7593 Subject: Re: 100 free SBC; VonNeumann Gametheory how to play StockMarket > Well, today I gained 100 free shares of SBC plus some cash extras. Fabbo, Mr. P., congratulations! Not to bad mouth the method, only to follow-up on prior posts: as of market close Friday Aug. 15 (such as it was) solar pee-vee peddler AstroPower closed down (yet another) 39% at $1.61, while solar pee-vee peddler zoomed ahead (another $14%) to close at $5.00. If there were somehow a way to have distinguised one from the other on those two days not so long ago when they closed at exactly the same price and the same change in price, oh how happy we would be to collect our 5/1.61 = 3.11x shares of AstroPower! (or is it 2.72x?) Anyway, who would want them? As you rightly pointed out before (to paraphrase): silly-con chips are NOT Blue Chips! -dl Subject: Re: 100 free SBC; VonNeumann Gametheory how to play StockMarket > Well, today I gained 100 free shares of SBC plus some cash extras. > Fabbo, Mr. P., congratulations! <... > solar pee-vee peddler zoomed ahead (another $14%) ^^^^ erps... Spire Corp. -dl Subject: Re: About Russell's first paradox |Consider the set M to be The set of all sets that do not contain |themselves as members. Formally: A is an element of M if and only |if A is not an element of A. In the sense of Cantor, M is a |well-defined set. No, from Cantor's point of view there are multiplicities that are not possible to gather together as sets. Frege's system was the one that foundered on the paradox. Keith Ramsay Subject: Re: About Russell's first paradox rats; I'll have to cut my hair, tomorrow. > No, from Cantor's point of view there are multiplicities that are > not possible to gather together as sets. Frege's system was > the one that foundered on the paradox. --UN HYDROGEN (sic; Methanex (TM) reformanteurs) ECONOMIE?... La Troi Phases d'Exploitation de la Protocols des Grises de Kyoto: (FOSSILISATION [McCainanites?] (TM/sic))/ BORE/GUSH/NADIR @ http://www.tarpley.net/aobook.htm. Http://www.tarpley.net/bushb.htm (content partiale, below): 17 -- L'ATTEMPTER de COUP D'ETAT, 3/30/81 23 -- Le FIN d'HISTOIRE 24 -- L'ORDEUR du MONDE NOUVEAU 25 -- THYROID STORK !?! Subject: Re: a comment on searching for 3 orthogonal 10x10 squares > While trying to find 3 such orthogonal squares, I used over 300 sets of duals, > but the best I could do was fit 26 squares of the 3rd layer into a > tri-orthogonal arrangment before the problem exhausted the possibilities. > Most of the duals were found on the net, as James Buddenhagen pointed out, but > transforming the row/columns/diagonals easily led to other pairs. Wendy Myrvold and I have been searching on and off for an orthogonal triple for a few years. Our code can test 2-3 squares per second as to whether they lie in a triple but the number of squares is much too large to allow testing of all of them. Instead we have focussed on the squares with non-trivial symmetries (autotopisms and autoparatopism). So far we checked all squares with a symmetry group of order 3 or more, and are part way through testing the squares with symmetry group of order 2. We should be finished later this year. No triples so far, alas. Brendan McKay. Subject: An equation to match a graph I need to create an equation (y=f(x)) that has a certain shape and qualities. I know exactly how I want it to look but can't figure out a good equation. It should satisfy: * Differentiable (Has first derivative at every point) * Slopes (think of y=x^2): slope(0)=0 slope(x > 0) > 0 (never = 0) slope(x < 0) < 0 (never = 0) *The derivative near the points where x is an ODD INTEGER should be far larger (in the absolute value sense of the word) than the derivative near the points where x is an EVEN INTEGER. The definition of near is flexible. A picture in this case is worth a 1000 words, but it doesn't seem appropriate to attach one; if anyone wants me to send them a graph (hand-drawn) of this function just tell me. I'll try to describe the graph: It should look like y = x^2, but twisted so that points near x=odd integer are steep, and points nearest x=even integer are ALMOST horizontal, with the values in-between sloping gently from steep to almost-horizontal. Note that these restrictions don't force a single equation; there are probably lots of possibilities which are all good. In particular, the scale is meaningless. Since this equation is going to be called millions of times in a tight loop, it should (as a secondary objective) be computationally inexpensive. The function can have some parameters, such as: T- By how much steeper (i.e. 8) a point near an ODD is, than a point in an EVEN. Z- The distance (i.e. 0.4) from an ODD integer in which the y is multiplied by T. I was thinking something along the lines of: if x is within Z from an ODD: y= T* (some-trig-function(x)) * x^2 else: y= (some-trig-function(x)) * x^2 Daniel Subject: Re: Any idea for a math career? > I think that since mathematicians (and logicians) deal with abstract > objects (which can represent pretty much anything, so long as an > axiomatic description can be formulated), they should be the most > adept at 'thinking outside the box.' They should but life doesn't work this way in my experience as well. It took me a couple of years to get what was necessary for the so-called real world solutions. Unfortunately, this is not taught in math departments (there is no time). With physics it's a bit different - they don't have the time to teach real world approaches either but at least the subject itself has certain art of compromise built right into it which is noticeably lacking in mathematics (for better or worse). > Look at the works of Emmy > Noether on symettry in physical systems or Kurt Goedel's works. I know but this is all ivory tower compared to what most company executives need and expect. Goedel is actually a good example: his personal habits and life were extremely incompetent in that sense. > If thinking outside the box is equivalent to uncritical and > unrigorous, a plain-jane english lit major is the ideal candidate. No, that's not at all what it is. I used to laugh at it when I was a student but later realized there was a non-trivial wisdom there. Jan Bielawski Subject: Re: Any idea for a math career? > I think that since mathematicians (and logicians) deal with abstract > objects (which can represent pretty much anything, so long as an > axiomatic description can be formulated), they should be the most > adept at 'thinking outside the box.' > They should but life doesn't work this way in my experience as well. > It took me a couple of years to get what was necessary for the > so-called real world solutions. Unfortunately, this is not taught in > math departments (there is no time). With physics it's a bit different > - they don't have the time to teach real world approaches either but > at least the subject itself has certain art of compromise built > right into it which is noticeably lacking in mathematics (for better > or worse). > Look at the works of Emmy > Noether on symettry in physical systems or Kurt Goedel's works. > I know but this is all ivory tower compared to what most company > executives need and expect. Goedel is actually a good example: his > personal habits and life were extremely incompetent in that sense. > If thinking outside the box is equivalent to uncritical and > unrigorous, a plain-jane english lit major is the ideal candidate. > No, that's not at all what it is. I used to laugh at it when I was a > student but later realized there was a non-trivial wisdom there. care to elaborate? I liken myself to The Pretender on the TV series, the man who can become anyone he wants to be, and he's a doctor one week, a racing car driver the next and sells hotdogs the day after that... after spending all his childhood in isolation and training. Since Application Developers get to be involved in virtually any industry, at small scale jobs you often end up testing the software on yourself as the employee, so I've had a lot a variety in work. The one thing I pin down to learning the 'real world' skills in a new industry is what I call the 'office politics'. Essentially this is what the company fills in the staff members in the know what information doesn't go around! Sounds simple having pinned it down but its usually 3 months of induction and taking on broader tasks before the politics get revealed. Usually related to avoiding legal dillemas or a sales technique or edging away competition but every industry has it! Herc Subject: Re: Any idea for a math career? Leigh Ann J > I'm just finishing up my PhD in mathematics (focus on stochastic > calculus) and I am looking at possible career paths. I've always had > some interest in economics, and actually taught a first year financial > mathematics course at university. Not knowing too much about math > careers in finance, does anyone here have any ideas? Consider the military. In my experience, ability can make its way in the forces. And the modern forces, including Australia's, have many different things going on at any one time. Much depends on the person, and a bit depends on the specific company or institution, not just the type of institution. Personally I'm rather sensitive to institutional climate. Banks and the like are much too regimented for me to be happy in one. Academia has its office politics, to be sure. I struggle by in a private sector niche; the money is marginal but I like the work and there and lots of smart and interesting people around. > Also, I seem to notice (in Australia at least) that math grads are > often pidgeon-holed by business recruiters as being too narrow minded, > not being able to think outside the square, good at quant analysis but > not so good with qualitative analysis, etc, etc. A recuiter who is a > personal friend has said that this is what her company has discovered > after some hiring some math grads. Does anyone here have anything to > say in regards to whether you think this is a fair/unfair generalisation? Pretty fair, IMO. Mathematicians are perfectionistic, at least in my experience. Each one is accustomed to being the smartest guy in the room, and they have some of their own habits of thinking and communicating. That doesn't mix well with the aims and protocols of business, generally speaking. Another poster talks about money. Academics don't die of malnutrition any more, but I have yet to meet a rich mathematician. (Wolfram is not a mathematician in my books.) People who are keen enough on math to get a Ph.D. in it are not the type to become rich, it seems to me. Most of the rich people I know are pretty dumb. I kid you not. As a rule they talk about nothing but their money and their illnesses, just as Ian Fleming said long ago. Larry Subject: Re: Are all mathematicians music lovers? > I don't think that has anything to do with arts or music. Correct me if I'm > wrong, but didn't these temples contain representations of God? If I am not > mistaken, representation of God in a pictural form is not allowed by Islam. Mohammad hated all things Vedic, thus trashed the statues. He was illiterate and of low social origin, thus detested anything that was associated with the Vedic elite of the time. Many Muslim rituals (e.g. going around the circle in a clockwise direction) are mirror images of Vedic rituals. > Conservative Muslims today refrain from listening to music. > I don't know where you hold this fact from. In fact, Muslim prayers are quite > melodic. It almost seems they are sung. The Taliban outlawed music. Afghan deserts are littered with bushes that have long ribbons waving in the breeze, because of music tapes confiscated from drivers and destroyed. There are also several African Muslim countries that outlaw music. Music is thought to lead souls astray into the wrong desires. Apparently, muezzin prayers are exempt because it is the message of God. The church of Medieval Europe also had similar attitudes concerning music; music had to be sanctioned by the church, or played in secret among aristocrats. Subject: Re: Are all mathematicians music lovers? > It seems every math/science type I meet is a Bach lover, having fallen for > the propaganda that music expands the mind. Actually, there are people > who go crazy from repetitive tunes that won't stop inside their head. > I've > been able to drive otherwise calm-and-collected math/science types berserk > by saying music is stupid. Are there any mathematicians today who have > the > courage to say music is stupid? > Define music. Axioms are not necessary. Music is that which has exactly zero to do with either Goedel, Escher, or Bach. Since none of them where musicians. Bach was an instrumentalist. > Norm Subject: Re: Are all mathematicians music lovers? on Sunday 17 >> I don't think that has anything to do with arts or music. Correct me if > I'm >> wrong, but didn't these temples contain representations of God? If I am > not >> mistaken, representation of God in a pictural form is not allowed by > Islam. > Mohammad hated all things Vedic, thus trashed the statues. He was > illiterate and of low social origin, thus detested anything that was > associated with the Vedic elite of the time. Many Muslim rituals (e.g. > going around the circle in a clockwise direction) are mirror images of Vedic > rituals. > Conservative Muslims today refrain from listening to music. >> I don't know where you hold this fact from. In fact, Muslim prayers are > quite >> melodic. It almost seems they are sung. > The Taliban outlawed music. Talibans are not a reference among muslims, not even among conservative ones; their vision of Islam is very crooked. > The church of Medieval Europe also had similar attitudes concerning > music; music had to be sanctioned by the church, or played in secret among > aristocrats. Apart from the obvious gregorian music (which is religious), troubadours were common and accepted. So I don't see what you are talking about; some music, though, was thought to be of evil origin; I recall Niccolo Paganini being accused of having sold his soul to the devil because of the effect his music had on some people, especially women (there are stories of faintings and such). But that's not extendable to all kinds of music; not even all kinds except religious. What you're saying is very probably to some extent true, but I don't think we can generalize to all kinds of music. Sam -- Don't be afraid, I'm gonna give you the choice I never had... - Lestat in Interview with the Vampire (Ann Rice, 1976) Subject: Re: Are all mathematicians music lovers? 7e3Z2dkQEBC5ubm60r3HAAACeUlEQVR4nIXTQW+ bMBQAYChLfcWzUq5pipUr7lPptUNGXBMC8zUZ qnt1aNz392c7pGXSpvkSyV/ee37POFr/Y0X/g56rxd+ gb2IQC8X79brdLtovaBgACIsRMVTr4hN4 Ah54xy0rtdbjFRCAupizIhUDB69X2IHAGKCrxyrxcJyA76CQKwZn2ckEXK6h/ YwoOld/zJUMEYds ghMIYhj0kjehxqGYFacgalk1PpM+jBOkvg0othFR4bjDVHyjQx+FZ+ b29fMEe114SIC6UB/RXoCX enAhGQORAdxp/TaN5N63lNqGJZACO2n9/ QL81qfVQjEXAkBKPyoPm7BfPqFLJ0qBt1fwmfTAFiYW MbAC1WmCnyEitcYImtI07fY3FwglBuwMUiqoiLv98gL+ 4PrNSmXdWGJK7Q4D8JBpWeXGxi6CpqOJ QoOrkOksKzMaRinN7LsI8OChsGOl+ogiE8SeaIAP9/ 9BRIs1xzgzESWteQ7gDgWQpmRdmcgtNFYJ / 5W4CWpRRsTyHE2UBgifjzvUQMGMOfYYO1CI7wFWfhwsWXSNq0qQcMQkgBvhADvI rOrduRArpW7D Rflb0gjHvO5kpUhX93h3DJ3v9VB+S8pMulW1kis73aAqh/ LpBYalbA2qiLZKP16G+KHLowIYojSm UTys+/ LrogaFVFTKpN2P45qT6UX58qrbEFcil81h9tQSfVSyIZ0v3zzNYEMPVtbR2e3XL 8v547w/ uDymk7wfXx7n8PBscrm1fW/G1a8/gKiz3GJeYbt5ncMJ+ Y1ckVxi19zMwaIcXXUreb6yc1gQ9ypr tLKSCueQjQ4qxFzKXo0zwNDbxv/U2ynkN+Dj/u9oA6vhAAAAAElFTkSuQmCC > Outside of the Western world, there are countries where people have > more freedom to say they don't like music, without risk of social > ostracism. Poor soul. You come into a mathematics newsgroup and proclaim music is stoopid and, for this, you are ostracized. Oh, the humanity. -- Sorry, wakeup to the real world. You're on your own dependent on me as your guide. Luckily for you, I'm self-correcting to a large extent, so if the proof were wrong, I'd tell you. It's not wrong. --- James Harris confirms that his proof is correct. Subject: Re: Are all mathematicians music lovers? > It seems every math/science type I meet is a Bach lover, having fallen for > the propaganda that music expands the mind. Actually, there are people > who go crazy from repetitive tunes that won't stop inside their head. I've > been able to drive otherwise calm-and-collected math/science types berserk > by saying music is stupid. Are there any mathematicians today who have the > courage to say music is stupid? I've known quite a number of non-mathematicians that you could probably drive berserk by saying music is stupid. And they're not trying to expand their minds, they just want some tunes. Subject: Re: Armand Borel dead >What exactly do you have against Serre? >Your animosity appears to me to be totally irrational. Ah! But is it infinite, too? Lee Rudolph Subject: Re: Armand Borel dead > I thought the idea was that free speech admits one to speak out about the > things that one considers indecent or wrong. I'm very tolerant about people's opinion on how good other people's mathematics are. But you've crossed the line between the mathematics and the people themselves: inserting Finally between Borel and dead, and comparing Armand Borel's demise to that of Idi Amin Dada?!? COME ON!!! So I'm doing exactly what you say above, i.e. I'm telling you that what you've done is indecent and wrong. It really tells us more about Hans Aberg than about either of these two... (That's *my* bit of free speech.) Besides, I note that you give an e-mail address and a home page affiliated with the Math Department of Stockholm University. OTOH, you're not listed among the faculty on that departments Web site. Anyway, a glance at their homepage shows that all that crap (free speech, right?) you've been telling us about Serre lately isn't exactly their official view... Hugo Subject: Re: Armand Borel dead >> I found the algebra and commutative algebra volumes useful, but the later >> homological algebra I found pointless. Also, the original idea of >> Bourbaki, which I take it was to bring logical accuracy to some selected >> messy early twentieth century mathematical fields, but continue beyond >> that seems pointless. >Who says that was the object of the game? I have a vague memory that some of the original members described how Bourbaki came about. It was mainly an Andre Weil idea that drove it. >I've always assumed the aim was to produce a complete logical account >of the central themes of pure mathematics. The motivation would be to develop that core so that it would be accessible in a clear way. This would explain the sometimes strange-looking Bourbaki generality. Clearly, Bourbaki itself influenced the development of math into greater clarity and generality, making it becoming unneeded. >The only part of Bourbaki that seems to fit your description >is the part on Set Theory, >which has always struck me as the weakest part of all. I also have vague memory of that the set theory was a weak part, even though it is said to have had some influence on the notation of set theory. If you go into books on metamathematics, like those by Kleene and Mendelson, their axiomatic set theory looks very different from what most mathematicians use today. But it would explain why this group ventured into that topic, as axiomatic set theory is a proper foundation for math such as the algebra they were perhaps better expert at. >Surely there were no logical inaccuracies in algebra >(or commutative algebra) by the time Bourbaki got to work ? There always has and still is a gap between the perceived logical accuracy of the written math and the formal math. That would be perceived as a lack of clarity in the description. For example, some of Andre Weil's very own writings on algebraic geometry are very difficult for modern mathematicians to read and ensure accuracy of, as concepts like generic are not formally defined as in say in Grothendieck schemes. That problem still exists today: For example, Mendelson in his book on metamathematics gives English, not mathematical, definitions of concepts such as definition and metaproof. >In any case, you haven't explained why you attack Serre >for his supposed contribution to Bourbaki, >when he has produced a vast number of works >which would seem a more logical target for your criticism -- >criticism which I for one find completely incomprehensible. I was merely commenting on the NYT description, which at the time seemed to be inaccurate. >What exactly do you have against Serre? >Your animosity appears to me to be totally irrational. Serre is mainly a math celebrity and not a mathematician that by his work shows the way into the future for science and mankind, and that is negative when it takes over the natural plurality governed by individual choice that should exist in science. Such a plurality and individual choice will, if allowed to come forth, provide proper the scientific developments that mankind will need. Serre is a traditionalist that defines in a narrow way tradition to mean his own interests, whatever they may be at a given time. Serre, for example, does not honor the ethical principles of mathematical publishing, but refuses to referee papers that are not within his current, temporary, interests, stating that the submitted paper would not be accepted even if demanding a proper refereeing (which I say, because I have a copy of such a letter, and also know of more copies). One also explained to me some of the methods in use in order to ensure that submitted papers would not become published if the editor does not want it. In other words, Serre uses his editorship as a form of payment to forward his self-interest. He was not the only guy in the field doing that; the practise seemed to be widespread in that circuit. When I was in Princeton, and Serre's name was brought up, Andre Weil looked as though he felt this guy Serre was not the great genius as his cohorts like to think. Such events caused me to think more about the quality of Serre's math. For example, Serre does not make conjectures as normal mathematicians do, but instead raises questions. One example I encountered at close range was about the rationality of Poincare series. The idea was that this should be an analogue of the Andre Weil's conjecture. Intuitively, such a conjecture could not possibly be true, as Weil's conjectures hinges on deep properties of number fields. However, despite officially merely being a question, Serre pushed a group of people into working on it (for several years), not even considering that it might be false. Eventually, it was disproved by a Berkeley student, Anick, who incidentally considered the opposite. When angry letter, demanding a correction. Clearly, there is no proper metamathematical definition of a question, but only of conjectures. So one must ask oneself why a mathematician would do such a thing. The answer can only be that Serre does not want to loose prestige, and for that reason goes as far as redefining the scientific principles that everybody else is using. In politics, such practises of altering the truth are typically traits of dictatorships. I have also seen some people that have being put at work on developing further some of the stuff in some of the papers that Serre has written. Then it turns out to be very difficult to do that, because of the method that Serre is using when writing those papers: He is picking together some facts together very elegantly, but he does not go into the depth. Thus, if one would start with one of those Serre papers and go into the depth, one will find that the work that properly should have been done has not been done. This technique of his became more apparent, the more I saw of his writings. Serre is also known for using people for his very own purposes as long as he is interested; then, when he looses interest, he dumps those without explanation. Clearly, such things could not happen if there was clearly defined and explained scientific objectives motivating Serre's temporary interests, because then people would be able to go on by their own without having the need for close contacts with Serre. So when I encountered people in that circuit, in the eighties, one did not speak about the science of Serre but about his interests whatever he now is interested in. So there is a sorry combination here, a celebrity status that is allowed to deflect much needed resources away from topics that would provide for a more proper scientific development. Serre is, of course, a great talent, but not a Nobel Prize level scientist or anything on that level. Hans Aberg * Anti-spam: remove remove. from email address. * Email: Hans Aberg * Home Page: Subject: Re: Armand Borel dead >>What exactly do you have against Serre? >>Your animosity appears to me to be totally irrational. >Ah! But is it infinite, too? I have noticed that Serre favors yes-men with little individual thinking. :-) Hans Aberg * Anti-spam: remove remove. from email address. * Email: Hans Aberg * Home Page: Subject: Re: Armand Borel dead |Clearly, there is no proper metamathematical definition of a |question, but only of conjectures. i personally would be all in favor of hearing stories about the evil deeds of j-p. serre if i could work up some slight reason to suspect that they might be true. i tend to lose interest though when they're coming from an absolute moron who doesn't even understand what the word question means. -- [e-mail address jdolan@math.ucr.edu] Subject: Re: Armand Borel dead >> I thought the idea was that free speech admits one to speak out about the >> things that one considers indecent or wrong. >I'm very tolerant about people's opinion on how good other people's >mathematics are. > But you've crossed the line between the mathematics and >the people themselves: inserting Finally between Borel and dead, Have you ever met Borel? :-) I of course met with Borel, in 1984, and he already then knew what I felt about his personality, namely that his and mine person chemistries seemed to be incompatible. And his seemed to be incompatible with most people that I knew that had encountered him. The best statement I ever encountered was that some felt they could cope with his person chemistry, but no more. Yesterday I was at a party, and the lady at the table said she felt honored to meet with me, so you can guess what I am as a person. :-) -- Some people just do not like that I tell openly what I feel is the truth. > and >comparing Armand Borel's demise to that of Idi Amin Dada?!? COME ON!!! So there is one set of ethics principles for famous mathematicians and another for other people? >It really tells us more about Hans Aberg than about either of these two... >(That's *my* bit of free speech.) You mean, that I stick to one set of ethics principles for everyone, not having special ones for famous mathematicians? >Besides, I note that you give an e-mail address and a home page affiliated >with the Math Department of Stockholm University. Is my home page broken? > OTOH, you're not listed >among the faculty on that departments Web site. I am only visiting. > Anyway, a glance at their >homepage shows that all that crap (free speech, right?) you've been telling >us about Serre lately isn't exactly their official view... There was a very strong Serre group here trying to prove his Poincare question until Anick disproved it. But that was in the eighties. I think they are retired by now. But that is the reason I (in the eighties) saw some those negative aspects. I have really not been interested in anything in that circuit since then. This stuff does not really concern me anymore. If some science can prove of value to mankind, then I will become interested and promote that, but not otherwise. So if you have something more to say, please hurry up before I loose interest. :-) Hans Aberg * Anti-spam: remove remove. from email address. * Email: Hans Aberg * Home Page: Subject: Re: Armand Borel dead >|Clearly, there is no proper metamathematical definition of a >|question, but only of conjectures. >i personally would be all in favor of hearing stories about the evil >deeds of j-p. serre if i could work up some slight reason to suspect >that they might be true. i tend to lose interest though when they're >coming from an absolute moron who doesn't even understand what the >word question means. So please explain your deep mathematical insights of the word question. Start with a metamathematical definition, please. How is that used in a formal proof? Hans Aberg * Anti-spam: remove remove. from email address. * Email: Hans Aberg * Home Page: Subject: Re: Armand Borel dead > In any case, you haven't explained why you attack Serre > for his supposed contribution to Bourbaki, > when he has produced a vast number of works > which would seem a more logical target for your criticism -- > criticism which I for one find completely incomprehensible. > What exactly do you have against Serre? > Your animosity appears to me to be totally irrational. you're quite right, so don't bother trying to reason with him. he's either a troll or has some big personal problems (jealousy?). tom Subject: Re: Armand Borel dead >> What exactly do you have against Serre? >> Your animosity appears to me to be totally irrational. >you're quite right, so don't bother trying to reason with him. he's either a >troll or has some big personal problems (jealousy?). It seems that in math, cons and pros of celebrities cannot be discussed openly as in the rest of society. Pretty strange. At the time I was engaged in the circuit in question, in the eighties, these things were however discussed vividly in private discussions. Hans Aberg * Anti-spam: remove remove. from email address. * Email: Hans Aberg * Home Page: Subject: Re: Armand Borel dead > For example, some of Andre Weil's very own > writings on algebraic geometry are very difficult for modern > mathematicians to read and ensure accuracy of, as concepts like generic > are not formally defined as in say in Grothendieck schemes. I don't have Weil's Foundations of Algebraic Geometry to hand, but I am quite sure the term generic as in generic point on a variety is defined clearly and accurately there, as the whole book -- contrary to what you say -- is written with complete rigour. (I imagine that was Weil's main motivation in writing it.) Off the top of my head, I would say that a point P on a variety V over k is said to be generic if it doesn't belong to any proper sub-variety of V, or equivalently, if V is the variety defined by P over k. That may be wrong; but the point is, Weil will have given a rigorous definition. -- Timothy Murphy e-mail: tim@birdsnest.maths.tcd.ie tel: +353-86-233 6090 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland Subject: Re: Armand Borel dead > Serre is also known for using people for his very own purposes as long as > he is interested; then, when he looses interest, he dumps those without > explanation. Although you have hinted at personal failings of Serre, this seems to be your only concrete criticism of his personality. I encountered Serre briefly many decades ago, and my recollection of him is exactly the opposite of yours. At the time, Grothendieck was giving a series of seminars on what was then his recent re-formulation of algebraic geometry. At the end of each lecture, Serre in effect explained what Grothendieck had just said. Serre seemed happy enough to act as Grothendieck's disciple, or publicist -- a far cry from your picture of him as self-seeking and self-important.b Also he went out of his way to explain points raised by those at the seminars. -- Timothy Murphy e-mail: tim@birdsnest.maths.tcd.ie tel: +353-86-233 6090 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland Subject: Re: Armand Borel dead > Have you ever met Borel? :-) No. Have you met Idi Amin? > I of course met with Borel, in 1984, and he already then knew what I felt > about his personality, namely that his and mine person chemistries seemed > to be incompatible. And his seemed to be incompatible with most people > that I knew that had encountered him. The best statement I ever > encountered was that some felt they could cope with his person chemistry, > but no more. > [...] > So there is one set of ethics principles for famous mathematicians and > another for other people? No, there should of course be only one set. So, do you actually put what you've said above about Borel in the same league as Idi Amin's legacy? > I am only visiting. That doesn't change the fact that you use your host's platform to say things your host obviously disagrees with. Does he know about that? > This stuff does not really concern me anymore. If some science can prove > of value to mankind, then I will become interested and promote that, but > not otherwise. And what science of value to mankind exactly *are* you promoting in the present thread? Hugo Subject: Re: Armand Borel dead >I don't have Weil's Foundations of Algebraic Geometry to hand, >but I am quite sure the term generic as in generic point on a variety >is defined clearly and accurately there, >as the whole book -- contrary to what you say -- >is written with complete rigour. >(I imagine that was Weil's main motivation in writing it.) >Off the top of my head, I would say that a point P on a variety V over k >is said to be generic if it doesn't belong to any proper sub-variety of V, >or equivalently, if V is the variety defined by P over k. >That may be wrong; but the point is, >Weil will have given a rigorous definition. If it was not the question of a generic _point_, it might have been question of parametrization of some sort. This was said to have been problematic in some other geometers work (Italian school?). I have a vague memory it was difficult to see exactly how he defined rationality and some such things. Deligne once pointed a reference to some of Weil's writings and tried to look it up, but found he could not read it. You will have to go look into those books yourself too see what you feel. The question is not really whether there was any lack of clarity in Weil's writings within the standards of his own subculture, as clarity was his trademark, but pointing out that different times have different standards as how far one should go. Clearly Bourbaki made an attempt to go all the way, but noticed that was not practically possible in view of needed abuse of notation. Some of the original Bourbaki members have described this early development and Andre Weil's role in it. The best would be that those interested would look that up. Hans Aberg * Anti-spam: remove remove. from email address. * Email: Hans Aberg * Home Page: Subject: Re: Armand Borel dead >> Serre is also known for using people for his very own purposes as long as >> he is interested; then, when he looses interest, he dumps those without >> explanation. >Although you have hinted at personal failings of Serre, >this seems to be your only concrete criticism of his personality. And scientific failings. Notice though that I have never met a person with a full spectrum of scientific capabilities: They all do different things and things differently designed around their personal traits and failings. One cannot be successful at a high level without doing that. An overly inflated celebrity status, though, may pretend a person have capabilities not in actuality present, a well known social phenomenon. >I encountered Serre briefly many decades ago, >and my recollection of him is exactly the opposite of yours. >At the time, Grothendieck was giving a series of seminars >on what was then his recent re-formulation of algebraic geometry. >At the end of each lecture, >Serre in effect explained what Grothendieck had just said. >Serre seemed happy enough to act as Grothendieck's disciple, or publicist -- >a far cry from your picture of him as self-seeking and self-important.b >Also he went out of his way to explain points >raised by those at the seminars. As for those negative sides, when I first encountered him I only saw that official picture of him, as the one you described. But there came by some other facts suggesting there was an other side of the coin as well. It took me and some of his close collaborators (which I never was) several years to make sense of this his behavior. As for Grothendieck, I heard that after he had resigned and later wanted to get a new job, those famous mathematicians refused to write him a letter of recommendation. I do not know if that is true. In the end, his seminars were said to consisted of two halves, a political one and a mathematical one, and one had to attend both. :-) So I was fed with a lot of strange anecdotes as well in those years. :-) Hans Aberg * Anti-spam: remove remove. from email address. * Email: Hans Aberg * Home Page: Subject: Re: Armand Borel dead >> Have you ever met Borel? :-) >No. Have you met Idi Amin? It seems that this discussion does not converge anymore. Even though the original title might have been viewed as disturbing to some, I think that those that really have met Borel would have taken it with a :-) and said that is just about it. >> I am only visiting. >That doesn't change the fact that you use your host's platform to say things >your host obviously disagrees with. Does he know about that? They do not have any such collective principles here. I notice that you use a fake address; you should reveal your true status before attempting to intimidate me: One good thing with my Borel encounter was that I learned how to not become intimidated. So you must try much harder if you want to succeed. :-) >> This stuff does not really concern me anymore. If some science can prove >> of value to mankind, then I will become interested and promote that, but >> not otherwise. >And what science of value to mankind exactly *are* you promoting in the >present thread? None; has this thread ever claimed to be of scientific value? Pure mathematicians seem to often be socially underdeveloped relative rest of society, and I thought by standing by with some replies that might help the community a bit. Hans Aberg * Anti-spam: remove remove. from email address. * Email: Hans Aberg * Home Page: Subject: Re: Armand Borel dead >Have you ever met Borel? :-) >>No. Have you met Idi Amin? > It seems that this discussion does not converge anymore. Ex - cuse - me: *you* brought up the comparison between Borel and Amin, not me. > Even though the original title might have been viewed as disturbing to > some, I think that those that really have met Borel would have taken it > with a :-) and said that is just about it. What a lousy way to try to get back onto your feet! You've posted to a *newsgroup*, and despite it being sci.math, the vast majority of people *haven't* met Borel. Besides, even if among the people who *have* met him there are some that think the same of him as you do, most of these will probably be as outraged as I am. > I notice that you use a fake address; you should reveal your true status No way, buddy. Besides, a closer inspection of the headers of your own posts shows that you don't even post from Stockholm University, so your address might be as fake as mine. > before attempting to intimidate me: I'm not trying to intimidate you. I'm just expressing my outrage about the way you talk of a man that just left our world, and in particular about your comparison with one of the worst criminals of the 20th century who happened to leave it around the same time. A comparison you now try to put to the side (see above) in a way that isn't even remotely elegant. >This stuff does not really concern me anymore. If some science can prove >of value to mankind, then I will become interested and promote that, but >not otherwise. >>And what science of value to mankind exactly *are* you promoting in the >>present thread? > None; Then why don't you stick to the principle you've so aptly described and shut up? Hugo Subject: Re: Armand Borel dead >> I notice that you use a fake address; you should reveal your true status >No way, buddy. The name you use is evidently a fake, too. So it is probably best to ignore the writings of this person in the future. > Have you ever met Borel? :-) >>>No. Have you met Idi Amin? > >> It seems that this discussion does not converge anymore. >Ex - cuse - me: *you* brought up the comparison between Borel and Amin, not >me. I suggested that the ethical principles applied should be the same. It was an idea introduced in the OT of the Bible, as opposed to the case in say older Mesopotamian law. Which is why the discussion isn't converging, as the replies refers to Hans Aberg * Anti-spam: remove remove. from email address. * Email: Hans Aberg * Home Page: Subject: Re: Armand Borel dead > So it is probably best to ignore the writings of this person in the future. Actually, I'm going to ignore your writings. You have shown yourself to be completely without decency; I have better things to do than be annoyed at and offended by you. *plonk* V. -- mail me at lastname at cs utk edu Subject: Re: Armand Borel dead >Actually, I'm going to ignore your writings. You have shown yourself to >be completely without decency; I have better things to do than be >annoyed at and offended by you. This reminds me of the release of the movie Un chien andaloe by Lois Bunuel, to whom one of the viewers said This movie made me to throw up, to which he replied: This movie was made in order to make people like you to throw up. Hans Aberg * Anti-spam: remove remove. from email address. * Email: Hans Aberg * Home Page: Subject: Astronomical Observations - Parts 1, 2, 3 Comments: This message probably did not originate from the above address. It was automatically remailed by one or more anonymous mail services. You should NEVER trust ANY address on Usenet ANYWAYS: use PGP !!! Get information about complaints from the URL below X-Remailer-Contact: http://80.65.224.85/POL/ In case my abuse address is unreachable: It is because it has been flooded by , please contact -----BEGIN PGP SIGNED MESSAGE----- ANY MODERN ASTRONOMY program will work for this lesson. I recommend using the freeware Astrolog 5.41G with the freeware JPL-DE406 Swiss Ephemeris, Carte du Ciel 2.75 which is also freeware, and includes links to download dozens of freeware catalogues and other plugin options, or check out the SkyMap 9 demo version on my links URL: http://groups.google.com/groups?selm=IHO72C3E37766.8210763889@ Gilgamesh-frog .org This is very basic, and will show you how every planet visible to the naked eye, which includes the Sun, Moon, Mercury, Venus, Earth, Mars, Jupiter, Saturn, & Uranus, this will show you how these planets move as seen from the Earth in conspicuously repetitious and predictable patterns which are easily counted by days, months, and years between repeating sidereal and synodic multiples. This absolutely destroys any and all arguments against the ancients being perfectly able to see the motion of the planets against the night sky and counting by days, months and years to predict sidereal & synodic periods for each planet at least out to Saturn and possibly to Uranus, since it rarely can be seen with the naked eye. This is a big deal because secular academia has closed their eyes to timeless science and its reproducibility. This clearly transcends simple astronomy, but includes astrology, metaphysics, and all spiritual implications. Limit your program to what is visible to the naked eye. No guesswork & no speculation. Your astronomy software reliably emulates what we'd see when viewing the night sky in that direction, at that time from that location, conveniently, efficiently and with impressive accuracy. Of course, the view is better through a good telescope, or through the unaided, human eye, since it is assumed that ancients didn't have other means to see the stars. That's a humongous ad hoc assumption, but I'm granting modern-day atheistic science that much and I still win. Accurate positions of planets and stars is all we need for this lesson. Your favorite software will work fine. No telescope needed. We can see this all with our eyes, so reduce your software's star magnitude limit to five, and assume Uranus, Neptune and Pluto to be nonexistent (not as Gods, but to pacify the unbelieving scientist). For this lesson, we're concerned only with heliacal ri- sings of each planet separately, which depends only on sufficient angle between the planet and the Sun, so it can be spotted against background stars before sunrise. The Sun must be about 18 degrees below the horizon for full darkness and a little less for heliacal phenomena. This angle varies with each planet, and each star, and time of year, temperature, pressure, how good your eye- sight is, the geographical latitude of observation and local horizon, obstructions and circumstances of light pollution, smog, haze from forest fires, volcanos, etc. While these conditions can vary to extremes, generally, provided reasonably good seeing conditions towards the eastern horizon about an hour or so before sunrise, as you look to the east (from moderate latitudes) you can barely make out a planet that you expect to see rising heliacally on or about that date. If you miss it, then try again in a couple of days and you're bound to spot the planet you're looking for if it's Mars, Jupiter or Saturn; or plan ahead and begin looking sooner if it's Mercury whose orbit you can see is eccentric. You know that each planet has predictable orbital patterns, and although these patterns vary over the short-term, over the long-term they become more and more predictable to fractions of a degree in sidereal longitude & latitude. That's how you know that Venus is the most predictable, since Venus has the least eccentric orbit. We see this behavior of Venus through heliacal risings or settings, especially at maximum elongations inferior or superior. If getting up at four in the morning is not your style, simply open your astronomy program and set it for your geographical location and voilla! You're ready to view to heliacal risings of every planet--against the stars. In the next part we focus on Saturn's heliacal risings. Open your favorite astronomy program. As always, I use Astrolog, so all examples given refer to JPL ephemeris DE-406 with Abramov's expanded version of fixstars.ast provided by S. Moshier using the Astronomical Almanach. All data is accurate to within several milliarcseconds, which is vastly better accuracy than the plus or minus half a degree or thirty arcminutes we can achieve with an extended pinky finger at arm's length measuring one arcdegree...twice the apparent diameter of a full Moon. Three closed middle fingers spans five degrees, or the whole hand equals about ten degrees. You can calibrate simple hand measurements by memorizing bright marking stars near the ecliptic by their approximate longitude on the caelestial zodiac. The constellations and their associated myths help us to easily locate and identify stars as we become familiar with their appearances and their order in the sky. This is where Carte du Ciel or SkyMap comes in handy, since they depict the stars and planets graphically, and include millions more objects and dozens of unabridged catalogues for the astronomer. However, only Astrolog can chart the marking stars and planets by their zodiacal, constellational coordinates as used by ancient stargazers for tracking the planets. The complete list of almost 1000 stars is posted on my website, but here's an abbreviated list for convenient reference with the values rounded off to whole degrees and favoring brighter stars in the northern hemisphere. Remember the goal is not to memorize every star but is to estimate a planet's position at its heliacal rising, setting, opposition and other repeating synodic phases against the fixed background of this caelestial sphere: Name Longit. Lat. Bayer Al Pherg : 2 Ari + 5 etPsc Sheratan : 9 Ari + 8 beAri Caph : 10 Ari +51 beCas Hamal : 13 Ari +10 alAri Shedir : 13 Ari +47 alCas Cih : 19 Ari +49 gaCas Ruchbah : 23 Ari +46 deCas Segin : 0 Tau +48 epCas Algol : 1 Tau +22 bePer Alcyone : 5 Tau + 4 etTau Mirphak : 7 Tau +30 alPer Aldebaran : 15 Tau - 5 alTau Rigel : 22 Tau -31 beOri Bellatrix : 26 Tau -17 gaOri Capella : 27 Tau +23 alAur Mintaka : 28 Tau -23 deOri Alnilam : 29 Tau -25 epOri Alnitak : 0 Gem -25 zeOri Saiph : 2 Gem -33 kaOri Polaris : 4 Gem +66 alUMi Betelgeuse: 4 Gem -16 alOri Menkalinan: 5 Gem +21 beAur Alhena : 14 Gem - 7 gaGem Sirius : 19 Gem -40 alCMa Castor : 25 Gem +10 alGem Pollux : 28 Gem + 7 beGem Procyon : 1 Can -16 alCMi Asellus Au: 14 Can + 0 deCnc Kochab : 19 Can +73 beUMi Dubhe : 20 Can +50 alUMa Subra : 29 Can - 4 omiLeo Alphard : 2 Leo -22 alHya Algieba : 5 Leo + 9 ga1Leo Regulus : 5 Leo + 0 alLeo Thuban : 13 Leo +66 alDra Dhur : 17 Leo +14 deLeo Denebola : 27 Leo +12 beLeo Vindemiatr: 15 Vir +16 epVir Spica : 29 Vir - 2 alVir Arcturus : 29 Vir +31 alBoo Menkent : 18 Lib -22 thCen Zubenelgen: 20 Lib + 0 al2Lib Dschubba : 8 Sco - 2 deSco Antares : 15 Sco - 5 alSco Rastaban : 17 Sco +75 beDra : 21 Sco -12 epSco Sabik : 23 Sco + 7 etOph Rasalhague: 28 Sco +36 alOph Sargas : 1 Sag -20 thSco Gal.Center: 2 Sag - 6 SgrA* Eltanin : 3 Sag +75 gaDra Sacred Tre: 5 Sag + 0 ----- Solar Apex: 7 Sag +53 HerA* Kaus Austr: 10 Sag -11 epSgr Nunki : 18 Sag - 3 siSgr Vega : 21 Sag +62 alLyr Altair : 7 Cap +29 alAql Dabih : 9 Cap + 5 beCap Sadr : 0 Aqu +57 gaCyg Enif : 7 Aqu +22 epPeg Fomalhaut : 9 Aqu -21 alPsA Deneb : 11 Aqu +60 alCyg Markab : 29 Aqu +19 alPeg Scheat : 5 Pis +31 bePeg Algenib : 14 Pis +13 gaPeg Alpheratz : 20 Pis +26 alAnd Since we're beginning with Saturn, set restrictions in Astrolog to restrict all then uncheck only the Sun and and you'll see Saturn at opposition in 15 Gemini. This is just one pinky finger in longitude from Alhena at 14 Gemini. With the Sun in 15 Sagittarius, then Saturn adding 378 days, which is January 13, 2005. But Saturn is a little slow in getting there, reaching opposition the next day January 14, in 29 Gemini. The oppositions, which we'll skip for Jupiter and Mars, prove to us the planets Mars, Jupiter and Saturn, are orbiting the Sun beyond Earth's orbit, and these orbits are predictable, especially over long-term observations. As with Saturn, by adding 3781 days to its synodic phase, we arrive at which is May 9, 2014, again missing exactitude by only one or two days, due to Saturn's moderate eccentricity and about 2.5 degrees inclination to the ecliptic. For long-term predictions, the ancient Babylonians noticed that 9 sidereal orbits of Saturn coincided with around 256 synodic periods and 265 tropical years speaking in have January 1, 2269. Sure enough, there's Saturn near opposition in 14 Gemini directly above Alhena and just two days from true opposition January 3, 2269, showing that the Babylonians knew what they were talking about two thousand years before Christ. It's no mystery, but is readily observable, predictable and reproducible in the laboratory of the night sky, like heliacal risings. The predawn risings of stars and planets have been the carefully watched and predicted since men could mark a cave wall with a piece of coal, blood or whatever else has handy. Primitive stone observatories emerged which had much greater longevity, and showed the teamwork of prehistoric stargazers, and the importance they placed on the ephemeris of the Sun, Moon & Stars to the Earth. Naturally, the Sun is the single most important object visible in the Earth's sky. Man has watched the Sun as it rises and sets every day since humankind has walked the Earth. All life forms follow the diurnal circadian rhythm of Earth's daily rotation in one way or another. Hence the Sun formed the fundamental basis of tracking time from the beginning of every civilization that has come and gone, from primitive tribes of early hominids to more advanced human cultures, most of which are too distant in the past for their records to have survived. More recently, the Egyptians, Babylonians, Mayans, and others around the post-deluvian world are close enough in time for many of their records to be extant, mostly bits and pieces, some fairly intact, like the pyramids. In mans present time, secular-religious archaeologists prefer to believe that civilization is basically under 7000 years old worldwide, due to their historical ties to the Roman church, and continued use of the language in their laws and their sciences. This is not to blame the ancient translation of the bible, the Vulgate, but has been the politics of religion, as men serve mammon. After all the bible predicted this would happen, so it isn't surprising that the schism of religious-apostasy should continue to rule the minds of men. Yet the Moon & Stars have continued to illuminate the night sky for geological aeons and shall continue to do so for aeons. So it is that Saturn has been rising and setting helia- cally in very predictable intervals and shall continue to do so for many long ages to come. Since the initial date and time for observation of Saturn before sunrise will vary, we know the Sun needs to be some 18 degrees below the horizon to ensure visibility of any brighter star or planet from moderate latitude any time of year, weather permitting. But in fixed locations, i.e. where ancient and antediluvian population centers flourished, the heliacal risings of stars and planets were readily estimated to within a few days time and by the seasons of the year, tied directly to planting, harvesting and every single aspect of their lives. Thus astrology was the natural result of watching and predicting when the stars and planets would rise and set, by knowing where the planets are day and night. This knowledge was made by simple observation, counting days, months and years between cycles and phases. When Saturn rose heliacally, it was always about 378 days give or take a day or two since the last time it was observed to rise heliacally. With each consecutive heliacal rising of Saturn, fixed stars in the background showed that Saturn moves about 13 degrees in keeping with the Sun's progress relative to the stars some 13 days later each year--again, give or take a day or two, talking about long-term averages rounded off to integer days since the whole premise is to show that ancient stargazers could and did see that the planets clearly orbit the Sun, and that they could readily observe and recognize the sidereal and synodic orbits by watching the heliacal risings of planets and stars. The accuracy of the ancient ephemeris increased commensurate with continued calibration by observation of heliacal phenomena over the centuries and millennia of that civilization from its rise until its fall. The quality of long-lost very ancient ephemeredes is known by mans inherent ability as a man to see the night sky and to notice patterns and repetition in nature. These are perfectly natural talents that all people are born with--at least most people are. Once again, this comes down to how much credit we give prehistoric man. There are anthropologists who have recognized that early man was smarter than modern day, secular-religious science had theretofore acknowledged. Likewise the recognition that at least semi-intelligent hominids have been here many millions of years earlier than the orthodoxy used to believe albeit some still cling to their hopelessly obsolete superstitions about the antiquity of man, etc, it is clear that man and man-like sentient beings have roamed the Earth for aeons. One might reasonably argue that dolphins or whales are smart enough to notice the planets and stars rising and setting, and to count the days and years of these events. Elephants are known to remember things very well. At a minimum, we can safely say that early man was intelligent enough to count the days, months or years of observable heliacal phenomena and we see that such observable events are predictable, simply counting these events by days, months and years. ______________________________________________________ I think this is what makes modern astronomers angry at those of us who have realized that planetary motion is not nearly as mysterious as they'd like you to believe. ______________________________________________________ The Egyptians, Babylonians and Mayans showed admirable levels of sophistication in their astronomical records and their ability to predict very long-term periodical events, the great year of precession being among these, since the Earth's axis of rotation visibly gyrates one degree against the fixed stars about every 26000 solar days, which is about 71 tropical years, two months and nine days, therearound. This is according to the Mayan astronomers, whose astronomical skills were comparable to those of the Babylonians. Both left records proving that they could see the night sky, and that they could accurately count and predict periodic planetary orbits against the starry background of the caelestial sphere. As in this case, we *see* Saturn observably progresses about twelve degrees every year against the stars seen from Earth. Every twenty-nine and a half years, Saturn goes full circle against the stars, and over centuries of observation we see that Saturn circles the Sun nine times every two hundred sixty-five years--meaning that Saturn advances closer to twelve and a quarter degrees longitude per year thereby making short-term estimates of Saturn's motion a little more accurate and reliable than our round number of twelve degrees per year. Thus we may safely predict that Saturn will have moved east by closer to forty-nine degrees every four years, plus our ephemeris for Saturn has improved significantly by repeated observation and simple mathematical deduction. We'll notice Saturn's thirteen degree advance at times of entering or leaving retrograde motion and that this retrograde lasts for about one hundred thirty-eight or so days centered on inferior conjunction or opposition to the Sun. Every three hundred seventy-eight days, we see these motions repeat, when Saturn appears to stand still in the sky then begin to move backwards for some four and a half months before standing still again and returning to normal motion. Every time we see it again, about 378 days have passed and Saturn is approximately 13 sidereal degrees from where it was last time around. Carte du Ciel is especially useful for animating these apparent synodic motions against the background of the stars, since you can fine-tune increments down to days, hours and minutes, and mark the locations with finder circles to readily observe a planet's motion relative to the stars & constellation figures, and to the other planets. Although the accuracy of the ephemeris is not very reliable beyond plus or minus four thousand years, especially for the Moon, you can view distant dates to circa 20,000 years BC / AD. While tropical seasons can be way off the mark the apparent motion of a planet to the stars may not be far off the mark for say, 9000 BC. You just won't know the season, or the Moon's position at such a distant date, but other planets are probably within a couple of degrees of where they actually were. Not that this matters much, since you are simply using the present-day ephemeris to view synodic and sidereal motion of the planets that are visible and predictable. For example, most of us'll probably be up and about at should remember to walk outside for a moment and check out Saturn in 15 Gemini--just above and east of Alhena, and right below Mebsuta which marks sidereal 15 Gemini just 2 degrees above the ecliptic. Your extended thumb at arm's length spans about two arcdegrees thus you'll see that Saturn is maybe a pinky fingernail's width or so (about 2/3's of a degree) below the ecliptic at the time of observation. Since Asellus Australis (see list above) marks 14 Cancer right on the ecliptic (actually +0:04'38 but round degrees are all a stargazer needs), and bright Regulus at 5 Leo is less than half a degree above the ecliptic, you can quickly visualize the line, rather the arc of the ecliptic across the sky. Jupiter at 24 Leo and about a degree above the ecliptic should be visible in the eastern sky. Sirius at 19 Gem and 40 degrees below the ecliptic will be hard to miss in the southern sky (unless you live north of Barrow, Alaska). If you live in the southern US or similar latitude you might spot bright Canopus at 20 Gem -76 degrees barely above the south horizon. Orion should be in clear view below right of Saturn. See if you can spot Al-debaranu, the prime fiducial of the caelestial zodiac at 15Tau00 and 5 degrees below the ecliptic. As you see, when you look at a planet in the night sky the background stars help you to locate the planet's longitude and latitude, hence confirming previous predictions, and calibrating future predictions. In ancient times this was done for centuries & millennia. Let's look at Saturn heliacally. Just to be on the safe side, we'll put 30 degrees past Saturn for the predawn Sun. That ought to make it easy to spot Saturn before sunrise, whether you're watching from the old, royal Greenwich observatory at 25 meters above sea level & 00E00:00 longitude 51N28:38 latitude, or viewing atop the Great Pyramid at 31E09:00 29N58:51, or from the Sun Pyramid in Teotihuacan, Mexico ~19:44N 98:50W or from the site of ancient Babylon 44E24 32N33. Use your own default observation location, set up your favorite astronomy program to watch the sky from there. I'm using my own location here in central Colorado USA. Saturn is plainly visible at heliacal rising August 14, with Saturn 27 Gemini and the Sun 27 Cancer. We'll add the 378 days for Saturn's synodic period, to August 27, 2005, with Saturn 10 Can and Sun 9 Leo. Like before we are just a day short, so on August 28, 2005, Saturn is some 13 degrees further along in the caelestial zodiac which is 756 days, and we have September 9, 2006 which is about two days shy of Saturn 30 sidereal degrees to the Sun, thus September 11 2006 finds Saturn rising at previous observations is closer to 3781 than 3780. The date is December 21, 2014. Low and behold, Saturn's at 5 Scorpio and the Sun is 5 Sagittarius, right where we expected it to be. Remember, Saturn was at 27 Gem back Saturn is heliacally risen we see that Saturn is 5 Sco and the Sun 5 Sag. That's near 128 degrees that Saturn has progressed in ten synodic periods or ten times our round figure of 13 degrees. Again, as observations are made over longer and longer periods of time, ephemeris calibration and improvements are the inevitable result. These long-term observations of the heliacal phenomena inevitably reveal the limits as to how far the planets can appear to stray from Earth's ecliptic with the Sun, revealing each planet's orbital inclination to Earth's, and also revealing other obvious limits, such as Venus and Mercury display their orbital eccentricity when at maximum elongation, Venus very little, Mercury a whole lot more. This plainly shows the observer that Venus & Mercury are closer in heliocentric orbit than Earth is, and of course the paths of Mars, Jupiter & Saturn show that they are further away from the Sun in their helio- centric orbits than Earth is. We'll cover more on this in later parts. Jupiter is next on the list of planets. at 13 Virgo, 30 sidereal degrees from the Sun 13 Libra. just below bright Venus at 7 Virgo. Zaniah (etaVir) is between them near 10 Virgo. Remember, we are measuring the sky with our naked eye and extended hand, so round degrees, maybe down to a sixth of a degree, or ten arc- minutes, is as good of accuracy as we can achieve. You can see that the modern accuracy of JPL's ephemeris is based on observations made by large observatories, and formulated using advanced knowledge of mathematics and These values are rounded off to the nearest arcseconds of longitude and latitude, while the internal accuracy of the software is good to milliarcseconds (JPL-DE406): Aldebaran : 15Tau00'00 -5:28'00 Venus : 6Vir55'17 +1:32'47 Zaniah : 9Vir30'15 +2:35'21 Jupiter : 12Vir35'49 +1:07'05 Sun : 12Lib32'48 +0:00'00 Ancient observers would commonly use a measuring stick or metal rod notched with linear increments calibrated by the observer which he or she could comfortably hold at arm's length between both hands, ensuring a uniform perspective of sidereal measurement. But we will limit our ancient observers as having nothing but themselves to view the heavens, since that's all that they needed to clearly view the predictable motions of the planets against the fixed background of stars. Easily accurate to plus or minus one degree, simple enough so children could be taught to do this and carry on the stargazing tradition, counting the days, weeks, months, and years, planting, harvesting, worshipping by the ephemeris and its religiously-observed calendar--the religion of the stars. As each civilization developed, and became more sophisticated, they organized and specialized, so that astronomical observation, astronomy, and their logical deductions based on astronomical observations--meaning mathematics--ergo astrology, became the disciplines of specialists so that others in their community could go about their business. In ancient times, the astrologer was synonymous with the mathematician, star-logician in the most literal sense. Even in our day and age, it was only within the last few centuries that astrologer and astronomer reached a schism, since astrologers had long-since ignored the proper mathematics of astrology, and astronomers became disenchanted with the illusions yet perpetuated by today's tropicillogical astrologers and other schisms of astrology,--all who've hopelessly lost their grasp on the ancient practice of star-logic. Since this schism, astronomers have changed their ways of measuring the sky such that constellations became synonymous with unequal boundaries associated with the asterisms or some 88 familiar groups of brighter stars instead of the ancient method which divides the entire caelestial sphere into twelve equal meridians as signs with meridians of latitude from the caelestial equator. Modern astronomers began referencing positions only to Earth's terrestrial equator by its intersection on her ecliptic. Next time there's a pole shift, or crustal displacement (or both?), that'll screw up their method of measuring the sky in a heartbeat. Meanwhile Earth's slow gyration of precession continues to change modern astronomer's coordinates. For example, look at Regulus at 5 Leo near the ecliptic. In 8000 BC, Regulus was at 5 Leo. In 8000 AD, Regulus will still be at 5 Leo. The position of Regulus is easy to see and easily recalled. Only the slight, very long-term wobble of the ecliptic itself affects how we chart latitude of stars near the ecliptic, and also the longitude of stars farther away from the ecliptic. As a result, Regulus might be close to a degree from the ecliptic at some remote epoch but it's still going to mark 5 Leo for a long time to come, irrespective of precession, pole-shift or annihilation of civilization. Any survivors can point up at Regulus and confidently say Look! There's Regulus 5 Leo, and any planet passing nearby will certainly be identified by its position--relative to a recognizable fixed star, and certainly not by its RA/Dec. As for this example, on Julian Day -1200514, 1-Jan--7999 (8000 BC Gregorian) Carte du Ciel shows Regulus at 0h46m35s +4*36'06, and Carte du Ciel shows Regulus at 15h29m45s -17*53'29 on Julian Day 4643000 1-Jan-8000 (8000 AD). For a caveman marking scores on a cave wall to remember positions of planets relative to nearby stars counting days, months and years between repeating heliacal risings and other predictable synodic phases relative to the Sun, anyone can see that the positions of planets are most readily and easily tracked by their positions to visible stars, and that those stars remain fixed in their position on Earth's caelestial sphere with subtle proper motion so slow that it takes millennia even to be noticed by the best of naked-eye astronomers. Hence Orion's Belt, for example, is very close to the same position in the sky as it was when they built the Great Pyramids 10,500 BC, since the three stars of the belt have very low proper motion. So Mintaka 28 Tau, Alnilam 29 Tau, and Alnitak the Great Pyramid star at 0 Gem have illuminated the same positions on Earth's caelestial zodiac ever since. we can easily see where it is in relation to the stars before sunrise, since the stars tell us where 13 Virgo is. In this case, Zaniah at 10 Vir is nearby, so it is easy to estimate Jupiter's position to plus or minus a degree of certainty. With this simple observation, the next heliacal rising of Jupiter is easily predicted by the average period that Jupiter has been seen for ages to repeat its synodic cycles. That is December 4, 2005, but Jupiter is about five days past the 30-degree mark from the Sun. November 29, 2005 finds Jupiter 12 Libra and the Sun in 12 Sco, and Jupiter will be rising near rising Jupiter, you can be sure where 29 Virgo is. But Kappa Virgo at 10 Lib and +3 latitude--although it's a lot closer to Jupiter--may be difficult to see at 4.18 magnitude. The star called 109 Vir is a bit brighter at 3.72 magnitude and marks 13 Libra near +17 latitude. The important thing is to know which stars that you're looking at, and their approximate longitude & latitude in the zodiac. In the 395 days between heliacal rising, Jupiter will complete one sidereal orbit approximately every 12 years. Jump ahead 4000 days from October 30th late. We must go back to October 5, 2015, with Jupiter has moved from 13 Vir to 17 Leo, 26 degrees before the completion of one sidereal year for Jupiter. Estimates that Jupiter would take about 12 years to complete one sidereal orbit. Jupiter's tenth heliacal rising showed us that Jupiter moved about 334 degrees over 3992 days. We might extrapolate off this, and figure that Jupiter will make about 360 degrees in another 311 days making a rough estimate 3992 + 311 = 4303 days for a sidereal year of Jupiter based on a total of three observations. Let's look at the next rising of Jupiter 400 days from October 5, 2015, November 8 2016. Now we're about four days late, so go back to November 4, 2016, for Jupiter my location. So for eleven heliacal risings 30 degrees from the Sun, it took 4388 days, and Jupiter transited Virgo on November 4 2016. That's fully 360 degrees and 4 extra degrees that Jupiter was observed to move over the course of 4388 days and a touch more than 12 years. Simple interpolation estimates a sidereal year at 4340 days, 37 days higher than our previous estimate but is now based on four observations not just three. Further observations empirically calibrate our rough estimates. takes a little less than 12 years to complete one side- real orbit, since we are plus 4 degrees after 12 years. Repeated observation refined our estimate to 4340 days. After centuries, the ancients were able to winnow this down to some 4332 days or about 11 tropical years plus around 316 days that it takes Jupiter to orbit the Sun. It doesn't take any rocket scientist but only common sense with a little simple addition and subtraction of round degrees, days and years. The stargazer could see Jupiter go retrograde for some 121 days centered on in- ferior conjunction (opposition), and see these synodic events repeat every 400 days by the long-term averages. Ancient Babylonian astronomers were sufficiently adept to notice that 36 sidereal orbits of Jupiter was quite close to 427 tropical years and 391 synodic periods of if they knew what they were talking about. Try October 30, 2431. Just 6 days later Jupiter is 30 degrees from the Sun with Jupiter 12 Vir and the Sun 12 Vir. That's ancient synodic multiple for Jupiter is right in there. End Parts 1, 2, 3. See Part 4 For Continuation... Daniel Joseph Min *Min's Planetary Awareness Technique (chapters 1 thru 6): http://groups.google.com/groups?selm=HFVIRNCU37838.7946990741@ Gilgamesh-frog. org *Min's Official PGP Public Key on the MIT server: http://groups.google.com/groups?selm=3XWB7QJO37766.971099537@ Gilgamesh-frog.o rg *Min's Home Page On The World Wide Web: http://groups.google.com/groups?selm=0XNKAO4L37773.8337962963@ Gilgamesh-frog. org -----BEGIN PGP SIGNATURE----- iQA/AwUBP0BSCZljD7YrHM/ nEQISkACgwQtuAWIqmCwLjHG4rsj9F3u9KXQAnjkT GkjhrgskyYff3s4c/uxK3kiD =UO1r -----END PGP SIGNATURE----- Subject: Re: Astronomical Observations - Parts 1, 2, 3 > ... > ... > So it is that Saturn has been rising and setting helia- > ... Beautifully formatted mate! What went wrong with the two lines starting Btw are you Miss Minny Banister, Henry Crun's friend? -- G.C. Subject: Re: Astronomical Observations - Parts 1, 2, 3 > ... > ... > So it is that Saturn has been rising and setting helia- > ... > Beautifully formatted mate! What went wrong with the two lines starting Nonsense, there were also: >>For this lesson, we're concerned only with heliacal ri- and >>that they are further away from the Sun in their helio- and >>takes a little less than 12 years to complete one side- and >>Jupiter go retrograde for some 121 days centered on in- as well as the following that one might not count: >>time of year, temperature, pressure, how good your eye- and >>degrees, maybe down to a sixth of a degree, or ten arc- is at the front of a line in an email message. Some mail- processing programs automatically add a - or > before each -jiw Subject: axioms of mathematics information on the axioms of mathematics. I can find that there are 6 (possibly 7) axioms from which all mathematical proofs can ultimately be derived, but which themselves are unproved, or unprovable. But not much more than that, before heavy duty talk of Godel. I am a complete layperson, and I cannot find out what these axioms actually are (I vaguely recall things like ab=ba, x+y=y+x, things as elementary as that, are axiomatic). Can anyone furnish me with a list (or URL that points to a list) of the axioms; can anyone tell me if they are indeed unproved or actually unprovable, and why that should be so? I repeat that I am a total layperson, who studied school mathematics to 16 years old, more than 20 years ago! sorry for the moron question - replying off list would be great if you don't want to pollute your inboxes! regards bruce Subject: Re: axioms of mathematics >information on the axioms of mathematics. I can find that there are 6 >(possibly 7) axioms from which all mathematical proofs can ultimately be >derived, but which themselves are unproved, or unprovable. But not much more >than that, before heavy duty talk of Godel. >I am a complete layperson, and I cannot find out what these axioms actually >are (I vaguely recall things like ab=ba, x+y=y+x, things as elementary as >that, are axiomatic). The really basic axioms you want are the axioms of set theory (see below). >Can anyone furnish me with a list (or URL that points to a list) of the >axioms; can anyone tell me if they are indeed unproved or actually >unprovable, and why that should be so? >I repeat that I am a total layperson, who studied school mathematics to 16 >years old, more than 20 years ago! >sorry for the moron question - replying off list would be great if you don't >want to pollute your inboxes! >regards >bruce Here is an online guide, written by Thomas Jech, the author of a famous textbook on set theory: http://www.seop.leeds.ac.uk/entries/set-theory/ --- Jeff Subject: Re: axioms of mathematics > information on the axioms of mathematics. I can find that there are 6 > (possibly 7) axioms from which all mathematical proofs can ultimately be > derived, but which themselves are unproved, or unprovable. But not much more I've not read Singh but he may have been referring to axioms for set theory of which there are two commonly used versions (Zermelo-Fraenkel and von Neumann-Bernays-G.9adel) and a number of others. Here is a link for the Zermelo-Fraenkel axioms: http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html > than that, before heavy duty talk of Godel. > I am a complete layperson, and I cannot find out what these axioms actually > are (I vaguely recall things like ab=ba, x+y=y+x, things as elementary as > that, are axiomatic). These aren't axioms of set theory. Perhaps Singh was talking about number theory. > Can anyone furnish me with a list (or URL that points to a list) of the > axioms; can anyone tell me if they are indeed unproved or actually > unprovable, and why that should be so? Not everything can be proved because either one would go round in circles or one would embark on an infinite regress. So one starts from axioms that one _chooses_ not to prove. How those axioms are chosen is a long story. > I repeat that I am a total layperson, who studied school mathematics to 16 > years old, more than 20 years ago! > sorry for the moron question - replying off list would be great if you don't > want to pollute your inboxes! > regards > bruce -- G.C. Subject: Re: axioms of mathematics > information on the axioms of mathematics. I can find that there are 6 > (possibly 7) axioms from which all mathematical proofs can ultimately be > derived, but which themselves are unproved, or unprovable. But not much more > than that, before heavy duty talk of Godel. > I am a complete layperson, and I cannot find out what these axioms actually > are (I vaguely recall things like ab=ba, x+y=y+x, things as elementary as > that, are axiomatic). > Can anyone furnish me with a list (or URL that points to a list) of the > axioms; can anyone tell me if they are indeed unproved or actually > unprovable, and why that should be so? > I repeat that I am a total layperson, who studied school mathematics to 16 > years old, more than 20 years ago! > sorry for the moron question - replying off list would be great if you don't > want to pollute your inboxes! The precurser to Godel's proof are the axioms for predicate calculus. 1. Ayi(F->(G->F)) 2. Ayi((F->(G->H))->((F->G)->(F->H))) 3. Ayi((~F->~G)->((~F->G)->F))) 4. Ayi(Ax(F->G)->(F->AxG)) x is not free in F 5. Ayi((F->G)->(AyiF->AyiG)) 6. Ayi(AxF(x)->F(y)) y is not quantified when substituted A simpler way of writing them courtesy G Frege (sci.logic) The following is a well know axiom system of PC: A1 A -> (B -> A) A2 (A -> B -> C) -> ((A -> B) -> (A -> C)) A3 (A -> (B -> C)) -> (B -> (A -> C)) A4 (A -> B) -> (~B -> ~A) A5 ~~A -> A A6 A -> ~~A (The other connectives of PC can be defined with ~ and ->.) Actually it is a list of axiom-schemas. A, B and C are meta-variables which have to be replaced with wffs (when forming axioms out of the axiom-schemas). The rule of derivation is (of course) MP: A -> B , A ---------- B Modus Ponens is a rule applied to exsisting axioms to form new formula. eg. if its raining then I will get wet, its raining -------------------------------------------- I will get wet or Rain -> Wet, Rain >> Wet _____________________ some meanings of the axioms.. 1. Ayi(F->(G->F)) If F is true then G->F 2. Ayi((F->(G->H))->((F->G)->(F->H))) Implication (->) distributes over itself 3. Ayi((~F->~G)->((~F->G)->F))) If both ~G and G are implied by ~F, then ~F cannot be true, in other words F is true. Also look up Peano postulates. Godels proof was actually that all of mathematics can *not* be derived from finite axioms. Herc Subject: Bezier curve :Lenght of [a,b] ? ===== In the following k, n (0 =< k =< n , n >= 2 ) are natural numbers and C(n,k)=n!/(k!(n-k)!) , b_{n,k}(t)=C(n,k)t^k(1-t)^{n-k} , P_k=(x_k,y_k) , x_k ,y_k in R , (k=0,1,...,n) , m_1 := |Sum_{k=0 to k=n}(-1)^{n-k}C(n,k)x_k| , m_2 := |Sum_{k=0 to k=n}(-1)^{n-k}C(n,k)y_k| . Suppose m_1> 0 , m_2>0 and put sqrt[n](x)= x^{1/n}=x**(1/n) . Consider the Bezier curve x=X(t)=Sum_{k=0 to k=n } b_{n,k}(t)x_k , t in [0,1] , y=Y(t)=Sum_{k=0 to k=n } b_{n,k}(t)y_k associated to points P_0,P_1,...,P_n . QUESTION: Let A>0 , B>0 , -inftyMy defintion of the real world would be everything within our >rationality. > Your definition of the real world includes a lot of things that most > people would exclude. For example, unicorns. Wow. I don't know what to say. You assume I believe in fairy tales to belittle my argument? > No. I disagree with your definition of rationality; I disagree with > your definition of real world; Ok, what is your definition of rationality then? If mine isn't good enough you should suggest another one. For what its worth a more descriptive explanation of my definition would be: The limits of our rationality are defined by the limits in our ability to observe and model the universe with our biological hardware (our senses of observation and our brains respectively). Whats rational to us would be the real world. What is outside the real world would also lie outside our rationality. In the case of the Calculus solutions in question, I'm not saying that the entire solution lies outside our rationality, but it is not completely within our rationality. Mike Helland Subject: Re: Calculus is irrational? Visiting Assistant Professor at the University of Montana. >Arturo Magidin >My defintion of the real world would be everything within our >rationality. > >> Your definition of the real world includes a lot of things that most >> people would exclude. For example, unicorns. >Wow. I don't know what to say. You assume I believe in fairy tales to >belittle my argument? You're trolling, aren't you? I am not belittling your argument and I am not assuming you believe in fairy tales. I am pointing out that the definition you give FORCES you to accept that unicorns are part of the real world. For unicorns are definitly within our rationality: we can describe them perfectly, we can paint them, we know how they behave, what they like and dislike, we pretty much know them as well as we know horses and cats and dogs. So why would they be exluded under your definition? What I am saying, in summary, is that your definition is a bad definition. >> No. I disagree with your definition of rationality; I disagree with >> your definition of real world; >Ok, what is your definition of rationality then? If mine isn't good >enough you should suggest another one. The reason your definition of rationality is a bad one is that it is entirely subjective, and depends entirely on how much knowledge, training, and experience the speaker has. Thus, you consider a lot of things to lie outside rationality because you are, quite simply, ignorant of what those things are and mean. That makes the term essentially synonimous with I understand it, and that's not a good meaning for the word. What exactly is wrong with the meaning given in dictionaries? Rational is relating to, based on, or agreeable to reason. > For what its worth a more >descriptive explanation of my definition would be: >The limits of our rationality are defined by the limits in our ability >to observe and model the universe with our biological hardware (our >senses of observation and our brains respectively). In that case, as I pointed out in the part that you removed in this reply, the question you asked is not within the limits of our rationality. The question you originally posed relates to a ball that bounces without end. That is not something that we can observe. Since the question lies outside the limits of our rationality, the problem lies in your assumption that (a) it lies within them; and (b) any answer to that question must itself lie within them. Simply put, the language you need to answer the question as I answered it (by saying that at any distance short of 1/9 the ball is still bouncing, but that at any distance of 1/9 or larger the ball never gets there) is the same language you need to even ASK the question in the first place. You are complaining that I am using that language for the answer, insinuating that the answer should not use that language somehow, just because you end of the question, Where does the ball stop, does not use any such language. But you cannot ask the question without context, and you cannot set up the context without using that same language. Yet you complain that the answer uses that language, which was required to ask the question. If there are any jumps in the way we treat the question, as you claim, then the jump occurs at the beginning, when you assume an answer must avoid the very language needed to ask the question in the first place. >Whats rational to us would be the real world. Again, why would this exclude unicorns under you definition? I'm not saying you believe in unicorns. I am saying that you have not fully explored the implications of the definition you offer, and that once you do so you will realize why that definition is no good. >What is outside the >real world would also lie outside our rationality. In the case of >the Calculus solutions in question, I'm not saying that the entire >solution lies outside our rationality, but it is not completely within >our rationality. And I am saying that if the solution lies outside our rationality, then so does the question. That your continual complaint about the solution and argument that the answer to a question should lie within certain boundaries, when the question cannot even be asked without transcending those boundaries, is what most people would deem irrational. It would be like demanding that one express the answer to 1/2 + 1/3 in integral terms. The question cannot be asked without going beyond the integers. Why would anyone object that the answer be given in a way that requires you to go beyond the integers as well? ============================================================== ======== It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) ============================================================== ======== Arturo Magidin magidin@math.berkeley.edu Subject: Re: Calculus is irrational? [...] > However, I can make my point without relying on rational or irrational > numbers at all, and stick strictly to the argument of what is and what > is not within human rationality. Infinity lies outside of human > rationality. In what sense? If you've shifted your use of the word rationality to now mean something like our ability to reason about something, then mathematical infinities are of course generally quite within human rationality. Precise reasoning about inifities is common in math -- as people keep trying to tell you as well as demonstrate in other postings in this thread. If you're using some other definition of human rationality, then, well, maybe you should make that clear, instead of equivocating furiously between the different usages and definitions of rational. >Depending on infinity to find a precise answer (whether > algebraically or using the short-cuts of calculus) depends on > something outside of human rationality. Calculus -- as has been pointed out here again and again, ad nauseam -- does not depend on infinity to find a precise answer. This is elementary math -- *please* come away with at least that from all this... And in any case, inifinity (in the mathematical sense) is not something outside of human rationality (at least as usually defined). It might be outside our real-world *experiences*, but then that's true of pretty much all math, no? > I only decided to add (ir)rational numbers into the mix because I was > getting the argument that results such as 1 or 1/9 are rational > numbers. While thats true, I'm more focused on the solutions that > provide these results, not the results themselves. > I assert that the solutions require one to wander outside of human > rationality. I'm not saying that these solutions are wrong, invalid, > useless, or even something we should avoid. I'm just saying that it > does step outside of rationality. Its interesting to see so much > resistence to this claim. There's a lot of resistence to this claim because it's trivially false using the normal mathematical and philosophical definitions of the terms used -- and because, frankly, whenever you post something specific and precise, it's been shown to be wrong. More commonly, you post vague-sounding things like step outside of rationality that either say nothing useful, or keep equivocating on the word rational. You keep saying we miss your point, but, frankly, your point seems to be that if one equivocates on the word rational and uses enough poorly-defined terms and a general lack of rigour, you can do cargo-cult math which sorta somehow sounds a bit like math but isn't. Oh wait, that's *my* point. Sorry. Your imprecision makes it hard to understand just what your point *is*. Hamish Subject: derivations of the simple continued fraction for e What methods are known for proving that e has the simple continued fraction [2,1,2,1,1,4,1,1,6,1,1,...], where the pattern continues with successive even numbers separated by pairs of 1's. Here are the methods I know: (1) http://research.microsoft.com/~cohn/Papers/e.ps uses a method (which it attibutes to Hermite) based on showing that integral(x^n (x-1)^n exp(x) / n!, x=0..1) = e p(3n) + q(3n), where the p's & q's are respective numerators & denominators of the required convergents. (2) http://www.nrich.maths.org.uk/mathsf/journalf/sep02/art1/ shows how to prove tanh(x) = [0, 1/x, 3/x, 5/x, ...] (x <> 0) using a hypergeometric function recurrence relation. I was then able to derive the scf for e from that of tanh(1/2) = (e-1)/e+1). (3) http://mathworld.wolfram.com/e.html claims the desired result follows immediately from re-writing the usual power series as e = 2 + (1/2)(1 + (1/3)(1 + (1/4)(1 + (1/5)(1 + ...)))). I would appreciate a hint on how that works, as I must be missing something obvious. English translation of Euler's paper, but I haven't managed to find it yet.) Other methods? Subject: Re: derivations of the simple continued fraction for e r.e.s. skrev i melding > What methods are known for proving that e has the simple continued > fraction [2,1,2,1,1,4,1,1,6,1,1,...], where the pattern continues > with successive even numbers separated by pairs of 1's. > Here are the methods I know: > (1) http://research.microsoft.com/~cohn/Papers/e.ps uses a method > (which it attibutes to Hermite) based on showing that > integral(x^n (x-1)^n exp(x) / n!, x=0..1) = e p(3n) + q(3n), > where the p's & q's are respective numerators & denominators of > the required convergents. > (2) http://www.nrich.maths.org.uk/mathsf/journalf/sep02/art1/ > shows how to prove tanh(x) = [0, 1/x, 3/x, 5/x, ...] (x <> 0) > using a hypergeometric function recurrence relation. I was then > able to derive the scf for e from that of tanh(1/2) = (e-1)/e+1). > (3) http://mathworld.wolfram.com/e.html claims the desired result follows immediately from re-writing the usual power series as > e = 2 + (1/2)(1 + (1/3)(1 + (1/4)(1 + (1/5)(1 + ...)))). > I would appreciate a hint on how that works, as I must be missing > something obvious. > English translation of Euler's paper, but I haven't managed to > find it yet.) > Other methods? If you look in Elementary Number Theory by David M. Burton, you will find several clues to both finite and infinite continued fractions, included how Euler developed the infinite continued fraction of the number e. Karl-Olav Nyberg konyberg@online.no Subject: Factor Analysis/Principle Components Analysis help Dear Math/Stat experts, I have been reading the method of Factor Analysis and I am confused about what exactly it is. Principle Components is pretty simple in my opinion. You have p random variables, build your p x p covariance matrix, grab the eigenvalues and eigenvectors and you have your principle components. Factor analysis, you have an error term, so I believe you have x_1 = q_11*y_1 + ... + q_1n*y_n + e_1 ... x_m = q_n1*y_1 + ... q_nn*y_n + e_n where the matrix Q = (q_ii) is the loading matrix. How is the Error matrix calculated exactly? Isn't it that Iteratively, the diagonal error matrix is guessed until _______ (I don't know what goes in the blank). We then have the equality S = QQ^T + E, where S is the observed covariance matrix, and Q is the loading matrix, and Q^T is Q transposed, and E is the error diagonal matrix. My colleague explained it to me like this : The error matrix is computed iteratively, and then it is subtracted from the covariance matrix, and then principle components is performed on the remaining matrix, which in this case would be QQ^T. Is this how factor analysis is performed? In this case, QQ^T would be real and symmetric, so it's eigenvectors would be orthogonal. However, the column vectors in Q would not necessarily be orthogonal. Am I way off? Tim Subject: Re: Factor Analysis/Principle Components Analysis help It is not Principle Component Analysis but Principal Component Analysis. Subject: Re: Factorial/Exponential Identity, Infinity <3c6b9c1e.0308100457.73ff5939@posting.google.com> <3c6b9c1e.0308101544.48f969ec@posting.google.com> <3c6b9c1e.0308111623.38637b13@posting.google.com> <3c6b9c1e.0308140003.4517c9c6@posting.google.com> :Subject: Re: Factorial/Exponential Identity, Infinity : :The above is riddled with error. The content of the subject or Ross A. Finlayson. (Not necessarily mutually exclusive - in fact, one might say equivalent.) Subject: Re: Factorial/Exponential Identity, Infinity > The above is riddled with error. > ... > What are implications of using an infinity as a number in the same > equations as finite numbers and infinitesimals? This thread started because I think that half of the sequences have equal numbers of ones and zeros. Yet, I am unable to prove that. What was shown, and it was known, or at least a variation of it was an exercise in a book, is that: lim n->oo n! sqrt(n) / ( (n/2)!^2 2^n) = sqrt(2 / (Pi n )) I wanted to get an expression for how many of the sequences (the infinitely many binary sequences of infinite length) had as a fraction of the population / probability of random occurrence xn or (1-x)n for 0oo n! / ( ((x)n)! ((1-x)n)! 2^n) = ( ((x)n)! ((1-x)n)! / (n/2)!^2 ) (sqrt(2/ (Pi n))) Here's a thought: it goes back to the binomial distribution where there is probability p of success and q of failure, each of p and q in the above is 1/2, what if p and q multiply to be 1/e, where their sum equals one? Then instead of 2^n it would be e^(n/2). p + q = 1 p * q = 1/e Is there more than one pair of possible values for p and q? Is there a solution? There isn't. The value of 1/e is 1/2.7182... or .36789.... , and (1/2)(1/2) = 1/4 = 0.25 is about the maximum value of pq for p+q=1 and 0 4/e =. Heh, 4/e = 1.472.... For example p*q can equal 1/(2e). Then instead of 2^n it would be (2e)^(n/2). This is only when k=n/2. Otherwise it would be p^k q^(n-k), where all of the consideration in this thread has been with p=q=1/2, as half of the sequence elements are one and the other half zero. p+q = 1 q = 1-p p*q = 1/2e p(1-p) = 1/2e p-p^2 = 1/2e p^2 - p + 1/2e = 0 p= -(-1 +- sqrt(1-2/e))/2 p = (1-sqrt(1-2/e))/2 n! / (( n/2)!^2 2^(n/2) e^(n/2) ) That's the expression for the probability of half successes and half failures of a sequence of length n where p+q=1 and p*q = 1/(2e). Anyways, from that a variety of different expressions for n! were shown vis-a-vis Stirling's formula. I looked to Euler's formula for Gamma, and got the bright idea of replacing the variable z with an expression of the variable n, z = n/x, to find more identities fo n! for basically infinite n, the limit of n! as n diverges. I soon found that it was only true for z being small finite values instead of presumably finite values that were expressions of n, assumed to be an arbitrarily large and then ever larger finite variable. That led into consideration of polynomials of the form (n+1)...(n+n), for the specific case of the expression (2n)! / ( n! n^n). Polynomials for other values of x would be about (3n/2)! / (n! n^(n/2)), etcetera, (n+1)...(n+ n/2), for ((1+1/x)n)! / (n! n^(n/x)). Thus I wanted to get an idea of what expressions would reducedly symbolize the polynomial coefficients of (n+1)...(n+n) as a way to get started. I figured out that a recurrence relation would give the coefficient in terms of n, and have yet to find a form for it. The idea of determing what those coefficients are is that they would form another polynomial as divided by n^n that equals (2n)! / (n! n^n). At that point I digressed about the variable as an infinite value and discussed scalar infinities and scalar infinitesimals. I think those might actually be visible in boundary value problems. They might certainly be useful in a variety of ways. I'll talk about that more later. Anyways, back to the polynomials, I note in a recent post that Ullrich says polynomials have to have a finite number of terms. Here I'm talking about a case where a polynomial would have n+1 many summands for infinite n. For example, n! is (n)(n-1)...(n-n+1), I am here more concerned with (n+1)...(n+n). So, as I hope to discover the recurrence relations for the coefficienct a_i for i=1 to n+1, it would have to be in the form where n is assumed to diverge, and I will call (2n)!/n! a polynomial. The univariate polynomial (n+1)...(n+n) has as its first coefficient at least a simple result: 1. The second coefficient is the sum of each value of n, from 1 to n. That value has a simple expression: (n+1)(n/2). In the form of a recurrence relation, f(1)=1, f(n) = f(n-1)+n. The coefficient a_3 is not so simple to compute. It is the the coefficient a_i is the sum of the product of each of the elements of each subset of length i-1 of N. One way to approach that is to sum all the n-1 pairs with 1's in them, then all the n-2 remaining pairs with 2's, then all the remaining pairs with n-3 3's, for each of the (n^2-n)/2 pairs. Then for the first set of pairs their products' sum is (n+1)(n/2)-1. The pairs with 2 have the product ((n+1)(n/2)-1-2)*2. The pairs with 3 have the product ((n+1)(n/2)-1-2-3)*3. f(x) = ((n+1)(n/2)-sum(x))*x sum x=1^n f(x) = a_3 (n+1)(n/2)(n+1)(n/2) - sum x=1^n sum(x)*x (n^2 + n)^2 / 4 - sum x=1^n sum(x)*x (n^2 + n)^2 / 4 - sum x=1^n (x+1)(x/2) * x (n^2 + n)^2 / 4 - sum x=1^n (x^3 + x^2)/2 I think the above is an expression for a_3. Then trying to find an expression for a_4, the sum of the product of each triple of N, is quickly more complicated, where in trying to find a recurrence relation to a_3, it is to start with a_3 minus something and call it the sum of each triple with 1. Then there are all of the triples with 2, etcetera. Here are a couple things I have come across here that I want to understand better: it's the plain old sum function that is the the sum of integers from 1 to n, much like the factorial is the product of integers from 1 to n. Then also I have the cumulative sum function, which is another recurrence relation which is the sum of the integers 1 to n plus the sum of the integers 1 to n-1 plus the sum of the integers 1 to n-2, etcetera. These are parts of the recurrence relations about summing the the products of pairs, triples, etcetera of sets of integers with contiguous values starting at one. There's a lot of available information about the recursive product of numbers, factorial, where's the information about recursive sums? I read in the algorithm books about Theta space and big-O, with the expression in few variables of more complicated recurrence relations, I'm talking about more exactitude. The sum of n integers 1 to n is f(n)=1, f(n+1)= f(n) + n+1, it's also (n+1)(n/2). What's a function for f(n)=1, f(n+1)=f(n)+f(n-1)+f(n-2)... + n+1? How about a recurrence relation for the products? Is there a compendium or dictionary of recurrence or difference equations? So, we're discussing factorial/exponential identities. Ross Subject: Re: Factorial/Exponential Identity, Infinity > This thread started because I think that half of the sequences have > equal numbers of ones and zeros. Yet, I am unable to prove that. If Ross means that of the collection of all infinite sequences all of whose terms are zeros or ones only half have equal numbers of zeros and ones, he underestimates. Any such sequence which has infinitely many zeros and infinitely many ones has equally (denumerably) many of each digit, and it may be shown that there are uncountably many of such sequences, but the set of sequences containing only finitely many zeros or finitely many ones is countable. Thus more sequences have equally many zeros and ones than have unequally many. Subject: Re: Field of rationals and pi In sci.math, James Harris <3c65f87.0308160905.4fcb9e1f@posting.google.com>: > Euler found out that he could define pi^2 in the following nifty way: > pi^2/6 = 1 + 1/4 + 1/9 + 1/16 +... I agree that that's nifty. :-) > Which means you can define it using members of the *field* of > rationals. However, pi is transcendant and is itself not a rational. > I'm curious about the rule mathematicians use to exclude pi^2/6 from > the field of rationals, as it itself is the result of an infinite sum > of members of that field. > Is that it? Mathematicians simply exclude infinite sums from the > field of rationals? Or do they rely on the definition of a rational > as the ratio of a/b, where 'a' and 'b' are integers? The latter, of course. Infinite sums such as 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... + 1/2^n + ... can be perfectly rational -- in fact, this one's an integer, 1. This despite the fact that the numerator and denominator of the partial sums both increase without limit (the partial sum is (2^n-1) / 2^n). But one has to be careful regarding infinite sums. I could just as easily have written 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + ... + 1/n! + ... which converges to Euler's constant, e. > Continuing in that direction, recently a leading mathematician at a > major university in the United States of America (a top 20 > university) sent me an email stating that my rule of no other integers > being units except -1 and 1 did not exclude pi if you used Z[pi]. > I said it did in the following reply (Professor's name omitted): > Professor ****: > > You assertion is easily proven false. Please consider the following. > > infinity. Pedant point: the (more or less) standard notation for this sort of thing is pi^2/6 = 1 + 1/4 + 1/9 + ... + 1/k^2 + ... The extra ellipsis serves as a reminder that this is an infinite sum. In a pinch one can leave out the k term if sufficiently obvious from context, although one has to be slightly careful in some cases. > > But then you have > > pi^2/6 = 1 + 1/4(1 + 1/4 + 1/9 +...1/k^2) + 1/9(1+1/4 + > 1/9+...)+...1/k^2(1+k^2), which is > > pi^2/6 = 1 + pi^2/24 + pi^2/54 +...pi^2/6k^2, What is this equality? Either you're trying to compute 1 = 1/(1 + 1/4 + ... + 1/k^2 + ...) + 1/(4*(1 + 1/4 + ... + 1/k^2 + ...)) + 1/(9*(1 + 1/4 + ... + 1/k^2 + ...)) + 1/(16*(1 + 1/4 + ... + 1/k^2 + ...)) + ... + 1/(m^2*(1 + 1/4 + ... + 1/k^2 + ...)) + ... which is valid, if a little odd-looking (note that m is independent of k), or you're attempting to compute pi^4/36 in a slightly peculiar fashion, or something even weirder. I can't fathom your equality. > > multiplying out and collecting to the left except for 1, > > 6(24)(54)...(6k^2) pi^2 [24(54)...(6k^2) - 6(54)...(6k^2) - > 6(24)...(6k^2) - ... -6(24)(54)...] = 1, > > which proves that you have an infinite number of units, some of which > are 6, 24, and 54, which is the result if you include pi in a ring > with integers, so my definition *does* exclude it. This ugly mess looks like infinity/infinity and may not be all that useful. > Well he replied: Actually, Z[pi] has no units save 1, -1, as a consequence of > the fact that pi is transcenddental (not algebraic). Z[pi] does not > contain the numbers you are considering above, which (as I read it) > are obtained by summing infinite series. The elements of > Z[pi] are just those real numbers that can be expressed as > f(pi) where f(x) is a FINITE polynomial with integer coefficients. > Do you agree with the professor, who I remind is a *leading* > mathematician? Assuming Z[x] is the ring of all numbers derivable from 1 and x, using the arithmetic operators +, -, and *, as in http://mathworld.wolfram.com/Ring.html and that a unit is described as in http://mathworld.wolfram.com/Unit.html which states that a unit is an element which has a multiplicative inverse in the ring, then yes, Z[pi] is such that 1 and -1 are the only units. Were pi a unit there would exist a polynomial with coefficients in Z, at least one of which is nonzero, with pi as the root; this is easily proven: Let ip !=0 be such that ip * pi = 1, and assume ip is in Z[pi]. Then there exists integers a_0, a_1, ..., a_n such that ip = a_0 + a_1 * pi + a_2 * pi^2 + ... + a_n * pi^n (it should be obvious that this is from our definition of Z[pi] above). We know at least one of the a_i is nonzero since neither pi nor ip is rational. Multiply this equation through by pi: 1 = a_0 * pi + a_1 * pi^2 + ... + a_n * pi^(n+1) and subtract 1: 0 = -1 + a_0 * pi + a_1 * pi^2 + ... + a_n * pi^(n+1) and we have an polynomial of which pi is a root, with integer coefficients. Since pi is transcendental according to http://mathworld.wolfram.com/TranscendentalNumber.html we have a contradiction; hence pi has no inverse in Z[pi] and is not a unit. I doubt there are any other units in Z[ip], either. I suspect that every element in the set Z[ip] - Z is transcendental. In fact, were this not the case one would have a number y in Z[ip] - Z which is algebraic, therefore a root of an equation 0 = b_0 + b_1 * y + ... + b_m * y^m with integer b_i, at least one of which is nonzero. Since y is a polynomial in pi with integer coefficients, one can grind out an equation with integer c_j: 0 = c_0 + c_1 * pi + ... + c_p * pi^p and derive yet another contradiction, as at least one of the c_j must be nonzero as well. The only reason this wouldn't work is if y is an integer, as opposed to a member of Z[ip] - Z; this gives the trivial equation 0 = c_0 + y or 0 = c_0 - y. Since the logic above to prove pi is not a unit in Z[ip] also works word for word to prove y is not a unit, and since 1 and -1 are the only units in Z, they are also the only units in Z[pi]. So yes, I for one agree with the professor; he's not exactly treading on new ground here, whoever he is. Note that transcendental numbers can be approximated with infinite series; in fact you yourself have such a series, defining pi^2/6, which of course is also transcendental. Or one can use the series (pretty from a theoretic standpoint, ugly from a practical one): pi/4 = 1 - 1/3 + 1/5 - ... + 1/(4k+1) - 1/(4k+3) + ... or, my personal favorite, the far more speedily converging pi/4 = 4(1 - 1/(3 * 5^2) + 1/(5 * 5^4) - 1/(7 * 5^6) ...) - (1 - 1/(3 * 239^2) + 1/(5 * 239^4) - 1/(7 * 239^6) ...) from the equality atan(1) = 4 * atan(1/5) - atan(1/239). There are probably better series but I like this one for its relative simplicity to implement. Since the subseries are absolutely convergent (one can easily bound the first one to 25/24, for example, by simply ignoring the (2k+1) multiplier in the denominator and changing all '-' signs to '+'), one can rewrite it without fear: pi/4 = (4 - 1) - (4/(3*5^2) - 1/(3*239^2)) + (4/(5*5^4) - 1/(5*239^4)) - (4/(7*5^6) - 1/(7*239^6)) ... and thereby compute pi to any precision up to about 60,000 digits, or to 4 billion digits if one uses a 32-bit long in place of a 16-bit int. (I don't know who first discovered this particular series. I've been playing with it since the early 1980's, although I can't say I did a lot with it beyond implement pi calculators with it. :-) ) Another transcendental number, e^(m/n) [*], m, n integers, m != 0, n != 0, is also approximated by the famous series e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + ... + x^n/n! + ... This series always converges for any x in the complex plane. > James Harris [*] If m = 0, then e^0 = 1, which is not transcendental, but every other case is. -- #191, ewill3@earthlink.net It's still legal to go .sigless. Subject: Re: Field of rationals and pi >> Infinite sums are not excluded from the rationals; >> every rational number _is_ an infinite sum. But there's >> no rule that says an infinte sum of rationals has to be >> rational, which is possibly what you meant. >There's a subtle point of confusion I hope you can clarify for me. In the >previous post by Arturo Magidin, he said (I think) that infinite sums *are* >excluded from the rationals. But evidently he was referring to the >operation of taking infinite sums, rather than the value of the sum itself. Of course that's what he meant. >Certainly any rational number *can* be expressed as an infinite sum, >whereas irrational and transcendental numbers may *only* be expressed that >way. >Could you be a little more specific about what constitutes a valid I'm not sure exactly what you mean. In case it helps, all I meant was that for example 1 _is_ an infinite sum (can be written as an infinite sum in various ways in fact) but that doesn't mean it's excluded from Q. >-- >There are two things you must never attempt to prove: the unprovable -- and >the obvious. >-- >Democracy: The triumph of popularity over principle. ************************ Subject: Re: Field of rationals and pi > Euler found out that he could define pi^2 in the following nifty way: > pi^2/6 = 1 + 1/4 + 1/9 + 1/16 +... > Continuing in that direction, recently a leading mathematician at a > major university in the United States of America (a top 20 > university) sent me an email stating that my rule of no other integers > being units except -1 and 1 did not exclude pi if you used Z[pi]. > I said it did in the following reply (Professor's name omitted): > Professor ****: > > You assertion is easily proven false. Please consider the following. > > infinity. > > But then you have > > pi^2/6 = 1 + 1/4(1 + 1/4 + 1/9 +...1/k^2) + 1/9(1+1/4 + > 1/9+...)+...1/k^2(1+k^2), which is > > pi^2/6 = 1 + pi^2/24 + pi^2/54 +...pi^2/6k^2, > > multiplying out and collecting to the left except for 1, > > 6(24)(54)...(6k^2) pi^2 [24(54)...(6k^2) - 6(54)...(6k^2) - > 6(24)...(6k^2) - ... -6(24)(54)...] = 1, That is incorrect. It should be pi^2 [24(54)...(6k^2)... - 6(54)...(6k^2)... - 6(24)...(6k^2)... - ... -6(24)(54)...] = 6(24)(54)...(6k^2)..... What I find interesting is that no one, not even the leading mathematician pointed out the error. While one poster did notice that my series should be followed by an additional .... I, of course, was simply careless as I've been before. > which proves that you have an infinite number of units, some of which > are 6, 24, and 54, which is the result if you include pi in a ring > with integers, so my definition *does* exclude it. It still does. Realizing why is not difficult. > Well he replied: Actually, Z[pi] has no units save 1, -1, as a consequence of > the fact that pi is transcenddental (not algebraic). Z[pi] does not > contain the numbers you are considering above, which (as I read it) > are obtained by summing infinite series. The elements of > Z[pi] are just those real numbers that can be expressed as > f(pi) where f(x) is a FINITE polynomial with integer coefficients. > Do you agree with the professor, who I remind is a *leading* > mathematician? Do you? James Harris Subject: Re: Field of rationals and pi >> Euler found out that he could define pi^2 in the following nifty way: > >> pi^2/6 = 1 + 1/4 + 1/9 + 1/16 +... > >> Continuing in that direction, recently a leading mathematician at a >> major university in the United States of America (a top 20 >> university) sent me an email stating that my rule of no other integers >> being units except -1 and 1 did not exclude pi if you used Z[pi]. > >> I said it did in the following reply (Professor's name omitted): > >> Professor ****: >> >> You assertion is easily proven false. Please consider the following. >> >> infinity. >> >> But then you have >> >> pi^2/6 = 1 + 1/4(1 + 1/4 + 1/9 +...1/k^2) + 1/9(1+1/4 + >> 1/9+...)+...1/k^2(1+k^2), which is >> >> pi^2/6 = 1 + pi^2/24 + pi^2/54 +...pi^2/6k^2, >> >> multiplying out and collecting to the left except for 1, >> >> 6(24)(54)...(6k^2) pi^2 [24(54)...(6k^2) - 6(54)...(6k^2) - >> 6(24)...(6k^2) - ... -6(24)(54)...] = 1, >That is incorrect. >It should be > pi^2 [24(54)...(6k^2)... - 6(54)...(6k^2)... - 6(24)...(6k^2)... - >... -6(24)(54)...] = 6(24)(54)...(6k^2)..... >What I find interesting is that no one, not even the leading >mathematician pointed out the error. The reason for that is that the error you're talking about here is an error in the algebraic manipulations. That error really doesn't matter, because there are much more serious conceptual errors: _Many_ of us have pointed out that you seem to be assuming that rings are closed under infinite sums - they're not. >While one poster did notice that my series should be followed by an >additional .... >I, of course, was simply careless as I've been before. > >> which proves that you have an infinite number of units, some of which >> are 6, 24, and 54, which is the result if you include pi in a ring >> with integers, so my definition *does* exclude it. >It still does. Realizing why is not difficult. Idiot. >> Well he replied: > >Actually, Z[pi] has no units save 1, -1, as a consequence of >> the fact that pi is transcenddental (not algebraic). Z[pi] does not >> contain the numbers you are considering above, which (as I read it) >> are obtained by summing infinite series. The elements of >> Z[pi] are just those real numbers that can be expressed as >> f(pi) where f(x) is a FINITE polynomial with integer coefficients. > >> Do you agree with the professor, who I remind is a *leading* >> mathematician? >Do you? Deaf and blind idiot. >James Harris ************************ Subject: Re: Field of rationals and pi > Euler found out that he could define pi^2 in the following nifty way: > pi^2/6 = 1 + 1/4 + 1/9 + 1/16 +... I don't think he *defined* pi^2 in that way; pi had been defined many centuries before. Euler showed that pi^/6 happens to equal the indicated sum. > Which means you can define it using members of the *field* of > rationals. However, pi is transcendant and is itself not a rational. transcendental; transcendant is not a word. EVERY positive number, whether rational, irrational, integer, transcendental, algebraic, or whatever can be represented as an infinite sum of reciprocals of integers. > I'm curious about the rule mathematicians use to exclude pi^2/6 from > the field of rationals, as it itself is the result of an infinite sum > of members of that field. The rule is: pi^2/6 cannot be expressed as the quotient of two integers. Another possible rule is that pi cannot be represented as a finite sum of reciprocals of integers. > Is that it? Mathematicians simply exclude infinite sums from the > field of rationals? No - rationals and integers themselves can be represented as infinite sums of raionals. Or do they rely on the definition of a rational > as the ratio of a/b, where 'a' and 'b' are integers? That's the usual definition (plus b <> 0). > Continuing in that direction, recently a leading mathematician at a > major university in the United States of America (a top 20 > university) sent me an email stating that my rule of no other integers > being units except -1 and 1 did not exclude pi if you used Z[pi]. > I said it did in the following reply (Professor's name omitted): > Professor ****: > > You assertion is easily proven false. Please consider the following. > > infinity. > Properly one might say pi^/6 = limit as k --> infinity of (1 + 1/4 + 1/9 + 1/16 + ... 1/k^2). > But then you have > > pi^2/6 = 1 + 1/4(1 + 1/4 + 1/9 +...1/k^2) + 1/9(1+1/4 + > 1/9+...)+...1/k^2(1+k^2), which is > Looks wrong to me. First, the right side is not an inifinite series; therefore it is rational. So there must be an implied limit. But then the right side is easily seen to be bigger than the left side. That last part in particular, 1/k^2(1 + k^2), is > 1. Add to that the 1 leading off the series, and the result is > 2. But pi^2/6 = 1.6449 ... < 2. > pi^2/6 = 1 + pi^2/24 + pi^2/54 +...pi^2/6k^2, > > multiplying out and collecting to the left except for 1, > > 6(24)(54)...(6k^2) pi^2 [24(54)...(6k^2) - 6(54)...(6k^2) - > 6(24)...(6k^2) - ... -6(24)(54)...] = 1, > > which proves that you have an infinite number of units, some of which > are 6, 24, and 54, which is the result if you include pi in a ring > with integers, so my definition *does* exclude it. This is an ill-defined confused mess. If you are implicitly taking limits, then the left side is (infinity) - (infinity), which is indeterminate. > Well he replied: Actually, Z[pi] has no units save 1, -1, as a consequence of > the fact that pi is transcenddental (not algebraic). Z[pi] does not > contain the numbers you are considering above, which (as I read it) > are obtained by summing infinite series. The elements of > Z[pi] are just those real numbers that can be expressed as > f(pi) where f(x) is a FINITE polynomial with integer coefficients. Since pi is transcendental, Z[pi] is isomorphic to Z[x], i.e., polynomials in an unknown x. > Do you agree with the professor, who I remind is a *leading* > mathematician? On this last part, yes, unless you are misquoting him. What difference does it make that he is *leading* ? Nora B. > James Harris Subject: Re: Field of rationals and pi Visiting Assistant Professor at the University of Montana. [.snip.] > transcendental; transcendant is not a word. > EVERY positive number, whether rational, irrational, integer, >transcendental, algebraic, or whatever can be represented as >an infinite sum of reciprocals of integers. Why this unwarranted discrimination against the negatives? Surely every negative real number x may also be obtained that way, just by changing the signs of the expression for -x? I guess 0 might be a bit trickier, of course, but luckily it is a finite sum of reciprocals of integers, so I vote we let it into the club as well. (-: [.snip.] ============================================================== ======== Why do you take so much trouble to expose such a reasoner as Mr. Smith? I answer as a deceased friend of mine used to answer on like occasions - A man's capacity is no measure of his power to do mischief. Mr. Smith has untiring energy, which does something; self-evident honesty of conviction, which does more; and a long purse, which does most of all. He has made at least ten publications, full of figures few readers can critize. A great many people are staggered to this extend, that they imagine there must be the indefinite something in the mysterious all this. They are brought to the point of suspicion that the mathematicians ought not to treat all this with such undisguised contempt, at least. -- A Budget of Paradoxes, Vol. 2 p. 129 by Augustus de Morgan ============================================================== ======== Arturo Magidin magidin@math.berkeley.edu Subject: Re: Four Color Graph. >> Graph X has chroma = 4, >> What is the probability that X is planar if n = 10? n = 100? > you are saying that every distinct graph is equally likely then, > but you haven't told us what a distinct graph is (specifically, are you > talking about labeled or unlabeled graphs? connected or [maybe-not]? My comments here are not going to be very helpful. But I have never much found picking one uniformly from some set of graphs very convincing as a random graph generator, even when the set is given by some fixed number of vertices and/or edges, labelled or not. It strikes me that the random generation of the graphs ought to have something to do with the context, and that is entirely possible here. The thought I have, necessarily considers the graphs as being labelled, (which is usual)but that flows naturally from the situation, as we will see. So I was playing around with a few small graphs, on this problem, and after noting that given 5 vertices and chroma = 4 there was ZERO probability that it would be nonplanar, ;-) , and that given 6 vertices ditto there was clearly positive probability, the following occurred to me. I had been considering graphs obtained by fixing the number (and labels) of the vertices, and keeping on adding edges at random, until the chroma first became 4, then checking for planarity. Then it occurred to me that one might also start with the *complete graph* (rather than the empty graph, as I had been doing), and *subtract* edges under the same conditions. It seemed clear that the probabilities would be different. So then I thought, well then, there's a stretch in between these two procedures, so we should choose randomly WITHIN it as well. So here's what I finished up with. Given n, and n vertices a,b,c,...,z: FIRST choose at random an ordering from the factorial(n choose 2) possible; imagine all these edges (from the left of the ordering list) being added to the empty graph, OR (equivalently) imagine the edges (from the right) being subtracted from the complete graph; and under each edge note the chroma of the graph thus obtained. It will rise monotonically from left to right, starting with 0,2,... and finishing with n. In between there will be an interval of case where the chroma is constantly 4. Now... SECOND choose one graph at random from this interval, & check its planarity. Its strikes me that that is the appropriate random process for determining the original desired probability. Of course it's a helluva job, even with machine help, and probably a helluva more of a job to program a fully automatic procedure for. But then, I SAID this wasn't going to be a very helpful contribution... -------------------------------------------------------------- -------------- -- Bill Taylor W.Taylor@math.canterbury.ac.nz -------------------------------------------------------------- -------------- -- Have you paid your random road tax recently? If not - then visit your local speed camera! -------------------------------------------------------------- -------------- -- Subject: Re: Four Color Graph. <962b628d.0308140951.1c2519a4@posting.google.com> <962b628d.0308161526.4cb0d4a@posting.google.com> > The vertices are labeled and connected in a closed ring; ie, polygon > by 'n' edges. This polygon is a subgraph in every graph. Distinct > graphs are created by unique combinations of diagonals. Each graph > will have exactly 3n-6 edges. Only simple graphs are considered. > The chromatic number for any specific graph may be 2, 3 or 4 but never >4. > To clarify the confusion about different paths; each specific diagonal > combination represents one and only one distinct graph. This is not the same problem you said before... you were asking about all planar graphs, and now you are restricting yourself to triangulated polygons. For a polygon on a fixed number of vertices (equivalently, edges) it is a homework problem to show that the number of distinct triangulations is equal to a certain Catalan number. Calculating Catalan numbers is much easier than couting all (connected) planar graphs. J Subject: Re: Four Color Graph. <962b628d.0308140951.1c2519a4@posting.google.com> <716e06f5.0308162047.5973b10d@posting.google.com> > It strikes me that the random generation of the graphs ought to have > something to do with the context, and that is entirely possible here. Indeed, there usually is a context and some associated informal understanding of what a properly randomly generated graph would be for whatever the issue at hand... there is a whole study around such ideas. > So I was playing around with a few small graphs, on this problem, and after > noting that given 5 vertices and chroma = 4 there was ZERO probability > that it would be nonplanar, ;-) , and that given 6 vertices ditto > there was clearly positive probability, the following occurred to me. Well, for something to be nonplanar, it has to have a subdivision of a K_5 or a K_{3,3} ... AND the graph needs to be 4-chromatic, so for these conditions to be met, the graph will have to have quite a few vertices. > SECOND choose one graph at random from this interval, & check its planarity. > Its strikes me that that is the appropriate random process for determining > the original desired probability. Is it also surprising if you have a collection of all 4-chromatic graphs and then randomly pick one and check its planarity, it would be a good representative of the desired probability? J Subject: Harmonic dynamics Is anybody familiar with the concept harmonic dynamics? It is claimed to relate to the overlapping of coins dropped in a cup. Thus it might be related to occupancy problems. It is said to be used in some form of physics . Is it part of probability theory? Many thanks in advance. Stig Holmquist Subject: Help! Algebraic Topology Question: Given two maps f,g: X -> Y show that f~g (homotopic) if Y is contractive. I can't come up with answer. please help me Subject: Re: Help! Algebraic Topology |Question: |Given two maps f,g: X -> Y show that f~g (homotopic) if Y is contractive. | |I can't come up with answer. |please help me it seems difficult to think of a good hint offhand without completely giving the answer away. it depends to some extent on what your mathematical background is, though; for example if you have any background in category theory or groupoid theory then a good hint is: first solve the analogous problem for groupoids in place of topological spaces. i suppose another hint would be: remember that the definition of contractive (or contractible as i usually hear it called) involves maps from y to y. so try to figure out how maps from x to y (like f and g in the problem) might interact with maps from y to y. -- [e-mail address jdolan@math.ucr.edu] Subject: Re: Help! Algebraic Topology > Question: > Given two maps f,g: X -> Y show that f~g (homotopic) if Y is contractive. > I can't come up with answer. > please help me The response by James Dolan included this sentence: it seems difficult to think of a good hint offhand without completely giving the answer away. which is a very good hint (although it does appear to say hardly anything at all). Once you understand the essence of contractibility for Y, the problem will be nearly trivial. At any rate, the problem is very simple, and if it's not appearing solvable, you may be looking either too hard, or in the wrong place altogether. Contractibility of Y is the only feature to be concerned with, and its application in this problem is virtually a direct citation of the definition. Perhaps a little hint can be provided, without completely throwing the solution out there: Rather than showing f~g directly, think of an obvious map h:X --> Y (keeping contractibility of Y in mind) to which every map from X to Y is homotopic. Then f~h, g~h, and you can get f~h~g ==> f ~ g. Dale. Dale Subject: Re: Help! Algebraic Topology >> Question: >> Given two maps f,g: X -> Y show that f~g (homotopic) if Y is contractive. > >> I can't come up with answer. >> please help me > >The response by James Dolan included this sentence: > it seems difficult to think of a good hint offhand > without completely giving the answer away. >which is a very good hint (although it does appear to say hardly >anything at all). You and James really should get directly to the point; there's no two ways about it. Lee Rudolph Subject: Re: HouseSale.NET Whoa $195 with 8 hours to go, no use staying up I guess, email now for prime virtual estate in real estate. Lets see 100,000 hits a day X 0.01% sales X 10% commision X average house price ($200,000) = ... $200,000 per day income, going for $195. _ _ _ _ _ _ _ _ _ _ _ _ _ _ and $20,000 of that goes to the 10 webmasters who referred the buyers! Herc www.A1Sites.com Subject: Re: HouseSale.NET Can I spot 'em, if a different snap registrar snapped it someone would have picked it up for $60. Herc Subject: How to prove (conclude) this? a -> b [alpha] is equivalent to ~a V b [beta] ...but how can one conclude the one from the other? I only have one rule for elimination of implication and that's: {1} a -> b {2} a {1,2} b and this doesn't get me where i want. Note that i don't want to use truth table for this. I'd like to deduce [alpha] from [beta] using the rules of introduction/elimination of conectives. By other words - how do we know that [alpha] <-> [beta] if we're not to use truth table? -- V.8anligen Konrad --------------------------------------------------- Sleep - thing used by ineffective people as a substitute for coffee Ambition - a poor excuse for not having enough sence to be lazy --------------------------------------------------- Subject: Re: How to prove (conclude) this? > a -> b [alpha] > is equivalent to > ~a V b [beta] > ...but how can one conclude the one from the other? > I only have one rule for elimination of implication > and that's: > {1} a -> b > {2} a > {1,2} b > and this doesn't get me where i want. > Note that i don't want to use truth table for this. > I'd like to deduce [alpha] from [beta] using the > rules of introduction/elimination of conectives. I need to know what rules you have for inferences. I don't remember them offhand. Less formally, the equivalence is shown as follows. Claim: a->b implies (~a or b) Proof: Suppose a->b. Case 1. Suppose ~a. Hence, ~a or b. Case 2. Suppose a. Then, b since a->b. Hence, ~a or b. Claim: (~a or b) implies a->b Proof: Suppose ~a or b. Suppose a. Then, b since ~a or b. Hence, a->b. -- Bill Hale Subject: Re: Mathematicians Living in the Past 7e3Z2dkQEBC5ubm60r3HAAACeUlEQVR4nIXTQW+ bMBQAYChLfcWzUq5pipUr7lPptUNGXBMC8zUZ qnt1aNz392c7pGXSpvkSyV/ee37POFr/Y0X/g56rxd+ gb2IQC8X79brdLtovaBgACIsRMVTr4hN4 Ah54xy0rtdbjFRCAupizIhUDB69X2IHAGKCrxyrxcJyA76CQKwZn2ckEXK6h/ YwoOld/zJUMEYds ghMIYhj0kjehxqGYFacgalk1PpM+jBOkvg0othFR4bjDVHyjQx+FZ+ b29fMEe114SIC6UB/RXoCX enAhGQORAdxp/TaN5N63lNqGJZACO2n9/ QL81qfVQjEXAkBKPyoPm7BfPqFLJ0qBt1fwmfTAFiYW MbAC1WmCnyEitcYImtI07fY3FwglBuwMUiqoiLv98gL+ 4PrNSmXdWGJK7Q4D8JBpWeXGxi6CpqOJ QoOrkOksKzMaRinN7LsI8OChsGOl+ogiE8SeaIAP9/ 9BRIs1xzgzESWteQ7gDgWQpmRdmcgtNFYJ / 5W4CWpRRsTyHE2UBgifjzvUQMGMOfYYO1CI7wFWfhwsWXSNq0qQcMQkgBvhADvI rOrduRArpW7D Rflb0gjHvO5kpUhX93h3DJ3v9VB+S8pMulW1kis73aAqh/ LpBYalbA2qiLZKP16G+KHLowIYojSm UTys+/ LrogaFVFTKpN2P45qT6UX58qrbEFcil81h9tQSfVSyIZ0v3zzNYEMPVtbR2e3XL 8v547w/ uDymk7wfXx7n8PBscrm1fW/G1a8/gKiz3GJeYbt5ncMJ+ Y1ckVxi19zMwaIcXXUreb6yc1gQ9ypr tLKSCueQjQ4qxFzKXo0zwNDbxv/U2ynkN+Dj/u9oA6vhAAAAAElFTkSuQmCC > It is as simple as this: if Cantor's theorems are really the absolute > truths some people hold them to be, then you will provide a proof of > them using nothing but the postulates of Euclid. But I defy you to do > this. > I say again, these posters live in the past. >> You have undoubtedly proved to me that some posters live in the past. > Wasn't it meant as irony, then? I thought it was a parody of the > recent thread about Cantor's influence on math. To be honest, I haven't any idea whether it was parody. Deeth is a Nathan the Great is not a serious pseudonym), but I think he genuinely has a bug up his butt regarding Cantor. I have no idea whether he really believes that mathematical axioms are supposed to be self-evident truths or not. If he was being funny and I stepped in like a doofus, then I'm embarrassed. -- So, at this time, I'd like to assure you that I am not interested in making sure mathematicians worldwide get fired. I've rethought my desire to go to Congress and try to get funding for mathematicians cut. -- James Harris is a reasonable man. Whew! Subject: Modular Arithmetic Hey Guys, I'm reading through some lecture notes and have come across something I don't understand.... 0 mod m. What is the point of 0 mod m? Surely the answer is just 0 regardless of the value of m? The notes state things like: Using the relation: (a-3)^2 = 0 mod m at one point. And for the equation Xn+1 = aXn mod p; If X0 != 0 mod p and a is a primitive root mod p then the sequence has period p-1. In my mind 0 mod p = 0 everytime. It also uses 0 mod 1 later on, but surely that is 0 as well? Am most confused, and trawled through the web looking for an answer but didn't come across one. Actually not true, there is some stuff on congruence which looked interesting, but it always used brackets in its notation whereas these notes never do and don't even mention this congruence relation. Ahh nevermind, if any of you can help I'd be most appreciative. J Subject: Re: Modular Arithmetic If x = 0 mod m, then x is a multiple of m. For example 16 = 0 mod 4, so 16 is a multiple of 4. hth Guy > Hey Guys, > I'm reading through some lecture notes and have come across something I > don't understand.... 0 mod m. > What is the point of 0 mod m? Surely the answer is just 0 regardless of the > value of m? The notes state things like: > Using the relation: > (a-3)^2 = 0 mod m > at one point. > And for the equation Xn+1 = aXn mod p; > If X0 != 0 mod p and a is a primitive root mod p then the sequence has > period p-1. > In my mind 0 mod p = 0 everytime. > It also uses 0 mod 1 later on, but surely that is 0 as well? > Am most confused, and trawled through the web looking for an answer but > didn't come across one. Actually not true, there is some stuff on > congruence which looked interesting, but it always used brackets in its > notation whereas these notes never do and don't even mention this congruence > relation. > Ahh nevermind, if any of you can help I'd be most appreciative. > J Subject: Re: Modular Arithmetic Visiting Assistant Professor at the University of Montana. >Hey Guys, >I'm reading through some lecture notes and have come across something I >don't understand.... 0 mod m. The short, colloquial answer, is that 0 mod m means multiple of m. The slightly longer explanation is the following: fix a nonnegative integer m. Then we define a binary relation between integers as follows: we say that a=b (mod m) (a and b are congruent modulo m; most books will replace the = with a symbol with 3 lines, but ASCII does not have it) if and only if m divides a-b. Then it is easy to verify that if m=0, this is just the usual equality; if m>0, then a=b (mod m) if and only if a and b leave the same residue when divided by m. Remember that the residue must be an integer between 0 and m-1, inclusively. Now, this is an equivalence relation; that is, for all integers a,b,c, a=a (mod m) [reflexivity]; if a=b (mod m) then b=a (mod m) [symmetry]; and if a=b (mod m) and b=c (mod m), then a=c (mod m) [transitivity]. It is also easy to verify that every integer is congruent modulo m to one and only one of 0,1,2,3,...,m-1 (namely, to its residue modulo m). a modulo m; or to mean the unique integer between 0 and m-1 which is congruent to a modulo m, depending on the context. >What is the point of 0 mod m? Surely the answer is just 0 regardless of the >value of m? The notes state things like: >Using the relation: >(a-3)^2 = 0 mod m >at one point. What they are saying is that you know that (a-3)^2 is a multiple of m, presumably from some hypothesis prior to that statement. >And for the equation Xn+1 = aXn mod p; >If X0 != 0 mod p and a is a primitive root mod p then the sequence has >period p-1. This is a bit more confusing to me... >In my mind 0 mod p = 0 everytime. I think you are confusing the notation with the meaning of mod in computer science. There, a mod p means divide a by p and take the integer between 0 and p-1 which is congruent to a modulo p. ============================================================== ======== It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) ============================================================== ======== Arturo Magidin magidin@math.berkeley.edu Subject: Re: Modular Arithmetic > Hey Guys, > I'm reading through some lecture notes and have come across something I > don't understand.... 0 mod m. > What is the point of 0 mod m? Surely the answer is just 0 regardless of the > value of m? The notes state things like: > Using the relation: > (a-3)^2 = 0 mod m > at one point. I think I see the source of the confusion. The mod m part modifies the entire equation, not just the right hand side. That is, the statement 6 = 1 (mod 5) means that 6 and 1 have the same remainder when divided by 5. If we were to say 5 = 0 (mod 5) we would NOT be saying that 5 = 0, but rather that 5 and 0 have the same remainder when divided by 5. If you were to apply your reasoning to 5 = 0 (mod 5) you would say that 0 mod anything is 0, so the statement reads 5 = 0. But that is of course false. The best way to think about mod is to go back to the definition, which is that a = b (mod 5) is true when a - b is divisible by 5. the common computer programming use of the mod operator, in which 6 mod 5 equals1. This is related to the mathematical usage, but is subtly different, as your example shows. Subject: Re: Near-integer values of polynomials on integers >I tested some small values of f(n) = a + b n and f(n) = a + b n + c n^2 >where a=pi, b=e, c=sqrt(2). It certainly does appear from a plot that >the points of the form ( log(n), n | { f(n) } | ) are uniformly >distributed in the region (0, infty) x (0, 1) (where {x} = x - round(x) >is the distance to the nearest integer). >Here are the values of n < 3.10^6 where f(n) is within 1/n of an >integer: > {1, 2, 3, 4, 8, 11, 29, 36, 75, 107, 178, 501, 572, 1037, 2038, 3039, > 4040, 11583, 20127, 29672, 47761, 65850, 83939, 256285, 446720, > 655244, 1054203, 1453162, 1852121, 2251080} >for the linear function Looking for n so that f(n) is within 1/sqrt(n) of an integer seems a lot easier. For example, n=37951980991 works. Rich Burge Subject: Re: Negative Binomial Distribution > Does anybody know the markov chain of a Negative Binomial Distribution with > given parameter (a,b)? If a=P{success} and b is a positive integer, the Markov chain has states {0,1,2,...,b} and the transition probabilities are go from i to i+1 with probability a, stay in i with probability 1-a for i Con(ZF + not AC + not CH)? When you deny AC, what is your statement of CH? Since cardinals could be incomparable, we don't know what you mean by CH unless you tell us. Subject: Re: not AC + not CH [Sorry if this is a repeat. My newsserver has been acting up lately.] > Con(ZF + not AC + not CH)? > >When you deny AC, what is your statement of CH? Since cardinals could >be incomparable, we don't know what you mean by CH unless you tell us. > Isn't CH the statement that, for all X, if there exist injections from N to X and from X to P(N), then there exists either an injection from P(N) to X or one from X to N? And is there any version of CH such that Con(ZF + not AC + not CH) is either false or unknown? -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu Subject: order of an element in a group [re-post] I got a message saying my post didn't go through (if this is a repeat I apologize). Let a,b,c,d be integers, and suppose <(c,d)> is a subgroup of Z/aZ x Z/bZ. What is | (Z/aZ x Z/bZ)/<(c,d)> | in terms of a,b,c,d? Subject: Re: order of an element in a group [re-post] in message <2e24e317.0308162228.70346a0d@posting.google.com>: > Let a,b,c,d be integers, and suppose <(c,d)> is a subgroup of Z/aZ x > Z/bZ. What is | (Z/aZ x Z/bZ)/<(c,d)> | in terms of a,b,c,d? I'll assume you're asking for the order of the quotient group, not of an element. Below, GCD is the greatest common divisor and LCM the least common multiple. |(c,0)| = a/GCD(a,c) == a/x, and likewise |(0,d)| = b/GCD(b,d) == b/y, so |(c,d)| = LCM(a/x,b/y). Since <(c,d)> is cyclic, |<(c,d)>| = |(c,d)|, and thus |(Z/aZ x Z/bZ)/<(c,d)>| = |Z/aZ x Z/bZ| / |(c,d)| = ab/LCM(a/x,b/y) = xy*GCD(a/x,b/y). -- Jim Heckman Subject: Orders in the group GL(n,2) For m>=2 what is the least n such that the group GL(n,2) has an element of order m ? Subject: Re: Orders in the group GL(n,2) > For m>=2 what is the least n such that the group GL(n,2) has an > element of order m ? Is seems that for odd n it is the multiplicative order of 2 mod m . Bill Subject: Re: Paraboloidal Coordinates Parameterize with respect to a single parameter. e.g.,for a parabola x= 2 a t ; y= a t^2 better than y^2 = 4 a x . Subject: Re: Paraboloidal Coordinates > I've been looking at some coordinate systems on Mathworld, and for some of > them we get x, y, z in terms like > x^2 = (a^2-lambda)(a^2-mu)(a^2-nu)/(b^2-a^2) > etc. > What's with the squared quantities on the left-hand side? We're only > supposed to work in an octant of three-dimensional space? Can't > distinguish x=10 from x=-10? This is true for many coordinate systems (bipolar for instance). Like all things in math (and life) choice of a suitable coordinate system requires some judgement. These systems are only useful for symmetrical graphs, so f(10) = f(-10) and the distinction just doesn't matter. If the symmetry is not present, just choose a different coordinate system. Darren Subject: Partition numbers divisible by n Is it true that for every integer n>=2 there is some partition number p(m) divisible by n ? Subject: Re: Physical repn in noneuclidean spaces? > > When you have a physical, curved object you want to do a construction > on, how about using a string made sticky with some kind of glue? > Assuming, in your words, the student has a steady hand. --- < snip > --- > Assuming the student has a > steady mind. (chuckle) Ted Shoemaker shoemakerted@yahoo.com Subject: Please Help With Simple Median Question I hope I have a correct newsgroup, I apologize if not. My question are: Given the following scores: 2,3,5,8,8,9,11 find the median. Several books (and Excel) suggest the answer is 8. However, one book I have suggest that the answer is 7.75, using the reasoning that there are 3 scores below the value of 8 and 2 scores above value 8; hence 8 is not a true median. This book solves this dilemma by subdividing 8 into two scores with the limits of the first score 7.50 - 8.00 and the second score 8.00 - 8.50. So now the array is 2, 3, 5, 7.75, 8.25, 9, 11 (7.75 & 8.25 are the mid points of the limits listed above) so now they book suggest the answer is 7.75. Which is the technically correct answer -- 8 or 7.75? I am preparing for a CLEP exam, which answer would I use for this type of exam? I thank you in advance!! Subject: Poincare's work in quantum mechanics Although it's rather well known that Henri Poincare anticipated a number of results in special relativity prior to Einstein's 1905 publication, it seems that fewer people are aware that Poincare also played a role in another revolution in physics at the beginning of the 20th century -- namely, quantum mechanics (I guess there are some people who might also argue that he participated in another revolution for his pioneering work in dynamical systems). Recently, I had the good fortune to come across Russell Volume 58 (191); 1967; pages 37-55). Besides discussing Poincare's work, it offers some fascinating glimpses into the first Solvay Conference that took place in October and November of 1911. Below are a few (six) excerpts that interested in the historical development of physics during this time period: -------------- 1. At the time, Maurice de Broglie remarked to F. A. Lindemann that of all those present Poincare and Einstein were in a class by themselves (p 40). 2. Lorentz recalled that in the discussions Poincare had shown all the vivacity and penetration of his spirit, and that one had admired the facility with which he entered vigorously into even those questions of physics which were new to him (p. 40). 3. ... it was Planck, however, who stimulated Poincare's most penetrating, questioning spirit .... He twice pressed Planck to give good grounds for deciding among the several possible ways of decomposing phase space into the finite elementary areas for the probability calculations. He wanted to know how the energy of a system of several degrees of freedom might be quantized, since the one-dimensional quantization procedure was incompatible with transformations of axes in higher-dimensional systems. Poincare regretted that as yet there had been no discussion of mechanisms for the interaction of fixed resonators; for in the absence of any definite mechanism there could be no exchange of energy between radiations of different frequencies, and therefore no final equilibrium. Planck had stressed quanta of action rather than quanta of energy . . . but he did not know what it means to speak of the conservation of action. Finally, he was skeptical of Planck's new formulation of the radiation theory, according to which the absorption of energy by the resonators varies continuously with time (p. 41). 4. In a descriptive essay he spelled out the essence of Planck's theory as it appeared to him: A physical system is capable of only a finite number of distinct states; it jumps from one of those states to another without going through a continuous series of intermediate states. The image of a physical system jumping from one discrete state to another put him in a speculative certain allowed paths in phase space, shifting discontinuously between them. And he supposed that the universe as well as an electron ought to experience quantum jumps. Since there would be no distinguishable instants within the motionless states between universal jumps, there should exist an atom of time. Such were the kinds of ideas going through Poincare's mind shortly before he died; there was nothing timid or grudging about his late acquaintance with the quantum theory (p 50). 5. ... above all it was the unquestioned authority of Poincare in mathematical matters which secured him an attentive audience. Jeans undoubtedly voiced a majority sentiment when he said that we shall probably feel inclined to trust to the accuracy of Poincare's mathematics. (pp 51-52). 6. Whereas Jeans had strongly opposed the quantum theory in Brussels . . ., he came out vigorously in support of quanta at the Birmingham meeting of the British Association in September 1913, fourteen months after Poincare's death. There is no doubt about what caused him to change his mind. Jeans had read Poincare's paper and been converted by it. ... The French scientist's arguments had been so completely persuasive that from this time on every theory would have to logically involve either the belief that Poincare is wrong, or the belief that he is right, together with all that this involves. . . . And Jeans himself felt compelled to accept the quantum hypothesis in its entirety. (p. 53). -------------- As an undergraduate, I had become aware that Poincare's paper Sur la 1912, while undergoing an operation) had been influential in gaining wider acceptance for Planck's then-controversial quantum theory of blackbody radiation. The brief historical excerpt I had read at the time implied that Poincare proved (essentially) that Planck's theory required the existence of discrete energy quanta. accurate statement. Instead, it would appear that Poincare's proof (modulo some necessary refinements [see page 52 of McCormmach]) was based on some debatable assumptions. My main questions regard the legitimacy of Poincare's proposed mechanisms whereby pairs of resonators exchanged energy (page 45 of McCormmach). There are other questions that one can raise as well (as some of Poincare's contemporaries did). I haven't read Poincare's original paper (my French not being particularly strong), so I'm wondering if anyone who is familiar with Poincare's work in quantum mechanics can comment on whether Poincare's legacy in quantum mechanics is either both A and B or just B for the following statements: (A) A legitimate proof that Planck's theory required the existence of discrete energy quanta (in spite of working off of a physical foundation that predated Heisenberg and Schroedinger's work in QM by 13-14 years). (B) Helped to gain further acceptance for the theory of quantum mechanics among physicists circa 1912. - Mark p.s. - While on the topic of Poincare, can anyone comment on whether there is general concurrence on whether Perelman's third paper successfully finishes off the Geometrization conjecture (and thereby the Poincare conjecture)? Subject: program in algebraic/topological automata hello, i am hoping the more knowledgable folks out there can help me out on this. which schools, if any, in north america, offers a cs grad program with emphasis on algebraic and topological considerations to the theory of computation? thanks in advance to anyone who takes time to reply. M.T. Subject: Re: Q: automorphisms of S_6 (was: finite groups) in message <3F3EA39B.B5FA3A27@utu.fi>: [...] > So it just is a remarkable coincidence that there are two > conjugacy classes of this same size, when n=6. > If my calculations are correct, then it seems to me that > another pair conjugacy classes of S_6 of the same size, > namely the classes of (1234)(56) and (1234) remain fixed > under the outer automorphisms. No reason why they should > change, but the classes of (123) and (123)(456) do get > swapped, so I was mildly surprised to see this pair stay > fixed. Nothing deep in their probably. Well, since A_6 is the unique subgroup of index 2 in S_6, no automorphism can swap odd and even elements such as (1234) and (1234)(56). There are also interesting swaps of subgroups, including one that doesn't swap any element classes: The class of the Klein 4-group <(12)(34),(13)(24)>, with support 4, swaps with that of <(12)(34),(12)(56)>, with support 6. [...] -- Jim Heckman Subject: Question: what is a conjecture? in order that those who may have killfiled the Borel thread will have the chance to killfile something else, or contribute if they wish. In , >For example, Serre does not make conjectures as normal mathematicians do, >but instead raises questions. One example I encountered at close range ... >Clearly, there is no proper metamathematical definition of a question, >but only of conjectures. Clearly I am not even an improper metamathematician, but I am having great trouble imagining (what you suppose to be) a proper metamathematical definition of a conjecture, and equally great trouble imagining why (you should suppose) that it's clear (or even true though obscure and difficult) that, given a proper metamathematical definition of a conjecture, it should not be possible to modify it into a proper metamathematical definition of a `question'. I need to know the answer very soon, since in a paper I am trying to get ready to submit (not, I must admit, to a journal edited by either J-P Serre or Hans Aberg) I have been bold enough to include several Questions. I call them that because I don't know the answers, and don't have much reason to favor any of the possible answers over the others--if I did, I might call them Conjectures. But my choice of words seems to me to be (quite properly) dictated by non-mathematical *and* non-metamathematical circumstances. Conjecture: I am not going to get a satisfactory answer to the question raised in this post, at least not from Hans Aberg (and I'm not in correspondance with J-P Serre, who also is not--it seems--an active participant in sci.math, possibly because he views it as a career-limiting move; though who knows, perhaps he is using some handle like Mike Deeth to while away his leisure hours). Lee Rudolph Lee Rudolph Subject: Re: Question: what is a conjecture? |in order that those who may have killfiled the Borel thread will |have the chance to killfile something else, or contribute if they |wish. | |In , | |>For example, Serre does not make conjectures as normal mathematicians do, |>but instead raises questions. One example I encountered at close range |... |>Clearly, there is no proper metamathematical definition of a question, |>but only of conjectures. | |Clearly I am not even an improper metamathematician, but I am |having great trouble imagining (what you suppose to be) a proper |metamathematical definition of a conjecture, and equally great |trouble imagining why (you should suppose) that it's clear (or |even true though obscure and difficult) that, given a proper |metamathematical definition of a conjecture, it should not be |possible to modify it into a proper metamathematical definition |of a `question'. | |I need to know the answer very soon, since in a paper I am trying to |get ready to submit (not, I must admit, to a journal edited by either |J-P Serre or Hans Aberg) I have been bold enough to include several |Questions. I call them that because I don't know the answers, |and don't have much reason to favor any of the possible answers |over the others--if I did, I might call them Conjectures. But |my choice of words seems to me to be (quite properly) dictated |by non-mathematical *and* non-metamathematical circumstances. | |Conjecture: I am not going to get a satisfactory answer to the |question raised in this post, at least not from Hans Aberg (and |I'm not in correspondance with J-P Serre, who also is not--it |seems--an active participant in sci.math, possibly because he |views it as a career-limiting move; though who knows, perhaps he |is using some handle like Mike Deeth to while away his leisure |hours). how about: a question is an unordered pair of conjectures, each the negation of the other. -- [e-mail address jdolan@math.ucr.edu] Subject: Re: Question: what is a conjecture? >how about: a question is an unordered pair of conjectures, each the >negation of the other. In case conjecture C is its own negation, does that make the corresponding question C or {C}, in your set theory? Of course, if your set theory is ill-founded, C and {C} might be identical! Lee Rudolph Subject: Re: Question: what is a conjecture? ... > |>Clearly, there is no proper metamathematical definition of a question, > |>but only of conjectures. > | > |Clearly I am not even an improper metamathematician, but I am > |having great trouble imagining (what you suppose to be) a proper > |metamathematical definition of a conjecture, and equally great > |trouble imagining why (you should suppose) that it's clear (or > |even true though obscure and difficult) that, given a proper > |metamathematical definition of a conjecture, it should not be > |possible to modify it into a proper metamathematical definition > |of a `question'. > | > |I need to know the answer very soon, since in a paper I am trying to > |get ready to submit (not, I must admit, to a journal edited by either > |J-P Serre or Hans Aberg) I have been bold enough to include several > |Questions. I call them that because I don't know the answers, > |and don't have much reason to favor any of the possible answers > |over the others--if I did, I might call them Conjectures. But > |my choice of words seems to me to be (quite properly) dictated > |by non-mathematical *and* non-metamathematical circumstances. ... > how about: a question is an unordered pair of conjectures, each the > negation of the other. ... I too make no claim of being a proper metamathematician, but nevertheless think that's an excellent definition of mathematical question.* However, try substituting issue or notion in place of question. The definition would fall down in either case, although all four terms might be applied to some mathematical ideas. -jiw * except for the drawback that the theoretically unordered elements of an unordered pair are in practice listed perforce in one order or another Subject: Re: Question: what is a conjecture? >how about: a question is an unordered pair of conjectures, each the >negation of the other. > In case conjecture C is its own negation, does that make > the corresponding question C or {C}, in your set theory? C or not-C, that is the question. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) Subject: Re: Question from Chaos |I was recently reading through Chaos by James Gleick (finally - after all |these years !), and in a paragraph on the dichotomy between mathematicians |and physicists, he says | |Smale could be happy with an example like this : take a number, a fraction |between zero and one, and double it. The drop the integer part, the part to |the left of the decimal point. Then repeat the process. *Since most numbers |are irrational and unpredictable in their fine detail*, the process will |just produce an unpredictable sequence of numbers. | |What does he mean by that ? - what is unpredicatable about numbers in their |*fine detail*. i don't know exactly what he meant by that (for a number of reasons including that i didn't read that book). but what he should have meant by it is something like this: think about the numbers involved here as being expressed in binary notation. since these numbers are between zero and one they have no digits (or bits, really, since we're using binary) to the left of the decimal point (or binary point, really, since we're using binary). then the dynamical process described here works like this: strip off the leading bit of the number, the one just to the right of the binary point, and push all the other bits one position to the left to make up for it. so for example: .01010010101010010101010101110101... turns into: .1010010101010010101010101110101.... which then turns into: .010010101010010101010101110101..... and so forth. notice that the strings of dots here are creeping leftward and will soon completely take over! that's really the whole point of what gleick is saying. those strings of dots represent our ignorance of the least significant bits of the numbers. the dynamical process described has the effect of promoting the previously least significant bits of the numbers into more significant bits and eventually into the most significant bits, and so the new most significant bits of the numbers seem completely unpredictable and random because in general there's no necessary connection between the most significant bits and the least significant bits of randomly chosen numbers. the least significant bits represent information on an extremely microscopic scale that's constantly being transferred to a more macroscopic scale by the particular dynamic process being studied, and so if you're watching a pictorial representation of this microscopically invisible random fluctuations are constantly magnified into macroscopically visible differences. if for some strange reason the number you're applying the dynamical process to _does_ exhibit some sort of correlation between its most significant bits and its least significant bits- say for example that the bits follow a simple eternally repeating pattern of some sort- way. but gleick's point is that such correlation between most significant bits and least significant bits is exceptional; there's just no reason why numbers in general are required to exhibit any such correlation. with randomly chosen numbers you just never know what the next bit is going to be until you look at it. -- [e-mail address jdolan@math.ucr.edu] Subject: Re: Question from Chaos |notice that the strings of dots here are creeping leftward and will |soon completely take over! that's really the whole point of what |gleick is saying. those strings of dots represent our ignorance of |the least significant bits of the numbers. ^^^^^^^^^^^^^^^^^^^^^^ aka the fine detail of the numbers. -- [e-mail address jdolan@math.ucr.edu] Subject: Re: questions about the Dedekind construction of R > 1) R was defined as the set of all cuts. Like most people, I normally > think of a real number as something with an infinite decimal > expansion. It appears to me that a decimal expansion is not the same > as a cut. How is this problem resolved? > I don't see the problem. First you construct R, then you show that every > element of R has a decimal expansion. Easier said than done. Hence, I would like to know how to prove something like: For every cut, there is a corresponding decimal expansion. > 2) Rudin showed that there is an isomorphism between Q, and the > ordered field Q*, whose elements are the rational cuts, which he > defined. He then argued that therefore Q is a subfield of R. I agree > that Q* is a subset of R, is a field, and hence is a subfield of R. I > don't understand why Q is therefore a subfield of R. How is Q a > subset of the set of all cuts? > no means the same as r but the properties we are concerned with (arithmetic > and order) are the same in the two fields and It is this identification > of Q with Q* that allows us to regard Q as a subfield of R. Are we both speaking English here? How is regarding Q as a subfield of R different from saying that Q is a subfield of R? The theorem Rudin is proving states that R contains Q as a subfield. The parts that you quoted I took to be the argument that therefore Q is a subfield of R. So what are you trying to say? My original question still stands. Subject: Re: questions about the Dedekind construction of R Content-transfer-encoding: 8bit > 1) R was defined as the set of all cuts. Like most people, I normally > think of a real number as something with an infinite decimal > expansion. It appears to me that a decimal expansion is not the same > as a cut. How is this problem resolved? > I don't see the problem. First you construct R, then you show that every > element of R has a decimal expansion. > Easier said than done. Hence, I would like to know how to prove > something like: > For every cut, there is a corresponding decimal expansion. > 2) Rudin showed that there is an isomorphism between Q, and the > ordered field Q*, whose elements are the rational cuts, which he > defined. He then argued that therefore Q is a subfield of R. I agree > that Q* is a subset of R, is a field, and hence is a subfield of R. I > don't understand why Q is therefore a subfield of R. How is Q a > subset of the set of all cuts? > no means the same as r but the properties we are concerned with (arithmetic > and order) are the same in the two fields and It is this identification > of Q with Q* that allows us to regard Q as a subfield of R. > Are we both speaking English here? How is regarding Q as a subfield > of R different from saying that Q is a subfield of R? The theorem > Rudin is proving states that R contains Q as a subfield. The parts > that you quoted I took to be the argument that therefore Q is a > subfield of R. So what are you trying to say? My original question > still stands. Think of it this way: he is REDEFINING Q to be Q*. That's what identification means. What's the difference between Q in English, and Q in French? Three over 7 vs. trois sur sept? Only the pronunciation. The ARITHMETIC properties, which is all that we're interested in, are the same. This is done all the time in mathematics. If you don't comprehend this, I suggest you switch majors. --Ron Bruck Subject: Re: questions about the Dedekind construction of R > 1) R was defined as the set of all cuts. Like most people, I normally > think of a real number as something with an infinite decimal > expansion. > What do you mean by an infinite decimal expansion? > The infinite decimal expansion is usuall defined in terms of limits. > But, the limit must pre-exist before you can say that the > infinite decimal expansion has a limit. Rudin used the supremum, but no limits to define the decimal expansion. The use of supremum should be on solid ground, since the Dedekind contruction defined the supremum to be the union of all the cuts in a particular subset of R. Thus it should not be necessary to use limits. However, it is still not clear to me how to convert a cut into a decimal expansion, or vice-versa. > Remember that you are trying to show the existence of real numbers > by defining what they are (in terms of Dedekind cuts of rational > numbers). > You might want to look at a second way of defining real numbers > by using Cauchy sequences of real numbers. This is closer to > your idea of infinite decimal expansion. > If you just try to define a real number to be an infinite > decimal expansion, you will see that you will have trouble > defining the addition, multiplication, etc of them. You also > have to worry about the problem that .999... = 1. Yes, I have heard of Cantor's construction based on Cauchy sequences. However, should not the Dedekind construction be equivalent? Can't we approach these questions directly? I am not suggesting that we should define R to be the set of all decimal expansions. Instead, I merely want to know how decimal expansions follow from the Dedekind construction. This leads me to a related question: The Dedekind and Cantor constructions of R from Q demonstrate the existence of a set which satisfies a list of certain properties. We know that these properties describe what we intend by the set of real numbers. Is there any proof that this set, which we call R, is unique? In other words, are there several (or infinitely many) possible sets which qualify as R? Or can all of the constructions of R be shown to equivalent? By equivalence, I mean that these sets are equal, not just isomorphic. For that matter, can all constructions of R at least be shown to all be isomorphic to one another? If you know of any good references on this matter, please let me know. Subject: Re: questions about the Dedekind construction of R Content-transfer-encoding: 8bit > 1) R was defined as the set of all cuts. Like most people, I normally > think of a real number as something with an infinite decimal > expansion. > What do you mean by an infinite decimal expansion? > The infinite decimal expansion is usuall defined in terms of limits. > But, the limit must pre-exist before you can say that the > infinite decimal expansion has a limit. > Rudin used the supremum, but no limits to define the decimal > expansion. The use of supremum should be on solid ground, since the > Dedekind contruction defined the supremum to be the union of all the > cuts in a particular subset of R. Thus it should not be necessary to > use limits. However, it is still not clear to me how to convert a cut > into a decimal expansion, or vice-versa. For a positive real (cut), x, Rudin gets a sequence of non-negative integers n_0, n_1, ..., n_k such that x = lub(n_0 + n_1/10 +...+n_k/10^k, k = 0, 1, ....} which, in this case, is the same as lim(n_0 + n_1/10 +...+n_k/10^k, k = 0 ,,, oo). However, at least in my ancient edition, Rudin hasn't defined limit yet. > Remember that you are trying to show the existence of real numbers > by defining what they are (in terms of Dedekind cuts of rational > numbers). > You might want to look at a second way of defining real numbers > by using Cauchy sequences of real numbers. This is closer to > your idea of infinite decimal expansion. > If you just try to define a real number to be an infinite > decimal expansion, you will see that you will have trouble > defining the addition, multiplication, etc of them. You also > have to worry about the problem that .999... = 1. > Yes, I have heard of Cantor's construction based on Cauchy sequences. > However, should not the Dedekind construction be equivalent? Can't we > approach these questions directly? I am not suggesting that we should > define R to be the set of all decimal expansions. Instead, I merely > want to know how decimal expansions follow from the Dedekind > construction. > This leads me to a related question: > The Dedekind and Cantor constructions of R from Q demonstrate the > existence of a set which satisfies a list of certain properties. We > know that these properties describe what we intend by the set of real > numbers. Is there any proof that this set, which we call R, is > unique? In other words, are there several (or infinitely many) > possible sets which qualify as R? Or can all of the constructions > of R be shown to equivalent? By equivalence, I mean that these sets > are equal, not just isomorphic. Not a chance. The set of equivalence classes of Cauchy sequences is definitely not equal to the set of cuts. Indeed, any set of caardinality 2^N can be considered as the set of real numbers. > For that matter, can all > constructions of R at least be shown to all be isomorphic to one > another? Yes, all complete ordered fields are isomorphic. A Google searrch on complete ordered field turns up 500+ hits. One might be what you are looking for. -- Paul Sperry Columbia, SC (USA) Subject: Re: questions about the Dedekind construction of R Content-transfer-encoding: 8bit > 1) R was defined as the set of all cuts. Like most people, I normally > think of a real number as something with an infinite decimal > expansion. It appears to me that a decimal expansion is not the same > as a cut. How is this problem resolved? > I don't see the problem. First you construct R, then you show that every > element of R has a decimal expansion. > Easier said than done. Hence, I would like to know how to prove > something like: > For every cut, there is a corresponding decimal expansion. > 2) Rudin showed that there is an isomorphism between Q, and the > ordered field Q*, whose elements are the rational cuts, which he > defined. He then argued that therefore Q is a subfield of R. I agree > that Q* is a subset of R, is a field, and hence is a subfield of R. I > don't understand why Q is therefore a subfield of R. How is Q a > subset of the set of all cuts? > no means the same as r but the properties we are concerned with (arithmetic > and order) are the same in the two fields and It is this identification > of Q with Q* that allows us to regard Q as a subfield of R. Compare that to my earlier edition: Rational cuts will be identified with rational numbers ( and will be called rational numbers)....This [referring to the properties of rational cuts] enables us to identify the rational cut r* with the rational number r. > Are we both speaking English here? How is regarding Q as a subfield > of R different from saying that Q is a subfield of R? The theorem > Rudin is proving states that R contains Q as a subfield. The parts > that you quoted I took to be the argument that therefore Q is a > subfield of R. So what are you trying to say? My original question > still stands. Well, Rudin is stuck since he doesn't have isomorphism (either algebraic or topological) to work with and doesn't want to take up a chapter introducing it. In the usual development of the reals from a set assumed to satisfy Peano's axioms, the naturals are regarded as a subset of the integers, and the integers are regarded as a subset of the rationals and the rationals are regarded as a subset of the reals although none of those is _really_ a subset of the other. To do otherwise would be a notational nightmare. -- Paul Sperry Columbia, SC (USA) Subject: quotient group of a cartesian product by a cyclic group I'm working on a representation theory problem that I've hit a sticking point with. Given a,b,c,d integers, is there a way to make (Z_a x Z_b)/<(c,d)>, where <(c,d)> is the cyclic subgroup of Z_a x Z_b generated by the element (c,d), isomorphic to a group of the form G = < x,y | a*x=0, b*y = e*x > where a=ord(x), b=[G : ] (the index of in G), and e is the unique integer such that (1) 0 <= e < a and (2) b*y = e*x ? Subject: Re: quotient group of a cartesian product by a cyclic group My last post had poor choice of subscripts; it should read: Given a,b,c,d integers, is (Z_a x Z_b)/<(c,d)>, where <(c,d)> is the cyclic subgroup of Z_a x Z_b generated by the element (c,d), isomorphic to a group of the form G = < x,y | Ax=0, By = Ex > where A=ord(x), B=[G : ] (the index of in G), and E is the unique integer such that (1) 0 <= E < A and (2) By = Ex ? If so, what are A,B,E in terms of a,b,c,d ? Subject: re: Signal nonlocality in orthodox quantum theory erratum Important typo error: The sentence fragment in original it says that signal nonlocality is not essential even in linear nonlocal micro-quantum theory Should be it says that signal locality is not essential even in linear nonlocal micro-quantum theory i.e. change nonlocality to locality - fixed below: Memorandum for the Record from Jack Sarfatti The author of this paper is competent and cites key past papers arguing against signal nonlocality in orthodox micro-quantum physics based essentially on the fact that the first reduced quantum density matrices distant choice of type of measurement (basis) as described by Henry Stapp and several others now almost 30 years ago. I have not had time to understand every detail of Srikanth's paper based on new technology that was not available at the time in the late 1970's when I came up with a roughly similar idea of using relative phase interference at the receiver that was published in the early editions of Gary Zukav's The Dancing Wu Li Masters. Under pressure from some physicists, Gary caved in and removed that entire section in later editions that are now of great historical interest showing I was generally ahead of the curve back then. Of course Srikanth's idea is much better than what I had originally more vaguely suggested, which would not have worked, but which, nevertheless, pointed in an interesting direction of inquiry that blossomed in 1999. If Srikanth's 1999 paper survives critical scrutiny it is very important because it says that signal locality is not essential even in linear nonlocal micro-quantum theory as distinct from more is different (PW Anderson) nonlinear local macro-quantum theory. The macro-quantum theory is to micro-quantum theory as general relativity is to special relativity with nonlinearity from Bose-Einstein condensation analogous to curvature in general relativity. This impacts Antony Valentini's work that signal locality is a consequence of sub-quantal heat death i.e. an equilibrium of the hidden variables giving the Born probability distribution P(x) ~ ||^2 for the pure micro-quantum state |psi>. My interferometer EPR gedankenexperiment was described on pages 311 to 314 of the original 1979 edition of The Dancing Wu Li Masters published by William Morrow. Note the remarks below were written several years before Aspect's actual Paris experiment was completed in 1982. I also did not have PW Anderson's 1967 More is different idea in mind at Goldstone broken symmetry in solid state physics (Jahn-Teller) with Marshall Stoneham at UKAEA Harwell (published in Proceedings of the Physical Society of London 1967). I had not yet connected those dots and would not do so until 2002 as in http://qedcorp.com/APS/Ukraine.doc It seems that the collapse of the micro-quantum wave in measurement always needs an irreversible amplification from micro to macro levels, which according to PW Anderson has something akin to a More is different BEC, preserving the micro-information in the amplification needed for a good measurement, i.e. a definite decohered pointer state. That is, the initial micro-quantum information must be reliably preserved in the amplification process leading to a macro signal that is the final measurement. Each quantum in the cascade amplification must keep that original micro-information intact which is analogous to a BEC where each quantum is in the same micro-quantum state. According to Sarfatti, however, in terms of information theory, the Aspect experiment, at best, will demonstrate a pure noise superluminal channel. This is because the Aspect experiment, as it is designed, uses Therefore, Sarfatti proposes that we replace Aspect's photomultipliers 'Heisenberg microscopes' at each slit. A double slit system is a wave a double slit system with Heisenberg microscopes ... gives us the option ... Sarfatti theorizes ... that the interference pattern that we see at one end of such a dual double slit system is inseparably linked, photon pair by photon pair, with the interference pattern that we see at the other end in a way that is beyond spacetime. Therefore, a modulation of the interference pattern at one end of the system will cause a similar modulation at the other end of the system even though no energy-momentum transporting signal is connecting the two processes. This is what Sarfatti calls a 'nonlocal phase lock over space-like intervals' ... pp. 312 - 313 1979 edition of Wu Li Masters. Now compare the above from 1978 to Srikanth 1999 aspect. Srikanth does not use a double slit at the sender but he has a functional equivalent to it. Two spatially seperated observers, Alice and Bob analyze the light. Alice is equipped with a device, labelled K, to measure momentum or position of the photons in x and y direction. Bob is equipped with a Young's double-slit interferometer Equation (5) shows the Young double slit interference pattern at the receiver that Bob sees locally provided that Alice makes a wave (transverse momentum) measurement at the sender. On the other hand if Alice at the sender chooses to measure both transverse position y and longitudinal momentum px, then eq. (9) for a single slit diffraction pattern is observed by Bob at the receiver. Srikanth's gedankenexperiment is, of course, far superior and more quantitative than what I suggested 20 years before him. It also uses technology not available in the 1970's. Should his scheme prove correct, I cannot say at this point, then it is a great achievement and it will again show how premature rejection of even half-baked ideas on the fringe can often slow the progress of fundamental physics. [BTW note in passing that I mention torsion in gravity theory in footnote on p. 199 of that 1979 edition.] I saw the reference to the Oak Ridge paper, and noticed that they referenced R Srikanth, a 'target' of conceptual RV'ing. See this paper: http://www.arxiv.org/abs/quant-ph/9904075 Noncausal Superluminal Nonlocal Signalling Authors: R. Srikanth Comments: Includes a refinement of the thought-experiment presented earlier, 4 figures (new); Two sections added: to explain how no-signalling arguments are circumvented; and to propose a plan for a possible practical realization of the thought-experiment Report-no: IIAp-99/7/1 Subject: Re: Signal nonlocality in orthodox quantum theory erratum just because it was detected?... as you note, Young's original experiment was a two *hole* job, which gives a nice moire pattern; the two slits just make a simpler patter to observe. I could go on, but why, ask the Original DWL Master, Why? > Therefore, Sarfatti proposes that we replace Aspect's photomultipliers > 'Heisenberg microscopes' at each slit. A double slit system is a wave > a double slit system with Heisenberg microscopes ... gives us the option > ... Sarfatti theorizes ... that the interference pattern that we see at > one end of such a dual double slit system is inseparably linked, photon > pair by photon pair, with the interference pattern that we see at the > other end in a way that is beyond spacetime. Therefore, a modulation of > the interference pattern at one end of the system will cause a similar > modulation at the other end of the system even though no energy-momentum > transporting signal is connecting the two processes. This is what > Sarfatti calls a 'nonlocal phase lock over space-like intervals' ... > pp. 312 - 313 1979 edition of Wu Li Masters. --Dec.2000 'WAND' Chairman Paul O'Neill, reelected to Board. Newsish? http://www.rand.org/publications/randreview/issues/rr.12.00/ http://members.tripod.com/~american_almanac Subject: Re: Solution to a Differential eq. ?!! >I have the following question: >Given a scalar differential equation > x'(t) = a x^2 + bx + c(t) > where a > 0, b < 0, and c(t) >= 0 has an upper bound ( c(t) <= k ) and > depends on previous values of x(t) (i.e., c(t) = q x(t-tau) and q >0). Are these statements consistent? It will require 0 <= x(t-tau) <= k/q. > Assume for the period [0,tau) the diff. eq. is > x'(t) = ax^2 + bx + k > let the roots of the quadratic term be such that r1 < r2 > r1,r2 = (-b -/+ sqrt(b^2 -4ak) ) / 2a > then we know if x(0) < r2 then x(t) --> r1 . OK, assuming b^2 > 4 a k. > Now for the period [tau, 2 tau) let the diff. eq. be > x'(t) = a x^2 + bx + c(t) , c(t) =q x(t-tau) > Since in [0,tau) x(t) was decreasing (when x(0) r1. Presumably x(0) = k/q. > c(t) is decreasing. So know the smaller root (r1) in this period > is smaller than its value in the previous period [0, tau). > Looking further at [2 tau, 3 tau) we notice that c(t) keeps decreasing > as x(t) decreases from period to period. > My question is : Can we conclude that c(t) will keep decreasing > such that x(t) --> r1 --> 0 asymptotically ? x'(t) < 0 for r1(t) < x(t) < r2(t), where r1(t) and r2(t) are the roots of a x^2 + b x + c(t), assuming b^2 - 4 a c(t) > 0, i.e. c(t) < b^2/(4 a). Thus r1(t) = (-b - sqrt(b^2-4 a c(t)))/(2 a). When c'(t) < 0, r1'(t) < 0 and r2'(t) > 0. At the first time (if any) when x(t) = r1(t), you'd have x'(t) = 0 but r1'(t) < 0, and that would mean x' < r1 slightly before that time. So if you start out with x > r1, this will continue forever. Now since x is decreasing and bounded below, it approaches a limit L as t -> infinity, and x' -> 0. Then c(t) = q x(t-tau) -> q L, r1(t) -> (-b - sqrt(b^2 - 4 a q L))/(2a) and r2(t) -> (-b + sqrt(b^2 - 4 a q L))/(2a) (which are the roots of a x^2 + b x + q L = 0) and x' = a x^2 + b x + c(t) -> a L^2 + b L + q L. So we must have a L^2 + b L + q L = 0. The solutions of this are L = 0 and L = -(b+q)/a. If q + b < 0, and you start out with x > -(b+q)/a, the limit will have to be be -(b+q)/a. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 Subject: Re: Solution to a Differential eq. ?!! >and x' = a x^2 + b x + c(t) -> a L^2 + b L + q L. So we must have >a L^2 + b L + q L = 0. The solutions of this are L = 0 and L = -(b+q)/a. >If q + b < 0, and you start out with x > -(b+q)/a, the limit will have >to be be -(b+q)/a. But in fact, that can't happen. If q + b < 0 and k > -(b+q)/a, then when x = k, a x^2 + b x + q k = k (a k + b + q) > 0 so x will not be decreasing. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 Subject: Re: tensor notation survey question > Yes, I realize your point on the empty slots -- my image was not intended to > be a real-life example. Even with the presence of empty slots, there is a > correspondence between script/slots in the superscript and subscript that is > indicated by horizontal position. My question is about the geometrical > nature of that correspondence in the case where the scripted item is italic. I'd recommend vertical alignment in all cases. Jan Bielawski Subject: Re: tensor notation survey question > The Donald Knuth's TeXbook is the first book I consulted but it says nothing > about this. TeX in fact got a few things a bit wrong in the mathematical typesetting. They are all fixable. 1. The function superposition symbol (the little circle) is too large. It should be about half its size. 2. The less-or-equal and greater-or-equal signs have the underline horizontal instead of parallel to the lower stroke of the main body. The horizontal look is an ugly throwback to the old electric typewriter days which didn't have a separate less-or-equal sign so it was faked with typing the underscore first (which didn't advance the carriage) followed by the less-than. 3. The big direct sum and tensor product signs are BOLD instead of standard weight which looks amateurish as it never matches the formula that follows. Jan Bielawski Subject: Re: tensor notation survey question > TeX in fact got a few things a bit wrong in the mathematical > typesetting. They are all fixable. And they all have NOTHING to do with TeX, but with the design of a particular font. Donald Arseneau asnd@triumf.ca Subject: Re: tensor notation survey question > TeX in fact got a few things a bit wrong in the mathematical > typesetting. They are all fixable. > And they all have NOTHING to do with TeX, but with the design of a > particular font. ... also by donald knuth. Subject: Re: tensor notation survey question >> The Donald Knuth's TeXbook is the first book I consulted but it says nothing >> about this. >TeX in fact got a few things a bit wrong in the mathematical >typesetting. They are all fixable. >1. The function superposition symbol (the little circle) is too large. >It should be about half its size. you mean, the symbol that knuth designed isn't as you like it. >2. The less-or-equal and greater-or-equal signs have the underline >horizontal instead of parallel to the lower stroke of the main body. >The horizontal look is an ugly throwback to the old electric >typewriter days which didn't have a separate less-or-equal sign so >it was faked with typing the underscore first (which didn't advance >the carriage) followed by the less-than. you mean, the symbols that knuth designed aren't as you like them. >3. The big direct sum and tensor product signs are BOLD instead of >standard weight which looks amateurish as it never matches the formula >that follows. you mean, the symbols that knuth designed aren't as you like them. you're at liberty to use different fonts, of course. for example, the slanted >= (etc) are available in the ams symbol fonts. none of the above has anything to do with the placement of sub- and superscripts, which is about the only area i'm seriously dissatisfied with knuth's maths setting, and which is the subject that was concerning the op of this thread. -- Robin Fairbairns, Cambridge Subject: Terminology: sgn, abs, normal, magnitude On real numbers, the abs and sgn are commonly defined as: abs(x) = x if x>0, 0 if x=0, -1 if x<0 sgn(x) = 1 if x>0, 0 if x=0, -1 if x<0 In many kinds of vector space (in this case, a 3d vector space), the magnitude and normalize operations are commonly defined as: magnitude(v) = v.x^2 + v.y^2 + v.z^2 normalize(v) = v / (v.x^2 + v.y^2 + v.z^2) if v<>0, 0 otherwise These operations obey the identities: abs(x) = x*sgn(x) sgn(x) = lim(y->x) y/abs(y) magnitude(v) = v*normalize(v) (where * is the inner product operation) normalize(v) = lim(w->v) w/magnitude(w) So, from a certain point of view these two sets of operations represent the same notion: absolute value and magnitude represent the directionless size of a number or vector, while sign and normalize represent the sizeless direction of a number or vector. QUESTION: Is there a name for or formalization of this generalized notion of size and direction? And exactly what is the minimal set of spaces upon which these operations are generally defined? normed vector spaces (http://mathworld.wolfram.com/NormedSpace.html) are what I'm looking for. If this is the answer, then would it be considered kosher to view the real numbers as a normed vector space (over the real numbers themselves), where the norm is simply the absolute value? The context of the question is this: I'm writing a computer math library, and am looking for the appropriate set of types over which the sgn/abs/magnitude/normalize functions can be defined generally, and am looking for an abstraction and nomenclature which will offend the fewest people possible. Subject: Re: Terminology: sgn, abs, normal, magnitude > On real numbers, the abs and sgn are commonly defined as: > abs(x) = x if x>0, 0 if x=0, -1 if x<0 I presume you intended above: -x if x<0 > sgn(x) = 1 if x>0, 0 if x=0, -1 if x<0 > In many kinds of vector space (in this case, a 3d vector space), the magnitude and normalize operations are commonly defined as: > magnitude(v) = v.x^2 + v.y^2 + v.z^2 > normalize(v) = v / (v.x^2 + v.y^2 + v.z^2) if v<>0, 0 otherwise > These operations obey the identities: > abs(x) = x*sgn(x) > sgn(x) = lim(y->x) y/abs(y) No. The above fails for x=0. Perhaps this is another typo. For example, it is true that, for all real x, sgn(x) = lim(y->x) x/abs(y). Maybe that's what you intended. The corresponding change should be made in the second statement below. > magnitude(v) = v*normalize(v) (where * is the inner product operation) > normalize(v) = lim(w->v) w/magnitude(w) David Cantrell Subject: Re: Terminology: sgn, abs, normal, magnitude Content-transfer-encoding: 8bit Besides the typos that David Cantrell pointed out, you are making things needlessly complicated. In R^n, |(x_1,...x_n)| = sqrt(x_1^2 + ... + x_n^2). This is true even if n = 1. If x =/= 0 is a vector the normalization of x is (1/|x|)x. > On real numbers, the abs and sgn are commonly defined as: > abs(x) = x if x>0, 0 if x=0, -1 if x<0 > sgn(x) = 1 if x>0, 0 if x=0, -1 if x<0 > In many kinds of vector space (in this case, a 3d vector space), the magnitude and normalize operations are commonly defined as: > magnitude(v) = v.x^2 + v.y^2 + v.z^2 > normalize(v) = v / (v.x^2 + v.y^2 + v.z^2) if v<>0, 0 otherwise > These operations obey the identities: > abs(x) = x*sgn(x) > sgn(x) = lim(y->x) y/abs(y) > magnitude(v) = v*normalize(v) (where * is the inner product > operation) > normalize(v) = lim(w->v) w/magnitude(w) > So, from a certain point of view these two sets of operations > represent the same notion: absolute value and magnitude represent > the directionless size of a number or vector, while sign and normalize represent the sizeless direction of a number or vector. > QUESTION: Is there a name for or formalization of this generalized > notion of size and direction? And exactly what is the minimal set > of spaces upon which these operations are generally defined? normed vector spaces (http://mathworld.wolfram.com/NormedSpace.html) > are what I'm looking for. It seems so unless you insist on an inner (dot) product in which case you need (guess what) an inner product space. > If this is the answer, then would it be > considered kosher to view the real numbers as a normed vector space > (over the real numbers themselves), where the norm is simply the > absolute value? Perfectly OK. R is also an inner product space. > The context of the question is this: I'm writing a computer math > library, and am looking for the appropriate set of types over which > the sgn/abs/magnitude/normalize functions can be defined generally, > and am looking for an abstraction and nomenclature which will offend > the fewest people possible. I'm guessing more people are familiar with inner product spaces than with normed linear spaces. If you don't want to use | |, I suggest norm rather than abs or magnitude. I don't know of any common notation for (1/|x|)x. -- Paul Sperry Columbia, SC (USA) Subject: Textbooks for the New Foundations Syllabus? While surfing the Web site of Professor Miles Reid of the University of Warwick, I came across his sermon titled New Foundations syllabus, which advocates a radical overhaul of a first-term university course in the foundations of mathematics. I find particularly provocative his suggestion that ABSOLUTELY NO set theory should be included in such a course. The Web page with his suggested syllabus http://www.maths.warwick.ac.uk/~miles/Sermons/Foundations makes clear what he thinks ought to be in such a foundations of mathematics course. Professor Reid's interesting suggestions raise a question for me: where can one find textbooks that thoroughly teach the kind of content he advocates? I have quite a few books about foundations of mathematics at home, for self-education, but all have the kind of set theory emphasis Professor Reid decries. By contrast, I hardly know where to look to find a thorough, accurate treatment of many of the topics he favors for a first-year university mathematics course. Do any of you have suggestions for texts on, say, Polynomials and power series Symmetries or Permutations as Professor Reid outlines those topics? I take it that his Arithmetic heading would be covered by a good course in number theory, right? And what he labels as Vectors and Matrices would be covered in texts on linear algebra, wouldn't they? Do you have suggestions for textbooks on those subjects? new foundations syllabus http://www.maths.warwick.ac.uk/~miles/Sermons/Foundations and how to study it. -- Karl M. Bunday Christ has set us free. Galatians 5:1 Learn in Freedom (TM) http://learninfreedom.org/ kmbunday AT earthlink DOT net (preferred email address) Subject: Re: The Fundamental Theorem of Fundamental Theorems > Friends, > I have recently stumbled upon a very interesting theorem which I > have named The Fundamental Theorem of Fundamental Theorems. You may be on to something. The Fundamental Theorem of Arithmetic is a theorem of arithmetic. The Fundamental Theorem of Calculus is a theorem of calculus. But the Fundamental Theorem of Algebra is not a theorem of algebra, it's a theorem of analysis. There is no known purely algebraic proof of FTA. Is there another example of a fundamental theorem whose proof lies in a different discipline? Subject: Re: The Fundamental Theorem of Fundamental Theorems > But the Fundamental Theorem of Algebra is not a theorem of algebra, it's > a theorem of analysis. There is no known purely algebraic proof of FTA. Just for the record: There is. It's using quadratic forms, and so could be called algebraic. You can find it in the book Quadratic forms with applications to algebraic geometry and topology by A. Pfister. Cheers Philipp Subject: Re: Two orthogonal Latin squares of order 10 > With more work one can convert the pair to the following list of lists > form, in case someone wants to play with them. And find, for example, a > third Latin square of order 10 orthogonal to these two. :-) You're going to have to work harder for this. I've already thrown a dual processor linux SMP system at it and spent lots of computer time, and nothing turned up. Subject: Re: uniformly continuous?? NNTP-Posting_Host: student-concordia-dorm.student.concordia.net NNTP-P0STING-HOST: 210.117.60.5 >Subject: uniformly continuous?? >hello.sir~ >show that f(x) = e^x is not uniformly conti on R >--------------------- >not uniformly conti ><=> given e>0, for each d>0, there is x,y in R >|x-y| |f(x)-f(y)| >= e >----------------------- Hmmm.... maybe you should stop using 'e' to mean two different things. adam Subject: Re: uniformly continuous?? > hello.sir~ > show that f(x) = e^x is not uniformly conti on R > --------------------- > not uniformly conti > <=> given e>0, for each d>0, there is x,y in R > |x-y| |f(x)-f(y)| >= e Corrections Start with there is e>0 ... and replace the logical connection of the formulas by > |x-y|= e Fortunately, in your problem the function is so emphatically non-uniformly continuous that you can pick e>0 as big as you please, say 73 or 1. I pick frivolously e=73: Consider sequences x(n) = log(n+73) and y(n) = log(n), what can you say about exp(x(n)) - exp(y(n)) (that's the easy part) and about x(n) - y(n) ? Use Mean Value Theorem applied to log(x). Remark: If your problem were harder, such as f(x) = sin(x^2), your freedom in picking e>0 would be reduced. Again, Mean Value Theorem applied to cleverly chosen x(n), y(n) would help. Cheers, ZVK(Slavek). Subject: Re: uniformly continuous?? we have common sense. Subject: Re: uniformly continuous?? In sci.math, hot-girl : > hello.sir~ > show that f(x) = e^x is not uniformly conti on R > --------------------- > not uniformly conti > <=> given e>0, for each d>0, there is x,y in R > |x-y| |f(x)-f(y)| >= e A slightly odd definition, but, if one wants to show that e^x is not uniformly continuous over R, then one picks an eps > 0, and shows that, for any delta > 0, one can pick x = (2*eps)/delta y = (2*eps)/delta + delta/2 and grind it out: e^((2*eps)/delta + delta/2) - e^(delta/2) = e^((2*eps)/delta) * e^(delta/2) - e^(delta/2) = (e^((2*eps)/delta) - 1) * e^(delta/2) > (1 + ((2*eps)/delta) - 1) * e^(delta/2) = 2*eps/delta * e^(delta/2) > 2*eps/delta * (1 + (delta/2)) > 2*eps/delta * (delta/2) = eps since e^x > 1 + x for any real x != 0. [*] QED. > ----------------------- > in the solution paper, > it have put the x=1/d y=(1/d) + (d/2) > i think that this is wrong choice Well, you're on the right track but you forgot the epsilon. :-) > it can apply to f(x)=x^2 That's another problem. If one picks x = eps/delta y = eps/delta + delta/2 then y^2 - x^2 = (y - x) * (y + x) = delta/2 * (2 * eps/delta + delta/2) > delta/2 * 2 * eps/delta = eps. so x^2 is not uniformly continuous. QED. I'm not at all sure about this definition of uniform continuity. Fortunately, f(x) = C_0 and f(x) = C_1 x + C_0 seem to be immune to this sort of thing. Unfortunately I don't see any other function that is, offhand. mathworld.wolfram.com doesn't seem to know about it either. Maybe it's a professor-specific thing. :-) > can it apply to f(x)=e^x ?? > if so, how put the x,y,e?? > help me, please. genius doctor [*] a quickie proof: e^x's derivative is e^x; therefore, the tangent line at x = 0 has a slope of 1 and an intercept of 1; the single common point is x=0, y=1. -- #191, ewill3@earthlink.net It's still legal to go .sigless. Subject: Re: uniformly continuous?? > we have common sense. ... responding to what? Please include at least a relevant part of the previous message. Cheers, ZVK(Slavek). Subject: Re: uniformly continuous?? i can your advice for problem. it's looks good. thank you very much to help for me. Subject: Re: uniformly continuous?? it looks good idea. thank you very much. Subject: Units in Z[pi], field forced In a previous thread I started talking about units in Z[pi]. In this post I use a limit approach to show that adding pi to the ring of integers forces you into a field. Euler found out that he could define pi^2 in the following nifty way: pi^2/6 = 1 + 1/4 + 1/9 + 1/16 +... Operations will be in the field of rationals with a translation to the ring Z[pi, pi_approx1, pi_approx2]. Let, in rationals, pi_approx1^2/ 6 = 1 + 1/4 + 1/9 + 1/16 +...+1/k^2, where k is a positive integer. Now still in rationals, let pi_approx2^2/6 = 1 + 1/4(1 + 1/4 + 1/9 +...1/k^2) + 1/9(1+1/4 + 1/9+...+1/k^2) ...1/k^2(1+1/4 + 1/9+...+1/k^2), which is pi_approx2^2/6 = 1 + pi_approx1^2/24 + pi_approx1^2/54 +...+pi_approx1^2/6k^2, multiplying out and collecting, pi_approx2^2 [24(54)...(6k^2)] - pi_approx1[6(54)...(6k^2)- 6(24)...(6k^2)-... -6(24)(54)...6k^2] = 6(24)(54)...(6k^2) where now I'm only using elements of the ring Z[pi, pi_approx1, pi_approx2], and note integers besides 1 and -1 are now units, for instance 6 is a unit. As k increases in size, more and more elements become units in the ring, while there is never a diminishing number of unit elements. But, in the limit as k goes out to infinity, I have pi^2 [24(54)...(6k^2)- 6(54)...(6k^2)- 6(24)...(6k^2)- ... -6(24)(54)...6k^2] = 6(24)(54)...(6k^2) and the ring is Z[pi], as pi_approx1 and pi_approx2 approach pi, in the limit. At which point you have an infinite number of units in the ring. Previously I've seen attempts to dispute that result by claiming that you can't use infinite sums, in particular cases, with integers or rationals, but introducing pi into the ring changes the rules, so you don't just have the ring of integers, as you have pi and the ring of integers. Notice that the *field* of rationals arbitrarily excludes certain elements based on the rule that to be rational a number can be written as b/a, where 'a' and 'b' are integers, and 'a' is nonzero. That is, that a rational by definition can be written as the ratio of integers, where the denominator is nonzero. So the field of rationals has elements that are simply excluded based on an ad hoc rule. However, Z[pi] does not have a similar rule, though some posters seem intent on adding one!!! Trying to exclude infinite sums with an infinite creation like pi, is just an arbitrary attempt at avoiding a conclusion that is not desired, and in fact, adding pi to the ring of integers introduces an infinite number of units into the ring, as it makes all non-zero integers units, so it creates a field. Interestingly, it forces the field of rationals *plus* pi itself, forcing a way around the arbitrary exclusion of pi from the field of rationals. James Harris Subject: Re: Units in Z[pi], field forced >In a previous thread I started talking about units in Z[pi]. >In this post I use a limit approach to show that adding pi to the ring >of integers forces you into a field. >Euler found out that he could define pi^2 in the following nifty way: > pi^2/6 = 1 + 1/4 + 1/9 + 1/16 +... >Operations will be in the field of rationals with a translation to the >ring > Z[pi, pi_approx1, pi_approx2]. >Let, in rationals, >pi_approx1^2/ 6 = 1 + 1/4 + 1/9 + 1/16 +...+1/k^2, >where k is a positive integer. >Now still in rationals, let > pi_approx2^2/6 = 1 + 1/4(1 + 1/4 + 1/9 +...1/k^2) + 1/9(1+1/4 + >1/9+...+1/k^2) ...1/k^2(1+1/4 + 1/9+...+1/k^2), which is > pi_approx2^2/6 = 1 + pi_approx1^2/24 + pi_approx1^2/54 >+...+pi_approx1^2/6k^2, >multiplying out and collecting, > pi_approx2^2 [24(54)...(6k^2)] - pi_approx1[6(54)...(6k^2)- >6(24)...(6k^2)-... -6(24)(54)...6k^2] = 6(24)(54)...(6k^2) >where now I'm only using elements of the ring Z[pi, pi_approx1, >pi_approx2], and note integers besides 1 and -1 are now units, for >instance 6 is a unit. >As k increases in size, more and more elements become units in the >ring, while there is never a diminishing number of unit elements. >But, in the limit as k goes out to infinity, I have > pi^2 [24(54)...(6k^2)- 6(54)...(6k^2)- 6(24)...(6k^2)- ... >-6(24)(54)...6k^2] = 6(24)(54)...(6k^2) >and the ring is Z[pi], as pi_approx1 and pi_approx2 approach pi, in >the limit. If you wanted to say that Z[pi] was the limit of that other ring as k -> infinity it's possible that someone somewhere would agree you were making sense. But that does not change what Z[pi] _is_ - that fact that that other family of rings includes this and that does not imply that Z[pi] does. >At which point you have an infinite number of units in the ring. >Previously I've seen attempts to dispute that result by claiming that >you can't use infinite sums, in particular cases, with integers or >rationals, but introducing pi into the ring changes the rules, Uh, no it doesn't. Z[pi] is the subring of the reals generated by Z and pi. That means it's the set of all finite sums of the form P(pi), where P is a polynomial with integer coefficients. That's the rules. And it's easy to show that Z[pi] contains no units except 1 and -1: Say P(pi) is a unit; then P(pi)*Q(pi) = 1, where P and Q are polynomials with integer coefficients. Hence (PQ)(pi) = 1; since pi is transcendental this shows that PQ = 1. So P and Q must both have degree zero, and their constant terms must be units in the integers, hence plus or minus 1. > so you >don't just have the ring of integers, as you have pi and the ring of >integers. That's true. And you have no units except 1 and -1. >Notice that the *field* of rationals arbitrarily excludes certain >elements based on the rule that to be rational a number can be written >as b/a, where 'a' and 'b' are integers, and 'a' is nonzero. That is, >that a rational by definition can be written as the ratio of >integers, where the denominator is nonzero. >So the field of rationals has elements that are simply excluded based >on an ad hoc rule. Right. Just like the set of dogs does not include my cat - my cat was simply excluded from the set of dogs by an ad hoc rule. And Philadelphia was simply excluded from the set of gerbils by an ad hoc rule... You really have _no_ idea what a _definition_ is, right? >However, Z[pi] does not have a similar rule, though some posters seem >intent on adding one!!! Z[pi] does not have any rules. It does include pi, not because Z[pi] has a rule, but because that's part of the definition of Z[pi]. It does not include any units other than 1 and -1, not because of any rule that says it doesn't, but because it simply _doesn't_. >Trying to exclude infinite sums with an infinite creation like pi, is >just an arbitrary attempt at avoiding a conclusion that is not >desired, and in fact, adding pi to the ring of integers introduces an >infinite number of units into the ring, as it makes all non-zero >integers units, so it creates a field. >Interestingly, it forces the field of rationals *plus* pi itself, >forcing a way around the arbitrary exclusion of pi from the field of >rationals. Interestingly, you're making a much bigger idiot of yourself than usual, because it's not just that the things you're saying are false, you're showing that you don't understand that words and notations mean what their definitions say they mean. >James Harris ************************ Subject: Re: Units in Z[pi], field forced > In a previous thread I started talking about units in Z[pi]. > ... and gave even more evidence of the paucity of your ideas. > In this post I use a limit approach to show that adding pi to the ring > of integers forces you into a field. > No, what you do is to convince yourself of something false. That's how you think you're doing mathematics: by producing a falsehood, building an argument that points to that falsehood, and then marching about as though you've just set the world on its ear. I've got news for you: *you* are the only one who's fooled by any of this. > Euler found out that he could define pi^2 in the following nifty way: > > pi^2/6 = 1 + 1/4 + 1/9 + 1/16 +... > > Operations will be in the field of rationals with a translation to the > ring > > Z[pi, pi_approx1, pi_approx2]. > > Let, in rationals, > > pi_approx1^2/ 6 = 1 + 1/4 + 1/9 + 1/16 +...+1/k^2, > > where k is a positive integer. > Are you actually taking the sum, multiplying by 6 & taking a square root to get that number? If so, you're adding irrationals. > Now still in rationals, let > > pi_approx2^2/6 = 1 + 1/4(1 + 1/4 + 1/9 +...1/k^2) + 1/9(1+1/4 + > 1/9+...+1/k^2) ...1/k^2(1+1/4 + 1/9+...+1/k^2), which is > > pi_approx2^2/6 = 1 + pi_approx1^2/24 + pi_approx1^2/54 > +...+pi_approx1^2/6k^2, > > multiplying out and collecting, > > pi_approx2^2 [24(54)...(6k^2)] - pi_approx1[6(54)...(6k^2)- > 6(24)...(6k^2)-... -6(24)(54)...6k^2] = 6(24)(54)...(6k^2) > > where now I'm only using elements of the ring Z[pi, pi_approx1, > pi_approx2], and note integers besides 1 and -1 are now units, for > instance 6 is a unit. > I don't see here that you've proven 6 a unit in this ring. Your notation suggests that pi_approx1 and pi_approx2 are supposed to be approximations to pi. I can imagine that for pi_approx1, provided you do the arithmetic it takes to produce the value (rather than leaving it at the form pi_approx1^2/6, as you have done here). The second value, pi_approx2, only appears to be what you get by substituting pi_approx1^2/6 as a factor in all but the first term of 1 + 1/2^2 + 1/3^2 + ... + 1/k^2. That process doesn't appear to yield an approximation for pi. > As k increases in size, more and more elements become units in the > ring, while there is never a diminishing number of unit elements. > obtained by virtue of arithmetic). Each of the two values (if I'm reading the somewhat imprecise description correctly) is the square root of some particular rational number, so your rings are of this form: Z[ sqrt(n1/d1), sqrt(n2/d2) ] where n1/d1 and n2/d2 are not really independent, but related by n2/d2 = 1 + n1/d1 ( 1/4 + 1/9 + ... + 1/k^2 ). So, for instance, we know that d1 is a divisor of 6 (k!)^2 and we know that both n1/d1 and n2/d2 are in the ring. Does that entail that d1 and d2 are units? Well, if the ring contains n1/d1 then it must contain 1-n1/d1, which is (d1-n1)/d1, and if n1/d1 is in lowest terms, then n1 and d1-n1 are relatively prime in Z. Thus one could produce the quotient 1/d1. Similarly with n2/d2: d2 will be a unit. However, you have yet to specify what those values are, so either one is left to guess, or do further calculation. I don't care to do the arithmetic you leave behind, just to get caught in a big flap of correcting your innumerable errors and be called a racist for my troubles, so it's back to you on that account. BTW, addition of elements to a ring cannot reduce the number of units. After all, if rs = 1 in a ring R, then adjoining elements to R while retaining the existing ring structure doesn't change the fact that rs = 1. > But, in the limit as k goes out to infinity, I have > > pi^2 [24(54)...(6k^2)- 6(54)...(6k^2)- 6(24)...(6k^2)- ... > -6(24)(54)...6k^2] = 6(24)(54)...(6k^2) > > and the ring is Z[pi], as pi_approx1 and pi_approx2 approach pi, in > the limit. I said before, pi_approx2 does not look like an approximation to pi; you'll need to do better to support any claim that it is such. > > At which point you have an infinite number of units in the ring. > So? What if all you're doing to this ring is adding 1/k! to the ring at the kth stage? That process simply inverts all integers from 2 to k, inclusive. It has nothing to do with pi. > Previously I've seen attempts to dispute that result by claiming that > you can't use infinite sums, in particular cases, with integers or > rationals, but introducing pi into the ring changes the rules, so you > don't just have the ring of integers, as you have pi and the ring of > integers. > Here's the golden statement: but introducing pi into the ring changes the rules Here's the truth: no it doesn't. When you add x to the ring Z to get the polynomial ring Z[x], the rule that Z[x] embodies is that ANY ring obtained from Z by the addition of a single element q is isomorphic to the ring Z[x], by the following isomorphism: Z[x] ----> Z[q] P(x) |---> P(q) That is: map Z to itself via the identity mapping (n |---> n), map x to q and complete the map as the ring structure requires. By this simple feature, Z[x] is isomorphic to Z[q] for any transcendental q. > Notice that the *field* of rationals arbitrarily excludes certain > elements based on the rule that to be rational a number can be written > as b/a, where 'a' and 'b' are integers, and 'a' is nonzero. That is, > that a rational by definition can be written as the ratio of > integers, where the denominator is nonzero. > You contradict yourself. In the first place, you state that certain numbers are excluded *arbitrarily*. Here's what Webster has to say about arbitrary: based on or determined by individual preference or convenience rather than by necessity or the intrinsic nature of something However, you also note that rational numbers are *defined* as the quotient of integers. Doesn't that concept of *definition* occur to you as constituting (to swipe words from the above definition of arbitrary) the intrinsic nature of something? What will it take for you to realize that definitions are not just things that people use to make other people appear stupid? I know that until you recognize this, you will always be made to appear stupid by the use of definitions. > So the field of rationals has elements that are simply excluded based > on an ad hoc rule. > Ad hoc (Webster's again): for the particular end or case at hand without consideration of wider application Again, you are shown to be speaking nonsense. Rational numbers are determined by the definition. Excluding things that do not meet the definition is anything BUT ad hoc. > However, Z[pi] does not have a similar rule, though some posters seem > intent on adding one!!! > Z[x] has its own definition. All mathematical constructions are determined by their definitions. It is a sad, but amusing, fact that you are unswayed by the appeal of precision (or accuracy, for that matter), that the judicious application of definition brings to mathematics. > Trying to exclude infinite sums with an infinite creation like pi, is > just an arbitrary attempt at avoiding a conclusion that is not > desired, and in fact, adding pi to the ring of integers introduces an > infinite number of units into the ring, as it makes all non-zero > integers units, so it creates a field. > No. Infinite sums are not included because they do not happen to be required by the definition. Z[x] is the *smallest* ring that contains both Z and x. Z[pi] is the *smallest* ring that contains Z and pi. In no case is an infinite sum required for any element of either ring. In some cases, it is possible, even useful, to include some infinite sums; the ring of power series (or when defined, the ring of convergent power series) comes to mind. > Interestingly, it forces the field of rationals *plus* pi itself, > forcing a way around the arbitrary exclusion of pi from the field of > rationals. > This is plain stupid. > > James Harris How about those non-unit divisors of 5 that I showed you? Have you gotten the point yet? Definitions count. They are not a ruse to evade discussion, rather they clarify what the discussion is all about. Dale. Subject: Re: Units in Z[pi], field forced Visiting Assistant Professor at the University of Montana. >In a previous thread I started talking about units in Z[pi]. >In this post I use a limit approach to show that adding pi to the ring >of integers forces you into a field. Limits are not ring operations. Taking a limit does not necessarily leave you inside the ring. Unless, of course, you are now redefining what ring means. By definition, Z[pi] is the smallest subring of C which contains both the integers and pi. It is easy to see that any such ring must contain all polynomial expressions in pi with integer coefficients must be elements of such a ring. And it is also easy to verify that the collection of all polynomial expressions in pi with integer coefficients, under the obvious multiplication and addition rules, form a ring. So Z[pi] consists exactly of all complex numbers which can be written as a polynomial expression in pi with integer coefficients. To be a bit more complete, one proves that every element of Z[pi] can be expressed in such a way in a UNIQUE way. To do this, assume that f = a_0 + a_1*pi + ... + a_n*pi^n and g = b_0 + b_1*pi + ... + b_n*pi^n are two polynomial expressions in pi with integer coefficients, and that f=g, with at least one of a_n and b_n nonzero. The claim is that a_i=b_i for i=0,...,n. Indeed, since f=g, f-g=0. And f-g = (a_0-b_0) + ... + (a_n-b_n)pi^n. Let h(x) be the polynomial with integer coefficients h(x) = (a_0-b_0) + (a_1 - b_1)x + ... + (a_n-b_n)x^n If there exists i with a_i not equal to b_i, then h(x) is not the zero polynomial. But h(pi)=0, which means that pi is the root of a polynomial with integer coefficients. This implies that pi is algebraic, which is false. This contradiction arises from assuming that some a_i is not equal to b_i, so we conclude that a_i=b_i for each i. This shows that Z[pi] is isomorphic to the ring of polynomials with integer coefficients, Z[x], by the map Z[x]->Z[pi] taking x to pi (evaluate at pi). Since the only units in Z[x] are 1 and -1, the only units in Z[pi] are 1 and -1. >Euler found out that he could define pi^2 in the following nifty way: > pi^2/6 = 1 + 1/4 + 1/9 + 1/16 +... >Operations will be in the field of rationals with a translation to the >ring > Z[pi, pi_approx1, pi_approx2]. >Let, in rationals, >pi_approx1^2/ 6 = 1 + 1/4 + 1/9 + 1/16 +...+1/k^2, >where k is a positive integer. Then you really have pi_approx1(k), not pi_approx1. >Now still in rationals, let > pi_approx2^2/6 = 1 + 1/4(1 + 1/4 + 1/9 +...1/k^2) + 1/9(1+1/4 + >1/9+...+1/k^2) ...1/k^2(1+1/4 + 1/9+...+1/k^2), which is > pi_approx2^2/6 = 1 + pi_approx1^2/24 + pi_approx1^2/54 >+...+pi_approx1^2/6k^2, >multiplying out and collecting, > pi_approx2^2 [24(54)...(6k^2)] - pi_approx1[6(54)...(6k^2)- >6(24)...(6k^2)-... -6(24)(54)...6k^2] = 6(24)(54)...(6k^2) >where now I'm only using elements of the ring Z[pi, pi_approx1, >pi_approx2], and note integers besides 1 and -1 are now units, for >instance 6 is a unit. That's because you have included rational numbers by throwing in pi_approx1 and pi_approx2. However, neither pi_approx1 nor pi_approx2 are elements of Z[pi]. >As k increases in size, more and more elements become units in the >ring, while there is never a diminishing number of unit elements. >But, in the limit as k goes out to infinity, I have Limits are not ring operations. If you take a limit, you must prove independently that the result stays in the ring. > pi^2 [24(54)...(6k^2)- 6(54)...(6k^2)- 6(24)...(6k^2)- ... >-6(24)(54)...6k^2] = 6(24)(54)...(6k^2) >and the ring is Z[pi], as pi_approx1 and pi_approx2 approach pi, in >the limit. No, it is not. Z[pi] contains no rationals, but every ring Z[pi_approx1(k),pi_approx2(k)] does. And you aren't really taking those rings, what you are TRYING to do is take Z[pi_approx1(1),pi_approx2(1),pi_approx1(2),pi_approx2(2),..., pi_approx1(k), pi_approx2(k),...] that is, an increasing union of rings. However, NONE of the rings you are dealing with contains Z[pi], nor is the union equal to Z[pi], because each of the elements in your increasing family of rings is a subring of the rational numbers, and Z[pi] includes pi, which is not a rational number. >At which point you have an infinite number of units in the ring. Even assuming this were true, note that you have not proven what you claimed: that it forces you into a field. For example, the ring Z[sqrt(2)] contains an infinite number of units, but is not a field. >Previously I've seen attempts to dispute that result by claiming that >you can't use infinite sums, in particular cases, with integers or >rationals, but introducing pi into the ring changes the rules, so you >don't just have the ring of integers, as you have pi and the ring of >integers. Nonsense and sophistry. >Notice that the *field* of rationals arbitrarily excludes certain >elements based on the rule that to be rational a number can be written >as b/a, where 'a' and 'b' are integers, and 'a' is nonzero. That is, >that a rational by definition can be written as the ratio of >integers, where the denominator is nonzero. >So the field of rationals has elements that are simply excluded based >on an ad hoc rule. The field of rationals includes all elements which are rational. What exactly is your problem? >However, Z[pi] does not have a similar rule, though some posters seem >intent on adding one!!! Z[pi] is by definition the smallest subring of C containing all integers and containing pi. Your construction includes way too many elements that are not needed to obtain a ring. [.snip.] ============================================================== ======== Why do you take so much trouble to expose such a reasoner as Mr. Smith? I answer as a deceased friend of mine used to answer on like occasions - A man's capacity is no measure of his power to do mischief. Mr. Smith has untiring energy, which does something; self-evident honesty of conviction, which does more; and a long purse, which does most of all. He has made at least ten publications, full of figures few readers can critize. A great many people are staggered to this extend, that they imagine there must be the indefinite something in the mysterious all this. They are brought to the point of suspicion that the mathematicians ought not to treat all this with such undisguised contempt, at least. -- A Budget of Paradoxes, Vol. 2 p. 129 by Augustus de Morgan ============================================================== ======== Arturo Magidin magidin@math.berkeley.edu Subject: Re: Units in Z[pi], field forced > In a previous thread I started talking about units in Z[pi]. > In this post I use a limit approach to show that adding pi to the ring > of integers forces you into a field. You, who have demonstrated repeatedly that you are incapable of learning, are now going to teach? Puleez! -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com Subject: Re: Units in Z[pi], field forced James Harris skrev i melding > In a previous thread I started talking about units in Z[pi]. > In this post I use a limit approach to show that adding pi to the ring > of integers forces you into a field. > Euler found out that he could define pi^2 in the following nifty way: > pi^2/6 = 1 + 1/4 + 1/9 + 1/16 +... > Operations will be in the field of rationals with a translation to the > ring > Z[pi, pi_approx1, pi_approx2]. > Let, in rationals, > pi_approx1^2/ 6 = 1 + 1/4 + 1/9 + 1/16 +...+1/k^2, > where k is a positive integer. > Now still in rationals, let > pi_approx2^2/6 = 1 + 1/4(1 + 1/4 + 1/9 +...1/k^2) + 1/9(1+1/4 + > 1/9+...+1/k^2) ...1/k^2(1+1/4 + 1/9+...+1/k^2), which is > pi_approx2^2/6 = 1 + pi_approx1^2/24 + pi_approx1^2/54 > +...+pi_approx1^2/6k^2, > multiplying out and collecting, > pi_approx2^2 [24(54)...(6k^2)] - pi_approx1[6(54)...(6k^2)- > 6(24)...(6k^2)-... -6(24)(54)...6k^2] = 6(24)(54)...(6k^2) > where now I'm only using elements of the ring Z[pi, pi_approx1, > pi_approx2], and note integers besides 1 and -1 are now units, for > instance 6 is a unit. > As k increases in size, more and more elements become units in the > ring, while there is never a diminishing number of unit elements. > But, in the limit as k goes out to infinity, I have > pi^2 [24(54)...(6k^2)- 6(54)...(6k^2)- 6(24)...(6k^2)- ... > -6(24)(54)...6k^2] = 6(24)(54)...(6k^2) > and the ring is Z[pi], as pi_approx1 and pi_approx2 approach pi, in > the limit. > At which point you have an infinite number of units in the ring. > Previously I've seen attempts to dispute that result by claiming that > you can't use infinite sums, in particular cases, with integers or > rationals, but introducing pi into the ring changes the rules, so you > don't just have the ring of integers, as you have pi and the ring of > integers. > Notice that the *field* of rationals arbitrarily excludes certain > elements based on the rule that to be rational a number can be written > as b/a, where 'a' and 'b' are integers, and 'a' is nonzero. That is, > that a rational by definition can be written as the ratio of > integers, where the denominator is nonzero. > So the field of rationals has elements that are simply excluded based > on an ad hoc rule. > However, Z[pi] does not have a similar rule, though some posters seem > intent on adding one!!! > Trying to exclude infinite sums with an infinite creation like pi, is > just an arbitrary attempt at avoiding a conclusion that is not > desired, and in fact, adding pi to the ring of integers introduces an > infinite number of units into the ring, as it makes all non-zero > integers units, so it creates a field. > Interestingly, it forces the field of rationals *plus* pi itself, > forcing a way around the arbitrary exclusion of pi from the field of > rationals. > James Harris Mr. Harris I'm confused. Can you explain your understanding of grouptheory. What are sets, rings and so on? What are the axioms which defines the different representations of these different algebraic (and aritmetric) classes? Can you give a clear explanation how you define these classes (and sub and sup of them)? Karl-Olav Nyberg konyberg@online.no Subject: Re: Units in Z[pi], field forced I meant groups instead of classes. Subject: Re: Units in Z[pi], field forced A correction: ... stuff deleted ... > When you add x to the ring Z to get the polynomial ring Z[x], > the rule that Z[x] embodies is that ANY ring obtained from Z > by the addition of a single element q is isomorphic to the ring > Z[x], by the following isomorphism: > Z[x] ----> Z[q] > P(x) |---> P(q) > That is: > map Z to itself via the identity mapping (n |---> n), > map x to q > and complete the map as the ring structure requires. The above is true only for q transcendental, or an indeterminate. If q is algebraic, then Z[q] is the quotient of Z[x] by the ideal spanned by Irred(q), where Irred(q) is the primitive polynomial over Z, of minimal degree, having q as a root. > By this simple feature, Z[x] is isomorphic to Z[q] for any > transcendental q. ... stuff deleted ... > Dale. My apologies for whatever confusion may have resulted from my error. Dale. Subject: Re: Units in Z[pi], field forced In sci.math, James Harris <3c65f87.0308170716.18e835ca@posting.google.com>: > In a previous thread I started talking about units in Z[pi]. > In this post I use a limit approach to show that adding pi to the ring > of integers forces you into a field. > Euler found out that he could define pi^2 in the following nifty way: > > pi^2/6 = 1 + 1/4 + 1/9 + 1/16 +... To be more pedantic, pi^2/6 = lim(n -> oo) sigma(i=1,n,1) (1/i^2) Because of the limit operation conclusions need to be drawn carefully. > Operations will be in the field of rationals with a translation to the > ring > Z[pi, pi_approx1, pi_approx2]. > > Let, in rationals, > pi_approx1^2/ 6 = 1 + 1/4 + 1/9 + 1/16 +...+1/k^2, > where k is a positive integer. > Now still in rationals, let > pi_approx2^2/6 = 1 + 1/4(1 + 1/4 + 1/9 +...1/k^2) + 1/9(1+1/4 + > 1/9+...+1/k^2) ...1/k^2(1+1/4 + 1/9+...+1/k^2), which is > pi_approx2^2/6 = 1 + pi_approx1^2/24 + pi_approx1^2/54 > +...+pi_approx1^2/6k^2, > multiplying out and collecting, > pi_approx2^2 [24(54)...(6k^2)] - pi_approx1[6(54)...(6k^2)- > 6(24)...(6k^2)-... -6(24)(54)...6k^2] = 6(24)(54)...(6k^2) > where now I'm only using elements of the ring Z[pi, pi_approx1, > pi_approx2], and note integers besides 1 and -1 are now units, for > instance 6 is a unit. OK, your notation is a bit confused. You have pi_approx1 and you jump to pi_approx2 without a horribly convincing explanation. Also, are you computing pi_approx2^2/6 = 1 + sigma(i=2,k,1) (1/(k^2 * sigma(j=1,k,1) (1/j^2) ) ) or pi_approx2^2/6 = 1 + sigma(i=2,k,1) ((1/k^2) * sigma(j=1,k,1) (1/j^2) ) or perhaps pi_approx2^2/6 = sigma(i=1,k,1) (1/(k^2 * sigma(j=1,k,1) (1/j^2) ) ) or pi_approx2^2/6 = sigma(i=1,k,1) ((1/k^2) * sigma(j=1,k,1) (1/j^2) ) ? In the third case the double sum is actually the value 1! > As k increases in size, more and more elements become units in the > ring, while there is never a diminishing number of unit elements. > But, in the limit as k goes out to infinity, I have > pi^2 [24(54)...(6k^2)- 6(54)...(6k^2)- 6(24)...(6k^2)- ... > -6(24)(54)...6k^2] = 6(24)(54)...(6k^2) > and the ring is Z[pi], as pi_approx1 and pi_approx2 approach pi, in > the limit. And theren lies your problem. The bracketed expression is a variant of infinity (k = +oo). Therefore, one has to approach things *very* carefully. If you can somehow prove that the bracketed expression, divided by the expression on the right side of the equals sign as k increases without limit, is unity (1), you might have something. Otherwise, you have something along the lines of pi^2 [oo - oo] = oo which isn't all that meaningful as an expression. > At which point you have an infinite number of units in the ring. > Previously I've seen attempts to dispute that result by claiming that > you can't use infinite sums, in particular cases, with integers or > rationals, but introducing pi into the ring changes the rules, so you > don't just have the ring of integers, as you have pi and the ring of > integers. Pi does not change the rules, where did you get that notion? Your operations are interesting, but your conclusion off base. I'm not sure precisely where the flaw is but it's obvious that one can approximate transcendental numbers easily using simple series; pi is but one example. Try your logic, for example, with the following sequence: e = 1 + 1/1! + 1/2! + 1/3! + ... + 1/k! + ... and you'll get exactly the same solution, that Z[e] has somehow gotten an infinite number of units and is therefore a field, which is of course as false as Z[pi] being a field, and probably easier to prove. > Notice that the *field* of rationals arbitrarily excludes certain > elements based on the rule that to be rational a number can be written > as b/a, where 'a' and 'b' are integers, and 'a' is nonzero. That is, > that a rational by definition can be written as the ratio of > integers, where the denominator is nonzero. This is arbitrary? The definition of a rational number is the quotient of two integers, the denominator of which is non-zero. Of course, one can use Dedekind sets with rational elements to define any real number. This proves that the field of rational numbers is not closed under infinite series. Or one can define Q_L, where Q_L is Q union the non-infinite supremum (or infimum, take your pick) of all subsets of Q, proper and improper, infinite and non-infinite. It turns out Q_L = R, not Q. pi is one element of R and of Q_L. > So the field of rationals has elements that are simply excluded based > on an ad hoc rule. > However, Z[pi] does not have a similar rule, though some posters seem > intent on adding one!!! Z[pi] has almost exactly the same rule. Elements of Z[pi] are such that [1] 1 is in Z[pi]. [2] pi is in Z[pi]. [3] For any x, y in Z[pi], x+y, x-y, and x*y are in Z[pi]. How ad hoc is that? You could of course claim that all of mathematics is ad hoc (there is some truth to that but it's not usually a problem; mathematicians have reached a good consensus on many axioms) but that way lies madness. In any event, I don't even have to postulate 0 in Z[pi]; that's automagic because 1-1 = 0 has to be in there. One could ask whether [3] can be applied an infinite number of times, though. (The short answer appears to be no, otherwise one runs into problems similar to the ones you are apparently having.) > Trying to exclude infinite sums with an infinite creation like pi, is > just an arbitrary attempt at avoiding a conclusion that is not > desired, and in fact, adding pi to the ring of integers introduces an > infinite number of units into the ring, as it makes all non-zero > integers units, so it creates a field. > Interestingly, it forces the field of rationals *plus* pi itself, > forcing a way around the arbitrary exclusion of pi from the field of > rationals. OK, silly one. If pi is rational then pi = m/n, m and n integers. What are m and n? (No, m = 22 and n = 7 is not a solution, although it's a surprisingly good approximation. Ditto for m=355, n=113.) > James Harris -- #191, ewill3@earthlink.net It's still legal to go .sigless. Subject: Re: Who here believes maths is all there is? Some see it as Literature, some see it as gospel. There are too many logical flaws to be gospel, IMHO. Lurch > cj-bubba@mindspring.com asks: >Where did Noah put the dinosaurs? > Good question, and well worth treating in a logical way; > That story has its value, but that value is not entirely historical fact, but > more preferably a story to illustrate ethics and conduct in a literary artistic > way. The authors(?) of the story did not know about dinosaurs. > Math occurs plainly in a few ways in that great story: pi, and the cubit. > Maybe other ways, too. > G C Subject: WOW! Is this known. WOW! Is this known? Is this known about the Collatz conjecture? Where if either integer is odd or even then this rule applies, 3n+1 if n is odd and n/2 if n is even. In the table below, read the 2 left columns in each row as one set and likewise the 2 right columns in the corresponding row as the other set. You will see the important relationship between the 2 sets. Left # of Right col. # of column children start # (n) children start # (n) including double of left +1 of col.2 start # col. start # children 1 1 2 2 2 2 4 3 3 8 6 9 4 3 8 4 5 6 10 7 6 9 12 10 7 17 14 18 8 4 16 5 9 20 18 21 10 7 20 8 11 15 22 16 12 10 24 11 13 10 26 11 14 18 28 19 15 18 30 19 16 5 32 6 17 13 34 14 18 21 36 22 19 21 38 22 20 8 40 9 21 8 42 9 22 16 44 17 n... Etc. I believe this has to happen -----> oo Why doesn't this prove the conjecture? ;-) Really, think about how this pattern could be broken! No way! The 2 set comparison is interesting also because the path going back to 4,2,1 is the same thus all the same children. Where doubling (n) in the second set just adds 1 more term to the path. Also the second set shows up later in another row in the first set. More supporting evidence that the conjecture is true! Dan Subject: Re: WOW! Is this known. >You will see the important relationship between the 2 sets. What, that the numbers in the last column are all one more the numbers in the second column? But that's obvious, because when you start with an even number N you take one step and then you have N/2, so you take one more step than you do for N/2. Or is there some other relationship you're referring to? -- Richard -- Spam filter: to mail me from a .com/.net site, put my surname in the headers. FreeBSD rules! Subject: Re: WOW! Is this known. > WOW! Is this known? Yes, it is known. If you define _any_ sequence using the rule that a(n+1) = a(n)/2 if a(n) is even plus any other rules then you will get exactly the same result. I sincerely hope you can find out yourself why that is the case. Subject: WWW: Notes On Methods of Proof Here's a link you, or JSH, might find useful. Funny, I don't see JSH in any current post on this reader. http://www.math.csusb.edu/notes/proofs/pfnot/pfnot.html Yours, Doug Goncz, Replikon Research, Seven Corners, VA Unequal distribution of apoptotic factors regulates embryonic neuronal stem cell proliferation