mm-351 === Subject: Re: some complex integration questions > I have a few quick questions about integration of complex functions. > first, how does one integrate over a curve that crosses a branch cut? > is it possible? if c(t)=2e^it, t in [-pi,pi], then can integration of > 1/(z^2-1)=1/2(1/(z-1)+1/(z+1)) be done as usual? what care do i have > to take when doing this? I do not understand your question. Your decomposition of 1/(z^2 - 1) is almost correct; it should be 1/(z^2 - 1) = 1/2(1/(z - 1) - 1/(z + 1)). So, all that is left for you to do is to calculate the integrals of 1/(z - 1) and 1/(z + 1) over your curve. For this, you use Cauchy's formula or the definition of index. There are no branch cuts here. === Subject: Re: the anticlassicalist }{ ii: the spectre continues |: > Oh, did you miss the thread on modality in language as well? |: |: What, Andrew Patterson's nonsense? That was (a) independent |: of your posturing and (b) obviously received largely with |: indifference. | |You see, that is the great evil laying at the heart of all your anger with |my post. There are people out there besides you, Brian, with many varied |interests. As long as we stay on topic to a particular group, everyone |should have the right to post their interests and questions. The usenet |does not follow office politics. Newcomers have all the same rights as |those who have been posting for years. It's not a newcomers-versus-oldtimers issue. Newcomers and oldtimers are not essentially different on this issue. It's a basic issue of shared resources. Bandwidth is very cheap indeed, and like with most resources, there's no simple maximum available amount of it. But like with most resources, the environment degrades as people use more and more of it beyond a certain point. Nearly every communication channel that's free for the sender is being overused in an unpleasant way. Now and then I get woken up at night by people honking their horns to get people's attention. As I pick up my newspaper in the morning, I find that junk mail has been added to it by sticking it under the rubber band, in addition to the usual advertising inside it. Or sometimes it's attached to the door knob, despite our no-soliciting policy. Outside, egocentric youngsters have spraypain their nicknames on buildings. At work, we've had a guy trying to sell Thomas Kincaid paintings from office to office, who didn't want to leave. Then there's the guy who thinks shouting to the receptionist at the other end of the office is easier than using the intercom. Our mailserver at work is pretty good at filtering out spam, but once in awhile somebody decides to pummel it with mail addressed to a large number of common names @ our domain, hoping that at least some of them will be somebody's user name. When I go to the bathroom, fairly often someone has attached advertisements for some work-from-home scheme to the wall of the stall. When I then leave work, I routinely find that advertising has been put under the windshield wiper. Back at home, every so often somebody repeats a message saying they've been assigned to review my mortgage and need to talk to me. And of course the usual messages from Nigeria and so on when I check my email. I'm not claiming that what you're doing is as obnoxious as those things, but it shares with them a certain selfish tendancy. How can we tell that it's selfish? Just honestly perform the thought-experiment of imagining what effect it would have on the netnews-reading experience if everybody were as casual about massive crossposting as you have been. It doesn't take too much imagination; overly crosspos threads are not so rare, and most of us have some idea how poorly they tend to work. A posting does not become on topic to a group just because it contains some paragraph that would be on topic on its own. If messages, with such weak ties to the groups they're initially pos to, initiate threads of any length, thread-drift usually causes them rather soon to become entirely off-topic except in one or two of the original groups. If people were really on their toes, they would stop crossposting at that point, but quite often that doesn't work. Sci.math has had prolonged discussions of all kinds of hot-button topics inflic on it. It seems that typically once a group such as talk.abortion gets involved, you can assume it will take months for the noise-to-signal ratio to go back down to normal again. I had a partizan on one side of the abortion flame war tell me point blank that he considered his extended criticism of someone's character to be on topic in sci.math, because his enemy happened to be a mathematician. One difference between a newcomer and an oldtimer, I suppose, is that if you are familiar with the way usenet used to be, you can see how much less pleasant it is now, due to the proliferation of noise. |: > Expense? |: |: Yes, expense. You are, for example, directly responsible |: for cluttering sci.lang with off-topic mathematics and |: complaints about the lack of physical content in your posts |: from sci.physics. | |One collection of people upset that I am posting mathematics, another upset |that there is not enough. Maybe it is these two groups that should be |arguing between each other and not be including me at all, I'd be a heck of a lot more sympathetic, if I didn't see you doing such disengenuous things. Here you probably don't realize how obvious it is that you're playing dumb in order to sound innocent. It would take some amazing degree of confusion for a person to suppose that people on sci.math complaining that they're having too many nonmathematical postings inflic on them, and people in sci.lang complaining that they're getting too many mathematical postings inflic on them, are disagreeing *with each other*. All this would also fail to be objectionable if there really wasn't any nicer way for you to call attention to your attemp interdisciplinary discussion than to crosspost so much of it. You could easily have chosen one place as home for the discussion, and pos only the bits actually relevant to various other groups, with a reminder of where the whole big thing was available. Doing it the way you did it instead serves only as an attention-getting move, an attempt to draw the attention of people (for example) reading sci.physics, but not interes enough in what you have to say to consider it worth checking out your home page or wherever you housed the rest of the discussion. You could just as well do it in a polite way, but you don't bother to. |but I think the |more prudent action would be for those who don't find content they are |interes in to just skip my threads. This is the standard excuse. Just delete the email, just give a few seconds of my time to the salesman, just wait a bit for the noise to go away, just toss out the junk mail, and so on, and so on, and so on, and so on, and so on, and so on. It's true that this is normally the best way of handling junk, but it's disengenuous coming from one of the sources of the junk, because it disregards the responsibility of the sender to have some shred of self-restraint. It just ignores how much cheesier life has become on account of so many communication channels now being half-filled with stuff that's there ONLY because the sender considers their interests in getting attention more important than the receiver's preferences in what to pay attention to. [...] |You start attacks, mister Scott, and |that puts you in error here. Massive crossposting is an attack. === Subject: Re: the anticlassicalist }{ ii: the spectre continues > |: > Oh, did you miss the thread on modality in language as well? > |: > |: What, Andrew Patterson's nonsense? That was (a) independent > |: of your posturing and (b) obviously received largely with > |: indifference. > | > |You see, that is the great evil laying at the heart of all your anger with > |my post. There are people out there besides you, Brian, with many varied > |interests. As long as we stay on topic to a particular group, everyone > |should have the right to post their interests and questions. The usenet > |does not follow office politics. Newcomers have all the same rights as > |those who have been posting for years. > It's not a newcomers-versus-oldtimers issue. Newcomers and oldtimers are > not essentially different on this issue. > It's a basic issue of shared resources. Bandwidth is very cheap indeed, > and like with most resources, there's no simple maximum available amount > of it. But like with most resources, the environment degrades as people > use more and more of it beyond a certain point. But, with scientists the message is always the same. It doesn't matter what *bandwidth* costs, since we are not paying for bandwidth. We are paying for information. As the same goes for water. It doesn't matter what water costs. Since we are not paying for water. We are paying for ice. === Subject: tutor sought for help in Sheldon Ross's Probability Models text mentioned above. Specifically with some joint distributions and conditional expectations. This is pretty basic so if someone who has a bit of patience would be willing to help I would be happy to pay a reasonable(??) rate. I have access to an online chat room to do the tuturial. === Subject: More dice help for the mathematically inept. Yep, it's me again; I've got yet another dice problem. Yes, I realise these things are childishly basic compared to a lot of the discussion that goes on here, but what the hell - you folks are marginally less likely to flame the hell out of me for my profound ignorance than most other forums I've tried, so I might as well. The system works like this: the player rolls a variable number of dice, each with (equally-weigh) sides numbered 1 through 10. For every die that comes up 7, 8, or 9, he scores one point; for every die that comes up 10, he scores two points. What I need to figure out is the probability of scoring *at least* a particular number of points, given that one is rolling a particular number of dice. I've already got the first couple of points. For example, the probability of scoring at least one point is 1 - (0.6^d), where d is the number of dice. Similarly, the probabilty of scoring at least two points is 1 - [(0.6^d) * (1 + 0.5d)], or, to summarise: P(1 or more) = 1 - (0.6^d) P(2 or more) = 1 - [(0.6^d) * (1 + 0.5d)] However, at this point I find myself at something of a loss. I can brute-force higher results with a simple algorithm easily enough, but I've been unable to come up with a formula that agrees with the results so obtained for higher scores (e.g., P(3 or more) and onward). I don't have an overwhelming amount of background in this area, so I'm probably missing something blindingly obvious here... Anyone feel like taking pity on me? ;) -- - Sir Bob. === Subject: Re: More dice help for the mathematically inept. >Yep, it's me again; I've got yet another dice problem. Yes, I realise these >things are childishly basic compared to a lot of the discussion that goes on >here, but what the hell - you folks are marginally less likely to flame the >hell out of me for my profound ignorance than most other forums I've tried, >so I might as well. >The system works like this: the player rolls a variable number of dice, each >with (equally-weigh) sides numbered 1 through 10. For every die that >comes up 7, 8, or 9, he scores one point; for every die that comes up 10, he >scores two points. What I need to figure out is the probability of scoring >*at least* a particular number of points, given that one is rolling a >particular number of dice. >I've already got the first couple of points. For example, the probability >of scoring at least one point is 1 - (0.6^d), where d is the number of dice. >Similarly, the probabilty of scoring at least two points is 1 - [(0.6^d) * >(1 + 0.5d)], or, to summarise: >P(1 or more) = 1 - (0.6^d) >P(2 or more) = 1 - [(0.6^d) * (1 + 0.5d)] >However, at this point I find myself at something of a loss. I can >brute-force higher results with a simple algorithm easily enough, but I've >been unable to come up with a formula that agrees with the results so >obtained for higher scores (e.g., P(3 or more) and onward). I don't have an >overwhelming amount of background in this area, so I'm probably missing >something blindingly obvious here... >Anyone feel like taking pity on me? ;) You are talking about the d-fold convolution of i.i.d. random variables with distribution p(0) = 0.6, p(1) = 0.3, p(2) = 0.1. My guess is the easiest way for you detemine the desired probabilities is by using probability generating functions. f(z) = 0.6 + 0.3z + 0.1 z^2. The generating function of the convolution is [f(z)]^d = [0.6 + 0.3z + 0.1 z^2]^d The probability that the sum is at least n is the sum of the coeffiicients of all terms of the nth power and greater. Play with the multinomial formula to get an exact formula for these. -- === Subject: Re: More dice help for the mathematically inept. > You are talking about the d-fold convolution of i.i.d. random variables > with distribution p(0) = 0.6, p(1) = 0.3, p(2) = 0.1. My guess is the > easiest way for you detemine the desired probabilities is by using > probability generating functions. > f(z) = 0.6 + 0.3z + 0.1 z^2. > The generating function of the convolution is > [f(z)]^d = [0.6 + 0.3z + 0.1 z^2]^d > The probability that the sum is at least n is the sum of the > coeffiicients of all terms of the nth power and greater. Play with the > multinomial formula to get an exact formula for these. You know, that's a lot simpler than I was trying to make it. I guess that's what I get for trying to do probabilities using half-remembered high-school statistics and a second-hand Math 100 textbook. Thank you muchly. - Sir Bob. === Subject: Re: Advice for future math majors? > I'm a high school senior planning on studying pure math as a major > this fall in college. What I'm wondering is, what kind of advice > could those of you who have already been through the experience offer > to people in my position? What kind of background is necessary (or > recommended) to begin college mathematics courses, and what could a > future math major do in order to get a head start? Are there any > books or other information sources that would be helpful? Any advice > is greatly apprecia. Pfui. Have some fun (and if it isn't fun, well, ... ). Work through one or two of Raymond Smullyan's puzzle books. I recommend To Mock a Mockingbird which starting from nothing ends at Godel's incompleteness theorem for combinators / lambda calculus (showing, along the way, how to use combinators to get arithmetic), or Forever Undecided which likewise starts with no assumptions and takes a slightly more conventional route to the incompleteness theorem. Both emhasize resoning and discovery over learning technique or material. The books seem to be out of print in the US but are available from amazon.co.uk. === Subject: Re: Advice for future math majors? > I'm a high school senior planning on studying pure math as a major > this fall in college. What I'm wondering is, what kind of advice > could those of you who have already been through the experience offer > to people in my position? What kind of background is necessary (or > recommended) to begin college mathematics courses, and what could a > future math major do in order to get a head start? Are there any > books or other information sources that would be helpful? Any advice > is greatly apprecia. > Pfui. Have some fun (and if it isn't fun, well, ... ). > Work through one or two of Raymond Smullyan's puzzle books. > I recommend To Mock a Mockingbird which starting from nothing > ends at Godel's incompleteness theorem for combinators / lambda > calculus (showing, along the way, how to use combinators to get > arithmetic), or Forever Undecided which likewise starts with > no assumptions and takes a slightly more conventional route to > the incompleteness theorem. Both emhasize resoning and discovery > over learning technique or material. > The books seem to be out of print in the US but are available > from amazon.co.uk. I haven't read Forever Undecided, but To Mock a Mockingbird must surely be the most mathematical book to not contain an equation. Your recommendation for To Mock a Mockingbird makes me want to get Forever Undecided. === Subject: Re: The coin problem > * Alexey007@hotmail-dot-com.no-spam.invalid > OK, I apologize for not being explicite enough. The number of tosses > must be bounded, procedures that might > go on indefinitely are not allowed. > Then no procedure is ever possible unless the coin is fair and the > number of participants is a power of two. > (Actually, there might be other solutions for a given unfair coin with > known probability of head. But I don't know if it is so, and my gut > feeling is no..) An unfair coin can be used to simulate a fair coin. Also, the procedure is very fast. Toss 2 coins or the same coin twice and let the guess be alike or different. Let both coins fall heads with probability 0.5+d. The probability that both will fall the same (heads heads or tails tails) is 0.5+2d^2 and for different it is 0.5-2d^2. Since d is greater than or equal to 0 and smaller than 0.5, 2d^2 is either smaller than d or equal to 0. As this process is repea, this difference decreases as follows for a value of d=0.1: 0.02 0.0008 0.00000128 3.2768e-12 This last one can be taken to be 0 for all practical purposes. The process can be considered to be naturally bounded. I am often an official umpire at table tennis tournaments. Before a match players have to make a guess so that they can decide on either choice of ends or who has to serve. Our South African coins do not have a head,except for an animal head that is on the tails side. A fellow umpire asks for animal or no animal instead of heads or tails. I have decided to use 2 coins at the next tournament for two reasons: the bias is less and the confusion is less. === Subject: e^i(pi) = -1 revisi with Doubly Infinites Re: infinite rightward strings tacked-on to p-adics serves as Orthogonality and makes Doubly-Infinites the points of Lobachevskian Geometry I would have liked to post this followup to a post I made several hours earlier tonight but Google has not yet correla that post to the newsgroup board and so I post this one as a non-sequitor followup. I remember the last time I visi e^i(pi) = -1 was in the early 1990s after learning that the p-adics of the 5-adics have a number that is truly i. And I remember Karl Heuer trying to help me see if that equation can be broken down into its number parts rather than a symbol-equation. So we had -1 as a Real, and we had e and pi as Reals and we had i as a 5-adic. But then no progress. Well, tonight I have something new and different that I did not have in the early 1990s. Tonight I have the idea that the Reals were a fake or fictional set and that the only true numbers are either p-adics or Doubly-Infinites. So let me see if progress now can be achieved with the equation e^i(pi) = -1. I prefer 10-adics because all of us are so used to seeing the decimal system but that my hinder us because i is in the 5-adics. So can we re-arrange the 5-adic i into a 10-adic? So in the equation e^i(pi) = -1, let us mark out what numbers we have. Since Reals are a fictional set then e and pi are Doubly-Infinites. That leaves i and -1 and they are thus p-adics. In 10-adics -1 is easily replaced as ....999999. But can we use the 5-adic i in 10-adics? If we can us the 5-adic i or transform it into a 10-adic i then it is easy to transfer straight across the Doubly Infinite for e and pi as .....00002.71...... and .....0003.14..... If we can, would leave the question has how one multiplies p-adic with Doubly-Infinites to achieve an end result of .....9999999 which is representative of -1. Archimedes Plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: e^i(pi) = -1 revisi with Doubly Infinites Re: infinite rightward strings tacked-on to p-adics serves as Orthogonality and makes Doubly-Infinites the points of Lobachevskian Geometry > I would have liked to post this followup to a post I made several > hours earlier tonight but Google has not yet correla that post to > the newsgroup board and so I post this one as a non-sequitor followup. > I remember the last time I visi e^i(pi) = -1 was in the early 1990s > after learning that the p-adics of the 5-adics have a number that is > truly i. And I remember Karl Heuer trying to help me see if that > equation can be broken down into its number parts rather than a > symbol-equation. So we had -1 as a Real, and we had e and pi as Reals > and we had i as a 5-adic. But then no progress. > Well, tonight I have something new and different that I did not have > in the early 1990s. Tonight I have the idea that the Reals were a fake > or fictional set and that the only true numbers are either p-adics or > Doubly-Infinites. > So let me see if progress now can be achieved with the equation > e^i(pi) = -1. The progress has already been done. Since Euler not only proved that the equation is true, but he also proved that you need Geometric a proof to prove it. Goedel proofs are insufficient. You need a pair of duel functions to prove that it's true. So that the consistency relationship: Jerk Mod Wanker == 0 is also preserved throughout the prove. Otherwise you consistently get the retarded answer: e^log(-pi) is bounded from above. === Subject: how we turn e^i(pi) = -1 into c = (pi)d Re: e^i(pi) = -1 revisi with Doubly Infinites (snipped) > I remember the last time I visi e^i(pi) = -1 was in the early 1990s > after learning that the p-adics of the 5-adics have a number that is > truly i. And I remember Karl Heuer trying to help me see if that > equation can be broken down into its number parts rather than a > symbol-equation. So we had -1 as a Real, and we had e and pi as Reals > and we had i as a 5-adic. But then no progress. > Well, tonight I have something new and different that I did not have > in the early 1990s. Tonight I have the idea that the Reals were a fake > or fictional set and that the only true numbers are either p-adics or > Doubly-Infinites. > So let me see if progress now can be achieved with the equation > e^i(pi) = -1. > I prefer 10-adics because all of us are so used to seeing the decimal > system but that may [sic] hinder us because i is in the 5-adics. So can we > re-arrange the 5-adic i into a 10-adic? > So in the equation e^i(pi) = -1, let us mark out what numbers we have. > Since Reals are a fictional set then e and pi are Doubly-Infinites. > That leaves i and -1 and they are thus p-adics. In 10-adics -1 is > easily replaced as ....999999. But can we use the 5-adic i in > 10-adics? If we can use [sic] the 5-adic i or transform it into a 10-adic i > then it is easy to transfer straight across the Doubly Infinite for e > and pi as .....00002.71...... and .....0003.14..... If we can, would > leave the question has how one multiplies p-adic with Doubly-Infinites > to achieve an end result of .....9999999 which is representative of > -1. Thinking about this, this morning if the above is correct in part or whole draws various implications, and geometrical ones which are often more easily tackled when a problem is turned from algebraic to geometrical. If the world consists of 2 and only 2 types of numbers where the P-adics and Doubly-Infinites are the only numbers. And where Reals and NaturalNumbers=Finite-Integers are a fictional or fake lot just as Newtonian absolute time and absolute space is fake. Thus the equation e^i(pi) = -1 would really be an equation talking about the circumference of the entire universe starting at point 0 and going out to 1 then 2 then 3 in the 10-adic string and ending up back at the same starting point of 0 having traversed to .....999997 then ....99998 then ....9999 and finally back to 0. So that every P-adic string starts at 0 and comes back around in some fantastic great circle to its starting point. So the P-adics circumscribe a great circle. And the equation e^i(pi) = -1 has two p-adic numbers of i and -1 but the -1 is not properly designa for it really is ....9999 in 10-adics and respectively in any other adic of its infinite leftward string. So the equation e^i(pi) = -1 has two p-adics and it has two Doubly-Infinites of e and pi. Now if the entire Universe is a atom of plutonium and specifically 231Pu then it would be circular in traversing the Universe. Atoms are circular in overall geometry and starting at 0 going out far enough you will end up at 0. So the equation e^i(pi) = -1 is really telling us that the circumference of the Universe is ...999999 in distance. We all know the familar geometrical equation of circumference of a circle is pi times diameter or c = pi(d). In the case of the Universe as an atom of 231Pu it may be circular or spherical in one route of P-adics such as for instance the 5-adics which has a i but in other p-adics such as 2-adics where there is no i the universe is lobelike and not circular or spherical. The portion of the equation e^i(pi) = -1 of e^i(pi) would then be similar to pi(d) where the diameter is tangled up with e and i and pi. So we have given in the equation e^i(pi) = -1 of the 10-adics the number ....9999999 and the i number of perhaps the 5-adics which we hope to transform into a 10-adic substitute. We then have the two Doubly-Infinites of e as .....00002.17...... and the Doubly-Infinite of .....000003.14...... When multiplying Doubly-Infinites with p-adics is not the same as multiplying with Reals. Doubly-Infinites are not Reals. Doubly-Infinites are more like p-adics except that they have a infinite rightward string that imposes a orthogonality compared to p-adics. So we can imagine ourselves getting on a rocket that can traverse the entire Universe starting at 0 in 10-adics and we eventually will come back around to ....99999. So the equation e^i(pi) = -1 of its term of e^i(pi) is a quick multiplier of the numbers e and i and pi to get us to the end point number of ....99999 Archimedes Plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies (www.iw.net/~a_plutonium) website of the science of AP under revision what used to be my old science website www.newphys.se/elektromagnum/physics/LudwigPlutonium from years 1993 === Subject: Re: Motivate Jim Brown asked, > Can someone please explain to me the usage of the word motivate, as in a > recent posting Motivation for e? >>This appears to be American mathematician's argot. It is unfamiliar to at >>least some of the British mathematicians who post here, and thus is probably >>fairly recent. I have seen this usage of the word motivate in quite a few > It is not mathematician's argot, it is educationists' > argot. There is this belief among those who do not > understand concepts that the only way to learn something > is to be sufficiently motiva. Learning something now > because it may be useful later is not acceptable. You're spending too much time reading K-12 curriculum reports (or soemthing) and it's having an evil effect. The term motivate (well-motiva, unmotiva) has been in use by mathematicians since at least the mid-70s, to describe exposition and presentation. Well-motiva means that the author/presenter is developing a theme, and the capable and diligent reader/listener can see it unfolding -- one sees that everything in the piece is there because it moves things along towards a goal. Unmotiva is roughly the opposite -- an unmotiva lemma may provide some crucial fact, but it seems to come out of thin air, with no discernable relation to whatever is going on other than the magical effec. For example (from memory 35 years ago so it may be a bit off) there a theorem along the lines that the lattice of subgroups of a group G is distributive iff every finitely genera subgroup is cyclic, One way the argument just makes sense (and is not too hard) -- it is well motiva -- but the othe way requires some little number-theoretic diddle that does not bear in any obvious way on the issue, there's no aha or of course when it appears -- it's an unmotiva trick. === Subject: Re: Tries to find proper factors > Consider the therm: x^a + y^b - z^c > where a;b;c are odd numbers > and x;y;z such integethat x+y-z is some odd number > does some c^2 could be the factor of the upper therm ? > ( according to my accounts c value could be only some > prime number beginning with c=5 ) > Compliments > Ro You'll have to strengthen that a little bit; c = 1 always works. Did you just want non-silly solutions, or was there supposed to be another condition on a, b, and c? Also, why c in particular, instead of a and b? === Subject: Re: Which Journal (If any) might publish this? > I have found derived a formula which approximates the area under the > integral of exp(-x^2). I have seen several such formulas on Mathworld > but none are similiar to the one I have found. > Which UK maths journal would I be advised to send my work off to? (If > at all) > I realise this is a very minor result but I would just appreciate the > advice of some experienced mathematicians on whether it has any value > mathematically. > Thank you for your time. > Dave What is the computational efficiency for a given accuracy? Is it in a special form that could be of any special theoretical use? If the answer to the above two questions is 'no,' then you want a recreational journal. If the answer to the first is 'yes,' you probably want to send it to an applied math or computer journal. If the answer to the second is yes, then send it to a journal in whatever area you used to get the answer. You will, of course, get the opinion of experienced mathematicians in the form of your paper either being accep or rejec, but if there's a university nearby you might want to set up an appointment with a professor there to talk it over, or you could post it here and somebody (not me) would probably be able to tell you if it looked useful. === Subject: Re: on the male alpha problem [snippi] > .... Read it if you care, it's a > : very spooky book about knowledge. > I _have_ read Foucault's Pendulum. It _was_ a very powerful book on the > nature of knowledge. Don't go around pretending like everyone is more > ignorant than you. It is quite a disgusting display. Ok. You missed the reference, I drew the wrong conclusion. The point is taken, after all I was acussing you of almost the same thing. > I would love to discuss Eco in depth with you, but you need to correct your > tone before that happens, and I would prefer to take that somewhere more > appropriate. Let's wait and see what the future brings. > (By the way, I'm sure you have seen, my dear Jorge de Burgos, that there is > a strange similarity of the heresy of Aristotle's work on humor and levity > and the heresy I am accused of by yourself and otheof being arrogant > enough to enjoy my work and my creativity. I know, I know... I should be > more solemn when walking through the halls of knowledge you wish to > control...) I'm no bitter blind old monk, I'm just a regular fool demanding some respect for plain common sense. And anyway, I haven't found out how to poison the edges of pernicious usenet postings; darn! > I pos a central question that was the focus of my towards a constructive > education thread. In fact, it's right there in the title. Should we > expand the education of logic? Restructure a bit, maybe move certain things > to younger ages? Include the nonclassical? Now, don't get falacious again, of course you asked a central question, but you also were proposing a whole programme (your own word) with it. The original was also heavily loaded with your personal opinion and feelings. By the point where you actually sta your question, you had lost most of your audience. [snip a bit of restating] > My entire point has already been sta quite a number of times, that the Yes therefore I allowed myself to snip a bit out, no censorship intended. > models we are constructing about the world around us, the languages we use > in natural languages, folk-science, and even quite a lot of modern science, > all make heavy use of nonclassical logics. There is still quite a tradition > to use classical logic as the metamodel, which then faces difficulties with > classification of what types of statements it can really be applied to. It > is not universally applicable to all statements, and the study of this is > fairly well established. And yet you have not produced a single tangible instance of a meaningful application of what you want. By now, you must have realised that your assumption that there would be people out there just waiting for someone to start discussing these things multidiciplinarily, was wrong. There were a very few ones, but rejection came in droves on all possible levels. So give them something to bite on. As you say yourself, there have been practically no refutations of the factual correctness of your assertions. However, you also haven't proven that they are factually right. Since you are the proponent, it is you who has to do the demonstration, in order to be more convincing. > So I also introduced my suspicion as to why classical logic gets presen > in the ways I have seen it presen, as the universal logic of everything. > I suspect that people really want things to be true or false always, > either-or, that they desire at least metaphysically for such a > classification to exist even when we have known at least for three quarters > of a century that we cannot always deduce such from our math. This I > associate with monotheism's popularity, and I am not the only one as I point > to the quote by R. M. Martin in his book on semiotics makes the same claim > and attributes its origin to Whitehead. It really doesn't matter who talked first about this suspicion as you call it. It is just a very bad idea to mix religion into a discussion about Science. Religion takes place in the domain of faith. Science in that of reason. You cannot oppose the one to the other. You cannot prove by faith, and you cannot believe by science. You can choose to embrace either or even both. > And I believe that if people understood the way we can formalise different > structures of reasoning about the world, how we can build various models and > test them to our experience, and along the way were shown that alot of our > most useful models today are built in structures of nonclassical logics, > maybe some of the fundamentalism I see in the world would fade. Just a > little. I'm don't expect major changes. People are very far from understanding these things for reasons that are totally out of topic. Don't show this kind of naivit.8e, it causes anger. Maybe it has a positive effect on a very few people, by giving for sheer candidness a cute side to your personality. But please realise that before any number of people significant enough to make a difference, however small, get to understand these things, a lot of much more real, serious and pressing problems, have to be solved first. > Its just that I think there is something inherently unhealthy about a > framework or world view that forces people to attribute an absolute truth or > falseness to all statements in their mind. It makes people build defensive > psychologies to hold on to whatever world-view they have construc when > logical consistency starts kicking in, and other ideas become excluded to > the point where hatreds are formed. And maybe it is just an inherently unhealthy world that forces, with brutality on people, the duality of its nature . I exchanged a cannonade with Mitch about this, unfortunately we didn't make anything out of it, besides a few ruffled feathers that is. Those defenses you talk about are built on a very low level and are absolutely necessary for one to function properly. Most of the time we dont go around reasoning but simply apprehending, discriminating, reacting and interacting according to the rules of the model of the world that we have built. Imagine if we had to reason everey step necessary to lift a glas of water to our lips. We would die of thirst. So in those simple rules lies the root of what so frightens you, not in unsecurity but in its opposite. When we actually come to reasoning, those mechanisms are still in place, and one does have to push people a bit. Hey wake up, don't discriminate by default, listen to me. Then of course you have to give them something they can listen to. As for getting the waking call when consistency problems arise, mankind has developed many escape mechanisms. There are healthier and unhealthier ones. You seem to consider the fact that we have to escape at all, unhealthy in itself. I understand you are proposing to make those problems go away by building better models. If you can, then do so, and teach us. But remember, the landlord won't go away without his rent check, no matter how cleverly you choose to look at the matter. And if you can't, then you are just escaping by the worst possible path, namely an unknown and untreaded. And if that were the case, you would be particullarly obnoxious, because life is hard enough, without someone telling me how to make it better, by doing weird stuff to the way I look at the world. > I like open, curious people who don't desire to post contentless negativity > when something doesn't fit inside their worldview, because that is a > prerequisite of true science. I believe that there is a certain mental > dynamic that quests for certainty. Some choose religion, particularly > monotheism, because it often screams out quite loudly its importance as the > source of the certain absolute truth and adds some nice, though unseen, > perks through Pascal's wager. Others choose science, which at first seems Don't keep telling us why people choose religion. You can't choose religion if you have lost the faith. And you don't choose religion because of any reason that is rela to that which we are discussing here. > Take, for instance, superstring theory. As with all ideas when they are > first introduced, it has been attacked quite a lot from many different > directions. With more research in the field, it has been able to establish I don't have a clue about superstring theory. I once read a stupid book about it. My personal conclusion is that it is absoluty worthless because of its complexity. Good science is as good as the model it produces. The quality of the model depends not how it reflects the cleverness of its creator, but on the benefit that can be derived from it. A theory that proposes several dimensions just for the sake of postulating wrigling strings as the building blocks of reality, and can therefore not be understood by anyone at all, is worthless. Of course most of us don't understand the theories of relativity and quantum physics either, I certainly don't, but there are enough scientists around who do, and who can derive benefit from them. > I have my suspicions that some of the anger response is residual > manifestations of the male alpha problem. In past societies, hierarchies of > power formed through displays of strength and aggressiveness. It was useful > because strength and aggressiveness were useful on a biological level. > Nowadays, I see the same quest for hierarchies in science, in particular the > same quest for absolute truth centered on one model. That is why, for > example, there is a standard view of quantum mechanics and alternate > interpretations are so commonly demeaned. I was wondering when you were going to come out with some feminism, I'd been expecting it for a while ;p Don't call it a problem, its natural you know, just like sh... O.K. forget it. I do believe I have given you a few good reasons for the angry responses, at leas a few good clues. Don't forget, most of us are animals out here. > So you see, Guenther, I have now expressed yet again the whole point of any > of this. Along the way, you will notice that I have had to fight my share > of alpha attacks. But I have always asser, as I do to you now, that I > have a right to ask the questions I have, in the forums I have, at the time > I have chosen. > Everyone here has that right. Agree as long as you don't say everyone is entitled to that. Big difference you know. regards === Subject: Re: Complete, but not Baire > Could someone give me an example of a complete uniform space which > is not a Baire space? I've tried to construct such a space, but in > vain. Furthermore, there's no exemple of such a space in Steen & > Seebach's Counterexamples in Topology. >> Because Q (the space of rational numbers) is metrizable the family >> of all neighbourhoods of the diagonal forms (a base for) a uniformity >> this uniformity is complete. >> However, Q is not a Baire space. > I see that the topology induced on Q by this uniform structure is the > usual one, but why is Q complete with respect to it? Is it obvious? > (for me it isn't; otherwise I wouldn't be asking it, right? ;-) ) Consider a Cauchy-filter F and look at the family of closures of elements of F, call that G. The family O={QA:A in G} consists of open sets. If it were a cover it would belong to the uniformity and so it would meet F. That is, you'd have A in G such that QA belongs to F; but A also belongs to F and we have a contradiction. So O does not cover, which means the intersection of the family G is nonempty, because F is Cauchy the intersection consists of one point only and F converges to that point. KP === Subject: Re: Complete, but not Baire >> Could someone give me an example of a complete uniform space which >> is not a Baire space? I've tried to construct such a space, but in >> vain. Furthermore, there's no exemple of such a space in Steen & >> Seebach's Counterexamples in Topology. > Because Q (the space of rational numbers) is metrizable the family > of all neighbourhoods of the diagonal forms (a base for) a uniformity > this uniformity is complete. > However, Q is not a Baire space. >> I see that the topology induced on Q by this uniform structure is the >> usual one, but why is Q complete with respect to it? Is it obvious? >> (for me it isn't; otherwise I wouldn't be asking it, right? ;-) ) > Consider a Cauchy-filter F and look at the family of closures of > elements of F, > call that G. The family O={QA:A in G} consists of open sets. > If it were a cover it would belong to the uniformity and so it would > meet F. > That is, you'd have A in G such that QA belongs to F; but A also > belongs to F and > we have a contradiction. > So O does not cover, which means the intersection of the family G is > nonempty, > because F is Cauchy the intersection consists of one point only and F > converges > to that point. Thanks. I was about to post here that I had already seen why your statement was true. BTW, my proof is distinct from yours. Thanksa lot, === Subject: Irreducible polynomial of Z_2[x] In Z_2[x], the polynomial x^6+x+1 is irreducible. It is easy to show that it cannot be factored in polynomials of degree 1. Which is the easiest/quickest method to find that it cannot be factored in 2*4 and 3*3 polynomial degree? Thanks === Subject: Re: Irreducible polynomial of Z_2[x] > In Z_2[x], the polynomial x^6+x+1 is irreducible. It is easy to show that it > cannot be factored in polynomials of degree 1. > Which is the easiest/quickest method to find that it cannot be factored in > 2*4 and 3*3 polynomial degree? > Thanks You only have to test for divisibility by _irreducible_ polynomials of degree 2 and 3. Over Z_2, aren't the quadratics x^2, x^2 + 1, x^2 + x all reducible leaving only x^2+x+1 to test for as a quadratic factor? Similarly, I believe that x^3 + x^2 + 1 abd x^3 + x + 1 are the only irreducible cubics. === Subject: Re: Irreducible polynomial of Z_2[x] > In Z_2[x], the polynomial x^6+x+1 is irreducible. It is easy to show that it > cannot be factored in polynomials of degree 1. > Which is the easiest/quickest method to find that it cannot be factored in > 2*4 and 3*3 polynomial degree? > Thanks compare (X^2+ax+1)(X^4+bx^3+cx^2+dx+1) and (x^3+ax^2+bx+1)(X^3+cx^2+dx+1) after multiplying out with x^6+x+1 === Subject: Re: Irreducible polynomial of Z_2[x] >> In Z_2[x], the polynomial x^6+x+1 is irreducible. It is easy to show that it >> cannot be factored in polynomials of degree 1. >> Which is the easiest/quickest method to find that it cannot be factored in >> 2*4 and 3*3 polynomial degree? >> Thanks >compare (X^2+ax+1)(X^4+bx^3+cx^2+dx+1) and >(x^3+ax^2+bx+1)(X^3+cx^2+dx+1) after multiplying out with x^6+x+1 forgot to say: each factor of degree 3 or fewer is irreducible or there is a linear factor, so in the first one a must in fact be 1, and in the second, only one of a and b can be 1, the other must be 0. === Subject: Re: Irreducible polynomial of Z_2[x] > In Z_2[x], the polynomial x^6+x+1 is irreducible. It is easy to show that > it cannot be factored in polynomials of degree 1. > Which is the easiest/quickest method to find that it cannot be factored in > 2*4 and 3*3 polynomial degree? I'm not sure if this is the best way, but if p(x) (your polynomial) consis of 3 irreducible quadratic factors (they would have to be different since p'(x) = 1) then it would divide U_2(x) = x^{2^2} - x. Similarly, if it consis of 2 irreducible factors of degree 3 it would divide U_3(x) = x^{2^3} - x. So if you compute gcd(p(x),U_2(x)) and gcd(p(x),U_3(x)) this will give you the answer. -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Minimally simple finite groups? [...] >Is it covered in Carter's Simple Groups of Lie Type? How about >the Atlas? (Not that a copy of that can be had for love nor >money, as far as I've been able to discover. :-( Thank you! It never occurred to me that they'd have something not available at amazon.com. -- Jim Heckman === Subject: Re: maxiize quadratic form under quadratic constraint X-ID: XLLBCmZLZeg1FDA1KcvpQeg9vc50R7GCMc9b8PrDXZbDmIESniWegv jammy tsai schrieb: > Daer all: > I have a question about optimization. > The description is as below: > max. (x-t)'R(x-t) > S.T. (x-m)'W(x-m)=c > t, m, R, W, c is given > t m x is a n*1 vector > R W is sym. matrix > c is a scalar this seems quite easy (if n is reasonable). See http://mathworld.wolfram.com/LagrangeMultiplier.html hth klaus > any body can solve this question?? > jammy === Subject: Re: Multiverse as a Macro-QUBIT Computer > Note #3 The QUBIT Frame of Reference >[big snip] The map 10.2.2 induces a local Minkowski frame at a given point ... > therefore the base space acquires a pseudo-Riemannian structure in > addition to a spinor structure. > to be continued Please don't. Your posts have the effect of removing knowledge from the universe. === Subject: Question about poitwise convergence Is there a norm on C[a,b] linear space such that convergence with respect to the norm is equivalent to poitwise convergence? Thanks. === Subject: Re: Question about poitwise convergence >Is there a norm on C[a,b] linear space such that convergence with >respect to the norm is equivalent to poitwise convergence? No. You could see this by considering properties of the _topology_ such that convergence in that topology is equivalent to pointwise convergence... Here's a direct proof. Take [a, b] = [0,1]. Let f_n be the piecewise-linear function interpolating f_n(0) = 0, f_n(1/n) = 1, f_n(2/n) = f(n) = 0 (so f_n = 0 except for a narrow spike near x = 0.) Now c_n * f_n -> 0 pointwise for _any_ sequence of scalars c_n. So if there were a norm as above we would have c_n * ||f_n|| -> 0 for any sequence c_n, which is impossible (because ||f_n|| > 0: let c_n = 1/||f_n||.) >Thanks. ************************ === Subject: f continuous f:R-->R surjective with the property ( for any x(n) real sequence f(x(n)) converge => x(n) converge ) Prove that f is continous === Subject: Re: f continuous > f:R-->R surjective with the property > ( for any x(n) real sequence f(x(n)) converge => x(n) converge ) > Prove that f is continous It is clear from the condition that f is injective so it is a bijection. Set g = f^-1. Then g is also a bijection and the given condition implies g is continuous. So it is enough to show a continuous bijection R->R has continuous inverse. This is easy - use the fact that a continuous injection R->R is monotone. === Subject: Re: f continuous >f:R-->R surjective with the property >( for any x(n) real sequence f(x(n)) converge => x(n) converge ) >Prove that f is continous What about the function f(x) = x except when x = 0 or 1 where f(0) = 1 and f(1) = 0? --Lynn === Subject: Re: f continuous >What about the function f(x) = x except when x = 0 or 1 where >f(0) = 1 and f(1) = 0? >--Lynn Woops, pos that too quickly. Never mind. --Lynn === Subject: Re: f continuous === > f:R-->R surjective with the property > ( for any x(n) real sequence f(x(n)) converge => x(n) converge ) > Prove that f is continous Apply the theorem that in a first countable topological space, functions which preserve limits are continuous. === Subject: Re: f continuous Apply the theorem that in a first countable topological space, What say this theorem ? === Subject: Re: f continuous > f:R-->R surjective with the property > ( for any x(n) real sequence f(x(n)) converge => x(n) converge ) > Prove that f is continous Apply the theorem that in a first countable topological space, > functions which preserve limits are continuous. But here f doesn't preserve limits, but instead does something in reverse of that. === Subject: Re: Simple question on product topology === Subject: Simple question on product topology >First he states that defining the product topology of spaces >(X,Tx) = (Y,Ty) = R by Bxy = {O1 x O2 | O1 in Tx, O2 in Ty} will >not work since although the sets (0,1) x (0,1), and (2,3) x (2,3) >are in Bxy, [(0,1) x (0,1)] U [(2,3) x (2,3)] is not. >If it were, then <0.5,2.5> in (0,1)x(2,3), but <0.5,2.5> is not in >[(0,1)x(0,1)] U [(2,3)x(2,3)]. Thus T is not closed under unions >and so is not a topology. >Does this guy simply not know what he's talking about or am I >missing something very obvious or have made some basic logical >error? He's jerking off with (0,1)x(2,3). Here's good version. For B = [(0,1)x(0,1)] U [(2,3)x(2,3)] to be in B_XY there's some open sets U,V with B = UxV Now (1/2,1/2) in B, hence 1/2 in U and (5/2,5/2) in B, hence 5/2 in V Thus (1/2,5/2) in UxV, but (1/2,5/2) not in B ---- === Subject: Re: Interactive Proof Writing Tutorial (Freeware) <95ou30tv1r3s3dqi07dlad0od54ulm5jl6@4ax.com> <4m1v3094a4ea2vetiakqr2l2ipps2ahmh9@4ax.com> <1sO%b.360$Xy3.1137@tor-nn1.netcom.ca> >> Who the did EVER write >> p | q >> when he MEANT >> p v q or p / q or p or q >> ??? Well, OTTER seems to do so IIRC, irritating me no end... > Somebody whose been taking CompSci for too many years? Wouldn't cs people rather use + (from switching theory, where | also means NAND)? Note that the | notation used in some computing languages doesn't really denote a logical operator but a truth-evaluation function for which it's usually assumed that the right side isn't evalua if the left side returns true. regards Stephan === Subject: Re: Interactive Proof Writing Tutorial (Freeware) X-ID: EYH6DMZewexTNOnivyQKy-r55WLAcUmIBNqPCJas38vrzjN8L40eo8 Imho, it's extremely misleading to use | instead of the > connective v. Who [...] did EVER write p | q when he MEANT p v q or p / q or p or q ??? Somebody whose been taking CompSci for too many years? Right. Actually, I KNOW that. ;-) But surely there's a world beyond the CS department, isn't it? Indeed, OTTER for example uses | to denote disjunction in its ASCII /input files/. But frankly, I would hardly call OTTER a teaching tool. (And there's a certain difference between an ASCII input file of a highly sophistica automatic proving software and an interactive proof writer intended as a teaching tool. ;-) > Honestly, I've done it before, particularly when I've been working on > programming and then go to take a break. Of course, my scratch work > contains all sorts of nonstandard symbols... :-/ Right. And I guess that's rather normal. Why should anyone care? For example, in a *personal* (e-mail) communication *I* recently proposed to use | for abbreviating xor. Rationale: It's just simpler than >-< or _v or xor, and esthetically more appealing. With other words, we are free to do that in _certain_ contexts. - This way, Either P or ~P would be expressed with P | ~P. Well, right, but *I* certainly would n e v e r consider to adopt | to express or. ;-) F. Subject: re:Interesting problem === Then again, it's impossible to split a field into two such sets. Pos Via Usenet.com Premium Usenet Newsgroup Services ---------------------------------------------------------- ** SPEED ** RETENTION ** COMPLETION ** ANONYMITY ** ---------------------------------------------------------- http://www.usenet.com Subject: re:Interesting problem === If by finite extension you mean extension by one element (or finitely many elements) then it's possible. If F+ can be split into two closed sets (under mult. and add.) then if we add x transcendental over F to obtain the field F(x) which consist of rational functions of x over F, we can consider those functions with leading coefficients in A or B. Doesn't quite work if x is algebraic over F. Pos Via Usenet.com Premium Usenet Newsgroup Services ---------------------------------------------------------- ** SPEED ** RETENTION ** COMPLETION ** ANONYMITY ** ---------------------------------------------------------- http://www.usenet.com === Subject: Re: Interesting problem >I'm still not convinced that a non-empty A can exist... > Neither am I. But I'm also not convinced that it can't exist. > Note that if you only had to worry about addition, not multiplication, > there would be examples: e.g. take an additive but nonlinear function > f on the reals (f(x+y) = f(x)+f(y) for all x,y in R, but not f(x) = cx. > Such exotic beasts are consequences of the Axiom of Choice (e.g. > using a Hamel basis for the reals over the rationals). Then > take A = {x > 0: f(x) >= 0} and B = {x > 0: f(x) < 0}. However, > these won't satisfy the multiplicative requirements. Here's a simpler version: Let U be a finite extension of the algebraic numbeand U+ = U n R+. Does there exist A,B meeting the requirements: U+ = A u B, A n B = {}, A and B closed under +, *? I _think_ the answer is yes, but I don't quite have a proof yet... Cheers - Chas (n == intersction, u == union) === Subject: Re: Number Theory Problem! by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1SD7en03941; by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i1S8mwi16155 by proapp.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 $, proapp) id i1S8mwq00509; >
 
 >Does anyone know how to prove the statement: 
 >Let p be a prime and let a,b be integers such that a^p is 
 congruent to b^p 
 modulo p. 
 >Prove that a^p is congruent to b^p mudulo p^2. 
 >Also, If n>4 and n is composite, prove that (n-1) is 
 congruent to 0 modulo 
 n. 
 >Another is, Prove that 1/5n^5 + 1/3n^3 + 7/15n is an integer 
 of every 
 integer n. 
 >I would appreciate any suggestion or solutions on how to 
 prove the above 
 problems/statement!. 
 >Thanks 
 >Ferdinand 
 >Please send to my e-mail sirferdz23i@yahoo.com 
 === 
 Subject: Re: Number Theory Problem! 
 Ferdinand Balmes  escribi.97: 
 >> 
 
 >> Does anyone know how to prove the statement: 
 >> Let p be a prime and let a,b be integers such that a^p is 
 congruent 
 >> to b^p modulo p. 
 >> Prove that a^p is congruent to b^p mudulo p^2. 
 a^p = a (mod p) if p is prime. Then a^p = b^p (mod p) ===> a = 
 b (mod p) 
 ==> 
 b = a + k*p 
 b^p = (a + k*p)^p = a^p + p*a^(p-1)k*p + ... + k^p*p^p = a^p + 
 p^2*M ==> 
 b^p = a^p (mod p^2) 
 -- 
 Saludos, 
 Ignacio Larrosa Ca.96estro 
 A Coru.96a (Espa.96a) 
 ilarrosaQUITARMAYUSCULAS@mundo-r.com 
 === 
 Subject: Re: Number Theory Problem! 
 >Also, If n>4 and n is composite, prove that (n-1) is 
 congruent to 0 modulo 
 n. 
 You probably mean prove that (n-1)! is divisible by n. 
 Hint: either n = p^2 with p > 2, or n = ab with 1 < a < b < n. 
 John Robertson 
 === 
 Subject: Re: Number Theory Problem! 
 Ferdinand Balmes  a .8ecrit dans le 
 message de 
 > 
 >Another is, Prove that 1/5n^5 + 1/3n^3 + 7/15n is an integer 
 of every 
 integer n. 
 > 
 24C(n,5)+48C(n,4)+32C(n,3)+8C(n,2)+C(n,1)=1/5n^5 + 1/3n^3 + 
 7/15n 
 === 
 Subject: Re: Number Theory Problem! 
 Ferdinand Balmes  a .8ecrit dans le 
 message 
 de 
 > 
 >Another is, Prove that 1/5n^5 + 1/3n^3 + 7/15n is an integer 
 of every 
 > integer n. 
 > 
 > 24C(n,5)+48C(n,4)+32C(n,3)+8C(n,2)+C(n,1)=1/5n^5 + 1/3n^3 + 
 7/15n 
 That's nice! Is it true that every polynomial with rational 
 coefficients f(n) such that f(n) is an integer for every 
 interger n is 
 a linear combination of binomial coefficients C(n,k)? 
 Mark Sapir 
 === 
 Subject: Re: Number Theory Problem! 
 Ferdinand Balmes  escribi.97: 
 >> Also, If n>4 and n is composite, prove that (n-1) is 
 congruent to 0 
 >> modulo n. 
 (n - 1) =/= 0 (mod n) always that n > 1!! 
 >> Another is, Prove that 1/5n^5 + 1/3n^3 + 7/15n is an 
 integer of 
 >> every integer n. 
 1/5n^5 + 1/3n^3 + 7/15n = n(3n^4 + 5n^2 + 7)/15 
 Study if n(3n^4 + 5n^2 + 7) is always multiple of 3 and 5. 
 Consider first n 
 = 0, +/- 1 (mod 3), and second n = 0, +/-1 and +/-2 (mod 5) 
 -- 
 Saludos, 
 Ignacio Larrosa Ca.96estro 
 A Coru.96a (Espa.96a) 
 ilarrosaQUITARMAYUSCULAS@mundo-r.com 
 === 
 Subject: Re: Tries to find proper factors 
 by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, 
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 1.9 primary) id i1SD7fO03963; 
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 $Revision: 
 1.9 primary) with ESMTP id i1SAZgi24702 
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 >> Consider the therm: x^a + y^b - z^c 
 >> where a;b;c are odd numbers 
 >> and x;y;z such integethat x+y-z is some odd number 
 >> does some c^2 could be the factor of the upper therm ? 
 >> ( according to my accounts c value could be only some 
 >> prime number beginning with c=5 ) 
 >> Compliments 
 >> Ro 
 >You'll have to strengthen that a little bit; c = 1 always 
 works. Did 
 >you just want non-silly solutions, or was there supposed to 
 be another 
 >condition on a, b, and c? Also, why c in particular, instead 
 of a and 
 >b? 
 You are very right: 
 first off all a;b;c are bigger or equal to 3 once also odd 
 numbers 
 (composite primes or primes) 
 Next there is my fault once I repeat c value instead of some 
 optiable m but such that m^2 is the factor of x^a + y^b - z^c. 
 One more condition is that x^a + y^b - z^c therm is retrieved 
 only in the shape 3M + 2 ( 2 mod3 ) 
 Thank You for Your attention, 
 Ro 
 === 
 Subject: Re: Cross product -- conceptual questions -- 4-space 
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 The messages here definitely deal with quaterions and there 
 relation to the cross-product/dot-product. Still, in my 
 searching I am yet to find an answer to the original 
 posting from http://mathforum.org/discuss/sci.math/a/t/573744 
 The question from that posting was: 
 Is (iii) or the first corollary, below, already known? 
 My relations are: 
 Let Q(A) be the quaternion algebra over the reals. 
 For x = (x_1)i + (x_2)j + (x_3)k + (x_4)1, define 
 res: Q(A) -> R^3, res(x) = (x_1, x_2, x_3) and 
 tes: Q(A) -> R , tes(x) = x_4 
 Then for all x, y, z in Q(A) with x_4 = y_4 = z_4 = 0 the 
 following holds: 
 (i) res(x) scalar-product res(y) = - tes(x*y) 
 (ia) ||res(x)|| = sqrt( - tes(x*x) ), euclidean norm 
 (ii) res(x) cross-product res(y) = res(x*y) 
 (iii) det( res(x), res(y), res(z) ) = -tes(x*y*z) 
 Corollaries: 
 (1) If tes(x*y) = tes(y*z) = 0, then 
 res(x) cross-product res(y) cross-product res(z) = res(x*y*z) 
 Proof of (1): 
 (res(x) cross-product res(y)) cross-product res(z) = 
 = res(x*y) cross-product res(z), by (ii) 
 = res((x*y)*z), again by (ii) ; since tes(x*y) = 0 
 = res(x*(y*z)) 
 = res(x) cross-product (res(y) cross-product res(z)), 
 by (ii), since tes(y*z) = 0, q.e.d. 
 (2) 
 res(x) scalar-product (res(y) cross-product res(y)) = 
 = det( res(x), res(y), res(z) ) 
 Proof of (2): 
 res(x) scalar-product (res(y) cross-product res(z)) = 
 = res(x) scalar-product res(y*z), from (ii) 
 = - tes( x*(y*z) ), from (i) 
 = - tes( x*y*z ) 
 = det( res(x), res(y), res(z) ), from (iii) 
 q.e.d. 
 ***** 
 A check: Let I be the (3 by 3) identity matrix, then 
 det(I) = det( res(i), res(j), res(k) ) = 
 = - tes( i*j*k ) = - tes( -1 ) = 1 
 ***** 
 C. Dement 
 === 
 Subject: Re: Cross product -- conceptual questions -- 4-space 
 > Is (iii) or the first corollary, below, already known? 
 In short: You reduce the*-product of quaternions to the 
 cross-product, 
 first by letting the scalar part of x,y,z equal zero (so they 
 are 
 3D-vectors), then you demand the dot-product of x and y equals 
 zero 
 (so one is giving no shadow on the other with 
 parallel-projection) and 
 then You ask only for the vector-part of - a vector (this 
 build x*y) 
 *-multiplied with another vector - , which is again a 
 cross-product. 
 If this is known, i don't know. 
 The determinant is well known, it gives the volume of a 
 crystal-shape 
 called spat: build of two of the three vectors a 
 parallelogramm and 
 translate it (scroll it) along the third. 
 Counter-question: Quaternion-addition one can think of as 
 translation, 
 but : 
 Do You have a geometric visualisation (picture, concept) of 
 *-multiplication of quaternions ? 
 Hero 
 === 
 Subject: Re: Coin-Flipping Machine 
 >Subject: Coin-Flipping Machine 
 >Message-id:  
 >Story from NPR: 
 >http://www.npr.org/display_pages/features/feature_1697475.html 
 >Apparently, coin-toss-outcomes are more a function of 
 >human-unpredictability than of the coin's unpredictabilty. 
 What a waste of time. I could have told him that. Oh, but then 
 he 
 wouldn't have gotten a grant to study a non-problem. 
 >So, perhaps it is more accurate to talk about, rather than a 
 biased 
 >coin, a biased HUMAN-BEING!... 
 >;) 
 towards heads). Of course, coins don't have a symmetrical mass 
 distribution, so I could have told him that also. And I didn't 
 see 
 any reference to the control test of an unstamped coin blank. 
 And is he conducting the tests in a vacuum chamber at a 
 controlled 
 temperature? I wonder how much of my tax money he fleeced out 
 of the government for this stupid project? 
 >(What!?!...Human-beings are BIASED!?...Huh???..) 
 Not biased, RANDOM. Unless he puts a lot of effort into it, the 
 human cannot reproduce the same conditions of the flip from 
 one try to the next. This uncertainty is what makes the flips 
 fair. If you knew exactly what the initial conditions were, you 
 would know exactly what the outcome would be. But you cannot 
 know this information, so you cannot predict the outcome. 
 >(I apologize if this link has been mentioned recently on 
 sci.math 
 >already.) 
 >Leroy Quet 
 -- 
 === 
 Subject: Re: Coin-Flipping Machine 
 > Story from NPR: 
 > 
 http://www.npr.org/display_pages/features/feature_1697475.html 
 > Apparently, coin-toss-outcomes are more a function of 
 > human-unpredictability than of the coin's unpredictabilty. 
 > So, perhaps it is more accurate to talk about, rather than a 
 biased 
 > coin, a biased HUMAN-BEING!... 
 > ;) 
 > (What!?!...Human-beings are BIASED!?...Huh???..) 
 > (I apologize if this link has been mentioned recently on 
 sci.math 
 > already.) 
 > 
 By the way, in my previous post, by you I don't mean Leroy of 
 course. He's cool. I'm talking to all the trolls who crosspost 
 all 
 this junk about osophy and the beginnings of the universe and 
 the 
 nature of learning. I *swear* we don't care; just go away. 
 === 
 Subject: Re: Coin-Flipping Machine 
 > Story from NPR: 
 > 
 http://www.npr.org/display_pages/features/feature_1697475.html 
 > Apparently, coin-toss-outcomes are more a function of 
 > human-unpredictability than of the coin's unpredictabilty. 
 > So, perhaps it is more accurate to talk about, rather than a 
 biased 
 > coin, a biased HUMAN-BEING!... 
 > ;) 
 > (What!?!...Human-beings are BIASED!?...Huh???..) 
 > (I apologize if this link has been mentioned recently on 
 sci.math 
 > already.) 
 > 
 A coin, when flipped the same way, will come up with the same 
 outcome 
 We now have *proof* for the vague osophical concept of 
 determinism! I'm going to go publicly pronounce the death of 
 free 
 will; brb. 
 (sarcasm, if you're too stupid to realize) 
 === 
 Subject: Re: rearrangement question.... 
 >hello......... 
 >i know that 
 >if sigma An is absolutely convergent, then any rearrangement 
 >is also absolutely convergent, 
 >and furthermore all rearrangement of this series have the 
 >same sum. 
 In this theorem a rearrangement means you take the 
 same terms but in a different order. Your manipulations 
 below are not just rearrangements in this sense. 
 >i saw the next infinite series 
 >sigma 1/{(2k-1)(2k+1)} = 1/(1*3) + 1/(3*5) + 1/(5*7) + ... 
 >thus 
 >sigma (1/2)[{1/(2k-1)}-{1/(2k+1)}] 
 >= (1/2)(1/1 - 1/3) + (1/2)(1/3-1/5) +......... 
 >= 1/2 (because rearrangement) 
 >and similary 
 >simga {k/(2k-1)} - {(k+1)/(2k+1)} 
 >= (1 - 2/3) + (2/3 - 3/5) +............ 
 >= 1 (because rearrangement) 
 >----------------------------------------------- 
 >um.......i can't understand...... 
 >why not same mutually ??? 
 >let me advice ......please.....thank you very much. 
 ************************ 
 === 
 Subject: Re: rearrangement question.... 
 En el mensaje:4cT%b.23020$A12.68@edtnps84, 
 Larry Hammick  escribi.97: 
 hot-girl 
 >> hello......... 
 >> i know that if sigma An is absolutely convergent, 
 >> then any rearrangement is also absolutely convergent, 
 >> and furthermore all rearrangement of this series have the 
 same sum. 
 >> i saw the next infinite series 
 >> sigma 1/{(2k-1)(2k+1)} = 1/(1*3) + 1/(3*5) + 1/(5*7) + ... 
 >> thus 
 >> sigma (1/2)[{1/(2k-1)}-{1/(2k+1)}] 
 >> = (1/2)(1/1 - 1/3) + (1/2)(1/3-1/5) +......... 
 >> = 1/2 
 > True. It is easy to prove that the partial sums 
 > of sigma 1/{(2k-1)(2k+1)} are k/(2k+1) 
 > and these have limit 1/2. 
 >> ... 
 >> sigma {k/(2k-1)} - {(k+1)/(2k+1)} 
 >> = (1 - 2/3) + (2/3 - 3/5) +............ 
 >> = 1 
 > but the series whose summands are 
 > ((-1)^k) k/(2k-1) 
 > is not absolutely convergent. 
 > If a series of real numbers converges, but not absolutely, 
 > its terms can be rearranged to produce _any_ real sum. 
 But a rearrengement implies to change the order of the 
 summands. If the 
 order of the summands isn't changed, the sum is the same. 
 The problem is in the partial sums: 
 S(n) = Sum(k/(2k-1) - (k+1)/(2k+1), k, 1, n) = (1 - 2/3) + 
 (2/3 - 3/5) + 
 (3/5 - 4/7) +... + (n/(2n-1) - (n+1)/(2n+1)) 
 = 1 - (n+1)/(2n + 1) 
 Then, Lim(S(n), n, inf) = 1 - 1/2 = 1/2, as expec. 
 In the first form, the nth partial sum is 
 S(n) = (1/2)Sum(1/(2k-1) - 1/(2k+1), k, 1, n) = (1/2)((1 - 
 1/3) + (1/3 - 
 1/5) + ... + (1/(2n-1) - 1/(2n+1)) 
 = (1/2)(1 - 1/(2n+1)) 
 And Lim(S(n), n, inf) = 1/2(1 - 0) = 1/2 
 The difference is in the last no canceled term in the nth 
 partial sum. In 
 one case go to 0 and in the other go to 1/2, as n go to 
 infinity. 
 -- 
 Ignacio Larrosa Ca.96estro 
 A Coru.96a (Espa.96a) 
 ilarrosaQUITARMAYUSCULAS@mundo-r.com 
 === 
 Subject: Re: Logic question #1 
 > Subject: Logic question #1 
 >Poy Rott, the great Thai detective, receives a call from 
 Inspector 
 >Goofball. Mr. Neil Diamond, the manager of Timpani's, the 
 famous 
 >jewelry store, had repor that he had had a massive robbery 
 during 
 >that morning. The police had rounded up the only three people 
 who 
 >had come in for the day, Messrs Agassi, Butler and Cooper. 
 > We shall allow R, G & T aren't suspects but only A,B,C,D 
 >Agassi is blind, and would have had to have an accomplice. 
 But he is 
 >a known crook who would never work with more than one person. 
 > A -> B or C or D 
 > A -> ~(B & C) and ~(C & D) and ~(D & B) 
 >Butler swears that Cooper is innocent, so 
 >if Butler is innocent, then so is Cooper. 
 > ~B -> ~C 
 >The police have found out that if exactly two of them 
 >are guilty, Agassi would have to be one of them. 
 > If one, then B or C or D 
 > If two, then A&B or A&C or A&D 
 > If three, then B&C&D 
 >They also know that if Cooper is innocent, Butler is innocent. 
 > ~C -> ~B 
 > B <-> C; B&C or ~B&~C 
 > Thus B&C&D or (~B & ~C) 
 > Hence conclude: D or A&D or B&C&D 
 >Poy Rott turns, and points to..... 
 >(A) Agassi (B) Butler (C) Cooper (D) Agassi and Cooper 
 >(E) Agassi and Butler (F) Inspector Goofball 
 >(G) Neil Diamond 
 > (G) 
 > ---- 
 Hi William, 
 Thanks for your help. When I attemp to work it before, I 
 deduced 
 that A, B, C, D and E were not the answers. But I didn't even 
 think 
 about the fact that Diamond could be a suspect. Thanks a lot. 
 I hope 
 you'll help with some others I have;) 
 Heidi 
 === 
 Subject: Logic question #2 
 Hey guys, 
 Here is question #2. I haven't the slightest idea with this 
 one. If 
 you can explain this to me, I'd appreciate it. Thanks. 
 -------------------------------------------------------------- 
 -------------- 
 - 
 The rest of this (except for the Roman numerals and the 
 letters to 
 select your choice) is in a simple code. 
 (i) Wkhuh duh vla fkrlfhv. Wlfn rii wkh rqh diwhu wkh vhfrqg. 
 (A) rqh (B) wzr (C) wkuhh (D) irxu (E) ilyh (F) 
 vla 
 (ii) Wkhuh duh qrz ilyh fkrlfhv. Wlfn rii wkh rqh diwhu wkh 
 irxuwk. 
 (A) rqh (B) wzr (C) wkuhh (D) irxu (E) ilyh 
 -------------------------------------------------------------- 
 -------------- 
 -- 
 === 
 Subject: Re: Logic question #2 
 > Hey guys, 
 > Here is question #2. I haven't the slightest idea with this 
 one. If 
 > you can explain this to me, I'd appreciate it. Thanks. 
 > 
 -------------------------------------------------------------- 
 ------------ 
 --- 
 > The rest of this (except for the Roman numerals and the 
 letters to 
 > select your choice) is in a simple code. 
 > (i) Wkhuh duh vla fkrlfhv. Wlfn rii wkh rqh diwhu wkh vhfrqg. 
 There are six choices. Tick off the one after the second. 
 > (A) rqh (B) wzr (C) wkuhh (D) irxu (E) ilyh (F) 
 one two three four five 
 > vla 
 > six 
 > (ii) Wkhuh duh qrz ilyh fkrlfhv. Wlfn rii wkh rqh diwhu wkh 
 irxuwk. 
 There are now five choices. Tick off the one after the fourth. 
 > (A) rqh (B) wzr (C) wkuhh (D) irxu (E) ilyh 
 one two three four five 
 > 
 -------------------------------------------------------------- 
 ------------ 
 ---- 
 The puzzle seems too simple, unless the point was to solve the 
 cipher. 
 === 
 Subject: Re: Logic question #2 
 > Hey guys, 
 Here is question #2. I haven't the slightest idea with this 
 one. If 
 > you can explain this to me, I'd appreciate it. Thanks. 
 -------------------------------------------------------------- 
 ------------ 
 > --- 
 The rest of this (except for the Roman numerals and the 
 letters to 
 > select your choice) is in a simple code. 
 (i) Wkhuh duh vla fkrlfhv. Wlfn rii wkh rqh diwhu wkh vhfrqg. 
 > There are six choices. Tick off the one after the second. 
 (A) rqh (B) wzr (C) wkuhh (D) irxu (E) ilyh (F) 
 > one two three four five 
 > vla 
 > six 
 > 
 > (ii) Wkhuh duh qrz ilyh fkrlfhv. Wlfn rii wkh rqh diwhu wkh 
 irxuwk. 
 > There are now five choices. Tick off the one after the 
 fourth. 
 (A) rqh (B) wzr (C) wkuhh (D) irxu (E) ilyh 
 > one two three four five 
 -------------------------------------------------------------- 
 ------------ 
 > ---- 
 > The puzzle seems too simple, unless the point was to solve 
 the cipher. 
 Richard, 
 How did you decipher the code? Did you just look at it and 
 know? Tell 
 me please. Or am I just stupid, or slow, or maybe both? LOL. 
 Anyway, 
 thanks a lot. Question # 3 is on it's way. 
 Heidi 
 === 
 Subject: Re: Logic question #2 
 > Hey guys, 
 Here is question #2. I haven't the slightest idea with this 
 one. If 
 > you can explain this to me, I'd appreciate it. Thanks. 
 > 
 -------------------------------------------------------------- 
 ------------ 
 > --- 
 The rest of this (except for the Roman numerals and the 
 letters to 
 > select your choice) is in a simple code. 
 (i) Wkhuh duh vla fkrlfhv. Wlfn rii wkh rqh diwhu wkh vhfrqg. 
 > There are six choices. Tick off the one after the second. 
 (A) rqh (B) wzr (C) wkuhh (D) irxu (E) ilyh (F) 
 > one two three four five 
 > vla 
 > six 
 (ii) Wkhuh duh qrz ilyh fkrlfhv. Wlfn rii wkh rqh diwhu wkh 
 irxuwk. 
 > There are now five choices. Tick off the one after the 
 fourth. 
 (A) rqh (B) wzr (C) wkuhh (D) irxu (E) ilyh 
 > one two three four five 
 > 
 -------------------------------------------------------------- 
 ------------ 
 > ---- 
 The puzzle seems too simple, unless the point was to solve the 
 cipher. 
 > Richard, 
 > How did you decipher the code? Did you just look at it and 
 know? Tell 
 > me please. Or am I just stupid, or slow, or maybe both? LOL. 
 Anyway, 
 > thanks a lot. Question # 3 is on it's way. 
 In simple English, the triad the is just about the most 
 common. Start 
 by 
 trying to find a fit for that. 
 === 
 Subject: Is string theory Cargo Cult pseudoscience? 
 SAUL-PAUL: 
 1) GREAT! I can tell you're making fine use of Dr Greene's 
 GREAT book 
 FABRIC OF THE COSMOS. Now, if we could only get the Doc to 
 open up 
 Greene's terrific new book so he could get himself up-to-date 
 on 
 leading-edge physics information like you, Captain Collins, Dr 
 Puthoff, 
 Stan Friedman, Gary Bekkum, etc., already are! Working 
 together, we can 
 bring the Doc up to speed! 
 Victor, you do not understand that string theory is pretty and 
 interesting, but it is mathematics not physics. String theory 
 has very 
 little contact with the pressing experimental mysteries of the 
 day - in 
 contrast to my theory BTW. 
 Ed Witten himself has said that string theory is not able to 
 explain 
 dark energy. 
 Saul-Paul correct me if I am wrong, but although string theory 
 predicts 
 a spin 2 graviton there is no direct mathematical derivation 
 of Einstein's 
 Guv + /guv = -8pi(G/c^4)Tuv 
 from the mathematics of string theory? 
 String theory is not background independent so has problems 
 with Diff(4)? 
 At best it corresponds only to the linear graviton, i.e. spin 2 
 perturbation theory on Minkowski space-time? 
 On the other hand. there does seem to be a string theory 
 derivation of S 
 = A/4Lp^2 with blackhole horizons? 
 Also is it not true that there is no mathematical derivation 
 from string 
 theory of the actual standard model 
 U(1)xSU(2)xSU(3) of lepto-quarks and gauge forces? 
 Furthermore there is an embarrasing non-uniqueness of the 
 theory - too 
 many solutions leading to Susskind's Landscape? 
 Please correct my misunderstandings above if there are any? 
 Remember in my theory I have a very simple explanation for 
 Witten's 
 alpha' for observable hadron resonances, not for speculative 
 stuff at 
 Planck scale that has never been seen nor is there any real 
 chance it 
 ever will be? Also I explain the stability of spatially 
 extended charged 
 partons. I explain the string tension on scale of 1 fermi as 
 strong 
 short range gravity. 
 There is a running gravity coupling that gets strong to 
 explain hadronic 
 and leptonic structure. There is no Planck scale at the current 
 cosmological epoch, no hiearchy problem. 
 Show me one experimental fact that Briane Greene is able to 
 explain in 
 any of his books on strings that do not have alternative 
 explanations? 
 The Achilles Heel of string theory is its lack of contact with 
 experiment and observation, it's lack of Popper 
 falsifiability. Indeed, 
 applying Feynman's test of Cargo Cult pseudoscience without 
 any double 
 standard to string theory, I am not sure if it would pass the 
 test? I do 
 not really know yet because I am not expert in it. 
 Gran that string theory is pretty math and seductive, but is it 
 really good physics? 
 2) Dr Sarfatti had queried you earlier about Greene's writings 
 on mirror 
 symmetry, so here are some very useful sites for everyone to 
 utilize. 
 Watch your mailbox next week,.. a small package is on its way 
 via NWO 
 black helicopters! -- 
 http://web.mit.edu/afs/athena.mit.edu/user/r/e/redingtn/www/ 
 netadv/ssym.html 
 www.voting.ukscientists.com/greene2.html 
 The Official String Theory Web Site: Welcome to the 
 Superstring Store! 
 http://superstringtheory.com/store/stringbooks.html 
 www.math.uiuc.edu/~katz/class/s03/ms.html 
 www.drury.edu/multinl/story.cfm?ID=1114&NLID=135 
 www.pbs.org/wgbh/nova/elegant/greene.html 
 http://phys.columbia.edu/faculty/greene.htm 
 Subject: Brian Green, Mirror Symmetry & ADEX-theory 
 On Brian Greene & Mirror Symmetry: 
 In defense of Brian Greene's work in string theory I note the 
 following. 
 Brian Greene and Ronen Plesser discovered a duality that 
 became known as 
 mirror symmetry. 
 In the book *Mirror Symmetry and Algebraic Geometry* by David 
 A. Cox and 
 Sheldon Katz (American Mathematical Society, 1999) we read: 
 Early evidence for mirror symmetry of Calabi-Yau threefolds was 
 given 
 by lists of Calabi-Yau hypersurfaces in weigh projective 
 spaces (or 
 their 
 quotients by finite groups). The Hodge numbers of these 
 hypersurfaces, 
 mirror symmetry was demonstra in [GPl] by first showing mirror 
 symmetry 
 for certian Landau-Ginzburg theories ... and then relating 
 these theories 
 to 
 the sigma models of the [Calabi-Yau] hypersurfaces. 
 [GPl] is the reference: B. Greene and M.R. Plesser, Duality in 
 Calabi-Yau 
 moduli space, Nucl. Phys. B 338 (1990), 15-37. 
 There are two other Greene references in the bibliography: 
 B.R. Green, C. Vafa & N.P. Warner, Calabi-Yau manifold and 
 renormalization 
 group flows, Nucl. Phys. B 324 (1989), 371-390. 
 B.R. Green, M.R. Plesser & S.S. Roan, New constructions of 
 mirror 
 manifolds: 
 Probing moduli space far from Fermat points, in *Essays on 
 mirror 
 manifolds* 
 (S.-T. Yau, ed.), Internat. Press, Hong Kong, 1992, pp. 
 347-389. 
 Mirror Symmetry has become a hot topic in string theory. A 
 recent tutorial 
 book on the this topic is 
 *Mirror Symmetry* by Kentaro Hori, Sheldon Katz, Albrecht 
 Klemm, Rahul 
 Pandharipande, Richard Thomas Cumrun VAfa, Ravi Vakil, and 
 Eric Zaslow, 
 published by the American Mathematical Society and the Clay 
 Mathematics 
 This book cites in addition to the Greene Plesser paper of 
 1990 two more 
 Greene coauthored papers: 
 B. Greene, D. Morrison, & A. Strominger, Black hole 
 condensation and the 
 unification of string Vacua, Nucl. Phys. B451 (1995) 109, 
 hep-th/9504145. 
 B. Greene & H. Ooguri, Geometry & quantum field theory: A brief 
 Although I have only recently star reading the two Mirror 
 Symmetry books 
 (via the A-D-E Coxeter-Dynkin graphs) plays a big role in this 
 topic. I 
 already knew that the 2-d superconformal field theories are 
 A-D-E 
 classified. But it was news to me that the these theories 
 correspond to 
 certain Landau-Ginzburg theories, and the correspondence is 
 via the A-D-E 
 book we read: 
 It turns out that (2,2) superconformal theoris with c < 3, or 
 equivalently D < 1, can be classified and all correspond to 
 Landau-Ginzburg 
 theories with quasi-homogeneous superpotential. Moreover they 
 are in 1-1 
 correspondence with ADE singularities of C^2/Gamma where Gamma 
 is a 
 discrete 
 subgroup of SU(2). 
 So the McKay correspondence of the subgroups of SU(2) with the 
 ADE 
 Kac-Moody Lie algebras is also entailed in the subject of 
 Mirror Symmetry. 
 I have long conjectured that the the A-D-E classifications will 
 provide 
 the underlying mathematical structure for string theory and 
 its later 
 development into M-theory, Mirror Symmetry (which is a kind of 
 T-duality) 
 and so on. There are 20 some mathematical objects already 
 A-D-E classified 
 (including now a potent kind of Landau-Ginzburg 
 superpotential). I call the 
 study of all the A-D-E classifications and their applications 
 to 
 mathematics 
 and physics by the name ADEX-theory. 
 some updating after I have absorbed this (new to me) Mirror 
 Symmetry stuff. 
 Apparently Brian Greene's peers consider him to be an 
 important early 
 developer of the Mirror Symmetry. Perhaps his next book will 
 be a popular 
 description of these ideas. 
 Nuff said ;-) 
 Saul-Paul 
 === 
 Subject: Re: Science Without Math? (model-free common sense 
 steering) 
 Isn't this simply the difference between intuitive reasoning 
 and 
 mathematical reasoning? I realize many here would say that all 
 intuitive 
 thinking can be reduced to mathematics (albiet impossibly 
 complica), but 
 isn't the real differece is one of quality, not quantity? It 
 is the 
 difference between an analogue computer and a digital one. 
 Certianly the 
 way a human reasons is closest to the former, not the latter. 
 Why do we have this need to measure human reasoning abilities 
 in terms of 
 computers anyway? Even to the extent of coming up with the 
 term fuzzy 
 logic to describe the human's comparatively weak reasoning 
 abilities as 
 opposed to the computer, which over the last 20 years or so 
 seems to have 
 become the accep standard that all things--even those things 
 human--are 
 measured by. 
 Scott 
 > Neural networks are model-free estimatoin that they do not 
 require 
 an 
 > in-depth understanding of the phenomena they are modeling. 
 > http://www.arcon.com/arconneu.html 
 > THE MATHEMATICS OF CROSSING THE STREET: 
 > You are at the curb deciding, Should I 
 > cross the street? Well, it depends. 
 > AT THE CURB 
 > The walk light is on, but you see a 
 > truck approaching fast. How fast? 
 > There is no exact number. Instead, there are an infinite 
 number of 
 > possibilities - from 1kph to over 100kph and everything in 
 between. You 
 > don't have a radar gun, so instead you watch the truck for a 
 second or 
 two, 
 > and sum its speed up in two words very fast. That is good 
 enough. 
 > Your senses have told you the truck is coming very fast, but 
 you need 
 more 
 > information before you can decide whether or not to risk 
 crossing. How 
 far 
 > down the street is the truck? Is it slowing down? Again, 
 there are 
 no 
 > exact numbeso you sum up the situation - close, not slowing 
 quickly 
 > enough. 
 > Somehow your brain adds fast + close + not slowing quickly 
 enough, 
 and 
 > warns you instantly that the risk is high. It is purely 
 cognitive 
 process. 
 > It involves a complex combination of sensory information and 
 experience. 
 > ...Since there are no exact numbers in this story, the 
 mathematical 
 version 
 > must be told with fuzzy numbers... 
 > But, the process is still not quite over. Should I wait or 
 cross? You 
 have 
 > to make the decision. Risk tolerance leads to different 
 spins and 
 endings. 
 > If you walk with a cane, you reason, The risk is high, so 
 I'll wait. 
 You 
 > watch as the truck runs the red light. If you are a jogger, 
 impatient to 
 > cross, you disregard the evidence, step into the 
 intersection, and jump 
 back 
 > just in time to save your life. 
 > http://www.decyde.com/crossingthestreet.html 
 > Fuzzy logic works the way that humans think as opposed to 
 the way that 
 > computers typically work. For example, consider the task of 
 driving a 
 car. 
 > You notice that the stoplight ahead is 
 > red and the car ahead is braking. Your 
 > mind might go through the thought process, 
 > I see that I need to stop. The roads are 
 > wet because it's raining and there is a 
 > car only a short distance in front of me. 
 > Therefore I need to apply a significant 
 > pressure on the brake pedal. 
 > This is all subconscious (in general), but that's the way we 
 think - in 
 > fuzzy terms. Do our brains compute the precise distance to 
 the car ahead 
 of 
 > us and the exact coefficient of friction between our tires 
 and the road, 
 and 
 > then use a Kalman filter to derive the optimal pressure 
 which should be 
 > applied to the brakes? Of course not. We use common-sense 
 rules and they 
 > seem to work pretty well. On the other hand, when we do 
 finally get 
 around 
 > to pressing the brake pedal there is some exact force that 
 we apply, say 
 > 1.326 pounds. So although we think in fuzzy, noncrisp ways, 
 our final 
 > actions are crisp. The process of translating the results of 
 fuzzy 
 reasoning 
 > to a crisp, nonfuzzy action is called defuzzification. 
 > http://www.innovatia.com/software/papers/fuzzy.htm 
 > ...In particularly vast networks in fast moving 
 environments, the split 
 > second it takes to traverse the circuit is greater than the 
 time it takes 
 > for the situation to change. In reaction, the last node 
 tends to 
 compensate 
 > by ordering a large correction. But this also is delayed by 
 the long 
 journey 
 > across many nodes, so that it arrives missing its moving 
 mark, birthing 
 yet 
 > another gratuitous correction. 
 > The same effect causes student drivers 
 > to zigzag down the road, as each late 
 > large correction of the steering wheel 
 > overreacts to the last late overcorrection. 
 > Until the student driver learns to tighten 
 > the feedback loop to smaller, quicker 
 > corrections, he cannot help but swerve down 
 > the highway hunting (in vain) for the center. 
 > This then is the bane of the simple auto-circuit. It is 
 liable to 
 flutter 
 > or chatter, that is, to nervously oscillate from one 
 overreaction to 
 > another, hunting for its rest. There are a thousand tricks 
 to defeat this 
 > tendency of overcompensation, one trick each for the 
 thousand advance 
 > circuits that have been inven. 
 > http://www.kk.org/outofcontrol/ch7-c.html 
 > Fuzzy systems are based on 
 > storage of common-sense rules. 
 > For example, a fuzzy Army-ant robot controller might have 
 the fuzzy 
 > association if load is heavy, then signal for help longer. 
 Fuzzy 
 phenomena 
 > admit degrees: some loads are heavier than others; some 
 signal durations 
 are 
 > longer then others. 
 > A single association (heavy,longer) 
 > encodes all combinations... 
 > Fuzzy systems reason with 
 > parallel associative inference. 
 > A fuzzy system reasons with multivalued sets, instead of 
 true or false 
 > propositions, and it may adaptively modify its fuzzy 
 associations from 
 > representative numerical samples. 
 > http://www-2.cs.cmu.edu/~unsal/thesis/thesisch2.html 
 > Wired: What is fuzzy logic and why do critics call it the 
 cocaine of 
 > science? 
 > Kosko: Fuzzy logic is Spock's worst nightmare - a way of 
 doing science 
 > without math. It's a new branch of machine intelligence that 
 tries to 
 make 
 > computers think the way people think and not the other way 
 around. You 
 don't 
 > write equations for how to wash clothes. Instead you load a 
 chip with 
 vague 
 > rules like if the wash water is dirty, add more soap, and if 
 very 
 dirty, 
 > add a lot more. All wash water is dirty and not dirty - to 
 some degree. 
 > It's just common sense. But it breaks the old either/or 
 logic of 
 Aristotle. 
 > That offends some scientists, who would like us to think and 
 talk like 
 > off/on switches. But they still haven't produced a statement 
 of fact like 
 the sky is blue or E=mc^2 that is 100 percent true or 100 
 percent 
 false. 
 > Fact ain't math. You can never get the science right to more 
 than a few 
 > decimal places. That's one reason we find chaos when we look 
 at things up 
 > close... 
 > ...Fuzzy systems are universal computers. I proved that as a 
 theorem - 
 the 
 > fuzzy approximation theorem. In theory, you can replace 
 every book on 
 > physics or economics with equivalent books that have fuzzy 
 systems where 
 the 
 > equations used to be. Fuzzy systems are model-free 
 estimators. You 
 don't 
 > have to guess at equations to build a bridge from inputs to 
 outputs. 
 Fuzzy 
 > rules build that bridge for you. There is math behind the 
 rules, but you 
 > don't need to know it to program a fuzzy system. You can 
 program it in 
 > English. If the air is cool, turn the AC down a little. But 
 the math 
 is 
 > not fuzzy. That's why you can capture fuzzy logic in a 
 digital chip. 
 > Most of the first fuzzy systems were in control - as in 
 adjusting a 
 camera 
 > lens or backing up a trailer truck to a loading dock. Now 
 we're applying 
 > fuzzy systems to wireless communications and multimedia. The 
 fuzzy rules 
 can 
 randomly spread signals over a wide bandwidth or teach an 
 intelligent 
 > agent the kind of houses or sunsets you prefer. The math 
 says we can 
 apply 
 > them anywhere. In practice, it may not be so easy. 
 > http://www.wired.com/wired/archive/3.02/kosko_pr.html 
 > Fuzzy logic is a superset of conventional(Boolean) logic 
 that has been 
 > extended to handle the concept of partial truth- truth 
 values between 
 completely true and completely false. As its name suggests, it 
 is 
 the 
 > logic underlying modes of reasoning which are approximate 
 rather than 
 exact. 
 > The importance of fuzzy logic derives 
 > from the fact that most modes of human 
 > reasoning and especially common_sense 
 > reasoning are approximate in nature. 
 > Boolean vs. Fuzzy: 300 years B.C., the Greek osopher, 
 Aristotle came 
 up 
 > with binary logic(0,1), which is now the principle 
 foundation of 
 > Mathematics. It came down to one law: A or not-A, either 
 this or not 
 this. 
 > For example, a typical rose is either red or not red. It 
 cannot be red 
 and 
 > not red. Every statement or sentence is true or false or has 
 the truth 
 value 
 > 1 or 0. This is Aristotle's law of bivalence and was 
 osophically 
 correct 
 > for over two thousand years. 
 > Two centuries before Aristotle, Buddha, had the belief which 
 contradic 
 > the black-and-white world of worlds, which went beyond the 
 bivalent 
 cocoon 
 > and see the world as it is, filled with contradictions, with 
 things and 
 not 
 > things. He sta that a rose, could be to a certain degree 
 completely 
 red, 
 > but at the same time could also be at a certain degree not 
 red. Meaning 
 that 
 > it can be red and not red at the same time. 
 > Conventional(Boolean) logic states that a glass can be full 
 or not full 
 of 
 > water. However, suppose one were to fill the glass only 
 halfway. Then the 
 > glass can be half-full and half-not-full. Clearly, this 
 disprove's 
 > Aristotle's law of bivalence. This concept of certain degree 
 or 
 multivalence 
 > is the fundamental concept which propelled Zader Lofti of 
 University 
 Berkely 
 > in the 1960's to introduce fuzzy logic. The essential 
 characteristics of 
 > fuzzy logic founded by him are as follows. 
 > In 1965, Lofti Zadeh formally developed multivalued set 
 theory, and 
 > introduced the term fuzzy into the technical literature. 
 Nowadays, the 
 > recent emergence of fuzzy commercial products, as well as 
 new theory, has 
 > genera a new interest in multivalued systems. Yet already 
 engineers 
 have 
 > successfully applied fuzzy systems in many commercial areas 
 : intelligent 
 > subways automation, emergency breakecement mixeKanji 
 characters 
 > recognition, control air conditioneautomatic washing 
 machines, guide 
 of 
 > robot-arm manipulatoand so on. 
 > Fuzzy systems store banks of fuzzy associations or 
 common-sense rules 
 such 
 > as IF traffic is heavy in this direction, THEN keep the 
 light green 
 longer 
 > that might be articula by an human expert. Some traffic 
 configuration 
 are 
 > heavier that others and some green-light duration are longer 
 than others, 
 so 
 > that, the single fuzzy association (HEAVY, LONGER) encodes 
 all these 
 > combinations. That is to say, fuzzy systems directly encode 
 structured 
 > knowledge but in a numerical framework : by entering the 
 fuzzy 
 association 
 > (HEAVY, LONGER) as a single entry in a rule database we are 
 defining an 
 > input-output transformation. 
 > http://www.etse.urv.es/~aoller/fuzzy/fuzzy_logic.htm 
 > Fuzzy Logic is a computational paradigm capable of modelling 
 the own 
 > uncertainness of human beings. Fuzzy reasoning is nothing 
 else than a 
 Fuzzy 
 > Logic-based formalism for encoding human knowledge or common 
 sense in a 
 > numerical framework. Indeed, the mathematical concepts on 
 which Fuzzy 
 Logic 
 > is suppor are very easy to understand. In a Fuzzy 
 Controller, human 
 > experience is codified by means of linguistic if-then rules, 
 which 
 compute 
 > control actions upon given conditions. Fuzzy Logic has been 
 applied to 
 > problems that are difficult to solve mathematically. One of 
 its main 
 > advantages lies in the fact that it offers a straightforward 
 methodology 
 for 
 > modelling and controlling non-linear systems, which are 
 difficult to face 
 by 
 > means of conventional techniques. 
 > http://www.wkap.nl/prod/b/1-4020-7359-3 
 > Fuzzy logic models itself on the pattern of human reasoning 
 in its use of 
 > approximate information and uncertainty to generate 
 decisions. It was 
 > designed (during late 1980s and early 1990s) to 
 mathematically represent 
 > vagueness and develop tools for dealing with imprecision 
 inherent in 
 several 
 > problems. Normally, in digital computers one uses the binary 
 logic 
 where 
 > the digital signal has two discrete levels : low (logic 
 zero) or high 
 (logic 
 > one); nothing in-between. Fuzzy systems use soft linguistic 
 variables 
 (e.g. 
 > hot, tall, slow, light, heavy, dry, small, positive, 
 ...etc.) and a range 
 of 
 > their weightage (or truth) values, called membership 
 functions, in the 
 > interval (0, 1), enabling the new computers to make 
 human-like decisions. 
 > Since human beings tend to use words rather than numbers to 
 describe 
 > behaviour patterns, fuzzy controls avoid the conventional 
 rigidity of 
 > computers and allow them to use parameters based on common 
 sense. 
 > http://www.tribuneindia.com/2002/20021024/science.htm 
 > Fuzzy logic best summed up by common sense 
 > Computer Corner 
 > John Boyd 
 > Fuzzy logic was introduced to the world 27 years ago by 
 Professor 
 > Lotfi Zadeh in his Fuzzy Sets paper published in Information 
 > Control magazine, though it is only recently that we've seen 
 it 
 > applied across a broad range of products. 
 > Some readers have asked for more explanation on fuzzy logic, 
 so 
 > here's an attempt to defuzzify the subject a little further. 
 > Simply put, fuzzy logic is aimed at enhancing our prissy 
 computer 
 > technology with a touch of common sense. 
 > One problem with the conventional digital computer is that 
 it is 
 > such a scrupulously either-or beast. It cannot be easily 
 coaxed 
 > to handle approximations or vague notions like young, a lot 
 and 
 > probably. 
 > Yet most of us rely on such terms daily because we happen to 
 be 
 > humans dealing with other humans, not robots building cars. 
 > It's an easy matter to arbitrarily program a computer so it 
 > designates everyone falling into the age-range 0f 15 to 18 
 > as being a youth. Such a precise category has come to be 
 called 
 > a crisp set since the emergence of fuzzy logic. 
 > Yet we all know some 14-year-olds can look older than some 
 > late-developers turning 20. Such exceptions, however, cannot 
 > be accoun for in conventional computing. Or at least not 
 > without an inordinate amount of additional programming and 
 > expense. 
 > As Tetsuya Yamada, a senior engineer at Hitachi Ltd., replied 
 > when I asked him if we couldn't just continue using 
 conventional 
 > programming and technology for controlling new products, 
 instead 
 > of fuzzy, Well, we could. And you could probably swim across 
 the 
 > Pacific if you got enough support from enough people. But ... 
 > To overcome this problem, Zadeh was inspired to develop his 
 fuzzy 
 > theory and the math to go with it that could be used to 
 create 
 > fuzzy sets based on imprecise natural language. 
 > Each member in a fuzzy set (such as the youths and others 
 considered 
 > in the above example) is assigned one of a continuous range 
 of values 
 > (called the membership value) between zero and one. 
 > Whereas in the above crisp set a 13-year-old going on 14 
 would still 
 > have to be considered a minor and thus be designa as zero in 
 > binary logic, fuzzy logic could assign him a membership 
 value of 
 > say 0.1. Likewise, an immature 20-year-old who would 
 normally fall 
 > outside our either-or crisp-set range could be assigned a 
 membership 
 > value of 0.9 depending upon the criteria we use to measure 
 youth. 
 > Working out just what criteria to use, what values should be 
 assigned 
 > each member and deciding what rules are necessary to govern 
 the 
 > relationships between members is the key to successfully 
 applying 
 > fuzzy control in products. 
 > In some applications, determining the optimum rules has 
 become so 
 > complex, some manufacturers have resor to employing the aid 
 of 
 > neural networks, which may be stretching a good thing too 
 far, given 
 > fuzzy logic's original purpose to get round complexity. 
 > Still, the flexibility in herent in fuzzy is clearly useful 
 in 
 > dealing with approximate calculations, such as about 100. 
 > It can be used in artificial intelligence to provide us with 
 an 
 almost true answer. It can also infer a common-sense result 
 even 
 > when the data is not precise. 
 > Our handwritten 5 in 250 would be trea as 5, not the letter 
 S, 
 > for instance, in Sony's fuzzy-based Palmtop computer. 
 > While we have all seen fuzzy logic-based products from the 
 likes of 
 > Matsua, Sanyo and Hitachi, one unlikely company that has made 
 > fuzzy technology a central part of its business strategy is 
 Omron 
 > Corp. 
 > It began its research into fuzzy logic in 1984 and has since 
 applied 
 > for over 700 patents. This puts it in the forefront of fuzzy 
 > applications in areas like factory and industry control, as 
 well as 
 > in medical equipment. 
 > In 1989, Omron also signed on lotfi Zadeh as a senior 
 advisor. 
 > Earlier this year at the Business Show in Harumi, Omron 
 demonstra 
 > its fuzzy workstation. Omron manufactures both standard 
 Motorola 
 > 68040-based and 88000 reduced-instruction or RISC-based 
 workstations 
 > that can be fit with a fuzzy inference board, turning them 
 into 
 > the world's first fuzzy workstations. 
 > Omron claims such a RISC-based workstation can achieve 4 
 billion 
 > operations per second, an incredible speed if they haven't 
 fuzzed 
 > on the number. Fuzzy logic is used in the workstations to 
 store 
 > and retrieve fuzzy information and make inferences. 
 > Ranging in price from Y2.5 million to almost Y4 million (a 
 US dollar 
 > is about 120 Yens -FM), these machines are not the kind of 
 products 
 > you will find down in Akihabara. (a section of Tokyo famous 
 for its 
 > quantity and variety of electronic goods -FM) Rather, they 
 are 
 > typically aimed at value-added resellers in niche markets, 
 and 
 > engineers who want to develop fuzzy applications, fuzzy 
 databases 
 > and expert systems, as well as fuzzy inference systems. 
 > However, the entrepreneurs among you may be interes in 
 Omron's 
 > FB-30AT fuzzy inference board for the IBM PC and compatible 
 wares. 
 > It features a 24 MHz FP-3000 fuzzy chip capable of 
 processing up 
 > to 128 rules, with five antecedents and 2 consequents. 
 Training 
 > software and a compiler is also available. 
 > Omron has also produced a fine little booklet on fuzzy called 
 Clearly Fuzzy that I dipped into when writing this column. 
 > Tadashi Katsuno, at Omron's public relations section, tells 
 me 
 > he still has a limi number of copies left that he will send 
 > to the first readers of Computer Corner who write to him with 
 > contact information. 
 > The address is Omron Corp., International Public Relations 
 Section, 
 > Omron Tokyo Bld., 3-4-10 Toranomon, Minato Ward, Tokyo 105. 
 > 
 -------------------------------------------------------------- 
 ------ 
 > - Farzin Mokhtarian 
 > farzin@apollo3.ntt.jp 
 http://www-cgi.cs.cmu.edu/afs/cs/project/ai-repository/ai/ 
 areas/fuzzy/doc/in 
 tro/j_times.tgz 
 > http://www.ece.utep.edu/research/webfuzzy/about.html 
 http://www.sztaki.hu/~viharos/homepage/Publications/1999_ICIMS 
 _NOE_ASI99/ASI 
 '99_ViharosMonostori.htm 
 > http://www.bjarne.ca/pmflp.pdf 
 > http://www.etse.urv.es/~aoller/fuzzy/fuzzy_logic.htm 
 > 
 http://www-pablo.cs.uiuc.edu/Project/PPFS/PPFSII/ 
 FuzzyLogicControl.htm 
 === 
 Subject: inverse galois problem 
 hello 
 I'm looking for help. 
 i study inverse galois problem for abelian group 
 i know that for every abelian group G we can found K in order 
 to 
 Gal(KQ)=G 
 i search a polynome P whose verify Gal(Q[x]/P(x))=Z/7Z 
 i know only Z/7Z is a quotient of (Z/29Z)* 
 sorry for my bad english and thank you for your help 
 === 
 Subject: Re: inverse galois problem 
 > hello 
 > I'm looking for help. 
 > i study inverse galois problem for abelian group 
 > i know that for every abelian group G we can found K in 
 order to 
 > Gal(KQ)=G i search a polynome P whose verify 
 Gal(Q[x]/P(x))=Z/7Z 
 > i know only Z/7Z is a quotient of (Z/29Z)* 
 Let z be a primitive 29-th root of unity. 
 Take a number a with muliplicative order 7 mod 29. Any 4-th 
 power will 
 do as long as it's not 1 mod 29. Then 
 let w = z + z^a + z^{a^2} + ... + z^{a^6}. 
 Find the minimum polynomial of w. 
 (you could express w^7 as an integer linear combination of 1, 
 w, ..., w^6 
 or you could do something smarter). 
 That's your P. 
 -- 
 Lacan, Jacques, 79, 91-92; mistakes his penis for a square 
 root, 88-9 
 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ 
 === 
 Subject: Re: inverse galois problem 
 >> hello 
 >> I'm looking for help. 
 >> i study inverse galois problem for abelian group 
 >> i know that for every abelian group G we can found K in 
 order to 
 >> Gal(KQ)=G i search a polynome P whose verify 
 Gal(Q[x]/P(x))=Z/7Z 
 >> i know only Z/7Z is a quotient of (Z/29Z)* 
 > Let z be a primitive 29-th root of unity. 
 > Take a number a with muliplicative order 7 mod 29. Any 4-th 
 power will 
 > do as long as it's not 1 mod 29. Then 
 > let w = z + z^a + z^{a^2} + ... + z^{a^6}. 
 > Find the minimum polynomial of w. 
 > (you could express w^7 as an integer linear combination of 
 1, w, ..., w^6 
 > or you could do something smarter). 
 > That's your P. 
 Doh! I'm getting my Gaussian periods mixed up :-( 
 Swap 4 and 7 in the above so .... 
 Take a number a with muliplicative order 4 mod 29. Any 7-th 
 power of 
 a quadratic nonresidue will do. Then let w = z + z^a + z^{a^2} 
 + z^{a^3}. 
 Well we can take a = 12, and get w = z + z^{12} + z^{-1} + 
 z^{-12} 
 = cos(2pi/29) + cos(24pi/29) when choosing the obvious z. 
 -- 
 Lacan, Jacques, 79, 91-92; mistakes his penis for a square 
 root, 88-9 
 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ 
 === 
 Subject: a little - big problem 
 solving it. 
 Let U be an open set in R^n. 
 Consider 
 f in C^{infty}(U, R) 
 such that 
 exists lim_{x -> x_0} f(x) / ||x-x_0||^{k-1} = 0 . 
 Then f belongs to I^k_{x_0}(U, R) . 
 -------------------------------------------------------------- 
 -------------- 
 -- 
 I^k_{x_0}(U, R) denotes the product of the ideal I_{x_0}(U, R) 
 k-times with itself and I_{x_0}(U, R) denotes the ideal in 
 C^{infty} (U,R) 
 of function vaniscing at p. 
 In other words, f belongs to I^k_{x_0}(U, R) if and only if f 
 is of the form: 
 f = h_0 g_01 * g_02 * .. * g_0{k-1}*g_0k + 
 +...+ 
 + h_m g_m1 * g_02 * .. * g_m{k-1}*g_0k 
 where m is a natural number , say m=1 or m >1 , 
 h_i belongs to C^{infty}(U, R) 
 and 
 g_ij belongs to I_{x_0}(U,R) <==> g_ij(x_0)= 0 in R. 
 Note that for m=0 , the proposition above is false, 
 consider for example k=2 and f(x,y)=x^2+y^2. 
 Then, exists lim_{(x,y) -> x_0} (x^2+y^2) / ||(x,y)||^{2-1} 
 = lim_{(x,y) -> x_0} (x^2+y^2)^{1/2}=0 
 ,on the other hand, it's impossible to write f as 
 f = h_0 g_01 * g_02 with g_01 and g_02 in C^{infty}(U, R) 
 My ask is: 
 Is the proposition above true with m=1 or m>1 ? 
 Regards for any help 
 Tern 
 === 
 Subject: Re: a little - big problem 
 i think it's a problem in taylor expansion: 
 you can show that all partial derivatives up to the (k-1)th 
 order vanish at 
 x_0, 
 next, look at the k-th order taylor residue of f around x_0 - 
 it's an expression exactly of the desired type (i.e., belongs 
 to I^k) 
 > solving it. 
 > Let U be an open set in R^n. 
 > Consider 
 > f in C^{infty}(U, R) 
 > such that 
 > exists lim_{x -> x_0} f(x) / ||x-x_0||^{k-1} = 0 . 
 > Then f belongs to I^k_{x_0}(U, R) . 
 > 
 -------------------------------------------------------------- 
 ------------ 
 -- 
 > -- 
 > I^k_{x_0}(U, R) denotes the product of the ideal I_{x_0}(U, 
 R) 
 > k-times with itself and I_{x_0}(U, R) denotes the ideal in 
 C^{infty} 
 (U,R) 
 > of function vaniscing at p. 
 > In other words, f belongs to I^k_{x_0}(U, R) if and only if f 
 > is of the form: 
 > f = h_0 g_01 * g_02 * .. * g_0{k-1}*g_0k + 
 > +...+ 
 > + h_m g_m1 * g_02 * .. * g_m{k-1}*g_0k 
 > where m is a natural number , say m=1 or m >1 , 
 > h_i belongs to C^{infty}(U, R) 
 > and 
 > g_ij belongs to I_{x_0}(U,R) <==> g_ij(x_0)= 0 in R. 
 > Note that for m=0 , the proposition above is false, 
 > consider for example k=2 and f(x,y)=x^2+y^2. 
 > Then, exists lim_{(x,y) -> x_0} (x^2+y^2) / ||(x,y)||^{2-1} 
 > = lim_{(x,y) -> x_0} (x^2+y^2)^{1/2}=0 
 > ,on the other hand, it's impossible to write f as 
 > f = h_0 g_01 * g_02 with g_01 and g_02 in C^{infty}(U, R) 
 > My ask is: 
 > Is the proposition above true with m=1 or m>1 ? 
 > Regards for any help 
 > Tern 
 === 
 Subject: Re: a little - big problem 
 Excuse me, there's a litlle mistake on the previous message. 
 .............. 
 ,on the other hand, it's impossible to write f as 
 f = h_0 g_01 * g_02 with g_01 and g_02 in vanishing at (0,0) 
 .............. 
 === 
 Subject: Quantum Conflicts 
 Sarfatti Savants, 
 As a member of 
 appreciative 
 audience of this mighty (Jack's word) metaphysical drama 
 coming in on ten 
 emails per day, I would like to express my thanks to all 
 participants for 
 providing the most sophistica and educational web 
 entertainment in town. 
 I am editing and serializing this tale of quantum conflicts on 
 my Combat 
 Diaries web site in order that at least some of it will be 
 preserved rather 
 than being lost in threads. At times the battle between Jack 
 Sarfatti and 
 and Hal Puthoff rivals that classic motion picture The Raven, 
 with Vincent 
 Price and Peter Lorre. We have had the drama of the hot tubs, 
 the story of 
 telephone conversations in the past with alien computeand the 
 story 
 threads through Marconi, Mussolini, and (yes!) now appear the 
 awesome names 
 of Puharich and Geller! With that pair, anything can happen. 
 Of late we 
 have 
 had beautiful women gun-toting agents, threats, insults and 
 denials, and 
 all 
 this to gain control over the high frontier of quantum 
 metaphor! 
 Reputations 
 character, and even the uncertainties of higher mathematics 
 all are at 
 stake. 
 Well it all has great class, 
 and Jack's brilliant new book as pos 
 http://qedcorp.com/destiny is 
 certainly better than the 
 lower-middle-class chatter of the plumbers and carpenters of 
 the 
 nut-and-bolt school. To hell with them and their narrow-nosed 
 docubox 
 language full of late Victorian steam-age legalise. Their 
 practical sober 
 books listing long-gone high school scientific 
 facts are enough to make 
 warthogs roll over and die cross-eyed with petite-bourgeois 
 grief. 
 Congratulations all round. 
 Physics as Media has arrived. 
 Colin Bennett 
 ** 
 Politics of the Imagination (the life, work, and ideas of 
 Charles Fort) 
 awarded Best Biography for 2002 
 === 
 Subject: Chaikin's Spheroid-Ellipsoid Packing Results Indicate 
 Superiority 
 of Growth-Expansion-Contraction Over Curvilinear Motion 
 by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, 
 $Revision: 
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 by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, 
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 by proapp.mathforum.org (8.11.6/8.11.6/The Math Forum, 
 $Revision: 
 1.9 $, proapp) id i1SElqT11390; 
 Paul Chaikin of Princton University and his coauthors Salvatore 
 Torquato of Princeton and several of their colleagues have 
 published the results of their computer simulations in the 
 February 
 sphere or spheroid or an ellipsoid, rather than one direction, 
 the 
 and they can reorient themselves before compacting into a 
 stable configuration, unlike spheres which are symmetric in all 
 directions. This occurs for nonrandom and random packings. 
 There is a definite improvement in packing density for 
 spheroids 
 and ellipsoids (e.g., M&Ms type candies) over the previously 
 indica dense packing of spheres. 
 Rare Event Theory (RET) is governed by the Riccati Differential 
 Equation and its subtypes the Logistic and Simple Exponential 
 Differential Equations which describe 
 Growth-Expansion-Contraction 
 in many directions simultaneously as opposed to the usual type 
 of 
 Curvilinear Motion in one direction at a time considered in 
 Mathematical Physics and most other fields outside Biology. An 
 interesting feature of the former type of motion is that growth 
 may be at different rates and to different degrees or extents 
 in 
 different directions, with or without systems of equations. 
 See my paper in B. N. Kursunuglu et al (Eds.) Quantum Gravity, 
 Generalized Theories of Gravitation, and Superstring 
 Theory-Based 
 Unification, Kluwer: N.Y. 2000, 89-97, for some more background 
 on RET and/or LEUT (Logical-Experimental Unification Theory, 
 based on RET). 
 Osher Doctorow 
 === 
 Subject: Curve Drawing Machines 
 === 
 by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, 
 $Revision: 
 1.9 primary) with ESMTP id i1SG9Bi21134 
 Greetings! 
 The site http://www.museo.unimo.it/theatrum/ gives excellent 
 references about machines (instruments) to draw conics 
 (the perfect compass is admirable) and some special curves like 
 cycloids and spirals. 
 Are there machines (instruments) that are able to draw 
 exponential, sine, tangent and arbitrary power graphs? 
 Any references? 
 Thanks in advance, Humberto. 
 === 
 Subject: Scimitar Theorem 
 by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, 
 $Revision: 
 1.9 primary) id i1SGVTv23566; 
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 Let y = f(x) be the shape of a (zero-width, finite, planar) 
 scimitar 
 blade, and also of its scabbard. The blade remains in contact 
 with the 
 scabbard for any amount of blade withdrawal. Prove that y must 
 be a 
 circular arc. 
 === 
 Subject: Re: Scimitar Theorem 
 > Let y = f(x) be the shape of a (zero-width, finite, planar) 
 scimitar 
 > blade, and also of its scabbard. The blade remains in 
 contact with the 
 > scabbard for any amount of blade withdrawal. Prove that y 
 must be a 
 > circular arc. 
 Need a clearer definition of in contact with. 
 === 
 Subject: Re: ellipse from 4 points 
 > I have the position of 4 points on a 2D plane. The points are 
 > unequally spaced. 
 > Is there anyway I could fit an ellipse (or any other 
 circular shape) 
 > to these points (it has to pass through the 4 points)? 
 > 
 > Tejas 
 Was thinking about this a bit more, and it seems to lead to 
 all sorts 
 of interesting questions in real algebraic geometry. (Admitly 
 may 
 not be very relevant to the OP's practical question.) 
 To set it up, note that each 5-tuple of points 
 P=(P1,P2,P3,P4,P5) in 
 (R^2)^5 determines a conic C_P in the plane. Well, need to 
 discard 
 some 5-tuples that are degenerate, but most 5-tuples give a 
 unique 
 conic. So we can divide (R^2)^5 into four distinct pieces: 
 E = points P for which C_P is an ellipse 
 R = points P for which C_P is a parabola 
 H = points P for which C_P is an hyperbola 
 D = points P for which C_P is degenerate (e.g., consists of 
 two lines, 
 or for which there is more than one conic through the five 
 points) 
 E and H are open sets, while R and D are lower dimensional. 
 Now look at the projection map, say onto the first four 
 coordinates: 
 F : (R^2)^5 --> (R^2)^4 
 F(P1,P2,P3,P4,P5) = (P1,P2,P3,P4) 
 Question: Is F(E) all of (R^2)4? 
 Of course, this is just a fancy way of asking if every four 
 points in 
 R^2 can be placed onto an ellipse. But it sets things up in a 
 natural 
 way to be generalized. If F(E) is not all of (R^2)^4, it would 
 be 
 interesting to describe what it looks like. Ditto for F(H), or 
 course. 
 And what does F(R) look like? 
 Finally, I'll mention that since the order of the points is 
 irrelevant, one should be really taking P as a point in 
 (R^2)^5/S_5, 
 that is, mod out by the symmetric group. Offhand I don't know 
 what 
 this quotient space looks like, but it's undoubly known. A 
 similar 
 problem that's often assigned as an exercise in algebra or 
 algebraic 
 geometry classes is to describe the quotient space C^n/S_n 
 (here C is 
 the complex numbers). The answer is that this quotient is 
 isomorphic 
 to C^n. The proof uses the fundamental theorem of algebra and 
 the fact 
 that every symmetric polynomial is itself a polymomial in the 
 elementary symmetric polynomials. 
 JS 
 === 
 Subject: Re: ellipse from 4 points 
 > Was thinking about this a bit more, and it seems to lead to 
 all sorts 
 > of interesting questions in real algebraic geometry. 
 (Admitly may 
 > not be very relevant to the OP's practical question.) 
 > To set it up, note that each 5-tuple of points 
 P=(P1,P2,P3,P4,P5) in 
 > (R^2)^5 determines a conic C_P in the plane. Well, need to 
 discard 
 > some 5-tuples that are degenerate, but most 5-tuples give a 
 unique 
 > conic. So we can divide (R^2)^5 into four distinct pieces: 
 > E = points P for which C_P is an ellipse 
 > R = points P for which C_P is a parabola 
 > H = points P for which C_P is an hyperbola 
 > D = points P for which C_P is degenerate (e.g., consists of 
 two lines, 
 > or for which there is more than one conic through the five 
 points) 
 D = points P for which at least 3 points in the 5-tuple are 
 collinear. 
 > E and H are open sets, while R and D are lower dimensional. 
 > Now look at the projection map, say onto the first four 
 coordinates: 
 > F : (R^2)^5 --> (R^2)^4 
 > F(P1,P2,P3,P4,P5) = (P1,P2,P3,P4) 
 > Question: Is F(E) all of (R^2)4? 
 > Of course, this is just a fancy way of asking if every four 
 points in 
 > R^2 can be placed onto an ellipse. But it sets things up in 
 a natural 
 > way to be generalized. If F(E) is not all of (R^2)^4, it 
 would be 
 > interesting to describe what it looks like. Ditto for F(H), 
 or course. 
 > And what does F(R) look like? 
 The 4-tuples in F(E) should consist of convex quadrilaterals. 
 The 
 4-tuples in F(R) should be convex quadrilaterals other than 
 parallelograms. (Note that a degenerate parabola consists of 
 two 
 parallel lines.) F(H) should simply be all quadrilaterals 
 (i.e., no 3 
 points collinear). 
 -- 
 Daniel W. Johnson 
 panoptes@iquest.net 
 http://members.iquest.net/~panoptes/ 
 039 53 36 N / 086 11 55 W 
 === 
 Subject: Re: random number generator for multivariate gaussian 
 distribution??? 
 >Hello! 
 >I am looking for a (free) random number 
 >generator written in C++ or C that 
 >generates vectors with a multivariate 
 >gaussian distribution. It should be 
 >possible to specify the mean vector and 
 >the covariance matrix. 
 I assume you can get independent normal (0,1) 
 random variables. So if m is the n-vector of 
 means, and the nxn matrix S of covariances is 
 equal to A*A' (any solution), and x is an n-vector 
 of independent (0,1) normal random variables, 
 use m+A*x. 
 -- 
 This address is for information only. I do not claim that 
 these views 
 are those of the Statistics Department or of Purdue University. 
 Herman Rubin, Department of Statistics, Purdue University 
 hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 
 === 
 Subject: Re: Bayesian Class and Math/Stat Teaching Techniques 
 boundary=------------050400060200000302000001 
 -------------------------------------------------------------- 
 ------- 
 But Radford, most word problems address situations far from the 
 studemt's potential field of application. They are seen to be 
 as 
 little more relevant that the more abstract problems. In 
 secondary 
 school a common trig problem involves a tree, a river, and the 
 tree's 
 shadow leading to determination of the tree's height. Neat 
 problem but 
 not too much rela to Fourier transforms. IMHO few stats 
 instructo 
 at least at the lower academic levels, have in fact analyzed 
 and 
 interpre much real data tied to real world problems. Anyhow a 
 blanket condemnation of the student reeks of the abdication of 
 responsibility or the attemp teaching of students unready or 
 unqualified for college. 
 > 
 >>department was just different. They liked theory and they 
 liked formulas. 
 >>They liked elegant solutions and proofs, even if they were 
 irrelevant to 
 >>application. I sensed a certain disdain for word problems 
 and real 
 world 
 >>analogies and explanations to help the students 
 conceptualize the theory 
 >>because real math students don't need those crutches. 
 >> 
 >My experience, and that of other math/stat instructors whom 
 I've 
 >talked to, is quite the opposite. It's the STUDENTS who don't 
 like 
 >word problems, and resist applications (eg, to physics), 
 because to do 
 >them they have to actually understand the mathematical 
 material (and 
 >even some physics!), rather than just applying formulas 
 without really 
 >knowing what they're doing. This may not be true of real math 
 >students, however, who ought to be able to do the word 
 problems (but 
 >who may find the standard ones to be too easy to be 
 interesting). 
 > Radford Neal 
 >------------------------------------------------------------- 
 --------- 
 === 
 Subject: Re: Give me that old time ontology: (was: the 
 anticlassicalist }{ 
 i: 
 | 
 |> Why don't you state one single mind bogling fact 
 | 
 |Classical propositional calculus has non-Boolean 
 |models! Rela to this and even more surprising 
 |is that classical propositional calculus can be 
 |modeled by a non-Boolean lattice [Reference 6], a 
 |fact apparently overlooked for over 100 years! 
 |Common intuition is that classical propositional 
 |calculus and Boolean algebra go hand-in-hand. 
 |Lattice O6 is a counterexample that shows this 
 |intuition is false. 
 Let me say first that I do find this amusing and worthy of 
 note. To 
 me, however, it falls somewhat short of mind-boggling. Being 
 amazing 
 is harder than it seems. 
 My first reaction was that it would have to depend on what 
 exactly one 
 meant by modeled. For example, since there's a translation of 
 classical 
 logic into intuitionist logic, any model (however you define 
 it) of 
 intuitionist logic will in a less direct or meaningful way 
 constitute 
 some kind of model of classical logic. So 
 . 1 
 | 
 . p 
 | 
 . 0 
 which is a Heyting algebra (where ~~p=1) could count as a 
 model in a 
 *weak* sense, where one has redefined the connectives to be 
 something 
 other than their normal meaning in a Heyting algebra. For 
 instance, 
 the classical x or y is defined as ~(~x & ~y), so that p or p 
 evaluates 
 to 1. Pretty soon, though, you notice it's a little silly to 
 keep 
 pretending to distinguish between elements with the same 
 double negation, 
 and when you identify them, you have a Boolean algebra again. 
 In the referenced example, 
 . 1 
 / 
 ~q . . p 
 | | 
 ~p . . q 
 / 
 . 0 
 X->Y is defined to mean ~X or Y, rather than being the minimal 
 element Z 
 with the property that X&Z <= Y. So in particular, it's 
 possible to have 
 X<->Y (i.e. (X->Y)&(Y->X)) evaluate to 1 even though X and Y 
 are distinct 
 elements of the lattice. 
 The example should serve as a reminder, then, that the sense 
 in which a 
 Boolean algebra incarnates classical logic depends on assuming 
 something 
 that ensures that equivalent elements (in the sense of X<->Y 
 evaluating 
 to 1) are equal, as elements of the algebra. Since p<->q 
 evaluates to 1, 
 we'd need to identify p with q, and ~p with ~q, which leaves 
 us with the 
 Boolean algebra 
 . 1 
 / 
 ~p . . p=q 
 / 
 . 0 
 having the property that the same expressions evaluate to 1 in 
 it as 
 evaluate to 1 in the given six-element example. 
 I'd categorize this as cute rather than mind boggling. 
 === 
 Subject: what is the z-transform of sinc function? 
 Can anybody tell me what is the z-transform of sinc function 
 and what 
 is 
 its region of convergence? 
 Thanks a lot, 
 -Joenyim 
 === 
 Subject: Re: min area to flip 2 hinged rods 
 ... 
 > 2 rigid rods of unit length are hinged to each other. 
 Initially they are 
 > parallel to each other, much like a closed pair of divider. 
 > This divider is placed on a piece of paper. What is the 
 minimum area of 
 the 
 > paper which allow the divider to be opened such that the 
 angle between 
 the 
 > two legs extend from 0 to 360 degrees. The divider is to 
 touch the paper 
 > and no part of the divider is to extend beyond the paper 
 throughout the 
 > whole process. 
 I assume these hypothetical rods have no thickness, and move 
 only 
 within the plane formed by the top flat surface of the paper. I 
 think around pi/3 should be enough area. Suppose the hinge C 
 is at 
 the origin and the rod tips A and B are at (1,0) to start. 
 While 
 raising A, slide C to the right past (1,0) until AC is 
 vertical. 
 I think area used so far is around pi/8, and pi/2 radians have 
 been 
 swept. Now pull B down while C goes up and left a little ways 
 and 
 A swings through part of the area already used, so area used 
 so far 
 is about pi/4, and swept angle is pi. Now push C to the right, 
 etc. 
 > What area of mathematics deal with such problem ? 
 Computational Geometry 
 -jiw 
 === 
 Subject: Re: min area to flip 2 hinged rods 
 > ... 
 > 2 rigid rods of unit length are hinged to each other. 
 Initially they 
 are 
 > parallel to each other, much like a closed pair of divider. 
 This divider is placed on a piece of paper. What is the 
 minimum area of 
 the 
 > paper which allow the divider to be opened such that the 
 angle between 
 the 
 > two legs extend from 0 to 360 degrees. The divider is to 
 touch the 
 paper 
 > and no part of the divider is to extend beyond the paper 
 throughout the 
 > whole process. 
 There is a rela problem about passing a ladder of length L 
 around a 
 right angled corner from a corridor of width A to another of 
 width B. 
 Assuming the ladder is kept horizontal and that its thickness 
 is 
 negligible, it transpires that if L^(2/3) <= A^(2/3) + B^2/3), 
 the 
 desired passage of the ladder is just possible. 
 Thus we may simplify the problem of the hinged rod by 
 providing a much 
 smaller definite upper bound on the area that has been 
 proposed yet. 
 Assuming each rod to be of length 1, the boundary curve need 
 be no 
 larger that (x^2)^(1/3) + (y^2)^(1/3) = 1 
 Since this bounds an area of 3*pi/32 in each quadrant, the 
 total area 
 required will be less than 3*pi/8 ~ 1.7881 square units. 
 Since the rods are not constrained by the axes but only by the 
 curve 
 itself, and not, as was the ladder, constrained by the walls, 
 a slightly 
 smaller boundary is possible, possibly one of the similar form 
 |x|^u + 
 |y|^u = 1 with u < 2/3. 
 Can anyone come up with an area less that 3*pi/8? 
 === 
 Subject: Re: min area to flip 2 hinged rods 
 > 2 rigid rods of unit length are hinged to each other. 
 Initially they are 
 > parallel to each other, much like a closed pair of divider. 
 > This divider is placed on a piece of paper. What is the 
 minimum area of 
 the 
 > paper which allow the divider to be opened such that the 
 angle between 
 the 
 > two legs extend from 0 to 360 degrees. The divider is to 
 touch the paper 
 > and no part of the divider is to extend beyond the paper 
 throughout the 
 > whole process. 
 Unless I'm horribly mistaken, it's just a circle of diameter 1 
 (or a 
 semicircle of diameter 1, if you count refolding back the way 
 you 
 star for the part about opening from 0 to 360), but I can't 
 prove 
 it offhand. For what I do know for a fact, I can state the 
 following: 
 the area of the minimal figure is between 1/2 (big enough to 
 hold both 
 rods when at right angles) and pi (circle, a working upper 
 bound), 
 must have two perpendicular chords of length at least 1 and a 
 chord of 
 length 2. Indeed, you must be able to fit ANY isoceles 
 triangle with 
 two sides of length 1 into the figure somehow. 
 Consider fixing one rod and swinging the other around; then 
 you trace 
 this circle out, so this is an upper bound. Now the only think 
 you 
 can do differently is not to fix the rod while swinging the 
 other arm 
 out; while I'm not going to do this for you, consider two 
 continuous 
 parametric paths through space p1(t) and p2(t) which represent 
 the 
 positions of each end of the previously fixed rod. The only 
 condition 
 on these is that the distance between them remains 1. Also, the 
 second rod rotates to be at an angle of t with the first two; 
 call its 
 location q(t). Then consider the area of the union of triangles 
 p1(t), p2(t), q(t) for all t from 0 degrees to 360 degrees, 
 which you 
 should be able to do with a clever path integral, find the 
 minimum 
 over all paths (which I assume is pi, given by p1, p2 
 constant) and 
 you'd have a rigorous proof of the fact. For one thing, the 
 area of 
 the ribbon traced out by the rod (p1, p2) would have to be 
 rather 
 small to avoid exceeding pi on its own; it is possible that a 
 solution 
 involving oscillation over a fixed, small area could exist, but 
 === 
 Subject: Re: min area to flip 2 hinged rods 
 I thought of this problem but have no idea how to tackle it. 
 2 rigid rods of unit length are hinged to each other. 
 Initially they 
 are 
 > parallel to each other, much like a closed pair of divider. 
 This divider is placed on a piece of paper. What is the 
 minimum area 
 of 
 > the paper which allow the divider to be opened such that the 
 angle 
 between 
 > the 
 > two legs extend from 0 to 360 degrees. The divider is to 
 touch the 
 paper 
 > and no part of the divider is to extend beyond the paper 
 throughout the 
 > whole process. 
 > Since no part of the divider is to extend beyond the paper, 
 we assume 
 that 
 > it is laid down on the paper, not stood up on one point. We 
 further assume 
 > that the whole of (the underside of) the divider must be in 
 contact with 
 > the paper throughout any rotation operation, and that the 
 paper does not 
 > move. 
 > With these fairly simple assumptions in place, it's hard to 
 see how the 
 > answer will be anything other than pi square units (area of 
 a circle of 
 > radius one unit). 
 I don't think so. Imagine you pull the ends of the divider away 
 from each other until the divider is straight, then nudge the 
 hinge past 
 vertical and push the ends back together. The envelope should 
 be a sort 
 of diamond with in-bowed sides, and it should fit inside a 
 circle of 
 unit radius with plenty of room to spare. 
 ----j7y 
 -- 
 ************************************************************** 
 ************ 
 jere7my tho?rpe / 734-769-0913 There is no spoon. SPOON! 
 There 
 > j7y@liws.org <<< is no spoon. SPOON! There is 
 no 
 invert liws to reply via email spoon. SPOON! -- The Tick vs. 
 Neo 
 === 
 Subject: Re: min area to flip 2 hinged rods 
 I thought of this problem but have no idea how to tackle it. 
 2 rigid rods of unit length are hinged to each other. 
 Initially they 
 >> are parallel to each other, much like a closed pair of 
 divider. 
 This divider is placed on a piece of paper. What is the 
 minimum area 
 >> of the paper which allow the divider to be opened such that 
 the angle 
 >> between the 
 >> two legs extend from 0 to 360 degrees. The divider is to 
 touch the 
 >> paper and no part of the divider is to extend beyond the 
 paper 
 >> throughout the whole process. 
 >> Since no part of the divider is to extend beyond the paper, 
 we 
 assume 
 >> that it is laid down on the paper, not stood up on one 
 point. We further 
 >> assume that the whole of (the underside of) the divider 
 must be in 
 >> contact with the paper throughout any rotation operation, 
 and that the 
 >> paper does not move. 
 >> With these fairly simple assumptions in place, it's hard to 
 see how the 
 >> answer will be anything other than pi square units (area of 
 a circle of 
 >> radius one unit). 
 > I don't think so. Imagine you pull the ends of the divider 
 away 
 > from each other until the divider is straight, then nudge 
 the hinge past 
 > vertical and push the ends back together. The envelope 
 should be a sort 
 > of diamond with in-bowed sides, and it should fit inside a 
 circle of 
 > unit radius with plenty of room to spare. 
 Ah, so the paper /is/ moving! (Or rather, the paper and the 
 divider hinge 
 move relative to each other -- and therefore one of my 
 assumptions is 
 incorrect.) Yes, that certainly makes the problem more 
 interesting. 
 Being no geometer, the most I can salvage out of this is to 
 say that pi 
 constitutes an upper limit on the minimum area of paper 
 required. 
 -- 
 Richard Heathfield : binary@eton.powernet.co.uk 
 Usenet is a strange place. - M Ritchie, 29 July 1999. 
 C FAQ: http://www.eskimo.com/~scs/C-faq/top.html 
 K&R answeC books, etc: http://users.powernet.co.uk/eton 
 === 
 Subject: Re: min area to flip 2 hinged rods 
 > With these fairly simple assumptions in place, it's hard to 
 see how the 
 > answer will be anything other than pi square units (area of 
 a circle of 
 > radius one unit). 
 > I don't think so. Imagine you pull the ends of the divider 
 away 
 > from each other until the divider is straight, then nudge 
 the hinge past 
 > vertical and push the ends back together. The envelope 
 should be a sort 
 > of diamond with in-bowed sides, and it should fit inside a 
 circle of 
 > unit radius with plenty of room to spare. 
 The shape I, Mister Math Whiz, was trying to describe is 
 apparently 
 an astroid. I believe its area should be 3/8 pi square units, 
 which 
 is less than half the area of the unit circle. 
 No idea if that's a minimum, though. 
 ----j7y 
 -- 
 ************************************************************** 
 ************ 
 jere7my tho?rpe / 734-769-0913 There is no spoon. SPOON! 
 There 
 > j7y@liws.org <<< is no spoon. SPOON! There is 
 no 
 invert liws to reply via email spoon. SPOON! -- The Tick vs. 
 Neo 
 === 
 Subject: Re: min area to flip 2 hinged rods 
 With these fairly simple assumptions in place, it's hard to 
 see how 
 the 
 > answer will be anything other than pi square units (area of 
 a circle 
 of 
 > radius one unit). 
 I don't think so. Imagine you pull the ends of the divider away 
 > from each other until the divider is straight, then nudge 
 the hinge past 
 > vertical and push the ends back together. The envelope 
 should be a sort 
 > of diamond with in-bowed sides, and it should fit inside a 
 circle of 
 > unit radius with plenty of room to spare. 
 > The shape I, Mister Math Whiz, was trying to describe is 
 apparently 
 > an astroid. I believe its area should be 3/8 pi square 
 units, which 
 > is less than half the area of the unit circle. 
 > No idea if that's a minimum, though. 
 > ----j7y 
 And if you are allowed to roll the divider over, while it still 
 touches the paper, you can cut the 3/8 pi in half by having 
 only half 
 of an astroid, plus a little more for the rolling operation. 
 Unfold the divider so it is straight, roll it over, and fold 
 it back 
 within the same half of the astroid you openned it within. 
 3/16 pi +epsilon, perhaps. 
 Leroy Quet 
 === 
 Subject: What are Bessel functions? 
 What are Bessel functions? What do they do? 
 What is their purpose? 
 === 
 Subject: Re: What are Bessel functions? 
 > What are Bessel functions? What do they do? 
 Bessel functions (among other things) describe vibration modes 
 of disk 
 (or anulus). (Recall how sin and cos describe vibrations of a 
 string.) Bessel functions come from a Sturm-Liouville set of 
 equations and hence enjoy many nice properties such as being 
 othongonal (wrt to the inner produce natural for disk/anulus 
 shaped 
 spaces). 
 > What is their purpose? 
 They are a nice orthogonal set of functions which span a 
 Hilbert 
 space. You can make Bessel transforms which act similarly to 
 Fourier 
 transforms. 
 They often appear when you deal with round or circular things. 
 -- 
 === 
 Subject: Re: What are Bessel functions? 
 > What are Bessel functions? What do they do? 
 > What is their purpose? 
 Expand exp(z(t - 1/t)/2) in a Laurent series in powers of t. 
 The 
 coefficient of t^n is J_n(z) the Bessel _coefficient_ of order 
 n 
 (n integral). 
 J_n was studied by Euler when investigating the vibration of 
 drum 
 skins. Lagrange also studied them. Bessel studied them in 
 connection 
 with planetary motion. The Bernoulli's studied them--Daniel in 
 connection with swinging chains. 
 J_n satisfies Bessel's equation: 
 d^2y/dz^2 + (dy/dz)/z + (1 - (n/z)^2)*y = 0 
 n does not need to be an integer in the DE. The general 
 solution is the 
 Bessel _function_: 
 J_n(z) = 1/(2*pi*i)*(z/2)^n * 
 int_{-infty}^0 t^{-n-1} exp(t - z^2/(4*t)) dt 
 = sum_{r=0}^infty [ (-1)^r * z^{n + 2*r} 
 /(2^{n + 2*r} r! Gamma(n + r + 1)) ] 
 One can expand arbitrary functions in a J_n series (n 
 integral) rather 
 like a Fourier expansion. 
 I don't know if they are in fashion now. You may wish to read 
 G N 
 Watson Theory of Bessel Functions CUP and older books on 
 calculus/analysis. 
 -- 
 G.C. 
 === 
 Subject: Re: What are Bessel functions? 
 >What are Bessel functions? What do they do? 
 >What is their purpose? 
 Try 
 http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html 
 for 
 starters. Google gives many more references. The first place I 
 ever 
 saw them as a student was as solutions to the heat equation in 
 a 
 circular rod. 
 --Lynn 
 Subject: Maths/Statistics no hoper 
 === 
 This is my first post, but reading through the other topics I 
 can see 
 you guys know what you're talking about as I can't understand a 
 thing. 
 :?: :D 
 Basically I have two problems to solve - one to do with 
 statistics and 
 the other with logarithms which I am having some bad trouble 
 with. 
 The two questions are long so would anybody please be able to 
 help me 
 via pm/email? - I don't expect the answers but if someone 
 could point 
 me in the right direction it would really be a big help. 
 The questions are not at the level you guys talk about - but 
 are very 
 difficult for me. 
 So if anyone is willing to help me please could they let me 
 know 
 Thanks everyone. 
 Pos Via Usenet.com Premium Usenet Newsgroup Services 
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 Subject: re:Maths/Statistics no hoper 
 === 
 OK :D 
 I'll try and explain please tell me if I need to tell you any 
 more as 
 the actual question is very long. 
 I then have some data for two different classes. such as A = 
 1, 50, 
 100 etc and then S = 200, 342, 478 etc 
 I have an equation log S = log C + z log A 
 where C is a constant and z determines the shape of the curve 
 I have to convert these to logarithms and then plot them on a 
 graph. I 
 also have to find the value of z. How do I do these things? 
 Pos Via Usenet.com Premium Usenet Newsgroup Services 
 ---------------------------------------------------------- 
 ** SPEED ** RETENTION ** COMPLETION ** ANONYMITY ** 
 ---------------------------------------------------------- 
 http://www.usenet.com 
 === 
 Subject: Re: Maths/Statistics no hoper 
 > This is my first post, but reading through the other topics 
 I can see 
 > you guys know what you're talking about as I can't 
 understand a 
 > thing. 
 > :?: :D 
 > Basically I have two problems to solve - one to do with 
 statistics and 
 > the other with logarithms which I am having some bad trouble 
 with. 
 > The two questions are long so would anybody please be able 
 to help me 
 > via pm/email? - I don't expect the answers but if someone 
 could point 
 > me in the right direction it would really be a big help. 
 > The questions are not at the level you guys talk about - but 
 are very 
 > difficult for me. 
 > So if anyone is willing to help me please could they let me 
 know 
 > Thanks everyone. 
 I have solved both your problems. Unfortunately, I pos my 
 answers in 
 the 
 same place you pos the problems. 
 === 
 Subject: how to clean up derivative and integral Re: cracks in 
 Euclidean 
 Geometry and why Reals are fake 
 The biggest evidence of cracks in the Real Numbers as a system 
 of 
 numbers is the fact of the plethora of integration and 
 differentiation 
 measures. We have Riemann integral and we have Lebesgue 
 integral and 
 Stieltjes integral and Radon integral and 50 other integrals. 
 Then we 
 have 50 or more types of differentiation. Messy, you say. That 
 is only 
 the starting of all the gaps and holes in Real Analysis. The 
 entire 
 subject of Real Analysis is gap ridden, hole ridden and 
 overall sloppy 
 and messy. The reason being is that the Real Numbers are a 
 nonexistant 
 entity. They are fakes. Just as the NaturalNumbers equals 
 Finite-Integers are nonexistant and a fake set. 
 What is real is the P-adics and the Doubly-Infinites. The 
 trouble with 
 both the Finite-Integers and the Reals is that no number 
 exists which 
 is not an infinite string. The P-adics are infinite strings 
 leftward 
 and the Doubly Infinites are leftward and rightward infinite. 
 The reason Number Theory had more unsolved problems 
 accumulating since 
 2,000 years ago is because they had no infinite componentry to 
 their 
 numbers. So Number Theory was even more lousy messed up than 
 REal 
 Analysis. At least REal Analysis had some infinite strings 
 such as 
 3.3333..... whereas finite-integers never had infinite strings 
 and so 
 the accumulation of unsolved problems in Number Theory was huge 
 compared with the messy gaps and holes in REal Analysis in 
 things such 
 as differentiation and integrals. Once that Integrals and 
 Derivatives 
 are given over to Doubly Infinites will all of the holes and 
 gaps in 
 REal Analysis begin to disappear and vanish. 
 Archimedes Plutonium 
 whole entire Universe is just one big atom where dots 
 of the electron-dot-cloud are galaxies 
 (www.iw.net/~a_plutonium) website of the science of AP under 
 revision 
 what used to be my old science website 
 www.newphys.se/elektromagnum/physics/LudwigPlutonium from 
 years 1993 
 === 
 Subject: Re: how to clean up derivative and integral Re: 
 cracks in Euclidean 
 Geometry and why Reals are fake 
 > The biggest evidence of cracks in the Real Numbers as a 
 system of 
 > numbers is the fact of the plethora of integration and 
 differentiation 
 > measures. 
 Selections from the website, where he proves that he is a 
 supergenius: 
 Archimedes Plutonium (my true legal name) 
 Below in chemistry I have a circular periodic table [...] God 
 is 231Pu 
 and the best bible is the best most up-to-date physics 
 textbook. 
 If the Brain Locus theory is correct, then through a single 
 atom in the 
 brain can all the thinking and thoughts be conduc. 
 I make this biological speculation that the source of my 
 supergenius 
 is that there is a Pu atom loca in my brain, the focus of my 
 mind. 
 The brain is a parabolic reflecting telescope which has one 
 atom as 
 the center focus. 
 They walk among us. A scary thing indeed. 
 Servo 
 === 
 Subject: Re: how to clean up derivative and integral Re: 
 cracks in Euclidean 
 Geometry and why Reals are fake 
 > The biggest evidence of cracks in the Real Numbers as a 
 system of 
 > numbers is the fact of the plethora of integration and 
 differentiation 
 > measures. 
 Welcome to my filter. 
 === 
 Subject: Re: how to clean up derivative and integral Re: 
 cracks in Euclidean 
 Geometry and why Reals are fake 
 X-SessionID: LWa0c-13332-Y4-33183@news.uchicago.edu 
 X-Hash-Info: post-filter,v:1.4 
 X-Hash: 9d6c2097 456c6e33 d9ba6f9c f4ba4829 7cd95c08 
 >> The biggest evidence of cracks in the Real Numbers as a 
 system of 
 >> numbers is the fact of the plethora of integration and 
 differentiation 
 >> measures. 
 >Welcome to my filter. 
 You're a quick learner. Congrats. 
 Mati Meron | When you argue with a fool, 
 meron@cars.uchicago.edu | chances are he is doing just the same 
 === 
 Subject: Re: how to clean up derivative and integral Re: 
 cracks in Euclidean 
 Geometry and why Reals are fake 
 > The biggest evidence of cracks in the Real Numbers as a 
 system of 
 > numbers is the fact of the plethora of integration and 
 differentiation 
 > measures. 
 > Welcome to my filter. 
 Most understandable. 
 I would not call the Reals fake, although there is a osophical 
 question about whether they're inven or discovered, and I 
 don't know 
 about cracks in Euclidean Geometry, but there are cracks in the 
 real number line. 
 Do I get to meet your filter? 
 George 
 PS Good luck with D'Inverno. 
 === 
 Subject: Field endomorphism 
 Suppose B is a field, h(x),k(x) are polynomials in B[x]. 
 Consider the 
 function 
 F: f(x) in B[x] |--> h(x)f(x)+k'(x) in B[x]. 
 Which condition must k(x) satisfy to make F an endomorphism? 
 Thanks. 
 === 
 Subject: Re: Field endomorphism 
 i believe k(x) ought to be a constant (i.e., k' = 0 ), in 
 order for F to 
 preserve addtion: 
 F(f+g) = h(f+g) + k' 
 F(f)+F(g)=hf + k' +hg +k' = h(f+g) + 2k' 
 equationg, you get 2k'=k', hence k'=0 
 this condition suffices for F to be a linear endomorphism, not 
 a ring 
 endomorphism, so i just assume that's your intention. 
 > Suppose B is a field, h(x),k(x) are polynomials in B[x]. 
 Consider the 
 > function 
 > F: f(x) in B[x] |--> h(x)f(x)+k'(x) in B[x]. 
 > Which condition must k(x) satisfy to make F an endomorphism? 
 > Thanks. 
 === 
 Subject: Re: Field endomorphism 
 ofek  ha scritto nel messaggio 
 > i believe k(x) ought to be a constant (i.e., k' = 0 ), in 
 order for F to 
 > preserve addtion: 
 > F(f+g) = h(f+g) + k' 
 > F(f)+F(g)=hf + k' +hg +k' = h(f+g) + 2k' 
 > equationg, you get 2k'=k', hence k'=0 
 > this condition suffices for F to be a linear endomorphism, 
 not a ring 
 > endomorphism, so i just assume that's your intention. 
 Maybe what you say is correct in a trivial case. Read my post 
 to Virgil. 
 === 
 Subject: Re: Field endomorphism 
 > Suppose B is a field, h(x),k(x) are polynomials in B[x]. 
 Consider the 
 > function 
 > F: f(x) in B[x] |--> h(x)f(x)+k'(x) in B[x]. 
 > Which condition must k(x) satisfy to make F an endomorphism? 
 > Thanks. 
 If k'(x) is the formal derivative of k(x), then wouldn't a 
 sufficient 
 condition would be that k(x) be constant? 
 === 
 Subject: Re: Field endomorphism 
 Virgil  ha scritto nel messaggio 
 > Suppose B is a field, h(x),k(x) are polynomials in B[x]. 
 Consider the 
 > function 
 F: f(x) in B[x] |--> h(x)f(x)+k'(x) in B[x]. 
 Which condition must k(x) satisfy to make F an endomorphism? 
 > Thanks. 
 > 
 > 
 > 
 > If k'(x) is the formal derivative of k(x), then wouldn't a 
 sufficient 
 > condition would be that k(x) be constant? 
 I think that depends on the characteristic of B. Suppose 
 char(B)=p, then 
 B[x^p]={a_n (x^p)^n+...+a_0 | a_i in B} 
 would be nice. For example, if char(B)=3, then x^3-c is a good 
 polynomial. 
 But I want to know a more general result. 
 === 
 Subject: Re: Sum{k=2 to n} 1/ln(k) and prime-counting 
 >I am reposting this because it has not propaga correctly 
 throughout the 
 Usenet. 
 >Leroy 
 >> I was wondering: 
 >> How good an approximation to pi(n), the number of primes <= 
 n, 
 >> is 
 >> sum{k=2 to n} 1/ln(k) ? 
 >> It overestimates integral{2 to n} dx/ln(x), 
 By less than 1/ln(2) ~ 1.44. This is an overestimate of 
 the difference. I have not done a careful calculation, 
 but I suspect the difference is less than 1. 
 >> and so I would bet that it is not very accurate, relatively. 
 >> But... 
 >> For which n's, if any, 
 >> is 
 >> sum{k=2 to n} 1/ln(k) < 
 >> pi(n) ? 
 As the error in the li approximation to the the number of 
 primes or so) 
 can get to be quite large (O(sqrt(n)) or so), the difference 
 is not 
 important. 
 >> 
 -- 
 This address is for information only. I do not claim that 
 these views 
 are those of the Statistics Department or of Purdue University. 
 Herman Rubin, Department of Statistics, Purdue University 
 hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 
 === 
 Subject: Finding out base of a number 
 I need help with the following problem: 
 749 in base 11 equals 279 in base b. What is base b? 
 === 
 Subject: Re: Finding out base of a number 
 >I need help with the following problem: 
 >749 in base 11 equals 279 in base b. What is base b? 
 If you are looking for an integer b perhaps you have copied the 
 problem incorrectly? 
 --Lynn 
 === 
 Subject: Re: Finding out base of a number 
 | Vikram Hegde asked: 
 | I need help with the following problem: 
 | 
 | 749 in base 11 equals 279 in base b. What is base b? 
 If you meant: 
 749 in base 11 equals 297 in base b... 
 then b would be nineteen. ________________________Gerard S. 
 === 
 Subject: Re: Finding out base of a number 
 > I need help with the following problem: 
 > 749 in base 11 equals 279 in base b. What is base b? 
 In base 10 
 7*11^2 + 4*11 + 9 = 2*b^2 + 7*b + 9 
 2*b^2 + 7*b - 11*(7*11 + 4) = 0 
 Can you solve quadratics? 
 -- 
 G.C. 
 === 
 Subject: Non-linear congruence in Z 
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 How can I solve the following non-linear congruence in Z? 
 Does there exist x,y from Z such that: 
 y(x^2-y)=7 (mod 17) 
 I don't think it's wise to try out all combinations for x,y 
 from {0,1,...16}. 
 Any good reference for such equations? 
 Hardy-Wright doesn't seem to discuss this type of congruences. 
 Cron 
 === 
 Subject: Re: Non-linear congruence in Z 
 Cron  escribi.97: 
 > How can I solve the following non-linear congruence in Z? 
 > Does there exist x,y from Z such that: 
 > y(x^2-y)=7 (mod 17) 
 > I don't think it's wise to try out all combinations for x,y 
 > from {0,1,...16}. 
 > Any good reference for such equations? 
 > Hardy-Wright doesn't seem to discuss this type of 
 congruences. 
 10 for X=0 to 16 
 20 for Y=0 to 16 
 30 if Y*(X^2-Y)@17=7 then print X,Y 
 40 next Y 
 50 next X 
 x y 
 == == 
 3 10 
 3 16 
 5 1 
 5 7 
 12 1 
 12 7 
 14 10 
 14 16 
 If you want to do it by hand, let y = 0, 1, ... ,16 and solve 
 the quadratic 
 congruence for x. 
 -- 
 Ignacio Larrosa Ca.96estro 
 A Coru.96a (Espa.96a) 
 ilarrosaQUITARMAYUSCULAS@mundo-r.com 
 === 
 Subject: Proposition for Euclidean geometry 
 Does anyone know a Godel proposition for Euclidian space? 
 JS 
 === 
 Subject: Re: Any news on odd perfect numbers? 
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 >> Anyone know of any recent work on the odd perfect number 
 >> question? 
 Oddly enough, a paper claiming to prove the conjecture has 
 >> http:// 
 www.arxiv.org/abs/he 
 p-th/0401052/ 
 >> I haven't got the background to tell whether it's good or 
 bad. 
 >> If I correctly read what is written in that paper, the 
 first error is 
 >> around the top of the first page. I do not know whether 1.1 
 can be 
 >> satisfied. 
 >It's certainly one of the worst-written maths papers I've 
 seen for a 
 while. 
 >He really is most allergic to actually explaining his 
 notations. :-( 
 >I nearly got to the end of section 1. 
 >Abstract. He might mention that it's easy to prove that an odd 
 >perfect number (indeed any odd number n with sigma(n) = 2 
 (mod 4)) 
 >has the form n = q^{4m+1} p_1^{2 r_1} ... p_s^{2 r_s} 
 >where q, p_1, ..., p_s are distinct primes, and q = 1 (mod 4). 
 >(He really should say that his 4k+1 is a *prime* distinct 
 >from any of his q_i s). 
 >Page 1. (1.1) is just a definition --- he's saying that 
 >if []/[] is put into lowest terms, it's a_i/b_i. 
 >(Here [] and [] are those gruesome fractions on either side 
 >which I won't copy). 
 >(1.2) is really an if and only if. As it stands it's just 
 >a complica way of saying that N is *not* perfect. 
 >(This is a bit loopy: his section heading talks of 
 >one for a number being not perfect ) 
 >I'm not sure why all those square roots are lying around... 
 >why didn't he write this as 2(4k+1)(b_1 ...b_l)/() 
 []^{l+1}.... =/= ... ? 
 >(1.4) Now here his notation starts to get a bit gothic. 
 >He has things like a_{13}. I reckon that doesn't mean the 
 >thirteenth a_i but rather a_1 a_3. Also he has .Square . 
 >It took me a while to realize that .Square means times a 
 square 
 >(presumably of a rational number). But realizing that, the 
 formula 
 >looks even more batty. Why include brackets like 
 >(a_1 a_3 b_2/b_1 b_3 a_2) when he could have had 
 >(a_1 a_2 a_3/b_1 b_2 b_3) without any difference in meaning? 
 >After that he talks about these things, but in his notation he 
 >shoves overlines onto his subscripts ... Why? 
 >He then claims that some of these fractions aren't square 
 >multiples of 2(4k+1) referring to his previous paper ..... 
 >In (1.5) he introduces some notation and talks about 
 fractions being 
 square-free.... why doesn't he keep things simple, and 
 >represent these quantities as a square-free integer times 
 >a square of a rational? 
 >Next page... (1.7) lacks a parenthesis, but (1.6) through to 
 (1.9) 
 >are unreadable as they stand. E.g., (1.9) contains 
 rho-hat_{2i,2} 
 >which has never been defined. ((1.5) defines rho-hat_{3j+2}). 
 >At this stage, I wonder whether there is any worth trying to 
 >keep second-guessing this geezer as to what he means. 
 >Anyway my guess as to the import of section 1 is that he 
 reckons 
 >with N of the form given in the abstract, sigma(N)/N can't 
 >even be twice a square of a rational and he reckons he's 
 proved 
 >that in the sequel. 
 >-- 
 > 
 Needless to say, I had the last laugh. 
 > Alan Partridge, _Bouncing Back_ (14 times) 
 The paper clearly has an algebraic mistake in (1.2). 
 With Definition 1.1, we can write 
 (qi^(2ai + 1) - 1)/(qi - 1) = (bi/ai)*((4k + 1)^(4m + 2) - 
 1)/4k. 
 Substituting that in the first stuff in (1.2) gives 
 (2(4k + 1))^(1/2)*(b1b2...bl/a1a2...al)^(1/2)*((4k + 1)^(4m + 
 2) - 
 1)/4k)^(l+1/2) 
 Clearly, l + 1/2 = (2l + 1)/2, not (l + 1)/2. 
 === 
 Subject: Re: Rotation through 90 degrees anticlockwise from 
 the first 
 quadrant 
 Hi good pals, I have a problem in transformation which has 
 genera a 
 > lot controversy in our group. Assuming you have the following 
 > cordinates (2 5)and (6 5). What will be the cordinates when 
 it is 
 > rota through 90 degrees ANTI-CLOCKWISE? Is it the same as -90 
 > degrees?. 
 > When it is rota? 
 > What is the antecedent of it -- are you rotating your two 
 vectors 
 > while keeping the coordinate system fixed, or are you 
 keeping the 
 > vectors fixed and rotating your coordinate axes? The answer 
 depends 
 > on which of these you mean. 
 > In either case, clockwise rotations are considered to be 
 negative and 
 > anticlockwise rotations are positive. (I guess clock faces 
 were not 
 > designed by mathematicians.) So no, a 90-degree 
 anticlockwise rotation 
 > is *not* the same as a -90 degree rotation. 
 > Assuming you are rotating the vectors while keeping the 
 coordinate 
 > system fixed, then the new coordinates after the rotation 
 will be 
 > (-5,2) and (-5,6). 
 I'm sorry, the rotation is about the origin and affects the 
 two vectors 
 === 
 Subject: Class vs. Dimension Equations 
 When I talk about the dimension equation of a finite group, I 
 mean the 
 one expressing the order of the group as the sum of the 
 squares of the 
 dimensions of the irreducible representations of the group 
 over the 
 field of complex numbers. 
 My question is: Is the dimension equation directly computable 
 (without 
 knowing the group) from the class equation or vice versa? 
 ---- David 
 === 
 Subject: Re: Switching Integers In Grid (Goal:Coprime-Grid) 
 > For a positive integer n, 
 > arrange 1 through n^2 in a square grid like so: 
 > 1 2 3 4....n 
 > n+1 n+2 n+3... 2n 
 > 2n+1 ... 3n 
 > ... 
 > n^2-n+1... n^2 
 > What is the minimum number of switches needed 
 > to get the n-by-n arrangement of the integers where 
 > EVERY pair of adjacent integers is coprime? 
 > By adjacent, I mean 
 > immediately next to in the directions of up, down, left, 
 right. 
 > By switch, I mean 
 > the exchanging of two adjacent integers' positions. 
 > So, we might have, for m=2, 
 > 1 2 
 > 3 4 
 > We can switch the 3 and 4, so we then have 
 > 1 2 
 > 4 3 
 > Since every adjacent pair is now coprime, only 1 switch is 
 needed for n = 
 2. 
 > Yes, this puzzle was inspired by Sam Loyd's 14-15 Puzzle. 
 > (Or was it the 15-14 Puzzle...) 
 Clarifications: 
 First, unlike many of my other coprime grid puzzles, the final 
 layout of the integers does *not* have to have numerically 
 consecutive 
 integers physically adjacent to each other, necessarily. 
 And we can, after moving an integer, move it later as well. 
 And, finally, it was Sam Loyd's 14-15 Puzzle. 
 Leroy Quet 
 === 
 Subject: Re: Alternative ways to solve a quadratic equation 
 === 
 by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, 
 $Revision: 
 1.9 primary) with ESMTP id i1SLrYi19049 
 > Many thanks William Hale. I will use his method. Thanks again 
 You can approximate the roots by using the trace function of a 
 graphing 
 calculator. 
 -- 
 Take out the trash to reply 
 === 
 Subject: Re: Chaikin's Spheroid-Ellipsoid Packing Results 
 Indicate 
 Superiority of Growth-Expansion-Contraction Over Curvilinear 
 Motion 
 by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, 
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 1.9 $, proapp) id i1SMbBg31044; 
 The title of the Chaikin et al paper is How Ellipsoids Pack: 
 Im- 
 proving the density of jammed disordered packings using 
 ellipsoids, 
 by A. Donev, I. Cisse, D. Sachs, E. A. Variano, F. H. 
 Stillinger, 
 R. Connelly, S. Torquist, P.M. Chaikin, Science Magazine Volume 
 The same issue contains a paper by David A. Weitz, PHYSICS: 
 Pack- 
 ing the spheres, pages 968-969 in the Perspectives section, 
 extreme importance of the results for the shpping and 
 manufacturing 
 industries. Weitz is at Harvard. 
 Osher Doctorow 
 === 
 Subject: Re: Congruence Involving 6 & Some Sums-of-Sums 
 > Let, for each nonnegative integer k, 
 > a(6k) = a(6k +1) = 1; 
 > a(6k +2) = a(6k +5) = 0; 
 > a(6k +3) =a(6k +4) = -1. 
 > Let A(0,m) = a(m); 
 > and for all positive integers n, and for nonnegative 
 integers m, 
 > A(n,m) = sum{k=0 to m} A(n-1,k); 
 > Then, for q and r = any nonnegative integers: 
 > m!*(A(q,m+1) -binomial(q+m,q-1)) 
 > is congruent to 
 > m!*A(r,m) (-1)^m (mod {m+q+r}). 
 > So, more specifically, from the above we get: 
 > For ODD m, 
 > (m-1)!*A(r,m) 
 > is congruent to 
 > (r+m)!/(m r!) (mod {m+2r+1}). 
 > For EVEN m, 
 > (m-1)!*A(r,m) 
 > is congruent to 
 > (r+m-2)!/(m (r-2)!) (mod {m+2r-2}). 
 > (Someone might enjoy confirming the above congruences...) 
 > I wonder if any of these congruences have any interesting 
 > number-theory implications... 
 > Leroy 
 > Quet 
 I must point out that 
 A(n,m) = 
 sum{k>=0} (binomial(m+n-1-6k,n-1) +binomial(m+n-2-6k,n-1) 
 -binomial(m+n-4-6k,n-1) -binomial(m+n-5-6k,n-1)), 
 where 
 binomial(q,j) = q!/(j!(q-j)!) 
 if q >=0, as before, 
 but, here, 
 binomial(q,j) = 0 
 for q < 0. 
 Leroy Quet 
 === 
 Subject: Re: Periodic Linear Recursions 
 > Well-known, I am sure... 
 > For a fixed positive integer n, 
 > Which conditions on the fixed coefficients {c(k)} and on the 
 initial a's, 
 > a(k) for 0<=k<=n, are necessary and sufficient for the 
 sequence 
 > {a(k)} to be periodic, where 
 > a(m) = sum{j=1 to n} c(j)*a(m-j) ? 
 > For example: 
 > a(m) = sum{k=1 to n} a(m-k)(-1)^(k+1) 
 > is always periodic, 
 > with period 
 > 2n+2 if n is even, 
 > and period 
 > n+1 if n is odd (>=3). 
 > 
 Has this message, let alone my more interesting posts, propaga 
 throughout Usenet properly? For this question *must* have a 
 well known 
 answer, I bet. 
 One thing, which could help find an answer: 
 a(m+1) -a(m-n)c(n)c(1) = 
 sum{k=1 to n-1} a(m-k) (c(k)c(1) +c(k+1)) 
 (Linear-algebra is not my strong-area. But this is easy 
 stuff...) 
 Leroy Quet 
 === 
 Subject: Prime factors of number near googolplexplex 
 Hello folks, 
 In (probably) the first page on Internet about numbers near 
 googolplexplex, that is, 10^googolplex, I put the prime 
 factors I 
 found using trial division of numbers in the range 
 googolplexplex-999 
 to googolplexplex+999. 
 The URL is: http://www.alpertron.com.ar/GOOGOLP.HTM 
 Dario Alejandro Alpern 
 Buenos Aires - Argentina 
 http://www.alpertron.com.ar/ENGLISH.HTM 
 === 
 Subject: Re: Prime factors of number near googolplexplex 
 > Hello folks, 
 > In (probably) the first page on Internet about numbers near 
 > googolplexplex, that is, 10^googolplex, I put the prime 
 factors I 
 > found using trial division of numbers in the range 
 googolplexplex-999 
 > to googolplexplex+999. 
 > The URL is: http://www.alpertron.com.ar/GOOGOLP.HTM 
 > 
 > Dario Alejandro Alpern 
 > Buenos Aires - Argentina 
 > http://www.alpertron.com.ar/ENGLISH.HTM 
 Very impressive. How'd you do that? I would think that 
 factoring numbers 
 that large would not be possible on any computing equipment or 
 software 
 available today. 
 === 
 Subject: Re: Differentiate many variables 
 > I have a system consisting of 6 atoms. They all interact 
 with one another 
 > (bonds and van der waals forces). 
 > I have the equations for the interactions between atoms. 
 > Each atom is in 3D space. I can alter the x,y,z for each 
 atom. 
 > I will have a function that depends on these 3*6 variables. 
 Constructing 
 > this function should be fairly easy. 
 > (For ease of work, I presume that I should fix one of the 
 atoms at the 
 > origin). 
 > I want to minimise this function (analytically). 
 > So, How do I differentiate this 18 variable function? 
 > I do have a degree in comp + math but it was a long time 
 ago, and I don't 
 > even know what this would be called. Multivariate 
 differentiation? 
 Partial 
 > differentiation? 
 > Anyone know how to do this (or have any links) 
 > Also, I've got some tools that I'll be using once I got the 
 method 
 straight 
 > in my head: Mathematica, mathcad and matlab. 
 > Anyone got any idea how I would use one of the packages to 
 help me? 
 > (preferably mathematica, as I heard that the strongest 
 symbolically) 
 > Thanks for any advice. 
 > Choca 
 There is the 3 body ( n- body) problem based on gravitational 
 forces 
 between any two bodies. The final solution leads to a number of 
 singular Lagrange points. E.g.,browse 
 Microsoft PowerPoint - CS395TF02-lect2.ppt 
 === 
 Subject: Re: Differentiate many variables 
 > I have a system consisting of 6 atoms. They all interact 
 with one another 
 > (bonds and van der waals forces). 
 > I have the equations for the interactions between atoms. 
 > Each atom is in 3D space. I can alter the x,y,z for each 
 atom. 
 > I will have a function that depends on these 3*6 variables. 
 Constructing 
 > this function should be fairly easy. 
 > (For ease of work, I presume that I should fix one of the 
 atoms at the 
 > origin). 
 > I want to minimise this function (analytically). 
 > So, How do I differentiate this 18 variable function? 
 > I do have a degree in comp + math but it was a long time 
 ago, and I don't 
 > even know what this would be called. Multivariate 
 differentiation? 
 Partial 
 > differentiation? 
 > Anyone know how to do this (or have any links) 
 > Also, I've got some tools that I'll be using once I got the 
 method 
 straight 
 > in my head: Mathematica, mathcad and matlab. 
 > Anyone got any idea how I would use one of the packages to 
 help me? 
 > (preferably mathematica, as I heard that the strongest 
 symbolically) 
 > Thanks for any advice. 
 > Choca 
 Hi Choca: 
 (Just trying to guide you to the right track but be aware that 
 my 
 knowledge of these subjects is small). 
 Supposing (just for economy of writing) that your function F 
 depended 
 only on the positions of 2 atoms (instead of 6) you should 
 first 
 define for Mathematica your function of those 3*2 variables. 
 For 
 example with: 
 F[x1_,y1_,z1_,x2_,y2_,z2_]:= (your analytical expression of 
 the 6 
 variables) 
 Next remember that the increase dF of a function F when their 
 variables are increased respectively by 
 dx1,dy1,dz1,dx2,dy2,dz2 is 
 called the total differential of the function and is given by: 
 dF = (@F/@x1)dx1 + (@F/@y1)dy1 + ... + (@F/@z2)dz2 [1] 
 where @F/@xi means the partial derivative of the function F 
 respect 
 to the variable xi. 
 Mathematica implements total differentials with the operator 
 Dt[function], so if you want to calculate the total 
 differential of 
 your F, you can input in Mathematica: 
 Dt[F[x1,y1,z1,x2,y2,z2]] 
 and you will obtain an expression with the pertinent partial 
 derivatives of the expression [1] already calcula and the 
 differential increments dx1, dy1,... of the variables written 
 in the 
 form Dt[x1], Dt[y1],.. 
 you are trying to solve a big problem (called in Physics a 
 many-body 
 problem) that AFAIK can only be solved analytically in very few 
 circumstances. So I can't guess how are you now going to 
 minimise your 
 function F. 
 Best regards 
 Carlos L 
 === 
 Subject: Re: Differentiate many variables 
 > I have a system consisting of 6 atoms. They all interact 
 with one another 
 > (bonds and van der waals forces). 
 > I have the equations for the interactions between atoms. 
 > Each atom is in 3D space. I can alter the x,y,z for each 
 atom. 
 > I will have a function that depends on these 3*6 variables. 
 Constructing 
 > this function should be fairly easy. 
 > (For ease of work, I presume that I should fix one of the 
 atoms at the 
 > origin). 
 > I want to minimise this function (analytically). 
 > So, How do I differentiate this 18 variable function? 
 > I do have a degree in comp + math but it was a long time 
 ago, and I don't 
 > even know what this would be called. Multivariate 
 differentiation? 
 Partial 
 > differentiation? 
 Same way you differentiate any other function of multiple 
 variables: 
 you find the gradient. The gradient of a function f is a 
 vector whose 
 i-th component (component in the direction of the coordinate 
 x_i) has 
 magnitude @f/@x_i. It points in the direction of steepest 
 ascent for 
 f -- in this case, a direction in 18-space. 
 If you back down the gradient direction, you may find a local 
 minimum, 
 you may find the global minimum, or you may find there is no 
 minimum. 
 And by the way, you are implicitely ignoring a possible 
 orientation of 
 each atom, and treating them as points. This may be adequate, 
 I'm not 
 sure. 
 === 
 - 
 On the planet Fingal live the Fingas. Fingas look human, 
 except that 
 they have no thumbs, and the number of fingers on each hand 
 may be 
 different. This gives them their surnames, so that a Finga 
 with three 
 fingers on his left hand and seven on his right might be named 
 Joseph 
 3-7. 
 There is a wake on Fingal, to mourn the death of Finnegan, a 
 Finga. 
 There is the beautiful ceremony of the touching of the 
 fingewhen 
 the whole population forms a chain, a Finga touching one 
 neighbour to 
 the right, finger to finger. Last night before Finnegan died, 
 he was 
 in the chain, and the chain formed a complete circle. How 
 beautiful! 
 Now Finnah 6-9 begins a chain, touching Joa 9-11, and so on, 
 and 
 forming one circle with some of the Fingers. Fella 9-6 begins a 
 separate chain, forming a second circle with the rest of the 
 Fingas. 
 What is Finnegan's name? 
 (A) Finnegan 6-9 (B) Finnegan 9-6 (C) Finnegan 9-9 (D) 
 Finnegan 11-9 (E) Finnegan 11-6 (F) Finnegan 6-11 
 -------------------------------------------------------------- 
 -------------- 
 -- 
 === 
 Subject: Re: Trying to unify axioms. 
 >> [snip] 
 >> Nothing. 
 You forever spewing ing imbecile, an axiom by definition is 
 >> irreducible and unprovable. 
 > 
 >False. 
 > 
 >First off, an axiom isn't necessarily irreducable, unless you 
 want to 
 >claim that reducing it makes the axiom not reducable and 
 therefore not 
 >an axiom. Sometimes we have an axiom that we find out can be 
 reduced 
 >into other axioms. Does that therefore disprove the axiom, or 
 destroy 
 >the usefulness of the axiom, or of taking it as such? No it 
 doesn't. 
 > 
 >Second, axioms are not unprovable. They CAN be proven. For an 
 example 
 >of this, consider the law of identity. Can you prove it? If 
 not, then 
 >how do we even know it's true? We do know it's true, and it 
 IS an 
 >axiom, so that just proves that axioms are not unprovable. 
 > 
 >(...Starblade Riven Darksquall...) 
 > If an axiom were reducible or proveable, it would be a 
 conclusion, not 
 >an 
 > axiom. The axioms would become the axioms used to reach that 
 >conclusion. 
 >>So, then, the axioms in the system of the human mind are 
 human 
 >>perception and automatic thought processes, and the ability 
 to learn? 
 >>The fact is, the very idea that we need axioms is false. 
 What we need 
 >>are principles. Those can be proven but that does not mean 
 they are 
 >>therefore reducable to other facts. 
 >The neat thing about building a math with axioms, is that you 
 can 
 >first build one, then go back and change just one axioms 
 slightly. 
 >Go through the exercise of building the math again, and see 
 the 
 >differences between the first and the second build. 
 >It's fun to do. The neat thing about math is one doesn't have 
 >to include a reality check. 
 I had a little fun with that when I was getting into general 
 relativity 
 and differential geometry. A metric space has some conditions 
 like a 
 distance being defined, the distance is >= zero, d(x,y)=0 iff 
 x=y, and the 
 triangle law, d(x,y)=d(y,z)>=d(x,z). And d(x,y)=d(y,x). If I 
 remember 
 rightly. 
 Relativity relaxes the condition d(x,y) >= 0. Distances can be 
 greater 
 than zero, less than zero, or zero, depending on whether 
 they're 
 time-like, space-like, or light-like. I think that's called a 
 pseudo-metric. 
 I wondered what other conditions we can relax, and Bilge 
 helped me find 
 out that people have tried relaxing each and every one of 
 them. And so 
 it's possible to talk about a semi-pseudo metric, which I 
 think removes 
 the requirement that d(x,y)=d(y,x) in addition to non-negative 
 distances,, 
 for instance. 
 -- 
 Experiments are the only means of knowledge at our disposal. 
 The rest is 
 poetry, imagination. -- Max Planck 
 === 
 Subject: eigenvalue must be defined only for square matrix? 
 I think the concept of eigenvalue can be defined also for 
 rectangular 
 nonsquare matrix... 
 but the equation |A-lamda*I|=0 only defined for square 
 matrix... 
 what's the problem? 
 Anybody clarify my confusion? 
 === 
 Subject: UK universities 
 Can anyone recomment UK universities or colleges with a 
 mathematics 
 department which is strong in (some of) the following fields: 
 - (differential) topology 
 - set theory 
 - logics 
 - differential geometry 
 - it has kind to connections to theoretical physics 
 (cosmology, field 
 theory) 
 R.M. 
 === 
 Subject: why the gradient of a multi-variate function is 
 column vector? 
 Why the first differential of a multi-variate function is row 
 vector and 
 the 
 gradient vector is defined to be the transpose of that first 
 differential 
 and be column vector? 
 I don't understand. can anybody clarify for me? 
 Thanks a lot, 
 === 
 Subject: Re: e is transcendental 
 >The following was agreed: 
 >Re(e^[iPi]) = Re(-1+i[0]) = -1 AND 
 >Im(e^[i pi]) = Im(-1+i[0]) = 0. 
 >>True. 
 >However,let me point out that this theory of 
 >complex notation 
 >being ,admily ,helpful for solving 
 >complica engineering 
 >>It is FAR MORE IMPORTANT than just being helpful 
 >>for solving complica engineering. And it is 
 >>not just a theory of complex notation. It is 
 >>a theory of complex NUMBERS. You appear to be 
 >>drastically underestimating the importance of 
 >>complex numbers 
 I was referring here to the MATHEMATICAL importance 
 of complex numbers. 
 > [not a panacea however ]problems 
 >[taking care simply angles of unity] 
 >in engineering it should not interfere with 
 >Eucledian Geometry. 
 >The Classical osophers were very strict ,in 
 >specifying that the only tools that should be allowed 
 >to be used,in solvinf Geometric Problems ,should be the 
 >ungradua straight edge and the compass [maximum]. 
 >>This is just a completely irrelevant comment which looks 
 >>designed to stop you having to retract your previous 
 >>statement that exp[i pi] = 0. 
 >>Just what do you think the relevance of your statement 
 >>above to complex numbers is? 
 >I, shall add this too. 
 >When the resultant of the two components of the complex 
 >number is calcula ,then -i , and +i are ignored . 
 >[I, would rather say that only +i is ignored ] 
 >>So what? What relevance does this have to do with the 
 >>fact the you should retract your statement that 
 exp[i pi] = 0? 
 >>Why don't you just retract your statement that 
 exp[i pi] = 0, and be done with it? 
 >What I, exactly sta is in the two formed crosspostings , 
 >including what I, accep . 
 You very dogmatically sta that exp[i pi] = i[0] = 0 as what 
 you called the imaginary part solution to exp[i pi] = -1+i[0]. 
 After you claimed to have your questions regarding the matter 
 cleared up, you did not retract that statement. I feel that 
 it is not absolutely certain whether or not you still hold to 
 exp[i pi] = i[0] = 0 as the imaginary part solut, since you 
 have 
 not retrac that claim. My question is: do you still hold to 
 your previous statements or don't you? A simple Yes, I do or 
 No, I don't will be a sufficient answer. 
 David McAnally 
 At the moment, they (the Time Lords) are far from being 
 all-powerful. 
 That's why it's been left up to me and me and me. 
 quote by: Troughton in The Three Doctors 
 -------