> Do negative numebrs exist in the real world or are they just a purely > symbolical and mathematical construct? I don't know. Do positive numbers exist in the real world? -- ==== > Do negative numebrs exist in the real world or are they just a purely >> symbolical and mathematical construct? I don't know. Do positive numbers exist in the real world? -- Excellent answer. Some numbers have direction. This even can be true in the real world. If something is countable or measured, then the opposite count or measure may also occur. One of them is the opposite of the other (obviously). G C ==== > intersection of the surface with the given plane. Find equations for the > line tangent to C at the point P. Plane x = 2; P (2, 1, 5) A solution I looked at first found f(2) = 5, then the partial derivative > of > f[sub-y](2,1) = 2. Then it turned it into the following two equations > x = 2 > The tangent line is in the plane x=2, which explains this equation. > equation > z - 5 = 2(y - 1) > is the line (in the yz plane) through (1,5) satisfying dz/dy=2. > LH > ==== Part of the problem involves this: ....C is the curve of intersection of the cylinder y - 2z^2 = 1 and the plane z = x + 1 from (0,3,1) to (1,9,2). Their solution to parametrizing it is: r(u) = (u - 1)i + (1 + 2u^2)j + uk u E [1,2] My question is what steps did they go to reach that. How did they find the intersection? ==== Part of the problem involves this: ....C is the curve of intersection of the cylinder y - 2z^2 = 1 and the >plane z = x + 1 from (0,3,1) to (1,9,2). Their solution to parametrizing it is: r(u) = (u - 1)i + (1 + 2u^2)j + uk u E [1,2] My question is what steps did they go to reach that. How did they find the >intersection? By renaming z to u. John Mitchell ==== ... > Should circles be drawn by simple rules that check which neighbor to > turn on next, or by using cosine and sine functions? ... I don't know. Of course, in computer graphics, one can use Bresenham's circle-drawing algorithm, which uses fast, simple, precise techniques much like those of his line-drawing algorithm. Here is another simple method that draws a round-looking thing but not a circle: Choose m so that m(m+1) is approximately equal to desired diameter. For the first quadrant, draw a line up n and left 1, then (each time from previous line's endpoint) a line up n-1 and left 2, then a line up n-2 and left 3, etc., and finally up 1 and left n. Reflect for other quadrants. -jiw ==== > >>I am not sure about this, but I was just wondering if anyone knows anything >>about this: if a function is bounded above, is the derivative bounded >>above? > I guess yes, if the function is differentiable continuously in every domain point. No - consider -abs(sqrt(abs(x))) at x=0. ==== > So far it seems that all solutions to B6 were based on a discrete version. The reduction to the discrete situation is trivial, opens up many extra proof techniques, and doesn't seem to rule out any proof techniques. So it's natural to express solutions in the discrete situation. Here, for example, is a discrete version of your proof. Fix real numbers r_1, r_2, etc. For t>0 define A(t) = sum_i[t=-r_i]. Then B(t)-A(t) = sum_i[t>=r_i>=-t], which is between 0 and n, so (B(t)-A(t))^2 <= n(B(t)-A(t)). Furthermore, |r_i| = int_{t>0} ([t0} (sum_i[t0} (A(t)+n-B(t)). Similarly, |r_i+r_j| = int_{t>0} ([t>=r_i][t<-r_j] + [t>=-r_i][t=r_j] + [t=-r_j]), so sum_{i,j} |r_i+r_j| = int_{t>0} (2(n-A(t))(n-B(t)) + 2A(t)B(t)) = int_{t>0} (n^2 + (B(t)+A(t)-n)^2 - (B(t)-A(t))^2) >= int_{t>0} (n^2 + 0 - n(B(t)-A(t))) = n sum_i |r_i|. ---D. J. Bernstein, Associate Professor, Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago ==== > So far it seems that all solutions to B6 were based on a discrete version. The reduction to the discrete situation is trivial, opens up many extra > proof techniques, and doesn't seem to rule out any proof techniques. So > it's natural to express solutions in the discrete situation. On the other hand, Pop's solution works quite generally to show that if X and Y are independent identically distributed random variables, then E|X+Y= E|X|. The discrete inequality is a special case of this inequality. -- A. ==== > >>So far it seems that all solutions to B6 were based on a discrete version. > The reduction to the discrete situation is trivial, opens up many extra > proof techniques, and doesn't seem to rule out any proof techniques. So > it's natural to express solutions in the discrete situation. > I never said the contrary. Ciprian hi, I need help with a question from an old exam at my school. Find all k-transposable numbers (and the corresponding k, where k is an integer, of course) where a k transposable number is a number where when you move the leftmost digit and move it all the way to the right, the number is k times as large. also part two of the question asked about the reverse question -- moving the rightmost digit all the way too the left. find all, say, kk - transposable numbers. Any help would be appreciated. Jason ==== > hi, I need help with a question from an old exam at my school. Find all k-transposable numbers (and the corresponding k, > where k is an integer, of course) > where a k transposable number is a number where when you move > the leftmost digit and move it all the way to the right, the > number is k times as large. also part two of the question asked about the reverse question -- > moving the rightmost digit all the way too the left. find all, > say, kk - transposable numbers. ... Have you already noted that k*(d*10^j+e)=10*e+d (where d is a leading digit and e is a j-digit number in base 10), from which e = d*(k*10^j -1)/(10-k) ? Also k*d < 10, e must be an integer with the same factors as d*(k*10^j -1)/(10-k), and when k=2, d must be even. You will need a few more ideas besides these, but should be able to readily characterize solutions at k=1, show there are no solutions at k=2, find solutions at k=3 with j=5, 11, 17, 23..., etc. -jiw ==== > >> Try talk.origins if you want to see the sparks fly. By comparison, >> sci.math is tame. And there are groups far worse that >> talk.origins. >Which ones? talk.abortion ? > Oooh... Good answer. ==== > > |Has been tried, it will not work. If he does not get attention he starts > |and so he must be correct, but does not receive the proper attention. > > when was this occasion that everyone ignored jsh for a period of a > couple of months? > | > |Long, long ago. > > when was it? was it after jsh started posting? was it during a time > period that's recorded in the google archive well enough to determine > whether you're telling the truth? ignored James. There are always people that will reply, and it would now be worse than at that time (a larger public you know). But if I remember right, almost all of the regular responders refrained from responding for JSH where he gleefully tells that he has silenced all objectors to his math (ignoring those he had not answered, and also ignoring that the answers to those he did answer were not really answers). In it, of course, was the complete exposition of his rambling theory. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ ==== |Has been tried, it will not work. If he does not get |claims that nobody disagrees, and so he must be correct, but |does not receive the proper attention. when was this occasion that everyone ignored jsh for a period of a couple of months? | |Long, long ago. when was it? was it after jsh started posting? was it during a time period that's recorded in the google archive well enough to determine whether you're telling the truth? | |everyone ignored James. There are always people that will reply, and |it would now be worse than at that time (a larger public you know). |But if I remember right, almost all of the regular responders |refrained from responding for some time. It did not stop the flow of |that he has silenced all objectors to his math (ignoring those he had |not answered, and also ignoring that the answers to those he did |answer were not really answers). In it, of course, was the complete |exposition of his rambling theory. so almost all of the regular responders refrained from responding for some time, and then found themselves unable to refrain any longer because they did not have the self-control to ignore kindergarten-level taunting. it would be hilarious to compare the actual length of time that they heroically managed to refrain from responding to the period of a couple of months that you seem to be claiming. -- ==== > so almost all of the regular responders refrained from responding for > some time, and then found themselves unable to refrain any longer > because they did not have the self-control to ignore > kindergarten-level taunting. it would be hilarious to compare the > actual length of time that they heroically managed to refrain from > responding to the period of a couple of months that you seem to be > claiming. He seems to be claiming a couple of months only because in one of your posts you said a couple of months. Dik didn't ever explicitly months and it was to this that Dik first responded. In any case, I won't defend or criticize those who respond to James Harris and in recent months, I've responded directly to him more often than previously. Certainly, I've poked fun at him passively for years, by using JSH quotations in my .sigs. But I know that the endless JSH activity is a great annoyance to many people that come here for loftier reading (Penthouse Letters would be an example of loftier reading). In any case, I don't find your ability to put words in Dik's mouth and to beat up strawmen particularly edifying either. -- Jesse F. Hughes That's what's annoying about Usenet as some loser will state a case, get their ass kicked, but STILL keep coming back as if nothing happened. -- James Harris explains his strategy. ==== >[...] But I know that the >endless JSH activity is a great annoyance to many people that come >here for loftier reading (Penthouse Letters would be an example of >loftier reading). He-heh. Do they still have those? I never thought I'd write a post to sci.math... ************************ David C. Ullrich <87u1477nex.fsf@phiwumbda.org> ==== >[...] But I know that the >>endless JSH activity is a great annoyance to many people that come >>here for loftier reading (Penthouse Letters would be an example of >>loftier reading). He-heh. Do they still have those? I never thought I'd write a > post to sci.math... How would I know? Since moving to Holland (Pornography Capital of the Free World, according to the promotional material I received), I've unfortunately must learn Dutch. -- Sale or rental of this disc is ILLEGAL. If you have rented or purchased this disc, please call the MPAA at 1-800-NO-COPYS. -- The MPAA begins a new anti-piracy program, found on a DVD purchased in China ==== ... > |everyone ignored James. There are always people that will reply, and > |it would now be worse than at that time (a larger public you know). > |But if I remember right, almost all of the regular responders > |refrained from responding for some time. It did not stop the flow of > |that he has silenced all objectors to his math (ignoring those he had > |not answered, and also ignoring that the answers to those he did > |answer were not really answers). In it, of course, was the complete > |exposition of his rambling theory. > > so almost all of the regular responders refrained from responding for > some time, and then found themselves unable to refrain any longer > because they did not have the self-control to ignore > kindergarten-level taunting. it would be hilarious to compare the > actual length of time that they heroically managed to refrain from > responding to the period of a couple of months that you seem to be > claiming. This is a gross misrepresentation. I have refrained from posting in his first entry to the newsgroup. I think I never did attack James on in a direct response to an answer by him (but sometimes I piggy-backed on other responders). (And I may have sinned of late.) What is happening is that James posts something new. There are a few people that question his findings. James answers, but actually does not answer. Similar questions are posted. James goes into insult mode, never even answering the questions, only providing examples. Some regulars step in clarifying the first questions were not the correct questions, and showing what the actual questions should be. And James comes in with more insult. Actually finding the gaps in James' reasoning is quite a challenge... -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ ==== [.snip.] >> Try talk.origins if you want to see the sparks fly. By comparison, > sci.math is tame. And there are groups far worse that > talk.origins. >Which ones? talk.abortion ? alt.revisionism Alan -- Defendit numerus ==== >[...] >> Also, you seem to know a lot more maths than most people, but judging by the >> threads you are no expert. Why don't you get some help, get yourself >> thinking straighter, then you might be able to learn a lot more about maths, >> and even do some interesting work for people to appreciate? I haven't claimed to be an expert!!! Guffaw. How many times have you explained that the only explanation is that the professional mathematicians' brains are just not wired right, somehow they're just unable to follow these proofs which are so simple to you? ************************ David C. Ullrich <3c65f87.0312091243.1feb11da@posting.google.com> <3c65f87.0312100650.6135dec8@posting.google.com> <55cftv4ff1aqnc5k8k15alnf4eeoihu84c@4ax.com> ==== >[...] > Also, you seem to know a lot more maths than most people, but judging by the > threads you are no expert. Why don't you get some help, get yourself > thinking straighter, then you might be able to learn a lot more about maths, > and even do some interesting work for people to appreciate? >>I haven't claimed to be an expert!!! Guffaw. How many times have you explained that the only explanation > is that the professional mathematicians' brains are just not wired > right, somehow they're just unable to follow these proofs which > are so simple to you? Ignoring right vs. wrong wiring, I think I agree with that assessment. Somehow, I'm just unable to follow proofs which are self-evident to James (not that I'm a professional mathematician). -- Yup, as far as I'm concerned, if you live out your lives smiling the entire time full of pride in your *believed* accomplishments, when you never had any, well that's ok with me. --James Harris, a man of remarkable accomplishments. ==== > Ignoring right vs. wrong wiring, I think I agree with that > assessment. Somehow, I'm just unable to follow proofs which are > self-evident to James (not that I'm a professional mathematician). The correct way to handle a proof from a crank: Start at the top, read up to the first sentence that is either wrong or where you don't understand what he means. Takes very little time usually. A technique I like but that doesn't work with cranks: Without bothering whether the proof is correct or incorrect or makes any sense at all, make slight changes to the proof that don't change its correctness (or incorrectness) but make it prove something that is well known to be wrong. Many proofs that a^n + b^n = c^n has no solutions in positive integers a, b, c and n >= 3 can easily be modified to the case n=2 as well, which is a strong indication to anyone but the crank that there is something wrong with the proof. <3c65f87.0312091243.1feb11da@posting.google.com> <3c65f87.0312100650.6135dec8@posting.google.com> <55cftv4ff1aqnc5k8k15alnf4eeoihu84c@4ax.com> <87r7zb7n5w.fsf@phiwumbda.org> ==== > Ignoring right vs. wrong wiring, I think I agree with that >> assessment. Somehow, I'm just unable to follow proofs which are >> self-evident to James (not that I'm a professional mathematician). The correct way to handle a proof from a crank: Start at the top, read > up to the first sentence that is either wrong or where you don't > understand what he means. Takes very little time usually. But what effect does this have? It doesn't convince the crank that his argument is wrong (not that I know of any technique that works in this regard). The only advantage that this technique has over the one below is: it takes less time to fail to achieve your goal than the other one. > A technique I like but that doesn't work with cranks: Without bothering > whether the proof is correct or incorrect or makes any sense at all, > make slight changes to the proof that don't change its correctness (or > incorrectness) but make it prove something that is well known to be > wrong. Many proofs that a^n + b^n = c^n has no solutions in positive > integers a, b, c and n >= 3 can easily be modified to the case n=2 as > well, which is a strong indication to anyone but the crank that there is > something wrong with the proof. -- Jesse Hughes Certainly he who can digest a second or third fluxion need not, methinks, be squeamish about any point in divinity. George Berkeley, 1734 <87ekvcc1g4.fsf@phiwumbda.org> ==== | >> If he does not get attention he starts |posting a multitude of >> be correct, but does not receive the proper attention. when >> was this occasion that everyone ignored jsh for a period of a >> couple of months? >> | >> |Long, long ago. > |> when was it? was it after jsh started posting? was it during a >> time period that's recorded in the google archive well enough to >> determine whether you're telling the truth? > | > |Like Dik, I remember something like the experiment you advocate who the hell said i was advocating something? this entire message of > yours seems to be based on a bizarre hallucination. Fair enough. I assumed you were the person who said earlier in the thread, If everyone would ignore him, he *might* go away after a couple weeks or months. Insulting him only inspires him. I assumed that, in part, because of the odd tone in your post (how do I know you're telling the truth?) but the assumption was wrong. You didn't advocate ignoring JSH as far as I can tell, so my post is a response to a claim you've never made. But if you do make it, I'm ready. -- Jesse Hughes But nothing's being Dr. Ullrich is a particular case of something's being such that nothing is it: (Ex)~(Ey)(y = x) -- John Correy on the failings of first order logic ==== |>> when was this occasion that everyone ignored jsh for a |>> period of a couple of months? |>> | |>> |Long, long ago. |>> |>> when was it? was it after jsh started posting? was it during a |>> time period that's recorded in the google archive well enough to |>> determine whether you're telling the truth? |> | |> |Like Dik, I remember something like the experiment you advocate |> |> who the hell said i was advocating something? this entire message |> of yours seems to be based on a bizarre hallucination. | |Fair enough. I assumed you were the person who said earlier in the |thread, If everyone would ignore him, he *might* go away after a |couple weeks or months. Insulting him only inspires him. I assumed |that, in part, because of the odd tone in your post (how do I know |you're telling the truth?) but the assumption was wrong. ok, so you think it's odd to be skeptical of laughably implausible bullshit. -- <87fzfsa1oo.fsf@phiwumbda.org> ==== >> when was this occasion that everyone ignored jsh for a > period of a couple of months? > | > |Long, long ago. >> |>> when was it? was it after jsh started posting? was it during a > time period that's recorded in the google archive well enough to > determine whether you're telling the truth? >> | >> |Like Dik, I remember something like the experiment you advocate > |> who the hell said i was advocating something? this entire message >> of yours seems to be based on a bizarre hallucination. > | > |Fair enough. I assumed you were the person who said earlier in the > |thread, If everyone would ignore him, he *might* go away after a > |couple weeks or months. Insulting him only inspires him. I assumed > |that, in part, because of the odd tone in your post (how do I know > |you're telling the truth?) but the assumption was wrong. ok, so you think it's odd to be skeptical of laughably implausible > bullshit. There's a difference between your belligerent tone and its skeptical motivations. How do I know you're telling the truth? [paraphrased] sounds like more than disinterested skepticism. Anyway, laughably implausible bullshit or not, my laughably implausible (but foggy) memory agrees with Dik's laughably implausible memory. -- Yup, you guessed it. If worse comes to worse, I *will* turn to the Army to help me with mathematicians. And then mathematicians don't think the NSA or CIA can save your asses, as generals LIKE me. -- James Harris's latest foray into mathematical logic. ==== snip > James, as one sufferer to another possible sufferer. I think you should get > some help for the mental state you seem to be in. You assume too much. I *have* consulted mental health professionals. Why are you accepting online diagnosis from posters or even my own > comments about my fears of mental illness? Shouldn't you be *more* sensitive on the subject? Worse, by accepting their diagnoses, you're simply giving posters who > like to use accusations of mental illness more power. Don't just assume that because you guess that another person is > mentally ill that you have the expertise to make that medical > diagnosis, and especially don't just assume that you know what their > medical history is--like whether or not they've consulted mental > health professionals. > Also, you seem to know a lot more maths than most people, but judging by the > threads you are no expert. Why don't you get some help, get yourself > thinking straighter, then you might be able to learn a lot more about maths, > and even do some interesting work for people to appreciate? I haven't claimed to be an expert!!! I've found some math that interests me, which I believe *should* > interest mathematicians if their own claims about pure math and > appreciation of beauty in mathematics are to be believed. However, they've responded with insults and personal attacks like > accusations of mental illness. But actual mental health professionals have never diagnosed me with > any mental illness other than a brief bit of depression back when I > was in college. It pains me that I need to share that information because of people > like you Pat, who open the door to spurious attacks by taking them > seriously. Just because I like to post about my math research on several > newsgroups, can get belligerent at times, and at times question my own > mental health, it doesn't make me mentally ill. Mental health professionals are tasked with making such diagnoses and > people who decide to act as psychiatrists or psychologists online to > give their own medical diagnoses, usually as a means to attack another > person, are a danger to others because unfortunately, people *do* take > them seriously. > James Harris James, I'm not necessarily saying you are diagnosed mentally ill or would receive such a diagnoses from a psychiatrist or psychiatrists if you were to make contact with mental health services now or in the future. That diagnosis is based on psychiatric concepts generally based on observations made in a clinical environment by someone with a license to practice psychiatry. And such concepts, it has been argued, are flawed on a number of levels. But, that does not discount the observation that some people do behave in a way that is seen as 'strange' or sometimes even unacceptable to other people, and some of the people who behave like that indeed do have diagnostic labels. All I said was I think you should get some help. You could get help from a friend. All, I've said is I'm diagnosed as mentally ill, and I've mixed with a lot of people closely who are also diagnosed as mentally ill, and I see in some of them to varying degrees behaviour that is similar to your behaviour from my impressions of reading your posts. Also, my impression is you are getting obsessional, and I recognize a bit of that in myself from time to time, and I try to guard against it. So, I do feel some empathy towards you because you do seem to have some of the problems that I have and other people I know have. Yes, of course I'm sensitive to these issues, there are a lot of stereotypes concerning the 'mad.' But, you are not doing me any favours, James, are you? You seemed to be patronizing me with your comments about people stepping on ants, referring to other posters having no concern for people 'like me,' which I feel, could have been based on some empathy towards people diagnosed like I am, but mainly I think you were trying to get me on your side. Do you think all people 'like me' are just morons that you can persuade so easily? Is evidence of this patronizing attitude also linked to your *assumption* that I can't come to an independent judgement about your behaviour? and I am so easily persuaded by others here? James, it is common for people to call each other nutter, psycho, etc. when someone is acting like your acting, and to a certain extent it fosters stereotypes about the 'mad' in the publics' mind. But, James, if you do have concern for people like me which your statement about stepping on ants was an attempt to convey to me, then you should realize that you are encouraging words like psycho to be banded about and directed at you by your unacceptable behaviour. So, if you do have some empathy towards people like me you should at least make some sort of effort to modify your behaviour and lessen the insults. Do you not agree? ==== > Also, my impression is you are > getting obsessional, Getting? GETTING?!?! V. -- homepage: cs utk edu tilde lastname ==== > No, that is wrong. The Indian philosophical thought - Sanatana > dharma, or the way of life beyond the scope of time - is completely > different from the modern and dominant Jewish [...] The Judeo-Christian ideology defines thought in the West today, but is India > the West? Clearly to an Eastern thinker, the ego/moral emphasis of > Christianity, Judaism and Islam is not a viable doctrine. contempt black skin. ==== contempt black skin. And thats why most important menifestation of god in hinduism like vishnu, shiva,ram,krishna have black skin.Stupid propoganda machine learn the facts right before spewing venom against hinduism. ==== contempt black skin. > And thats why most important menifestation of god in hinduism like > vishnu, > shiva,ram,krishna have black skin.Stupid propoganda machine learn the > facts right before spewing venom against hinduism. That may be so. But, one is disadvantage if one's skin color is kala in India. Why? Dan ==== > No, that is wrong. The Indian philosophical thought - Sanatana > dharma, or the way of life beyond the scope of time - is completely > different from the modern and dominant Jewish thinking [...] This frame of mind, of course, serves to lend additional credence to the otherwise unbelievable notion that the Swastika actually originated in India. ==== > Knowledge is wasted upon low-minded and deliberate fools. > Well, 5000 years without flight in the presence of all that > knowledge: clearly it must have been wasted on the whole > subcontinent all those millennia. > Mortals neither designed nor made vimans, though some aspects of their > design were known [...] Whatever. Clearly the knowledge must have been wasted on the whole subcontinent all those millennia. It's a good thing the Wright Brothers were not mortal, or we'd still be taking the chariot to work. > Meanwhile, the first airplanes used by Deccan Airways were made > and purchased from whom? > We are talking about the past, with reference to the future. The present is > just a bad idea. Wow. I've seen people with a touch of nostalgia before. But don't you think calling the 1940's the future or present is going a little bit overboard? Meanwhile, I wonder how many temples have been erected to the Gods known as McDonald and Douglas; the savior of India's 5000 year flight-impairment whose insights were communicated to the masses by their emissary: Deccan Airways? Oops, did I get that spelling right? We don't want to get the golden arches in the mix here. What about temples devoted to the Gods known as Apollo -- Apollo 17, Apollo 16, Apollo 15, etc., that is? ==== > What about temples devoted to the Gods known as Apollo -- Apollo 17, > Apollo 16, Apollo 15, etc., that is? Rockets are for fireworks. ==== theorem. Here is the problem: integral over C (counter-clockwise) of [ (3xy + y^2)dx + (2xy + 5x^2)dy ] C: (x - 1)^2 + (y + 2)^2 = 1 I got as far as: (double integral) 7x dxdy My book found the solution by taking the centroid of the circle and then multiplying by the area to get 7[pi] I was wondering if there was another way to compute the double integral without using centroids and how would I do that. ==== Its messy, but You solve the curve to get y1=f(x),y2 =f(x) and the limits of x Then bung them into the double integral integrating over y from y1 to y2 then over x. The think about maths problems is finding the neatest solution, thats what the author has doone. -- Bruce Harvey bruce@bearsoft.co.uk The Alternative Physics Site http://users.powernet.co.uk/bearsoft > theorem. Here is the problem: integral over C (counter-clockwise) of [ (3xy + y^2)dx + (2xy + > 5x^2)dy ] > C: (x - 1)^2 + (y + 2)^2 = 1 I got as far as: (double integral) 7x dxdy My book found the solution by taking the centroid of the circle and > then multiplying by the area to get 7[pi] I was wondering if there was another way to compute the double > integral without using centroids and how would I do that. > ==== Its messy, but You solve the curve to get y1=f(x),y2 =f(x) and the limits of x Then bung them into the double integral integrating over y from y1 to y2 > then over x. The think about maths problems is finding the neatest solution, thats what > the author has doone. -- Bruce Harvey > bruce@bearsoft.co.uk > The Alternative Physics Site > http://users.powernet.co.uk/bearsoft > theorem. Here is the problem: integral over C (counter-clockwise) of [ (3xy + y^2)dx + (2xy + > 5x^2)dy ] > C: (x - 1)^2 + (y + 2)^2 = 1 I got as far as: (double integral) 7x dxdy My book found the solution by taking the centroid of the circle and > then multiplying by the area to get 7[pi] I was wondering if there was another way to compute the double > integral without using centroids and how would I do that. > X-Received: (from approve@localhost) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hB9KwL510666; ==== >> Now what if some person says they found a brilliant gem, should they >> go through college courses, and learn a lot of techniques in the >> analysis of gems? >> >> Or can't they just holler out that they found something? >> >> Bad analogy. >> >> Someone says they've found a gem. Lots of people look at it. Everyone >> who knows about gems says, That's just a lump of coal. >> >> Would that mean that it really was just a lump of coal, or does it mean >> that the discoverer can carry on shouting that he's being ignored? Yeah, but here I can show the find: 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). Reference: http://mathforprofit.blogspot.com /25, which is the number of primes up to 100 or p(10,3) = 4, and those >primes are 2, 3, 5 and 7. So it's more like someone finding a gem, and having experts claim it's >glass, only to then have that person cut glass with it. I want critical thinkers to search on prime counting function and >see the methods that mathematicians have on record for counting prime >numbers to compare with what I just gave and see for yourselves how >obvious it is that what I have isn't just junk. Remember, mathematicians are fighting to totally dismiss my result as >unimportant to justify not putting it in math references. >James Harris My math discoveries, found for profit >http://mathforprofit.blogspot.com / That's just a lump of coal. X-Received: (from approve@localhost) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBADQQO17430; ==== >prove if x is an irrational number then sqroot of x is irrational?? Any rational number has a finite number of digits, so when multiplied by itself it would have at most twice a many while any irrational number requires infinite digits. ==== >>prove if x is an irrational number then sqroot of x is irrational?? >Any rational number has a finite number of digits You mean, like 1/3 = 0.3333333333333333...? Doug ==== >prove if x is an irrational number then sqroot of x is irrational?? Any rational number has a finite number of digits, so when > multiplied by itself it would have at most twice a many while any > irrational number requires infinite digits. What do you mean any rational number has a finite number of digits? For example, in base 10, the number 1/3 has infinitely many digits. Or if you mean finitely many different digits (so that 1/3 has one digit), then pi has only digits. Thomas X-Received: (from approve@localhost) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBAEB6g20736; ==== Does anyone know a little about the history of this problem (that {n*x} is dense in [0,1) if x is irrational)? I've seen in some places that it was first considered by Chebyshev. Was he the first to prove it? And how did Kronecker get involved in approximation theory with his beliefs about irrational numbers? Any information would be appreciated. Marcus X-Received: (from approve@localhost) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBAHQtO03689; ==== Sorry to take this discussion further afield than it already is, but I just wanted to comment on this point. I disagree with Herman that theory must always come before computation. Many times computation provides an understanding of what motivates the theory. It's a lot prettier to present the theory first as if it was discovered fully formed. But that often obscures the computations and historical developments that motivated it. Pedagogically that is not helpful for the student. As for the claim that all mathematics courses should be rigorous, well that depends on what you mean by rigorous. People use mathematics for very different purposes. A course should be taught appropriate to the needs of the students. For some students, proficiency in computation is far more important than absolute rigor of every single step - assuming that is even possible. No doubt young students should be exposed to more of the facets of mathematics than just computation. But a good early education in mathematics should introduce them to all of it's aspects; it's history, computational techniques, foundations, and even philosophy. Marcus >There are rigorous books, starting with high school Euclid, >Landau's _Foundations of Analysis_ (number systems), texts >on logic, set theory, abstract algebra, and real analysis, >all of which are readable. Those who can learn these >subjects should learn them before computation without any >idea of what it means, and should. I would put off real >analysis until proofs are understood; otherwise, both have >to be learned at the same time. But proofs HAVE BEEN taught >to fifth graders, and I believe it can be done earlier. One does not always have to be complete, but all mathematics >courses need to be rigorous, with no exceptions. X-Received: (from approve@localhost) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBANttD02210; ==== . . . what is a mouse set? X-Received: (from approve@localhost) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hBB4DuM21249; ==== hi could you please help me with these problems please.the answers. >> 1. y = -x y = 2x 2. 3x+y=6 y+2=x 3. x-y=6 2x-4y+28 4. 4x=3y+44 x+y=-3 THANKS.. > ==== > hi could you please help me with these problems please.the answers. > 1. y = -x > y = 2x 2. 3x+y=6 > y+2=x 3. x-y=6 > 2x-4y+28 4. 4x=3y+44 > x+y=-3 > THANKS.. > hi could you please help me with these problems please.the answers. >> For all the problems do this. Solve for y in both equations. This has already been done in #1. Set the two solutions for y equal. Solve the resulting equation for x. Use that value for x and one of the solutions for y, to get the value of y. > 1. y = -x > y = 2x > For example, setting the two solutions = gives -x = 2x Which I leave for you to solve. > 2. 3x+y=6 Thus y = ??1 > y+2=x and y = ??2 setting the two equal, solve ??1 = ??2 3. x-y=6 > 2x-4y+28 4. 4x=3y+44 > x+y=-3 > THANKS.. ==== > What can you add in this context for Jove, Saturn, Uranus and Neptune? The perihelion advance is too small, and the orbits too poorly determined, for any verification. > I would be more interested in cases where GR predicts a perihelion advance, but there is none observed. Are there any such cases to the best of your knowledge Dr. Van Flandern? Yes, a few binary star systems have observed pericenter motions different from GR predictions. These are mainly cases of two contributing-but-unequal masses. However, other effects such as tidal forces or mass transfer have not yet been ruled out. See, e.g., http://www.gsanctuary.com/general_relativity.html. -|Tom|- Tom Van Flandern - Washington, DC - see our web site on replacement astronomy research at http://metaresearch.org ==== What can you add in this context for Jove, Saturn, Uranus and Neptune? The perihelion advance is too small, and the orbits too > poorly determined, for any verification. Dear Dr. Tom Van Flandern Why so strongly have varied the legitimated quantities of planetary masses between 1980 and 1990? Or it can be, that the modern legitimated quantities of planetary masses too poorly determined, for any modern verification... ? What can you tell about changes in the measuring techniques of quantities of planetary masses between 1980 and 1990? Sincerely yours, Aleksandr I would be more interested in cases where GR predicts a perihelion > advance, but there is none observed. Are there any such cases to the > best of your knowledge Dr. Van Flandern? Yes, a few binary star systems have observed pericenter > motions different from GR predictions. These are mainly cases of two > contributing-but-unequal masses. However, other effects such as tidal > forces or mass transfer have not yet been ruled out. See, e.g., > http://www.gsanctuary.com/general_relativity.html. -|Tom|- > Tom Van Flandern - Washington, DC - see our web site on replacement > astronomy research at http://metaresearch.org ==== What can you add in this context for Jove, Saturn, Uranus and Neptune? The perihelion advance is too small, and the orbits too > poorly determined, for any verification. I would be more interested in cases where GR predicts a perihelion > advance, but there is none observed. Are there any such cases to the > best of your knowledge Dr. Van Flandern? Yes, a few binary star systems have observed pericenter > motions different from GR predictions. These are mainly cases of two > contributing-but-unequal masses. However, other effects such as tidal > forces or mass transfer have not yet been ruled out. See, e.g., > http://www.gsanctuary.com/general_relativity.html. -|Tom|- > Dr. Van Flandern: To what extent you think the website you quote ( http://www.gsanctuary.com/general_relativity.html) can be a reliable source of information, when it is asserted there that the universe is only 6,000 years old and the speed of light is dependent on the position of its source and path? I may as well read the Bible then and forget about science... Tom Van Flandern - Washington, DC - see our web site on replacement > astronomy research at http://metaresearch.org ==== > Define: f(0)=0 > f(x+1)=(f(x)+1)/2 > f(1/x)=1-f(x) I think this function is defined for every integer n and for every > number of the form 1/n, n<>0, n integer. It's increasing for x>=0, but > I don't see why it's defined for every rational >=0. > Anyway, this is not a counter example, you didn't show a monotonic > function such that the set of points where it's not differentiable is > not null. > Amanda Once you define it for every number of the form 1/n, then you can define it for every number 1+1/n, 2+i/n, ... 1 / (1 + 1/n), ect. Fact is, he has defined f for all rationals. ==== > >It's well known that if a real valued function f is monotonic on a >compact interval I then f is integrable over I and the set of >discontinuities of f on I is countable. I'm not sure, but it seems to >me that if we combine this fact with the Fundamental Theorem of >Integral Calculus (in the form that deals with the derivative of an >integral)and consider Lebesgue Integrability Criterion, then we come >to the conclusion that the set of elements of I at which f is not >differentiable has measure zero.Is this conclusion true? The _conclusion_ is indeed true - a monotonic function is > differentiable except on a set of measure zero. But I don't > follow how you think you've proved this at all. (if you can > give a proof that's that simple you can publish it. But you're > going to need to explain the argument a little more carefully > first...) IIRC, there's a proof very early on in Riesz-Nagy. Since it IS very early on, it can't use much measure theory. I suppose it is possible to prove a theorem a.e. without having measure theory--you just directly define a set of measure zero--but I don't have the book in front of me and I don't know how they handle this. They also have one of the standard examples of an everywhere continuous, nowhere differentiable function. --Ron Bruck ==== > >It's well known that if a real valued function f is monotonic on a >compact interval I then f is integrable over I and the set of >discontinuities of f on I is countable. I'm not sure, but it seems to >me that if we combine this fact with the Fundamental Theorem of >Integral Calculus (in the form that deals with the derivative of an >integral)and consider Lebesgue Integrability Criterion, then we come >to the conclusion that the set of elements of I at which f is not >differentiable has measure zero.Is this conclusion true? The _conclusion_ is indeed true - a monotonic function is > differentiable except on a set of measure zero. But I don't > follow how you think you've proved this at all. (if you can > give a proof that's that simple you can publish it. But you're > going to need to explain the argument a little more carefully > first...) Well, when I looked more carefully into what I thought might be a conclusions I cited, the conclusion does not follow....(at least, I can't show it follows). Anyway, I had a good guess... Well, that happened to Fermat, too, right? He thought he had proved that famous theorem and when he got carefully into the details, he saw there was a mistake. nothing...And, still, FLT is named after him, who never proved that theorem... Amanda ==== >> >>It's well known that if a real valued function f is monotonic on a >>compact interval I then f is integrable over I and the set of >>discontinuities of f on I is countable. I'm not sure, but it seems to >>me that if we combine this fact with the Fundamental Theorem of >>Integral Calculus (in the form that deals with the derivative of an >>integral)and consider Lebesgue Integrability Criterion, then we come >>to the conclusion that the set of elements of I at which f is not >>differentiable has measure zero.Is this conclusion true? >> >> The _conclusion_ is indeed true - a monotonic function is >> differentiable except on a set of measure zero. But I don't >> follow how you think you've proved this at all. (if you can >> give a proof that's that simple you can publish it. But you're >> going to need to explain the argument a little more carefully >> first...) Well, when I looked more carefully into what I thought might be a >conclusions I cited, the conclusion does not follow....(at least, I >can't show it follows). Anyway, I had a good guess... >Well, that happened to Fermat, too, right? He thought he had proved >that famous theorem and when he got carefully into the details, he saw >there was a mistake. nothing...And, still, FLT is named after him, who >never proved that theorem... Yup. Too bad this theorem is already named for someone else. (Just teasing...) >Amanda ************************ David C. Ullrich ==== im trying to find all groups of order eight. i already know what these groups are, but i am trying to derive it, mainly from the sylow theorems. will this approach work? for example, I was able to work with groups of order 21=3*7, because I was able to find a semidirect product representation of the group, but 8=2^3 doesnt really lead me anywhere. is there another method? thanks ==== > im trying to find all groups of order eight. i already know what these > groups are, but i am trying to derive it, mainly from the sylow > theorems. will this approach work? Trouble is, as you say later, is that 2 is a prime power so Sylow tells you nowt. > for example, I was able to work > with groups of order 21=3*7, because I was able to find a semidirect > product representation of the group, but 8=2^3 doesnt really lead me > anywhere. is there another method? Two possible attacks (i) the exponent of the group: this is the smallest n such that a^n = 1 for all a in the group. For a group of order 8 it is 2, 4 or 8. Divide into these three cases. (ii) Use the centre. A standard theorem states that the centre Z of a p-group G is nontrivial. Split into cases according to the structure of the centre. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) ==== |>Lastly, is there any converse of the It can be approximated too well |>so it must be transcendental? Can all transcedentals be approximated |>too well or are there known examples of transcendentals which cannot |>be approximated any better than an algebraic? | |No. Consider the numbers whose classical continued |fraction has only 1's and 2's. These cannot be |better approximated than sqrt(2), and yet there must |be transcendentals, as there are continuum many. For each irrational r, let a(r) be the greatest lower bound of the C>0 such that there are only finitely many rationals p/q satisfying |r-p/q| aleph0. But each infinitely long sequence of 01 notations can be mapped with some natural number, for example: ...000 <--> 1 ...001 <--> 2 ...010 <--> 3 ...011 <--> 4 ...100 <--> 5 ...101 <--> 6 ...110 <--> 7 ...111 <--> 8 ... Therefore we can conclude that 2^aleph0 = aleph0, and we come to contradiction. (2^aleph0 >= aleph0) = {}, and we have a proof saying that Boolean Logic cannot deal with infinitely many objects. Doron ==== > Because I don’t know how to write my idea in the common formal > way, I am going to do it in a non-formal way, but I will do my best to > write it in the clearest way (PLEASE READ ALL OF IT, OTHERWISE IT > CAN'T BE UNDERSTOOD). So here it is: Let us check these lists. P(2) = {{},{0},{1},{0,1}} = 2^2 = 4 and also can be represented as: 00 > 01 > 10 > 11 > P(3) = {{},{0},{1},{2},{0,1},{0,2},{1,2},{0,1,2}} = 2^3 = 8 and also can be represented as: 000 > 001 > 010 > 011 > 100 > 101 > 110 > 111 Let us call any full 01 list, combinations list. Now, let us use Cantor's Diagonalization method on some finitely long > combinations list, for example, the combinations list of number 3: 000 > 001 > 010 > 011 > 100 > 101 > 110 > 111 We can change the order of the rows, and then use Cantor's > Diagonalization method, for example: 001 > 011 > 010 > 000 > 101 > 100 > 111 > 110 The input for Cantor's Diagonalization method in the first example is > 000 and the output is 111. of course that choice was completely arbitrary. you could have started with any string. and any ordering of the strings. The input for Cantor's Diagonalization method in the second example is > 010 and the output is 101. > fine, so you're ignoring the bottom 2^n - n rows then? > In both examples we find that the result is already in the > combinations list, and this combination, which is already in the list, > is one of the combinations that Cantor's Diagonal does not cover. The number of the combinations, which are out of the range of Cantor's > diagonal is: 2^n - n > so that's where that number came from. now i understand what it means. incidentally, you'd (still be wrong but) be better off saying domain. > Every column, which belongs to some combinations list is a sequence of > 01 notations, based on some periodic frequency changes, for example: the right column of number 3 combinations list, is based on 2^0(=1). Therefore the periodic frequency changes are 1, and the result in this > case is: > 01010101. > but a completely arbitrary ordering of the rows wouldn't look so nice would it? i mean 01110100 is possible under some ordering, so what's your point? > The result of the middle column is based on 2^1(=2), therefore the > sequence is: > 00110011. The result of the left column is based on 2^2(=4), therefore the > sequence is: > 00001111. and we get the full combinations list of number 3: 000 > 001 > 010 > 011 > 100 > 101 > 110 > 111 We can get a combinations list of infinitely many places, by using the > ZF Axiom of infinity induction, on the left side of our combinations > list, by using the induction on the power_value of each column, for > example: 2^0, 2^1, 2^2, 2^3, ... > still passing to infinite constructions without demonstrating it's valid. > In this stage we have proven, by induction, that Cantor's diagonal > cannot cover any full 01 combinations list, finite or infinite. > what do you mean cover? at best you'd need transfinite induction wouldn't you to pass from a finite cardinal to a countably infinite cardinal? otherwise: the sum of 1/r from 1 to n is finite. therefore, using your assumption, the infinte sum is finite. prove that's a problem if you don't already know > Therefore its result is not a new combination (that has to be added to > the list). Because Cantor's diagonal cannot cover the full 01 combinations list, > (of aleph0 places for each combination) we can conclude that 2^aleph0 >> aleph0. But each infinitely long sequence of 01 notations can be mapped with > some natural number, for example: ...000 <--> 1 > ...001 <--> 2 > ...010 <--> 3 > ...011 <--> 4 > ...100 <--> 5 > ...101 <--> 6 > ...110 <--> 7 > ...111 <--> 8 > ... > what does the string entirely made of ones get sent to? i can see how that works for strings with a finite number of ones in them: the string (x_i) i in N gets sent to sum x_i.2^i but where do the ones with infinitely many 1's get sent to? > Therefore we can conclude that 2^aleph0 = aleph0, and we come to > contradiction. (2^aleph0 >= aleph0) = {}, and we have a proof saying that Boolean > Logic cannot deal with infinitely many objects. > Doron I think that at best you've proven the set of finite subsets of a countable set is countable. But that's not surprising is it? You've not got the infinite subsets in your enumeration. if you insist on doing this. note that it is not required to talk about all those sets and strings. just think about all strings of 0's and 1's, forget that set stuff, it's just confusing and unnecessary. ==== A simple question about mutual information: mutual information is defined as I(X;Y) = H(X) - H(X|Y) If X and Y aren't simple discrete random variables, but are defined on set of random discrete variables (X in {X1,X2, ....,Xn}, Y in {Y1,Y2, ....,Yn}, with Xi and Yi discrete random variables), how can i calculate I(X;Y)? ==== I am working on the so called twisted Hall algebra of a Dynkin quiver, and I have a problem proving a fundamental relation among the genereating elements. I am able to solve the problem if the following more general holds: Let R be a ring and let M be a finite (yes: finite) R-module. Assume that M is a direct sum of two simple R-modules S_1 and S_2. Then the number of submodules N of M isomorphic to S_1 with M/N isomorphic to S_2 is the same as the number of submodules N' of M isomorphic to S_2 with M/N' isomorphic to S_1. Is this true? Perhaps it is trivial but I am not able to prove it. -- Michael Knudsen [knudsen@imf.au.dk] ==== I am working on the so called twisted Hall algebra of a Dynkin quiver, and > I have a problem proving a fundamental relation among the genereating > elements. I am able to solve the problem if the following more general > holds: Let R be a ring and let M be a finite (yes: finite) R-module. Finite = finitely many elements? > Assume that > M is a direct sum of two simple R-modules S_1 and S_2. Then the number of > submodules N of M isomorphic to S_1 with M/N isomorphic to S_2 That's redundant by the Jordan-Holder theorem. > is the same > as the number of submodules N' of M isomorphic to S_2 with M/N' isomorphic > to S_1. If S_1 is not isomorphic to S_2, then S_1 is the only submodule of M isomorphic to S_1 (consider the projection onto S_2). Both numbers are one. If S_1 and S_2 are isomorphic, then both numbers must be the same :-) -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) ==== > Finite = finitely many elements? Yes. > That's redundant by the Jordan-Holder theorem. I do not understand that. The Jordan-Holder theorem states that two composition series of M are equivalent, right? > If S_1 is not isomorphic to S_2, then S_1 is the only > submodule of M isomorphic to S_1 (consider the projection onto S_2). > Both numbers are one. Oh! Is the following correct? Let N be a submodule of M isomorphic to S_1. Assume that N doesn't equal S_1. Then the projection N --> S_2 is non-zero. Since S_2 is simple the projection is surjective and since M is the direct sum of S_1 and S_2 the projection is injective. Thus N is isomorphic to S_2. $qedsymbol$ Do you know about Ext groups? I am having trouble finding out their relation to finite length modules. I am working over a K-algebra, K a finite field. It know that Ext(S_1,S_2)=K, so there must be an indecomposable module N of length 2 and an exact sequence 0 --> S_2 --> N --> S_1 --> 0 Taking into account the assumptions Ext(S_1,S_1)=Ext(S_2,S_1)=0, we see that there are just two isomorphism classes of modules of length 3 with 2 composition factors of the form S_1 and one of the form S_2, namely X=N+S_1 and Y=2S_1+S_2. I have not seen any litterature explaining this stuff in detail but I am sure that I can work something more general out myself, if somebody can help me with the example above. I cannot see a clear connection between Ext groups and modules of finite length. -- Michael Knudsen [knudsen@imf.au.dk] -- Michael Knudsen ==== > >> Finite = finitely many elements? Yes. > >> That's redundant by the Jordan-Holder theorem. I do not understand that. The Jordan-Holder theorem states that two > composition series of M are equivalent, right? Yes, M has a composition series 0 <= S_1 <= M with factor groups S_1 and M/S_1 = S_2. So any composition series has S_1 on the bottom and S_2 up top or vice versa. >> If S_1 is not isomorphic to S_2, then S_1 is the only >> submodule of M isomorphic to S_1 (consider the projection onto S_2). >> Both numbers are one. Oh! Is the following correct? Let N be a submodule of M isomorphic to S_1. Assume that N doesn't equal > S_1. Then the projection N --> S_2 is non-zero. Since S_2 is simple the > projection is surjective And so S_1 has a quotient isomorphic to S_2; only possible if S_1 and S_2 are isomorphic (S_1 is simple). Do you know about Ext groups? I am having trouble finding out their > relation to finite length modules. I am working over a K-algebra, K a > finite field. It know that Ext(S_1,S_2)=K, so there must be an > indecomposable module N of length 2 and an exact sequence 0 --> S_2 --> N --> S_1 --> 0 right? OK ... there are |K|-1 non-splitting isomorphism classes of such shortexactsequences. > Taking into account the assumptions Ext(S_1,S_1)=Ext(S_2,S_1)=0, we see > that there are just two isomorphism classes of modules of length 3 with 2 > composition factors of the form S_1 and one of the form S_2, namely > X=N+S_1 and Y=2S_1+S_2. If we have composition series 0 <= A <= B <= M exactly one quotient is S_2. If A is isomorphic to S_1 then B is a direct sum of S_1 and a simple. So B is S_1 (+) S_1 or S_1 (+) S_2. In the former case ext(S_2,B)=0 so M is S_1 (+) S_1 (+) S_2. In the fomer case ext(S_1,B) = ext(S_1,S_2) so the S_1 in A splits from M and M is S_1 (+) an extension of S_2 by S_1. If A is S_2, then M/A is S_1 (+) S_1. Now ext(S_1(+)S_1,S_2) = K^2 so there are nontrivial extensions. Now S_1(+)S_1 has to have lots of submodules isomorphic to S_1. If each element of this ext comes from ext(C,S_2) for some C which is a direct summand of S_1(+)S_1 we're in business (a complementary direct summand will split off).... I'm sure that's true, but I don't have the time/inclination to work that out in deatil. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) ==== > In the fomer case ext(S_1,B) > = ext(S_1,S_2) so the S_1 in A splits from M and M is S_1 (+) > an extension of S_2 by S_1. Hmmmm...shouldn't it be: M is S_1 (+) an extension of S_1 by S_2? This is what we would like to prove. Anyway, how do you get that? We know, as you assumption. How do you use that to get the desired decomposition? -- Michael Knudsen [knudsen@imf.au.dk] ==== > >> In the fomer case ext(S_1,B) >> = ext(S_1,S_2) so the S_1 in A splits from M and M is S_1 (+) >> an extension of S_2 by S_1. Hmmmm...shouldn't it be: M is S_1 (+) an extension of S_1 by S_2? This is > what we would like to prove. Anyway, how do you get that? We know, as you > But the last group is K by > assumption. How do you use that to get the desired decomposition? The point is that the characteristic class of the ses in Ext(S_1,B) comes from one in Ext(S_1,S_2). This comes from an ses looking like 0 -> S_1 (+) S_2 -> S_1 (+) N -> S_1 -> 0 where 0 -> S_2 -> N -> S_1 -> 0 is an ses. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) ==== > If we have composition series 0 <= A <= B <= M exactly > one quotient is S_2. If A is isomorphic to S_1 then B is a > direct sum of S_1 and a simple. Why is that true? Do you say that the canonical sequence 0 --> A --> B --> B/A --> 0 splits? Is that always true? > So B is S_1 (+) S_1 or > S_1 (+) S_2. In the former case ext(S_2,B)=0 so > M is S_1 (+) S_1 (+) S_2. To conclude M = S_1(+)S_1(+)S_2 from Ext(S_2,B)=0 you need an exact sequence 0 --> B --> M --> S_2 --> 0, right? Do you know that such a sequence exists? -- Michael Knudsen [knudsen@imf.au.dk] ==== > >> If we have composition series 0 <= A <= B <= M exactly >> one quotient is S_2. If A is isomorphic to S_1 then B is a >> direct sum of S_1 and a simple. Why is that true? 'cos Ext(S_1,S_1) = 0 and Ext(S_2, S_1) = 0. No matter what B/A is, then Ext(B/A,A) = 0. > splits? Is that always true? > >> So B is S_1 (+) S_1 or >> S_1 (+) S_2. In the former case ext(S_2,B)=0 so >> M is S_1 (+) S_1 (+) S_2. To conclude M = S_1(+)S_1(+)S_2 from Ext(S_2,B)=0 you need an exact > sequence 0 --> B --> M --> S_2 --> 0, right? Do you know that such > a sequence exists? Yes. (By J-H if B = S_1(+)S_1, M/B is S_2). -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) ==== hi im a high school student and i was browsing some good ol' math sites geared for prepping for the Putnam competition and there was this one question that said: evaluate the following integral: INT[1/(a^2+(x+1/x)^2),x,0,positive infinity]....(the integral is taken over the positve reals) for a>= 2 I finally was able to evaluate the integral using non-complex analysis methods but out of plain mathematical curiosity i started to wonder if this integral could be evaluated by summing complex residues (residue theorem). Im pretty knew to complex analysis since I have been teaching myself it (pretty cool field of math) so i would be very much obliged if someone could help me out by suggesting how to get started. what kind of contour should i choose of contour integration? etc thanks~ sam r. ==== > hi im a high school student and i was browsing some good ol' math > sites geared for prepping for the Putnam competition and there was > this one question that said: > evaluate the following integral: > INT[1/(a^2+(x+1/x)^2),x,0,positive infinity]....(the integral is taken > over the positve reals) for a>= 2 I finally was able to evaluate the integral using non-complex analysis > methods but out of plain mathematical curiosity i started to wonder if > this integral could be evaluated by summing complex residues (residue > theorem). Im pretty knew to complex analysis since I have been > teaching myself it (pretty cool field of math) so i would be very much > obliged if someone could help me out by suggesting how to get started. > what kind of contour should i choose of contour integration? etc You have INT[1/(a^2 + (x + 1/x)^2),x,0,+oo] = (1/2).INT[1/(a^2 + (x + 1/x)^2),x,-oo,+oo] = (1/2).INT[x^2/(a^2.x^2 + (x^2 + 1)^2),x,-oo,+oo] = (1/2).INT[x^2/(x^4 + (a^2 + 2)x + 1),x,-oo,+oo] Now, choose a big positive real number R and consider the path that goes from -R to R along a straight line and then goes along a semicircle, through the complex numbers with positive imaginary part, from R into -R. I hope that this helps, Jose Carlos Santos <3FD6F7A6.F52A40E@ix.urz.uni-heidelberg.de> <1Ff9oJOLyx1$Ew6T@baesystems.com> ==== In message <3FD6F7A6.F52A40E@ix.urz.uni-heidelberg.de>, Bjoern [snip] >> sqrt(2) is the >> result of turing machine(x), where x is some integer, probably >> under a million with any crude mapping technique. That's incomprehensible. Could you rephrase this, please? He means that any Turing machine can be represented by a tape fed to a > universal Turing machine, and the contents of that tape can be > represented by some number x. How can the content of the whole tape be represented with a single number x? And where does he get the probably under a million from? [snip] >> Non computable numbers is not a proof that irrationals exist, IMO, >> that no halting function exists does not clearly define what the halting >> number is, it states it is impossible. There is no gap on the number >> line from non computable numbers. That's incomprehensible again. I think he's trying to argue that because there exist TMs for which we > can't determine to which of the sets halts or does not halt they > belong, the does not halt set must be empty. *scratches head* Does anyone understand this logic? Bye, Bjoern ==== >> >> In message <3FD6F7A6.F52A40E@ix.urz.uni-heidelberg.de>, Bjoern [snip] > sqrt(2) is the > result of turing machine(x), where x is some integer, probably > under a million with any crude mapping technique. >>That's incomprehensible. Could you rephrase this, please? >> >> He means that any Turing machine can be represented by a tape fed to a >> universal Turing machine, and the contents of that tape can be >> represented by some number x. How can the content of the whole tape be represented with a single > number x? Because all but finitely many characters on the tape will be blank. That means that there are only countably many possible configurations of the input tape. So we can associate each one with a unique integer x. That there are only finitely many non-blank characters on the input tape is normally just a conventional assumption. So this argument is a bit weak for the general case. In this case, the input tape encodes a Turing machine, and all Turing machines are finitely describeable. Accordingly the set of Turing machines is countable and it follows that the set of canonical encodings on tape is countable. Therefore each canonical encoding can be associated with a unique integer x. John Briggs ==== > >> In message <3FD6F7A6.F52A40E@ix.urz.uni-heidelberg.de>, Bjoern [snip] > sqrt(2) is the > result of turing machine(x), where x is some integer, probably > under a million with any crude mapping technique. >>That's incomprehensible. Could you rephrase this, please? >> He means that any Turing machine can be represented by a tape fed to a >> universal Turing machine, and the contents of that tape can be >> represented by some number x. How can the content of the whole tape be represented with a single > number x? Because all but finitely many characters on the tape will be blank. > That means that there are only countably many possible configurations > of the input tape. So we can associate each one with a unique integer > x. That there are only finitely many non-blank characters on the input > tape is normally just a conventional assumption. So this argument > is a bit weak for the general case. In this case, the input tape encodes a Turing machine, and all > Turing machines are finitely describeable. Accordingly the set of > Turing machines is countable and it follows that the set of canonical > encodings on tape is countable. Therefore each canonical encoding can > be associated with a unique integer x. Bye, Bjoern ==== -- www.StealthHostiing.com You rule Truman. http://tinyurl.com/iky4 Hey Trueman...love the show. YOU ARE the Truman I heard him. Very spooky! >Is the truman living in Townsville? I've been hearing stuff, yeah. Webmasters help the TRUEman by joining www.theBanner.net Current:1 Goal:1000 ---------------------------------------------------------------------------- ------ >> In message <3FD6F7A6.F52A40E@ix.urz.uni-heidelberg.de>, Bjoern > [snip] > > sqrt(2) is the > result of turing machine(x), where x is some integer, probably > under a million with any crude mapping technique. >>That's incomprehensible. Could you rephrase this, please? >> He means that any Turing machine can be represented by a tape fed to a >> universal Turing machine, and the contents of that tape can be >> represented by some number x. > How can the content of the whole tape be represented with a single > number x? Because all but finitely many characters on the tape will be blank. > That means that there are only countably many possible configurations > of the input tape. So we can associate each one with a unique integer > x. That there are only finitely many non-blank characters on the input > tape is normally just a conventional assumption. So this argument > is a bit weak for the general case. In this case, the input tape encodes a Turing machine, and all > Turing machines are finitely describeable. Accordingly the set of > Turing machines is countable and it follows that the set of canonical > encodings on tape is countable. Therefore each canonical encoding can > be associated with a unique integer x. Any encoding technique that represents all TMs will suffice. Here you can see a TM in action counting in binary, only 3 states : 1 0 1 r 2 1 1 1 l 3 . 2 0 0 l 1 2 1 0 r 2 . 3 0 1 l 1 3 1 1 l 99 . To represent this as an integer is not difficult, each transition link from state to state has a limited number of options which can be encoded. The integer does not have to be parsed directly on a standard TM (UTM), as long as there is *some* defined mapping from integers to the associated TM's output number. There are 2 * 2 * 4 possible links on a 3 state TM, that makes 6^16 possible TMs, just count through them, although for theory a UTM applied to a tape is elementary. Herc ==== What's the obsession with computers? A number being irrational has a perfectly clear definition and had one long before anyone had the idea that machines could do any amount of logic. ~ Chris ==== > What's the obsession with computers? A number being irrational has a > perfectly clear definition and had one long before anyone had the idea > that machines could do any amount of logic. > That definition is that they cannot be represented as integer over integer. The definition is *not* uncountable. Look at a typical irrational number, it CAN be counted to. Here you can see complicated numbers being output from simple Turing machines. I have calculated the index *number* of these machines, I would hazard a guess that sqrt(2) would be encapsulated on a small TM, whose index number could be found. Aren't all the equations you write algorithmic? It seems odd Information Theory is wiped out of existence from a single 'proof' that is self referential, can anyone give me an actual number that isn't computable? Herc ==== Why pick sqrt(2)? By your logic, 1/3 is irrational because no Turing machine can generate its decimal representation in finite time. And don't say decimal representation is the problem. There is nothing intrinsically natural about base 10, or even about base n (positional notation). Lots of other representations of numbers exist. If I am free to pick the representation of the number, I can generate sqrt(2) in finite time. I will use as my number representation one which uses the sequence of numbers in the polynomial which produces the number as its root. So 1/3 is the root of 3x - 1 =0, its representation is (3,1). You can easily change this representation to be a single number by mapping any arbitrary finite sequence of natural numbers to a single natural number (you might already know how to do this using prime multipliers). Similarly, sqrt(2) is the root of x^2 - 2 = 0; its representation is (1,0-2). I can get a Turing machine to generate the single number corresponding to this sequence. So in this representation, 1/3, sqrt(2), and all other roots of polynomials are computable by Turing machines. And this number representation is just as valid as base 10 positional notation. With a little more work, I can make pi and a whole lot of other irrational numbers (but not all) computable in finite time by a Turing machine, by picking an appropriate representation for the number and asking the TM to generate it. 0.3333... may be imposible to write out in full, but 1/3 is computable. 1.414.... may be impossible to write out in full, but sqrt(2) is computable. You are talking about limitations of arbitrary representations of numbers, not the numbers themselves. Peter Webb What is an irrational number? Can you count to it? > Can you pin point it? There is no such thing. All numbers are the result of computable functions, sqrt(2) is the > result of turing machine(x), where x is some integer, probably > under a million with any crude mapping technique. That only leaves two types of numbers left that qualify for irrational, > non computable and random numbers. Non computable numbers is not a proof that irrationals exist, IMO, > that no halting function exists does not clearly define what the halting > number is, it states it is impossible. There is no gap on the number line > from non computable numbers. That leaves all numbers on the number line that cannot be encapsulated within > a computing indexing system are just..... random. Irrational numbers are just random sequences, otherwise a computer will map to them. And what exactly is an infinite sequence of random numbers? nothing, they > don't exist except for rationalisations about their finite limits. In all eventuality > with a small probability, on any representation system, each digit will flip to > a different digit, Shannon's noise demon at work. As an infinite sequence any > random number is no longer defined. Cantor's proof is cyclic, given a finite space to represent all possibilities of infinite sequences > there are more sequences than combinations, but they are only random anyway. Herc > -- > www.StealthHostiing.com You rule Truman. http://tinyurl.com/iky4 > Hey Trueman...love the show. YOU ARE the Truman I heard him. Very spooky! > >Is the truman living in Townsville? I've been hearing stuff, yeah. > Webmasters help the TRUEman by joining www.theBanner.net Current:1 Goal:1000 > -------------------------------------------------------------------------- -------- > ==== > Why pick sqrt(2)? By your logic, 1/3 is irrational because no Turing machine > can generate its decimal representation in finite time. And don't say decimal representation is the problem. There is nothing > intrinsically natural about base 10, or even about base n (positional > notation). Lots of other representations of numbers exist. If I am free to > pick the representation of the number, I can generate sqrt(2) in finite > time. I will use as my number representation one which uses the sequence of > numbers in the polynomial which produces the number as its root. So 1/3 is > the root of 3x - 1 =0, its representation is (3,1). You can easily change > this representation to be a single number by mapping any arbitrary finite > sequence of natural numbers to a single natural number (you might already > know how to do this using prime multipliers). Similarly, sqrt(2) is the root > of x^2 - 2 = 0; its representation is (1,0-2). I can get a Turing machine to > generate the single number corresponding to this sequence. So in this > representation, 1/3, sqrt(2), and all other roots of polynomials are > computable by Turing machines. And this number representation is just as > valid as base 10 positional notation. With a little more work, I can make pi > and a whole lot of other irrational numbers (but not all) computable in > finite time by a Turing machine, by picking an appropriate representation > for the number and asking the TM to generate it. 0.3333... may be imposible to write out in full, but 1/3 is computable. > 1.414.... may be impossible to write out in full, but sqrt(2) is computable. > You are talking about limitations of arbitrary representations of numbers, > not the numbers themselves. Peter Webb This is the basis of the universal count to all numbers. 1/3 and sqrt(2) can appear on the same list of counted numbers, so what types of numbers are we missing? It doesn't matter if 0.3333. goes on indefinately or 1.414... When we count rationals we just index the *formula* to *make* that number. We encode the numerator and denominator, not 0.3333333.. Counting over the UTM parameter is no different to counting the numerator / denominator grid. Is it impossible to count all square roots, because most of them take up infinite space? No, sqrt(1), sqrt(2), sqrt(3), sqrt(4), sqrt(5)... will suffice. The square roots of integers are countable, only this list will *not* count out all rationals aswell. But UTM(Z) will count out *both* lists and more. There are no numbers themselves they are abstract concepts, we can only ever deal with representations. Herc > What is an irrational number? Can you count to it? > Can you pin point it? There is no such thing. All numbers are the result of computable functions, sqrt(2) is the > result of turing machine(x), where x is some integer, probably > under a million with any crude mapping technique. That only leaves two types of numbers left that qualify for irrational, > non computable and random numbers. Non computable numbers is not a proof that irrationals exist, IMO, > that no halting function exists does not clearly define what the halting > number is, it states it is impossible. There is no gap on the number line > from non computable numbers. That leaves all numbers on the number line that cannot be encapsulated > within > a computing indexing system are just..... random. Irrational numbers are just random sequences, otherwise a computer will > map to them. And what exactly is an infinite sequence of random numbers? nothing, they > don't exist except for rationalisations about their finite limits. In all > eventuality > with a small probability, on any representation system, each digit will > flip to > a different digit, Shannon's noise demon at work. As an infinite sequence > any > random number is no longer defined. Cantor's proof is cyclic, given a finite space to represent all > possibilities of infinite sequences > there are more sequences than combinations, but they are only random > anyway. Herc ==== > >> Why pick sqrt(2)? By your logic, 1/3 is irrational because no Turing machine >> can generate its decimal representation in finite time. >> And don't say decimal representation is the problem. There is nothing >> intrinsically natural about base 10, or even about base n (positional >> notation). Lots of other representations of numbers exist. If I am free to >> pick the representation of the number, I can generate sqrt(2) in finite >> time. I will use as my number representation one which uses the sequence of >> numbers in the polynomial which produces the number as its root. So 1/3 is >> the root of 3x - 1 =0, its representation is (3,1). You can easily change >> this representation to be a single number by mapping any arbitrary finite >> sequence of natural numbers to a single natural number (you might already >> know how to do this using prime multipliers). Similarly, sqrt(2) is the root >> of x^2 - 2 = 0; its representation is (1,0-2). I can get a Turing machine to >> generate the single number corresponding to this sequence. So in this >> representation, 1/3, sqrt(2), and all other roots of polynomials are >> computable by Turing machines. And this number representation is just as >> valid as base 10 positional notation. With a little more work, I can make pi >> and a whole lot of other irrational numbers (but not all) computable in >> finite time by a Turing machine, by picking an appropriate representation >> for the number and asking the TM to generate it. >> 0.3333... may be imposible to write out in full, but 1/3 is computable. >> 1.414.... may be impossible to write out in full, but sqrt(2) is computable. >> You are talking about limitations of arbitrary representations of numbers, >> not the numbers themselves. >> Peter Webb This is the basis of the universal count to all numbers. 1/3 and sqrt(2) can appear > on the same list of counted numbers, so what types of numbers are we missing? It doesn't matter if 0.3333. goes on indefinately or 1.414... When we count > rationals we just index the *formula* to *make* that number. We encode the > numerator and denominator, not 0.3333333.. Counting over the UTM parameter is no different to counting the numerator / denominator grid. It is different in a very important way. Not all UTM parameters encode numbers. Some Turing machines both fail to halt and fail to produce the i'th digit of their output. And it is provable that there is no computable procedure to tell with certainty which do and which do not. That means that if you try to make a list of numbers in UTM parameter order, that list will not be computable. By contrast, a list of all numerator/denominator pairs _is_ computable. Even if you decide to eliminate redundant pairs. John Briggs ==== > Why pick sqrt(2)? By your logic, 1/3 is irrational because no Turing machine >> can generate its decimal representation in finite time. >> And don't say decimal representation is the problem. There is nothing >> intrinsically natural about base 10, or even about base n (positional >> notation). Lots of other representations of numbers exist. If I am free to >> pick the representation of the number, I can generate sqrt(2) in finite >> time. I will use as my number representation one which uses the sequence of >> numbers in the polynomial which produces the number as its root. So 1/3 is >> the root of 3x - 1 =0, its representation is (3,1). You can easily change >> this representation to be a single number by mapping any arbitrary finite >> sequence of natural numbers to a single natural number (you might already >> know how to do this using prime multipliers). Similarly, sqrt(2) is the root >> of x^2 - 2 = 0; its representation is (1,0-2). I can get a Turing machine to >> generate the single number corresponding to this sequence. So in this >> representation, 1/3, sqrt(2), and all other roots of polynomials are >> computable by Turing machines. And this number representation is just as >> valid as base 10 positional notation. With a little more work, I can make pi >> and a whole lot of other irrational numbers (but not all) computable in >> finite time by a Turing machine, by picking an appropriate representation >> for the number and asking the TM to generate it. >> 0.3333... may be imposible to write out in full, but 1/3 is computable. >> 1.414.... may be impossible to write out in full, but sqrt(2) is computable. >> You are talking about limitations of arbitrary representations of numbers, >> not the numbers themselves. >> Peter Webb This is the basis of the universal count to all numbers. 1/3 and sqrt(2) can appear > on the same list of counted numbers, so what types of numbers are we missing? It doesn't matter if 0.3333. goes on indefinately or 1.414... When we count > rationals we just index the *formula* to *make* that number. We encode the > numerator and denominator, not 0.3333333.. Counting over the UTM parameter is no different to counting the numerator / denominator grid. It is different in a very important way. Not all UTM parameters encode > numbers. Some Turing machines both fail to halt and fail to produce the > i'th digit of their output. And it is provable that there is no > computable procedure to tell with certainty which do and which do not. That means that if you try to make a list of numbers in UTM parameter > order, that list will not be computable. By contrast, a list of all numerator/denominator pairs _is_ computable. > Even if you decide to eliminate redundant pairs. > True, there's no bijection, but as long as computable numbers cover reals there are *not* infinitely more reals than computables. Herc ==== > All numbers are the result of computable functions, sqrt(2) is the > result of turing machine(x), where x is some integer, probably > under a million with any crude mapping technique. That doesn't mean that sqrt(2) is not irrational. Some Turing machines can compute irrational numbers. > That only leaves two types of numbers left that qualify for irrational, > non computable and random numbers. But there are only countably many Turing machines, so there are only countably many computable numbers. But why does that mean there are only countably many numbers? Who says that all numbers are the result of computable functions? Indeed, easy proof that they are not: there are only countably many computable numbers, but uncountably many numbers. So some are not computable. Thomas ==== ---------------------------------------------------------------------------- ------ All numbers are the result of computable functions, sqrt(2) is the > result of turing machine(x), where x is some integer, probably > under a million with any crude mapping technique. That doesn't mean that sqrt(2) is not irrational. Some Turing > machines can compute irrational numbers. Right! rational is Z/Z That only leaves two types of numbers left that qualify for irrational, > non computable and random numbers. But there are only countably many Turing machines, so there are only > countably many computable numbers. But why does that mean there are > only countably many numbers? Who says that all numbers are the result > of computable functions? Process of elimination : if its not computable it is : random or non computable function Neither of which are well defined numbers. Indeed, easy proof that they are not: there are only countably many > computable numbers, but uncountably many numbers. So some are not > computable. > The theory is to refute there are uncountable numbers, so the deduction is not substantiated here. Herc ==== > But there are only countably many Turing machines, so there are only > countably many computable numbers. But why does that mean there are > only countably many numbers? Who says that all numbers are the result > of computable functions? Process of elimination : if its not computable it is : random or non computable function Neither of which are well defined numbers. I don't know what a random number is. But the existence of non computable numbers is perfectly well defined. Unless you want to define well defined as computable, and then we're back where we began. > Indeed, easy proof that they are not: there are only countably many > computable numbers, but uncountably many numbers. So some are not > computable. The theory is to refute there are uncountable numbers, so the deduction is > not substantiated here. But then what you have done is not to prove that there are uncountably many numbers. If you are correct, then you have uncovered a contradiction in ZFC, and I would urge you to put it in straightforward FOL form so that it is incontestable. Thomas ==== > But there are only countably many Turing machines, so there are only > countably many computable numbers. But why does that mean there are > only countably many numbers? Who says that all numbers are the result > of computable functions? Process of elimination : if its not computable it is : random or non computable function Neither of which are well defined numbers. I don't know what a random number is. But the existence of non > computable numbers is perfectly well defined. Computers can't output random sequences of digits. What you consider the uncountable real numbers I call random superfluous useless non repeatable digit sequences that will succomb to Shannons noise theory and corrupt. Unless you want to define well defined as computable, and then > we're back where we began. not at all, if you can identify the position on the number line I will accept that as a number, if you can also show that no computable function shares that position then you have established a proof that computable numbers C reals. (proper subset here) > Indeed, easy proof that they are not: there are only countably many > computable numbers, but uncountably many numbers. So some are not > computable. The theory is to refute there are uncountable numbers, so the deduction is > not substantiated here. But then what you have done is not to prove that there are uncountably > many numbers. If you are correct, then you have uncovered a > contradiction in ZFC, and I would urge you to put it in > straightforward FOL form so that it is incontestable. > perhaps, there are other avenues for developing applications in the search space of functions. Herc ==== > I don't know what a random number is. But the existence of non > computable numbers is perfectly well defined. Computers can't output random sequences of digits. What you > consider the uncountable real numbers I call random superfluous > useless non repeatable digit sequences that will succomb to Shannons > noise theory and corrupt. I have no idea (still) what you mean by random. It seems to me though that you are saying useless because not computable. Well, maybe useless to you, but very useful to others. I'm sure that some people find quaternions useless. That doesn't make them useless to others. > not at all, if you can identify the position on the number line I > will accept that as a number, if you can also show that no > computable function shares that position then you have established a > proof that computable numbers C reals. (proper subset here) Do you agree that the set of positions on the number line has the least upper bound property? Thomas ==== ... > I don't know what a random number is. But the existence of non > computable numbers is perfectly well defined. May I introduce a new question. Suppose we have a computable system where we can define some computable irrational numbers. Call the set of numbers that can be defined with this computable system, the set C. Question: - Is there a computable system as described above, for which all the axioms concering numbers, as layed down by the Ancient Greeks (I have to look this up), are valid for the set C? If the answer is NO, then Cantor's system is rather unavoidable, but if the answer is YES... Lucas ==== To extend, basically I'm reopening a 75 post thread from several months ago with Daryl McCollough (I recognise Jim Burns and BruceS my skeptic counterparts too). Richard was correct : He means that any Turing machine can be represented by a tape fed to a universal Turing machine, and the contents of that tape can be represented by some number x. Namely x is countable (an integer) Straight after I posted I realised a glaring flaw! Apply Cantors proof directly to the TM ordering, instead of his usual random list. DIGIT 1 2 3 4 5 6 __________________________ UTM(1) 4 3 6 4 2 4 UTM(2) 7 4 3 4 3 2 UTM(3) 0 1 0 1 1 1 UTM(4) 1 2 2 2 2 2 UTM(5) 7 7 7 7 7 7 I am suggesting here all computable numbers covers all possible and all types of numbers, but then we can take the diagonal : 4 4 0 2 7 and deduce a new 'irrational' not on the list 5 5 1 3 8 I don't think this is a contradiction though more an error of reasoning on infinite lists. The diagonal proof is nothing more than you can always add one more, regardless of what its applied to, completeness and infinite lists have to be more carefully reasoned about. Essentially real numbers are divided into 2 types : Rational (integer over integer) and not Rational Not Rational can be divided into 2 types, computable (eg.sqrt(2)) and not computable. Computable can again be divided into 2 types : non computable functions and random sequences of digits. My conclusion is that 'not computable' are not numbers, they are just random sequences that for any length can be computed and tallied in the computable number list. Otherwise Non Computable Numbers are just theoretic assertions of non existence, .... Here is the beginning of a long thread about Non Computable Numbers, essentially 5 5 1 3 8 (the diagonal with a transformation applied) is non computable since it doesn't fit on the TM list. Uncle Al, sqrt(2) isn't random, check your calculator. > More seriously, though, the reason it can't be computable is > because if it were, then we could compute the sequence H(n) > by computing its digits. We know that it's impossible to > compute whether the n-th Turing machine halts on input n > for all n, so this number r isn't computable. > please go through my above post, the sequence H(n) isn't computable. that has a different meaning to its result isn't computable. According to your logic, a non computable function that has a range onto the integers implies that integers are not computable. Ok, the busy beaver function values so far are 1, 2, 6, 2012, options for the 5th value are up to 555654434 (all figures made up) bb(n) is a non computable sequence but the values are all integers, they have to be integers because they are all finite and they are all a count of the number of 1s the TM outputs before it terminates. So why can you say irrationals cannot be calculated by a TM because of non computable functions that map to irrationals, yet non computable functions also map to integers, we cannot compute these values either by your logic. I have a 3 state TM that outputs all integers in binary if you want to check. Herc http://tinyurl.com/yo9w ==== To extend, basically I'm reopening a 75 post thread from several months ago > with Daryl McCollough (I recognise Jim Burns and BruceS my skeptic > counterparts too). I'm surprised you remember me. I assumed at the time that you did not read my post. > Richard was correct : > He means that any Turing machine can be represented by a tape > fed to a universal Turing machine, and the contents of that > tape can be represented by some number x. Namely x is countable (an integer) Straight after I posted I realised a glaring flaw! > Apply Cantors proof directly to the TM ordering, instead of > his usual random list. Cantor does not require a random list. Cantor does not require anything of the list of reals other than it be a list of reals. Since your TM ordering, below, is a list of reals, Cantor's argument applies to it. The fact that it's (imagined to be) the output of each Turing Machine makes the list more interesting than a random list, but it doesn't make the argument invalid. This is the DIGIT 1 2 3 4 5 6 > __________________________ > UTM(1) 4 3 6 4 2 4 > UTM(2) 7 4 3 4 3 2 > UTM(3) 0 1 0 1 1 1 > UTM(4) 1 2 2 2 2 2 > UTM(5) 7 7 7 7 7 7 I am suggesting here all computable numbers covers all possible > and all types of numbers, but then we can take the diagonal : 4 4 0 2 7 and deduce a new 'irrational' not on the list 5 5 1 3 8 I don't think this is a contradiction though more an error of > reasoning on infinite lists. The diagonal proof is nothing more than you can always add one > more, regardless of what its applied to, completeness and > infinite lists have to be more carefully reasoned about. You make a good point that one must be careful about completeness, and infinite things. We are probably both thinking of the various Tales of the Infinite Hotel, all rooms full, yet still room for an infinite number of additional guests. However, I think that this argument has been carefully made. The argument isn't really you can always add one more. It's more like All the guests cannot fit in the hotel. I'm going to make some parts of the argument more explicit. I claim there are no complete lists of the real numbers. Let's look at _all_ the lists of real numbers and see that _none_ of them are complete. Let Lists = the set of all lists of real numbers (that is, all one-to-one functions from the natural numbers to the reals). Define the function Cantor: Lists -> Reals as in Cantor's proof. Note that for all L in Lists, Cantor(L) is a real number that is not listed in L (is not in the image of the function L). Remember that this is not defined to be true. Cantor first defined his function and then showed it had this property. We have shown that there are no complete lists of reals numbers. We have _not_ shown (so far) that there are more reals than integers. That requires one last step, where I think Cantor's brilliance really shines: define what we shall mean by sets (finite or infinite) being the same or different sizes. (Here I depend on vague recollection. I may have details wrong, but I think my formulation should be functionally equivalent.) -- Two sets A and B are the same size, |A| = |B|, if and only if there exists a bijection between the two sets. -- If B is a subset of A, then A is at least the same size as B, that is, |B| =< |A|. (There is more that needs to be shown. For example, for all sets A and B, either |A| =< |B| or |B| =< |A|, and that |A| =< |B| and |A| =< |B| together imply |A| = |B|. I think those two points are all we have to assume, though.) We have shown _there does not exist_ a bijection between the natural numbers and the reals. Remember we have looked at _all_ the potential bijections and _all_ of them failed at at least one point. As we understand the size of infinite sets, that means N and R are different sizes. It's easy to show |N| =< |R|, since N is a subset of R. Thus |N| < |R|. > Essentially real numbers are divided into 2 types : > Rational (integer over integer) and not Rational Not Rational can be divided into 2 types, computable (eg.sqrt(2)) > and not computable. Computable can again be divided into 2 types : non computable > functions and random sequences of digits. Random is, I think, not well-defined here, unless you just mean not-computable, along the same lines as irrational. > My conclusion is that 'not computable' are not numbers, they are > just random sequences that for any length can be computed and > tallied in the computable number list. Otherwise Non Computable > Numbers are just theoretic assertions of non existence, .... Your conclusion just seems to be the assertion that non-computable numbers are not real numbers. I think what you have done is re-define the reals as the set of all computable numbers. There is nothing wrong with defining a set of all computable numbers, and there is nothing logically wrong with calling it anything you want, the real numbers or Philip, if you want. But what the rest of us call the real numbers are defined completely separately from notions of computability. For example, I might not know whether the least upper bound of some complicated set is computable, but I do know that, if the set is bounded, the LUB exists. Under your scheme, I might not know whether the LUB exists, but I would know that, if it existed, it would be computable. You might prefer to know the second and not the first, but generally speaking, knowing the first and not the second is more useful in real analysis. Jim Burns ==== For example, I might not know whether the least upper bound > of some complicated set is computable, but I do know that, > if the set is bounded, the LUB exists. Under your scheme, > I might not know whether the LUB exists, but I would know > that, if it existed, it would be computable. You might > prefer to know the second and not the first, but generally > speaking, knowing the first and not the second is more > useful in real analysis. > Its not more useful, its a superimposed mutually sustaining impression that mathematics lies outside the realMs of computability. Your proof jumps to the fact of a contradiction, what is this contradiction? Is it self referencing? YES Does the formulation of extra numbers occur purely depending on the original formulation of all numbers? YES Does this contradiction rely on specifying the new number is *different* from the original forumlation at finite set points? YES Does the extra number reinsert into the original list to demontrate the imcompatibility? YES The extra number is simply this : Given a number on a list, change it and put it back in the list without changing the list. Does the diagonal transform form a well defined number? NO -> computables = reals YES -> computables C reals There is only 1 infinity type, the answer is NO, here's a similar exposition of diagonalisation. > Physicists hold the stage on the accepted view > because they are corrupt. hundred names attached to this discovery. We > don't need new science or maths its all there, > we already know everything. All that is > incomplete is the dispertion of facts. The > world expert on such topic KNOWS certain facts > but another expert on a different topic will > never union the information. Accepting peer > review is too trivial as a confirmation. Our society > is established enough now for public review. > our knowledge base? Only a few percent have > do we have to find to assume matter is just forces. What don't we know about the universe, everything > about the atom, our composition, our life, large > scale matter, forces. Hailing a unifying theory > to be when we already have all the mathematics for > the respective domains. What physical problem > can't we calculate? Physics is 50 years old, > the only way they retain their authoritative view > is by dispersing to the public a flawed assertion > on limitations of information systems. Within > a couple years your OS will download an upgrade > to Outlook with a third window under this text > body, being an Artificial Intelligence comment on > the post. That will be one way to stop the incessant > error of the majority of educated people perpetuating > the repurcussions of Godels Flawed Proof. Here is proof that it is the most elaborate dead end > in science. Most versions of the proof start with the assertion > that finding a contradiction in the negation of a > statement is a proof of the statement. Then the G > statement is formulated, and a mathematics is given > to formulate the self referencing statement. The G > statement is 'this theorom does not belong to the > universal model' or 'this statement has no proof'. > NOW assume 'this has no proof' is false, then it is > proven, so it must be true, uh oh a contradiction, that > means its true. Somehow it MEANS its true but everyone > accepts that as completely different to proving truth > and our minds are superior to logic. The statement > is taken as true however, and a systematic prover by > virtue of its expressive power is incapable of proving > it. STUPID. Set theory, natural language, number > theory all allow the expression of paradoxical statements. > But our thoerom prover is dissalowed this privalege, > the proof by contradiction rule is all encompassing > and theorom provers must give absolute truth or false > values to all theoroms. This is rubbish, the self > referencing statement 'this statement is false' is not > false. Dont run around yelling its a true statement. This is a second version of Godels limit popular with > physisits : > Enter Godel The man who showed once and for all that Russell's aim was impossible was, > Undecidable Propositions of Principia Mathematica and Related Systems. In > it he showed that a statement in a system could be made to refer to itself > was very complicated involving the mapping of prime numbers onto statements. > For example, Godelese for (x)(x=x) is the unique prime number code 28 X 311 > X 58 X 78 X 1111 X 135 X 1711 X 199. A Godelian proof Here is a simpler proof that no number system can generate all the > statements which might be true within it. This proof is based on the > writings of A. W. Moore and Roger Penrose. #1. POINT TO PROVE: IT IS IMPOSSIBLE TO DERIVE ALL MATHEMATICAL TRUTH FROM > ANY SET OF SELF-EVIDENT AXIOMS. #2. IF ALL MATHEMATICAL TRUTHS CAN BE DERIVED FROM A CHOSEN SET OF AXIOMS, > THEN, IN PRINCIPLE, AN ALGORITHM A CAN BE CREATED TO TEST WHETHER OR NOT > ANY GIVEN THEOREM DERIVES FROM THE CHOSEN AXIOMS.84I.E.: WHETHER OR NOT IT IS > TRUE OR FALSE. #3. AT PRESENT WE DO NOT HAVE SUCH AN ALGORITHM. IF A CAN BE SHOWN TO BE > IMPOSSIBLE, THEN #1 IS ESTABLISHED. #4. LIST THE FACTUAL STATEMENTS WHICH CAN BE MADE ABOUT NUMBERS. EXAMPLES OF > SUCH STATEMENTS ARE X IS EVEN, X IS ODD, X IS PRIME,X IS LESS THAN > 100, ETC. #5. CREATE A TABLE OF SUCH STATEMENTS, BEGINNING WITH THE SIMPLEST AND > MOVING TO THE MORE COMPLEX. WE WILL CALL OUR STATEMENTS 1, 2, 3, 4... NOW WE > NOTE THAT OUR TABLE CAN REFER TO ITS OWN STATEMENTS. SUPPOSE STATEMENT 0 > MEANS: X IS EVEN, STATEMENT 1 X IS ODD ETC... WE LET THE VERTICAL AXIS > REPRESENTS THE STATEMENT NUMBER. THE HORIZONTAL AXIS REPRESENTS ALL NUMBERS > FROM 0 TO INFINITY. WE THEN ASK OURSELVES FOR EACH NUMBER IN THE HORIZONTAL > AXIS, IS THE VERTICAL STATEMENT TRUE OF THIS NUMBER? WE WRITE Y BELOW IT > IF IT IS TRUE, AND N IF IT ISN'T: 0 1 2 3 .... 0 (EVEN) N N Y N... 1 (ODD) N N Y Y 2 (PRIME) N N Y Y... 3 (x<100 ) Y Y Y Y.... ... .................. #6. FOR ANY NATURAL NUMBER (HORIZONTAL LINE) WE NOW HAVE A METHOD OF > DECIDING IF THE VERTICAL STATEMENT IS TRUE. SINCE EVERY POSSIBLE STATEMENT > OF THE SYSTEM CAN APPARENTLY BE LISTED AND SINCE EVERY NATURAL NUMBER CAN > ALSO BE LISTED, IT APPEARS WE HAVE A COMPLETE SYSTEM OF NATURAL NUMBERS AND > AXIOMS. NOTICE THAT EACH STATEMENT ON THE VERTICAL AXIS PRODUCES ITS OWN > UNIQUE HORIZONTAL LINE OF Ys AND Ns. #7. CREATE A NEW WELL-DEFINED SEQUENCE OF Ys AND Ns BY FOLLOWING A DIAGONAL > ON THE CHART WE HAVE JUST CREATED. DO THIS BY TURNING EACH DIAGONAL ELEMENT > INTO ITS OPPOSITE. THE N AT 0/0 ON THE TABLE BECOMES A Y. THE Y AT 1/1 > BECOMES AN N. THE Y AT 2/2 BECOMES AN N. THE Y AT 3/3 BECOMES AN N AND SO > FORTH. WE GET YNNN... DOES ANY STATEMENT WHICH HAS ALREADY BEEN GIVEN > PRODUCE THIS NEW SEQUENCE? #8. STATEMENT 0 DOESN'T BECAUSE IT HAS AN N WHERE THE NEW STATEMENT HAS Y. > 1, 2, AND 3 DON'T BECAUSE THEY HAVE Ys WHERE THE NEW STATEMENT HAS Ns. THIS > WOULD HOLD TRUE TO INFINITY IF WE COULD MAKE OUR TABLE THAT LONG, #9. WE KNOW WE LEGITIMATELY CREATED THIS NEW Y & N PATTERN, IE: IT IS TRUE. > YET NONE OF THE EXISTING AXIOM STATEMENTS PRODUCE THIS DIAGONAL STATEMENT. A > NEW AXIOM IS NEEDED TO EXPRESS THE DIAGONAL. 10. IF WE WRITE A NEW STATEMENT (CALL IT R) THAT INCLUDES A PROCEDURE FOR > MAKING THIS DIAGONAL , AT SPACE R/R A NEW DIAGONAL LETTER WILL APPEAR AND WE > WILL HAVE TO ADD STATEMENT S TO REPRESENT THIS NEW SEQUENCE. BUT AT S/S A > NEW DIAGONAL NUMBER WILL APPEAR, REQUIRING A STATEMENT T AND SO ON, > INFINITELY. 11. THEREFORE ALGORITHM A IS IMPOSSIBLE, WHICH IS THE PROOF REQUIRED BY #2. > IT IS IMPOSSIBLE TO AUTOMATICALLY DERIVE ALL POSSIBLE MATHEMATICAL TRUTH. Step 9 is erronous : #9. WE KNOW WE LEGITIMATELY CREATED THIS NEW Y & N PATTERN, > IE: IT IS TRUE. YET NONE OF THE EXISTING AXIOM STATEMENTS > PRODUCE THIS DIAGONAL STATEMENT. A NEW AXIOM IS NEEDED TO > EXPRESS THE DIAGONAL. The pattern is not legitimately created, it is obviously > self referencing and a has a paradoxical bit when it evaluates > its own number. Just because there's two steps in seeing the > plausibility in a theorom, one of the steps fails so the > theorem fails, not the whole encapsulation of theoroms. Herc > that's my 2 cents Wonderful.. I picked up on the self referential paradox right away but I find it to be revelatory. Mathematics begins from the very self reference of consciousness as unity self divided and thus self equal which is an assertion of a negation that has no actuality in fact. So all mathematics is self referentially paradoxical. In fact consciousness itself is so. Only by negation can any divisive discernment take place even if this negation is entirely synthetic. Only with the introduction of a No can the Yes be apprehended and in no other way. It is important that I point out that this by no means therefore negates mathematics or logic or reason at all, but grounds them in our fundamental nature. -- *á.ü_ü.á'¬¬) (_ü.á' Raan ==== ... > -- Two sets A and B are the same size, |A| = |B|, if and only if > there exists a bijection between the two sets. If size is a natural number, then you can't take the size of an infinite set (without additional axioms and definitions). At the moment you say that infinite sets are of the same size when there exists a bijection, then you already introduce some of the choices Cantor made. But those are choices, not proofs. > -- If B is a subset of A, then A is at least the same size > as B, that is, |B| =< |A|. > (There is more that needs to be shown. For example, for > all sets A and B, either |A| =< |B| or |B| =< |A|, and > that |A| =< |B| and |A| =< |B| together imply |A| = |B|. I think > those two points are all we have to assume, though.) We have shown _there does not exist_ a bijection between > the natural numbers and the reals. Remember we have looked > at _all_ the potential bijections and _all_ of them failed > at at least one point. This is indeed proven. > As we understand the size of infinite sets, that means N > and R are different sizes. It's easy to show |N| =< |R|, > since N is a subset of R. Thus |N| < |R|. This is Cantor's choice, not a proof. You may also conclude that the logical system you are currently using is not capable of definining all irrational numbers and leave it there. So, there are at least two choices you can make: a. You may refer to R even when there is no logical system that can list all its elements (this is Cantor's choice). b. You must be aware that R is never be complete in your logical system (alternative to Cantor). I don't think choice b is very attractive, but to my opinion it is a way you can try to go. However, I always questioned if saying that |N| < |R| has any more meaning than saying that irrational numbers are green. Lucas ==== > I am suggesting here all computable numbers covers all possible and all types > of numbers, Surely that is false. There are more real numbers than there are computations. ==== To extend, basically I'm reopening a 75 post thread from several months ago >with Daryl McCollough (I recognise Jim Burns and BruceS my skeptic >counterparts too). Sorry, I don't recall what it was about. -- Daryl McCullough ==== ---------------------------------------------------------------------------- ------ To extend, basically I'm reopening a 75 post thread from several months ago >with Daryl McCollough (I recognise Jim Burns and BruceS my skeptic >counterparts too). Sorry, I don't recall what it was about. I think the Cantor diagonalisation across UTM(Z) is more a concern anyway, but the thread is here : http://tinyurl.com/yo9w I said reals are countable by UTM(Z) You said that it misses non computable numbers, like r = Halt(1) + 1/10 Halt(2) + 1/100 Halt(3)... since Halt(n) is unknown for all n. i.e. r is not computable and will not be listed in UTM(Z). I tried to argue r is just not known, you tried to show r is not computable at all (standard view), now I am now suggesting r it not existent! so Computables = Reals. Herc ==== > In message <3FD6F7A6.F52A40E@ix.urz.uni-heidelberg.de>, Bjoern <...> sqrt(2) is the >> result of turing machine(x), where x is some integer, probably >> under a million with any crude mapping technique. That's incomprehensible. Could you rephrase this, please? He means that any Turing machine can be represented by a tape fed to a > universal Turing machine, and the contents of that tape can be > represented by some number x. Well, let's forget about the Turing machine for a second. Let's just note that sqrt(2) is computable in the sense that, armed with some finite algorithm and an indefinite amount of storage, we can compute an arbitrary number of digits of it. Ok ... that does sound a lot like a Turing machine after all. ;-) This stirs some vague recollection: there are some privileged irrational numbers which can be specified with a finite amount of information. Obviously _all_ irrational numbers cannot be so specified, or they in fact would be countable. So most irrational numbers are poor lost souls which not only have non-repeating decimal representations, but can't even be named in any meaningful way -- they are unknowable. (This was the subject of some (IBM?) news release within the past decade, possibly one of those over-hyped news releases which appear regularly, repeating essentially known results as if were fresh revelation). Can somebody remind me what this concept is called? Do these unknowable irrational numbers correspond to non-computable functions? >> That only leaves two types of numbers left that qualify for irrational, >> non computable and random numbers. Well, I would say that irrational numbers are not computable; why do >you think otherwise? Because he's confused about the halting problem. So am I, evidently -- would you refresh my memory of its significance here? And, repeating the argument above, surely _some_ irrational numbers are in fact computable? >> Non computable numbers is not a proof that irrationals exist, IMO, >> that no halting function exists does not clearly define what the halting >> number is, it states it is impossible. There is no gap on the number line >> from non computable numbers. That's incomprehensible again. I think he's trying to argue that because there exist TMs for which we > can't determine to which of the sets halts or does not halt they > belong, the does not halt set must be empty. Ah yes ... irrational numbers correspond to the output of a machines which neither halts nor loops. As to what he is trying to say, OTOH, I would feel comfortable leaving it in the incomprehensible bin. Ah, knowledge, ah, intellect, ah, thought, I miss you ... you jade. ==== In message <3FD6F7A6.F52A40E@ix.urz.uni-heidelberg.de>, Bjoern <...> sqrt(2) is the >> result of turing machine(x), where x is some integer, probably >> under a million with any crude mapping technique. >That's incomprehensible. Could you rephrase this, please? He means that any Turing machine can be represented by a tape fed to a > universal Turing machine, and the contents of that tape can be > represented by some number x. Well, let's forget about the Turing machine for a second. Let's just > note that sqrt(2) is computable in the sense that, armed with some > finite algorithm and an indefinite amount of storage, we can compute > an arbitrary number of digits of it. Ok ... that does sound a lot like a Turing machine after all. ;-) Well, the lecture on computer science I once heard was several years ago, so I don't know exactly what computable means... > This stirs some vague recollection: there are some privileged > irrational numbers which can be specified with a finite amount of > information. Obviously _all_ irrational numbers cannot be so > specified, or they in fact would be countable. So most irrational > numbers are poor lost souls which not only have non-repeating decimal > representations, but can't even be named in any meaningful way -- they > are unknowable. (This was the subject of some (IBM?) news release > within the past decade, possibly one of those over-hyped news releases > which appear regularly, repeating essentially known results as if were > fresh revelation). Can somebody remind me what this concept is called? Do these > unknowable irrational numbers correspond to non-computable functions? Does sound a bit like transcendental (sp?) numbers: irrational numbers which aren't solutions of polynomial equation with rational coefficients. [snip rest] Bye, Bjoern ==== In sci.math, |-|erc What is an irrational number? Can you count to it? > Can you pin point it? There is no such thing. Dedekind cuts or Cauchy sequences are AFAIK adequate for the strict mathematician; most others are content with 53-bit approximations. (53? That's the number of bits in a standard 8-byte floating point's mantissa, including the hidden '1'. There's an IEEE spec for this approximation but I'd have to dig for it.) These approximations are always multiples of a power of 2. It gets worse: the number of approximations is finite -- there are only 64 bits to play with, after all, in total. (1 sign, 11 bits for the excess-0x200 exponent, and 52 mantissa bits, the '1' being hidden. By convention, a value of 0x0000000000000000 is taken to be exactly 0.0 by most floating-point processors.) All numbers are the result of computable functions, sqrt(2) is the > result of turing machine(x), where x is some integer, probably > under a million with any crude mapping technique. An interesting viewpoint from an engineering standpoint; mathematicians, however, aren't constrained by Turing machines. :-) Most use a 64-bit integer with the IEEE mapping. (A 32-bit integer can also be mapped, with 1 sign bit, 8 bits for the exponent, 23 bits for the mantissa.) That only leaves two types of numbers left that qualify for irrational, > non computable and random numbers. There are no irrational numbers, really, in the standard double. Of course this hasn't stopped engineering professionals from using M_PI as though it really were pi; if one constructs a bridge with an arch of 100 m radius using M_PI, one gets an error of about 100 * 2^-53 = 1.11 * 10^-14 m. Since this is about the size of 10 atomic nuclei placed end-to-end this is more than adequate for the vast majority of applications. It's also worth noting that 2^53/100 = 90 trillion (we do want to be able to represent 1 cent :-) ), although the guardbanding approximations might give headaches over about 22 trillion. Since the GWP (gross world product, http://www.cia.gov/cia/publications/factbook/geos/xx.html) is about 49 trillion this usually isn't much of a concern. In a pinch one can count pennies -- literally -- using long longs instead. Non computable numbers is not a proof that irrationals exist, IMO, Define exist. *No* numbers exist. However, given the usual rules of mathematics, irrational numbers can be defined in a consistent fashion. > that no halting function exists does not clearly define what the halting > number is, it states it is impossible. Halting number? > There is no gap on the number line from non computable numbers. The standard double precision floating point approximation is little more than a scattering of 2^64 points among the ocean of the numberline. That leaves all numbers on the number line that cannot be > encapsulated within a computing indexing system are just..... random. That sentence didn't parse too well. Could you clarify? Irrational numbers are just random sequences, otherwise a > computer will map to them. Far from random. Cauchy sequences, in particular, will converge to a specific point. Which point, of course, may be a matter of debate, on occasion. And what exactly is an infinite sequence of random numbers? nothing, > they don't exist except for rationalisations about their finite > limits. In all eventuality with a small probability, on any > representation system, each digit will flip to a different digit, > Shannon's noise demon at work. As an infinite sequence any > random number is no longer defined. Cantor's proof is cyclic, given a finite space to represent all > possibilities of infinite sequences there are more sequences > than combinations, but they are only random anyway. Cantor's proof has apparently been thrown out, but replaced by a more rigorous demonstration of an uncountable infinity of real numbers within [0,1]. However, I'd have to look for the details. [.sigsnip] -- #191, ewill3@earthlink.net It's still legal to go .sigless. ==== What is an irrational number? Can you count to it? > Can you pin point it? There is no such thing. sqrt(2) http://mathforum.org/library/drmath/view/57117.html http://medialab.dyndns.org/bignum/ > All numbers are the result of computable functions, sqrt(2) is the > result of turing machine(x), where x is some integer, probably > under a million with any crude mapping technique. The concepts irrational and transcendental are rigorousy defined and rigorously provable. Do some reading. If you think sqrt(2) is not irrational, supply integers for the ratio. Note that sqrt(2) has been taken to ridiculous lengths - though not to the 1.241 trillion places of pi by Yasumasa Kanada - to do statistical correlations. ftp://metalab.unc.edu/pub/docs/books/gutenberg/etext94/2sqrt10a.zip sqrt(2) to 5 million decimal places http://antwrp.gsfc.nasa.gov//htmltest/gifcity/sqrt2.10mil sqrt(2) to 10 million decimal places http://www.sciencenews.org/20021214/mathtrek.asp sqrt(A): 1) Choose a rough approximation G of sqrt(A). 2) Divide A by G and then average the quotient with G, G* = ((A/G)+G)/2 3) If G* is sufficiently accurate, stop. Otherwise, let G = G* and return to step 2. The number of correct decimal places roughly doubles with each repetition of step 2. > That only leaves two types of numbers left that qualify for irrational, > non computable and random numbers. Jesus you do whine. The digit occurances of sqrt(2) are demonstrated to be random (with one exception) with a huge battery of statistical qualifiers including string poker hands (no flushes), but fractionally less so than those of pi (also with the obvious excpetion). [snip] -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) Quis custodiet ipsos custodes? The Net! ==== Dear friends, How to express in elliptic functions sn, cn, or dn if the denominator of our integral is not easily being transform into a standard from ? For example, the denominator of the integral is sqrt of 1 + c^3*x + c^2*x^2 - 1/4*x^4, where c is a constant. Natanael ==== > Dear friends, How to express in elliptic functions sn, cn, or dn if the denominator of our > integral is not easily being transform into a standard from ? > For example, the denominator of the integral is sqrt of > 1 + c^3*x + c^2*x^2 - 1/4*x^4, where c is a constant. Natanael Transform the plane so that the zeros of the polynomial are the vertices of a rectangle. ==== I am looking for an example of a supermartingale (Xn)n=0,1,2,... such that Sup E[|Xn|]<+oo and Xn-> 0 a.s. , and yet Xn<0 with non-zero probability for some n. Any help is appreciated. Noel. ==== I am looking for an example of a supermartingale (Xn)n=0,1,2,... such that >Sup E[|Xn|]<+oo and Xn-> 0 a.s. , and yet Xn<0 with non-zero probability >for some n. Is that really all you want, or is there another condition you left out? It seems to me very easy to give an example of a martingale doing all this. For example take [0,1] as the probability space, and let X_n be -2^n times the characteristic function of [0,2^(-n)]. >Any help is appreciated. Noel. ************************ David C. Ullrich ==== Is that really all you want, or is there another condition you left > out? It seems to me very easy to give an example of a martingale > doing all this. No, there is no mistake. This is exactly what I need. Noel. ==== Is that John Ennis again, (better known as soggy on sci.skeptic)? -jcr ==== Is that John Ennis again, (better known as soggy on sci.skeptic)? > Naaah, it's a kook called Herc. Ignore him, he pops up every now and then when he forgets to take his meds. Really. -- Find out about Australia's most dangerous Doomsday Cult: http://users.bigpond.net.au/wanglese/pebble.htm You can't fool me, it's turtles all the way down. ==== Can someone explain to me what Jackson's supposed Gricean defense of assert the stronger means? It seems like assert the stronger would sometimes conflict with the maxim of quantity, e.g., if you only need to say THIS giraffe has a long neck yet assert the stronger holds that you say ALL giraffes have long necks if you know this second statement to be true. Is this what Jackson means when he talks about asserting the stronger? This couldn't be it, because it would so often obviously conflict with the maxim of quantity, right? Someone please clarify... ==== How do you produce the material conditional symbol that looks kind of like supset but is longer and thinner? ==== > How do you produce the material conditional symbol that looks kind of like > supset but is longer and thinner? I'm not sure to have understood your question. If you want to know how to get it in a LaTeX document, the answer is Rightarrow or Longrightarrow. But, if I got it right, then the next time you want to know how to produce a symbol in a LaTeX document, read Scott Pakin's The Comprehensive LATEX Symbol List. You'll find it at CTAN at /info/symbols/comprehensive/ Jose Carlos Santos ==== > How do you produce the material conditional symbol that looks kind of like > supset but is longer and thinner? > 'P implies Q' in ascii? ->, -->, ==> ==== (also posted in sci.fractals but it seems to be dead in there) Please forgive my ignorance, but I've only just been introduced to the world of fractals. I already have the box counting method understood, but I'mstuck with something else... If I were to randomly select a particular point in a given area (whether or not that point contains the particular thing I'm looking at eg. say I was looking at stars in an area of space, the point wouldn't necessarily have to have a star there), how would I go about calculating the fractal dimension for that particular point? I've seen ways of calculating the fractal dimension for a range of other situations, but nothing that could help me with this. Is it possible (or have I just confused you all)? If not, why not? If yes, I would appreciate it if you could let me know how or point me in the direction of some relevant literature. I've searched google, but can't find anything that may help with this situation. ==== > (also posted in sci.fractals but it seems to be dead in there) > Please forgive my ignorance, but I've only just been introduced to the world > of fractals. I already have the box counting method understood, but > I'mstuck with something else... If I were to randomly select a particular point in a given area (whether or > not that point contains the particular thing I'm looking at eg. say I was > looking at stars in an area of space, the point wouldn't necessarily have to > have a star there), how would I go about calculating the fractal dimension > for that particular point? I've seen ways of calculating the fractal > dimension for a range of other situations, but nothing that could help me > with this. Is it possible (or have I just confused you all)? If not, why not? If yes, > I would appreciate it if you could let me know how or point me in the > direction of some relevant literature. I've searched google, but can't find > anything that may help with this situation. > I don't know whether this answers your question, but here goes: A. Dimension is a property of a set of points, not of individual points. By restricting the set under consideration to a neighborhood of a given point, a local dimension can be calculated (or estimated, for sampled data). The dimension is determined by the number of neighborhoods of a given diameter are needed to contain the set, as a function of the diameter of those neighborhoods, as that diameter tends towards zero. Note that for sampled data, you don't really have the requisite uncountable number of points, so the discrete nature of your sample set has to be taken into account as you choose the diameters of your neighborhoods that cover the set. B. If you choose a point not in the set under consideration, the dimension is that of the empty set. After all, at the chosen level of detail, you can't see the set, so it may as well be empty. My recollection is that (-1) is the convention for the dimension of the empty set, but I may be mistaken. C. If you choose a point that is isolated in the set (that is, there are no nearby points of that set other than the one you've chosen), the dimension is that of a single point: 0. Once again, at the resolution of interest, that point may as well be the only point in the set. D. If you choose a point that is in a cluster of points, then take a neighborhood, and look at this URL: http://life.csu.edu.au/fractop/doc/notes/ It contains a number of definitions that can probably be converted to useful algorithms without too much pain, and there is also a link to a software page, which I haven't done more with than to verify its existence. Dale. ==== > ... stuff deleted ... > ... look at this URL: http://life.csu.edu.au/fractop/doc/notes/ It contains a number of definitions that can probably be converted > to useful algorithms without too much pain, and there is also a link > to a software page, which I haven't done more with than to verify > its existence. Dale. > I just noted a bit of text located in the above-referenced page that gives me pause, and suggests one look at the page with at least a mild degree of care: BEGIN QUOTED TEXT: The fractal dimension is a geometric description of an image. It has an integer value for topological sets and a non-integer value for fractal sets. Central to fractal geometry is the concept of self-similarity. END QUOTED TEXT One of the measures, the Hausdorff dimension, is well-suited to what the author calls topological sets, and does, in fact, yield non-integral dimension for the standard middle-thirds) Cantor set: dim_H(CantorSet) = log(2)/log(3). The reader will note that the various Cantor sets are indeed topological sets, inasmuch as that term has any meaning at all. In addition, while self-similarity is a feature that is very useful for the dimension calculation to be tractable, it is by no means *central* to the concept of fractional dimension. It is true that many, if not most, of the sets of non-integral dimension that we've managed to construct have been done via iteration of some basic construction (the middle-thirds Cantor set is an example). However, it is a vast overstatement (so vast that it is simply incorrect) to suggest that this self-similarity is central to the existence of sets of non-integral dimension. That's my caveat wrt the above web page. Dale. ==== Assume b=a^t and study this function of t. -- Maxi ==== By chance I found his page, A Treatise on Class Theory this morning, and was quite impressed by it - and not least by the fact that it was written by a 15/16-year old! So now I'm really curious.. four years have passed since he used to hang out at sci.math, but some of you might still know; what is he doing these days? Is he still into math, and if so, is he having a splendid career? Again, I'm just so curious. -Leander ( Seraph-sama's page on Class Theory: http://members.tripod.com/~SeraphSama/class5.html ) ==== Too weird to read. > Here is an example of an emerging digital conceptual art form. > I don't know the real identity of the author below citing my writings. [time posted on SSS 12-07-03 06:08p.m.c.s.t.] [04:21p.m. 12-07-03] TITLE : UFOS : Jung and Pauli's - Einstein Dream of Extra - Dimensions and Radioactivity GOLDEN SECTION AS PSYCHOVISUAL ARCHETYPE ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ DATABASE : Meier, C.A., (ed.), Atom and Archetype, The Pauli/Jung Letters 1932- > 1958, Princeton University Press, New Jersey, 2001 Wolfgang Pauli, Carl Jung and the Challenge of the Unified > Psychophysical Reality > ALSO SEE > BACKGROUNDER :http://groups.yahoo.com/group/SarfattiScienceSeminar/me > ssage/4561 Synchronicity Trivial Coincidence : [09:57p.m. 12-06-03] .61803398875 falls on 1953.13+ Wolfgang Pauli : Theory Of Relativity Sept. 1921 1953 - Pauli's Pauli : 1921_____________________1953.13+_____________1973 Sarfatti - Abdus salam [MWT 1973!] [ time now is 10:02:36p.m. 12-06-03] +++++++++++++++++++++++++++++++++++ It appears my Really Nutty theory is not so and/or maybe nutty > afterall? The below from Remo Roth's webpage Seems to confirm in a manner of > speaking? what I writ. > SUPPORTING EVIDENCE? SEE BELOW : http://www.psychovision.ch/synw/paujubw_e.htm Carl Jung said : Hence as a working hypothesis one can state that radioactivity > (interpreted on a psychophysical level) is connected with psychic > relativity of space-time, wherein empirical psychic or even > psychophysical experience is possible, that with the help of physical > concepts no longer can be explained. REMO ROTH : Since the explosion of the first atomic bomb in 1945, reports of UFO > encounters and abductions have increased at an exponential rate. > Further, more and more people begin to have the distinct feeling that > the development of the atomic bomb and the appearance of UFO > phenomena somehow belong together. If so, the question arises whether > UFOs are physically real. The answer is: yes and no. ETC. THE GODDESS Eros - understood in its original definition as the principle of > relation - can be compared with the wave principle of quantum > physics. The Copenhagen interpretation of Niels Bohr tells us, that > what we call matter is also a complementary phenomenon: It depends on > our observation's instrument, which aspect of it we see, the wave or > with Logos, we can conclude that there is a great possibility that, > by the repression of the psychic observation's instrument of Eros, we > have projected the bipolarity of the consciousness into the > complementarity of the macrocosmic aspect of matter. However, Eros consciousness, as I call this specific, deeply > introverted altered state, wants to be redeemed. My experience has > shown me, that UFO encounter and abduction victims are unconsciously > forced into this specific sort of an altered consciousness, which > compensates our far too one sided use of Logos. In it, they observe a > today constellated real creation and incarnation out of the > psychophysical universe (unus mundus): The World Soul, the ruling > principle of this intermediate world, wants to give birth to a new > child. The way this novel creation happens, is called creatio > continua, the possibility of continuous creation. It is an archetypal > compensation to the Jewish/Christian creatio ex nihilo, the unique > bang!. GODDESS Because the Eros principle is the background of this today > constellated creation myth, it is completely natural that the > archetype behind it is the so-called coniunctio, the sexual union of > a god and a goddess. Therefore, it is this archetype, which will > compensate the Jewish/Christian Genesis. > ++++++++++++++++++++++++++++++++++++ BACKGROUNDER : http://groups.yahoo.com/group/SarfattiScienceSeminar/message/4554 NOTE : for the Record : I DID NOT READ REMO ROTHS website pages in > #4554. http://www.psychovision.ch/synw/paujubw_e.htm As a confirmation, in the same year, 1934, another dream urged > Wolfgang Pauli, to take into consideration that behind quantum > physics there could be yet another hidden dimension of reality. In > this dream a man who looked like Einstein said to him that quantum > physics was but a one-dimensional part of a deeper reality. It was > Einstein in fact who always emphasized that quantum physics was not > the last word and that beneath it laid yet another dimension (that > he, however, deducted was purely physical and not psychic). This shows that Wolfgang Pauli made quite an effort to understand > radioactivity, so central an event in the every day life of today, at > a deeper level, which he would later - according to Jung's view - > call psychophysical or psychoid. The idea of another dimension behind quantum physics was eventually > developed further by David Bohm [Wholeness and Implicate Order, 1980] > as the concept of the implicate order, out of which the visible > world, the explicate order, materializes. But on one hand, Bohm's > hypothesis is not verifiable empirically, and, on the other hand, > individual consciousness and the collective unconscious play no role > in his theory. This is reasonable since he saw his vision as an > expansion of physics. As we have seen, the relevant messages in Pauli's dreams are: > Beta > radioactivity (antineutrino) and synchronicity are connected in a > manner yet not understood and Behind the world of quantum physics > another dimension is hidden. Summarizing both statements one must > come to the following conclusions: a) This deeper reality behind quantum physics must have something to > do with the observation of the phenomenon of synchronicity (or an > extension thereof). b) Where, on the one hand, with the principle of synchronicity the > physical and the psychic worlds unite, and on the other hand > Pauli's > dreams want to convince us that the observation of synchronicities > implies the production of the radioactive substance, the postulated > antineutrino and through it Beta radioactivity reaches into a world, > that transcends the world of physics and includes both matter as well > as psyche. Therefore, it would seem that the above-guessed violation > of the physical law of the conservation of energy is no longer out of > the realm of possibility in processes which obey the principle of > synchronicity (or an extension thereof). +++++++++++++++++++++++++++ 04:45p.m. 12-07-03] Implicate [Esoteric] Explicate [Exoterisized] Bit from It and It from > Bit. IN THE OPERATION CROSSROADS CASE THE GODDESS SPOKEN OF BY REMO > ROTH WOULD BE EXTERIORIZED IN THE PHYSICAL GILDA ABLE [ RITA > HAYWORTH - SEX GODDESS] OPERATION CROSSROADS PEACE TIME ATOMIC BOMB > TESTS. GILDA ABLE 07-01-1946 - and - HELEN 7-24-1946. CHARLIE WAS CANCELLED SEE: > http://groups.yahoo.com/group/SarfattiScienceSeminar/message/4554 MORE INSANITY FROM A TRUE LUNATIC MIND : PK AND THE CONNECTION > BETWEEN 1946 AND 1947. This is as crazy as it gets. I put together this super nutty theory that the 1947 Roswell Flying > Saucer Crash was a manifestation, the result of the imaginational > power [string theory, supermind connection] of men and women > globally in time frame history 1946 1947 who were brand spankin NEW > TO THE IDEA OF WHAT ATOMIC meant. This phenomena manifest by means > of a PK Jungian Collective Unconscious energy bank generated by a two > way retarded / advanced tachyonic exchange of sympathetic self re- > generating energy backacting in a resonating feedback loop with all > strings in concert moving together playing the flying saucer > manifests in 1947 song. SEE BELOW FOR : CONTINUATION [SIDEBAR INCIDENTAL : TRIVIAL NAME COINCIDENCE BETWEEN - > MARGHERITA SARFATTI - ANDMARGHERITA [RITA] HAYWORTH - HAWORTH, > changed to Hayworth. Mussolini. Sarfatti was Mussolini's lover? > Crossroads followed which led into a 40 year period of Cold War Which according to Jung and Pauli is related to the incidence of UFO > sightings. M. Sarfatti was also involved with the great artists paintings the > Masters one being Michaelangelo. Michealangelo painted the CREATION > OF ADAM, which Jack Sarfatti has said is a metaphor for ICE - 9. This > relates forward to Sir Martin Ree's OUR FINAL HOUR, and parallels > means to travel FTL. Sarfatti says atomic, nuclear energy pales in > comparison to exotic energy. Thus the historical progression is : > Determinism - indeterminism - with its nuclear spin off applications > forward in time to WMAP'S discovery of exotic energy in the form of > Dark matter /energy. Sarfatti claims this is WHY the UFOS are here. ENIAC : Jack Sarfatti is related to Margherita Sarfatti. > More coincidence : 1953 Jack Sarfatti had a technical book on ENIAC > under his arm, just home from the library when he answered the > telephone from a Spacecraft Computer from the Future.] http://www.angelfire.com/ga/goldengodesses/rita.html Rita was contracted by Fox At the age of 15. she shortened her name > to Rita Cansino and appeared as a dancer in Dante's Inferno. By 1937, > she had taken her mother's maiden name, Haworth, and was appearing > regularly in supporting roles. IS RITA'S MOTHER VOLGA HAWORTH, and thereby Rita related to Lee > Haworth who worked on the Eniac? Rita was born in Brooklyn as was Sarfatti. Also Rita's parents came > from Spain. Coincidence : The rabbi De Troyes relatives migrated to > Spain. Is there a connection between the Cansio's [Volga Haworth > married a Cansio] and De Troyes? Lee Haworth worked on the ENIAC SEE : ... I was associate head of Division 6 until Lee Haworth left. ... of > vacuum tube circuitry, > which enabled him to put together a much more reliable ENIAC than he > could ... > www.ieee.org/organizations/history_center/ > oral_histories/transcripts/chance.html - http://www.ieee.org/organizations/history_center/oral_histories/transc > ripts/chance.html > ,,,,,which enabled him to put together a much more reliable ENIAC > than he could otherwise have done. ] +++++++++++++++++++++++++++++++++++++++ : CONTINUATION The players in this Pychokinetic Theatre, the witnesses if you > will, would consist of a highly excited world population some > billions of human beings fresh out of 5.94 years of some of the > greatest? human suffering known to humankind. The GILDA ABLE bomb named after the beloved sex symbol [sex and > death, with associated emotions] and major WWII pinup women Rita > Hayworth's movie GILDA was detonated and Radio Broadcast to the > entire world to hear SIMULTANEOUSLY. If there is a Jungian Collective Unconscious and/ or Geist / > noosphere and it is EFFECTED by humans and animals, there were > 1,000s of animals on those target boats at Kwajalein, think of > the mental charge the living charge if you will, that would have > been generated, registered and imparted at the instant of detonation > into GAIA'S memory. [Valis?] GILDA was detonated on - [06-30-1946] - 07-01-1946 almost exactly one > year to the day > before the infamous Roswell Flying Saucer crash found by some of > the same men who had observed GILDA exploding to kingdom come. The > newspapers worldwide reported that a seductive picture of Rita would > be painted on the side of the bomb, this picture was he famous > Bob Landry photo of Rita in LIFE MAGAZINE, 11 August 1941, this > conditioning had the effect > of creating Anticipation and a corresponding TENSION that would be > released at the instant of detonation. HELEN another women's name was detonated on 07-24-1946. Both bombs were named after women. The 1940s was a man's world. Both > detonations were RADIO BROADCAST WORLDWIDE TO a worldwide audience of > countess millions WHO WOULD SIMULTANEOUSLY HEAR AT THE VERY SAME TIME > AND INSTANT GILDA ABLE AND HELEN'S RELEASE OF ATOMIC NUCLEAR ENERGY. SPECTRUM OF EMOTIONS : These men women and childern of the world having just endured the > hardships and great sufferings of WWII, a long hard bloody war would > at the instant of detonation experience an incredible cross section > of conflicting emotions. At the same time 1,000,000s would be > ecstatic and 1,000,000s would be deeply mourning. Some of these same people who were at Kwajalien would later be at > the 509th Roswell Walker airfield and who in JULY 4-5th 1947 would > again become highly excited by the story of a crashed flying saucer > with rumors of Little Green Men having been captured. In additon > the worlds population was hearing about and seeing Flying Saucers > practically everywhere, except Kansas it was a dry state:-) Anticipation, tension then release, these are the main elements I > want to focus on. This so-called Flying Saucer [with rumors of captured Little Green > Men] story was printed in the bases newspaper by none other by the > bases commander Colonel Blanchard. This alien intrusion into the > worlds affairs had coincidentally taken place almost a year to the > day after the highly publisized and exciteable Operation Crossroads. > Operations Crossroads had been a public peace time demonstration of > America's might. Crossroads demonstrated the winning weapons power > that won the war and now their sex object their beloved sex symbol > the goddess the symbol of the sacred feminine Rita Hayworth's GILDA > picture had been painted on the side of the Atomic bomb. > Weird huh? Pinups were a regular normal effect of the times just as > they are now. > Princeton Physicist Thomas Lanahan told me there was no painted > picture of Margherita Hayeworth on the abomb only a black stencil > with the word GILDA the name of Rita's hottest new movie. How did he > know? He stenciled it there himself. Also General Woodrow Swancutt > who flew Dave's Dream the B-29 which dropped GILDA ABLE said there > was NADA NO picture of Rita on the bomb either. This was or still is > somewhat controversial. Anway as my so-called theory goes the energy > generated from the detonations of 1946 somehow manifest as the > Kenneth Arnold Flying Saucer sightings cira 06-24-1947 and by > extension the 07-05-1947 Flying Saucer incident thru an HYSTERIA > [law of contagion] which had its beginnings at Operation Crossroads. > Actually the beginning and endings were controvertible.Time flowed in > both directions remember Sarfattian advanced and retarded waves ++++++++++++++++++++++++++++++++++++++++ REMO ROTH : [04:10p.m. 12-07-03] http://www.psychovision.ch/synw/paujubw_e.htm We find a considerable amount of dreams in Pauli's correspondence > that described how he was urged to tie the so-called Beta > radioactivity - in which the antineutrino plays an important role - > together with the depthpsychological phenomenon of synchronicity > postulated by C.G. Jung. So Pauli in 1949 used the example of > Jung's > famous Scarab Synchronicity (see also Synchronicity principle) to > show that his dreams since 1934 insisted on the following scenario: > Through the observation of this and similar synchronicities the > famous psychoanalyst .b3succeeded in producing a radioactive > substance. In synchronicity the outer and inner, the physical and psychic, > worlds unite for a short time. This is why these dreams indicate that > the antineutrino cannot simply be categorized as another physical > physics and enters the world of depth psychology. If so, a > transformation from physical to psychic energy (and vice versa) is > possible - a fact that both Pauli and Jung rejected. As a confirmation, in the same year, 1934, another dream urged > Wolfgang Pauli, to take into consideration that behind quantum > physics there could be yet another hidden dimension of reality. In > this dream a man who looked like Einstein said to him that quantum > physics was but a one-dimensional part of a deeper reality. It was > Einstein in fact who always emphasized that quantum physics was not > the last word and that beneath it laid yet another dimension (that > he, however, deducted was purely physical and not psychic). > ++++++++++++++++++++++++++++++++ REMO ROTH : http://www.psychovision.ch/rfr/roth_e_ufos.htm Since the explosion of the first atomic bomb in 1945, reports of UFO > encounters and abductions have increased at an exponential rate. > Further, more and more people begin to have the distinct feeling that > the development of the atomic bomb and the appearance of UFO > phenomena somehow belong together. If so, the question arises whether > UFOs are physically real. The answer is: yes and no. In an attempt to help mankind find its way back to a more > comprehensive view of the totality of his soul the depth psychologist > Carl G. Jung and the physicist and Nobel laureate Wolfgang Pauli > postulated a world behind or beyond the split into inner and outer > world, into psyche and matter. Pauli called it the unified > psychophysical reality while Jung referred to this realm as the unus > mundus (united world) - a concept he obtained from the alchemist and > student of Paracelsus, Gerardus Dorneus. With the help of this new concept, which reaches, however, back to > Medieval alchemy, the above paradox can be solved: UFOs incarnate out > of this intermediate world, which is the same as the realm of the > alchemist's World Soul, their Goddess of equivalent worth to the > Christian God. In the beginning was the Logos, and the Logos was with God, and the > Logos was God. (John 1:1). As a result of this definition of the > Christian God as the Logos, our consciousness developed - especially > since the birth of natural science in the 17th century - in an > extremely one sided way. Therefore we are not used anymore, to see > the world and the universe with the help of the Eros, as Medieval > people yet were able. We have repressed this principle, complementary > to the Logos consciousness. Eros - understood in its original definition as the principle of > relation - can be compared with the wave principle of quantum > physics. The Copenhagen interpretation of Niels Bohr tells us, that > what we call matter is also a complementary phenomenon: It depends on > our observation's instrument, which aspect of it we see, the wave or > with Logos, we can conclude that there is a great possibility that, > by the repression of the psychic observation's instrument of Eros, we > have projected the bipolarity of the consciousness into the > complementarity of the macrocosmic aspect of matter. However, Eros consciousness, as I call this specific, deeply > introverted altered state, wants to be redeemed. My experience has > shown me, that UFO encounter and abduction victims are unconsciously > forced into this specific sort of an altered consciousness, which > compensates our far too one sided use of Logos. In it, they observe a > today constellated real creation and incarnation out of the > psychophysical universe (unus mundus): The World Soul, the ruling > principle of this intermediate world, wants to give birth to a new > child. The way this novel creation happens, is called creatio > continua, the possibility of continuous creation. It is an archetypal > compensation to the Jewish/Christian creatio ex nihilo, the unique > bang!. Because the Eros principle is the background of this today > constellated creation myth, it is completely natural that the > archetype behind it is the so-called coniunctio, the sexual union of > a god and a goddess. Therefore, it is this archetype, which will > compensate the Jewish/Christian Genesis. The purely masculine god of the Genesis created the world by the > principle of separation and discrimination, i.e. Logos. The novel > future world, however, will be incarnated by its contrary, the > unification. In Medieval alchemy this synthesis is represented by the > sexual union of the king and the queen, for example in the Rosarium > Philosophorum. In a modern view we can interpret this today newly > constellated creation myth as the union of the Logos Self (Carl Jung) > with the Eros Self (the World Soul). As in quantum physics, the necessary condition for the incarnation of > this new creation would be its observation by human consciousness. > Contrasting to natural science, such a renewed researcher of the > future must however learn first to dwell consciously into the altered > state of Eros. Up until today Western science, however, has developed in the > opposite direction. Since the 17th century, the deification of the > Logos ego led to an omission of the development of the introverted > Eros consciousness. Therefore, the conscious observation of the > microcosmic coniunctio in one's own body and soul, which is > autonomous and absolutely independent of our consciousness, is not > possible anymore. Instead of concentrating on the resulting birth of > the World Soul's children, the empirically observable > natural inner quantum leaps, Western science was and is yet misled > by a deeply unconscious power complex which enforced it to produce > artificial quantum leaps in outer matter. The destructive > consequences of this behaviour are the atomic bomb and the nuclear > plants (UFOs are so interested in!). Since the invention of the > atomic bomb, will power, a Logos principle, produces artificially > liberated radioactivity of outer matter, i.e. in the macrocosm: The > World Soul is raped by the will-power of Western science and forced > to give birth to children She does not want to incarnate at all. Because of the repression of the Eros consciousness, the divine > feminine principle is not, as in Medieval times, able anymore to > present her creations to mankind in the introverted world of the > microcosm. As a logical conclusion of this repression, She is now > forced to incarnate her children in the macrocosm, as UFOs and > their inhabitants. Further, as a raped woman, She seeks for revenge > and rapes her victims, too, by forcing them into an unconscious > change from the will-powered Logos into the will-less Eros. Like this > She pummels or even destroys the extraverted Logos consciousness: UFO > encounter and abduction happen spontaneously (creatio continua) and > must therefore be observed by the victims against their conscious > will. With this brief sketch of the archetypal background - derived as a > result of the continuation of Carl Jung's, Wolfgang Pauli's and Marie- > Louise von Franz' research - one need not be a prophet to expect that > in the near future the UFO encounter and abduction problem will > increase to an unimaginable extent. Only a radical change of our > consciousness and of its old-fashioned paradigm can prevent us from > participating in the most horrible effect of this development - of > the destruction of mankind, be it psychic, be it physical, or both. ++++++++++++++++++++++++++++++++++ [04:16p.m. 12-07-03] The Connection between Radioactivity and Synchronicity in the > Pauli/Jung Letters > http://www.psychovision.ch/synw/paujubw_e.htm Wolfgang Pauli (1900 - 1958) the well-known physicist and Nobel Prize > laureate taught at the Federal Institute of Technology > (Eidgenoessische Technische Hochschule, ETH) in Zurich, Switzerland > where C.G. Jung also was a Professor of Psychology. In 1931 - shortly > after his invention of the so called antineutrino - Wolfgang Pauli > extreme affects and the emotional ups and downs he underwent caused > him to have a great deal of trouble with women. He therefore turned > to the already well-known psychiatrist and depth psychologist > (psychoanalyst) C.G. Jung, who sent him to his student, Erna > Rosenbaum, ostensibly because he didn't want Pauli's genuine > unconscious process to be disturbed by his knowledge and related > advice. After about four years of psychoanalysis (1931 to 1934) which, for > the most part dealt with his emotional problems, he apparently > resolved these problems and married a second time to Franca Bertram. > Later he admitted that in his interpersonal relationships he could be > compared to a cold devil. Despite this, the letters show that Pauli > was able to go to great lengths to put his emotional world in order. Although the patient had been released as cured from analysis, his > unconscious continued to produce deep archetypal dreams. These dreams > had less to do with Pauli's set of personal problems than with > the > archetypal basis of the theoretical assumptions of physics and > natural science. Some of these dreams have been published in 1992 in > the book Wolfgang Pauli und C.G. Jung - Ein Briefwechsel 1932-1958 > [Meier, 1992] which is now also published in the U.S. as Atom and > Archetype, The Pauli/Jung Letters 1932-1958 [Meier, 2001]. In the > years 1993, 1996 and 1999 many more of Pauli's important letters > were > published (Letters from 1940 to 1954) [Pauli, 1993], [Pauli, 1996], > [Pauli, 1999], that were enhanced in 2001 (Letters from 1955 to 1956) > [Pauli, 2001]. When one considers the dreams published in these letters on the basis > of C.G. Jung's psychology, one comes to the conclusion that it is > very likely that they reacted to Pauli's .b3invention of the > hypothesis in his famous letter to the Liebe radioaktive Damen und > Herren (.b3Dear radioactive ladies and gentlemen) for the > recovery of > one of the most fundamental hypotheses of physics: The law of > conservation of energy. For physicists of those days it was a big > shock to accept that this dogma of modern science was violated by so- > called radioactive beta decay, the process of the transformation of > the neutron into a proton and an electron. With the .b3invention > of a > supposed to be eliminated. Pauli formulated later on that: Physical > energy is, without exception, indestructible; it does not change into > hidden, nonphysical forms of energy (such as 'psychic energy', for > example). We find a considerable amount of dreams in Pauli's correspondence > that described how he was urged to tie the so-called Beta > radioactivity - in which the antineutrino plays an important role - > together with the depthpsychological phenomenon of synchronicity > postulated by C.G. Jung. So Pauli in 1949 used the example of > Jung's > famous Scarab Synchronicity (see also Synchronicity principle) to > show that his dreams since 1934 insisted on the following scenario: > Through the observation of this and similar synchronicities the > famous psychoanalyst .b3succeeded in producing a radioactive > substance. In synchronicity the outer and inner, the physical and psychic, > worlds unite for a short time. This is why these dreams indicate that > the antineutrino cannot simply be categorized as another physical > physics and enters the world of depth psychology. If so, a > transformation from physical to psychic energy (and vice versa) is > possible - a fact that both Pauli and Jung rejected. As a confirmation, in the same year, 1934, another dream urged > Wolfgang Pauli, to take into consideration that behind quantum > physics there could be yet another hidden dimension of reality. In > this dream a man who looked like Einstein said to him that quantum > physics was but a one-dimensional part of a deeper reality. It was > Einstein in fact who always emphasized that quantum physics was not > the last word and that beneath it laid yet another dimension (that > he, however, deducted was purely physical and not psychic). In 1948, stimulated by an impressive paranormal Pauli-Effect at the > foundation ceremony of the C.G. Jung-Institute (see Wolfgang Pauli's > Fludd/flood Synchronicity and the Future Development of > discussion and not for publication, he desired to find a neutral > language for some physical expressions that would have permitted him > to explain the phenomena not only in physical terms but also > according to their symbolic, depth psychological essence. He cited > these .b3symbolically interpreted physical expressions. The manner > in > which he wanted to define radioactivity in a neutral language he > described in a letter dated December 12, 1950 to Jung: .b3A process of transmutation of an active center, ultimately > leading > to a stable state, is accompanied by self-duplication > (`multiplying') > and expanding phenomena, associated with further transmutation that > are brought about through an invisible reality. > This shows that Wolfgang Pauli made quite an effort to understand > radioactivity, so central an event in the every day life of today, at > a deeper level, which he would later - according to Jung's view - > call psychophysical or psychoid. The idea of another dimension behind quantum physics was eventually > developed further by David Bohm [Wholeness and Implicate Order, 1980] > as the concept of the implicate order, out of which the visible > world, the explicate order, materializes. But on one hand, Bohm's > hypothesis is not verifiable empirically, and, on the other hand, > individual consciousness and the collective unconscious play no role > in his theory. This is reasonable since he saw his vision as an > expansion of physics. Bohm's idea of an implicate order behind quantum physics appears > to > be similar to what Wolfgang Pauli and C. G. Jung sought in their > later years, the unified psychophysical or unified psychoid reality, > the unus mundus of the Medieval alchemist Gerardus Dorneus (see > Wolfgang Paulis psychophysischer Monismus), except for one major > difference. Pauli and Jung felt that individual consciousness is able > to come into relation with this implicate order. Or, as the possible > scenario expressed in David Bohm's terminology, the question > arises: > Does a possibility exist that the implicate order could unfold with > the help of an act of consciousness related to the unified > psychophysical reality beyond the split in quantum physics and depth > psychology? We do not know the answer yet, but one thing seems certain: For the > solution of this problem, which will really be a unique .b3act of > creation in time [Jung, 1971, ¤ 955] in the sense of a creatio > continua, radioactivity (seen from a psychophysical level) along with > synchronicity must be included. As we have seen, the relevant messages in Pauli's dreams are: > Beta > radioactivity (antineutrino) and synchronicity are connected in a > manner yet not understood and Behind the world of quantum physics > another dimension is hidden. Summarizing both statements one must > come to the following conclusions: a) This deeper reality behind quantum physics must have something to > do with the observation of the phenomenon of synchronicity (or an > extension thereof). b) Where, on the one hand, with the principle of synchronicity the > physical and the psychic worlds unite, and on the other hand > Pauli's > dreams want to convince us that the observation of synchronicities > implies the production of the radioactive substance, the postulated > antineutrino and through it Beta radioactivity reaches into a world, > that transcends the world of physics and includes both matter as well > as psyche. Therefore, it would seem that the above-guessed violation > of the physical law of the conservation of energy is no longer out of > the realm of possibility in processes which obey the principle of > synchronicity (or an extension thereof). As mentioned, Wolfgang Pauli leaned with dogmatic vehemence toward > the possibility of a transformation of energy in the above-mentioned > sense. Nevertheless, he sought a union of physics with depth > psychology, that he later - after a heavy battle with Marie-Louise > von Franz - changed with the demand for a fusion of physics and > parapsychology (see Wolfgang Pauli and Parapsychology). He sought > empirical examples for that which he termed .b3background physics. > He > did not succeed in the breakthrough to this, hence my opinion, > because he dogmatically supported the physical law of the > conservation of energy at the same time. Much earlier, in the year 1918, Emmy Noether showed that the law of > conservation of energy is equivalent to the hypothesis of the > isotropy of time. The violation of the former in the case of Beta > radioactivity, which also means the replacement of the antineutrino > through a psychophysical equivalent, would, as a consequence, > disprove the honored physical law of uniform passing (isotropy) of > time. We know that Wolfgang Pauli's dreams point in this direction. In > a > psychology the bridge to yet unknown creative ideas in the collective > unconscious - manifests her conception of time with the assistance > of .b3odd oscillation symbols, that belong to the same category of > periodic symbols as .b3the light and dark stripes and the > .b3pendulum > and the `little men' from the earlier material. As the > preceding > letter shows, the latter relates to his so-called world-clock vision > of 1934, which Jung had interpreted and published in 1936 [Jung, CW > 12, ¤¤ 307] - but in the year 1950 Pauli discerned that Jung's > interpretation was unsatisfactory. This world-clock vision generated > in Pauli a sensation of sublime harmony, which shows that its > content - understood on a psychophysical level - would have been the > solution of the oscillation problem of Pauli's Anima. Wolfgang Pauli was not able to find this solution. This is why the > symbolism of oscillations, frequencies, rhythms, spectrums, light and > dark stripes, along with his phobia of wasps (yellow (i.e. bright) > and black stripes!) pursued him even until the end of his life. > Therefore we find countless references to this sort of dream in his > correspondence. On the basis of the above, the hypothesis that this oscillative > conception of time of Wolfgang Pauli's Anima has to be brought > into > conjunction with the negation of the isotropy of time, cannot be > denied. Such a change, however, is not possible within the boundaries > of physics, for this leads to qualitative and therewith depth > psychological (or even psychophysical) statements about the > phenomenon of time. In our daily lives, we do not experience this phenomenon in the sense > of an isotropy. Intense times alternate with times of Lange-weile > (= long-times, i.e. boredom), where the flow of time manifests as a > much slower experience, as in the first instance. When one is in > agony or mortal fear it seems that time takes on an unbelievable > velocity, as cited in reports by survivors of life-threatening > situations, as, for example, in falls from mountain precipices, > wherein the victims see their entire lives pass before their inner > eyes within seconds. In addition to the problem of the psychic relativity of time, the > psychic relativity of space also played a large role in Wolfgang > Pauli's dreams. He had already mentioned a .b3constant > criticism of the > space-time-concept in relation with a .b3close fusion of > psychology > with the scientific experience of the processes in the material > physical world in a letter to Jung dated December 23, 1947. This > formulation came out of a dream he had on October 28, 1946 > about .b3objectified rotation. It is likely that this dream > referred > to the so called spin of the electron he postulated in 1927. > The .b3preconscious knowledge (C.G. Jung) of the collective > unconscious wanted to make it clear to him, that this objectification > of the rotation, that is, the mathematical formulation of the spin, > could be wrong on the level of the psychophysical reality. Pauli > sensed that, contrary to the physicist's point of view, the dream > postulated that this rotation has something to do > with radioactivity at the level of the unified psychophysical > reality and with .b3the relativity of the concept of space in > relation > to the psyche. In a very deciding dream from September 28, 1952 the above-mentioned > oscillating movement of Wolfgang Pauli's (Chinese) Anima caused > space > to contract and to begin to rotate. This shows that the > .b3preconscious > knowledge of the collective unconscious also wanted to point out a > new concept of space, that not only is subjected to a contraction - > as specified in Einstein's Theory of Relativity - but must be > brought > into connection with the concept of rotation as well. One can formulate generally that the constellated problem somehow is > connected with a process in which an oscillation turns into a > rotation and in a yet unknown manner remains related to > radioactivity. It seems to be this development, the future > interpretation of which on a psychophysical level will bring > extremely important insights into the processes in the unified > reality that Wolfgang Pauli and C.G. Jung searched for more than > fifty years ago. My above-mentioned thoughts bring the psychic (and not the physical) > relativity of space and time into connection with the phenomenon of > synchronicity, which is shown through Pauli's dreams as connected to > radioactive decay, interpreted at a deeper, the psychophysical level. > Hence as a working hypothesis one can state that radioactivity > (interpreted on a psychophysical level) is connected with psychic > relativity of space-time, wherein empirical psychic or even > psychophysical experience is possible, that with the help of physical > concepts no longer can be explained. The theoretical description and the empirical proof of such > hypothetical connections between radioactivity and space-time on a > psychophysical level is a challenge for the future, yet it seems to > me, that the above-mentioned working hypothesis is a fruitful first > starting point. Wolfgang Pauli's question regarding a physical- > symbolic dimension of radioactivity showed us a first step in the > right direction. But he was painfully conscious of the limits of his > future challenge must include finding .b3that other, more > comprehensive > coniunctio (union of opposites), that transcends the artificial > separation of psyche and matter and the smaller coniunctio of > wave. > He thus stated in a letter to Markus Fierz with a sigh: .b3M.9age > eine `gl.9fcklichere Nachkommenschaft' dies erreichen (May > a 'more > fortunate offspring' achieve this). The term for this unified psychophysical reality behind psyche and > matter C.G. Jung borrowed from a student of Paracelsus, Gerhard Dorn > (Dorneus), who called it the unus mundus, the Unified World. The > energetic principle of this unus mundus was known as the world soul > (anima mundi) (link2; link3) in the natural philosophy of the Middle > Ages. Synchronicity is based upon this medieval concept of the world > soul and the unus mundus, that corresponds to Wolfgang Pauli's > world > of .b3background physics. To understand this world theoretically > and > to prove it empirically, we are forced into the development of a new > scientific discipline, wherein a union of quantum physical theory and > depth psychological experience is necessary. It will be based on the > non-physical role of the antineutrino and its yet unknown connection > with the individual experienced relativity of space and time. Thereby > it appears, above all, that the experience of time is connected with > the phenomenon of variable frequencies that symbolizes itself in the > individually experienced psychic intensity of time. As we have seen, this frequency and vibrational symbolism belongs in > the greater context of the transformation of oscillative phenomena > into a rotation. Oscillation and rotation constitute on the other > hand two of the most empirically observable phenomena in UFO > sightings and abductions. Therefore, there is a good possibility that > scientific research of these frightening UFO abduction phenomena (see > UFO Network), in which exactly such psychophysical events are > experienced, can be an important contribution to the understanding of > these modern phenomena with the help of an enhanced theory of > radioactivity at the deeper level of the unified psychophysical > reality. Through this a unification of quantum physics and Jungian > depth psychology can take place. ---------------------------------------------------------- Jung, C.G., Gesammelte Werke, vol. 8, Walter, Olten, Switzerland, 1971 Jung, C.G., Gesammelte Werke, vol. 12, Walter, Olten, Switzerland, > 1972 Laurikainen, K.V., Beyond the Atom, The Philosophical Thought of > Wolfgang Pauli, Springer, Berlin, 1988 Meier, C.A. (ed.), Wolfgang Pauli und C.G. Jung - Ein Briefwechsel > 1932-1958, Springer, Berlin, 1992 Meier, C.A., (ed.), Atom and Archetype, The Pauli/Jung Letters 1932- > 1958, Princeton University Press, New Jersey, 2001 Pauli, Wolfgang, Wissenschaftlicher Briefwechsel mit Bohr, Einstein, > Heisenberg, u.a, ed. Karl v. Meyenn, vol. 2, Springer, Berlin, 1985 Pauli, Wolfgang, Wissenschaftlicher Briefwechsel mit Bohr, Einstein, > Heisenberg, u.a, ed. Karl v. Meyenn, vol. 3, Springer, Berlin, 1993 Pauli, Wolfgang, Wissenschaftlicher Briefwechsel mit Bohr, Einstein, > Heisenberg, u.a, ed. Karl v. Meyenn, vol. 4/I, Springer, Berlin, 1996 Pauli, Wolfgang, Wissenschaftlicher Briefwechsel mit Bohr, Einstein, > Heisenberg, u.a, ed. Karl v. Meyenn, vol. 4/II, Springer, Berlin, 1999 Pauli, Wolfgang, Wissenschaftlicher Briefwechsel mit Bohr, Einstein, > Heisenberg, u.a, ed. Karl v. Meyenn, vol. 4/III, Springer, Berlin, > 2001 > ------------------------------------------------------------------ see also > Wolfgang Pauli und die Parapsychologie (1. Teil) (in German) Wolfgang Pauli und die Wiederkehr der Weltseele (in German) http://www.psychovision.ch/synw/synfrsch.htm +++++++++++++++++++++++++++++++++ :trixcleverspacealien [12-07-03 - 06:03p.m.c.s.t] ** Patronize any Yahoo! Group Sponsor at your own risk. > below) - - - - - - - - - - - > To change any characteristic of your online membership access, visit via > web: > http://groups.yahoo.com/subscribe/SarfattiScienceSeminar Join in our ongoing discussions and theoretical science writings: > http://groups.yahoo.com/messages/SarfattiScienceSeminar http://stardrive.org > http://www.1st-books.com - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - > - - - SarfattiScienceSeminar@YahooGroups.com > Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/ ==== this rule: in pseudo-LaTeX d/dt(vec(A)dotvec(B))=d/dt(vec(A))dotvec(B)+d/dt(vec(B))dot vec(A) P.S. Wouldn't you like to have LaTex in this forum ? It looks so damn ugly unformatted ! /Lasse ==== > P.S. Wouldn't you like to have LaTex in this forum ? > It looks so damn ugly unformatted ! It looks ugly unformatted, which is why you shouldn't use it again. ==== > this rule: > in pseudo-LaTeX > d/dt(vec(A)dotvec(B))=d/dt(vec(A))dotvec(B)+d/dt(vec(B))dot vec(A) > It's the same thing you learned with the product rule for functions d(f*g)/dt = f'*g + f*g' > P.S. Wouldn't you like to have LaTex in this forum ? No! > It looks so damn ugly unformatted ! Exactly why, no! What's worse, it verges upon unreadable when much use as in above line. ==== I just wanted to check my solution for one more task ... ;-) ### task: express the following term through an infinite sequence. solution: 2/x/(x+3) = A/x + B/(x+3) <=> 2 = A(x+3) + Bx => A = 2/3; B = -2/3 => 2/x/(x+3) = 2/3 * ( 1/x - 1/(x+3) ) = 2/3 * ( (1-(-x)-1)^(-1) - ( 1+(2+x) )^(-1) ) = 2/3 * ( ( 1--(x-1) )^(-1) - ( 1--(2+x) )^(-1) ) = 2/3 * ( ( 1-(1-x) )^(-1) - ( 1-(-x-2) )^(-1) ) = 2/3 * ( sum( (1-x)^k, k, 0, infinity - sum( (-x-2)^k, k, 0, infinity) ) = 2/3 * ( sum( (1-x)^k - (-x-2)^k, k, 0, infinity) ) This equation is only valid for |1-x| < 1 and |-x-2| < 1 => |x+2| < 1 ### Is this correct? Karl ==== I just wanted to check my solution for one more task ... ;-) > ### > task: express the following term through an infinite sequence. > solution: > 2/x/(x+3) = A/x + B/(x+3) > <=> 2 = A(x+3) + Bx => A = 2/3; B = -2/3 > => 2/x/(x+3) = 2/3 * ( 1/x - 1/(x+3) ) > = 2/3 * ( (1-(-x)-1)^(-1) - ( 1+(2+x) )^(-1) ) > = 2/3 * ( ( 1--(x-1) )^(-1) - ( 1--(2+x) )^(-1) ) > = 2/3 * ( ( 1-(1-x) )^(-1) - ( 1-(-x-2) )^(-1) ) > = 2/3 * ( sum( (1-x)^k, k, 0, infinity - sum( (-x-2)^k, k, 0, > infinity) ) > = 2/3 * ( sum( (1-x)^k - (-x-2)^k, k, 0, infinity) ) This equation is only valid for |1-x| < 1 and |-x-2| < 1 => |x+2| < 1 > ### Is this correct? > There are no numbers x that satisfy both |1-x| < 1 and |x+2| < 1. So I would say it is not correct. How about expanding both terms in powers of the same thing? Both in powers of x, say, or powers of (x+1)? ==== F{} is a linear operator/transform if: F{A + B} = F{A} + F{B} But what about this? G{A*B} = G{A} * G{B} Are there operators for which this is true? are they well known? ==== F{} is a linear operator/transform if: >F{A + B} = F{A} + F{B} >But what about this? >G{A*B} = G{A} * G{B} Are there operators for which this is true? are they well known? Depends on which multiplication(s) you use. If o is the convolution operator (sort of multiplication between functions) and * is the normal multiplication operator, then L(f(x) o g(x)) = L(f(x)) * L(g(x)) F(f(x) o g(x)) = F(f(x)) * F(g(x)) where L(f(x)) is the Laplace transform and F(f(x)) is the Fourier transform of f(x). ==== > F{} is a linear operator/transform if: >F{A + B} = F{A} + F{B} I would call that additive not linear. >But what about this? >G{A*B} = G{A} * G{B} I would call that multiplicative. Are there operators for which this is true? are they well known? Depends on which multiplication(s) you use. If o is the convolution > operator (sort of multiplication between functions) and * is the > normal multiplication operator, then L(f(x) o g(x)) = L(f(x)) * L(g(x)) > F(f(x) o g(x)) = F(f(x)) * F(g(x)) where L(f(x)) is the Laplace transform and F(f(x)) is the Fourier > transform of f(x). ==== Is there a better group for computational geometry ? ==== > Is there a better group for computational geometry ? comp.graphics.algorithms or comp.theory are probably better than here, not so much because they're more topical but because they have better snr. -- David Eppstein http://www.ics.uci.edu/~eppstein/ Univ. of California, Irvine, School of Information & Computer Science ==== I never know what to do to fill out the AMS Standard Cover Sheet. I think my research interests don't fit neatly into any of the subject classifications. In the past, when I've tried to choose what seems like the nearest category and hope for the best, I've found that people who seriously attach themselves to that category mean something entirely different by it and have expectations for others of that category that I never thought I was committing myself to by selecting it. As noted in the marginalia of the AMS Notices, the cover sheet is intended to help departments process job applications. As such, it serves a purely bureaucratic purpose and could perhaps be forgiven on that basis if it were then not also used to determine how one is to be treated. I don't know what they should do instead, but I am finding the codification of primary and secondary interests to be a primary and secondary obstacle to putting together job applications. The one category that sounds sufficiently ambiguous for me to feel I'm not misrepresenting myself is 00, i.e. General. Perhaps by its nature, after looking at publications with that classification in Math-Sci, I can't figure out what it means, but that doesn't matter. What does matter is what assumptions a hiring committee will make about someone who describes his/her primary interest as 00. And what would they assume? Maybe the real purpose of the AMS Standard Cover Sheet is to help weed out people with a low tolerance for things that don't make sense, things which they would certainly encounter in more severe forms after they get hired. If someone can look over my publications on Math-Sci and figure out what the appropriate codes are for my primary and secondary interests for the AMS Standard Cover Sheet, I would appreciate the advice. I don't seem to have been programmed with this capability. Allan Adler ara@zurich.ai.mit.edu **************************************************************************** * * * Intelligence Lab. My actions and comments do not reflect * * in any way on MIT. Moreover, I am nowhere near the Boston * * metropolitan area. * * * **************************************************************************** ==== > I never know what to do to fill out the AMS Standard Cover Sheet. I think > my research interests don't fit neatly into any of the subject > classifications. > In the past, when I've tried to choose what seems like the nearest category > and hope for the best, I've found that people who seriously attach themselves > to that category mean something entirely different by it and have expectations > for others of that category that I never thought I was committing myself to > by selecting it. As noted in the marginalia of the AMS Notices, the cover sheet is intended > to help departments process job applications. As such, it serves a purely > bureaucratic purpose and could perhaps be forgiven on that basis if it were > then not also used to determine how one is to be treated. I don't know what they should do instead, but I am finding the codification > of primary and secondary interests to be a primary and secondary obstacle > to putting together job applications. The one category that sounds sufficiently ambiguous for me to feel I'm > not misrepresenting myself is 00, i.e. General. Perhaps by its nature, > after looking at publications with that classification in Math-Sci, I can't > figure out what it means, but that doesn't matter. What does matter is what > assumptions a hiring committee will make about someone who describes his/her > primary interest as 00. And what would they assume? Maybe the real purpose of the AMS Standard Cover Sheet is to help weed out > people with a low tolerance for things that don't make sense, things which > they would certainly encounter in more severe forms after they get hired. If someone can look over my publications on Math-Sci and figure out what > the appropriate codes are for my primary and secondary interests for the > AMS Standard Cover Sheet, I would appreciate the advice. I don't seem to > have been programmed with this capability. How about this: Take your papers published in the last 10 years, and see what classification they have in Math Reviews. I did that, and it seems you are a 14, with 11 in second place. Now look up what that is... 14=algebraic geometry, 11=number theory. Here we get hundreds (maybe even thousands?) of applications. Obviously no single person will read them all. A committee of ten subdivides the applications (how? perhaps by that number you are writing there), one person reads your application...and unless that person thinks it is exceptional it may be the end for you. If he/she DOES think it is exceptional, it will get additional readings. So... if an algebraic geometer is the one who reads your application, would that make you happy or not? -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ ==== Problem : Use the implicit function theorem to show that as a subspace of R^{n+1} an n-surface M is locally homeomorphic to an open subset of R^n. That is, for each p in M, there exists a neighborhood of p (call it O, which is a subset of M) and an open set U (subset of R^n) and a homeomorphism f : U --> O Steven Rossi ==== >Problem : Use the implicit function theorem to show that as a subspace of >R^{n+1} an n-surface M is locally homeomorphic to an open subset of R^n. >That is, for each p in M, there exists a neighborhood of p >(call it O, which is a subset of M) and an open set U (subset of R^n) >and a homeomorphism f : U --> O >Steven Rossi > The details depend on how your surface is defined. If it's defined (at least locally) by an equation of the form F(x) = 0, with x in R^{n+1} and F: R^{n+1} -> R being non-singular on the zero-set M, the implicit function theorem says that F locally looks like a linear function L: R^{n+1} -> R. Clearly, the zero-set of a linear function is globally (and therefore locally) diffeomorphic to an open subset of R^{n+1}, so the same is true for F. The phrase looks like means that there are locally diffeomorphisms of the domain and range space that transform F to L. John Mitchell ==== Problem : Use the implicit function theorem to show that as a subspace of >R^{n+1} an n-surface M is locally homeomorphic to an open subset of R^n. >That is, for each p in M, there exists a neighborhood of p >(call it O, which is a subset of M) and an open set U (subset of R^n) >and a homeomorphism f : U --> O >Steven Rossi > The details depend on how your surface is defined. If it's defined (at > least locally) by an equation of the form F(x) = 0, with x in R^{n+1} > and F: R^{n+1} -> R being non-singular on the zero-set M, the implicit > function theorem says that F locally looks like a linear function L: > R^{n+1} -> R. Clearly, the zero-set of a linear function is globally > (and therefore locally) diffeomorphic to an open subset of R^{n+1}, so > the same is true for F. The phrase looks like means that there are > locally diffeomorphisms of the domain and range space that transform F > to L. My surface is defined as follows : An n-surface M (which is a subset of R^{n+1}) is a non-empty subset such that M = f^{-1}(c) where f : U ----> R (U is a subset of R^{n+1}) , f is C-infinity, and such that for all p in M, p is a regular point of f (i.e. it's not a critical point of f). That's all I am given. John Mitchell ==== I found this on a web search, the result was stated without the details of derivation: gravitational field around a central mass m, the eccentricity (e) is given by e=sqrt(1 + 2Eh^2 / m^2)where E = v^2 / 2 - m/r is the total energy (kinetic plus potential), h = rv_t is the angular momentum, v is the total speed, v_t is the tangential component of the speed, and r is the radial distance from the center of the mass. (Note v_t is v subscript t). ==== > I found this on a web search, the result was stated without the > details of derivation: gravitational field around a central mass m, the eccentricity (e) is > given by e=sqrt(1 + 2Eh^2 / m^2)where E = v^2 / 2 - m/r is the total > energy (kinetic plus potential), h = rv_t is the angular momentum, v > is the total speed, v_t is the tangential component of the speed, and > r is the radial distance from the center of the mass. (Note v_t is v > subscript t). > I'd suggest you reference Becker's book Introduction to Theoretical Mechanics, Section 10-5 in which he addresses Equation of the Orbit by the Energy Method. While the equations of the derivation are a bit too much to post on Usenet, essentially Becker begins with expressions for the total energy and angular-momentum : T + V = W mr^2*Theta(dot) = J After a bit of integration and routine substitutions, Becker imports a previously derived relationship derived earlier in chapter 10: r = 1/((1/ep) - (1/p)Cos(Theta)) and obtains: e = sqrt((2WJ^2)/mk^2) + 1) This is a remarkably similar to form as the equation that you posted, and by expanding to equivalent terms as that you posted, I suspect that they will or should turn out to be equivalent. I wasn't sufficiently ambitious to perform this final step. Harry C. p.s. My copy of Becker's text is dated 1954 (Lord, how time flies!), so you may have a difficult time locating a copy. If so, I'd be happy ==== > I found this on a web search, the result was stated without the > details of derivation: gravitational field around a central mass m, the eccentricity (e) is > given by e=sqrt(1 + 2Eh^2 / m^2)where E = v^2 / 2 - m/r is the total > energy (kinetic plus potential), h = rv_t is the angular momentum, v > is the total speed, v_t is the tangential component of the speed, and > r is the radial distance from the center of the mass. (Note v_t is v > subscript t). > Your m in the above is actually mu = G*M, the gravitational parameter. It's possible that you've chosen a canonical system of units where G = 1. You will find a derivation of this expression in Fundamentals of Astrodynamics by Bate et al. The text is only about ten bucks on Amazon; it's a great buy. ==== Blindly stumbling around with a spreadsheet and the Mathworld website, I came across the following O(1) approximation to the prime counting function pi(x): If we let: L = ln(x)/ln(10) And then: a = L^[2/(3L)] b = 1 + 2/(17*[cosh(L-e^2)^(1/32)]) (where e is the Euler number 2.71828...) Then using a and b, let: k = 1 + ln(a/b) Then pi(x) ~= f(x) = ceiling[ k.x/ln(x) ] The actual values of pi(x) for the first 23 powers of ten are listed here: http://mathworld.wolfram.com/PrimeCountingFunction.html For comparison, the table below lists values of this approximation for the same powers of ten. n: f(10^x) 1: 4 2: 25 3: 166 4: 1226 5: 9632 6: 78921 7: 666544 8: 5768728 9: 50891480 10: 455331213 11: 4119162101 12: 37603713448 13: 345905216269 14: 3202537348491 15: 29816016445424 16: 278937171003645 17: 2620625350176597 18: 24713263774290675 19: 233830772554480467 20: 2219050621287889610 21: 21115275766387628492 22: 201408873490832120876 23: 1925378259117770127176 24: 18442719150950880371139 Carl ==== >Blindly stumbling around with a spreadsheet and the Mathworld website, I >came across the following O(1) approximation to the prime counting >function pi(x): If we let: > L = ln(x)/ln(10) And then: > a = L^[2/(3L)] b = 1 + 2/(17*[cosh(L-e^2)^(1/32)]) > (where e is the Euler number 2.71828...) Then using a and b, let: > k = 1 + ln(a/b) Then > pi(x) ~= f(x) = ceiling[ k.x/ln(x) ] The actual values of pi(x) for the first 23 powers of ten are listed >here: http://mathworld.wolfram.com/PrimeCountingFunction.html For comparison, the table below lists values of this approximation for >the same powers of ten. The error for 10^23 appears to be 57867510952120449. Why do you call this an O(1) approximation? You have some reason to think the error is never larger than 57867510952120450 or what? > n: f(10^x) > 1: 4 > 2: 25 > 3: 166 > 4: 1226 > 5: 9632 > 6: 78921 > 7: 666544 > 8: 5768728 > 9: 50891480 >10: 455331213 >11: 4119162101 >12: 37603713448 >13: 345905216269 >14: 3202537348491 >15: 29816016445424 >16: 278937171003645 >17: 2620625350176597 >18: 24713263774290675 >19: 233830772554480467 >20: 2219050621287889610 >21: 21115275766387628492 >22: 201408873490832120876 >23: 1925378259117770127176 >24: 18442719150950880371139 Carl ************************ David C. Ullrich ==== > The error for 10^23 appears to be 57867510952120449. Why do > you call this an O(1) approximation? You have some reason to > think the error is never larger than 57867510952120450 or what? My apologies. By training I am a computer scientist, and so had forgotten that the O(x) notation is more commonly interpreted as an indication of the size of difference from some required value. By O(1), I was defining the _computational_ complexity with regard to the variable 'x' and not defining the size of an error term. Simple formulae are not normally used for calculating, say, the Riemann Prime Counting function or the Logarithmic integral, as both are usually calculated by summing an infinite series. Summing an infinite series is computationally O(p) where p is some chosen limit of precision. Again, sorry for any confusion. Carl ==== >There are various versions of this. I give three possible solutions at the end. Please comment. > >Infinitely many balls, each numbered (#1,#2,#3, etc.) are to be placed into a bucket, ten at a time, by the scheme given below. Immediately after each group of ten are placed in the bucket, one is removed and discarded. The process is as described below. >> >>11am: Balls #1 - #10 placed into the bucket. Ball #1 is removed and discarded. >> >>11:30am: Balls #11-#20 placed into the bucket. Ball #2 removed and >>discarded. >> >>11:45am: Balls #21-#30 placed into the bucket. Ball #3 removed and >>discarded. >> >>11:52.5am: Balls #31-#40 placed into the bucket. Ball #4 is removed and >>discarded. >> >>Etc. >> >>The process continues by halving the remaining time until 12 noon. Then ten are placed in and one is removed and discarded by the above scheme. The remaining time is halved again, etc. There is a flurry of activity just prior to 12 noon. The process does not continue at or beyond 12 noon. >> >>Question: How many balls remain in the bucket at 12 noon? >> >>There are three common, though not necessarily correct, replies. >> >>1) In that the net gain is +9 balls per event, and there are infinitely >>many events, there are infinitely remaining balls in the bucket. >> >>2) None remain. Any given ball, say ball #k, is removed and discarded at a specific time prior to 12 noon. >> >>3) The question is meaningless as the process can not be extended to or >>beyond 12 noon. >> >>Comments? >> You are trying to deal with infinity, or limits to infinity, as if they were finite numbers. The ambiguity lies in what limit is requested. Let's say a transaction includes of both actions, putting in the ten new balls and removing the least numbered ball. Let x_n be the total cumulative number of balls placed in the basket through the nth transaction. Let y_n be the number of balls remaining in the basket just after the nth transaction. Let z_n be the smallest amongst the numbers on the balls remaining in the basket just after the nth transaction. (As others have noted, the time of the events is a red herring.) Then z_n = n + 1, and y_n = x_n - z_n + 1. Since z_n approaches infinity as n does, in (2) you want to conclude that y_n approaches 0. Yet you are playing with the devil (as I think Cauchy said): you are subtracting one infinite limit from another. For it is clear that x_n = 10n, whence y_n = 9n -> infinity as n -> infinity (your conclusion 1). In other words, it is simultaneously true that both the number of balls in the basket and the number of balls removed grow arbitrarily large as time approaches noon. It makes no sense to talk about what is in the basket at noon, as time is outside the realm of your thought experiment (your conclusion 3), which is not physically realizable. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu ==== > In other words, it is simultaneously true that both the number of balls > in the basket and the number of balls removed grow arbitrarily large as > time approaches noon. It makes no sense to talk about what is in the > basket at noon, as time is outside the realm of your thought experiment ^^^^^^^ > (your conclusion 3), which is not physically realizable. I meant to type as *that* time, i.e., noon. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu ==== I was wondering whether there be a name for a faithful functor F: C --> D such that given any isomorphism f': F(a) --> b' in D, there is unique object b of C such that for some isomorphism f: a --> b, F(f) = f'? In other words, when F is into the category of sets, given a structure on a and a bijection f from a into b, there is a unique structure on b such that f is an isomorphism from a into b. This seems to be the categorical notion with which you would want to replace Bourbaki's notion of structure and morphism (as defined in Theories des Ensembles, Ch. IV) if you want all his results to work out yet (understandably) want to use a categorical approach. It is what I prefer to think of when he uses the concepts of morphism and structure. ==== |I was wondering whether there be a name for a faithful functor F: C |--> D such that given any isomorphism f': F(a) --> b' in D, there is |unique object b of C such that for some isomorphism f: a --> b, F(f) = |f'? In other words, when F is into the category of sets, given a |structure on a and a bijection f from a into b, there is a unique |structure on b such that f is an isomorphism from a into b. This |seems to be the categorical notion with which you would want to |replace Bourbaki's notion of structure and morphism (as defined in |Theories des Ensembles, Ch. IV) if you want all his results to work |out yet (understandably) want to use a categorical approach. It is |what I prefer to think of when he uses the concepts of morphism and |structure. if i'm not misunderstanding what you said too badly, then it sounds like you really want to consider only the case where c and d are groupoids rather than arbitrary categories. (in particular when you take the objects of d to be the sets it seems that you want the morphisms to be the bijections rather than the arbitrary functions.) in that case i think the standard name for a functor of the type you're describing would be something like discrete fibration, and yes they're used for exactly the purpose that you're describing. -- ==== Consider the operator -d^2/dx^2 - F x + p(x) + q(x) on the L_2(0, +infty), where F>0; p(x) -- real periodic function from L_{1, loc}, p(x+1) = p(x); q(x) -- decreasing function. Does the spectrum of such operator? What condition on the p and q for preservation absolutely continuous cpectrum? ==== Please help me with my problem ,,,, Consider : X_1, X_2, ... is a sequence of independent random variables with finite variances and a common distribution F such that F(0) = 0. What is the relationship between 2*E[(X_n)^2] and E[1 / ( (X_{n-1})^2 + (X_{n-2})^2 ) ]? Is one always at least as large as the other? (E denotes expectation) If I replaced 2*E[(X_n)^2] by C*E[(X_n)^2], what value of C would still make this true? (C is a real number) Henrique ==== (axiom-schema) is introduced...along with power set...pairing is > redundant, they advocate keeping it because it is elementary and > essential for the sequential development and even for a scheme that > shuns replacement, pairing makes sense. I think they are suggesting > that set theory even without replacement is strong enough for almost > all practical purposes. What do you think? Oh, I see, I'm used to the shunning replacement version, because I'm > used to GBN set theory. So I think of replacement+comprehension as > just GBN's comprehension, but of course, that isn't quite right, and > this is an example why. I believe that in GBN set theory, pairing *is* required, is it not? Thomas Ha ha...I'm having trouble reading just ZF (Zermelo-Fraenkel, I think), haven't yet reached ZFC (with Choice)...let alone ZFC+ (with construction...the authors refer to Godel...and I've heard scary things about that guy)...but I would very much like to know what GBN stands for. VNB starts with von Neuman, I forgot the other guy's name. Quine appears all the time in the footnotes. What did he do to logic? So much for the summer vacasion. ==== > Ha ha...I'm having trouble reading just ZF (Zermelo-Fraenkel, I > think), haven't yet reached ZFC (with Choice)...let alone ZFC+ (with > construction...the authors refer to Godel...and I've heard scary > things about that guy)...but I would very much like to know what GBN > stands for. VNB starts with von Neuman, I forgot the other guy's name. > Quine appears all the time in the footnotes. What did he do to logic? Goedel, Bernays, von Neuman. It's a different formulation of set theoretic axioms, now proved equivalent in the relevant ways, which explicitly has proper classes as a part of the system. Thomas ==== forgot what the little numbers at the top and botom of the integral sign are called. One of my problems has a triangle with verticies at (0,0), (1,1), and (0,1) and I had to do something to it to end up with this: [int from 0 to 1] {[integral from 0 to y] x dx} dy My book showed it to me this way and I figured why couldn't I switch the order of integration: [int from 0 to 1] {[int from x to 1] x dy} dx I solved both and the answers matched (I hope I integrated correctly). This was easy because the shape I was integrating over was simple and had simple numbers. My question is do you have any hints/tips/sites that show how to determine the correct numbers/variables in the integrand. ==== <4ab51$3fd8af60$80dcc27d$31393@msgid.meganewsservers.com>, > forgot what the little numbers at the top and botom of the integral sign are > called. One of my problems has a triangle with verticies at (0,0), (1,1), and (0,1) > and I had to do something to it to end up with this: [int from 0 to 1] {[integral from 0 to y] x dx} dy My book showed it to me this way and I figured why couldn't I switch the > order of integration: [int from 0 to 1] {[int from x to 1] x dy} dx I solved both and the answers matched (I hope I integrated correctly). This > was easy because the shape I was integrating over was simple and had simple > numbers. My question is do you have any hints/tips/sites that show how to > determine the correct numbers/variables in the integrand. > It usually helps me a lot to first draw a sketch of the region of integration and label each boundary piece with its equation. ==== I have the following application: I have a force sensor probe that touches 3 points on a plate. At each of the points, I can calculate an XYZ point representing the point that the probe detectes the plate. I can easily calculate the equation of the plane Ax+By+Cz+D = 0. What I really need is the rotation matrix/Euler angles as represented by the plane so I can correctly translate positions along the slope of the plane. Internally, I am storring all my locations and frames of reference as XYZ points and Euler angles in terms of DZ, DX, DZ rotations. Any thoughts would be appreciated ==== | | [...] | | I hope this helps! Yes, very much. This got me interested. Do you know any good books and/or webpages that explain these two different viewpoints on tensors and their relationship. Boris -- boris@uncommon-sense.net - He who hesitates is last. ==== I am trying to find a primitive function for y = (1 - x^2)^2 but have now idea how to do. Now I am wondering if there is a formula describing how to find primitive functions for this and similar functions. -- /Torbj.9arn Svensson Diaz Please visist this site. http://www.againsttcpa.com/ ==== I am trying to find a primitive function for y = (1 - x^2)^2 but have > now idea how to do. Now I am wondering if there is a formula > describing how to find primitive functions for this and similar > functions. Note (1 -x^2)^2 = 1 - 2*x^2 + x^4, and can be integrated term by term. ==== : I am trying to find a primitive function for y = (1 - x^2)^2 but have Typo. It's supposed to be y = (1 - x^2)^(1/2). -- /Torbj.9arn Svensson Diaz Please visist this site. http://www.againsttcpa.com/ ==== > : > I am trying to find a primitive function for y = (1 - x^2)^2 but have Typo. It's supposed to be y = (1 - x^2)^(1/2). So this is an example where f and g both have algebraic primitives, but their composition does not. Integrals of square-roots of cubic or quartic polynomials are elliptic integrals. ==== >I am trying to find a primitive function for y = (1 - x^2)^2 but have Typo. It's supposed to be y = (1 - x^2)^(1/2). Trig substituion: let x = sin(t), etc. ==== >: >I am trying to find a primitive function for y = (1 - x^2)^2 but have Typo. It's supposed to be y = (1 - x^2)^(1/2). Mathematica gives (x * sqrt(1-x^2) + arcsin(x)) / 2. Looks correct. Maybe a clever trigonometric substitution like x = sin t: Int[ (1 - x^2)^(1/2) dx ] = Int[ (cos t)^2 dt ] = Int[ 1/2 * ( 1 + cos 2x) ] = 1/2 * [ t + 1/2 * sin 2x ] = t/2 + sin(2x)/4 = t/2 + (sin t cos t) / 2 Substituting back gives: = arcsin(x) / 2 + (x cos(arcsin x)) / 2 = arcsin(x) / 2 + x * Sqrt(1-x^2) / 2 Presto! ==== Helping my son with his probability theory course, we have a problem with the following exercise: A rat is in a chamber in a maze. There are 3 doors to the chamber. Door_1 will make the rat return to the chamber after 3 minutes. Door_2 will make the rat find its way out after 2 minutes. Door_3 will make the rat return to the chamber after 5 minutes. The probabilities that the rat takes the doors 1, 2 and 3 are resp. 1/2, 1/6 and 1/3. Find the expected value and the variance of the total time spent in the maze. This is an exercise on the theorem on conditional expected values: E[X] = Sum( all t, E[X|T=t] * P[T=t] ) It happens very early in the course, just before the chapter on Markov Chains. If we label the total time spent in the maze as X, and the first door chosen as T, we have P[T=1] = 1/2 P[T=2] = 1/6 P[T=3] = 1/3 We can calculate the expected value of X as follows: E[X|T=1] = 3+E[X] 3 minutes, then back to square one E[X|T=2] = 2 2 minutes and leaving the maze E[X|T=3] = 5+E[X] 5 minutes, then back to square one Using this, we get the equation E[X] = (3+E[X])*1/2 + 2*1/6 + (5+E[X])*1/3 which produces E[X] = 21 Then we *tried* the following to calculate E[X^2]: E[X^2|T=1] = (3+E[X])^2 E[X^2|T=2] = 4 E[X^2|T=3] = (5+E[X])^2 Using the value E[X] = 21 , we get E[X^2] = (3+21)^2*1/2 + 4*1/6 + (5+21)^2*1/3 = 514 and thus VAR[X] = E[X^2] - E[X]^2 = 514 - 21^2 = 73 I have ran a simulation (producing 10^8 values of X) of this and I got average[X] = 21.00034 average[X^2] = 878.86246 average[(X-21)^2] = 437.84824 and indeed average[X^2] - (average[X])^2 = 437.84824 This 437.84824 should be very close to the variance VAR[X] = E[X^2] - E[X]^2 but it is not very close to 73 at all, so something is obviously wrong and I suspect the error is in the second calculation. Anyone any idea how to correctly calculate E[X^2] and/or VAR[X]? Dirk (and Bert) Vdm ==== You have misstake when you calculate E[X^2]. This formula wrong E[X^2] = (3+21)^2*1/2 + 4*1/6 + (5+21)^2*1/3. You have to write following Let P -- measure on your probability space, s -- time of walking E[X] = int s(w) P(w) = 21 Here you absolutely right. E[X^2] = int s^2(w) P(w) = int (s + 3)^2(w) P(w) * (1/2) + 2^2 * (1/6) + int (s + 5)^2(w) P(w) * (1/3) = int s^2(w) P(w) * (1/2) + int 6 s(w) P(w) * (1/2) + int 3^2 P(w) * (1/2) + 2/3 + int s^2(w) P(w) * (1/3) + int 10 s (w) P(w) * (1/3) + int 25 P(w) * (1/3) = E[X^2] * (1/2) + 3 * E[X] + 9/2 + 2/3 + E[X^2] * (1/3) + (10/3) * E[X] + 25/3 Thus, E[X^2] * (1/6) = 3*9 + 27*2 + (19/3) * E[X], E[X^2] = 879, E[X^2] - E[X]^2 = 438. > Helping my son with his probability theory course, we have a problem > with the following exercise: A rat is in a chamber in a maze. There are 3 doors to the chamber. > Door_1 will make the rat return to the chamber after 3 minutes. > Door_2 will make the rat find its way out after 2 minutes. > Door_3 will make the rat return to the chamber after 5 minutes. > The probabilities that the rat takes the doors 1, 2 and 3 are > resp. 1/2, 1/6 and 1/3. > Find the expected value and the variance of the total time spent > in the maze. This is an exercise on the theorem on conditional expected values: > E[X] = Sum( all t, E[X|T=t] * P[T=t] ) > It happens very early in the course, just before the chapter on > Markov Chains. If we label the total time spent in the maze as X, and the first > door chosen as T, we have > P[T=1] = 1/2 > P[T=2] = 1/6 > P[T=3] = 1/3 We can calculate the expected value of X as follows: > E[X|T=1] = 3+E[X] 3 minutes, then back to square one > E[X|T=2] = 2 2 minutes and leaving the maze > E[X|T=3] = 5+E[X] 5 minutes, then back to square one > Using this, we get the equation > E[X] = (3+E[X])*1/2 + 2*1/6 + (5+E[X])*1/3 > which produces > E[X] = 21 Then we *tried* the following to calculate E[X^2]: > E[X^2|T=1] = (3+E[X])^2 > E[X^2|T=2] = 4 > E[X^2|T=3] = (5+E[X])^2 > Using the value > E[X] = 21 , > we get > E[X^2] = (3+21)^2*1/2 + 4*1/6 + (5+21)^2*1/3 = 514 > and thus > VAR[X] = E[X^2] - E[X]^2 = 514 - 21^2 = 73 I have ran a simulation (producing 10^8 values of X) of this and I got > average[X] = 21.00034 > average[X^2] = 878.86246 > average[(X-21)^2] = 437.84824 > and indeed > average[X^2] - (average[X])^2 = 437.84824 This 437.84824 should be very close to the variance > VAR[X] = E[X^2] - E[X]^2 > but it is not very close to 73 at all, so something is obviously > wrong and I suspect the error is in the second calculation. Anyone any idea how to correctly calculate E[X^2] and/or > VAR[X]? Dirk (and Bert) Vdm ==== Awhile back I posted this linier equation --- e^2 - (1/(n-2))/(1/n) = 0 I had no closed form for n at the time. The closed form for (n) in this case I just discovered --- n = (1/((1/((1/(e^2 - 7))*2))+3))+2 where n satisfies this equation --- e^2 - (1/(n-2))/(1/n) = 0 where n = 2.3130352854993313036361612469... Where the cf of n = [2:3,5,7,9,11,13,15,17,19,21,23,25,27,...] This cf contains all the odd integers ---->oo after even integer 2. As you will note, the last two terms on the right in the closed form for (n) are the first two terms in its cf in reverse order. This cf. is a simple ordered progression of an irrational (n) even more significant than e's cf. because of its simplicity. Where the cf. of e = [2:1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1...] Strange how (e^2) has no order in its cf but the (n) in my equation has. This constant (n) probably disserves a spot in OEIS because of its well ordered continued fraction derived from e^2. Dan ==== Dan schrieb: Strange how (e^2) has no order in its cf but the (n) in my equation > has. > cf(e^2) = [ 7, 2, 1, 1, 3, 18, 5, 1, 1, 6, 30, 8, 1, 1, 9, 42, 11, 1, 1, 12, 54, 14, 1, 1, 15, 66, 17, 1, 1, 18, 78, 20, 1, 1, 21, 90, 23, 1, 1, 24, 102, 26, 1, 1, 27, 114, 29, 1, 1, 30, 126, 32, 1, ... ] computed by the gnu-program maxima Gottfried Helms ==== Suppose I want to studdy a function f : G -> H where H is the quaternion division algebra, hypercomplex numbers of dimension 4 and G is a given (commutative) group, for example a finite one e.g. Z/nZ. Is there a kind of 'representation theory' with values on H ? What can be said for the 'caracter' of Z/nZ with walue in H ? I think they are of the form phi_k : l -> e^{ j*2*pi*k*l/n } for j a unit pure quaternion. Once you have fixed j, you get a complete set of caracter, and you can compute Fourier-like decomposition, with an inversion formula. Is it that simple ? I would like some pointer on classical studies of this theory. G. ==== > As can be seen by the number of posts in this thread, > and the references to his web site in thousands of other posts, > a computer programmer, who took some data processing classes > at a third rate California college, has become a highly regarded expert > in math, physics, and other science disciplines, and > many people, who pretend to be rational, intelligent, open-minded > scientists (Or at least, pretend to have a scientific mind.), > frequently use this programmer as a major reference. So what-if it is true? What is your point? Suppose x is a real number and 0=0 compute: 1: a[i]:=the largest integer k s.t. floor(k*x[i])=1 2: x[i+1]:=a[i]*x[i]-1 It's not too difficult to see: (1) 2<=a[i]<=5 if i>0, and (2) x=1/a[0]+1/(a[0]*a[1])+... So a[i] is reasonably called the ith digit of x. In a very nice post from several years ago, Rob Johnson showed the expected distribution of the digits of a random x is [206,81,30,20]/337 (tinyurl.com/ys8t). Thus p(a[i]=5) is 20/337 or about .059347. The exact distribution of the first n digits of log(2) is as follows (according to Pari/GP): n freq 1000:[587,276,87,50] 2000:[1179,537,178,106] 3000:[1783,793,267,157] 4000:[2404,1035,360,201] 5000:[3008,1274,448,270] 6000:[3618,1529,528,325] 7000:[4240,1753,625,382] 8000:[4886,1943,727,444] 9000:[5490,2194,912,562] 10000:[6113,2413,912,562] 20000:[12236,4817,1780,1167] 30000:[18337,7266,2678,1719] 40000:[24449,9716,3545,2290] 50000:[30564,12105,4443,2888] 60000:[36692,14483,5368,3457] 70000:[42847,16819,6291,4043] It is kind of curious to note the observed frequency of the digit 5 is *always* less than expected. Is there an explanation for this wierdness? rich ==== > Suppose x is a real number and 0=0 compute: 1: a[i]:=the largest integer k s.t. floor(k*x[i])=1 > 2: x[i+1]:=a[i]*x[i]-1 It's not too difficult to see: (1) 2<=a[i]<=5 if i>0, and (2) > x=1/a[0]+1/(a[0]*a[1])+... So a[i] is reasonably called the ith digit of x. In a very nice post from several years ago, Rob Johnson showed the expected > distribution of the digits of a random x is [206,81,30,20]/337 > (tinyurl.com/ys8t). Thus p(a[i]=5) is 20/337 or about .059347. The exact > distribution of the first n digits of log(2) is as follows (according to > Pari/GP): n freq 1000:[587,276,87,50] > 2000:[1179,537,178,106] > 3000:[1783,793,267,157] > 4000:[2404,1035,360,201] > 5000:[3008,1274,448,270] > 6000:[3618,1529,528,325] > 7000:[4240,1753,625,382] > 8000:[4886,1943,727,444] > 9000:[5490,2194,912,562] > 10000:[6113,2413,912,562] > 20000:[12236,4817,1780,1167] > 30000:[18337,7266,2678,1719] > 40000:[24449,9716,3545,2290] > 50000:[30564,12105,4443,2888] > 60000:[36692,14483,5368,3457] > 70000:[42847,16819,6291,4043] It is kind of curious to note the observed frequency of the digit 5 is > *always* > less than expected. Is there an explanation for this wierdness? > Always? Or just when n ends in 000? ==== Is there an arithmetic compression formula or algorithm that can be used to store a numeric 64bit or numeric 32bit quantity in a compressed format into a numeric 16bit quantity? Then, when needed, decompress back into a 32bit or 64bit quantity? Is this a totally ridiculous request?? :-) :-) Guru ==== Is there an arithmetic compression formula or algorithm that can be used > to store a numeric 64bit or numeric 32bit quantity in a compressed format > into a numeric 16bit quantity? Then, when needed, decompress back into a 32bit or 64bit quantity? The number of different 16bit quantities is 65536. Applying your decompress to each of those will result in only one 32bit or 64bit quantity. In the case of decompressing 16bit to 32bit, this means that at least 4294901760 32bit quantities can not result from your decompress. -- Daniel W. Johnson panoptes@iquest.net http://members.iquest.net/~panoptes/ 039 53 36 N / 086 11 55 W ==== Is there an arithmetic compression formula or algorithm that can be used to store > a numeric 64bit or numeric 32bit quantity in a compressed format into a numeric > 16bit quantity? Then, when needed, decompress back into a 32bit or 64bit quantity? Is this a totally ridiculous request?? :-) :-) > The VAX instruction set allows you to a subset of the 32, 64 or 128 bit floating point values (32 bit F_Floating, 64 bit D_floating or G_floating and 128 bit H_floating) as six bit literals in the instruction stream. You get a 3 bit exponent and a 3 bit (plus the free bit from normalization) exponent. That allows you to encode 1/2 through 15/16 in units of 1/16, 1 through 1 and 7/8 in units of 1/8 and so on up to integers 64 through 120 in steps of 8. If you need to encode a number outside of that set of 64 values, you need to use a different encoding. John Briggs ==== > Then, when needed, decompress back into a 32bit or 64bit quantity? Lossless compression can't be achieved unless your number have some special properties. If your number is taken from a series with some temporal coherence / scale coherence / spectral coherence (all these 3 are of course somewhat related !) you can perform a lossy compression with ratio 1/4 and store the result in an array of 16 bit float. This will be much more efficient than making a raw quantization of the data (e.g. cast your data from double to 16bit !). try google with 'dct' or 'wavelet transform'. Gabriel ==== Is there an arithmetic compression formula or algorithm that can be used > to store a numeric 64bit or numeric 32bit quantity in a compressed format > into a numeric 16bit quantity? Then, when needed, decompress back into a 32bit or 64bit quantity? Is this a totally ridiculous request?? :-) :-) Yes :-) -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) ==== Goedel is to a discussion about the meaning of life and fate of the universe as Newton is to a discussion about how I got to the grocery store and what I am having for dinner. Painting with mathematics, logic, computer science, and cosmology, and then trying to fit Life, the Universe, and Everything onto that canvas just won't work. There is something else in the room - You. The perciever. That which sees. That which is asking the questions. That which is holding the brush. Turn towards the East, and seek the Vedas, the Buddhist texts, and Zen writings. They have a lot to say about that-which-is-holding-the-brush. Artists and philosophers and even the laymen of the era use the current knowledge of the day to fit their life, and draw conclusions about the world around them. It happened with Aristotle (logic, rhetoric, taxonomy), with Copernicus (Man is Not the measure of all things), with Einstein (there are no absolute truths), with quantum physics (there is no objective reality). We read about these discoveries, we think about our lives and what they mean, and then we try to correlate. We try to live as if we know what is really going on. We try to draw conclusions about life, the universe, and everything. ==== It was clear to me what Paul was talking about. Suppose you have > an algorithm that allows you to compute the state of the universe > at time t given the state of the universe at time 0. You still may > not be able to answer questions of the form If the universe is in state s0 at time 0, then will there ever > be a time in which the universe is in state s1? The quantification over all future times implicit in the question > can make the question undecidable even if the evolution is perfectly > algorithmic. Sure. Now how does an evolutionary process solve this? Paul makes an analogy with a Turing machine. Given a description of > a Turing machine and the tape at time 0, you can compute the state at > any future time. But you still may not be able to answer the question: > Does there exist a time at which the Turing machine halts? Paul seems to think that a nondeterministic Turing machine, or an > evolutionary process, are somehow able to solve this problem, which > they are not. There is some axiom that solves every halting problem. If you insist on a single approach to developing mathematics you will run up against a Goedel limit and there will be some of these axioms that you never consider. Alternatively you can consider every possible axiom (at least in a potentially infinite universe). Many of these axioms are false but every halting problem will be solved by one of these. When I talk about a nondeterministic process I am talking about a process that explores, not every possible axiom, but an ever expanding set of mutually incompatible axioms. The crucial difference from a single path approach is that you never come to agreement about which of these ever expanding number of alternatives is correct. This is the price you pay for avoiding a Goedel limit. Such a process resembles biological evolution with its great and (given the resources) expanding number of species. I think the similarity of these structures is not coincidental. I think unlimited creativity requires unlimited diversity and this is an implication of Goedel's theorem. -- Paul Budnik Mountain Math Software http://www.mtnmath.com 408 353 3824 ==== > There is some axiom that solves every halting problem. If you insist on a > single approach to developing mathematics you will run up against a Goedel > limit and there will be some of these axioms that you never consider. > Alternatively you can consider every possible axiom (at least in a > potentially infinite universe). Many of these axioms are false but every > halting problem will be solved by one of these. But solved is a far cry from solved correctly, you know. > When I talk about a nondeterministic process I am talking about a process > that explores, not every possible axiom, but an ever expanding set of > mutually incompatible axioms. The crucial difference from a single path > approach is that you never come to agreement about which of these ever > expanding number of alternatives is correct. This is the price you pay for > avoiding a Goedel limit. But we can do that already, and model it right in ZFC. We don't have to exceed any bounds to talk about that. Moreover, folks working in the subject do. Thomas ==== Torkel Franzen says... > I take it you have in mind theorems of the form R is a >well-ordering, where R specifies a recursive (we may assume primitive >recursive) relation. How do you prove that it is not the case >that there is for every recursive ordinal alpha a primitive recursive >well-ordering R of order type alpha such that ZFC proves R is a >well-ordering? Is this question specifically for Paul, or can anyone play? Let R_n be the nth primitive recursive relation on naturals such that ZFC proves that R_n is a well-ordering. Assume some primitive recursive pairing function mapping pairs of naturals onto naturals. Define a well-ordering R on naturals as follows: To compute R(n,m), decode n into a pair (i,j), and decode m into a pair (k,l). If i < k, then R(n,m) = true. If i > k, then R(n,m) = false. If i=k, then R(n,m) = R_i(j,l) It seems to me that R will be a primitive recursive well-ordering, but R will be larger than any well-ordering provable in ZFC. So (unless ZFC is inconsistent) ZFC does not prove that R is a well-ordering. -- Daryl McCullough Ithaca, NY ==== > It seems to me that R will be a primitive recursive well-ordering, > but R will be larger than any well-ordering provable in ZFC. This looks good except that there is no reason to think that R is primitive recursive. But this can be fixed by modifying the construction or by appealing to general results. ==== |anyone? anyone at all have a post thats against the grain in sci.math that |didn't get them verbally abused? I don't know exactly what you mean by against the grain, or what kind of reaction you count as verbal abuse. Bill Taylor and I have both explained why we don't entirely believe the axiom of choice without getting anything that I would call verbal abuse. The axiom of choice is standard enough a part of mainstream mathematics to make disbelief in it eccentric. In my case, it's the credibility that constructivism as a philosophy of mathematics that leads me not to wholly embrace nonconstructive assumptions like the axiom of choice. I've posted about this many times without ever getting what I would call abuse. Only a tiny minority of mathematicians or people who've examined the philosophy of mathematics are actually constructivists or even take it very seriously, so I would say that this runs against the grain. Bill Taylor has different reasons; he's more interested in things like predicativity. One could argue that if it doesn't get much of a negative response, it must not be so much against the grain after all. This is one reason why I wonder what exactly you mean by against the grain and abuse. |such a nice crowd according to you all must be 1000s I don't know what you mean by a 1000. Someone who bats 1000, as in basebasll? Keith Ramsay ==== [...] |> What third rate California college? Who rated it? What criteria? |> |Hey Wormley, |as you use this programmer's web site as your primary rederence, |it seems to me that you should know what college your resident expert |attended. So you want the credibility of sources to be *more* dependent on their credentials and occupation than they are now? Keith Ramsay ==== >Something is proven only >when there is concensus that the supplied evidence is >valid and the conclusions made are, by concensus, agreed too. Logic is not a popularity contest. -- Wolf Kirchmeir, Blind River ON Canada Nature does not deal in rewards or punishments, but only in consequences. (Robert Ingersoll) ==== Let a_n = n*Pi^2/6 - sum(1/(k*C(n+k-1,n)),k=1..infinity), where C(n+k-1,n) is binomial coefficient n+k-1 take n. Then {a_n} is a sequence of rational numbers which starts out {0,2,15/4,49/9,1025/144,5269/600,...}. Is there a formula, perhaps involving Bernoulli or Stirling numbers for a_n? Jim Buddenhagen -- To reply copy jbuddenh@REMOVEtexas.net to address bar and edit out REMOVE ==== ... > of solutions for n=3...7; some solution times in seconds; > and some solution-sizes, ie, number of rectangles in solutions: > n #sols time Solution-sizes ... > 8 >2330 >5400 14 16-27 > (n=8 has run about 1.5 hours so far, on my 450MHz Athlon) n=8 finished this morning, after 48.5 cpu hours (1000 times longer than at n=7), with 3434 solutions. I don't plan to devote the years of computer time that exhaustive searches for n>8 would take; however, I ran n=9 to 12 for a few seconds each with following results: n Smallest-solution-size-seen Run-time and sol-count in that time 9 9 (10s, 91) 10 14 (10s, 34) 11 19 (10s, 12) 12 16 (70s, 46) I also tested 13 to 20 at about 15 minutes each but got no solutions. Here's the reported 9-cover at n=9 -- Cells Prime Corner Form 18 743 0 6 2x9 18 739 0 4 2x9 14 577 1 2 2x7 10 223 0 0 2x5 9 373 0 8 1x9 6 401 6 0 2x3 2 151 8 2 2x1 2 107 5 0 2x1 2 7 0 2 2x1 (Corner = top left corner's row and column numbers) -jiw ==== > ... > the 3-by-3 case has at least 6 solutions (where I stopped counting); > but the 4-by-4 grid has (I believe I proved, perhaps) only 2 solutions. > (I am referring to the back-and-forth variation of the puzzle.) (You can try to find the 4-by-4 solutions yourself. > More fun, and not too hard, is trying to prove that there > are ONLY 2 solutions for the grid. > Extra credit if you find more than 2 solutions!) There are 4 solutions at n=4: 1 2 3 4 aabb aabb aabb aabb > 8 7 6 5 aacc cdee aacd cdef > 9 10 11 12 aacc cdee aacd cdef > 16 15 14 13 ddcc ffee eecf ggeh Yeah, I missed that the combo of primes in the upper left could be added into a single prime. of solutions for n=3...7; some solution times in seconds; > and some solution-sizes, ie, number of rectangles in solutions: > n #sols time Solution-sizes > 3 9 0.001 3 5 6 > 4 4 0.005 4 6 8 > 5 154 0.15 3 5 7 9-14 > 6 837 10 8 10 11-19 > 7 24 175 11 13 15 17 19 > 8 >2330 >5400 14 16-27 > (n=8 has run about 1.5 hours so far, on my 450MHz Athlon) > -jiw Oh, never mind then about n=8!... :) 9, 4, 154, 837, 24,... Hmmm... not in the EIS yet. ... . thanks, Leroy Quet ==== > The Schoenfeld Theorem: > An infinite bounded sequence of random numbers contains all finite > bounded sequences of numbers. > [snip] Trivially disproven by example in Hofstadter's Godel, Escher, Bach. > It's only 777 pages long. If you look at each page for one second you > can find the table within 12 minutes. What are you babbling on about now? An infinite random sequence of integers bound by [n,m] contains all > finite sequences of integers within [n,m] (not necessarily bound by > n,m though). This is proven. It's called the SCHOENFELD THEOREM. Well, I am truly glad it is not called the Heymann theorem, because it has by now been discredited thoroughly by more than one contributor. Why don't you get out whilr the boing is good? Franz Heymann ==== Friday April 26th 2002 116/249 16504 R I Z Z U T O 18 9 26 26 21 20 15 = 135 I am posting Vito Rizzuto's stats again because Shriner/Freemason Don Ocean and/or his friend James Takayama is repeatedly calling me a pedophile. I have never sexually assaulted anybody, I have never been charged with sexual assault, I have never been arrested for sexual assault, and I am not sexually attracted to children. In 1988 I criticized the phallic worship in the Protestant and Catholic churches and was repeatedly arrested and tortured at the U of S for years. Ruby would have me arrested for failing to kiss her God-damned ass and wear the clothes she was always trying to force upon me (and because Protestants and Catholics were upset with my words and lobbied her to shut me up), Ruby would have me arrested and chemically lobotomized, and then she would come to the psychiatric ward and force her choice of clothes upon me there. Eventually I found myself internet access at the U of S and posted Collecting Mail For The Coming Anti-Christ, in the essay I spoke in defense of people to wear their own choice of clothes, and this included defending the rights of women and girls to go topless if they wanted. Now Don Ocean calls me a pedophile, and as a result of Don Ocean, James Takayama in Hawaii has begun doing the same. First the fellow libels me in the WaxyOrg website using the name of Nospam, then using the name of Thomas (aka MauiCop), James Takayama quotes his own material posted as Nospam. Now another website has begun to quote the libel as being truthful, saying that I have been repeatedly arrested for sexual assault and that I was a pedophile (and it is possible that James Takayama is responsible for this website and may start others where he will claim I have been repeatedly arrested for sexual assault). Now as a result, people in Saskatoon are calling me a pedophile and are threatening to beat me up and kill me. The Saskatoon police are not in the habit of charging people for assault when they give me beatings (in fact they arrested me after James De Witt brutally assaulted me at the Seventh Day Adventist Church, I was brutally assaulted and then the police arrested me and took me to the U of S relievers when these people crack and break my ribs. The situation is quite unfair, and now Vito Rizzuto will have his stats posted daily until the situation is resolved. I get tortured year after year and begged people in futility for assistance to flee the country, now watch Vito spend huge piles of cash again this year turning trees into decorated idols, for his compassion is limited to traditions. 203 Nicolo Nicolo 68 Rizzuto 135 225 Libertina Libertina 90 Rizzuto 135 201 Vito 21 2 46 52/313 +4014 Vito 66 Rizzuto 135 177 Maria Maria 42 Rizzuto 135 Mom's first name adds to 90 (66th non-prime), Exodus contains 1213 verses (66+66+66th prime) and terminates at chapter 90 (66th non-prime). Vito adds to 66, mom and the little sister have first names averaging 66. The kids have first names adding together for 108 (the first 6 primes in prime positions). The kids have first names differing in value by the 24 chapters of Bible Books 6 and 10 (6th non-prime). Dad has a 6 lettered first name, there are 24 letters in all the first names, the number of chapters in Books 6 and 10 (6th non-prime). All first names add together for 266. Dad and Vito have first names adding together for 134, corresponding to Numbers 17 with 13 verses (the 6th prime). Mom and Vito have first names averaging 78 (6 times the 6th prime). Dad and Vito have full names adding together for the 404 verses of Bible Book 66, Revelation. Vito was born on the 21st, corresponding to Ecclesiastes (chapters 660 to 671). Vito has consonants adding to 132 (66+66). Primes In Prime Primes Positions 1 2 2 3 <- 3 3 5 <- 5 4 7 5 11 <- 11 6 13 7 17 <- 17 8 19 9 23 10 29 11 31 <- 31 12 37 13 41 <- 41 --- 108 Vito Rizzuto (135) was hot for Cammalleni (83), the names differ in value by 52, pretty as Vito was born on the 52nd day of the year. Giovanna (83) Cammalleni (83) was born with names adding to the 23rd prime and to the 23rd prime, together for 46, and she gets a husband that was born in 46. 166 Giovanna 48 Giovanna 83 Cammalleni 83 Vito married Giovanna (83) Cammalleni (83). She was born with 18 (6+6+6) letters adding to 166. Both of her names added to the 83 verses of Second Timothy (the 16th Book of the New Testament). Her names added to 83 and 83, corresponding to Exodus 33 and Exodus 33, together for 66. At birth, Vito and Giovanna had 29 letters in their names, or 6 plus the 6th prime (13) plus the 6th non-prime (13), there are 29 (6+6p+6np) chapters in Bible Book 13 (the 6th prime), there are 29 (6+6p+6np) verses in chapter 666 (Ecclesiastes 7). At birth Vito and Giovanna had 29 letters adding together for 367 (First Chronicles 29). Vito's sister's first name adds to 42 (the 29th non-prime), Bible Book 29 is Joel (42). Now Giovanna's name adds to 218 (twice the 29th prime). Nicolo's name adds to 203 (7x29). Leonardo's name adds to 219, or 3 times the 73 verses of Bible Book 29. Libertina was born in 73 (the length of Book 29 and is the Lucas numbers up to 29). Leonardo and Libertina have first names adding together for 174 (6x29). Dad and the first two kids have first names adding together for 218 (twice the 29th prime). The males were born in years adding to 182 (Deuteronomy 29 with 29 verses). 201 Vito 21 2 46 52/313 +4014 Vito 66 Rizzuto 135 218 Giovanna 48 Giovanna 83 Rizzuto 135 203 Nicolo 67 Nicolo 68 Rizzuto 135 219 Leonardo 69 Leonardo 84 Rizzuto 135 225 Libertina 73 Libertina 90 Rizzuto 135 Lucas 1 3 4 7 11 18 29 -- 73 <-the Lucas numbers up to 29 add to the 73 verses of Bible Book 29 J O E L <-Bible Book 29 10 15 5 12 = 42 <-29th non-prime C O P P E R <-29th element 3 15 16 16 5 18 = 73 <-Book 29 and is the Lucas numbers up to 29, there is a copper riding a horse on the 1973 Canadian 25 cent piece C E N T <-made out of 29th element 3 5 14 20 = 42 <-29th non-prime They have three kids, the first and last of the kids bear Vito's parent's names, so the daughter's first name adds to 90 (66th non-prime). The kids have first names adding together for 242 (First Samuel 6), all names in the family add together for 1066. The kids were born in 67, 69 and 73, these are the 19th prime, 50th non-prime and the 21st prime, together for 90 (66th non-prime and the value of the daughter's first name). Mom and her sons have first names adding together for 235... the 184th prime (1097) and the 184th non-prime (235) averages 666. The brothers have names averaging 211, it is approximately 66.6% of the 66th prime (317) and is the terminating chapter of Bible Book 6. Book 6 chapter 6 (193) plus the terminating chapter of Book 6 (211) adds together for the 404 verses of Bible Book 66. Daughter's first name not only adds to 90 (66th non-prime), but her first name adds to 66.666...% of her last name. This is a family of 5, the 4014 days dad is older than me is the 959 verses of Bible Book 5 short of the 666th prime (4973). Primes Non-Primes 2 1 3 4 5 6 7 8 11 9 13 10 17 12 19 14 23 15 29 16 31 18 37 20 41 21 43 22 47 24 53 25 59 26 61 27 67 28 71 30 73 32 79 33 83 34 89 35 97 36 101 38 103 39 107 40 109 42 113 44 127 45 131 46 137 48 139 49 149 50 151 51 157 52 163 54 167 55 173 56 179 57 181 58 191 60 193 62 197 63 199 64 211 65 223 <-48th-> 66 227 68 229 69 233 70 239 72 241 74 251 75 257 76 263 77 269 78 271 80 277 81 281 82 283 84 293 85 307 86 311 87 313 88 317 <-66th-> 90 Vito adds to 66 (48th non-prime). Vito's first name adds to 48.888...% of his last name, God gives him a wife that was born in 48. Vito and his wife have first names adding together for the 149 verses of Bible Book 48, Galatians. Vito's 201 valued name exceeds his 52nd day of birth by the 149 verses of Bible Book 48. Rizzuto (135) was hot for Cammalleni (83), the 83 value of mom's maiden name is 61.48% of the 135 value of Rizzuto. The males were born in years adding to 182, the females in years adding to 121 (66.48%). The kids are missing 15 letters from their first names, these missing letters add to 248. The 4014 days dad is older than me is 18 times the 48th prime (223), prettier as I was born on the 48th day of the year and with 317 days remaining in the year (66th prime). 1-50 - Genesis 51-90 - Exodus 91-117 - Leviticus 118-153 - Numbers 154-187 - Deuteronomy 188-211 - Joshua 930-957 - Matthew 958-973 - Mark 974-997 - Luke 998-1018 - John 1019-1046 - Acts 1047-1062 - Romans 123 <-Numbers 6, it is three times the 13th prime (41+41+41), keeping in mind that 13 is the 6th prime 188 <-the opening chapter of Book 6 is 6x6x6 short of the 404 verses of Bible Book 66, it is the 6th prime squared (13x13) short of the 357 verses of Daniel (also in part about 666) 193 <-Book 6 chapter 6 is the 44th prime, while 44 is in turn 66.666...% of 66 211 <-the terminating chapter of Book 6 is approximately 66.6% of the 66th prime (317) 357 <-the opening chapter of Book 6 plus the 6th prime squared is the 357 verses of Daniel (in part about 666) 404 <-the 6th prime squared (13x13) plus the 6th prime squared (13x13) plus 66 adds to the 404 verses of Bible Book 66 1062 <-666 plus 6x66 is a combination of the 658 verses of Bible Book 6 plus the 404 verses of Bible Book 66, and is the terminating chapter of New Testament Book 6 1070 <-666 plus the 404 verses of Book 66 is the 1070 verses of Job (Book 6+6+6) 1213 <-Exodus terminates at chapter 90 (66th non- prime) with 1213 verses (the 198th or the 66+66+66th prime) 1292 <-the 658 verses of Book 6 plus twice the 66th prime (317) is the 1292 verses of Isaiah (the Book contains 66 chapters) Vito is 4014 days older than me, his name adds to 201 (Joshua 14), his mom and daughter share the same 225 valued name (Judges 14). The kids have odd valued letters in their first names adding together for 140. The letters that are neither prime nor square in the kids' given names add to 196 (14x14). Vito and his daughter have names differing in value by 14. All first names add to 391 (314th non-prime). Dad's name adds to 67+67+67. The first of the kids was born in 67 and has names differing in value by 67. These first three family members have first names adding to 66, 83 and 68, corresponding to Exodus 16, 33 and 18, together for 67. This 67 is the 19th prime, the second kid was born in 69 (Exodus 19) and has a name adding to 219. The brothers were born in years adding to 136 (Numbers 19). Vito and the last of the kids were born in years adding to 119. Perhaps mom was 19 years old when she gave birth in 67 (the 19th prime). Mom's name adds to twice 109 (Leviticus 19) while dad's name adds to 201 (3 times the 19th prime). The parents have names adding together for 419 (Nehemiah 6 with 19 verses). Vito and his kids were born in years adding to 255 (First Samuel 19). Vito and his kids have first names averaging 77 (the primes up to 19). The kids were born in years adding to 5909 (19x311). Libertina's last name adds to 150% of her first name (150 chapters in Bible Book 19). Vito was born a multiple of 19 days into the century (16853=19x887). Vito's vowels add to 69 (Exodus 19). The 2460 verses of Bible Book 19 is 19x19+19x19+19x19+19x19+19x19+19x19+19x19 minus 67 (the 19th prime). Primes Non-Primes 2 1 3 4 5 6 7 <-4th-> 8 <-Bible Book 8 contains 11 4 chapters, pretty as 13 8 is the 4th non-prime 17 19 -- 77 <-the first 8 primes plus 8 more adds to the 85 verses of Bible Book 8 Primes In Prime Primes Positions 1 2 2 3 <- 3 3 5 <- 5 4 7 5 11 <- 11 6 13 7 17 <- 17 8 19 9 23 10 29 11 31 <- 31 12 37 13 41 <- 41 14 43 15 47 16 53 17 59 <- 59 18 61 19 67 <- 67 <-the 8th prime in prime position R U T H <-Book 8 18 21 20 8 = 67 Vito's names differ in value by 69 and his second kid was born in 69. Vito was born on the 21st and his third kid was born in 73 (21st prime). Vito was born on the 21st and his name adds to 201, and he was likely 21 years old when the first of the kids was born. Vito was born in 46, his 201 valued name exceeds it's 155th non-prime position by 46. The kids have prime and square valued letters in their first names adding together for 46. Vito was born on day 52, it's the number of chapters in Bible Book 24, while his daughter gets a name that exceeds his by 24. Vito and I were together born 530 days closer to the beginning of our years than to the end of our years (Psalm 52). This 530 is a combination of the first three perfect numbers (6, 28 and 496), they have factors that add to form themselves: Perfects 6 - 1, 2, 3 28 - 1, 2, 4, 7, 14 496 - 1, 2, 4, 8, 16, 31, 62, 124, 248 The males have first names adding together for 218, and mom's name adds to 218. Mom was born with 18 letters and took a last name that begins with the 18th letter of the alphabet, it is a last name that adds to 135 (Numbers 18). Vito and his sister have first names adding together for 108 (Leviticus 18). Rizzuto adds to 135 (the 103rd non-prime), the 11 different letters utilized in the construction of the first names for the kids add together for 103. 3x3x3x3x3x3 3x3x3 103 <- the 3x3x3rd prime ----------- 859 <- the number of verses in Bible Book 3 The parents were born in 46 and 48, Bible Books 46 and 48 contain 437 and 149 verses. These Books contain an average of 293 verses while the 437 is 293.28% of the 149. The 437 and 149 are the 353rd non-prime and 35th prime, and so it is pretty that there would be 35 letters in the family first names, and pretty that the last name would add to 135. Mom and her sons have first names adding together for 235. The brothers have first names adding together for 152 (Numbers 35). Libertina Kabatoff adds to 152 (Numbers 35), prettier as 293 is the 62nd prime while Kabatoff adds to 62. Rizzuto adds to 135, or 57 plus the 57th non-prime (78). The main Books of end-times prophecy are Daniel with 357 verses and Revelation with 404 verses, or 57 plus the 57th prime (269) plus the 57th non-prime (78). Primes Non-Primes 2 1 3 4 5 6 7 8 11 <-5th-> 9 -- -- 28 28 Vito and his sister Maria have first names adding to 66 and 42, together these Bible Books contain 1555 verses. Vito has 11 letters and a first name adding to a multiple of 11, his kids have first names adding together for 11x11+11x11 (11 is the 5th prime). Then Vito marries Giovanna (83) Cammalleni (83), there are 83 verses in Bible Book 55, and she soon finds herself in a family of 5. The 5th of 5 family members has a name adding to 225, prettier as 25 is not only 5x5 but is 5 plus the 5th prime (11) plus the 5th non-prime (9), keeping in mind that Bible Book 5 contains 959 verses. Perhaps mom was 25 (5x5 or 5+5p+5np) years old when she gave birth to the 5th of 5 family members. Libertina's first name adds to a multiple of 5 and also a multiple of 9 (5th non-prime). Libertina's (the 5th of 5 family members) first 5 letters add together for dad's 46th year of birth. My name adds to 187 (the terminating chapter of Bible Book 5), it is 5x5x5 plus twice the 5th prime in prime position (31). If Libertina took my 62 (twice the 5th prime in prime position) valued last name, then her names would have an average value of 76 (55th non-prime). Lamentations is Bible Book 25 with 154 verses, the 154th prime is 887 while Vito was born on the 19x887th day of the century, pretty because if Libertina married me then he would be lamenting. Vito and his kids already have first names adding together for 308, or twice the 154 verses of Lamentations. Note that Bible Books 5 and 5x5 differ in length by 805 verses (the 666th non-prime). And Old Testament Book 9 (the 5th non-prime) and New Testament Book 9 (the 5th non-prime) together contain 959 verses (the number of verses of Book 5). Primes In Prime Primes Positions 1 2 2 3 <- 3 3 5 <- 5 4 7 5 11 <- 11 6 13 7 17 <- 17 8 19 9 23 10 29 11 31 <- 31 12 37 13 41 <- 41 14 43 15 47 16 53 17 59 <- 59 --- 167 Esther Book 17 Vito (66) was born on day 52, it is an average of 59 (the 17th prime). have names averaging 189 (the first 17 primes minus the first 17 non-primes). Vito and Giovanna have names differing in value by 17 (7th prime, the primes up to 7 add to 17). Giovanna's name adds to 218 (Book 7 chapter 7). Vito and Giovanna were born with last names adding together for 218 (Book 7 chapter 7). The first of the kids gets a first name adding to a multiple of 17 and he was born in 67 (Exodus 17). The second gets a first name adding to 7 times the 7th non-prime (12). The brothers have first names adding to 68 and 84 (a span of 17). The daughter gets a name adding to 225 (Book 7 chapter 7+7). The kids were born in 67, 69 and 73, corresponding to Exodus 17, 19 and 23, together for 59 (the 17th prime). Libertina's names differ in value by 45 (45 chapters contain the length of 17 verses), she might take my name and end up with 17 letters. Primes Non-Primes 2 1 3 4 5 6 7 8 11 9 13 10 17 12 19 14 23 15 29 16 31 18 37 20 41 21 43 22 47 24 53 25 59 <-17th-> 26 --- --- 440 251 I wanted 17 French fry girls at my weddink, they would throw French fries high up into the air, the french fries would fall to the ground and get dirty, and nobody would ever eat french fries again. And I wanted 59 big nosed Greeks with their big noses throwing hamburglers at us from across the street. And I wanted American Noel Nibblett blowing his bugle and leading a marching regiment of cadets up and down the street while American Don Ocean and his Shriner friends ride circles around them on their little motorscooters. Scientists have recently discovered that tomatoes contain properties that help to prevent prostate cancer, in men, and since A&W allows one to take as much ketchup as they wanted, the Great A&W Rootbear was on the fast track to becoming an international symbol of health and fertility... so of course I wanted the Great A&W Rootbear to be the best man at my weddink. My weddink was soon approaching and it was goink to be a glorious affair, probababbly. 187 Dar 17 2 57 48/317 00 Daryl 60 Shawn 65 Kabatoff 62 187 Marcia 6 8 80 219/147 8571 Marcia 45 Veronica 87 Acevedo 55 207 Libertina 73 Libertina 90 Acevedo-Kabatoff 55-62 Anyway, if you people think that you have the right to use my abusive parents as tools and arrest and torture me, then I think that I should have the right to ask women to marry me, or to marry Marcia and me, our last names add together for the 117 verses of Song of Solomon, it's the Bible's Book of Love. The nubile sweety was born on the 6th and has a 6 lettered first name). Isaiah is the Book with 66 chapters, pretty as it is Book 23, or the 6th prime plus the 6th non-prime (13+10=23). Isaiah 4, 12 and 20 (adds to 6x6) each contains 6 verses. Isaiah 4:1 is about Marcia (and me), and 6 other women who are capable of feeling shame rather than pride, greed or lust, or who limit their love for traditions and for people who abide by their traditions. You people have Egyptian penises on the roofs of your churches and lined city streets with representations of penises, and had me tortured for years for saying so, others just sat back in silence while they were doing this to me, and similarly you remain silent and compassionless now that the arrests and torture have ceased. You people spent millions of dollars having me tortured, and then annually you spend billions on your decorated trees, I begged and begged for assistance to flee the country (they tortured me for years at the U of S) and you people are so cheap that you can't even offer to buy me a cookie when I bust my ass to show you evidence that your very name is a gift from God!!! Should Libertina marry me, great, but if Marcia marries me and then Libertina marries Marcia and me, then Nicolo and Leonardo are both goink to win themselves a shiny new Cadillac!!! And if Libertina turns out to be some sort of sexual acrobat then Vito and Giovanna are both goink to win themselves a shiny new Cadillac too. Or if Melinda Jarocki sleeps with me then Vito will win himself a used Pontiac. I have Scripture to support taking seven brides (Isaiah 4:1) and I have Scripture to support sleeping with Melinda Jarocki outside of wedlock (First Kings 1:1-5), while you people have a vast multitude of Scriptures condemning your decorated trees, phallic-capped churches and your violence against me for daring to point out your pagan traditions. Good luck and may God bless you!!! Daryl Shawn Kabatoff Box 7134 Saskatoon Saskatchewan Canada S7K 4J1 Isaiah 45:4, Ephesians 3:15 - God gives you your name!!! What a wonderful weddink there will be What a wonderful day for you and me Church bells will chime You will be mine In apple blossom time... Note that every day Freemason Don Ocean (or his friend James Takayama) libels me and calls me a pedophile (or such) on the usenet, then I will repost the stats for the following people: Daniel Bruner, Angel Cadwell, Brittanie Cecil, Carl Koopang, Jay Larson, Brian Lott, Adam Millikan, Cody Milliken, Christopher Ridsdale, Vito Rizzuto, Melissa Schultz, Erin Sorenson, Ann Wigdahl, Michael Winkler, or maybe more (more or less). ==== James Takayama posts under the names of Thomas, Nospam and Maui Cop. He posted libel against me using the name of Nospam in the WaxyOrg web site and then quotes the material using the name Thomas, quoting your libel and attributing it to someone else is just more libel. He has repeatedly cited my interest in Callie to support his unfounded allegations that I am a pedophile but Callie is not a child but is instead a woman. I have never been hospitalized in mental institutions for accosting women (nor do I accost women as he purports), but only for daring to criticize Protestant and Catholic churches and daring to try to show people that God provides them with their names (and for failing to abide by my parent's good advice). -Daryl S. Kabatoff HALOHA!!! ==== : at 01:28 PM, George Greene said: : >That is not only not interesting, it is the opposite of reasonable. : >If the object is itself something as trivial and structureLESS as a : >point, to begin with, asking ANYbody to think of it as something : >BIGGER (e.g., the collection of all arrows ending at it) is simply : >RIDICULOUS. : The duality principal in Projective Geometry was quite productive. It : is RIDICULOUS to pretend otherwise. : : -- : Shmuel (Seymour J.) Metz, SysProg and JOAT I have NEVER pretended otherwise, and it is WORSE than ridiculous -- it is just slanderous -- to suggest that I have. We were NOT TALKING about a projective geometry. The path category in question was over the AFFINE plane, NOT the projective one. The examples from statistics that DE was talking about represent the independent variable as the x-axis and the dependent variable as the y-axis of an AFFINE plane, NOT a projective one. Exploiting projective duality in THAT context is NOT automatic; it requires some preliminary WORK. You have to transform the affine plane INTO a projective one first. DE talked about vertical lines but the affine plane DOESN'T HAVE any of those. One way of making an affine plane projective is to add a line at infinity, each of point of which becomes the new-intersection-point for a parallel-class of lines of the original affine plane. One of these classes will be parallel to the new line at infinity, which means in some sense that the lines in this class won't fit into the new geometry. THOSE become the vertical lines for purposes of the duality that DE was talking about. They essentially have to be defined out of the original affine plane EVEN though the points in them are still there. For a given parallel-class, the point-on-the-line-at-infinity where its lines now intersect can be located as the tangent of the angle that the class forms relative to the line at infinity (perpendicular=0, parallel = +/- pi/2, which has no tangent). Of course, adding a completely independent vertical line and forgetting about all the ones you had before doesn't seem to have much of an effect on all the other original points and lines in the affine plane, so you could (and working statisticians, apparently, do) speak of the point-line duality as applying directly to them. But in the context of the original discussion, which was about representing more complex categories by simpler canonical ones, that is going to CAUSE MORE complexity than it cures. -- --- It's difficult ... you need to be united to have any strength, but internal issues have to be addressed. --- E. Ray Lewis, on liberalism in America ==== >>Actually category theory started out as a way to define ``natural >>transformation'' [There is a quote by Mac Lane to this effect. I don't >>have the precise reference handy.] > ... > >>The fundamental importance of naturality in homology and >>homotopy is impossible to overstate. > Well,if the path category in the Euclidean plane is > prototypical, what are the simplest natural transformations > that are most directly relevant to it? Natural transformations come only with functors. Anyway, I didn't mean to claim that path categories are prototypical categories (they are the test cases for weak infinity-categories though, in case, the suitable weak natural transformations will involve homotopies), but only as categories found in nature whose objects are not generally thought of as sets with structures with morphisms preserving such structures. ==== The problem really only has one answer, despite the fact that it appears as a paradox. If we merely redescribe the same problem using only mathematical and set terms, we get the answer right away. Consider the following redescription: Let the set S0 = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } {1}. Now recursively define the set Sn as follows: Sn = Sn+1 U { 10n+1, 10n+2, ..., 10n+10 } {10n+1}. Let S = lim n->inf (Sn). If you agree with the above as a mathematical description of the problem, then Solution #2 is the only correct answer, for the reasons given. Solution #1 is incorrect because it confuses Card(lim n->inf Sn) with lim n->inf Card(Sn). Solution #3 is avoided because no laws of physics are involved in this redescription, and these set operations are defined on countable unions. Jonathan Hoyle Gene Codes Corporation ==== >There are various versions of this. I give three possible solutions at the >end. Please comment. >>Infinitely many balls, each numbered (#1,#2,#3, etc.) are to be placed into >a bucket, ten at a time, by the scheme given below. Immediately after each >group of ten are placed in the bucket, one is removed and discarded. The >process is as described below. >>11am: Balls #1 - #10 placed into the bucket. Ball #1 is removed >and discarded. >>11:30am: Balls #11-#20 placed into the bucket. Ball #2 removed and >discarded. >>11:45am: Balls #21-#30 placed into the bucket. Ball #3 removed and >discarded. >>11:52.5am: Balls #31-#40 placed into the bucket. Ball #4 is removed and >discarded. >>Etc. >>The process continues by halving the remaining time until 12 noon. Then ten >are placed in and one is removed and discarded by the above scheme. The >remaining time is halved again, etc. There is a flurry of activity just >prior to 12 noon. The process does not continue at or beyond 12 noon. >>Question: How many balls remain in the bucket at 12 noon? >>There are three common, though not necessarily correct, replies. >>1) In that the net gain is +9 balls per event, and there are infinitely >many events, there are infinitely remaining balls in the bucket. >>2) None remain. Any given ball, say ball #k, is removed and discarded at a >specific time prior to 12 noon. >>3) The question is meaningless as the process can not be extended to or >beyond 12 noon. >>Comments? > Almost 4 years ago, there was a very long thread on this very topic. >> See the thread starting at > http://groups.google.com/groups?threadm=38b0da2f.5767305@news.globalcenter.ne t > As you mention, there are three ways to approach this problem; each >> boils down to a question of convergence. If we have a sequence {a_n}, >> we might ask what happens to that sequence as n tends to infinity, and >> thus, pass to the limit to define what happens at infinity. However, >> to define a limit, we must first define a topology. But the question can be answered without appealing to any sort of >topology. > ------- >> There are standard topologies on both the reals and the integers, so >> that if the a_n are reals or integers, we use those topologies. >> For example, let a_n be the number of balls in the bucket after the n^th >> placement/removal. That sequence converges to infinity as n tends to >> infinity. But that has nothing to do with the question that was asked. The question asked was, how many balls? Counting how many balls are in the bucket at any given time would seem to have something to do with the question. Just because we don't know which balls are in the bucket does not mean there are no balls in the bucket. Suppose the problem were asked in a slightly different way. Suppose that instead of ball #n being removed at step n, we remove ball #10n, that is, the last ball added. Here I don't think you will disagree that there are 9 balls added to the bucket at each step that stay there for all time. Thus, at noon, there are infinitely many balls. Suppose one ball is removed at each step, but we are not told which ball that is. What happens at noon. All we know is that at step n, there are 9n balls in the bucket. How many balls are in the bucket at noon? Does our knowledge of which balls are in the bucket alter how many balls are in the bucket? >> ------- >> Let us define a_n to be the function, after the n^th placement/removal, >> which maps the set of balls to {0,1}, where 1 means that ball is in the >> bucket and 0 means it is not. If we use the topology of pointwise >> convergence on these functions, the sequence {a_n} converges to the >> function which maps all the balls to 0, since at some point, each ball >> is removed from the bucket, never to be put back. If we view a_n to be a function defined in the time domain rather than in >the domain of natural numbers, such that a_n(t) = 1 if ball n is in the >bucket at time t and a_n(t) = 0 otherwise, then we find that a_n(t_0) = 0 >for each n, where t_0 = noon. Thus a(t_0) = sum_n=1^oo a_n(t_0) = 0. Although this reasoning leads to the same conclusion as your argument, >there is an essential difference. Your argument uses limits as t->t_0, >but mine does not. The only limit that appears in my argument is the one >that says the sum of an infinite collection of zeros is zero. Your >argument depends on justifying the step of introducing a pointwise >topology and using it to answer the original question, while my argument >depends on no such artifice. You are summing an infinite number of zeroes. Just because they are zeroes does not excuse you from having to take a limit, trivial as it may be. Taking a limit involves a topology. >> ------- >> let us use the discrete topology on this sequence. With this topology, >> the functions converge to a function f if after some point, all the a_n >> are equal to f. Thus, using this topology, the {a_n} do not converge. Again, I don't see that this argument has anything to do with the >question that was asked. It does in the sense that although we know that past step n there are more that 9n balls in the bucket, the set of balls keeps changing. Thus, the set of balls in the bucket never settles to any limit. It doesn't settle to the empty set since there is an increasing number of balls in the bucket, yet no particular ball is in the bucket forever. >> Thus, each of the replies is correct depending on how you define the >> convergence. Since the question asks, how many balls, it seems to me >> I would say an infinite number. If the question were which balls, >> balls. And this is strong evidence that there is something wrong in your >reasoning. If you know which balls, then you automatically know how >many balls. Every set has a cardinality. If your reasoning leads to >inconsistent answers, then your reasoning is wrong. If you know which balls, then you know how many balls; but if you know how many balls, you don't necessarily know which balls. Looking only at the cardinality of the balls in the bucket, we get that there are an infinite number of balls at noon. However, looking at which balls are in the bucket, we cannot say as that information keeps changing. Let us pose a different problem. Suppose we have two balls, #1 and #2. At each step above we swap which ball is in the bucket. At each step there is one ball in the bucket. How many balls are in the bucket at noon? Which ball is in the bucket at noon? Although we can answer the first question pretty easily, we can't answer the second. Rob Johnson take out the trash before replying ==== > Almost 4 years ago, there was a very long thread on this very topic. > See the thread starting at > http://groups.google.com/groups?threadm=38b0da2f.5767305@news.globalcenter.ne t > As you mention, there are three ways to approach this problem; each > boils down to a question of convergence. If we have a sequence {a_n}, > we might ask what happens to that sequence as n tends to infinity, and > thus, pass to the limit to define what happens at infinity. However, > to define a limit, we must first define a topology. >>But the question can be answered without appealing to any sort of >>topology. > ------- > There are standard topologies on both the reals and the integers, so > that if the a_n are reals or integers, we use those topologies. > For example, let a_n be the number of balls in the bucket after the n^th > placement/removal. That sequence converges to infinity as n tends to > infinity. >>But that has nothing to do with the question that was asked. > The question asked was, how many balls? Counting how many balls are > in the bucket at any given time would seem to have something to do with > the question. Just because we don't know which balls are in the bucket > does not mean there are no balls in the bucket. Counting the balls at some time before noon has nothing to do with the bucket's contents at noon, if there are still balls to be moved in and out. Counting the balls at noon is what matters. Your approach reminds me of the old joke that when you have a hammer, every problem begins to resemble a nail. Here you have three hammers, slightly different in design, but they are all basically the same tool. Yet nowhere do you address the question of whether your tool is the appropriate one for the job. Not only that, your tools give conflicting answers, which is evidence that at least some of them are being incorrectly applied. > Suppose the problem were asked in a slightly different way. Suppose > that instead of ball #n being removed at step n, we remove ball #10n, > that is, the last ball added. Here I don't think you will disagree that > there are 9 balls added to the bucket at each step that stay there for > all time. Thus, at noon, there are infinitely many balls. Right answer, wrong reason. The reason there are infinitely many balls at noon is that a_n(t_0) = 1 for each n that is not a multiple of 10, and there are infinitely many such n. Therefore a(t_0) = sum_{n=0}^oo a_n(t_0) = +oo. > Suppose one ball is removed at each step, but we are not told which ball > that is. What happens at noon. All we know is that at step n, there > are 9n balls in the bucket. How many balls are in the bucket at noon? > Does our knowledge of which balls are in the bucket alter how many balls > are in the bucket? If the ball to be removed is selected at random at each step, then I can conclude that with probability 1, there are no balls left at noon. > ------- > Let us define a_n to be the function, after the n^th placement/removal, > which maps the set of balls to {0,1}, where 1 means that ball is in the > bucket and 0 means it is not. If we use the topology of pointwise > convergence on these functions, the sequence {a_n} converges to the > function which maps all the balls to 0, since at some point, each ball > is removed from the bucket, never to be put back. >>If we view a_n to be a function defined in the time domain rather than in >>the domain of natural numbers, such that a_n(t) = 1 if ball n is in the >>bucket at time t and a_n(t) = 0 otherwise, then we find that a_n(t_0) = 0 >>for each n, where t_0 = noon. Thus a(t_0) = sum_n=1^oo a_n(t_0) = 0. >>Although this reasoning leads to the same conclusion as your argument, >>there is an essential difference. Your argument uses limits as t->t_0, >>but mine does not. The only limit that appears in my argument is the one >>that says the sum of an infinite collection of zeros is zero. Your >>argument depends on justifying the step of introducing a pointwise >>topology and using it to answer the original question, while my argument >>depends on no such artifice. > You are summing an infinite number of zeroes. Just because they are > zeroes does not excuse you from having to take a limit, trivial as it > may be. Taking a limit involves a topology. Didn't I already say that? Actually, I do see a way to formulate the argument that does not involve a limit at all, but that was not the point I was trying to make in that paragraph, where I specifically said I was taking a limit and I identified exactly where it was. The point is not whether a topology is involved, but *which* topology is involved, and how it relates to the stated problem. > ------- > let us use the discrete topology on this sequence. With this topology, > the functions converge to a function f if after some point, all the a_n > are equal to f. Thus, using this topology, the {a_n} do not converge. >>Again, I don't see that this argument has anything to do with the >>question that was asked. > It does in the sense that although we know that past step n there are > more that 9n balls in the bucket, the set of balls keeps changing. > Thus, the set of balls in the bucket never settles to any limit. It > doesn't settle to the empty set since there is an increasing number of > balls in the bucket, yet no particular ball is in the bucket forever. If no particular ball remains in the bucket until noon, and no ball is returned to the bucket having once been removed, then no ball is in the bucket at noon. > Thus, each of the replies is correct depending on how you define the > convergence. Since the question asks, how many balls, it seems to me > I would say an infinite number. If the question were which balls, > balls. >>And this is strong evidence that there is something wrong in your >>reasoning. If you know which balls, then you automatically know how >>many balls. Every set has a cardinality. If your reasoning leads to >>inconsistent answers, then your reasoning is wrong. > If you know which balls, then you know how many balls; but if you know > how many balls, you don't necessarily know which balls. Oh, come, now. You know better than that. If I demonstrate if A then B, it is useless to retort that I have not proved the converse, since I have not even stated the converse or relied on it in any way. If the answer to which balls is none, then the answer to how many balls cannot possibly be infinitely many. This demonstrates that something is wrong with the application of your chosen tools. >Looking only at > the cardinality of the balls in the bucket, we get that there are an > infinite number of balls at noon. No, we do not get that. Not unless you add an unwarranted assumption concerning continuity at noon. The function is obviously discontinuous at a great many points (each time balls are added or removed), so why should we expect continuity at noon? After all, noon (t=t_0) is a limit point of the set of discontinuities of the function a = sum a_n, and therefore it should not be surprising that there is also a discontinuity at t_0. >However, looking at which balls are > in the bucket, we cannot say as that information keeps changing. Each of the changes takes place before noon, and the state of each ball is unchanging once that ball is removed. > Let us pose a different problem. Suppose we have two balls, #1 and #2. > At each step above we swap which ball is in the bucket. At each step > there is one ball in the bucket. How many balls are in the bucket at > noon? Which ball is in the bucket at noon? Although we can answer the > first question pretty easily, we can't answer the second. If we use b_1(t) and b_2(t) as the characteristic functions of the two balls, then we find that in this case (unlike the original problem) b_1(t_0) and b_2(t_0) cannot be determined from the stated information, and therefore the approach that I described fails. But so what? This problem obviously is not well-posed, but the original problem does not have that defect. In that case we can determine a_n(t_0) for each n. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. ==== >Depends on whether you pay by the ball or by the minute... Wait, both ways they get the same pay! The only guy I can find who is willing to work that hard wants to be paid by the ball in real time. Unfortunately, he got snared in the Wal Mart roundup and is back in Mexico now. --Lynn