mm-999 Subject: Number-Placement Game For some even positive integer m, we have a m-by-m grid. In this 2 player game, each player has (m^2/2) counters, each counter numbered with a distinct integer from 1 to (m^2/2). The players take turns placing the counters into the grid¹s squares in any order the players wish. (We do not actually need the counters, for players can simply write the numbers in the grid¹s squares. But the counters make it easy to know which numbers each player has already used, for each integer is to be used one per player.) (Or we can simply play this on a computer.) Scoring: One player is rows, the other player is columns. For, say, rows, every set of adjacent integers, where each immediately adjacent (to left/right) pair is coprime, is multiplied, then these groups of multiplied integers are all added up to get the row-player¹s score. For columns, we do the same, but we consider immediately adjacent pairs which are adjacent above/below for multiplication if coprime. As to help explain what I mean, here is an example (of a game played against myself without using any strategy): (Who plays which number is unimportant.) 8 5 8 3 6 1 2 6 7 3 1 5 2 4 7 4 Rows gets: 8*5*8*3 + 6*1*2 + 6 + 7*3*1*5 + 2 + 4*7*4 Columns gets: 8 + 6*7*2 + 5*1*3*4 + 8 + 2*1*7 + 3 + 6*5*4 I would guess that higher m than 4 would be more interesting. Leroy Quet === Subject: Re: Statue of Elvis found on Mars Jack $arfatti > re: http://qedcorp.com/APS/StarGate1.mov > proper citation of identity of author. > This is an excerpt from the book Super Cosmos I am writing now, the > third in the Space-Time and Beyond Series. [12K of spam snipped.] The next time one of those UFO¹s þies over, buttonhole one of the aliens for me, would you Jack? I have a couple of questions about science I would like to ask them. LH === Subject: Martingale Central limit theorem? Could someone please help me with this martingale/random variable problem? I can¹t Þgure it out and it doesn¹t seem to be too trivial (I don¹t think). Consider a martingale M_k , k = 1,2,... Let M_1 = X_1 where X_k = 1 with probability 1/2 and -1 with probability 1/2 Let M_k = [ (X_1 + 3)/2 ]*(X_2 + X_3 + ... + X_k) + X_1 , k >= 1 So the game I think we play is : Bet a dollar and þip a coin and if you win the Þrst game, you bet 2 for the rest of the games, otherwise you bet 1 for the rest of the games... Then, what is lim (k --> inÞnity) of P [ (M_k / (E((M_k)^2)^1/2) <= x ] I am thinking to relate E((M_k)^2)^1/2 to M_k somehow and then use conditional expectations and then use the central limit theorem. The limit should probably be a weighted average of normals. This is similar to the central limit theorem I think...I am trying to expand out the stuff inside the probability but can¹t get anywhere...any thoughts or help? Steve (P = probability, E = expectation, <= is less or equal to, >= is greater or equal to) === Subject: Rows Ever = Columns? This math question is based upon a game I have just posted (Number-Placement Game). For some even positive integer m, we have a m-by-m grid. We place the integers 1 through m^2 in the grid-squares in some order. (Different than the game, where in the game: 1 through (m^2/2) are each placed in grid twice, once per player.) I am wondering if it is possible to get the *same* row-score as column-score, and in how many ways, for a particular m. Scoring: For, say, rows, every set of adjacent integers, where each immediately adjacent (to left/right) pair is coprime, is multiplied, then these groups of multiplied integers are all added up to get the rows¹ score. For columns, we do the same, but we consider immediately adjacent pairs which are adjacent above/below for multiplication if coprime. Here is an example (where I have no idea what the scores are). 8 5 16 3 6 1 10 14 7 11 9 13 2 4 15 12 Rows gets: 8*5*16*3 + 6*1*10 + 14 + 7*11*9*13 + 2 + 4*15 +12 Columns gets: 8 + 6*7*2 + 5*1*11*4 + 16 + 10*9 + 15 + 3*14*13*12 Leroy Quet I would guess that higher m than 4 would be more interesting. Leroy Quet === Subject: Re: Interesting math news story, hard to say whether it¹s overrated or not <3FBD1E68.4080309@farir.com> X-Cise: tanbanso@iinet.net.au X-CompuServe-Customer: Yes X-Coriate: admin@interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Pose: george_cox@btinternet.com X-Punge: Micro$oft X-Sanguinate: themvsguy@email.com X-Terminate: SPA(GIS) X-Tinguish: Mark GrifÞth X-Treme: C&C,DWS at 08:17 PM, W. Dale Hall said: >I think that¹s a bit harsh. You forget that the typical elementary or >junior-high math teacher has a degree in education, rather than >mathematics. Which is consistent with his description. >In addition, the education major is a haven for people >with little conÞdence in their mathematical abilities. And yet enough conÞdence to declare something revolutionary without asking anybody with a better education. >All told, I would imagine that the technique was novel *to the >teacher*, If the teacher were not a cretin, he would have understood that there were a lot of old things that he was unaware of. >and certainly to the student. The student was not the cretin. >The amount of ingenuity or creativity that >this modest step took should be recognized, Certainly; she was entitled to honest recognition: You¹ve found something that most children don¹t understand until they¹re older. >it only makes them people who are easily impressed. Not cretins. When they declare it revolutionary, that goes beyond being easily impressed. >When your child shows you a drawing she made, do you crumple it and >sneer, Rembrandt cries bitter tears over your pitiful attempts at >art, or do you make use of your refrigerator magnets? Do you say Rembrant would be jealous, or do you just praise her and make use of your refrigerator magnets? That¹s the real decision; the question that you asked was disingenuous. >In short, which is the approach one *should* take over the efforts >of children: celebration, or ennui? In short, why do you present phony choices instead of the real choices available? Why do you rule out praise constrained by honesty? Do you routinely lie to your children? -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail will be subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamtrap@library.lspace.org === Subject: Re: Interesting math news story, hard to say whether it¹s overrated or not >Do you routinely lie to your children? About beating your wife? === Subject: Re: Interesting math news story, hard to say whether it¹s overrated or not Shmuel (Seymour J.) Metz > Do you routinely lie to your children? Doesn¹t everybody? -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Interesting math news story, hard to say whether it¹s overrated or not <3FBD1E68.4080309@farir.com> <873ccijslr.fsf@becket.becket.net> X-Cise: tanbanso@iinet.net.au X-CompuServe-Customer: Yes X-Coriate: admin@interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Pose: george_cox@btinternet.com X-Punge: Micro$oft X-Sanguinate: themvsguy@email.com X-Terminate: SPA(GIS) X-Tinguish: Mark GrifÞth X-Treme: C&C,DWS >The real reason teachers are incompetent is that school systems pay >lousy for teaching, but great for babysitting. No, that¹s one of the reasons, but far from the only reason and not the most important reason. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail will be subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamtrap@library.lspace.org === Subject: Re: Interesting math news story, hard to say whether it¹s overrated or not > But the _teacher_ said that she¹d never seen this method - she > said that she called people from college and they¹d never seen > it either. The teacher actually seems to have learned something > that she never would have been able to Þgure out on her own, > and that¹s the other, much sadder point to the story for many > of us - a teacher to whom the kid¹s discovery appears to be an > actual _discovery_ has no business teaching math even at that > level - she can¹t possibly understand it well enough to be > explaining how it works and why. > (Yes, it _was_ a big deal to the teacher - she says that > she¹s going to teach her students this method from now on, > which is great, but which also shows it honestly was news > to her...) The teacher was a he, not a she. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Interesting math news story, hard to say whether it¹s overrated or not > The story is at: > http://www.af.mil/stories/story.asp?storyID=123006043 > Basically, a 12 year old girl Þgures out that > A - (B/C) = (A-1) + (C/C) - (B/C), > and uses it as an improved way to subtract fractions. > However, the artical comes off making it sound like she has invented > calculus or something. Obviously if a grown adult were to discover > the time. > On the other hand, the fact that she is 12 does lend her some credit. > I¹m not sure how much it lends, though. Certainly if she were 3 or > even 6, it would lend a lot more, but 12 is pushing it. When I was 12 > I discovered that you could do those damned division problems where > you¹re required to give the remainder, using a calculator: > A/B = þoor(A/B) with a remainder of B*[A/B-þoor(A/B)] > (I used the QBasic int rather than þoor though) > When I showed that to my math teacher her reply was you¹re trying to > do algebra that you don¹t understand. She was an ignorant person to > one extreme, but the teacher of the girl in this story seems to be an > ignorant person to the opposite extreme: his claim that this girl¹s > discovery is some kind of revolution does a lot to make him look like > an idiot. The fact that it got so far (the principal called it a day > of mathematical rejoicing and some ignorant USAF reporter put it on > the Air Force news) without anyone noticing that it is not really all > that big a deal, shines a very poor light on all parties involved. > Of course it would be cruel and heartless to express such sentiments > to the girl, but on the other hand, is it no less cruel or heartless > to decieve her so utterly? Suppose, hypothetically, that she had > independantly discovered the quadratic equation: certainly she would > deserve great praise and honors, but to tell her you are the Þrst > person to ever solve this problem would be an outright black lie. > It seems as though in the entire journey from the classroom to the > news, noone ever once thought of actually consulting a mathematician. > (I guess a lot of ignorant folk would think the teacher in the story > is a mathematician... sigh) I like the following lines: She uses negative numbers, which McCabe said he had never seen before. How about that -- a math teacher who has never seen negative numbers. Her process was not used in any of McCabe¹s reference materials. Here we get close to the heart of the problem. The textbooks used in primary/secondary/introductory college math classes are horribly written -- full of color photographs and loaded with errors and poor methods. The topics are organized by a blender using frappe. I went home and tried to Þnd fault with it, but I couldn¹t. That¹s good. I wonder if he broke out the Oija board and consulted Gauss, just to make sure. - Brett === Subject: Re: Interesting math news story, hard to say whether it¹s overrated or not > I like the following lines: > She uses negative numbers, which McCabe said he had never seen > before. > How about that -- a math teacher who has never seen negative numbers. I don¹t think so. You¹re quoting the caption of a picture - in the She used the concept of negative numbers in a way that has never been done before, as far as her seventh-grade teacher has been able to ascertain. I¹m not real impressed by the teacher, but I think it¹s fair to say that he had seen negative numbers before. The caption was poorly written. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Interesting math news story, hard to say whether it¹s overrated or not I have a question, how would the teacher have calculated 143 - 2/37? That¹s what I don¹t get. Would they have converted 143 to 2291/37? === Subject: Re: Interesting math news story, hard to say whether it¹s overrated or not >I have a question, how would the teacher have calculated 143 - 2/37? >That¹s what I don¹t get. Would they have converted 143 to 2291/37? Convert? Pshaw! Regardez: +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ | 1 | | 4 | | 3 | | - | | 2 | |-:-| | 3 | | 7 | | = | +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ And for goodness¹ sake do not confuse the subtraction and unary-minus keys! dave === Subject: Re: Interesting math news story, hard to say whether it¹s overrated or not > I have a question, how would the teacher have calculated 143 - 2/37? > That¹s what I don¹t get. Would they have converted 143 to 2291/37? That is what an unþexible and preprogrammed mind would do, making this ten times harder than necessary. I think the natural solution would be to convert 143 to 142 + 1, convert 1 to 37/37, and the result is 142 35/37 without any effort. === Subject: Re: Interesting math news story, hard to say whether it¹s overrated or not > I have a question, how would the teacher have calculated 143 - 2/37? > That¹s what I don¹t get. Would they have converted 143 to 2291/37? 143 - 2/37 = 142 + 37/37 - 2/37 = 142 + 35/37 === Subject: Re: Interesting math news story, hard to say whether it¹s overrated or not > I have a question, how would the teacher have calculated 143 - 2/37? > That¹s what I don¹t get. Would they have converted 143 to 2291/37? > 143 - 2/37 = 142 + 37/37 - 2/37 = 142 + 35/37 Well, of course, but are you a 7th grade math teacher? How would this particular math teacher do this problem? Jon Miller === Subject: Re: Interesting math news story, hard to say whether it¹s overrated or not >> The story is at: >> >> http://www.af.mil/stories/story.asp?storyID=123006043 >> >> Basically, a 12 year old girl Þgures out that >> A - (B/C) = (A-1) + (C/C) - (B/C), >> and uses it as an improved way to subtract fractions. >> >> The fact that it got so far (the principal called it a day >> of mathematical rejoicing and some ignorant USAF reporter put it on >> the Air Force news) without anyone noticing that it is not really all >> that big a deal, shines a very poor light on all parties involved. >> Of course it would be cruel and heartless to express such sentiments >> to the girl, but on the other hand, is it no less cruel or heartless >> to decieve her so utterly? I would suspect that a 12 year old girl who is capable of Þguring out some maths like this on her own is not necessarily a genius, but probably quite bright. So it seems very likely that she _knows_ that her maths teacher is a bit dim, and that this has been blown out of all proportion, and that she isn¹t really deceived at all. === Subject: Re: Vedic Mathematics --- Myth and Reality > Children will be freed from the torture of > learning arithmetic the modern bad way, and they should have a more > positive learning attitude towards mathematics as a result of > incorporating Vedic arithmetic in primary schools. Let¹s not forget that the Vedic principle crosswise and vertical > does also lie at the heart of this modern, bad, clumsy way... > Because the Vedic principle does not talk about the order of > the additions, afaik. Herman Jurjus > I don¹t think you have fully grasped the subtleties of the method. > I don¹t think you have fully grasped the meaning of what i said. > The methods are different, but both are based on crosswise vertical. > The expression crosswise vertical _could_ lead to your method, > but just as well to the usual one. > In short, i didn¹t say the methods are the same, but that they are both based > on crosswise vertical. In any case, the importance of place value in arithmetic is something people like you seem to have totally missed. I have given a rigorous deÞnition for the long multiplication method, following what I understood of the Vedic multiplication method. Looks like you are totally ignoring that. > I suppose you have to be a primary school teacher or a primary school > child to understand that! Not talking about the order of the > additions, that being implicit, is the real strength of the method, > where there is independence of columns and thus avoidance of carrying > till the very last stage where it is done in one go. Thus, this > method is admirable for parallel processing, and can be implemented > with very few lines of code. I could easily multiply two ten thousand > digit numbers using simple C code with the deÞnition I have given of > this algorithm. I could never do such a thing so simply with other > methods. > Yes, the gist of it is single register you only have to remember one > number, all along the process. fact. > But is _that aspect_ of the method Vedic? The expression crosswise vertical > doesn¹t say much about it, does it? Oh yes, it does. That is the beauty of Sanskrit. It is not verbose; it just deÞnes the heart of things, as brieþy as possible, and leaves the rest to those with vision. Vedic means secret - only the seer can understand the Vedas. They are meaningless to others. > Please remember that very learned people can also be shown to be very > foolish. Like, all those who believed in Aristotle, and those today > who believe in the Special Theory of Relativity! > Arindam Banerjee. > It is certainly true that the stupidest mistakes are often > made by the smartest people. No, this is illogical. Smart people by deÞnition cannot be stupid - they would hardly be considered smart if they did do stupid things. But learned people can look very stupid, when new knowledge comes to upset whatever they had learnt and believed in. Arindam Banerjee. > Herman Jurjus === Subject: Re: Vedic Mathematics --- Myth and Reality > ... restricting to numbers only > doesn¹t make maths very exciting for laymen, i¹m afraid. > Personally, i would never have gotten any interest in mathematics > if not for geometry, reasoning from axioms, etc. Historically, > mathematical progress always ended fairly soon, in all cultures > that restricted maths to Œdoing calculations¹. > When they do calculations properly, they succeed. The success of > modern Western cultures lies very largely in their ability to do > calculations very fast, using computers for number crunching, and > terriÞc use of calculations in all aspects of engineering, Þnance, > banking, selling, etc. > The progress of modern mathematics in the past 300 years > is due mainly to its advancement beyond calculus. And please don¹t > call it Western. It¹s fully global. How nice. But I don¹t agree. The triumph of Western world comes from highly efÞcient calculation. The higher mathematics is mainly irrelevant and simply a pastime. > The alternative multiplication method is absolutely welcome. > I don¹t know the rest of Vedic arithmetic, and I don¹t think I will > know till I get that book. Which may take a few years, I am afraid, > unless someone gives me a copy. > If you didn¹t read the book, how do you know that your multiplication > method is Vedic? Crosswise vertical is not much of a description, > is it? There is a Sanskrit word for the method, which has been translated to crosswise vertical perhaps by the Western mathematicians. Similarly, there are Sanskrit words that describe the core of the division and square root processes. I did read the book for 15-30 minutes. Enough time for me to grasp the multiplication method, and also glance at the other methods. Actually, I was not interested so much in the book, I was more interested in books on Kalidas, Sanskrit prosody, grammar, etc. > Herman Jurjus === Subject: Re: Vedic Mathematics --- Myth and Reality >Wrong! You have multiplied 804 with 1000000, which is 10 >multiplications. Then again you have multiplied 32823 with 1000 which >is again 9 multiplications. So totally you have done 22+19 or 41 >multiplications. Which is more than 25! I know that the last 18 are >easily done, but they are multiplications all the same. Even if we >count the multiplies with 10^x as 1 operation, then we still come down >to 25 operations, just as O(n^2). > Well, no, we aren¹t multiplying 804 by 1000000 etc; we¹re just moving > it over six places. That is a matter of practice. In theory you are multiplying it by 10^x, and we were talking about theory, when people start to debunk this method on the ground that it is theoretically not better than O(n*n). In practice I could hardwire all the multiplications using a ROM (an electronic device) and the whole question of O(f(n)) would simply not arise. I could have just as well written the sum > 279105 > 32823 > 804 > --------- > 837102105 Yes, so the whole argument is hinging around the importance of place value. We avoid its complications using Vedic arithmetic. When you do multi-line arithmetic, as you do, the whole business about the carries comes in. Your writing in multi-line structure clearly assumes that you have actually multiplied the 804 by 10^x. So you *did* multiply, don¹t evade the issue. > Be that as it may it is quite clear that you still haven¹t grasped > what is meant by the O() notation or you wouldn¹t have said anything You have *not* shown FFT to be a better method. Let us take the example 12999 * 23999. You will Þnd that using FFT as you yourself did we need to do *28* multiplications! Which is more than 25! If FFT was a better method, then why should it take *more* multiplications - plus lots of other complications, as you show! > Seemingly you do not understand what the O() notation signiÞes. When > one says that the FFT method is O(n*log(n)) one is saying that there > is some constant C such that for n sufÞciently large, > > (# of required multiplies) is less than C*n*log(n) > > This does not mean that the cost of the FFT method is less than n*n > for all n, just that it is for n sufÞciently large. Thus, your > proposed test is irrelevant to the point under discussion. What is sufÞciently large? To be above criticism, being too complex seeming, and thus requiring *belief*? Like, Planck¹s law works for sufÞciently large frequencies and not for smaller frequencies? If a method is really good, it should work equally well for all values of n. Your pathetic failure to prove your own point is most revealing. If you cannot show it more efÞcient for small n, why should anyone believe you that it will work for large n? After all, we are talking not about statistics here, but of standard arithmetic processes. > Ponder that statement. Think about it. Please don¹t follow up to > this posting until you understand it. I shall reply to this posting, to show what an abject fool you show yourself to be. Like the dogmatic Einstein-worshippers, I suppose. >And all these are pretty complex operations, always needing the carry >no means you have shown that this is simpler than the Vedic method. > I didn¹t say that it was simpler than the Vedic method. And, of > course, it is not simpler. However it is much more efÞcient if one > is multiplying large numbers, large being dozens or hundreds of digits > in the multiplicands. A statement for the believers. Really, mathematics is not religion. Here, you need to demonstrate what you are asserting. And do that clearly and credibly. >In fact, it is dreadful. I am sure that in due course people will >multiply using the Vedic arithmetic. So much easier. Properly done >(with hard-wired single digit computation, that will make >multiplication efforts meaningless) I have no doubt that it will beat >FFT hands down. In fact, FFT as you show it is a clumsy approximation >to the elegant Vedic multiplication. >> Be that as it may, the cross product method for multiplication is >> quite obvious and is regularly rediscovered. I discovered it myself >> as a child and even then was under no illusion that I had done >> anything remarkable. >But no one demonstrated that here before I did, with the example. >People came up with all sorts of ideas, but no one could do it. And I >did give them the chance! I wanted others to show it, and none did. >To say now that you knew it, does not convince. Did you also know >about the one-line division method? Supposing someone were to explain >that with an example (I may do that after I get that book within an >year or so) would you then say that you knew that method all along? > very venue (rec.arts.books) on Jan 19, 1997 in a thread entitled > idiot savants. A copy also be found on my web site. I cannot believe anything you say. Discussion between us is pointless. Arindam Banerjee. > Richard Harter, cri@tiac.net > http://home.tiac.net/~cri, http://www.varinoma.com > We have people from every planet on the earth in this State. > -- California Governor Gray Davis === Subject: Re: Simple proof of polynomial distribution of primes? > The result would follow easily if we could prove that there is a prime > between n and n+sqrt(n), for every big enough n. Maybe so, but as the Riemann Hypothesis follows from the existence of a prime between n and n + sqrt(n).... > (Or weaker: between n > and n+n^(1-epsilon) for some epsilon.) Still requires some heavy duty analytic number theory. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Simple proof of polynomial distribution of primes? > the Riemann Hypothesis follows from the existence > of a prime between n and n + sqrt(n).... Really? Seems quite unlikely. What do you believe to be the most precise result along these lines? === Subject: Re: Simple proof of polynomial distribution of primes? > the Riemann Hypothesis follows from the existence > of a prime between n and n + sqrt(n).... > Really? Seems quite unlikely. > What do you believe to be the most precise result > along these lines? Baker, R. C.; Harman, G.; Pintz, J. The difference between consecutive primes. II. Proc. London Math. Soc. (3) 83 (2001), no. 3, 532--562. The authors prove that for all sufÞciently large n there¹s a prime between n and n + n^c for c = 0.525. On the Riemann Hypothesis, it¹s known that for any positive e and for any sufÞciently large n (that is, n > n_0(e)) there¹s a prime between n and n + n^(0.5 + e). It¹s conjectured that if p_n is the n-th prime then p_(n + 1) - p_n is big-oh of the square of log n. Of course, this is much, much stronger than the n^c results above. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Simple proof of polynomial distribution of primes? >> the Riemann Hypothesis follows from the existence >> of a prime between n and n + sqrt(n).... >> Really? Seems quite unlikely. >> What do you believe to be the most precise result >> along these lines? >Baker, R. C.; Harman, G.; Pintz, J. The difference between consecutive >primes. II. Proc. London Math. Soc. (3) 83 (2001), no. 3, 532--562. >The authors prove that for all sufÞciently large n there¹s a prime >between n and n + n^c for c = 0.525. >On the Riemann Hypothesis, it¹s known that for any positive e and >for any sufÞciently large n (that is, n > n_0(e)) there¹s a prime >between n and n + n^(0.5 + e). >It¹s conjectured that if p_n is the n-th prime then >p_(n + 1) - p_n is big-oh of the square of log n. >Of course, this is much, much stronger than the n^c >results above. I assumed that what he thought was unlikely was the statement that RH _follows_ from the existence of a prime between n and n + sqrt(n) (for large n, of course). Do you have a reference for that? >-- >Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) David C. Ullrich === Subject: Re: Simple proof of polynomial distribution of primes? >> the Riemann Hypothesis follows from the existence >> of a prime between n and n + sqrt(n).... >Really? Seems quite unlikely. Really? Why does that seem unlikely? >What do you believe to be the most precise result >along these lines? David C. Ullrich === Subject: Compuation Complexity: Multiplication and Division I would like to compare two methods (A and B) in terms of computation complexity - total number of operations done by each method. Method A and B both are doing some multipications, additions, divisions...I would like to get total number of operations done by each method - say method A has done M operations and method B has done N operations. So if M>N => B is better else A better... So basically I need to give Œweight¹ to each operations... - Multiplication takes Œr¹ clock cycles (or some unit, pls advice) - Division takes Œd¹ clock cycles - Addition takes... The numbers are Œþoat¹. I am using Pentium-4 machine. Please advice me how should I do it. I am trying to Þnd some relevant material but not able to Þnd. Please send me some relavant links or key words to search for. -ashish === Subject: Re: Compuation Complexity: Multiplication and Division > I would like to compare two methods (A and B) in terms of computation >complexity - total number of operations done by each method. > Method A and B both are doing some multipications, additions, >divisions...I would like to get total number of operations done by >each method - > say method A has done M operations and method B has done N >operations. > So if M>N => B is better else A better... > So basically I need to give Œweight¹ to each operations... > - Multiplication takes Œr¹ clock cycles (or some unit, pls advice) > - Division takes Œd¹ clock cycles > - Addition takes... > The numbers are Œþoat¹. I am using Pentium-4 machine. > Please advice me how should I do it. If you want to predict realistic timings on Pentium-4 machines, I think it¹s a good deal more complicated than numbers of each operation. The Pentium-4 has a pipelined superscalar architecture and SSE (Streaming SIMD Extensions); depending on how the calculations are organized, this can in effect do several operations at the same time. A very important factor is that access to main memory (outside the cache) is slow. So method A might be faster than B if it makes more efÞcient use of the cache, even though it uses more arithmetic operations. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Compuation Complexity: Multiplication and Division > I would like to compare two methods (A and B) in terms of computation > complexity - total number of operations done by each method. > Method A and B both are doing some multipications, additions, > divisions...I would like to get total number of operations done by > each method - > say method A has done M operations and method B has done N > operations. > So if M>N => B is better else A better... > So basically I need to give Œweight¹ to each operations... > - Multiplication takes Œr¹ clock cycles (or some unit, pls advice) > - Division takes Œd¹ clock cycles > - Addition takes... > The numbers are Œþoat¹. I am using Pentium-4 machine. > Please advice me how should I do it. > I am trying to Þnd some relevant material but not able to Þnd. > Please send me some relavant links or key words to search for. > -ashish Go to The Source! Consult The Art of Computer Programming, Volume 2, Seminumerical Algorithms by Donald Knuth. He discusses a large number of different approaches and algorithms including very detailed analysis of computing time. Not for the feint at heart, but then again, this is a complex subject. Good luck! -Michael. === Subject: Re: Question on generation of large prime numbers Phil, all ... 1 a the whole amount, quantity, or extent of (all the upheaveal; all this clutter; waited all day; all his life), b (with pl.) the entire number of (all the others left; all ten men) ... Merriam Webster¹s Collegiate Dictionary, Eleventh Edition, deÞnes ``all¹¹ similarly. HTH. John === Subject: Re: Question on generation of large prime numbers > Phil, > all ... 1 a the whole amount, quantity, or extent of (all the upheaveal; all > this clutter; waited all day; all his life), b (with pl.) the entire number of > (all the others left; all ten men) ... > Merriam Webster¹s Collegiate Dictionary, Eleventh Edition, deÞnes ``all¹¹ > similarly. > HTH. Well duh! All of _what_, though. HAND, Phil -- Unpatched IE vulnerability: mhtml wecerr CAB þip Description: Delivery and installation of an executable Reference: http://msgs.securepoint.com/cgi-bin/get/bugtraq0305/48.html === Subject: Re: Question on generation of large prime numbers [...] |Forget the largest prime thing. That¹s what¹s confusing everyone. I¹m sorry, but I don¹t agree that I¹ve been confused, nor have I seen Dik Winter show signs of confusion. |Magnitude of the primes is _utterly_ irrelevant to Euclid¹s proof. We¹re discussing the validity of a proof which starts if there is a largest prime..., not Euclid¹s proof. |In black and white - and you may quote me on this - |EUCLID DID NOT MAKE REFERENCE TO THERE BEING A LARGEST PRIME IN HIS PROOF |OF THE INFINITUDE OF PRIMES. (Of course, Euclid had no concept of inÞnite, |as such, so he didn¹t word it that way.) |The only time he refers to magnitude is that of the set size, not the |numeric size of the elements. i.e the set with 4 elements is larger in |magnitude than the set with 3 elements. I have no problem with this. What I Þnd puzzling is why you think this helps to indicate that Richard¹s proof is invalid, as opposed to not being as elegant as possible, or as well formulated as possible. |> |This is why I was blathering about what all meant. |> The meaning is plain; all primes means all of them: {n : n is prime}. | |I think the meaning is plain. However, I, like Euclid, think that {2,3,7} |is a perfectly valid example for what could be posited as a Þnite set of |all primes. i.e. in taht case {2,3,7} _is_ all primes. (And that leads to |a contradiction, and therefore it isn¹t.) There seems to be as muc a |fundamental misunderstanding of how proof by contradiction works as well |as a misunderstanding of Euclid¹s proof. | |If you have a problem with {2,3,7} being posited as all primes, then you |have a problem with the mechanics of proof by contradiction. The problem is that you were trying to use this to demonstrate that one of the steps in the posted proof was invalid, and I don¹t agree with that. Yes, there are occasions when, in order to illustrate a qualm with a proof, people will discuss hypothetical situations which may even be contrary to fact, such as {2,3,7} being the set of all primes. Questions asked during colloquium talks are often vague questions of this kind. Suppose we didn¹t actually know that lemma 5 was true.... But once it serves its function as a hint at what the person¹s qualm is, once the gesture is made, it hardly helps to try to base a formal complaint on it. In my experience, mathematicians seldom claim that a step in a proof isn¹t valid if all that is needed to repair it is to insert a single step, especially if the step is of this kind. Almost always, something else is said, like it would be clearer if you had this step in the proof here. You know, I know, and Richard knows what kinds of details would smooth over this particular qualm. You merely have the idea that the implicit reasoning (which was not Þlled in in this way) is a much more serious omission than I think it is. |> |> But Richard¹s conclusion that P has been shown not to be the largest |> |> prime number is also wrong; what has been shown is that the assumption |> |> of a Þnite number of primes is wrong. |> | |> |Yup, which is why I said Nope.. |> But his Þrst step was to say that if P is the largest prime, then there |> are Þnitely many primes. That¹s a correct deduction. If the conclusion of |> that step then leads to a contradiction, it is valid to conclude that the |> premise of it was false too. | |Or one of the other premises. Richard introduced other assumptions, not |realising that he¹d done so. I think what you are calling other assumptions (plural?) are nothing but obvious inferences from the one he made. |> |> The only reason to assume that |> |> at least one of the additional primes discovered must be larger than |> |> P is that we *thought* P was the largest prime, before we discovered |them. |> | |> |Euclid permits me to assume {2,3,7,13,43,139,3263443} is the Þnite set |> |of primes, with maximum prime P=3263443. In what way does |> |{2,3,7,13,43,139,3263443}, P=3263443 violate the premise Let P be the |> |largest prime? |> 3263443 is not the largest prime. | |I¹m sorry, I thought we were playing a game of proof by contradiction. |As far as I cen tell immediate gainsaying is not a valid move in that game. I was simply answering the question as stated. I don¹t know what actual question you might have in mind instead. The usual game of proof by contradiction is can I prove this result by contradiction?. The game we are playing now is can Phil invalidate someone else¹s proof by contradiction?. The example is a Þnite set of primes, and it is not the set of all primes <=3263443, which serves to highlight the step which goes from 3263443 being the largest prime to the list of all primes being the same as the list of all primes <=3263443. That¹s Þne so far as it goes, but then this is so small a step, in my opinion, that it hardly makes sense to highlight it in this way. |If the set of all primes is {2,3,7,13,43,139,3263443} then |3263443 _is_ the largest prime. That¹s a simple unassailable |mathematical fact. It¹s also a simple unassailable mathematical fact that if 3263443 is the largest prime, then every prime <= 3263443 is among the set of all primes, and that the smallest prime divisor of their product is >3263443. It¹s a fact of course merely because it¹s of the form X->Y where X is false, and unassailable because provable, but simple as well because the reasoning is very simple. To invalidate a proof by contradction, you surely realize that it¹s not enough to Þnd an example (which fails to satisfy the initial premise) and discover that it also fails some later step. The question is what additional virtue this particular Þction has. |Euclid¹s proof does not say: | Let PP be the set of all primes, but PP musn¹t be |{2,3,7,13,43,139,3263443}. |does it? It says (when reworded in more modern language): | Let PP be the set of all primes. | |Why do you have a problem with {2,3,7,13,43,139,3263443}? |Euclid didn¹t. Kummer didn¹t. I don¹t. I don¹t see that you¹ve been very speciÞc in saying what kind of problem you think I have with it. If you had decided instead to critique the original proof of Fermat by introducing the example of a=125, b=153, c=155, and p=13, and Þnding (say) that y^2=x(x-125^11)(x+153^11) isn¹t actually semistable (I haven¹t checked whether it is or not), and demanded to know what problem I had with a=125, b=153, c=155, p=13. what should I say? Other than the obvious 125^13+153^13 != 155^11, and that I don¹t see it as all that relevant that this particular Þction falls apart in that manner at that particular step in the proof. [...] |> Compare the proof with what is in Hardy and Wright, which they also call |> Euclid¹s proof: |> Let 2,3,5,...,p be the aggregate of primes up to p, and let |> q = 2.3.5. ... .p + 1. |> Then q is not divisible by any of the numbers 2,3,5,...,p. It is |> therefore either prime, or divisible by a prime between p and q. |> In either case, there is a prime greater than p, which proves |> the theorem. | |That¹s quite a way from Euclid¹s formulation. Quite a way is a fairly relative term. In practice people reformulate proofs in bigger ways and keep attributing them to the original authors anyway. As I recall V.I.Arnold attributes to Gauss the calculation of some cohomology group or K group, though that¹s maybe an extreme case. |It presupposed that you can generate all primes up to P. |Euclid¹s proof doesn¹t. |I don¹t care that one can generate all primes up to P trivially, and |can prove it can be done pretty trivially, it¹s just _unnecessary_ |as part of a proof of the inÞniteness of the set of primes. | |Euclid¹s proof was that the set of primes is larger than any Þnite |set. At no point did it make reference to the magnitude of any of the |elements in the set. (except that primes aren¹t units, of course.) These are all points in its favor. Note however that people don¹t always aim for these virtues. I don¹t think it¹s just sloppiness or laziness. I tend to like the more economical proof myself, but people also have a tendancy to prefer at times to have a degree of irrelevant concreteness, like assuming that the set of primes in question is all of the primes up to a given bound. |> It is valid to reason in this way if the set of primes in question is all |> of the primes up to a given prime p. This is mainly what Richard does. |> It appears to me, then, that a lot of your objection hinges on the reader |> not being ready to regard as equivalent the fact that a set of natural |> numbers is Þnite, that it has a largest element P, and that it consists of |> (all) those elements which are <= P. I think anybody who is unable to |> recognize these as equivalent is not really ready to be reading Euclid¹s |> proof. And these equivalences continue to hold even when we¹re considering |> a counterfactual condition such as P being the largest prime. | |No. My objection is to people presupposing they know what all means, |when they¹ve not considered what assumptions they¹ve made in order to |come up with that meaning. Is it that you think Richard is not consciously enough aware of the step of considering his list to consist of all primes <= P? I just don¹t see any reason to think that anyone has gotten a mistaken or confused notion of what all means here. One can speak loosely of a property Q acquiring a meaning for someone on account of their having gotten the idea that Q is equivalent to various differently-expressed properties Q2, Q3,.... If the problem really is that these are not equivalent, but that they only mistakenly think they are, then I don¹t think it helps to describe the problem as one of their presupposing they know what Q means. For such a basic term as all, especially, I would say it¹s a bit presumptuous to claim they really don¹t know what it means, unless they¹ve shown more serious a pattern of misusing it. |> I don¹t know whether Richard had this consciously in mind when he |> his version of it, but since his P is introduced as a hypothetical largest |> prime, references to it only make sense so long as we are still under the |> assumption that there is such a P. So in particular, the statement that |> there is a prime greater than P is still among the consequences of P being |> the largest prime. That¹s valid. True, the same reasoning can be used to |> show that there is a prime > P without assuming that P is the largest prime |> number. |> His last step, asserting that his original assumption is false, is correct |> too. It refers to P again, but simply denies the original assumption. | |But he had extra unstated assumptions. That¹s what I¹ve been jumping on. |Repeatedly. Proof by contradiction denies one of the assumptions, but |doesn¹t tell you which one is denied. As I have said repeatedly. You¹ve repeated the accusation, yes, but it all seems to be based on a rather uncharitable interpretation of what he of drawing the needed intermediate inferences. |> |If Richard has simply added let all primes <=P be known to his premise |> |I wouldn¹t have jumped on it that way. |> I don¹t think this would help. Talking about what primes are known is |> subjective. | |Not really, would assigned make you happy? That only seems more vague. At this point, I wonder why this is any better than saying plainly that the set (which in Richard¹s proof goes nameless) of all primes is the same as the set of primes <= P. |The set of all primes is the set of all known primes in this proof. |I¹ve said that repeatedly. The only point I can see for introducing the issue of whether we know a prime or not, is to make the inference that the set of primes is equal to the set of primes <= P seem more nontrivial than it is. |The set-theoretic notation for what my sentences expressed would be |no different if I included or excluded the word known. But the set-theoretic notation would actually mean one of the actually mathematical properties here, like being prime or being a prime <= P, not is a prime known to some unspeciÞed person or persons. | I was |simply trying to avoid the naked word all as people immediately |misinterpret that based on their knowledge about the primes. I think you are underestimating your audience hugely. |> Without the subjective reference, what¹s the alleged extra assumption? |> That all the primes <= P are actually among the set of all primes? | |This is why I jump on people¹s wording - the above is vacuously |true as worded (and uses the naked term all twice, which immedately |biases the inexpert reader as to what it might refer to). But yes, |that is the assumption. It is possible to deny that clause and still |prove the inÞniteness of the primes using Euclid¹s proof. Redundancy does not made a proof invalid. [...] |> |However, if he had, then it would |> |_not_ have been Euclid¹s proof (and he claimed what followed was Euclid¹s |> |proof so I would have jumped on that instead), |> I think you would have been better off saying that instead. | |if he had... . He hadn¹t, so I didn¹t. I think the audience is more intelligent and informed than you think, and that whether or not it¹s desirable to add such a clause, it¹s not a real necessity. |> Of course, an |> error of a similar nature is in Hardy and Wright. They don¹t start with a |> counterfactual assumption, but they do ask us to consider all the primes |> up to p, and as I understand it Euclid doesn¹t. | |Yup, Euclid asks us to consider an arbitrary Þnite list. | |Euclid, like Kummer (whose proof hasn¹t been distorted over time) |_doesn¹t_ even require _2_ to be in the list of primes. | |This comes a shock to many people, but it¹s God¹s honest truth. |Euclid¹s proof doesn¹t presume _any_ particular number is prime, not even 2. | |When people see Let p1 I¹m sure Ockham would have appreciated an elegant argument, but I doubt he |> would say that an argument becomes invalid if it contains redundancies. | |But, as I said before, if they do introduce new assumptions, which this did, |then it throws a spanner into the works when it comes to proof by |contradiction. |With the assumption, you prove A |= B, without the assumption you prove |= B. |In order to get |= B from the Þrst you need |= A. I.e. it¹s _not_ a proof of |B until you add a proof of A. The whole issue as far as I¹m concerned is whether the interpolated extra step is nontrivial enough for it to be necessary to spelled out further. [...] |> I admit that for someone lacking the mathematical competence to recognize |> that a set of natural numbers being Þnite, its having a largest element, |> and its consisting of precisely those elements <= its maximum element are |> equivalent, this presentation would not be enough of a proof. | |I¹m glad I don¹t have the mathematical competence to think that a set |of natural numbers | a) being Þnite | b) having a largest element | c) consisting of precisely those elements <= its maximum element |are equivalent. | |{2,3,7} satisÞes (a), and (b), but does not satisfy (c). I didn¹t say natural numbers <= its maximum element; I said elements <= its maximum element. I had considered giving the set a name, S and phrasing (c) in a wordier way, but it seemed like unnecessary pedantry. 5 is not an element of {2,3,7}, so it¹s not an example of an element <=7 but not in the set. |So - are you mathematically competent enough to think (a), (b), and (c) |are equivalent? Feel free to make further arbitrary guesses as to my abilities-- it will provide entertainment to those more familiar with me. The philosophers have a principle they call the principle of charity, which roughly says it¹s preferable to interpret others as making sense when such an interpretation is available. A number of the points in this thread strike me as being examples of your making needlessly uncharitable interpretations of what others write. Charitable interpretation of proofs means not jumping at the opportunity to assert their invalidity, unless you actually cannot interpret the problem as a lesser one like a typo or an implicit step or two which needs to be would mean at _least_ not automatically assuming that the most natural interpretation of elements in (c) was something other than elements of the set itself. Something like the same principle weighs also against assuming your discussion partners have inferior and incoherent understanding of phrases like all primes. |> I¹m just having a hard time seeing any of these critiques as a meaningful |> objections to the proof. | |What is the proof? The original leaves us with A |= B. That¹s _not_ a |proof of B. Sure, we know that A is provable, but as it stands without a |proof of A, we don¹t have a proof of B. (e.g. all things that are |dependent on RH aren¹t proven yet.) If you appeared to have a nontrivial A, this would be a problem. But a proof that B follows from a trivial enough A counts as a proof of B. |> We should not be trying to create the impression |> that in mathematics, it¹s normal to engage in hair-splitting like that. |> You already have Richard chalking up a new, phony reason to think he¹s not |> quite cut out to be a mathematician (not that that necessarily makes any |> difference). That strikes me as silly. | |It¹s unnecessary to introduce things into a proof that are not necesary |for the proof. That strikes me as silly. I saw it, I said it. I agree that it¹s unnecessary, but not especially harmful. |If it takes hair-splitting to separate an unconditional proof from a |conditional proof reliant on an unproved assumption, then split hairs I |will. As I said above, mathematicians do not in my experience very often go around declaring steps invalid when the implicit intermediate step is of such an elementary nature. It¹s one of a number of ways in which real mathematical discussions tend to me much more pleasant than usenet ones. They also, depending on their style, introduce a greater or fewer number of superþuous picturesque ingredients in their proofs. Someone once described a project he assigned to a student which involved examining the proofs in a book. The student coined the phrase proof by deletion, owing to the volume of material in proofs in the book which could be deleted without harm, whole pages in some places. Keith Ramsay === Subject: Re: Question on generation of large prime numbers > [...] > |> I admit that for someone lacking the mathematical competence to > recognize > |> that a set of natural numbers being Þnite, its having a largest > element, > |> and its consisting of precisely those elements <= its maximum > element are > |> equivalent, this presentation would not be enough of a proof. > |I¹m glad I don¹t have the mathematical competence to think that a > set > |of natural numbers > | a) being Þnite > | b) having a largest element > | c) consisting of precisely those elements <= its maximum element > |are equivalent. > |{2,3,7} satisÞes (a), and (b), but does not satisfy (c). > I didn¹t say natural numbers <= its maximum element; I > said elements <= its maximum element. I had considered > giving the set a name, S and phrasing (c) in a wordier way, > but it seemed like unnecessary pedantry. The elements of S are deÞned as a subset of |N, thus by a predicate deÞned over |N. The largest element statement is again making reference to properties within |N and therefore I assumed the subsequent predicate would also be over |N. One man¹s unnecessary pedanty is another man¹s necessary pedantry. If it had been expressed in maths rather than English, then there would have been no such problem. is an element of being a binary operator never leaves an ambiguity of what set is being refered to. Yes this has been a thread entirely about pedantry. I genuinely feel that if everyone had been more pedantic right from the start the thread would have been 2 posts long. Everyone has used ambiguous terminology at some point, myself included, and everyone¹s been stung by that. I trim your post just to that one point as I think it neatly contains an example of what I¹ve been whittering on about for the last week. On the whole I agree with most of what you say. I wasn¹t aware of the charity courtesy - I¹ve grown up in the Bernstein/Lenstra NFS era :-) . Phil -- Unpatched IE vulnerability: window.open search injection Description: cross-domain scripting, cookie/data/identity theft, command execution Reference: http://safecenter.net/liudieyu/WsFakeSrc/WsFakeSrc-Content.HTM Exploit: http://safecenter.net/liudieyu/WsFakeSrc/WsFakeSrc-MyPage.htm === Subject: Re: Question on generation of large prime numbers > I have looking at a few web pages dealing with the largest known > calculated primes and a great deal of computational time is taking > into searching for these numbers and verifying they are primes. I have > seen the Euclid¹s proof of inÞnitude primes and it occurs that me > that super-large prime numbers can be calculated using the following: > > p1=2 < p2=3 > A large prime number, p, can be generated using > p = (p1 * p2) + 1 = 7 > > p1=2 < p2=3 < p3=5 > p = (p1*p2*p3) + 1 = 31 > > p1=2 < p2=3 < p3=5 < p4=7 > p = (p1*p2*p3*p4) + 1 = 211 > > p1=2 < p2=3 < p3=5 ..... < pn=... > p = (p1*p2*p3*p4*....*pn) + 1 = .... > > It seems to me that using the above method, super large prime numbers > exceeding currently known largest primes can be generated rather > quickly. > Excellent! Why not look at > q = (p_1 p_2 p_3 .... p_n) - 1 > also. That must be prime, by the same argument, and you have a proof > that there are inÞnitely many twin primes :-) * That proof, if valid, would win you a Field¹s medal. earle * === Subject: Re: Question on generation of large prime numbers >> Excellent! Why not look at >> q = (p_1 p_2 p_3 .... p_n) - 1 >> also. That must be prime, by the same argument, and you have a proof >> that there are inÞnitely many twin primes :-) > That proof, if valid, would win you a Field¹s medal. Excellent! Now I just hope they don¹t Þnd out I plagiarized it from an old posting in this group :-) -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) === Subject: Set proof Can someone please outline a simple proof for: For all m>k, and non-increasing E_j, m U (E_jE_(j+1)) = E_kE_(m+1) j=k === Subject: Re: Set proof === Subject: Set proof >Can someone please outline a simple proof for: >For all m>k, and non-increasing E_j, > m > U (E_jE_(j+1)) = E_kE_(m+1) > j=k By induction, / / union intersection /(j=k..m+1) EjE_(j+1) = /(j=k..m) EjE_(j+1) / E_(m+1)E_(m+2) = EkE_(m+1) / E_(m+1)E_(m+2) ... = EkE_(m+2) ---- === Subject: Re: Set proof === > Subject: Set proof > >Can someone please outline a simple proof for: > >For all m>k, and non-increasing E_j, > > m > > U (E_jE_(j+1)) = E_kE_(m+1) > > j=k > By induction, / / union intersection > /(j=k..m+1) EjE_(j+1) > = /(j=k..m) EjE_(j+1) / E_(m+1)E_(m+2) > = EkE_(m+1) / E_(m+1)E_(m+2) > ... > = EkE_(m+2) === Subject: Re: Set proof >For all m>k, and non-increasing E_j, I assume you mean E_{j+1} is a subset of E_j. > m > U (E_jE_(j+1)) = E_kE_(m+1) > j=k Hint: if x is in E_jE_{j+1}, is it in E_k? Is it in E_{m+1}? Conversely, if x is in E_kE_{m+1}, take the biggest j such that ... Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Set proof >For all m>k, and non-increasing E_j, > I assume you mean E_{j+1} is a subset of E_j. > m > U (E_jE_(j+1)) = E_kE_(m+1) > j=k > Hint: if x is in E_jE_{j+1}, is it in E_k? Is it in E_{m+1}? > Conversely, if x is in E_kE_{m+1}, take the biggest j such that ... === Subject: is this easy to put in a spreadsheet? I can describe this problem in words, but not put it into any mathematical form. Can someone help by suggesting a method for solving it with xl spreadsheet? A client of mine sells lengths of aluminium window parts at 6.5 m length. He has customers who want to buy it at various lengths, in various quantities, and want to reduce the amount of wastage. E.g.: 30 lengths at 1 m, 25 at 1.5, 25 at 2. How can you work out the smallest amount of 6.5 m lengths to buy to achieve this? I guess this also means you have to determine the combinations? Sorry for the lameness of the question, but he¹s a good client and thinks I¹m a whizz with computers just because I design his web-site. I hate to disappoint. -- Ben (check my email address before sending) === Subject: Re: is this easy to put in a spreadsheet? > I can describe this problem in words, but not put it into any > mathematical form. Can someone help by suggesting a method for solving > it with xl spreadsheet? > A client of mine sells lengths of aluminium window parts at 6.5 m length. > He has customers who want to buy it at various lengths, in various > quantities, and want to reduce the amount of wastage. E.g.: 30 lengths > at 1 m, 25 at 1.5, 25 at 2. How can you work out the smallest amount of > 6.5 m lengths to buy to achieve this? Although there are integer programming methods to solve problems like this, this problem is simple enough to solve directly. First, you need at least 30*1m + 25*1.5m + 25*2m= 117.5m of material so you need at least 19*6.5m lengths. Seeing that if we want to completely use a 6.5m length we must cut either 1 or 3 1.5m lengths out of it, let us start by taking 19 1.5m pieces out of the 19 6.5m pieces. Now we need to get 6 more 1.5m pieces, so let us cut them from three of the Þve meter pices. So we have cut our 25 1.5 meter pieces and have 16x5m pieces and 3x2m pieces. Set aside the Three 2m pieces, and cut two two meter pieces from 11 of the Þve meter pieces. We now have all the 1.5 and 2 meter pieces cut and 11 1m pieces, as well as 5 5m pieces. Cut the remaining 19 1m pieces, leaving waste of 6m. In summary 3*1.5m + 1*2m (three times) 1*1.5m + 2*2m + 1*1m (11 times) 1*1.5m + 5*1m (four times) (one 1m piece extra) 1*1.5m + 1*5m (once) (one 5 m piece extra) One can get excel¹s optimization package to do this for you, but unless the problems become much more difÞcult pencil and paper is going to be a whole lot easier. > I guess this also means you have to determine the combinations? > Sorry for the lameness of the question, but he¹s a good client and > thinks I¹m a whizz with computers just because I design his web-site. I > hate to disappoint. === Subject: Int. J. of Algebra and Computation - Contents alert International Journal of Algebra and Computation (IJAC) Articles are available at http://www.worldscinet.com/ijac.html Contents: On the ProÞnite Topology on Coxeter Groups by R. Gitik pp393 Bounded Generation and Linear Groups by M. Abert, A. Lubotzky and L. Pyber pp401 The Andrews.89¥Curtis Conjecture and Black Box Groups by A. V. Borovil, E. I. Khukhro and A. G. Myasnikov pp415 The Structure of Residuated Lattices by K. Blount and C. Tsinakis pp437 Varieties of Equality Structures by D. Fearnley-Sander and T. Stokes pp463 Dualisability of Finite Semigroups by M. Jackson pp481 Fast Isomorphism Testing in Arithmetical Varieties by T. A. Gorazd pp499 For more information, go to http://www.worldscinet.com/ijac.html === Subject: Mensanator dishonesty paragraph in which I claim that either Mensanator cannot interpret what he reads, or he is simply a dishonest person. He probably was offended by some observations I made, accusing him of not understanding basic geometry and probability notions. I will update the page to show his errors, meanwhile I maintain that there is evidence of dishonesty on his part, perhaps based in this personal offense. http://www.thequantummachine.com/mounds.php === Subject: Re: Mensanator dishonesty > paragraph in which I claim that either Mensanator cannot interpret > what he reads, or he is simply a dishonest person. He probably was > offended by some observations I made, accusing him of not > understanding basic geometry and probability notions. I will update > the page to show his errors, meanwhile I maintain that there is > evidence of dishonesty on his part, perhaps based in this personal > offense. > http://www.thequantummachine.com/mounds.php Why is P seven pixels below, and one pixel to the right of the mound you claim it¹s on. Why is D six pixels below and 3 pixels to the right of the mound you claim it¹s on? Mansanator doesn¹t need the likes of me sticking up for his character, but the evidence is that you are spouting gibberish and displaying þagrant disregard for principles of probability, and Mensanator is displaying a perfectly sufÞcient knowledge of geometry to interpret and invalidate your illusions, a likwise sufÞcient knowledge of probability to puncture your probabilistic posturing, and to be perfectly frank, is being perfectly honest and reasonable with you when he criticises your nonsense. More honest than you appear to deserve. *_PLONK_* Phil -- Unpatched IE vulnerability: Timed history injection Description: cross-domain scripting, cookie/data/identity theft, command execution Reference: http://safecenter.net/liudieyu/BackMyParent2/BackMyParent2- Content.HTM Exploit: http://www.safecenter.net/liudieyu/BackMyParent2/ BackMyParent2-MyPage.HTM === Subject: Re: Mensanator dishonesty Better to take it with humour. This guy will not be an image processing analyst: http://www.thequantummachine.com/images.php === Subject: Re: difference rings and Þelds > Hey, does anyone know what the difference between a ring and a Þeld > is? > A Þeld is a commutative ring with unit, different from {0}, in which every > non-zero element is invertible. > If you throw away the commutativity, one speaks of a division ring. > A commutaitve ring can be considered to be a subring of a Þeld (i.e. is > isomorphic to a subring of a Þeld) if, and only if, it is integral (that > is, different from {0} and the product of two non-zero elements is non-zero, > which is clearly a necessary conditions). Necessary, but also sufÞcient? I see that it¹s sufÞcient if the ring has a unit; but if the ring doesn¹t have a 1, are there any further conditions? For example, suppose for some distinct a,b,c,d in R, and some integer n, we have that a*b = n*b, and c*d = n*d. In any Þeld with unit 1 containing R, it would follow that a = c = n*1, so this would contradict a,c distinct; but I don¹t see how this situation is prevented purely by the integral property of R if there is no 1 in R. === Subject: Re: difference rings and Þelds >Hey, does anyone know what the difference between a ring and a Þeld >is? > >>A Þeld is a commutative ring with unit, different from {0}, in which every >>non-zero element is invertible. >>If you throw away the commutativity, one speaks of a division ring. >>A commutaitve ring can be considered to be a subring of a Þeld (i.e. is >>isomorphic to a subring of a Þeld) if, and only if, it is integral (that >>is, different from {0} and the product of two non-zero elements is non-zero, >>which is clearly a necessary conditions). >> >Necessary, but also sufÞcient? I see that it¹s sufÞcient if the ring >has a unit; but if the ring doesn¹t have a 1, are there any further >conditions? Given an integral domain R, can we not construct a Þeld in the same way we construct Q from Z? I.e., equivalence classes on R x (R{0}), where (a, b) ~ (c, d) iff a d = b c . (Integral domains are by deÞnition commutative, no? So says Herstein.) -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu === Subject: Re: difference rings and Þelds > >Hey, does anyone know what the difference between a ring and a Þeld >is? > >A Þeld is a commutative ring with unit, different from {0}, in which every >>non-zero element is invertible. >>If you throw away the commutativity, one speaks of a division ring. >>A commutaitve ring can be considered to be a subring of a Þeld (i.e. is >>isomorphic to a subring of a Þeld) if, and only if, it is integral (that >>is, different from {0} and the product of two non-zero elements is non-zero, >>which is clearly a necessary conditions). >> >Necessary, but also sufÞcient? I see that it¹s sufÞcient if the ring >has a unit; but if the ring doesn¹t have a 1, are there any further >conditions? > Given an integral domain R, can we not construct a Þeld in the same > way we construct Q from Z? I.e., equivalence classes on R x > (R{0}), where (a, b) ~ (c, d) iff a d = b c . (Integral domains > are by deÞnition commutative, no? So says Herstein.) If R has a 1, then the mapping f(r) = (r,1) gives the desired embedding; but without the 1, how do we embed R? === Subject: Re: difference rings and Þelds Adjunct Assistant Professor at the University of Montana. >> Given an integral domain R, can we not construct a Þeld in the same >> way we construct Q from Z? I.e., equivalence classes on R x >> (R{0}), where (a, b) ~ (c, d) iff a d = b c . (Integral domains >> are by deÞnition commutative, no? So says Herstein.) >If R has a 1, then the mapping f(r) = (r,1) gives the desired >embedding; but without the 1, how do we embed R? You are not really mapping to (r,1), you are mapping to the class of (r,1). So map r to the class of (r^2,r) if r is nonzero, and map 0 to the class of (0,r) for arbitrary r different from 0. It¹s not denial. I¹m just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: problem A broken line passes through all n2 points in a square array. Show that it must have at least 2n-2 links. [In other words show that it is not possible to Þnd 2n-2 points P1, P2, ... , P2n-2 so that the 2n-3 segments P1P2, P2P3, ... , P2n-3P2n-2 contain all n2 points (x,y) where x,y = 1,2, ... , n.) === Subject: Help / ideas with statistical prediction problem Hi. I have the problem described below and would be grateful for any solutions, comments, ideas or direction - thank you for your time. Problem statement: There is an unbounded set of events. Each event can occur more than once. The events occur sequentially, so that no two events occur at the same time. The events started occuring a Þnite time ago. If it helps, we can assume that we know the entire history of which events have occured at which times. Not a great deal is known about the distribution of the events¹ occurences. Some events will occur many times, others will occur only once or never. When events occur multiple times, it is expected that the occurences will come in clusters that are fairly evenly spaced. How can I compare two event histories to Þnd out which of the two is most likely to occur Þrst after the present time? Can I calculate a crude soon-ness index based on the event history which estimates how soon we expect the event to occur again? I¹m looking for crude measures which might be fairly quick to calculate. Example: The last event time was 15. Here are some events and the times at which they occured: Event A: 3, 4, 5, 9, 13 Event B: 1, 2, 6 Event C: 7, 8, 10, 11 Intuitively, I would expect to order the events A, C, B in decreasing order of expected soon-ness. I would give them soon-ness indices of A(4), C(?), B(+inf). This is just from looking at the example data and guessing. Toby. === Subject: Re: Help / ideas with statistical prediction problem > Hi. I have the problem described below and would be grateful for any > solutions, comments, ideas or direction - thank you for your time. > Problem statement: > There is an unbounded set of events. Each event can occur more than > once. The events occur sequentially, so that no two events occur at > the same time. The events started occuring a Þnite time ago. If it > helps, we can assume that we know the entire history of which events > have occured at which times. > Not a great deal is known about the distribution of the events¹ > occurences. First you need to deÞne the random variable you are interested in. Candidates for random processes include: Number of events [of type X] in a given time interval, or times between events of type X. > Some events will occur many times, others will occur only > once or never. When events occur multiple times, it is expected that > the occurences will come in clusters that are fairly evenly spaced. Does that mean you think the next value of T_X depends on the previous value? Independent arrival models are most common, but you can certainly model dependency if you think that¹s necessary. > How can I compare two event histories to Þnd out which of the two is > most likely to occur Þrst after the present time? You can develop sample statistics of the random variable(s) you are interested in. These will be approximations of the true or population statistics. Given those, you can estimate various functions of those random variables. > Can I calculate a > crude soon-ness index based on the event history which estimates how > soon we expect the event to occur again? This would be the expectation value of T_X. Certainly given an estimated distribution of T_X, you can calculate its expection value. In fact this is about the easiest estimate to calculate: the sample mean of T_X is the average of the values in your sample. > I¹m looking for crude > measures which might be fairly quick to calculate. > Example: > The last event time was 15. Here are some events and the times at > which they occured: > Event A: 3, 4, 5, 9, 13 > Event B: 1, 2, 6 > Event C: 7, 8, 10, 11 If I were assuming independence, I¹d just note down the interarrival times, the times between events: A: 1, 1, 4, 4. Mean = 2.5 C: 1, 4. Mean = 2.5 B: 1, 2, 1 Mean = 1.3 But are they independent? Does event A depend on when B or C occurred, or how many times A has occurred? > Intuitively, I would expect to order the events A, C, B in decreasing > order of expected soon-ness. Why? Look at the inter-event spacings. Why do you have this expectation? Do you have some prior knowledge that is relevant? > I would give them soon-ness indices of > A(4), C(?), B(+inf). This is just from looking at the example data and > guessing. Based on what? What is leading to that guess of +inf for this score for B? - Randy === Subject: A question on differential geometry Suppose that I¹ve got a three-dimensional manifold equipped with a positive deÞnite Riemannian metric. If the manifold has spherical symmetry I can write the line element as ds^2 = g_(rr)dr^2 + R^2*d(Omega)^2 where d(Omega)^2 is the line element on a unit 2-sphere, g_(rr) is some scalar, and R = R(r) is some areal radial coordinate. I¹m taking r to be some radial coordinate. My question is: what is the form of the unit normal vector to surfaces of constant R? === Subject: Morse-Thue-style sequence puzzle This might appeal to you if you get an IQ of 300 in Whats-the-next-term-of-the-sequence puzzles :-) (Leroy, do you take? :-) If I were 20 years younger, I had cracked this nut by myself, as I¹m very good at pattern recognition. But now...Anyway, this sequence arose with rational knots. I had to compute it by hand (I could provide two terms more, but leave me something to check your bold hypotheses :-) and would me most interested to have a recursion formula. Preliminaries: To each digit of a binary number, a letter (N,Z,X) and an integer will be assigned. Up to n=5 (from 7-crossing knots) it goes like: 1 X+2 10 Z-2Z-2 11 N-3N-3 100 X+4Z+2Z+2 101 N+1Z+0N+1 110 Z+2Z+2X+4 111 X+4X+4X+4 1000 Z-4Z-2Z-2Z-2 1001 N-3Z+0Z+0N-3 1010 X+0Z+2Z+2X+0 1011 Z-4Z-2X+0X+0 1100 N-5N-5N-5N-5 1101 X+0X+0Z-2Z-4 1110 Z-2Z-2Z-2Z-4 1111 N-5N-5N-5N-5 10000 X+6Z+2Z+2Z+2Z+2 10001 Z+1Z+0Z+0Z+0Z+1 10010 X+4Z-2Z-2Z+2Z+0 10011 X+6Z+2Z+2X+4X+4 10100 N+3Z+0N-1N-1N-1 10101 Z+0Z-2X-4Z-2Z+0 10110 Z+0Z+2Z-2Z-2X+4 10111 N+3Z+0N-1N-1N-1 11000 Z+4Z+4X+6X+6X+6 11001 X+4X+4Z+2Z+2X+6 11010 N-1N-1N-1Z+0N+3 11011 Z+4Z+4X+6Z+4Z+4 11100 X+6X+6X+6Z+4Z+4 11101 N-1N-1N-1Z+0N+3 11110 Z+2Z+2Z+2Z+2X+6 11111 X+6X+6X+6X+6X+6 Easy patterns (these are LAWS): - Adjacent same bit => have same symbol. - Bitstring A=reverse(B) =>sequence(A)=reverse(sequence(B)) - Bitstring A=reverse(B)+all bits þipped => ditto. - N goes with odd, ZX with even integers. For the last bit, I have an exact recursion formula: - Integer($0) and ($1) = -Integer($) plus 0 or 1 or -1 (depending on the letter NZX - details left as homework) Note: In case your super elegant formula collides with a single item, I probably made a transfer typo :-) -- Hauke Reddmann <:-EX8 Private email:fc3a501@math.uni-hamburg.de For our chemistry workgroup,remove math from the address For spamming, remove anything else === Subject: New, jobs and news services from EEVL, the Internet guide to engineering, mathematics and computing These two new, free services from EEVL are so simple and easy to use we¹ve named them ŒOneStep¹. http://www.eevl.ac.uk/onestepnews/ and http://www.eevl.ac.uk/onestepjobs/ OneStep Industry News gathers together, in one place, engineering, mathematics and computing news headlines from a number of top sources. OneStep Jobs does the same for jobs announcements. One click takes you to the full news item or job details at the source site. Sources include: e4engineering.com, Buildingtalk, Manufacturingtalk, Electronicstalk, Nature - Materials Update, Moreover, LTSN Engineering, LTSN Materials, the Institute of Physics (Optics.org News, Fibers.org News, Nanotechweb.org News, Compoundsemiconductor.net News), scenta, LTSN Maths, The Register, Slashdot, Nanodot, Jobsite, theengineerjobs.co.uk, jobs.ac.uk, Nanotechweb.org Jobs, Perl Jobs, and general technology newsfeeds such as BBC Tech News and CNN Technology. More sources will be added in the future. The OneSteps utilise a data format called RSS to aggregate content. The default displays intermingle headlines from the various sources. It is also possible to view headlines by subject cluster, or individually. These two new services should make it a lot easier to Þnd industry news and job announcements in the subjects covered. The focus, overall, is on UK sources. A press release giving more information is at: http://www.eevl.ac.uk/pressrelease/pressrelonesteps.htm EEVL is a UK-based not-for-proÞt free guide. It is is funded by JISC through the Resource Discovery Network (RDN). === Subject: Re: The pure math lie, Discussion, linux) > ... > > I¹ll admit it. I don¹t go in for the pure math hooey, and am > > honestly in it for the money. I get mathematicians to acknowledge my > > discovery, get in the papers, sell my story, get paid. > If you are in it for the money, do *not* attempt anything in mathematics. > There is no money in it. Not for mathematicians, no. But that¹s where JSH differs from the usual approach. He¹s assured that he is *not* mathematically trained, so when he takes the mathematical world by storm, people will pay to hear his story. And the money is in entertainment, not mathematics. It¹s a foolproof plan for becoming wealthy and famous, see? -- If you have a really big idea, you can get a measure of how big it is REALLY, REALLY, *REALLY*, BIG DISCOVERY!!! --James Harris, on being ignored === Subject: Re: (statistics)how to make date more like Laplacian distribution? walala > The non-invertible transform > introduces less distortion than that quantization... Entirely depends on just how badly you truncate stuff in the DCT/iDCT. I have no way of telling, but it was you who brought up that round off errors in your transform were signiÞcant in another thread. > It is not simple for me. Can you tell me how to make the coefÞcient > values more like Laplacian? As I said before, it isnt about making them laplacian ... you assume the distribution is laplacian, and Þt it (using the variance). If you look at the paper Blocking Artifact Detection and Reduction in Compressed Data on page 880, at the left bottom they give a formula to determine the offset down from the center of the quantization range. The formula only needs the quantization step and the per coefÞcient variance. The variance is estimated by the variance of the center reconstructed quantized coefÞcients (that is all the information the decoder has). Concretely, say for coefÞcient (3,3) ... you take all the (3,3) coefÞcient magnitudes dequantized in the standard way and compute their variance, you feed that together with Q into formula 10. Then you subtract the result from all the (3,3) coefÞcient magnitudes. Alternative you can use the interpixel variance from the paper they mention, [40], which has the advantage that the offsets dont have to be uniform over the entire image (you could compute the variance for a single block and determine the per coefÞcient variances from that to compute the offsets) and it doesnt suffer from poor Þtting due to low numbers of samples (for high frequencies there will be very few quantized coefÞcients >0). There are lots of papers written on this topic, search for laplacian dct on ieeexplore for instance. This is a relatively easy way (computationally) to get a modest improvement in DCT coding, it wont do wonders though ... you will still need to apply additional deblocking. Marco === Subject: Re: Hard College Algebra/Trig Problems, misc. For challenging Œcollege algebra¹ problems Œwritten like they used to be¹, take a look at the book: Higher Algebra by Hall and Knight. --Jim Buddenhagen To reply copy jbuddenh@REMOVEtexas.net to address bar and edit out REMOVE === Subject: Patenting Numbers? http://www.theregister.co.uk/content/28/34148.html === Subject: Re: Patenting Numbers? > http://www.theregister.co.uk/content/28/34148.html In some sense, numbers have been granted patents already -- a computer program is really just one big number. - Brett === Subject: Re: Patenting Numbers? >http://www.theregister.co.uk/content/28/34148.html http://www.theonion.com/onion3311/microsoftpatents.html Doug === Subject: Re: Patenting Numbers? > http://www.theregister.co.uk/content/28/34148.html Surely, this is a joke, right? Right? Please say it is a joke. -Michael. === Subject: Re: Patenting Numbers? > http://www.theregister.co.uk/content/28/34148.html > Surely, this is a joke, right? Right? Please say it is a joke. I think it is as The Register journalists use real email addresses, Verity Strop, or whoever the journo was supposed to be called, simply had a nameless email address. i.e. I /think/ Verity Strapon doesn¹t actually exist. (Verity=truth, too much of a coincidence to accompany somtehing that isn¹t a wind-up, surely?) However, it¹s not a particularly well researched one because numbers have already been patented. I think in 1996 Roger Schlaþy patented 2 numbers for the purposes of RSA encryption. I believe that he was hoping to demonstrate the invalidity of the RSA patent, and got a silly patent as a side-effect. (And alas failed to demonstrate the invalidity of the RSA enough.) Phil -- Unpatched IE vulnerability: Alexa Related Privacy Disclosure Description: Unintended disclosure of private information when using the Related feature Reference: http://www.secunia.com/advisories/8955/ Reference: http://www.imilly.com/alexa.htm === Subject: Re: Patenting Numbers? >> http://www.theregister.co.uk/content/28/34148.html >Surely, this is a joke, right? Right? Please say it is a joke. It¹s a joke. If in doubt, do a Google search for the author¹s name. -- Richard -- Spam Þlter: to mail me from a .com/.net site, put my surname in the headers. FreeBSD rules! === Subject: Re: Patenting Numbers? >> http://www.theregister.co.uk/content/28/34148.html >Surely, this is a joke, right? Right? Please say it is a joke. > It¹s a joke. > If in doubt, do a Google search for the author¹s name. At any rate, they¹d have to deal with Microsoft¹s lawyers for patent violation. http://www.rfcafe.com/miscellany/humor/1n0_patent.htm I should note that the above carries no attribution. It is copyrighted material originally published in The Onion, but I couldn¹t locate any of the original Onion material online anywhere. Please don¹t reproduce it without crediting The Onion (I¹m a fan and like to see credit where credit is due). - Randy === Subject: Re: Patenting Numbers? >I should note that the above carries no attribution. It is >copyrighted material originally published in The Onion, >but I couldn¹t locate any of the original Onion material >online anywhere. I¹ve already posted the Onion material (actually, I¹ve posted the link to the material fron the Onion¹s own website) elsewhere in this thread. Doug === Subject: Re: Patenting Numbers? > http://www.theregister.co.uk/content/28/34148.html The next step is patenting air and water. === Subject: Re: Patenting Numbers? >>http://www.theregister.co.uk/content/28/34148.html > The next step is patenting air and water. Nope. That is prior art. However taxing folks on their exusions and bodily productions is not far in the future. The jisim tax. It¹s the coming thing. Bob Kolker === Subject: Re: Patenting Numbers? > http://www.theregister.co.uk/content/28/34148.html > The next step is patenting air and water. Patently absurd! David === Subject: AB and BA have same eigenvalue A and B have different dimensions, but AB and BA are both square. Can anyone provide a proof showing the eigenvalues are the same for AB and BA ? === Subject: Re: AB and BA have same eigenvalue >A and B have different dimensions, >but AB and BA are both square. >Can anyone provide a proof showing the eigenvalues are the same for AB and BA ? The non-zero ones, including multiplicity. Consider the determinant of the matrix I xA B I If one subtracts xA times the second row from the Þrst, the determinant is seen to be that of I-xAB. But if one subtracts B times the Þrst row from the second, it is I-xBA. -- 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: AB and BA have same eigenvalue >A and B have different dimensions, >but AB and BA are both square. >Can anyone provide a proof showing the eigenvalues are the same for AB and >BA ? If A and B are not square, then A*B and B*A are not of the same size, and the larger of A*B and B*A will have 0 as an eigenvalue, which need not be an eigenvalue of the smaller. === Subject: Re: AB and BA have same eigenvalue Le x e eigenvalue of AB, then there exists a vector u different from 0 s.t. ABu=xu. Since x =/= 0 we have A(Bx)=/=0 if x=/=0 so Bx=/=0 and BA(Bx)=xBu. But this is nothing that x is an eigenvalue of BA (and Bx the eigenvector!). Now if 0 is an eigenvalue of AB then 0=det(AB)=det(A)det(B)=det(BA)=0... Finaly the A and B play the same role here, so eigenvalues of AB = eigenvalues of BA! > A and B have different dimensions, > but AB and BA are both square. > Can anyone provide a proof showing the eigenvalues are the same for AB and BA ? === Subject: Re: AB and BA have same eigenvalue We can resume my calculation by this two diagrams: let A : E ---> F and B : F ---> F be two linear operators, then we have (BA-xId) : F ---> F ^ ^ (A) : | | : (A) (AB-xId) : E ---->E & (BA-xId) : E ---> E ^ ^ (B) : | | : (B) (AB-xId) : F ---->F Ps : this method is correct even if we have inÞnite dimension vetcor spaces. > Le x e eigenvalue of AB, then there exists a vector u different from 0 s.t. > ABu=xu. > Since x =/= 0 we have A(Bx)=/=0 if x=/=0 so Bx=/=0 and BA(Bx)=xBu. But this > is > nothing that x is an eigenvalue of BA (and Bx the eigenvector!). Now if 0 is > an eigenvalue of AB then 0=det(AB)=det(A)det(B)=det(BA)=0... > Finaly the A and B play the same role here, so eigenvalues of AB = > eigenvalues of > BA! > A and B have different dimensions, > but AB and BA are both square. > Can anyone provide a proof showing the eigenvalues are the same for AB and > BA ? === Subject: Re: AB and BA have same eigenvalue > A and B have different dimensions, > but AB and BA are both square. > Can anyone provide a proof showing the eigenvalues are the same for AB and BA The larger size one could have extra eigenvalues zero. === Subject: Re: 0/0 > Witt grava .88 la saucisse et au marteau: [...] > Another problem is: > Given a fraction a/b with a different from 0. > Assume 0/0 is deÞned. > a/b = a/b * (0/0) = 0/0 > The only possibility for it to be true is 0/0 = 0 Wow! Assuming that result, check out a/b + 0/0 = (0*a + b*0)/0*b = 0/0 So, a/b + 0/0 = 0/0 and since 0/0 = 0 (according to your post quoted above), this means that a/b + 0 = 0 a/b = 0 So anything divided by anthing equals zero! Wow!!! :-) :-) :-) :-) :-) Is this silly or what? Disclaimer: JMHO Alan E. Feldman === Subject: Re: 0/0 Alan E. Feldman grava .88 la saucisse et au marteau: > So anything divided by anthing equals zero! Wow!!! :-) :-) :-) :-) :-) I just said that if 0/0 should have a value, the operation with the fractions needed it to be 0. As you proved after, this value is totally aberrant for other use. Therefore, the only conclusion is that 0/0 can not have a deÞned value. My goal was not to say that we could give a value for 0/0 but that this was undeÞned (at least other than by a set). But I did only one part. If I had done your part too, I could have concluded without your sarcasm, but it¹s always cheerful thinking someone cares about you :) -- Nicolas === Subject: Re: 0/0 > Alan E. Feldman grava .88 la saucisse et au marteau: > So anything divided by anthing equals zero! Wow!!! :-) :-) :-) :-) :-) > I just said that if 0/0 should have a value, the operation with the > fractions needed it to be 0. As you proved after, this value is totally > aberrant for other use. > Therefore, the only conclusion is that 0/0 can not have a deÞned value. > My goal was not to say that we could give a value for 0/0 but that this > was undeÞned (at least other than by a set). But I did only one part. > If I had done your part too, I could have concluded without your > sarcasm, but it¹s always cheerful thinking someone cares about you :) I thought that maybe you were doing that, but couldn¹t tell for sure as you yourself say that you didn¹t follow through. I just wanted to do the follow-through and thought it was neat to be able to conclude that a/b always equals zero! Converting a/b * (0/0) to (a*0)/(b*0) does assume an intermediate step of it being (a/b)*0*(x/0) and that gets you into trouble. I intended to complete what I thought was the ultimate point of your post; I did not intend to be sarcastic to your post. The smileys just meant the fun of reaching such an absurd conclusion by being purposefully silly to complete the point. A picture¹s worth a thousand words, but just which of those thousand words is meant when the picture is being read is anyone¹s guess. The same goes for smileys. What makes normal division useful is the fact that it yields a unique number. Since division by zero doesn¹t do that, it is not useful in normal algebra. NTL, there are cases, like Ohm¹s Law -- which is V=IR -- that if V and R are both zero (as in a superconductor), then any current (except a current so large that it destroys the sample) is allowed. (V is voltage or if one prefers electromotive force, I is the current, and R is the resistance.) In this case we get 0 = I*0 which just means that any reasonable current is allowed. It doesn¹t mean you can divide by zero. Disclaimer: JMHO &-) <---------- (Bill de Cat smiley) Alan E. Feldman === Subject: Re: Max area of rectangle inscribed in quarter-disc? Originator: tchow@lagrange.mit.edu.mit.edu (Timothy Chow) >So maybe you want the vertices on the straight sides to be adjacent, >rather than diagonally opposite? Yes, sorry, that is what I meant. I was too terse in my description of the set of allowable rectangles. The other two vertices need to be on the curved arc. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences === Subject: Re: JSH: All the dumb crap in journals Sci.math has nothing to do with what gets published in math journals, > for the simple reason that most of the Œprestigious¹ journals are > owned by, and run in the interest of, a handful of large corporations, > or professional guilds like the ASL, which have no reason whatsoever > to publish anything which departs from prevailing mathematical norms; > whereas sci.logic and sci.math provide forums (in principle at least), > for work in progress which þaunts these norms, and in so doink > challenges the credibility and authority of gate-keepers... Some people think that the mathematics I do indeed belongs on the > usenet. -Dar S. Kabatoff > Name one, other than your alter egos. > Your god isn¹t powerful enough to... > a) give you a Book of instruction > b) provide you with your name > c) know how to count You did NOT answer my question: Name one person (other than one of your alter egos) who thinks that the mathematics you do belongs on the usenet. === Subject: Re: JSH: All the dumb crap in journals > ... > ... > Sci.math has nothing to do with what gets published in math journals, > for the simple reason that most of the Œprestigious¹ journals are > owned by, and run in the interest of, a handful of large corporations, > or professional guilds like the ASL, which have no reason whatsoever > to publish anything which departs from prevailing mathematical norms; > whereas sci.logic and sci.math provide forums (in principle at least), > for work in progress which þaunts these norms, and in so doing > challenges the credibility and authority of gate-keepers like yourself. > Even if what you say about a the manipulating powers behind journals > were true, þaunting the norms of mathematics would still be no better > method for advancing mathematics. You use the term prevailing as if > the norms of mathematics could be something else, but they just are > period, regardles of anybody¹s personal, political or whatsoever > interests. Mathematics is like art and fashion, it is deÞned by its producers and consumers and it has its trends and staples. Everything changes in mathematics, some faster than others. --Vaughan Pratt, FOM > The conþict between you and JSH (among others) is brought about > by factors that are evident in Budd Schulberg¹s contrast of the > businessman with the artist. > and the artist: > > A good businessman...aims to please as many people as possible while > minimizing risk and standardizing production. The aim of the good > artist, on the other hand, is exactly the opposite: he turns his > back on every formula, keeps breaking new ground, risks everything, > and whether he succeeds or fails, prepares to risk again. > (Budd Schulberg, Movies in America: After Fifty Years, _The > Atlantic Monthly_, November 1947; cited in Leo Gurko, _Highbrows > and the Popular Mind_, Charter Books, 1953, p. 165) > This is quite ridiculous. A good businessman aims to maximize his > proÞt and the aim of an artist is not deÞnable in a few lines. Businessman maximize proÞt by controlling markets and stifÞng competitors. What is more, nothing is more foreign to the business mentality than JSH¹s Œgo-for-broke¹ approach to FLT. > However it is pretty clear that rebellious, revolutionary > avant-gardism, is only one of a multitude of artistic attitudes which > becomes fashionable from time to time, and is in itself no guarantee > for artistic creation. > The implication of your posting is, that the mere act of rebellion > against norms has in itself a positive effect towards whatever purpose > you are intent on attaining, but anyone can see that this is not the > case most of the time. > You wouldn¹t want to þy in a plane piloted by someone whose only > credentials as pilot are that he refuses to see any beneÞt in > actually learning to þy. > A pilot who revolutionizes aviation will Þrst have to learn the > prevailing norms or he won¹t have a chance of contributing anything > new to his craft (unless he Þnds a way to do so in afterlife). > And by the way, JSH openly states that he is into revolutionizing math > for proÞt, as a businessman in this regard, he has failed both > pleasing many people, as your Budd Schulberg requires, and making > any proÞt. More relevant to the sci.math treatment of JSH (who has never expressed an interest in Œrevolutionizing aviation¹), is how society treats inventors. Writing in 1955, Gurko says: Inventors are the real professionals of a mechanical society; one would expect them, of all people, to be regarded with respect, if not actual reverence. Yet few groups are more scorned. Until, that is, their inventions succeed, at which time they become garlanded heroes instantly escorted to the hall of fame, where they are set up as exemplars for future generations of boys to envy and emulate. Before an invention is commercialized, however, the inventor must be prepared for calumny. He is a crank, crackpot, dabbler, nut; he is a fool slightly tilted toward the bughouse; he is an irresponsible loafer probably neglecting his wife and children; he just isn¹t right in the head. But the minute he hits the jackpot, these accusations are washed away and forgotten in a þood of acclaim. His eccentricity has been known all along to be a sign of genius. His impracticality has become vision; his neglect of loved ones a desire to win for them a greater security; his indifference to criticism is no longer a mark of footlessness but of courage. So long as the inventor has only his brains, he will be labeled with contemptuous epithets, from harmless crank to raving lunatic. Only the successful marketing of his product can save him... Leo Gurko, _Heroes, Highbrows and the Popular Mind_ (New York: Charter Books, 1953), pp. 54-55) === Subject: Re: JSH: All the dumb crap in journals Excuse me John, can you, aside from quoting others, please give an answer to what I said or is this conversation pointless? I gave you some thoughts on the fact that you are wrong assuming that asuming a maverick attitude is a positive step in itself towards attaining a purpose. You come to me with ...JSH (who has never expressed an interest in Œrevolutionizing aviation¹)..., have you never heard of metaphores? Then you are implying that JSH is receiving the treatment of inventors as described by Gurko. Well, that description does not include the treatment he has been given by many here, who have in the past and still keep even now, taken the trouble of reading his posts and pointing out his mistakes on the mere ground of human cordiality. You know, math is a Þeld where you are either right or wrong but theres nothing in between. The nice thing is you can prove you¹re right. The real trouble with JSH or your defending him, is the use of wrong reasons. His postulates are not going to become right by arguing about how he is treated, this is not politics. > ... > ... > Sci.math has nothing to do with what gets published in math journals, > for the simple reason that most of the Œprestigious¹ journals are > owned by, and run in the interest of, a handful of large corporations, > or professional guilds like the ASL, which have no reason whatsoever > to publish anything which departs from prevailing mathematical norms; > whereas sci.logic and sci.math provide forums (in principle at least), > for work in progress which þaunts these norms, and in so doing > challenges the credibility and authority of gate-keepers like yourself. > > Even if what you say about a the manipulating powers behind journals > were true, þaunting the norms of mathematics would still be no better > method for advancing mathematics. You use the term prevailing as if > the norms of mathematics could be something else, but they just are > period, regardles of anybody¹s personal, political or whatsoever > interests. > Mathematics is like art and fashion, it is deÞned by its producers > and consumers and it has its trends and staples. Everything changes > in mathematics, some faster than others. > --Vaughan Pratt, FOM > > The conþict between you and JSH (among others) is brought about > by factors that are evident in Budd Schulberg¹s contrast of the > businessman with the artist. > and the artist: > > A good businessman...aims to please as many people as possible while > minimizing risk and standardizing production. The aim of the good > artist, on the other hand, is exactly the opposite: he turns his > back on every formula, keeps breaking new ground, risks everything, > and whether he succeeds or fails, prepares to risk again. > (Budd Schulberg, Movies in America: After Fifty Years, _The > Atlantic Monthly_, November 1947; cited in Leo Gurko, _Highbrows > and the Popular Mind_, Charter Books, 1953, p. 165) > > This is quite ridiculous. A good businessman aims to maximize his > proÞt and the aim of an artist is not deÞnable in a few lines. > Businessman maximize proÞt by controlling markets and stifÞng > competitors. What is more, nothing is more foreign to the business > mentality than JSH¹s Œgo-for-broke¹ approach to FLT. > However it is pretty clear that rebellious, revolutionary > avant-gardism, is only one of a multitude of artistic attitudes which > becomes fashionable from time to time, and is in itself no guarantee > for artistic creation. > The implication of your posting is, that the mere act of rebellion > against norms has in itself a positive effect towards whatever purpose > you are intent on attaining, but anyone can see that this is not the > case most of the time. > You wouldn¹t want to þy in a plane piloted by someone whose only > credentials as pilot are that he refuses to see any beneÞt in > actually learning to þy. > A pilot who revolutionizes aviation will Þrst have to learn the > prevailing norms or he won¹t have a chance of contributing anything > new to his craft (unless he Þnds a way to do so in afterlife). > And by the way, JSH openly states that he is into revolutionizing math > for proÞt, as a businessman in this regard, he has failed both > pleasing many people, as your Budd Schulberg requires, and making > any proÞt. > More relevant to the sci.math treatment of JSH (who has never > expressed an interest in Œrevolutionizing aviation¹), is > how society treats inventors. Writing in 1955, Gurko says: > Inventors are the real professionals of a mechanical society; one > would expect them, of all people, to be regarded with respect, > if not actual reverence. Yet few groups are more scorned. Until, > that is, their inventions succeed, at which time they become > garlanded heroes instantly escorted to the hall of fame, where > they are set up as exemplars for future generations of boys to > envy and emulate. Before an invention is commercialized, however, > the inventor must be prepared for calumny. He is a crank, crackpot, > dabbler, nut; he is a fool slightly tilted toward the bughouse; he > is an irresponsible loafer probably neglecting his wife and children; > he just isn¹t right in the head. But the minute he hits the jackpot, > these accusations are washed away and forgotten in a þood of acclaim. > His eccentricity has been known all along to be a sign of genius. His > impracticality has become vision; his neglect of loved ones a desire > to win for them a greater security; his indifference to criticism > is no longer a mark of footlessness but of courage. So long as > the inventor has only his brains, he will be labeled with contemptuous > epithets, from harmless crank to raving lunatic. Only the successful > marketing of his product can save him... Leo Gurko, _Heroes, > Highbrows and the Popular Mind_ (New York: Charter Books, 1953), > pp. 54-55) === Subject: Re: JSH: All the dumb crap in journals Yes, visionaries are sometimes considered cranks. But far more often cranks are considered cranks. The reason we don¹t grab every nutty doofus that comes along and follow them around as though they were the next coming of jesus is that most are in fact cranks. So, human experience is vindicated. Every once and awhile some lunatic will also be a genius (or occasionally lucky) and revolutionalize a Þeld. The vast majority, however, will just give their neighbors something to gawk at. There are alot of inept people all over the place. Alot of them make outrageous claims. For example, bag ladies, bums, religious wackos, etc. We don¹t listen, because they are nuts. To sum it up for you John, JSH is not a revolutionary. He is doesn¹t know anything about the math he claims to use. One more point. Most people who have changed the world through their inventions weren¹t considered cranks. Most quietly laboured away for years. The crank-genius is an exception, of course their stories are far more interesting than the rest. (Euler, Fourier, Gauss, Hilbert, Legendre, Leibnitz, Mandelbrot, Noether, Poincare, Riemann, Stokes, Taylor, Weyl, Abel, Bessel, Boole, Des Cartes, Dirichlet, Kronecker, Lie, Lobachevsky, Markov, Mobius, von Neumann, Newton, Maxwell, Einstein, Lagrange, Laplace, Hamilton, Faraday, Planck, Bohr, Fermi, Schrodinger, Heisenberg, Pauli, Dirac. This is just a short list of those that weren¹t considered during cranks. What was your point again? Which revolutionaries did I miss? Apparently there must be some, since that was the point of your post. Oh, btw, Galois was never considered a crank, so let¹s not try to use him. Alot of the people in this list were quirky, but they were all recognized based on their merits without reference to their quirks. The fact is, mathematicians don¹t care how crazy you are, they only are about your mathematics.) Justin Van Winkle > ... > ... > Sci.math has nothing to do with what gets published in math journals, > for the simple reason that most of the Œprestigious¹ journals are > owned by, and run in the interest of, a handful of large corporations, > or professional guilds like the ASL, which have no reason whatsoever > to publish anything which departs from prevailing mathematical norms; > whereas sci.logic and sci.math provide forums (in principle at least), > for work in progress which þaunts these norms, and in so doing > challenges the credibility and authority of gate-keepers like yourself. > Even if what you say about a the manipulating powers behind journals > were true, þaunting the norms of mathematics would still be no better > method for advancing mathematics. You use the term prevailing as if > the norms of mathematics could be something else, but they just are > period, regardles of anybody¹s personal, political or whatsoever > interests. > Mathematics is like art and fashion, it is deÞned by its producers > and consumers and it has its trends and staples. Everything changes > in mathematics, some faster than others. > --Vaughan Pratt, FOM > The conþict between you and JSH (among others) is brought about > by factors that are evident in Budd Schulberg¹s contrast of the > businessman with the artist. > and the artist: A good businessman...aims to please as many people as possible while > minimizing risk and standardizing production. The aim of the good > artist, on the other hand, is exactly the opposite: he turns his > back on every formula, keeps breaking new ground, risks everything, > and whether he succeeds or fails, prepares to risk again. > (Budd Schulberg, Movies in America: After Fifty Years, _The > Atlantic Monthly_, November 1947; cited in Leo Gurko, _Highbrows > and the Popular Mind_, Charter Books, 1953, p. 165) > This is quite ridiculous. A good businessman aims to maximize his > proÞt and the aim of an artist is not deÞnable in a few lines. > Businessman maximize proÞt by controlling markets and stifÞng > competitors. What is more, nothing is more foreign to the business > mentality than JSH¹s Œgo-for-broke¹ approach to FLT. > However it is pretty clear that rebellious, revolutionary > avant-gardism, is only one of a multitude of artistic attitudes which > becomes fashionable from time to time, and is in itself no guarantee > for artistic creation. > The implication of your posting is, that the mere act of rebellion > against norms has in itself a positive effect towards whatever purpose > you are intent on attaining, but anyone can see that this is not the > case most of the time. > You wouldn¹t want to þy in a plane piloted by someone whose only > credentials as pilot are that he refuses to see any beneÞt in > actually learning to þy. > A pilot who revolutionizes aviation will Þrst have to learn the > prevailing norms or he won¹t have a chance of contributing anything > new to his craft (unless he Þnds a way to do so in afterlife). > And by the way, JSH openly states that he is into revolutionizing math > for proÞt, as a businessman in this regard, he has failed both > pleasing many people, as your Budd Schulberg requires, and making > any proÞt. > More relevant to the sci.math treatment of JSH (who has never > expressed an interest in Œrevolutionizing aviation¹), is > how society treats inventors. Writing in 1955, Gurko says: > Inventors are the real professionals of a mechanical society; one > would expect them, of all people, to be regarded with respect, > if not actual reverence. Yet few groups are more scorned. Until, > that is, their inventions succeed, at which time they become > garlanded heroes instantly escorted to the hall of fame, where > they are set up as exemplars for future generations of boys to > envy and emulate. Before an invention is commercialized, however, > the inventor must be prepared for calumny. He is a crank, crackpot, > dabbler, nut; he is a fool slightly tilted toward the bughouse; he > is an irresponsible loafer probably neglecting his wife and children; > he just isn¹t right in the head. But the minute he hits the jackpot, > these accusations are washed away and forgotten in a þood of acclaim. > His eccentricity has been known all along to be a sign of genius. His > impracticality has become vision; his neglect of loved ones a desire > to win for them a greater security; his indifference to criticism > is no longer a mark of footlessness but of courage. So long as > the inventor has only his brains, he will be labeled with contemptuous > epithets, from harmless crank to raving lunatic. Only the successful > marketing of his product can save him... Leo Gurko, _Heroes, > Highbrows and the Popular Mind_ (New York: Charter Books, 1953), > pp. 54-55) === Subject: Re: JSH: All the dumb crap in journals > Yes, visionaries are sometimes considered cranks. But far more often cranks > are considered cranks. The reason we don¹t grab every nutty doofus that > comes along and follow them around as though they were the next coming of > jesus is that most are in fact cranks. > So, human experience is vindicated. Human experience is vindicated? Oh, dahling! Oh, oh, o-o-oh! > Every once and awhile Every once and awhile? Oh, dahling! Oh, oh, o-o-oh! > some lunatic will also be a genius (or occasionally > lucky) and revolutionalize Oh, dahling! Oh, oh, o-o-oh! > a Þeld. The vast majority, however, will just > give their neighbors something to gawk at. There are alot of inept people > all over the place. Alot Right. Inept. Alot. > of them make outrageous claims. For example, bag > ladies, bums, religious wackos, etc. We don¹t listen, > because they are nuts. If YOU get listened to, who shouldn¹t be? > To sum it up for you John, JSH is not a revolutionary. He is doesn¹t know > anything about the math he claims to use.¹ He sure is doesn¹t! > One more point. Can I pass? > Most people who have changed the world through their > inventions weren¹t considered cranks. > Most quietly laboured away for > years. Oh, dahling! Oh, oh, o-o-oh! > The crank-genius is an exception, > of course their stories are far > more interesting than the rest. (Euler, Fourier, Gauss, Hilbert, Legendre, > Leibnitz, Mandelbrot, Noether, Poincare, Riemann, Stokes, Taylor, Weyl, > Abel, Bessel, Boole, Des Cartes, Dirichlet, Kronecker, Lie, Lobachevsky, > Markov, Mobius, von Neumann, Newton, Maxwell, Einstein, Lagrange, Laplace, > Hamilton, Faraday, Planck, Bohr, Fermi, Schrodinger, Heisenberg, Pauli, > Dirac. > This is just a short list of those that weren¹t considered during > cranks. No, dahling! Not during! Oh, oh, o-o-oh! > What was your point again? > Which revolutionaries did I miss? Which ones did you hit on? > Apparently there must be some, > since that was the point of your post. > btw, Galois was never considered a crank, so let¹s not try to use him. Not Galois. Please. Not Galois. Please, no. > Alot > of the people in this list were quirky, Quirky. In this list. Alot, > but they were all recognized based > on their merits without reference to their quirks. > The fact is, > mathematicians don¹t care how crazy you are, they only are about your > mathematics.) > Justin Van Winkle === Subject: Re: JSH: All the dumb crap in journals >The chance that John Correy is a sock puppet of James Harris is >vanishingly small. <... And there¹s a big contrast between their pet theories As if nobody *other* than John Correy has a Œpet logic¹ (FOL=), or huge holdings in wholly -owned and -operated FOL= subsidiaries like (Z(F(C)) and NBG/MK. >: The things > James says are typically simply wrong, You, who have never dared to stray from well-beaten paths, have some gall to address such a comment to James. > while the things John says > (when he¹s talking about logic instead of just farting online) > are typically not demonstrably wrong in the same way - while > many of us don¹t see why anyone would care about the things he > seems to think are hugely important revolutionary ideas, Your ceaseless attempts to derail non-reþexive identity having come to nothing, your fall-back strategy is to make it seem as if there is nothing of interest at stake in the conþict between (e.g.) MKC and other set theories--and between non-reþexive and reþexive identity. Have you given any *thought* to this matter? Of course not! > and mislead > don¹t see why his axioms make any particular sense, he does > typically state his axioms clearly (eventually), and once the > context is clariÞed, So your blunders were not blunders, but guesses about axioms whose context I had failed to clarify? Another of your deceptions, as I will presently show. --John Membership in the Industry Standard Sock Puppet set is no longer axiomatic. John Correy === Subject: Re: JSH: All the dumb crap in journals >>The chance that John Correy is a sock puppet of James Harris is >>vanishingly small. ><...> And there¹s a big contrast between their pet theories >As if nobody *other* than John Correy has a Œpet logic¹ (FOL=), or >huge holdings in wholly -owned and -operated FOL= subsidiaries >like (Z(F(C)) and NBG/MK. >>: The things >> James says are typically simply wrong, >You, who have never dared to stray from well-beaten paths, have some >gall to address such a comment to James. >> while the things John says >> (when he¹s talking about logic instead of just farting online) >> are typically not demonstrably wrong in the same way - while >> many of us don¹t see why anyone would care about the things he >> seems to think are hugely important revolutionary ideas, >Your ceaseless attempts to derail non-reþexive identity having >come to nothing, your fall-back strategy is to make it seem as >if there is nothing of interest at stake in the conþict >between (e.g.) MKC and other set theories--and between >non-reþexive and reþexive identity. >Have you given any *thought* to this matter? Of course not! >> and mislead Huh - a new tactic, intentional misquotation. Congratulations, you¹ve become Archimedes Plutonium. >> don¹t see why his axioms make any particular sense, he does >> typically state his axioms clearly (eventually), and once the >> context is clariÞed, >So your blunders were not blunders, but guesses about axioms >whose context I had failed to clarify? Another of your >deceptions, as I will presently show. Uh, what you should do is answer the question I asked recently. You know, the one I repeated when you asked for an examle of questions you didn¹t answer, in reply to your request for an example of same. Another question, which I¹ve asked and you haven¹t answered - you really should reply, lest your denial of my statement that you don¹t answer questions you don¹t feel like answering look silly: When James said recently that I¹d lied to my department head you agreed with him. Question: Exactly what lie is it I told the department head here? Another question you haven¹t answered: Exactly what beneÞt would my students derive from me putting class-related material on my website, that I¹m not already providing them? >--John >Membership in the Industry Standard Sock Puppet set >is no longer axiomatic. >John Correy ***************************** John Correy says: Christian (what an oxymoron!): Degrade, demean, goad and bait me as Ullrich and the Boyz have done to JSH, and I won¹t triþe with writing your employer: I¹ll come after you with an AK-47! David C. Ullrich === Subject: Re: closed subset of positive measure Originator: grubb@lola >I am investigating the following proposition and could use some help, >either a conÞrmation/rewriting of the idea behind the proof or a >Let E be a subset of [a,b] with mE > 0, where m is the Lebesgue >measure. Then there exists a closed subset F of E with mF > 0. If you are assuming that E is measurable, this is true. Otherwise, it is not. --Dan Grubb === Subject: Re: closed subset of positive measure >I am investigating the following proposition and could use some help, >either a conÞrmation/rewriting of the idea behind the proof or a >Let E be a subset of [a,b] with mE > 0, where m is the Lebesgue >measure. Then there exists a closed subset F of E with mF > 0. > If you are assuming that E is measurable, this is true. Otherwise, > it is not. You¹re saying there¹s a non-measurable set E with m(E) > 0? --Ron Bruck === Subject: Re: closed subset of positive measure >I am investigating the following proposition and could use some help, >either a conÞrmation/rewriting of the idea behind the proof or a > >Let E be a subset of [a,b] with mE > 0, where m is the Lebesgue >measure. Then there exists a closed subset F of E with mF > 0. > > If you are assuming that E is measurable, this is true. Otherwise, > it is not. > You¹re saying there¹s a non-measurable set E with m(E) > 0? > --Ron Bruck m = Lebesgue outer measure: if E is non-measurable, then certainly m(E) > 0. The sup of the measures of the closed sets inside E is the inner measure of E. So, in the original question, if E is measurable and m(E) > 0 [outer measure], then [since E is measurable] the inner measure of E is > 0, so there is a closed set F inside E with m(F) > 0. === Subject: Re: closed subset of positive measure > >I am investigating the following proposition and could use some help, >either a conÞrmation/rewriting of the idea behind the proof or a > >Let E be a subset of [a,b] with mE > 0, where m is the Lebesgue >measure. Then there exists a closed subset F of E with mF > 0. > > If you are assuming that E is measurable, this is true. Otherwise, > it is not. > > You¹re saying there¹s a non-measurable set E with m(E) > 0? > > --Ron Bruck > m = Lebesgue outer measure: if E is non-measurable, then certainly > m(E) > 0. Direct quote from the OP: ...where m is the LEBESGUE MEASURE. My emphasis. He didn¹t say Lebesgue outer measure. --Ron Bruck === Subject: Re: closed subset of positive measure > Direct quote from the OP: ...where m is the LEBESGUE MEASURE. My > emphasis. He didn¹t say Lebesgue outer measure. In some texts, the term Lebesgue measure means what I call Lebesgue outer measure. The OP did not say what text he was using. === Subject: Re: Big Number Game 0038 >> A trillion seems a little high. (I hear that billion isn¹t a word >> in the UK? Does that mean that trillion isn¹t either?) >In the UK, these words do exist, but with different meanings than those to >which you might be accustomed. Billion means 10^12 (a million squared), >trillion means 10^18 (a million cubed), quadrillion means 10^24 (a >million to the power 4), etc etc (except in the hands of ignorant people >such as politicians, businessmen, and the media). > And then for the other numbers, they say thousand billion, thousand > trillion, etc.? > never sounded right to my ears that a million squared was a trillion, > not a billion. > -- > dgates@spamfreelinkline.com === Subject: Re: Comment welcome -- martian mounds anomaly === >Subject: Re: Comment welcome -- martian mounds anomaly > You just can¹t _prove_ it¹s not random. Probability has no meaning > for something that has already happened. You claim the odds are > 200,000,000,000 to 1 and then you say that no other conÞguration > has this property. > Duh. You are one of those who confuse the probability of an event occuring with the event itself. Duh... In the mounds problem, the probabilistic experiment is having 6 mounds formed by some process. The process has a great number of pausible outcomes, depending on its particular physical nature. If you assume some type of a random formation process, then the probability of getting the formation shown is extremely low. Instead of wasting your time uploading gifs and accusing people, why don¹t you program a simulation of 6 mounds forming at random, set your accuracy range for parallel and perpendicular lines and see of you can get that or similar formations and what the frequency of those are. That will convince you the probability of getting that formation or similar ones is extremely low. That¹s all there is to it. By the way, has you ever been to a casino? The number you see the roullette ball sitting on of course just happended but you can¹t bet on it. If you think talking about its probability has no meaning because it just happened, then if you have just won, why should they pay you back 36 times your bet? Just take your money back and that¹s it.:) === Subject: Re: Comment welcome -- martian mounds anomaly > You just can¹t _prove_ it¹s not random. Probability has no meaning > for something that has already happened. To clarify my answer in the other post, what is always objected is that you need a priory hypothesis, because if not, your conclusions are tricky, but not necesarily wrong. But this is absurd. I could have predicted that the mounds would display a very basic geometry and even calculate which geometry. Does that change the odds of them being random or not? No. It only would granted that I did not choose among thousands of possible conÞgurations, thus biasing my selection criteria. That¹s all. And I claim that the geometry is so basic, and so similar in properties and p to that of straigh line, which is displayed by other geological features (crater chains), that there is a geological mechanism and that the p calculations makes sense. If the conÞguration would be different, but equally striking, I would possibly not make that claim. If the conÞguration showed a duck, even if striking to the eye, I would SURELY not make that claim. There are millions of possible conÞgurations resembling some known object. Not so many, but still a lot with a geometrical regularity, and only a few showing a very basic geometry. I think that this is the point. === Subject: Re: Comment welcome -- martian mounds anomaly === >Subject: Re: Comment welcome -- martian mounds anomaly > Nothing. What do the mounds have to do with math? >Well, take a look at the polygon BAGED. It displays the maximum number >of DIFFERENT parallel and perpendicular directions of lines drawn >among their vertex. The only other polygon that display this property >is a square with a Þfth point being (as D in BAGED) in the >intersection of the diagonals of the square (parallelogram for BAGED) >drawn from the adjacent vertex. > So what? That is a quite simple concept.I wouldn¹t expect for random points to display so easy geometrical arrangements. Even if the chances are the same for them being in a straigh line and similar distances each other, that is dull. This is elegant, and 2-D. Whatever mechainsm, it is strange. >It gets very much complex to say than to draw it. > Well, if it¹s that easy to draw, then maybe there¹s nothing remarkable about > it. If it were too complex to draw I would shout numerology and piramidology. Being so simple is more than intriguing. >Notice that this is quite puzzling since the BAGED polygon had been > I see nothing puzzling about it. Well, then even if there were 1000 mounds aligned in a straigh line you wouldn¹t Þnd it strange. And if a geologist would say that does not know how they can be ordered that way, I bet that you would still say that it is not puzzling. >discovered in 1994, and in 1996-1997 I found the parallel and >perpendicular relationships. I made the claim that this conÞguration >seemed to display such maximum parallel/perpendicular relationship >property, and using very basic geometry it is demonstrated so, and >that this Þgure is identical to that by Crater&McDaniel. (the >demonstration of this point is clear, but the Þrst point depends on >the drawing, I think nothing is missed and that this claim is valid, >i.e., not other conÞgurations show that property). >However, it still may be an odd coincidence, > Why would the coincidence be odd? Why do you think there is no > order in randomness? What do you think the point was in that link > I posted? There is order in randomness. But to a different level. Here the order would directly mean that there is a geological mechanism which align mounds in that (2-D) layout. And if you are too strict, nothing is really random. >granting that there are >geological lineaments in the region and that the mounds may tend to >follow these lineaments (this objection has also been made by critic >Ralph Greenberg). >I am not talking of little green martians. > And the people who talk about Intelligent Design don¹t say god either. I am not talking of intelligent desing. Even so, in the time and context, there was a scenario in which intelligent desing was plausible. Of course, if the mounds are not independent each other and geology fails, that should be an option. While I am inclined for the posibility of geological mechanism, there is nothing illogical in the artiÞcial hypothesis. >I am saying (and that is >objectable, but I will just consider scientiÞc objections) that the >mounds distribution is not random. Nothing else. > You just can¹t _prove_ it¹s not random. Probability has no meaning > for something that has already happened. You claim the odds are > 200,000,000,000 to 1 and then you say that no other conÞguration > has this property. check in some websites. In astrophysics, certain clusters of galaxies aligned are detected, and the statistical tests are conducted AFTER the photo has been taken. In crater chains, you Þnd the crater chain, and generally you don¹t even make a statistical analysis. But you are doing a quickly check. If the alignment is not very clear, you would apply Kolmogorov-Smirnov. Probability has meaning for something that has happened. You can extract consequences. If you would Þnd not 6, but 3000 mounds, all aligned in a straigh line, you would call the geology department immediately. The same for 1500, for 1000, for 100... for 30... for 20??? uhmmm... for 10??? uhmmmm.... and for 6???? See what I mean? In fact, you mentally say... 3000 mounds, p = 1*10-20 1500 mounds, p = 10-10 100 mounds, p = 1/1000000 30 mounds, p = 1/100000 ... 6 mounds, p = 1/1000 ... uhmmm... Please answer to this claim. This point is quite tricky, but you will recognize that the example is pretty obvious. Of course, there is a limit, and this is clear: I found a B letter in Mars. What are the chances of it happening? Well, there is a big area to factor, and even so, the chances are small. If I would Þnd and the chances were 1 / 1000, I could accept that it was just a coincidence, and that, obviously, the chances had instantly converted to 1. However, if the chances were 1 / 1000 000 000, I would say that the photo had been doctored. > Duh. > How many conÞgurations would you have to inspect to Þnd a second > occurrence? 200 000 000 000. >And this pattern of maximality of parallel/perpendicular may suggest >some unknown geological mechanism. Of course, in order to discover it, >maths have to be applied (maths is applied in astrophysical alignments >of galaxies, for example, in a way similar to the Kolmogorov-Smirnov >test that was applied to the mounds. Seeing that the pattern may be >non-random, you go further in your analysis). > Don¹t you understand that a _single_ occurrence at high odds _is_ > evidence of randomness? If you want to claim a non-random process, > you have to present enough cases so as to defy probability. Again a claim that can be applied to both crater chains and alignments of articial pyramids in the Earth. As for the PRIORY hypothesis, it is even present, but I give not too much credit, reasons below. But it was argued that the area was crowded with tetrahedral geometry. The mounds display such a geometry. A simple coincidence, but if you adhere to scientiÞc process, it means a priory. After that, 6 more mounds which had been previously identiÞed, were tested to see if they followed the geometry found in the diagram of 6. They did. Again a priori. The Þrst a priori I don¹t take it too seriously because was based in piramidology measurements by a non scientist. === Subject: Re: Comment welcome -- martian mounds anomaly === >Subject: Re: Comment welcome -- martian mounds anomaly === >>Subject: Re: Comment welcome -- martian mounds anomaly >> Nothing. What do the mounds have to do with math? >>Well, take a look at the polygon BAGED. It displays the maximum number >>of DIFFERENT parallel and perpendicular directions of lines drawn >>among their vertex. The only other polygon that display this property >>is a square with a Þfth point being (as D in BAGED) in the >>intersection of the diagonals of the square (parallelogram for BAGED) >>drawn from the adjacent vertex. >> So what? >That is a quite simple concept.I wouldn¹t expect for random points to >display so easy geometrical arrangements. Even if the chances are the >same for them being in a straigh line and similar distances each >other, that is dull. This is elegant, and 2-D. Whatever mechainsm, it >is strange. >>It gets very much complex to say than to draw it. >> Well, if it¹s that easy to draw, then maybe there¹s nothing remarkable >about >> it. >If it were too complex to draw I would shout numerology and >piramidology. >Being so simple is more than intriguing. >>Notice that this is quite puzzling since the BAGED polygon had been >> I see nothing puzzling about it. >Well, then even if there were 1000 mounds aligned in a straigh line >you wouldn¹t Þnd it strange. And if a geologist would say that does >not know how they can be ordered that way, I bet that you would still >say that it is not puzzling. >>discovered in 1994, and in 1996-1997 I found the parallel and >>perpendicular relationships. I made the claim that this conÞguration >>seemed to display such maximum parallel/perpendicular relationship >>property, and using very basic geometry it is demonstrated so, and >>that this Þgure is identical to that by Crater&McDaniel. (the >>demonstration of this point is clear, but the Þrst point depends on >>the drawing, I think nothing is missed and that this claim is valid, >>i.e., not other conÞgurations show that property). >>However, it still may be an odd coincidence, >> Why would the coincidence be odd? Why do you think there is no >> order in randomness? What do you think the point was in that link >> I posted? >There is order in randomness. But to a different level. Here the order >would directly mean that there is a geological mechanism which align >mounds in that (2-D) layout. And if you are too strict, nothing is >really random. >>granting that there are >>geological lineaments in the region and that the mounds may tend to >>follow these lineaments (this objection has also been made by critic >>Ralph Greenberg). >>I am not talking of little green martians. >> And the people who talk about Intelligent Design don¹t say god either. >I am not talking of intelligent desing. Even so, in the time and >context, there was a scenario in which intelligent desing was >plausible. Of course, if the mounds are not independent each other and >geology fails, that should be an option. While I am inclined for the >posibility of geological mechanism, there is nothing illogical in the >artiÞcial hypothesis. >>I am saying (and that is >>objectable, but I will just consider scientiÞc objections) that the >>mounds distribution is not random. Nothing else. >> You just can¹t _prove_ it¹s not random. Probability has no meaning >> for something that has already happened. You claim the odds are >> 200,000,000,000 to 1 and then you say that no other conÞguration >> has this property. >check in some websites. >In astrophysics, certain clusters of galaxies aligned are detected, >and the statistical tests are conducted AFTER the photo has been >taken. >In crater chains, you Þnd the crater chain, and generally you don¹t >even make a statistical analysis. But you are doing a quickly check. >If the alignment is not very clear, you would apply >Kolmogorov-Smirnov. >Probability has meaning for something that has happened. You can >extract consequences. If you would Þnd not 6, but 3000 mounds, all >aligned in a straigh line, you would call the geology department >immediately. The same for 1500, for 1000, for 100... for 30... for >20??? uhmmm... for 10??? uhmmmm.... and for 6???? >See what I mean? In fact, you mentally say... >3000 mounds, p = 1*10-20 >1500 mounds, p = 10-10 >100 mounds, p = 1/1000000 >30 mounds, p = 1/100000 >... >6 mounds, p = 1/1000 ... uhmmm... >Please answer to this claim. This point is quite tricky, but you will >recognize that the example is pretty obvious. Now you¹re saying the odds are 1/1000? I thought the web page claimed the odds were 1/200000000000. You said you were claiming the distribution is not random. Now you¹re saying that the alignment of 6 mounds is not an anomoly. What exactly are you claiming? >Of course, there is a limit, and this is clear: >I found a B letter in Mars. What are the chances of it happening? >Well, there is a big area to factor, and even so, the chances are >small. If I would Þnd and the chances were 1 / 1000, I could accept >that it was just a coincidence, and that, obviously, the chances had >instantly converted to 1. Why don¹t you ask instead what are the chances of Þnding any letter of the alphabet on Mars? Or any ASCII character? And how about including every alphabet ever created? Odds keep going down, don¹t they? Here¹s a game for you. Put 64 coins in a cigar box and give it a good shake. Open it up and note the pattern of heads and tails. The odds of that _particular_ pattern occuring were 1/18446744073709551616 yet it came up, didn¹t it? Spooky, eh? >However, if the chances were 1 / 1000 000 000, I would say that the >photo had been doctored. >> Duh. >> How many conÞgurations would you have to inspect to Þnd a second >> occurrence? >200 000 000 000. Incorrect. The number of trials expected for m successes is m/p where p is the probability, so you would need to check 2/(1/200000000000) or 400000000000 cases. Do try to get your probability theory straight. >>And this pattern of maximality of parallel/perpendicular may suggest >>some unknown geological mechanism. Of course, in order to discover it, >>maths have to be applied (maths is applied in astrophysical alignments >>of galaxies, for example, in a way similar to the Kolmogorov-Smirnov >>test that was applied to the mounds. Seeing that the pattern may be >>non-random, you go further in your analysis). >> Don¹t you understand that a _single_ occurrence at high odds _is_ >> evidence of randomness? If you want to claim a non-random process, >> you have to present enough cases so as to defy probability. >Again a claim that can be applied to both crater chains and alignments >of articial pyramids in the Earth. >As for the PRIORY hypothesis, it is even present, but I give not too >much credit, reasons below. >But it was argued that the area was crowded with tetrahedral geometry. >The mounds display such a geometry. One would think that the folly of trying to read too much into an observation as exempliÞed by Percival Lowell would have been a lesson learned. Apparently not. >A simple coincidence, but if you >adhere to scientiÞc process, it means a priory. >After that, 6 more mounds which had been previously identiÞed, were >tested to see if they followed the geometry found in the diagram of 6. >They did. Again a priori. I can make the same conclusions about the pebbles in my driveway. >The Þrst a priori I don¹t take it too seriously because was based >in piramidology measurements by a non scientist. And I don¹t take any of your anomolies seriously either. -- Mensanator Ace of Clubs === Subject: Re: Comment welcome -- martian mounds anomaly > Now you¹re saying the odds are 1/1000? I thought the web page claimed the odds > were 1/200000000000. You said you were claiming the distribution is not random. > Now you¹re saying that the alignment of 6 mounds is not an anomoly. What > exactly are you claiming? Those were Þgures, nothing else. The computer simulations showed 1/200000000000 of THAT pattern being found. Other analysis show that such easy distributions are quite unique. The exact probabilities of them being random are simply unknown, but because there is just a few geometrical patterns with so much regularity, the claim is made that that number is signiÞcant. Also there is the conÞrming 6 other mounds, but I still maintain that the 6 mounds are enough, as I tried to demonstrate with this example and with the crater chain example (there you could use random simulations because you are looking for a deÞnite pattern, a straight line. Here the properties are very similar to that). >Of course, there is a limit, and this is clear: >I found a B letter in Mars. What are the chances of it happening? >Well, there is a big area to factor, and even so, the chances are >small. If I would Þnd and the chances were 1 / 1000, I could accept >that it was just a coincidence, and that, obviously, the chances had >instantly converted to 1. > Why don¹t you ask instead what are the chances of Þnding any letter of the > alphabet on Mars? Or any ASCII character? And how about including every > alphabet ever created? Odds keep going down, don¹t they? Yes, that is precisely the point I make. If you discovered a A letter, you would make the claim of Þnding a B. Doing so you would have validated your hypothesis, if the pixel componding the B letter were enough. I claim that Þnding a perfect A letter would be quite a puzzling thing, if they are exact to some A letter in some typographic model, pixel by pixel in your image and with an enormous high resolution. In other words, you could believe there were martians with the same alphabet than us, or shout You have doctored the image!!!. But notice that you would do it without a priori statement. That is what I claim. > Here¹s a game for you. Put 64 coins in a cigar box and give it a good shake. > Open it up and note the pattern of heads and tails. The odds of that > _particular_ pattern occuring were > 1/18446744073709551616 > yet it came up, didn¹t it? Spooky, eh? But we are talking of the same all the time. What I would Þnd extraordinaty is to Þnd all of them as heads. And the chances would be 1/2^64. I would immediately believe that they had heads in both sides!!!!... This is what I claim. Notice not a priory bet is made... >However, if the chances were 1 / 1000 000 000, I would say that the >photo had been doctored. >> Duh. >> >> How many conÞgurations would you have to inspect to Þnd a second >> occurrence? >200 000 000 000. > Incorrect. The number of trials expected for m successes is > m/p where p is the probability, so you would need to check > 2/(1/200000000000) or 400000000000 cases. Do try to get your probability theory > straight. I did not talk of p number. The Þgure of 1/200 000 000 000 was extracted from the number of times in the computer simulation. Also notice that you may need 100 more times that number to Þnd an ocurrence, or Þnd 2 in the half of those cases. If you refer to 50% chances of Þnding that pattern, the Þgure is less, not more as you say (not p in 1 / 200 000 000 000). >>And this pattern of maximality of parallel/perpendicular may suggest >>some unknown geological mechanism. Of course, in order to discover it, >>maths have to be applied (maths is applied in astrophysical alignments >>of galaxies, for example, in a way similar to the Kolmogorov-Smirnov >>test that was applied to the mounds. Seeing that the pattern may be >>non-random, you go further in your analysis). >> >> Don¹t you understand that a _single_ occurrence at high odds _is_ >> evidence of randomness? If you want to claim a non-random process, >> you have to present enough cases so as to defy probability. >Again a claim that can be applied to both crater chains and alignments >of articial pyramids in the Earth. >As for the PRIORY hypothesis, it is even present, but I give not too >much credit, reasons below. >But it was argued that the area was crowded with tetrahedral geometry. >The mounds display such a geometry. > One would think that the folly of trying to read too much into an observation > as exempliÞed by Percival Lowell would have been a lesson learned. Apparently > not. >A simple coincidence, but if you >adhere to scientiÞc process, it means a priory. >After that, 6 more mounds which had been previously identiÞed, were >tested to see if they followed the geometry found in the diagram of 6. >They did. Again a priori. > I can make the same conclusions about the pebbles in my driveway. Do it and publish it. >The Þrst a priori I don¹t take it too seriously because was based >in piramidology measurements by a non scientist. > And I don¹t take any of your anomolies seriously either. That is because you don¹t understand basic geometry and probability issues. You have not coreectly answered to my crater chain or 1000 aligned mounds example. Neither you have heard of Kolmogorov-Smirnov tests for random pattern study. They test for correspondence among different subsets in a population, and can check the randomness without a priory judgement. === Subject: Re: Comment welcome -- martian mounds anomaly >> I can make the same conclusions about the pebbles in my driveway. >Do it and publish it. I could have easily reproduced your parallel lines from a picture of my driveway... ...HAD THERE BEEN ANY PARALLEL LINES! Exhibit 1 http://members.aol.com/mensanator666/cydonia/symm2.gif added my own lines to check your geometry. What kind of dumbass uses a lossy image format for technical drawings? Or were you hoping that the smearing of detail would hide your fraud? The white line A-G that I drew is coincident to the original green line A-G. It is also perfectly vertical with respect to the image grid. The white line B-P that I drew is also vertical with respect to the image grid, and thus parallel to my line A-G. Finally, I drew a line G-P. I didn¹t go any further as by this time, there was no point. See details below. Exhibit 2 http://members.aol.com/mensanator666/cydonia/detail1.gif First, note that the green line B-P deviates from being coincident with my white line B-P. Since the white lines B-P and A-G are parallel by construction, this proves that the original green lines are not parallel. Second, note that the original white line A-D exhibits aliasing, indicating that it is not horizontal with respect to the image grid, thus it is not perpendicular to either of the white lines A-G or B-P. It _may_ be perpendicular to the original green line B-P, but that is irrelevant as the original green line is not parallel to A-G. Third, note that the lines converging on mound D miss it by a mile, so there is no point in trying to verify any other line connecting to D. Exhibit 3 http://members.aol.com/mensanator666/cydonia/detail2.gif Note that the original red line G-P also misses mound P by a mile and any claims of parallelism to the other red lines is just a Þgment of the imagination. Summarizing, upon close examination, the parallelism of the mounds completely evaporates and the claimed anomoly does not, in fact, exist. -- Mensanator Ace of Clubs === Subject: Re: Comment welcome -- martian mounds anomaly > Note that the original red line G-P also misses mound P by > a mile and any claims of parallelism to the other red lines > is just a Þgment of the imagination. You are simply a LIAR. Even in the by hand adjusted image, the deviation is of 7 pixels. It has been enlarged 40%, which corresponds to less than 5 pixels in the original image. With a resolution of less than 50 meters/pixel, that gives: 5 * 50 = 250 meters. That is several times less than 1 mile, and also must be noticed that the adjust by computer has more precission, being the deviation of less than 3.5 pixels, AS CLEARLY stated in my webpage. You are simply a LIAR, and even so, you CRIMINALLY acusse me of fraud. I may be wrong, or not, but you are a disgusting dishonest person. === Subject: Re: Comment welcome -- martian mounds anomaly > added my own lines to check your geometry. What > kind of dumbass uses a lossy image format for > technical drawings? Or were you hoping that the > smearing of detail would hide your fraud? You are simply pathetic. First of all, it is expicitly said in my web use orthorectiÞed images. It is stated too that the diagram image is not orthorectiÞed. So you come with measurements. That is a stupid way of acting. I say don¹t make measurements, and you make measurements. Also, given that you are a slow thinker, you will ask then why do you make measurements? I didn¹t, the image is a diagram showing the parallel lines with can be Þtted by computer in an orthorectiÞed image, with a margin of 0.2 degrees in angle. It just show the relations in colored segments (didn¹t you object that they are segments instead of lines?). It is also stated that the image has been rotated, enlarged (this is the fuzzy aspect, due to enlargment), and colorized. Also it is stated that it has been Þtted manually. Haven¹t you seen that PG lines do not touch each other? Did you think my computer run out of green and red pixels? The explanation is much simpler, I simply adjusted manually, an in an non-orthorectied image because the orthorectiÞed did get more blur due to the processing with such a low resolution. So because all of this, I didn¹t take the adjustment extremely careful. Also notice that you choose slightly different starting points for your lines. I must remeber you that, as stated, in the ORIGINAL image, the average deviation is of 1.5 pixels, and goes to a maximum of 3.5 for mound P. There is plenty of room to adjust. Alternatively, you can make perfect parallel - perpendicular lines, without touching vertices, and let the deviation to be visually detectable. It is an alternative. I never received the accusation of fraud. If, after reading this, don¹t retire this false acussation, I will try to sue you. And I am very serious, am very upset by your bad intentions. (I give a lot of detail in the webpage, to have DISHONEST people like you accusing me of fraud.) I will clarify also in my webpage soon, givin links to your webpage. You will be acussed of dishonesty. You can sue me also, if you want. I am wishing it. === Subject: Re: Comment welcome -- martian mounds anomaly > added my own lines to check your geometry. What > kind of dumbass uses a lossy image format for > technical drawings? Or were you hoping that the > smearing of detail would hide your fraud? > You are simply pathetic. First of all, it is expicitly said in my web > use orthorectiÞed images. It is stated too that the diagram image is > not orthorectiÞed. > So you come with measurements. That is a stupid way of acting. I say > don¹t make measurements, and you make measurements. > Also, given that you are a slow thinker, you will ask then why do you > make measurements? > I didn¹t, the image is a diagram showing the parallel lines with can > be Þtted by computer in an orthorectiÞed image, with a margin of 0.2 > degrees in angle. It just show the relations in colored segments > (didn¹t you object that they are segments instead of lines?). > It is also stated that the image has been rotated, enlarged (this is > the fuzzy aspect, due to enlargment), and colorized. > Also it is stated that it has been Þtted manually. Haven¹t you seen > that PG lines do not touch each other? Did you think my computer run > out of green and red pixels? > The explanation is much simpler, I simply adjusted manually, an in an > non-orthorectied image because the orthorectiÞed did get more blur > due to the processing with such a low resolution. So because all of > this, I didn¹t take the adjustment extremely careful. Also notice that > you choose slightly different starting points for your lines. I must > remeber you that, as stated, in the ORIGINAL image, the average > deviation is of 1.5 pixels, and goes to a maximum of 3.5 for mound P. > There is plenty of room to adjust. Alternatively, you can make > perfect parallel - perpendicular lines, without touching vertices, and > let the deviation to be visually detectable. It is an alternative. > I never received the accusation of fraud. If, after reading this, > don¹t retire this false acussation, I will try to sue you. And I am > very serious, am very upset by your bad intentions. (I give a lot of > detail in the webpage, to have DISHONEST people like you accusing me > of fraud.) Ok, I apologize. I did not say it was fraud, I _asked_ if it was. You say it was not, that answers the question. It¹s just that extraordinary claims require extraordinary evidence. What you present on your web site cannot even be considered evidence, let alone extraordinary evidence. I¹m supposed to take your word that if presented correctly, the lines would be parallel? That¹s not how science is done. And let¹s not forget that you solicited comments. Do you think that I am the only one who sees problems with what you presented? You should be happy to see such comments so that you can correct the situation. > I will clarify also in my webpage soon, Good. > givin links to your webpage. Don¹t you have your priorities wrong? Shouldn¹t you focus on what _you_ are presenting? > You will be acussed of dishonesty. Dishonesty? I just presented what I saw. You admit that the original image was not orthorectiÞed, so I was honest in saying that the lines do not appear parallel. Now if I made a mistake, it was an honest one, because I used the image that _you_ provided. Rather than explaining in the text why the image was incorrect, wouldn¹t it have been better to post a correct image? > You can sue me also, if you want. I am wishing it. Sue you for what? Calling me dishonest? Calling me pathetic? Calling me a slow thinker? Calling me stupid? These comments won¹t make you look good. === Subject: Re: Comment welcome -- martian mounds anomaly That is just a by-hand diagram. Your accusation of fraud may be retired or confronted in a court. I am completely serious. === Subject: Differentiate I have the question: Find f¹(x) when f(x) is given by sin^-1(x^2) And the answer (given to me): f¹(x)= 2x / (sqrt(1 - x^4)) ...but I don¹t know how to achieve this. Could someone please show me the steps because I have a lot of similar questions and I need a full example to set me off...? TIA === Subject: Re: Differentiate > I have the question: > Find f¹(x) when f(x) is given by sin^-1(x^2) > And the answer (given to me): > f¹(x)= 2x / (sqrt(1 - x^4)) > ...but I don¹t know how to achieve this. > Could someone please show me the steps because I have a lot of similar > questions and I need a full example to set me off...? > TIA State the Chain Rule. Write sin^(-1)(x^2) as the composite of two functions. State the derivatives of each of those two functions. Apply the Chain Rule. If the result is not what you thought, then come back here, with the answers to the four items above, and we will provide further help. === Subject: Re: Differentiate > If the result is not what you thought, then come back here, with the > answers to the four items above, and we will provide further help. How do you differentiate sin^(-1)(x)? I don¹t know the rule for these kind of functions. === Subject: Re: Differentiate > If the result is not what you thought, then come back here, with the > answers to the four items above, and we will provide further help. > How do you differentiate sin^(-1)(x)? I don¹t know the rule for these kind > of functions. If f(x) = arcsin(x) then sin(f(x)) = x and cos(f(x))*f¹(x) = 1. FDrom this you should be able to deduce the form for the derivative of the arcsine. === Subject: Re: Differentiate > If the result is not what you thought, then come back here, with the > answers to the four items above, and we will provide further help. > How do you differentiate sin^(-1)(x)? I don¹t know the rule for these kind > of functions. An easy way to remember how to Þnd derivatives of inverse functions: f(x) = sin^(-1)(x) <=> sin(f(x)) = x Differentiate both sides: cos(f(x)) * f Œ(x) = 1 from which you can calculate f Œ(x). Of course you¹ll have to Þnd a way to express the cos(f(x)) as a function of sin(f(x)) [which happens to be x], but that shouldn¹t be too difÞcult ;-) Dirk Vdm === Subject: Re: Differentiate If the result is not what you thought, then come back here, with the > answers to the four items above, and we will provide further help. > How do you differentiate sin^(-1)(x)? I don¹t know the rule for these kind > of functions. > An easy way to remember how to Þnd derivatives of inverse > functions: > f(x) = sin^(-1)(x) > <= sin(f(x)) = x > Differentiate both sides: > cos(f(x)) * f Œ(x) = 1 > from which you can calculate f Œ(x). > Of course you¹ll have to Þnd a way to express the cos(f(x)) > as a function of sin(f(x)) [which happens to be x], but that > shouldn¹t be too difÞcult ;-) Is there a set rule I can apply for differentiating: sin^(-1)(x) or sin^(-2)(x) or sin^(2)(x) or sin^(4)(x) etc.? === Subject: Re: Differentiate If the result is not what you thought, then come back here, with the > answers to the four items above, and we will provide further help. How do you differentiate sin^(-1)(x)? I don¹t know the rule for these > kind > of functions. > An easy way to remember how to Þnd derivatives of inverse > functions: > f(x) = sin^(-1)(x) > <= sin(f(x)) = x > Differentiate both sides: > cos(f(x)) * f Œ(x) = 1 > from which you can calculate f Œ(x). > Of course you¹ll have to Þnd a way to express the cos(f(x)) > as a function of sin(f(x)) [which happens to be x], but that > shouldn¹t be too difÞcult ;-) > Is there a set rule I can apply for differentiating: > sin^(-1)(x) > or > sin^(-2)(x) > or > sin^(2)(x) > or > sin^(4)(x) > etc.? Ha, I see, you are a bit confused about the notation. Sometimes sin^(-1)(x) means arcsin(x) and sometimes sin^(-1)(x) means 1 / sin(x). I know, it¹s annoying, but it happens very often. obvious (for me) that the former was meant, so I gave the way to always Þnd the derivative of arcsin(x) when you have forgotten. The examples you give here, are examples of powers of sin(x): sin^(-1)(x) = 1 / sin(x) sin^(-2)(x) = 1 / [sin(x)]^2 sin^2(x) = [sin(x)]^2 etc... in general: sin^n(x) = [sin(x)]^n These are a bit easier to Þnd with the chain rule: D[ sin^n(x) ] = n * sin^(n-1)(x) * cos(x) So the -1-case would give D[ 1/sin(x) ] = -cos(x) / sin^2(x) But as I said, in this case they were clearly using the (-1) as the inverse-sign, like in... arcsin(x) = sin^(-1)(x) arctan(x) = tan^(-1)(x) log(x) = exp^(-1)(x) etc... Careful out there! hth. Dirk Vdm === Subject: Re: Differentiate > Ha, I see, you are a bit confused about the notation. > Sometimes sin^(-1)(x) means > arcsin(x) > and sometimes sin^(-1)(x) means > 1 / sin(x). > I know, it¹s annoying, but it happens very often. Is there actually a textbook somewhere that uses sin^(-1)(x) = 1/sin(x)? It is common, of course, to use sin^(n)(x) to mean (sin(x))^n, but only in the case where n is not equal to -1. I have never seen sin^(-1)(x) used with that meaning. -- Dave Seaman Judge Yohn¹s mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Differentiate > Ha, I see, you are a bit confused about the notation. > Sometimes sin^(-1)(x) means > arcsin(x) > and sometimes sin^(-1)(x) means > 1 / sin(x). > I know, it¹s annoying, but it happens very often. > Is there actually a textbook somewhere that uses sin^(-1)(x) = 1/sin(x)? > It is common, of course, to use sin^(n)(x) to mean (sin(x))^n, but only > in the case where n is not equal to -1. I have never seen sin^(-1)(x) > used with that meaning. I didn¹t mean to say that the notation sin^(-1)(x) for 1/sin(x) is common, but the confusion about it ;-) I do recall having seen it a few times (at most) in text books but many times on Usenet and websites. Dirk Vdm === Subject: Re: Differentiate <45twb.41773$0C3.1088646@phobos.telenet-ops.be> Sometimes sin^(-1)(x) means > arcsin(x) > and sometimes sin^(-1)(x) means > 1 / sin(x). > I know, it¹s annoying, but it happens very often. > Is there actually a textbook somewhere that uses sin^(-1)(x) = 1/sin(x)? > It is common, of course, to use sin^(n)(x) to mean (sin(x))^n, but only > in the case where n is not equal to -1. I have never seen sin^(-1)(x) > used with that meaning. > I didn¹t mean to say that the notation sin^(-1)(x) for 1/sin(x) is > common, but the confusion about it ;-) > I do recall having seen it a few times (at most) in text books > but many times on Usenet and websites. Most calculators use sin^(-1) to mean arcsin. There is a different notation for the reciprocal: cosec(x) = 1/(sin(x)) (The notation csc(x) is sometimes also seen.) -- P.A.C. Smith The vast majority of Iraqis want to live in a peaceful, free world. And we will Þnd these people and we will bring them to justice. === Subject: Re: Differentiate > How do you differentiate sin^(-1)(x)? I don¹t know the rule for these kind > of functions. Use: sin(arcsin(x)) = id and the chain rule (for real functions): (f(g(x)))¹ = f Œ(g(x))*g¹(x). === Subject: Re: Differentiate > How do you differentiate sin^(-1)(x)? I don¹t know the rule for these kind > of functions. > Use: sin(arcsin(x)) = id I¹ve never seen this before. > and the chain rule (for real functions): (f(g(x)))¹ > = f Œ(g(x))*g¹(x). Yep, but what two functions do I break it down to? What is f(x) and what is g(x) if I¹m using the chain rule? === Subject: Re: Differentiate > How do you differentiate sin^(-1)(x)? I don¹t know the rule for these > kind > of functions. Use: sin(arcsin(x)) = id > I¹ve never seen this before. > and the chain rule (for real functions): (f(g(x)))¹ > = f Œ(g(x))*g¹(x). > Yep, but what two functions do I break it down to? What is f(x) and what is > g(x) if I¹m using the chain rule? If h(x) = f(g(x)), then h¹(x) = f¹(g(x)) * g¹(x), so what¹s your problem? === Subject: Mathematical Argument For the Existence of God This is an interpretation of Christopher Langan¹s CTMU, www.ctmu.org , and Saint Anslem¹s ontological argument. 1.] If it is possible for a mind to perfectly understand[model] every aspect and detail of reality, then the mind that perfectly models reality is a super-intelligence, for all intents and purposes, the super-intelligence is God. 2.]If the perfect correspondence can be approached via a convergent analytic-synthetic propositional limit, then the limit exists, even though a sentient mind within reality can only approach the limit. 3.] If the limit exists, the exact mental correspondence exists in the mind of a super-intelligence. 4.] That is to say, if the limit exists then a description exists. 5.] If the description exists then the describer exists, since the description is isomorphic. 6.]The describer is a super-intelligence. 7.] By deÞnition, the super-intelligence is God. The burden of proof becomes the burden of proving the convergence, to an exact correspondence, between the mental construct[inÞnite number of axioms] and reality At the limit [MIND]<--->[REALITY] M = R [axiomatic method]--->[exact correspondence]<---[scientiÞc method] === Subject: Re: Mathematical Argument For the Existence of God > This is an interpretation of Christopher Langan¹s CTMU, www.ctmu.org > , and Saint Anslem¹s ontological argument. Anselm¹s argument is an excercise in begging the question. It is -worthless-. Nothing exists by deÞnition. Existence has to be proven or observed. Bob Kolker === Subject: Re: Mathematical Argument For the Existence of God > This is an interpretation of Christopher Langan¹s CTMU, www.ctmu.org > , and Saint Anslem¹s ontological argument. > Anselm¹s argument is an excercise in begging the question. > It is -worthless-. Nothing exists by deÞnition. Existence has to be > proven or observed. > Bob Kolker Bull Bob. A beginner course in logic of religion pretty much states what this dude says using his 8 cylinder lingo. In a nutshell: I have an idea of a being that is all loving, all powerful, all knowing Therefore this being must exist Case closed class dismissed NEXT..... === Subject: Re: Mathematical Argument For the Existence of God >7.] By deÞnition, the super-intelligence is God. That isn¹t any deÞnition that I have heard before. In fact, is it even appropriate to use the word deÞne and God in the same sentence? === Subject: Re: Mathematical Argument For the Existence of God >>7.] By deÞnition, the super-intelligence is God. > That isn¹t any deÞnition that I have heard before. In fact, is it even > appropriate to use the word deÞne and God in the same sentence? The problem is that God means different things to different people - hence any claims of proof are irrelevant. === Subject: Re: Logic missing from Proofs from THE BOOK >Goedel¹s Þrst incompleteness theorem has a large part to it that is >totally ugly algebra/number theory/encoding, but the gist of it is simple. >>The algebra/number theory/encoding is totally unnecessary. >>All that is needed is to have every formula/proposition/ >>theorem/proof have a positive integer assigned to it; one >>way to do this is to use a direct ASCII (or other such) >>representation. >Hold on a moment...Þrst of all, assigning a positive integer to syntactic >objects is what people often mean by encoding, so this doesn¹t show that >encoding is totally unnecessary. Encoding of some sort is necessary; it need not have any connection to number theory. Mostowski¹s very readable book on Goedel¹s theorem and generalizations does not use any kind of encoding similar to Goedel¹s, and does not use any number theory. >But more importantly, one needs more than just the assignment; one needs >to represent the relevant number-theoretic functions (that correspond to >rules of inference in your sequent calculus, for example) in the language >of the system whose incompleteness you¹re proving. One does need the Peano Postulates, and a few other things of that nature. It is necessary to be able to identify the steps in a proof, for example. But the number of things which have to be identiÞed is small. The trick used by Mostowski is a particular method of expressing ordered pairs of integers. However, no use is made of the Chinese Remainder Theorem, or even of factorization. -- 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: Logic missing from Proofs from THE BOOK >> at 04:13 PM, Mitch Harris said: >topology is really just a fancy word for geometry, right? >> Wrong. There are many issues in Geometry beyond the scope of Topology. >> Before the last few centuries, hardly any work in Geometry had >> anything to do with Topology. >I meant by my short comment about topology and geometry that, by >whatever route one is lead to topology (category theory, point-set >analysis, differential manifolds, etc) that one is doing generalized >geometry. Point-set topology is in no way generalized geometry, nor even related to it. -- 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: Logic missing from Proofs from THE BOOK By the way, I wouldn¹t describe this part of the proof as ugly. It¹s > quite beautiful that arbitrarily complex recursive functions can be > represented in such weak systems. >> sure, the -fact- is wonderful but the -justiÞcation- using CRT or >> whatever else is kind of messy algebraically. I guess one of my metrics >> is shortness of proof. >roþ - um, an odd metric to choose, in light of the discussion of >godel numbering..... lmao.... I¹m not saying it is the only metric. Otherwise, I¹d feel bound to say that modus ponens is the onliest and beautifulest (!) proof of all (and then support my other contentious statement that math -is- logic). >otoh, per that metric, does that big book o¹ proofs have the world¹s >simplest proof of the isosceles triangle theorem (look at it from the >front, from the back, and cite cpctc)? do you mean I.5 Pons Asinorum? and does cpctc mean SAS Side-Angle-Side). If so, THE BOOK (the one you get to see when you die/reach Nirvana/etc.) probably has it, but the book written by mere mortals, does not mention it (at least not by name; as witnessed by my previous gaffe, I could easily have missed it; it could easily be referenced (too obliquely for me) in terms of superposition). The book is just a collection that pleases the authors, any inclusion or exclusion is academically arguable. Most of the proofs are 20th century with a few 19th c. ones (many thms were proved previously, it¹s just the beautiful proofs were discovered later. The Þrst proof given in the book is Euclid¹s for the inÞnitude of primes, but that seems to be the only ancient proof. Mitch === Subject: Re: Logic missing from Proofs from THE BOOK > surely sci.logic would be a good place for the OP to get expert info > on this kinda thing..... lol - i hereby rescind that suggestion... I just checked sci.logic for the Þrst time in like 5 years - lol - it¹s gone to hell with cranks even worse than sci.math has..... (I guess corporatization was NOT is fact the biggest threat to the internet...) sorry for the bad idea, cdj === Subject: Re: Logic missing from Proofs from THE BOOK >>surely sci.logic would be a good place for the OP to get expert info >>on this kinda thing..... > lol - i hereby rescind that suggestion... I just checked sci.logic for > the Þrst time in like 5 years - lol - it¹s gone to hell with cranks > even worse than sci.math has..... (I guess corporatization was NOT > is fact the biggest threat to the internet...) > sorry for the bad idea, No problem. I took the advice but no one seems to have bitten there. OK here¹s some relevant content: Proofs without words are beautiful. Purely algebraic proofs are not. Discuss. Mitch, I know, nobody cares. === Subject: Re: Logic missing from Proofs from THE BOOK <3fba8c5c$8$fuzhry+tra$mr2ice@news.patriot.net> <3FBC7EA7.5090709@tcs.inf.tu-dresden.de> X-Cise: tanbanso@iinet.net.au X-CompuServe-Customer: Yes X-Coriate: admin@interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Pose: george_cox@btinternet.com X-Punge: Micro$oft X-Sanguinate: themvsguy@email.com X-Terminate: SPA(GIS) X-Tinguish: Mark GrifÞth X-Treme: C&C,DWS >I meant by my short comment about topology and geometry that, by >whatever route one is lead to topology (category theory, point-set >analysis, differential manifolds, etc) that one is doing generalized >geometry. I understood what you meant: it was wrong. Topology is just one aspect of Geometry. Topology is totally irrelevant to, e.g., projective spaces over Þnite Þelds. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail will be subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamtrap@library.lspace.org === Subject: Poll: Is Linear Algebra Restricted to Finite-Dimensional Vector Spaces? I know this is not a very deep question, but as I¹ve come accross rigorous texts like Halmos¹s and Lax¹s on linear algebra and noticed their emphasis on Þnite dimensional vector spaces (certainly to be expected in Halmos¹s case), it makes me wonder if linear algebra is regarded mainly as the study of these spaces. Also, are there linear algebra books (as opposed to functional analysis texts) that treat inÞnite dimensional vector spaces? Martin Garzon === Subject: Re: Poll: Is Linear Algebra Restricted to Finite-Dimensional Vector Spaces? >I know this is not a very deep question, but as I¹ve come accross >rigorous texts like Halmos¹s and Lax¹s on linear algebra and noticed >their emphasis on Þnite dimensional vector spaces (certainly to be >expected in Halmos¹s case), it makes me wonder if linear algebra is >regarded mainly as the study of these spaces. Mainly, yes. >Also, are there linear algebra books (as opposed to functional >analysis texts) that treat inÞnite dimensional vector spaces? There are many that mention them, though most, especially the elementary ones, don¹t do much with them. You might look at Kaplansky, Linear Algebra and Geometry: A Second Course, Allyn & Bacon 1969, which does treat inÞnite dimensional spaces extensively. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: teminology > i Œve found the following word while studiing geometry: > I have found: Let N be either a Þnite dimensional vector space V or else a > half space in V. > What does half space mean? Given a vector space V, in order for the term half space in V to make sense, the Þeld of scalars must be the real numbers, R. In that case, take a nonzero linear map: L: V ---> R Given a scalar c, the sets {v in V | L(v) > c} and {v in V | L(v) < c} are called open half-spaces in V. Similarly, the sets {v in V | L(v) >= c} and {v in V | L(v) <= c} are called closed half-spaces in V. A half-space in V is such a subset. I do think the closed half-spaces chose that (closed half-space) as the deÞnition, accordingly, but seemed to restrict the discussion to those half-spaces that are given by some coordinate being non-negative or non-positive. That is unduly restrictive, and doesn¹t allow one to prove at least one important result: every convex subset of V is the intersection of a family of half-spaces of V. The situation in 2 or 3 dimensions is easily visualized. In the plane, a half-space is one or the other of the halves into which a line divides the plane. The half-space is closed or open, respectively, according as whether the line does or does not belong to the half-space. In three dimensions, one is given a plane and then chooses one or the other of the halves of 3-space into which the plane divides it. As before, the half-space is closed if it contains that dividing plane, and open if it doesn¹t. That¹s all I know. Dale. === Subject: Re: teminology >> i Œve found the following word while studiing geometry: >> I have found: Let N be either a Þnite dimensional vector space V or else a >> half space in V. >> What does half space mean? >Given a vector space V, in order for the term half space in V to make >sense, the Þeld of scalars must be the real numbers, R or any other ordered Þeld. >In that case, >take a nonzero linear map: > L: V ---> R >Given a scalar c, the sets > {v in V | L(v) > c} >and > {v in V | L(v) < c} >are called open half-spaces in V. >Similarly, the sets > {v in V | L(v) >= c} >and > {v in V | L(v) <= c} >are called closed half-spaces in V. >A half-space in V is such a subset. I do think the closed half-spaces >chose that (closed half-space) as the deÞnition, accordingly, but >seemed to restrict the discussion to those half-spaces that are given >by some coordinate being non-negative or non-positive. That is unduly >restrictive, and doesn¹t allow one to prove at least one important >result: every convex subset of V is the intersection of a family of >half-spaces of V. >The situation in 2 or 3 dimensions is easily visualized. In the plane, >a half-space is one or the other of the halves into which a line divides >the plane. The half-space is closed or open, respectively, according as >whether the line does or does not belong to the half-space. >In three dimensions, one is given a plane and then chooses one or the >other of the halves of 3-space into which the plane divides it. As >before, the half-space is closed if it contains that dividing plane, >and open if it doesn¹t. >That¹s all I know. >Dale. === Subject: Re: teminology >i Œve found the following word while studiing geometry: >I have found: Let N be either a Þnite dimensional vector space V or else a >half space in V. >What does half space mean? >>Given a vector space V, in order for the term half space in V to make >>sense, the Þeld of scalars must be the real numbers, R > or any other ordered Þeld. oops, yes of course. >>Dale. Dale. === Subject: Virginia Tech Regional Math Competition Distribution: world The results of the 25th Virginia Tech Regional Math Contest are now posted. For more information, go to http://www.math.vt.edu/events/competitions/Vtregional/ info.html Peter A. Linnell http://www.math.vt.edu/people/linnell/ === Subject: How to compute this one (simultanous density function) Suppose you¹ve got a stick of length 1. Now, following happens. 1) The stick is broken at X (where X is the distance between the point of impact and the left end of the stick). 2) The left part of the stick gets hit again and brakes at Y (where Y is the distance between the point of impact and the left end of the stick). Now, provided that the both hits are uniformly distributed, what is the conditional distribution function for Y given that X = x? That¹s the problem itself. My attempt to solve it (feel free to tell me if it¹s wrong direction) is: P( Y=y | X=x ) = f(x,y) / f_Y(y). The problem is that i can¹t Þnd f(x,y). Any tips? -- Kindly Konrad --------------------------------------------------- May all spammers die an agonizing death; have no burial places; their souls be chased by demons in Gehenna from one room to another for all eternity and more. Sleep - thing used by ineffective people as a substitute for coffee Ambition - a poor excuse for not having enough sence to be lazy --------------------------------------------------- === Subject: Polygonal Path I¹m looking for an ofÞcial deÞnition of a polygonal path (in an open set G in the complex plane.) I¹m guessing that: if there exists a polygonal path between points a and b in G, then there exists a Þnite set of straight lines such that a connects to b. Of course, this is inadequate, because how do a and b connect? I¹m assuming that the set G needs to be connected as well. (In the traditional sense.) GREG === Subject: Re: Polygonal Path >I¹m looking for an ofÞcial deÞnition of a polygonal path (in an open set G >in the complex plane.) >I¹m guessing that: if there exists a polygonal path between points a and b >in G, then there exists a Þnite set of straight lines such that a >connects to b. A polygonal path from a to b is a Þnite sequence of straight line segments [a_j, a_{j+1}] for j = 0...n-1 such that a_0 = a and a_n = b. This path is in the open set G if each segment is a subset of G. >I¹m assuming that the set G needs to be connected as well. (In the >traditional sense.) No. Of course if it¹s not connected, there will be some pairs a,b in G such that there is no polygonal path from a to b in G. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Polygonal Path >I¹m looking for an ofÞcial deÞnition of a polygonal path (in an open set G >in the complex plane.) >I¹m guessing that: if there exists a polygonal path between points a and b >in G, then there exists a Þnite set of straight lines such that a >connects to b. > A polygonal path from a to b is a Þnite sequence of straight line > segments [a_j, a_{j+1}] for j = 0...n-1 such that a_0 = a and a_n = b. > This path is in the open set G if each segment is a subset of G. >I¹m assuming that the set G needs to be connected as well. (In the >traditional sense.) > No. Of course if it¹s not connected, there will be some pairs a,b > in G such that there is no polygonal path from a to b in G. Ok. But if a set is connected, then any two points should have a polygonal path between them, right? GREG > Robert Israel israel@math.ubc.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia > Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Polygonal Path > Ok. But if a set is connected, then any two points should have a polygonal > path between them, right? No. But if the set is open and connected in R^2 as you said earlier, then yes. === Subject: Mensanator horrendous image processing techniques Can someone provide with a mathematical algorithm which explains the horrendous pixel-crime perpetred by the infamous Mr. Mensanator? http://www.thequantummachine.com/images.php I thought that my TFT screen had been degraded by some air pollutant, but then I realized that it was only his image which had some fungus on it... :-) === Subject: Quadratic Formula/ae Hi all, I have seen the quadratic formula derived in my textbook, but the working is no short of cryptic(the author seems to miss step or do some weird things without apparent explanation). Can someone please explain to me how it is derived? === Subject: Re: Quadratic Formula/ae > Hi all, > I have seen the quadratic formula derived in my textbook, but the > working is no short of cryptic(the author seems to miss step or do > some weird things without apparent explanation). Can someone please > explain to me how it is derived? Recall: (y + d)^2 = y^2 + 2yd + d^2 Hence, y^2 + ky = (y + k/2)^2 - k^2/4 (This is known as Œcompleting the square¹.) Starting from ax^2 + bx + c = 0: divide by a: x^2 + bx/a + c/a = 0 Complete the square: (x + b/(2a))^2 - b^2/(4a^2) + c/a = 0 Move terms to right-hand side: (x + b/(2a))^2 = b^2/(4a^2) - c/a Put right-hand side over common denominator (4a^2): (x + b/(2a))^2 = (b^2 - 4ac)/(4a^2) Take square root: (x + b/(2a)) = +/- sqrt(b^2 - 4ac)/(2a) subtract b/2a: x = - b/(2a) +/- sqrt(b^2 - 4ac)/(2a) -- P.A.C. Smith The vast majority of Iraqis want to live in a peaceful, free world. And we will Þnd these people and we will bring them to justice. === Subject: Re: Quadratic Formula/ae Adjunct Assistant Professor at the University of Montana. >I have seen the quadratic formula derived in my textbook, but the >working is no short of cryptic(the author seems to miss step or do >some weird things without apparent explanation). Can someone please >explain to me how it is derived? Suppose you want to Þnd the values of x for which ax^2 + bx + c = 0, where a, b, c are constants, and a>0. Recall that (r+s)^2 = r^2 + 2rs + s^2. We want to think of ax^2+bx+c as being of the form (rx+s)^2 - t =0, with r different from 0, because then we can solve it: (rx+s)^2 = t rx+s = sqrt(t) or rx+s = -sqrt(t) so either x = (1/r)*(sqrt(t)-s) or else x = (1/r)*(-sqrt(t)-s). To do that, we think of ax^2+bx as the Þrst part of (rx+s)^2. We set r = sqrt(a), so that r^2*x^2 will be equal to ax^2. Then we choose s so that 2rs*x = bx; that is, 2rs = b; so 2*sqrt(a)*s = b; so s = b/2sqrt(a). And then that means that we are missing s^2. So we add and subtract it: 0 = ax^2 + bx + c = ax^2 + bx + [b^2/4a] - [b^2/4a] + c = [sqrt(a)x]^2 + 2*[sqrt(a)][b/2*sqrt(a)] + [b/2*sqrt(a)]^2 + ( c - [b/2sqrt(a)]^2) = (sqrt(a)x + b/2*sqrt(a))^2 + c-(b^2/4a) Now proceed as outlined before: solving this for x gives: (sqrt(a)*x + [b/2*sqrt(a)])^2 = (b^2/4a) - c = (b^2-4ac)/4a so sqrt(a)*x + [b/2*sqrt(a)] = sqrt(b^2-4ac)/2*sqrt(a) or sqrt(a)*x + [b/2*sqrt(a)] = -sqrt(b^2-4ac)/2*sqrt(a). Thus sqrt(a)*x = (-b + sqrt(b^2-4ac))/2*sqrt(a) or sqrt(a)*x = (-b - sqrt(b^2-4ac))/2*sqrt(a). Now dividing by sqrt(a) and remembering that sqrt(a)*sqrt(a)=a gives x = (-b+sqrt(b^2-4ac))/2a or x = (-b-sqrt(b^2-4ac))/2a, yielding the quadratic formula: x = (-b +/- sqrt(b^2-4ac))/2a. It¹s not denial. I¹m just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Why is it so hard to solve nonlinear differential equations? Firstly what exactly are NONLINEAR differential equations? Secondly why are they considered so difÞcult to solve? Answers in ENGLISH please. === Subject: Re: Why is it so hard to solve nonlinear differential equations? > Firstly what exactly are NONLINEAR differential equations? > Secondly why are they considered so difÞcult to solve? > Answers in ENGLISH please. Hello Mary, this is a very good question. Lets see if the following paragraph from the wonderful book by Jordan and Smith (Nonlinear Ordinary Differential Equations) help answer the second part of your question. It is not in general possible to obtain analytic solutions to an arbitrary differential equation. This is not simply because ingenuity fails, but because the repertory of standard functions (polynomials, exp, sin and so on) in terms of which solutions may be expressed is too limited to accommodate the variety of differential equations encountered in practice. Because of this, we typically try to investigate the qualitative behavior of NL DEQs. We try to understand important characteristics of the solutions of the equations without actually solving them. Such areas of study include things like phase plane analysis to look at equilibrium, periodicity, unlimited growth, stability and so on. Today, this has become a wonderful and fun topic and is called dynamics (linear and nonlinear and look at topics like bifurcations, perturbations, structural stability and chaos). There are various very good books on the subject from Springer-Verlag. The Þrst part of your question ... what exactly are NL-DEQs? Please see the site: http://csep1.phy.ornl.gov/ode/node2.html Perhaps a cleaner deÞnition can be found in the following presentation (covers many other DEQs too and is a nice review). http://wwwmaths.anu.edu.au/DoM/secondyear/MATH2061/lect-07-4. pdf HTH, Flip It gives examples and tells you the difference between linear and nonlinear equations. === Subject: Re: Why is it so hard to solve nonlinear differential equations? >Firstly what exactly are NONLINEAR differential equations? A linear differential equation (with dependent variable y and independent variable x) can be written as a_n(x) d^n y/dx^n + ... + a_1(x) dy/dx + a_0(x) y = F(x) A nonlinear differential equation is a differential equation that is not linear. In particular, any terms involving some more complicated function of y and/or its derivatives makes the equation nonlinear. >Secondly why are they considered so difÞcult to solve? Because they are not easy to solve. There are very simple-looking Þrst-order nonlinear differential equations such as dy/dx = x y^2 + x + 1 for which AFAIK no closed-form solution is known, even involving an integral. By contrast, there is a well-known formula (involving integrals) for solving Þrst-order linear differential equations. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: James Harris - Challenge problem > Somehow I doubt it. Even if your proof were to be found to be entirely > correct, nobody would care. Why? Because the thing you assert is > completely irrelevent. Why don¹t you know this? Because you don¹t know any > mathematics. You don¹t know any mathematics. Seriously, think about this. > You don¹t know any of the mathematics you are trying to do. Generally, > proofs, at Þrst glance, either seem right or not. You lack this intuition. Mathies are creatures of habit, who--because they combine hermetically sealed minds with Bush-esque egos--are frightfully resentful of anything that is not hand-me-down, cookie-cutter mathematics. > You don¹t know any of the mathematics you are trying nto do. You could > study, but you seem to think that that is a rediculous idea. So, you will > continue to be an internet crank. By the time we are in a position to be > anywhere in mathematics you will still not know any mathematics, and you¹ll > spend your time posting to newsgroups as a crank. Have a nice life. Go back to your hand-me-down, cookie-cutter mathematics and leave JSH alone. === Subject: determining basis & linear independence Hallo, I have two tasks here. For the Þrst task i¹m not sure, whether my solution is right, for the other one I don¹t know what to do. :-( Task 1: ^^^^^^^^ Prove or disprove: If the vectors v_1, v_2, v_3 in the vectorspace V are linearly independent. Then the vectors v_1+v_2+v_3, v_1-v_2+v_3, v_1+v_2-v_3 are linearly independent, too. My approach: ^^^^^^^^^^^^^^^^ a(v_1+v_2+v_3)+b(v_1-v_2+v_3)+c(v_1+v_2-v_3)=0 <=> a*v_1+a*v_2+a*v_3+b*v_1-b*v_2+b*v_3+c*v_1+c*v_2-c*v_3 = 0 <=> v_1(a+b+c)+v_2(a-b+c)+v_3(a+b-c)=0 Since v_1, v_2 and v_3 are linearly independent, the vectors v_1+v_2+v_3, v_1-v_2+v_3 and v_1+v_2-v_3 are linearly independent, too. Is that correct? Task 2: ^^^^^^^^^ Determine a basis for L(v_1, v_2, v_3) intersecting L(w_1, w_2, w_3). v_1 = (1,1,1,1) from|R^4 ; w_1 = (3,4,5,6) from |R^4 v_2 = (1, -2, -5, -8) from |R^4 ; w_2 = (1, 0, -1, -2) from |R^4 v_3 = (3, 4, 5, 0) from |R^4 ; w_3 = (0, 0, 1, 2) from |R^4 My approach: ^^^^^^^^^^^^^^^^ Unfortunately, I didn¹t Þnd an approach so far. But I think the following two facts are true: v_1, v_2, v_3 and w_3 are linearly independent. You can express w_1 as a linear combination of v_1, v_2, v_3 and w_3. Would somebody be so kind and help me? Please! Karl. === Subject: Re: determining basis & linear independence > Hallo, > I have two tasks here. For the Þrst task i¹m not sure, whether my solution > is right, for the other one I don¹t know what to do. :-( > Task 1: > ^^^^^^^^ > Prove or disprove: If the vectors v_1, v_2, v_3 in the vectorspace V are > linearly independent. Then the vectors v_1+v_2+v_3, v_1-v_2+v_3, > v_1+v_2-v_3 are linearly independent, too. > My approach: > ^^^^^^^^^^^^^^^^ > a(v_1+v_2+v_3)+b(v_1-v_2+v_3)+c(v_1+v_2-v_3)=0 > <=> a*v_1+a*v_2+a*v_3+b*v_1-b*v_2+b*v_3+c*v_1+c*v_2-c*v_3 = 0 > <=> v_1(a+b+c)+v_2(a-b+c)+v_3(a+b-c)=0 > Since v_1, v_2 and v_3 are linearly independent, the vectors v_1+v_2+v_3, > v_1-v_2+v_3 and v_1+v_2-v_3 are linearly independent, too. Is that correct? Only if you now show that a+b+c = a-b+c = a+b-c = 0 requires also that a = b = c = 0. An alternate method: [[ (v_1 + v_2 + v_3) ] [[ 1 1 1 ] [[ v_1 ] [ (v_1 - v_2 + v_3) ] = [ 1 -1 1 ] * [ v_2 ] [ (v_1 + v_2 - v_3) ]] [ 1 1 -1 ]] [ v_3 ]] where the numerical matrix is invertable (determinant = 4 <> 0) > Task 2: > ^^^^^^^^^ > Determine a basis for L(v_1, v_2, v_3) intersecting L(w_1, w_2, w_3). > v_1 = (1,1,1,1) from|R^4 ; w_1 = (3,4,5,6) from |R^4 > v_2 = (1, -2, -5, -8) from |R^4 ; w_2 = (1, 0, -1, -2) from |R^4 > v_3 = (3, 4, 5, 0) from |R^4 ; w_3 = (0, 0, 1, 2) from |R^4 > My approach: > ^^^^^^^^^^^^^^^^ > Unfortunately, I didn¹t Þnd an approach so far. But I think the following > two facts are true: > v_1, v_2, v_3 and w_3 are linearly independent. You can express w_1 > as a linear combination of v_1, v_2, v_3 and w_3. > Would somebody be so kind and help me? Please! We are looking for only those vectors which are expressible as linearly conbinations of v_1, v_2 and v_3} and also as linear combinations of m_1, w_2, w_3}, i.e., are in the space spanned by both sets of vectors. If v_1, v_2, v_3 and w_3 are linearly independent then w_3 cannot be a linear combination of v_1, v_2 and v_3, so consider the space spanned by {w_1, w_2}. Now try to solve x*w_1 + y*w_2 = v_1, or x*w_1 + y*w_2 = v_2, or x*w_1 + y*w_2 = v_3. It will transpire that 2 of these are solvable, and the corresponding v_i also span the intersection space, and the other v_i is not, so the intersection is the space spanned by w_1 and w_2 and also by two of v_1, v_2 and v_3. Hope this helps. === Subject: Re: determining basis & linear independence Adjunct Assistant Professor at the University of Montana. >Hallo, >I have two tasks here. For the Þrst task i¹m not sure, whether my solution >is right, for the other one I don¹t know what to do. :-( >Task 1: >^^^^^^^^ >Prove or disprove: If the vectors v_1, v_2, v_3 in the vectorspace V are >linearly independent. Then the vectors v_1+v_2+v_3, v_1-v_2+v_3, >v_1+v_2-v_3 are linearly independent, too. >My approach: >^^^^^^^^^^^^^^^^ >a(v_1+v_2+v_3)+b(v_1-v_2+v_3)+c(v_1+v_2-v_3)=0 ><=> a*v_1+a*v_2+a*v_3+b*v_1-b*v_2+b*v_3+c*v_1+c*v_2-c*v_3 = 0 ><=> v_1(a+b+c)+v_2(a-b+c)+v_3(a+b-c)=0 You are not done. What you want to prove is that if a(v_1+v_2+v_3) + b(v_1-v_2+v_3) + c(v_1+v+2-v_3) = 0 then a=b=c=0 (i.e., linearly independent). >Since v_1, v_2 and v_3 are linearly independent, the vectors v_1+v_2+v_3, >v_1-v_2+v_3 and v_1+v_2-v_3 are linearly independent, too. Is that correct? Since v_1,v_2,v_3 You are not done. Now you know that, since v_1, v_2, and v_3 are linearly independent, you must have a+b+c = 0 a-b+c = 0 a+b-c = 0 from which you can (but have not yet) deduced what you want: that a=b=c=0 (->provided<- the characteristic of the Þeld is not 2; otherwise, the collection is not linearly independent). >Task 2: >^^^^^^^^^ >Determine a basis for L(v_1, v_2, v_3) intersecting L(w_1, w_2, w_3). >v_1 = (1,1,1,1) from|R^4 ; w_1 = (3,4,5,6) from |R^4 >v_2 = (1, -2, -5, -8) from |R^4 ; w_2 = (1, 0, -1, -2) from |R^4 >v_3 = (3, 4, 5, 0) from |R^4 ; w_3 = (0, 0, 1, 2) from |R^4 >My approach: >^^^^^^^^^^^^^^^^ >Unfortunately, I didn¹t Þnd an approach so far. But I think the following >two facts are true: >v_1, v_2, v_3 and w_3 are linearly independent. You can express w_1 >as a linear combination of v_1, v_2, v_3 and w_3. >Would somebody be so kind and help me? Please! First, Þgure out the dimension of the intersection (hint: unless the two spaces are equal, the dimension must be either 0, 1, or 2). Find a nonzero vector in the intersection. If the dimension is 1, you are done; that¹s your basis. If the dimension is 2, then there must be some other vector, not a multiple of the Þrst, in the intersection. Take the two vectors you have found, and you are done. To Þgure out the dimension of the intersection, Þrst show that the two spaces are not equal: that should be easy. Shbow, for example, that you cannot get w2+w3 =(1,0,0,0) as a linear combination of v1, v2, and v3. If you could Þnd a, b, c such that av_1 + bv_2 + cv_3 = (1,0,0,0), then a+b+3c = 1 a-2b+4c = 0 a-5b+5c = 0 a-8b = 0. So 8b = a = 2b-4c, so 6b=-4c or 3b=-2c; But also 8b=5b-5c, so 3b=-5c, hence c=0. Therefore b=0; therefore a=0, and that is impossible. Is the intersection of dimension at least 1? If so, will every vector in the intersection be a scalar mulitple of the one you found? My further hint would be to replace w1, w2, and w2 with easier vectors; e.g., replce w2 with w2+w3 = (1,0,0,0); and then replace w_1 with w1-3(w2+w3)-5w3 = (3,4,5,6) + (-3,0,0,0) + (0,0,-5,-10) = (0,4,0,-4) and then divide it by 4 to get (0,1,0,-1). So what you have is the vector space spanned by (1,0,0,0), (0,1,0,-1), and (0,0,1,2). This is much easier to handle, since we can describe all vectors there: they are all of the form (a,b,c,2c-b). Do something similar with v1, v2, v3 to get an easier description before proceeding. It¹s not denial. I¹m just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: f(x)=? if lim delta->0 [f(x+delta)-f(x)]/(delta)^2 exists f(x)=? if lim delta->0 [f(x+delta)-f(x)]/(delta)^2 exists. wondering if there are functions that the following limit does exist: lim delta->0 [f(x+delta)-f(x)] / [(delta)^alpha] exists. where alpha>1 Any references would be nice too. TIA === Subject: Re: f(x)=? if lim delta->0 [f(x+delta)-f(x)]/(delta)^2 exists > f(x)=? > if > lim delta->0 [f(x+delta)-f(x)]/(delta)^2 exists. > wondering if there are functions that the following limit does exist: > lim delta->0 [f(x+delta)-f(x)] / [(delta)^alpha] exists. where alpha>1 > Any references would be nice too. That limit exists at every point x (in an interval)? f(x) = Constant. That¹s it. === Subject: math Insert the missing letter: B/F G/M N/V P/? What is the missing letter i don¹t know? -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: math 4 letters 6 letters 8 letters what¹s 10 letters from P? > Insert the missing letter: > B/F G/M N/V P/? > What is the missing letter i don¹t know? -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: math << Insert the missing letter: B/F G/M N/V P/? What is the missing letter i don¹t know? Count the distances in the English alfabet between the given letters. What hirt my eyes, that whyt the Þrst letter in the last group is P instead of W -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Beware of the Al-gebra movement (JOKE) At New York¹s Kennedy airport today, an individual later discovered to be a public school teacher was arrested trying to board a þight while in possession of a ruler, a protractor, a set square, a slide rule, and a calculator. At a morning press conference, Attorney general John Ashcroft said he believes the man is a member of the notorious Al-gebra movement. He is being charged by the FBI with carrying weapons of math instruction. Al-gebra is a fearsome cult, Ashcroft said. They desire average solutions by means and extremes, and sometimes go off on tangents in a search of absolute value. They use secret code names like Œx¹ and Œy¹ and refer to themselves as Œunknowns,¹ but we have determined they belong to a common denominator of the axis of medieval with coordinates in every country. As the Greek philanderer Isosceles used to say, ŒThere are 3 sides to every triangle,¹ Ashcroft declared. When asked to comment on the arrest, President Bush said, If God had wanted us to have better weapons of math instruction, He would have given us more Þngers and toes. I am gratiÞed that our government has given us a sine that it is intent on protracting us from these math-dogs who are willing to disintegrate us with calculus disregard. Murky statisticians love to inþict plane on every sphere of inþuence, the President said, adding: Under the circumferences, we must differentiate their root, make our point, and draw the line. President Bush warned, These weapons of math instruction have the potential to decimal everything in their math on a scalene never before seen unless we become exponents of a Higher Power and begin to factor-in random facts of vertex. Attorney General Ashcroft said, As our Great Leader would say, ŒRead my ellipse!¹ Here is one principle he is uncertain of: though they continue to multiply, their days are numbered as the hypotenuse tightens around their necks. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Can¹t Þgure this out... Spent a few hours on this one... prove (sin x)/(1+cos x)+(1+cos x)/(sin x)=2csc x i can¹t Þgure it out -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Can¹t Þgure this out... > Spent a few hours on this one... > prove (sin x)/(1+cos x)+(1+cos x)/(sin x)=2csc x > i can¹t Þgure it out There are surely other ways equally short or shorter, but here¹s the Þrst thing that came to mind and only took a minute: Clear fractions, ie multiply both sides by the LCD sin(x)*(1+cos(x)) and reduce, and you get: sin^2(x) + 1 + 2cos(x) + cos^2(x) = 2 + 2cos^2(x) 1 + 1 + 2cos(x) = 2 + 2 cos^2(x) ...since sin^2(x)+cos^2(x) = 1 2 + 2 cos(x) = 2 + 2 cos(x) At this point it is sometimes challenged (usually by teachers who want you to do it their way and only their way) that it was illegal to multiply both sides by the LCD. It was perfectly legal and the resulting known true statement implies the proposition is true to begin with (why?) If by chance you are still bothered by this at all, change all = to <> and get a contradiction (ie proof by contradiction). -- Darrell -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Can¹t Þgure this out... > Spent a few hours on this one... > prove (sin x)/(1+cos x)+(1+cos x)/(sin x)=2csc x Common denominator: (sin x)/(1+cos x)+(1+cos x)/(sin x) = ((sin x)^2 + (1 + cos x)^2)/ ((sin x)(1+cos x)) = ((sin x)^2 + 1 +2(cos x) +(cos x)^2)/ ((sin x)(1+cos x)) = 2(1+ (cos x)) / ((sin x)(1+cos x)) = 2 / (sin x) = 2 csc x Constraint: cos(x)!=-1, to avoid division by 0. -- Kevin Karplus karplus@soe.ucsc.edu http://www.soe.ucsc.edu/~karplus life member (LAB, Adventure Cycling, American Youth Hostels) Effective Cycling Instructor #218-ck (lapsed) Professor of Computer Engineering, University of California, Santa Cruz Undergraduate and Graduate Director, Bioinformatics AfÞliations for identiÞcation only. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Can¹t Þgure this out... > Spent a few hours on this one... > > prove (sin x)/(1+cos x)+(1+cos x)/(sin x)=2csc x > Common denominator: > (sin x)/(1+cos x)+(1+cos x)/(sin x) > = ((sin x)^2 + (1 + cos x)^2)/ ((sin x)(1+cos x)) > = ((sin x)^2 + 1 +2(cos x) +(cos x)^2)/ ((sin x)(1+cos x)) > = 2(1+ (cos x)) / ((sin x)(1+cos x)) > = 2 / (sin x) > = 2 csc x > Constraint: cos(x)!=-1, to avoid division by 0. What is necessary is that sin(x) != 0, that is, x != k(pi), where k is any integer. That is, all integer multiples of pi are excluded as x values. This subsumes the constraint cos(x) != -1, which happens for x being an *odd* integer multiple of pi. Notes: 1. != is used for is not equal to. 2. It¹s probably a 50-50 chance that a typical trig or pre-calc book would not bother mentioning any constraint for this trig identity, since both sides of the (proposed) identity have the same constraint. --- Joe --- Joe -- Delete the second o to e-mail me. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Can¹t Þgure this out... > Common denominator: > (sin x)/(1+cos x)+(1+cos x)/(sin x) > = ((sin x)^2 + (1 + cos x)^2)/ ((sin x)(1+cos x)) > = ((sin x)^2 + 1 +2(cos x) +(cos x)^2)/ ((sin x)(1+cos x)) > = 2(1+ (cos x)) / ((sin x)(1+cos x)) > = 2 / (sin x) > = 2 csc x > Constraint: cos(x)!=-1, to avoid division by 0. Would you not also need a constraint: sin x !=0, for the same reason? Rich -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: Re: Can¹t Þgure this out... > Spent a few hours on this one... > prove (sin x)/(1+cos x)+(1+cos x)/(sin x)=2csc x > i can¹t Þgure it out > There are surely other ways equally short or shorter, but here¹s the Þrst > thing that came to mind and only took a minute: Clear fractions, ie > multiply both sides by the LCD sin(x)*(1+cos(x)) and reduce, and you get: > sin^2(x) + 1 + 2cos(x) + cos^2(x) = 2 + 2cos^2(x) Sorry. This and subsequent occurances of the lhs should be 2+2cos(x), not 2+2cos^2(x). To spell it out: 2csc(x) * sin(x) * (1+cos(x)) = 2(1/sin(x)) * sin(x) * (1+cos(x)) = 2(1+cos(x)) = 2+2cos(x) Of course the end result is correct. > 1 + 1 + 2cos(x) = 2 + 2 cos^2(x) > ...since sin^2(x)+cos^2(x) = 1 > 2 + 2 cos(x) = 2 + 2 cos(x) -- Darrell -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: heron¹s formula I am a junior in highschool doing a geometry report on Heron¹s Formula...if anyone is able to help me Þnd resources I thank you very much now! Please and thank you to all those we read this! -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: handhelds in k-12 settings Has anyone personally had experience with handhelds in mathematics classes in k-12. I¹m interested in your experience. -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html === Subject: re-arranging formulaes what is the answer to this question s=1/2gtsquared you have to make t the subject -- submissions: post to k12.ed.math or e-mail to k12math@k12groups.org private e-mail to the k12.ed.math moderator: kem-moderator@k12groups.org newsgroup website: http://www.thinkspot.net/k12math/ newsgroup charter: http://www.thinkspot.net/k12math/charter.html