mm-1309 === Subject: Bad random number generator : Microsoft Excel (office XP) I was conducting some tests with the built-in random number generator of Excel. I asked 4 runs of 30,000 random numbers between 0 and 1. It has some limitations of course since a column can hardly have more rows anyway, if you ask for 1 million entries the program is too dumb to figure out to spread the results on many columns, so you have to split the thing in many parts. In principle this works since it apparently make a different run each time BUT...! There is a little surprise with this. Actually it does not look really a good random number generator because of this. - There are 120,000 entries but only 3037 are different, in other words it did pick up the same number up to 15 times! - The precision is suppossedly of 15 digits but according to the choice of different values I bet one could find the generator by doing a reverse engineering on it, ;-) - It did pick up the number 1 , 2 times. I think that this is BAD, how a random number generator can pick up the same values so many times at a precision of 15 digits : this is impossible. Did someone ever tested this particular bad program and if so, does anybody has any idea what is the formula or algorithm they (Bill Gates and Co.) use? I suspected that Excel and Microsoft are not so advanced , I had no idea it was so bad, this is Mickey Mouse Mathematics! I use the latest version of Excel with Windows XP, service pack 1 and Office XP. If you want to do a simulation and you use Excel? : bad idea. ps: I should send this to D.E. Knuth! Simon Plouffe === Subject: Re: Bad random number generator : Microsoft Excel (office XP) simon.plouffe@sympatico.ca says... > I was conducting some tests with the built-in > random number generator of Excel. > I asked 4 runs of 30,000 random numbers between 0 and 1. > It has some limitations of course since a column can hardly > have more rows anyway, if you ask for 1 million entries the > program is too dumb to figure out to spread the results on > many columns, so you have to split the thing in many parts. > In principle this works since it apparently make a different > run each time BUT...! > There is a little surprise with this. > Actually it does not look really a good random number generator > because of this. > - There are 120,000 entries but only 3037 are different, in other > words it did pick up the same number up to 15 times! > - The precision is suppossedly of 15 digits but according to the > choice of different values I bet one could find the generator by > doing a reverse engineering on it, ;-) > - It did pick up the number 1 , 2 times. > I think that this is BAD, how a random number generator > can pick up the same values so many times at a precision > of 15 digits : this is impossible. > Did someone ever tested this particular bad program and > if so, does anybody has any idea what is the formula > or algorithm they (Bill Gates and Co.) use? > I suspected that Excel and Microsoft are not so advanced > , I had no idea it was so bad, this is Mickey Mouse Mathematics! > I use the latest version of Excel with Windows XP, service > pack 1 and Office XP. > If you want to do a simulation and you use Excel? : bad idea. > ps: I should send this to D.E. Knuth! > Simon Plouffe There's a (highly critical) review by McCullough and Wilson of Excel 2000 and XP 's statistical facilities at: http://portal.acm.org/citation.cfm?id=635312&dl=ACM&coll=portal referring to a paper in Computational Statistics and Data Analysis, Oct 2002. The paper is available from there, too. The summary contends, The problems that rendered Excel 97 unfit for use as a statistical package have not been fixed in either Excel 2000 or Excel 2002 (also called Excel XP). Microsoft attempted to fix errors in the standard normal random number generator and the inverse normal function, and in the former case actually made the problem worse. Philip A. Viton Ohio State University === Subject: Re: Bad random number generator : Microsoft Excel (office XP) === >Subject: Bad random number generator : Microsoft Excel (office XP) > I was conducting some tests with the built-in >random number generator of Excel. >I asked 4 runs of 30,000 random numbers between 0 and 1. >It has some limitations of course since a column can hardly >have more rows anyway, if you ask for 1 million entries the >program is too dumb to figure out to spread the results on >many columns, so you have to split the thing in many parts. >In principle this works since it apparently make a different >run each time BUT...! >There is a little surprise with this. >Actually it does not look really a good random number generator >because of this. >- There are 120,000 entries but only 3037 are different, in other >words it did pick up the same number up to 15 times! >- The precision is suppossedly of 15 digits but according to the >choice of different values I bet one could find the generator by >doing a reverse engineering on it, ;-) >- It did pick up the number 1 , 2 times. I just did 131,072 random numbers and got 131,061different numbers. There were 11 numbers repeated once each. How many should I have gotten? >I think that this is BAD, how a random number generator >can pick up the same values so many times at a precision >of 15 digits : this is impossible. >Did someone ever tested this particular bad program and >if so, does anybody has any idea what is the formula >or algorithm they (Bill Gates and Co.) use? >I suspected that Excel and Microsoft are not so advanced >, I had no idea it was so bad, this is Mickey Mouse Mathematics! >I use the latest version of Excel with Windows XP, service >pack 1 and Office XP. >If you want to do a simulation and you use Excel? : bad idea. >ps: I should send this to D.E. Knuth! >Simon Plouffe -- Mensanator Ace of Clubs === Subject: Re: Bad random number generator : Microsoft Excel (office XP) > I suspected that Excel and Microsoft are not so advanced > , I had no idea it was so bad, this is Mickey Mouse Mathematics! Well, that's par for the course for MS. === Subject: Re: Bad random number generator : Microsoft Excel (office XP) > I was conducting some tests with the built-in >random number generator of Excel. >I asked 4 runs of 30,000 random numbers between 0 and 1. >It has some limitations of course since a column can hardly >have more rows anyway, if you ask for 1 million entries the >program is too dumb to figure out to spread the results on >many columns, so you have to split the thing in many parts. >In principle this works since it apparently make a different >run each time BUT...! >ps: I should send this to D.E. Knuth! >Simon Plouffe Maybe you should think of a way of capitalizing on this. I mean Microsoft will certainly capitalize on you finding the bug -- they will fix it and hence have a (slightly) better product, which could potentially bring them more money. But what are you going to get out of this? Knuth, for example, is offering money for people finding erros in his works. Maybe it is time for users to start getting money for finding bugs in commerical software. Microsoft has had a horrible history of bugs and patches. Users would buy their products for insane amounts of money, find bugs and vulnerabilites, and Microsoft will respond with a patched new version, which the users can buy for even more insane amount of money, find more bugs, and so on. If nothing else users of Microsoft products, should be getting paid as testers. Vladimir === Subject: Re: Bad random number generator : Microsoft Excel (office XP) >Maybe you should think of a way of capitalizing on this. I mean >Microsoft will certainly capitalize on you finding the bug -- [...] >Maybe it is time for users to start getting money for finding bugs >in commerical software. Microsoft has had a horrible history of bugs Huh?!? Should M$ pay for bugs found in its sw, it would be soon out of business!! ;-) Michele #!/usr/bin/perl -lp BEGIN{*ARGV=do{open $_,q,<,,$/;$_}}s z^z seek DATA,11,$[;($, =ucfirst)=~s x .*x q^~ZEX69l^^q,^2$;][@,xe.$, zex,s e1e q 1~BEER XX1^q~4761rA67thb ~eex ,s aba m,P..,,substr$&,$.,age __END__ === Subject: Re: Bad random number generator : Microsoft Excel (office XP) > I was conducting some tests with the built-in >random number generator of Excel. >I asked 4 runs of 30,000 random numbers between 0 and 1. >It has some limitations of course since a column can hardly >have more rows anyway, if you ask for 1 million entries the >program is too dumb to figure out to spread the results on >many columns, so you have to split the thing in many parts. >In principle this works since it apparently make a different >run each time BUT...! >ps: I should send this to D.E. Knuth! >Simon Plouffe > Maybe you should think of a way of capitalizing on this. I mean > Microsoft will certainly capitalize on you finding the bug -- > they will fix it and hence have a (slightly) better product, which > could potentially bring them more money. But what are you going > to get out of this? Knuth, for example, is offering money for > people finding erros in his works. > Maybe it is time for users to start getting money for finding bugs > in commerical software. Microsoft has had a horrible history of bugs > and patches. Users would buy their products for insane amounts of > money, find bugs and vulnerabilites, and Microsoft will respond with > a patched new version, which the users can buy for even more insane > amount of money, find more bugs, and so on. If nothing else users of > Microsoft products, should be getting paid as testers. > Vladimir Apologies for coming to the defense of Mr Microsoft but I needed a robust Random Number Generator to simulate large numbers of DNA profiles. I was checking for matches in randomly generated 20 digit strings. I had to throw out 2 other versions as they produced too many repeats. As I was calling the RNG 80 million times in sequence and other runs sampling from 200 million and checking for matches. The last thing I wanted was the RNG throwing up repeats. This simple RNG function has not let me down - sorry about VB as I am not a programmer and my maths properly stopped 25 years ago *********** Randomize a = 214013 c = 2531011 x0 = Timer temp = x0 * a + c temp = temp / z x1 = (temp - Fix(temp)) * z x0 = x1 phj = x1 / z *********** The above generator using VB and Word97 sort macro etc (again apologies ) simulation of large DNA profile database http://www.nutteing2.freeservers.com/dnas5.htm or nutteingd in a search engine email nonarevers@yahoo.co.....uk (remove 4 of 5 dots) === Subject: Re: Bad random number generator : Microsoft Excel (office XP) > *********** > Randomize > a = 214013 > c = 2531011 > x0 = Timer > temp = x0 * a + c > temp = temp / z > x1 = (temp - Fix(temp)) * z > x0 = x1 > phj = x1 / z > *********** What value did you use for 'z'? -Michael. === Subject: Re: Bad random number generator : Microsoft Excel (office XP) > *********** > Randomize > a = 214013 > c = 2531011 > x0 = Timer > temp = x0 * a + c > temp = temp / z > x1 = (temp - Fix(temp)) * z > x0 = x1 > phj = x1 / z > *********** > What value did you use for 'z'? > -Michael. I tried the Kaner/ Vokey RNG with z = 2^ 40 trying each a= 27182819621,c = 3 and a = 8413453205,c = 99991 but both were useless for repeats I also downloaded http://sunny-beach.net/random_numbers/ but it didn't like my sound card and i settled for the RND i mentioned before that is used in some Microsoft product IIRC. What they aren't telling you about DNA profiles and what Special Branch don't want you to know. http://www.nutteing2.freeservers.com/dnapr.htm or nutteingd in a search engine email nonarevers@yahoo.co.....uk (remove 4 of 5 dots) === Subject: Re: Bad random number generator : Microsoft Excel (office XP) > I tried the Kaner/ Vokey RNG with z = 2^ 40 > trying each > a= 27182819621,c = 3 > and a = 8413453205,c = 99991 > but both were useless for repeats > I also downloaded > http://sunny-beach.net/random_numbers/ > but it didn't like my sound card and i settled > for the RND i mentioned before that is > used in some Microsoft product IIRC. > What they aren't telling you about DNA profiles > and what Special Branch don't want you to know. > http://www.nutteing2.freeservers.com/dnapr.htm > or nutteingd in a search engine > email nonarevers@yahoo.co.....uk (remove 4 of 5 dots) I probably found the well-behaved RNG on here http://support.microsoft.com/default.aspx?scid=kb;EN-US;28150 === Subject: Re: Bad random number generator : Microsoft Excel (office XP) > *********** > Randomize > a = 214013 > c = 2531011 > x0 = Timer > temp = x0 * a + c > temp = temp / z > x1 = (temp - Fix(temp)) * z > x0 = x1 > phj = x1 / z > *********** > What value did you use for 'z'? > -Michael. Sorry about that - sloppy cut and paste z = 2 ^ 24 === Subject: Re: Bad random number generator : Microsoft Excel (office XP) Sounds very odd ... are you sure you made no handling error (sorry ... i can not check on earlier Excel)? For your Q on testing: try to search for Leo Knuesel (with an 'Umlaut') at Department of Statistics, University of Munich. The short note 'On the Reliability of Microsoft Excel XP ...' covers the random number generator of Excel as being 'questionable' [i dont have the link at hand]. A reasonable way may be to use NtRand (a 'free' add in) which uses Mersenne Twister: http://www.numtech.com/NtRand/, http://www.math.keio.ac.jp/~matumoto/emt.html Axel -- use mail for mail not nonail > I was conducting some tests with the built-in > random number generator of Excel. > I asked 4 runs of 30,000 random numbers between 0 and 1. > It has some limitations of course since a column can hardly > have more rows anyway, if you ask for 1 million entries the > program is too dumb to figure out to spread the results on > many columns, so you have to split the thing in many parts. > In principle this works since it apparently make a different > run each time BUT...! > There is a little surprise with this. > Actually it does not look really a good random number generator > because of this. > - There are 120,000 entries but only 3037 are different, in other > words it did pick up the same number up to 15 times! > - The precision is suppossedly of 15 digits but according to the > choice of different values I bet one could find the generator by > doing a reverse engineering on it, ;-) > - It did pick up the number 1 , 2 times. > I think that this is BAD, how a random number generator > can pick up the same values so many times at a precision > of 15 digits : this is impossible. > Did someone ever tested this particular bad program and > if so, does anybody has any idea what is the formula > or algorithm they (Bill Gates and Co.) use? > I suspected that Excel and Microsoft are not so advanced > , I had no idea it was so bad, this is Mickey Mouse Mathematics! > I use the latest version of Excel with Windows XP, service > pack 1 and Office XP. > If you want to do a simulation and you use Excel? : bad idea. > ps: I should send this to D.E. Knuth! > Simon Plouffe === Subject: Fractionally dimensioned Space This may sound strange, but.. Is fractional dimension of a space meaningful? like e.g. R^2.5, 2.5, 3.5 dimensions of manifold/space ? Fractional differentiation has meaning in calculus,real numbers in between integers are well known, a polygon of 2.5 sides(a five cornered star completed in 2 revolutions)is OK in geometrical conception or interpretation and fabrics are woven 2.5 dimensions.It occurs as a thought while pondering about Klein Bottle and Knots in Topology,(a subject I like to learn properly) but may bear no relavence to it at all. === Subject: Re: Fractionally dimensioned Space >This may sound strange, but.. >Is fractional dimension of a space meaningful? There are various definitions of the dimension of a topological space or the dimension of a metric space, and with many of them a space can certainly have a fractional dimension. For example, the standard middle-thirds Cantor set has hausdorff dimension equal to log(2)/log(3). >like e.g. R^2.5, 2.5, >3.5 dimensions of manifold/space ? But those notions of dimension are _not_ the same as the notion of the dimension of a vector space (although there are of course relations between the two, in particular the dimension of R^n is n for all the notions of dimension I know). I'm not aware of anything that might reasonably be called R^2.5; the dimension of a manifold is an integer (for every notion of dimension I'm aware of.) >Fractional differentiation has >meaning in calculus,real numbers in between integers are well known, a >polygon of 2.5 sides(a five cornered star completed in 2 >revolutions)is OK in geometrical conception or interpretation and >fabrics are woven 2.5 dimensions.It occurs as a thought while >pondering about Klein Bottle and Knots in Topology,(a subject I like >to learn properly) but may bear no relavence to it at all. I _doubt_ that it has much relevance to the sort of thing you're thinking about here; spaces with fractional dimension are non-smooth fractal spaces, as opposed to nice smooth spaces like knots and Klein bottles. ************************ David C. Ullrich === Subject: Re: Fractionally dimensioned Space > This may sound strange, but.. > Is fractional dimension of a space meaningful? like e.g. R^2.5, 2.5, 3.5 > dimensions of manifold/space ? Fractional differentiation has meaning in > calculus,real numbers in between integers are well known, a polygon of > 2.5 sides(a five cornered star completed in 2 revolutions)is OK in > geometrical > conception or interpretation and fabrics are woven 2.5 dimensions.It > occurs as a thought while pondering about Klein Bottle and Knots in > Topology,(a subject I like to learn properly) but may bear no relavence > to it at all. Ouch, mixed metaphors. The existence of fractions does not necessarily have anything to do with the existence of fractional dimensions (that's not to say no one did this already; look up fractals and fractional dimensions on Mathworld). And I take issue with your characterization of a 5-pointed star as a 2.5-sided polygon. It's not a polygon at all, since its edges intersect; you could of course draw it, say, on a projective surface, which would cure that problem, but then it would have only five sides (straight line segments intersecting only at their endpoints). The dimension of a vector space is defined to be the cardinality of certain subsets (sets of basis vectors), which if they are finite are always integers, so R^2.5 doesn't even make sense as a symbol without some serious generalization. The question of dimension was probably originally algebraic (relating to vector spaces) but got taken over by topology, where it seems to me it gets redefined depending on context and convenience. For example, the Krull dimension (important in algebraic geometry) is defined in terms of irreducible subsets of a topological space; if you work with the definition a little, you'll notice that the only irreducible subsets of R^n are points for any n, so that they all have Krull dimension 0. Dimension is a context-sensitive quantity, though for an intuitive definition it tends to be related to R^n somehow (like in manifolds). Ryan Reich ryanr@uchicago.edu === Subject: Re: Centroid for exp(-t) > This doesn't seem right. Can you define the centroid of a function defined > on (0,oo) please? > Imagine the shape cut out of cardboard. Even though it extends to infinity, > it has a finite area, a finite moment about the x-axis and a finite moment > about the y-axis. You are talking about the centroid of the region bounded by the given curve, the x-axis, and the y-axis. That is not the same as the centroid of a function. I was trying to get the OP to state the problem precisely. Also the OP has Cy = integral((h(t))^2 dt)/integral(h(t)dt) = 1/(2*tau), when it should be 1/2 that. (In dA = h(t)dt, Cy is 1/2*h(t)). So the centroid is actually in the region for every tau. === Subject: Re: What ever happended to the French avoirdupois > Donald G. Shead: > >When the French instituted the metric system, what happened to > >avoirdupois? > It went the way of the atticstater and atticmina. I thought physics and philosophy sought truth; apparently not huh? === Subject: Re: What ever happended to the French avoirdupois Donald G. Shead: >> Donald G. Shead: >> >When the French instituted the metric system, what happened to >> >avoirdupois? >> >> It went the way of the atticstater and atticmina. >I thought physics and philosophy sought truth; apparently not huh? No apparently about it. The only reason to become a physicist is to be able to afford several ferrarri F50's, a 40,000 sq ft house with a swimming pool, the staff to run it and constantly be distracted from any truth seeking by the throngs of naked women ringing the doorbell at all hours. === Subject: excel Bad random number generator, findings of Leo Knusel This is what professor Leo Knuesel have found on this a while ago (as Fred Helenius pointed out). http://www.stat.uni-muenchen.de/~knuesel/elv/excelxp.pdf , there are potentially 32768 random numbers, all of the form k/32767, 0 <= k <= 32767. His findings are interesting, he concludes that Excel and the random number generator cannot be used for any serious scientific work. Simon Plouffe === Subject: Re: excel Bad random number generator, findings of Leo Knusel >http://www.stat.uni-muenchen.de/~knuesel/elv/excelxp.pdf , >there are potentially 32768 random numbers, all of the form >k/32767, 0 <= k <= 32767. >His findings are interesting, he concludes that Excel and >the random number generator cannot be used for any serious >scientific work. Well, this sounds somewhat odd to me in the sense that I cannot imagine anyone thinking of using Excel for doing any serious scientific work. I guarantee that I'm not joking: for sure I don't like products made in Redmond as a general rule, but as you can see I do use at least one of them, and while I have *no* version of Excel installed and I think I will never install one, I *do* see how it can be useful in some situations, though scientific work is not one of them. So the question is: does this really come as a surprise?!? Michele > Comments should say _why_ something is being done. Oh? My comments always say what _really_ should have happened. :) - Tore Aursand on comp.lang.perl.misc === Subject: Re: excel Bad random number generator, findings of Leo Knusel >This is what professor Leo Knuesel have found on this >a while ago (as Fred Helenius pointed out). >http://www.stat.uni-muenchen.de/~knuesel/elv/excelxp.pdf , >there are potentially 32768 random numbers, all of the form >k/32767, 0 <= k <= 32767. >His findings are interesting, he concludes that Excel and >the random number generator cannot be used for any serious >scientific work. Do people ever use it for serious scientific work? That seems like a silly thing to do a priori... >Simon Plouffe ************************ David C. Ullrich === Subject: Re: excel Bad random number generator, findings of Leo Knusel === >Subject: excel Bad random number generator, findings of Leo Knusel >This is what professor Leo Knuesel have found on this >a while ago (as Fred Helenius pointed out). >http://www.stat.uni-muenchen.de/~knuesel/elv/excelxp.pdf , >there are potentially 32768 random numbers, all of the form >k/32767, 0 <= k <= 32767. >His findings are interesting, he concludes that Excel and >the random number generator cannot be used for any serious >scientific work. >Simon Plouffe Wouldn't Excel then fail the DieHard tests if it was that bad? I'll be the first to admit I don't quite understand the output of the DieHard suite, but it seems very obvious when a PRNG fails. I don't have my results in front of me, but when I was playing with DieHard, I ran it against perl, Excel and Python. The perl PRNG appeared to obviously fail, but the Python and Excel (to my surprise) didn't fail in the obvious way that perl did. Maybe I'm not interpreting the results correctly, but it didn't seem to me that Excel is as bad as everyone claims it is. -- Mensanator Ace of Clubs === Subject: Re: excel Bad random number generator, findings of Leo Knusel === >>Subject: excel Bad random number generator, findings of Leo Knusel >>This is what professor Leo Knuesel have found on this >>a while ago (as Fred Helenius pointed out). >>http://www.stat.uni-muenchen.de/~knuesel/elv/excelxp.pdf , >>there are potentially 32768 random numbers, all of the form >>k/32767, 0 <= k <= 32767. >>His findings are interesting, he concludes that Excel and >>the random number generator cannot be used for any serious >>scientific work. >>Simon Plouffe >Wouldn't Excel then fail the DieHard tests if it was that bad? >I'll be the first to admit I don't quite understand the output of >the DieHard suite, but it seems very obvious when a PRNG fails. >I don't have my results in front of me, but when I was playing with >DieHard, I ran it against perl, Excel and Python. >The perl PRNG appeared to obviously fail, but the Python and >Excel (to my surprise) didn't fail in the obvious way that perl did. >Maybe I'm not interpreting the results correctly, but it didn't seem >to me that Excel is as bad as everyone claims it is. It was just a few weeks ago that someone showed me a plot of some data he'd generated in Maple to illustrate the Central Limit Theorem. My immediate reaction was that the data showed the Maple RNG was no good, the data should have been closer to a smooth bell-shaped curve. But I did some back-of-envelope calculations and decided the deviations he had from an actual gaussian were about the size one would expect. When I did the same experiment in Python the results were much worse. (Say a trial consists of flipping a coin 1000 times and counting the number of heads. The experiment consists of doing 1000 trials and then looking at a histogram of how many times we got n heads for various values of n...) ************************ David C. Ullrich === Subject: Re: excel Bad random number generator, findings of Leo Knusel === >>Subject: excel Bad random number generator, findings of Leo Knusel >>This is what professor Leo Knuesel have found on this >>a while ago (as Fred Helenius pointed out). >>http://www.stat.uni-muenchen.de/~knuesel/elv/excelxp.pdf , >>there are potentially 32768 random numbers, all of the form >>k/32767, 0 <= k <= 32767. >>His findings are interesting, he concludes that Excel and >>the random number generator cannot be used for any serious >>scientific work. >>Simon Plouffe >Wouldn't Excel then fail the DieHard tests if it was that bad? >I'll be the first to admit I don't quite understand the output of >the DieHard suite, but it seems very obvious when a PRNG fails. >I don't have my results in front of me, but when I was playing with >DieHard, I ran it against perl, Excel and Python. >The perl PRNG appeared to obviously fail, but the Python and >Excel (to my surprise) didn't fail in the obvious way that perl did. >Maybe I'm not interpreting the results correctly, but it didn't seem >to me that Excel is as bad as everyone claims it is. > It was just a few weeks ago that someone showed me a plot of > some data he'd generated in Maple to illustrate the Central > Limit Theorem. My immediate reaction was that the data showed > the Maple RNG was no good, the data should have been closer > to a smooth bell-shaped curve. > But I did some back-of-envelope calculations and decided the > deviations he had from an actual gaussian were about the size > one would expect. When I did the same experiment in Python > the results were much worse. Are you still using Python version 1? With version 2 the random number generator is a Mersenne Twister (I don't recall which version I used for my test, but it was something greater than 2 because I never had an earlier version). In the other thread: === >Subject: Bad random number generator in excel (follow up) >The rand() functions seems ok, >the faulty one is found using >Tools -> Data Analysis -> Random Number generator : choose >between 0 and 1, 1 variable and pick 30,000 entries and >THEN you will see how bad it is. So I tried that and got only 19,645 unique numbers out of 30,000. A little better than 3037, but there were still lots of duplicates, occuring as many as 6 times. By comparison, when I had Python 2.3 spit out 30,000 random numbers, I got 30,000 unique values. So it looks like Excel does suck after all. Now I'll have to run a DieHard test on the Excel rnd(), the Data Analysis random numbers and the VBA random numbers in order to find out which ones I can trust, if any. > (Say a trial consists of flipping a coin 1000 times and > counting the number of heads. The experiment consists of > doing 1000 trials and then looking at a histogram of how > many times we got n heads for various values of n...) > ************************ > David C. Ullrich === Subject: Re: excel Bad random number generator, findings of Leo Knusel Perhaps Microsoft copied code from a FORTRAN program. Remember 8 byte precision ? > This is what professor Leo Knuesel have found on this > a while ago (as Fred Helenius pointed out). > http://www.stat.uni-muenchen.de/~knuesel/elv/excelxp.pdf , > there are potentially 32768 random numbers, all of the form > k/32767, 0 <= k <= 32767. > His findings are interesting, he concludes that Excel and > the random number generator cannot be used for any serious > scientific work. > Simon Plouffe === Subject: Re: Non-parametrical method > Group: > A lurker here! Can someone give me a simple explanation of what is mean by > a 'non-parametrical statistical method'? > Den Google nonparametric. One such hit is http://www.statsoftinc.com/textbook/stnonpar.html === Subject: Re: Non-parametrical method > A lurker here! Can someone give me a simple explanation of what is mean > by > a 'non-parametrical statistical method'? Do you know what is meant by a statistical distribution? For example, a bell curve, or Gaussian distribution, is completely specified by the population mean and standard deviation. These two quantities are parameters. Other statistical distributions, such as the exponential, uniform, Poisson, and Gamma, for example, are also completely specified by certain parameters, depending on the specific distribution. Typically, statistical methods estimate some parameter an assumed distribution or tests between hypotheses about the parameters of some distributions. Nonparametric methods, also known as distribution-free methods, are typically analyzed without making any assumptions of the form of underlying distributions. These methods do not test between the parameters of some underlying distribution. Nonparametric methods are typically based on percentiles, such as the median, estimated from the data. Examples of non-parametric methods include the Kolmogorov-Smirnov goodness-of-fit test, the Mann-Whitney-Wilcoxon statistic, a method for Analysis Of Variance invented by Milton Friedman, and I-m-surprised-at-how-much-I've forgotton. Nonparametric methods are usually not as powerful (this has a specific technical meaning in statistics) as the corresponding parametric method under the exact assumptions of the parametric methods. But non-parametric methods are applicable under weaker assumptions, and the stronger assumptions of a parametric method may not be easily tested. There are also robust parametric methods, such as trimmed or windorized means. These methods are applicable under assumptions not as weak as nonparametric methods, but weaker than those needed for traditional parametric methods. I presented the above all from a classical perspective. Maybe a Bayesian would present things differently. Try http://csf.colorado.edu/pkt/pktauthors/Vienneau.Robert/Bukharin.html To solve Linear Programs: .../LPSolver.html r c A game: .../Keynes.html v s a Whether strength of body or of mind, or wisdom, or i m p virtue, are found in proportion to the power or wealth e a e of a man is a question fit perhaps to be discussed by n e . slaves in the hearing of their masters, but highly @ r c m unbecoming to reasonable and free men in search of d o the truth. -- Rousseau === Subject: Re: Non-parametrical method Robert for me!! D > A lurker here! Can someone give me a simple explanation of what is mean > by > a 'non-parametrical statistical method'? > Do you know what is meant by a statistical distribution? For example, > a bell curve, or Gaussian distribution, is completely specified by > the population mean and standard deviation. These two quantities > are parameters. Other statistical distributions, such as the > exponential, uniform, Poisson, and Gamma, for example, are also > completely specified by certain parameters, depending on the > specific distribution. > Typically, statistical methods estimate some parameter an assumed > distribution or tests between hypotheses about the parameters of > some distributions. > Nonparametric methods, also known as distribution-free methods, > are typically analyzed without making any assumptions of the > form of underlying distributions. These methods do not test > between the parameters of some underlying distribution. Nonparametric > methods are typically based on percentiles, such as the > median, estimated from the data. > Examples of non-parametric methods include the Kolmogorov-Smirnov > goodness-of-fit test, the Mann-Whitney-Wilcoxon statistic, a > method for Analysis Of Variance invented by Milton Friedman, > and I-m-surprised-at-how-much-I've forgotton. > Nonparametric methods are usually not as powerful (this has > a specific technical meaning in statistics) as the > corresponding parametric method under the exact assumptions > of the parametric methods. But non-parametric methods are > applicable under weaker assumptions, and the stronger assumptions > of a parametric method may not be easily tested. > There are also robust parametric methods, such as trimmed or > windorized means. These methods are applicable under assumptions > not as weak as nonparametric methods, but weaker than those > needed for traditional parametric methods. > I presented the above all from a classical perspective. Maybe > a Bayesian would present things differently. > -- > Try http://csf.colorado.edu/pkt/pktauthors/Vienneau.Robert/Bukharin.html > To solve Linear Programs: .../LPSolver.html > r c A game: .../Keynes.html > v s a Whether strength of body or of mind, or wisdom, or > i m p virtue, are found in proportion to the power or wealth > e a e of a man is a question fit perhaps to be discussed by > n e . slaves in the hearing of their masters, but highly > @ r c m unbecoming to reasonable and free men in search of > d o the truth. -- Rousseau === Subject: Re: Algorithms and math-history? > So I'm wondering if all those novel algorithms also represent 'streams' >> that might someday be merged with the old mathematical streams, and >> whether there's an abstract science of algorithms already searching in >> that direction...? Every computer program is a mathematical algorithm. On the math side, > you can study set theory, discrete mathematics, and combinatorics. On > the computing side, you can study computability and algorithmic > complexity - you'll see the connection. > This is not true. One of the defining features of an algorithm is that it > must terminate in a finite time. Some computer programs are designed to > run forever. For example, a word processing program will keep going until > the user exits (or the computer crashes). This is nitpicking, when all known computer implementations run inside a finite universe. What importance if you have a proof that your algorithm finishes in a _finite_ time of 40 billion years, when the end of the universe is in 15 billion years? __Pascal_Bourguignon__ http://www.informatimago.com/ There is no worse tyranny than to force a man to pay for what he doesn't want merely because you think it would be good for him.--Robert Heinlein http://www.theadvocates.org/ === Subject: Re: Algorithms and math-history? === >Subject: Re: Algorithms and math-history? >Message-id: <87k6yf5erw.fsf@thalassa.informatimago.com So I'm wondering if all those novel algorithms also represent 'streams' > that might someday be merged with the old mathematical streams, and > whether there's an abstract science of algorithms already searching in > that direction...? >> Every computer program is a mathematical algorithm. On the math side, >> you can study set theory, discrete mathematics, and combinatorics. On >> the computing side, you can study computability and algorithmic >> complexity - you'll see the connection. >> This is not true. One of the defining features of an algorithm is that it >> must terminate in a finite time. Some computer programs are designed to >> run forever. For example, a word processing program will keep going until >> the user exits (or the computer crashes). >This is nitpicking, when all known computer implementations run inside >a finite universe. What importance if you have a proof that your >algorithm finishes in a _finite_ time of 40 billion years, when the >end of the universe is in 15 billion years? The importance is that it's finite. What isn't important is whether you can demonstarte that it's finite. >-- >__Pascal_Bourguignon__ http://www.informatimago.com/ >There is no worse tyranny than to force a man to pay for what he doesn't >want merely because you think it would be good for him.--Robert Heinlein >http://www.theadvocates.org/ -- Mensanator Ace of Clubs === Subject: Re: Algorithms and math-history? <87k6yf5erw.fsf@thalassa.informatimago.com> So I'm wondering if all those novel algorithms also represent 'streams' >> that might someday be merged with the old mathematical streams, and >> whether there's an abstract science of algorithms already searching in >> that direction...? > Every computer program is a mathematical algorithm. On the math side, > you can study set theory, discrete mathematics, and combinatorics. On > the computing side, you can study computability and algorithmic > complexity - you'll see the connection. >> This is not true. One of the defining features of an algorithm is that it >> must terminate in a finite time. Some computer programs are designed to >> run forever. For example, a word processing program will keep going until >> the user exits (or the computer crashes). > This is nitpicking, when all known computer implementations run inside > a finite universe. What importance if you have a proof that your > algorithm finishes in a _finite_ time of 40 billion years, when the > end of the universe is in 15 billion years? > -- > __Pascal_Bourguignon__ http://www.informatimago.com/ > There is no worse tyranny than to force a man to pay for what he doesn't > want merely because you think it would be good for him.--Robert Heinlein > http://www.theadvocates.org/ Globs! Since when it is know that the end of the universe will come in 15 000 000 000 years? http://www.telecable.es/personales/gamo/ perl -e'print phone .hex(825475818),n;' perl -e'print 111_111_111*111_111_111,n;' === Subject: Re: Algorithms and math-history? > Globs! Since when it is know that the end of the universe will come in > 15 000 000 000 years? At least 20 years. __Pascal_Bourguignon__ http://www.informatimago.com/ There is no worse tyranny than to force a man to pay for what he doesn't want merely because you think it would be good for him.--Robert Heinlein http://www.theadvocates.org/ === Subject: Re: Algorithms and math-history? >>Globs! Since when it is know that the end of the universe will come in >>15 000 000 000 years? > At least 20 years. Somehow the decimal point seems to have slipped about 90 places to the left (current guess is around 10^100 years until start of dark era). If you're anthropocentric, habitable places will start getting harder to find in about 10^11 years when everything but the local cluster becomes too remote to observe, and pretty well gone in 10^12 when there are no live stars left. === Subject: Re: Algorithms and math-history? >So I'm wondering if all those novel algorithms also represent >'streams' that might someday be merged with the old mathematical >streams, and whether there's an abstract science of algorithms already >searching in that direction...? >>Every computer program is a mathematical algorithm. On the math side, >>you can study set theory, discrete mathematics, and combinatorics. On >>the computing side, you can study computability and algorithmic >>complexity - you'll see the connection. >This is not true. One of the defining features of an algorithm is that >it must terminate in a finite time. Some computer programs are >designed to run forever. For example, a word processing program will >keep going until the user exits (or the computer crashes). >>Welcome to the halting problem. Not all mathematical algorithms >>terminate. The corresponds to the idea of having functions that have >>domains which are proper subsets of the standard domain. >>Some of the open problems in mathematics, such as the Collatz >>Conjecture, deal with the question of whether an algorithm is guaranteed >>to halt. http://en.wikipedia.org/wiki/Collatz_conjecture > By definition, an algorithm is finite: algorithm - A step-by-step > problem-solving procedure, especially an established, recursive > computational procedure for solving a problem in a finite number of steps. > (source > ) > The halting problem does not change the definition of an algorithm. A different definition: http://mathworld.wolfram.com/Algorithm.html A specific set of instructions for carrying out a procedure or solving a problem, usually with the requirement that the procedure terminate at some point. Specific algorithms sometimes also go by the name method, procedure, or technique. The word algorithm is a distortion of about algebraic methods. The process of applying an algorithm to an input to obtain an output is called a computation. Note the word usually. Will Twentyman email: wtwentyman at copper dot net === Subject: Is pure mathematics worth spending tax money for it? I think every ecconomically advanced nation spends money anually, if not much, supporting researchers for pure mathematics like number theory, algebraic geometry, differential topology, etc. I heard arithmetic geometry on elliptic curves has an application for cryptology. But I don't know many examples which justify spending people's money for pure mathematics. === Subject: Re: Is pure mathematics worth spending tax money for it? > I think every ecconomically advanced nation spends money anually, > if not much, supporting researchers for pure mathematics > like number theory, algebraic geometry, differential topology, > etc. I heard arithmetic geometry on elliptic curves > has an application for cryptology. But I don't know many examples > which justify spending people's money for pure mathematics. I believe, as some math-lovers (such as Hardy), that math is MORE interesting when it is *not* applied directly to some real-world problem. Math is an art at least as much as it is a science. But I would propose that mathematics, even pure math, does always have at least one purpose: to exercise the minds of those studying it, possibly resulting in smarter minds. And this, ultimately, will hopefully benefit mankind somehow....hopefully. ;) Leroy Quet === Subject: Re: Is pure mathematics worth spending tax money for it? > I think every ecconomically advanced nation spends money anually, > if not much, supporting researchers for pure mathematics > like number theory, algebraic geometry, differential topology, > etc. I heard arithmetic geometry on elliptic curves > has an application for cryptology. But I don't know many examples > which justify spending people's money for pure mathematics. What stupid question!! Not only the advanced nations. Venezuela spends millions of dollars in Juvenile Orquestras. And I heartly agree . I spended thousands of dollars in the apprentisage of music of my sons. And myself spended thousands in my mathematical library. How many millions spend Las Vegas County in electricity for maintaining gambling and vice? === Subject: Re: Is pure mathematics worth spending tax money for it? > I think every ecconomically advanced nation spends money anually, > if not much, supporting researchers for pure mathematics > like number theory, algebraic geometry, differential topology, > etc. I heard arithmetic geometry on elliptic curves > has an application for cryptology. But I don't know many examples > which justify spending people's money for pure mathematics. Economically advanced nations spend money supporting artists and composers. You wouldn't ask for applications for a concerto, or for a sculpture - why do you need them for a theorem? Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Is pure mathematics worth spending tax money for it? >> I think every ecconomically advanced nation spends money anually, >> if not much, supporting researchers for pure mathematics >> like number theory, algebraic geometry, differential topology, >> etc. I heard arithmetic geometry on elliptic curves >> has an application for cryptology. But I don't know many examples >> which justify spending people's money for pure mathematics. > Economically advanced nations spend money supporting artists > and composers. You wouldn't ask for applications for a concerto, > or for a sculpture - why do you need them for a theorem? more people (though a decreasing set) enjoy, say, Martha Argerich than Andrew Wiles. (though she hardly needs subsidy) as far as funding compositions which sound like theorems being typed, (but without the logic) well, that's another story. === Subject: Re: Is pure mathematics worth spending tax money for it? > I think every ecconomically advanced nation spends money anually, > if not much, supporting researchers for pure mathematics > like number theory, algebraic geometry, differential topology, > etc. I heard arithmetic geometry on elliptic curves > has an application for cryptology. But I don't know many examples > which justify spending people's money for pure mathematics. > Economically advanced nations spend money supporting artists > and composers. You wouldn't ask for applications for a concerto, > or for a sculpture - why do you need them for a theorem? Aaah, yes, but sculptors and composers do not get anywhere near as much money from the National Endowment for the Arts, say, as mathematicians working on really useless things get from the NSF and other organizations. I agree with the sentiment of what you're saying, Gerry, but I would say the interesting aspect of Nobuo's statement is that while funding for mathematics of every kind can be justified, the current justification for it is completely misleading. Mathematicians' lobbyists have done a good job of convincing important folks in the government that funding research on things like the Andrews-Curtis conjecture, Virtual Fibration Conjecture, etc., is crucial for the U.S. to keep its technological advantage. I've always thought that if the average tax-payer understood what it was that pure mathematicians do, s/he would be outraged at the amount of money going to them. People's ignorance of mathematics is ultimately the cash-cow by which we survive. === Subject: Re: Is pure mathematics worth spending tax money for it? > I think every ecconomically advanced nation spends money anually, > if not much, supporting researchers for pure mathematics > like number theory, algebraic geometry, differential topology, > etc. I heard arithmetic geometry on elliptic curves > has an application for cryptology. But I don't know many examples > which justify spending people's money for pure mathematics. > Economically advanced nations spend money supporting artists > and composers. You wouldn't ask for applications for a concerto, > or for a sculpture - why do you need them for a theorem? I don't think composers get as much money as pure mathematicians get from their government. Anyway, you don't need special knowledge to enjoy music. So it's easier for people to pay for composers than for pure mathmaticians. === Subject: Re: Is pure mathematics worth spending tax money for it? > I think every ecconomically advanced nation spends money anually, > if not much, supporting researchers for pure mathematics > like number theory, algebraic geometry, differential topology, > etc. I heard arithmetic geometry on elliptic curves > has an application for cryptology. But I don't know many examples > which justify spending people's money for pure mathematics. That's probably because you actually don't know much pure math. Number theory and algebraic geometry have applications in cryptography among othe things, and they all (well, those three you mentioned) have applications in physics (especially differential topology). But seriously, if you have to ask whether or not pure math has any useful applications, you obviously don't know enough mathematics. === Subject: Re: Is pure mathematics worth spending tax money for it? > I think every ecconomically advanced nation spends money anually, > if not much, supporting researchers for pure mathematics > like number theory, algebraic geometry, differential topology, > etc. I heard arithmetic geometry on elliptic curves > has an application for cryptology. But I don't know many examples > which justify spending people's money for pure mathematics. > That's probably because you actually don't know much pure math. Number > theory and algebraic geometry have applications in cryptography among othe > things, and they all (well, those three you mentioned) have applications > in physics (especially differential topology). But seriously, if you have > to ask whether or not pure math has any useful applications, you obviously > don't know enough mathematics. While tax payers have right to ask their government if their tax money is spent properly, they don't have to know much pure math. I mentioned the applications of pure math in cryptography Could you please tell us about the applications of those three pure math in physics? What are they? === Subject: Re: Is pure mathematics worth spending tax money for it? > Could you please tell us about the applications of those three pure math > in physics? What are they? Number theory - Connection between Riemann zeta function and statiscal physics (Gaussian Unitary Ensemble Hypothesis and such). Also, by way of the Langlands program, connection with representation theory and automorphic representations (also used in physics). Algebraic Geometry - Extremely useful in string theory and quantum mechanics among other things. Differential topology - Also extremely useful in string theory and quantum. Manifolds (differential geometry) play an important role in relativity, the concept of Lie groups, algebras, etc also very useful in physics. These are just a few examples, there are others of course. Remember, at one time the concept of imaginary numbers was completely a part of pure mathematics ;) The pure math of today is the applied math of tomorrow! === Subject: Re: Is pure mathematics worth spending tax money for it? > I think every ecconomically advanced nation spends money anually, > if not much, supporting researchers for pure mathematics > like number theory, algebraic geometry, differential topology, > etc. I heard arithmetic geometry on elliptic curves > has an application for cryptology. But I don't know many examples > which justify spending people's money for pure mathematics. > That's probably because you actually don't know much pure math. Number > theory and algebraic geometry have applications in cryptography among othe > things, and they all (well, those three you mentioned) have applications > in physics (especially differential topology). But seriously, if you have > to ask whether or not pure math has any useful applications, you obviously > don't know enough mathematics. > While tax payers have right to ask their government if their tax money > is spent properly, they don't have to know much pure math. > I mentioned the applications of pure math in cryptography > Could you please tell us about the applications of those three pure math > in physics? What are they? Pure mathematics is by definition the part of mathematics for which no applied use has been found. The bit that has been found useful is called Applied Maths. There is typically some lag between a use being found and the branch of maths moving from Pure to Applied. You want example of the use of Pure Mathematics? How about the computer industry. This is totally dependent upon the mathematical theory of bases, Boolean logic, and computational theory which was all Pure Maths prior to 1940 or so. Or the use of prime numbers in e-commerce, as a more recent example. What do you think the world's computer industry is worth, in dollar terms? Not a bad payoff, eh? Still think mathematical research isn't a good investment of taxpayer dollars? === Subject: Re: Is pure mathematics worth spending tax money for it? ..................... >> While tax payers have right to ask their government if their tax money >> is spent properly, they don't have to know much pure math. >> I mentioned the applications of pure math in cryptography >> Could you please tell us about the applications of those three pure math >> in physics? What are they? >Pure mathematics is by definition the part of mathematics for which no >applied use has been found. The bit that has been found useful is called >Applied Maths. There is typically some lag between a use being found and >the branch of maths moving from Pure to Applied. At a meeting more than half a century ago, I argued that there is no real difference between pure and applied mathematics. There is mathematics which has been applied, and mathematics which has not yet been applied. As for mathematics which the mathematicians of the time did not think would be applied, there are analytic function theory, group theory, matrix theory, the Fast Fourier Transform, number theory. When did the government start spending tax money on it, other than through establishing and funding universities? The answer is right after WWII, when the military decided to fund it; they had found that, during the war, pure mathematicians and pure scientists could solve problems which applied people could not. Even within mathematics itself, understanding often comes from considering situations far more general than those of importance, as the unnecessary properties tend to be confusing, rather than simplifying. >You want example of the use of Pure Mathematics? How about the computer >industry. This is totally dependent upon the mathematical theory of bases, >Boolean logic, and computational theory which was all Pure Maths prior to >1940 or so. Or the use of prime numbers in e-commerce, as a more recent >example. Would anyone have even attempted to come up with public key codes if the idea of trap door functions had not been considered? This one was applied almost immediately. >What do you think the world's computer industry is worth, in dollar terms? >Not a bad payoff, eh? Still think mathematical research isn't a good >investment of taxpayer dollars? This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Is pure mathematics worth spending tax money for it? > I think every ecconomically advanced nation spends money anually, if not > much, supporting researchers for pure mathematics like number theory, > algebraic geometry, differential topology, etc. I heard arithmetic > geometry on elliptic curves has an application for cryptology. But I > don't know many examples which justify spending people's money for pure > mathematics. You're right. Mathematics is a scam. It has no real applications. Those fat cat mathematicians get to suck on the public teat because Joe Taxpayer is too stupid with numbers to figure out he's being ripped. Lance Lamboy === Subject: Re: Is pure mathematics worth spending tax money for it? > I think every ecconomically advanced nation spends money anually, > if not much, supporting researchers for pure mathematics > like number theory, algebraic geometry, differential topology, > etc. I heard arithmetic geometry on elliptic curves > has an application for cryptology. But I don't know many examples > which justify spending people's money for pure mathematics. Americans can break other countries secret codes; they spend much money on mathematicians and research. === Subject: Re: Is pure mathematics worth spending tax money for it? > I think every ecconomically advanced nation spends money anually, > if not much, supporting researchers for pure mathematics > like number theory, algebraic geometry, differential topology, > etc. I heard arithmetic geometry on elliptic curves > has an application for cryptology. But I don't know many examples > which justify spending people's money for pure mathematics. The problem is: you never know *which* research item will end up having important consequences for the other sciences. The fact that a result in one branch can provide the key for an important problem in another branch makes things that much more convoluted. In general, there is a history of math being generated and useful applications being found much later. Will Twentyman email: wtwentyman at copper dot net === Subject: (c+y)^2 + (c-x-y)^2 + (c+x)^2 = (c-y)^2 + (c+x+y)^2 + (c-x)^2 > (c+y)^2 + (c-x-y)^2 + (c+x)^2 = (c-y)^2 + (c+x+y)^2 + (c-x)^2 > [arranged avoiding windows specific characters] > Note to Rasinger : avoid those weird characters î ó © ... and > PROHIBIT them absolutely in the subject (objet in fr). > However set up your outlook express so it says > Sum of 3 squares... sum of squares... > It might be used to find some numbers which are sum of 3 squares > in more than one way. > Unfortunately this doesn't give all of them. > 41 = 1^2 + 2^2 + 6^2 = 5^2 + 4^2 + 0^2 > but 41 = 3^2 + 4^2 + 4^2 not of the form (1) You mean using a phytoagorean substitution 5^2 + 0^2 = 3^2 + 4^2 I dont know, if it is necessary or possible to implant this feature. > Even worse 59 = 1^2 + 3^2 + 7^2 = 3^2 + 5^2 + 5^2 can't be written > as an identity (1). False, use ( x, c, y ):=( 6, 1, -2 ) You mean using a sign substitution 59 = (-1)^2 + (-3)^2 + (+7)^2 = (+3)^2 + (+5)^2 + (-5)^2 then (-2 + c)^2 + (-4 + c)^2 + (+6 + c)^2 = (+2 + c)^2 + (+4 +c)^2 + (-6 +c)^2 with c := 1; > An integer is sum of 3 squares (including 0^2) if and only if it > is not a 4^n.(8k+7) (Gauss). Surprising Am I allowed to take complex numbers in squares (negative squares)? But what means n, k in 4^n.(8k+7)? > Out of this case, the identity (1) is just an identity as > there are so many... That's what I try to find out. And I found out other interesting things, Euler studied 3-square identities too ... In general I dont think it is a good way to say: 59 is result. How can it be calculated? Helmut === Subject: Re: Relationship between a function and its inverse > Yes, in fact they do carry through geometric invariants. That is much better wording for what I was inquiring about. It just seemed that a function and its inverse were just different arrangements of the same underlying geometrical object. Also, if constants are taken into account, different functions can be seen to show the same underlying object at different distances, angles, etc, if you imagine viewing the graph of the function in 3d and moving it around. I find that most interesting. So far, learning about functions and inverses have given me the most interesting questions. > What looks to be y= Exp(x) becomes Y=Ln(X) by looking at it from the > back side , after swapping (x,y) axes. Also y = x^2/a becomes Y = > Sqrt(a X). > This is mapping as mirroring with respect to line x = y [or also X=Y > to get back the original curve]. That would be an interesting way to analyze the relationships at first. You might then be able to translate and rotate the transparency of Exp(x), plot the points, and see what function fits those points. Then it might be seen what sorts of formula abstractions give rise to rotations and things. I hope that makes sense. For instance, take f(x) = sin(x). It is then seen that if you move the graph up by some amount P, then the equation becomes f(x) = sin(x) + P. So, f(x) = sin(x) + P might be the more abstract viewing of the sin(x) object. It is hard to find proper words to express what I mean. > As you rightly observed, shape does not change by desribing the > function in an inversive manner.It is natural to ask what does not > change in reflection. Among the intrinsic shape Invariants associated > with any curves are: Arc length , Tangential Rotation, Curvature, > Cornu arc derivative of curvature. They are all intrinsic, remain > constant under Euclidean motions Viz., translation, rotation, > mirroring/reflection about any line y = mx + C, in particular, y = + x > and y = -x. [I was not lucky enough to get like help from Lee Rudolph, > Ken Pledger and others, there was no Internet,so helped myself after > seeing plots of functions and inverse functions in text-books]. I need to learn calculus again using what my new knowledge of set it. > For mirroring about x = y case, I had derived the differential > relationships : > F(x,y,y',y'',y''',... ) = 0 as equivalent to > F(Y,X,1/Y',-Y''/ Y'^3,(3 Y''^2-Y'Y''')/Y'^5,...) = 0. Using these > relations, one can obtain differential equation in another form in > mirrored axes. > In the example of above parabola, y'= 2x/a -> 1/Y'= 2 Y /a ; Integrate > to get Y^2= a X . > Integrating these first,second and third order ODE equations with > swapped boundary conditions will give you an insight of what comes > through. One problem is that when the curve turns back, infinte slope, > part of the curve is lost, and continuation should be manipulated. > Hope that it helps. A possible problem that I encountered when looking at functions and inverses was when a function like f(x) = x^2 was graphed. An inverse would be f^(-1)(x) = +SQR(x) and -SQR(x). So rotations of a function can sometimes break it up into cases it seems. x^2 is a perfectly fine function, but when the hyperbola object is rotated to the right by 90 degrees it has to be expressed as two function cases because each x will have two y points. The same underlying object is there, the hyperbola, but now it is awkward to express. === Subject: Re: (c+y)^2 + (-y+c-x)^2 + (x+c)^2 = (c-y)^2 + (y+c+x)^2 + (-x+c)^2 Sorry, mea culpa I didnt know about charset problem in newsgroups William Elliot a .8ecrit dans le message de > (c+y)î + (-y+c-x)î + (x+c)î = (c-y)î + (y+c+x)î + (-x+c)î > I take it that what looks like quotation marks on my screen > are really supposed to be superscript twos, so we're talking > about a sum-of-squares identity. See Gloden, Mehrgradige Gleichungen. > On my screen those quotation marks appeared as shaded rectangles. === Subject: Re: Number by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i5ALnMt29525; >Alas, I have gone back to my first definition of parity. It was >par(n) = (-1)^{ f( c( b(n), b(n) )) } >Multiplication is now given by >n*m = (-1)^{ f( c( b(|n|), b(|m|) ) } nm, |n|, |m| > 1 > = nm otherwise >>P(2) = 1 P(31) = 1 P(60) = -1 P(89) = 1 >>P(3) = 1 P(32) = 1 P(61) = 1 P(90) = -1 >>P(4) = 1 P(33) = -1 P(62) = -1 P(91) = 1 >>P(5) = 1 P(34) = -1 P(63) = 1 P(92) = 1 >>P(6) = -1 P(35) = -1 P(64) = 1 P(93) = -1 >>P(7) = 1 P(36) = 1 P(65) = -1 P(94) = -1 >>P(8) = 1 P(37) = 1 P(66) = -1 P(95) = -1 >>P(9) = 1 P(38) = -1 P(67) = 1 P(96) = -1 >>P(10) = -1 P(39) = -1 P(68) = 1 P(97) = 1 >>P(11) = 1 P(40) = -1 P(69) = -1 P(98) = 1 >>P(12) = 1 P(41) = 1 P(70) = -1 P(99) = 1 >>P(13) = 1 P(42) = -1 P(71) = 1 P(100) = 1 >>P(14) = -1 P(43) = 1 P(72) = 1 >>P(15) = -1 P(44) = 1 P(73) = 1 >>P(16) = 1 P(45) = 1 P(74) = -1 >>P(17) = 1 P(46) = -1 P(75) = 1 >>P(18) = 1 P(47) = 1 P(76) = 1 >>P(19) = 1 P(48) = -1 P(77) = -1 >>P(20) = 1 P(49) = 1 P(78) = -1 >>P(21) = -1 P(50) = 1 P(79) = 1 >>P(22) = -1 P(51) = 1 P(80) = 1 >>P(23) = 1 P(52) = 1 P(81) = 1 >>P(24) = -1 P(53) = 1 P(82) = -1 >>P(25) = 1 P(54) = -1 P(83) = 1 >>P(26) = -1 P(55) = -1 P(84) = -1 >>P(27) = 1 P(56) = -1 P(85) = -1 >>P(28) = 1 P(57) = 1 P(86) = -1 >>P(29) = 1 P(58) = -1 P(87) = 1 >>P(30) = -1 P(59) = 1 P(88) = -1 >Note: See comment, above. >Define Dar(m) = {n in N | par(mn) = par(m)par(n) } >Dar(2) seems interesting: >Dar(2) = {2, 4, 8, 9, 15, 16, 18, 21, 22, 24, 25, 30, 32, 33, ...} >Simple Property One. Dar(2) contains no primes greater than 2. >Proof. p prime -> par(p) = 1 >Since p > 2, it also follows that par(2p) = -1 (check) >But then par(2p) = -1 != 1 = par(2)par(p) >q.e.d. >Harder Property Two. Choose u in N. Then >forall m, n in Dar(u): par(m) and par(n) not both -1 -> mn in Dar(u) >(This one could indeed be wrong... but if relatively few >exceptions existed, it would still seem somewhat strange to me- >as it already seems strange enough to me now.) >Property Three: Choose u in N. Then u in Dar(u). >Proof. par(u)par(u) = 1. It must be shown that par(u^2) = 1. >Assume that b(u) = (p_1, p_2, ..., p_m) >Then par(u^2) = (-1)^{ f( c( b(u^2), b(u^2) )) } > = (-1)^{ f( c( (p_1, p_1, p_2, p_2, ..., p_m, p_m) , > (p_1, p_1, p_2, p_2, ..., p_m, p_m) ))} >= -1^{ f( p_1, p_1, ..., p_m, p_m, p_1, p_1, ..., p_m, p_m )) >Now, to order this, the p_1 appearing after p_m, above, >apparently has to be shifted an even number of times to the left. >So does every other p_n following it. It follows that >f( p_1, p_1, ..., p_m, p_m, p_1, p_1, ..., p_m, p_m ) is an even >number. Thus, >= -1^{ f( p_1, p_1, ..., p_m, p_m, p_1, p_1, ..., p_m, p_m )) = 1. >q.e.d. It could be possible to extend the ideas discussed above to other cyclic groups. Take the group {i, -i, 1, -1} where i is imaginary and define par(n) = (i)^{ f( c( b(n), b(n) )) } Multiplication would be given by n*m = (i)^{ f( c( b(|n|), b(|m|) ) } nm, |n|, |m| > 1 = nm otherwise It is now easy to see, for ex., that (12*4)*6 = -i288 = 12*(4*6) >C. Dement === Subject: Name this function? I stumbled across this function a couple of years ago. I was wondering if has been discovered before and if it is particularly interesting. define T:Z^+->Z by T(p)=p for p prime. T(nm)=T(n)+T(m) It can also be extended to a homomorphism from Q^+->Z by defining T(a/b)=T(a)-T(b) Does this look familiar to anyone. -- Dale Henderson Imaginary universes are so much more beautiful than this stupidly- constructed 'real' one... -- G. H. Hardy === Subject: Re: Name this function? > I stumbled across this function a couple of years ago. I was wondering if > has been discovered before and if it is particularly interesting. > define T:Z^+->Z by > T(p)=p for p prime. > T(nm)=T(n)+T(m) > It can also be extended to a homomorphism from Q^+->Z by defining > T(a/b)=T(a)-T(b) > Does this look familiar to anyone. It's basically a logarithm, from the last two properties. Actually, it's a combined logarithm (that's not a technical term) in that if you restrict it just to the multiplicative submonoids of integers which are powers of a prime p, then it is really just the function p log_p (p times the logarithm base p). Then you observe that Z^+ is the direct sum of these multiplicative submonoids (by unique factorization, essentially) and your function T just splits into the sum of these logarithms on each term of the sum (it's actually a product; i.e. every integer is a finite product of powers of primes, uniquely). Of course, this is just a fancy way of saying it's a homomorphism. It's more interesting if you extend it to Q^+ since then you get a group homomorphism, and the kernel is exactly those fractions a/b with T(a) = T(b). For example, 5/6: T(5) = 5, and T(6) = T(2) + T(3) = 2 + 3 = 5. Q^+ is of course not a cyclic group (unique factorization again, on the denominator of whatever prospective generator you have in mind) so you expect this thing to have a kernel, but the kernel does say something about the group structure of Q^+. In fact, since Z is a free group, you get Q^+ = ker(T) + Z, where the + is a direct sum. You might even try a similar trick on ker(T). More helpful, actually, would be redefining T so that T(p) = 0 except for a single prime p. Then the kernel is just the set of fractions without p in their factorizations, and the image is pZ, the multiples of p. Then you get Q^+ = ker(T) + Z again, but this time ker(T) is easy to describe. In fact, you could define another function T': ker(T) -> Z which whittles off a different prime and repeat, and in the end you get a representation of Q^+ as a countable direct sum of copies of Z. In fact, you can even say what the isomorphism is: given a fraction a/b you peel off the factors of 2 and apply T to this, then the factors of 3, and so on, and whatever sequence you get is the corresponding term in the direct sum. Incidentally, your original T is then the homomorphism which comes from then summing the sequence (which is always finite). That didn't have much of a point, but it does make the function pretty interesting. Ryan Reich ryanr@uchicago.edu === Subject: Re: Name this function? > I stumbled across this function a couple of years ago. I was wondering > if has been discovered before and if it is particularly interesting. > define T:Z^+->Z by > T(p)=p for p prime. > T(nm)=T(n)+T(m) > It can also be extended to a homomorphism from Q^+->Z by defining > T(a/b)=T(a)-T(b) > Does this look familiar to anyone. It looks familiar to the OEIS: David Cantrell === Subject: Re: Name this function? >I stumbled across this function a couple of years ago. I was wondering >if has been discovered before and if it is particularly interesting. >define T:Z^+->Z by >T(p)=p for p prime. >T(nm)=T(n)+T(m) I'd call it sum of prime factors. Maybe adding a word that indicates if a factor is present multiple times, it gets added in multiple times. --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: re:Which math college in New York City? NYU (Courant Institute) and Columbia U. have first rate math departments. For undergraduate study, CUNY or Cooper Union would also serve. ---------------------------------------------------------- ** SPEED ** RETENTION ** COMPLETION ** ANONYMITY ** ---------------------------------------------------------- http://www.usenet.com http://www.newsfeed.com The #1 Newsgroup Service in the World! >100,000 Newsgroups ---= 19 East/West-Coast Specialized Servers - Total Privacy via Encryption =--- === Subject: Re: primitive elements Can you please provide me an example of field with 125 elements? > How do I decide how many primitive elements there is in a field with > 125 elements. > I need to find the number of elements with order 124, how can I do > that? === Subject: Re: primitive elements days. My association with the Department is that of an alumnus. >Can you please provide me an example of field with 125 elements? There is one and only one field of 125 = 5^3 elements, though there are many ways of representing it. One way is to start with the field of five elements GF(5), and adjoin the root of an irreducible cubic. For example, x^3 + x +1 is irreducible over GF(5), since it has no roots. Thus, one way to describe the field GF(5^3) of 125 elements is that it is the field of all expressions a + b*z + c*z^2 where a, b, c, are elements of GF(5), and z is an element that satisfies z^3 = -(z+1). It is not hard to verify that this is a field, and that it has 125 elements. It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: primitive elements > Can you please provide me an example of field with 125 elements? There are no examples -- there is precisely one such field up to isomorphism. Some might call it GF(5^3), others might call it the field with 125 elements. Construction of it is pretty much the same as any other finite field (with non-trivial prime power order). Phil 1st bug in MS win2k source code found after 20 minutes: scanline.cpp 2nd and 3rd bug found after 10 more minutes: gethost.c Both non-exploitable. (The 2nd/3rd ones might be, depending on the CRTL) === Subject: Re: primitive elements It's ok, but just curious what is the inverse for say 100 in GF(5^3)? It should be such element 100^1 * 100^(-1) == 1, so it should be 124, right? But it is also inverse for 100 and so on... is it normal, or do I miss something? > Can you please provide me an example of field with 125 elements? > There are no examples -- there is precisely one such field up > to isomorphism. Some might call it GF(5^3), others might call it > the field with 125 elements. Construction of it is pretty much > the same as any other finite field (with non-trivial prime power > order). > Phil > -- > 1st bug in MS win2k source code found after 20 minutes: scanline.cpp > 2nd and 3rd bug found after 10 more minutes: gethost.c > Both non-exploitable. (The 2nd/3rd ones might be, depending on the CRTL) === Subject: Re: primitive elements > It's ok, but just curious what is the inverse for say 100 in GF(5^3)? > It should be such element > 100^1 * 100^(-1) == 1, > so it should be 124, right? > But it is also inverse for 100 and so on... is it normal, or do I miss > something? '100' hs no meaning until you define the structure. (16:58) gp > p=Mod(1,5)*Mod(x+4,x^3+x+1) %23 = Mod(Mod(1, 5)*x + Mod(4, 5), x^3 + x + 1) (16:58) gp > for(i=1,125,pp=p^i;if(p==pp,print(i))) 1 125 (17:00) gp > p^-1 %28 = Mod(Mod(3, 5)*x^2 + Mod(3, 5)*x + Mod(1, 5), x^3 + x + 1) So here we see the element that could be called any of 'x+4', or '14'_5 or 9, has multiplicative order 124, and its inverse is '3*x^2+3*x+1' or '331'_5 or 91 Using the same structure (using polynomials modulo x^3+x+1), 100^-1 is (17:03) gp > p=Mod(1,5)*Mod(4*x^2,x^3+x+1) %31 = Mod(Mod(4, 5)*x^2, x^3 + x + 1) (17:04) gp > p^-1 %32 = Mod(Mod(4, 5)*x^2 + Mod(1, 5)*x + Mod(4, 5), x^3 + x + 1) or 109. Using a different structure, (17:04) gp > p=Mod(1,5)*Mod(4*x^2,x^3+2*x+1) %33 = Mod(Mod(4, 5)*x^2, x^3 + 2*x + 1) (17:05) gp > p^-1 %34 = Mod(Mod(3, 5)*x^2 + Mod(1, 5)*x + Mod(1, 5), x^3 + 2*x + 1) 100^-1 = 81 i.e. the numbers are _meaningless_ unless accompanied by a definition of the structure used (the polynomial used to define the elements). I appreciate my explanations are utterly lousy. I can't pretend to Galois Fields (that's where the 'GF' in GF(5^3) comes from) would help. Phil 1st bug in MS win2k source code found after 20 minutes: scanline.cpp 2nd and 3rd bug found after 10 more minutes: gethost.c Both non-exploitable. (The 2nd/3rd ones might be, depending on the CRTL) === Subject: Re: primitive elements > i.e. the numbers are _meaningless_ unless accompanied by a definition > of the structure used (the polynomial used to define the elements). === Subject: Re: primitive elements Dmitriy Samsonov TOP-POSTED: > It's ok, but just curious what is the inverse for say 100 in GF(5^3)? > It should be such element > 100^1 * 100^(-1) == 1, > so it should be 124, right? GF(125) has characteristic 5, and so 100 = 0 in GF(125). If by inverse you mean reciprocal then 100 hasn't one. > But it is also inverse for 100 and so on... is it normal, or do I miss > something? >> Can you please provide me an example of field with 125 elements? >> There are no examples -- there is precisely one such field up >> to isomorphism. Some might call it GF(5^3), others might call it >> the field with 125 elements. Construction of it is pretty much >> the same as any other finite field (with non-trivial prime power >> order). Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: primitive elements > Dmitriy Samsonov TOP-POSTED: > It's ok, but just curious what is the inverse for say 100 in GF(5^3)? > It should be such element 100^1 * 100^(-1) == 1, so it should be 124, right? > GF(125) has characteristic 5, and so 100 = 0 in GF(125). > If by inverse you mean reciprocal then 100 hasn't one. Very good point. I was stripping the traditional numerical meaning off the digit string right at the outset. (And so '100' was not divisible by '5' (there is no '5'!). Sorry Dmitriy if this compounded any confusion. Phil 1st bug in MS win2k source code found after 20 minutes: scanline.cpp 2nd and 3rd bug found after 10 more minutes: gethost.c Both non-exploitable. (The 2nd/3rd ones might be, depending on the CRTL) === Subject: A Non-standard Theory of Prime Numbers (NSTPM) It is posible to construct a non-standard theory for the distribution of primes oriented to the solution of the main non solved conjectures . I propose the following axioms. (The identifications are arbitrary) AXIOM I.- (BOREL'S AXIOM) If from an X1 suffiently large, we take equal intervals D of a size of the order of (Log(X1))^2, then the frequencies of primes included in D, behaves as a random variable with Mean Value = D / Log(X1) and Binomial Distribution. AXIOM II.- (EULER'S AXIOM) Accepting Borel's Axiom and the use of Probabilty Theory.(n=Natural Number) If one event concerning prime numbers is represented by a Function F(n), and if it's Mean Value, from an n > N onwards, is greater than Log F(n). then making indefinitely groups of n trials (That is between F(n) and F(2n)) the event will occur infintely many times. To be applied to the solution of Twin primes Conjecture and Goldbach's. Accepting Borel's Axiom an the use of Probability Theory. If one event concerning prime numbers is represented by a function F(n), and if it's Mean Value, from an n > N onwards, if lesser than i /(Log Fn))^2 then making indefinitely groups of n trials, the event will never occur beyond F(2N). To be applied to Cramer's Conjecture, Fermat's Numbers and similar. AXIOM IV .- (LUDOVICUS AXIOM) Accepting Borel's Axiom and the use of Probability Theory. If by the uniform application of an algorithm, a monotonous increasing integer sequence results ,which its divisibilty properties don't rule out the presence of primes, then the sequence falls under the scrutiny of Axioms II ans III. To be applied to the problem of primes in Fibonacci , Quadratic , Mersenne , Fermat etc. sequences. The solution of Riemann's Hypothesis results of the application of Borel's Axiom, Landau's Theorem and Khinch.92n's Theorem. I invite the curious amteurs and matematicians to develop the consequences of that four axioms in order to find contraditions. Luis Rodriguez sequences results, === Subject: Re: A Non-standard Theory of Prime Numbers (NSTPM) > It is posible to construct a non-standard theory for the distribution > of primes > oriented to the solution of the main non solved conjectures . > I propose the following axioms. (The identifications are arbitrary) Yes, but there is even an easier way: for each conjecture (after a suitable period of tests to check it is true), add a new axiom. Of course, for some conjectures, like Gauss pi(n) AXIOM I.- (BOREL'S AXIOM) > If from an X1 suffiently large, we take equal intervals D of a size of > the order > of (Log(X1))^2, then the frequencies of primes included in D, behaves > as a random variable with Mean Value = D / Log(X1) and Binomial > Distribution. > AXIOM II.- (EULER'S AXIOM) > Accepting Borel's Axiom and the use of Probabilty Theory.(n=Natural > Number) > If one event concerning prime numbers is represented by a Function > F(n), and > if it's Mean Value, from an n > N onwards, is greater than Log F(n). > then > making indefinitely groups of n trials (That is between F(n) and > F(2n)) > the event will occur infintely many times. (with probability 1). For instance, take a random integer. The probability it is smaller than 5 is 0. So 0, 1,2,3 and 4 don't exist. > To be applied to the solution of Twin primes Conjecture and > Goldbach's. > Accepting Borel's Axiom an the use of Probability Theory. > If one event concerning prime numbers is represented by a function > F(n), and if it's Mean Value, from an n > N onwards, if lesser than i > /(Log Fn))^2 then > making indefinitely groups of n trials, the event will never occur > beyond F(2N). idem. For instance, use the event n is prime and n-1 is a 2-power. This proves there is no Mersenne prime above 2^127-1 > To be applied to Cramer's Conjecture, Fermat's Numbers and similar. > AXIOM IV .- (LUDOVICUS AXIOM) > Accepting Borel's Axiom and the use of Probability Theory. > If by the uniform application of an algorithm, a monotonous increasing > integer sequence results ,which its divisibilty properties don't rule > out the presence > of primes, then the sequence falls under the scrutiny of Axioms II ans > III. > To be applied to the problem of primes in Fibonacci , Quadratic , > Mersenne , Fermat etc. sequences. > The solution of Riemann's Hypothesis results of the application of > Borel's > Axiom, Landau's Theorem and Khinch.92n's Theorem. > I invite the curious amteurs and matematicians to develop the > consequences of > that four axioms in order to find contraditions. > Luis Rodriguez > sequences > results, === Subject: Re: A Non-standard Theory of Prime Numbers (NSTPM) ) > Yes, but there is even an easier way: for each conjecture (after a suitable > period of tests to check it is true), add a new axiom. Of course, for some > conjectures, like Gauss pi(n) so what? A little bit of incoherence is surely a small price to pay for such > a simplification of the proof business... > This is nonsense and a dirty boycott. Luis Rodriguez === Subject: Re: how to check if 2^n-1 is prime for some n ? ETAuAhUAlClqm0pK9Y1LxNfmOJJw9CxM6YsCFQCGNz4BlhCT69GQgtFK8YJcx8FgZQ== If 3 and 2*3+1 are both prime then 2^3-1 is not prime? --OL === Subject: Re: how to check if 2^n-1 is prime for some n ? OL> If 3 and 2*3+1 are both prime then 2^3-1 is not prime? OL> --OL That's embarrassing. I had just recently found this theorem and just assumed it was true. Shame on me. The real statement (given at ) is apparently: Let p = 3 (mod 4) be prime. 2p+1 is also prime if and only if 2p+1 divides 2^p-1. Mea Culpa. Dale Henderson Imaginary universes are so much more beautiful than this stupidly- constructed 'real' one... -- G. H. Hardy === Subject: Re: Article: Greatest maths problem 'solved' > Greatest maths problem 'solved' > By Dr David Whitehouse > BBC News Online science editor > A mathematician at Perdue University Purdue. Perdue sells disassembled dead chickens as food. > in the US claims to have proved the > Riemann Hypothesis - called the greatest unsolved problem in maths. > The hypothesis concerns prime numbers and has stumped the world's > mathematicians for more than 150 years. > Now, Professor Louis De Branges de Bourcia has posted a 23-page paper on the > internet detailing his attempt at a proof. > There is a $1m prize for whoever solves the hypothesis. > Cult following > I invite other mathematicians to examine my efforts, says de Branges. > While I will eventually submit my proof for formal publication, due to the > circumstances I felt it necessary to post the work on the internet > immediately. 23 pages is not a very long proof. Given its importance if it is valid (and the attendant embarassment if it is not), it makes perfect sense to post it to the Web and let hundreds of folks look it over. > http://news.bbc.co.uk/2/hi/science/nature/3794813.stm Like a programmer, a mathmatician only has to get it right once. Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) Quis custodiet ipsos custodes? The Net! === Subject: Re: Article: Greatest maths problem 'solved' > 23 pages is not a very long proof. Well, originally it was 24 pages, but there were some important factors to consider. === Subject: Re: Article: Greatest maths problem 'solved' >> 23 pages is not a very long proof. > Well, originally it was 24 pages, but there were some important factors to consider. the 23 pages is a personal description and tour. The actual manuscript on his home page (entitled Riemann Zeta Functions) or something like that is over 100. Somewhere in that big one there was a theorem that looked like it might, but I'm hardly an expert. But according to mathworld.wolfram.com: Riemann Hypothesis Proof Much Ado About Nothing A June 8 Purdue University news release reports a proof of the Riemann Hypothesis by L. de Branges. However, both the 23-page preprint cited in the release Branges's home page seem to lack an actual proof. Furthermore, a counterexample to de Branges's approach due to Conrey and Li has been known since 1998. The media coverage therefore appears to be much ado about nothing. === Subject: Re: Article: Greatest maths problem 'solved' > Riemann Hypothesis Proof Much Ado About Nothing A June 8 Purdue > University news release reports a proof of the Riemann Hypothesis by > L. de Branges. However, both the 23-page preprint cited in the release > Branges's home page seem to lack an actual proof. Furthermore, a > counterexample to de Branges's approach due to Conrey and Li has been > known since 1998. The media coverage therefore appears to be much ado > about nothing. If the resolution to the Riemann Hypothesis has not been announced in the New York Times, it never happened. Bob Kolker === Subject: Re: Article: Greatest maths problem 'solved' 23 pages is not a very long proof. Given its importance if it is > valid (and the attendant embarassment if it is not), it makes perfect > sense to post it to the Web and let hundreds of folks look it over. > http://news.bbc.co.uk/2/hi/science/nature/3794813.stm > Like a programmer, a mathmatician only has to get it right once. The actual proof is 124 pages long. === Subject: Re: Article: Greatest maths problem 'solved' > Like a programmer, a mathmatician only has to get it right once. Apparently de Branges is the real McCoy, so his work should receive due care and review. He might be right. He might be wrong, but he is no Crank. Bob Kolker === Subject: Re: Article: Greatest maths problem 'solved' > Apparently de Branges is the real McCoy, so his work should receive due > care and review. He might be right. He might be wrong, but he is no Crank. You seem to imply that no-one's work deserves due care and review unless they are the real McCoy. Or am I misunderstanding you? Alec McKenzie === Subject: Re: Article: Greatest maths problem 'solved' > You seem to imply that no-one's work deserves due care and review unless > they are the real McCoy. Or am I misunderstanding you? You misunderstand. Bob Kolker === Subject: Re: Pitchfork bifurcation? > Can somebody explain (if possible, in beginner-friendly-way) what are the > conditions for pitchfork bifurcation appearance? > I tried to figure it out by myself, but i failed : ( ... Please, keep it > simple. Does this help? Or would you prefer an explanation that is not mathematical? Jim Meiss === Subject: Re: Pitchfork bifurcation? > Does this help? > Or would you prefer an explanation that is not mathematical? It helps to confuse me even more ; ) but it helps... === Subject: Re: Pitchfork bifurcation? >> Does this help? >> Or would you prefer an explanation that is not mathematical? > It helps to confuse me even more ; ) but it helps... What about http://www.enm.bris.ac.uk/staff/hinke/courses/NDC/notes/set12.pdf ? === Subject: Re: Pitchfork bifurcation? > Can somebody explain (if possible, in beginner-friendly-way) what are the > conditions for pitchfork bifurcation appearance? > I tried to figure it out by myself, but i failed : ( ... Please, keep it > simple. When you see a fork in the road, take it. - Yogi Berra Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: The principle of equivalence One of the most important observations in physics is that: 'The response of any body - with any weight (w) - to gravity is the same as its resistance to acceleration': That is the ratio of any body's weight (w) divided by the acceleration (g) at which it will freefall is equal to the ratio of the net force (f) exerted on and/or by it, divided by the acceleration (a) that is caused. For any given body, these ratios are equivalent: Equal and constant; as well as being a measure of the body's mass, and/or its inertia. Mass and/or inertia is expressed in Units of Mass! === Subject: Re: The principle of equivalence The principle of idiocy: > some text === Subject: Re: The principle of equivalence > One of the most important observations in physics is that: Dumb Donny Head doesn't know the difference among inertial, gravitational, active, and passive mass. Dumb Donny Head cannot be educated. Dumb Donny Head is a ing imbecile. http://www.apa.org/journals/psp/psp7761121.html http://insti.physics.sunysb.edu/~siegel/quack.html Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) Quis custodiet ipsos custodes? The Net! === Subject: Re: The principle of equivalence One of the most important observations in physics is that: > Dumb Donny Head doesn't know the difference among inertial, > gravitational, active, and passive mass. Dumb Donny Head cannot > be educated. Dumb Donny Head is a ing imbecile. I hope you're wrong about that last one. It would certainly be bad if there's any possibility that he's spreading his germ. === Subject: Re: The principle of equivalence > One of the most important observations in physics is that: > 'The response of any body - with any weight (w) - to gravity is the > same as its resistance to acceleration': That is the ratio of any > body's weight (w) divided by the acceleration (g) at which it will > freefall is equal to the ratio of the net force (f) exerted on and/or > by it, divided by the acceleration (a) that is caused. > For any given body, these ratios are equivalent: Equal and constant; > as well as being a measure of the body's mass, and/or its inertia. > Mass and/or inertia is expressed in Units of Mass! Elk === Subject: Re: The principle of equivalence > One of the most important observations in physics is that: Equivalence Principle of Gravitation http://scienceworld.wolfram.com/physics/EquivalencePrincipleofGravitation.ht m l Equivalence Principle of Special Relativity http://scienceworld.wolfram.com/physics/EquivalencePrincipleofSpecialRelativ i ty.html === Subject: Re: Can't wait to tackle the Riemann Hypothesis > I really hope that the proof is correct. That means the Goldbach > Conjecture is also true, right? To the best of my knowledge, no one has ever proved that the Riemann Hypothesis implies the Goldbach Conjecture. There is some connection between RH and the *ternary* GC, but that's a different matter. Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: DeBranges and RH > FYI Jun 9 DeBranges has just claimed a proof of RH...... (but of course, he has claimed this before) > Nor has he actually posted the proof on the web page provided by Eamon > Warnock, but merely an apology for the proof (in the Platonic sense, I > suppose, though if history repeats itself it may acquire a more literal > meaning). As I mentioned elsewhere in this thread, there's also a much longer paper on de Branges' website, over 100 pages. That's the paper with the purported proof. Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: DeBranges and RH http://mygate.mailgate.org/mynews/sci/sci.math/15b74bbe8623ab051b6853d43e730 b cc.61944%40mygate.mailgate.org > The solution of the Riemann Hypothesis is indeed the Holy > Grail of analysis, and the principal investigator is by most > accounts the most capable person in the world to attempt > a solution. Perhaps somebody could explain to me why, if he is employed full-time by Purdue University in order to teach and do research with pen and paper, an award is necessary? How did that 400 grand aid him in his heretofore fruitless Search? === Subject: Re: DeBranges and RH >Perhaps somebody could explain to me why, if he is employed full-time by >Purdue University in order to teach and do research with pen and paper, >an award is necessary? How did that 400 grand aid him in his heretofore >fruitless Search? I cannot speak for de Branges, but a typical university faculty is paid for 9 months (an Academic Year). The summer summer months may be spent sunning on the beach (with no salary) or can be spent on campus doing research while being paid from a funding agency. The university continues to provide office space, library, janitorial services, yard-maintenance, parking, air-conditioning, secretarial and administrative help, computing, networking and copy machines during the summer, therefore it takes a cut, typically around 1/3, of the NSF grant, so not all the money goes to the grantee's pocket. Does this answer your question? rr === Subject: Re: DeBranges and RH >Perhaps somebody could explain to me why, if he is employed full-time by >Purdue University in order to teach and do research with pen and paper, >an award is necessary? How did that 400 grand aid him in his heretofore >fruitless Search? > I cannot speak for de Branges, but a typical university faculty is > paid for 9 months (an Academic Year). The summer summer months may be > spent sunning on the beach (with no salary) or can be spent on campus > doing research while being paid from a funding agency. > The university continues to provide office space, library, janitorial > services, yard-maintenance, parking, air-conditioning, secretarial > and administrative help, computing, networking and copy machines > during the summer, therefore it takes a cut, typically around 1/3, > of the NSF grant, so not all the money goes to the grantee's pocket. And the money that does go into the grantee's pocket actually goes to pay for overseas visitors to come to collaborate, for graduate students and postdocs to work as research assistants, to pay for travel to conferences, and so on. Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Please check my math - kinematics > Could someone check my math on this? (No this is not my homework, I promise.) So, is it someone else's homework that you're doing for them and want us to do the doing for you? ;-) > A projectile is launched at 30 degrees above horizontal. > The initial speed is 10 m/s. How far does the projectile travel? > Assume no wind resistance and a level field of fire. > My answer: 8.8332 > I'm told this is incorrect. > Here's my work: > Theta = 30 degres > Vi = 10 m/s > Vix = Vi * Cos(Theta) = 8.66 > Viy = Vi * Sin(Theta) = 5 > Vfx = Vix = 8.66 > Vfy = -(Viy) = -5 > ax = 0 > ay = -9.8 m/s/s > Get time in flight (t): > Vfy = Viy + ay * t > -5 = 5 + (-9.8 * t) > -10 = -9.8 t > t = 1.02 seconds > Get distance traveled (x) > x = vix*t + 0.5*ax*t2 > x=8.66*1.02 + (.5 * 0 * 1.04) > x=8.8332 m Looks fine to me. Who told you it is wrong? (is it possible that you're expected to asssume g = 10m/s^2? -- it is a typical approximation to simplify the numbers, and in that case the correct answer would be 8.66m, instead of 8.83) Carlos -- === Subject: Re: Please check my math - kinematics > So, is it someone else's homework that you're doing for them > and want us to do the doing for you? ;-) LOL. Actually it's kind of a physics scavenger hunt. source, they realized they had written 10 m/s but they really meant 1000 m/s. The answer they were looking for was 884. === Subject: Re: Please check my math - kinematics > Could someone check my math on this? (No this is not my homework, I promise.) > A projectile is launched at 30 degrees above horizontal. > The initial speed is 10 m/s. How far does the projectile travel? > Assume no wind resistance and a level field of fire. > My answer: 8.8332 > I'm told this is incorrect. > Here's my work: > Theta = 30 degres > Vi = 10 m/s > Vix = Vi * Cos(Theta) = 8.66 > Viy = Vi * Sin(Theta) = 5 > Vfx = Vix = 8.66 > Vfy = -(Viy) = -5 > ax = 0 > ay = -9.8 m/s/s > Get time in flight (t): > Vfy = Viy + ay * t > -5 = 5 + (-9.8 * t) > -10 = -9.8 t > t = 1.02 seconds > Get distance traveled (x) > x = vix*t + 0.5*ax*t2 > x=8.66*1.02 + (.5 * 0 * 1.04) > x=8.8332 m Your kinematic theory is O.K., but the answer isn't quite right because of rounding-off errors. You use 8.66 and 1.02 as three-figure approximations to 10*sqrt(3) and 10/(9.8) respectively, but then give your answer 8.8332 to five significant figures. If you carry through the same argument without approximating, you should get (50*sqrt(3))/(9.8). *Then* get out your calculator and you'll find the answer to five significant figures is 8.8370. That's the mathematics, but how about the physics? Is your value g = 9.8 m/s/s really correct to five figures? Shouldn't the answer 8.8370 be pruned a bit? Ken Pledger. === Subject: Re: Please check my math - kinematics > Could someone check my math on this? (No this is not my homework, I promise.) > A projectile is launched at 30 degrees above horizontal. > The initial speed is 10 m/s. How far does the projectile travel? > Assume no wind resistance and a level field of fire. > My answer: 8.8332 > I'm told this is incorrect. > Here's my work: > Theta = 30 degres > Vi = 10 m/s > Vix = Vi * Cos(Theta) = 8.66 > Viy = Vi * Sin(Theta) = 5 > Vfx = Vix = 8.66 > Vfy = -(Viy) = -5 > ax = 0 > ay = -9.8 m/s/s > Get time in flight (t): > Vfy = Viy + ay * t > -5 = 5 + (-9.8 * t) > -10 = -9.8 t > t = 1.02 seconds > Get distance traveled (x) > x = vix*t + 0.5*ax*t2 > x=8.66*1.02 + (.5 * 0 * 1.04) ^ | Where did this zero | come from? > x=8.8332 m === Subject: Re: Please check my math - kinematics >>Could someone check my math on this? (No this is not my homework, I > promise.) >>A projectile is launched at 30 degrees above horizontal. >>The initial speed is 10 m/s. How far does the projectile travel? >>Assume no wind resistance and a level field of fire. >>My answer: 8.8332 >>I'm told this is incorrect. >>Here's my work: >>Theta = 30 degres >>Vi = 10 m/s >>Vix = Vi * Cos(Theta) = 8.66 >>Viy = Vi * Sin(Theta) = 5 >>Vfx = Vix = 8.66 >>Vfy = -(Viy) = -5 >>ax = 0 >>ay = -9.8 m/s/s >>Get time in flight (t): >>Vfy = Viy + ay * t >>-5 = 5 + (-9.8 * t) >>-10 = -9.8 t >>t = 1.02 seconds >>Get distance traveled (x) >>x = vix*t + 0.5*ax*t2 >>x=8.66*1.02 + (.5 * 0 * 1.04) > ^ > | > Where did this zero | come from? ax = 0 >>x=8.8332 m Will Twentyman email: wtwentyman at copper dot net === Subject: Injectuve and Surjective function difficulty that some of you may know the terms as one-to-one/monic and onto/epic, respectively. I understand the general strategy to use to prove whether or not a function is injective, surjective, or bijective. The difficulty that I experience is with the following problem in the proof for surjectivity; an element is in (0, INF) and the function takes (1, INF). Problem. Is the following function injective, surjective, both or neither? Let t: (1, INF) -> R be defined by t(x) = ln x for all x in (1, INF). Proof of injectivity. Assume t(x) = t(y); we will show this implies x = y and hence t is injective. Let x, y in (1, INF). Then t(x) = t(y) implies ln x = ln y. Since ln x in (-INF, INF), EXP(ln x) is well-defined, and likewise for EXP(ln y). Therefore, EXP(ln x) = EXP(ln y) gives x = y, as desired. QED. Proof of surjectivity that is in question. Let b in R; we will show that there exists some a in (1, INF) such that t(a) = b, and hence t is surjective. Since b in R, EXP(b) in (0, INF). Therefore, t(EXP(b)) = b, as desired. QED? I don't feel the proof of surjectivity is correct since EXP(b) is in (0, INF) and the domain of t is (1, INF), but I don't know how to work around this, or if this means that t is not surjective. === Subject: Re: Injectuve and Surjective function difficulty days. My association with the Department is that of an alumnus. >that some of you may know the terms as one-to-one/monic and onto/epic, >respectively. Be a bit careful with monic and epic. They have meanings in category theory that are not always equivalent to 'injective' and 'surjective'. Otherwise, yes. > I understand the general strategy to use to prove whether or >not a function is injective, surjective, or bijective. The difficulty that I >experience is with the following problem in the proof for surjectivity; an >element is in (0, INF) and the function takes (1, INF). Huh? >Problem. Is the following function injective, surjective, both or neither? >Let t: (1, INF) -> R be defined by t(x) = ln x for all x in (1, INF). Ah. INF is infinity and you are talking about open intervals (since you've been posting mostly about set theory, I thought you were talking about ordered pairs!). And function takes (1,INF) means that the domain of the function is (1,infinity)? >Proof of injectivity. Assume t(x) = t(y); we will show this implies x = y >and hence t is injective. Let x, y in (1, INF). Then t(x) = t(y) implies ln >x = ln y. Since ln x in (-INF, INF), EXP(ln x) is well-defined, and likewise >for EXP(ln y). Therefore, EXP(ln x) = EXP(ln y) gives x = y, as desired. >QED. Have you seen anything about the relation between injectivity, surjectivity, bijectivity, one sided inverses, and inverses? >Proof of surjectivity that is in question. Let b in R; we will show that >there exists some a in (1, INF) such that t(a) = b, and hence t is >surjective. Since b in R, EXP(b) in (0, INF). Therefore, t(EXP(b)) = b, as >desired. QED? No, incorrect. You cannot apply t to EXP(b) unless exp(b)>1; the image of the natural log, if you only allow values greater than 1, is only the positive reals: a negative real will never be the image of ln(x) for some x>1. >I don't feel the proof of surjectivity is correct since EXP(b) is in (0, >INF) and the domain of t is (1, INF), but I don't know how to work around >this, or if this means that t is not surjective. Come up with an example. Say, y = -1. Now, if t(x)=y, i.e., ln(x)=-1, that means that e^{-1} = x; so x = 1/e < 1. Which means that x is not in the domain. Thus, no element of (1,infinity) maps to -1, which means t is not surjective. It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Injectuve and Surjective function difficulty > that some of you may know the terms as one-to-one/monic and onto/epic, > respectively. You have to be careful: monic is often not the same as 1-1 and epic often is not equivalent to onto. In general, the definitions are as follows: A map f is called monic, if for any g_1 and g_2 that can be composed with f, it holds: f o g_1 = f o g_2 ==> g_1 = g_2 h_2 composable with f, we have h_1 o f = h_2 o f ==> h_1 = h_2 It should be clear that, when you consider arbitrary maps between sets, that f monic <==> f 1-1 f epic <==> f onto But in general, e.g. when the allowed maps are ring-homomorphisms, this is not the case. For a more exact definition check out any introduction to category theory. hang my head drown my fear till you all just disappear reverse my forename for mail! - saibot === Subject: Re: Injectuve and Surjective function difficulty > that some of you may know the terms as one-to-one/monic and onto/epic, > respectively. I understand the general strategy to use to prove whether or > not a function is injective, surjective, or bijective. The difficulty that > I experience is with the following problem in the proof for surjectivity; > an element is in (0, INF) and the function takes (1, INF). > Problem. Is the following function injective, surjective, both or neither? > Let t: (1, INF) -> R be defined by t(x) = ln x for all x in (1, INF). > Proof of injectivity. Assume t(x) = t(y); we will show this implies x = y > and hence t is injective. Let x, y in (1, INF). Then t(x) = t(y) implies > ln x = ln y. Since ln x in (-INF, INF), EXP(ln x) is well-defined, and > likewise for EXP(ln y). Therefore, EXP(ln x) = EXP(ln y) gives x = y, as > desired. QED. > Proof of surjectivity that is in question. Let b in R; we will show that > there exists some a in (1, INF) such that t(a) = b, and hence t is > surjective. Since b in R, EXP(b) in (0, INF). Therefore, t(EXP(b)) = b, as > desired. QED? > I don't feel the proof of surjectivity is correct since EXP(b) is in (0, > INF) and the domain of t is (1, INF), but I don't know how to work around > this, or if this means that t is not surjective. The way you have stated the problem, t isn't surjective. In fact, t takes only positive values since if x > 1, then ln x > 0. So if you make the codomain of t R, you will never get a negative value. Conversely, if you restrict the codomain of t to just (0, INF) then your argument is fine, since EXP: (0, INF) -> (1, INF), and that is the domain of t. Ryan Reich ryanr@uchicago.edu === Subject: Question on the proof of the Prime Number Theorem in Hardy by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i5B0jCw12195; I am trying to understand a proof of the prime number theorem found in An Introduction to the Theory of Numbers 5th ed, by Hardy and Wright. Please see the first line in pp 365 on the beta and the sixth line in the same page. I think beta is bounded by alpha. To see this, consider the area for the rectangle alpha*@ ( I couldn't read the Greek letter for the upper bound of the integral. ) But geometrically, it is clear that alpha*@ geq int_{0}^{@}|V(#)|d# where # denotes another Greek letter which I couldn't read. alpha geq (1/@)int_{0}^{@}|V(#)|d# . Clearly, letting @-> infty on the right is equivalent to the expression on the right of the first line given in pp 365. But from this, the inequality alpha leq beta doesn't make sense at all. (It does if alphy = beta.) Furthermore, the authors say that they take a contradiction by deducing beta < alpha under the assumption alpha > 0. Could anyone help me out? H. Shinya === Subject: Re: Question on the proof of the Prime Number Theorem in Hardy and Wright > I am trying to understand a proof of the prime number theorem found > in An Introduction to the Theory of Numbers 5th ed, by Hardy and > Wright. > Please see the first line in pp 365 on the beta and the sixth > line in the same page. > I think beta is bounded by alpha. To see this, consider the area > for the rectangle > alpha*@ ( I couldn't read the Greek letter for the upper bound > of the integral. ) xi > But geometrically, it is clear that > alpha*@ geq int_{0}^{@}|V(#)|d# No. For the benefit of readers without H & W to hand, V is a bounded function on the positive reals, alpha is limsup of |V(x)| as x -> infinity and beta is limsup of (1/x)integral_0^x |V(t)| dt as x -> infinity. The hypotheses don't rule out the possibility that V(x) -> 0 as x -> 0. In that case, alpha would be zero, but then your inequality would be absurd. > where # denotes another Greek letter which I couldn't read. eta Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Method for Determining Info System Relational Transformation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i5B0jC012209; Hello Again: Recently I posted a request on Math Forum for information on a Information System Relational Transformation. Last time I checked I found no response to this post. The method I came up for determining a Relational Transformation for a Information System is at: http://www.zimmathematics.com/app.htm#CxApplication This relational transformation can be used with my System/God Transformation to make it work Mathematically. This System/God Transformation can be found at: http://www.zimmathematics.com/app.htm#Transformation This System/God Transformation is usefull in determining System Informational functionality. Zim Olson http://www.zimmathematics.com === Subject: Re: Number by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i5B0jBV12163; >It could be possible to extend the ideas discussed above >to other cyclic groups. Take the group {i, -i, 1, -1} >where i is imaginary and define >par(n) = (i)^{ f( c( b(n), b(n) )) } >Multiplication would be given by >n*m = (i)^{ f( c( b(|n|), b(|m|) ) } nm, |n|, |m| > 1 > = nm otherwise Sorry, such a definition could only be valid for natural numbers. Should have stated: Let mg and nh be elements of the ring generated from {i, -i, 1, -1} where m, n are whole numbers and g and h in {i, 1} Multiplication would be given by (mg)*(nh) = (i)^{ f( c( b(|n|), b(|m|) ) } nmgh, |n|, |m| > 1 = nmgh , otherwise (One could, perhaps, go into defining a similar multiplication between any two ring elements using the distributive property- but I am merely trying to convey the general idea at this point...) >It is now easy to see, for ex., that >(12*4)*6 = -i288 = 12*(4*6) C. Dement === Subject: Re: Color quiz > (defun nats (n) > cons (n, nats(n+1)) > ) > this outputs (1,2,3,4,5,6...) and never stops. > Actually, it fills up memory and never gets around to outputting anything. > At least, such is my understanding of evaluation. No > (defun first (n, l) > (if (equal n, 1) (car l) > (first (n-1, cdr(l))) > ) > this outputs the first n members of a list E&OE. > So > (first 3, nats(1)) = <1, 2, 3 Sneaky or what? This is functional programming and YES IT WORKS. > Pity none of you learn about handling infinite streams you all have > Cantor's rubbish driveling out of your mouths. > I'm not up on my Lisp, but it's clear that it does in fact work, > in a sufficiently sophisticated system. practically any functional parser will handle this, its just lazy evaluation, it evaluates parameters as they are needed, so the infinite loop in nats is no problem since it truncates off a finite section. > I won't. But {0.3, 0.33, 0.333, ... } doesn't contain (/ 1 3) either. Your usage of contain is very limited. Say one process draws a line from 0 to 1/2 1/2 to 3/4 3/4 to 7/8 etc. Another process starts at 2 going down 2 to 1 1/2 1 1/2 to 1 1/4 1 1/4 to 1 1/8 etc. Considering the infinite processes, do the lines meet? 0 0.9 0.99 0.999 ... what is the diagonal on this list? How many recurring 9s are part of the diagonal? Herc === Subject: About rational numbers and their decimal form I have the following conjecture about rational numbers and their decimal form: Let N be a rational number and not an integer (hence its decimal representation has a least one digit), and let d be the denominator of their irreducible rational (fraction) form. Then: - N has a finite decimal representation if and only if d has no prime factors other than 2 and 5. - N has a pure periodic decimal form if and only if d is neither divisible by 2 nor by 5, or it is not divisible by 3. - N has a non-pure periodic decimal form if and only if d is divisible by 3 and it is also either divisible by 2 or by 5. If you can prove (or disprove) this conjecture I will be glad you did, since, as you can see, this is a characterization of rational numbers according to their decimal forms. DO NOT FORGET: d is the divisor of the IRREDUCIBLE rational form... === Subject: Re: About rational numbers and their decimal form > I have the following conjecture about rational numbers and their > decimal form: > Let N be a rational number and not an integer (hence its decimal > representation has a least one digit), and let d be the denominator of > their irreducible rational (fraction) form. Then: > - N has a finite decimal representation if and only if d has no prime > factors other than 2 and 5. > - N has a pure periodic decimal form if and only if d is neither > divisible by 2 nor by 5, or it is not divisible by 3. > - N has a non-pure periodic decimal form if and only if d is divisible > by 3 and it is also either divisible by 2 or by 5. Should it happen that you define pure as at http://www.spd.dcu.ie/johnbcos/download/2nd_year/2nd%20BA%20Cantorian/decima l%20expansions2.html then your conjecture is wrong. Take N = 1/14. Then d = 14 and the assertion d is neither divisible by 2 nor by 5, or it is not divisible by 3 is clearly true; it would follow from your conjecture that N has a pure periodic decimal, but, in fact, 1/14 = 0.0714285714285714285... and so it's not pure. Jose Carlos Santos === Subject: Re: About rational numbers and their decimal form > - N has a pure periodic decimal form if and only if d is neither > divisible by 2 nor by 5, or it is not divisible by 3. > - N has a non-pure periodic decimal form if and only if d is divisible > by 3 and it is also either divisible by 2 or by 5. 3 does not behave any different from other prime divisors that are neither 2 nor 5. Correct is: - N has a pure periodic decimal form if and only if d is neither divisible by 2 nor by 5. - N has a non-pure periodic decimal form if and only if d is divisible by a prime number other than 2 or 5 and it is also divisible by 2 or by 5. See also: http://www.lrz-muenchen.de/~hr/numb/period.html The first paragraphs are not really a proof but should contain the essential ideas of a proof. Helmut Richter === Subject: Re: About rational numbers and their decimal form > I have the following conjecture about rational numbers and their > decimal form: > Let N be a rational number and not an integer (hence its decimal > representation has a least one digit), and let d be the denominator of > their irreducible rational (fraction) form. Then: > - N has a finite decimal representation if and only if d has no prime > factors other than 2 and 5. This is true. Let N=c/d, d>0, gcd(c,d)=1. Then N has a finite decimal repr, iff there is an integer k>0 such that (10^k)c/d is an integer, iff there is k>0 such that any prime dividing d also divides 10^k, iff the only primes dividing d are 2, 5. > - N has a pure periodic decimal form if and only if d is neither > divisible by 2 nor by 5, or it is not divisible by 3. By pure periodic I assume you mean N has a decimal form that looks like c.(a_1a_2...a_m)(a_1a_2...a_m)..., and by non-pure periodic that it looks like c.b_1b_2...b_n(a_1a_2...a_m)(a_1a_2...a_m)..., where not all of the digits a_j are zero? If so, this is false. If d=2, then the statement d is neither divisible by 2 nor by 5, or it is not divisible by 3 is true, since 2 is not divisible by 3; however N=1/d does not have a pure periodic form. > - N has a non-pure periodic decimal form if and only if d is divisible > by 3 and it is also either divisible by 2 or by 5. This is also false. Take d=14, then N=1/d has a non-pure periodic form, but d is not divisible by 3. > If you can prove (or disprove) this conjecture I will be glad you did, > since, as you can see, this is a characterization of rational numbers > according to their decimal forms. > DO NOT FORGET: d is the divisor of the IRREDUCIBLE rational form... === Subject: Re: Godel DISPROOF > QUESTION 1 > ----1----------------------------------------------------------------------- - --- > Randi will test you when you properly apply to be tested. Sign up here: > http://www.randi.org/research/challenge.html > Quick question, have you done this? he didn't reply. > QUESTION 2 > ----2----------------------------------------------------------------------- - ---- > It really all depends on the situation. > QUESTION 3 > ----3----------------------------------------------------------------------- - ---- > If ever I actually found myself in that situation, I'd hold it upright, > with the intent of attacking my assailant's knife hand. > ANSWER OPTIONS > ----------------- > A cliff86 > B Rust > C Shanx > D NormDePloom > E Rich Shewmaker > F CNote > G Jeff > H See You In Hell My Friend. > I Someone > J Greg Neill > I'll guess 1-I, 2-B, 3-H Nope > If you stumble across a million to one correlation, then you check for > further samples of paranormal, if you don't guess the authors names > by what they write, then you wasted 5 minutes in your leap of faith, move on. > A single random guess proves nothing. It has almost no statistical > significance. Also, your quotes may have been selected for a particular > quality you felt was tied to the person's name, causing it to not be random. LOOKING FOR There's 3 questions not 1, 10 options is 1 in 1000 chance of false positive. The posts are fairly randomly selected because they are all on the one day. If you can get 3 replies to you on one day that consistently beat 1000 to 1 odds like this then I would be very surprised. the world communicates to me symbolically. One way to capture the macro quantum field in action is to analyse responses to me. If my theory proves correct you'll go down in history as having the will of 20 men. Herc === Subject: Re: limitation to induction on finite bounds > Try again. > x the 1st row is finite length > o > oo > oox > ooxx the nth row is finite length (o is some finite section of x's) > o > oo > oox > ooxx n > ooxxx if the nth row is finite length, the n+1th row is finite length > By induction > x > xx > xxx > ... > Every row is finite length. > -------------------------------------------- > Is that valid induction? > Very valid, although your notation might be better expressed: Try again. x the 1st column is finite length o oo oox ooxx the nth column is finite length (o is some finite section of x's) o oo oox ooxx n ooxxx if the nth column is finite length, the n+1th column is finite length By induction x xx xxx ... Every column is finite length. -------------------------------------------- Is that valid induction? Herc === Subject: departure process of M/M/c/c For the classical erlang loss queueing system M/M/c/c, where the first 'M': the Possion arrival process; the second 'M': the exponential service time; the first 'c': the number of servers the second 'c': no buffer. 1. I need to know the departure process of this system.How to model the departure process? Poisson or IPP or MMPP or other process? 2. If the system becomes M/G/c/c where 'G' represents the general service time. Then, how to model the departure process? I have checked a lot of books and papers and surprisingly to find that no conclusion can find. Plz give some suggestions or keyworks. Many === Subject: Re: departure process of M/M/c/c [posted and mailed] > For the classical erlang loss queueing system M/M/c/c, where > the first 'M': the Possion arrival process; > the second 'M': the exponential service time; > the first 'c': the number of servers > the second 'c': no buffer. > 1. I need to know the departure process of this system.How to model > the departure process? Poisson or IPP or MMPP or other > process? > 2. If the system becomes M/G/c/c where 'G' represents the general > service time. Then, how to model the departure process? > I have checked a lot of books and papers and surprisingly to find that > no conclusion can find. Plz give some suggestions or keyworks. Many See Ken Mitchell, Member, IEEE, Khosrow Sohraby, Senior Member, IEEE, Appie van de Liefvoort, Member, IEEE,and Jerry Place, Member, IEEE,Approximation Models of Wireless Cellular Networks Using Moment Matching, IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 19, NO. 11, NOVEMBER 2001. Good luck, Jerry Harder === Subject: The pound-force The pound-force is a unit of force or weight (properly abbreviated lbf or #). The U.S. pound-force is equal to a mass of 32.17405# sec^2 per foot, divided by the acceleration (g) at which it will free fall (which varies slightly over earth's surface and averages about 32.17405'/sec^2, or approximately 9.80665 m/sec^2): The formula is m = f/a = w/g Therefore: One pound-force, or weight = m/g = (32.17405# sec^2/foot)/32.17405'/sec^2 = 1# One newton force or weight = m/g = (9.806605 N sec^2/m)/9.805505 N/sec^2 = 1N A pound-force is equal to about 4.4482216 newtons of force. We cannot arbitrarily define _exact_ values for g, or w because they vary slightly at various locations over Earth's surface: That is g is location dependent, and is the reason why weight varies depending on _its_ location! === Subject: Re: The pound-force pound-rectum. === Subject: Re: The pound-force Nothing. http://www.apa.org/journals/psp/psp7761121.html http://insti.physics.sunysb.edu/~siegel/quack.html Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) Quis custodiet ipsos custodes? The Net! === Subject: Re: The pound-force >The pound-force is a unit of force or weight (properly abbreviated >lbf or #). No, that # is just an informal, produce-section handwritten-sign symbol for the pound mass, not the pound force. Nothing proper about its use. The # symbol for a unit of measure in a science newsgroup is the sure sign of a crackpot. I'm happy that you continue to use it, Donald. Yes, you should use lbf for this unit, to distinguish it from the lb for normal pounds as units of mass. Some people also use lbm for the mass units, but that is not recommended by or followed by any of the national standards laboratories. >The U.S. pound-force is equal to a mass of 32.17405# >sec^2 per foot, That's nonsense, a circular definition. If you already know what a pound force is and use it to define a pound mass, why in the world would you bother with all that bull? But how do you know what that pound force is, before you know what a pound is? >divided by the acceleration (g) at which it will free >fall (which varies slightly over earth's surface and averages about >32.17405'/sec^2, or approximately 9.80665 m/sec^2): The formula is m = >f/a = w/g Now you are getting away from defining a pound force, and instead are telling us that the force which a pound (a unit of mass, or in your usage an object with a mass of 1 lb) will exert will vary with location. >Therefore: > One pound-force, or weight = m/g = (32.17405# >sec^2/foot)/32.17405'/sec^2 = 1# > One newton force or weight = m/g = (9.806605 N sec^2/m)/9.805505 >N/sec^2 = 1N >A pound-force is equal to about 4.4482216 newtons of force. >We cannot arbitrarily define _exact_ values for g, or w because >they vary slightly at various locations over Earth's surface: That is >g is location dependent, and is the reason why weight varies >depending on _its_ location! It isn't g which is arbitrarily defined in exact values. Rather that is g_n (the normal acceleration of gravity). For the purpose of defining a kilogram force this has the official value of 9.80665 m/sî. For the purpose of defining a pound force, nobody has ever bothered setting an official value. The ones used are usually either 32.16 ft/sî or 32.2 ft/sî or 386 in/sî or the official one for defining kilograms force is borrowed for this purpose as well. There is also the cousin of g_n, the dimensionless constant g_c, which is the ratio of the acceleration used to define a pound force and the acceleration to define a poundal (1 ft/sî), so the units Gene Nygaard === Subject: Re: The pound-force > The pound-force is a unit of force or weight (properly abbreviated > lbf or #). The U.S. pound-force is equal to a mass of 32.17405# > sec^2 per foot, divided by the acceleration (g) at which it will free > fall (which varies slightly over earth's surface and averages about > 32.17405'/sec^2, or approximately 9.80665 m/sec^2): The formula is m = > f/a = w/g I thought I smelled units being cooked. Bob Kolker === Subject: Re: The pound-force the usual crap. La-de-ing-da. Google Obsessive-Compulsive Disorder to understand Shead's post. Jim Pennino === Subject: how many math courses I begin 3rd year math studies this fall. I'm wondering how many math classes is practical. I have finished all my general ed, and each quarter I have 1-2 elective courses and want to take math electives. I'm wondering if 4 math courses per quarter is too much--does anyone do it? On the other hand, I had to take a German course and I ended up spending more time in that course that differential equations... so maybe its no issue at all. Anyway, I know talking to my counselor would be the best place to seek advice, and I will do that when the time comes for that. However, I am a bit anxious now to at least get an idea of how my next two years will be planned. === Subject: Re: how many math courses > I begin 3rd year math studies this fall. I'm wondering how many math > classes is practical. I have finished all my general ed, and each quarter I > have 1-2 elective courses and want to take math electives. I'm wondering > if 4 math courses per quarter is too much--does anyone do it? On the other > hand, I had to take a German course and I ended up spending more time in > that course that differential equations... so maybe its no issue at all. > Anyway, I know talking to my counselor would be the best place to seek > advice, and I will do that when the time comes for that. However, I am a > bit anxious now to at least get an idea of how my next two years will be > planned. People do take 4 or more math courses per semester. It depends on how much you like math (would you enjoy spending most of your time studying it?), and on what you want to do with your career / life later. In college (where I assume you are), I've found that having another course (perhaps a philosophy / music / whatever class) brings some balance and helps me focus better on the actual math courses. On the other hand, if in a given semester the university offers 4 math courses that you really want to take, by all means, go for it. Don't take the courses if you have no interest in the actual topics, however -- you will be miserable and might end up hating math altogether! === Subject: Re: how many math courses > I begin 3rd year math studies this fall. I'm wondering how many math > classes is practical. I have finished all my general ed, and each quarter I > have 1-2 elective courses and want to take math electives. I'm wondering > if 4 math courses per quarter is too much--does anyone do it? On the other > hand, I had to take a German course and I ended up spending more time in > that course that differential equations... so maybe its no issue at all. > Anyway, I know talking to my counselor would be the best place to seek > advice, and I will do that when the time comes for that. However, I am a > bit anxious now to at least get an idea of how my next two years will be > planned. You would have to consider the specifics of the courses: their difficulty as relates to your background and abilities, whether their contents overlap each other's and whether their contents overlap the content of your previous work. You also have to consider your learning style and the teaching style of the instructors and the materials and textbooks assigned. You are right to suppose that the number of courses is as important or more so than the number of credits involved, as task-switching is a big cost. I would say that not many people can make A's in all of four hard technical courses, and you are pushing the limit when you try it, even if you are very smart. Talk to the instructors and to the previous year's students. Look at the previous year's class notes and books and papers. Sometimes you might want to prepare for an upcoming course during vacation. See how you do taking three at a time and remember that math courses are not like cans of beans. If you happen to make straight A's in four courses once, don't assume you can do it every time. Each course is different. === Subject: Re: how many math courses > You would have to consider the specifics of the courses: their > difficulty as relates to your background and abilities, whether their > contents overlap each other's and whether their contents overlap the > content of your previous work. You also have to consider your learning > style and the teaching style of the instructors and the materials and > textbooks assigned. > You are right to suppose that the number of courses is as important or > more so than the number of credits involved, as task-switching is a big > cost. > I would say that not many people can make A's in all of four hard > technical courses, and you are pushing the limit when you try it, even > if you are very smart. This was what I was wondering. I suppose I'll go for 3 coures the first quarter to be safe and feel things out. I don't want my GPA to suffer for being overly ambitious. > Talk to the instructors and to the previous year's students. Look at > the previous year's class notes and books and papers. Sometimes you > might want to prepare for an upcoming course during vacation. Yes, I know upper division math is more logic and proof based. I took a logic course of propositional calculus and predicate calculus a semester ago in hopes of picking up some basic ideas. I'm also planning on reading Concepts of Modern Mathematics by Ian Stewart and How to Solve It by Polya this summer. I was also considering picking up the text book my university uses for their intro to abstract math course: Foundations of Higher Mathematics. A reviewer on amazon.com said he read it the summer before beginning upper graduate studies and that it helped a lot. > See how you do taking three at a time and remember that math courses are > not like cans of beans. If you happen to make straight A's in four > courses once, don't assume you can do it every time. Each course is > different. Very true. One other new question. My catalog has the courses Mathematical Logic, Set Theory, Topology and Modern Geometry in its undergraduate section and also graduate section. Do you recommend I try and fit these courses in my senior year, or wait until grad school? What is the usual? I've always wanted to study set theory and such because I like to understand things at the lowest level. However, I can wait until grad. school if it is so recommended unless it is recommended that I take them before grad school. === Subject: Re: how many math courses > I don't want my GPA to suffer for > being overly ambitious. But think about more than your immediate grades. You are laying a foundation for your later studies. I found that following the proofs in my first calculus courses didn't do much for my near-term grades, but prepared me for advanced courses. > I'm also planning on reading It's good that you are so to speak sending out advance guards. That way you identify what you like, what is difficult for you, what the richest research areas are, and what is most generative of further understanding. > Do you recommend I try and fit these courses in my senior > year, or wait until grad school? I didn't do it, but I've often thought it would be good to spend an extra year taking undergrad math beyond the graduation requirements. That way you can get a very broad view of math so you are unlikely to feel ignorant of any major area. And you will be better oriented and more confident and better able to choose a good research area. And I think it takes some time for your subconcious to work out the consequences of your concious learning. Things usually make more sense and are much easier the second time over. You should ask your counselor and other profs about this idea. I seriously doubt that anybody would frown on it if you can afford the time and money. Some people pass (make B's or better) their grad courses but fail their PhD qualifying exams and have to settle for an MS. So it pays to understand rather than to just get the grades in your courses. === Subject: Re: how many math courses >I begin 3rd year math studies this fall. I'm wondering how many math >classes is practical. I have finished all my general ed, and each quarter I >have 1-2 elective courses and want to take math electives. I'm wondering >if 4 math courses per quarter is too much--does anyone do it? On the other >hand, I had to take a German course and I ended up spending more time in >that course that differential equations... so maybe its no issue at all. >Anyway, I know talking to my counselor would be the best place to seek >advice, and I will do that when the time comes for that. However, I am a >bit anxious now to at least get an idea of how my next two years will be >planned. The answer depends on things we can't possibly know, having to do with how sharp you are. The courses do get harder as you get higher up; a lot of people who have no trouble with DE find more advanced courses difficult. A lot of people don't. Even better than asking your advisor about this would be asking the people who you took your previous math courses from - _they_ will have a better idea than anyone how difficult you may find more advanced courses. ************************ David C. Ullrich === Subject: Re: resolving Will's misunderstanding >y = 1 >for n = 1 to 3 > y = y * x >next that should be fine for exp. we disallow x*x but allow x^2. simple theory but I'm >keeping your constructions on file until I can spend some time to go over them. Herc >>x^2 is defined by x*x. You can't keep one and not the other. >> > No its not. y = 1 > for n = 1 to 2 > y = y * x > next > GOOD y = x * x > BAD Ok, First: > y=x > z=y*x Second: > z=x*x How are they different? Don't they both make z = x*x = x^2? 2/ Modify the halt function TMm(x, y) loops IFF TMh(x, y) != 0 GOOD > (no halting contradiction) >2/ Modify the halt function TMm(x) loops IFF TMh(x, x) != 0 BAD >What if G lays down F1 on the tape followed by x, x, then enters the UTM >>This appears to do what you said is impossible. BAD > See a pattern? My concern is that by making such restrictions, you are going to > 1) eliminate from consideration certain computable functions > 2) give you something more restrictive than the partial recursive > functions, which don't halt either. > Let F be a computable function of one variable (given by F(n)). > Define G(x,y) by F(x + y). Suppose that G(x,x) is undefined for every > x. Then so is F(x + x). So no computable functions are defined on > even inputs. Similarly, no function is defined at any square. Heck, > let H(x,y) be any computable function. Then F(H(x,x)) is undefined, > even if H(x,y) = x! (which implies that F(H(x,x)) = F(x). So no > computable function is defined anywhere. > Damn straight this elminates computable functions. ;-) Of course, I mean H(x,y) = x! to mean H(x,y) is a projection function, not a factorial function. :) 'cid 'ooh === Subject: Re: resolving Will's misunderstanding >y = 1 >for n = 1 to 3 > y = y * x >next >>that should be fine for exp. we disallow x*x but allow x^2. simple theory but I'm >keeping your constructions on file until I can spend some time to go over them. >>Herc >>x^2 is defined by x*x. You can't keep one and not the other. > No its not. y = 1 >for n = 1 to 2 > y = y * x >next >GOOD y = x * x >BAD >>Ok, >>First: >>y=x >>z=y*x >>Second: >>z=x*x >>How are they different? Don't they both make z = x*x = x^2? >2/ Modify the halt function TMm(x, y) loops IFF TMh(x, y) != 0 GOOD >(no halting contradiction) >>2/ Modify the halt function TMm(x) loops IFF TMh(x, x) != 0 BAD >>What if G lays down F1 on the tape followed by x, x, then enters the UTM >>This appears to do what you said is impossible. BAD >See a pattern? >>My concern is that by making such restrictions, you are going to >>1) eliminate from consideration certain computable functions >>2) give you something more restrictive than the partial recursive >>functions, which don't halt either. > ok, but its this type of operation to avoid > f(x) = g(x, x) > You can't avoid it and still work with primitive recursive functions. > See below. > if you want to trick the compiler with > y=x > f(x) = g(x,y) > y is free in f so I don't think you can refute Halt with that. > 2/ Modify the halt function TMm(x) loops IFF TMh(x, y) != 0 > Its not in the lines of code its the function defn! And that can only use > the parameters available. ;-) > As much as you dislike it, x*x is primitive recursive, and ALL primitive > recursive functions are implementable by TMs. You are working on having > something so limited that you won't have to discuss the halting problem, > you'll have to discuss how to do arithmatic. > Proof that x*x is primitive recursive: > 1) f(x) = x+1 is primitive recursive > 2) f(x1,x2,...,xn) = m is primitive recursive where n and m >= 0 > 3) f(x1,x2,...,xn) = xi is primitive recursive where 0 < i <= n > 4) If g1, g2, .. gm, are primitive recursive and n-ary, and h is > primitive recursive and m-ary then so is n-ary f defined as > f(x1,x2,...,xn) = h(g1(x1,x2,...,xn),...,gm(x1,x2,...,xn)) > 5) For n>=1, if f is n-ary, g is (n-1)-ary, h is (n+1)-ary, and g and h > are primitive recursive, then so is f defined by the following two rules: > a) f(0,x2,x3,...,xn) = g(x2,x3,...,xn) > b) f(x1+1,x2,x3,...,xn) = h(x1,f(x1,x2,...,xn),x2,...,xn) > Define: > Ident(x) = x by rule 3 > Mult(x,y) = x*y (as generated in another post in this thread) > f(x) = Mult(Ident(x),Ident(x)) by rule 4 > Result: f(x) = x*x > f(x) is primitive recursive. Which rule do you want to throw out that > will prevent f(x) but still let you do addition? Way ahead of you Robin! > 4) If g1, g2, .. gm, are primitive recursive and n-ary, and h is > primitive recursive and m-ary then so is n-ary f defined as > f(x1,x2,...,xn) = h(g1(x1,x2,...,xn),...,gm(x1,x2,...,xn)) Improvement : 4) If g1, g2, .. gm, are primitive recursive and n-ary, and h is primitive recursive and m-ary then so is n-ary f defined as f(x1,x2,...,xn) = h(g1(z1*x1,z2*x2,...,zn*xn),...,gm(zm1*x1,zm2*x2,...,zmn*xn)) where z1 or z11 or z21 or ... or zm1 = 1 (exclusively) z2 or z12 or z22 or ... or zm2 = 1 ... otherwise zxy = 0 That way each parameter of f is only used once. Herc === Subject: Re: resolving Will's misunderstanding >>y = 1 >for n = 1 to 3 >y = y * x >next >>that should be fine for exp. we disallow x*x but allow x^2. simple theory but I'm >keeping your constructions on file until I can spend some time to go over them. >>Herc >>x^2 is defined by x*x. You can't keep one and not the other. >No its not. >>y = 1 >for n = 1 to 2 > y = y * x >next >GOOD >>y = x * x >BAD >>Ok, >>First: >>y=x >>z=y*x >>Second: >>z=x*x >>How are they different? Don't they both make z = x*x = x^2? >2/ Modify the halt function TMm(x, y) loops IFF TMh(x, y) != 0 >>GOOD >(no halting contradiction) >2/ Modify the halt function TMm(x) loops IFF TMh(x, x) != 0 >>BAD >What if G lays down F1 on the tape followed by x, x, then enters the UTM >>This appears to do what you said is impossible. >>BAD >See a pattern? >>My concern is that by making such restrictions, you are going to >>1) eliminate from consideration certain computable functions >>2) give you something more restrictive than the partial recursive >>functions, which don't halt either. >ok, but its this type of operation to avoid >f(x) = g(x, x) >>You can't avoid it and still work with primitive recursive functions. >>See below. >if you want to trick the compiler with >y=x >f(x) = g(x,y) >y is free in f so I don't think you can refute Halt with that. >2/ Modify the halt function TMm(x) loops IFF TMh(x, y) != 0 >Its not in the lines of code its the function defn! And that can only use >the parameters available. ;-) >>As much as you dislike it, x*x is primitive recursive, and ALL primitive >>recursive functions are implementable by TMs. You are working on having >>something so limited that you won't have to discuss the halting problem, >>you'll have to discuss how to do arithmatic. >>Proof that x*x is primitive recursive: >>1) f(x) = x+1 is primitive recursive >>2) f(x1,x2,...,xn) = m is primitive recursive where n and m >= 0 >>3) f(x1,x2,...,xn) = xi is primitive recursive where 0 < i <= n >>4) If g1, g2, .. gm, are primitive recursive and n-ary, and h is >>primitive recursive and m-ary then so is n-ary f defined as >> f(x1,x2,...,xn) = h(g1(x1,x2,...,xn),...,gm(x1,x2,...,xn)) >>5) For n>=1, if f is n-ary, g is (n-1)-ary, h is (n+1)-ary, and g and h >>are primitive recursive, then so is f defined by the following two rules: >>a) f(0,x2,x3,...,xn) = g(x2,x3,...,xn) >>b) f(x1+1,x2,x3,...,xn) = h(x1,f(x1,x2,...,xn),x2,...,xn) >>Define: >>Ident(x) = x by rule 3 >>Mult(x,y) = x*y (as generated in another post in this thread) >>f(x) = Mult(Ident(x),Ident(x)) by rule 4 >>Result: f(x) = x*x >>f(x) is primitive recursive. Which rule do you want to throw out that >>will prevent f(x) but still let you do addition? > Way ahead of you Robin! >4) If g1, g2, .. gm, are primitive recursive and n-ary, and h is >primitive recursive and m-ary then so is n-ary f defined as > f(x1,x2,...,xn) = h(g1(x1,x2,...,xn),...,gm(x1,x2,...,xn)) > Improvement : > 4) If g1, g2, .. gm, are primitive recursive and n-ary, and h is > primitive recursive and m-ary then so is n-ary f defined as > f(x1,x2,...,xn) = h(g1(z1*x1,z2*x2,...,zn*xn),...,gm(zm1*x1,zm2*x2,...,zmn*xn)) > where > z1 or z11 or z21 or ... or zm1 = 1 (exclusively) > z2 or z12 or z22 or ... or zm2 = 1 > ... > otherwise zxy = 0 > That way each parameter of f is only used once. Ok, this works for +, *, ^, ^2. I'll have to look and see if it causes problems with - and /. Also, since squaring is equivalent to self multiplication, what have you gained? Wouldn't it be easier to stick to primitive-recursive functions as they are and just ignore the halting problem completely? Will Twentyman email: wtwentyman at copper dot net === Subject: How to find an upperbound of a constant via inequality constraint I met the following question during the proof of one research question: Assume we know the following three conditions: (1) int_{0}^{a} (1/(b+f(x)) dx <= c (2) int_{0}^{a} (1/(f(x)) dx >= d (3) d > c Here a,b,c,d are positive constants, and f(x) is a positive function of x. The integration is for x during the interval [0,a]. My questions is: how to find the range of b that satisfing the above conditions? Esp, I am interested in the upperbound of b. === Subject: Re: How to find an upperbound of a constant via inequality constraint >Assume we know the following three conditions: >(1) int_{0}^{a} (1/(b+f(x)) dx <= c >(2) int_{0}^{a} (1/(f(x)) dx >= d >(3) d > c >Here a,b,c,d are positive constants, and f(x) is a positive function >of x. The integration is for x during the interval [0,a]. >My questions is: how to find the range of b that satisfing the above >conditions? Esp, I am interested in the upperbound of b. Those inequalities do not suggest any upper bound for b. If b is raised to a very, very high value, (1) is more easily satisfied. The 3 together do imply b>0, but you already stated that it was a positive constant. --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Re: How to find an upperbound of a constant via inequality constraint I am sorry. I really interested in the lower-bound of b. === Subject: Re: Analytically Isomorphic Singularities > I was going over Hartshorne Algebraic Geometry Section I.5 and came across > problem 5.14 concerning isomorphic singularities of plane curves. It sounds > like an interesting idea to study the relationship between singularities, > but I cannot even see how to do part a): If Y = { (x,y): f(x,y) = 0} and Z = > { (x,y): g(x,y) = 0} are plane curves where f and g are polynomials over a > field k, and Pin Y and Qin Z are analytically isomorphic, prove that the > lowest order terms in f have the same degree as the lowest order terms in g. Just curious: in what way are the singularities of f(x,y) = x^2-y^2 and g(x,y) = x^4-y^4 not isomorphic (where the field is R) ? === Subject: Re: Analytically Isomorphic Singularities > I was going over Hartshorne Algebraic Geometry Section I.5 and came across > problem 5.14 concerning isomorphic singularities of plane curves. It sounds > like an interesting idea to study the relationship between singularities, > but I cannot even see how to do part a): If Y = { (x,y): f(x,y) = 0} and Z = > { (x,y): g(x,y) = 0} are plane curves where f and g are polynomials over a > field k, and Pin Y and Qin Z are analytically isomorphic, prove that the > lowest order terms in f have the same degree as the lowest order terms in g. > Just curious: in what way are the singularities of > f(x,y) = x^2-y^2 and g(x,y) = x^4-y^4 not > isomorphic (where the field is R) ? Hartshorne always assumes k is algebraically closed. -Nathan === Subject: Re: Terminal Velocity without drag? > Why is c the speed limit? > approaches c? > Is Cerenkov radiation ever faster than c? It's just faster than light > when light is slowed, nearer to c. > Two point light sources coincidentally start emission at the same time > towards each other. Does not each photon hurtle at c thus that their > relative velocity is 2c? Yes ... that's corect, and two physical bodies can have a relative closing velocity as seen by a third observer as arbitrarily close to 2c. I can also prearrange to have a long line of flash bulbs go off at 10c. So? > The sources disappear and the photons > remain, why is not one at rest and the other moving at 2c? Nothing moving at c every defines a proper rest frame... but we soon start talking in slogans at this rate. > The > visible universe has many light sources. > How would you define a tachyon, and what are its properties? > http://scienceworld.wolfram.com/physics/Tachyon.html > and tachyon speed are reciprocal. Yeah. It's kinda like that. :-) === Subject: Re: Terminal Velocity without drag? > Yo ho ho, and a bottle of rum. > I speak in a variety of English language accents. But not sensibly in any of them. > Ross F. > Here the interruption came from three or four points at once. === Subject: Mathematical Models and Methods in Applied Sciences - Vol. 14, No. 6 Mathematical Models and Methods in Applied Sciences View table-of-contents and abstracts at http://www.worldscinet.com/m3as.html Contents: Spatial Decay Bounds In The Channel Flow Of An Incompressible Viscous Fluid Changhao Lin and Lawrence E. Payne Wave Propagation In A 3-D Optical Waveguide Oleg Alexandrov And Giulio Ciraolo On The Asymptotic Analysis Of Kinetic Models Towards The Compressible Euler And Acoustic Equations A. Bellouquid A Parallel Solver For Reaction.9aDiffusion Systems In Computational Electrocardiology Piero Colli Franzone and Luca F. Pavarino On A Lubrication Problem With Fourier And Tresca Boundary Conditions Mahdi Boukrouche And Grzegorz Lukaszewicz ********** Authors are invited to submit their papers for publication in M3AS. For more information, go to http://www.worldscinet.com/m3as.html View other Mathematics journals at http://www.worldscinet.com/maths.shtml ********** === Subject: Re: The flight of a spinning table-tennis ball > Asger Grunnet When the ball is spinning sideways (say counter-clockwise > as seen from above) the ball will curve (in this case to the > left when looking in the direction of travel of the ball). > : > : Overpressure > : _ . _ > : .' ' > : Rotation ' ' > : | . . <------- Flight direction > : | | | > : V . . > : . . > : '. . > : ' - ' > : Underpressure > : Your diagram only appeared in the quoted version at my end. The reason you are getting the wrong answer is that your diagram is > wrong! Assuming that V is supposed to be the head of a velocity > vector, the overpressure would appear at the bottom of the diagram, > where the air piles up, not the top. Why would you think it's the > other way around? > My experience with putting spins on actual ping pong balls is thay they > curve away from the forward moving side and towards the rearward moving > side, regardless of what the theory predicts they should do. Well, that's what theory as I understand it predicts they should do. === Subject: Re: Pi Shop I know this is something of a shameless plug, but I've just put > together a Pi shop at Cafe Press with shirts, stickers, and other > items with a Pi theme. If anyone is interested, you can visit the >site > at http://www.cafeshops.com/joyofpi Shouldn't the gal's thong have a nu instead? If nothing else, it > will break the monotony. > Well, if you are a white male you are an acceptable target. Now that I > think of it, I am a red-head with blue eyes. I represent a minority in the > total human population! Where is my scholarship and recompense?! :-) Look at the politically correct side, Mike. If you were female you'd be making a fortune selling RCHs to machine shops as primary width standards. That's what feminazi liberation is all about - outlawing the RCH, then demanding compensatory subsidies for loss of income. Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) Quis custodiet ipsos custodes? The Net! === Subject: Re: Cranks > If you want to see *real* cranks, visit sci.physics and > sci.physics.relativity. > For every crank you find here, there's 50 over there. > An appetizer: > http://users.pandora.be/vdmoortel/dirk/Physics/ImmortalFumbles.html > Bring a supply of diapers. On sci.econ, its the mainstream economists, generally, who don't know what they are talking about and go on and on completely clueless: === Subject: Re: Cranks > If you want to see *real* cranks, visit sci.physics and > sci.physics.relativity. > For every crank you find here, there's 50 over there. > An appetizer: > http://users.pandora.be/vdmoortel/dirk/Physics/ImmortalFumbles.html > Bring a supply of diapers. > On sci.econ, its the mainstream economists, generally, who don't know > what they are talking about and go on and on completely clueless: > I know - I've seen it before. (b.t.w. it has no link to your home page) Dirk Vdm === Subject: Re: Cranks >This would seem to be a good theory of cranks...but how do you explain >all those mathematicians who presumably were taught it properly and >well but at some point in their career, become cranks? Sadly, this >does not seem to be exceedingly rare. Personally I think it should be important to stress the difference between a crank and a crackpot: a serious mathematician (or physicist or...) can actually be a bizzarre person, a crank indeed. But a crackpot is many orders of magnitude more crankish than a plain crank IMHO... Michele > Comments should say _why_ something is being done. Oh? My comments always say what _really_ should have happened. :) - Tore Aursand on comp.lang.perl.misc === Subject: Re: Cranks >This would seem to be a good theory of cranks...but how do you explain >all those mathematicians who presumably were taught it properly and >well but at some point in their career, become cranks? Sadly, this >does not seem to be exceedingly rare. > Personally I think it should be important to stress the difference > between a crank and a crackpot: a serious mathematician (or physicist > or...) can actually be a bizzarre person, a crank indeed. But a > crackpot is many orders of magnitude more crankish than a plain > crank IMHO... I think I'm using the phrase crank entirely in the way it's been intended so far in the discussion. And I'm not using it to mean merely a bizarre person. What I mean is someone who is a mathematical crank (cf Underwood Dudley's _Mathematical Cranks_). Perhaps it's hard for you to believe(?) but there are plenty of mathematicians, who having spent most of their lives not being mathematical cranks, suddenly become one. Of course, there are degrees of mathematical crankhood. Quite common seems to be the mathematician who insists he (and I don't mean s/he; this seems entirely a phenomenon involving only men) has solved such and such and keeps pestering his colleagues with yet another proof, only to be shot down again and again. And I'm not even talking about honest real attempts. I'm talking about attempts that seem transparently clueless. Then further along the spectrum, we have the more crankish variety that turn out proofs that nobody can make heads or tails of. It's too incoherent to crank much more infrequently than the previous variety. === Subject: simple math problem High school math class was a long time ago. Now I'm afraid that my husband and I will divorce before we figure out if we will be able to get our new (custom ordered!) sofa into the house! For the record, it's not yet actually ordered because of this. Please help. Or send a divorce attorney. Quickly. The sofa is 40 deep, 38 high and 93 long. It has to come through a door that's 30 wide, down a flight of stairs (really don't know what to tell you there!) and through a second door that's RIGHT at the bottom of the stairs (as in, no wiggle or twisting room for manovering). We've drawn diagrams, twisted paper into bits and can't follow each other's logic. Good thing the rest of the marriage works! Suffice it to say, we won't ever be building a house!I hope I've provided enough info for you to help. For that matter, I just hope I'm posting in an appropriate group. TIA === Subject: Re: simple math problem >The sofa is 40 deep, 38 high and 93 long. It has to come through a >door that's 30 wide, down a flight of stairs (really don't know what >to tell you there!) and through a second door that's RIGHT at the >bottom of the stairs (as in, no wiggle or twisting room for >manovering). The sofa of greatest area which can be moved around a right-angled hallway of unit width has an area of 2.21953167 http://mathworld.wolfram.com/MovingSofaProblem.html Even tilted as suggested in another response, the area of the footprint of your sofa is 2.73394 in normalized units. You will never get the sofa up the stairs. I suggest Douglas Adams' book Dirk Gently's Holistic Detective Agency http://www.douglasadams.com/creations/0671746723.html for a timely solution. A search of Google, using the key words: sofa constant yields quite a bit of information for background. John Bailey http://home.rochester.rr.com/jbxroads/mailto.html === Subject: Re: simple math problem > The sofa is 40 deep, 38 high and 93 long. It has to come through a > door that's 30 wide, down a flight of stairs (really don't know what > to tell you there!) and through a second door that's RIGHT at the > bottom of the stairs (as in, no wiggle or twisting room for > manovering). We've drawn diagrams, twisted paper into bits and can't > follow each other's logic. Good thing the rest of the marriage works! The side view of the sofa looks like a fat L. If you tip it forward far enough, so it looks like this: |<| going through the door, the front to back distance _might_ be less than 30. The legs might also unscrew from the sofa and buy you a couple more inches. If you had access to the sofa, or its twin, you need to take some masking tape and run a piece from the back to the tip of the arm and a piece from the back to the front of the seat. (Doing this as close to one end of the sofa as possible.) Then measure from the back lower corner of the sofa to the closest point on each tape. If either measurement is bigger than 30, you're screwed. (Remember to take the cushions off.) You also have to drag the upholtery past all protruding objects, so the fact that the sofa is pliable and helps gain you an inch here or there, also hurts when you tear a big gash in your leather/fancy material on the light switch/banister/ door latch/etc. If it's close, you can take the door off the hinges and take a couple of pieces of wood off the my life, I can pretty much guarantee that Murphey's Law will kick in somehow, somewhere. If it's a sofa bed, there's good news and bad news. the back might lay down, and the thing will fit through easily. Bart === Subject: Re: simple math problem You may be ok. To know for sure the cross-sectional shape of the sofa is needed, and also the height of the doors / ceiling, and the steepness of the stairs. > High school math class was a long time ago. Now I'm afraid that my > husband and I will divorce before we figure out if we will be able to > get our new (custom ordered!) sofa into the house! For the record, > it's not yet actually ordered because of this. Please help. Or send a > divorce attorney. Quickly. > The sofa is 40 deep, 38 high and 93 long. It has to come through a > door that's 30 wide, down a flight of stairs (really don't know what > to tell you there!) and through a second door that's RIGHT at the > bottom of the stairs (as in, no wiggle or twisting room for > manovering). We've drawn diagrams, twisted paper into bits and can't > follow each other's logic. Good thing the rest of the marriage works! > Suffice it to say, we won't ever be building a house!I hope I've > provided enough info for you to help. For that matter, I just hope I'm > posting in an appropriate group. > TIA === Subject: Re: MAN WAKENS FROM 20 YEAR COMA > ...looks out the hospital window, sees the flag at half mast, > and says > What's that all about? > The nurse replies > Ronald Reagan just died. > To which he reponds > You mean that bastard Bush is now president?! I remember that same joke when it was about Eisenhower and Nixon. Now where did I put that slide rule? http://hertzlinger.blogspot.com === Subject: Re: MAN WAKENS FROM 20 YEAR COMA === >Subject: Re: MAN WAKENS FROM 20 YEAR COMA >Message-id: <6wayc.5592$Y3.5146@newsread2.news.atl.earthlink.net> ...looks out the hospital window, sees the flag at half mast, >> and says >> What's that all about? >> The nurse replies >> Ronald Reagan just died. >> To which he reponds >> You mean that bastard Bush is now president?! >I remember that same joke when it was about Eisenhower and Nixon. >Now where did I put that slide rule? Look underneath your stack of National Lampoons. That's where I stole it from. >-- >http://hertzlinger.blogspot.com -- Mensanator Ace of Clubs === Subject: Re: Approximating the surface area of n-ellipsoids, a la Muir appeared in the Math Forum's archive. I hope this post will appear in the > A simple and effective method, possibly new, for approximating the > surface area of ellipsoids in n dimensions is presented. > In 1883, T. Muir presented an approximation for the perimeter of an > ellipse. With its semiaxes of lengths a and b, the approximation has the > form > 2 pi (1/2 (b^p + a^p))^(1/p) . > Muir's value of p = 3/2 optimizes an approximation in this form for > a 3-dimensional ellipsoid with semiaxes of lengths a, b, and c, Knud > Thomsen proposed an approximation which may be regarded as a > generalization of the form used by Muir > 4 pi (1/3 ((bc)^p + (ac)^p + (ab)^p))^(1/p) > together with the value p = 1.6075, which he chose in order to minimize > worst |relative error|. I then suggested that p = 8/5 exactly may be used > in order, among other things, to optimize the approximation for nearly > spherical ellipsoids. (It is fortuitous that p = 8/5 is quite close to > Thomsen's value, for it means that, in optimizing for nearly spherical > ellipsoids, we do not then substantially increase worst |relative > error|.) > This generalization can be extended nicely to provide approximations for > surface areas of ellipsoids in n dimensions. Let S_n denote the surface > area of the unit n-sphere: > If n is even, S_n = 2 pi^(n/2)/(n/2 - 1)! > If n is odd, S_n = 2^((n+1)/2) pi^((n-1)/2)/(n-2)!! > [Note the double factorial. Example: 7!! = 7*5*3*1, rather than (7!)!] > With semiaxes of lengths a_1, a_2, a_3,..., and a_n, the form of the > approximation is > (to be viewed in a fixed-width font, please) > ( 1 n p ) 1/p > S_n ( - sum (product a_j) ) > ( n i=1 j<>i ) > Thus, for example, for a four-dimensional ellipsoid with semiaxes of > lengths a, b, c, and d, the surface area would be approximated by > 2 pi^2 (1/4 ((bcd)^p + (acd)^p +(abd)^p +(abc)^p))^(1/p) . > The question remaining is: In order to optimize such approximations for > nearly spherical ellipsoids, what values of p must be used? > Answer: p = 2(n + 1)/(n + 2) > [That answer is based on calculations of the optimal values of p up to > n = 7. The pattern seems quite obvious, but I have not done a proof.] > How well do these approximations work for ellipsoids which are not almost > spherical? > I have not had the opportunity to examine worst relative error for n>3. > However, we may use the data presented by Garry Tee in Table 2 on page 17 > of his Surface area and capacity of ellipsoids in n dimensions at > to make a few > comparisons. I also give the relative error provided by the YNOT-style > approximations proposed recently by Knud Thomsen (see Gerard Michon's > ) and, independently, by me at > . [Areas computed by > Tee are for ellipsoids having semiaxes lengths in arithmetic progression > from a_1 = 1 to the stated a_n. I give the computed areas truncated to > only 5 significant digits, although Tee gave them to 12 significant > digits.] > Computed Relative errors > n a_n area YNOT-style Muir-style > ---------------------------------------------- > 4 2 6.3922*10^1 -0.14% 0.025% > 10 2 8.2918*10^2 -0.31% 0.015% > 20 13 2.5891*10^14 -3.11% 1.773% > 256 3 1.6264*10^-78 -0.70% 0.005% > 256 36 8.1862*10^147 -5.80% 0.749% > 256 100 7.5617*10^254 -7.65% 2.629% > [Looking at this data, readers may be tempted to surmise that the surface > area is always underestimated by the YNOT-style approximations (except in > the degenerate and spherical cases) and always overestimated by the > Muir-style approximations (except in the spherical case). Such a surmise > would be incorrect. (For example: Muir's own approximation > _under_estimates the perimeter of noncircular ellipses. And for n=3, both > YNOT- and Muir-style approximations both over- and underestimate surface > area at various places.)] > We might also consider the singly degenerate case, in which exactly one > axis is of length zero. The relative error, > pi (n-1)!!/(2^(n/2) n^((n+2)/(2(n+1))) (n/2 - 1)!) - 1 if n is even > and > 2^((n-1)/2) ((n-1)/2)!/(n^((n+2)/(2(n+1))) (n - 2)!!) - 1 if n is odd, > is an increasing function, negative for n=2 and positive for n>=3. As n > increases, relative error seems to approach a value slightly larger than > 1/4. [Is this limit perhaps Sqrt(pi/2) - 1 exactly?] Yes, it is. It was silly of me to have written relative error as I did above, simplified depending on the parity of n. Writing relative error instead as Sqrt(pi)/2 (n-1)/n^((n+2)/(2(n+1))) Gamma((n-1)/2)/Gamma(n/2) - 1, the desired limit can be obtained. Indeed, for large n, relative error is approximately Sqrt(pi/2) (1 - (1+2log(n))/(4n)) - 1. By the way, it might also be noted what happens in the singly degenerate case according to error relative to the area of the circumscribing sphere (as mentioned near the end of ). At its worst (occurring when all of the nondegenerate axes have the same length), this alternative measure of error is 1/n^((n+2)/(2(n+1))) + 2/Sqrt(pi) Gamma(n/2)/((n-1)Gamma((n-1)/2)), which reaches its maximum at n = 16 and goes to 0 as n increases, being approximately (1 - Sqrt(2/pi))/Sqrt(n) for large n. David W. Cantrell > This is not good; > but of course, one may then use the YNOT-style approximations instead, > which work perfectly in this particular case. > Thoughtful comments are always invited. > David W. Cantrell === Subject: Re: Interesting GRE Limit Problem >> Do you ... ASCII-art? >> oo >> 1 --- >> lim - > ln(k/n) >> n->oo n --- >> k=1 >Nice. This limit however is oo. > Whether I copied it correctly or not, were you able to read it easily? > That's what I'm on about... Help! Help! I'm bein' redressed! Yes indeed, with speed and ease unlike those who have caught disease x^4+4=x^4+4x^2+4-4x^2=(x^2+2)^2-4x^2=(x^2+2-2x)(x^2+2+2x) === Subject: a simple two-node queueing network 1. The first node is M/G/c/c with Poisson input, general service, c server without buffer. 2. The output of the first node will attempt to enter the second node which has general service time, c2 server without buffer. I am greatly appreciated if you could please give some suggestions upon analyzing this queueing network. === Subject: Perfect numbers and slightly imperfect numbers I'm looking for slightly imperfect numbers. These are integers which equal the sum of their divisors, minus one. Does anyone have a BASIC program (sorry, it's the only programming language I know!) for finding perfect numbers by brute force which could be adapted to find these numbers? === Subject: Re: Perfect numbers and slightly imperfect numbers >I'm looking for slightly imperfect numbers. These are integers which equal >the sum of their divisors, minus one. Does anyone have a BASIC program >(sorry, it's the only programming language I know!) for finding perfect >numbers by brute force which could be adapted to find these numbers? These are called quasiperfect numbers: http://mathworld.wolfram.com/QuasiperfectNumber.html Apparently there are none known, and all numbers of fewer than 36 digits have been checked. I think the following are additional necessary conditions: 1) n must be of the form m^2 where m is odd; 2) if 3|m, the highest power of 3 dividing m must be even; 3) if 5|m, the highest power of 5 dividing m must be divisible by 4. There are similar necessary conditions for most other prime divisors. -- Erick === Subject: Re: Perfect numbers and slightly imperfect numbers > These are called quasiperfect numbers: Yes, the number is equal to the sum of its non-trivial divisors. I always thought this was more perfect than the standard definition, that (IMHO artificially) includes 1 as a divisor. > Apparently there are none known, and all numbers of fewer than 36 digits have > been checked. Right. But oddly enough, quasi-amicable pairs abound, relatively speaking; unlike amicables, which are somewhat rarer. The first few quasi-amicables are... 48 75 140 195 1575 1648 2024 2295 5775 6128 8892 16587 9504 20735 62744 75495 186615 206504 196664 219975 199760 309135 266000 507759 312620 549219 They appear to show no sign of any substantial thinning out. AFAIK there are no quasi-sociable groups of numbers, i.e. sets of three or more in a cycle of aliquot-sums(less 1) of one another. But that's only AFAIK. Does anyone have better info on quasi-aliquotery?