mm-3879 === Subject: Re: Algebraically closed Yes, but since there are fewer fields of finite degree over E is easier to verify only for them. === Subject: Re: Linear Span > Suppose you are given > S = {V_1,V_2,...V_r} > T = {V_1,V_2,...V_r,V_r+1} > does span(S) = span(T)? This seems like it would obviously be no. The spans can be equal under suitable conditions. > under what condition would span(S) = span(T)? I am not sure where to go with > this one. A good place to start is with the definitions involved. What is the span of a set of vectors? === Subject: approximating intersecting hyperplanes Apologies if this is too bizzare a question! I was wondering if there was a technique that would do the oppposite of what piecewise linearization does. For instance, if I had two intersecting hyperplanes, and I wanted a curve that best approximates it, can that be done? In the simplest case, lets say I had two lines that intersected as shown below, then, can I find a curve that approximates it as shown? Any general concepts/algorithms/resources you can point me to would be much appreciated! / / / /_ // // // (the double lines indicates a curve that approximates a portion of the intersecting lines). === Subject: Re: approximating intersecting hyperplanes Hi. Why don't you give NURBS a shot? (non uniform rational b-splines) You can define an arbitrary number of control points, supply a weight value for each one, and the curve tries to interpolate them in a continuous manner. (It doesn't pass through the control points, unless of course for the first and last one) They can also be generalized to surfaces. === Subject: Re: JSH: Extreme mathematics reminder > Extreme mathematics involves brainstorming for new ideas, and then, > eventually, getting around to checking them thoroughly, tossing out > what doesn't work. > Apparently it means systematically avoiding actually learning any > mathematics. http://en.wikipedia.org/wiki/Brainstorming I think it sad that a technique that is so useful for generating ideas is so unknown to most of you. > You have actually learned some things in the last few years. But it's > very inefficient to > 1) Make a wrong guess. > 2) Derive an elaborate theory from that. > 3) Repeatedly insult anyone who attempts to set you straight. > 4) Go round and round for weeks. > 5) Eventually understand the counterexamples people are showing you. > 6) Go to 1. > It would be much more efficient to simply learn some abstract algebra. Check that link on brainstorming, and learn how it really works. I think for most of you brainstorming is that thing they made you do a couple of times in school, which you really didn't like, and didn't see the point of, and you have convinced yourselves that sitting down and carefully thinking through an idea here or there is the way to go. It's not. I explain what I do, and most of you never seem to get it. That is what's remarkable as it's not really so complex and extraordinary an activity that intelligent individuals can't understand it. Brainstorming involves a LOT of ideas, generated without much critiquing or careful thought. That's part of the process and it's a key part of what I call extreme mathematics. James Harris === Subject: Re: JSH: Extreme mathematics reminder > Extreme mathematics involves brainstorming for new ideas, and then, > eventually, getting around to checking them thoroughly, tossing out > what doesn't work. Apparently it means systematically avoiding actually learning any > mathematics. > http://en.wikipedia.org/wiki/Brainstorming > I think it sad that a technique that is so useful for generating ideas > is so unknown to most of you. I doubt it's unknown to any of us. Except you. > You have actually learned some things in the last few years. But it's > very inefficient to > 1) Make a wrong guess. > 2) Derive an elaborate theory from that. > 3) Repeatedly insult anyone who attempts to set you straight. > 4) Go round and round for weeks. > 5) Eventually understand the counterexamples people are showing you. > 6) Go to 1. > It would be much more efficient to simply learn some abstract algebra. > Check that link on brainstorming, and learn how it really works. It works by getting a bunch of people in a room to come up with as many ideas as possible in a short period of time. Afterward, you take a little more time to prune out the truly crackpot notions. Then you pursue what's left. It doesn't work by taking one idea, insisting it's right for six months, finally giving up, then coming up with a new idea which you hold for six months arguing against all critics, then giving up.... The point of brainstorming is that you've got your basic list of ideas in a couple of hours, and within a day or two you've already gone through the evaluation process. I think you're stuck on the part that says criticism is forbidden and unusual ideas are welcome. You missed the part that says focus on quantity. If your brainstorming session has come up with only one idea, and that idea has still not been critically examined from all sides a week later, your brainstorming has failed. You also missed the short period of time concept. Any business that took your approach to brainstorming and evaluation would be dead before getting around to generating idea number 2. Not really what the brainstorming consultants have in mind, is it? - Randy === Subject: Re: JSH: Extreme mathematics reminder >> Extreme mathematics involves brainstorming for new ideas, and then, >> eventually, getting around to checking them thoroughly, tossing out >> what doesn't work. >> Apparently it means systematically avoiding actually learning any >> mathematics. >http://en.wikipedia.org/wiki/Brainstorming >I think it sad that a technique that is so useful for generating ideas >is so unknown to most of you. As far as we can see from the evidence in your case, the ideas that are generated by this technique are all total crap. >> You have actually learned some things in the last few years. But it's >> very inefficient to >> 1) Make a wrong guess. >> 2) Derive an elaborate theory from that. >> 3) Repeatedly insult anyone who attempts to set you straight. >> 4) Go round and round for weeks. >> 5) Eventually understand the counterexamples people are showing you. >> 6) Go to 1. >> It would be much more efficient to simply learn some abstract algebra. >Check that link on brainstorming, and learn how it really works. Uh, you seem to miss a few points here (big suprise). Brainstorming is great. But brainstorming is not the same as having a wacky idea, deciding for no valid reason that it must be right, and proceeding to insult anyone who says it's wrong. >I think for most of you brainstorming is that thing they made you do a >couple of times in school, which you really didn't like, and didn't see >the point of, and you have convinced yourselves that sitting down and >carefully thinking through an idea here or there is the way to go. >It's not. >I explain what I do, and most of you never seem to get it. >That is what's remarkable as it's not really so complex and >extraordinary an activity that intelligent individuals can't understand >it. >Brainstorming involves a LOT of ideas, generated without much >critiquing or careful thought. >That's part of the process and it's a key part of what I call extreme >mathematics. And the other part is having an appropriate _filter_ to decide on the value of those ideas - that's the part that's lacking here. >James Harris ************************ David C. Ullrich === Subject: Re: JSH: Extreme mathematics reminder > Extreme mathematics involves brainstorming for new ideas, and then, > eventually, getting around to checking them thoroughly, tossing out > what doesn't work. > Apparently it means systematically avoiding actually learning any > mathematics. > You have actually learned some things in the last few years. But it's > very inefficient to > 1) Make a wrong guess. > 2) Derive an elaborate theory from that. > 3) Repeatedly insult anyone who attempts to set you straight. > 4) Go round and round for weeks. > 5) Eventually understand the counterexamples people are showing you. > 6) Go to 1. > It would be much more efficient to simply learn some abstract algebra. You forgot: 2.5) Post your elaborate theory in twenty different threads. 3.5) After responding to 3 replies in a thread, add it to your killfile. JSH would gain a lot of credibility if he added: 1.5) CAREFULLY check your work, look for counterexamples. I could also make a comment about half-cocked, but he might interpret it as a medical reference. --- Christopher Heckman === Subject: test test === Subject: calculation... hello.....doctor long time no see... a = 1/100 + 1/101 + 1/102 + ..... + 1/198 + 1/199 b = 1/(100*199) + 1/(101*198) + ..... + 1/(149*150) find a/b ------------------------------------------------ i need your hint, please... thank you very much. === Subject: Re: calculation... > long time no see... Indeed! :-) > a = 1/100 + 1/101 + 1/102 + ..... + 1/198 + 1/199 > b = 1/(100*199) + 1/(101*198) + ..... + 1/(149*150) > find a/b 1/100 + 1/199 = 299/(100*199) 1/101 + 1/198 = 299/(101*198) 1/102 + 1/197 = 299/(102*198) ... Jose Carlos Santos === Subject: Re: A Probability Question I don't know what should I say. The problem was solved yesterday, by using both of total probability and poisson THEOREM. I describe the answer of this problem, and the usage of poisson theorem in this solution. maybe you saw this theorem with different name or ... First we should partition the sample space in another way, So we say that when we insert remaining balls from second basket into first and then pick up a ball from final basket, the ball that we take have two conditions: 1) The ball that we picked up is one of the balls that beloneg to first basket. 2) The ball that we picked up is one of the balls that beloneg to second basket. We define I := Event of the ball belongs to the first basket and II := Event of the ball belongs to the second basket. We also define W := Event of the ball was white. so the answer of question is: P(W|I).P(I) + P(W|II).P(II) in other words, the answer is the probability that the ball was white if we know that the ball belongs to the first basket [ P(W|I) ] * the probability that the ball belongs to the first basket [ P(I) ] + The ptobability that the ball was white if we know that the ball belongs to the second basket [ P(W|II) ] * the probability that the ball belongs to the second basket [ P(I) ]. OK, in this equation we know the P(W|I), P(I) and P(II). P(W|I) := (6/10) - P(I) := (10/18) - P(II) := (8/18) but we don't know the value of P(W|II). now we can use poisson theorem to find this. By Poisson theorem, if we pickup some balls from a basket and without seeing their color take them away, the probability of being white doesn't change. in other words if we have a basket with five white balls and seven balck balls and pick up a random ball, the probability of being white is (5/12). now if we take 4 balls away without seeing their color and select a ball randomly, the probability is (5/12) yet. So the P(W|II) is (5/12) and the final answer is : 0.51851851. This answer is equal to the answer by previous solution. OK, know our exercise is to proof the Poisson theorem, and also we should find that why we cann't use this teorem from first. in other words why we SHOULDN'T SAY that when we insert remaining balls from second basket to first basket, the probabilty is equal to when we dosn't pick any balls from second basket? === Subject: Re: A Probability Question <14659641.1145599811615.JavaMail.jakarta@nitrogen.mathforum.org I don't know what should I say. The problem was solved yesterday, by using both of total probability and poisson THEOREM. I describe the answer of this problem, and the usage of poisson theorem in this solution. maybe you saw this theorem with different name or ... > First we should partition the sample space in another way, So we say that when we insert remaining balls from second basket into first and then pick up a ball from final basket, the ball that we take have two conditions: > 1) The ball that we picked up is one of the balls that beloneg to first basket. > 2) The ball that we picked up is one of the balls that beloneg to second basket. > We define I := Event of the ball belongs to the first basket and II := Event of the ball belongs to the second basket. > We also define W := Event of the ball was white. > so the answer of question is: P(W|I).P(I) + P(W|II).P(II) > in other words, the answer is the probability that the ball was white if we know that the ball belongs to the first basket [ P(W|I) ] * the probability that the ball belongs to the first basket [ P(I) ] + The ptobability that the ball was white if we know that the ball belongs to the second basket [ P(W|II) ] * the probability that the ball belongs to the second basket [ P(I) ]. > OK, in this equation we know the P(W|I), P(I) and P(II). > P(W|I) := (6/10) - P(I) := (10/18) - P(II) := (8/18) > but we don't know the value of P(W|II). now we can use poisson theorem to find this. By Poisson theorem, In a previous post, you said that there was no Poisson Theorem. You said you meant Poisson distribution, or Poisson process, or something (it was not clear what you were saying). > if we pickup some balls from a basket and without seeing their color take them away, the probability of being white doesn't change. This has nothing at all to do with Poisson. It is simply a way of looking at the sample space of the probabilistic experiment. > in other words if we have a basket with five white balls and seven balck balls and pick up a random ball, the probability of being white is (5/12). now if we take 4 balls away without seeing their color and select a ball randomly, the probability is (5/12) yet. > So the P(W|II) is (5/12) and the final answer is : 0.51851851. > This answer is equal to the answer by previous solution. > OK, know our exercise is to proof the Poisson theorem, There is no Poisson Theorem, at least not one that has anything to do with this problem. > and also we should find that why we cann't use this teorem from first. in other words why we SHOULDN'T SAY that when we insert remaining balls from second basket to first basket, the probabilty is equal to when we dosn't pick any balls from second basket? I cannot understand what you are trying to say. You described the problem one way before, and it seemed to be important that you DO pick balls from the second basket, then put the ones remaining into the first basket. If you think otherwise, you need to prove it; just SAYING it is not good enough. RGV === Subject: Re: Cohomology, homology, cup product, spheres, question > My question is to show that any map S^4 ---> S^2 x S^2 must induce > the zero homomorphism on H_4. > Using cup product arguments, I have shown that any map S^4 ----> S^2 > x S^2 induces the zero homomorphism on H^4. I want to somehow now > use the naturality of the Universal coefficients theorem, but I just > can't get it. > Of course all Ext groups vanish in the Universal Coefficients > theorem. The question is : In this case, if I showed that the zero > homomorphism is induced on H^4, does that mean we get a zero > homomorphism on H_4? Why? > James Let f : S^4 ---> S^2 x S^2. Ext(H_3(S^2 x S^2), Z) >--> H^4(S^2 x S^2) -->> Hom(H_4(S^2 x S^2), Z) | | | f% | f* | . o f_*| | | | V V V Ext(H_3(S^4), Z) >---> H^4(S^4) --->> Hom(S^4 , Z) where the vertical homomorphisms are induced from the map f (the first and last are induced from f_* by contravariance of Ext and Hom, wrt the first variable, the middle is f*, the homomorphism on cohomology. As you noticed, the groups on the left are both zero, so this diagram is: H^4(S^2 x S^2) >-->> Hom(H_4(S^2 x S^2), Z) | | f* | . o f_*| | | V V H^4(S^4) >--->> Hom(S^4 , Z) Since the homomorphism f* is zero (as you noticed, by virtue of H^4(S^2 x S^2) being generated by the cup product of 2-dimensional classes, the homomorphism . o f_* (which is, as the notation suggests, composition of f_* with an element of Hom(H_2(S^2 x S^2),Z)) is also zero. Now, let f_* take the generator [S^4] to the multiple k[S^2 x S^2] of H_4(S^2 x S^2). Then, taking a nonzero homomorphism h of H_4(S^2 x S^2) to Z, it would take the generator [S^2 x S^2] to some nonzero integer m. Composing h with f_*, we find [S^4] |----> k [S^2 x S^2] |----> km Since m is assumed nonzero, k*m is zero (as required) only if k = 0. Done. Dale. === Subject: Re: Cohomology, homology, cup product, spheres, question === Subject: Re: Topological vector space > A topological vector space is regular. Are there any simple example of > a non-normal topological vector space ? Take the product of uncountable many copies of R. Jose Carlos Santos === Subject: Re: Principia Book 1 <9ch1g.4461$V73.3435@trnddc06> I likewise admire the reach of Sir Isaac's mind, >He couldn't have got far without the surrender monkeys finding Paris. >I wonder if the French think that the term cheese eating surrender >monkeys is an insult. (Coming as it does from a nation that prizes the >cheeseburgher and voted for a chimpanzee.) One tends to forget that the inroads into scientific research and the measure of longitude and all the rest of it was due to French research. Newton stood on the shoulders of French giants. And however the French came to the conclusion that Emperor Bush was a lunatic and dirty trickster, the fact remains they were absolutely correct in their prognosis about Iraq and Afghanistan. They were right about Vietnam too. Perhaps that's why they stayed out of the evil emperor's camp. === Subject: Re: Perfect fields The perfect fields appearing in the author's statement need not be algebraic extensions of each other. You could for example have a countable chain F_1subset F_2subset .... of perfect fields such that each step F_ksubset F_(k+1) is not algebraic. H === Subject: Re: Perfect fields maybe the author needs only the case of fields over F_{p} for further discussion. === Subject: Re: Ring inside a field > Hello again, > If E is an algebraic extension of a field F, and if > F C R CE where R > is a ring, then R is a field. > I think the hypotheses can be weakened to an integral > domain R which is > algebraic over a field F. In this case, R would still > be a field. Right > Julien Santini If by >>R is algebraic over F<< you mean that every element of R is a root of a monic polynomial with coefficients in F, then the answer is NO. Take F=rational numbers and R=Z[sqrt(2)] and E the field of fractions of R. H === Subject: Re: Ring inside a field >> Hello again, >> If E is an algebraic extension of a field F, and if >> F C R CE where R >> is a ring, then R is a field. >> I think the hypotheses can be weakened to an integral >> domain R which is >> algebraic over a field F. In this case, R would still >> be a field. Right >> ? >> Julien Santini >If by >>R is algebraic over F<< you mean that every >element of R is a root of a monic polynomial with >coefficients in F, then the answer is NO. >Take F=rational numbers and R=Z[sqrt(2)] and E the >field of fractions of R. We do not have F C R in that example! I have to admit I am not completely sure exactly what Julien Santini is asking. Derek Holt. === Subject: Re: Ring inside a field > Hello again, > If E is an algebraic extension of a field F, and if > F C R CE where R > is a ring, then R is a field. > I think the hypotheses can be weakened to an integral > domain R which is > algebraic over a field F. In this case, R would still > be a field. Right ? Yes. The inverse can be obtained by a trivial rearrangement of the minimal polynomial. === Subject: Re: Does this error of logic have a Logical Fallacy name to it?; Re: page 3 <4076016.1145566956577.JavaMail.jakarta@nitrogen.mathforum.org> Do you even read posts before replying to them? The whole point of my post was that the proofs given above actually use the definition of a prime, no one has ever said anything about not using #1. Martin Wanvik Perhaps I have been too combative in arguing. Let me try a different tact. Dik acknowledges that 5(a) is correct. Where Dik and I disagree is that Dik believes 5(b) and 5(c) can also be correct. Whereas I believe 5(b) and 5(c) cannot be correct and must be invalid. Martin, do you accept 5(a) as a correct proof? The problem I have with 5(b) and 5(c) is that, if 5(a) is correct, then 5(b) and 5(c) are contradictory to the definition of prime by considering W+1 as composite. So it is what I would call a pseudo-contradiction. In other words, if the proof is easily begot from 5(a), then why take an opposite claim that W+1 is composite and not necessarily prime and indulge in 5(b) and 5(c). Another question Martin. Do you accept the idea that with 5(a) is an easy proof that W+1 and W-1 are necessarily prime which leads to an easy proof of the Infinitude of Twin Primes. This is an argument of Economy. If 5(a) is easier than 5(b) or 5(c) and that only 5(a) leads to a proof of the Infinitude of Twin Primes. Then does this power tell us that only 5(a) must be truly correct and 5(b) and 5(c) are invalid. Sort of like this argument in real life. Given a choice of 3 ladies to marry, (a), (b) and (c) where (a) is easy to marry and has a large fortune for which (b) and (c) only have marriage and no power of fortune. And finally, there is the issue of the Symbolic Logic format in which only 5(a) discharges all the previous steps whereas 5(b) and 5(c) leave open previous steps, namely the definitional step. I suspect there is alot more to learn about IP proof for which I am not done and finished with and that some more surprizes are awaiting to be discovered. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: Does this error of logic have a Logical Fallacy name to it?; Re: page 3 > Do you even read posts before replying to them? The > whole point of my > post was that the proofs given above actually > use the definition of a prime, no one has ever said > anything about not > using #1. > Martin Wanvik > Perhaps I have been too combative in arguing. Let me > try a different > tact. > Dik acknowledges that 5(a) is correct. Where Dik and > I disagree is that > Dik believes 5(b) and 5(c) can also be correct. > Whereas I believe 5(b) > and 5(c) cannot be correct and must be invalid. > Martin, do you accept 5(a) as a correct proof? Yes. > The problem I have with 5(b) and 5(c) is that, if > 5(a) is correct, then > 5(b) and 5(c) are contradictory to the definition of > prime by > considering W+1 as composite. So it is what I would > call a > pseudo-contradiction. In other words, if the proof is > easily begot from > 5(a), then why take an opposite claim that W+1 is > composite and not > necessarily prime and indulge in 5(b) and 5(c). If you use assumption (2), you immidiately get that W+1 is composite, since it is larger than any of the numbers on the list and is not divisible by any of them. Using assumption (1) first instead, implies that W+1 is prime. It is a question of order, nothing else. > Another question Martin. Do you accept the idea that > with 5(a) is an > easy proof that W+1 and W-1 are necessarily prime > which leads to an > easy proof of the Infinitude of Twin Primes. No. As it stands, the proof only gives you infinitude of primes. In order to do twin primes you'd have to add the assumption that there are only finitely many twin primes. Notice that these are a proper subset of the primes, so if some W+1 and W-1 are not divisible by any of the twin primes on your list, (1) can't give you primality any more, since both W+1 and W-1 may be divisible by other primes than those on your list. > This is > an argument of > Economy. If 5(a) is easier than 5(b) or 5(c) and that > only 5(a) leads > to a proof of the Infinitude of Twin Primes. Then > does this power tell > us that only 5(a) must be truly correct and 5(b) and > 5(c) are invalid. No, you're mixing things up here. The correctness of a proof is not determined by whether or not it is hard or what the proof method may or may not imply. But certainly, if all this were true, 5a would be a more economical proof. > Sort of like this argument in real life. Given a > choice of 3 ladies to > marry, (a), (b) and (c) where (a) is easy to marry > and has a large > fortune for which (b) and (c) only have marriage and > no power of > fortune. > And finally, there is the issue of the Symbolic Logic > format in which > only 5(a) discharges all the previous steps whereas > 5(b) and 5(c) leave > open previous steps, namely the definitional step. As I've been trying to say just about a zillion times, none of these proofs leave anything open, /all/ assumptions are used in /all/ of the proofs, it is just a matter of order. > suspect there is > alot more to learn about IP proof for which I am not > done and finished > with and that some more surprizes are awaiting to be > discovered. > Archimedes Plutonium > www.iw.net/~a_plutonium > whole entire Universe is just one big atom > where dots of the electron-dot-cloud are galaxies === Subject: Re: Does this error of logic have a Logical Fallacy name to it?; Re: page 3, History of Euclid Proof of Infinitude of Primes... Basically I am saying that if a proof of the Infinitude of Primes, reductio ad absurdum method leads to a proof of the Infinitude of Twin Primes, but that all other variants of the proof of Infinitude of Primes such as Dik Winter's 5(b) and 5(c) cannot yield a Infinitude of Twin Primes proof, then there is some logical flaw in those variants. A flaw that makes those variants invalid arguments. And basically where I have traced where that flaw is from Symbolic Logic is that the definition of prime in step #1 is not discharged. Logic calls such a fallacy as a NonSequitur. That is the best I can do till now. Perhaps there is even more of a flaw, and bigger than what I have done so far. If so, well, there is more to learn from this adventure. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: Does this error of logic have a Logical Fallacy name to it?; Re: page 3, History of Euclid Proof of Infinitude of Primes... <444722D6.1000701@dtgnet.com> You never state what a number is in your proof. Does that make your proof invalid ? I think you can assume that anyone who wants to know whether the primes are infinite or not knows what a prime is and, indeed, what a natural number is. Another thing you do not understand about mathematics, is that mathematics can be written as one long book starting with axioms and definitions and theorems all in a long string. And when we come to theorems that are complex and long and advanced, example MiniMax theorem (vonNeumann), we do not have to define everything all over again and redo Calculus from scratch because those things were done earlier in this long stringed book of all of mathematics. That is why I do not need to consider number in the Infinitude of Primes proof. But I need to have included the definition of prime to differentiate a prime from a composite. Your statement belies the fact that mathematics is probably not your major and that you only dabble in mathematics. And I just do not have anymore time for someone who cannot post with his real name. A person with a pseudoname does not take himself serious enough and why should I take him serious. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: Does this error of logic have a Logical Fallacy name to it?; Re: page 3, History of Euclid Proof of Infinitude of Primes... <444722D6.1000701@dtgnet.com You never state what a number is in your proof. > Does that make your proof invalid ? > I think you can assume that anyone who wants to know > whether the primes are infinite or not knows what a prime > is and, indeed, what a natural number is. > Another thing you do not understand about mathematics, is that > mathematics can be written as one long book starting with axioms and > definitions and theorems all in a long string. And when we come to > theorems that are complex and long and advanced, example MiniMax > theorem (vonNeumann), we do not have to define everything all over > again and redo Calculus from scratch because those things were done > earlier in this long stringed book of all of mathematics. > That is why I do not need to consider number in the Infinitude of > Primes proof. But I need to have included the definition of prime to > differentiate a prime from a composite. > Your statement belies the fact that mathematics is probably not your > major and that you only dabble in mathematics. > And I just do not have anymore time for someone who cannot post with > his real name. A person with a pseudoname does not take himself serious > enough and why should I take him serious. All this is just obfuscation on your part. Your proof does not follow through. Are you assuming pn is the last prime ? Yes or No If pn is the last prime, is every number greater than pn composite? Yes or No Is w+1 > pn? Yes or No Is w+1 composite ? Yes or No The answers to these questions are Yes, Yes, Yes , Yes Is w+1 divisible by a prime, p1 to pn ? Yes or No Is every composite divisible by a prime ? Yes or No Is w+1 a composite with no prime factors ? Yes or No The answers to these questions are No, Yes, Yes What are your answers to these questions ? No, Yes, No, Yes, No, No, No ? Where do you actually use the notion of primeness rather than irreducibility in your exegesis ? === Subject: Testing if sign(a.x)=sign(b.x) Suppose a,b in R^n, is there an efficient algorithm to tell if sign(a.x)=sign(b.x) forall x in {0,1}^n ? === Subject: Re: Testing if sign(a.x)=sign(b.x) On 21 Apr 2006 00:39:39 -0700, Yaroslav Bulatov >Suppose a,b in R^n, is there an efficient algorithm to tell if >sign(a.x)=sign(b.x) forall x in {0,1}^n ? I'm not familiar with the notation. What is sign(a,x)? quasi === Subject: Re: Testing if sign(a.x)=sign(b.x) >On 21 Apr 2006 00:39:39 -0700, Yaroslav Bulatov >>Suppose a,b in R^n, is there an efficient algorithm to tell if >>sign(a.x)=sign(b.x) forall x in {0,1}^n ? >I'm not familiar with the notation. >What is sign(a,x)? sign(r) = 0 for r=0 1 for r>0 -1 for r<0 I think Yaroslav is talking about the sign of a dot product. Here are some filters that might help: * If sign(a_i) != sign(b_i) for 1 or more i, the statement is false. * If a = k * b for a positive scalar k, the statement is true. * If sign(a_i) = sign(b_i) for all i and sign(a_i) != -sign(a_j) for any (i,j), the statement is true. (That is, positive and negative components are not mixed.) If these filters all fail you might need to exhaustively check all 2^n values of x. --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Re: Point Inside Polygon - Ray Method > If the boundary of a plane region is a simple closed curve, then one ray > from any point not on the boundary is enough to determine whether that > point is inside or outside the region. The point is inside or outside > according to whether the parity of the number of intersections of the > ray with the boundary is odd or even (with points of multiple contact of > the ray with the boundary counted with their multiplicities, though a > slight dispacement of the ray will usually eliminate all such multiple > points). Quite a naive response. (Apparently, you have never tried to _actually_ work out and computer-code this yourself) The following page contains theory, free (Windows) executables and all of accompanying source code: http://huizen.dto.tudelft.nl/deBruijn/grondig/crossing.htm Han de Bruijn === Subject: Re: Point Inside Polygon - Ray Method On Fri, 21 Apr 2006 09:55:00 +0200, Han de Bruijn >> If the boundary of a plane region is a simple closed curve, then one ray >> from any point not on the boundary is enough to determine whether that >> point is inside or outside the region. The point is inside or outside >> according to whether the parity of the number of intersections of the >> ray with the boundary is odd or even (with points of multiple contact of >> the ray with the boundary counted with their multiplicities, though a >> slight dispacement of the ray will usually eliminate all such multiple >> points). >Quite a naive response. (Apparently, you have never tried to _actually_ >work out and computer-code this yourself) The following page contains >theory, free (Windows) executables and all of accompanying source code: >http://huizen.dto.tudelft.nl/deBruijn/grondig/crossing.htm So how does your code handle the case when the ray intersects infinitely many times? === Subject: Re: Point Inside Polygon - Ray Method >>http://huizen.dto.tudelft.nl/deBruijn/grondig/crossing.htm > So how does your code handle the case when the ray intersects > infinitely many times? Read the page. If you mean coincident with a line segment, then it is solved in the code (as explained somewhere on the page). Hope you don't mean something that cannot be realized in the domain of computation ... Han de Bruijn === Subject: Re: Point Inside Polygon - Ray Method > On Fri, 21 Apr 2006 09:55:00 +0200, Han de Bruijn >> >> If the boundary of a plane region is a simple closed curve, then one ray >> from any point not on the boundary is enough to determine whether that >> point is inside or outside the region. The point is inside or outside >> according to whether the parity of the number of intersections of the >> ray with the boundary is odd or even (with points of multiple contact of >> the ray with the boundary counted with their multiplicities, though a >> slight dispacement of the ray will usually eliminate all such multiple >> points). >Quite a naive response. (Apparently, you have never tried to _actually_ >work out and computer-code this yourself) The following page contains >theory, free (Windows) executables and all of accompanying source code: >http://huizen.dto.tudelft.nl/deBruijn/grondig/crossing.htm > So how does your code handle the case when the ray intersects > infinitely many times? Rotate the ray until it doesn't, or even move the point a small enough distance so as not to cross any boundaries and ray again. === Subject: Re: Point Inside Polygon - Ray Method > > If the boundary of a plane region is a simple closed curve, then one ray > from any point not on the boundary is enough to determine whether that > point is inside or outside the region. The point is inside or outside > according to whether the parity of the number of intersections of the > ray with the boundary is odd or even (with points of multiple contact of > the ray with the boundary counted with their multiplicities, though a > slight dispacement of the ray will usually eliminate all such multiple > points). > Quite a naive response. (Apparently, you have never tried to _actually_ > work out and computer-code this yourself) Computer coding to get mathematical results is mostly for those who can't do the mathematics without it. === Subject: Re: Point Inside Polygon - Ray Method larryhammick@telus.net says... > Even for a simple closed curve, a ray may hit it infinitely many times. > For a polygon, a whole side may lie on the ray. Messy. I like my way > better :) Could you elaborate on your method; I did not follow. Also, why do your rays (three of them, right?) not have the same problem? -- Christer Ericson http://realtimecollisiondetection.net/ === Subject: Re: Point Inside Polygon - Ray Method <4447F62D.A6CF7A29@pat7.com> do your rays (three of them, right?) not have the same problem? I was too brief, sorry. Any of the 3 rays can hit the curve countably or uncountably many times. But if we follow the curve, and keep track of the intersections, say A B C B B CCCC(inf) B AAAA(inf) B C it cannot go from one ray to a _different_ ray infinitely many times. The sum of f(x,y) where f(A,B) = f(B,C) = f(C,A)= 1/3 f(x,y) = -f(y,x) for all x,y is a finite integer. LH === Subject: Re: Point Inside Polygon - Ray Method larryhammick@telus.net says... > Could you elaborate on your method; I did not follow. Also, why > do your rays (three of them, right?) not have the same problem? > I was too brief, sorry. Any of the 3 rays can hit the curve countably > or uncountably many times. But if we follow the curve, and keep track > of the intersections, say > A B C B B CCCC(inf) B AAAA(inf) B C > it cannot go from one ray to a _different_ ray infinitely many times. > The sum of f(x,y) > where > f(A,B) = f(B,C) = f(C,A)= 1/3 > f(x,y) = -f(y,x) for all x,y > is a finite integer. > LH I'm afraid I still have no idea what your method is. Do you think you could start describing it from scratch, without any assumptions of my knowledge? I.e. I'm given a (possibly concave) polygon P (defined by points P_i) and a query point Q, both in the same plane. I want to find out if Q lies inside P. Now what? (I'm hoping that your algorithm description would answer questions such as: what are the rays, where do they originate from and what direction do they have, why are there three rays, what do these rays tell me, and how does that relate to the posed problem. Currently I can't say I even begin to understand your algorithm.) -- Christer Ericson http://realtimecollisiondetection.net/ === Subject: Re: Should be easy Let P be a set of propositions, which does not all have the same truth > value. Define a relation R on P by: > p R q if p --> q holds > Show that R is reflective and transitive, but neither symmetric nor > antisymmetric .... > The book defines antisymmetric as: > p R q AND q R p => p=q > So I still do not see how I am supposed to choose values for p and q > that satisfies .... > The book's writer has been careless here. The problem should have > said that the set contains at least three propositions. In that case > you can choose p and q to be different propositions having the same > truth value, and get your counter-example. > (If the set contains only two propositions, one true and one > false, then the relation is actually antisymmetric.) > Ken Pledger. Will the following modification do? Let P be a set of propositions, which do no all have the same truth values. Define a relation R on P by: p R q if p --> q holds Show that R is reflexive and transitive, but may not be symmetric and may not be antisymmetric. I think the author wanted the student to realize that p if and only if q does not make p = q. Muhammad === Subject: Re: Should be easy > The books defines antisymmetric as: > p R q AND q R p => p=q > So I still does not see how I am suppose to choose values for p an q > that satiesfies > -- > Let P be a set of propositions, which does no all have the same truth > value, Define a relation R on P by: > p R q if p --> q holds > Show that R is reflective and transitive, but neither symmetric nor > antisymmetric > -- > Any ideas ? Yes, include context so it's easy to flow flow of thought and make reply. I've not the time now to back track the thread and review your question. -- To Google and MathForum users: Reply only if adequate context is included _within_ the reply. Otherwise all contexts are removed from my view, the flow of thought disrupted and chaos reigns. In particular for Google users: Instead of simply hitting the prominent Reply link, which doesn't include a copy of the post to which one is replying, click the Show Options link (toward the top of an item in the thread), which causes a shaded area of links to appear next to the top of the item, including Reply (first) that does introduce a copy of the previous text (offset by > signs in the usual fashion). ---- === Subject: Re: Should be easy > Yes, include context so it's easy to flow flow of thought and make reply. > I've not the time now to back track the thread and review your question. Why do you keep complaining about this? Get a real newsreader like Thunderbird, where the messages are duly threaded and easy to backtrack. Including a lot of quoted material is a waste of bandwidth IMHO. My newsserver actually refuses to post a reply, whenever more than 50 per cent of the text is quoted. IMVHO this is a well thought out policy. Jyrki === Subject: Re: Should be easy I've not the time now to back track the thread and review your question. > Why do you keep complaining about this? Get a real newsreader like > Thunderbird, where the messages are duly threaded and easy to backtrack. Great, send me at your expense new computer system, removed of all toxic odors and software that you demand I use that's also not ill designed, and teach me to use it on your time while compensating me for my lost time. Anything else must I do? Use the metric system instead of the English system. Use Euros instead of dollars? > Including a lot of quoted material is a waste of bandwidth IMHO. > My newsserver actually refuses to post a reply, whenever more than > 50 per cent of the text is quoted. IMVHO this is a well thought out > policy. It isn't at all. It's like those answering machines that if you dare to pause a moment, it starts bitching are you done yet with your message? Press one if you are otherwise press two to continue or press three to ... Just another example of mandated automation. So don't complain it won't quote. Pad your reply with enough random words to exceed the 50% forced quote limit. -- humans are obsolete The more enforced automation, the more computers force us to accept them thinking for us, the more dehumanized, mechanized and not designed for people our societies will, are becoming. Too bad excellent technologies when implemented by stupid people for stupid people have become worse than not having them. Yet nay you fear, soon everybody will be updated to be computers with mandated electronic implants and you won't be bother by old fossils like me who refuse the inconvenience of machines that think for better me than I can. === Subject: Re: Should be easy >Yes, include context so it's easy to flow flow of thought and make reply. >I've not the time now to back track the thread and review your question. >>Why do you keep complaining about this? Get a real newsreader like >>Thunderbird, where the messages are duly threaded and easy to backtrack. > Great, send me at your expense new computer system, removed of all toxic > odors and software that you demand I use that's also not ill designed, and > teach me to use it on your time while compensating me for my lost time. No need for a new computer system. There are threaded newsreaders that will run on old systems with only a text interface. No one is deamanding that you use one, but it may well make your time with usenet more productive, and they can be learned in matter of minutes. > Anything else must I do? Use the metric system instead of the English > system. Use Euros instead of dollars? False analogy... I prefer Imperial measures too, but do not post demands that all others must use my preferred system. >>Including a lot of quoted material is a waste of bandwidth IMHO. >>My newsserver actually refuses to post a reply, whenever more than >>50 per cent of the text is quoted. IMVHO this is a well thought out >>policy. > It isn't at all. It's like those answering machines that if you dare to > pause a moment, it starts bitching are you done yet with your message? > Press one if you are otherwise press two to continue or press three to > ... Just another example of mandated automation. > So don't complain it won't quote. Pad your reply with enough random words > to exceed the 50% forced quote limit. See below :) > -- humans are obsolete > The more enforced automation, the more computers force us to accept them > thinking for us, the more dehumanized, mechanized and not designed for > people our societies will, are becoming. Too bad excellent technologies > when implemented by stupid people for stupid people have become worse > than not having them. > Yet nay you fear, soon everybody will be updated to be computers with > mandated electronic implants and you won't be bother by old fossils like > me who refuse the inconvenience of machines that think for better me than > I can. === Subject: Re: Should be easy <4448c54f$0$9237$ed2619ec@ptn-nntp-reader01.plus.net> On Fri, 21 Apr 2006, it was written: >Yes, include context so it's easy to flow flow of thought and make reply. >I've not the time now to back track the thread and review your question. >>Why do you keep complaining about this? Get a real newsreader like >>Thunderbird, where the messages are duly threaded and easy to backtrack. > Great, send me at your expense new computer system, removed of all toxic > odors and software that you demand I use that's also not ill designed, and > teach me to use it on your time while compensating me for my lost time. > No need for a new computer system. There are threaded newsreaders that > will run on old systems with only a text interface. No one is deamanding > that you use one, but it may well make your time with usenet more > productive, and they can be learned in matter of minutes. I have a threaded newsreader and I've not the time to retrace and reconstruct the thread, especially the raw discussion without unneeded context removed. > Anything else must I do? Use the metric system instead of the English > system. Use Euros instead of dollars? > False analogy... I prefer Imperial measures too, but do not post demands > that all others must use my preferred system. No context, no intelligent reply, less work for me, more and better work for those who include context. The choice is their's. Do the quick no quote and save you time; save me much more time. === Subject: Closing the Intersection Ok, here is something that I've been wondering since my old bachelor days but I never got to know to answer. Let X be a topological space. Is there a characterization that tells us when a given A,B subset of X can have cl(A / B)=cl(A)/cl(B) or one could ask the dual question, when int(A/B)=int(A)/int(B) Jose Capco === Subject: Re: Closing the Intersection > Let X be a topological space. Is there a characterization that tells us > when a given A,B subset of X can have > cl(A / B)=cl(A)/cl(B) When A and B are closed. > or one could ask the dual question, when > int(A/B)=int(A)/int(B) The dual answer, A and B open. === Subject: Hausdorff and compact Does there exist a Hausdorff space which cannot be continiously and bijectively mapped onto compact ? Message was edited by: eugene === Subject: Re: Hausdorff and compact >Does there exist a Hausdorff space which cannot be continiously and bijectively mapped onto compact ? >(...) This question can be reduced to: is there a Hausdorff topology T on some set X such that no topology on X coarser than T makes X compact ? (T' coarser than T means: T finer than T') If you mean by 'compact' what is called 'quasi-compact' in French topology litt. (i.e. if Hausdorff it is not demanded for compactness), then the answer is no: any topology on X is finer than the topology on X that has only X and the empty set as open sets, and this one is (quasi-) compact. If you mean by 'compact' Hausdorff *and* 'quasi-compact' (as in French litt. - after Bourbaki ...), then the question seems to me really non trivial. I tried some simple ideas, but found no answer. Something related is the question: is a minimal Hausdorff topology (minimal in the ordered set of topologies on a given set) always compact ? (the converse is true: a [Hausdorff] compact topology is a minimal Hausdorff topology). If the answer was yes, *and* if Zorn's lemma was applicable in this situation (but it seems that it isn't ...) then the answer to your question - with 'compact' defined as including Hausdorff - would be no ... So these questions are left to others. Someone has an idea ? === Subject: Re: Hausdorff and compact bijectively mapped onto compact ? > This question can be reduced to: is there a Hausdorff topology T > on some set X such that no topology on X coarser than T makes > X compact ? (T' coarser than T means: T finer than T') What if image space is compact Hausdorff and space itself irregular Hausdorff space. Then image space is normal. Is it coarser? What if the map isn't the identity map, which seems necessary for your line of thought? > Something related is the question: is a minimal Hausdorff topology > (minimal in the ordered set of topologies on a given set) always > compact ? (the converse is true: a [Hausdorff] compact topology > is a minimal Hausdorff topology). If the answer was yes, *and* if > Zorn's lemma was applicable in this situation (but it seems that > it isn't ...) then the answer to your question - with 'compact' > defined as including Hausdorff - would be no ... What about a minimal Hausdorff topology for Q? > So these questions are left to others. Someone has an idea ? The first question is to OP. Do you mean compact Hausdorff? === Subject: Re: Hausdorff and compact > So these questions are left to others. Someone has > an idea ? > The first question is to OP. Do you mean compact > Hausdorff? Actually in the problem condition it wasn't stated that compact must also be Hausdorff. But, i'm interested surely in the case when compact=(quasi-compact) +Hausdorff=compact+Hausdorff, because otherwise the prob;em becomes trivial and the anser is to my question would be no. === Subject: Re: Hausdorff and compact bijectively mapped onto compact ? This question can be reduced to: is there a Hausdorff topology T > on some set X such that no topology on X coarser than T makes > X compact ? (T' coarser than T means: T finer than T') > What if image space is compact Hausdorff and space itself irregular > Hausdorff space. Then image space is normal. Is it coarser? > What if the map isn't the identity map, which seems necessary for your > line of thought? > Something related is the question: is a minimal Hausdorff topology > (minimal in the ordered set of topologies on a given set) always > compact ? (the converse is true: a [Hausdorff] compact topology > is a minimal Hausdorff topology). If the answer was yes, *and* if > Zorn's lemma was applicable in this situation (but it seems that > it isn't ...) then the answer to your question - with 'compact' > defined as including Hausdorff - would be no ... > What about a minimal Hausdorff topology for Q? Steen's Counterexamples in Topology, space #100 is countable minimal Hausdorff non-compact. I don't see how you can make that into a counter example of OP's conjecture for the reason I gave above, aren't you assuming the bijection is the identity map? It's almost compact, ie every open cover has a finite subcover whose closures cover the space. > So these questions are left to others. Someone has an idea ? > The first question is to OP. Do you mean compact Hausdorff? === Subject: Re: Hausdorff and compact Porter, Jack R.; Stephenson, Robert M. Minimal Hausdorff spaces---then and now. Handbook of the history of general topology, Vol. 2 (San Antonio, TX, 1993), 669--687, Hist. Topol., 2, Kluwer Acad. Publ., Dordrecht, 1998. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: Hausdorff and compact > Does there exist a Hausdorff space which cannot be continiously and > bijectively mapped onto compact ? What does that mean? Onto a compact space? No. Map the space onto itself with the indiscrete topology. > Message was edited by: eugene How about proof reading and spell checking? === Subject: Re: Hausdorff and compact > Does there exist a Hausdorff space which cannot be continiously and > bijectively mapped onto compact ? > What does that mean? Onto a compact space? No. > Map the space onto itself with the indiscrete topology. > Message was edited by: eugene > How about proof reading and spell checking? improve the question: Does there exist a Hausdorff space which cannot be continiously and bijectively mapped onto a compact Hausdorff space? === Subject: Re: Hausdorff and compact <210420060759062664%anniel@nym.alias.net.invalid Message was edited by: eugene > How about proof reading and spell checking? > improve the question: > Does there exist a Hausdorff space which cannot be continiously and > bijectively mapped onto a compact Hausdorff space? Nice proof reading and improvement. Spell checking still missing. Here's positive definite version instead of double negative stuff. For all Hausdorff S, there's a compact Hausdorff space K and a continuous bijection from S onto K. Question, is there a compact Hausdorff space K with a continuous bijection from Q onto K? === Subject: Re: Including the constant term in this model? > I have created a regression formula relating the market cap of a > company to it's book value and also its earnings. The formula has a > constant term, therefore, this implies that even if the book value and > the earnings were 0, the company would still be worth something. > Should I include this constant term into my equation? When I eliminate > the constant term, the model becomes slightly more inaccurate. No model is going to be perfect over the entire range of inputs. If including the term makes the model more useful over the set of *realistic* inputs (how many companies have book value zero and earnings zero, anyway?) then keep it. A mathametical analogy might be: n / ln n is a useful first approximation to pi(n), even though it fails to get pi(2) right. -- Larry Lard Replies to group please === Subject: Re: Including the constant term in this model? > I have created a regression formula relating the market cap of a > company to it's book value and also its earnings. The formula has a > constant term, therefore, this implies that even if the book value and > the earnings were 0, the company would still be worth something. In general a linear formula is an approximation that is valid for a range of values. If that range of values excludes zero then you do not need to worry about including the constant. > Should I include this constant term into my equation? When I eliminate > the constant term, the model becomes slightly more inaccurate. Adding more constants will always give you a better fit, so you have to ask if the inclusion of a particular constant is justified by performing a number of other tests. === Subject: Re: Reality check: Counting prime numbers >> [Rick Decker, on JSH's prime-counting formula] >> The key is that for y <= x, p(x, y) is the cardinality of the set >> consisting of (1) the primes <= y and (2) the numbers <= x that >> are not divisible by any prime <= y. For example p(25, 3) is >> the cardinality of {2, 3, 5, 7, 11, 13, 17, 19, 23, 25}. In simple >> terms, p(x, y) is the number of elements remaining at the end >> of the appropriate step in the Sieve of Eratosthenes. >> For the heck of it, I tried to clean room derive his formula, and turns >> out it's almost as easy to derive from thinking about how the venerable old >> sieve works as Legendre's phi is to derive from straightforward >> inclusion/exclusion. >> The second key is that the >> p(x/k, k-1) - p(k-1, sqrt(k-1)) = >> p(x/k, k-1) - pi(k-1) [1] >> term counts, when k is prime, the number of new integers crossed off >> during the sieve step that first discovers k _is_ prime. IOW, it's the >> number of integers in 1..x, excluding k itself, whose smallest prime divisor >> is k. Deriving [1] from that characterization does make a pleasant little >> exercise. >Then do it. Uh, he did. Believe it or not, not everyone is as slow as you are, regarding is no surprise - it doesn't follow that anything significant is missing, for readers with a tiny bit of mathematical ability. What? How could that be? We all know that you're the world's greatest number theorist... Uh, no. Remember that silliness about p-2 a few days ago? It's either a prime, the product of two primes, or has a factor less than the cube root of p? You _stated_ that it took you several days to get that worked out. Everyone who read your post realized _immediately_ that it was trivial. Wake up. >[...] >The key point--the reality test--is whether or not any poster actually >starts at a begining and proceeds to go all the way to a complete >derivation of my prime counting function. >I know that most human brains are incapable of doing it because the >necessary wiring just isn't there. >That makes it an interesting test, as if you do not have the necessary >wiring you simply cannot do it. Not even to save your life. The fact that you continue to say things like this, in spite of the fact that more or less every day we see evidence of your utter mathematical incompetence, is truly fascinating. >You can't do it any more than a blind person could see. You just will >not be able to accomplish the task, but for some people their brains >will simply tell them they can as that reality is not acceptable to the >brain. >And whether you realize it or not, your mind is just a function of your >brain. >Your actual capabilities are limited by the hardware between your ears. >James Harris ************************ David C. Ullrich === Subject: Re: Reality check: Counting prime numbers <4agtilFt15guU1@individual.net> <9NKdnfb387tekdXZRVn-uw@comcast.com> corpus collosums, which is the connection between your left and right > brains, cut because it was necessary because of their epilepsy. > Well, the researchers noticed that if they blocked the left brain from > seeing something, and let the right brain see it and react, the left > brain would explain the reaction. not be able to accomplish the task, but for some people their brains > will simply tell them they can as that reality is not acceptable to the > brain. > And whether you realize it or not, your mind is just a function of your > brain. > Your actual capabilities are limited by the hardware between your ears. > James Harris Doctor Jimmy and Mister Jim When I'm pilled You don't notice him He only comes out when I drink my gin The Who Doctor Jimmy , Quadrophenia === Subject: Re: Reality check: Counting prime numbers <4agtilFt15guU1@individual.net> <9NKdnfb387tekdXZRVn-uw@comcast.com> <9pqdnb_urJ6cz9XZnZ2dnUVZ_vidnZ2d@comcast.com [Rick Decker, on JSH's prime-counting formula] > The key is that for y <= x, p(x, y) is the cardinality of the set > consisting of (1) the primes <= y and (2) the numbers <= x that > are not divisible by any prime <= y. For example p(25, 3) is > the cardinality of {2, 3, 5, 7, 11, 13, 17, 19, 23, 25}. In simple > terms, p(x, y) is the number of elements remaining at the end > of the appropriate step in the Sieve of Eratosthenes. > [Tim Peters] >> For the heck of it, I tried to clean room derive his formula, and turns >> out it's almost as easy to derive from thinking about how the venerable >> old sieve works as Legendre's phi is to derive from straightforward >> inclusion/exclusion. >> The second key is that the >> p(x/k, k-1) - p(k-1, sqrt(k-1)) = >> p(x/k, k-1) - pi(k-1) [1] >> term counts, when k is prime, the number of new integers crossed off >> during the sieve step that first discovers k _is_ prime. IOW, it's the >> number of integers in 1..x, excluding k itself, whose smallest prime >> divisor is k. Deriving [1] from that characterization does make a >> pleasant little exercise. > [jstevh@msn.com] > Then do it. > else here who understands the Sieve of Eratosthenes to do it for themself: > assume that p(m, n) counts the number of integers in 1..m remaining after > the Sieve of Eratosthenes has processed all primes <= n. Assume that pi(n) > returns the number of primes <= n. Then given an integer m and a prime k, > derive a formula (in terms of p() and pi()) giving one more than the number > of integers in 1..m whose smallest prime divisor is k. Any undergrad with a > smattering of introductory number theory should be able to do that in less > than an hour; I expect that the brighter kids in my high school could have > done it too. Your brain will lie to you in order to convince you of something that is not true. The point here isn't that you can understand the gist of how it all works, but can you put down a complete derivation from beginning to end, where at the end you have the formula I show at http://en.wikipedia.org/w/index.php?title=Prime_counting_function&oldid=9142 249 and by now, with all your replies, you could have accomplished that, if you were capable, while instead you keep repeating pieces, here and there. And note again, the need to diminish the importance of the task by claiming any undergrad could do it. Why don't YOU just do it? > I've looked and waited, but not seen a single post that just goes step > by step showing a derivation. > Why would we bother, James? Just because you challenged us with childish > insults and accusations? Ptui -- doesn't mean squat to me, bubba. Rick > Decker suggested earlier that it _sounds_ like you're trying to goad someone > into giving a coherent writeup, so that you can pass it off as your own. > That sounded plausible to me. By your own admission, your own explanation > for how to derive it is so convoluted that reading it doesn't help anyone > <0.1 wink>. Dodge and rationalize. To date, I have not seen anyone besides me do it, at all, despite my giving this challenge for years. My point is that for some special reasons, despite the inherent simplicity of the derivation, the human brain for most people has a wiring lack, which doesn't allow you to step through a derivation. You can prove the equation true. You can understand a derivation if given to you. But you cannot personally step through a full derivation as the circuitry needed to do so is missing from your brain. > If someone on alt.math.undergrad is truly interested, I'd be happy to help > them work out a full derivation. > Posters keep claiming it's easy, and they've done it somewhere else, > but no one just steps through a derivation. > So you try again, and I'll _help_ you make it comprehensible. I won't do it > for you. I like my own derivation, and need it for credit for having had the first derivation known. And hey, remember, my equations could have been found by mathematicians thousands of years ago, or hundreds of years ago, if it were really easy. But they weren't, and I think I know why: wiring of the human brain. > I'll tell readers of some interesting research where people had their > corpus collosums, which is the connection between your left and right > brains, cut because it was necessary because of their epilepsy. > Well, the researchers noticed that if they blocked the left brain from > seeing something, and let the right brain see it and react, the left > brain would explain the reaction. > That is, that part of the brain to maintain its own sense of > consistency would make up an explanation for something it actually had > no explanation for. > Posters can't accept that they can't step through a derivation of my > prime counting function, so they make up a story, just telling > themseves they do it. > But notice, no one actually does it. > They don't write it up here just because you order them to. >> Then that's really all there is to it, except for the obfuscations of >> spelling pi(n) long-windedly as p(n, sqrt(n)) three times in the formula, >> and using pi(k)-pi(k-1) as a 0-1 multiplier to throw away useless >> instances of [1] (those where k is composite). > Critical here also is, notice the criticisms. > Yoda you again like speaking? Yes, relentlessly using p(n, sqrt(n)) instead > of pi(n) needlessly complicates the presentation; and you're off in your own > world about the importance of the pi(k)-pi(k-1) term. > Not only do posters routinely rationalize, but they make sure to try > and diminish my work in some way. > I've always said I like your prime-counting formula. As to diminish, do > correct me if I'm wrong, but I believe you haven't yet proved that it's > _important_ in any way. So what's to diminish? > The essential point is that they have decided ahead of time what to > believe about me and my research, so when reality doesn't fit in, their > brains just make something up. > Huh? > But there is characteristic behavior showing it happening. > What? >> I wonder whether James derived it that way. I gave up trying to follow >> one of his derivations some time ago, but that one expressed his formula >> as a system of mutually recursive functions, with odd names like >> dS (which I guess were supposed to remind the reader of differential >> equations). It's possible he made it much harder for himself than it >> needed to be. > The key point--the reality test--is whether or not any poster actually > starts at a begining and proceeds to go all the way to a complete > derivation of my prime counting function. > OK, you won't believe this, but I don't care that you won't: it's obvious > (honestly, it is) to anyone with enough mathematical chops to follow this > argument that I did derive the whole thing, just based on the bit of mine > you quoated at the start. Nothing difficult is missing. Do you really > think it takes a genius to start from there and fill in: It can be almost painful to watch after a while, as a person will do any number of rationalizations and use all kinds of avoidance behavior. Now to me, it would be quite ok for other people to just step through a derivation, as then I could get some progress in getting my research acknowledged. But people can't, so my research is ridiculed not because it's wrong or unimportant, but because fully comprehending it escapes most people. Discussion becomes meaningless, as people can't accept what they can't fully comprehend, and the reality check here is that NO ONE besides me is stepping through that derivation. So you have this huge thing that jumps out at you, if you want to see it. But for many of you, it would be the last thing you'd admit. James Harris === Subject: Re: [geom] 3 min-max triangles Hi Rainer, >> 2.c transform line TU -> tu >> X = EQ.TU, x = PX.eq >> Y = VQ.TU, y = PY.vq >> line XY = TU is transformed into line xy > I saw a construction with less lines to draw: > tu = (TU)' = (TU.MN)(PT.vd)) Fine ! Choosing M = VD.vd = VT.vd and N = VE.ve = VU.ve of course, instead of my EQ.eq and VQ.vq which requires more not allready drawn lines. However I didn't try to get the absolute minimum elements to draw, just the principle, and also points and lines which fit in the paper sheet. Searching for your own construction is great : finding one shows that finally [you] made [your] way into the [jungle of] projective world. I'll add or even simpler, as noted by Rainer, draw... on my web site. -- philippe mail : chephip at free dot fr site : http://chephip.free.fr/ === Subject: Re: why Riesz Representation Theorem not hold for L-infinity On Thu, 20 Apr 2006 16:17:30 -0700, Ronald Bruck On Wed, 19 Apr 2006 21:31:41 -0700, Ronald Bruck continuous functions f such that lim_x->infty f(x) exists. Call this > space X. Define a bounded linear functional T on X by > T(f)=lim_x->infty f(x). Extend it by Hahn-Banach. You can then show > that if there is a gin L^1 such that T(f)=int fg, then g must be 0. > But clearly T is nonzero. > >>Because the dual of L^infty is not L^1? >> No, because there exist functions in L^infinity that have a >> non-zero limit at infinity. >Hmmm. This whole thread is confusing me. So I found the OP's post, >where she explicitly asks for an example of an element of L^infty * >which isn't in L^1. That's what ceckhard was replying to, and he >correctly gave an example. I **thought** he was asking why the >contradiction?, since I saw no context (mistaking it as an original >post). >Now I'm confused by YOUR response. He's explicitly defined T on >L^infinity functions which have a limit at infinity, and then appealed >to HB to extend it to L^infinity continuously; what does it matter that >there exist functions in L^infinity that have a non-zero limit at >infinity? Clearly T is non-zero was the last sentence appearing in the bit you quoted, before you said Because...?. So I assumed that your Because...? was a comment/ question about the reason it's clear that T is non-zero. The reason that T is clearly non-zero is that there exist L^infinity functions that have a non-zero limit. ************************ David C. Ullrich === Subject: Re: Anyone read The Topos of Music by Guerino Mazzola? >> Music used to be one of the four (?) parts of Mathematics, > Perhaps you're thinking about the Quadrivium: arithmetic, geometry, music, > astronomy. >> and I'm sure Pythagoras would have approved of using topos theory >> in the study of music. > Maybe. We don't know very much, with certainty, about Pythagoras. (Cf. the > recent thread Pythagoras and beans.) Well, everyone seems agreed that Pythagoras thought there was a close link between music and mathematics, which is not surprising since he (or his cult) seem to have discovered the connection between harmony and ratios. I must say I would take the opposite tack to you; breaking the link between mathematics and music may well have damaged both. -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Fwd: by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id k3LBFY300918 for ; Fri, 21 Apr 2006 07:15:35 -0400 by support2.mathforum.org (8.12.11.20060308/The Math Forum, $Revision: 1.6 secondary) with SMTP id k3LBCOtP004267 for ; Fri, 21 Apr 2006 07:13:01 -0400 for ; Fri, 21 Apr 2006 14:07:37 +0200 -Sensattional revolution in medicine! -Enlarge your penis up to 10 cm or up to 4 inches! -It's herbal solution what hasn't side effect, but has 100% guaranted results! -Don't lose your chance and but know wihtout doubts, you will be i`mpressed with results! Clisk here: http://urbancastaway.info slouch stimulate swung ark brine berkowitz parlance finery seminar culbertson condemnate occult sandalwood absentia beecham sumac insight obnoxious douglas monocotyledon ratify wrest hydrangea caulk comfort shutout cane flaxseed manatee debbie applicable bottom benelux pose biz aorta biota gastrointestinal blat grammarian derelict chateau fireproof advise hereto quadruple knee boyd option split sod you'd casual dissuade === Subject: Re: A topological property <4aaclpFq4p1mU1@news.dfncis.de> <4acjh1Fsol83U1@news.dfncis.de> <4akjluFs67b8U1@news.dfncis.de> <4apur4FucdngU1@news.dfncis.de consider for a space X and a point x in X the set W(X,x) := > { n in N | x is in a a pseudo-connected subset of size } > and let W(X) be the union of all W(X,x). > Can every subset of N be realised as some W(X,x) or at least > some W(X) ? Perhaps already for X=N with some suitable topology? > Actually, it is rather easy (indicationg that my question > was not so deep at all): > Given any set X, fix some element p and take as nontrivial open > sets thoses subsets that have p as an element. Then every subset > of X is pseudo-connected. > This is the include point topology. > Ah o.k., I did not know that. > How many elements does X have? W(X,x) = |X| > How do you realize { 3,4,5 } or { 3,5 } as W(X,x) for some X & x? > Sorry, I confused my own question with something different. Then what is your question? > Accurate statements > If A is pc, then some a in A with pc Aa (1) > negation > A is pc and for all a in A, Aa not pc (2) > Do you mean forall A in (1) ( and exists A in (2) ) ? Yes. Statement and negation. (1) and (2) for all assembled A subset S, some a in A with assembled Aa some assembled A subset S with for all a in A, Aa disassembled (1) is false and (2) is true as nulset is assembled. Thus to modify, restate (1) and (2): for all assembled A nonnul subset S, some a in A with assembled Aa some assembled A nonnul subset S with for all a in A, Aa disassembled Recall connected and assembled are equivalent within completely normals spaces. Hence For discrete spaces, (1) is true. For the reals, (1) is false. > (1) ==> W(X,x) = { |A| in N : some pc A with x in A } > = { n in N : n <= |A| } Correct statement when assembled-C(x) = /{ assembled A : x in A } is finite is (1) ==> W(X,x) = { n in N : n <= |assembled-C(x)| } > Anyway as singletons and nulset is pc, always 0,1 in W(X,x). > Yes, at least one reason why my original question was shallow. > I suspect that (1) ist true but was unable to prove it. If it is, some limitions of finiteness is needed. -- > This week I seem to prove my ability to write nonsense beyond doubt. > The correct statement should read > A is assembled > iff > A is a connected subset of every open set U that contains A. As connected is intrinsic property of the (sub)space, if A is connected subset of some open superset, it's connected subset of every open superset. Now as whole space is open, that is equivlant to A is connected. Thus may I ask, is your week over yet? ;-) ---- === Subject: Hereditarily Homogeneous As space S is Hereditarily Homogeneous when for all open nonnul U, S homeomorphic U. Examples of hereditarily homeogenous spaces Indiscrete spaces Infinite cofinite spaces The rationals Any other suggestions? -- Free Ultra filter space Let S be a set and F a filter on S. Then S with the topology F / { nulset } is called a filter space. If S is countable and F a free ultra filter space, is the filter space (S,F) hereditarily homogeneous? If S is uncountable, is there a free ultra filter that makes (S,F) hereditarily homogeneous? In part, is there a free ultra filter on S with, for all U in F, U and S are equinumerous? In contrast, is there also a free ultra filter on uncountable S, whose sets aren't all equinumerous? ---- === Subject: Re: continuum hypothesis >> a student question: >> Using CH one can construct a subset A of R^2 such that all horizontal >> cross-secitons are countable and all vertical cross-sections are >> co-countable. >> Is the existence of such a set EQUIVALENT to CH? > If your set exists then there is a map f:R ->R such that > 1. |f(R)| <= aleph_1 > 2. for every u in R, |f^-1(u)| <= aleph_0 > I would assume the existence of such a map implies CH. > So for each r in R let H(r) and V(r) be the horizontal and vertical > cross-sections through (r,r), H(r) = {x : (x,r) in A}, > V(r) = {y : (r,y) in A}. So H(r) is countable, V(r) co-countable all r. > Let U be any aleph_1 subset of R. U meets every co-countable set, > so there is a function f:R -> R such that > f(r) in U cap V(r), all r. > Then f(R) is a subset of U, so 1) is satisfied. f^-1(u) is a subset of > H(u) so 2) is also satisfied. > Robert Sheskey >> containing equivalents to CH, as well as consequences of CH. >> For a list of the contents: >> http://matwbn.icm.edu.pl/kstresc.php?wyd=10&tom=4&jez=en >> David Bernier >> P.S.: I browsed through some chapters in PDF format, but >> could't find anything which I could immediately >> relate to the question above. > Interesting reference. My French is pretty weak, but isn't Proposition P_1 > of Chapter 1 relevant? (Sierpinski's 'deux ensembles' would be Edgar's > A and its complement.) Hmmm. Does 'somme' mean 'sum' or 'union'? [...] I believe you're right. The expression somme de deux ensembles used by Sierpinski has fallen out of use. We can formulate two conjectures D2 and D2' as follows: D2: R^2 is the disjoint union of two subsets A, B such that: (a) every horizontal line in R^2 contains at most countably many points of A. (b) every vertical line in R^2 contains at most countably many points of B. D2': R^2 is the union of two subsets A, B such that: (a) every horizontal line in R^2 contains at most countably many points of A. (b) every vertical line in R^2 contains at most countably many points of B. --- Depending on how one interprets somme de deux ensembles in Sierpinski's Proposition P1, one can use either D2 or D2'; then all possible interpretations of somme d'ensembles are covered. D2 => D2' is trivial; D2' => D2 is easy. Then Sierpinski's Proposition 1 can be taken to mean the same thing as conjecture D2. Sierpinski shows that CH is equivalent to D2. For CH => D2, he uses the well-ordering theorem. In modern notation, one can index the real numbers by t_0, ... t_alpha, ... for ordinals alpha less than the initial ordinal of cardinality 2^{aleph_0}. If CH holds, 2^{aleph_0} = omega_1 (or aleph_1). All ordinals < omega_1 are countable. Assuming CH, one defines: A = {(x,y): index(x)<=index(y)} and B = {(x,y): index(x)> index(y)}. So R^2 is the disjoint union of A and B. Also, A contains at most countably many points on any horizontal line in the plane. This establishes part (a) of D2. Next, suppose we have a vertical line in the plane. Then by the definition of B, plus properties of omega_1, part (b) of conjecture D2 is proven. This concludes Sierpinski's proof of CH => D2. For his proof of D2=> CH, see pages 11, 12 of Chapter I, starting at numero 2, which looks like a 2^0 . I'm on my way to understanding his proof in numero 2, but haven't gotten there yet. --- We can think of colouring the elements of A red, and the elements of B blue (this refers to conjecture D2.) Then conditions (a) and (b) imply that any horizontal line contains no more than countably many red points. Similarly, any vertical line contains no more than countably many blue points. Obviously, because A union B = R^2, all points in R^2 have been colored either red or blue. --- This made me wonder whether it is possible to color all points in R^2 either red or blue in such a way that any horizontal line contains no more than finitely many red points, and that every vertical line contains no more than finitely many blue points. (CH is being assumed, but no more than ZFC for the rest of the axioms). David Bernier === Subject: Re: continuum hypothesis On Thu, 20 Apr 2006 10:20:32 -0400, G. A. Edgar >a student question: >Using CH one can construct a subset A of R^2 such that all horizontal >cross-secitons are countable and all vertical cross-sections are >co-countable. >Is the existence of such a set EQUIVALENT to CH? I must be missing something simple. Are the following definitions correct? A horizontal cross-section of A means the intersection of A with a horizontal line. A vertical cross-section is defined analogously, A set is co-countable if its complement is countable. Assuming the above definitions are correct, we can construct a set A as follows: Let B be any countable subset of R and let C be any co-countable subset of R. Let A = C x B. Then any horizontal cross-section has the same cardinality as B and any vertical cross-section has the same cardinality as C. No use was made of CH. What am I missing? quasi === Subject: Re: continuum hypothesis a student question: >Using CH one can construct a subset A of R^2 such that all horizontal >cross-secitons are countable and all vertical cross-sections are >co-countable. >Is the existence of such a set EQUIVALENT to CH? > I must be missing something simple. > Are the following definitions correct? > A horizontal cross-section of A means the intersection of A with a > horizontal line. A vertical cross-section is defined analogously, > A set is co-countable if its complement is countable. > Assuming the above definitions are correct, we can construct a set A > as follows: > Let B be any countable subset of R and let C be any co-countable > subset of R. Let A = C x B. Then any horizontal cross-section has the > same cardinality as B and any vertical cross-section has the same > cardinality as C. > No use was made of CH. > What am I missing? For one thing, I think you have horizontal and vertical mixed up. For another, has the same cardinality as a co-countable set is not the same thing as is co-countable. Mainly: every cross-section does not mean every nonempty cross-section. To restate the OP's proposition: the plane can be partitioned into two sets A and B, so that every horizontal line has countable intersection with A, while every vertical line has countable intersection with B. === Subject: Re: continuum hypothesis On 21 Apr 2006 02:12:19 -0700, Butch Malahide >> On Thu, 20 Apr 2006 10:20:32 -0400, G. A. Edgar >>a student question: >>Using CH one can construct a subset A of R^2 such that all horizontal >>cross-secitons are countable and all vertical cross-sections are >>co-countable. >>Is the existence of such a set EQUIVALENT to CH? >> I must be missing something simple. >> Are the following definitions correct? >> A horizontal cross-section of A means the intersection of A with a >> horizontal line. A vertical cross-section is defined analogously, >> A set is co-countable if its complement is countable. >> Assuming the above definitions are correct, we can construct a set A >> as follows: >> Let B be any countable subset of R and let C be any co-countable >> subset of R. Let A = C x B. Then any horizontal cross-section has the >> same cardinality as B and any vertical cross-section has the same >> cardinality as C. >> No use was made of CH. >> What am I missing? >For one thing, I think you have horizontal and vertical mixed up. >For another, has the same cardinality as a co-countable set is not >the same thing as is co-countable. >Mainly: every cross-section does not mean every nonempty >cross-section. To restate the OP's proposition: the plane can be >partitioned into two sets A and B, so that every horizontal line has >countable intersection with A, while every vertical line has countable >intersection with B. quasi === Subject: Interesting problem on linear independence Let V be a vector space of dimension m over a finite field. Let B be a subset of V satisfying the following two conditions: 1) Any 4-element subset A of B is linearly independent. 2) The elements of B span V. Now, by (2) B needs to contain at least m elements, but the question is, could it contain more? And moreover, what is the maximum number of elements in B? === Subject: Re: Interesting problem on linear independence On Fri, 21 Apr 2006 08:58:41 EDT, Trevor Sykes >Let V be a vector space of dimension m over a finite field. Let B be a subset of V satisfying the following two conditions: >1) Any 4-element subset A of B is linearly independent. >2) The elements of B span V. >Now, by (2) B needs to contain at least m elements, but the question is, could it contain more? And moreover, what is the maximum number of elements in B? It's easy to see that it can contain more than m elements. Let K be any field, and V=K^4, so m=4. Let {e1, e2, e3, e4} be the standard basis for V, and let v = e1 + e2 + e3 + e4. Then B = {e1, e2, e3, e4, v} is a set of m+1 elements such that any 4 are linearly independent. quasi === Subject: 0EM Software boundary=----=_NextPart_000_0008_01C66543.DAB2379F by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with SMTP id k3LD2I315455 for ; Fri, 21 Apr 2006 09:02:19 -0400 by wsip-24-248-9-162.br.br.cox.net (8.9.3/8.9.3) with ESMTP id eDKlaR1ma42m for ; Fri, 21 Apr 2006 06:02:46 -0700 by inbound.stocksrising.com.emailmx.com with SMTP for ; Fri, 21 Apr 2006 06:02:46 -0700 --------------------------------------------------------------------- Special Offer Adobe Video Collection Adobe Premiere 1.5 Professional Adobe After Effects 6.5 Professional Adobe Audition 1.5 Adobe Encore DVD 1.5 $149.95 More Info >> Microsoft 2 in 1 MS Windows XP Pro $99.95 More Info >> Microsoft + Adobe 3 in 1 MS Windows XP Pro Adobe Acrobat 7.0 Professional $149.95 More Info >> Bestsellers Rating: 6 reviews Retail price: $550.00 You save: $480.05 (87%) Our price: $69.95 [Add to cart] Microsoft Windows XP Professional Rating: 8 reviews Retail price: $200.00 You save: $150.05 (75%) Our price: $49.95 [Add to cart] Adobe Photoshop CS2 V 9.0 Rating: 3 reviews Retail price: $599.00 You save: $529.05 (88%) Our price: $69.95 [Add to cart] === Subject: fwd: Financial Market Traderr news by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id k3LDSk318742 for ; Fri, 21 Apr 2006 09:28:46 -0400 by support2.mathforum.org (8.12.11.20060308/The Math Forum, $Revision: 1.6 secondary) with ESMTP id k3LDPdcc006304 for ; Fri, 21 Apr 2006 09:26:40 -0400 Large Marketing Campaign running this weekend! 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As of year-end 2005, the health care industry consumed an astonishing 17.9% of U.S. Gross Domestic Product making Healthcare and related industries a market of staggering size with tremendous growth potential for all of its participants. === Subject: Re: Harvey Friedman on Cantorian pseudomathematics david petry says... >> When I say to your AI that the property of being a prime less than >> 100 is satisfied by exactly 25 natural numbers, it will understand, but >> if I then suggest that the phrase being satisfied by exactly 25 >> natural >> numbers denotes a property it will say no. >I'm not sure what your point is. >Certainly having 25 elements is a property for finite sets of >positive integers. What's an algorithm that given a property, returns true if the property is satisfied by exactly 25 naturals, and returns false otherwise? -- Daryl McCullough Ithaca, NY -- NewsGuy.Com 30Gb $9.95 Carry Forward and On Demand Bandwidth === Subject: Re: Harvey Friedman on Cantorian pseudomathematics Patricia Shanahan says... >> It's unlikely that I will respond to your future posts. >I would rather have an interesting debate than win by default An interesting debate is really not possible with David Petry. In his mind, if you disagrees with him, it has to be because you are an evil Cantorian propagandist (or the brain-washed victims of the propagandists). There is, for David Petry, no such thing as respect for alternative views. -- Daryl McCullough Ithaca, NY -- NewsGuy.Com 30Gb $9.95 Carry Forward and On Demand Bandwidth === Subject: Re: Harvey Friedman on Cantorian pseudomathematics > Can we even, with certainty, identify any > piece of mathematics that will never turn out to be useful, as distinct > from things for which we have not yet found uses. >> Yes. Any mathematical statement that has no observable implications >> will never turn out to be useful. > Number theory, at one time, had no observable implications! > So by Petry's standards it cannot be useful. I think that what David is saying (not that I agree with him by the way) is that observable means that it yields a pi-0-1 statement that can, for each natural number, be (computationally) tested. There are an abundance of pi-0-1 statements in number theory, like the Goldbach conjecture which make observable predictions in the sense described above. As far as I can tell David's rule for good mathematics doesn't depend on the availability of applications. Although that may be Han's rule, which IMO is more extreme and perhaps confusing the issue in this thread. Although David alludes to the computer as a microscope into the world of computation I don't think he means to suggest that a physical computer is required as a means to validate good mathematics; pencil & paper computation is enough to do that. The physical computer is just a tool that enhances our ability to investigate that world. Maybe I've got David's argument all wrong. Other posters have given summaries of David's position only to be told by him later that they didn't get it. I don't think that he is doing a good job of communicating his ideas clearly. === Subject: Re: Harvey Friedman on Cantorian pseudomathematics and false in the first place. > Of course not. Nature decides that. As Virgil has pointed out, Nature is, but *truths about Nature* can only be grasped and distinguished by human minds. In other words, we may grant that there is something independent of the human mind that we call Nature, but any characterization of Nature that we humans can understand and appreciate will not be independent of the human mind. Take a simple example. Suppose we assert that The sun rises in the east Is the truth of this assertion (even assuming we are confined to the earth) independent of the human mind?. You will be tempted to argue that it is so. For you might assert that if all human beings are wiped out, the sun will continue to rise in the east. But think about it. I claim that east and west (or top and bottom, right and left, etc.) require human minds to make sense of. Your thought experiment where all humans are wiped out, and you consider the sun rising, say, to the east of a stone placed at some location, is still *your thought experiment*, and you determine east of the stone by *putting yourself in place of the stone* and determining which direction east is. You can say that a compass placed at the stone will determine the eastern direction without any humans required, but I claim that this is still an illusion. The direction that the compass needle points to is still a human perception, in my view. At the very least this issue requires very deep thought, and I honestly believe that all truths for formal propositions are axiomatic declarations of the human mind, as my logic NAFL dictates. If at all there are any truths independent of the human mind, these are not formal propositions of NAFL theories. For example. truths ABOUT axiomatic theories (such as, say, the consistency of the NAFL version of Peano Arithmetic) are not formalizable in NAFL theories and could possibly be Platonic truths independent of the human mind (i.e., unknowable). === Subject: Re: Harvey Friedman on Cantorian pseudomathematics >. Take a simple example. Suppose we assert that > The sun rises in the east > Is the truth of this assertion (even assuming we are confined to the > earth) independent of the human mind?. You will be tempted to argue > that it is so. For you might assert that if all human beings are wiped > out, the sun will continue to rise in the east. But think about it. I > claim that east and west (or top and bottom, right and left, etc.) > require human minds to make sense of. Your thought experiment where all > humans are wiped out, and you consider the sun rising, say, to the east > of a stone placed at some location, is still *your thought experiment*, > and you determine east of the stone by *putting yourself in place of > the stone* and determining which direction east is. You can say that a > compass placed at the stone will determine the eastern direction > without any humans required, but I claim that this is still an > illusion. The direction that the compass needle points to is still a > human perception, in my view. > At the very least this issue requires very deep thought, and I honestly > believe mumble If a tree falls in the woods, and there is no-one to hear it, do they give a damn? === Subject: Re: Harvey Friedman on Cantorian pseudomathematics >After all, it is only human minds that can distinguish between true >and false in the first place. >>Of course not. Nature decides that. > Nature just is, it does does not make statements about itself which can > be called true or false. There are two kinds of nature: living nature and dead nature. Humans can more or less decide whether they want to live or die. We define true as what promotes living nature. And we define false as what promotes death. Nature decides what the difference is. A statement like 1+1 = 3 is false because in the end it causes bridges to collapse, which promotes death. 1+1 = 2 is true, because in the end it enables us to build bridges that do not collapse, which promotes life. Deny gravity (false) and jump from a tall building: suicide. Accept gravity (true), invent a parachute and: get a life. Disclaimer: in a nutshell. Is this too far fetched for the most of us? Han de Bruijn === Subject: Re: Harvey Friedman on Cantorian pseudomathematics > >After all, it is only human minds that can distinguish between true >and false in the first place. >>Of course not. Nature decides that. > > Nature just is, it does does not make statements about itself which can > be called true or false. > There are two kinds of nature: living nature and dead nature. Humans can > more or less decide whether they want to live or die. We define true as > what promotes living nature. And we define false as what promotes death. What HdB calls true and false, many people will call good and evil. They are not the same thing, at least in English. > Nature decides what the difference is. A statement like 1+1 = 3 is false > because in the end it causes bridges to collapse, which promotes death. > 1+1 = 2 is true, because in the end it enables us to build bridges that > do not collapse, which promotes life. Deny gravity (false) and jump from > a tall building: suicide. Accept gravity (true), invent a parachute and: > get a life. > Disclaimer: in a nutshell. Is this too far fetched for the most of us? At least too far fetched for me. === Subject: Re: Harvey Friedman on Cantorian pseudomathematics >>The problem I have with this approach is a serious doubt about whether >>the more complicated, damped version would ever be invented without >>first understanding unlimited real function composition, in which the >>original function is a valid mathematical object. [ ... snip ... ] > In the end I think Han has only one agenda, he wants to be quoted with > his mantra as he calls it. Of course he's missing the fact that > witty quotes are remembered if they were made by people famous for > other reasons then for giving a witty quote. (Plus the quote has to > actually be witty.) I have no comment on Patricia's thoughtful responses. But I think _you_ don't quite understand what my agenda looks like. Han de Bruijn === Subject: Re: factor polynomial <25874070.1145544710876.JavaMail.jakarta@nitrogen.mathforum.org> <124g1tg3iqmur8d@corp.supernews.com ferg skrev i melding >> How do I factor: >> x^3 + x^2 - 1 >> If w is a root of x^3+x^2-1, then the polynomial >> factors as >> x^2+x^2-1 = (x-w)(x^2+(w+1)x+(w+1)w) >> You can take w to be the unique real root if you >> like, in which case w >> is approximately >> 0.754877666. >> The problem I'm dealing with says: >> x^3 + x^2 -1 / x^2 - 1 = x + 1 + x / >> x^2 -1 >> If you multiply that out, you can see that it is true >> but I don't >> understand how they got it. >> Don't you have to factor the LHS? >> You don't need to factor LHS; you just need >> to apply long division. See: >> http://mathworld.wolfram.com/LongDivision.html >> Also: you need to put parentheses in your >> formulas in order to be understood. >> Where you have written >> x^3 + x^2 -1 / x^2 - 1 = x + 1 + x / x^2 -1 >> write instead >> (x^3 + x^2 -1) / (x^2 - 1) = x + 1 + (x / (x^2 -1)) >> to make clear the order of operations. > The question I was asked was: > Factor: (x^3 + x^2 -1) / (x^2 - 1) > Now I know the answer is:x + 1 + (x / (x^2 -1)) > But what if I didn't know that? How could I factor this? > Hi. > No, your original question was: How do I factor (x^3+x^2-1). > Then you introdused the division by (x^2-1) and the correct solution to the > division and asked how to do that. We have shown you URLs where long > division is treated. > Karl-Olav === Subject: Re: how to proof noneuclid geometry Suppose you had a rectangle in the hyperbolic plane. Cut it into two triangles by connecting opposite corners with a geodesic. Then the angle sum of the two triangles must add up to 2*pi. Therefore you can say something about one of the triangles, leading to a contradiction. === Subject: Rings of strictly rational numbers with the usual add'n. and mult. Does anybody know of any results pertaining to a ring R with the following properties: 1. Addition and multiplication in R is the usual addition and multiplication of rational numbers; 2. For all X and Y in R, X and Y are strictly rational (i.e. non-integral); 3. For all X and Y in R, X + Y is also strictly rational; 4. For all X and Y in R, (1/X) and (1/Y) are strictly rational; 5. For all X and Y in R, (1/X) + (1/Y) is also strictly rational; and 6. (X * Y) = k, k an integer constant? === Subject: Re: Rings of strictly rational numbers with the usual add'n. and mult. : Does anybody know of any results pertaining to a ring R with the : following properties: : 1. Addition and multiplication in R is the usual addition and : multiplication of rational numbers; So a priori R is a subset of Q? : 2. For all X and Y in R, X and Y are strictly rational (i.e. : non-integral); So if m/n is in R then m/n + ... + m/n (n times) = m is in R. Oh dear. : 3. For all X and Y in R, X + Y is also strictly rational; : 4. For all X and Y in R, (1/X) and (1/Y) are strictly rational; : 5. For all X and Y in R, (1/X) + (1/Y) is also strictly rational; and : 6. (X * Y) = k, k an integer constant? Moot. Justin === Subject: Re: Rings of strictly rational numbers with the usual add'n. and mult. > Does anybody know of any results pertaining to a ring R with the following properties: > 1. Addition and multiplication in R is the usual addition and multiplication of rational numbers; > 2. For all X and Y in R, X and Y are strictly rational (i.e. non-integral); > 3. For all X and Y in R, X + Y is also strictly rational; > 4. For all X and Y in R, (1/X) and (1/Y) are strictly rational; > 5. For all X and Y in R, (1/X) + (1/Y) is also strictly rational; and > 6. (X * Y) = k, k an integer constant? Doesn't a ring have to contain 0? Properties 2-5 all forbid 0 from being in R. So this is no ring. Are you just looking for a subset of the rationals that has all these properties? You didn't specify whether or not R is closed under addition. If it is then it's empty, since adding X to itself n times, where n is the positive denominator of X, produces an integer, in contradiction to properties 2 and 3. If R isn't closed under addition then there are lots of examples, all of which are sets with two elements, if I understand property 6 correctly. === Subject: Re: Rings of strictly rational numbers with the usual add'n. and mult. > Does anybody know of any results pertaining to a ring R with the following properties: > 1. Addition and multiplication in R is the usual addition and multiplication of rational numbers; > 2. For all X and Y in R, X and Y are strictly rational (i.e. non-integral); This isn't a ring: it doesn't have an additive identity. Even if you decide allow zero in as a special case, you still can't have any other members: suppose r in R expressed in lowest terms is p/q, p <> 0, then since R is closed under addition, 2r, 3r, ... qr, ... are all members of R. But qr = p, an integer, contradiction; so there is no non-zero r in R. Oh well. Also, 'strictly rational' is to my mind not a particularly good way of saying 'non-integral'. -- Larry Lard Replies to group please === Subject: Re: Rings of strictly rational numbers with the usual add'n. and mult. > Does anybody know of any results pertaining to a ring R with the following properties: > 1. Addition and multiplication in R is the usual addition and multiplication of rational numbers; So R is a subring of Q? > 2. For all X and Y in R, X and Y are strictly rational (i.e. non-integral); A ring has at least an additive identity, and 0 is an integer. Perhaps you want to say every _non zero_ element of R is not an integer? Even this doesn't work though. Let r be a (non zero) member of R. Then we can write r=a/b for some integer a and positive integer b. Then r+r+...+r (b many times) is a member of R. i.e. a is a member of R. > 3. For all X and Y in R, X + Y is also strictly rational; > 4. For all X and Y in R, (1/X) and (1/Y) are strictly rational; > 5. For all X and Y in R, (1/X) + (1/Y) is also strictly rational; and > 6. (X * Y) = k, k an integer constant? Bertie === Subject: Re: Rings of strictly rational numbers with the usual add'n. and mult. > Does anybody know of any results pertaining to a ring R with the following properties: > 1. Addition and multiplication in R is the usual addition and multiplication of rational numbers; > 2. For all X and Y in R, X and Y are strictly rational (i.e. non-integral); > 3. For all X and Y in R, X + Y is also strictly rational; > 4. For all X and Y in R, (1/X) and (1/Y) are strictly rational; > 5. For all X and Y in R, (1/X) + (1/Y) is also strictly rational; and > 6. (X * Y) = k, k an integer constant? I assume that in (4) and (5) you intend to exclude 0. I'm also going to assume that by (6) you mean something like For all X and Y, there exists an integer k such that XY = k. If not (that is, if you mean that there exists a k such that for all X and Y, XY = k), then you get the following situation: Suppose there exists a rational X in R; then both 1/X and X + 1/X = (X^2 + 1)/X are in R. But then X * 1/X = 1 = k, but X * (X^2 + 1)/X = (X^2 + 1) = k as well. That forces X^2 + 1 = 1 and X = 0. But 1/0 can't be a rational with the usual ... multiplication of rational numbers. If I've interpreted (4), (5), and (6) correctly, it looks R is the zero ring (the ring containing only 0). Here's why: If there is a number other than 0 (say, x) in R, then its inverse 1/x is also in R; that gives x * (1/x) = 1 in R. But then 2 (1 + 1) is in R, so that 1/2 is in R. This implies (1/2) * (1/2) = 1/4 is in R. But we said that for all x,y in R, xy is an integer. Contradiction. 0 is allowed to be in R, though, because it's not subject to rules (4) and (5), and (if it's the only number in the ring) it fits rules (1),(2),(3), and (6). So I think you found the zero ring. Matt === Subject: Any algorithm for: There is the following expression Min ( Sum ((A - B) ^ 2) ) m = 0-P j = 0-Q n = 1 to R P R are constant and > 30000. Q are content and < 100 Is there any alrithm to reduce the computer time. Say normal loop will take P * Q * R times. === Subject: Re: Any algorithm for: ... > Min ( Sum ((A - B) ^ 2) ) > m = 0-P j = 0-Q > n = 1 to R > P R are constant and > 30000. > Q [is constant] and < 100 > Is there any [algorithm] to reduce the computer time. > Say normal loop will take P * Q * R times. I think mehdi.avdi meant to suggest overlapping data input time with computation time. It's true that you can work with in half but not affecting computation time. Does your data have any special characteristics (Eg, monotonicity within A or B, or a model such as g*j^2 + h*j + k > A_mj)? Even if it doesn't, you could precompute sum_j(A ^2) = S_m and sum_j(B ^2) = T_n and max values U_m of A and V_n of B, and then use sum_j ((A - B) ^ 2) = D_mn = S_m + T_n - 2*sum_j(A*sum_j(B) >= E_mn = S_m + T_n - 2*Q*U_m*V_n. If E_mn is less than your best previously found min D_.., then compute the exact value of D_mn. This of course could be sharpened a little by dividing your max value numbers more-finely; for example, if U'=max over first 1/3 of A, U=max over second 1/3, and U' max over last 1/3, and similarly for B, and then compute, compare, etc with D_mn >= F_mn = S_m + T_n - (2/3)*Q*(U'*V' + U*V + U'*V'). -jiw === Subject: Re: Any algorithm for: <44491A0B.25E88567@pat7.com> === Subject: Re: Any algorithm for: well I dont know if it will be helpful but one thing you can do is you already have A then start reading B one by one from inout and by the time reduce from A. then get the min. > There is the following expression > Min ( Sum ((A - B) ^ 2) ) > m = 0-P j = 0-Q > n = 1 to R > P R are constant and > 30000. > Q are content and < 100 > Is there any alrithm to reduce the computer time. > Say normal loop will take > P * Q * R times. === Subject: Re: Any algorithm for: Could you please explain more? === Subject: Re: what is self-consistent equation? > Does anyone know the actual definition of what self-consistent > equations are? Couldn't find anywhere... > > A set of equations with one or more solutions is consistent. > A set of equations with no solutions is inconsistent. No, this seems to be a term the physicists use. I have no idea what it > means, but I think it implies more than just consistent. Can someone who knows chip in their 2 cents? -- > Ron Bruck No it is not a term physicists use. For example if you have two lines > in the plane. Then either they cross at one point, they never cross, or > they overlap. But is never heard of self-consistent, and i read quite a > I read quite a bit of math books too. It's my profession. > OK, you don't accept my general knowledge of mathematics, so let's > Google it. Under self-consistent I find references to atomic > physics, self-consistent field computations; to the Hartree-Fock > electronic wave-function, on self-consistent one-electron orbitals; > self-consistent pseudopoentials in a generalized eigenvalue formalism, > Flory Theory and self-consistent field theory. > There's also some stuff on high-resolution shoreline databases, and > helping one deal, learn, know and master self. Too touchy-feely for > me; I've known who I was since I was 17 years old. > Bah. Ignorance is bliss, but it is a fundamental error of discourse to > assume that just because YOU have only seen a term used a certain way, > that's the only way ANYBODY uses it. > -- > Ron Bruck > What is your problem dude? Didn't take you medicine that day? Anyway he > was refering to self consistent equations not fields or electrons. And > i made no idication that you are necessarily wrong. If i would have i > would have said that the idiot above me with self esteem issues is > completely wrong. Typical for some one who thinks high of himself to > throw a fit for nothing. Eat my shorts. No it is not a term physicists use seems to be pretty categorical. You're saying I'm WRONG, based on an example (two lines in the plane) which any high-school sophomore knows. Given your invitation to eat my shorts, I have to wonder if you ARE a high-school sophomore. The physicists also use the term self-consistent equations. In the papers I've looked at I don't seem to find an actual definition; it seems to be a term of art rather than a technical term. I'm still waiting for the 2 cents worth OF SOMEONE WHO KNOWS. -- Ron Bruck === Subject: How Do I Find This Function? I'm trying to figure out what constraints the assumption of factorability places on an arbitrary polynomial of degree 2n, ie. Sum(i=1,n)[a_i v^2i] = Prod(i=1,n)[lambda_i + b_i*v^2i] I know there will be an equation a_i=g_i(lambda_j,b_k), but I just can't see how to get g_i. I thought it involved collapsing the outer product lambda_i*b_k, but that doesn't account for the powers of lambda, ie. lambda_i*lambda_j*...*lambda_n, and similar terms with multiplicities of lambda's and b's. I'm overextended by the size of this table, because it seems to be larger than an outer product. It's some kind of table of all possible outer products, or something like that, and I don't know how to write that down in standard notation. All I know about this thing is that when lambda_i=b_j=1, it reduces to Pascal's triangle. === Subject: Re: Help me understand an analysis problem which makes no sense >> The context is Borel functions on the real line. Let f be in L^2 and >> g be in L^1. Let T:L^2->L^2 be defined by the convolution Tf=g*f. >> Then T is a continuous linear transformation. For each polynomial p >> define mu(p)=. Then mu defines a Radon measure and hence is >> given by a Borel measure on the line. Find this measure explicitly, as >> the image of an absolutely continuous measure. > (SNIP) > Now, what we are trying to find is a Radon measure mu such that > = int p d mu > (SNIP) > -- > Ahh, this was the biggest help. > Why didn't the problem just SAY this? The author would probably say, But I DID say it. > This is actually a very simple > and easy problem to state, as you just stated, but the problem ran an > entire paragraph and was utterly indecipherable to me. Math is > supposed to be about clarity, but some authors go out of their ways to > make simple problems like this impenetrable :( > Well, I let F denote the inverse fourier transform of f, G the inverse > fourier transform of G, and I let U be the map sending h to G*h, then > it's easy to see p(T)f is the inverse fourier transform of p(U)F. So > an easy calculation gives > = 1/2pi int hat{f}(k) (p(U)F)(k) dk. > Is this the sort of answer the problem is asking for? Definitely on the way. I assume that when you say G*h you mean multiplying h by G. Then it becomse possible to express the r.h.s. as an integral where p is involved as a plain ol' function, contributing a numeric factor to a numeric integrand. Hint: see if you can make G appear explicitly in the integrand. -- Chris Henrich http://www.mathinteract.com God just doesn't fit inside a single religion. === Subject: Mortgage - refinance it! boundary=----=_NextPart_000_0000_F297B2F2.8AF31658 by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with SMTP id k3LFjw305849 for ; Fri, 21 Apr 2006 11:45:59 -0400 --------------------------------------------------------------------- Let the banks compete for your refinance or debt consolidation! No obligation. Just a quick form to fill out: http://www.deltacav.com/form - If you OWN real estate - LOWER your monthly payments! - Programs for every credit situation. - Credit history is NOT a factor. __________________ To be taken out, go here === Subject: Re: Retraction >Which is the difference between the term retraction and the locution left >inverse? Just consider the typical usage: Once again James Harris has issued a retraction of his previous claim. The mathematicians came in prose but left in verse. HTH! === Subject: Re: Retraction : Which is the difference between the term retraction and the locution left : inverse? If A is a subset of B, then a retraction is a left inverse of the inclusion map from A into B. However, retraction is usually only used in geometric/topological contexts, where A and B are topological spaces (perhaps with constraints) and the map is required to be continuous (or even stronger conditions, depending on context.) But we can talk about left inverses (if they exist) of maps which are not inclusion maps, so left inverse is more general. Ted === Subject: Re: Beliefs Create Reality, Even In Mathematics. >My take on that is that wishing is passive. You don't take any action >towards reaching the goal, you just sort of hope it comes to you all by >>Well, then I still don't see the difference between willing something >>and wishing for it. >Didn't you comprehend my sentence? Probably not, since you interrupted >it to respond. If you read the whole thing from start to finish without >distracting yourself, you might find that it expresses quite clearly and >unambiguously what that difference is. >> Except that because you have no insifght on the matter, and have >> consistently refused to learn from those who do (dull boy), you only ever >> trot out the opinions of other people which you have read. >> A > And what have you trotted out.... nothing but bluster and blather and > egotistical bull. > Give am some meat to chew on... because thus far you supply nothing > showing you have any experience or knowledge worth listening to > whatsoever. > -Douglas so this means that my comment about Tom is untrue. Right? A === Subject: Re: Beliefs Create Reality, Even In Mathematics. >The plural of anecdote isn't data. You still haven't provided good >>reason to believe it's impossible, just that you haven't done it. >>Yet, anecdotal evidence is all you've presented, too. If his isn't >persuasive, neither is yours. >>at least they present anecdotes Tom, >Anybody can tell a story. That doesn't make it true. >A public, verifiable demonstration is way better than a story. For >instance, you tell a story about your ability to psychometrise at a >distance, but I predict, without any psychic powers whatsoever, that no >such demonstration of your alleged power will ever be performed. >> So when will you tell us about your experience in Magic Tom? A simple >> enough request, after all, you did say you dont have any except your >> arguments... >> Didn't you? >> Oh... sorry. you now deny it of course. Well, here is your own passage >> from your post on 16/4... >My authority (such as it is) springs solely from my arguments and the >evidence upon which they rest, not from some club I may or may not have >belonged to. >> Simple enough. You dont have any *actual* experience at all it seems. Can >> we be sure you were telling the truth? I suspect we can, but in case you >> want to clarify... what *exactly* is your experience in Magic Tom? >> A > You stated previously you would discuss your experience... but you keep > dodging doing so by attacking Tom. > Well put up or shut up. > What is your experience and why should anyone believe anything you say > without knowing what that experience is? They should not but you seem > to think otherwise, which appears to put forth that you think everyone > but yourself is a dolt... when it is the other way around. > -Douglas Does this mean Tom *does* actually have some practical experience of Magic? Its just that he doesn't agree with you. A === Subject: Re: Beliefs Create Reality, Even In Mathematics. >His belief in his worth is *way* above his >>actual market value. >>A >The same can be said of you aiwASS/archASS/apotheosisASS. >> ApotheosAss? > Heh, I've been playing with how to exactly spell that one... > I last settled on apotheoASS but I'm not sure if I like it > better than apotheosisASS.... Apo the o ASS or Apotheos is ASS. > Either works when considering the subject of the matter but > I'm now leaning toward the latter. > -Douglas If you find yourself leaning dodgy, I recommend you alternate arms from time to time. A === Subject: Re: Beliefs Create Reality, Even In Mathematics. >> The difference is both simple in principle and complex in application,. >> The Will is a function of Godhead speaking through the Soul (Neschemah). > De headbone connected to de neckbone. Praise de name ob de Lawd. Well done Tom. Music therapy is supposed to be quite effective in extreme cases. A === Subject: Re: Beliefs Create Reality, Even In Mathematics. >> So when will you tell us about your experience in Magic Tom? > I won't, unless there seems to be a good reason to cite some specific > example, and there very seldom is such a reason. Your demands for it do > not amount to one. You mean you didnt state that you dont actually have any then? > And when will you tell us about *your* experience in magic? I will show you mine after you have shown me yours. A === Subject: Re: Beliefs Create Reality, Even In Mathematics. >> Nice try at diversion. Still not owning up to having no experience? OK > Since my claim of superior magical experience is not the basis of my > arguments, it's irrelevant for me. Since your claim of superior > experience is solely what you base your arguments upon, it is highly > relevant for you. But you don't seem to actually have any, so every > argument you make is based on nothing at all. No evidence, no > rationality, and no verifiable experience. All you have are empty and > unsupported claims. >> You can influence the probability of any event. > With your mind alone? Can you demonstrate that? >> Did I say that? No. > So that magical current you supposedly unleashed which was supposed to > destroy me actually is just bull, eh? Seeing as how it's been a > year since you proclaimed this demonstration of your magical powers, and > I'm still right here doing just as I've always done, the conclusion that > you've merely been spouting bull seems pretty accurate. So this means I *did* say it right? > So you can't actually influence events at a distance by magick? Or do you > subscribe to Crowley's definition of magick as any intentional act? Or > what? Well I tried to influence you to tell us all about your *actual* experience in Magic Tom. That appears to have failed (for reasons I am sure we all understand). So I suppose Magic doesn't work eh? A === Subject: Re: Beliefs Create Reality, Even In Mathematics. >> You got your Magical knowledge from Playboy? > I knew you'd say something as stupid as this. > You're are predictable as clockwork. Hmmm. The trick is to predict it *before* I say it Tom. That is the whole point. I doubt that your continually predicting *after* the event will impress many people. A === Subject: Re: Beliefs Create Reality, Even In Mathematics. >> But you don't believe you will ever achieve this, so you believe you are >> indeed wasting your time. > Way to tell me what I actually think. You told me first. I'm just feeding back the same message I'm getting from you. You say your dreams of success at this are nonexistent. Doesn't that mean you have no dreams of success? Doesn't the lack of even a dream of success indicate your conviction that you will fail? If not, what *does* it mean? >> Since you don't dream of this (it's nonexistent dreams for you), you >> believe it to be a waste of time. So, on the one hand, you decry a waste >> of time and on the other you assiduously waste your time. Seems pretty >> crazy to me. Have you ever been hospitalized and medicated for these >> crazily contradictory feelings? > /me can't catch sarcasm the size of a basketball Does that mean no? >we all have our definitions of crazy. >> Yes, we do. They differ widely. Yet you seem to believe that *your* >> definition should be sufficient to have people locked up and drugged >> against their will simply because they believe in what they assiduously >> practice while you assiduously practice the same thing even though you >> say you don't believe in it. > I didn't say by my definition, I just said that I've noticed that people > who think belief is more important than action end up hospitalized and > drugged against their will. I didn't say I make up the rules or > definitions, I just observe them. My point is that your conclusion is in error and this is not how things work at all. >I once read in a psych book that crazy was any behaviour significantly >different than the average >> That's because crazy is not a clinical psychological term. It's a >> social term for anything that doesn't make sense to you. Now fearing >> people who believe or act differently from what you're used to and >> treating them as if their differences were a disease has a label in the >> terminology of psychology. It's called xenophobia. > How do you feel about the DSM having about sixteen hundred too many > classifications of behaviour as abnormal? How do you feel about the > mistakes that clinical psych has made and then had to recant? Like > homosexuality for example... The DSM reflects the current thought and the results of current research at the time of its publication. Science is a constant process of refinement and revision. I am not at all discouraged by changes in the DSM over time. In fact, I am encouraged by it. I would be much more discouraged if the DSM never changed at all. That would indicate an altogether unwarranted certainty that the authors were infallible. >I've personally noticed that in north america if you don't jump on the >dogmatic bandwagon in some respects (the market is the king, private >property rules all), you will quickly be labelled crazy. >> Do you agree with that outlook? Isn't that what *you're* doing? > I'm again pointing out that this is something I've observed. Is it what > I'm doing? I don't think so. I would ask for some compelling evidence. I > don't see that as dogmatic, but as something a reasonable person would > demand. If you choose to ignore reason, I would say we've got nothing to > talk about. conformity with the view you were expressing. Prhaps you were not sufficiently clear. > I don't agree that beliefs create reality, > I do believe that beliefs can alter the perception of reality. > I agree that actions create reality > and that actions can alter the perception of reality. I agree. It can go even further, though. Actions based upon beliefs can result in an outcome consistent with the belief in many cases, even when the belief was initially false. This is especially true in social situations. A belief that you are being plotted against, for instance, may cause you to act in a hostile and suspicious manner towards others, which will influence others to act in a hostile and suspicious manner towards you, resulting in their plotting against you. Acting upon a false belief may very well cause that belief to become true. There's nothing necessarily paranormal about it, but it works pretty well. === Subject: Re: Beliefs Create Reality, Even In Mathematics. >>A little gonzo is not nearly gonzo enough to get someone locked up. >>Everybody is a little gonzo. It's what keeps us from being boring >>replicas of one another. >I'm not certain I can agree here... I've become decidedly more boring and >less gonzo in the last 10 years, and I'm really enjoying the boring >replica stuff. >> Except for your bizarre attempts to float pencils in the air and practice >> Hindu pop-meditation. >> But maybe you were a whole lot more gonzo than that a few years ago. > indeed I was. I'm surprised so few vestiges remain. A little nonsense now and then is cherished by the wisest men. === Subject: Re: Beliefs Create Reality, Even In Mathematics. > I don't know what you mean by unlike any seen before, unless you > mean, unlike any seen before by me, in which case you're simply > making an arguement from ignorance; it doesn't follow from your not > having experienced something, that thing doesn't exist. The best > explanation of the current body of parapsychological research is that > PK on random number generators, PK on living systems, ESP on the > ganzfeld, and Remote Viewing are real psi phenomena. Nonsense. In decades of lab work, Parapsychology has yet to come up with a single reproducible demonstration that any PK phenomenon even exists. Parapsychology cannot meet the standard we expect of laboratory science. Instead, they offer excuses and special pleading. -- --Bryan === Subject: Re: Beliefs Create Reality, Even In Mathematics. >> I don't know what you mean by unlike any seen before, unless you >> mean, unlike any seen before by me, in which case you're simply >> making an arguement from ignorance; it doesn't follow from your not >> having experienced something, that thing doesn't exist. The best >> explanation of the current body of parapsychological research is that >> PK on random number generators, PK on living systems, ESP on the >> ganzfeld, and Remote Viewing are real psi phenomena. > Nonsense. In decades of lab work, Parapsychology has yet > to come up with a single reproducible demonstration that any > PK phenomenon even exists. Parapsychology cannot meet the > standard we expect of laboratory science. Instead, they offer > excuses and special pleading. Not exactly true. Dean Radin, for example, gets consistent positive results, but they don't seem to be reproducible independently of his team. Of course, the effects he's studying are very, very small and could be nothing more than normal variance writ large. === Subject: Re: Beliefs Create Reality, Even In Mathematics. >>Nonsense. In decades of lab work, Parapsychology has yet >>to come up with a single reproducible demonstration that any >>PK phenomenon even exists. Parapsychology cannot meet the >>standard we expect of laboratory science. Instead, they offer >>excuses and special pleading. > Not exactly true. Dean Radin, for example, gets consistent positive > results, but they don't seem to be reproducible independently of his team. Yeah, that would be non-reproducible. > Of course, the effects he's studying are very, very small and could be > nothing more than normal variance writ large. He has a huge amount of data from his on-line tests. I wonder why he doesn't report it. -- --Bryan === Subject: Re: Beliefs Create Reality, Even In Mathematics. > well, TM as a 200$ course taught over 5 days may be gone... I've > explained it to people in about 15-20 minutes... sometimes less. > With some people I even give them a secret word to mentally chant > over and over and over... >> Some folks need the secrecy. The same is true in magick as well. >> With any luck, they get over it eventually, though some never do. > The problem is being caught up in the word secret. I suggest the > word private is a better fit to what is happening. But of course I > am also well aware of the thought vigilantes here in the US. Private seems much more appropriate, I agree. >> TM, I suppose, at least the kind that could be taught in 15-20 >> minutes, might be of benefit to both mind and body, but it fails to >> address the fundamental problem of man and his relationship to the >> universe. That's the only drawback I can see. > Satyr, Charity is just not easy for everyone yet. So it would seem. === Subject: Re: Expected area of a triangle >Select three points from a uniform distribution on the unit square. >What is the expected area of the triangle spanned by the three points? Some followups have been posted, but none get at the simplicity of the problem. Independence properties and uniformity properties can be used to great advantage. Three points are selected at random. This means that their X and Y coordinates are independent and have a uniform distribution from 0 to 1. Now if one has k independent observations on the unit interval, the k+1 intervals, the distribution on the unit simplex is uniform. All we use from this is that the expected values are equal. Now the interval between the smallest and largest is 2 of the 4 intervals, and so has expected value 1/2. Similarly, the expected value of the vertical range is 1/2, so the expected area of the smallest parallel rectangle containing the points is 1/4. Then the expected area is 1/4 the expected area of a random inscribed triangle in the unit square; that means that there is a point on each boundary side, and the two unrestricted coordinates are independent. Also, there must be at least one point at a corner, and the probability of two points at diagonally opposite corners is 1/3, as the Y coordinates of the points are independent of the X. We may then assume that in the transformed problem one vertex is at (0,0), and that with probability 1/3 another vertex is at (1,1) with the third point at random in the interior, and with probability 2/3, one vertex is at (a, 1) and the other at (1, b), and and b random. If one has a triangle with vertex at (0,0) in the positive quadrant, and the lower other vertex is at (u, v) and the upper vertex at (r, s), the area is .5*((u*s - v*r). In the case where one vertex is at (1, 1), we may by reflection assume that the third point is at (u, v), where u > v. Then the area is .5*(u-v), and as u and va are independent uniform (0, 1), E(u-v) = 1/3. In the other case, the area is .5*(1 -a*b), and its expectation is .5*(3/4) = 3/8. So the expected area for the reduced problem is 1/3*1/6 + 2/3*3/8 = 1/18 + 1/4 = 11/36, and so for the original problem it is 11/144. -- 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: Expected area of a triangle >On Fri, 21 Apr 2006 00:06:13 +0300, Toni Lassila Select three points from a uniform distribution on the unit square. >>What is the expected area of the triangle spanned by the three points? >See if this is what you asked for: > Very nice. I was wondering if there was some trick to doing the integrals with a CAS. === Subject: Re: Expected area of a triangle <5lhg42hhov196p6heank7h2dcgh2obfbtg@4ax.com> Select three points from a uniform distribution on the unit square. >>What is the expected area of the triangle spanned by the three points? >See if this is what you asked for: > integrals with a CAS. Well, that reference talks about dividing up into 142 pieces which sounds scary. In fact I think you can do it a lot more easily by ordering the points (x1,y1), (x2,y2), (x3,y3) such that x3 <= x2 <= x1 (remembering to multiply the end result by 6 to account for all permutations). Then you need to consider y2 below the line (x1,y1) to (x3,y3) or above it, but by symmetry the two cases must contribute the same expectation, so just consider one case and multiply the end result by a further factor of two. Laying it out this way and taking the integrals over the appropriate ranges results in integrals than can be computed exactly (and fairly easily) to yield the same answer of 11/144. === Subject: Re: Charity addresses the fundamental problem of humanitys relationship to the universe? >Some folks need the secrecy. The same is true in magick as well. With >any luck, they get over it eventually, though some never do. >> The problem is being caught up in the word secret. I suggest the word >> private is a better fit to what is happening. But of course I am also >> well >> aware of the thought vigilantes here in the US. >TM, I suppose, at least the kind that could be taught in 15-20 >minutes, might be of benefit to both mind and body, but it fails to >address the fundamental problem of man and his relationship to the >universe. That's the only drawback I can see. >> Satyr, Charity is just not easy for everyone yet. > Charity with no strings attached is doled out far too often to people who > refuse to take responsibility for themselves. > Creating the situation where the safety net is a convenient hammock, and > no contribution to society is required. You don't quite get that Charity is Love on purpose, not just giving out material contributions. 1 Corinthians 13 1 Though I speak with the tongues of men and of angels, and have not charity, I am become as sounding brass, or a tinkling cymbal. 2 And though I have the gift of prophecy, and understand all mysteries, and all knowledge; and though I have all faith, so that I could remove mountains, and have not charity, I am nothing. 3 And though I bestow all my goods to feed the poor, and though I give my body to be burned, and have not charity, it profiteth me nothing. 4 Charity suffereth long, and is kind; charity envieth not; charity vaunteth not itself, is not puffed up, 5 Doth not behave itself unseemly, seeketh not her own, is not easily provoked, thinketh no evil; 6 Rejoiceth not in iniquity, but rejoiceth in the truth; 7 Beareth all things, believeth all things, hopeth all things, endureth all things. 8 Charity never faileth: but whether there be prophecies, they shall fail; whether there be tongues, they shall cease; whether there be knowledge, it shall vanish away. 9 For we know in part, and we prophesy in part. 10 But when that which is perfect is come, then that which is in part shall be done away. 11 When I was a child, I spake as a child, I understood as a child, I thought as a child: but when I became a man, I put away childish things. 12 For now we see through a glass, darkly; but then face to face: now I know in part; but then shall I know even as also I am known. 13 And now abideth faith, hope, charity, these three; but the greatest of these is charity. (KJV) -- meltdarok http://hometown.aol.com/meltdarok/ === Subject: Re: Charity addresses the fundamental problem of humanitys relationship to the universe? >Some folks need the secrecy. The same is true in magick as well. With >any luck, they get over it eventually, though some never do. >> The problem is being caught up in the word secret. I suggest the word >> private is a better fit to what is happening. But of course I am also >> well >> aware of the thought vigilantes here in the US. >TM, I suppose, at least the kind that could be taught in 15-20 >minutes, might be of benefit to both mind and body, but it fails to >address the fundamental problem of man and his relationship to the >universe. That's the only drawback I can see. >> Satyr, Charity is just not easy for everyone yet. > Charity with no strings attached is doled out far too often to people who > refuse to take responsibility for themselves. > Creating the situation where the safety net is a convenient hammock, and > no contribution to society is required. Quite right. Good observation. A === Subject: Re: Charity addresses the fundamental problem of humanitys relationship to the universe? >>Some folks need the secrecy. The same is true in magick as well. With >>any luck, they get over it eventually, though some never do. > The problem is being caught up in the word secret. I suggest the word > private is a better fit to what is happening. But of course I am also > well > aware of the thought vigilantes here in the US. >>TM, I suppose, at least the kind that could be taught in 15-20 >>minutes, might be of benefit to both mind and body, but it fails to >>address the fundamental problem of man and his relationship to the >>universe. That's the only drawback I can see. > Satyr, Charity is just not easy for everyone yet. >> Charity with no strings attached is doled out far too often to people >> who refuse to take responsibility for themselves. >> Creating the situation where the safety net is a convenient hammock, and >> no contribution to society is required. > Quite right. Good observation. Get real, that is so much Western bull it's not funny. http://www.cartercenter.org/healthprograms/healthpgm.htm -- meltdarok http://hometown.aol.com/meltdarok/ === Subject: Re: Charity addresses the fundamental problem of humanitys relationship to the universe? > Charity with no strings attached is doled out far too often to people who > refuse to take responsibility for themselves. In order to be effective, charity requires as much thought as any other kind of problem-solving. The flaw to most giving is it's thoughtlessness. It's our inability or unwillingness to deal with the real problem. === Subject: Re: Charity addresses the fundamental problem of humanitys relationship to the universe? >> Charity with no strings attached is doled out far too often to people >> who refuse to take responsibility for themselves. > In order to be effective, charity requires as much thought as any other > kind of problem-solving. The flaw to most giving is it's > thoughtlessness. It's our inability or unwillingness to deal with the > real problem. That's one of the reasons personal contributions may be best if it is to some of the watch dog approved organizations. -- meltdarok http://hometown.aol.com/meltdarok/ === Subject: Re: Charity addresses the fundamental problem of humanitys relationship to the universe? >TM, I suppose, at least the kind that could be taught in 15-20 >minutes, might be of benefit to both mind and body, but it fails to >address the fundamental problem of man and his relationship to the >universe. That's the only drawback I can see. >> Satyr, Charity is just not easy for everyone yet. > Charity with no strings attached is doled out far too often to > people who refuse to take responsibility for themselves. > Creating the situation where the safety net is a convenient hammock, > and no contribution to society is required. Heh. I doubt that is what he meant by Charity: Though I speak with the tongues of men and of angels, and have not charity, I am become as sounding brass, or a tinkling cymbal. And though I have the gift of prophecy, and understand all mysteries, and all knowledge; and though I have all faith, so that I could remove mountains, and have not charity, I am nothing. And though I bestow all my goods to feed the poor, and though I give my body to be burned, and have not charity, it profiteth me nothing. Charity suffereth long, and is kind; charity envieth not; charity vaunteth not itself, is not puffed up, doth not behave itself unseemly, seeketh not her own, is not easily provoked, thinketh no evil; rejoiceth not in iniquity, but rejoiceth in the truth; beareth all things, believeth all things, hopeth all things, endureth all things. Charity never faileth: but whether there be prophecies, they shall fail; whether there be tongues, they shall cease; whether there be knowledge, it shall vanish away. For we know in part, and we prophesy in part. But when that which is perfect is come, then that which is in part shall be done away. When I was a child, I spake as a child, I understood as a child, I thought as a child: but when I became a man, I put away childish things. For now we see through a glass, darkly, but then face to face: now I know in part; but then shall I know even as also I am known. And now abideth faith, hope, charity, these three; but the greatest of these is charity. - 1 Corinthians 13 The word here translated as charity is in some more modern translations rendered as love. The fact that you would immediately jump to the conclusion that some freeloader was skulking about in search of a handout rather handily illustrates my point. Your life was a gift bestowed upon you without any strings attached, as is the air you breathe and the sun that shines upon your face. No one charges us fees for daisies and dandelions, sunrise and sunset, the winter snow or a gentle rain, a mother's smile for the newborn at her breast. Some of the greatest things in life are like that, given freely by the universe we inhabit, simply because we are here and alive. My spiritual practice has left me accutely aware of these things, the value of these things, and I am most grateful. This is a major part of our relationship to the world around us. Without internalizing this principle and giving of ourselves in the same spirit, you miss out on one of the greatest joys of being alive. === Subject: Re: Charity addresses the fundamental problem of humanitys relationship to the universe? > Your life was a gift bestowed upon you without any strings attached, > as is the air you breathe and the sun that shines upon your face. No > one charges us fees for daisies and dandelions, sunrise and sunset, > the winter snow or a gentle rain, a mother's smile for the newborn at > her breast. Some of the greatest things in life are like that, given > freely by the universe we inhabit, simply because we are here and > alive. My spiritual practice has left me accutely aware of these > things, the value of these things, and I am most grateful. Yet there are plenty of examples where these things aren't true. Children growing up in foster homes from birth comes to mind. Or AIDS babies doomed to die very young, but old enough that they can start to feel ripped off. Or children drafted as soldiers... I suppose one can find grotesque beauty in forcing the existential issues associated with killing other humans on people not even sexually mature... I'm better dollars to doughnuts that they don't interpret existence as something with no strings attached. The fee is the aspect of fate (where you're born, the power structure you're born in to). For some this isn't a fee and plays out rather well as you indicate. Not everyone is happy to be alive, and at the same time not everyone ought to be happy at being alive. It's quite pessimistic, but I don't see a whole lot else for the unpleasant situations. > This is a major part of our relationship to the world around us. > Without internalizing this principle and giving of ourselves in the > same spirit, you miss out on one of the greatest joys of being alive. This makes the assumption that fate deals you the hand that allows this perception. === Subject: Re: Charity addresses the fundamental problem of humanitys relationship to the universe? >>TM, I suppose, at least the kind that could be taught in 15-20 >>minutes, might be of benefit to both mind and body, but it fails to >>address the fundamental problem of man and his relationship to the >>universe. That's the only drawback I can see. > Satyr, Charity is just not easy for everyone yet. >> Charity with no strings attached is doled out far too often to >> people who refuse to take responsibility for themselves. >> Creating the situation where the safety net is a convenient hammock, >> and no contribution to society is required. > Heh. I doubt that is what he meant by Charity: > Though I speak with the tongues of men and of angels, and have not > charity, I am become as sounding brass, or a tinkling cymbal. And > though I have the gift of prophecy, and understand all mysteries, and > all knowledge; and though I have all faith, so that I could remove > mountains, and have not charity, I am nothing. And though I bestow all > my goods to feed the poor, and though I give my body to be burned, and > have not charity, it profiteth me nothing. > Charity suffereth long, and is kind; charity envieth not; charity > vaunteth not itself, is not puffed up, doth not behave itself > unseemly, seeketh not her own, is not easily provoked, thinketh no > evil; rejoiceth not in iniquity, but rejoiceth in the truth; beareth > all things, believeth all things, hopeth all things, endureth all > things. > Charity never faileth: but whether there be prophecies, they shall > fail; whether there be tongues, they shall cease; whether there be > knowledge, it shall vanish away. For we know in part, and we prophesy > in part. But when that which is perfect is come, then that which is in > part shall be done away. When I was a child, I spake as a child, I > understood as a child, I thought as a child: but when I became a man, > I put away childish things. For now we see through a glass, darkly, > but then face to face: now I know in part; but then shall I know even > as also I am known. And now abideth faith, hope, charity, these three; > but the greatest of these is charity. > - 1 Corinthians 13 > The word here translated as charity is in some more modern > translations rendered as love. The fact that you would immediately > jump to the conclusion that some freeloader was skulking about in > search of a handout rather handily illustrates my point. > Your life was a gift bestowed upon you without any strings attached, > as is the air you breathe and the sun that shines upon your face. No > one charges us fees for daisies and dandelions, sunrise and sunset, > the winter snow or a gentle rain, a mother's smile for the newborn at > her breast. Some of the greatest things in life are like that, given > freely by the universe we inhabit, simply because we are here and > alive. My spiritual practice has left me accutely aware of these > things, the value of these things, and I am most grateful. > This is a major part of our relationship to the world around us. > Without internalizing this principle and giving of ourselves in the > same spirit, you miss out on one of the greatest joys of being alive. Couldn't have said it better myself. -- meltdarok http://hometown.aol.com/meltdarok/ === Subject: Re: Charity addresses the fundamental problem of humanitys relationship to the universe? >>TM, I suppose, at least the kind that could be taught in 15-20 >>minutes, might be of benefit to both mind and body, but it fails to >>address the fundamental problem of man and his relationship to the >>universe. That's the only drawback I can see. >Satyr, Charity is just not easy for everyone yet. >>Charity with no strings attached is doled out far too often to >>people who refuse to take responsibility for themselves. >>Creating the situation where the safety net is a convenient hammock, >>and no contribution to society is required. > Heh. I doubt that is what he meant by Charity: > Though I speak with the tongues of men and of angels, and have not > charity, I am become as sounding brass, or a tinkling cymbal. And > though I have the gift of prophecy, and understand all mysteries, and > all knowledge; and though I have all faith, so that I could remove > mountains, and have not charity, I am nothing. And though I bestow all > my goods to feed the poor, and though I give my body to be burned, and > have not charity, it profiteth me nothing. > Charity suffereth long, and is kind; charity envieth not; charity > vaunteth not itself, is not puffed up, doth not behave itself > unseemly, seeketh not her own, is not easily provoked, thinketh no > evil; rejoiceth not in iniquity, but rejoiceth in the truth; beareth > all things, believeth all things, hopeth all things, endureth all > things. > Charity never faileth: but whether there be prophecies, they shall > fail; whether there be tongues, they shall cease; whether there be > knowledge, it shall vanish away. For we know in part, and we prophesy > in part. But when that which is perfect is come, then that which is in > part shall be done away. When I was a child, I spake as a child, I > understood as a child, I thought as a child: but when I became a man, > I put away childish things. For now we see through a glass, darkly, > but then face to face: now I know in part; but then shall I know even > as also I am known. And now abideth faith, hope, charity, these three; > but the greatest of these is charity. > - 1 Corinthians 13 > The word here translated as charity is in some more modern > translations rendered as love. The fact that you would immediately > jump to the conclusion that some freeloader was skulking about in > search of a handout rather handily illustrates my point. > Your life was a gift bestowed upon you without any strings attached, > as is the air you breathe and the sun that shines upon your face. No > one charges us fees for daisies and dandelions, sunrise and sunset, > the winter snow or a gentle rain, a mother's smile for the newborn at > her breast. Some of the greatest things in life are like that, given > freely by the universe we inhabit, simply because we are here and > alive. My spiritual practice has left me accutely aware of these > things, the value of these things, and I am most grateful. > This is a major part of our relationship to the world around us. > Without internalizing this principle and giving of ourselves in the > same spirit, you miss out on one of the greatest joys of being alive. How nice of you to lecture on *the* *fundamental problem* experienced by the *whole species*. You stuffed shirt. === Subject: Re: Charity addresses the fundamental problem of humanitys relationship to the universe? >TM, I suppose, at least the kind that could be taught in 15-20 >minutes, might be of benefit to both mind and body, but it fails to >address the fundamental problem of man and his relationship to the >universe. That's the only drawback I can see. >>Satyr, Charity is just not easy for everyone yet. >Charity with no strings attached is doled out far too often to >people who refuse to take responsibility for themselves. >Creating the situation where the safety net is a convenient hammock, >and no contribution to society is required. >> Heh. I doubt that is what he meant by Charity: >> Though I speak with the tongues of men and of angels, and have not >> charity, I am become as sounding brass, or a tinkling cymbal. And >> though I have the gift of prophecy, and understand all mysteries, and >> all knowledge; and though I have all faith, so that I could remove >> mountains, and have not charity, I am nothing. And though I bestow all >> my goods to feed the poor, and though I give my body to be burned, and >> have not charity, it profiteth me nothing. >> Charity suffereth long, and is kind; charity envieth not; charity >> vaunteth not itself, is not puffed up, doth not behave itself >> unseemly, seeketh not her own, is not easily provoked, thinketh no >> evil; rejoiceth not in iniquity, but rejoiceth in the truth; beareth >> all things, believeth all things, hopeth all things, endureth all >> things. >> Charity never faileth: but whether there be prophecies, they shall >> fail; whether there be tongues, they shall cease; whether there be >> knowledge, it shall vanish away. For we know in part, and we prophesy >> in part. But when that which is perfect is come, then that which is in >> part shall be done away. When I was a child, I spake as a child, I >> understood as a child, I thought as a child: but when I became a man, >> I put away childish things. For now we see through a glass, darkly, >> but then face to face: now I know in part; but then shall I know even >> as also I am known. And now abideth faith, hope, charity, these three; >> but the greatest of these is charity. >> - 1 Corinthians 13 >> The word here translated as charity is in some more modern >> translations rendered as love. The fact that you would immediately >> jump to the conclusion that some freeloader was skulking about in >> search of a handout rather handily illustrates my point. >> Your life was a gift bestowed upon you without any strings attached, >> as is the air you breathe and the sun that shines upon your face. No >> one charges us fees for daisies and dandelions, sunrise and sunset, >> the winter snow or a gentle rain, a mother's smile for the newborn at >> her breast. Some of the greatest things in life are like that, given >> freely by the universe we inhabit, simply because we are here and >> alive. My spiritual practice has left me accutely aware of these >> things, the value of these things, and I am most grateful. >> This is a major part of our relationship to the world around us. >> Without internalizing this principle and giving of ourselves in the >> same spirit, you miss out on one of the greatest joys of being alive. > How nice of you to lecture on *the* *fundamental problem* experienced by > the *whole species*. > You stuffed shirt. Gee Martin, as smart as you are, could you be a tad Jealous? -- meltdarok http://hometown.aol.com/meltdarok/ === Subject: Re: Charity addresses the fundamental problem of humanitys relationship to the universe? >TM, I suppose, at least the kind that could be taught in 15-20 >minutes, might be of benefit to both mind and body, but it fails >to address the fundamental problem of man and his relationship to >the universe. That's the only drawback I can see. >>Satyr, Charity is just not easy for everyone yet. >Charity with no strings attached is doled out far too often to >people who refuse to take responsibility for themselves. >Creating the situation where the safety net is a convenient >hammock, and no contribution to society is required. >> Heh. I doubt that is what he meant by Charity: >> Though I speak with the tongues of men and of angels, and have not >> charity, I am become as sounding brass, or a tinkling cymbal. And >> though I have the gift of prophecy, and understand all mysteries, >> and all knowledge; and though I have all faith, so that I could >> remove mountains, and have not charity, I am nothing. And though I >> bestow all my goods to feed the poor, and though I give my body to >> be burned, and have not charity, it profiteth me nothing. >> Charity suffereth long, and is kind; charity envieth not; charity >> vaunteth not itself, is not puffed up, doth not behave itself >> unseemly, seeketh not her own, is not easily provoked, thinketh no >> evil; rejoiceth not in iniquity, but rejoiceth in the truth; >> beareth all things, believeth all things, hopeth all things, >> endureth all things. >> Charity never faileth: but whether there be prophecies, they shall >> fail; whether there be tongues, they shall cease; whether there be >> knowledge, it shall vanish away. For we know in part, and we >> prophesy in part. But when that which is perfect is come, then that >> which is in part shall be done away. When I was a child, I spake as >> a child, I understood as a child, I thought as a child: but when I >> became a man, I put away childish things. For now we see through a >> glass, darkly, but then face to face: now I know in part; but then >> shall I know even as also I am known. And now abideth faith, hope, >> charity, these three; but the greatest of these is charity. >> - 1 Corinthians 13 >> The word here translated as charity is in some more modern >> translations rendered as love. The fact that you would >> immediately jump to the conclusion that some freeloader was >> skulking about in search of a handout rather handily illustrates my >> point. >> Your life was a gift bestowed upon you without any strings >> attached, as is the air you breathe and the sun that shines upon >> your face. No one charges us fees for daisies and dandelions, >> sunrise and sunset, the winter snow or a gentle rain, a mother's >> smile for the newborn at her breast. Some of the greatest things in >> life are like that, given freely by the universe we inhabit, simply >> because we are here and alive. My spiritual practice has left me >> accutely aware of these things, the value of these things, and I am >> most grateful. >> This is a major part of our relationship to the world around us. >> Without internalizing this principle and giving of ourselves in the >> same spirit, you miss out on one of the greatest joys of being >> alive. > How nice of you to lecture on *the* *fundamental problem* > experienced by the *whole species*. > You stuffed shirt. Ah, the twat queefs. === Subject: Re: Torkel =?ISO-8859-1?Q?Franz=E9n_is_dead?= > Torkel Franz.8en, well known for his many Usenet posts, died of skeleton > cancer at Wednesday, April 19, at the age of 56. > Torkel Franz.8en worked as a university lecturer at the department of > Computer Science and Electrical Engineering, at Lule.8c University of > Technology, Sweden. > He taught programming courses, mostly using Java and Prolog. He earned > Guide to Its Use and Abuse, which appeared in 2005. > Usenet posts on this and related subjects, but he did also write posts > on many other subjects. > Torkel's too early death is a great loss for his family, colleagues, > and Usenet friends. Jeezus! Bummer! Torkel was one of the few voices of sanity on Usenet. Bob Kolker === Subject: =?iso-8859-1?q?Re:_Torkel_Franz=E9n_is_dead?= I was greatly sadenned to read this. I am a huge fan of his approach to the philosophy of mathematics and of his writing style; I learned a lot from his books and posts. Like most people here, I never knew him in person, but I will cherish our few e-mail exchanges. My sincerest condolences to his family and friends. - Alon === Subject: =?iso-8859-1?q?Re:_Torkel_Franz=E9n_is_dead?= erland@bredband.net skrev: > He earned Ooops, wrong. He earned his PhD in 1987. Sorry for the mistake. Erland Gadde === Subject: analytic? I read this on a physics website about trigonometry: What we would like to have is a way of relating the angles in the triangle to the lengths of the sides. It turns out that there's no simple analytic way to do this. Can someone please explain what is meant by the term analytic in this context? === Subject: Re: analytic? > I read this on a physics website about trigonometry: > What we would like to have is a way of relating the angles in the > triangle to the lengths of the sides. It turns out that there's no > simple analytic way to do this. > Can someone please explain what is meant by the term analytic in this > context? It certainly doesn't mean analytic in the mathematical sense, because the trig functions and inverse trig functions are analytic on their respective domains. In mathematics, a function is called analytic on a region if it can be represented there by a power series. Since the quote is from a physics website, perhaps you should ask in sci.physics. -- Dave Seaman U.S. Court of Appeals to review three issues concerning case of Mumia Abu-Jamal. === Subject: Re: prime > > LOL. Speaking of *the* Bob Silverman = a legendary former > > poster, > > > > Not _former_ poster. 'tween occasional and regular poster. > > One of the sci.math must read list for anyone with more > > than a passinginterest in discrete mathematics. > > > > worked for RSA, quite knowledgeable about number theory, > > > > Woh!?!?!?!?!? > I did not know he ever worked at RSA. Mitre I know. Microsoft I know. > And indeed one of the leading people in number theory. I believe my first response to Bob on Usenet was flaming him back in his RSA days, about RSA. Phil -- What is it: is man only a blunder of God, or God only a blunder of man? -- Friedrich Nietzsche (1844-1900), The Twilight of the Gods === Subject: Re: prime > > LOL. Speaking of *the* Bob Silverman = a legendary former > > poster, > > > > Not _former_ poster. 'tween occasional and regular poster. > > One of the sci.math must read list for anyone with more > > than a passinginterest in discrete mathematics. > > > > worked for RSA, quite knowledgeable about number theory, > > > > Woh!?!?!?!?!? > I did not know he ever worked at RSA. Mitre I know. Microsoft I know. > And indeed one of the leading people in number theory. IIRC he once had 'rsa' as part of his e-mail address. May be I'm wrong. I once read a report written by him, where he compared the computational resources required to break RSA or ECC. The upshot of that report was that many researchers treat RSA unfairly, because they forget to consider the memory requirements of attacks on RSA. IIRC RSA was listed as his affiliation in that report. I agree with Phil here. He is on the must read list. He must not (and frankly cannot) be confused with Joseph Silverman, the author of two Springer GTM series textbooks on elliptic curves, who also posts here occasionally (and also has an interest/investment in public-key cryptography). Jyrki === Subject: delta functions and test functions I'm a bit confused about what makes a test function. Basically, I have a physics question that's asking me to integrate f(x)delta(x-1/2) between 0 and 1 where f(x) = x for 0 = x = 1/2 and (1-x) for 1/2 = x = 1. However, I was under the impression that the test function has to have a finite derivative of all orders, or something like that, or a continuous first derivative, or seemingly many other definitions. Does that mean this integral is undefined, or am I overcomplicating things? Nat === Subject: Re: delta functions and test functions Originator: grubb@lola >I'm a bit confused about what makes a test function. >Basically, I have a physics question that's asking me to integrate >f(x)delta(x-1/2) between 0 and 1 where f(x) = x for 0 = x = 1/2 and >(1-x) for 1/2 = x = 1. However, I was under the impression that the >test function has to have a finite derivative of all orders, or >something like that, or a continuous first derivative, or seemingly >many other definitions. >Does that mean this integral is undefined, or am I overcomplicating >things? If your test functions are as stated, then you are right, f is not a test function. However, for integration against a delta function, it is possible to extend the test functions to include any continuous function. This is what the exercise almost certainly intends you to do. There will be an unambiguous answer. In the same way, if you are dealing with the derivative of the delta function, you can extend the workable test functions to include every function with a continuous derivative (even if the higher order derivatives are not defined). --Dan Grubb === Subject: Re: delta functions and test functions > I'm a bit confused about what makes a test function. > Basically, I have a physics question that's asking me to integrate > f(x)delta(x-1/2) between 0 and 1 where f(x) = x for 0 = x = 1/2 and > (1-x) for 1/2 = x = 1. I'll assume the above is a typo for f(x) = {x, if 0<=x<1/2} (1) {1-x, if 1/2<=x<1} (or f(x) = {x, if 0<=x<=1/2} {1-x, if 1/2 However, I was under the impression that the > test function has to have a finite derivative of all orders, or > something like that, or a continuous first derivative, or seemingly > many other definitions. Right. A test (or bump) function, is constructed differently. You start with f(x) = {exp(-1/x), if x>0} {0, if x <=0} A test function would then be something like: f(1-x^2). This function looks like a bell-shaped curve. If you simplify it, you end up with: f(1-x^2) = {exp(1/(x^2-1)), if |x|<1} {0, if |x|>1} It is infinitely differentiable, even at the points x=1 and x=-1. Perhaps your professor is using the function f(x) of definition (1) as a pseudo-test function, even though it is not a proper test function. > Does that mean this integral is undefined, or am I overcomplicating > things? You are overcomplicating it. Dirac delta functions are distributions and they have the property: int(Dirac(x),x=-infinity..infinity)=1. Accordingly, int(f(x)*Dirac(x-1/2),x=-infinity..infinity)=1/2. f(x) vanishes outside the interval [0,1], thus, int(f(x)*Dirac(x-1/2),x=0..1)=1/2. > Nat -- Ioannis === Subject: Re: +/- The primes >Starting with prime (2) >2-3+5-7+11-13+17... >-1,4,-3,8,-5,12.. >In OEIS as A008347 >Then pairing each product with its corresponding prime >to form sets >-1 3 ,4 5 , -3 7 , 8 11 ,-5 13 , 12 17 , >-7 19 , 16 23 ,-13 29 ,18 31 ,-19 37 , 22 41 , >-21 43 , 26 47 ,-27 53 , 32 59 ,-29 61 , 38 67, >-33 71 , 40 73 ,-39 79 , 44 83 ,-45 89 , 52 97 , I think you're misusing the term product here. What you have is a difference of odd-numbered primes and even-numbered primes. >Show for each abs set (no negative integer in set) >where the second integer in that set is prime then using >abs value of first integer in (previous) set = n. >Where first integer of each abs set = (prod) and second >number of same set is prime = (p) >p== n( mod (prod)) ( i'm a poet and didn't know it) ;-) Let me translate that conjecture into something the rest of sci.math can understand. Define p(n) = nth prime f(n) = (sum of odd-numbered primes up to nth) -- (sum of even-numbered primes up to nth) = sum_i=1^n [p(i) * (-1)^(i+1)] Your conjecture is that: p(n) == |f(n-1)| mod |f(n)| That should be easy to prove for cases where f(n-1) and f(n) have opposite signs, because then p(n) = |f(n) - f(n-1)| p(n) = |f(n)| + |f(n-1)| p(n) == |f(n-1)| mod |f(n)| Now prove that the signs always alternate and you're done. >Also show for each negative set (first integer in set >negative)where its abs value > previous first integer >(n) in set then for that set-- >p==n(mod(prod)) also. >Unlike the abs sets this only happens occasionally. >Whereas with the abs sets -- p==n(mod(prod)). >I believe happens every other set! >Can a counter example be found for any abs set or >if not can a proof be found for this conjecture? >Dan --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Re: Generic term for row/column/coordinate? > Don't know about spreadsheets, but when I want to refer to > a row or column of a matrix, but don't want to commit as to > whether it's a row or it's a column, I use the word line. replied! -- Always look on the bright side of life. To reply by email, replace no.spam with my last name. === Subject: Re: induction problem for proving that (n+k)!/n! is divisible by k! >> 6 and 8 are both factors of 24, but 24 is not divisible by 6x8. > I was presuming that (n+k)! > n! * k!. 6 and 8 are both factors of 120, but 120 is not divisible by 6x8. -- Ben === Subject: Re: aritmetic and postulates >hi there, this is the first time I've posted anything. I was going >through some work on the history of maths, and was wondering if there >was any postulates in arithmetic. I have only found Peones 5 >postulates on natural numbers that could be close, but otherwise >nothing yet. Could somebody let me know if there are any or not and if >advance for the time spent on this subject, if any One could get a set of postulates for all integers by assuming they are an ordered integral domain, with the induction property for non-negative integers with addition by 1 as the successor operator. It is somewhat harder for rationals and reals, as it is needed to get the positive integers. I believe this can be done arithmetically for the rationals, but not the reals. Essentially, you have to get the Peano Postulates for the integers, and be able to develop from that. -- 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: Why are infinite series so difficult for students? >On 20 Apr 2006 12:59:26 -0700, Proginoskes One thing that I've wondered, having taught Calculus II, is why >>students have so much trouble with infinite series. I asked one student >>yesterday during my office hours, and he said they were too abstract. >>I don't think this is the complete answer, though. What do you all have >>to say? >> --- Christopher Heckman >First, they don't understand sequences. Then the sum of a series being >the limit of the sequence of partial sums is too much for them to >parse. The successful ones will learn to do the ratio, comparison, >and integral tests without coming to grips with the theory much like >many students learn to calculate derivatives from formulas without >understanding what they represent. >My 2 cents worth. I agree. They do not understand the concept of limit, which is needed BEFORE getting on with calculus computations. For most of them, their entire mathematical education is in how to solve certain types of problems, not in any understanding of concepts. It would be better to require them to take a course in mathematics, not computation, before starting calculus. Most of an introductory real variables course can be so taught. Newton's derivation of (1+x)^u, for fixed u not necessarily an integer and |x| < 1, preceded Taylor series. -- 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: Why are infinite series so difficult for students? >> One thing that I've wondered, having taught Calculus II, is why >> students have so much trouble with infinite series. I asked one student >> yesterday during my office hours, and he said they were too abstract. >> I don't think this is the complete answer, though. What do you all have >> to say? >Previously, students never had to understand any theory >to be able to complete the vast majority of problems, >which they do by correctly manipulating symbols. Here, the >old order doesn't work. >Now, to proceed, a student would need a rudimentary grasp >of what sequences and their limits are, in contrast to what a >series is, and the interplay between the two. Their first roadblock >is that many never developed an informal grasp of limits. (Consider >the ad nauseam threads on 0.999...) In recognition of this, many >math departments have put series in the Calculus III curriculum, Limits belong when infinite decimals are first introduced, if not earlier. It is customary to have children see the argument that 1/3 = .3333 ... without showing it as a formal argument, etc. -- 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: Why are infinite series so difficult for students? >> One thing that I've wondered, having taught Calculus II, is why >> students have so much trouble with infinite series. I asked one student >> yesterday during my office hours, and he said they were too abstract. >> I don't think this is the complete answer, though. What do you all have >> to say? >> It's possible the pattern recognition game, whereby Johnny gets >> a 75 on the test without really knowing anything, fails him here. >Then why don't they have as much trouble with integration techniques? > --- Christopher Heckman Techniques, not concepts. Ask them to integrate exp(-|x|) over the entire real line, or maybe from -1 to 2. Or to show that sin(1/x) is integrable from 0 to 1. Techniques are memorized; concepts are understood. -- 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: Why are infinite series so difficult for students? >One thing that I've wondered, having taught Calculus II, is why >students have so much trouble with infinite series. I asked one student >yesterday during my office hours, and he said they were too abstract. >I don't think this is the complete answer, though. What do you all have >to say? > --- Christopher Heckman This is partly correct; nowadays the standard curriculum discourages using any abstract concepts, and assumes that someone can only learn them by accumulation examples and generalizing. If you checked your students carefully, you will find that they have little idea of what a derivative is, and most think that integral is defined by anti-derivative. That said, one could point out that a polynomial can be written with the terms by increasing degree, not as they are used to having them by decreasing degree, and that a formal power series is an infinite degree polynomial, and the value of a power series is the limit of that of the finite degree polynomials. But do they have any idea of what limit means? -- 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: Why are infinite series so difficult for students? What a coincidence. Am currently taking calculus II and we're about to go on to infinite sequences and series next week. Interestingly enough, the professor is starting out with Taylor polynomials and applications. So far from what I gather a Taylor series is a polynomial representation of a function that can be represented as a power series. === Subject: Re: Why are infinite series so difficult for students? On 20 Apr 2006 12:59:26 -0700, Proginoskes students have so much trouble with infinite series. I asked one student >yesterday during my office hours, and he said they were too abstract. >I don't think this is the complete answer, though. What do you all have >to say? I think what the student said and what Wade said are exactly right. The way I'd put it is that they've never had to deal with the idea that they're supposed to be making _assertions_ about things, and reasoning about whether various facts are true or false. That simply doesn't come up in the math they're used to, where all they have to do is learn how to do it. It gets worse in something like beginning linear algebra (don't ask why this example springs to mind right now...) Of course most of the material is very simple, and many of the students have no problem. But some of them can't seem to grasp the idea that they need to read the question and do what the instructions ask them to do - if we've been diagonalizing matrices and you ask them to find a basis for the nullspace of a matrix they begin by trying to find the eigenvalues... the problem is that they've learned some algorithms, but the algorithms are _filed_ under the label what to do when you see a matrix in this chapter. > --- Christopher Heckman ************************ David C. Ullrich === Subject: Re: Why are infinite series so difficult for students? students have so much trouble with infinite series. I asked one student > yesterday during my office hours, and he said they were too abstract. > I don't think this is the complete answer, though. What do you all have > to say? > Have you ever tried teaching them about infinite products? > Even today ring theorists still struggle with them, e.g. see > Jim Coykendall: Some remarks on infinite products > http://math.ndsu.nodak.edu/faculty/coykenda/chapelhill3a.pdf > --Bill Dubuque This is mean, sitting behind an assumed name and commenting. Now, pray tell what are the ring theorists struggling with? Muhammad === Subject: Re: Why are infinite series so difficult for students? > One thing that I've wondered, having taught Calculus II, is why > students have so much trouble with infinite series. I asked one student > yesterday during my office hours, and he said they were too abstract. > I don't think this is the complete answer, though. What do you all have > to say? >> Have you ever tried teaching them about infinite products? >> Even today ring theorists still struggle with them, e.g. see >> Jim Coykendall: Some remarks on infinite products >> http://math.ndsu.nodak.edu/faculty/coykenda/chapelhill3a.pdf > This is mean, sitting behind an assumed name and commenting. Why do you think I sit behind an assumed name? What's mean? > Now, pray tell what are the ring theorists struggling with? Everyone struggles with infinite products -- they're much more difficult than infinite series. That was the point of my remark. I mentioned Jim's paper as evidence in support of this claim, and because I thought readers might find it of interest (as I did). I often sprinkle my posts with references to interesting related work (including yours). And if you want to see some really mean struggles with infinite products have a look at Bill Gosper's matrix infinite products. He humorously (and very modestly) rates their difficulty on a scale of milli-Ramanujans. --Bill Dubuque === Subject: Re: Why are infinite series so difficult for students? students have so much trouble with infinite series. I asked one student > yesterday during my office hours, and he said they were too abstract. > I don't think this is the complete answer, though. What do you all have > to say? >> Have you ever tried teaching them about infinite products? >> Even today ring theorists still struggle with them, e.g. see >> Jim Coykendall: Some remarks on infinite products >> http://math.ndsu.nodak.edu/faculty/coykenda/chapelhill3a.pdf > This is mean, sitting behind an assumed name and commenting. > Why do you think I sit behind an assumed name? What's mean? Sorry, if I like someone I usually check out what kind of stuff he/she has worked on. In your case I could not find any reference to the name Dubuque after 1949 in Mathematics and that is P. Dubuque. And you cannot be that Dubuque. Besides at a couple of occasions I did need to get in touch with you and you were nowhere to be found. There could be other reasons that led me to believe that you may not be the one who you claim to be. For the other question look at the mix Have you ever tried teaching them about infinite products? Even today ring theorists still struggle with them,... You did single out ring theorists and concentrate/focus on Even today. > Now, pray tell what are the ring theorists struggling with? > Everyone struggles with infinite products -- they're much > more difficult than infinite series. That was the point of > my remark. I mentioned Jim's paper as evidence in support > of this claim, and because I thought readers might find it > of interest (as I did). I often sprinkle my posts with > references to interesting related work (including yours). (Now this is coming from the person I began to like two or three years ago.) They become much more difficult if you are dealing with elements that cannot be inverted. > And if you want to see some really mean struggles with > infinite products have a look at Bill Gosper's matrix > infinite products. He humorously (and very modestly) > rates their difficulty on a scale of milli-Ramanujans. I am sure he has a good reason to study them. If I have a good reason > --Bill Dubuque Now that I have stuck my neck out let me say a word on the difficulty of infinite series. They are and they will remain a difficult to teach topic if they are crammed, as a topic, in an already heavily laiden course. Pull them out of there and teach them at a decent pace with infinite products or some other suitable topic and teachers and students would both like it. But the trouble is the other departments may not appreciate the idea. They want only n credits of Mathematics and they want everything. So, grin and bear it. (When I could make decisions I did introduce the idea of extra contact hours (three-credit course, four contact hours). I did try to do that at an American school, got my way after a big fight, some courses got extra contact hours and showed good results but I was booted out the next year. (Hey, no regrets.) My point is some concepts need time to sink in, if you cannot get the students that needed time do not blame them. Muhammad === Subject: Re: Why are infinite series so difficult for students? Back to the original question: Why do students have problems with infinite series, but not with infinite integration? For me personally (I'm NTh amateur and studied math/physics 20 years ago) it was the fact of a) infinity and b) discreteness. An Integral is something continuous, and there exist functions to do the job: Integral [f'(t) dt] = f(t) + C, end of the story, period. Finite integration is even easier. And this one was taught at school for years! A Sum is something that seems well-known, but infinite sums are not! Calculus of infinite sums, dealing with oo made me a little frighten, you know. And the convergence criteria are just not that easy, are they? Therefore the integration criterion was very helpful for me. Looking at sums as a series of partial sums and adapting the convergence criteria to them was a little confusing for me. In addition, we never dealt with infinite sums at school, so the subject was almost entirely new for me. And sums are dealt with at the beginning of the University. No time to fill up the gaps being left over from school. Nevertheless calculating infinite sums and dealing with infinity is still fascinating. It was a hard job, but finally I got it. Today I can even prove for p <= x Sum(1/p) = ln ln x + C -> oo (as x -> oo). I was proud when I understood how to do this. The technique is not easy and the result is amazing. I had erroneously expected this sum to be convergent ... a typical amateur error. You just don't have the math experience and the feeling for it! Hope this helps teaching your students! === Subject: Re: Why are infinite series so difficult for students? Little error correction: Looking at _infinite_ sums as _sequences_ (not series!) of partial sums, then adapting the convergence criteria for sequences and calculate the lim for the upper bound running to infinity was confusing for me. Sorry for this one, I was a little sloppy there ... and my English isn't too good, as I'm German. :-) CU, Wolfgang. === Subject: Re: Why are infinite series so difficult for students? >On 20 Apr 2006 12:59:26 -0700, Proginoskes One thing that I've wondered, having taught Calculus II, is why >>students have so much trouble with infinite series. I asked one student >>yesterday during my office hours, and he said they were too abstract. >>I don't think this is the complete answer, though. What do you all have >>to say? >> --- Christopher Heckman >First, they don't understand sequences. Then the sum of a series being >the limit of the sequence of partial sums is too much for them to >parse. The successful ones will learn to do the ratio, comparison, >and integral tests without coming to grips with the theory much like >many students learn to calculate derivatives from formulas without >understanding what they represent. >My 2 cents worth. >--Lynn I would not suspect that most calculus students do understand sequences since they usually aren't introduced until just before series are introduced. I find this interesting because when I took advanced calc, the very first thing that was introduced was sequences and their limits. I am interested in others' views why freshmen calc does not introduce sequences in the beginning? Why do you suppose that is? Brian === Subject: Re: Why are infinite series so difficult for students? >>On 20 Apr 2006 12:59:26 -0700, Proginoskes students have so much trouble with infinite series. I asked one student >yesterday during my office hours, and he said they were too abstract. >I don't think this is the complete answer, though. What do you all have >to say? > --- Christopher Heckman >>First, they don't understand sequences. Then the sum of a series being >>the limit of the sequence of partial sums is too much for them to >>parse. The successful ones will learn to do the ratio, comparison, >>and integral tests without coming to grips with the theory much like >>many students learn to calculate derivatives from formulas without >>understanding what they represent. >>My 2 cents worth. >>--Lynn >I would not suspect that most calculus students do understand >sequences since they usually aren't introduced until just before >series are introduced. >I find this interesting because when I took advanced calc, the very >first thing that was introduced was sequences and their limits. I am >interested in others' views why freshmen calc does not introduce >sequences in the beginning? Why do you suppose that is? >Brian Most books have them, but they are almost skipped over. If one does not understand limits, calculus merely becomes a course in memorizing mechanical rules to compute derivatives and anti-derivatives. This method of learning such rules to solve routine problems is all the students are likely to have seen in elementary and high school; no understanding whatever. -- 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: least square problem > Hi all, > it looks like there is something strange in the result I jut got. > I have a set of data points (xp(i), yp(i)) for i=1..N that I want to > approximate by a polynomial > P(x). At the end I compute the distance between the data points such as > d=sum((P(xp(i))-yp(i))^2); > It looks like up to a polynomial of order 6, d decreases. Then for > polynomials of higher degree, d increases. In other words, the best > approximation in a least square sense is achieved with a polynomial of > order 6. > Is this a normal behavior ? > thank you, > Pluton [A better forum for this question: sci.math.num-analysis] As noted elsewhere, d should decrease (or stay ultimately constant), and round-off can introduce errors that distort the picture. There are remedies: avoid the power form (basis 1, x, x^2, ... x^n) for higher degree polynomials, and use a different basis. An ideal basis would come from Gram-Schmidt orthogonalization of the power basis with inner product (u(x), v(x)) = sum(u(xp(i))*v(xp(i))) but that may involve too much extra calculation. A near-orthogonal set can be set up as follows: say all the data points have xp(i) in the interval [a,b], then modified Chebyshev polynomials T_k ((2*x - a - b)/(b - a)) (recall: T_k(t) = cos(k*arccos(t))) offer reasonably good behavior under round-off (even if they are not exactly orthogonal under the above inner product). (Use Google to find out about Chebyshev polynomials.) Hope it helps, ZVK(Slavek). === Subject: Re: Chess boards & connections. Mail-To-News-Contact: abuse@dizum.com >Trying to calculate if I can write a Chess AI. I need to define all possible boards. I have a total of 64 different pieces, 16 for Black >and 16 for White to start, and since each pawn >can be promoted, to either a Queen or Knight, a > They can't be promoted to a bishop or a rook? >> LOL, then you'll totally my math, actually >> I have a patch, but you're right ;-) . >what about in passing for pawns? >and castling? I don't see how either of those would affect the total number of board positions, which is what the OP was after. All that they'd affect would be the set of allowed transitions in the state diagram. -- Michael F. Stemper #include Visualize whirled peas! === Subject: Re: Chess boards & connections. Mail-To-News-Contact: abuse@dizum.com >>Trying to calculate if I can write a Chess AI. >>I need to define all possible boards. >>providing 64*65=4160 boards. >> Nope. >> Place the first of the 64 pieces. How many choices for a square do >> you have? 65. Now, for each of those possibilities, you have 64 choices >> for placing the second piece. So, two pieces already uses up your 4160 >> boards. If you place a third piece, you'll have 63 available squares >> for each of those 4160 possibilities, or 65*64*63 = 262080. To place >> 64 pieces on 65 squares -- even with disallowing two pieces on the same >> square from the start -- you're going to have 65! possible boards. As Proginoskes pointed out elsewhere, this number (65!) is actually low. According to: there are about 10^50 possible boards. >You are too complicated. All I needed was the 2D Array >I defined by matrix (64,65) defining all boards. That only gives you 4160 boards, or about one out of every 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 >>(I'll skip using exclusion right now, like two pieces >>can't occupy the same square, but all that means >>is there are < 4160 possible boards). >> Nope. Like I pointed out above, even if you build in exclusion from >> the start, you'll need to add a lot of zeros to the right of 4160 >Well then provide the needed Array, I gave you an estimate as to its size: about 10^50. > I'll repeat >I get a (64,65) for ALL configurations, including >many that are impossible. Repeating it doesn't make it true. -- Michael F. Stemper #include Life's too important to take seriously. === Subject: Re: Chess boards & connections. Mail-To-News-Contact: abuse@dizum.com >>Trying to calculate if I can write a Chess AI. >>I need to define all possible boards. >>I have a total of 64 different pieces, >>There are 65 locations on the board, 8*8 + 1 for >>non-existance called the side bar. >>So I dimension an Array, (64,65) where the 64 >>provides the *serial number* for all possible >>pieces and each of those can be in 65 locations, >>providing 64*65=4160 boards. >> Place the first of the 64 pieces. How many choices for a square do >> you have? 65. Now, for each of those possibilities, you have 64 choices >> for placing the second piece. [...] >Not if the first piece is placed off the board! Then there are 65 >possibilities for the second piece. You're right, of course. So my upper bound of 65! boards was a little low, then. -- Michael F. Stemper #include Life's too important to take seriously. === Subject: Re: Chess boards & connections. <200604201803.k3KI38D25580@walkabout.empros.com> <44481995$0$84074$892e7fe2@authen.yellow.readfreenews.net>Trying to calculate if I can write a Chess AI. >>I need to define all possible boards. >>I have a total of 64 different pieces, 16 for Black >>and 16 for White to start, and since each pawn >>can be promoted, to either a Queen or Knight, a >> They can't be promoted to a bishop or a rook? > LOL, then you'll totally my math, actually > I have a patch, but you're right ;-) . > what about in passing for pawns? > and castling? >>further 16 for B&W's 8 pawns for another 32. >>There are 65 locations on the board, 8*8 + 1 for >>non-existance called the side bar. >>So I dimension an Array, (64,65) where the 64 >>provides the *serial number* for all possible >>pieces and each of those can be in 65 locations, >>providing 64*65=4160 boards. >> Nope. >> Place the first of the 64 pieces. How many choices for a square do >> you have? 65. Now, for each of those possibilities, you have 64 choices >> for placing the second piece. So, two pieces already uses up your 4160 >> boards. If you place a third piece, you'll have 63 available squares >> for each of those 4160 possibilities, or 65*64*63 = 262080. To place >> 64 pieces on 65 squares -- even with disallowing two pieces on the same >> square from the start -- you're going to have 65! possible boards. >> I'd say that, since each player can only have 16 pieces at any time, >> you number the pieces 1-16 (or 0-15 if you prefer), and keep track of >> what type of piece (Q, K, Kt, p, etc.) each one is. This way, you only >> have 32 pieces to deal with. Then, you probably should get rid of your >> sidebar and just not necessarily have all of the pieces included in >> the board at any time. > You are too complicated. All I needed was the 2D Array > I defined by matrix (64,65) defining all boards. > too limited. try checkers instead. Ok, you might be right, I was thinking we could go through all 64 Pieces (as defined in the OP) and output a Location (0-64) like, Location (P) = L where P => 1 to 64 and L=>0 to 64, to describe any possible chess board. Is that true? >> This strategy will reduce the number of configurations to (64!)/(32!) >> time some multiplier based upon how many duplicate pieces a player >> has. >>(I'll skip using exclusion right now, like two pieces >>can't occupy the same square, but all that means >>is there are < 4160 possible boards). >> Nope. Like I pointed out above, even if you build in exclusion from >> the start, you'll need to add a lot of zeros to the right of 4160 > Well then provide the needed Array, I'll repeat > I get a (64,65) for ALL configurations, including > many that are impossible. > if your math skills indicate your programming skills, you are a starting > hack-er. Yup, trying to learn algebra. Ken === Subject: Re: Chess boards & connections. >Ok, you might be right, I was thinking we >could go through all 64 Pieces (as defined >in the OP) and output a Location (0-64) like, >Location (P) = L >where P => 1 to 64 and L=>0 to 64, >to describe any possible chess board. >Is that true? I'm not sure where 64 pieces came from. There can never be more than 32 pieces. And if you count pawn-morphs as new pieces there are 96 potential pieces at the beginning of the game, because a pawn can be promoted into any of {queen, rook, bishop, knight}. But knowing the location of each piece isn't the whole story. You also need the following info: * Which player is next to move? 1 bit. * King-side castle possible? 1 bit per player, zeroed when king or king's rook has moved. * Queen-side castle possible? 1 bit per player, zeroed when king or queen's rook has moved. * En passent target file 0..8 -- 0 unless last move was a pawn moving 2 spaces, then it's the file of that pawn. That's enough info to define the legal *physical* moves. If you want to consider the procedural moves of claiming a draw due to the 50-move rule or the third-time repeat, you should keep a 50-move history. --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Re: Maclaurin series question > I need to find a Maclaurin series for f(x) = x^3/ (x-2)^2 > I'm supposed to be able to start with a known representation and make > the appropriate substitutions/changes to get the desired > representation. > The problem? That form doesn't look familiar to me. The closest I've > got is 1/1-x but despite several attempts, I just can't get that to > work out. > Any ideas? When in trouble, change the variable. What does (x^3) * 1/(2-x)^2 look like when you replace x with 2*t (and later return to t=x/2) ? === Subject: polynomial... hello...doctor~ f(x) = a.x^4 + b.x^3 + c.x^2 + d.x + e f(0)=0, f(1)=1/2, f(2)=2/3, f(3)=3/4, f(4)=4/5 find f(5). -------------------------------------------- is this simultaneous equations ?? but that is complex... so, i need your advice. thank you very much for your advice.