Possible proof of Gabriel's Theorem?
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From: David C. Ullrich on 8 Mar 2005 07:20 On Mon, 07 Mar 2005 15:41:37 +0000, Angus Rodgers <angus_prune(a)bigfoot.com> wrote: >[...] >You're saying that, in spite of the strange presentation on Gabriel's >website, and in spite of Jason's many hilarious misconceptions (about >constant functions not being differentiable, and so on), there is a >coherent conjecture here that hasn't been settled? > >I don't suppose you could save me the effort of deciphering Gabriel's >handwriting by pointing me to an article where this conjecture has >been competently formulated, could you? Suppose that f is differentiable on [0,1] (or differentiable on (0,1) and continuous on [0,1]). Does it follow that f(1) - f(0) = lim_N sum_1^N f'(j/N)/N ? Of course this is clear from the fundamental theorem of calculus if f' is continuous. It seems unlikely that it's true under the stated hypotheses, but "seems unlikely" is not quite a counterexample. (Also it's not entirely clear whether this is exactly Gabriel's claim, because of confusion over the definition of "derivative"; we've been given several mutually inconsistent definitions, together with the assertion that we mean the same thing as usual by f'... The question I was talking about takes the standard definition of the derivative.) >Sorry I haven't been paying attention. I wasn't expecting actually >to have to look at the maths, and I only hope my rusty analysis is >up to the job. I'd better get that hat and humble pie ready! ************************ David C. Ullrich
From: Jason on 8 Mar 2005 12:21 > In fact I don't find your messages annoying, I find them hilarious. > In case you didn't know, so do a lot of people - every time you > make a post here it leads to people all over the planet rolling > on the floor. Honest. Hate to tell you that *you* are the one being laughed at. :-) Why? Well, I don't claim to know. That's why I post messages on this board soliciting information from *those who claim to know*. Now you claim to know yet not once have you been able to answer any of the questions I asked you. Your responses are hilarious and non-commital, e.g. Nope. Honest. etc. > I have. The fact that you keep saying I'm wrong doesn't make me wrong. You have been wrong two times I am aware of. The rest of your commitments say nothing so how can you say too much which is wrong? You are supposed to be a mathematician, no? Well, show me through mathematics where and why in a detailed explanation how Gabriel's theorem is wrong. You keep changing your tone! Well, what exactly are you saying? > No. The reason is that it was clear long ago that there's no > possibility anyone will ever convince you that you're wrong > about more or less everything, so there would be no point to > an offline "discussion". Nothing is clear from what you have written. You have a non-comittal approach and you have made at least two mistakes which you refuse to admit. Once again David, you need to admit your mistakes and support your criticisms. You have not been able to convince me of anything because you have not said much that makes sense yet. You pick out some vague topic and then present an example which is irrelevant. As a mathematician you would need to show in detail (giving reasons) why a proof is incorrect. Just remember this is not my work and I am in fact answering questions which gabriel himself should be answering. I may not be answering correctly all the time but I am trying... My occupation is *hobby mathematician*. I openly admit this. You are supposed to be a professional?! > I don't believe that anyone has _said_ that his theorem is false > as stated. But nobody has given a proof of it, and people > _suspect_ it's false, for example if there does indeed exist > a non-constant differentiable f such that f'(r) = 0 for all > rational r, as I suspect, then the theorem's certainly false. Giggle, giggle. About 90% of the contributors on this forum have said his theorem is false! Can you *read* ?! Have you understood any of the posts? Now I am rolling in laughter. As for a proof: I attempted a proof and you have not been able to show me anywhere that this proof is incorrect. I don't know if it is indeed correct or not. I outlined my concerns which once again you failed to address. Probably because this is how you teach your classes too: you simply dismiss questions as irrelevant when you don't have answers or you are afraid to commit yourself. So David, if indeed the planet is laughing at me, I am at least accomplishing some good: laughter is good for the bones! Here's to the *good health* of all those reading my funny posts!! ha, ha. Please continue to read and improve your health! Your bill will be in the mail soon. What good are you doing? Dr. Jason Wells. Hee, hee.
From: John Gabbriel on 8 Mar 2005 12:48 David C. Ullrich wrote: > On Mon, 07 Mar 2005 15:41:37 +0000, Angus Rodgers > <angus_prune(a)bigfoot.com> wrote: > > >[...] > > Suppose that f is differentiable on [0,1] (or differentiable > on (0,1) and continuous on [0,1]). Does it follow that > > f(1) - f(0) = lim_N sum_1^N f'(j/N)/N ? > > Of course this is clear from the fundamental theorem of calculus > if f' is continuous. It seems unlikely that it's true under the > stated hypotheses, but "seems unlikely" is not quite a counterexample. This is just the first part of Gabriel's theorem. Part ii) states the Integral (x to x+w) f'(t)dt = some limit on the right. We have many counterexamples which show that if f is differentiable, then f' need not be Riemann (or even Lebesgue) integrable. So the theorem as stated is wrong. Part i) might be true, but I am confident there will be counterexamples to this. I read somewhere that there are strictly increasing differentiable functions g such that g'(x) = 0 almost everywhere, (supposedly this is given as Example 2.1.2.1 in B.R. Gelbaum and J M H Olmstead, Theorems and counterexamples in mathematics, Springer 1990) which leads me to believe there will be functions such that f(0) < f(1) and f'(r) = 0 for rational r.
From: John Gabbriel on 8 Mar 2005 13:05 Jason wrote: > Hate to tell you that *you* are the one being laughed at. :-) Why? > Well, I don't claim to know. That's why I post messages on this board > soliciting information from *those who claim to know*. Now you claim to > know yet not once have you been able to answer any of the questions I > asked you. Your responses are hilarious and non-commital, e.g. OK. Let's start a vote here. I am laughing at Jason Wells. Score: Ullrich: 1. Jason: 0. > You are supposed to be a mathematician, no? Well, show me through > mathematics where and why in a detailed explanation how Gabriel's > theorem is wrong. You keep changing your tone! Well, what exactly are > you saying? He did not say that the theorem is wrong. He just pointed out that the proof is wrong. Do you get that, Jason? Or do you think that any attack on the proof is an attack on the theorem (conjecture/whatever) itself? > have not said much that makes sense yet. You pick out some vague topic > and then present an example which is irrelevant. As a mathematician you Just because you cannot understand what he is saying does not make it irrelevant. And *you* talk about being open minded... > would need to show in detail (giving reasons) why a proof is incorrect. No. It is the responsibility of the theorem 'prover' to come up with coherent definitions and arguments as to why the theorem is true. btw, you were given detailed reasons as to why the proof is incorrect. You chose to ignore them. > > > I don't believe that anyone has _said_ that his theorem is false > > as stated. But nobody has given a proof of it, and people > > _suspect_ it's false, for example if there does indeed exist > > a non-constant differentiable f such that f'(r) = 0 for all > > rational r, as I suspect, then the theorem's certainly false. > > Giggle, giggle. About 90% of the contributors on this forum have said > his theorem is false! Can you *read* ?! Have you understood any of the > posts? The theorem as stated is false! As I can see on the webpage, there are three parts to the theorem and part ii is clearly false. I think David is talking about part i. > Now I am rolling in laughter. Me too, but at you. > > As for a proof: I attempted a proof and you have not been able to show > me anywhere that this proof is incorrect. He has. Read carefully and try to understand this time. > > Dr. Jason Wells. Hee, hee. I see you are getting better :-)
From: Jason on 8 Mar 2005 13:26
You are an imposter! As promised I will be reporting you! What a coward you are! Too afraid to post anything under your real name? John Gabbriel wrote: > Jason wrote: > > Hate to tell you that *you* are the one being laughed at. :-) Why? > > Well, I don't claim to know. That's why I post messages on this board > > soliciting information from *those who claim to know*. Now you claim > to > > know yet not once have you been able to answer any of the questions I > > asked you. Your responses are hilarious and non-commital, e.g. > > OK. Let's start a vote here. > > I am laughing at Jason Wells. > > Score: Ullrich: 1. Jason: 0. > > > > You are supposed to be a mathematician, no? Well, show me through > > mathematics where and why in a detailed explanation how Gabriel's > > theorem is wrong. You keep changing your tone! Well, what exactly are > > you saying? > > > He did not say that the theorem is wrong. He just pointed out that the > proof is wrong. Do you get that, Jason? Or do you think that any attack > on the proof is an attack on the theorem (conjecture/whatever) itself? > > > > have not said much that makes sense yet. You pick out some vague > topic > > and then present an example which is irrelevant. As a mathematician > you > > Just because you cannot understand what he is saying does not make it > irrelevant. And *you* talk about being open minded... > > > > would need to show in detail (giving reasons) why a proof is > incorrect. > > > No. It is the responsibility of the theorem 'prover' to come up with > coherent definitions and arguments as to why the theorem is true. > > btw, you were given detailed reasons as to why the proof is incorrect. > You chose to ignore them. > > > > > > > I don't believe that anyone has _said_ that his theorem is false > > > as stated. But nobody has given a proof of it, and people > > > _suspect_ it's false, for example if there does indeed exist > > > a non-constant differentiable f such that f'(r) = 0 for all > > > rational r, as I suspect, then the theorem's certainly false. > > > > Giggle, giggle. About 90% of the contributors on this forum have said > > his theorem is false! Can you *read* ?! Have you understood any of > the > > posts? > > The theorem as stated is false! As I can see on the webpage, there are > three parts to the theorem and part ii is clearly false. I think David > is talking about part i. > > > > Now I am rolling in laughter. > > Me too, but at you. > > > > > As for a proof: I attempted a proof and you have not been able to > show > > me anywhere that this proof is incorrect. > > He has. Read carefully and try to understand this time. > > > > > > Dr. Jason Wells. Hee, hee. > > I see you are getting better :-) |