v2.v3) = u^3.f do you get
what you
get. There is however *no* justication for that equality.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam,
nederland,
+31205924131
home: bovenover 215, 1025 jn amsterdam, nederland;
http://www.cwi.nl/~dik/
===
Subject: Re: New paper, algebraic integers, Galois Theory
> Commented on Galois Theory at end. ___JSH
>
-------------------------------------------------------------
---------
> I. First section
> The following are in a commutative ring.
> Start with
> P(m) = f^2 ((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1 + mf^2)
xu^2 + u^3 f)
> with the factorization
> P(m) = (a_1 x + uf)(a_2 x + uf)(a_3 x + uf)
> and note that at
> m=0, P(0) = u^2 f^2(3x + uf),
> which gives you terms that do not vary as m varies.
> So what about (a_1 x + uf), (a_2 x + uf), and (a_3 x + uf)?
> (a_1 x + uf)(a_2 x + uf)(a_3 x + uf) = u^2 f^2 (3x + uf)
> which shows that at least two of the as have to equal 0 at
m=0, while
> one equals 3.
> Since, at m=0, two of the as must equal 0, its convenient
to just
> arbitrarily select a_1 and a_2 as those two.
> Then you have uf for the rst, uf for the second and 3x +
uf for the
> third as terms that do not vary when m varies.
> Now then, if m=1, what are the *constant* terms?
> They are uf, for the rst, uf for the second, and 3x + uf
for the
> third.
> Thats logical because they do not vary with m, so if
m=1003909273,
> what are the constant terms?
> They are uf, for the rst, uf for the second, and 3x + uf
for the
> third.
> Now divide f^2 from both sides, which gives
> P(m)/f^2 = (m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1 + mf^2)
xu^2 + u^3 f
> P(m)/f^2 = (a_1 x + uf)(a_2 x + uf)(a_3 x + uf)/f^2
> and you note that P(0)/f^2 = u^2(3x + uf), which means that
now your
> constant terms are u for the rst, u for the second and 3x
+ uf for
> the third.
> Now then, if m=1, what are the constant terms now?
> They are u for the rst, u for the second, and 3x + uf for
the third.
> If m = 2938479378, what are the constant terms now?
> They are u for the rst, u for the second, and 3x + uf for
the third.
> How can the constant terms of the rst two go from uf to u?
> They must be divided by f.
> Now, the constant term of a_1 x + uf, is uf, but when f^2
is divided
> from P(m), it is u; therefore, a_1 x + uf is divided by f,
and you
> have
> a_1 x/f + u
> and the constant term of a_2 x + uf is uf, but when f^2 is
divided
> from P(m), it is u; therefore, a_2 x + uf is divided by f,
and you
> have
> a_2 x/f + u
> while the constant term of a_3 x + uf is 3x + uf, and after
f^2 is
> divided off, it is 3x + uf, so you have
> a_3 x + uf
> so, dividing P(m) by f^2 gives
> P(m)/f^2 = (a_1 x/f + u)(a_2 x/f + u)(a_3 x + uf).
> II. Second section
> Now take
> P(m)/f^2 = (a_1 x/f + u)(a_2 x/f + u)(a_3 x + uf)
> and multiply inside the parentheses by f^2/(a_1 a_2 a_3),
and outside
> by f^2(a_1 a_2 a_3) and you have
> P(m)/f^2 = ((a_1 a_2 a_3)/f^2)(x + uf/a_1)(x + uf/a_2)(x +
uf/a_3)
> and since a_1 a_2 a_3 = f^2(m^3 f^4 - 3m^2 f^2 + 3m), that
is
> P(m)/f^2 =
> (m^3 f^4 - 3m^2 f^2 + 3m)(x + uf/a_1)(x + uf/a_2)(x +
uf/a_3).
> Now consider the case that m, f, and u are algebraic
integers, then I
> have the ratios of algebraic integers:
> uf/a_1, uf/a_2, and uf/a_3,
> and now let
> v_1/w_1 = uf/a_1, v_2/w_2 = uf/a_2, and v_3/w_3 = uf/a_2
> where the vs and ws are algebraic integers in each case
coprime to
> each other.
> Making the substitutions I have
> P(m)/f^2 =
> (m^3 f^4 - 3m^2 f^2 + 3m)(x + v_1/w_1)(x + v_2/w_2)(x +
v_3/w_3).
> And I have from before that
> P(m)/f^2 = (m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1 + mf^2)
xu^2 + u^3 f
> so
> (m^3 f^4 - 3m^2 f^2 + 3m)(v_1 v_2 v_3)/(w_1 w_2 w_3) = f
> as that is the last coefcient from the last term u^3 f,
which proves
> that
> (m^3 f^4 - 3m^2 f^2 + 3m) has w_1, w_2 and w_3 as factors,
so let
> (m^3 f^4 - 3m^2 f^2 + 3m) = w_1 w_2 w_3
> then I have
> P(m)/f^2 = (w_1 x + v_1)(w_2 x + v_2)(w_3 x + v_3)
> but I still have that
> P(m)/f^2 = (a_1 x/f + u)(a_2 x/f + u)(a_3 x + uf).
> III. Third section
> So, even if a_1/f is not an algebraic integer, you can nd
w_1 an
> algebraic integer.
> But if a_1/f is an algebraic integer and w_1 is not, they
cannot be
> equal.
> But I have
> P(m)/f^2 = (w_1 x + v_1)(w_2 x + v_2)(w_3 x + v_3)
> and
> P(m)/f^2 = (a_1 x/f + u)(a_2 x/f + u)(a_3 x + uf)
> so how do you reconcile a case where a_1 x/f is not an
algebraic
> integer?
> There must exist z_1, z_2, and z_3 such that
> w_1 = (a_1 x z_1)/f, w_2 = (a_2 x z_2)/f and w_3 = a_3 x z_3
> and z_1 z_2 z_3 = 1,
> so algebraically the zs are units, but z_1, z_2 and z_3
are not units
> in the ring of algebraic integers, if a_1/f is not.
> Ive often faced arguments over the result from Section 1,
and at
> times Ive dealt with people claiming that Galois Theory
proves
> something about the factors of roots of monic polynomials
with integer
> coefcients.
> The basic claim is that *each* of the roots of a monic
polynomial with
> integer coefcients that is irreducible over rationals must
share
> non-unit factors with ALL of the prime factors of the last
> coefcient.
Note that this claim does not require Galois Theory, and is
easily
proven using the very basic results of eld theory, together
with
the elementary properties of the integers. Briey stated, one
takes
such a polynomial P (i.e., monic integral polynomial,
irreducible
over rationals) and constructs the eld Q(a), where a is a
root of
P. This is done without reference to which root a really is,
and
as a result, one nds that the two eld extensions Q(a) and
Q(b),
where a and b are distinct roots of P, are isomorphic in such
a way
that the elements of Q are held xed.
In particular, the ring of integers in Q is held xed, and
further:
*anything you can express* using arithmetic operations and
integers and the root a can be rewritten *simply by replacing
all occurrences of the symbol a with the symbol b* to yield
an equivalent statement about the root b.
This directly implies that if k is an integer, then if a and
k are
coprime, so are b and k. Why? Because if a and k are coprime,
I
can nd polynomials A, B, and C, with integer coefcients,
such that
A(x)*x + B(x)*k = C(x)*P(x) + 1.
If this is true, then I can substitute b in for the variable
x in
that equation, and nd integers u,v in Q(b) for which
u b + v k = 1.
No Galois Theory, no fancy dancing with eld extensions, no
Galois
group. Just a pair of eld extensions that we can prove
directly are
isomorphic, xing the eld of rationals (and its ring of
integers).
> For instance, with P(x) = x^2 + x + 6, the claim would be
that the two
> roots:
> (-1 + sqrt(32))/2 and (-1 - sqrt(32))/2
Um, the roots of P(x) are given by the quadratic formula:
For the equation
ax^2 + bx + c = 0
the roots are given by the formula
x = (-b +/- sqrt(b^2 - 4ac))/2a
Here, we have a = 1, b = 1, c = 6.
The discriminant is b^2 - 4ac = 1 - 4*1*6 = 1 - 24 = -23
So the roots are
x = (-1 +/- sqrt(-23))/2
Not, as you have incorrectly written,
> (-1 + sqrt(32))/2 and (-1 - sqrt(32))/2
> would, supposedly, each have to share non-unit factors with
2 and 3.
So, lets see about this. Does each share a factor with 2 and
3?
Let r be either of these roots. Note that the numbers
g = -(r + 2)
h = r - 1
k = 2r + 3
are algebraic integers, and that
g h = -(r+2)(r-1) = 2 - r - r^2
= 2 - (r^2 + r) = 2 - (-6) = 8,
and
g k = -(r + 2)(2r + 3)
= -(2r^2 + 7r + 6)
= -2r^2 - 7r - 6
= -2(-r - 6) - 7r - 6
= 2r + 12 - 7r - 6
So,
g k = -5r + 6.
But, since r^2 = -r - 6, we have
r^3 = -r^2 - 6r
= -(-r - 6) - 6r
= r + 6 - 6r = -5r + 6.
Thus, gk = r^3.
In other words, the numbers g,h,k are all algebraic integers,
and satisfy
gh = 2^3,
gk = r^3.
If we take cube roots
u^3 = g,
v^3 = h,
w^3 = r,
well have algebraic integers u,v,w with
uv = 2
uw = r.
A similar exercise yields common algebraic integer
divisors for r and 3. In this case, instead of
the above choices of g,h,k, one uses the following:
g = -2r - 3
h = 2r - 1
k = r + 2.
and one obtains
gh = 3^3
gk = r^3.
Ill leave the details up to the reader, since Im tired of
typing all
this stuff in. YOu will need to take cube roots again, as the
ideals
 and  are both of order 3 in the ideal class group
of the
extension Q(r).
> My works shows that its possible that actually neither
does and you
> have to check using advanced polynomial factorization
techniques.
Sorry, wrong answer.
> Faced with the algebra, certain people simply claimed that
Galois
> Theory *forces* that result, when in fact, it does not.
Oh, really? Why does elementary eld theory force it to be
the case, then?
Where is your fact, anyhow? All youre doing is making
baseless
assertions, unsupported by *any* evidence. Show an example,
why
dont you? I mean an *actual example*, with numbers for the
coefcients, and a *specic* claim.
> Thats kind of obvious as consider
> P(x) = x^2 + 5x + 6 = (x+2)(x+3)
> and if Galois Theory forced the previous on irrationals,
why wouldnt
> it force it on rationals as well?
You still havent gotten it through that thick skull that
reducibility makes a real difference, have you?
The fact that the polynomial splits already means that the
factors
can be independent in Q[x]. If the polynomial P is
irreducible in Q[x],
then 
, the ideal it generates in Q[x], is prime; Q[x] is
of dimension
one, so nonzero prime ideals are maximal, thus Q[x]/
 forms
a eld.
In this eld, which contains a canonical copy of Q itself,
the class
represented by x + 
 is a root of the polynomial P(x).
You might note that this formal extension of Q doesnt really
depend
on which root you are considering. It shows the arithmetic
properties
of Q(a) for *any* root of the irreducible polynomial P.
The same thing just does not hold for a reducible polynomial
P. For one
thing, the construction above just does not produce a eld.
When you
perform that construction using a reducible polynomial, you
actually
introduce zero-divisors and destroy any likelihood of
reaching a eld
until those zero-divisors are eliminated.
> It doesnt force anything on either. They were just wrong.
Please show my error in the arithmetic regarding the quadratic
equation above, fool.
> James Harris
The more you imagine you can fool anyone with your meaningless
blathering about stuff you clearly know nothing about, the
less
likely it is that youll ever get to a point of knowledge.
Whats the real problem with sitting down and *learning*
something?
Are you mentally decient? Scared? Too proud of your own
ignorance
and too ashamed of your own shortcomings?
Dale
===
Subject: Re: New paper, algebraic integers, Galois Theory
>
> 
>
> Now divide f^2 from both sides, which gives
>
> P(m)/f^2 = (m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1 +
mf^2) xu^2 +
u^3 f
>
> P(m)/f^2 = (a_1 x + uf)(a_2 x + uf)(a_3 x + uf)/f^2
>
> and you note that P(0)/f^2 = u^2(3x + uf), which
means that now
your
> constant terms are u for the rst, u for the
second and 3x + uf
for
> the third.
>
>
> True enough, but if what you want at the end is a
factorization
> with algebraic integer coefcients, this is the
wrong way to
> divide f^2 out of the right side. It has been proved
that
> there is a RIGHT way. By putting unjustied stress
on preserving
> the constant terms AFTER division by f^2, you have
lost sight of
> the primary goal of ending up with a factorization
with algebraic
> integer coefcients. That is the root of some of
your subsequent
> problems with this paper. But see below also.
>
>
> The terms constant with respect to m, are just that,
constant with
> respect to m.
>
> In this exposition Ive stepped through an argument
quite drily
> without extra.
>
> Now youre adding extra, but giving no mathematics.
>
>
> No - YOU added extra. The original requirement was ONLY
that you
> nd a factorization of P(m)/f^2 with algebraic integer
coefcients.
> To that you have added the requirement that the constant
terms
> of the factors (after the division by f^2) must be
constant -
> that in turn adds the constraint that you can split up
f^2 in only
> one way. And that is what leads to your problem.
>Youve now added your own requirement, though anyone who
bothers to
>read through my original post or the latest draft can see
that
>requirement isnt there.
>
> The constant terms (i.e., uf, uf, and 3*x + uf), after
division
> by f^2, must indeed multiply together to produce the
constant
> term of P(m)/f^2. However that does NOT mean that
individually,
> they need to be constant after division. Three
nonconstant
> functions can multiply together to produce a constant.
>
> Right?
>
>So, youre saying the the constant terms are actually NOT
constant
>terms, right?
> FINALLY I think I see what you have been thinking about
> this. Sorry to take so long in discerning it.
Hmmm...
> I will be as brief as I can, but it takes a little
> explaining. Please bear with me. In fact, please
> humor me. Its not that long or difcult.
> We claim that f^2 must be split up into three factors,
> v_1, v_2, and v_3, and that in fact they are functions of
> m:
> f^2 = v_1(m) v_2(m) v_3(m).
> You think this cannot work because of concerns that
> the constant term will not remain constant.
Youre asserting a dependency that doesnt follow from
anything.
Its just yanked out of the air because you wish it to be so.
One could just as easily have
f^2 = j_1(k) j_2(k) j_3(k)
because its an independent variable.
> Consider g_1(m) = a_1(m)*x + uf.
> The constant term is g_1(0) = uf, because a_1(0) = 0.
> We agree on that.
Then why do you keep arguing?
> Now suppose you divide g_1(m) by v_1(m).
> Let h_1(m) = g_1(m)/v_1(m).
> By denition, the constant term of h_1(m) is h_1(0).
> This is:
> h_1(0) = g_1(0)/v_1(0)
> = (a_1(0)/v_1(0))*x + u*f/v_1(0).
> = 0 + u*f/v_1(0).
> = u*f/v_1(0).
> Now, we both agree that v_1(0) has to equal f. Right?
Yes, but the value of m is irrelevant.
Remember, what youre doing is dividing f^2 from P(m) where
P(m) = f^2 ((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1 + mf^2) xu^2
+ u^3 f)
and
P(m) = (a_1 x + uf)(a_2 x + uf)(a_3 x + uf)
so f^2 is a *multiple* of P(m).
Why does it matter what value m has?
The division of f^2 from P(m) has nothing to do with m, and
it doesnt
go one way or another depending on what value m has, despite
your
protestations to the contrary.
It happens one way; therefore, as m=0 is a value at which you
can
actually see how it happens, that value is convenient for
doing so to
the mathematician, without being a special case.
The *logic* of it is trivial.
> So when I write the constant term of h_1(m), it is
> actually
> h_1(0) = u*f/v_1(0) = u*f/f = u.
> No problem with that, right ?
Well yeah, thats what Ive been saying. The constant term
with
respect to m from before is uf, and it becomes u after f^2 is
divided
from P(m).
Its so beautifully simple.
> But it is right here that you are making a big
> mistake.
Sigh.
> You think that if v_1 = v_1(m) is the function by
> which you are dividing, the constant term AFTER THE
> DIVISION has to be
> u*f/v_1(m).
> You say: this constant term must equal u. We
> agree on that.
> You say, therefore, v_1(m) must equal f, for ALL m.
> You say, theres no other way it can work out. Right?
> No. Wrong. You are miscalculating the constant
> term of h_1(m). It is NOT
> u*f/v_1(m).
> Instead it is
> u*f/v_1(0) = u*f/f = u.
I know it is, but you argue instead that it is the other, not
me.
The reason why you argue that is that with
a_1 x + uf
the constant term IS uf, and its clear that if the constant
term is u
after f^2 is divided off from P(m), then you have to divide
by f, so
that you have
a_1 x/f + u
which is what follows algebraically.
BUT, you can nd values where m and f are algebraic integers,
where
a_1/f is not, which is where some people get confused.
But it can all be explained mathematically without a lot of
fuss.
> These two little equations are the key.
> Its not the *position* of the constant term that
> determines it. It is the VALUE of it.
Well, um, there we can agree, and I dont have anything extra
to add.
> To compute its value you MUST substitute 0 for m EVERYWHERE.
> THATS THE DEFINITION OF CONSTANT TERM. You have been
thinking
> v_1(m), not v_1(0). You have made the wrong substitution.
Its
> that simple!
Setting m=0, allows you to see the constant term. But,
logically, the
value of m is irrelevant to terms constant to m.
Like, look at P(x) = x + 2, and the constant term is 2.
If you let x=0, then P(0) = 2, but if you let x=3, P(x) = 3,
but the
constant term is STILL 2.
It doesnt change; its constant.
In the case of P(m), you have u^2 f^2 (3x + uf) as the
constant term,
so there are variables, but they are constant with respect to
m, so
the principle remains the same.
Their value does not change as m changes.
> No contradiction. No problem. The constant term
> *stays constant* if you make the correct substitution.
Well, Ive made the correct substitutions. I was trashing
*your*
argument.
> There is nothing to prevent v_1(m) from being a
> *variable* function. Even if it is, the constant
> term stays constant. You have thought not because you
> have been computing it incorrectly. Its not
> uf/v_1(m). Its uf/v_1(0) = u.
>
> Once you see this, all the rest should come together.
>
Well, lets put that to the test Nora Baron and see if the
algebra
reaches you.
Consider a_1 x + uf, and you claim it has a factor that is
v_1(m) that
varies with m, now just divide through by that factor and you
get
a_1 x/v_1(m) + uf/v_1(m)
and you can see that the term constant with respect to m is
now
uf/v_1(m), which is a basic contradiction, as now it *varies*
with m.
Understand?
Your position necessarily must be that there is no actual
constant
term, and that is a position that other posters have actually
taken,
claiming that m=0, is in fact a special case and for the
factors like
a_1 x + uf of P(m), no actual constant term exists.
Now I gured you understood that, but here youre posting as
if you
think there is a constant term.
Simple question: What happens to the constant term as m
changes?
Simple answer is nothing, as its constant.
But, for two of the as the constant term goes from uf to u,
which
requires that they be divided by f.
Its simple.
Ive actually been intrigued by the ability to argue in this
area, as
some of you have argued for well over a year, and Ive tried
to see it
from your point of view.
I think what happens at least for some of you is that you
dont
understand mathematically what constancy means!!!
Next you focus on the as and your belief that *functions*
are the
most important elements, and that constants are
trivial--though you
dont quite understand them.
So to you, it makes sense that since the functions are whats
important--because in your mind functions are a BIG DEAL while
constants are trivial--it makes sense to you that the
FUNCTIONS
determine what the factor is, and push the constants around!
Its like theres some weird hierarchy in your brain where
constants
are peons and functions are king, so the functions must be
able to
push the trivial constants around.
The issue never comes up with something trivial like P(x) =
2(x+2), so
you never need worry about the function--2x--versus the
constant
term--4--so youre not prepared.
To their credit, Ive yet to get the same argument from an
experienced
math professor.
Its like they almost immediately adjust, like when I went to
Vanderbilt University I tossed that objection at the
professor I
personally visited to explain the paper Advanced Polynomial
Factorization to, and he paused for a second like maybe he
was going
to go for it, and then he shook his head, and said it was
nonsense.
So I guess some of you do have some fairly steady mathematical
training.
I must admit I was a bit surprised, if only because it makes
the
situation slightly more complicated.
For a while I gured maybe understanding this work was beyond
the
wiring of the human brain, but, nope, its not.
>Since Nora Baron decided to talk rather than give math,
Ill explain
>to those who may now be confused.
> See above. Please, dont dismiss it out of hand. Its my
> honest attempt to explain. I should have apprehended your
thinking
> earlier.
> Nora B.
Fascinating.
James Harris
===
Subject: Re: New paper, algebraic integers, Galois Theory
>
> 

>So, youre saying the the constant terms are actually NOT
constant
>terms, right?
>
> FINALLY I think I see what you have been thinking about
> this. Sorry to take so long in discerning it.
>Hmmm...
> I will be as brief as I can, but it takes a little
> explaining. Please bear with me. In fact, please
> humor me. Its not that long or difcult.
> We claim that f^2 must be split up into three factors,
> v_1, v_2, and v_3, and that in fact they are functions of
> m:
> f^2 = v_1(m) v_2(m) v_3(m).
> You think this cannot work because of concerns that
> the constant term will not remain constant.
>Youre asserting a dependency that doesnt follow from
anything.
I am assuming for the sake of argument that v_1(m),
etc., are functions of m. I want to analyze your
conclusion that they must be CONSTANT functions of m.
>Its just yanked out of the air because you wish it to be so.
No, at this point it is just an assumption. I want
to see where it leads.
>One could just as easily have
>f^2 = j_1(k) j_2(k) j_3(k)
>because its an independent variable.
k is not a variable of interest here. m is.
> Consider g_1(m) = a_1(m)*x + uf.
> The constant term is g_1(0) = uf, because a_1(0) = 0.
> We agree on that.
>Then why do you keep arguing?
Jeez, let me nish!
> Now suppose you divide g_1(m) by v_1(m).
> Let h_1(m) = g_1(m)/v_1(m).
> By denition, the constant term of h_1(m) is h_1(0).
> This is:
> h_1(0) = g_1(0)/v_1(0)
> = (a_1(0)/v_1(0))*x + u*f/v_1(0).
> = 0 + u*f/v_1(0).
> = u*f/v_1(0).
>
> Now, we both agree that v_1(0) has to equal f. Right?
>Yes, but the value of m is irrelevant.
Youre not listening to what I am saying. Youre
listening to one of your own old arguments. Try to
be a little more open-minded.
>Remember, what youre doing is dividing f^2 from P(m) where
>P(m) = f^2 ((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1 + mf^2)
xu^2 + u^3 f)
>and
>P(m) = (a_1 x + uf)(a_2 x + uf)(a_3 x + uf)
>so f^2 is a *multiple* of P(m).
I wouldnt say that. I would say P(m) is a multiple
of f^2.
>Why does it matter what value m has?
Keep reading ...
>The division of f^2 from P(m) has nothing to do with m, and
it doesnt
>go one way or another depending on what value m has, despite
your
>protestations to the contrary.
>It happens one way; therefore, as m=0 is a value at which
you can
>actually see how it happens, that value is convenient for
doing so to
>the mathematician, without being a special case.
>The *logic* of it is trivial.
> So when I write the constant term of h_1(m), it is
> actually
> h_1(0) = u*f/v_1(0) = u*f/f = u.
> No problem with that, right ?
>Well yeah, thats what Ive been saying.
No, its not what youve been saying.
>The constant term with
>respect to m from before is uf, and it becomes u after f^2
is divided
>from P(m).
That part is true.
>Its so beautifully simple.
> But it is right here that you are making a big
> mistake.
>Sigh.
> You think that if v_1 = v_1(m) is the function by
> which you are dividing, the constant term AFTER THE
> DIVISION has to be
> u*f/v_1(m).
> You say: this constant term must equal u. We
> agree on that.
> You say, therefore, v_1(m) must equal f, for ALL m.
> You say, theres no other way it can work out. Right?
> No. Wrong. You are miscalculating the constant
> term of h_1(m). It is NOT
> u*f/v_1(m).
> Instead it is
> u*f/v_1(0) = u*f/f = u.
>I know it is, but you argue instead that it is the other,
not me.
No - you need to show that v_1(m) must be a constant
function, and therefore it must equal f for all values
of m. Here is the logic:
-------------------------------------------------------------
The constant term is u*f/v_1(m). Because we know that the
constant term must be equal to u, we conclude that
v_1(m) = f
*for all m*.
-------------------------------------------------------------
So whats wrong with that logic?
Whats wrong is saying the constant term is
u*f/v_1(m).
Its not. It is u*f/v_1(0). No m is involved.
So you can say u*f/v_1(0) = u*f/f = u.
But it tells you NOTHING about v_1(m) for m <> 0.
Thats the point.
>The reason why you argue that is that with
>a_1 x + uf
>the constant term IS uf, and its clear that if the constant
term is u
>after f^2 is divided off from P(m), then you have to divide
by f, so
>that you have
>a_1 x/f + u
>which is what follows algebraically.
>BUT, you can nd values where m and f are algebraic
integers, where
>a_1/f is not, which is where some people get confused.
>But it can all be explained mathematically without a lot of
fuss.
> These two little equations are the key.
> Its not the *position* of the constant term that
> determines it. It is the VALUE of it.
>Well, um, there we can agree, and I dont have anything
extra to add.
Actually I think we dont agree. I am afraid you are not
getting it.
> To compute its value you MUST substitute 0 for m
EVERYWHERE.
> THATS THE DEFINITION OF CONSTANT TERM. You have been
thinking
> v_1(m), not v_1(0). You have made the wrong substitution.
Its
> that simple!
>Setting m=0, allows you to see the constant term. But,
logically, the
>value of m is irrelevant to terms constant to m.
In a way, you are saying the right thing.
>Like, look at P(x) = x + 2, and the constant term is 2.
>If you let x=0, then P(0) = 2, but if you let x=3, P(x) = 3,
but the
>constant term is STILL 2.
>It doesnt change; its constant.
Try this example:
Let h(m) = 5*(m + 2) + 7.
Divide h(m) by a variable function: say, v(m) = m + 2.
Thus h(m)/v(m) = 5 + 7/(m + 2).
Multiple choice quiz:
What is the constant term of h(m)/v(m) ?
a) 5
b) 7/v(m)
c) 7/(m + 2)
d) 7/2
e) 8.5
f) Something else, specify: __________________________
[more than one answer may be right]
>In the case of P(m), you have u^2 f^2 (3x + uf) as the
constant term,
>so there are variables, but they are constant with respect
to m, so
>the principle remains the same.
>Their value does not change as m changes.
> No contradiction. No problem. The constant term
> *stays constant* if you make the correct substitution.
>Well, Ive made the correct substitutions. I was trashing
*your*
>argument.
Youre simply not getting it.
> There is nothing to prevent v_1(m) from being a
> *variable* function. Even if it is, the constant
> term stays constant. You have thought not because you
> have been computing it incorrectly. Its not
> uf/v_1(m). Its uf/v_1(0) = u.
>
> Once you see this, all the rest should come together.
>
>Well, lets put that to the test Nora Baron and see if the
algebra
>reaches you.
>Consider a_1 x + uf, and you claim it has a factor that is
v_1(m) that
>varies with m, now just divide through by that factor and
you get
>a_1 x/v_1(m) + uf/v_1(m)
>and you can see that the term constant with respect to m is
now
>uf/v_1(m), which is a basic contradiction, as now it
*varies* with m.
thought you were thinking.
By denition, if you divide one variable function by
another variable function, say h(m)/k(m), the constant
term is h(0)/k(0). Thats YOUR denition. I dont disagree
with it; its a perfectly good denition.
Lets apply this exactly to a_1(m)*x + uf. The constant
term must be
(a_1(0)/v_1(0)) + uf/v_1(0) = 0 + uf/v_1(0) = uf/v_1(0).
But YOU now say the constant term is uf/v_1(m) !!!
Right? Thats what you said just above.
Right there is the problem. See, you are substituting in
m instead of 0 ! That is NOT the denition of the constant
term. Yes, it is in the POSITION occupied by the constant
term, but it is NOT the constant term. Position vs. value:
value wins.
>Understand?
Youve made what you are thinking exceedingly clear. I
could not have stated it more clearly myself. And it is
exactly what I thought. And it is totally wrong.
>Your position necessarily must be that there is no actual
constant
>term, and that is a position that other posters have
actually taken,
>claiming that m=0, is in fact a special case and for the
factors like
>a_1 x + uf of P(m), no actual constant term exists.
Not at all. i am perfectly happy with YOUR denition of
constant term: if h(m) is a function, the constant term
is h(0).
>Now I gured you understood that, but here youre posting as
if you
>think there is a constant term.
Right. See just above.
>Simple question: What happens to the constant term as m
changes?
Nothing. h(0) is not changed when m changes.
>Simple answer is nothing, as its constant.
>But, for two of the as the constant term goes from uf to u,
which
>requires that they be divided by f.
Yes, but you cannot equate the constant term to
uf/v_1(m), and thereby conclude that v_1(m) must be
cannot conclude anything about v_1(m).
>Its simple.
True enough.
>Ive actually been intrigued by the ability to argue in this
area, as
>some of you have argued for well over a year, and Ive tried
to see it
>from your point of view.
>I think what happens at least for some of you is that you
dont
>understand mathematically what constancy means!!!
We agree on what you mean by constant term of a
function. I applied your perfectly good denition
quite literally to obtain: uf/v_1(0).
You however want to go farther. Because uf appears
as the last term in the expression a_1(m)*x + uf,
you want to say that the constant term is actually
uf/v_1(m).
Please try to get this. There is just a teeny level
of abstraction here. The constant term can be computed
sheerly by substitution. It is uf/v_1(0). This is
true no matter what kind of variable function v_1(m)
is. It is NOT true for variable functions v_1(m) in general
that
uf/v_1(m) = uf/v_1(0).
If you think so, you are assuming what you want to prove.
>Next you focus on the as and your belief that *functions*
are the
>most important elements, and that constants are
trivial--though you
>dont quite understand them.
We are in complete agreement about the denition of
constant term. But when, as above, you say:
and you can see that the term constant with respect
to m is now uf/v_1(m)
you are making a serious error. I do NOT say that the
constant term is uf/v_1(m). That does NOT follow from
Thats ALL it is. You are not free to simply plug in
m instead of 0. There is no variable m involved.
The constant term is simply about the VALUE of the
function when m = 0. It is not about the POSITION of
certain components of the function. I think that is
what is confusing you: position vs. value.
>So to you, it makes sense that since the functions are whats
>important--because in your mind functions are a BIG DEAL
while
>constants are trivial--it makes sense to you that the
FUNCTIONS
>determine what the factor is, and push the constants around!
>Its like theres some weird hierarchy in your brain where
constants
>are peons and functions are king, so the functions must be
able to
>push the trivial constants around.
You are extrapolating far, far beyond anything I have
said. When I talk about the constant term, I am strictly
following YOUR denition. You however are not.
>The issue never comes up with something trivial like P(x) =
2(x+2), so
>you never need worry about the function--2x--versus the
constant
>term--4--so youre not prepared.
>To their credit, Ive yet to get the same argument from an
experienced
>math professor.
Not true ...
>Its like they almost immediately adjust, like when I went to
>Vanderbilt University I tossed that objection at the
professor I
>personally visited to explain the paper Advanced Polynomial
>Factorization to, and he paused for a second like maybe he
was going
>to go for it, and then he shook his head, and said it was
nonsense.
>So I guess some of you do have some fairly steady
mathematical
>training.
>I must admit I was a bit surprised, if only because it makes
the
>situation slightly more complicated.
>For a while I gured maybe understanding this work was
beyond the
>wiring of the human brain, but, nope, its not.
>Since Nora Baron decided to talk rather than give math,
Ill
explain
>to those who may now be confused.
>
> See above. Please, dont dismiss it out of hand. Its my
> honest attempt to explain. I should have apprehended your
thinking
> earlier.
> Nora B.
>Fascinating.
Go back and look at the example I gave above,
h(m) = 5*(m + 2) + 7 and v(m) = m + 2.
Of course the constant term of h(m) is h(0) = 17. I am
sure you got that right. You can say
h(m) = 5*m + 17.
h(m)/v(m) would be 17/v(m) = 12/(m + 2).
See, this is exactly analogous to uf/v_1(m), which is
what ***you said above*** is the constant term of
(a_1(m)*x + uf)/v_1(m).
But its not. Obviously 17/(m + 2) is not a constant.
It is a variable. The true constant term is
17/(0 + 2) = 17/2.
The fact this IS a constant clearly does NOT imply
that
17/v(m) = 17/(m + 2),
is a constant, or that v(m) = (m + 2) is a constant.
See, its not the POSITION of 17 in h(m) = 5*m + 17
which determines the constant function of h(m)/v(m). Its
just the VALUE of
h(0)/v(0) = (5*0 + 17)/(0 + 2) = 8.5.
Is there any chance you are understanding this?
Nora B.
>James Harris
===
Subject: Re: New paper, algebraic integers, Galois Theory
> Like, look at P(x) = x + 2, and the constant term is 2.
> If you let x=0, then P(0) = 2, but if you let x=3, P(x) = 3,
Oops. P(x) = 5.
> but the
> constant term is STILL 2.
===
Subject: Re: New paper, algebraic integers, Galois Theory
I have gotten an acute case of deja vu.
...
> By denition, the constant term of h_1(m) is h_1(0).
Yup, see for instance:
by Arturo Magidin (although at that time James still used the
constant
5 rather than the variable x and he also used the constant 7
in stead
of now the variable f, and also he has now introduced an
additional u).
And, yes, at that time Arturo was still replying to James as
James had
not yet asked Arturo not to reply to him.
Also interesting in that thread I nd:
yup, by me, that is why I found it interesting ;-). More so,
because
I uked at the end.
Later on now he will come up with the distributive property
like he did
slightly less than a year ago in the thead with subject:
JSH: Mathematical clarity
Alas, his own contributions to that thread are deleted...
James appears to try to revise history and in the process
forgets what
has been refuted, so he goes back to a previous stage.
reasoning is wrong. Starting with the polynomial
Q(m,f,x) = f^2((4 m^2 - 3 m).x^3 - 3(-1 + m).x + f)
and a factorisation of that to:
Q(m,f,x) = (a1(m,f).x + f)(a2(m,f).x + f)(a3(m,f).x + f)
I found that the as are the negatives of the roots of:
a^3 + 3(-1 + m).a^2 - f^2.(4 m^2 - 3m).
When m = 0, I nd:
a1(0, f) = a2(0, f) = 0 and a3(0, f) = -3,
so we nd the factorisation:
Q(0, f, x) = (0x + f)(0x + f)(3x + f),
so exactly two of the factors are divisible by f. (And this is
precisely the case James has in his hands.)
When m = 1, I nd the a-polynomial:
a^3 - f^2,
so now the as are the negatives of the three cube roots of
f^2. Lets
dene W = (-1 + sqrt(-3))/2, and Z = - cbrt(f^2).
Q(1, f, x) = (Z.x + f)(Z.W.x + f)(Z.W^2.x + f),
and *none* of the factors is divisible by f. Rather all three
factors
share Z with f (and that is all), but because Z^3 = f^2, also
in this
case Q(1, f, x) is divisible by f^2.
When m = 2, I nd the a-polynomial:
a^3 + 3.a^2 - 10.f^2
and when f = 7 this reduces to
a^3 + 3.a^2 - 490 = (a - 7)(a^2 + 10.a + 70)
Where *both* of the latter two roots are not coprime to 7.
(Note that
this can be done with every odd integer f...)
What I think is James problem is that he abuses notation. He
talks
about a_1 as if it is a constant, but it is a function that
depends on
m and f...
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam,
nederland,
+31205924131
home: bovenover 215, 1025 jn amsterdam, nederland;
http://www.cwi.nl/~dik/
===
Subject: Re: New paper, algebraic integers, Galois Theory
Some additional editing, nothing major. ___JSH
-------------------------------------------------------------
---------
I. First section
The following are in a commutative ring.
Start with
P(m) = f^2 ((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1 + mf^2) xu^2
+ u^3 f)
with the factorization
P(m) = (a_1 x + uf)(a_2 x + uf)(a_3 x + uf)
and note that at
m=0, P(0) = u^2 f^2(3x + uf),
which gives you terms that do not vary as m varies.
So what about (a_1 x + uf), (a_2 x + uf), and (a_3 x + uf)?
(a_1 x + uf)(a_2 x + uf)(a_3 x + uf) = u^2 f^2 (3x + uf)
which shows that at least two of the as have to equal 0 at
m=0, while
one equals 3.
Since, at m=0, two of the as must equal 0, its convenient
to just
arbitrarily select a_1 and a_2 as those two.
Then you have uf for the rst, uf for the second and 3x + uf
for the
third as terms that do not vary when m varies.
Now then, if m=1, what are the *constant* terms?
They are uf, for the rst, uf for the second, and 3x + uf for
the
third.
Thats logical because they do not vary with m, so if
m=1003909273,
what are the constant terms?
They are uf, for the rst, uf for the second, and 3x + uf for
the
third.
Now divide f^2 from both sides, which gives
P(m)/f^2 = (m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1 + mf^2) xu^2
+ u^3 f
P(m)/f^2 = (a_1 x + uf)(a_2 x + uf)(a_3 x + uf)/f^2
and you note that P(0)/f^2 = u^2(3x + uf), which means that
now your
constant terms are u for the rst, u for the second and 3x +
uf for
the third.
Now then, if m=1, what are the constant terms now?
They are u for the rst, u for the second, and 3x + uf for
the third.
If m = 2938479378, what are the constant terms now?
They are u for the rst, u for the second, and 3x + uf for
the third.
How can the constant terms of the rst two go from uf to u?
They must be divided by f.
Now, the constant term of a_1 x + uf, is uf, but when f^2 is
divided
from P(m), it is u; therefore, a_1 x + uf is divided by f,
and you
have
a_1 x/f + u
and the constant term of a_2 x + uf is uf, but when f^2 is
divided
from P(m), it is u; therefore, a_2 x + uf is divided by f,
and you
have
a_2 x/f + u
while the constant term of a_3 x + uf is 3x + uf, and after
f^2 is
divided off, it is 3x + uf, so you have
a_3 x + uf
so, dividing P(m) by f^2 gives
P(m)/f^2 = (a_1 x/f + u)(a_2 x/f + u)(a_3 x + uf).
II. Second section
Now take
P(m)/f^2 = (a_1 x/f + u)(a_2 x/f + u)(a_3 x + uf)
and multiply inside the parentheses by f^2/(a_1 a_2 a_3), and
outside
by f^2(a_1 a_2 a_3) and you have
P(m)/f^2 = ((a_1 a_2 a_3)/f^2)(x + uf/a_1)(x + uf/a_2)(x +
uf/a_3)
and since a_1 a_2 a_3 = f^2(m^3 f^4 - 3m^2 f^2 + 3m), that is
P(m)/f^2 =
(m^3 f^4 - 3m^2 f^2 + 3m)(x + uf/a_1)(x + uf/a_2)(x + uf/a_3).
Now consider the case that m, f, and u are algebraic
integers, then I
have the ratios of algebraic integers:
uf/a_1, uf/a_2, and uf/a_3,
and now let
v_1/w_1 = uf/a_1, v_2/w_2 = uf/a_2, and v_3/w_3 = uf/a_2
where the vs and ws are algebraic integers in each case
coprime to
each other.
Making the substitutions I have
P(m)/f^2 =
(m^3 f^4 - 3m^2 f^2 + 3m)(x + v_1/w_1)(x + v_2/w_2)(x +
v_3/w_3).
And I have from before that
P(m)/f^2 = (m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1 + mf^2) xu^2
+ u^3 f
so
(m^3 f^4 - 3m^2 f^2 + 3m)(v_1 v_2 v_3)/(w_1 w_2 w_3) = f
as that is the last coefcient from the last term u^3 f,
which proves
that
(m^3 f^4 - 3m^2 f^2 + 3m) has w_1, w_2 and w_3 as factors, so
let
(m^3 f^4 - 3m^2 f^2 + 3m) = w_1 w_2 w_3
then I have
P(m)/f^2 = (w_1 x + v_1)(w_2 x + v_2)(w_3 x + v_3)
but I still have that
P(m)/f^2 = (a_1 x/f + u)(a_2 x/f + u)(a_3 x + uf).
III. Third section
So, even if a_1/f is not an algebraic integer, you can nd
w_1 an
algebraic integer.
But if a_1/f is an algebraic integer and w_1 is not, they
cannot be
equal.
But I have
P(m)/f^2 = (w_1 x + v_1)(w_2 x + v_2)(w_3 x + v_3)
and
P(m)/f^2 = (a_1 x/f + u)(a_2 x/f + u)(a_3 x + uf)
so how do you reconcile a case where a_1 x/f is not an
algebraic
integer?
There must exist z_1, z_2, and z_3 such that
w_1 = (a_1 x z_1)/f, w_2 = (a_2 x z_2)/f and w_3 = a_3 x z_3
and z_1 z_2 z_3 = 1,
so algebraically the zs are units, but z_1, z_2 and z_3 are
not units
in the ring of algebraic integers, if a_1/f is not.
Next, Galois Theory.
James Harris
===
Subject: Re: New paper, algebraic integers, Galois Theory
In sci.math, James Harris

> Some additional editing, nothing major. ___JSH
>
-------------------------------------------------------------
---------
> I. First section
> The following are in a commutative ring.
> Start with
> P(m) = f^2 ((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1 + mf^2)
xu^2 + u^3 f)
Side issue:
In GP/Pari, one can write this
P(m,f,u,x)=f^2*((m^3*f^4 - 3*m^2*f^2 + 3*m)*x^3 - 3*(-1 +
m*f^2)*x*u^2 +
u^3*f)
which GP/Pari interprets as an ad hoc multivariate function
def.
I cant speak for Maple or other such but presumably they have
similar capabilities.
One can also write
P(m)=f^2*((m^3*f^4 - 3*m^2*f^2 + 3*m)*x^3 - 3*(-1 +
m*f^2)*x*u^2 + u^3*f)
but thats less exible, as f, u, and x cannot be substituted
for.
Pari is included with many Linux distros, or one can obtain
the source
or a Windows binary from http://www.math.u-bordeaux.fr .

We now return you to our scheduled discussion... :-)
> with the factorization
> P(m) = (a_1 x + uf)(a_2 x + uf)(a_3 x + uf)
> and note that at
> m=0, P(0) = u^2 f^2(3x + uf),
> which gives you terms that do not vary as m varies.
> So what about (a_1 x + uf), (a_2 x + uf), and (a_3 x + uf)?
> (a_1 x + uf)(a_2 x + uf)(a_3 x + uf) = u^2 f^2 (3x + uf)
> which shows that at least two of the as have to equal 0 at
m=0, while
> one equals 3.
Be *very* careful here. One ring (nite eld, actually),
for example, is J mod 5 [*]. In J mod 5, 1 * 1 = 2 * 3 = 1.
(I suppose 2 = sqrt(-1) in J mod 5, but thats a topic
element in J mod 5 is a unit*; certainly, theyre easily
paired (1 * 1 = 2 * 3 = 4 * 4 = 1).
The best one can do is show that in an arbitrary ring,
P(0)/(u^2*f^2)
= 3*x + u*f, and Im not entirely certain of that, since J
mod 6
in particular has some peculiar properties (2 * 2 = 2 * 5 =
4).
Division in a ring is obviously *not* a given.
> Since, at m=0, two of the as must equal 0, its convenient
to just
> arbitrarily select a_1 and a_2 as those two.
Im not entirely sure of *that*, either. In J mod N, where N
is
nonprime, one can nd p and q such that p * q = 0, but p and q
are nonzero. Reprising J mod 6, for example, p = 2, q = 3, 2
* 3 = 0.
> Then you have uf for the rst, uf for the second and 3x +
uf for the
> third as terms that do not vary when m varies.
> Now then, if m=1, what are the *constant* terms?
P(1,f,u,x) = (f^6 - 3*f^4 + 3*f^2)*x^3 + (-3*u^2*f^4 +
3*u^2*f^2)*x +
u^3*f^3
The term associated with x^0 is u^3*f^3 here. Im not
entirely sure
what youre getting at.
> They are uf, for the rst, uf for the second, and 3x + uf
for the
> third.
> Thats logical because they do not vary with m, so if
m=1003909273,
> what are the constant terms?
> They are uf, for the rst, uf for the second, and 3x + uf
for the
> third.
> Now divide f^2 from both sides, which gives
Oops, you fell through the ice. Division in a ring is not at
all
guaranteed.
Had you specied a eld, youd be on rmer ground, of course,
and
many rings (J mod p for any prime p among them) are elds.
[rest snipped]
[*] I have *no* idea what the standard notation is for the
ring
of integers modulo 5; hopefully this is clear. Z[5] = Z,
making the [] notation useless here.
--
#191, ewill3@earthlink.net
Its still legal to go .sigless.
===
Subject: Schwarz Lemma problem (I think)
X-RFC2646:  Original
Suppose |f(z)|<=1 for |z|<1 and f is a non-constant analytic
function. By
considering the function g : D ---> D
(where D = {z : |z| < 1}) dened by
g(z) = [f(z)-a]/[1- /a f(z)]
where a = f(0) and /a denotes complex conjugate of a (is
there better
newsgroup notation for this?)
Then show that
[|f(0)| - |z|] / [1+|f(0)||z|] <= |f(z)| <= [|f(0)| + |z|] /
[1-|f(0)||z|]
This is denetely a Schwarz lemma problem. Here is what I
have so far :
First of all, g(0) = 0 since 1 - /af(0) = 0 if and only if 1
- |a|^2 = 0
iff |a| = 1 iff |f(0)| = 1. However, this cant happen
because we assumed f
is non-constant analytic, and so if |f(0)| = 1, by Maximum
modulus theorem f
is constant.
Thus, g is analytic on D, |g(z)| <= 1 for z in D, and g(0) =
0. Thus, we
can apply Schwarz Lemma! That is, |g(0)| <= 1 and |g(z)| <=
|z|. But now
I am stuck. I have tried to play around with the inequality
|g(z)| <= |z|,
but have gotten nowhere. Also |g(0)| <= 1 doesnt really
help me (should
it?)
But! I found the following very interesting :
Certainly (I believe I am right here)
[|f(0)| - |z|] / [1+|f(0)||z|] <= |g(z)| <= [|f(0)| + |z|] /
[1-|f(0)||z|]
is true (all I changed in the inequality was the middle term
f(z) was
changed to g(z)).
Thus, if I could prove that |g(z)| = |f(z)|, then I would be
done. But this
seems very unlikely to be true.
Any ideas?
Isaac
===
Subject: Questioning assumptions
Now I can show how my argument steps out in extreme detail,
and even
show how it is that posters could nd algebraic integers that
they
claim are counterexamples to my argument, and show why they
are not,
but it might help to point out a few assumptions other
posters are
making without giving mathematical proof.
Like one BIG assumption is that if a number is not a unit in
the ring
of algebraic integers, then its not meaningfully a unit,
which may
sound odd, but let me explain how people have made this issue
convoluted.
It was decided in math circles some time ago that algebraic
integers
fully contained all numbers that had certain key properties
where the
most important one is the ability to be coprime.
Like 2 and 3 are coprime in the ring of integers and in the
ring of
algebraic integers, but not in the eld of complex numbers as
there
2(3/2) = 3, so 2 is a factor of 3.
Thats basic.
The problem is that part of their assumption then becomes
that a unit,
is either an algebraic integer, or it is part of a eld, or
some
ring that has algebraic numbers added to it, like if you add
1/2 to
integers, or something like that.
That is, to put it simply, they assume that either a unit is
an
algebraic integer, or it is in a ring that contains what most
people
would call fractions, like 1/2, being in the ring makes 2 a
unit.
Ive shown that there is a middle way.
That middle way is to accept that there are other rings where
units
exist that are not algebraic integers, where there are also
not any
fractions.
Ive talked about that before as the ring of objects, where no
rational units exist other than -1 or 1.
Such a simple requirement!!! Yet one that mathematicians never
bothered to put forward, partly because they erroneously
believed that
algebraic integers covered all the bases, but you see, they
werent
really mathematicians because they couldnt see the truth.
I say rational unit because some sci.mathers would jump up
and down
when I said integer units, as theyd claim that irrationals
could be
integer units...but I digress, after all, sci.mathers are not
exactly, lets say, mathematical.
Now I know that several posters have made claims that Galois
Theory
has something to do with it, but in looking over Galois
Theory, I see
that mathematically, it doesnt.
Which leads me to conclude that certain people have gilded
the lily,
tossed on their own false assumptions on to the mathematics,
and ran
wild, which is how they can use Galois Theory all over the
place.
But theres whats mathematically true, and theres what
people
believe.
Some of you may ask yourselves questions here, while most of
you
probably will not.
Im not interested in most of you. Im trying to see if any
of you
are mathematicians.
Can you work your way through it to the foundations of
mathematics
itself?
Yes, Im talking about the foundations of number, and no, you
cant be
a mathematician, not really, and not get it.
What Ive been doing is presenting certain tests, as I work
through
the details myself, as Im kind of stuck like everyone else
in a world
where people dont quite understand number. Im looking for
people
like me who can get past the errors theyve been taught, and
actually
see numbers themselves, as they are the only people who can
truly be
mathematicians.
None of you may make the cut. I may be waiting for another
generation.
James Harris
===
Subject: Re: Questioning assumptions
> Now I can show how my argument steps out in extreme detail,
and even show
> how it is that posters could nd algebraic integers that
they claim are
> counterexamples to my argument, and show why they are not,
Then do so. Start here:
newssvr13.news.prodigy.com&oe=UTF-8&output=gplain>
Show specically what is wrong with the example there, rather
than just
talking about how you *can* show it.
--
--Tim Smith
===
Subject: Re: Questioning assumptions
Would you like some cheese with that whine?
> Now I can show how my argument steps out in extreme detail,
and even
> show how it is that posters could nd algebraic integers
that they
> claim are counterexamples to my argument, and show why they
are not,
> but it might help to point out a few assumptions other
posters are
> making without giving mathematical proof.
> Like one BIG assumption is that if a number is not a unit
in the ring
> of algebraic integers, then its not meaningfully a unit,
which may
> sound odd, but let me explain how people have made this
issue
> convoluted.
> It was decided in math circles some time ago that algebraic
integers
> fully contained all numbers that had certain key properties
where the
> most important one is the ability to be coprime.
> Like 2 and 3 are coprime in the ring of integers and in the
ring of
> algebraic integers, but not in the eld of complex numbers
as there
> 2(3/2) = 3, so 2 is a factor of 3.
> Thats basic.
> The problem is that part of their assumption then becomes
that a unit,
> is either an algebraic integer, or it is part of a eld, or
some
> ring that has algebraic numbers added to it, like if you
add 1/2 to
> integers, or something like that.
> That is, to put it simply, they assume that either a unit
is an
> algebraic integer, or it is in a ring that contains what
most people
> would call fractions, like 1/2, being in the ring makes 2 a
unit.
> Ive shown that there is a middle way.
> That middle way is to accept that there are other rings
where units
> exist that are not algebraic integers, where there are also
not any
> fractions.
> Ive talked about that before as the ring of objects, where
no
> rational units exist other than -1 or 1.
> Such a simple requirement!!! Yet one that mathematicians
never
> bothered to put forward, partly because they erroneously
believed that
> algebraic integers covered all the bases, but you see, they
werent
> really mathematicians because they couldnt see the truth.
> I say rational unit because some sci.mathers would jump up
and down
> when I said integer units, as theyd claim that irrationals
could be
> integer units...but I digress, after all, sci.mathers are
not
> exactly, lets say, mathematical.
> Now I know that several posters have made claims that
Galois Theory
> has something to do with it, but in looking over Galois
Theory, I see
> that mathematically, it doesnt.
> Which leads me to conclude that certain people have gilded
the lily,
> tossed on their own false assumptions on to the
mathematics, and ran
> wild, which is how they can use Galois Theory all over the
place.
> But theres whats mathematically true, and theres what
people
> believe.
> Some of you may ask yourselves questions here, while most
of you
> probably will not.
> Im not interested in most of you. Im trying to see if any
of you
> are mathematicians.
> Can you work your way through it to the foundations of
mathematics
> itself?
> Yes, Im talking about the foundations of number, and no,
you cant be
> a mathematician, not really, and not get it.
> What Ive been doing is presenting certain tests, as I work
through
> the details myself, as Im kind of stuck like everyone else
in a world
> where people dont quite understand number. Im looking for
people
> like me who can get past the errors theyve been taught,
and actually
> see numbers themselves, as they are the only people who can
truly be
> mathematicians.
> None of you may make the cut. I may be waiting for another
> generation.
> James Harris
===
Subject: Re: Questioning assumptions
> None of you may make the cut. I may be waiting for another
> generation.
Careful there. You might be mistaken for someone who is
arrogant.
-paul
===
Subject: determining the GCF
given a pair of terms such as:
x(x + 3) and 2(x + 3)
what are the steps of determining the GCF?
the GCF is (x + 3) but what is the process to get
from x(x + 3) and 2(x + 3) to (x + 3) ?
I can work out the GCF for any form
except one involving a pair of terms
in distributive form. I also understand
what GCF means and how to nd it, as I have said,
for other forms. But when I get to a form such
as above, that involves a pair of terms
in distributive form, I am unable to make sense
of it and nd the GCF.
Ive tried many variations, for example
x(x + 3) = x^2 + 3x
2(x + 3) = 2x + 6
and then I combined like terms: x^2 + 5x + 6
but to no avail, no matter which way I do it,
I cant nd a process to determine the GCF
which is (x + 3) in the example problem above.
So I want to ll in the gaps.
Currently I have:
x(x + 3) and 2(x + 3)

GCF: (x + 3)
Now what should be substituted with 
that will produce the result (x + 3) as the GCF?
===
Subject: Re: determining the GCF
> given a pair of terms such as:
> x(x + 3) and 2(x + 3)
> what are the steps of determining the GCF?
> the GCF is (x + 3) but what is the process to get
> from x(x + 3) and 2(x + 3) to (x + 3) ?
You should make it clear to the audience whether youre
talking
about gcfs of polynomials or about gcfs of the integers x(x
+ 3)
and 2(x + 3) for various values of x.
As for nding the gcf of polynomials, Ive sketched a method
in a reply
to your other thread upon this topic. Have you looked at that
reply?
> I can work out the GCF for any form
> except one involving a pair of terms
> in distributive form. I also understand
> what GCF means and how to nd it, as I have said,
> for other forms. But when I get to a form such
> as above, that involves a pair of terms
> in distributive form, I am unable to make sense
> of it and nd the GCF.
> Ive tried many variations, for example
> x(x + 3) = x^2 + 3x
> 2(x + 3) = 2x + 6
> and then I combined like terms: x^2 + 5x + 6
> but to no avail, no matter which way I do it,
> I cant nd a process to determine the GCF
> which is (x + 3) in the example problem above.
> So I want to ll in the gaps.
> Currently I have:
> x(x + 3) and 2(x + 3)
> 
> GCF: (x + 3)
> Now what should be substituted with 
> that will produce the result (x + 3) as the GCF?
===
Subject: Prime ideals of C[x]?
X-RFC2646:  Original
Let C = complex numbers. Please excuse my ignorance...Im
trying to
understand the prime ideals (let alone ideals) of C[x] /
(x^m) where m >=2
I can see that C[x] / (x^m) is isomorphic to just polynomials
of degree at
most m-1 with coefcients in C. But what are ideals of this
ring? What
are prime ideals in this ring?
Related to this is C[x] has a prime ideal I know, namely (x),
since C is an
integral domain. Is that the only prime ideal of C[x]?
So Im also trying to understand ideals in C[x]. For
instance, is there
anything different between (x^2) and (x^2+x^4)? Im not sure.
Firstly,
x^2+x^4 = x^2(x^2+1), so every element in the second ideal
can be written as
an element in the rst ideal, so clearly (x^2+x^4) is
contained in (x^2).
However, (x^2) I am not sure if it is contained in (x^2+x^4).
In
particular, x^2 isnt in (x^2+x^4), right? But if this is the
case, what is
the difference between C[x] / (x^2) and C[x] / (x^2+x^4) ??
Finally, are there any other prime ideals of C[x] other than
(x)?
Isaac
===
Subject: Re: Prime ideals of C[x]?
|I can see that C[x] / (x^m) is isomorphic to just
polynomials of degree at
|most m-1 with coefcients in C.
Well, thats a little imprecise. The polynomials of degree at
most m-1
arent closed under multiplication. Each element of
C[x]/(x^m) is a
set of the form P+(x^m). Each such set has a representative
thats
a polynomial of degree <=m-1 (subtract off the terms in P of
degree
>=m).
| But what are ideals of this ring? What
|are prime ideals in this ring?
Start with a simple special case. What are the ideals of
C[x]/(x^2)
or C[x]/(x^3)? The set {0} is an ideal of C[x]/(x^2), more
often
written as (0). One hint I can give you is that a lot of the
elements
of each of these are units. The ring generated by a single
polynomial
can be simplied by multiplying the polynomial by another one
that
makes the result (reduced mod x^2 or x^3 and so on) simple.
Remember the denition of prime ideal. If a product of two
elements
ab is in the ideal, then one of a or b is in the ideal. By
induction, if
some product a1...an is in the ideal, one of the a_i is.
Notice that
each ideal in C[x]/(x^n) contains 0 and in C[x]/(x^n) we have
x^n=0.
|Related to this is C[x] has a prime ideal I know, namely
(x), since C is an
|integral domain. Is that the only prime ideal of C[x]?
No, (x-1) is another prime ideal.
|So Im also trying to understand ideals in C[x]. For
instance, is there
|anything different between (x^2) and (x^2+x^4)? Im not
sure. Firstly,
|x^2+x^4 = x^2(x^2+1), so every element in the second ideal
can be written
as
|an element in the rst ideal, so clearly (x^2+x^4) is
contained in (x^2).
|However, (x^2) I am not sure if it is contained in
(x^2+x^4). In
|particular, x^2 isnt in (x^2+x^4), right?
Yes. Remember the denition of (a) where a is an element of a
ring.
Its the set of elements of the form ax where x is an element
of the
ring. (Im assuming your rings are rings with 1.) So the
question is
whether there exists a polynomial P(x) with coefcients in C,
such
that x^2=(x^2+x^4)*P.
|But if this is the case, what is
|the difference between C[x] / (x^2) and C[x] / (x^2+x^4) ??
For one thing, the rst ring is a two-dimensional vector space
over C, and the second ring is a four-dimensional vector
space over C.
Do you know what the rings Z/(m) are like, where m is an
integer? Did you know that they can sometimes be written
as products of simpler rings, e.g. Z/(6) is isomorphic to
Z/(2) x Z/(3) by the function 1->(1,1). This works out
smoothly
because the ideals in Z are all of the form (m) and Z has
unique factorization. The situation in C[x] is parallel, and
is well worth understanding well.
Keith Ramsay
===
Subject: Re: Prime ideals of C[x]?
: Let C = complex numbers. Please excuse my ignorance...Im
trying to
: understand the prime ideals (let alone ideals) of C[x] /
(x^m) where m
>=2
: I can see that C[x] / (x^m) is isomorphic to just
polynomials of degree at
: most m-1 with coefcients in C. But what are ideals of this
ring? What
: are prime ideals in this ring?
: Related to this is C[x] has a prime ideal I know, namely
(x), since C is
an
: integral domain. Is that the only prime ideal of C[x]?
There is also (x-a) for any complex number a.
To nd the prime ideals of C[x]/(x^m), recall the
correspondence
between ideals of a ring R and ideals of quotients of R.
: So Im also trying to understand ideals in C[x]. For
instance, is there
: anything different between (x^2) and (x^2+x^4)? Im not
sure. Firstly,
: x^2+x^4 = x^2(x^2+1), so every element in the second ideal
can be written
as
: an element in the rst ideal, so clearly (x^2+x^4) is
contained in (x^2).
: However, (x^2) I am not sure if it is contained in
(x^2+x^4). In
: particular, x^2 isnt in (x^2+x^4), right?
Well, (x^2+x^4) contains all elements of the form
(x^2+x^4)*p(x). Is x^2
such an element?
Ted
===
Subject: Re: Analytic Number Theory and A Problem of Geometry
anonymous@mathforum.org
(A.B.Smack.them.out!Next.time@mathforum.org,
> Could you see the following page
> http://www.cs.umb.edu/~asi/analysis2000/papers/montgomery
> _problems.pdf
> by Hugh L. Montgomery and Ulrike M.A. Vorhauer? Its title is
> Some Unsolved Problems in Harmonic Analysis as found in
> Analytic Number Theory. In their list, the authors put the
> equichordal point problem. I have been trying to nd out
information
> on relationship between the equichordal point problem and
number theory in
> Google, but can not nd any, other than this. Would
> anyone know about this?
Hugh Montgomery published a book, Ten Lectures on the
Interface
Between Analytic Number Theory and Harmonic Analysis, as #84
in
the CBMS Regional Conference Series in Mathematics, published
by
the American Mathematical Society. The last chapter is
Appendix:
Some Unsolved Problems. The last section of this chapter is
16. Miscellaneous. Problem 72 is the equichordal point
problem.
Its not at all clear to me that it is related to Number
Theory,
or that Montgomery is claiming it is related to Number Theory.
He gives three references. If there is a relation to Number
Theory,
I bet its in one or another of these references. These are
Michelacci and Volcic, Arch Math 55 (1990) 599-609; Schafke
and
Volkmer, J Reine Angew Math 425 (1992) 9-60, and Wirsing,
Arch Math
9 (1958) 300-307. Wirsing is well-known for his work in
Number Theory.
--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
===
Subject: Re: Methods that count primes without counting
primes or referring
to them...
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9C3OlX08620;
it looks like it should cover much of what Im interested in.
You are right, of course, that that expression of pi(n)
satises the
criteria I listed, and its denitely mildly interesting at
best
without having any explanatory power or particular point. Id
actually come across that formula previously on mathworld, in
fact.
Its hard not to feel like that formulas cheating just a bit,
though :) Because [ ((j - 1)! + 1) / j ] - [ (j - 1)! / j ] is
ultimately just an extremely curious equivalent to writing [
is_prime
[ j ] ] (assuming Im reading that correctly), it doesnt
seem to
shed any light on the nature of primes or their distribution
at all.
Hmm. Well, hopefully Cradalls book will answer what Im
looking
>No.
>Heres a formula from p. 19 of (the highly recommended book)
Crandall
>and Pomerance, Prime Numbers:
>pi(n) = sum from 2 to n of [ ((j - 1)! + 1) / j ] - [ (j -
1)! / j ],
>where [x] is the greatest integer not exceeding x.
>I think this formula meets your criteria, it is not a
noteworthy
>mathematical achievement, and it has probability 1.
>More examples are given in the books exercises.
>--
>Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
===
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===
Subject: Easiest way to discover differential forms from rst
principles?
Whats the most elegant (read: using 18th century
mathematics) proof
for the anticommutative property for differentials: dx dy =
-dy dx. I
have a pretty good idea involving transforming the plane from
x-y to
y-x coordinates and using the reversed orientation as proof
of this,
but its hardly rigorous.
Any ideas?
===
Subject: just wondering
Math have come to a conclusion, that from n=n+1 in Peanos
axiom follow
that
N is innity allthough that n=n+1 grows up as much bigness as
amount of
numbers.
Math has put to set of Natural numbers something that does
not comes from
Peanos axiom. Contrast.
Because everything is relative you can allways multiple the
amount by 2.
The more far away you are the more you have to go. Innity is
nonsense
concept.
Of course, im not saying that N is nite in a way that there
is biggest
number.
Just N->oo but not N=oo.
--
amount and bigness
1+1+1+...= innite and nite in math today
Petri Keckman
===
Subject: before Cantor
Today, speaking of mathematical objects that exist
but cannot possibly be dened is accepted by a large
majority of mathematicians; also existence proofs
that I know of for Hamel basises and non-measurable subsets
of the reals dont really dene them.
Before Cantor, did anyone speak of objects (not variables...)
that were not dened in the usual sense?
( I might want to go over the proof of the Riemann mapping
theorem in complex analysis.)
David Bernier
===
Subject: Many algebra questions
posting-account=4XHZpAwAAAB6b7qV0WqNfmj98QBHBa2z
I am just going to put this out there to see if there is
anything
someone can help me with. Basically, I am an undergraduate
taking a
graduate algebra class, and everything was going smoothly
until the
last 2 classes. I have no problem working at a highly
theoretical
level, but the professor just assumes just a little too much
(I am not
the only behind, and I hope to help others if I understand).
1) The rst problem is the easiest (in the sense that I
actually
understand it). Im looking to describe the Sylow subgroups of
GL(2,Z_5). I know the order: o(GL(2,Z_5))=480=(2^5)*3*5 .
I found some 5-groups, namely matrices of the form ((1 a)(0
1)), where
(1 a) is the rst row, and of course there are 6 of these.
Could there
be more?... I am not sure.
Next I found some 8-groups, matrices of the form ((a 0)(b
0)), and
there are 15 of these. Not sure if there can be more.
Finally, I need some 3-groups. These seem to be quite
difcult because
they are going to have nonzero values in all entries. Somehow
my
professor thought the hint H_3subset Z_5subset F_25* would
help
(where H_3 is a 3-subgroup, F_25 is the eld of 25 elements,
and F_25*
is the multiplicative group i.e. nonzero elements).
Anyone have any advice?
2) Now start the real problems, most of which I do not
understand their
statement. How many nonisomorphic groups are there with the
decomposition series Z_5/Z_7/Z_5 ? Z_7/Z_3/Z_2 . I put that in
quotations because that is exactly as it appears (with the
question
mark), and I do not understand his notation. Can we really
have a
composition series of the form:
1 --> Z_5 --> Z_7 --> Z_5 --> G ?
I do not see how that could begin to make sense.
3) Show that S_5=Aut A_5.
4) Show that an extension H=A_5subset G has a section if
G/A_5 is
cyclic group of odd order. Considering I was never told what a
section is, this problem seems hopeless. I could not nd that
term
an any literature, on/off-line. Does anyone have a denition?
5) I know that the rotation group of the cube is S_4 (not
sure how to
prove it though), and from a problem I worked by hand a while
ago, I
know that V_4 and A_4 are the only normal subgroups. So is the
following:
1 --> V_4 --> A_4 --> S_4
The desired extension?
6) I know, you probably hate me by now. What is the
automorphism group
of Aut(Z_p + Z_{p^2}). Hint: Use the associated ltration of
Z_p+Z_{p^2}-subgroup of elements of order p. I have never
heard my
professor use the word ltration, so not sure what this means.
That is pretty much it... I know, it is quite ridiculous. Any
hints or
help? Please take note that none of this is for grades... we
do not
even have homework grades, only exams. I just started feeling
lost
suddenly when previously I was thinking this would be my
easiest class.
Also possibly books that cover these advanced topics? The
ofcial book
for this class is Artins Algebra, but it does not cover any
of this
material.
===
Subject: Re: no comments?
> Well, so far youve failed to prove much of anything, and
there is
> a bijection: N <-> 2N is perfectly reasonable, and
establishes,
> by your logic, that N is innite, at least in the
theoretical
> sense.
No. It is innite by your logic.
Math have come to a conclusion, that from n=n+1 in Peanos
axiom follow
that
N is innity allthough that n=n+1 grows up as much bigness as
amount of
numbers.
Math has put to Natural numbers something that does not
comes from
Peanos axiom.
Because everything is relative you can allways multiple the
amount by 2.
The more far away you are the more you have to go.
Of course, im not saying that N is nite in a way that there
is biggest
number.
Just N->oo but not N=oo.
--
amount and bigness
1+1+1+...= innite and nite in math today
Petri Keckman
===
Subject: Re: no comments?
> n(N)=oo.
> I just cant believe that person who has learn so much
about math
> Like Will Twentyman seems can not follow this simpple proof
that
> there is contradiction saying that N has innite member and
> they are all nite.
> *************
> Talking about natural numbers:
> There must be number atleast 6 in set that has 6 item.
> 6 {1,2,3,4,5,6}
> Could be bigger of course
> 8 {1,2,3,4,5,8}
> but atleast there must be 6.
> There must number atleast n in set that has n item.
> n {1,2,3,...,n}
> Could be bigger but atleast there must be number that is n.
> So what number _atleast_ should be in set that has oo item?
> Atleast there should be number that is oo.
> *************
You are making a false assumption: namely that there must be
such a
number. You are claiming to nd something that simply does
not exist.
You can argue it, but your argument is wrong.
--
Will Twentyman
email: wtwentyman at copper dot net
===
Subject: Re: no comments?
> Because the Amount of Naturals is not *approaching*
anything. That
> would assume that it is somehow changing. n(N)=oo.
> We have to be satised for that. In every actual situation
we use set
> Naturals in computers, in our minds or whatever, it has to
be stopped
> somewhere
You are thinking about applications. That is not the same as
the
theoretical object N. There is a difference between
application and
theory.
> Algorithm have to stop.
Actually, algorithms dont have to stop. If you want bigger
headaches
you can look up the Halting Problem and Turing Machines.
> Amount=1
> loop
> Amount=Amount*n
> forever
> Where n is any n in N.
> It is so hopeless even try to catcj the innite. Do you
hear INFINITE!
> Not just something that you are used to use and manipulate
handly in
> mathemathics marking it oo and so on, but I N F I N I T E.
We arent trying to *reach* the innite, we are asserting
that the
number of numbers in N *is* innite. This is the distinction
you keep
missing. You cannot *construct* N in nite time. You can
*dene* it
and determine its properties in nite time, however. One of
those
properties is that it has innitely many numbers in it.
> Maybe not lately, but as a child, you perhaps has tried to
think innite
> big. And no matter how big ig big you it is allways the
same as it have
> the size of set {1}. We can not take the rst step an a way
to innite.
I can easily think of a set {1,2}, which is larger than {1}.
I cannot
achieve the thought of innity. I dont have to to work with
it.
> Same as in limit(1+1/2+1/4^2+...1/n^2) is just a limit.
> It never actually si the same.
> You are talking about two completely different elds of
mathematics.
> No no. Zeno Akhilles. Finite sums of innite many litlle
numbers.
> They are just the same inte we are handling here
everywhere.
But notice you are claiming to have an innite number of
non-zero
numbers. Then looking at the inverses gives you innitely many
different nite numbers.
> Limits do not apply to the cardinality of sets. However,
innite
> sums are *dened* to be the limit of nite sums.
> And there must be innite of those sums in 1+1/2+1/4+1/8...
> Not nite.
But notice, each thing you are adding is nite. Yet you claim
you are
adding innitely many of them. Do you think any of them is
equal to 0?
--
Will Twentyman
email: wtwentyman at copper dot net
===
Subject: Re: no comments?
...
> They can been formalized and let you see that they are
clear.
> But you seems not to understand what presumptions there are
behind
> those formalizations. The one is that N is innite. If it
is not,
> then nothing is.
When N is nite, there is a last element. What is the
successor of
that last element?
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam,
nederland,
+31205924131
home: bovenover 215, 1025 jn amsterdam, nederland;
http://www.cwi.nl/~dik/
===
Subject: Re: no comments?
> Heres what you dont seem to understand: *innity is not
a natural
> number*.
> I agree. it is not. It is nonsense too what comes to the
amount of n in
N.
> You do not seem to understand what i have tryied to say.
Perhaps you are saying N must either have nitely many
elements or
include innity to have an innite number of elements.
How many fractions are between 0 and 1? Are any of them
innite?
Note: for the rst question, I am looking for an integer or
innity.
If you say nite, tell me which nite value.
--
Will Twentyman
email: wtwentyman at copper dot net
===
Subject: Re: Average Length of a line
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9C5q6020558;
>Hi;
> Suppose we have an isosceles triangle, with the base
angles equaling 30
degrees and a height of 1. Naturally the apex angle is equal
to 120 degrees.
If I draw a random line from the apex of the triangle to the
base what is the
average length of that line? Now obviously the minimum length
is equal to 1
and the maximum length is one of the sides of the triangle
which is 2. I get
an average value of the line of about 1.25. I do not know how
to get the
exact analytical answer. Can someone help?
>Angela
>First you need to know the probability density function of
angles or
>of termination points on the baseline.
>phil
Hi Phil;
Yes you are right I should have stated that all the lines are
equally
likely. It seems that correctly stating a problem is just as
difcult for me
as solving it.
Angela
===
Subject: How tall is this tree?
This question is about Cantors transnite theory. I want
to know the height of a binary tree, given that it contains a
countably innite number of branches (nodes). Each branch is
located one foot above the previous branch. (see diagram)
Height tree branch #
------------------------------------------------
??? oo
...
... / / / / / / / /
4 ft 8 9 10 11 12 13 14 15
/ / / /
3 ft 4 5 6 7
/ /
2 ft 2 3
/
1 ft 1
| |
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= ground level
So, how high is the tree that contains a countably innite
number of branches?
===
Subject: rescaling list of numbers
This is a very simple question.
Suppose I have a list of numbers, say a_1,..., a_n and I want
to
rescale them to sum to one. The obvious thing to do is divide
by the
sum. However, if this is large, the sum of the resulting
numbers may
not be one, due to rounding error. The question is, what is
the best
way to correct for this. I can use various ad-hoc methods,
but am
wondering if numerical analysis has a `best practices
approach.