mm-240 === Subject: Re: Old homework: Stable and unstable valuesJoona I Palaste scribbled the following:> W. Dale Hall scribbled the following:>Here's one homework problem from my Dimension Theory course. The course>is over now, but this one homework problem keeps haunting me.>>Assume X and Y are metric spaces and f: X->Y is a continuous function.>A value x in f(X) is called unstable if for all epsilon>0 there exists>a continuous function g: X->Y so that for all y in f(X), the distance>between f(y) and g(y) is smaller than epsilon, and x is not in g(X).>Otherwise the value x is called stable.>The problem: Assume the spaces are both the closed unit ball B^2, and>the function is the identity function id: B^2->B^2. Prove that the>points on the boundary, i.e. those whose distance from 0 is 1, are>unstable, and the othei.e. those whose distance from 0 is smaller>than 1, are stable.>I was able to do the first part, but not the second. Here's the proof>for the first part:>Assume x is a boundary point of B^2, i.e. its distance from 0 is 1, and>epsilon is a positive real. Now the function g: B^2->B^2, g(x) =>(1-epsilon/2)*x, is continuous and all its values are less than>epsilon away from the original points, but g(B^2) does not include x.>Any hints on the second part? It was so hard that only one person in>the entire course could do it, but I don't remember his answer.>> If f:B^2 ---> B^2 is the identity mapping, then any map g within epsilon>> of f must contain the closed ball of radius 1-epsilon. If x is distance>> r from (0,0), then as long as 1-epsilon > r, x will be in the ball of>> radius (1-epsilon). Thus, x must be stable, since for a map g within>> epsilon of f we have this:>> x in r*B^2 subset (1-epsilon)*B^2 subset g(B^2).>> Doesn't that do it?> I don't understand where you get your first premise. Suppose epsilon => 0.5. Now the function g: B^2->B^2, g(x) = x/2 + (0.5, 0), is a> continuous function, and as far as *I* can see, it's no more than> epsilon (=0.5) away from the identity function. Yet it does not contain> the (0, 0)-centric ball of radius 1-epsilon = 0.5.> If you can explain how your first premise works then I guess I> understand the rest of your answer.Grumph. Not thinking clearly. Suppose g is defined as above. Theng((-1, 0))=(0, 0) which is clearly more than 0.5 away from (-1, === and unstable values> Assume X and Y are metric spaces and f: unstable if for all epsilon>0 there exists> a continuous function g: X->Y so that for all y in f(X), the distance> between f(y) and g(y) is smaller than epsilon, and x is not in g(X).> Otherwise the value x is called stable.I'm confused by this definition. if X is not Y, then f(y) and g(y) are not === valuesAlex Gittens <( swiftset (at) imap (dot) cc )> scribbled the following:>> Assume X and Y are metric spaces and f: X->Y is a continuous function.>> A value x in f(X) is called unstable if for all epsilon>0 there exists>> a continuous function g: X->Y so that for all y in f(X), the distance>> between f(y) and g(y) is smaller than epsilon, and x is not in g(X).>> Otherwise the value x is called stable.> I'm confused by this definition. if X is not Y, then f(y) and g(y) are not > defined.Yes, I made a typo. I should have written: for all y in X, the distancebetween f(y) and g(y) is smaller than === A mathematical proof of the Cantors goof?Nicolas tries to play at being Galileo. The principal difference is thatNicolas gives every impression of being an idiot. Galileo did not.>I would like to know if this is a mathematical proof of the Cantor's >goof, or conversely it is just another mathematical goof about the >Cantor's proof.> A MATHEMATICAL PROOF IN THREE ACTS>PROLEGOMENON>[The naturals]>A direct outcome of the definition of the natural numbers is that >* Point 1: There are not naturals with infinite nonzero digits.>* Point 2. As a result of Point 1, it is not possible to assume the >bijection N <-> R. There is no justification given for Point 2, therefore this fails to be a proof at this point. As others have poin out, there exist rational numbers whose decimal expansions are infinite in length. This does not stop there being a bijection between N and Q.>HC: That's great! It sounds elegant, logical and intuitive. However, >I think we will have a little problem using it with R.>C: Why?>HC: Because Point 2 tells us that we cannot propose that bijection >with R. So, your criterion doesn't work in this case. We will have to >look for another criterion. It's a pity, because it was nice.>C: Yes, that's true.A mathematician would not fall for such a badly suppor argument.Try again. Point 2 was sta as an assertion without you even tryingto bother giving it any supporting proof.> # SECOND ACT>[Mrs. Perfect Proof1, Mrs. Perfect Proof2, Mrs. Perfect Proof_n and >Human Curiosity]>PP1: Hello Mr. HC. Do you know what? I have found out myself that the >answer to your curiosity is that the reals are uncountable, because >it is no possible to achieve the bijection N <-> R.But there has not been a proof given. lot for your interest, but it's not >possible to conclude that R is not countable from that criterion, >because it cannot be used with R.This is more of the R does not satisfy the criterion for countabilityso it is impossible to tell whether or not R is uncountable rubbish.An infinite set is countable iff there exists a bijection between N andthe set. There is no such bijection between N and R. R is an infiniteset. Therefore R is not countable.> # FINAL ACT>[Mr. Mathman and Human Curiosity]>MR. MATHMAN: Do you know, Mr. HC? If we admit Cantor's criterion to >count R, as we positively know that N <-> R is not possible (it >doesn't matter why), it is clear that R will be necessarily >uncountable. What do you think about it?>HC: Well Mr. Mathman, we can do so, but it will not convince anybody. What you mean is that it won't convince anybody who can't use theirbrains for themselves. People who can actually think will be convinced. It is a consequence of the logic.>The problem is that we know that Point 2 invalidates Cantor's >criterion. Point 2 does not invalidate Cantor's criterion. Any suggestion thatit does is just a product of your vivid imagination.Also, you sta Point 2 as if it followed immediately from Point 1. It does not.>Therefore, R would be certainly uncountable, but no >because R be intrinsically uncountable, but because we don't have a >valid criterion to count it.R is uncountable because there exists a proof of its uncountability.>MR. MATHMAN: Well then, perhaps there are other different criteria to >count the reals.No real mathematician would ever make such a concession to your delusions.Obviously, Mr. Mathman does not have a genuine mathematical background.>HC: I don't know it, but if we put a simple analogy perhaps we could >clarify our thoughts. Let's try it. Let's not. It will only be more of your delusions.A ridiculous analogy which bears absolutely no relation to the question of an existence between N and R followed here. The ridiculous analogy also included further evidence that Mr. Mathmanis a mathematical ignoramus, and not a genuine mathematician.>***************************>Proof of Point 2>INITIAL ASSERTIONS>1) We can represent any real number using a given positional system >of numeration. For ease we will use the decimal notation.>2) Cantor uses a transfinite method (the diagonal argument) to prove >that the premise N <-> R is false, and we will use another >transfinite constructive method to prove that the transfinite >construction N <-> R is impossible, and therefore we cannot assume >the bijection N <-> R.This does not invalidate Cantor's argument.>Proposition: The transfinite construction N <-> R is not possible>Proof: >Every real number within the interval (0, 1) has as first decimal >digit one of the ten digits of the decimal system of numeration, i. >e. it must be of the form 0.0, 0.1, 0.2, ..., 0.9. As you can see for >this first digit there are 10 ^ 1 possibilities. For the second digit >we have 0.00, 0.01, 0.02, ..., 0.99. Consequently, there are 10 ^2 >possible combinations for the two first digits, 10 ^3 for the first >three digits, and so on. When the number of digits is infinite we get >R.>Now, when we have only a digit of the decimal expansion of the reals, >we make the one-to-one correspondence 0 <-> 0.0, 1 <-> 0.1, 2 <-0.2, ..., 9 <-> 0.9. Next, with two digits we begin the 1-1 >correspondence 0 <-> 0.00, 1 <-> 0.01, ..., 10 <-> 0.10, 11 <-> 0.11, ...,>99 <-> 0.99. With three digits, we continue doing the same, and so on.>And now is when the impossible transfinite construction appears. >While we keep doing the 1-1 correspondence within the finite decimal >expansion, everything will go fine. No, it won't. There is no limiting bijection between N and numbers in[0,1) with finite decimal expansions. This is easily fixed: 0 <-> 0, 1 <-> 0.1, 2 <-> 0.2, ..., 9 <-> 0.9, 10 <-> 0.01, 11 <-> 0.11, 12 <-> 0.21, ..., 45892 <-> 0.29854, .... Hopefully, you will get theidea.>But, what natural number would >correspond to the decimal expansion of the number e? I have to assume that you mean the fractional part of e, i.e. e-2.>By induction it >is obvious that it should be 71828182..., That would be ...828182817, using the correct attempt at bijection.>i. e., a number with infinite >nonzero digits, which doesn't belong to N (Point 1). This ignores the fact that N can be placed in bijection with a proper subset of itself. This means that there are sets, whose elements include all numbers with finite decimal expansion and also some setswith infinite decimal expansion, and which can be placed in bijection with N (this is because the proper subset of all natural numbers which correspond to numbers of finite decimal expansion can itself be placed in bijection with N). This step in your so-called proof required the assumption that N cannot be placed in bijection with any proper subset of N. This assumption is false.>Conclusion:>The transfinite construction N <->R is not possible if we do not >admit naturals with an infinite number of nonzero digits. This conclusion was based on a premise which is known to be false. Theconclusion was therefore not proven in the above argument.>If some day >we admit that, then the reals would be countable.This contradicts the definition of countability. R is, and always has been, uncountable. None of === Re: A mathematical proof of the Cantors goof?> The proof (plus analogy) in three acts has been written in this way > just for those who do not want to understand the proof. As it is not > in a correct and formal mathematical language, I understand that you > don't want to refute it.As it is written at great length but without the benefits of any logic or pattern compatible with mathematical exposition, it is more a drama that a proof, and of that particular type of drama called farce.However, the proof of Point 2 fulfils all the requirements of a > mathematical proof. Only in your own mind.> Therefore, as I suppose that most of you are professional > mathematicians, instead of talking and talking, the only thing you > have to do is to refute this proof. This is called the mathematical > method. We would have to find your proof first before we would attempt to refute it..As the conclusion of this proof clearly states that the natural > numbers fail in counting the reals because they cannot undertake the > infinity of R, it is obvious that the naturals cannot be used to > count the reals, and therefore the Cantor's criterion (that you call > definition) cant be used with R.Claims are easy to make. You make a lot of them, but your idea of a proof simply does not convince anyone that you know anything.In short, refute the proof of Point 2 and every thing will be as it > was before.PD: If someone else wants to use the criterion that Q is countable, > please show me a proof using the decimal expansion of n/m. I'll try > to find out why naturals succeed in counting rationals in this way, > and however they fail doing the same with the reals.Using decimal expansions would be a remarkably ugly approach to something easily proved (and proved in this thread several times) without using that approach. What is the point of trying to do === L^pOriginator: grubb@lola>I have a problem solving the following inequality in L^p spaces over the reals >with 2<=p|(f+g)/2|^p + |(f-g)/2|^p <= (|f|^p + |g|^p)/2>when p=2 I know I can use the parallelogram law, but I don't know what to do >about the other p's. I tried for example >to use the Minkowsky inequality and binomial expansions.. but I still can't >solve it. I would appreciate any suggestions or >comments.This is called Clarkson's Inequality. Proofs have a tendency to bea bit tricky. The best I have seen is the following (not due to me)It is enough to show this for complex numbers z and w. Let M_rbe (1/2 |z|^r + 1/2 |w|^r )^(1/r). Then Holder shows that M_r isnon-decreasing in r.Thus|1/2(z+w)|^p + |1/2(z-w)|^p = |1/2(z+w)|^2 |1/2(z+w)|^(p-2) + |1/2(z-w)|^2 |1/2(z-w)|^(p-2) <= |1/2(z+w)|^2 [1/2(|z|+|w|)]^(p-2) +|1/2(z-w)|^2 [1/2(|z|+|w|)]^(p-2) = [1/2(|z|+|w|)]^(p-2) [1/2(|z|^2 + |w|^2)]^(p-2) = (M_1 )^(p-2) (M_2 )^2 <= (M_p )^(p-2) (M_p )^2 =(M_p)^p as desired.This first equality is a factorization, the first inequality is a triangleinequality. The second equality is a factorization and parallelogram law.The third equality is definitional. The second inequality is the non-decreasingnature === numbers> @abouthugo.de:>> There is an old Fortran I was a little disappoin that it's not purely > binary, but I'll test it for speed and see if it matters or not.Not purely binary? Surely, that's what compilers are for.Or did you mean something different?-- Dave SeamanJudge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. the best way to do it?I all depends on your definition of a bunch and on what you want to obtain.Should you be interessed in getting byproducts as well (solutions ofLegendre's and Dirichlet's problem for your numbers), then a (simple) datastructure is needed along with the numbers.In any case, I would recommend the following approach: recursively Mod() thesecond largest number out of the first, until you are left with{x,0,0,...,0,0} where you're done: x is your GCD. Yes, this implies more orless a decreasing sort (or carefull insert in list or array), but I found itto be well worst the pain. This choice also heavily depend on the languageyou will be using: some language allow a very fast implementation of such amanipulation.Anyway, almost any reasonable implementation will be fast enough for anypractical purpose: Euclid was well aware of the future price of computerscycles and, accordingly, inven a very efficient algorithm ;-))Thus, you shouldn't care that much, except if you are handling a very largenumber of === case, I would recommend the following approach: recursively Mod()> the second largest number out of the first, until you are left with> {x,0,0,...,0,0} where you're done: x is your GCD. Yes, this implies more> orThat seems backwards. Won't this be faster? 1. Sort into increasing order, a[1], a[2], ..., a[n] 2. Set d = a[1], i = 2 3. If d = 1, gcd is d 4. Set d = gcd(d,a[i]) 5. If i = n, gcd is d 6. Set i = i+1 7. Goto === backwards. Won't this be faster?|> |> 1. Sort into increasing order, a[1], a[2], ..., a[n]|> |> 2. Set d = a[1], i = 2|> |> 3. If d = 1, gcd is d|> |> 4. Set d = gcd(d,a[i])|> |> 5. If i = n, gcd is d|> |> 6. Set i = i+1|> |> 7. Goto step #3Certainly in case you're only interessed in computing the overall out any possibility to solve Dirichlet'sproblems as it doesn't perform ExtendedGCD at each step. If you do, (andkeep track of the intermediate results) one you might end up with acomparable complexity.I have experimen with initial sort both increasing and decreasing andI've found that an initial decreasing sort produces *much* smallercoefficients in the extended GCD problem. This can be due to a particularproperty of the typical inputs I use.-- Here's where you can reach me === multiple numbers> But I can't stop thinking that there's a better way. Is it worthwhile to> sort the list of numbers?I'd expect it to help to start with the smaller numbers first. It'sprobably also worthwhile to specifically check for hitting === Einstein[snip Dinky's trash]Quite a CV {:-))Franz> Apart from your subjective opinions of any people involved, Franz, do you> have any> objective opinion on the mathematics at> http://www.androc1es.pwp.blueyonder.co.uk/Fundamental_rv_2.0. htm> or are you completely enraptured by Dinky's silly game of one-up-manship?> Androcles Jl3.4986923@phobos.telenet-ops.beI can testify that among relativists you said the truest thing everabout the 1905 paper,you did'nt recognise the excerpt from the 1905paper and this makes it all the more funny.Perhaps Androcles recognises that it is from the kinematics section ofthe 1905 paper and I am deligh that you agree with him that it is's' that is not worth reading.----------------->Let us take a system of co-ordinates in which the equations of> Newtonian mechanics hold good. In order to render our presentation> more precise and to distinguish this system of co-ordinates verbally> from others which will be introduced hereafter, we call it the> ``stationary system.''What is this?Some kind of quote of some post?An introduction to the s you produce later on?s that you expect someone will bother reading?Dirk === Dinky's trash]Quite a CV {:-))Franz> Apart from your subjective opinions of any people involved, Franz, do you> have any> objective opinion on the mathematics at> http://www.androc1es.pwp.blueyonder.co.uk/Fundamental_rv_2.0. htm> or are you completely enraptured by Dinky's silly game of one-up-manship?> Androcles Jl3.4986923@phobos.telenet-ops.beDirk Vdm> I can testify that among relativists you said the truest thing ever> about the 1905 paper,you did'nt recognise the excerpt from the 1905> paper and this makes it all the more funny.> Perhaps Androcles recognises that it is from the kinematics section of> the 1905 paper and I am deligh that you agree with him that it is> 's' that is not worth reading.> ----------------->Let us take a system of co-ordinates in which the equations of> Newtonian mechanics hold good. In order to render our presentation> more precise and to distinguish this system of co-ordinates verbally> from others which will be introduced hereafter, we call it the> ``stationary system.''> What is this?> Some kind of quote of some post?> An introduction to the s you produce later on?> s that you expect someone will bother reading?> Dirk> -----------------I already said this on boi.hp.combut I will repeat it here:You still don't seem to understand why I asked youthese 4 questions.So let me try to explain in simple words.Let's have a close look at the message you are referringto here: a system of co-ordinates in which the equations of | Newtonian >mechanics hold good.2 In order to render our presentation | more precise and to >distinguish this system of co-ordinates verbally | from others which will be >introduced hereafter, we what you gave us.As you see, - There is something severely wrong with the format. - If this was a quotation from Einstein, you left out the quotes. - You write a paragraph I will kindly ask again, and I will clarify what I mean:1) What is this?2) Some kind of quote of some post?Clarification: Something you want us to believe you inven? Something you found somewhere? Something you want to tell us? Something you want to tell us something about? Something you forgot to Al,?3) An introduction to the s you produce later on?4) s that you expect someone will bother reading?Clarification: The 's' in question 4 is a reprise of the 's' in question 3. This is what we call a 'style figure'.Didn't they teach you to write === Defending Einstein[snip Dinky's trash]Quite a CV {:-))Franz> Apart from your subjective opinions of any people involved, Franz, do you> have any> objective opinion on the mathematics at> http://www.androc1es.pwp.blueyonder.co.uk/Fundamental_rv_2.0. htm> or are you completely enraptured by Dinky's silly game of one-up-manship?> Androcles Jl3.4986923@phobos.telenet-ops.beAndrocles will never understand that c-v and c+v in Einstein's paperare algebraic expressions which are not physical speeds measured byany observer, but rates of travelled ** relative ** distances by === trash]Quite a CV {:-))Franz> Apart from your subjective opinions of any people involved, Franz, do you> have any> objective opinion on the mathematics at> http://www.androc1es.pwp.blueyonder.co.uk/Fundamental_rv_2.0. htm> or are you completely enraptured by Dinky's silly game of one-up-manship?> Androcles Jl3.4986923@phobos.telenet-ops.beDirk Vdm> Androcles will never understand that c-v and c+v in Einstein's paper> are algebraic expressions which are not physical speeds measured by> any observer, but rates of travelled ** relative ** distances by the> light rays.Yep, but I'd call them rates of change of distances between someobject and the front of a lightray, distances as calcula by a thirdparty observer who is assumed to measure c to be the speed of thefront of the === EinsteinDirk Van de moortel Dinky's trash]Quite a CV {:-))Franz> Apart from your subjective opinions of any people involved, Franz,do you> have any> objective opinion on the mathematics at> http://www.androc1es.pwp.blueyonder.co.uk/Fundamental_rv_2.0. htm> or are you completely enraptured by Dinky's silly game ofone-up-manship?> .109057$Jl3.4986923@phobos.telenet-ops.beDirk VdmAndrocles will never understand that c-v and c+v in Einstein's paper> are algebraic expressions which are not physical speeds measured by> any observer, but rates of travelled ** relative ** distances by the> light rays.> Yep, but I'd call them rates of change of distances between some> object and the front of a lightray, distances as calcula by a third> party observer who is assumed to measure c to be the speed of the> front of the lightray w.r.t. himself.>As this:But the ray moves relatively to the initial point of k, when measured inthe stationary system, with the velocity c-v, so that ...in which we have the observer (k), and the initial point of k.But that is somehow confusing and for a similar calculation I got a zero inan exam, as the teacher believed that I meant:But the ray moves relatively to [observer] k with the velocity c-vwhich ignores when measured in the stationary system.It would have been easier just to make c*t = v*t + L with L the initialseparation among lightray front and origin of k, than write directly L =(c-v)*t and talk of c-v as a speed (it is a speed dimensionally, but notphysicall, i.e., cannot be adscribed to === Dinky's trash]Quite a CV {:-))Franz> Apart from your subjective opinions of any people involved, Franz,> do you> have any> objective opinion on the mathematics at> http://www.androc1es.pwp.blueyonder.co.uk/Fundamental_rv_2.0. htm> or are you completely enraptured by Dinky's silly game of> one-up-manship?> .109057$Jl3.4986923@phobos.t> elenet-ops.beDirk VdmAndrocles will never understand that c-v and c+v in Einstein's paper> are algebraic expressions which are not physical speeds measured by> any observer, but rates of travelled ** relative ** distances by the> light rays.Yep, but I'd call them rates of change of distances between some> object and the front of a lightray, distances as calcula by a third> party observer who is assumed to measure c to be the speed of the> front of the lightray w.r.t. himself.Dirk Vdm> As this:> But the ray moves relatively to the initial point of k, when measured in> the stationary system, with the velocity c-v, so that ...> in which we have the observer (k), and the initial point of k.> But that is somehow confusing and for a similar calculation I got a zero in> an exam, as the teacher believed that I meant:> But the ray moves relatively to [observer] k with the velocity c-v> which ignores when measured in the stationary system.> It would have been easier just to make c*t = v*t + L with L the initial> separation among lightray front and origin of k, than write directly L => (c-v)*t and talk of c-v as a speed (it is a speed dimensionally, but not> physicall, i.e., cannot be adscribed to any observer).Hm, to me the initial separation between lightray front and originof k seems to be 0. It grows to x', and upon reflection, shrinksback to 0.It is clear on the spacetime diagram: http://users.pandora.be/vdmoortel/dirk/Stuff/ tau-equation.gifThere you clearly see from the geometry that the 'horizontal distances'(the dashed blue lines) between the red k-worldline (tau-axis) andthe grey light-worldline first grow from 0 at time t0 (event Flash)to x' at time t1 (event Reflection), and then shrink back to 0 attime t2 (event === Van de moortel Dinky's trash]Quite a CV {:-))Franz> Apart from your subjective opinions of any people involved,Franz,> do you> have any> objective opinion on the mathematics at> http://www.androc1es.pwp.blueyonder.co.uk/Fundamental_rv_2.0. htm> or are you completely enraptured by Dinky's silly game of> one-up-manship?> .109057$Jl3.4986923@phobos.t> elenet-ops.beDirk VdmAndrocles will never understand that c-v and c+v in Einstein's paper> are algebraic expressions which are not physical speeds measured by> any observer, but rates of travelled ** relative ** distances by the> light rays.Yep, but I'd call them rates of change of distances between some> object and the front of a lightray, distances as calcula by a third> party observer who is assumed to measure c to be the speed of the> front of the lightray w.r.t. himself.Dirk VdmAs this:But the ray moves relatively to the initial point of k, when measuredin> the stationary system, with the velocity c-v, so that ...in which we have the observer (k), and the initial point of k.> But that is somehow confusing and for a similar calculation I got a zeroin> an exam, as the teacher believed that I meant:But the ray moves relatively to [observer] k with the velocity c-vwhich ignores when measured in the stationary system.It would have been easier just to make c*t = v*t + L with L the initial> separation among lightray front and origin of k, than write directly L => (c-v)*t and talk of c-v as a speed (it is a speed dimensionally, but not> physicall, i.e., cannot be adscribed to any observer).> Hm, to me the initial separation between lightray front and origin> of k seems to be 0. It grows to x', and upon reflection, shrinks> back to 0.> It is clear on the spacetime diagram:> http://users.pandora.be/vdmoortel/dirk/Stuff/tau-equation.gif> There you clearly see from the geometry that the 'horizontal distances'> (the dashed blue lines) between the red k-worldline (tau-axis) and> the grey light-worldline first grow from 0 at time t0 (event Flash)> to x' at time t1 (event Reflection), and then shrink back to 0 at> time t2 (event Flashecho).>Yes, I was thinking of the mirror as the origin of k (I didn't have thepaper with me).In any case, Einstein made a quite difficult and lenghty derivation.It would have been much easier to derive the Lorentz transforms by using thetime dilation and length contraction effects. However, I guess thatpsychologically it WAS more convenient the other way: get the Lorentztransform. and from them, derive these effects.You provided a reference with such a kind of derivationhttp://www.courses.fas.harvard.edu/~phys16/Textbook/ ch10.pdfbut one thing I am at odds with, namely that the relative speeds of the twosystems of reference are not necessarily equal and opposite (a strangewording, btw). Without a proof of such a thing, the demo is not completelycorrect.Notice how Einstein was very careful about it, and instead of assuming it (Iknow PhD's who make that), he showed that the relative velocities were vand -v respectively [ === Van de moortel Dinky's trash]Quite a CV {:-))Franz> Apart from your subjective opinions of any people involved,> Franz,> do you> have any> objective opinion on the mathematics at> http://www.androc1es.pwp.blueyonder.co.uk/Fundamental_rv_2.0. htm> or are you completely enraptured by Dinky's silly game of> one-up-manship?> .109057$Jl3.4986923@phobos.t> elenet-ops.beDirk VdmAndrocles will never understand that c-v and c+v in Einstein's paper> are algebraic expressions which are not physical speeds measured by> any observer, but rates of travelled ** relative ** distances by the> light rays.Yep, but I'd call them rates of change of distances between some> object and the front of a lightray, distances as calcula by a third> party observer who is assumed to measure c to be the speed of the> front of the lightray w.r.t. himself.Dirk VdmAs this:But the ray moves relatively to the initial point of k, when measured> in> the stationary system, with the velocity c-v, so that ...in which we have the observer (k), and the initial point of k.> But that is somehow confusing and for a similar calculation I got a zero> in> an exam, as the teacher believed that I meant:But the ray moves relatively to [observer] k with the velocity c-vwhich ignores when measured in the stationary system.It would have been easier just to make c*t = v*t + L with L the initial> separation among lightray front and origin of k, than write directly L => (c-v)*t and talk of c-v as a speed (it is a speed dimensionally, but not> physicall, i.e., cannot be adscribed to any observer).Hm, to me the initial separation between lightray front and origin> of k seems to be 0. It grows to x', and upon reflection, shrinks> back to 0.> It is clear on the spacetime diagram:> http://users.pandora.be/vdmoortel/dirk/Stuff/tau-equation.gif> There you clearly see from the geometry that the 'horizontal distances'> (the dashed blue lines) between the red k-worldline (tau-axis) and> the grey light-worldline first grow from 0 at time t0 (event Flash)> to x' at time t1 (event Reflection), and then shrink back to 0 at> time t2 (event Flashecho).Dirk Vdm> Yes, I was thinking of the mirror as the origin of k (I didn't have the> paper with me).> In any case, Einstein made a quite difficult and lenghty derivation.> It would have been much easier to derive the Lorentz transforms by using the> time dilation and length contraction effects. However, I guess that> psychologically it WAS more convenient the other way: get the Lorentz> transform. and from them, derive these effects.> You provided a reference with such a kind of derivation> http://www.courses.fas.harvard.edu/~phys16/Textbook/ch10.pdf> but one thing I am at odds with, namely that the relative speeds of the two> systems of reference are not necessarily equal and opposite (a strange> wording, btw). Without a proof of such a thing, the demo is not completely> correct.As far as I can see the phrase equal and opposite does not appearin the derivation, but merely in an example.> Notice how Einstein was very careful about it, and instead of assuming it (I> know PhD's who make that), he showed that the relative velocities were v> and -v respectively [ phi(v)*phi(-v)=1]And as far as I can see here, he did not *show* that the relativevelocities are v and -v, but rather *used* that trivial fact to showthat === EinsteinDirk Van de moortel moortel messagemessage> [snip Dinky's trash]Quite a CV {:-))Franz> Apart from your subjective opinions of any people involved,> Franz,> do you> have any> objective opinion on the mathematics at>http://www.androc1es.pwp.blueyonder.co.uk/Fundamental_rv_2.0 .htm> or are you completely enraptured by Dinky's silly game of> one-up-manship?> .109057$Jl3.4986923@phobos.t> elenet-ops.beDirk VdmAndrocles will never understand that c-v and c+v in Einstein'spaper> are algebraic expressions which are not physical speeds measuredby> any observer, but rates of travelled ** relative ** distances bythe> light rays.Yep, but I'd call them rates of change of distances between some> object and the front of a lightray, distances as calcula by athird> party observer who is assumed to measure c to be the speed of the> front of the lightray w.r.t. himself.Dirk VdmAs this:But the ray moves relatively to the initial point of k, whenmeasured> in> the stationary system, with the velocity c-v, so that ...in which we have the observer (k), and the initial point of k.> But that is somehow confusing and for a similar calculation I got azero> in> an exam, as the teacher believed that I meant:But the ray moves relatively to [observer] k with the velocity c-vwhich ignores when measured in the stationary system.It would have been easier just to make c*t = v*t + L with L theinitial> separation among lightray front and origin of k, than write directlyL => (c-v)*t and talk of c-v as a speed (it is a speed dimensionally, butnot> physicall, i.e., cannot be adscribed to any observer).Hm, to me the initial separation between lightray front and origin> of k seems to be 0. It grows to x', and upon reflection, shrinks> back to 0.> It is clear on the spacetime diagram:> http://users.pandora.be/vdmoortel/dirk/Stuff/tau-equation.gif> There you clearly see from the geometry that the 'horizontaldistances'> (the dashed blue lines) between the red k-worldline (tau-axis) and> the grey light-worldline first grow from 0 at time t0 (event Flash)> to x' at time t1 (event Reflection), and then shrink back to 0 at> time t2 (event Flashecho).Dirk VdmYes, I was thinking of the mirror as the origin of k (I didn't have the> paper with me).> In any case, Einstein made a quite difficult and lenghty derivation.It would have been much easier to derive the Lorentz transforms by usingthe> time dilation and length contraction effects. However, I guess that> psychologically it WAS more convenient the other way: get the Lorentz> transform. and from them, derive these effects.You provided a reference with such a kind of derivation> http://www.courses.fas.harvard.edu/~phys16/Textbook/ch10.pdf> but one thing I am at odds with, namely that the relative speeds of thetwo> systems of reference are not necessarily equal and opposite (a strange> wording, btw). Without a proof of such a thing, the demo is notcompletely> correct.> As far as I can see the phrase equal and opposite does not appear> in the derivation, but merely in an example.No, it is in the table summarizing the effects derived from the constancy ofc.x=0 ---> x' = -v*t'Here v' should be used. v'=v is not a trivial fact. See below:> Notice how Einstein was very careful about it, and instead of assumingit (I> know PhD's who make that), he showed that the relative velocities were v> and -v respectively [ phi(v)*phi(-v)=1]> And as far as I can see here, he did not *show* that the relative> velocities are v and -v, but rather *used* that trivial fact to show> that phi(v)*phi(-v)=1.>No, he needed to prove phi(v)phi(-v) = 1, but just before the conclusion,Einstein states:For this purpose we introduce a third system of co-ordinates , K' whichrelatively to the system k is in a state of parallel translatory motionparallel to the axis of X, such that the origin of co-ordinates of system kmoves with velocity -v on the axis of X...andSince the relations between x', y', z' and x, y, z do not contain the timet, the systems K and K' are at rest with respect to one another, and it isclear that the transformation from K to must be the identicaltransformation. Thus phi(v)*phi(-v)=1.I do not believe it is a trivial fact if you use just 2 reference frames. Ifyou use 3, by symmetry, you can demonstrate it.However, notice that, without going into more details, the fact that a speedwhich in Galilean transformation of velocities should be c-v (or c+v) is c,means that the new (relativistic) transf. of velocities is not trivial. Inparticular, it is not guaranteed that an observer moving with v will see theother observer moving with -v.In fact, I commen this to a PhD, but he just used the apparent symmetryof the problem, without invoking a third observer. This didn't satisfied me.When I first bought the book by Pauli on relativity, in a trip to London(there was not Internet to buy books abroad by at that remote epoch!!!), Iread on the original derivation by Einstein (I later bought the Doveredition of The principle of relativity). I was a little dissapoin to seethat Einstein had taken that non-trivial fact into consideration and hesucceded on making a perfect derivation. Notice the convolu way ofsaying that apparently trivial fact. It is not a trivial fact indeed, and a3rd system === toward 0, not toward 1>If the things don't commute, yes. Your logic is inconsistent if the>things don't commute.WHY is there an inconsistency if things don't commute?Much of the rest of Gary's posting was an arrogant instruction to correct mathematics to correct the flaw that HE perceived in the fact === Squares of 0.999... tend toward 0, not toward 1> I think you like 'i' because it stands for IMAGINARY -- the value of your> EPSILON. karl mNo karl, I like i because nothing squared plus one equals zero.Garry Denke, GeologistDenoco Inc. of === toward 1> I think you like 'i' because it stands for IMAGINARY -- the value of your> EPSILON. karl mNo karl, I like i because nothing squared plus one equals zero.Garry Denke, Geologist> Denoco Inc. of TexasGarry likes 'i' because he is a solipsist === not toward 1> Or even more trivially, lim(x->0) lim(y->0) (x-y)/(x+y) = 1> lim(y->0) lim(x->0) (x-y)/(x+y) = -1But, perhaps Denke, being such a rockhound, has only rocks in his head?I see you excel in kindness Virgil. Your logic is as fake as yourname. Are you in hiding? Get real Virgil. Define your 1. Define your0. What is your derivation of -1?Garry Denke, === 0.999... tend toward 0, not toward 1> Or even more trivially, lim(x->0) lim(y->0) (x-y)/(x+y) = 1> lim(y->0) lim(x->0) (x-y)/(x+y) = -1But, perhaps Denke, being such a rockhound, has only rocks in his head?I see you excel in kindness Virgil. Your logic is as fake as your> name. Are you in hiding? Get real Virgil. Define your 1. Define your> 0. What is your derivation of -1?Garry Denke, Geologist> Denoco Inc. of TexasWho says that everything commutes to justify his wearing his sox outside his shoes and his undies outside hhis === slacks.Subject: Re: Squares of 0.999... tend toward 0, not post.newsfeed.com ***> Define your 1. Define your> 0. What is your derivation of -1?These notations stem from COUNTING. When you COUNT, ZERO is where youstart. You reach ONE after you TAKE your first UNIT from what is coming atyou. I'll leave putting UNITS back as an excercise for you. karl m -----= Pos via Newsfeed.Com, Uncensored Usenet News =-----http://www.newsfeed.com - The #1 Newsgroup Service in the World!-----== 100,000 Groups! - 19 Servers! - Unlimi === toward 0, not toward 1> Or even more trivially, lim(x->0) lim(y->0) (x-y)/(x+y) = 1> lim(y->0) lim(x->0) (x-y)/(x+y) = -1But, perhaps Denke, being such a rockhound, has only rocks in his head?I see you excel in kindness Virgil. Your logic is as fake as your> name. Are you in hiding? Get real Virgil. Define your 1.1 means the same thing here that it means whenever any actualmathematician uses it.> Define your> 0.See above.> What is your derivation of -1?First, proofs of internal limits using the epsilon-delta definition.To prove: lim(y->0) (x-y)/(x+y) = 1 for any x != 0For any epsilon > 0, take delta = min(epsilon*abs(x)/4,abs(x)/2)If abs(y - 0) < delta, we haveabs(y) < epsilon*abs(x)/4abs(x)/2 < abs(x+y) < 3*abs(x)/2abs ((x-y)/(x+y) - 1) = abs (-2*y/(x+y)) = 2 * abs(y) / abs(x+y) < 2 * epsilon*abs(x)/4 / (abs(x)/2) = epsilon Q.E.D.To prove: lim(x->0) (x-y)/(x+y) = -1 for any y != 0For any epsilon > 0, take delta = min(epsilon*abs(y)/4,abs(y)/2)If abs(x - 0) < delta, we haveabs(x) < epsilon*abs(y)/4abs(y)/2 < abs(x+y) < 3*abs(y)/2abs ((x-y)/(x+y) + 1) = abs (2*x/(x+y)) = 2 * abs(x) / abs(x+y) < 2 * epsilon*abs(y)/4 / (abs(y)/2) = epsilon Q.E.D.For any x != 0, lim(y->0) (x-y)/(x+y) = 1Therefore, lim(x->0) lim(y->0) (x-y)/(x+y) = lim(x-)0) 1 = 1For any y != 0, lim(x->0) (x-y)/(x+y) = -1Therefore, lim(y->0) lim(x->0) (x-y)/(x+y) = lim(y->0) -1 = -1 -- Daniel W. Johnsonpanoptes@iquest.nethttp://members.iquest.net/~panoptes/ === tend toward 0, not toward 1They should commute. Why the contradiction? Would you also expect the limits to commute in the following case?lim(x->0) lim(y->0) arctan((x-y)/(x+y))> lim(y->0) lim(x->0) arctan((x-y)/(x+y))Everything should commute, so if it doesn't, fix the error that causedit. Define zero, and define the placeholder 0. What do you mean byzero === Re: Squares of 0.999... tend toward 0, not toward 1They should commute. Why the contradiction? Would you also expect the limits to commute in the following case?lim(x->0) lim(y->0) arctan((x-y)/(x+y))> lim(y->0) lim(x->0) arctan((x-y)/(x+y))Everything should commute, so if it doesn't, fix the error that caused> it. Define zero, and define the placeholder 0. What do you mean by> zero Daniel?Zero is the sublime result of thousands of years of careful mathematical creativity, and is thus far beyond Denke's understanding.Garry Denke, Geologist> Denoco Inc. of TexasWho wears his underpants outside is pants because he commutes the order in which he puts them on and then says === toward 0, not toward 1They should commute. Why the contradiction? Would you also expect the limits to commute in the following case?lim(x->0) lim(y->0) arctan((x-y)/(x+y))> lim(y->0) lim(x->0) arctan((x-y)/(x+y))Everything should commute,I couldn't resist this OBVIOUS troll. How's your checking account? Ihope it's never in the red, because SUBTRACTION COMMUTES! We don'tneed negative numbers any more!Oh wait, I didn't eat 1/6 of a pie! I ate 6 PIES, because (drum rollplease .... bahdadadadadada ching!) DIVISION COMMUTES! Fantastic!Let's pack it up boys, get rid of those pesky real numbers,integration, matrices. All we need is the natural numbeaddition,and multiplication. Happy Day!!!!Go eat some rocks, troll!> so if it doesn't, fix the error that caused> it. Define zero, and define the placeholder 0. What do you mean by> zero Daniel?Garry Denke, Geologist> Denoco Inc. of === toward 1They should commute. Why the contradiction? Would you also expect the limits to commute in the following case?lim(x->0) lim(y->0) arctan((x-y)/(x+y))> lim(y->0) lim(x->0) arctan((x-y)/(x+y))Everything should commuteGood, then you should have no difficulty putting your socks on overyour shoes. Or, maybe you should put the gas in your car before it ispumped out of the ground? Of course, you could digest your food beforeyou eat it. Since EVERYTHING should commute, after all. Idiot!> so if it doesn't, fix the error that caused> it. Define zero, and define the placeholder 0. What do you mean by> zero Daniel?Garry Denke, Geologist> === toward 0, not toward 1 permission for an emailed response.> Everything should commute, so if it doesn't, fix the error that caused> it. Define zero, and define the placeholder 0. What do you mean by> zero Daniel?Everything should commute? Why?In fact, limits don't commute in general for exactly the === 0.999... tend toward 0, not toward 1>> Everything should commute, so if it doesn't, fix the error that caused>> it. Define zero, and define the placeholder 0. What do you mean by>> zero Daniel?>Everything should commute? Why?Should everything commute? works. Neither Everything commuteshould, Commute should everything, Should commute everything norCommute everything should make any sense.Even Denke's sentence structures don't commute so obviously he must === tend toward 0, not toward 1 <1g7g6ts.daqmb299jniwN%panoptes@iquest.net> <874quyibv7.fsf@becket.becket.net> Discussion, limits don't commute in general for exactly the same reason> that quantifiers don't.Exactly the same reason? Really? What reason is that?-- Even if [...] a communistic regime should come [to China], the oldtradition [...] will break Communism and change it beyond recognition,rather than Communism [...] break the old tradition. It must be so. -- Lin Yutang on === Squares of 0.999... tend toward 0, not toward 1>> In fact, limits don't commute in general for exactly the same reason>> that quantifiers don't.> Exactly the same reason? Really? What reason is that? The Vogon constructor ship hovered in midair in exactly the same way that bricks don't. - The Hitchhiker's Guide to the Galaxy (las Adams)-- Dave SeamanJudge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.Everything should commuteWhy?-- Richard-- Spam filter: to mail me from a .com/.net site, put === Squares of 0.999... tend toward 0, not toward 1> What contradiction?lim(n->+oo) lim(m->+oo) (1-1/10^m)^n = 1lim(m->+oo) lim(n->+oo) (1-1/10^m)^n = 01/10 = 0.1, 1 - 0.1 = 0.9, 1 = 1, 0.999... = 0.999..., 0 = 0. > There are all kinds of things in mathematics that do not commute.> Multiplication of matrices or quaternions, to name two examples.> Are matrices and quaternions contradictory?If the things don't commute, yes. Your logic is inconsistent if thethings don't commute. Perhaps you should try to solve your problemswith a system that is not contradictory, where things do commute. Theyshould, and they will, once you trash your errors.> Learning which things commute and which things don't is a part of> mathematics.Learn your mistakes is what you are suggesting. Okay, learn them, thencorrect the things that don't so your system is consistent. The thingsthat don't commute are the fallout from your system errors. That'sreally what we are doing here, fixing your mistake 0.999... = 1.> What contradiction? Which definition?Your notation zero is a good start. Define zero.Garry Denke, GeologistDenoco Inc. of === toward 1> What contradiction?lim(n->+oo) lim(m->+oo) (1-1/10^m)^n = 1> lim(m->+oo) lim(n->+oo) (1-1/10^m)^n = 01/10 = 0.1, 1 - 0.1 = 0.9, 1 = 1, 0.999... = 0.999..., 0 = 0. There are all kinds of things in mathematics that do not commute.> Multiplication of matrices or quaternions, to name two examples.> Are matrices and quaternions contradictory?If the things don't commute, yes. Your logic is inconsistent if the> things don't commute. Your criticism is illogical if the things don't commute.Putting on sox and and then putting on shoes do not commute. Do you put them on in reverse order?If so I suggest that you commute the operations of opening a window and then sticking your head out of it.> Perhaps you should try to solve your problems> with a system that is not contradictory, where things do commute. They> should, and they will, once you trash your errors.Then try the commuting the window experiment described above. With a double paned window.Learning which things commute and which things don't is a part of> mathematics.Learn your mistakes is what you are suggesting. Yes, he is suggesting that you learn your mistakes and learn from them. You apparently have little or no talent in that direction.> Okay, learn them, then correct the things that don't so your system > is consistent. The things that don't commute are the fallout from > your system errors. That's really what we are doing here, fixing your > mistake 0.999... = 1.We? We are doing fine You are the one making all the mistakes, which you are singularly unable or unwilling to correct.What contradiction? Which definition?Your notation zero is a good start. Define zero.I would ask you to define sanity, but it is obviously something with which you have hade === Re: Squares of 0.999... tend toward 0, not toward 1>Putting on sox and and then putting on shoes do not commute.They do if === tend toward 0, not toward 1>> What contradiction?>lim(n->+oo) lim(m->+oo) (1-1/10^m)^n = 1>lim(m->+oo) lim(n->+oo) (1-1/10^m)^n = 0>1/10 = 0.1, 1 - 0.1 = 0.9, 1 = 1, 0.999... = 0.999..., 0 = 0. What is the contradiction in that?>Your logic is inconsistent if the things don't commute.Why is not commuting inconsistent?-- Richard-- Spam filter: to mail me from a .com/.net site, put my surname in the headers.FreeBSD === toward 1>> What contradiction?> lim(n->+oo) lim(m->+oo) (1-1/10^m)^n = 1> lim(m->+oo) lim(n->+oo) (1-1/10^m)^n = 0> 1/10 = 0.1, 1 - 0.1 = 0.9, 1 = 1, 0.999... = 0.999..., 0 = 0. So answer the question. I explained to you that a contradiction is astatement of the form P and not P. Where do you see a controdiction inthose lines above? What you have instead is P and not Q, where P and Qare different statements. That is not a contradiction.>> There are all kinds of things in mathematics that do not commute.>> Multiplication of matrices or quaternions, to name two examples.>> Are matrices and quaternions contradictory?> If the things don't commute, yes. Prove it. Show me how to derived P and not P from noncommutativity ofcertain objects.>Your logic is inconsistent if the> things don't commute. Logic is inconsistent if it leads to a contradiction. You have failed topoint out a contradiction.>Perhaps you should try to solve your problems> with a system that is not contradictory, where things do commute. They> should, and they will, once you trash your errors.The mere fact that noncommutativity makes you uncomfortable is not acontradiction and is not a reason to abandon mathematics.>> Learning which things commute and which things don't is a part of>> mathematics.> Learn your mistakes is what you are suggesting. Okay, learn them, then> correct the things that don't so your system is consistent. The things> that don't commute are the fallout from your system errors. That's> really what we are doing here, fixing your mistake 0.999... = 1.>> What contradiction? Which definition?> Your notation zero is a good start. Define zero.Let C(Q) be the ring of Cauchy sequences of rationals, and let I be theideal consisting of sequences that converge to 0 in the rationals. Thenthe real number 0 is the neutral element of the quotient ring C(Q)/I.That is, it's the equivalence class that contains the constant sequence 0, 0, 0, 0, ...as one of its members.-- Dave SeamanJudge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.> No, limits should not in general commute. By considering a matrix with>> infinitely many rows and columns you should be able to construct doubly>> indexed sequences with lim_m (lim_n a(n,m)) being vastly different from>> lim_n (lim_m a(n,m)) at will.So just make it up as you go, like your fake name.> I'm sorry, Garry, but my name is not entries. One a mathematician atBristol (PhD program) and the other a Foley Artist and actor who worked onsuch films as Iris. You can decide which one is me, if any.I don't have this complete thread anymore, but I'm presuming you arereplying to me as you've included that quote. > Now I understand you.Garry === of 0.999... tend toward 0, not toward 1> No, limits should not in general commute. By considering a matrix with> infinitely many rows and columns you should be able to construct doubly> indexed sequences with lim_m (lim_n a(n,m)) being vastly different from> lim_n (lim_m a(n,m)) at will.So just make it up as you go, like your fake name.Now I understand you.Garry === 0.999... tend toward 0, not toward 1> No, limits should not in general commute. By considering a matrix with> infinitely many rows and columns you should be able to construct doubly> indexed sequences with lim_m (lim_n a(n,m)) being vastly different from> lim_n (lim_m a(n,m)) at will.So just make it up as you go, like your fake name.Now I understand you.> As you clipped his name, one can only suppose you do not believe === Squares of 0.999... tend toward 0, not toward 1Hi Garry,You missed one small point, I said that the difference between 1 and0.999... cannot be zero if they are different numbers. You just told methat you found a difference between 1 and 0.999... and that the differencethat you found is zero (since you sta that the difference was e and that0.999... + e = 1).I will try to explain why you told me that you found a difference of zerobetween 0.999... and 1 (because I don't think that you realize that you saidthat):Let's say that the difference between 0.999... and 1 is e (like you said)and furthermore let's assume that e is greater than zero.0.999... + 0.1 = 1.0999... > 10.999... + 0.01 = 1.00999... > 10.999... + 0.001 = 1.000999... > 1And on and on. Give me a number, e > 0, and I'll add it 0.999... and get anumber that is greater than 1, therefore e must be zero.> Garry,In mathematics we say that two numbea and b, differ if there is anumber> e <> 0 such that a + e = b. I challenge you to find such an e such that> 0.999... + e = 1.> 0.999... + i = 1> I like i better, so do most here, but if you like e, fine, we shall callit e.> 0.999... + e = 1> This will convince you that 0.999... = 1> You just said 0.999... + e = 1, now you want to change it?> Another way to show this:1) x = 0.999...> 2) 10x = 9.999...Subtract 1) from 2):9x = 9> No. 9 * 0.999... = 8.999... + e = 9> x = 1> No. x = 0.999... (see above).> The number of 9's in 1) has the same cardinality as the number of 9's in2)> since there is a 1 to 1 function from the # of 9's being the first to> publish the new definition for i with an e.> i^2 + 1 = 0> Anonymously:> e^2 + 1 = 0> Garry Denke, === 0.999... tend toward 0, not toward 1> Hi Garry,You missed one small point, I said that the difference between 1 and> 0.999... cannot be zero if they are different numbers.So you are saying, Anonymous, that you want 0 as your notation foryour solution of your problem 1 - 0.999... That is going to confusethe issue even further since your 0 is also your placeholder. I thinka vowel is less deceptive, since your 0 is your placeholder. Have youconsidered using an underscore _ for your placeholder? That woulddifferentiate it for you, and others.> You just told me that you found a difference between 1 and 0.999... and that the difference that you found is zero (since you sta that the difference was e and that 0.999... + e = 1).Do you want zero to be your notation for your solution of your problem1 - 0.999... or do you want your 0 to be your notation? If you wantyour 0 to be your notation for your solution of your problem 1 -0.999... then you will need a notation for your placeholder. Do youlike the underscore _ for your placeholder, Anonymous? Perhaps ahyphen - would be better since the hyphens -- don't connect like theunderscores __.> I will try to explain why you told me that you found a difference of zero> between 0.999... and 1 (because I don't think that you realize that you said> that):It appears that you prefer your zero notation, with your zero beingyour notation for your solution of 1 - 0.999... and with your 0 beingyour placeholder. Is that true, Anonymous?> Let's say that the difference between 0.999... and 1 is e (like you said)> and furthermore let's assume that e is greater than zero.Now you're back to using your e notation for your zero notation? Okay.We'll toss out your zero notation and use your e notation for yoursolution of your problem 1 - 0.999... and we'll toss out your zeronotation and use your 0 notation for your placeholder. Your definitionfor your zero notation is now your problem below. What is yournotation zero equal to under your system?> 0.999... + 0.1 = 1.0999... > 1> 0.999... + 0.01 = 1.00999... > 1> 0.999... + 0.001 = 1.000999... > 1And on and on. Give me a number, e > 0, and I'll add it 0.999... and get a> number that is greater than 1, therefore e must be zero.What is your notation zero equal to under your system, Anonymous?Garry Denke, GeologistDenoco Inc. of === toward 1> Hi Garry,You missed one small point, I said that the difference between 1 and> 0.999... cannot be zero if they are different numbers.So you are saying, Anonymous, that you want 0 as your notation for> your solution of your problem 1 - 0.999... That is going to confuse> the issue even further since your 0 is also your placeholder. I think> a vowel is less deceptive, since your 0 is your placeholder. Have you> considered using an underscore _ for your placeholder? That would> differentiate it for you, and others.As only Garry seems to have any problem with zero, there is no point in changing things.You just told me that you found a difference between 1 and 0.999... and > that the difference that you found is zero (since you sta that the > difference was e and that 0.999... + e = 1).Do you want zero to be your notation for your solution of your problem> 1 - 0.999... or do you want your 0 to be your notation? If you want> your 0 to be your notation for your solution of your problem 1 -> 0.999... then you will need a notation for your placeholder. Do you> like the underscore _ for your placeholder, Anonymous? Perhaps a> hyphen - would be better since the hyphens -- don't connect like the> underscores __.As only Garry seems to have any problem with zero, there is no point in changing things.I will try to explain why you told me that you found a difference of zero> between 0.999... and 1 (because I don't think that you realize that you > said> that):It appears that you prefer your zero notation, with your zero being> your notation for your solution of 1 - 0.999... and with your 0 being> your placeholder. Is that true, Anonymous?As only Garry seems to have any problem with zero, there is no point in changing things.Let's say that the difference between 0.999... and 1 is e (like you said)> and furthermore let's assume that e is greater than zero.But, according to you, such assumptions contrary to known fact are against the rules!Now you're back to using your e notation for your zero notation? Okay.> We'll toss out your zero notation and use your e notation for your> solution of your problem 1 - 0.999... and we'll toss out your zero> notation and use your 0 notation for your placeholder. Your definition> for your zero notation is now your problem below. What is your> notation zero equal to under your system?it appears that gary doesn't know from nothing!0.999... + 0.1 = 1.0999... > 1> 0.999... + 0.01 = 1.00999... > 1> 0.999... + 0.001 = 1.000999... > 1And on and on. Give me a number, e > 0, and I'll add it 0.999... and get a> number that is greater than 1, therefore e must be zero.What is your notation zero equal to under your system, Anonymous?0 = 0 in everyone else's systems. What does 0 equal in your system, gary?Garry Denke, Geologist> Denoco Inc. of Texas Gary must work for one of those oil companys that Dubya was into and that then had to be bailed out several times. If these threads are any kind of evidence, maybe it wasn't all Dubya's fault === 0.999... tend toward 0, not toward 1Garry,I admire your persistance, but the arguments that you are making are notmathematically sound. You certainly have the right to reject either theaxioms and definitions of a certain field of mathematics or the axioms ofaristotilian logic - but it would be more productive to explain why youreject these things then to try to argue that 0.999... <> 1. I am arguingthat 0.999... = 1 based upon the definition of convergence which stems fromthe analysis of the real numbers. A sequence of numbers converges to anumber, n, iff for any e > 0 there exists a point on the sequence such thatpast that point, all elements of the sequence, a, are such that |a-n| < e -that is, there is a point on the sequence such that past that point, allelements are within a distance of e from n for any e > 0. Also, two realnumbers are equivalent iff their difference is zero.Based upon these definitions:The sequence {0.999... + (0.1)*10^i} converges to 1 as (0.1)*10^i convergesto 0. Therefore, the difference between the number 0.999... and 1,(0.1)*10^i, is zero. I can now conclude (by the definition of equivalencesta above) that 0.999... = 1.If you are going to reply to this then please don't try to argue inmathematical terms (as you attemp below) - it ends up being very hard toread and meaningless in the end (I promise that I am not trying to insultyou, I simply don't understand what you're thinking when you're talkingabout O and 0). Please argue against these ideas of convergence, let meknow why you think that the whole science of mathematics took a wrong turnwith Calculus.> Hi Garry,You missed one small point, I said that the difference between 1 and> 0.999... cannot be zero if they are different numbers.> So you are saying, Anonymous, that you want 0 as your notation for> your solution of your problem 1 - 0.999... That is going to confuse> the issue even further since your 0 is also your placeholder. I think> a vowel is less deceptive, since your 0 is your placeholder. Have you> considered using an underscore _ for your placeholder? That would> differentiate it for you, and others.> You just told me that you found a difference between 1 and 0.999... andthat the difference that you found is zero (since you sta that thedifference was e and that 0.999... + e = 1).> Do you want zero to be your notation for your solution of your problem> 1 - 0.999... or do you want your 0 to be your notation? If you want> your 0 to be your notation for your solution of your problem 1 -> 0.999... then you will need a notation for your placeholder. Do you> like the underscore _ for your placeholder, Anonymous? Perhaps a> hyphen - would be better since the hyphens -- don't connect like the> underscores __.> I will try to explain why you told me that you found a difference ofzero> between 0.999... and 1 (because I don't think that you realize that yousaid> that):> It appears that you prefer your zero notation, with your zero being> your notation for your solution of 1 - 0.999... and with your 0 being> your placeholder. Is that true, Anonymous?> Let's say that the difference between 0.999... and 1 is e (like yousaid)> and furthermore let's assume that e is greater than zero.> Now you're back to using your e notation for your zero notation? Okay.> We'll toss out your zero notation and use your e notation for your> solution of your problem 1 - 0.999... and we'll toss out your zero> notation and use your 0 notation for your placeholder. Your definition> for your zero notation is now your problem below. What is your> notation zero equal to under your system?> 0.999... + 0.1 = 1.0999... > 1> 0.999... + 0.01 = 1.00999... > 1> 0.999... + 0.001 = 1.000999... > 1And on and on. Give me a number, e > 0, and I'll add it 0.999... and geta> number that is greater than 1, therefore e must be zero.> What is your notation zero equal to under your system, Anonymous?> Garry Denke, Geologist> Denoco === 0, not toward 1Please change (0.1)*10^i to (1)*(1/10)^i and we'll make i range from zeroto infinity for the arguments that you are making are not> mathematically sound. You certainly have the right to reject either the> axioms and definitions of a certain field of mathematics or the axioms of> aristotilian logic - but it would be more productive to explain why you> reject these things then to try to argue that 0.999... <> 1. I am arguing> that 0.999... = 1 based upon the definition of convergence which stemsfrom> the analysis of the real numbers. A sequence of numbers converges to a> number, n, iff for any e > 0 there exists a point on the sequence suchthat> past that point, all elements of the sequence, a, are such that |a-n| that is, there is a point on the sequence such that past that point, all> elements are within a distance of e from n for any e > 0. Also, two real> numbers are equivalent iff their difference is zero.> Based upon these definitions:> The sequence {0.999... + (0.1)*10^i} converges to 1 as (0.1)*10^iconverges> to 0. Therefore, the difference between the number 0.999... and 1,> (0.1)*10^i, is zero. I can now conclude (by the definition of equivalence> sta above) that 0.999... = 1.> If you are going to reply to this then please don't try to argue in> mathematical terms (as you attemp below) - it ends up being very hardto> read and meaningless in the end (I promise that I am not trying to insult> you, I simply don't understand what you're thinking when you're talking> about O and 0). Please argue against these ideas of convergence, let me> know why you think that the whole science of mathematics took a wrong turn> with Calculus.> Hi Garry,You missed one small point, I said that the difference between 1 and> 0.999... cannot be zero if they are different numbers.So you are saying, Anonymous, that you want 0 as your notation for> your solution of your problem 1 - 0.999... That is going to confuse> the issue even further since your 0 is also your placeholder. I think> a vowel is less deceptive, since your 0 is your placeholder. Have you> considered using an underscore _ for your placeholder? That would> differentiate it for you, and others.You just told me that you found a difference between 1 and 0.999...and> that the difference that you found is zero (since you sta that the> difference was e and that 0.999... + e = 1).Do you want zero to be your notation for your solution of your problem> 1 - 0.999... or do you want your 0 to be your notation? If you want> your 0 to be your notation for your solution of your problem 1 -> 0.999... then you will need a notation for your placeholder. Do you> like the underscore _ for your placeholder, Anonymous? Perhaps a> hyphen - would be better since the hyphens -- don't connect like the> underscores __.I will try to explain why you told me that you found a difference of> zero> between 0.999... and 1 (because I don't think that you realize thatyou> said> that):It appears that you prefer your zero notation, with your zero being> your notation for your solution of 1 - 0.999... and with your 0 being> your placeholder. Is that true, Anonymous?Let's say that the difference between 0.999... and 1 is e (like you> said)> and furthermore let's assume that e is greater than zero.Now you're back to using your e notation for your zero notation? Okay.> We'll toss out your zero notation and use your e notation for your> solution of your problem 1 - 0.999... and we'll toss out your zero> notation and use your 0 notation for your placeholder. Your definition> for your zero notation is now your problem below. What is your> notation zero equal to under your system?0.999... + 0.1 = 1.0999... > 1> 0.999... + 0.01 = 1.00999... > 1> 0.999... + 0.001 = 1.000999... > 1And on and on. Give me a number, e > 0, and I'll add it 0.999... andget> a> number that is greater than 1, therefore e must be zero.What is your notation zero equal to under your system, Anonymous?Garry Denke, Geologist> Denoco === 0, not toward 1>Say Garry, have you ever noticed that>.9 < .999...>(.99)^2 < (.999...)^2>(.999)^3 < (.999...)^3>(.9999)^4 < (.999...)^4Garry doesn't believe in the Squeeze Theorem (or === Push for JavaWhile I know C and C++, and have at times felt hampered by certainlimitations of Java, I've concluded that it's the best programminglanguage for wideranging discussions, as it's accessible to people whohaven't programmed before who aren't familiar with Unix, but areWindows people.I need as many people capable of checking things like programs that Ipost as I can get, so I'm making a push for Java.It turns out that for most of you it's as simple as going to the Sunwebsite, where you can get a free download:http://java.sun.com/j2se/1.4.2/download.htmlI'd suggest Sun's tutorials for those who aren't at all familiar withJava and running Java programs.To see some of my work, and a program that you can try running, see myblog archives:You can also do a search at Google Groups for PrimeCountH.java,The point for me here is to show you first that what I've discoveredworks, and then I think it'll be easier for me to explain uniquefeatures of my discovery, as I look towards finding more efficient andeffective ways of getting the word === C++, and have at times felt hampered by certain> limitations of Java, I've concluded that it's the best programming> language for wideranging discussions, as it's accessible to people who> haven't programmed before who aren't familiar with Unix, but are> Windows people.I saw some examples of your Java programming, and I must tell you that it is clearly beyond you. > To see some of my work, and a program that you can try running, see my> blog archives:> You can also do a search at Google Groups for PrimeCountH.java,I saw it, and it almost made me throw up. You'll never be a mathematician, and you'll never get a job as a programmer either. May I remind you that === Groups for PrimeCountH.java,You have already repudia that program in your recent posts to 'sci.math'where you insis that the sqrt function is not a function and cannot be afunction because it contains an *inherent* ambiguity which cannot beescaped. Your algorithm, unfortunately, uses the sqrt function and failsfor negative values of the sqrt.> James (Always-good-for-a-laugh) Harris--There are two things you must never attempt to prove: the unprovable -- andthe === theory-how to proof that?Does anybody know how to do === === I'm new with induction and would like some help with a problem, please. Ineed some general direction on how to approach this problem (a little morespecific than use induction though). If I have N circles in a plane, and each circle has a single chord (ingeneral position), how can I go about showing that only 3 colors are neededto color each neighboring region differently? I'm suppose to be usinginduction, but I am not getting very far. Another part of the problem is to show that if they are not in generalposition, that the 3-color scheme will not work. Not sure if that has to bedone with induction.I truly appreciate any/all === I have N circles in a plane, and each circle has a single chord (in>general position), how can I go about showing that only 3 colors are needed>to color each neighboring region differently? I'm suppose to be using>induction, but I am not getting very far.I assume the circles do not intersect each other. Assume it's true for n circles. What do you know about the region external to all of these?> Another part of the problem is to show that if they are not in general>position, that the 3-color scheme will not work. Not sure if that has to be>done with induction.I don't understand what you mean by not in general position. Do you mean that they can intersect? (That's certainly not what the phrase means.) All you need to === Generalization of Joint GausianI was wondering if there was a simple Tensor Generalization of theJoint Gauasian, where each ellement of the tensor is a differentcummulant of the formfor example R_{1,2,...,n}=E[(x1-mu1)(x2-mu2) ... (xn-mun)]recall the standard joint gausian is:fx(x)=1/(2*pi)^(n/2)/sqrt(|R|)*exp((X-MU)^T*inv(R)*(X-MU)) === construction of the> auatic decision support system for my strategy game, I'm looking for an> equation for estimating the monopolization factor.Company1 produces 3 units of goods, Company2 - 2, Company3 - 2. What is the> market monopolization factor for Company1, and how will it change when it> increases depends on whether you're the top hat or the little racingcar. Also, if you have Boardwalk and Park Place with hotels, thosefactors just don't mean the same thing as when you are sitting therewith Baltic and Mediterranean Avenues.Just kidding.BTW, you might be more likely to get good answers if you were to defineyour terms. Either that, or ask an economics === into construction of theauatic decision support system for my strategy game, I'm looking for anequation for estimating the monopolization factor.Company1 produces 3 units of goods, Company2 - 2, Company3 - 2. What is themarket monopolization factor for Company1, and how will it change when itincreases === Eilenberg Maclane Space K(G,1) cohomology and Group Cohomology--urgentCan anyone please tell me why the singular cohomology of the EilenbergMaclane Space K(G,1) is the same as the group cohomology groups H^n(G,Z)? I think I know this relies on showing you can get thecohomology of K(G,1) through a free resolution of ZG modules, but Iwas unable to determine how (Also seemed to involve the universalcover somehow, and the fact that it was contractible so all itshomology groups were 0, thus the semi-exact sequence of chaincomplexes was an exact one). The source I used, which is availablehere:www.math.uni-frankfurt.de/~johannso/ Scripts/topology2.ps (page 131 actually, but page 129 if you go by the page numbers on thepostscript page, result 9.11)seemed a little incomprehensible. I think I'm close conceptually butam having some problems making the final intuitive leap. I wouldappreciate a description of why this is the case; doesn't seem like itis very hard.It would be best if you could e-mail me any ideas; my e-mail addressis === linkwww.math.uni-frankfurt.de/~johannso/Scripts/topology2.ps> Can anyone please tell me why the singular cohomology of the Eilenberg> Maclane Space K(G,1) is the same as the group cohomology groups H^n> (G,Z)? I think I know this relies on showing you can get the> cohomology of K(G,1) through a free resolution of ZG modules, but I> was unable to determine how (Also seemed to involve the universal> cover somehow, and the fact that it was contractible so all its> homology groups were 0, thus the semi-exact sequence of chain> complexes was an exact one). The source I used, which is available> here:> www.math.uni-frankfurt.de/~johannso/Scripts/topology2.ps> (page 131 actually, but page 129 if you go by the page numbers on the> postscript page, result 9.11)> seemed a little incomprehensible. I think I'm close conceptually but> am having some problems making the final intuitive leap. I would> appreciate a description of why this is the case; doesn't seem like it> is very hard.> It would be best if you could e-mail me any ideas; my e-mail address> is === I think I figured it out. Good old Whitehead's Homotopy theoryseems to carry the result...> www.math.uni-frankfurt.de/~johannso/Scripts/topology2.ps> Can anyone please tell me why the singular cohomology of the Eilenberg> Maclane Space K(G,1) is the same as the group cohomology groups H^n> (G,Z)? I think I know this relies on showing you can get the> cohomology of K(G,1) through a free resolution of ZG modules, but I> was unable to determine how (Also seemed to involve the universal> cover somehow, and the fact that it was contractible so all its> homology groups were 0, thus the semi-exact sequence of chain> complexes was an exact one). The source I used, which is available> here:www.math.uni-frankfurt.de/~johannso/Scripts/topology2.ps( page 131 actually, but page 129 if you go by the page numbers on the> postscript page, result 9.11)seemed a little incomprehensible. I think I'm close conceptually but> am having some problems making the final intuitive leap. I would> appreciate a description of why this is the case; doesn't seem like it> is very hard.It would be best if you could e-mail me any ideas; my e-mail address> is === all, I know that the vector space R^n over R(real) is not isomorphic tothe vector space R over R. A book told me that if we regard R^n and R asvector spaces over Q(rational), R^n is isomorphic to R. Is that true? Howdoes the book come up with === all,> I know that the vector space R^n over R(real) is not isomorphic to> the vector space R over R. A book told me that if we regard R^n and R as> vector spaces over Q(rational), R^n is isomorphic to R. Is that true? How> does the book come up (but countable) dimension; so isR. Take B (resp. C) a basis of R^n (resp. R). You can find a bijection f:B -> C (since B and C have same cardinality). The linear map h such thath(b)=f(b) for === vector space R^n over R(real) is not isomorphic to> the vector space R over R. A book told me that if we regard R^n and R as> vector spaces over Q(rational), R^n is isomorphic to R. Is R^n is a Q-vector space with infinite (but countable) dimension; so is> R. Take B (resp. C) a basis of R^n (resp. R). You can find a bijection f:> B -> C (since B and C have same cardinality). The linear map h such that> h(b)=f(b) for any b in B is an isomorphism.--> Julien SantiniOn the contrary, Both spaces are of uncountable dimension.See, for example,http://en2.wikipedia.org/wiki/Hamel_dimensionAssuming one can well order a basis for each of R^n and R over Q, they can be shown isomorphic by an order isomorphism of their === contrary, Both spaces are of uncountable dimension.> See, for example,> http://en2.wikipedia.org/wiki/Hamel_dimensionYes, and better. As I mentionned some months ago:R is a K-vector space with finite dimension (where K is included in R) <=> K= === bcuPqGZL8e6B5grKRwvHaq+pdjWy3wRUqu10sAEzI2XNIIwD0CBfczTrue: R^n is a Q-vector space with infinite (but countable) dimension; No, because then R^n itself would be countable.R^n over Q is isomorphic to R over Q because an infinite cardinal does notchange when multiplied with a finite cardinal.-- Just because you're paranoidDon't mean they're not after youreverse === True: R^n is a Q-vector space with infinite (but countable) dimension; > No, because then R^n itself would be countable.> R^n over Q is isomorphic to R over Q because an infinite cardinal does not> change when multiplied with a finite cardinal.Or even when raised to the power of a finite cardinal.-- Dave SeamanJudge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. unit radius can fit into a bigger circle of radius 'n' units.> Example: How many circles of radius 1m can fit in a circle of radius> 9m?I appreciate any pointers. best known packings of equal circles in the unit circle (up to N => 500)> at http://hydra.nat.uni-magdeburg.de/packing/cci/cci.htmlHugo anything similar to this for spheres. How many unit spheres> can fit in a sphere of n units radius.-LlipticNot too much, seehttp://www.research.att.com/projects/OEIS?Anum=A084827http: //www.research.att.com/projects/OEIS?Anum=A084828http:// www.research.att.com/projects/OEIS?Anum=A084829Would be nice to get a result for the n=5 sphere (somewherebetween 65 and 70 === sum of 2 Beta rv's will generally NOT be close to Normal. > For example, here is a quick 2-line derivation and pdf plot using> mathStatica: [...]Robert Dodier replied:> Well, if NUMERICAL approximations are allowed, here's a short > construction in Octave which amounts to the same thing [ snip ]Neat code. Since you have raised the stakes, I shall see your Numerical solution, and raise you an EXACT SYMBOLIC solution(given parameter values). See: http://www.mathStatica.com/Sumof2Betas/ Method 2: Transform Method Exact Symbolic solutions for sum of 2 BetasThis yields some rather scrumptuous pdf plots. RosemathStatica Pty LtdEmail: colin@mathStatica.comWeb: === need old dawnI am looking for dish wash soap called dawn made by P&G before 1998.If Anyone has those kind of soap either dawn or ultra dawn,you maycontact me by jie_z@hotmail.com.I will buy those soap at the price 10 times high /bottle with theshipping and handle fee. Price is negotiable too( I can pay === evenhigher).Subject: Computing the rank of the jacoby variety is able to compute therank of the jacoby variety of a curve of genus >=2 (at least forhyperelliptic curves). If a program is not known, Id also take an(explicit) algorithm or a link to a === on counting prime numbersMy analysis continues to indicate that my research on counting primenumbers *should* be the most accessible than my other math researchwhich is more abstract, and clearly more difficult. Still I alsorecognize that significant parts of my prime counting research arebeyond a lot of people simply because that research extends intopartial differential equations.To me the starting point is simple enough:dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1,sqrt(y-1))],S(x,1) = 0, p(x, y) = floor(x) - S(x, y) - 1, and S(x,y) is the sum of dS from dS(x,2) to dS(x,y). But I'm increasingly aware that what looks simple to me, leavesmathematicians all over the world befuddled.Not surprisingly, faced with a difficult mental challenge, peoplecling to what's known, and with the less sophistica audience ofsci.math that has usually meant looking towards Legendre's Method,while when I've contac mathematicians more expert, it has usuallymeant looking towards Meissel's formula.And in fact in contacts with mathematicians at universities, forinstance my alma mater Vanderbilt University, I heard that what I'dfound was some variant on Meissel's formula.However, anyone who actually looks over known methods will find thatwhat I have above is astonishing in its conciseness. It's justamazingly short for a way to count prime numbers. And then you mightnotice that it has a partial difference equation at its heart, butthat's something I've emphasized only to see it apparently sail overthe heads of readeso I'm less interes in emphasizing it now, ashey, it's just a tad bit beyond most of you.Now then, on to the more expert commentary on my work looking likeMeissel's formula, which has helped me to understand that yes, my worksomehow is beyond most mathematicians ability to handle, as indeed youcan relate from it back to Meissel's formula, but you can't get to itfrom Meissel's formula.It's actually easy to show what I mean, as if you know anything aboutMeissel's formula, you know that there's one aspect of it where yousum something like pi(x/p_j) - (j-1)where you iterate through primes p.That follows from the root prime counting function which I discoveredeasily enough withdS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1,sqrt(y-1))],by considering special cases.First if you specifically use primes for y, you getdS(x,p_j) = [p(x/p_j, p_j - 1) - j-1], where you can see a lot of information from the root mathematics canbe dropped.It also is true that if p_j-1 >= sqrt(x/p_j), you can use[p(x/p_j, sqrt(x/p_j)) - j-1], and since with my function p(x,sqrt(x) = pi(x), you can finally get to[pi(x/p_j) - j-1].Now though, notice that you can't go back the other way!!!So my work in less space encodes more information that relates back towhat mathematicians already discovered!However, in looking at it, even experts seem to get lost from whatI've gathered in communicating with mathematicians worldwide since May2002.Now looking atdS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1,sqrt(y-1))],I can get some sense why even experts, like mathematicians bydefinition, would find it intimidating and difficult to comprehend,but then again, I find myself surprised at how clear it is that theexpression is completely overwhelming.It gets more interesting, and rather than move into the calculus bytalking bout the partial differential that follows, I'll talk moreabout practical matters.Facing a daunting expression, I've seen a tendency by posters to tryand simplify it, as if their minds are desperate to find somethingmore familiar. Beyond what I already mentioned, for instance, manyposters would keep deleting off the second factor and emphasizingusing primes!So they'd always push something likedS(x,p_j) = [p(x/p_j, p_j - 1) - j-1], where using just primes makes things look simpler. That behavior is in line with what I mentioned before where postersgrasping for the familiar looked to Legendre's method if novice in thefield, or Meissel's formula if more sophistica in their knowledge.Now here's something for fun.For a while I pursued fast prime counting programs to see if Icouldn't get progress by that route, and eventually I really just gotbored with figuring out fast algorithms, as I'm more interes in themath, but along the way I found some of the fastest expressionspossible for certain counts:With even N,N/2 - floor((N-4)/6) - floor((N-16)/10) + floor((N-16)/30) -floor((N-36)/14) + floor((N-22)/42)is basically the explicit result of summing an algorithmic form of thedS(x,y) function, where evens are auatically handled, from 2 up toand including 10, which represents the primes 2, 3, 5 and 7, so itgives a count of primes up to and including 120.For instance, N=100, gives 50 - 16 - 8 + 2 - 4 + 1 = 25as expec.But beyond counting primes in a small range it works to give N minusthe count of composites up to and including N that have 2, 3, 5, or 7as a factor out to positive infinity for even N.It is the smallest expression possible for that job.Want more on prime counting? === counting prime numbersX-DMCA-Notifications: http://www.giganews.com/info/dmca.html>My analysis continues to indicate that my research on counting prime>numbers *should* be the most accessible than my other math research>which is more abstract, and clearly more difficult. Still I also>recognize that significant parts of my prime counting research are>beyond a lot of people simply because that research extends into>partial differential equations.Your research extends into pde's? Right. What's _one_ exampleof something you've proved about pde's? I mean an example ofsomething you've proved about pde's in general, or about thatthing of yours that you call a pde even though that's not what itis. All I've ever seen you do with that pde is write it down - inparticular you've never given _any_ evidence that the solutionhas anything whatever to do with counting primes.>[...]>Now here's something for fun.>For a while I pursued fast prime counting programs to see if I>couldn't get progress by that route, and eventually I really just got>bored with figuring out fast algorithms, Meaning you got bored with people pointing out how youramazing algorithms were so much slower than well-knownmethods.>as I'm more interes in the>math, but along the way I found some of the fastest expressions>possible for certain counts:>With even N,>N/2 - floor((N-4)/6) - floor((N-16)/10) + floor((N-16)/30) ->floor((N-36)/14) + floor((N-22)/42)>is basically the explicit result of summing an algorithmic form of the>dS(x,y) function, where evens are auatically handled, from 2 up to>and including 10, which represents the primes 2, 3, 5 and 7, so it>gives a count of primes up to and including 120.>For instance, N=100, gives >50 - 16 - 8 + 2 - 4 + 1 = 25>as expec.>But beyond counting primes in a small range it works to give N minus>the count of composites up to and including N that have 2, 3, 5, or 7>as a factor out to positive infinity for even N.>It is the smallest expression possible for that job.Except for the many smaller expressions that several peoplepos the last time you decided to make this claim. Not thatanyone sees why you care...>Want more on prime counting? Why do you think _anyone_ wants more of _any_ of this stuff?I mean really: has _anyone_ expressed any interest? I meaneven one person?> Then see my blog === on counting prime numbers> My analysis continues to indicate that my research on counting prime> numbers *should* be the most accessible than my other math researchThat should have been that it should be more accessible than my othermath research.> which is more abstract, and clearly more difficult. Still I also> recognize that significant parts of my prime counting research are> beyond a lot of people simply because that research extends into> partial differential equations. > With even N,N/2 - floor((N-4)/6) - floor((N-16)/10) + floor((N-16)/30) -> floor((N-36)/14) + floor((N-22)/42)That's fine within a certain range. > is basically the explicit result of summing an algorithmic form of the> dS(x,y) function, where evens are auatically handled, from 2 up to> and including 10, which represents the primes 2, 3, 5 and 7, so it> gives a count of primes up to and including 120.For instance, N=100, gives 50 - 16 - 8 + 2 - 4 + 1 = 25as expec.But beyond counting primes in a small range it works to give N minus> the count of composites up to and including N that have 2, 3, 5, or 7> as a factor out to positive infinity for even N.Actually I forgot that beyond 106 you have more terms, and theexpression then isN/2 - floor((N-4)/6) - floor((N-16)/10) + floor((N-16)/30) -floor((N-36)/14) + floor((N-22)/42) + floor((N-106)/70) -floor((N-106)/210) + 2. > It is the smallest expression possible for that job.You can roll that 2 back into it--carefully--but it's just as easy === counting prime numbers> My analysis continues to indicate that my research on counting prime> numbers *should* be the most accessible than my other math research> which is more abstract, and clearly more difficult. Still I also> recognize that significant parts of my prime counting research are> beyond a lot of people simply because that research extends into> partial differential equations.To me the starting point is simple enough:dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1,> sqrt(y-1))],S(x,1) = 0, p(x, y) = floor(x) - S(x, y) - 1, and S(x,y) is the sum of dS from dS(x,2) to dS(x,y). But I'm increasingly aware that what looks simple to me, leaves> mathematicians all over the world === Focusing on counting prime numbers> My analysis continues to indicate that my research on counting prime> numbers *should* be the most accessible than my other math research> which is more abstract, and clearly more difficult. Still I also> recognize that significant parts of my prime counting research are> beyond a lot of people simply because that research extends into> partial differential equations.> To me the starting point is simple enough:> dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1,> sqrt(y-1))],> S(x,1) = 0,> p(x, y) = floor(x) - S(x, y) - 1,> and S(x,y) is the sum of dS from dS(x,2) to dS(x,y).> But I'm increasingly aware that what looks simple to me, leaves> mathematicians all over the world befuddled.Well... in July 2002 I pos an explanation of your algorithm:http://makeashorterlink.com/?X58722117as you seemed incapable of explaining it yourself.-- Clive === counting prime numbers> My analysis continues to indicate that my research on counting prime> numbers *should* be the most accessible than my other math research> which is more abstract, and clearly more difficult. Still I also> recognize that significant parts of my prime counting research are> beyond a lot of people simply because that research extends into> partial differential equations.You have been asked repealy to produce results with your so-called'partial differential equation' but to date have produced *none*. Thissuggests that the claim you have made about is beyond YOU to support, notbeyond others to comprehend.> To me the starting point is simple enough:> dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1,> sqrt(y-1))],Yes, but you have repudia this result in prior posts because it containsan *inherently* ambiguous use of 'sqrt'. This alleged flaw in the 'sqrt'cannot be avoided, according to you. By the arguments you presenearlier, this equation fails for negative values returned by the squareroot. (By your argument, 'sqrt' is *not* a function because it alwaysreturn two values, both of which must be honored.)[snip self-aggrandizing monologue exhorting the standard claim of James'brilliance and the incompetence of others]Keep it up, James! I've taken to inviting friends and acquaintances over toread your posts and laugh at them.--There are two things you must never attempt to prove: the unprovable -- andthe === works Consider,7(25x^2 + 30xy + 2y^2) = 7(x^2 + xy)(5^2) + 7(xy - y^2)(5) + 7^2 y^2so(5a_1(x,y) + 7y)(5a_2(x,y) + 7y) = 7(25x^2 + 30xy + 2y^2) > where the a's are roots of > a^2 - (xy - y^2)a + 7(x^2 + xy).> That should be a^2 - (x - y)a + 7(x^2 + xy)> Oh yeah, you're right. My previous post saying you were wrong, was wrong.I'll reply to it when it comes up in === space with |x|,|y|=1 that |(x+y)/2|=1 => x=y. I am assuming === x, y in lp space with |x|,|y|=1 that |(x+y)/2|=1 => x=y. I >am assuming here that>1 = sum_j (x_j + y_j) z_j/2 = 1. You also must have = 1and = 1. When do you have equality in Holder's inequality?Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia === posting-host=adsl-64-166-60-173.dsl.frsn01.pacbell.net; posting-account=48257; posting-date=1073438340X-URL: http://mygate.mailgate.org/mynews/comp/comp.theory/ 02f97b20a10e77778806fc87fe0751c2.48257%40mygate.mailgate.org> I will the the first to admit that the output of TM2 is> paradoxical.No, it isn't the least bit paradoxical. The paradox is onlyin your faulty mental model of what it means for someprocess to be repea an infinite number of times; once youget that fixed, everything else becomes simple. The problemis that so far you have refused to face that your mentalmodel is plain _wrong_, trying to work it back into theargument in a number of different disguises. It is easy totell all by yourself, without me ranting at you, when youhave done this: you again get to the _wrong answer_. Sinceyour answer is well known to be wrong, when you arrive at itvia another route, it is _your_ responsibility to confrontthat you have once again done something wrong, and go findthe place where you have let your erroneous model intrudeonce more to lead you astray. It is not appropriate for youto come back here to insist that because of your newlyincorrect formulation, what was false is somehow suddenlymagically made true. Math doesn't work that way.> There can't be a 0 at the end of the tape,A right-infinite tape doesn't _have_ an other end, so yourstatement is _meaningless_. This is part of your faultymental model of infinity, and has tripped you up at leasthalf a dozen times so far in this thread. Making argumentsusing meaningless concepts won't steer you away from yourwrong answer.Logicians tell us from false premises may any answerwhatsoever be deduced, so larding your arguments with falsepremises like infinite lists have last members isn'thelping you learn to think.> yet there must be at least one 0 somewhere on the tape.That is not true because it is not a complete description.You are trying to describe a dynamic process as if it were atermina process.There need only be a 0 after the current 0 for the current 0to be overwritten, but that is a _transient_ condition, notone that holds for that zero for all time. At some latertime it is some later 0, and still later, still a laterzero, but never is it a _last_ zero; with an infinite numberto be overwritten, one is never overwriting either the lastzero, or the next to the last zero. Each and every zerogets overwritten, thus it is a contradiction in terms to saythere is somehow one left (nonsense) when the process isfinished (meaningless). _Any_ specific zero (triviallyobviously) gets overwritten with a 1, therefore byinduction, all of them do.> Other posters have sugges looking at the limit of the> output tapes.I haven't. I've sugges you realize that process is allyou have available for consideration, not some finalproduct. You consistently confuse in the limit withafter the process is done; they aren't rela, and withthat confusion, you should avoid any path that goes past anend point of the calculation. You don't need it to solvethe problem.> If we give TM1 a finite string of 0's it will output a> finite string of 1's. TM2 will output a finite string of> 1's followed by exactly one 0.That is not the problem you posed, and is unworthy ofconsideration. It feeds your false mental model, butprovides nothing of value to make up for misleading you.Put it down, don't pick it up again, even if you dress itin a new suit of clothes.> We can say that TM1 will produce an infinite string of 1's> in the limit. But, the 0 at the endThere is no end; by your own statement of the problem,TM2's tape contains an infinite number of zeros. Aninfinite list doesn't _have_ an end. That's what it _means_to be infinite, after all: without end. Any chain ofthinking you do that implicitly or explicitly contains thatend is pure nonsense.> of TM2's tape doesn't go away in the limit.There is no limit, an infinite process (at least at asteady pace) goes on forever, and you cannot step _past_forever to look at what comes after.> The limit of TM2's tape is still finite.It didn't even start out finite, much less does it end thatway. You said:: Assume we give TM2 a tape that contains: an infinite string of 0's.That isn't some triviality, that is _the definition of theproblem_, and considering any other situation, such asstarting with a finite string of zeros, is _meaningless_in finding the answer to _this_ problem. Stop doing that,it is annoying and wastes time and patience.> If TM1 can write an infinite number of 1's then we have> to assume that TM2 can as well.Unamazingly, that is the conclusion any competentmathematician _does_ reach: TM2 will write an infiniteit falls farther and farther behind TM1 writing a 1 at thecorresponding spot on the other tape, but that doesn'tmatter, because TM2 _also_ has an infinite number of stepsavailable, and will sooner or later write a 1 in each spotbe able with a fairly simple calculation, given the step Nthat for any N, TM2 must write a 1 everywhere that TM1use it to talk yourself out of your delusions.Please! I've already asked this at least twice:1) Do not answer _at all_ until you understand how thisworks, you have been told many times by many correspondentsexactly where your errors occur, and you have ignored eachand every such input, simple repeating your stale argumentsin new words, never confronting that if the previousargument was demolished by an argument that _proves_ theexact opposite of what you maintain, that eliminates allchances that you can somehow derive that false conclusioncorrectly with a newly worded argument.That way lies insanity.2) Do answer (only) when you at long last catch a clue.As you correctly sta, you aren't very good at this mathstuff. Eventually you will learn, as probablynever will, that a math argument isn't won with debatingskills, but only with math skills, and that merely talkingyour opponents into a bored stupor doesn't give you a win,it just leaves you in firm possession of an erroneousopinion, to the benefit of no one, and the amusement ofeveryone but you.xanthian.You might want to consider a hobby where your skills are amatch for what you want to do. Just a thought. I'm not anygood at math (any more, I was once quite competent), so Itry to stick to arguing people out of errors in the limiparts of math I understand, as a hobby, and don't waste mytime trying to do innovative research, completely beyond myskills.-- Pos via Mailgate.ORG === Always FiniteOriginator: rp@win.tue.nl (Reinier Post)[...]>> It is simple to show that the number of 1's>> written by TM1 is some multiple of the number>> of 1's written by TM2.>> Inductively.>Turing doesn't say that computable numbers require induction.>It is easy to prove that the output of TM2 is finite.>TM2's tape will always have a string of 1's followed by a blank.>The blank must be at a finite position.>It is impossible for TM2 to write an infinite string of 1's.As xanthian said, you need to be more specific on the meaning ofinfinite. Mathematicians usually define the concept of infinity interms of limits, and this is exactly how a TM can write an infinitetape: after any (finite) number of steps, a TM has written a finitepiece of tape, but it may be the case that as the number of steps grows,the sequence of (finite) pieces of tape written tends to a specific,infinite piece of tape. This is the case, for example, if over timethe TM ceases to write on an ever increasing first part of the tape;it is undecidable whether an arbitrary TM does this, but particulartypes of TMs can be used that warrant this. E.g. you could write yourreal number generators in such a way that they systematically developthe numbers' decimals.>Russell>- 2 many === [...]>> It is simple to show that the number of 1's>> written by TM1 is some multiple of the number>> of 1's written by TM2.>> Inductively.>>Turing doesn't say that computable numbers require induction.>>It is easy to prove that the output of TM2 is finite.>TM2's tape will always have a string of 1's followed by a blank.>The blank must be at a finite position.>It is impossible for TM2 to write an infinite string of 1's.> As xanthian said, you need to be more specific on the meaning of> infinite. Mathematicians usually define the concept of infinity in> terms of limits, and this is exactly how a TM can write an infinite> tape: after any (finite) number of steps, a TM has written a finite> piece of tape, but it may be the case that as the number of steps grows,> the sequence of (finite) pieces of tape written tends to a specific,> infinite piece of tape. This is the case, for example, if over time> the TM ceases to write on an ever increasing first part of the tape;> it is undecidable whether an arbitrary TM does this, but particular> types of TMs can be used that warrant this. E.g. you could write your> real number generators in such a way that they systematically develop> the numbers' decimals.It makes sense to define infinity in terms of limits.We can say that the limit of TM1 is a tape withan infinitely long string of 1's.TM2 makes sure there will be a blank left on thebe an infinite string of 1's followed by a blank?The string produced by TM1 represents .111... (base 2).Does the tape produced by TM2 represent the same number?(I guess I should say the limit of TM1 represents a === posting-host=adsl-64-166-60-173.dsl.frsn01.pacbell.net; posting-account=48257; posting-date=1073235481X-URL: http://mygate.mailgate.org/mynews/comp/comp.theory/ 733d9c4ed35f5db839af3d010e86e8be.48257%40mygate.mailgate.org> TM2 always makes sure there is a trailing blankYes, for finite definitions of always.> There must be at least one blank on the tape.After any finite step, yes, but you are consideringthe behavior after infinite steps, which doesn'tfollow your intuition.> If the blank is not at a finite position,> then where is it?Exactly.That is the crux of the matter.You consider that an argument for its continued existence,but it is just the opposite.Your faulty intuition tells you it is there somewhere,because you just saw it, but a math proof by inductionshows that it has mysteriously vanished, alwaysoverwritten almost as soon as it is encountered.The _next_ zero merely enables the overwriting of the_previous_ zero, it _doesn't_ thereby protect its ownlocation, which remains squarely in the path of destruction.You are still trying to envision an infinite _process_ as amere _product_, and that doesn't work because there is _nopoint in time_ at which that _product_ *exists* or is_complete_, so you never have a _product_ to consider.You can only treat or consider the process as _itself_, adynamic thing, and any argument trying to describe thatunattainable product must be an argument from dynamicbehavior, not one from behavior that has stopped. That zeroyou think you see can only exist if the process stops, butthe process is _infinite_, so that never happens, and eachzero in turn, all infinitely many of them, gets overwritten.Since there is no last zero, there is no place the processstops and leaves one stray zero because there is not anenabling zero beyond it; the process eats them _all_.For any named value of any place, there is no any placethat zero can be, the _process_ will _always_ eat it at _any_any place, therefore it is no place, the only remainingchoice.xanthian.-- Pos via Mailgate.ORG Server - === important works availablewell, hopefully by now you see how. j. hughes got caught in a trap of> pre-emp assumptions that led him to wrong conclusions. as you can> see at present, he had no choice but to abdicate.Oh, bite me.I didn't abdicate.ok, you did not abdicate. you only stopped because you could notsucceed in debate.> So far, your only defense has been that perhaps you are so> passage which every person has agreed was intended to insult Arturo,> but that you had no such intention.so far, my only defence has been to crush your arguments to pieces bypointing out your flawed assumptions.> A rela defense has been the repea claim that only a moron speaks> to Maky, and I finally agreed with that claim (this post, then, is> some evidence I'm a moron).true you might well be a moron. though, there is plenty of evidencethat points to different === RepresentationX-Abuse: abuse@usq.edu.au>We almost all are aware that, for n = integer >= 2, we can write a>non-integer real with base-n digits (0 through {n-1}), some digits>following after a decimal-point if necessary.>>But what about in base-1?>>But what about non-integers?>>Have you any clever schemes for writing, say, 1/2 or pi in base-1??> How about separately converting the parts before and after the decimal> to another base? Using decimal, for example, 3.14 is represen by:> 111.11111111111111> It does get quite long-winded if there are a lot of decimal places.> 3.1415927 has 1415930 digits, plus the decimal point.It's also ambiguous - 3.1 and 3.01 would have the same === RepresentationDefending myself:By the way, I am very aware that base-1 is not a base in the samesense that we typically refer to our commonly-used number-system asbase-10and to binary as base-2.(Although some repliers seem to believe I am unaware of myless-than-literal use of the word base.)My particular definition of the term base-1 is not my own, yet Icannot recall where else I have seen it used in the sense I use it inmy original post, but I have seen the term used this way in severaldifferent (and reputable) places, I am sure.Leroy Quet> I am posting this as more a fun challenge rather than a serious> question.> {So, that is why I have cross-pos this to rec.puzzles AND> sci.math.}We almost all are aware that, for n = integer >= 2, we can write a> non-integer real with base-n digits (0 through {n-1}), some digits> following after a decimal-point if necessary.But what about in base-1?Integers are easy (though base-one representations are not exactly> analogous to higher bases, since we do not write base-1 integers using> only zeros).Example: 7 (base 10) => 1111111 (base 1)But what about non-integers?Have you any clever schemes for writing, say, 1/2 or pi in base-1??[The best I can come up with right now is to write the continued> fraction of the real, with each term consisting of a base-1 positive> integer. But this is really a list of base-1 === Reals RepresentationIn Base-0 the integers exist, but you can't tell two integers apart. ObPuzzle: Do non-integers exist?You might be able to use some geometric method, or the === Reals Representation> In Base-0 the integers exist, but you can't tell two integers apart. ObPuzzle: Do non-integers exist?Yes. Either of these newsgroups constitutes a constructive proof.-- Aatu Koskensilta (aatu.koskensilta@xortec.fi)Wovon man nicht sprechen kann, daruber muss man schweigen - Ludwig Wittgenstein, Tractatus === Nonfactual Statement> The Liar Paradox as a Nonfactual Statement> PROBLEM> We are given the statement, This statement is false. It is poin out> to us that if the statement is true, then it is false as it claims; and> if the statement is false, as it claims, then it is true. This is a> paradox. How can the statement be both true and false at the same time?That doesn't mean that it's both. It means that it's neither.This is false. is simply a program expressed in English that getsinto an infinite loop, and so it has no value. See:http://www.cs.nyu.edu/pipermail/fom/2002-November/006080. htmlFor a formal derivation of the Liar Paradox, including 1,000variations, see:http://www.cs.nyu.edu/pipermail/fom/2002-November/006100. htmlwhich links the 1st 1,000 variations.Also note that the paradox that is directly genera is 'It is nottrue of itself.' is true of itself. which, after the application ofadditional rules of inference, generates This is false. This is aspecial case of my more general theorem -~P(x,x)[P(a,b)].[BTW: For those of you who can't stand the idea of someone solving aproblem that nobody else has, rather than wasting people's time byciting references that don't contain what you claim, how about givinghere the FORMAL representation of This is false. and, using thatformalism, generate a few additional expressions and their Englishequivalent, as I do in the above (where the formal wff is IFIDBA andthe formal program in the PHP programming language is: functioniifoi($a) { return !$a($a) } ; iifoi(iifoi) ).Formal representation of This is false. = ?Formal rules of inference = ?Some additional expressions and their English equivalent = ?]2,000 years is long enough time spent trying to solve this simpleproblem.Cambridge, MAPS I have considered posting the Rules of English which are used totranslate the derivation into This is false. and ask if anyone coulduse my formalism to derive the rules for other natural languages (e.g.French), but never got around to it. If anyone would like to derivethe rules for some other language (that they speak fluently), then Iwill post the Rules of === connection is and why it exists? ;)>> Alexander,> Without going into a lot of detail... the connection is based on a> ratio that is shown in the Fibanacci sequence 1,1,2,3,5,8,13,21...etc.> each number is the sum of the two previous numbers. The ratio of the> higher to the adjacent lower number approaches 1.61803.... as the> sequence continues. The proportions of the human figure follow this> ratio (illustra in Da Vinci's painting Vitruvian Man) including> facial features and shape of the ear. The ratio appears in nature in> the spacing of petals of a rose and the placement of seeds in the head> of a sunflower and in the shape of a pine cone, the shape of conical> shells. Mozart's concertos use the ratio in the chords and timing of> notes as well as others. The ratio of female bees to male bees in a> hive approaches 1.6.^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^Hold the Phone WHAT male bees??All workes have no sex and the Queen in the only female and male die rightafter hatching in spring.a 1.6 ratio, I dont see it. The ratio can also be illustra by dividing a> perfect square in half and connecting opposite corners by a line and> using that line as a radius to draw another circle and extending a> side of the square to the point of intersection of the circle. The> circle can be defined by PI which leads to PHI. Toroids are based> entirely on PI. Images based on toroids and fractals resemble> galaxies, frost on a window and many things in nature. This is my> connection and I am constantly finding new things that fit in with> this connection.> There's always work to be done.> Its human nature to push the envelope. If it were possible to measure> the expansion of that envelope I'll bet it increases === Mystery Solved> What the connection is and why it exists? ;)>>Alexander,> Without going into a lot of detail... the connection is based on a> ratio that is shown in the Fibanacci sequence 1,1,2,3,5,8,13,21...etc.> each number is the sum of the two previous numbers. The ratio of the> higher to the adjacent lower number approaches 1.61803.... as the> sequence continues. The proportions of the human figure follow this> ratio (illustra in Da Vinci's painting Vitruvian Man) including> facial features and shape of the ear. The ratio appears in nature in> the spacing of petals of a rose and the placement of seeds in the head> of a sunflower and in the shape of a pine cone, the shape of conical> shells. Mozart's concertos use the ratio in the chords and timing of> notes as well as others. The ratio of female bees to male bees in a> hive approaches 1.6. The ratio can also be illustra by dividing a> perfect square in half and connecting opposite corners by a line and> using that line as a radius to draw another circle and extending a> side of the square to the point of intersection of the circle. The> circle can be defined by PI which leads to PHI. Toroids are based> entirely on PI. Images based on toroids and fractals resemble> galaxies, frost on a window and many things in nature. This is my> connection and I am constantly finding new things that fit in with> this connection.There's always work to be done.> Its human nature to push the envelope. If it were possible to measure> the expansion of that envelope I'll bet it increases by a ratio of> 1.61803 per generation.Since e is rela to pi, where does that fit in your scheme?> DaveLDave, glad you asked. This is where the beauty of the mystery unfoldsin the math.PHI=1.618033...i = SQRT(-1)PI=3.141592..e=2.7182818...e^(i*PI)= -1 and (PHI^2)-PHI=1 (PHI^2)-PHI = -(e^(i*PI)) 1 = 1If you solve next to last equation above for i, this will give you thevalue of the imaginary number (i) in terms of the Golden Ratio (PHI),PI, and e!! Everything is rela through an imaginary number. Howthis translates to the physical universe? My guess is possibly blackholes... which I believe are the centers of === Uncountable <1g70f54.rzp2su1gjc8uwN%panoptes@iquest.net> <3ffadc52$20$fuzhry+tra$mr2ice@news.patriot.net> <1g75muz.1d66n4i14gjvdlN%panoptes@iquest.net> <1g75yr4.1si6z861lg1554N%panoptes@iquest.net> <97adneZXNdeRbWCi4p2dnA@comcast.com> <40009fc8$16$fuzhry+tra$mr2ice@news.patriot.net> <40044b66$29$fuzhry+tra$mr2ice@news.patriot.net> h X-Treme: C&C,DWS at 03:43 PM, Russell Easterly said:>I previously gave Turing's definition of a computable number. How can>we write out a real number if we don't allow infinitely long tapes?The tape itself is infinite. But there are only a finite number ofnon-blank symbols on it.>If this is your definition of a TM then I can write a TM that>performs the operations I have described.Please do.>I don't understand what you mean by this.Typo. It should have been Only in the sense that p^~p=>q is trivially true.>Please demonstrate a system that enumerates the rational numbers>using only a finite number of computations.A finite number at each stage; the number of stages is infinite.>According to your argument, such a machine could never finish.Correct. However, it would produce each of the digits in turn.-- Shmuel (Seymour J.) Metz, SysProg and JOATUnsolici bulk E-mail will be subject to legal action. I reservethe right to publicly post or ridicule any abusive E-mail.Reply to domain Patriot dot net user shmuel+news to contact me. Donot reply to === Uncountable <1g70f54.rzp2su1gjc8uwN%panoptes@iquest.net> <3ffadc52$20$fuzhry+tra$mr2ice@news.patriot.net> <1g75muz.1d66n4i14gjvdlN%panoptes@iquest.net> <3ffd4579$1$fuzhry+tra$mr2ice@news.patriot.net> <1g78quy.1a1bzig1e5uy44N%panoptes@iquest.net> <400098e9$10$fuzhry+tra$mr2ice@news.patriot.net> <1g7epkg.1b14mvqgglvt4N%panoptes@iquest.net> <40044865$26$fuzhry+tra$mr2ice@news.patriot.net> <1g7ijlh.1lmkddr1ggrnryN%panoptes@iquest.net>h X-Treme: C&C,DWS at 06:39 PM, panoptes@iquest.net (Daniel W. Johnson) said:>You mean the way he was oscillating between the series for e-2>(whenever he claimed that it wasn't on the list) and the sum of an>initial segment of that series (whenever he claimed that it was a>rational number)?Yes.-- Shmuel (Seymour J.) Metz, SysProg and JOATUnsolici bulk E-mail will be subject to legal action. I reservethe right to publicly post or ridicule any abusive E-mail.Reply to domain Patriot dot net user shmuel+news to contact me. Donot reply to === Uncountable h X-Treme: C&C,DWS>I even give such a mapping for the set [0,1).No.>I am defining a method to compute x.No. You've defined a method to compute a sequence of rational numbers,not a method to compute a specific rational number.>Is this really an open set?Is what? Certainly {x: xS is supposed to contain ALL of the rational approximations of e-2.What is a rational approximation to e-2?>Isn't this the same thing as saying S contains e-2?No. Does {1/i:1=1 to oo} contain 0? No, but it contains numbersarbitrarily close.>I define a method to calculate the largest rational approximation of>e-2 in S:No. There is no such element.>For i=0 to ?:What is ?? If it's finite, then the method is wrong, and if it'sinfinite then the method is meaningless.>Clearly, if I can examine every member of S then I can compute x.FSVO Clearly. That may be clear to you, but it's wrong.>One of my three assumptions must be false.Number 3 is meaningless, but can be reworded as 3) S contains every rational number of the form .111...1 in base!, and e-2 is a limit point of that subset.The problem is not with your assumptions. The problem is that yourconclusion does not follow from your assumptions. -- Shmuel (Seymour J.) Metz, SysProg and JOATUnsolici bulk E-mail will be subject to legal action. I reservethe right to publicly post or ridicule any abusive E-mail.Reply to domain Patriot dot net user shmuel+news to contact me. Donot reply to === Uncountable> No such number exists. You might as well talk about the smallest> rational number greater than 0.You must mean the smallest rational number not in S.> Easy enough to do.> This proof can be easily conver to find> the smallest positive rational number not in S.He means just what he says. There is no such thing as a smallest positive rational over all rationals since half of any positive rational is yet a smaller rational, but there is a smallest positive rational not in [0,1), namely 1.So the cases are quite === is of the form .111...1 and> its length is equal to or greater than x> then set x to a string of 1's one longer than S(i).i = ....So?What, pray tell, is i supposed to be in the quo sentence?> I am assuming that every member of S> can be examined. Is there some reason> I shouldn't make this assumption?Besides the fact that you have not defined examined?> 0.0 <= x < 1x = 1/2! + 1/3! + ... + 1/k!Then x's position on your list is (k-1)! + (k-2)! + ... + 2! + 2 (and> obviously does not differ from that member of S).x is not in S so the position of S(i)=x is undefined.True or false:S((k-1)! + (k-2)! + ... + 2! + 2) = 1/2! + 1/3! + ... + 1/k! > I make three assumptions:1) Every member of S can be examined.> 2) I can determine if S(i) is less than, equal to, or greater than S(j)> 3) S contains every rational number in [0,1)If you think the proof is wrong please point out> which of my assumptions is false.x is not in S-- Daniel W. Johnsonpanoptes@iquest.nethttp://members.iquest.net/~panoptes/ === Uncountable> I make three assumptions:1) Every member of S can be examined.> 2) I can determine if S(i) is less than, equal to, or greater than S(j)> 3) S contains every rational number in [0,1)If you think the proof is wrong please point out> which of my assumptions is false.> x is not in SA truly === Rationals are Uncountable>Well, it's not a proof since it contains the errors that others have>poin out. But I do wish people would stop saying things like you>can't prove false things or you shouldn't attempt to. I'm not very sure>that mathematics is consistent. So, if there is a proof of statement>S and someone comes along with a proof of the negation of S and both>proofs are valid, then mathematics is doomed. Doomed? I would think finding two such proofs would be the begining of a most> enlightening period for mathematics. Unless, of course, *every* valid proof of> S could somehow be turned into a valid proof of ~S. A small dose of> inconsistency could be a very good thing. richIf we take some set theory as the foundation of mathematics (much more than 99% of current mathematics can be formula in ZFC for example), thenthe existence of correct proofs of some statement S and its negation immediately implies that ALL statements formula in the system areprovable. In that sense, there is no such thing as a small dose ofinconsistency. To reiterate:Read an introductory mathematical logic text, and you willsee that if a theory is inconsistent, then that theory contains ALLstatements in the language of that theory, i.e., if one contradictioncan be found then ALL statements are provable. So, if Russell's proofsare correct, then I can prove 1 = 2 and any other statement you coulddream of. Sounds like doom to me. But of course, we have already shown that Russell is wrong.-Leonard === X-Treme: C&C,DWS at 11:02 PM, jesse@phiwumbda.org () said:>No? If (1) was beyond doubt, there's no reason to list it as a>question to be settled.There is always reason to list a question if the answer has alreadybeen dispu. Which was why I lis it.-- Shmuel (Seymour J.) Metz, SysProg and JOATUnsolici bulk E-mail will be subject to legal action. I reservethe right to publicly post or ridicule any abusive E-mail.Reply to domain Patriot dot net user shmuel+news to contact me. Donot reply to === <87smiudtrj.fsf@phiwumbda.org> <3ffadc1f$19$fuzhry+tra$mr2ice@news.patriot.net> <87smisrg0b.fsf@phiwumbda.org> <3ffd41a6$22$fuzhry+tra$mr2ice@news.patriot.net> <871xqa1n9s.fsf@phiwumbda.org> <40009ae6$11$fuzhry+tra$mr2ice@news.patriot.net> <87smim7rqf.fsf@phiwumbda.org> <4001e592$19$fuzhry+tra$mr2ice@news.patriot.net> <87brp98ys1.fsf@phiwumbda.org> <40044d8d$31$fuzhry+tra$mr2ice@news.patriot.net> <87lloby0ht.fsf@phiwumbda.org>h X-Treme: C&C,DWS at 10:58 PM, jesse@phiwumbda.org () said:>And yet, as it turns out, it was an apt question.No. What does that paper have to do with it? would have been an aptquestion.-- Shmuel (Seymour J.) Metz, SysProg and JOATUnsolici bulk E-mail will be subject to legal action. I reservethe right to publicly post or ridicule any abusive E-mail.Reply to domain Patriot dot net user shmuel+news to contact me. Donot reply to === UncountableWell, it's not a proof since it contains the errors that others have>>poin out. But I do wish people would stop saying things like you>>can't prove false things or you shouldn't attempt to. I'm not very sure>>that mathematics is consistent. So, if there is a proof of statement>>S and someone comes along with a proof of the negation of S and both>>proofs are valid, then mathematics is doomed. Doomed? I would think finding two such proofs would be the begining of a>most>> enlightening period for mathematics. Unless, of course, *every* valid>proof of>> S could somehow be turned into a valid proof of ~S. A small dose of>> inconsistency could be a very good thing. rich>If we take some set theory as the foundation of mathematics (much more than >99% of current mathematics can be formula in ZFC for example), then>the existence of correct proofs of some statement S and its negation >immediately implies that ALL statements formula in the system are>provable. In that sense, there is no such thing as a small dose of>inconsistency. >To reiterate:>Read an introductory mathematical logic text, and you will>see that if a theory is inconsistent, then that theory contains ALL>statements in the language of that theory, i.e., if one contradiction>can be found then ALL statements are provable. So, if Russell's proofs>are correct, then I can prove 1 = 2 and any other statement you could>dream of. Sounds like doom to me. Sure it is bad news for, say, ZFC. But is mathematics in its entirety doomed? There are *no* alternatives to ZFC within which 95% of current mathematicscouldn't be re-formula (without the inconsistency)? I don't know. My guessis that mathematicians, being the very clever folk they are, would find somesystem that avoided the inconsistency. And rather quickly. Like I stainitially, it would be a very enlightening period. Something to look forwardto even (unless your work is likely to be part of the 4% that gets lost!). A *little* inconsistency would be a good thing for mathematics, just as === It raises an interesting|question: are there important results in Analysis or Topology that|depend on axioms for large cardinals?Not so far as I know, but I wouldn't take my word for it if I wereyou. My understanding is that certain aspects of point-settopology have become relatively set-theoretical, involvingpropositions independent of ZF. (Isn't there a volume titledSet-Theoretical Topology?) But I don't know that largecardinal axioms play a role.Here's an example of a statement of analysis independentof ZFC. Say that a set S of reals has _absolute measurezero_ if for any sequence a0,a1,... of positive reals, thereexist intervals of length a0, a1,... respectively whose unioncontains S. That every set of absolute measure zero iscountable is independent of ZFC. But as far as I can recallthere's no large cardinal axiom which decides it.Harvey Friedman has been trying to exhibit normalmathematical theorems that require axioms for largecardinals, and the last time I looked he was making someprogress. My impression, though, is that we're still somedistance away from the situation where one could call anyof it an important result in analysis or topology. But Ihaven't been following his progress lately so maybe === Uncountableh X-Treme: C&C,DWS>Here's an example of a statement of analysis independent of ZFC. Say>that a set S of reals has _absolute measure zero_ if for any sequence>a0,a1,... of positive reals, there exist intervals of length a0,>a1,... respectively whose union contains S. That every set of>absolute measure zero is countable is independent of ZFC. But as far>as I can recall Shmuel (Seymour J.) Metz, SysProg and JOATUnsolici bulk E-mail will be subject to legal action. I reservethe right to publicly post or ridicule any abusive E-mail.Reply to domain Patriot dot net user shmuel+news to contact me. Donot reply to === UncountableX-DMCA-Notifications: background post. It raises an interesting>|question: are there important results in Analysis or Topology that>|depend on axioms for large cardinals?>Not so far as I know, but I wouldn't take my word for it if I were>you. My understanding is that certain aspects of point-set>topology have become relatively set-theoretical, involving>propositions independent of ZF. (Isn't there a volume titled>Set-Theoretical Topology?) But I don't know that large>cardinal axioms play a role.>Here's an example of a statement of analysis independent>of ZFC. Say that a set S of reals has _absolute measure>zero_ if for any sequence a0,a1,... of positive reals, there>exist intervals of length a0, a1,... respectively whose union>contains S. That every set of absolute measure zero is>countable is independent of ZFC. But as far as I can recall>there's no large cardinal axiom which decides it.Huh - this is news to me.Not that it has anything to do with large cardinals, butthe result in analysis that I know that's independentof ZFC is this: Every homomorphism from one Banachalgebra to another is continuous. (For context, recallthat every homomorphism from a Banach algebra toC is auatically continuous...)>Harvey Friedman has been trying to exhibit normal>mathematical theorems that require axioms for large>cardinals, and the last time I looked he was making some>progress. My impression, though, is that we're still some>distance away from the situation where one could call any>of it an important result in analysis or topology. But I>haven't been following his progress lately so maybe sha1:04ecQAK3xdD+gDVZAyrVCiDitIg=> No, and no. That's not the basic objection and it does not satisfy the> basic objection. The basic objection is that an auaton with an> infinite input is not a TM at all.Like it or not, others use such liberal notions of Turing machines.For instance, Chaitin uses a tape with infinite input in hisdefinition of algorithmic information. He calls the machine readingthis tape a Turing machine (even though it uses three tapes total,including a work tape and an output tape). He explicitly allows infinite output, too.I am not altogether positive, but I think that Weihrauch has a tapewith infinite output, too, in /Computable Analysis/ and that he callsthe machine producing the infinite tape a Turing machine.Perhaps it's an abuse of terminology, since the commonest definitionof Turing machine does not allow for an infinite amount of data to beon the tape when the machine starts, and there is usually no means forproducing an infinite amount of output (since machines must halt fortheir output to be meaningful). Nonetheless, if it is an abuse ofterminology, it's a common one, commit by respectable researchers.In the end, however, I agree that Russell shouldn't add to theconfusion regarding what a TM is. If he wants to play with infiniteoutput from a machine, he should call it something else (and becareful about a clear specification of the machine).-- Jesse Hughes That is just froth and semantics. -- Nora Baron critiques JSH's === sha1:dP9lQ4ScaC7l7ZertgLm6+V7bxI=> at 09:05 PM, jesse@phiwumbda.org () said:>>As a result of pretty sure, you advised me thus?> to. You seemed to be rejecting the whole idea of modeling one formal> system in another.Nonsense. I was rejecting no such thing. You obviously weremis-reading me. (I have had some exposure to proof theory, after all,and know well that one can reduce one theory to another.)But, supposing I *was* rejecting that idea implicitly, that does notmean your response was appropriate. I was explicitly asking aboutcame about from mis-reading your post --- I thought that you, like theOP, had referred to the fact we model PA in ZFC, not the other wayaround).-- We are happy that you agree that cusers need to know that OpenSource is legal and stable, and we heartily agree with that sentenceof your letter. The others don't seem to make as much sense, but wefind the dialogue === Rationals are Uncountable permission for an emailed response.X-Zippy-Says: Yow! Are we laid back yet?> If you need to track down papers to reply to the question of> which step in the proof a few posts above is wrong then it> follows that you really don't know anything at all about this> stuff. Really. The statement that ZF |- Con(PA) is incredibly> obvious, whether you can find it in a paper or not - it's > utterly trivial. And the fact that Godel did in fact prove> that ZF |- Con(ZF) implies that ZF is inconsistent is> about the most well-known result in logic - it's the one> technical result in logic that people who know nothing> about logic know.I agree with the rest of your post, but I suspect that the mostwell-known result in logic might well be modus ponens. :)As for modern logic, I would think that Godel's first incompletenesstheorem is better known than the === UncountableX-DMCA-Notifications: http://www.giganews.com/info/dmca.html>> If you need to track down papers to reply to the question of>> which step in the proof a few posts above is wrong then it>> follows that you really don't know anything at all about this>> stuff. Really. The statement that ZF |- Con(PA) is incredibly>> obvious, whether you can find it in a paper or not - it's >> utterly trivial. And the fact that Godel did in fact prove>> that ZF |- Con(ZF) implies that ZF is inconsistent is>> about the most well-known result in logic - it's the one>> technical result in logic that people who know nothing>> about logic know.>I agree with the rest of your post, but I suspect that the most>well-known result in logic might well be modus ponens. :)I said most well-known _technical_ result...>As for modern logic, I would think that Godel's first incompleteness>theorem is better known than the second.Well, === Uncountable permission for an emailed response.>> If you need to track down papers to reply to the question of>> which step in the proof a few posts above is wrong then it>> follows that you really don't know anything at all about this>> stuff. Really. The statement that ZF |- Con(PA) is incredibly>> obvious, whether you can find it in a paper or not - it's >> utterly trivial. And the fact that Godel did in fact prove>> that ZF |- Con(ZF) implies that ZF is inconsistent is>> about the most well-known result in logic - it's the one>> technical result in logic that people who know nothing>> about logic know.>>I agree with the rest of your post, but I suspect that the most>well-known result in logic might well be modus ponens. :)I said most well-known _technical_ result...Ah, I missed that half of the sentence, sorry. I wasn't trying to === Rationals are UncountableX-DMCA-Notifications: http://www.giganews.com/info/dmca.html> If you need to track down papers to reply to the question of> which step in the proof a few posts above is wrong then it> follows that you really don't know anything at all about this> stuff. Really. The statement that ZF |- Con(PA) is incredibly> obvious, whether you can find it in a paper or not - it's > utterly trivial. And the fact that Godel did in fact prove> that ZF |- Con(ZF) implies that ZF is inconsistent is> about the most well-known result in logic - it's the one> technical result in logic that people who know nothing> about logic know.I agree with the rest of your post, but I suspect that the most>>well-known result in logic might well be modus ponens. :) I said most well-known _technical_ result...>Ah, I missed that half of the sentence, sorry. I wasn't trying to be>jumpy but good natured anyhoo. ;)This is sci.math - you wanna be good natured go somewhereelse, darnit. (Hmm, maybe it's time === Rationals are Uncountable>But the fact that you're now doubting (1 You have given me the prefect answer to David's question>> about where you misrepresen me, since you've just done it again.>> No, I am *NOT* doubting 1. I lis 1. solely because a large part of>> the debate revolved around it.>No? If (1) was beyond doubt, there's no reason to list it as a>question to be settled. If (1) is the assertion that proofs in ZFC can be transla into proofs in PAthen, yes, I had doubts about it. I've since learned that just because you canfind a proof for some statement, the statement isn't *always* true. So I'm nowOK with the translation process Shmuel described. Happy === <400442d2$24$fuzhry+tra$mr2ice@news.patriot.net> h X-Treme: C&C,DWS at 03:03 PM, Russell Easterly said:>It is impossible if the rabbit>moves any measurable, non-zero distance.The points are ordered by the distance.>Assume the rabbit starts at point 0.000...>and goes to point 1.000...>Which real number represents the point the rabbit>goes to from 0.000... ?It goes to *every* real number in [0,1]; that's the nature ofcontinuous motion. There is no the point, only the points.-- Shmuel (Seymour J.) Metz, SysProg and JOATUnsolici bulk E-mail will be subject to legal action. I reservethe right to publicly post or ridicule any abusive E-mail.Reply to domain Patriot dot net user shmuel+news to contact me. Donot reply to === Continuous?|> As another poster sugges, you should read the latest issue of SciAm.|> Some quantum loop theories quantize time and space.|> Plank's time is about 10^(-44) tricky to say just what it means to say one is quantizingspace and time, however. It's not a model in which time consists ofmoments of time separa by Planck time gaps (note the spellingof Planck). It's not a model in which the points in space are arrangedin a lattice with separations of Planck distance between them. Sucha model would be inconsistent with relativity for one thing. The modelhas some element of discreteness, but also some element ofcontinuity. Any two possible states can be continuously interpolaby quantum === the reals over the rationals>>Could anyone give a (Hamel) basis of the reals over the rationals?>> No. If the Axiom of Determinacy is consistent, and I>> believe even otherwise, there are models where there>> are no Hamel bases for the reals over the rationals.>> If the reals can be well-ordered, a Hamel basis can>> be the set of all reals which are not finite linear>> combinations of previous reals (in the well-ordering).>>How can these be classified?>> This question is unclear.>Could some Borel set be a Hamel basis for the reals over Q ?No. The set of all finite linear combinations withrational coefficients of a Borel set must be Lebesguemeasurable. If one drops one element from a Borel set itis still a Borel set. But if one drops one element from aHamel basis, the resulting finite linear combinations withrational coefficients must be a set of inner measure 0 andwith a complement of inner measure 0 as well.-- This address is for information only. I do not claim that these viewsare those of the Statistics Department or of Purdue University.Herman Rubin, Department of Statistics, Purdue Universityhrubin@stat.purdue.edu Phone: (765)494-6054 FAX: === <>sSHfTy;{Dhe&:+?b`9fUj5A~$gIYlYT0/$-asR-K~3S3[]q.R3YSmpR|$- GiZp>UN2a}!Fmw+%h}YL`!h_XXr5Q>_nGsY2_Could some Borel set be a Hamel basis for the reals over Q ?Re: where's Tex? + a problem of topology [Hamel basis]> But a Hamel basis could be a Lebesgue measurable set.-- === the reals over the rationals>>Could some Borel set be a Hamel basis for the reals over Q ?>> Re: where's Tex? + a problem of topology [Hamel basis]>But a Hamel basis could be a Lebesgue measurable set.A Hamel basis could be a set of Lebesgue measure 0.The well-ordering of the reals gives an easy constructionof a Hamel basis as a subset of the Cantor set, as anyreal number in the unit interval is a sum of two such.-- This address is for information only. I do not claim that these viewsare those of the Statistics Department or of Purdue University.Herman Rubin, Department of Statistics, Purdue Universityhrubin@stat.purdue.edu Phone: (765)494-6054 FAX: === rationals|>>Could anyone give a (Hamel) basis of the reals over the rationals?I think the answer is no, but it's independent of ZFC. According tohttp://www.math.wisc.edu/~miller/res/infcomb.ps Shelah has provenit's relatively consistent with ZFC that no Hamel basis is definable.This would imply in particular that any definition of a set of realscouldn't be _proven_ in ZFC to be a Hamel basis.On the other hand, Goedel proved that if ZF is consistent, then so isZF+V=L where V=L is an axiom implying that the axiom of choice istrue in quite a strong form: it implies that a particular formula gives awell-ordering on all sets. If this were true, then it would follow thata Hamel basis could be defined (like someone else said, by includingeach real number that's Q-linearly independent of the reals that comeearlier in the well-ordering).I don't know that anybody believes V=L, though, and I've seen papersexplaining why people who have an opinion on it tend to disbelieve it.|>> No. If the Axiom of Determinacy is consistent, and I|>> believe even otherwise, there are models where there|>> are no Hamel bases for the reals over the rationals.The axiom of determinacy is inconsistent with the axiom ofchoice, however. The existence of a Hamel basis follows inZFC. I don't see how ZF+AD implying that one doesn't existwould imply that nobody can give one.Perhaps some weaker form of determinacy is enough to implythat no Hamel basis is definable? Large cardinal axioms implydeterminacy for some classes smaller than the whole universe.|>> If the reals can be well-ordered, a Hamel basis can|>> be the set of all reals which are not finite linear|>> combinations of previous reals (in the well-ordering).|>> |>>How can these be classified?|>> |>> This question is unclear.|>|>Could some Borel set be a Hamel basis for the reals over Q ?||Surely not (where of course the word surely indicates|I don't know for sure...) Surely one can prove this by|something analogous to the standard proof that there|exists a non-Lebesgue-measurable set.http://www.math.niu.edu/~rusin/known-math/99/hamelcontains an argument that a Hamel basis is not Borelor even === each metal mean (irrational) starting with Phi is --m_1 = 1.618033989...m_2 = 2.414213562...m_3 = 3.302775638...m_4 = 4.236067977...m_5 = 5.192582404...m_6 = 6.16227766...Etc.All have this property -- (m_1) - 1 = 1/(m_1) (m_2) - 2 = 1/(m_2) (m_3) - 3 = 1/(m_3) (m_4) - 4 = 1/(m_4) Etc. Phi also has this property where the others dont, where ---(m_1)^2 - m_1 = 1. With a slight variation, starting with every odd (m_n) includingPhi --((m_1)-0)^2 - ((m_1)-0) = 1((m_3)-1)^2 - ((m_3)-1) = 3((m_5)-2)^2 - ((m_5)-2) = 7((m_7)-3)^2 - ((m_7)-3) = 13((m_9)-4)^2 - ((m_9)-4) = 21((m_11)-5)^2 - ((m_11)-5) = 31((m_13)-6)^2 - ((m_13)-6) = 43Etc.I am sure there are other irrationals that have this property andwill produce the missing integers below .2,4,5,6,8,9,10,11,12,14,15,16,17,18,19,20,22,23,24,25,26,27,28 ..First removing 2,6,12,20,30,42,56,72... because these have onlyinteger solutions -- 2^2 - 2 = 2,3^2 - 3 = 6, 4^2 - 4 = 12,5^2 - 5 = 20 ... so no irrationals possible with these numbers.Below is the edi list .4,5,8,9,10,11,14,15,16,17,18,19,22,23,24,25,26,27,28,29,32... Is there a simple closed form to find each of these irrationalsfor each integer in the list above?By brute force the first irrational is ---(2.561552812..)^2 - (2.561552812..) === mean (irrational) starting with Phi is --> m_1 = 1.618033989...> m_2 = 2.414213562...> m_3 = 3.302775638...> m_4 = 4.236067977...> m_5 = 5.192582404...> m_6 = 6.16227766...> Etc.> All have this property -- (m_1) - 1 = 1/(m_1)> (m_2) - 2 = 1/(m_2)> (m_3) - 3 = 1/(m_3)> (m_4) - 4 = 1/(m_4)> Etc.> Phi also has this property where the others don't, where ---> (m_1)^2 - m_1 = 1.> With a slight variation, starting with every odd (m_n) including> Phi --> ((m_1)-0)^2 - ((m_1)-0) = 1> ((m_3)-1)^2 - ((m_3)-1) = 3> ((m_5)-2)^2 - ((m_5)-2) = 7> ((m_7)-3)^2 - ((m_7)-3) = 13> ((m_9)-4)^2 - ((m_9)-4) = 21> ((m_11)-5)^2 - ((m_11)-5) = 31> ((m_13)-6)^2 - ((m_13)-6) = 43> Etc.> I am sure there are other irrationals that have this property and> will produce the missing integers below.> 2,4,5,6,8,9,10,11,12,14,15,16,17,18,19,20,22,23,24,25,26,27,28 ..> First removing 2,6,12,20,30,42,56,72... because these have only> integer solutions -- 2^2 - 2 = 2,3^2 - 3 = 6, 4^2 - 4 = 12,> 5^2 - 5 = 20 ... so no irrationals possible with these numbers.> Below is the edi list.> 4,5,8,9,10,11,14,15,16,17,18,19,22,23,24,25,26,27,28,29,32...> Is there a simple closed form to find each of these irrationals> for each integer in the list above?> By brute force the first irrational is ---> (2.561552812..)^2 - (2.561552812..) ~ 4I'm not sure what you're asking. Just find all m_n where m_n^2 - m_n = n?Just solve the quadratic:x^2 - x - n = 0;[1 +/- sqrt (1 -4(-n))]/2 = (1 +/- sqrt (4n + === irrationals!> With a slight variation, starting with every odd (m_n) including> Phi --((m_1)-0)^2 - ((m_1)-0) = 1> ((m_3)-1)^2 - ((m_3)-1) = 3> ((m_5)-2)^2 - ((m_5)-2) = 7> ((m_7)-3)^2 - ((m_7)-3) = 13> ((m_9)-4)^2 - ((m_9)-4) = 21> ((m_11)-5)^2 - ((m_11)-5) = 31> ((m_13)-6)^2 - ((m_13)-6) = 43> Etc.I am sure there are other irrationals that have this property and> will produce the missing integers below .2,4,5,6,8,9,10,11,12,14,15,16,17,18,19,20,22,23,24,25,26,27,28 ..First removing 2,6,12,20,30,42,56,72... because these have only> integer solutions -- 2^2 - 2 = 2,3^2 - 3 = 6, 4^2 - 4 = 12,> 5^2 - 5 = 20 ... so no irrationals possible with these numbers.Below is the edi list .4,5,8,9,10,11,14,15,16,17,18,19,22,23,24,25,26,27,28,29,32... > Is there a simple closed form to find each of these irrationals> for each integer in the list above?Let n be any one of the integers in your list. You are trying to solve u^2 - u = n. This is a quadratic equation. You can use the quadratic formula to solve it. You get u = [1 pm sqrt(4n + 1)] === alt.sci.physicsX-NewsOnePostHost: MGBLLKOLMKMBPBBBCPILILEKOKDDLCHBINLMNLPK> (Do something naughty to physics)You're trying to Unk; every day, but only have common sense enough toantagonize those who understand the subject; but don't let that discourageyou. ----- Pos via NewsOne.Net: Free (anonymous) Usenet News via the Web ----- http://newsone.net/ -- Free reading and anonymous posting to If this or other postsmade through NewsOne.Net violate posting === someone check them please?)Hi allAre the following proofs correct?1) Let n be an integer greater than 1. In a ring whichx^n=x for all x, show that ab=0 implies ba=0.Proof:Consider ba=(ba)^n=b*(ab)^(n-1)*a=b*0*a=0.so ba=0.(I already proved that a*0=0*a=0, for any a in a ring).2) Let R be the ring of all real-valued functions ofa single variable under pointwise addition and multiplication.The subset S of R of functions whose graphs pass throughthe origin forms a subring of R. Prove S is a subring of R.Proof:It suffices to show that S is closed under substraction andmultiplication.Clearly S is non-empty since f(x)=x is in S.So assume f(x)=x*h(x) and g(x)=x*z(x) for some h(x)and z(x) in R[x].But then f(x)-h(x)=x[h(x)-z(x)]=x*q(x) hence x=0 is a rootthus S is closed under substraction.Similarly f(x)*h(x)=x^2*h(x)*z(x)=x*p(x) thus S is closed undermultiplication and by the subring test this implies that === proof (can someone check them please?) Adjunct Assistant Professor at the University of Montana.>Are the following proofs correct?>1) Let n be an integer greater than 1. In a ring which>x^n=x for all x, show that ab=0 implies ba=0.>Proof:>Consider ba=(ba)^n=b*(ab)^(n-1)*a=b*0*a=0.>so ba=0.>(I already proved that a*0=0*a=0, for any a in a ring).Looks right.>2) Let R be the ring of all real-valued functions of>a single variable under pointwise addition and multiplication.>The subset S of R of functions whose graphs pass through>the origin forms a subring of R. Prove S is a subring of R.>Proof:>It suffices to show that S is closed under substraction and>multiplication.Hmmm... Either your definition of ring does not include amultiplicative identity, or else your definition of subring does notrequire that the multiplicative identity of the subring be the same asthe one for the ring (when the latter exists)... Otherwise, you wouldbe in trouble, since the multiplicative identity of R is the constantfunction 1, which is not in S. But S has its own multiplicativeidentity, given by the function whose value is 1 everywhere except at0, and 0 at 0.So just be careful there...>Clearly S is non-empty since f(x)=x is in S.This line should probably go before it suffices to show.>So assume f(x)=x*h(x) and g(x)=x*z(x) for some h(x)>and z(x) in R[x].>But then f(x)-h(x)=x[h(x)-z(x)]=x*q(x) hence x=0 is a root>thus S is closed under substraction.>Similarly f(x)*h(x)=x^2*h(x)*z(x)=x*p(x) thus S is closed under>multiplication and by the subring test this implies that S>is a subring of R.This is a complete mess. First, R[x] is of course ambiguous here inASCII. Second, what makes you think that you can write f and g as polynomials? Not all functions from the reals to the reals that gothrough 0 are polynomials! Nowhere in the statement are polynomialssingled out. Your functions need not be polynomials: they need noteven be continuous! R is the ring of ALL functions; the functionf(x) = 1 if x is irrational 0 if x is rationalis in S. How do you write it as x*h(x) with h(x) a polynomial?No, you went completely astray here. What you proved was that the setof polynomial with integer coefficients which have 0 as a root are asubring of the ring R[x]; you did not prove what you were === please?) Adjunct Assistant Professor at the University of Montana.Rats. Too much reading certain poster's arguments...>No, you went completely astray here. What you proved was that the set>of polynomial with integer coefficients which have 0 as a root are a>subring of the ring R[x]; you did not prove what you were asked.Replace polynomial[s] with integer coefficients with polynomialswith === questionCan someone explain to me why the quotient groupC* / (where C* is the non-zero complex numbeand a is a non-zerocomplex number)has a unique representative in the half-open annulus{z : 1 <= |z| < |a|}?I don't quite understand...to understand C* / , we are looking for asubset B of C* so that each coset z = {za^n : n in Z} contains exactlyone element of B. (Z = === questionX-DMCA-Notifications: http://www.giganews.com/info/dmca.html>Can someone explain to me why the quotient group>C* / (where C* is the non-zero complex numbeand a is a non-zero>complex number)>has a unique representative in the half-open annulus>{z : 1 <= |z| < |a|}>?It's not hard to show that each _element_ of that quotientgroup has a unique representative in that annulus.(At least if |a| is not equal to 1; it's not true if |a| = 1.)>I don't quite understand...to understand C* / , we are looking for a>subset B of C* so that each coset z = {za^n : n in Z} contains exactly>one element of B. (Z = integers)What is the absolute value of za^n? For a given z, assuming that |a| <> 1, how many values of n are there such that 1 <= |za^n| < === someone explain to me why the quotient group>C* / (where C* is the non-zero complex numbeand a is a non-zero>complex number)>has a unique representative in the half-open annulus>{z : 1 <= |z| < |a|}In order for this to make sense, you must be assuming |a| > 1.If |a| < 1 you can use {z: |a| <= |z| < 1}, but if |a| = 1 you'vegot trouble.>I don't quite understand...to understand C* / , we are looking for a>subset B of C* so that each coset z = {za^n : n in Z} contains exactly>one element of B. (Z = integers)... and this is true for B = {z: 1 <= |z| < |a|}. Namely, for any nonzerocomplex number w, there is some integer m such that |a|^(m-1) <= |w| < |a|^m. Then w a^(1-m) is in B, but |wa^n| > |a| ifn > 1-m and |wa^n| < |a| if n < 1-m. So the coset w does containexactly one element of B.Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia === slippery gooiness of biology is a consequence of its incrediblecomplexity, consisting as it does of complex systems based upon chemistry.And chemistry obeys the rules of physics, which exists because of, and isconsequently best described by, mathematics. Mathematics is the ur-fluid ofReality (gad, how poetic), and our symbollic attempts to representmathematics have given us windows through which our mushy grey-matter canpeer, and with which this same mushy grey-matter becomes altered, and wecall this alteration understanding (a frequently generous appellation).[...to press our noses against some of the windows that look upon biologicalsystems...]http://www.bio.brandeis.edu/biomath/ top.html---------------Why do many flowers have five or eight petals, but very few have six orseven? Why do snowflakes have six-fold symmetry? Why do tigers have stripes,but leopards spots? Science writer Ian Stewart suggests mathematicalregularities in natural forms and explains why math is the best tool yet forunderstanding the world around us.http://www.amazon.com/exec/obidos/tg/detail/-/0465072747/-- -----------------Life's first secret, Stewart says, is the molecular structure of DNA. Theother secret, he believes, is mathematical control of a growing organism.(Mathematician Stewart's activities include conducting Scientific American'sMathematical Recreations department.) Arguing that life is a partnershipbetween genes and mathematics, he embarks on an absorbing study of whatlife is, how it origina and how the search for mathematical laws thatunderlie the behavior of living organisms will illuminate those deepquestions. Along the way, he examines mathematical patterns in flowebirdfeatheanimal locomotion and many other features of life. But he hopesfor much more profound findings in biomathematics. A full understanding ofmolecules to ecosystems, we find mathematical patterns in innumerableaspects of life. It is time we put the mathematics and the biologytogether.-------------------What is life? Why is the world of living creatures so different from theinorganic world? The discovery of the first secret of life, the molecularstructure of DNA, in the middle of this century, showed that Life is a formof chemistry - but chemistry unlike any that ever graced a test tube. Somesecrets, however, lie deeper that the genetic code. It is the mathematicallaw of physics and chemistry that control the growing organism's response toits genetic instructions.That is Life's OTHER Secret.Its full understanding will come only when we combine the mathematical andphysical sciences with biochemistry, genetics, and developmental biology.One of the most exciting growth areas of twenty-first century science willbe biomathematics. The next century will withness an explosion of newmathematical concepts, of new kinds of mathematics, brought into being bythe need to understand the patterns of the living world.http://www.ima.umn.edu/stewart.html------------------- Biomathematics is a reaction against the worship of DNA,and the opinion thatgenes govern all aspects of of life [Genetic Determinism -LB]. DNA isessentially a complex set of instructions,and in recent decades there hasbeen a growing belief that it is this strand of molecules which guides allaspects of life.http://www.fortunecity.com/emachines/e11/86/natsums.html- ------------------Is mathematics the grand design for the Universe, or merely a figment of thehuman imagination, asks Ian StewartWhere does mathematics come from? Is it already out there, waiting for us todiscover it, or do we make it all up as we go along? Plato held thatmathematical concepts actually exist in some weird kind of ideal realityjust off the edge of the Universe. A circle is not just an idea, it is anideal. We imperfect creatures may aspire to that ideal, but we can neverachieve it, if only because pencil points are too thick. But there are thosewho say that mathematics exists only in the mind of the beholder. It doesnot have any existence independent of human thought, any more than language,music or the rules of football do.http://www.fortunecity.com/emachines/e11/86/ thinkmat.htmlBiomathematics is the use of mathematical models to help understandphenomena in biology.Modern experimental biology is very good at taking biological systems apart(at all levels of organization, from genome to global nutrient cycling),into components simple enough that their structure and function can bestudied in isolation. Dynamic models are a way to put the pieces backtogether, with equations that represent the system's components, processes,and the structure of their interactions.http://www.stat.ncsu.edu/biomath/whatis/ whatis.html> In Life's Other Secret, you make the argument> that biologists spend too much time studying> DNA and not enough searching for mathematical> underpinnings of biological processes. While> I found much of the research convincing, you> admit early on that the math is 'fragmen,> piecemeal, and open to dispute.' How much has> changed in biomathematics since Life's Other> Secret was published?Not as much as I'd like, but less than I'd feared. The point was always thatDNA and genetics is an extraordinarily significant part of biology, but thatthe subject also needs a substantial dose of mathematical insight (if onlyto work out what the DNA does, and how). I detect a growing consensus alongthese lines, but I do feel that many biologists still have too simple-mindedan idea of what DNA sequences can really tell us. They've sequenced thehuman genome, sort of, and the main thing that's been learned is that genesare a lot more subtle than anyone thought. There are only half as many aswas expec. So it was a brilliant piece of basic research, but it's takenus only a little closer to curing major diseases or understanding howorganisms work. There's a lot of talk now about 'proteomics' - working outthe functions of proteins. Knowing the amino acid sequence of a protein,which is all that the DNA code tells you, is fairly useless when you ask'yes, but what does the protein do?'The math needed for biology is getting more organized and integra, but ithas a long way to go to before it achieves its full potential. However,Biomathematics is now a very hot topic worldwide, and lots of agencies -including NSF and NIH - are actively pushing for more of it, and provingincreased funding. And there is far more dialog between biologists andmathematicians, nowadays. The younger scientists are much less territorial.> Have any guesses as to the reason> why this rift exists?It's partly an educational thing: mathematicians often learn physics, butnot much biology; biologists learn various mathematical techniques likestatistics (usually only the elementary stuff and minus warnings aboutlimitations) but they don't get a fell for the modeling side of math. Sothey expect mathematical theories in biology to be in agreement with everysingle known fact, and if not, they must be wrong. but - it isn't like thateven in physics, the most mathematical science. For instance, a lot ofgravitational models assume that planets are point masses. They're not! Butthe models are better because of this counterfactual assumption, not wrong.The assumption keeps the math simple and the predictions work as long as youdon't ask 'what shape is your planet?'.http://www.frontwheeldrive.com/ian_stewart.html------ -------------lack of books of mathematics dedica to undergraduate students of biologyand the explosive growth of mathematical and theoretical biology as alegitimate branch of science. Since the apparition in the early seventies ofthe texts by Batschelet and Clow & Urquhart, there have been no noticeableefforts to write textbooks of mathematics specifically devo to studentsof natural or life sciences. Paradoxically, we are experiencing the besttime of biomathematics:Mathematics is no longer just a tool or a method to be applied on biologicalproblems; mathematics and biology are engaged in a creative and mutuallybeneficial interplay and the later has became a source of mathematicalideas: The spreading of chaos theory was enormously boos by the work ofBob May in theoretical population dynamics, the mathematical theories ofneural networks, cellular auata, evolutionary programming and geneticalgorithms, just to mention a few, came from biology to mathematics. Most ofthe professionals of biology are missing this revolution: a fast look to theindices of the journals specialized in biomathematics and theoreticalbiology shows that physicists and mathematicians dominate the field. It isnot farfetched to say that the negligence in producing high-qualityeducative tools for the future biologists is to be blamed for thispernicious trend.In these circumstances, the appearance of the book Matem.87ticas para lasCiencias Naturales by Jos.8e Luis Guti.8errez S.87nchez and Faustino S.87nchezGardu.96o must be gree. This it is a book that comes to fill a gap and itdoes it very well. The aforementioned classical texts by Batschelet and Clow& Urquhart are essentially books of mathematics containing a lot ofbiological examples but their construction followed the 70s fashion ofdevoting a lot of effort to discuss set theory and then relations andfunctions to establish the foundations of differential and integralcalculus.This way of presentation could be (I doubt it) correct for the students ofmathematics but less attractive for physicists and engineers and, softlyspeaking, questionable for biologists. The work of Guti.8errez S.87nchez andS.87nchez Gardu.96o (GS-SG) is a novel and original contribution: it is rather abook of highly mathematized biology in which the mathematical concepts arisewhen they are necessary to advance in the knowledge of some biologicalhis book: A few decades ago mathematics played a modest role in lifesciences. Today, however, a great variety of mathematical methods is appliedin biology and medicine, the osophy of GS- SG seems to be the sameafter changing is applied in by is coming from. The authors have been verycareful in not to yield to the common temptation to write senselessexercises of the type suppose to biological phenomenon follows the lawy=1/3 x3 - x2 -3x + 3, locate the highest and lowest points of ecurvesthat are abundant in other texts, instead they try to teach mathematics oncethe student first engages in the biology of the exercises. The book is notfree of problems: The publishing work is poor, the figures can be improved alot and, by incredible that it seems; there are no subject or authorsindices. The sections dedica to the laws of Newton and Kepler as well asthe section about the notion of work in physics are a well-intentionedattempt to increase the scientific culture of the readers but I wonderwhether or not the effect will not be the opposite; regrettably, most of thestudents of biology are there because they do not like physics.The book is pleasant and it is written in good Latin American Spanish but tothe readers who do not know the Castilian language it will be perhaps oflittle motivation to know that this is the language spoken as a firstlanguage by the most people in the world. It is evident to me that the bookdeserves a careful edition and, urgently, a translation to English. Thefirst edition is limi to 2500 copies and its distribution will hardlytrespass the borders of Mexico.This is a pity because this work might contribute to wake up a mathematicalinterest among the students and, eventually, to contribute to incorporateorrows biologists to the construction of a much-needed theoreticalbiology.http://matematicas.fciencias.unam.mx/biomat /comentarios.htmlhttp://www.wolframscience.com/toc/http:// === Sorry, I meant to ask: what day of the year is the longest?>>This question is usually given a wrong or incomplete answer.>All the days are 24 hours except when a leap second is added.>If you mean to ask which day has the longest daylight at a given>location, that would be the day of the summer solstice. >>That is true if you define the day to be from sunrise to sunset.>>But if you define the day to be from sunrise to sunrise, then>>the day is longest when the Earth is after winter solstice).>I define the day to be from midnight to midnight. Perhaps you meant to>ask about the solar day, which is the time between transits of the>sun. However, your answer is incorrect. Perihelion occurs in early>January (around January 8, IIRC), and therefore the answer is *not* the>perihelion date. In general, the solar day is longer than average near>the solstices (both of them) and shorter than average near the>equinoxes.>The length of the solar day is determined by the sun's daily movement>in right ascension, which determines the difference between the>sidereal day and the solar day. What's confusing you is that the sun>moves fastest *in longitude* at perihelion, but fastest movement in>longitude does not equal fastest movement in right ascension. The>obliquity of the ecliptic causes the fastest movement in right>ascension to occur near the two solstices, despite the fact that one of>them happens to be near aphelion.>> Why is Dec. 19 the longest solar day instead of perihelion Jan. 4th?>It is neither. For the years in which I have compu it, the longest>solar day has been either Dec. 22 or Dec. 23. That is, it seems to fall>within a day or so of the December solstice.>The reason is that the length of the solar day is affec more by the>solstice than it is by perihelion, as I explained.>The longest solar day is presently achieved near the December solstice,>because that one is closer to perihelion than the June solstice is.>Because of precession of the equinoxes, there will come a time when>perihelion is not particularly close to either solstice, and therefore>nowhere near the longest solar day.The rate of change of the right ascension of the Sun is cos(obl) asc' = ----- lon' 1 - sin^2(obl) sin^2(lon)where asc is right ascension, obl is the obliquity of the ecliptic, andlon is longitude (obl ~ 23.45 degrees).Thus, even if the rate of change of longitude were constant, when thelongitude is pi/2 (summer solstice) or 3pi/2 (winter solstice), asc' isat a maximum; 9.00% faster than average. With this simplification, thesolar days are longest at the solstices; about 21 seconds longer thanaverage (24 hours).Due to the eccentricity of the Earth's orbit, the solar longitude isnot constant. It's rate of change, relative to the mean, is 1 1+e sec^2(E/2) lon' = ------------ sqrt( --- ) ------------------ 1 - e cos(E) 1-e sec^2((lon-per)/2)where per is the longitude of perihelion and E is the eccentric anomalywhich satisfies 1-e tan(E/2) = sqrt( --- ) tan((lon-per)/2) 1+eThus, the solar longitude is changing 3.47% faster at perihelion thanaverage (e ~ .017).Now for some rough estimation. Let a = lon-pi/2 and b = per-pi/2. Dueto the obliquity of the ecliptic, we have that asc' exceeds the mean by0900 cos(2a). Due to the eccentricity of our orbit, we have that lon'exceeds the mean by .0347 cos(a-b). Finding the a which maximizes .0900 cos(2a) + .0347 cos(a-b)gives a ~ .0347/.3947 b. b is about 12.9 degrees which means about 13.1days between solstice and perihelion. Therefore, the solar day would belongest about 1.15 days after solstice. At that time, the Sun's rightascension would be changing about 12.39% faster than mean, which wouldgive the solar day about 29 seconds extra.That is, unless I made a mistake somewhere.Rob Johnson take out the trash before === Seaman >Due to the eccentricity of the Earth's orbit, the solar longitude is>not constant. It's rate of change, relative to the mean, is> 1 1+e sec^2(E/2)> lon' = ------------ sqrt( --- ) ------------------> 1 - e cos(E) 1-e sec^2((lon-per)/2)>where per is the longitude of perihelion and E is the eccentric anomaly>which satisfiesOf course, I meant _the rate of change of_ the solar longitude is notconstant.Rob Johnson take out the trash === I thought that at least my simplest results, like the>prime counting function, or that funny little error with algebraic>integers might be generally accessible, but after years of arguing I>realize that your intellects are too limi to fully grasp my work.Maybe our limi intellects are just too short to remember that you'vealready admit that your simplest results were, in fact, simple enoughto be false. Silly === it. It is quite remarkable, though - you'd think that> there would be _someone_ _somewhere_ on the planet whose brain was able to> handle mathematics the way yours can. I mean those guys who solve> problems that nobody else could solve for hundreds of yeaguys like> that. But we've been harassing, I mean contacting, famous mathematicians> all over the world and it seems to be true, there's not one single person> anywhere on the planet with a brain like yours.Or, taking it from the other direction, if he was really so much moreadvanced than the rest of us, you'd expect him to be able to grasp ourprimitive mathematics and at least present his results in correct === be it. It is quite remarkable, though - you'd think that> there would be _someone_ _somewhere_ on the planet whose brain was able to> handle mathematics the way yours can. I mean those guys who solve> problems that nobody else could solve for hundreds of yeaguys like> that. But we've been harassing, I mean contacting, famous mathematicians> all over the world and it seems to be true, there's not one single person> anywhere on the planet with a brain like yours.Or, taking it from the other direction, if he was really so much more> advanced than the rest of us, you'd expect him to be able to grasp our> primitive mathematics and at least present his results in correct form.A few years ago I pos that the difference between James' IQ and mine is at least thirty. I'll raise that number to fourty. Seems he gets more and more stupid every === the problem I'm facing is that the human brain> isn't built to handle Mathematics.Well, I take it you are now claiming that === intellects> I've concluded that the problem I'm facing is that the human brain> isn't built to handle Mathematics.Well, I take it you are now claiming that you don't have a 'human === intellects> I've concluded that the problem I'm facing is that the human brain> isn't built to handle Mathematics.Well, I take it you are now claiming that you don't have a 'human brain'?> Think about it some more.> How about this:1) The human brain is not built to handle Mathematics, (your hypothesis)2) ' brain can handle Mathematics, (your repea claim)3) does *not* have a human brain -- read: is not human.(conclusion)QEDNow YOU think about some more.--There are two things you must never attempt to prove: the unprovable -- and the === intellects> How about this:1) The human brain is not built to handle Mathematics, (your hypothesis)> 2) ' brain can handle Mathematics, (your repea claim)> 3) does *not* have a human brain -- read: is not human.> (conclusion)QEDYou have an unsta assumption: ?) Human brains cannot do anything they are not built for.I see no evidence that this unsta assumption is true. It is quite commonfor people to be able to do things that they aren't built to do. E.g., thehuman wrist is not built for typing (hence, RSI), yet most of us manage.-- === human brain is not built to handle Mathematics, (your hypothesis)> 2) ' brain can handle Mathematics, (your repea claim)> 3) does *not* have a human brain -- read: is not human.> (conclusion)QED> You have an unsta assumption:> ?) Human brains cannot do anything they are not built for.> I see no evidence that this unsta assumption is true. It is quite common> for people to be able to do things that they aren't built to do. E.g., the> human wrist is not built for typing It is clear that James was complaining that the 'human brain' wasunable to handle Mathematics. (Look at the title of the thread.) I believe the unstaassumption you have identified is implicit in the context. How do you interpret James'post?--There are two things you must never attempt to prove: the unprovable -- and === ones> Can you brilliant and gif Mathematicians and Physicists help out> this poor soul and justify your acceptance of relativistic and quantum> mechanics by providing the axioms of a simultaneously discrete and> continuous space.Sure! I have a program that you can download and view (the source inincluded as the PRG file of the ZIP) that demonstrates exactly whatyou are asking for.The idea is that by taking a determinate system and adding a rule thatgroups several of the results of this system into a single result, wecan quantize whats happening in the model and produce a subset of datathat changes indeterminately.In other words, if I need to model General Relativity in a determinatesystem, its possible for me to model Quantum Mechanics in an uncertainsubsystem of this model. The subset is made of a discrete space, andthe entire set is a continious space (well, in this simple prototypeits not completely continiouis for computer resource reasons, but itis less discrete than the other version).You see, there isn't one spacetime domain that is simultaneouslydiscrete and continuous, instead there are actually two differentspacetime domains.Its loca here:http://www.techmocracy.net/science/time.zip(run time.exe to view the model)You'll need the VFP8 runtimes to execute : Re: The gif ones> Can you brilliant and gif Mathematicians and Physicists help out> this poor soul and justify your acceptance of relativistic and quantum> mechanics by providing the axioms of a simultaneously discrete and> continuous space.> Sure!Awwww... ain't that cute? The two little cranks are === X-Treme: C&C,DWS at 07:49 PM, j.schoenfeld@programmer.net (John Schoenfeld) said:>Can you brilliant and gif Mathematicians and Physicists help out>this poor soul and justify your acceptance of relativistic and>quantum mechanics by providing the axioms of a simultaneously>discrete and continuous space.First, the issue is one of Physics and not Mathematics. Second,physicists don't normally do axioms. Third, there is nothing in eitherSR or QT that would require, or even allow, a simultaneously discreteand continuous space. QFT requires a spacetime that is a realmanifold. The perceived discreteness is in certain measuredquantities, not in spacetime.Now, there is some research work into theories of a very differentcharacter, but it is much too early to speak of acceptance orjustification for them. A lot of that work is motiva by the desireto accommodate GR and QT in a single theory. If you are curious, thereshould be people in sci.physics who can direct you to appropriatesources.-- Shmuel (Seymour J.) Metz, SysProg and JOATUnsolici bulk E-mail will be subject to legal action. I reservethe right to publicly post or ridicule any abusive E-mail.Reply to domain Patriot dot net user shmuel+news to contact me. Donot reply to === you brilliant and gif Mathematicians and Physicists help out> this poor soul and justify your acceptance of relativistic and quantum> mechanics by providing the axioms of a simultaneously discrete and> continuous space.To go that route, you should shift focus away from point sets tofunction algebras. For instance, a circle is equivalentlydescribed by the following algebra genera from the elements: ...,z_{-3}, z_{-2}, z_{-1}, z_0, z_1, z_2, z_3,...with the multiplication rule: z_a z_b = z_{a+b}the corresponding functions being z_a <-> Z_a(x) = exp(2 pi i a x).In general, the algebra consisting of all C^{infinity} functions(i.e. functions continuously differentiable to all orders) fora compact Hausdorff space comprises a commutative C^* algebra-- and vice versa (all commutative C^* algebras are functionspaces for some compact Hausdorff space).To generalize to something that encompasses both continuous anddiscrete spaces, you generalize the space of functions fromthe C^{infinity) functions to a suitable class of generalizedfunctions -- in particular, a class large enough to includethe delta functions.Such a class, likewise, is to be closed under products (andsums and multiplication by constants, as usual), closedunder differentiation, and is to include both the deltafunctions and the C^{infinity} === gif Mathematicians and Physicists help out> this poor soul and justify your acceptance of relativistic and quantum> mechanics by providing the axioms of a simultaneously discrete and> continuous space.Your answer in a single word, NO.Here's a physics question for you. Will a conventional spring wound,balance wheel alarm clock gain or lose time when it floatsunconstrained in gravity free space, vs. the time it keeps whilesitting on a desk or table top here on earth. State the equationexpressing the gain or loss in time. (Be thoughtful in answering,because this is a classic homework problem in mosttheoretical/classical mechanics courses.)If you wish to play with physics guys, at laeast demonstrate someknowledge of physics as the stakes required to enter the game.You won't learn this stuff from your coffee table books. Harry === Mathematicians and Physicists help out> this poor soul and justify your acceptance of relativistic and quantum> mechanics by providing the axioms of a simultaneously discrete and> continuous space. Relativity star out as an exercise in imagination, but proved to be applicable to reality because it is _predictive_. Quantum Mechanics star out as a way to explain some shortcomings of Classical Electrodynamics (among other things) and proved to be applicable to reality because it is _predictive_. The Universe at large obviously does not give a s about axioms. Mathematical physicists care solely because they can use them to tie the experimentalists' results together, and experimentalists care because they can use them to suggest new experiments. Not only that, but different sets of mutually exclusive axioms are used in different situations when applicable. Frinst Newtonian Mechanics is perfectly acceptable at low velocities, but blatantly wrong at higher velocities. QM isn't much use at human scale, but absolutely essential at very small scales. In case you hadn't noticed, Relativity and QM are seldom applicable together to any specific situation. You seem to think that science works solely from the bot up; that one starts with First Principles and builds a huge, tottering edifice upwards which will collapse catastrophically when any Principle is falsified. That's extremely naive. It is constantly being rebuilt from both ends toward the middle and from the middle outwards in both directions, all simultaneously. The Universe did not come with documentation. We have to write it as we go along. Since you seem to have problems with both QM and Relativity, why don't you practice what you preach; come up with a consistent set of axioms that: resolve the discrete/continuous debate: do not contradict any experiment: _and_ predict conclusive === gif onesCan you brilliant and gif Mathematicians and Physicists help out> this poor soul and justify your acceptance of relativistic and quantum> mechanics by providing the axioms of a simultaneously discrete and> continuous space.Screw your personal problems. *You* provide a single reproducible> empirical contradiction to classical field theory as it now exists. > If it has no problems it needs no changes.Try buying a telescope.> http://arXiv.org/abs/hep-th/0307140> GR structure, especially Part 4/p. 7> Relativity in the GPS system > http://arXiv.org/abs/astro-ph/0401086> http://arxiv.org/abs/astro-ph/0312071> Deeply relativistic neutron star binariesUncle Al has moun a powerful empirical contradiction to metric> theories of gravitation in favor of affine theories complete with> scholarly references, pure geometric and mathmetical models, and an> inexpensive reduction to practice in existing apparatus. It also> reconciles continuous field theories with discrete processes while> cosmology,http://www.mazepath.com/uncleal/qz.pdfPookie pookie. See what being able to use a library will get === help:Where did you hear/see these statements, or how did you come to such a> conclusion? What was the context of the statements? Are you talking> about a vector space or a similar structure here or real space?It's called basic reasoning. Quantum mechanics dictates discretespace, Relativity dictates continuous space, if they are to co-exist,then space must be simultaneously discrete and continuous -- too badyou can't define such a space axiomatically.> Also, I should point out that most physicists don't like axiomizing> things -- especially things like the universe that no one yet fully> understands!If physicists don't like axioms then they should === X-Treme: C&C,DWS at 09:21 PM, j.schoenfeld@programmer.net (John Schoenfeld) said:>Quantum mechanics dictates discrete space,No, it dictates continuous space-time. Discreteness comes in with-- Shmuel (Seymour J.) Metz, SysProg and JOATUnsolici bulk E-mail will be subject to legal action. I reservethe right to publicly post or ridicule any abusive E-mail.Reply to domain Patriot dot net user shmuel+news to contact me. Donot reply to === called basic reasoning. Quantum mechanics dictates discrete> space,It does? A quantum wave function can be well defined on a continuous space time manifold. Quantum electrodynamics === onesX-SessionID: d6d7e580>> If you want real help: Where did you hear/see these statements, or how did you come to such a>> conclusion? What was the context of the statements? Are you talking>> about a vector space or a similar structure here or real space?>It's called basic reasoning. Quantum mechanics dictates discrete>spaceIt doesn't. Come back when you have bothered to learn a bit bout whatever it is you would like to discuss. Till then, ta.Mati Meron | When you argue with a fool,meron@cars.uchicago.edu | chances are he is doing just === Where did you hear/see these statements, or how did you come to such a>> conclusion? What was the context of the statements? Are you talking>> about a vector space or a similar structure here or real space?>>It's called basic reasoning. Quantum mechanics dictates discrete>spaceIt doesn't. Come back when you have bothered to learn a bit bout > whatever it is you would like to discuss. Till then, ta.Mati Meron | When you argue with a fool,> meron@cars.uchicago.edu | === The gif onesCan you brilliant and gif Mathematicians and Physicists help out> this poor soul and justify your acceptance of relativistic and quantum> mechanics by providing the axioms of a simultaneously discrete and> continuous space.It is old news that General Relativity and Quantum Mechanics are not> consistent with each other (as they are currently formalized).ThomasNo you didn't bilge.You are saying that space does not need to be simultaneously discreteand continuous in order to support Quantum and Relativistic mechanicssimultaneously.It's one or the other, but not both. Unless you can define the axiomsof such a space mathematically, that is (which you obviously === gif Mathematicians and Physicists help out> this poor soul and justify your acceptance of relativistic and quantum> mechanics by providing the axioms of a simultaneously discrete and> continuous space.> It must be poin out that Schoenfeld is a crackpot, and a troll.It must be poin out that Varney can't tell the difference betweendeceleration and -acceleration (and === that Varney can't tell the difference between> deceleration and -acceleration (and he has a PhD).There is no essential difference. Deceleration is just acceleration multiplied by -1. Either one means a change in velocity wrt time.Since acceleration is a vector all that counts is magnitude and direction. A vector is a vector is a vector.Bob === Mathematicians and Physicists help out> this poor soul and justify your acceptance of relativistic and quantum> mechanics by providing the axioms of a simultaneously discrete and> continuous space.>It must be poin out that Schoenfeld is a crackpot, and a troll.> It must be poin out that Varney can't tell the difference between> deceleration and -acceleration (and he has a PhD).It must be poin out that you are an ignorant twit. I do not yet have myPhD. and it is you who seem to have trouble grasping the concepts involvedin physics.Go === brilliant and gif Mathematicians and Physicists help out> this poor soul and justify your acceptance of relativistic and quantum> mechanics by providing the axioms of a simultaneously discrete and> continuous space. My shot is that it is a discrete continuum composed of infinitesimals.I base this on the assertion that there is a smallest quantity that isnonzero. The infinitly small is near(is the nearest thing to)zerowithout actually being zero. Just as there is a largest(infinity)with nothing larger; there is a smallest or an - indivisible.Zero = the abscence of quatity or no quantity at allinfinitesmal = the smallest of all quantitiesinfinity = the largest quantityI know these are shadow definitions but they serve to prove my point.The Space-time-curvature(gravity)-continuum is quantized bythe infinitesmal. An infinitesimal space-time-geometry interval.So we arive at the curved discrete continuum by using the === The gif ones> Zero = the abscence of quatity or no quantity at allZero is the identity element of an additive group or semi-group.> infinitesmal = the smallest of all quantitiesThere isn't just one infinitesimal in a non Archimedian field.> infinity = the largest quantityThere is an infinite hierarchy of infinities. Google transfinite numbers.I know these are shadow definitions but they serve to prove my point.The Space-time-curvature(gravity)-continuum is quantized by> the infinitesmal. An infinitesimal space-time-geometry interval.So we arive at the curved discrete continuum by using the infinitesimal.Sounds like nonsense to me.Bob === quatity or no quantity at allZero is the identity element of an additive group or semi-group.> infinitesmal = the smallest of all quantitiesThere isn't just one infinitesimal in a non Archimedian field.infinity = the largest quantityThere is an infinite hierarchy of infinities. Google transfinite numbers.I know these are shadow definitions but they serve to prove my point.The Space-time-curvature(gravity)-continuum is quantized by> the infinitesmal. An infinitesimal space-time-geometry interval.So we arive at the curved discrete continuum by using the infinitesimal.Sounds like nonsense to me.Bob KolkerUse your imagination. If you can think of the infinitesimal(first order?) as asmallest - nonzero - quantity you might See its usefullness in quantizingthe spacetime continuum.I am no mathematician but Where is the nonsense?I have just take the === onesmessage> New idiot on board.> Like you would know.......With you as a template, the observation is a === algorithms, code ?Hi !I am looking for a good refference and possibly code for the followingextensions of maximum (weigh/cardinality) matching problem onto: - hypergraphs - regular graphs where each node can be involved in UP TO pesha @ ai D0T mit D0T edu