> Is there a comprehensive classification system for mathematics? > I'm referring to something akin to the Dewey system for libraries, > magazines, etc. David Kinston > melbourne.au I don't think there's anything official. My suggestion for the best place to look for a classification system is the MathWorld site at http://mathworld.wolfram.com/. .... Bob ==== yeah; join the Found Tribe of Sir David, the Wolframites! so, now you know, why Conway never criticizes him; Ka-nights just don't do that!... is it treasonable? > look for a classification system is the MathWorld site at > http://mathworld.wolfram.com/. --les ducs d'Enron! http://members.tripod.com/~american_almanac/ ==== What's the name for the family of topologies in which every open set is also a closed set? Ted Shoemaker ==== > What's the name for the family of topologies in which every open set > is also a closed set? > The partition topology. Which is characterized by the fact that every open set is also closed. You form it by partitioning X into disjoint subsets and adding the empty set to get a basis. It is T3 but not necessarily T0 or T1. ==== > >What's the name for the family of topologies in which every open set >>is also a closed set? >> > >The partition topology. Which is characterized by the fact that every open >set is also closed. You form it by partitioning X into disjoint subsets and >adding the empty set to get a basis. It is T3 but not necessarily T0 or T1. > Huh? T1 is part of the definition of T3 in every book I ever looked at. (Generally, T_n ==> T_m for mWhat's the name for the family of topologies in which every open set >>is also a closed set? > >The partition topology. Which is characterized by the fact that every open >set is also closed. You form it by partitioning X into disjoint subsets and >adding the empty set to get a basis. It is T3 but not necessarily T0 or T1. Huh? T1 is part of the definition of T3 in every book I ever looked > at. (Generally, T_n ==> T_m for m such a space is regular? (It is.) Tangentially, what is an example of a regular T0 space which is not T3 > (i.e., not T1)? > Just to make sure we are talking the same language. T0: If a,b in X, there exists an open set O such that either a in O and b not in O or b in O and a not in O. T1: If a,b in X, there exists open sets O_a, O_b containing a and b respectively such that b not in O_a and a not in O_b. T3: If A is a closed set and b a point not in A there exists disjoint open sets O_A, O_b containing A and b respectively. A regular space is defined as a space which is both T0 and T3. So I assume the counterexample you want would be a T3 space which is not T0. A partition topology is a counterexample (provided one of the sets in the partition has at least two points, which you take to as your a and b in the definition of T0). You also asked for a T0 that's not T1 or T3 This called the particular point topology: Let X be a set and p a point in it. Open sets are those subsets of X which contain p. (Obviously in the definition of T1 you choose p to be one of the points). In general T2 =>T1=>T0 where T2 mean Hausdorff There are definitions for T4 and T5. T5=>T4 but T4=/=>T3 so you can't just go by the numbers and assume Tn=>Tm if n>m. Though of course it depends on your definitions. ==== Scott: >Just to make sure we are talking the same language. >T0: If a,b in X, there exists an open set O such that either a in O and b >not in O or b in O and a not in O. T1: If a,b in X, there exists open sets O_a, O_b containing a and b >respectively such that b not in O_a and a not in O_b. T3: If A is a closed set and b a point not in A there exists disjoint open >sets O_A, O_b containing A and b respectively. A regular space is defined as a space which is both T0 and T3. > I agree with T0 and T1. What you identify as T3 was called the T3 axiom by Alexandroff and Hopf, but most authors these days use that as the definition of regular. A T3-space is one that is T1 and regular (by this definition). (I see that Munkres fudges this, but even he is not clear - for example, he first states that every normal space is regular; *afterwards,* he remarks parenthetically that T1 is needed for this implication. See p. 195 in the 2nd. ed.) >There are definitions for T4 and T5. T5=>T4 but T4=/=>T3 so you can't just >go by the numbers and assume Tn=>Tm if n>m. Though of course it depends on >your definitions. > The standard definitions (though not clear in Munkres) are: completely regular: For each x and closed A not containing x, there exists a continuous function f into [0,1] such that f(x) = 1 and f vanishes on A. T3.5 (terminology ridiculed by Munkres): T1 + completely regular normal: disjoint closed sets can be separated by disjoint open sets T4: T1 + normal completely normal: separated sets (each set disjoint from the closure of the other) can separated by disjoint open sets T5: T1 + completely normal With these definitions, a T_n impiles T_m for m <3F75FA11.2000401@rutcor.rutgers.edu> ==== Definitions for the separation axioms vary from author to author. Thus to be safe say regular T1 regular sans T1 normal T1 normal sans T1 Also look out for the minor separation axioms. Sometimes there's agreement, other times not. -- for your enjoyment, humor or perplexment. Topology Course Lecture Notes by Aisling McCluskey and Brian McMaster http://at.yorku.ca/i/a/a/b/23.htm Definition 3 A space (X, T) is called T[3] or regular provided :- (i) it is T[1], and (ii) given x not in closed F, there exist disjoint open sets G and H so that x in G, F subset H. (v) Warning: Some books take T[3] to mean Definition [3]5.3(ii) alone, and regular to mean Definition [4]5.3(i) and (ii); others do exactly the opposite! Definition 5 A space (X, T) is T[4] or normal if (i) it is T[1], and (ii) given disjoint non-empty closed subsets A, B of X, there exist disjoint open sets G, H such that A subset G, B subset H. ==== > T0: If a,b in X, there exists an open set O such that either a in O > and b > not in O or b in O and a not in O. T1: If a,b in X, there exists open sets O_a, O_b containing a and b > respectively such that b not in O_a and a not in O_b. T3: If A is a closed set and b a point not in A there exists > disjoint open > sets O_A, O_b containing A and b respectively. A regular space is defined as a space which is both T0 and T3. I agree with T0 and T1. What you identify as T3 was called the T3 > axiom by Alexandroff and Hopf, but most authors these days use that as > the definition of regular. A T3-space is one that is T1 and regular > (by this definition). So, with the latter definition, what is an example of a regular T0-space which is not T1? -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu <3F75FA11.2000401@rutcor.rutgers.edu> ==== > So, with the latter definition, what is an example of a regular T0-space > which is not T1? > They're aren't any as regular T0 ==> Hausdorff, T1 if x /= y, then either x not in cl {y} or y not in cl {x} etc. We also have normal T1 ==> Hausdorff, regular An example of a T0, not T1 normal space S is the Sierpinski topology { nulset, {0}, {0,1} } ==== An almost discrete space S is characterized by { cl {x} | x in S } being both a base for S and a partition of S. In other words an almost discrete space is a bunch of open disjoint bags of points. For example: { {0}, {1,2}, {3,4}, {5,6,7,8}, ... } -- Theorems: almost discrete is hereditary discrete or indiscrete S ==> S almost discrete (for all s in S, cl {s} is open) ==> S almost discrete and as discussed by Stephen J. Herschkorn almost discrete T0 space S ==> S discrete -- Almost discrete spaces are exactly those with a discrete T0 identification or reduction That reduction is the equivalence sets of the equivalence relation x~y when for all open U, (x in U iff y in U) which always produces a T0 space. Note x~y iff x,y violate the T0 criteria. BTW, x~y iff cl {x} = cl {y} ---- ==== >What's the name for the family of topologies in which every open set >is also a closed set? >Ted Shoemaker > > If such a space is T1, it is discrete. Does T0 suffice? -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu ==== > What's the name for the family of topologies in which every open set >> is also a closed set? >> Ted Shoemaker >> > If such a space is T1, it is discrete. Does T0 suffice? Yes. Given x in the space, let A = (int ({x}'))', where ' indicates complement and int inicates interior. Then A is an open set containing x. (Note that all closed sets must be open.) Suppose there exists y in A, y distinct from x. Then, by definition of A , there is no neighborhood of y which excludes x. Thus, there must be an open set containing x but not y. But then the complement of this neighborhood is closed, hence open, and contains y, a contradiction. Seems to me that any such topology must have a base of pairwise disjoint sets. Of course, I still have not given a name. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu ==== What is the formula for drawing a sin curve? I want, for example, to show a 20 day cycle, with 10 days positive and 10 days negative. ==== >What is the formula for drawing a sin curve? >I want, for example, to show a 20 day cycle, >with 10 days positive and 10 days negative. address. Some questions simply don't deserve having the answers taking up space on thousands of servers alkl over the world. No to the question: Well THINK!!!! You want 20 units to represent 2Pi So 1 unit is Pi/10 so you want y = sin( Pi*x/10) ==== > What is the formula for drawing a sin curve? > I want, for example, to show a 20 day cycle, > with 10 days positive and 10 days negative. y = a * sin(2*pi*t/P) where a is amplitude P is period For your example, plug in P = 20. -- 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 ==== > Of Jews you say : > IMHO , you guys are the funniest tribe around . > Yea , There's nothing funnier than > obsessive-compulsive insulting . For example Sigmund Freud and Howard Stern ... Both Jewish . Don Rickles, Uncle Al, ........... ==== You point out two funny Jews : Don Rickles , Uncle Al Is it in the genes ? Or is it a cultural thing ? Maybe Al can't help himself . He want's to be nice ... But he just can't do it . ==== > You point out two funny Jews : > Don Rickles , Uncle Al > Is it in the genes ? Or is it a cultural thing ? > Maybe Al can't help himself . > He want's to be nice ... But he just can't do it . *pathy. ==== > You point out two funny Jews : > Don Rickles , Uncle Al > Is it in the genes ? Or is it a cultural thing ? I suspect that unc-al has a higher percentage of neanderthal DNA than most people. Neanderthals supposedly interbread to a certain extent with Homosapiens and unc got the worse side of that DNA Maybe Al can't help himself . He want's to be nice .. Sorry but that's impossible to believe. He has no desire to be nice. Just ask me and he'll tell you - after he insults you that is. Pmb <1cr8g04h6kpqi$.dlg@__.Jeff.Relf> ==== Of Uncle Al , You say : He has no desire to be nice . Just ask him and he'll tell you - after he insults you that is . I'm not going to ask him ... You ask him . :) ==== <__.Jeff-Relf@NCPlus.NET> Gave us: > Of Uncle Al , You say : > He has no desire to be nice . > Just ask him and he'll tell you - > after he insults you that is . >I'm not going to ask him ... You ask him . :) The question itself is an insult, you stupid bastard. It would appear to me that you deserve every insult that can be laid upon you. You troll fuck! ==== You say : The question itself is an insult , you stupid bastard . You're so right . ( Not ) How could I be so insensitive ! A thousand Apologies . ( Not ) You and Al are merely projecting your own self loathing . ==== <__.Jeff-Relf@NCPlus.NET> Gave us: > You say : > The question itself is an insult , you stupid bastard . >You're so right . ( Not ) How could I be so insensitive ! > A thousand Apologies . ( Not ) You and Al are merely projecting your own self loathing . You're an idiot, boy. It's so obvious too. ==== us: >Sorry but that's impossible to believe. He has no desire to be nice. Just >ask me and he'll tell you - after he insults you that is. Just ask ME, and HE'LL tell you? HAhahahahaha... You are stupid, even if Al IS mean. Bwuahahahhaahahahahahahaha.... So, if he called you that, he was correct. ==== us: I suspect that unc-al has a higher percentage of neanderthal DNA than most >people. Neanderthals supposedly interbread to a certain extent with >Homosapiens and unc got the worse side of that DNA When referring to the man, neanderthal is correct. When referring to attributes, neandertal is correct. Looks like you posses some neandertal attributes. Homo Sapiens is two words. You aren't doing very well at this baby bullshit, BOY! ==== To Pete , You say : Homo Sapiens is two words . You obviously understood him ... But who can understand your insensitivity ? ==== > Of Jews you say : > IMHO , you guys are the funniest tribe around . > Yea , There's nothing funnier than > obsessive-compulsive insulting . For example Sigmund Freud and Howard Stern ... Both Jewish . _That_ is an example of obsessive-compulsive insulting? I can't immagine a gun-toting, girl-bashing, radio talk-show Freud or a Stern borrowing terms from Chemistry to explain the inner workings of the human mind :-) Mark (Your ego and id need a tune-up, NOW. Are you on a Warranty?) ==== You say : Your ego and id need a tune-up , NOW . Are you on a Warranty ? Freud insulted people , and he was funny too . <23i4u4abj7go$.dlg@__.Jeff.Relf> <6b70c71c.0309260727.21531a9d@posting.google.com> ==== You say : > Your ego and id need a tune-up , NOW . > Are you on a Warranty ? Freud insulted people , and he was funny too . Hey stooopid boring troll Relfie-boy, go to a translation engine page and render Freud into English. Idiot. Uncle Al would direct you to an English-Deutsches dictionary, but that would be hate language directed against the illiterate. You never did learn how to type a line of text, did you? We see you did not. One looks at your spew and sees a fool who failed a matchbook cover computer school: Instructor, Now Relfie-boy, you must tabulate. All real programmers tabulate. Relfie-boy, Does it make a difference what I tabulate Instructor, Not if you work for Microsoft. -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) Quis custodiet ipsos custodes? The Net! ==== you mean English -Deutsch? >> You say : >> Your ego and id need a tune-up , NOW . >> Are you on a Warranty ? >> Freud insulted people , and he was funny too . Hey stooopid boring troll Relfie-boy, go to a translation engine page >and render Freud into English. Idiot. Uncle Al would direct you to >an English-Deutsches dictionary, but that would be hate language >directed against the illiterate. You never did learn how to type a >line of text, did you? We see you did not. One looks at your spew and sees a fool who failed a matchbook cover >computer school: Instructor, Now Relfie-boy, you must tabulate. All real programmers >tabulate. >Relfie-boy, Does it make a difference what I tabulate >Instructor, Not if you work for Microsoft. ==== you mean English -Deutsch? Not if you know any German grammar, no. There are eight possibilities (two languages times two forms times two orders): 1) English-Deutsches 2) English-German 3) Englisch-German 4) Englisch-Deutisches 5) Deutsch-Englisches 6) Deutsch-English 7) German-Englisches 8) German-English Relfie-boy, of course, is too fucking stupid to know that Sigmund Freud in German is English-equivalent Sigmund Joy as in freudestrahlend. The given name is the same either way. Ein Volk, ein Reich, ein Infobahn, etc. -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) Quis custodiet ipsos custodes? The Net! > You say : >> Your ego and id need a tune-up , NOW . >> Are you on a Warranty ? >> Freud insulted people , and he was funny too . Hey stooopid boring troll Relfie-boy, go to a translation engine page >and render Freud into English. Idiot. Uncle Al would direct you to >an English-Deutsches dictionary, but that would be hate language >directed against the illiterate. You never did learn how to type a >line of text, did you? We see you did not. One looks at your spew and sees a fool who failed a matchbook cover >computer school: Instructor, Now Relfie-boy, you must tabulate. All real programmers >tabulate. >Relfie-boy, Does it make a difference what I tabulate >Instructor, Not if you work for Microsoft. ==== Uncle Al says : Sigmund Freud in German is English-equivalent Sigmund Joy The only Freud Uncle Al seems to know is Schadenfreude . i.e. The Joy of seeing someone else in pain . The Germans just happen to have a word for that . ==== > The only Freud Uncle Al seems to know is Schadenfreude . > > i.e. The Joy of seeing someone else in pain . > > The Germans just happen to have a word for that . No they don't. Schadenfreude does not mean the joy of seeing someone else in pain, but joy in the misfortunes of others - a very human phenomenon. <112o7niv16j7q.dlg@__.Jeff.Relf> ==== You say the word Schadenfreude means : Joy in the misfortunes of others . That's a very important distinction , thanks . Fate must be the origin of the misfortune . So I sand corrected , Al is merely Sadistic . ==== > You say the word Schadenfreude means : > Joy in the misfortunes of others . That's a very important distinction , thanks . > Fate must be the origin of the misfortune . > So I sand corrected , Al is merely Sadistic . > Jeff, even if you sand corrected, then sadistic Al might still take that as a sign from you, as a sub, who is deeply in love with him and adores Al as his true & unconditional master. Fate, fortune or misfortune; when are the wedding bells, Jeff? ahahaha.......ahahahahanson ==== hanson@quick.net says... >Jeff, even if you sand corrected, then sadistic Al might still >take that as a sign from you, as a sub, who is deeply in love >with him and adores Al as his true & unconditional master. >Fate, fortune or misfortune; when are the wedding bells, Jeff? >ahahaha.......ahahahahanson Al is so sadistic that when he encounters a masochiost he does nothing. ---------------------------------------------------------------------- | Just Another Internet Wise Guy Macon, GA USA | ---------------------------------------------------------------------- ==== Gave us: >hanson@quick.net says... >Jeff, even if you sand corrected, then sadistic Al might still >>take that as a sign from you, as a sub, who is deeply in love >>with him and adores Al as his true & unconditional master. >>Fate, fortune or misfortune; when are the wedding bells, Jeff? >>ahahaha.......ahahahahanson Al is so sadistic that when he encounters a masochiost he does nothing. > More retarded baby bullshit from the internet retard that can't even leave the fukin' headers alone. Has to blow his horn. Get lost, ya fukin' troll. <112o7niv16j7q.dlg@__.Jeff.Relf> <6ojybtqoky31.dlg@__.Jeff.Relf> ==== Referring to Al and me , You say : When are the wedding bells , Jeff ? Any day now ... Al is going to be my bitch . ==== > Referring to Al and me , You say : > When are the wedding bells , Jeff ? > Any day now ... Al is going to be my bitch . Be careful what you wish for....... ==== > Referring to Al and me , You say : > When are the wedding bells , Jeff ? Any day now ... Al is going to be my bitch . Be careful what you wish for....... > Bill, I am afraid you are right, but maybe Jeff likes bitches like this: http://www.mazepath.com/uncleal/bung.jpg .......any day now, Jeff, any day now....... ahahahahanson ==== >> Referring to Al and me , You say : >> When are the wedding bells , Jeff ? >> Any day now ... Al is going to be my bitch . >> Be careful what you wish for....... >Bill, I am afraid you are right, but maybe Jeff likes bitches like this: >http://www.mazepath.com/uncleal/bung.jpg >.......any day now, Jeff, any day now....... >ahahahahanson Fukin' children, you three are... nothing more. This proves that numerical age does NOT an adult make. With Jeff Relf as chairperson of immaturity. He graduated from Stupidity University, not summa cum laude, he hit it off as: thissa wun cum whimper Good goin', you immature, retarded Usenet twit! ==== >> Referring to Al and me , You say : >> When are the wedding bells , Jeff ? > Any day now ... Al is going to be my bitch . Be careful what you wish for....... Yer a couple of fukin' wussy boys. ==== > Referring to Al and me , You say : > When are the wedding bells , Jeff ? >Any day now ... Al is going to be my bitch . >>Be careful what you wish for....... > Yer a couple of fukin' wussy boys. F-troupe has spoken. ==== <__.Jeff-Relf@NCPlus.NET> Gave us: snipped retarded baby troll bullshit. Hey, dipshit. You are a certifiable troll now. Certified retard Certified loon Certified twit Let it go, dipshit. ==== By the powers ( ? ) invested in Dark Matter , He proclaims me a : Certifiable troll now . Funny , I don't remember ever seeing his license . ==== > Referring to Al and me , You say : > When are the wedding bells , Jeff ? Any day now ... Al is going to be my bitch . > AHAHAHA.........hahahaha... touch.8e....(not douche).......not bad , Jeff ahahaha........hahahahsnson ==== > Relfie-boy, of course, is too fucking stupid to know that Sigmund > Freud in German is English-equivalent Sigmund Joy as in > freudestrahlend. The given name is the same either way. > Ein Volk, ein Reich, ein Infobahn, etc. > Uncle Al > etc, so, was Freud born in, live in or did he just frequent a Freudenhaus? hahaha....hahahanson <3F74E7EC.82C4BE0D@hate.spam.net> ==== You suggest that I : Render ' Freud ' into English Dictionary.COM's German to English translator says : Freud -> Freud ==== For any real numbers, x_0, x_1, ..., x_n, find with the proof, the minimum value of f[x]=sum_{k=1}^{n} |x-x_k|. Of course we can monotonize so that x_0<=x_1<=...<=x_n, and it is easily seen that the minimum lies within the interval [x_0,x_n]. After experimenting, I've come up with the conjecture that f[x] attains its minimum if and only if x=x_{n/2}+1 (n even) and on the interval [x_{Floor[n/2]},x_{Ceiling[n/2]}] (n odd), but I am unable to prove this. ==== > For any real numbers, x_0, x_1, ..., x_n, find with the proof, the minimum > value of f[x]=sum_{k=1}^{n} |x-x_k|. Of course we can monotonize so that x_0<=x_1<=...<=x_n, and it is easily > seen that the minimum lies within the interval [x_0,x_n]. After > experimenting, I've come up with the conjecture that f[x] attains its > minimum if and only if x=x_{n/2}+1 (n even) and on the interval > [x_{Floor[n/2]},x_{Ceiling[n/2]}] (n odd), but I am unable to prove this. The median of {x_0,...,x_n}is known to mimimize f[x]. Some, especially older, statistics texts provide proofs of this. ==== >For any real numbers, x_0, x_1, ..., x_n, find with the proof, the minimum >value of f[x]=sum_{k=1}^{n} |x-x_k|. Of course we can monotonize so that x_0<=x_1<=...<=x_n, and it is easily >seen that the minimum lies within the interval [x_0,x_n]. After >experimenting, I've come up with the conjecture that f[x] attains its >minimum if and only if x=x_{n/2}+1 (n even) and on the interval >[x_{Floor[n/2]},x_{Ceiling[n/2]}] (n odd), but I am unable to prove this. > Show that, for an x in any other interval [x_k, x_(k+1)], you can choose another x which makes f(x) strictly smaller. This problem can be translated into a linear program, by the way. -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu ==== There's the secretary problem, or marriage problem, where the goal is to inteview n candidates, one after another, and try to pick the best one. You can only pick the candidate you just interviewed, no interviewing anyone else after you pick, no going back to previous candidates. This comes up in hiring a secretary, choosing a spouse, etc. Except, instead of picking the best one, I tried to maximize the goodness of the choice. If the distribution isn't uniform, you could normalize it by ranking the candidates. If you've got one candidate left, you pick them, and on average they're average (0.5 on a scale of 0 to 1, 1 best, 0 worst). If you've got two candidates left, if you don't pick the second to last, you'll pick the last, which has expected goodness 0.5. So you pick the second to last if they're better than half the candidates you've seen so far. If you pick them, they're uniformly distributed in that upper range, so (.5+1)/2 = .75. Average goodness for the process on the last two candidates is then .75*.5 + .5*.5 = .625. If you've got n candidates left, and the expected goodness of the n-1 beyond the current one is x, then you pick the current one if they're better than x. That gives an average goodness for the process for these n candidates of (x+1)/2*(1-x) + x*x That series is: 1 : 0.5 2 : 0.625 3 : 0.6953125 4 : 0.7417297 5 : 0.7750815 6 : 0.8003757 7 : 0.8203006 8 : 0.8364465 9 : 0.8498214 10 : 0.8610982 11 : 0.8707451 12 : 0.8790985 13 : 0.8864071 14 : 0.8928587 15 : 0.8985984 16 : 0.9037395 17 : 0.9083726 18 : 0.9125704 19 : 0.9163923 20 : 0.9198874 If you have n candidates and don't know their distribution, you first have to learn the distribution. Once you've interviewed m, you have a good approximation of what goodness (m-1)/m is. Once (m-1)/m is higher than the expected goodness of the process on the remaining candidates you can consider choosing someone. I've seen this problem elsewhere on the web, but they gave different answers than the one I got. ==== > There's the secretary problem, or marriage problem, where the goal is > to inteview n candidates, one after another, and try to pick the best > one. You can only pick the candidate you just interviewed, no > interviewing anyone else after you pick, no going back to previous > candidates. This comes up in hiring a secretary, choosing a spouse, > etc. reject candidate no.1, pick the first candidate that is better than no. 1. ==== > There's the secretary problem, or marriage problem, where the goal is > to inteview n candidates, one after another, and try to pick the best > one. You can only pick the candidate you just interviewed, no > interviewing anyone else after you pick, no going back to previous > candidates. This comes up in hiring a secretary, choosing a spouse, > etc. reject candidate no.1, pick the first candidate that is better than no. 1. > The best candidate out of n, has a 1/n chance of being the first interviewed. Thus there's a 1/n possibility that your strategy will result in no selection. ==== There's the secretary problem, or marriage problem, where the goal is > to inteview n candidates, one after another, and try to pick the best > one. You can only pick the candidate you just interviewed, no > interviewing anyone else after you pick, no going back to previous > candidates. This comes up in hiring a secretary, choosing a spouse, > etc. reject candidate no.1, pick the first candidate that > is better than no. 1. The best candidate out of n, has a 1/n chance of being the first > interviewed. Thus there's a 1/n possibility that your strategy > will result in no selection. ...otherwise, pick the last available candidate. So by this method, you'd take one of the candidates at random, and then *pick* (as your secretary/spouse) a random candidate better than the original one taken. If your original one taken was already the best, then pick one of the others at random. So, very roughly, I guess it'd work out to an average first candidate value of 0.5, and an average pick value of 0.75 (assuming candidate goodness is ranked from 0 to 1). Could someone with more knowledge (or more sleep) please post a rigorous expected value of goodness for this algorithm, and/or explain *why* it's the best -- if in fact it is? -Arthur ==== >There's the secretary problem, or marriage problem, where the goal is >to inteview n candidates, one after another, and try to pick the best >one. You can only pick the candidate you just interviewed, no >interviewing anyone else after you pick, no going back to previous >candidates. This comes up in hiring a secretary, choosing a spouse, >etc. Except, instead of picking the best one, I tried to maximize the >goodness of the choice. > What does this mean? Do you want to maximize the *expected* goodness? -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu ==== I have a 4 questions about a couple of issues that are confusing me. Any answers to them would be very helpful: 1- Let the field extension F(alpha) be constructed by adjoining a root alpha of the irreducible polynomial f(x) of degree n in F[x]. Suppose that F(alpha) contains more than 1 root of f(x) (possible splitting field). Is there any way to tell whether the other roots other than alpha will be powers of alpha, c*alpha, or an F-linear combination of alpha^(n-1),.....,alpha, 1? 2- Suppose I have a cubic irreducible inseperable polynomial f(x). Let alpha be a root of multiplicity 2 and beta the other root of f(x) amd suppose that beta is not contained in F(alpha). Now [ F(beta) : F ] = 3 and in F(beta), f(x) is a product of the factor (x-beta) and a quadratic. Now, [ F(alpha, beta) : F(beta) ] = 2, so [ F(alpha, beta) : F ] = 6. Let us now reverse the steps of adjoining the roots. Now [ F(alpha) : F ] = 3 and in F(alpha), f(x) is a product of the factor (x-alpha)^2 and a factor (x-b). This implies that b = beta and that F(alpha, beta) = F(alpha) so [ F(alpha, beta) : F] = 3. Please somebody tell me whats wrong with my argument??! It shouldn't matter in what sequance I adjoin the roots?! 3- Let f(x) = x^3 - 2 and let cbr(2) denote the cubic root of 2, z(3) the 3rd primitive root of unity and let Q denote the rationals. Now, Q(z(3), cbr(3)) is the splitting field of f(x) and the roots of f(x) are cbr(2), z(3)*cbr(2) z(3)^2*cbr(2). I have 2 questions about this: a) In Q(cbr(3)) : f(x) = (x - cbr(2))(x^2 + x + 1). This clearly shows that cbr(2) is a root of f(x), but doesn't this also show that z(3) is also a root, since it is a root of x^2 + x +1 (the above equation shows that f( z(3) ) = 0). This is seriously confusing me! b) Again, in Q(cbr(3)) : f(x) = (x - cbr(2))(x^2 + x +1). This clearly shows that cbr(2) is a root of f(x), but I can't see how we can prove that for example z(3)*cbr(2) is also a root of f(x). This is seriously confusing me. Let me generalize my question: let K = F(b1, b2, b3, b4) be the splitting field for the irreducible g(x). Now, can we in general say what the roots of the g(x) will be?? If we adjoined b1 first then clearly b1 must be a root of g(x), but In f(b1), we have g(x) = (x - b1)*h(x) (suppose f(x) has only one root in f(b1)) and b2 will be a root of h(x) and we continue this process until g(x) splits in K. Now what are the roots of g(x) in K?? Are they some Q-linear combination of b1,b2,b3,b4?? How will this linear combination look like?? Can we tell in general?? Will they be products of b1, b2, b3, b4 as in the case of f(x) = x^3 - 2. 4- The following was part of a proof I read: In K/F is Galois, then K is the splitting field of some seperable polynomial f(x) in F[x]. Let E/F be any extension, then KE/E is the splitting field of f(x) viewed as a polynomial in E[x]. This doesn't make sense: THE splitting field of f(x) is the smallest field which f(x) splits. But K < KE. So, how can we have 2 splitting fields of a polynomial one propery conatined in the other? I'm sure there are pretty easy answers to these questions, that will clear up a lot of confusion. Any helpful clear answers will be highly appreciated. ==== AAA a .8ecrit dans le message de [snip] > 3- Let f(x) = x^3 - 2 and let cbr(2) denote the cubic root of 2, z(3) the > 3rd primitive root of unity and let Q denote the rationals. Now, Q(z(3), > cbr(3)) is the splitting field of f(x) and the roots of f(x) are cbr(2), > z(3)*cbr(2) z(3)^2*cbr(2). I have 2 questions about this: a) In Q(cbr(3)) : f(x) = (x - cbr(2))(x^2 + x + 1). f(x) factors into: (x-cbr(2))(x^2 + x.cbr(2) + cbr(2)^2) in Q(cbr(2)) > This clearly shows that > cbr(2) is a root of f(x), but doesn't this also show that z(3) is also a > root, since it is a root of x^2 + x +1 (the above equation shows that f( > z(3) ) = 0). This is seriously confusing me! f(z(3)) = z(3)^3 - 2 = 1 - 2 = 1 Alain ==== > 2- Suppose I have a cubic irreducible inseperable polynomial f(x). Let > alpha be a root of multiplicity 2 and beta the other root of f(x) amd > suppose that beta is not contained in F(alpha). An irreducible polynomial f(x) cannot have a multiple root unless it is separable, ie of the form g(x^p) in characteristic p. To see this, note that a multiple root of f(x) is a root of f'(x), and so of gcd(f(x),f'(x)), which divides f(x). This leads to a contradiction, unless f'(x) vanishes identically, which can only happen in finite characteristic p, with f(x) = g(x^p). In your case, you would have to have p = 3, and f(x) = x^3 + c, in which case f(x) would have 3 identical roots. -- Timothy Murphy tel: +353-86-233 6090 ==== > I have a 4 questions about a couple of issues that are confusing me. Any > answers to them would be very helpful: [cut] > 4- The following was part of a proof I read: In K/F is Galois, then K is the > splitting field of some seperable polynomial f(x) in F[x]. Let E/F be any > extension, then KE/E is the splitting field of f(x) viewed as a polynomial > in E[x]. This doesn't make sense: THE splitting field of f(x) is the > smallest field which f(x) splits. But K < KE. So, how can we have 2 > splitting fields of a polynomial one propery conatined in the other? K is the splitting field for f(x) over F. KE is the splitting field for f(x) over E. You need the phrase over either implicitly or explicitly. When you say M is the splitting field for g(x) over L, it is understood that M contains L and g(x) is in L[x]. You can't say K is the splitting field for f(x) over E, since K may not contain the elements of E. However, if K does contain the elements of E, then KE equals K and everything is still OK. Note that E cannot be any extension of F, since KE would not make much sense. There must be some superfield that contains both E and K lurking in the background somewhere. -- Bill Hale ==== I'm studying algebraic topology, which I find fascinating. However, I have noticed that many proofs use the fact that a certain space in homoemorphic to another space, or the space has the same homotopy type as another, or the space is contractible. I have run into many proofs that the author will assume the reader is knows these standard facts about certain spaces. Can anyone give me a list (or partial list) of some important: a) Spaces which are contractible b) Spaces which are homeomorphic to each other c) Spaces which have the same homotopy type. Also, It seems obvious that the 1-simplex (convex combination of 2 points) is contactible. Is the border (homomorphism in singular homology from S2 -->S1) of a 2-simplex also contractible. I was told no. Can someone tell me why? Does this hold for all n-simplexes (n>=1)? Are all n-simplexes contractible? ==== > I'm studying algebraic topology, which I find fascinating. However, I have > noticed that many proofs use the fact that a certain space in homoemorphic > to another space, or the space has the same homotopy type as another, or the > space is contractible. I have run into many proofs that the author will > assume the reader is knows these standard facts about certain spaces. Can > anyone give me a list (or partial list) of some important: a) Spaces which are contractible b) Spaces which are homeomorphic to each other c) Spaces which have the same homotopy type. Also, It seems obvious that the 1-simplex (convex combination of 2 points) > is contactible. Is the border (homomorphism in singular homology from > S2 -->S1) of a 2-simplex also contractible. I was told no. Can someone tell > me why? Does this hold for all n-simplexes (n>=1)? Are all n-simplexes > contractible? See the recent sci.math thread S^n is not contractible initiated by non-contractibility of spheres in one of my responses. John Mitchell ==== Also, It seems obvious that the 1-simplex (convex combination of 2 points) > is contactible. Is the border (homomorphism in singular homology from > S2 -->S1) of a 2-simplex also contractible. I was told no. Can someone > tell me why? It's homoemorphic to a circle. There are many ways of seeing the noncontractibility of a circle, e.g., it has nontrivial fundamental group. > Does this hold for all n-simplexes (n>=1)? Are all > n-simplexes contractible? Yes. Any convex set in R^n is contractible (I hope this is obvious). -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) ==== > Also, It seems obvious that the 1-simplex (convex combination of 2 points) > is contactible. Is the border (homomorphism in singular homology from > S2 -->S1) of a 2-simplex also contractible. I was told no. Can someone > tell me why? It's homoemorphic to a circle. There are many ways of seeing the > noncontractibility of a circle, e.g., it has nontrivial > fundamental group. > True enough, but if the O.P. is having difficulty with the more basic fact that the circle is not contractible, it's unlikely that he would understand the non-triviality of the circle's fundamental group at this stage. In fact, the most straightforward way to prove the non-triviality of PI_1(S^1) is to prove the non-contractibility of the circle! John Mitchell ==== > I'm studying algebraic topology, which I find fascinating. However, I have > noticed that many proofs use the fact that a certain space in homoemorphic > to another space, or the space has the same homotopy type as another, or the > space is contractible. I have run into many proofs that the author will > assume the reader is knows these standard facts about certain spaces. Can > anyone give me a list (or partial list) of some important: a) Spaces which are contractible > R^n and balls and boxes there in. Circles and spheres aren't. Contractible spaces are both path connected and simply connected. > b) Spaces which are homeomorphic to each other > R^n and (0,1)^n. The plane and a sphere with a single point removed. > c) Spaces which have the same homotopy type. Also, It seems obvious that the 1-simplex (convex combination of 2 points) > is contactible. Is the border (homomorphism in singular homology from > S2 -->S1) of a 2-simplex also contractible. I was told no. Can someone tell > me why? Does this hold for all n-simplexes (n>=1)? Are all n-simplexes > contractible? > Dare I ask what a simplex is? ==== The German words ACHT (8) and EINS (1) represent numbers and have their letters in alphabetical order. I can think of only one definite English example, excluding one-letter answers (e, i) (but have two other English examples if we get pushy). Roman numerals such as CCIV (204) are not acceptable, but a letter may be duplicated if all occurrences are adjacent. Don't forget that some numbers have multiple names, such as FOURTH or QUARTER for 1/4. For readers whose native language is not English, how many examples can you find in your language? -- Wanted: Experts at choosing the best of 100+ applicants for a position. Register as a California voter by September 22, and vote on October 7. Peter-Lawrence.Montgomery@cwi.nl Home: San Rafael, California Microsoft Research and CWI ==== > The German words ACHT (8) and EINS (1) represent numbers and have > their letters in alphabetical order. I can think of only > one definite English example, excluding one-letter answers (e, i) > (but have two other English examples if we get pushy). > Roman numerals such as CCIV (204) are not acceptable, > but a letter may be duplicated if all occurrences are adjacent. > Don't forget that some numbers have multiple names, > such as FOURTH or QUARTER for 1/4. For readers whose native language is not English, > how many examples can you find in your language? In Klingon loS (4) [CapitalEth]bIp (100,000) and [CapitalEth]'uy' (1,000,000) http://www.klingonska.org/ref/num.html and it makes about as much sense. ==== > The German words ACHT (8) and EINS (1) represent numbers and have > their letters in alphabetical order... For readers whose native language is not English, > how many examples can you find in your language? In Klingon loS (4) -bIp (100,000) and -'uy' (1,000,000) > http://www.klingonska.org/ref/num.html > and it makes about as much sense. Klingon alphabetical order puts the apostrophe (a consonant) last. The number-forming element {-'uy'} both starts and ends with an apostrophe, so it doesn't fit the rule. It and {-bIp} aren't really numbers anyway; they're something like the -ty suffiix in English. So {loS} is all there is. ==== Peter L. Montgomery escribi.97 en el > The German words ACHT (8) and EINS (1) represent numbers and have > their letters in alphabetical order. I can think of only > one definite English example, excluding one-letter answers (e, i) > (but have two other English examples if we get pushy). > Roman numerals such as CCIV (204) are not acceptable, > but a letter may be duplicated if all occurrences are adjacent. > Don't forget that some numbers have multiple names, > such as FOURTH or QUARTER for 1/4. For readers whose native language is not English, > how many examples can you find in your language? > -- In spanish only DOS (2). -- Ignacio Larrosa Ca.96estro A Coru.96a (Espa.96a) ilarrosaQUITARMAYUSCULAS@mundo-r.com ==== The German words ACHT (8) and EINS (1) represent numbers and have > their letters in alphabetical order. [snip] French has also a few similar ones. M. K. Shen ==== Mok-Kong Shen scribbled the following: >> The German words ACHT (8) and EINS (1) represent numbers and have >> their letters in alphabetical order. > [snip] > French has also a few similar ones. Which? Let's see: Un: no. Deux: no. Trois: no. Quatre: no. Cinq: yes. Six: no. Sept: no. Huit: no. Neuf: no. Dix: yes. So far I've found two. I don't remember what the French words for eleven through nineteen were. Vingt, anyway, is not such a word. Cent, however, qualifies, while mille does not. Three. That's more than I got out of Finnish and Swedish put together! Whee! -- /-- Joona Palaste (palaste@cc.helsinki.fi) --------------------------- | Kingpriest of The Flying Lemon Tree G++ FR FW+ M- #108 D+ ADA N+++| | http://www.helsinki.fi/~palaste W++ B OP+ | ----------------------------------------- Finland rules! ------------/ C++ looks like line noise. - Fred L. Baube III ==== > [snip] French has also a few similar ones. Which? Let's see: > Un: no. Deux: no. Deux: yes > Trois: no. Quatre: no. Cinq: yes. Six: > no. Sept: no. Huit: no. Neuf: no. Dix: yes. > So far I've found two. three > I don't remember what the French words for > eleven through nineteen were. they don't work because the six first end with ze or se: Onze:no, Douze:no, Treize:no, Quatorze:no, Quinze:no, Seize:no, The last three ones don't work because sept, huit,neuf don't Dix-sept:no, Dix-huit:no, Dix-neuf:no. Vingt, anyway, is not such a word. > Cent, however, qualifies, while mille does not. > Three. Four. namely (2, 5, 10, 100) >That's more than I got out of Finnish and Swedish put together! > Whee! > Alain ==== Peter L. Montgomery scribbled the following: > The German words ACHT (8) and EINS (1) represent numbers and have > their letters in alphabetical order. I can think of only > one definite English example, excluding one-letter answers (e, i) > (but have two other English examples if we get pushy). > Roman numerals such as CCIV (204) are not acceptable, > but a letter may be duplicated if all occurrences are adjacent. > Don't forget that some numbers have multiple names, > such as FOURTH or QUARTER for 1/4. > For readers whose native language is not English, > how many examples can you find in your language? Finnish has, AFAIK, no such numbers. Swedish, OTOH, has a few: EN (maybe ETT too if you count multiple consecutive same letters), TV.81, and erm, that's about it. -- /-- Joona Palaste (palaste@cc.helsinki.fi) --------------------------- | Kingpriest of The Flying Lemon Tree G++ FR FW+ M- #108 D+ ADA N+++| | http://www.helsinki.fi/~palaste W++ B OP+ | ----------------------------------------- Finland rules! ------------/ He said: 'I'm not Elvis'. Who else but Elvis could have said that? - ALF ==== [Followups set to rec.puzzles, for obvious reasons] The German words ACHT (8) and EINS (1) represent numbers and have > their letters in alphabetical order. I can think of only > one definite English example, excluding one-letter answers (e, i) > (but have two other English examples if we get pushy). > Roman numerals such as CCIV (204) are not acceptable, > but a letter may be duplicated if all occurrences are adjacent. > Don't forget that some numbers have multiple names, > such as FOURTH or QUARTER for 1/4. For readers whose native language is not English, > how many examples can you find in your language? (English spoiler follows) ........ ....... ...... ..... .... ... .. . One English number (and the only English integer; probably the only English number at all) that works is FORTY. -Arthur ==== If the polynomial 7 SUM a[n]x^n = 0 n=0 Has introduced before one of its coefficients the factor (1+E), where E is some small error, then the roots might be found by solving the matrix, LIM E->0 | | | (1+E)a[7] a[6] a[5] ... a[1] | -a[0] | | a[7] (1+E)a[6] a[5] ... a[1] | -a[0] | | a[7] a[6] (1+E)a[5] ... a[1] | -a[0] | | . | | | . | | | . | | | a[7] a[6] a[5] ...(1+E)a[1] | -a[0] | | | This is solved as though it were the system of linear equations, 7 SUM a[n]z[n]= -a[0] such that when row=column, the element n=1 is multiplied by (1+E). Knowing that a point in 7-space satisfies the system, it remains what point on the curve (z,z^2,..,z^n) is closest to it. ==== Plonk. ==== > [snip] The reason logic has compactness theorems is because language is > topological. You've stated something like that before. > What exactly do you mean with 'language is topological'? > Can you give a few examples? BTW, i'd say that the reason logic has compactness theorems is > because we want our arguments to have only finitely many steps, > hence every valid proof needs only finitely many assumptions. > But apparantly that thought is too simplistic? > It is not that our arguments have only finitely many steps.... Our ability to convey information to one another using language by any act whatsoever is intrinsically finite. Having used the word information, let me direct you toward a specific application before directing you to some papers that might give you an idea of what I am trying to say. The work of Barry Smith will probably give you the best examples. The most complex modern exposition to which I can refer you is Jerry Seligman's Perspectives in Situation Theory. My references on situation theory are about a decade old--so there have certainly been refinements. But here is a short summary. Seligman uses a combination of Barwise and Perry's situation theory and Dretske's data semantics. The situation theory is an abstraction of formal models such that information holds under a stituation rather than propositions hold under a model. In order to characterize persistence (objecthood), Seligman needed a concept of information flow between perspectives. That is where the data semantics becomes important. He use an information-theoretic source-receiver model to introduce dynamical contexts into the framework. I only mention Seligman's work because of two results concerning shifts between perspectives: the identity map is a strong and weak shift the inclusion shift from a subperspective to a perspective is strong and weak Mathematically, one can see the relationship of these predicates in the toposes of category theory, for example. One-to-one maps associated with these objects individuate on the basis of inclusion. What I want to focus on here are inclusions/parts/partial orders. Peter Gardenfors has written on the geometry of language with his conceptual spaces. I first became aware of his work at the link http://nb.vse.cz/kfil/elogos/logpoint/93-3/3GARD.htm If you look at this, you will see that he invokes betweenness and dimensionality. Betweenness is a geometric and lattice-theoretic property--in fact, Birkhoff's Lattice Theory is specifically cited here. And, dimension theory is a topological discipline. In the particular case of a language, one would look at combinatorial topology. The abstract complexes are realized geometrically if they can be expressed with linear combinations. When this is formalized in a Boolean context, one is speaking of linearly separable switching functions (See Threshold Logic by Hu). Lattice theory, of course, is a theory of partial orders. For specific examples, I would suggest looking at Barry Smith's work in mereotopology. Mereology is a theory of part and whole. Its formalizations are based on partial orders. In http://nb.vse.cz/kfil/elogos/logpoint/93-3/3GARD.htm you will find remarks like It is, however, in work in the area of cognitive linguistics on the part of Lakoff, Talmy, Langacker, Jackendoff and others that topological notions have been most explicitly and systematically applied. (See especially Talmy 1977ff., Brugman and Lakoff 1988, Lakoff 1989, Jackendoff 1991, and the related work of Petitot 1982ff., and Wildgen 1982f.) The importance of topology to the conceptual structuring effected by language is illustrated most easily in the case of prepositions. This is not an area in which I have expertise. I make the claim that I do on the basis of mathematical intuitions that I have investigated (but, unfortunately I have not communicated well). So, any examples I might try to give you would seem too formal. If you run searches on mereology and/or mereotopology, you should be able to find some good examples to consider. :-) mitch ==== > http://tinyurl.com/ou1f http://tinyurl.com/ou1h > The one the original poster was referring to was the > singer/songwriter/movie director guy who died in 1997 of liver disease. > The original poster said ANother one bites the dust in his first > Rolling Stones magazine from the early 70's until late 80's. For those who would like to read up on the fabulous life of one of the > music industries greatest ever contributors, here is a website dedicated > to Robert Palmer. http://www.arjazz.org/artists/hof/2002/2002_palmer.html the lights are on, but you're not home Herc Hey Herc, I'm watching you on television right now! I'm sure 100s in the pentagon are but you're not one of them. > my mind is not my own. Herc OK, then how come I know you are wearing a short sleeved blue shirt? how come Duggy said after some evasion yeah I've heard of him when > I asked about the Truman? every Townsvillite who answered all joked yes hey? even The Charming One > from Cairns isn't denying it. They've all been poisened by the frame artists > yanky truman company who force me to reply to their perversions and quote > me ad infinitum, and all conspire to refuse assisting me so I get years of absolutely > putrid around the clock psycho fuckwit intel ops, depriving me of thought itself > year after year so my retaliation livens up their fuckwit political domination show. Round of applause just for typing that, you are on the truman show. > Herc Wow, you are good. You make me look just ordinary. I'd give away the truman show in a second, its just billions of dollars spent by yanky cowards going out of their way to see to it I get tortured. they do anything and everything possible to hurt me. I get movies made about me and the Tv talks to me anywhere I go. That's tolerable but that's 1% of it. Its the greatest suffering in all humanity and they push everyone into continuing it nonstop. Noone will even listen to me. 2 years, I can't crzp in the morning without 5 psychiatrists yelling out at me with a massive PA and all my neighbours hearing me. 100,000 people in townsville have known for 2 years, and every second week I still end up eating stale bread for a week. I can't walk 2 blocks to the shop without atleast 10 people yelling at me telling me the fuck off. You've all got no idea, you all think this is a joke or I 'believe' it. Everyone who listens in to me being tortured thinks it right because I keep telling everyone to fuck off. you wan't to make something of your life be the 1st decent person in all history by actually helping me get away from the SICK FUKING PERVERT TRUMAN COMPANY Herc ==== > > http://tinyurl.com/ou1f http://tinyurl.com/ou1h > The one the original poster was referring to was the > singer/songwriter/movie director guy who died in 1997 of liver disease. > The original poster said ANother one bites the dust in his first > Rolling Stones magazine from the early 70's until late 80's. For those who would like to read up on the fabulous life of one of the > music industries greatest ever contributors, here is a website dedicated > to Robert Palmer. http://www.arjazz.org/artists/hof/2002/2002_palmer.html the lights are on, but you're not home Herc Hey Herc, I'm watching you on television right now! I'm sure 100s in the pentagon are but you're not one of them. > my mind is not my own. Herc OK, then how come I know you are wearing a short sleeved blue shirt? how come Duggy said after some evasion yeah I've heard of him when > I asked about the Truman? every Townsvillite who answered all joked yes hey? even The Charming One > from Cairns isn't denying it. They've all been poisened by the frame artists > yanky truman company who force me to reply to their perversions and quote > me ad infinitum, and all conspire to refuse assisting me so I get years of absolutely > putrid around the clock psycho fuckwit intel ops, depriving me of thought itself > year after year so my retaliation livens up their fuckwit political domination show. Round of applause just for typing that, you are on the truman show. > Herc Wow, you are good. You make me look just ordinary. I'd give away the truman show in a second, its just billions of dollars spent by yanky cowards > going out of their way to see to it I get tortured. they do anything and everything possible > to hurt me. I get movies made about me and the Tv talks to me anywhere I go. That's tolerable > but that's 1% of it. Its the greatest suffering in all humanity and they push everyone into > continuing it nonstop. Noone will even listen to me. 2 years, I can't crzp in the morning > without 5 psychiatrists yelling out at me with a massive PA and all my neighbours hearing me. 100,000 people in townsville have known for 2 years, and every second week I still end up > eating stale bread for a week. I can't walk 2 blocks to the shop without atleast 10 people > yelling at me telling me the fuck off. You've all got no idea, you all think this is a joke or I > 'believe' it. Everyone who listens in to me being tortured thinks it right because I keep telling > everyone to fuck off. you wan't to make something of your life be the 1st decent person in > all history by actually helping me get away from the SICK FUKING PERVERT TRUMAN COMPANY Herc I'll do everything in my power to help you Herc. I feel for you. ==== http://tinyurl.com/ou1f http://tinyurl.com/ou1h > The one the original poster was referring to was the > singer/songwriter/movie director guy who died in 1997 of liver disease. > The original poster said ANother one bites the dust in his first > Rolling Stones magazine from the early 70's until late 80's. For those who would like to read up on the fabulous life of one of the > music industries greatest ever contributors, here is a website dedicated > to Robert Palmer. http://www.arjazz.org/artists/hof/2002/2002_palmer.html the lights are on, but you're not home Herc Hey Herc, I'm watching you on television right now! I'm sure 100s in the pentagon are but you're not one of them. > my mind is not my own. Herc OK, then how come I know you are wearing a short sleeved blue shirt? how come Duggy said after some evasion yeah I've heard of him when > I asked about the Truman? every Townsvillite who answered all joked yes hey? even The Charming One > from Cairns isn't denying it. They've all been poisened by the frame artists > yanky truman company who force me to reply to their perversions and quote > me ad infinitum, and all conspire to refuse assisting me so I get years of absolutely > putrid around the clock psycho fuckwit intel ops, depriving me of thought itself > year after year so my retaliation livens up their fuckwit political domination show. Round of applause just for typing that, you are on the truman show. > Herc Wow, you are good. You make me look just ordinary. I'd give away the truman show in a second, its just billions of dollars spent by yanky cowards > going out of their way to see to it I get tortured. they do anything and everything possible > to hurt me. I get movies made about me and the Tv talks to me anywhere I go. That's tolerable > but that's 1% of it. Its the greatest suffering in all humanity and they push everyone into > continuing it nonstop. Noone will even listen to me. 2 years, I can't crzp in the morning > without 5 psychiatrists yelling out at me with a massive PA and all my neighbours hearing me. 100,000 people in townsville have known for 2 years, and every second week I still end up > eating stale bread for a week. I can't walk 2 blocks to the shop without atleast 10 people > yelling at me telling me the fuck off. You've all got no idea, you all think this is a joke or I > 'believe' it. Everyone who listens in to me being tortured thinks it right because I keep telling > everyone to fuck off. you wan't to make something of your life be the 1st decent person in > all history by actually helping me get away from the SICK FUKING PERVERT TRUMAN COMPANY Herc I'll do everything in my power to help you Herc. I feel for you. :) well I'm sure you mean take the happy pills but the sentiment is appreciated. Herc ==== |I'm also wondering if |momentum, tension, or compression could affect space (or time, or |space-time) like it effects gravity. These are mainly physics questions rather than math questions, but I'll give you a few leads. Einstein's theory of gravitation says that gravitation is a matter of the structure of space-time influencing objects, so whether in that theory they have gravitational effects and whether they affect space-time aren't separate questions. (I don't know how many of these effects have been confirmed, and I'm guessing that this one has not been, but the theory as a whole seems pretty well confirmed.) In the theory, energy and mass, momentum, stress and tension are all aspects of one composite thing called the energy-momentum-stress tensor. (ObMath: an application of tensor algebra!) One way to describe it is as the thing which tells you what energy density is at each point, in the reference frame of every observer passing by. |For example, could applying |enough momentum, tension, or compression on a suitably shaped mass in |a suitable way bend or even compress space (even if the energy levels |don't seem reachable with our technology)? Beware over reliance on the metaphor of space as a fabric which might be stretched, compressed, or whatever. Space is not cloth or rubber. But in a rough manner of speaking, yes, theoretically. |I really want to know |because if the answer is yes then I guess in theory you could make a |warp drive for a spaceship or something.... The hazard of searching for the natural keywords is that you tend to find stuff by Jack Sarfatti instead of what you want! But here are a couple of references to this kind of thing. Nick Kaiser at U. Hawaii has some lecture notes on his web site, and a few remarks in chapter 29, www.ifa.hawaii.edu/~kaiser/lectures/ chapters/chapter_29.ps are relevant: The essence of inflation is to assume that at early times the Universe passed through a phase with a strongly negative pressure (i.e. positive tension). The state of negative pressure hypothesized in inflation is not just space under tension; it's tension due to a matter field, although it's a state of the field which gets called a false vacuum. So generally you might be interested in cosmological inflation. And yes, one speculation has been that reigniting inflation locally could create just a kind of exotic matter enough to get one around the usual obstacles to creating things like warp fields or wormholes. See for example http://seds.lpl.arizona.edu/nodes/NODEv4n3-5.html and maybe references it mentions. Be warned that although not kooky, this stuff is speculative. Another way of looking at this, which I think makes a lot of sense, is that the possibility of this kind of thing is evidence against a theory. This is not just being a wet blanket; a theory which permits faster-than-light travel is liable to be inconsistent or have instabilities in it. That's because what is faster-than-light in one roughly inertial reference frame is travel into the past in another one. If a ship travelling faster than light relative to its surroundings and backward in time stopped, it would be the same as two ships appearing out of empty space together, one warping off in one direction and the other just sitting there. If such a thing is possible, it's not clear how one is going to rule out it just spontaneously happening in a vacuum anyway. It's also plausible that being unable to time-travel or go faster than light is related to some natural conditions on the nature of matter, like the nonexistence of negative energy of a certain kind. Have fun. Keith Ramsay ==== I am not quite familiar with spectral method. I know that when Spectral method is applied, we have to consider the problem in frequency domain. However, the numerical fourier transformation will lead to some error which is in complex form and in the time doamin it's something like a oscillating wave. I wonder how to absorb the oscillating wave and keep the numberical result as accurate as possible. ==== VonNeumann Gametheory Optimal Strategy of StockMarket Archimedes Plutonium NOdtgEMAIL whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies sci.econ, sci.math Portfolio of PAF as of 26SEP03 50 BCE 21.34 $1,067.00 50 BLS 23.87 $1,193.50 100 BMY 25.45 $2,545.00 50 DT 14.46 $723.00 51,050 Q 3.56 $181,738.00 11,450 SBC 22.16 $253,732.00 50 VZ 32.80 $1,640.00 50 WYE 46.05 $2,302.50 realestate land 3APR03 of 3 lots $19,000 art of science-lithographs & porcelain JAN-JUN03 for $12,000 realestate land 30JUL03 another lot $11,500 Last several weeks have had very very little to cheer about. Except for the prices and so today I took some petty cash from the portfolio and added 100 shares of SBC at 22.15 per share and added 1,000 shares of Qwest at 3.55 per share. Interesting reflection of the past several weeks is the question of which is worse for a company and industry for that matter? The periodic union hassles and fights such as the telecom industry or the troubles of regulation of the drug industry by the federal government? Which is worse-- unions or federal regulation, respective of telecoms to drug industry. whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies ==== It certainly is not hard to show what every textbook gives as a simple exercise but probably I am not bright enough to think up proof of implication I mentioned. I'd be grateful for some hint. olej ==== In another thread the notorious distinction between C^infty-ness and analyticity for *real* functions was reminded once again. I said that exp(-1/x^2) wouldn't have been my nightmare for that night, but actually I've been thinking about this issue... Now, my *actual* question is: how large the set of points E where a C^infty (say, on an interval) function f fails to be analytic can be? On the one hand I'm sure I've never been taught anything about this particular problem, OTOH I've been thinking about such functions as exp(-1/sin^2(1/x)), x*exp(-1/sin^2(1/x)), exp(-1/x^2)*exp(-1/sin^2(1/x))=exp(-1/x^2-1/sin^2(1/x)) but as a matter of a fact I'm too rusty on real analysis to study their behaviour in a neighbourhood of 0: intuitively it seems possible that E has an accumulation point in E[*], but I find it hard to see how it can contain a non-empty open subset. As usual what seems intuitive is not granted to be also true... [*] At first I thought that E should consist of isolated points. 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 ==== >In another thread the notorious distinction between C^infty-ness and >analyticity for *real* functions was reminded once again. I said that >exp(-1/x^2) wouldn't have been my nightmare for that night, but >actually I've been thinking about this issue... Now, my *actual* question is: how large the set of points E where a >C^infty (say, on an interval) function f fails to be analytic can be? Very. An infinitely differentiable function can be nowhere analytic. Let f(x) = sum a_n cos(2^n x). Then f is an example, for a suitable choice of a_n > 0; I'm pretty sure that a_n = 2^(-n^2) works, for example. (First verify that if you differentiate term by term k times the resulting series converges absolutely; hence f is infinitely differentiable. Now verify that the Taylor series centered at the origin has radius of convergence 0; this is the part I think is clear, if not it's true for a different choice of a_n. It follows that f is not analytic at 0, hence not analytic at multiples of pi. But now let f_N = f - S_N, where S_n is the sum of the first N terms in the series defining f. Now the Taylor series for f_N at 0 has radius of convergence 0, but f_N has a small period, p_N = 2*pi*2^(-N) or whatever, hence the Taylor series of f_N at multiples of p_N has radius of convergence 0, and hence the Taylor series for f at those points has radius of convergence 0 since S_N is analytic. So f is non-analytic on a dense set, hence nowhere analytic.) >On the one hand I'm sure I've never been taught anything about this >particular problem, OTOH I've been thinking about such functions as exp(-1/sin^2(1/x)), >x*exp(-1/sin^2(1/x)), >exp(-1/x^2)*exp(-1/sin^2(1/x))=exp(-1/x^2-1/sin^2(1/x)) but as a matter of a fact I'm too rusty on real analysis to study >their behaviour in a neighbourhood of 0: intuitively it seems possible >that E has an accumulation point in E[*], but I find it hard to see >how it can contain a non-empty open subset. As usual what seems >intuitive is not granted to be also true... >[*] At first I thought that E should consist of isolated points. >Michele ************************ David C. Ullrich ==== > Let f(x) = sum a_n cos(2^n x). Then f is an example, for a suitable choice of a_n > 0; I'm pretty > sure that a_n = 2^(-n^2) works, for example. Nice example. ==== > Let >> f(x) = sum a_n cos(2^n x). >> Then f is an example, for a suitable choice of a_n > 0; I'm pretty >> sure that a_n = 2^(-n^2) works, for example. Nice example. It's more or less a traditional example of a function analytic in the disk, smooth up to the boundary, with the entire boundary for a natural boundary. Or at least it is if the coefficients are right... (Just call me Mr. Lacunary.) ************************ David C. Ullrich ==== > In another thread the notorious distinction between C^infty-ness and > analyticity for *real* functions was reminded once again. I said that > exp(-1/x^2) wouldn't have been my nightmare for that night, but > actually I've been thinking about this issue... Now, my *actual* question is: how large the set of points E where a > C^infty (say, on an interval) function f fails to be analytic can be? On the one hand I'm sure I've never been taught anything about this > particular problem, OTOH I've been thinking about such functions as exp(-1/sin^2(1/x)), > x*exp(-1/sin^2(1/x)), > exp(-1/x^2)*exp(-1/sin^2(1/x))=exp(-1/x^2-1/sin^2(1/x)) but as a matter of a fact I'm too rusty on real analysis to study > their behaviour in a neighbourhood of 0: intuitively it seems possible > that E has an accumulation point in E[*], but I find it hard to see > how it can contain a non-empty open subset. As usual what seems > intuitive is not granted to be also true... There was a thread here recently containing an example of a function that is C^oo but nowhere analytic. Is that what you wanted to know? as an area or a volume. > ---------------------------------------------- > But what represents eigenvectors or eigen values ? Why are they useful ? (in 3D ?) > Eigenvectors are those vectors which remain unchanged (apart from their length) when you apply the matrix. In a 3D rotation for example the axis of rotation is an eigenvector. In a more general case there could also be some stretching involved which is where the eigenvalue come in, it tells you by how much the eigenvector is stretched. To find eigenvectors and eigenvalues you need to solve the following: Mv=lambda v This says I apply the matrix M to some vector v and all it does is change its length by some factor lambda. To solve the equation to find lambda notice that the equation can be rewritten (M-lambda I)v=0 where I is the identity matrix. What we have is some matrix (M-lambda I) applied to some vector v giving zero, that can only happen if the determinant of that matrix is zero. In other words det(M-lambda I)=0 That's a polynomial equation in lambda. In the 3D case it will (in general) be a cubic. Cubic equations have at least one real root. So (in general) there will be at least one eigenvector corresponding to this eigenvalue. I keep saying 'in general' because there can be matrices which do things like collapse 3D space to a plane, or a line or even a point (the zero matrix does this). All these type of matrices are characterised by the fact that their determinant is zero (so one of the eigenvalues is zero). Eigenvector and eigenvalues are useful in breaking down what a matrix does, in 3D if I find one, I know all it does is stretch things in that direction so I can concentrate on what it does in a plane perpendicular to that eigenvector. ==== On the other hand, producing a derivation in some system of > logic to show that A is a consequence of B is much like producing > multiplications to show that 802342034023 * 19109101=15332034964690943323. > Probably you meant to write: On the other hand, producing a derivation in some system of > _propositional_ logic to show that A is a consequence of B is much like > producing multiplications to show that 802342034023 * 19109101 = > 15332034964690943323. Applying a rule of inference to a wff(s) is much like using multiplication to show that 802342034023 * 19109101 = 15332034964690943323. What he actually meant to say was: On the other hand, producing a derivation in some system to show that A is a consequence of B is much like showing that 15332034964690943323 = 802342034023 * 19109101. (He just got the sides of his equation mixed up.) Charlie Volkstorf Cambridge, MA (Note: I am talking about factoring an arbitrary number.) (What you both need to do is to formalize the analogy. Then apply it to some other domain, e.g. solving equations in Recursion Theory or Hyperset Theory.) (I can show how to solve equations in each of these domains, although I have yet to combine my various algorithms into one system. The formalizations are the same but so far the algorithms are different!) ==== This is s o silly that it's not even ridiculous. I don't think producing formal derivations is particularly silly - > it's just a trivial activity. If it is so trivial, then how about showing us a formal derivation of the unsolvability of the Halting Problem? That's a pretty simple theorem and proof. [ If you give up, you can consult my arxiv paper below. :) ] Charlie Volkstorf Cambridge, MA http://www.mathpreprints.com/math/Preprint/CharlieVolkstorf/20021008.1/1 http://www.arxiv.org/html/cs.lo/0003071 ==== ÀDoes any body knows where does the sci op-research group gone? ÀHas it dissapear? Santiago Cerisola ==== ÀDoes any body knows where does the > sci op-research group gone? > ÀHas it dissapear? > Santiago Cerisola It's called sci.op-research and still there (wherever that may be:-). Have you changed your ISP since you last saw it? Maybe you need -- G.C. ==== Let S be the set of non-invertible matrices in Mn(R). Can someone help me to prove that [Micro](S)=0, where [Micro] is the Lebesgue measure on Mn(R) (considered as the standard nÓ-dimensional euclidean space) ? ==== > Let S be the set of non-invertible matrices in Mn(R). Can someone help me to prove that [Micro](S)=0, where [Micro] is the Lebesgue > measure on Mn(R) (considered as the standard nÓ-dimensional euclidean > space) ? > One approach is to use the fact that S is the set of matrices with zero determinant. The determinant is a polynomial on Mn(R) = R^(n^2), so it is sufficient to show that the set of zeros of a non-zero polynomial has Lebesgue measure zero. To do this, you can use the implicit function theorem to show that the set of non-singular zeros has codimension one (i.e., it locally looks like a coordinate hyperplane of dimension n^2 - 1), and hence Lebesgue measure zero. The set of singular zeros is defined by more polynomials of lower degree, so you can use induction on the degree. John Mitchell ==== > One approach is to use the fact that S is the set of matrices with > zero determinant. The determinant is a polynomial on Mn(R) = R^(n^2), > so it is sufficient to show that the set of zeros of a non-zero > polynomial has Lebesgue measure zero. To do this, you can use the > implicit function theorem to show that the set of non-singular zeros > has codimension one (i.e., it locally looks like a coordinate > hyperplane of dimension n^2 - 1), and hence Lebesgue measure zero. The > set of singular zeros is defined by more polynomials of lower degree, > so you can use induction on the degree. But if you just want Lebesgue measure 0, the proof is much more elementary. ==== > Let S be the set of non-invertible matrices in Mn(R). Can someone help me to prove that [Micro](S)=0, where [Micro] is the Lebesgue > measure on Mn(R) (considered as the standard nÓ-dimensional euclidean > space) ? > Forget about matrices. Prove this more general fact: If you have any nonzero polynomial p in k variables, then the set of x with p(x)=0 has measure zero in R^k. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ ==== >> >>>Wrong. The most persecuted Semitic population are the Palestinians, > and >>>ironically most persecuted by the Israelis. >>>> Oh great. Another bigot flaunts himself in front of the world. >>>> The one clearly demonstrating bigotry is yourself, I have no objections > to >>> Jews and every objection to Israeli foreign policy. Unlike you, I am not >>> so bigoted that I can't tell one from the other. >>It would be an interesting scientific study to determine why >>there is such aggressive, hostile reaction >>to criticism that involving Israel, Jews, a Jew, >>or some claimed accomplishment of a Jew, >>and why critics are personally attacked, >>and the issue in play is avoided and obscured. >>It is okay to demean most ethnic groups with words like >>raghead, redneck, wop, wet back, chink, polock, etc. >>and little is made of it, but suggest that something Jewish is negative >>and the shit hits the fan and goes flying in all directions, >>mostly toward the critic. >>As this intense negative reaction is counter productive >>to Jews over the long run, and harmful to the larger society, >>it seems that an effort should be made to examine this problem, >>and to see what causes it, and how to solve it. >>It may be that the intensive religious and ethnic indoctrination >>of Jewish children, which is far more intense than that of Muslims, >>Catholics, etc. is the root cause of the problem, >> It may be your brain not working is the root cause of yours. It is interesting to see that Lloyd Parker's post >tends to confirm my suggestion that many people are brainwashed >to react with aggression and hostility, when exposed to >ideas that conflict with their conditioning. It would certainly be in the interests of mankind >for a scientific study to be conducted, to see how >intense religious/ethnic conditioning causes social problems. As can be seen, Jews have often been in the eye of the >hurricane and at times have been drawn into the heavy winds. >Most of the instigators of the class wars of the 1900's were Jews, >and most of the instigators of the religious wars of the 2000's >are Jews. The question is, does the intense ethnic/religious conditioning >of Jewish children, bring them into conflict with their >communities, and other religious and ethnic groups? If so, how can this be prevented? >I suggest that governments should limit the amount of time >that a child can be subjected to ethnic/religious brainwashing. Perhaps we should limit inbreeding that produced scum like you. Lloyd Parker raises a good point! Considering that Jews frown on Jews breeding with non-Jews, it may be that inbreeding plays a significant role in the conflict that Jews have with the larger breeding population. But as I indicated, I think that the reason that Jews come into conflict with their neighbors is caused by the intense religious/ethnic brainwashing that Jewish children are subjected to, and considering that Jewish conflict with their neighbors has been the root cause of so many of mankind's problems, including the class wars of the 1900's, the religious wars of the 2000's, etc. I suggest that government should limit the amount of time that children can be subjected to religious/ethnic dogma. Tom Potter ==== > at 07:51 AM, tdp@earthlink.net (Tom Potter) said: It would certainly be in the interests of mankind >for a scientific study to be conducted, to see how >intense religious/ethnic conditioning causes social problems. May I suggest you as the first guinea pig. It would be interesting to > research the origins of your delusions. The origins of delusions are my observations that there are numerous references by Jews, to thier perception that many Americans, Muslims, Germans, and all other races and religions, and many posters in the newsgroups, are anti-Semetic, and the fact that Jews have come into conflict with almost all of their neighbors throughout history. As I indicated, this problem is of critical interests to mankind, as Jewish conflict has been at the root of many wars, and as can be clearly seen, Jewish conflict with their Palestinian neighbors has been the root cause of 911, the Middle East problem, the Iraqi War, the movement of America toward a police state, and the enormous encrease in the cost of government in America. I suggest that the root cause is the intense religious/ethnic brainwashing that Jewish children are subjected to, and that governments must limit the amount of time that children can be subjected to religious/ethnic brainwashing. Tom Potter ==== The question is, does the intense ethnic/religious conditioning > of Jewish children, bring them into conflict with their > communities, and other religious and ethnic groups? In the United States, Jewish children live in a peacful and law abiding > way among and with their Gentile neibhbors. What conflict? The occurence > of violence and criminality is much lower among Jews, than in the > country taken as a whole. What conflict? Bob Kolker Of course, one point in time and space does not negate the fact that Jews have come into conflict with their non-Jewish neighbors more often than not trhoughout history. And of course, as can be seen by the many posts asserting that many people in America, and many posters in the newsgroups are anti-Semetic, it is clear that many (Most?) Jews sense that conflict exists. The question that is of vital inportance to mankind is why Jews come into conflict with theitr neighbors, and what can be done to eliminate this problem. I suggest that the root cause of the problem lies in the fact that Jews are far more intensely brainwashed with religious and ethnic dogma, than Muslims, Catholics, Protestents, Buddhists, etc. and that a solution would be for governments to limit the amount of time that a child can be subjected to religious/ethnic brainwashing. Tom Potter ==== tdp@earthlink.net says... >the fact that Jews are far more intensely >brainwashed with religious and ethnic dogma, >than Muslims What planet are you posting from? It sure ain't earth! ---------------------------------------------------------------------- | Just Another Internet Wise Guy Macon, GA USA | ---------------------------------------------------------------------- ==== I'd like comments on the following problem solution which appears in the solutions manual of James Stewart's _Calculus_, section 7.2*, both 4th and 5th editions. I believe the solution is flawed. The problem is Prove that ln(x^r) = r*ln(x). The book's solution is Differentiating the left side with respect to x, we get r*x^(r-1)/x^r = r/x. Differentiating the right side with respect to x, we get r/x. Since the derivatives are equal, the two sides of the proposed identity differ by at most a constant. Evaluating both sides at x=1, we see that the constant is 0. My proposed correction is ... Since the derivatives are equal, the difference of the two sides of the proposed identity does not depend on x (it might still depend on r). Evaluating at x=1, we get that the difference is 0 for any r. One of my students evaluated at r=1, not x=1. ==== > I'd like comments on the following problem solution which appears in the > solutions manual of James Stewart's _Calculus_, section 7.2*, both 4th and > 5th editions. I believe the solution is flawed. The problem is Prove that ln(x^r) = r*ln(x). The book's solution is Differentiating the left side with respect to x, we get r*x^(r-1)/x^r = > r/x. Differentiating the right side with respect to x, we get r/x. > Since the derivatives are equal, the two sides of the proposed identity > differ by at most a constant. Evaluating both sides at x=1, we see that > the constant is 0. This is not flawed at all if the assumptions are made clear. You want to show ln(x^r) = r*ln(x) for all x > 0 and all r in R. It suffices to show that for any fixed r in R, ln(x^r) = r*ln(x) for all x in (0,oo). Now follow the book's proof. ==== >One of my students evaluated at r=1, not x=1. Explain to your student that evaluating at r=1 only works in one case; namely, when r=1. Doug ==== I'd like comments on the following problem solution which appears in the >solutions manual of James Stewart's _Calculus_, section 7.2*, both 4th and >5th editions. I believe the solution is flawed. The problem is Prove that ln(x^r) = r*ln(x). The book's solution is Differentiating the left side with respect to x, we get r*x^(r-1)/x^r = >r/x. Differentiating the right side with respect to x, we get r/x. >Since the derivatives are equal, the two sides of the proposed identity >differ by at most a constant. Evaluating both sides at x=1, we see that >the constant is 0. My proposed correction is >... Since the derivatives are equal, the difference of the two sides of >the proposed identity does not depend on x (it might still depend on r). If we're regarding r as a constant, which evidently we are, then something that depends on r is still a constant. >Evaluating at x=1, we get that the difference is 0 for any r. One of my students evaluated at r=1, not x=1. Well explain to him that he can't do that, because r is a _given_ constant which might not equal 1. ************************ David C. Ullrich ==== > Prove that ln(x^r) = r*ln(x). >>My proposed correction is >>... Since the derivatives are equal, the difference of the two sides of >>the proposed identity does not depend on x (it might still depend on r). If we're regarding r as a constant, which evidently we are, then >something that depends on r is still a constant. Actually I think he assumed r was a variable, just not dependant on x. I guess it's a case of unclear declaration, or then he omitted some details (like the obvious x>0 requirement) of the example. But then you could just as well assume that r is the variable and that x is a constant, like the student did, except that you'd have to dr instead of dx. ==== Carl Devore schrieb im Newsbeitrag I'd like comments on the following problem solution which appears in the > solutions manual of James Stewart's _Calculus_, section 7.2*, both 4th and > 5th editions. I believe the solution is flawed. The problem is Prove that ln(x^r) = r*ln(x). x^r = e^(ln(x)*r) = x^r How do you know that dln(x)=1/x*dx? ==== > Carl Devore schrieb im Newsbeitrag > Prove that ln(x^r) = r*ln(x). x^r = e^(ln(x)*r) = x^r > How do you know that dln(x)=1/x*dx? In this book, ln is defined before exp. ln(x) is defined to the antiderivative of 1/x whose value at 1 is 0. exp is defined as the inverse of ln. ==== Carl Devore schrieb im Newsbeitrag > In this book, ln is defined before exp. ln(x) is defined to the > antiderivative of 1/x whose value at 1 is 0. exp is defined as the > inverse of ln. This is no good book. After a meaningfull Definition of exp(x), ln(x) can be defined as the inverse function. Usually exp(x) is defined as a sum. exp(x) = sum(k from 0 to infinity) x^k/k! The partial sum can be defined: Sn(x) = sum(k from 0 to n) x^k/k! The derivative of Sn(x) converges to the same limit as Sn(x). The derivative of the inverse function can be defined using the inverse function theorem of calculus: f(g(x))=x so f'(g(x))g'(x)=1 exp'(ln(x))ln'(x)=1 ln'(x)=1/(exp'(ln(x))=1/x Are there any words about the begining of calculus in ancient times (method of exhaustion, calculating the area and volume) or the middle ages like Wallis, Leibnitz (geometric aproach finding the tangent of a function) or Newton (calculating the speed of a body)? This should be in any book about calculus. ==== > In this book, ln is defined before exp. ln(x) is defined to the > antiderivative of 1/x whose value at 1 is 0. exp is defined as the > inverse of ln. This is no good book. And why is that? One can certainly define ln(x) = int_[1,x] dt/t for x > 0, and then derive exp(x) as the inverse function to ln(x). The mathematics works just fine. Are you commenting on something other than the mathematics? ==== > In this book, ln is defined before exp. ln(x) is defined to the >> antiderivative of 1/x whose value at 1 is 0. exp is defined as the >> inverse of ln. >> This is no good book. And why is that? Because he said so. I mean, duh. >One can certainly define ln(x) = int_[1,x] dt/t for x > 0, >and then derive exp(x) as the inverse function to ln(x). The mathematics >works just fine. Are you commenting on something other than the mathematics? ************************ David C. Ullrich ==== > Carl Devore schrieb im Newsbeitrag >> In this book, ln is defined before exp. ln(x) is defined to the >> antiderivative of 1/x whose value at 1 is 0. exp is defined as the >> inverse of ln. > This is no good book. After a meaningfull Definition of exp(x), ln(x) > can be defined as the inverse function. Usually exp(x) is defined as a sum. That is certainly true, but in what way does this demonstrate that the ln-before-exp approach is flawed? > Are there any words about the begining of calculus in ancient times > (method of exhaustion, calculating the area and volume) or the middle > ages like Wallis, Leibnitz (geometric aproach finding the tangent of > a function) or Newton (calculating the speed of a body)? This should > be in any book about calculus. Is there some particular reason for thinking that a ln-before-exp approach precludes discussions of history? -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. ==== > Prove that ln(x^r) = r*ln(x). Differentiating the left side with respect to x, we get r*x^(r-1)/x^r = > r/x. Differentiating the right side with respect to x, we get r/x. > Since the derivatives are equal, the two sides of the proposed identity > differ by at most a constant. Evaluating both sides at x=1, we see that > the constant is 0. My proposed correction is > ... Since the derivatives are equal, the difference of the two sides of > the proposed identity does not depend on x (it might still depend on r). > Evaluating at x=1, we get that the difference is 0 for any r. > It's not a correction, it's just a slightly less clear restatement. > One of my students evaluated at r=1, not x=1. > Geesh, turn the solution around just to accommodate a student? Nay, tutor instead the student to discern how r isn't a variable. ==== : I'd like comments on the following problem solution which appears in the : solutions manual of James Stewart's _Calculus_, section 7.2*, both 4th and : 5th editions. I believe the solution is flawed. The problem is : Prove that ln(x^r) = r*ln(x). : The book's solution is : Differentiating the left side with respect to x, we get r*x^(r-1)/x^r = : r/x. Differentiating the right side with respect to x, we get r/x. : Since the derivatives are equal, the two sides of the proposed identity : differ by at most a constant. Evaluating both sides at x=1, we see that : the constant is 0. : My proposed correction is : ... Since the derivatives are equal, the difference of the two sides of : the proposed identity does not depend on x (it might still depend on r). : Evaluating at x=1, we get that the difference is 0 for any r. No, the book's solution does not appear to me to be flawed. The two derivatives are equal, so their antiderivatives differ by at most a constant, as the book says. When you say that the difference does not depend on x, you are saying the same thing. : One of my students evaluated at r=1, not x=1. which would still show that the constant = 0. ==== [cut] > : Prove that ln(x^r) = r*ln(x). [cut] > No, the book's solution does not appear to me to be flawed. The two > derivatives are equal, so their antiderivatives differ by at most a > constant, as the book says. When you say that the difference does > not depend on x, you are saying the same thing. : One of my students evaluated at r=1, not x=1. which would still show that the constant = 0. Consider proving that ln(x^r) = r*ln(x) + r*(r-1). The student's proof would still go through, but the result is wrong. Carl Devore is trying to explain why evaluating at r =1 is a mistake. I think his explanation is ok. I think Ullrich's explanation that r is a constant and can't be set to 1 is also ok. -- Bill Hale ==== mu(N) = Moebius function: If N has a repeated factor mu(N) = 0 If the number of different factors is even, mu(N) = 1, if odd mu(N) = -1 M(N) = Mertens function : Sum of mu(i) i = 1, 2, 3, ...N The values of Mertens function forms a pseudorandom sequence,( Due to the mixing effect of multiplication of primes) with Mean = 0 and follows the law of the iterated logarithm: With probability 1, only for finitely many events: M(N) > L(.5N LogLog(N))^(1/2) ; For some L > 1. But there is a theorem that says that if Riemann's Hypothesis is true then Max M(n) < K*N^(1/2+e) ; K > 1 ; e > 0 . e can be LogLogLog(N)/Log(n) Then it is possible that in finitely many cases Riemann Hypothesis can fail. ==== how can one base a proof upon pseudorandomness? the only randomness that exists is, That whose period is too long for you to compute. > mu(N) = Moebius function: If N has a repeated factor mu(N) = 0 > If the number of different factors is even, mu(N) = 1, if odd mu(N) = > -1 > M(N) = Mertens function : Sum of mu(i) i = 1, 2, 3, ...N > The values of Mertens function forms a PSEUDORANDOM sequence,( Due to > the mixing effect of multiplication of primes) with Mean = 0 and > follows the law of the iterated logarithm: > With probability 1, only for finitely many events: > M(N) > L(.5N LogLog(N))^(1/2) ; For some L > 1. > But there is a theorem that says that if Riemann's Hypothesis is true > then > Max M(n) < K*N^(1/2+e) ; K > 1 ; e > 0 . e can be LogLogLog(N)/Log(n) > Then it is possible that in finitely many cases Riemann Hypothesis can > fail. --les ducs d'Enron! http://members.tripod.com/~american_almanac/ ==== If I remember correctly, RP^3 is a ball D^3 with antipodal points > identified. OK. I must have remembered that incorrectly. What topology do we get when we take the ball D^3 and identify antipodal points? By the ball D^3 (is that the correct notation for a disk in 3 dimensions?) I mean the set of points (x,y,z) in R^3 such that x^2 + y^2 +z^2 < or = 1. Eugene Shubert http://www.everythingimportant.org ==== >> If I remember correctly, RP^3 is a ball D^3 with antipodal points >> identified. OK. I must have remembered that incorrectly. I was going to say that you remembered it just fine, when RP^3 is a ball D^3 with antipodal points OF ITS BOUNDARY identified. Maybe that's what you meant. If, on the each (x,y,z) of norm less than or equal to 1 is identified to (-x,-y,-z)), then you WERE incorrect: that space is the cone on RP^2 (so it isn't a manifold at the vertex of the cone; in particular, it isn't the manifold RP^3). Lee Rudolph ==== > If I remember correctly, RP^3 is a ball D^3 with antipodal points >> identified. I was going to say that you remembered it just fine, when RP^3 is a ball D^3 with antipodal points OF ITS BOUNDARY > identified. Maybe that's what you meant. I meant identify (x,y,z) to (-x,-y,-z) iff x^2 + y^2 +z^2 = 1. When I said antipodal I was thinking that the antipodal points were precisely the opposing pairs of points on the boundary of D^3. Did I get the notation right? D^3= (x,y,z) in R^3 such that x^2 + y^2 +z^2 < or = 1? But since D^3 is a flat manifold with boundary, does that mean that RP^3 is a flat manifold without curvature? > If, on the > each (x,y,z) of norm less than or equal to 1 is identified > to (-x,-y,-z)), then you WERE incorrect: that space is > the cone on RP^2 (so it isn't a manifold at the vertex > of the cone; in particular, it isn't the manifold RP^3). Lee, I don't have that much mathematical imagination. Eugene Shubert http://www.everythingimportant.org ==== >>> If I remember correctly, RP^3 is a ball D^3 with antipodal points >>> identified. and now clarifies that he *did* remmeber correctly: I meant identify (x,y,z) to (-x,-y,-z) iff x^2 + y^2 +z^2 = 1. When I said antipodal I was thinking that the antipodal points were >precisely the opposing pairs of points on the boundary of D^3. Did I get the notation right? >D^3= (x,y,z) in R^3 such that x^2 + y^2 +z^2 < or = 1? But since D^3 is a flat manifold with boundary, does that mean that >RP^3 is a flat manifold without curvature? No--because the *boundary* isn't flat. If you feel your mathematical imagination isn't (yet) up to dealing with the 3-dimensional situation, practice with the 2-dimensional situation. There, the real projective plane RP^2 is the identification space obtained by identifying antipodal points (x,y), (-x,-y) of S^1 = {(x,y) : x^2+y^2=1 }, the boundary of D^2 = {(x,y) : x^2+y^2=<1 }. But even though the 2-disk is flat, the real projective plane *cannot* be given a (Riemannian) metric which is flat everywhere! The reason is (ultimately) that S^1, as the boundary of D^2, has curvature. Even before getting to that point, you should grapple with the fact that when RP^2 is presented in this way, as this particular identification space, it *isn't equipped with a smooth structure* (in any natural way) in a neighborhood of any point that comes from a pair of antipodal points of S^1. So it certainly *isn't equipped with a Riemannian metric* thereabouts. (John Mitchell can explain what kind of metric it does have, I suspect.) About all you have, a priori, is a metric space which is a topological manifold (with a big, open, dense, flat, smooth subset, corresponding to the interior of the original D^2). This problem disappears if you model RP^2 as the quotient space of S^2 by the antipodal involution. Then you see that, in fact, RP^2 has a smooth structure and a Riemannian metric for which it is locally spherical, with just half as much total curvature as the S^2 you started with: by changing the radius of this RP^2 you can make the total curvature as small as you like, but it's always strictly positive. Lee Rudolph ==== HE'S BACK! format=flowed > Memorandum for the Record My letter to Skeptical Inquirer is published on p. 69 of their > comment by Robert Sheaffer which directly contradicts what I say in > summary of someone else's position with which he says he does not > those words in the text about Saddam's Iraq using zero point energy > to become a leading power , but I know I did not. Since the issue > of WMD in Iraq is a hot one I do not want to be accused of something I > did not do. with my comments is reproduced here for the record. This was the only I cull my actual words for the record in my reply to Art Wagner's > message: If you want to make sure I get your messages cc to sarfatti@well.com. I don't believe this story below, but I could be wrong. > Of course the Occult Nazi SS has probably been in Iraq since WWII and > if Nick Cook is right - maybe. > But I think this story is all really silly disinformation and > misinformation. > But what do I know. I'm just a warped macro-quantum mechanic. > Consider the source -- Art Bell? These radio talk show hosts don't > know which way is up. > It's like reading the National Inquirer and thinking it's the Wall > Street Journal. > New Age fast food for the mind is like transfatty acids for your heart > and tobacco for your lungs. :-) I think that is only a curious coincidence. I found out from Parvez at > SARA in Huntington Beach that he was called also. > So apparently they are calling lots of people who run corporations and > will ask for a whopping donation to go to White House dinner. Curious > that > DeLay's point man mentioned Iraq twice on the phone. I doubt that he > knew about my UFO investigations. > Not without my physics he won't. Both Hal Puthoff and I generally agree that UFO energy technology > probably does not require huge investment, but is mainly some kind of > electrical engineering including microwave plasma technology. While > Jim Corum's specific model does not work something in that general > area probably will. We had a secret engineering project at ISSO that > suggests this. The connection of spin 2 gravity to spin 1 gauge forces > is explicitly written down in my technical papers in terms of the > macro-quantum vacuum cohering of the zero point fluctuations, i.e. > (0|e+e-|0). That is like the E = mc^2 of metric engineering of the > Iranians and the Chinese are all smart enough to do something IF my > theory is on right track. Of course I could be wrong. I do not know > yet. Too soon to tell. The story below is clearly some kind of psyop connected with > Spielberg's Taken most likely. > greatest fear is that Saddam will reverse-engineer the crashed > alien spacecraft ... The > craft allegedly crashed during the Gulf War (1990 - 1991) or more > recently (possibly December 1998). This will be Iraq's Roswell. The > US > is currently reverse-engineering the Roswell craft and fears Saddam's > scientists will catch up with or even go beyond the U.S. in one or > more > areas. These areas of research include zero point, over-ratio or > gravimetric technology, which would allow for a tremendous advance, > allowing Iraq to become a leading power. It is obvious from the original below that I did not write the above > remarks. Sheaffer also falsely alleges that I promoted similar accounts about > saucer crashes and reverse-engineered alien technology. What does he > mean promoted and similar? I have never promoted any accounts of > saucer crashes in Iraq. I have mentioned stories of Roswell by Colonel > Corso and also some stories we heard at ISSO. I have never promoted > them as facts. My position has always been that of the > gedankenexperiment. If such devices are really here, how do they work? > I have never said I know for a fact that a saucer has crashed. I have > never said I believe stories, some by people claiming to be insiders, > told to me about reverse-engineered alien technology. > Here is the original sent to me by Art Wagner: >> --------------------------------------------------------------- >> Article by: Art Bell caller >> Friday 06 Dec 2002 >> Summary:A caller into the Art Bell show, who claims to have >> connections to the military, said a UFO crashed in Iraq in recent >> years. The US is searching for any public pretext to invade Iraq, >> but its greatest fear is that Saddam will reverse engineer a crashed >> alien spacecraft. Not without my physics he won't. > Weblink: >> http://ufos.about.com/gi/dynamic/offsite.htm?site=http://ufoinfo.com/ >> roundup/v03/rnd03%5F51.html%231 >> Reference at indymedia website: >> Article: >> [Posters Note: I heard this call to Art Bell on the Friday morning >> program. There are a few stories on google search, about UFOs in >> Iraq. But the most detailed appear to be taken down. Below is one >> which is still accessible. According to the logic suggested by the >> caller, the greatest fear is NOT the use of conventional weapons of >> mass destruction by Iraq-- chemical, bio, or even nuclear. US >> military planners fear something far more powerful. Both Hal Puthoff and I generally agree that UFO energy technology > probably does not require huge investment, but is mainly some kind of > electrical engineering including microwave plasma technology. While > Jim Corum's specific model does not work something in that general > area probably will. We had a secret engineering project at ISSO that > suggests this. The connection of spin 2 gravity to spin 1 gauge forces > is explicitly written down in my technical papers in terms of the > macro-quantum vacuum cohering of the zero point fluctuations, i.e. > (0|e+e-|0). That is like the E = mc^2 of metric engineering of the > Iranians and the Chinese are all smart enough to do something IF my > theory is on right track. Of course I could be wrong. I do not know > yet. Too soon to tell. The story below is clearly some kind of psyop connected with > Spielberg's Taken most likely. >> The US inner circle apparently fears that Iraqi scientists will >> harness some of the potential of technology taken from a spacecraft >> crashed in Iraq. The craft allegedly crashed during the Gulf War, >> or more recently. This would be Iraq's Roswell. The US is currently >> at work on reverse engineering the Roswell craft, and fears Saddams >> scientists will catch up, or even go beyond the US in one or more >> areas. These areas of research include zero point, over unity, ( or >> gravimetric ) technology, which would allow for a tremendous advance, >> allowing Iraq to become a leading power. >> According to this theory, the US is preparing to invade Iraq to >> capture this extraterrestial technology. All other reasons given >> publicly are said to be mere pretexts, to prevent Iraq from >> benefitting from its find.] >> Here is one story on UFOs in Iraq: >> UFOs INTERVENE IN OPERATION DESERT FOX >> Large triangular UFOs were seen in Iraq and upstate New York before >> and during Operation Desert Fox. >> On Thursday, December 16, 1998, at 2:31 a.m. local time, a >> triangle-shaped pattern of lights appeared over downtown Baghdad >> and was picked up by CNN's night-vision video camera. The lights >> hovered in position and moved slowly to the right, as Iraqi >> anti-aircraft tracer fire streaked away into the night. >> The scene appeared on CNN's live broadcast, which aired in the USA >> at 6:31 p.m. Eastern time, Wednesday, December 15. >> According to ufologist Ignatius Graffeo, I did see a triangular >> formation of lights moving very slowly at about 12:55 a.m. Baghdad >> time that Thursday on an NBC news report of the bombing...The light >> was steady, and it was definite and very striking. He described it >> as a V-shaped formation like the one at Phoenix, Arizona on March >> 13, 1997. >> [Posters Note: Several reports are circulating about the Phoenix >> events. A craft was alleged to have been found by the military, with >> three dead bodies. ] >> The UFO was against a black night sky and very different from the >> 'greenish' tracer fire moving across the sky, which did not hold >> their position for any length of time. >> According to the Boston Herald, Fort Drum, the U.S. Army post in >> upstate New York, was a staging area for the four-day Operation >> Desert Fox. The Herald reported, Along with the additional air >> assets, the Army is sending a battalion of light infantry troops from >> Fort Drum, N.Y. and nuclear-biological-chemical experts from several >> bases to watch for any attack by Iraq against Kuwait and other >> neighbors in the region. (See the Boston, Mass. Herald for December >> 17, 1998, U.S. to pump up military muscle, page 22.) >> Three weeks ago, around November 30, 1998, MUFON New York received >> reports of triangular UFO activity over Evan Mills, N.Y. and nearby >> Perch Lake, just west of Fort Drum. >> According to Larry Clark of MUFON New York, a woman living on Perch >> Lake reported heavy UFO activity. First there were triangular craft >> that have appeared in the local sky. The woman (witness) reported the >> sighting of a 'Pine Bush' triangle that moved at treetop level and >> continued out over the lake before disappearing. >> A teenager from the local Civil Air Patrol (CAP) told the woman that >> they have also spotted a craft over Evan Mills...Within the last few >> weeks, the woman has regularly observed a large bluish light with a >> small red streak at its center. She had reportedly seen the UFO >> hovering for hours over her property and a large undeveloped >> tract of land nearby. >> north of Syracuse, N.Y. (Many thanks to Errol Bruce-Knapp, Ignatius >> Graffeo, and MUFON Eastern director George A. Filer for the news >> items.) > --Apple-Mail-13-413141824 Memorandum for the Record My letter to Skeptical Inquirer is published on p. 69 of their comment by Robert Sheaffer which directly contradicts what I say in summary of someone else's position with which he says he does not those words in the text about Saddam's Iraq using zero point energy to become a leading power , but I know I did not. Since the issue of WMD in Iraq is a hot one I do not want to be accused of something I did not do. with my comments is reproduced here for the record. This was the only I cull my actual words for the record in my reply to Art Wagner's message: 1998,1998,FFFEsarfatti@pacbell today at all. If you want to make sure I get your messages cc to 1998,1998,FFFEsarfatti@well.com. I don't believe this story below, but I could be wrong. Of course the Occult Nazi SS has probably been in Iraq since WWII and if Nick Cook is right - maybe. But I think this story is all really silly disinformation and misinformation. But what do I know. I'm just a warped macro-quantum mechanic. Consider the source -- Art Bell? These radio talk show hosts don't know which way is up. It's like reading the National Inquirer and thinking it's the Wall Street Journal. New Age fast food for the mind is like transfatty acids for your heart and tobacco for your lungs. :-) I think that is only a curious coincidence. I found out from Parvez at SARA in Huntington Beach that he was called also. So apparently they are calling lots of people who run corporations and will ask for a whopping donation to go to White House dinner. Curious that DeLay's point man mentioned Iraq twice on the phone. I doubt that he knew about my UFO investigations. Not without my physics he won't. Both Hal Puthoff and I generally agree that UFO energy technology probably does not require huge investment, but is mainly some kind of electrical engineering including microwave plasma technology. While Jim Corum's specific model does not work something in that general area probably will. We had a secret engineering project at ISSO that suggests this. The connection of spin 2 gravity to spin 1 gauge forces is explicitly written down in my technical papers in terms of the macro-quantum vacuum cohering of the zero point fluctuations, i.e. (0|e+e-|0). That is like the E = mc^2 of metric engineering of the Iranians and the Chinese are all smart enough to do something IF my theory is on right track. Of course I could be wrong. I do not know yet. Too soon to tell. The story below is clearly some kind of psyop connected with Spielberg's Taken most likely. greatest fear is that Saddam will reverse-engineer the crashed alien spacecraft ... The craft allegedly crashed during the Gulf War (1990 - 1991) or more recently (possibly December 1998). This will be Iraq's Roswell. The US is currently reverse-engineering the Roswell craft and fears Saddam's scientists will catch up with or even go beyond the U.S. in one or more areas. These areas of research include zero point, over-ratio or gravimetric technology, which would allow for a tremendous advance, allowing Iraq to become a leading power. It is obvious from the original below that I did not write the above remarks. Sheaffer also falsely alleges that I promoted similar accounts about saucer crashes and reverse-engineered alien technology. What does he mean promoted and similar? I have never promoted any accounts of saucer crashes in Iraq. I have mentioned stories of Roswell by Colonel Corso and also some stories we heard at ISSO. I have never promoted them as facts. My position has always been that of the gedankenexperiment. If such devices are really here, how do they work? I have never said I know for a fact that a saucer has crashed. I have never said I believe stories, some by people claiming to be insiders, told to me about reverse-engineered alien technology. Here is the original sent to me by Art Wagner: --------------------------------------------------------------- Article by: Art Bell caller Friday 06 Dec 2002 Summary:A caller into the Art Bell show, who claims to have connections to the military, said a UFO crashed in Iraq in recent years. The US is searching for any public pretext to invade Iraq, but its greatest fear is that Saddam will reverse engineer a crashed alien spacecraft. Not without my physics he won't. Weblink: 1998,1998,FFFEhttp://ufos.about.com/gi/dyna mic/offsite.htm?site=http://ufoinfo.com/roundup/v03/rnd03%5F51.html%231 Reference at indymedia website: Article: [Posters Note: I heard this call to Art Bell on the Friday morning program. There are a few stories on google search, about UFOs in Iraq. But the most detailed appear to be taken down. Below is one which is still accessible. According to the logic suggested by the caller, the greatest fear is NOT the use of conventional weapons of mass destruction by Iraq-- chemical, bio, or even nuclear. US military planners fear something far more powerful. Both Hal Puthoff and I generally agree that UFO energy technology probably does not require huge investment, but is mainly some kind of electrical engineering including microwave plasma technology. While Jim Corum's specific model does not work something in that general area probably will. We had a secret engineering project at ISSO that suggests this. The connection of spin 2 gravity to spin 1 gauge forces is explicitly written down in my technical papers in terms of the macro-quantum vacuum cohering of the zero point fluctuations, i.e. (0|e+e-|0). That is like the E = mc^2 of metric engineering of the Iranians and the Chinese are all smart enough to do something IF my theory is on right track. Of course I could be wrong. I do not know yet. Too soon to tell. The story below is clearly some kind of psyop connected with Spielberg's Taken most likely. The US inner circle apparently fears that Iraqi scientists will harness some of the potential of technology taken from a spacecraft crashed in Iraq. The craft allegedly crashed during the Gulf War, or more recently. This would be Iraq's Roswell. The US is currently at work on reverse engineering the Roswell craft, and fears Saddams scientists will catch up, or even go beyond the US in one or more areas. These areas of research include zero point, over unity, ( or gravimetric ) technology, which would allow for a tremendous advance, allowing Iraq to become a leading power. According to this theory, the US is preparing to invade Iraq to capture this extraterrestial technology. All other reasons given publicly are said to be mere pretexts, to prevent Iraq from benefitting from its find.] Here is one story on UFOs in Iraq: UFOs INTERVENE IN OPERATION DESERT FOX Large triangular UFOs were seen in Iraq and upstate New York before and during Operation Desert Fox. On Thursday, December 16, 1998, at 2:31 a.m. local time, a triangle-shaped pattern of lights appeared over downtown Baghdad and was picked up by CNN's night-vision video camera. The lights hovered in position and moved slowly to the right, as Iraqi anti-aircraft tracer fire streaked away into the night. The scene appeared on CNN's live broadcast, which aired in the USA at 6:31 p.m. Eastern time, Wednesday, December 15. According to ufologist Ignatius Graffeo, I did see a triangular formation of lights moving very slowly at about 12:55 a.m. Baghdad time that Thursday on an NBC news report of the bombing...The light was steady, and it was definite and very striking. He described it as a V-shaped formation like the one at Phoenix, Arizona on March 13, 1997. [Posters Note: Several reports are circulating about the Phoenix events. A craft was alleged to have been found by the military, with three dead bodies. ] The UFO was against a black night sky and very different from the 'greenish' tracer fire moving across the sky, which did not hold their position for any length of time. According to the Boston Herald, Fort Drum, the U.S. Army post in upstate New York, was a staging area for the four-day Operation Desert Fox. The Herald reported, Along with the additional air assets, the Army is sending a battalion of light infantry troops from Fort Drum, N.Y. and nuclear-biological-chemical experts from several bases to watch for any attack by Iraq against Kuwait and other neighbors in the region. (See the Boston, Mass. Herald for December 17, 1998, U.S. to pump up military muscle, page 22.) Three weeks ago, around November 30, 1998, MUFON New York received reports of triangular UFO activity over Evan Mills, N.Y. and nearby Perch Lake, just west of Fort Drum. According to Larry Clark of MUFON New York, a woman living on Perch Lake reported heavy UFO activity. First there were triangular craft that have appeared in the local sky. The woman (witness) reported the sighting of a 'Pine Bush' triangle that moved at treetop level and continued out over the lake before disappearing. A teenager from the local Civil Air Patrol (CAP) told the woman that they have also spotted a craft over Evan Mills...Within the last few weeks, the woman has regularly observed a large bluish light with a small red streak at its center. She had reportedly seen the UFO hovering for hours over her property and a large undeveloped tract of land nearby. north of Syracuse, N.Y. (Many thanks to Errol Bruce-Knapp, Ignatius Graffeo, and MUFON Eastern director George A. Filer for the news items.) --Apple-Mail-13-413141824-- ==== I want to use the ADI method with 4 space variables and time. I dont want to have to invent the ADI scheme to do this, is there some papers somehow which show some methods to use the ADI scheme in 4 space variables and time? Tony ==== > I want to use the ADI method with 4 space variables and time. I dont > want to have to invent the ADI scheme to do this, is there some papers > somehow which show some methods to use the ADI scheme in 4 space > variables and time? Just do a product of 4 of those alpha I + d^2/dx_i^2 operators? Btw, you know that ADI as a solver for time-dependent problems is provably unstable for 3 dimensions? V. PS also try posting this to sci.math.num-analysis. -- ==== > In message , jabriol >What are the chances of atoms collecting together to form the simplest >self-reproducing cell? In his book A Guided Tour of the Living Cell, Nobel >Prize-winning scientist Christian de Duve admits: If you equate the >probability of the birth of a bacterial cell to that of the chance assembly >of its component atoms, even eternity will not suffice to produce one for >you. This is easy to answer. Over time quantum mechanics built the chemical > mechanisms of the cell bit by bit. It took millions of years, but back > in the old Hadean days there was nothing else to do. > so Quantum Mechanincs is God now.... understood ==== In message , jabriol > In message , jabriol >>What are the chances of atoms collecting together to form the simplest >>self-reproducing cell? In his book A Guided Tour of the Living Cell, >Nobel >>Prize-winning scientist Christian de Duve admits: If you equate the >>probability of the birth of a bacterial cell to that of the chance >assembly >>of its component atoms, even eternity will not suffice to produce one for >>you. >> This is easy to answer. Over time quantum mechanics built the chemical >> mechanisms of the cell bit by bit. It took millions of years, but back >> in the old Hadean days there was nothing else to do. > >so Quantum Mechanincs is God now.... understood No. -- http://www.earthpoetry.demon.co.uk RC ==== > How can one prove the consistency of field?? How to prove that > there are no self contradiction?? How can one prove the consistency of wombat?? How to prove that First catch your wombat... But if you can't find a wombat may you construct your own a la Frankenstein? So I have some legs, a body, etc, and I construct my own wombat... only to hear someone question the consistency of the legs. > there are no self contradiction?? -- > Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html > The League of Gentlemen -- G.C. ==== I'm taking a couple of math courses, and unfortunately, due to the very high number of foreign students in class, many of them have breath that smells like a sewage plant. I cannot think straight when I have to sit next to one of them. Can anyone suggest strategies for coping with this? I know they won't take breath mints because it's not their custom. ==== > I'm taking a couple of math courses, and unfortunately, due to the very high > number of foreign students in class, many of them have breath that smells > like a sewage plant. I cannot think straight when I have to sit next to one > of them. Can anyone suggest strategies for coping with this? I know they > won't take breath mints because it's not their custom. Check out WebMD for this problem and a solution. They will have to change a few things, but it is a curable problem as I've been there! HTH ==== with this -- the step which I (and probably those of you who also hit a dead end) completely missed was the obvious one that 10^((y-Ax)/10) = 10^(y/10)*10^(-A*x/10). Once you've got that it's easy, thanks again. Andrew Milne I need to express this equation in terms of y=f(x), but I simply don't know > how. Any help would be much appreciated (A and B are constants): x = (1/A) * (y - 10*log(B - 10^(y/10))) Andrew Milne ==== the integral which comes up (see below). Anyway I didn't find it so my answer is not complete.... >> x = (1/A) * (y - 10*log(B - 10^(y/10))) Ax = y - 10*log(B - 10^(y/10)) 10^(Ax) = 10^y * 10^(- 10*log(B - 10^(y/10))) 10^(y-Ax) = (B - 10^(y/10))^10 10^[(y-Ax)/10] = B - 10^(y/10) B = 10^[(y-Ax)/10] + 10^(y/10) since B = constant dB/dx =0 so ln10*10^[(y-Ax)/10]*(dy/dx - A)/10 + ln10*10^(y/10)*(dy/dx)/10 = 0 10^[(y-Ax)/10]*(dy/dx - A) + 10^(y/10)*(dy/dx) = 0 [10^(y/10)]*[10^(-Ax/10)]*(dy/dx - A)+ 10^(y/10)*(dy/dx) =0 [10^(-Ax/10)]*(dy/dx - A) + (dy/dx) = 0 (dy/dx)* [1+10^(-Ax/10)] = A*10^(-Ax/10) (dy/dx)* [10^(Ax/10)+1] = A dy/dx =A/ [10^(Ax/10)+1] If you intergrate the above equation you'll get y = f(x) + c and since you know that x = (1/A) * (y - 10*log(B - 10^(y/10))) let's say for y=0, x'= -(10/A)*log(B - 1) finally you can calculate c = -f(x'). I appologise for not giving the final solution but I guess if you have access to tables of integrals you'll find it easily. Hope I've helped Greetings, Spiros ==== I bought most of these when I was in grad school -- now they are just taking up shelf space. All copies are in new condition -- no highlighting or dog-eared pages. Shipping in the U.S. (via USPS) is included in the price. Checks or Paypal accepted. my address. Bill Topology: A First Course (Munkres) $45 Modular Forms and Fermat's Last Theorem (Cornell, et al) $35 An Introduction to Sympbolic Dynamics and Coding (Lind and Marcus) $30 Russian-English Dictionary of the Mathematical Sciences (Boas) $12 Analysis (Lieb and Loss) $40 Algebraic Geometry (Miyanshi) $15 Functions of One Complex VAriable I (Conway) $30 Algebraic Number Fields, Second Edition (Janusz) $20 An Invitation to Algebraic Geometry (Lorenzini) $35 Abstract Algebra, Second Edition (Dummit and Foote) $45 Contemporary Mathematics Series (AMS): Finite Fields: Theory, Applications, and Algorithms (Conference Proceedings, 1993) $6 Finite Fields: Theory, Applications, and Algorithms (Conference Proceedings, 1997) $6 DIMACS Series in Discrete Math and Computer Science: Volume 2: Distributed Computing and Cryptography (Feigenbaum, et al) $5 Volume 6: Discrete and Computational Geometry (Goodman, et al) $5 Volume 13: Advances in Computational Complexity Theory $5 ==== The measure of the amount of substance in any object; body and or mass of matter, is a mathematical ratio of the magnitude of the net thrust, to - divided by - the acceleration caused by that net thrust; where the thrust is either a push or a pull. A _net_ thrust [f] is the total thrust [F] exerted on and/or by a body, minus any resistance to acceleration; such as a body's weight and/or friction: Where the magnitude of resistance is the product of a body's weight [w], and a coefficient of friction [u]; so that the _net_ thrust is: f = [F-uw], and is known as the net force! It is an empirically known - or derivable - fact that the net force [f] exerted on and/or by any body, to - divided by the acceleration [a] that it causes is a constant: Which constant is a ratio, and a measure of the body's inertia, and/or mass: A measure of the quantity of matter in it. For any body resting on Earth's or some other similar planet's terra firma surface; the net force [f] exerted on and/or by any body on that surface or a support thereon; such as a weight-scale, is known as its weight [w]; which is proportional to the acceleration [g] at which it will free fall; at that location: This proportion [w/g] is a constant ratio for any given body, and is equal to the ratio of the net force [f] exerted on and/or by it, to - divided by the acceleration [a] that it causes: So that for any given body, its inertia and/or mass [m] can be expressed mathematically as: m = w/g = f/a Where w/g is its gravitational mass; which applies only when resting on terra firma, and f/a is its _inertia_; which applies anytime and anyplace! Units of mass and/or inertia are commonly given as slugs, grams and/or kilograms. ==== I have always been interested in the idea of the inertial frame. When Newton spun a bucket and saw that the water did not spin with it, he started a question that I think has never been answered. When I took Relativity I hoped that the subject would be discussed, but all I heard was that relativity works in an inertial frame. An inertial frame is stationary or moving uniformly with respect to the fixed stars. The question is simple: How the heck do the molecules of water in that bucket know where the fixed stars are? > The measure of the amount of substance in any object; body and or mass of > matter, is a mathematical ratio of the magnitude of the net thrust, to - > divided by - the acceleration caused by that net thrust; where the thrust is > either a push or a pull. A _net_ thrust [f] is the total thrust [F] exerted on and/or by a body, > minus any resistance to acceleration; such as a body's weight and/or > friction: Where the magnitude of resistance is the product of a body's > weight [w], and a coefficient of friction [u]; so that the _net_ thrust is: > f = [F-uw], and is known as the net force! It is an empirically known - or derivable - fact that the net force [f] > exerted on and/or by any body, to - divided by the acceleration [a] that it > causes is a constant: Which constant is a ratio, and a measure of the body's > inertia, and/or mass: A measure of the quantity of matter in it. For any body resting on Earth's or some other similar planet's terra firma > surface; the net force [f] exerted on and/or by any body on that surface or > a support thereon; such as a weight-scale, is known as its weight [w]; which > is proportional to the acceleration [g] at which it will free fall; at that > location: This proportion [w/g] is a constant ratio for any given body, and is equal > to the ratio of the net force [f] exerted on and/or by it, to - divided by > the acceleration [a] that it causes: So that for any given body, its inertia and/or mass [m] can be expressed > mathematically as: m = w/g = f/a Where w/g is its gravitational mass; which applies only when resting on > terra firma, and f/a is its _inertia_; which applies anytime and anyplace! Units of mass and/or inertia are commonly given as slugs, grams and/or > kilograms. ==== oo 1 S(x)= S ------------------- =? n=1 4n^2 + x^2 ==== oo 1 > S(x)= S ------------------- =? > n=1 4n^2 + x^2 S(x)= (pi/x*cosh(pi/2*x)/sinh(pi/2*x)-2/x^2)/4 More generally pi/(2*x)*cosh((pi-|y|)*x)/sinh(pi*x) =1/(2*x^2)+sum_{n=1}^oo cos(n*y)/(n^2+x^2) Hoping this helped, Raymond ==== where can i find a proof ? oo 1 > S(x)= S ------------------- =? > n=1 4n^2 + x^2 > S(x)= (pi/x*cosh(pi/2*x)/sinh(pi/2*x)-2/x^2)/4 More generally pi/(2*x)*cosh((pi-|y|)*x)/sinh(pi*x) > =1/(2*x^2)+sum_{n=1}^oo cos(n*y)/(n^2+x^2) Hoping this helped, > Raymond ==== TCL escribi.97 en el mensaje > where can i find a proof ? Fourier series develop of cosh(mx): cosh(mx) = (2m*sinh(mx)/pi)(1/(2m^2) - cos(x)/(1^2 + m^2) + cos(2x)/(2^2 + m^2) - cos(3x)/(3^2 + m^2) + ...) Let x = pi, cosh(m*pi) = (2m*sinh(m*pi)/pi)(1/(2m^2) + 1/(1^2 + m^2) + 1/(2^2 + m^2) + 1/(3^2 + m^2) + ...) ===> Sum(1/(k^2 + m^2), k, 1, inf) = (pi/(2m))coth(m*pi) - 1/(2m^2) = (m*pi*coth(m*pi) - 1)/(2m^2) -- Ignacio Larrosa Ca.96estro A Coru.96a (Espa.96a) ilarrosaQUITARMAYUSCULAS@mundo-r.com oo 1 > S(x)= S ------------------- =? > n=1 4n^2 + x^2 > S(x)= (pi/x*cosh(pi/2*x)/sinh(pi/2*x)-2/x^2)/4 More generally pi/(2*x)*cosh((pi-|y|)*x)/sinh(pi*x) > =1/(2*x^2)+sum_{n=1}^oo cos(n*y)/(n^2+x^2) Hoping this helped, > Raymond ==== NICE WORK, Ignacio and Raymond!!!! Now, a bit of harder sum: oo n^2 S(x)= S ------------------- =? n=1 n^4 + x^4 TCL escribi.97 en el mensaje >> where can i find a proof ? Fourier series develop of cosh(mx): cosh(mx) = (2m*sinh(mx)/pi)(1/(2m^2) - cos(x)/(1^2 + m^2) + cos(2x)/(2^2 + >m^2) - cos(3x)/(3^2 + m^2) + ...) Let x = pi, cosh(m*pi) = (2m*sinh(m*pi)/pi)(1/(2m^2) + 1/(1^2 + m^2) + 1/(2^2 + m^2) + >1/(3^2 + m^2) + ...) === >Sum(1/(k^2 + m^2), k, 1, inf) = (pi/(2m))coth(m*pi) - 1/(2m^2) = (m*pi*coth(m*pi) - 1)/(2m^2) -- Ignacio Larrosa Ca.96estro >A Coru.96a (Espa.96a) >ilarrosaQUITARMAYUSCULAS@mundo-r.com > oo 1 >> S(x)= S ------------------- =? >> n=1 4n^2 + x^2 >> S(x)= (pi/x*cosh(pi/2*x)/sinh(pi/2*x)-2/x^2)/4 >> More generally >> pi/(2*x)*cosh((pi-|y|)*x)/sinh(pi*x) >> =1/(2*x^2)+sum_{n=1}^oo cos(n*y)/(n^2+x^2) >> Hoping this helped, >> Raymond > > ==== Series expansion. > where can i find a proof ? > oo 1 > S(x)= S ------------------- =? > n=1 4n^2 + x^2 > S(x)= (pi/x*cosh(pi/2*x)/sinh(pi/2*x)-2/x^2)/4 More generally pi/(2*x)*cosh((pi-|y|)*x)/sinh(pi*x) > =1/(2*x^2)+sum_{n=1}^oo cos(n*y)/(n^2+x^2) Hoping this helped, > Raymond ==== I've never been very good at combinatorics, so its no surprise that I can't get anywhere with the following problem: (lottery problem) 6 balls are drawn without replacement from an urn containing 49 balls labelled 1 to 49. A player selects 6 numbers and wins if 3 or more match. How many sets of 6 must a player choose to be absolutely sure of at least 1 win. Obviously this needs some strategy on the part of the player - no point in selecting the same set of 6 each time, on the other hand there does need to be some degree of overlap. (If you are familiar with the UK lottery its how much do you need to spend to guarantee a win). I've seen the answer and as far as I remember its about 64 (give or take a bit). In other words there are N sets of 6 which can be so chosen as to make any other set of 6 have at least 3 numbers in common with at least 1 of the set of N, where N (to be found) is something like 64. More difficult question: Lets call the answer N, how many different ways can this set of N be chosen. ==== I want to clarify what I'm trying to do.My goal is to show the following theorem: In a metric space X, a subset S is closed iff S^c is open. To do this I define first open ball and interior and limit point of a set: open ball U(x_0,r) with radius r and center x_0 is a set of points in X whose distance to a fixed point x_0 is less than some positive number r i.e. U(x_0,r)={Ax in X:d(x,x_0)0,Ay in X: y in U(x,r) => y in S. x is a limit point of set S iff for every r>0 there exists some y in X such that y is in the intersection of U(x,r) and S i.e. Ar>0,Ey in X:y in ( U(x,r) n S ). Then I define open and closed set: A set S is open iff every point of set S is an interior point of set S i.e. {Ax in S,Er>0,Ay in X:y in U(x,r) => y in S } A set S is closed iff every limit point of a set S is in S i.e. {Ax in X,Ar>0,Ey in X:Ar>0,Ey in X:y in ( U(x,r) n S ) => x in S } Do you agree? ==== In a metric space X, a subset S is closed iff S^c is open. Proof (=>): 1 S is closed 2 x in S^c --3 S^c is not open (assumption that gives contradiction) | 4 Ae>0,Ey in X:y in (U(x,e) n S) (negation of line 3) | 5 x in Cl(S) (by def and line 4) | 6 x in S (by line 1,5) | 7 contradiction (by line 2,6) -------------------------------------- 8 S^c is open Do you agree?Is everything correct? ==== > In a metric space X, a subset S is closed iff S^c is open. > Proof > (=>): > 1 S is closed > 2 x in S^c > --3 S^c is not open (assumption that gives contradiction) > | 4 Ae>0,Ey in X:y in (U(x,e) n S) (negation of line 3) > | 5 x in Cl(S) (by def and line 4) > | 6 x in S (by line 1,5) > | 7 contradiction (by line 2,6) > -------------------------------------- > 8 S^c is open Do you agree?Is everything correct? I am puzzled if by S^c you mean the compliment of S in X then the statement is trivially true. A Metric space is a special kind of topological space and from topology, closed sets are defined as the compliment of open sets. I think it would help if it was a little less cryptic. I have no problem until line 4. Line 4 says 'negation of line 3', you're not entitled to assume line 3 is false, also if line 4 implies that you think 'not open' means closed, that is not true either. 'Not open' does not mean closed. A set may be neither open nor closed. ==== In a metric space X, a subset S is closed iff S^c is open. Proof: (<=): 1 S^c is open 2 x in Cl(S) 3 Ae>0,Ey in X:y in (U(x,e) n S) (by definition and line 2) --4 x in S^c (Assumption that gives later contradiction) | 5 Ee>0,Ay in X:y in U(x,e) => y in S^c (because line 1 and def. of open set) | 6 contradiction (because lines 3,5) ----------------------------------------- 7 x in S 8 S is closed (by definition and lines 2,7) ==== SIMPLER VERSION solve > y''+ a*sqrt(y)=0 [...] (Assuming derivatives of the above WRT x ...) If a = 0 then the equation is trivial. Otherwise replace y by a^2.y, so we can work with y'' + sqrt(y) = 0 and save some typing. Denoting z := y'^2, so that z' = 2.y'.y'', your equation amounts to: (dz/dx)^2 = 4.y.z In this substituting: y, z = u(t).e^(2t), e^(2t) gives: (dt/dx)^2 = u(t) [1] Thus: e^(2t) = (dy/dx)^2 = (dy/dt)^2.(dt/dx)^2 = (du/dt + 2u)^2.e^(4t).u [2] Letting u = v^2 and taking square roots gives from [1] and [2] resp: dt/dx = v [1'] 2.v^2.d/dt(v.e^t) = 1 [2'] where the sign option of [1'] is absorbed into x, and that of [2'] into both v and x (into x to leave [1'] fixed). Multiplying [2'] by e^(2t) and taking w = v.e^t gives: 2.w^2.dw/dt = e^(2t) of which the solution, with arbitrary constant C, is: 4.w^3 = 3.e^(2t) + 3.C i.e. plugging w = v.e^t in the RHS and rearranging: (w/v)^2 = (4/3).w^3 - C i.e. with W, C' := (4/3).w, (4/3)^2.C (W/v)^2 = W^3 - C' I presume the trivial solution mentioned in other posts corresponds to the case C = 0. But the general symbolic solution requires elliptic functions: For some independent parameter s we have: W/v, W = dP/ds, P [3] where P is a Weierstrass function with suitable argument. But P' = W/v = (4/3).(w/v) = (4/3).e^t implies: t = (3/4).log(P') [4] whence: dt/ds = (3/4).P''/P' Also, by [1'] and [3] : dx/dt = 1/v = P'/P so that: dx/ds = dx/dt . dt/ds = P'/P . (3/4).(P''/P') = (3/4).(P''/P) so that: x = (3/4) . integral{P''/P)}.ds + D [5] and [4], [5] constitute a symbolic (although probably not very useful) solution of the original, in terms of the parameter s. --------------------------------------------------------------------------- John R Ramsden (jr@adslate`rm -rf *`.com) (remove spambot trap before use) --------------------------------------------------------------------------- A stockbroker is someone who invests your money until it's all gone. Woody Allen ==== A slug is a _unit_ of mass and/or inertia; in that it is a mathematical ratio of the particular net force [f] that will give to any object; body and/or mass of matter, an acceleration [a] that is numerically equal to its weight [w]; as measured with a weight-scale at any particular location on Earth's or some similar planet's terra firma surface: Mathematically: One slug = f/a = 1 (lbf) secÓ/foot = w/g = 32 (lbf) secÓ/32 feet...! ==== A gram is a _unit_ of mass and/or inertia; in that it is a mathematical ratio of the particular net force [f] that will give to any object; body and/or mass of matter, an acceleration [a] that is numerically equal to its weight [w]; as measured with a weight-scale at any particular location on Earth's or some similar planet's terra firma surface: Mathematically: One gram = f/a = 1 dyne secÓ/cm = w/g = 980 dynes secÓ/980 cm...! ==== A kilogram is a _unit_ of mass and/or inertia; in that it is a mathematical ratio of the particular net force [f] that will give to any object; body and/or mass of matter, an acceleration [a] that is numerically equal to its weight [w]; as measured with a weight-scale at any particular location on Earth's or some similar planet's terra firma surface: Mathematically: One kilogram = f/a = 1 newton secÓ/meter = w/g = 9.8 newtons secÓ/9.8 meters...! The kilogram artifact at the archives in Severs' is a physical representation of a kilogram; which may or may not be more mathematicaly precise than a cubic decimeter of water; which was formerly known as a liter, and weighed 9.8 newtons - about 2.2 (lbf). ==== > A kilogram is a _unit_ of mass and/or inertia; in that it is a mathematical > ratio of the particular net force [f] that will give to any object; body > and/or mass of matter, an acceleration [a] that is numerically equal to its > weight [w]; as measured with a weight-scale at any particular location on > Earth's or some similar planet's terra firma surface: The weight-scale must be calibrated somehow, therefore this cannot be used to realize the kilogram. Mathematically: One kilogram = f/a = 1 newton secÓ/meter = w/g = 9.8 newtons > secÓ/9.8 meters...! The kilogram artifact at the archives in Severs' is a physical > representation of a kilogram; which may or may not be more mathematicaly > precise than a cubic decimeter of water; which was formerly known as a > liter, and weighed 9.8 newtons - about 2.2 (lbf). The artifact after BIMP-cleaning (actually many countries have their own copies of the cylinder made at the same time as the one held in Severs, which define the kilogram used in those countries) is precicely the exact representation of a kilogram. If someone went and cleaned it up a bit too roughly and changed it, a number of other units (such as the Ampere and Newton) depending on it would change as well. That's the whole problem and the reason why they're looking for a new way to define kilogram, it looks like the kilogram is going to be defined based on the ampere, which could be defined using Quantum-Hall, Josephson and single electron phenomena. Also, it's faulty to say which may or may not be more mathematicaly precise than a cubic decimeter of water; that's the old definition of a liter and one cubic decimeter of water is not a kilogram anymore. -Sunny, currently taking a metrology course ==== > A kilogram is a _unit_ of mass and/or inertia; in that it is a mathematical > ratio of the particular net force [f] that will give to any object; body > and/or mass of matter, an acceleration [a] that is numerically equal to its > weight [w]; as measured with a weight-scale at any particular location on > Earth's or some similar planet's terra firma surface: The weight-scale must be calibrated somehow, therefore this cannot be > used to realize the kilogram. > I think Sunny that you'll find that weight scales were calibrated long before the kilogram ever was. The newton though didn't exist until somebody got to thinking - about a hundred years after the metric system's debute - duh uh maybe we should have a special unit for force(;^) Mathematically: One kilogram = f/a = 1 newton secÓ/meter = w/g = 9.8 newtons > secÓ/9.8 meters...! The kilogram artifact at the archives in Severs' is a physical > representation of a kilogram; which may or may not be more mathematicaly > precise than a cubic decimeter of water; which was formerly known as a > liter, and weighed 9.8 newtons - about 2.2 (lbf). The artifact after BIMP-cleaning (actually many countries have their > own copies of the cylinder made at the same time as the one held in > Severs, which define the kilogram used in those countries) is > precicely the exact representation of a kilogram. I beg to disagree, but it's a physical impossibility to make an artifact that's precisely the exact representation of a kilogram. Manufacturing tolerances and all that you know. They made several, none of which precisely conformed, and they kept the best and put it in a vault in Sevres' France. Many countries got the less precise ones. If someone went and > cleaned it up a bit too roughly and changed it, a number of other > units (such as the Ampere and Newton) depending on it would change as > well. That's the whole problem and the reason why they're looking for > a new way to define kilogram, it looks like the kilogram is going to > be defined based on the ampere, which could be defined using > Quantum-Hall, Josephson and single electron phenomena. Also, it's > faulty to say which may or may not be more mathematicaly precise than > a cubic decimeter of water; that's the old definition of a liter and > one cubic decimeter of water is not a kilogram anymore. > Yes you're right about the cubic decimeter: They decided _water_ wasn't good enough: One of the biggest mistakes they _ever_ made. > -Sunny, currently taking a metrology course Getting dogmatized more huh(:^! ==== A kilogram is a _unit_ of mass and/or inertia; in that it is a mathematical > ratio of the particular net force [f] that will give to any object; body > and/or mass of matter, an acceleration [a] that is numerically equal to its > weight [w]; as measured with a weight-scale at any particular location on > Earth's or some similar planet's terra firma surface: A kilogram is, by definition, a mass of a unique standard cylinder of platinum-iridium alloy kept at Sevres. Does your claim follow from the definition? Mathematically: One kilogram = f/a = 1 newton secÓ/meter = w/g = 9.8 newtons > secÓ/9.8 meters...! The kilogram artifact at the archives in Severs' is a physical > representation of a kilogram; which may or may not be more mathematicaly > precise than a cubic decimeter of water; which was formerly known as a > liter, and weighed 9.8 newtons - about 2.2 (lbf). -- G.C. ==== A kilogram is a _unit_ of mass and/or inertia; in that it is a mathematical > ratio of the particular net force [f] that will give to any object; body > and/or mass of matter, an acceleration [a] that is numerically equal to its > weight [w]; as measured with a weight-scale at any particular location on > Earth's or some similar planet's terra firma surface: A kilogram is, by definition, a mass of a unique standard cylinder of > platinum-iridium alloy kept at Sevres. Does your claim follow from the > definition? > No, my claim _leads to_, and is a more precise mathematical definition: They couldn't quite get the platinum-irridium artifact to weigh exactly 9.866 65 newtons - nohow - so they gave up and adopted it by proclamation: Now they're looking to redefine that adopted kilogram: Funny the slug doesn't need to be redefined(:^) Mathematically: One kilogram = f/a = 1 newton secÓ/meter = w/g = 9.8 newtons > secÓ/9.8 meters...! The kilogram artifact at the archives in Severs' is a physical > representation of a kilogram; which may or may not be more mathematicaly > precise than a cubic decimeter of water; which was formerly known as a > liter, and weighed 9.8 newtons - about 2.2 (lbf). > -- > G.C. ==== > A kilogram is a _unit_ of mass and/or inertia; in that it is a > mathematical > ratio of the particular net force [f] that will give to any object; body > and/or mass of matter, an acceleration [a] that is numerically equal to > its > weight [w]; as measured with a weight-scale at any particular location > on > Earth's or some similar planet's terra firma surface: A kilogram is, by definition, a mass of a unique standard cylinder of > platinum-iridium alloy kept at Sevres. Does your claim follow from the > definition? No, my claim _leads to_, and is a more precise mathematical definition: > They couldn't quite get the platinum-irridium artifact to weigh exactly > 9.866 65 newtons - nohow - so they gave up and adopted it by proclamation: Of course they couldn't, any lump of metal will weigh (i.e. be pulled down by gravity) a different amount in different places (because gravity varies). That would be silly for an _international_ standard! (Btw, I think you meant 9.80665 N.) -- G.C. ==== > message A kilogram is a _unit_ of mass and/or inertia; in that it is a > mathematical > ratio of the particular net force [f] that will give to any object; body > and/or mass of matter, an acceleration [a] that is numerically equal to > its > weight [w]; as measured with a weight-scale at any particular location > on > Earth's or some similar planet's terra firma surface: A kilogram is, by definition, a mass of a unique standard cylinder of > platinum-iridium alloy kept at Sevres. Does your claim follow from the > definition? No, my claim _leads to_, and is a more precise mathematical definition: > They couldn't quite get the platinum-irridium artifact to weigh exactly > 9.866 65 newtons - nohow - so they gave up and adopted it by proclamation: Of course they couldn't, any lump of metal will weigh (i.e. be pulled > down by gravity) a different amount in different places (because gravity > varies). That would be silly for an _international_ standard! > It's the _ratio_ of force to acceleration [f/a], and weight to the acceleration of free fall [w/g], that is (the) constant! Silly. > (Btw, I think you meant 9.80665 N.) > Yes thanks, you are right: I meant 9.80665 N. > -- > G.C. ==== > They couldn't quite get the platinum-irridium artifact to weigh exactly > 9.866 65 newtons - nohow - so they gave up and adopted it by proclamation: > Now they're looking to redefine that adopted kilogram: I believe it's 9.80665 N. (I may be wrong). m. ;-) ==== >> They couldn't quite get the platinum-irridium artifact to weigh exactly >> 9.866 65 newtons - nohow - so they gave up and adopted it by proclamation: >> Now they're looking to redefine that adopted kilogram: I believe it's 9.80665 N. (I may be wrong). m. ;-) You are wrong, but in a different way than you think you might be. That value has nothing whatsoever to do with the definition of a kilogram as a unit of mass. Furthermore, if you stick to SI units, any standard acceleration of free fall is worthless as tits on a boar. All that 9.80665 number is good for, as the magnitude of a standard acceleration of gravity (as it is commonly called, or more properly, of free fall), is to define a kilogram force, an obsolete unit that is not a part of the modern metric system, the International System of Units (SI). The same value, of course, is sometimes also used ot define a pound force, which doesn't have an official definition as the kilogram force does. If you don't use kilograms force or pounds force, you have no need for a standard acceleration of gravity. -- Gene Nygaard http://ourworld.compuserve.com/homepages/Gene_Nygaard/ It's not the things you don't know what gets you into trouble. It's the things you do know that just ain't so. Will Rogers ==== > A kilogram is a _unit_ of mass and/or inertia; in that it is a mathematical > ratio of the particular net force [f] that will give to any object; body > and/or mass of matter, an acceleration [a] that is numerically equal to its > weight [w]; as measured with a weight-scale at any particular location on > Earth's or some similar planet's terra firma surface: A kilogram is, by definition, a mass of a unique standard cylinder of ^ Sorry, meant the. > platinum-iridium alloy kept at Sevres. Does your claim follow from the > definition? > Mathematically: One kilogram = f/a = 1 newton secÓ/meter = w/g = 9.8 newtons > secÓ/9.8 meters...! The kilogram artifact at the archives in Severs' is a physical > representation of a kilogram; which may or may not be more mathematicaly > precise than a cubic decimeter of water; which was formerly known as a > liter, and weighed 9.8 newtons - about 2.2 (lbf). -- > G.C. -- G.C. ==== 2 books to look up: Forster, Riemannsche Flaechen (also available in English) will give you a clean, but technical definition and puts it in the context of sheaves (you may find basics on divisors as well). The other is Peter W Michor et al, Natural Operations in Diff Geom (available online, i forgot the URL, but try Google) and more general. As the others already said: it is a equivalence class through shrinking open neighbourhoods around the points a and f(a)=b (an inverse limit if you like 'abstract nonsense'). One has to care which functions are to be handeled (hence the protest to use Taylor series only: analytic is more than that). For holomorphic (=analytic) functions it goes _like_ this: Let be f1, f2 holomorphic functions both defined 'around' a. (f1,a) and (f2,a) give the same holomorphic germ iff b = f1(a) = f2(a) and there are arbitrary small open neighbourhoods around a where f1 and f2 coincide. Due to the identity theorem you can restrict to choose an open circle around a for f1 and f2. [By the 'open mapping' for holomorphic fcts it is a (connected) open neigh- bourhood of b, so you can do that even for morphisms between spaces in the category.] But on disks a _holomorphic_ fct is determined through its Taylor series. Thus you have (with some handwaving) got an isomorphism {holomorphic fct Germs in a} -> C, (f,a) |-> Taylor series where C{x} denotes those formal powerseries, which do have a positive radius of convergence in 0 (write t = x-a where x is your indeterminant for f). ==== intuitively, a germ of function encapsulates not just the value of the function at a point, but also it's behaviour near that point. for simplicity, let's work in a complex analytic manifold M. Let x in M. Let F={f:U -> C | f is holomorphic and U an open nbd of x}. define an equivalence relation on F by f~g if there exists a nbd V of x st f and g agree on V. then F/~ is the set of germs at x. in the complex case there is an exact correspondence between holomorphic fns and power series, so f and g define the same germ at x if and only if their taylor series at x are equal, i.e. if and only if all their derivatives agree at x. good reference might be gunning & rossi on several complex variables? tom ==== >intuitively, a germ of function encapsulates not just the value of the function >at a point, but also it's behaviour near that point. Where near has to be understood as arbitrarily near, of course. (I'm spelling that out for M. Dondi, not for you.) >for simplicity, let's work in a complex analytic manifold M. Let x in M. Let >F={f:U -> C | f is holomorphic and U an open nbd of x}. define an equivalence >relation on F by f~g if there exists a nbd V of x st f and g agree on V. >then F/~ is the set of germs at x. in the complex case there is an exact correspondence between holomorphic fns >and power series, so f and g define the same germ at x if and only if their >taylor series at x are equal, i.e. if and only if all their derivatives >agree at x. good reference might be gunning & rossi on several complex variables? Definitely a good reference, particularly in what I still assume to be M. Dondi's context. But it can't be emphasized enough that the situation is really, really different for non-analytic germs. For instance, the well-known examples of not-identically-zero smooth functions from (R,0) to (R,0) which have identically-zero Taylor series give examples of germs which are not captured by Taylor series alone. Lee Rudolph ==== intuitively, a germ of function encapsulates not just the value of the function >at a point, but also it's behaviour near that point. Where near has to be understood as arbitrarily near, of course. > (I'm spelling that out for M. Dondi, not for you.) for simplicity, let's work in a complex analytic manifold M. Let x in M. Let >F={f:U -> C | f is holomorphic and U an open nbd of x}. define an equivalence >relation on F by f~g if there exists a nbd V of x st f and g agree on V. >then F/~ is the set of germs at x. in the complex case there is an exact correspondence between holomorphic fns >and power series, so f and g define the same germ at x if and only if their >taylor series at x are equal, i.e. if and only if all their derivatives >agree at x. good reference might be gunning & rossi on several complex variables? Definitely a good reference, particularly in what I still assume > to be M. Dondi's context. But it can't be emphasized enough that > the situation is really, really different for non-analytic germs. > For instance, the well-known examples of not-identically-zero > smooth functions from (R,0) to (R,0) which have identically-zero > Taylor series give examples of germs which are not captured by > Taylor series alone. Lee Rudolph Another reference, for smooth germs in the real case, is Th. Brocker (with an umlaut over the o), Differentiable Germs and Catastrophes, London Mathematical Society Lecture Notes Series 17, Cambridge University Press, 1976 (translated by L. Lander). John Mitchell ==== >intuitively, a germ of function encapsulates not just the value of the function >>at a point, but also it's behaviour near that point. Where near has to be understood as arbitrarily near, of course. >(I'm spelling that out for M. Dondi, not for you.) Well, fortunately even if I don't (didn't!) know what a germ is, I got to that point... :-) >>good reference might be gunning & rossi on several complex variables? Definitely a good reference, particularly in what I still assume >to be M. Dondi's context. But it can't be emphasized enough that Then thank you both for the good reference! >the situation is really, really different for non-analytic germs. >For instance, the well-known examples of not-identically-zero >smooth functions from (R,0) to (R,0) which have identically-zero >Taylor series give examples of germs which are not captured by >Taylor series alone. I know I started this thread saying more or less Hey! I'm too ignorant: explain me what a germ is, but hey! I'm not that ignorant after all... and exp(-1/|x|) won't be my nightmare tonight... :-) 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 ==== >I'm studying some notes in small divisors problems, esp. complex >>analytical dynamics and I see the concept of germ is used as >>something very basic, that doesn't need to be defined... [...] >>I guess that *in this context* a germ f:(C,a)->(C,b) is simply an >>analytical function such that f(a)=b. Am I right? >and I do know stuff about complex analytical topology). To >me and mine, a germ of a complex-analytic function f:(C,0)->(C,0) >would determine, and be determined by, a convergent power series in >the complex variable z, and you'd get a germ of a complex-analytic It may well be that I'm really dumb... and feel ashamed since this will regard my thesis' work, but I can't understand *exactly* what the >function f:(C,a)->(C,b) in the usual way (by translations in the >domain and range). In other words (*in your context*), not only >must you have f(a)=b, but you must have the value of every derivative >of f at a. (For a general smooth function, that won't be enough >to determine the germ; it is, for a complex-analytic function.) >In yet other words, a germ of a complex-analytic function f:(C,0)->(C,0) >is what Ahlfors's textbook (for instance) calls a function element. Then could you be so kind and supply an actual definition? Since I spoke about analytic function without further specifications I guess that the key concept is, loosely speaking, locality: that is a germ, if I understand correctly, should be given by a couple consisting of a neighbourood of the given point *and* an analytic function defined on it satisfying the obvious equality. >It's an equivalence class. Two maps are identified if they >agree on some neighborhood of a. You can add two germs by >restricting them both to a neighborhood where they are both >defined, and adding them there. Then again it seems to me that this circumstance stresses the local charachter of a germ. Am I right? However this is slightly different As I said before, I'd like to read a definition and/or to be pointed to some good (introductory) reference... TIA, 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 ==== |Since I spoke about analytic function without further specifications |I guess that the key concept is, loosely speaking, locality: that is |a germ, if I understand correctly, should be given by a couple |consisting of a neighbourood of the given point *and* an analytic |function defined on it satisfying the obvious equality. Yes, and two are considered equal if there is a neighborhood of the point (contained in the neighborhoods on which each has been given) such that the restrictions of the two functions to that neighborhood are the same. In general a germ of an 'X' at the point p is an equivalence class of pairs consisting of a neighborhood U of p and an X defined on U, where the equivalence relation is (U,x)~(U',x') if there exists a neighborhood V of p such that V is a subset of U and U', and the restriction of x to V equals the restriction of x' to V. This definition only makes sense in situations where we have such notions as an X defined on an open set, and the restriction of one to a smaller open set. A precise definition is then usually presented in the context of sheaf theory. A sheaf is a kind of structure which defines for each open set the set of sections of the sheaf (typically a particular kind of function) defined on the open set, and restriction functions from those defined on one open set to those defined on an open subset, where they are required to satisfy a few axioms. The thing I called an equivalence relation above, for instance, only has to be an equivalence relation because I'm implicitly assuming that if we take an x defined on U and restrict it to V then restrict it to W, we get the same result as if we restricted it straight from U to W. I guess actually we don't need all the axioms of a sheaf to hold; we can define germs of sections of presheaves.... That theory is not so hard, but it's a little abstract, and presumably you don't need that much generality, which I think is why people have been telling you how the idea works once it's been specialized to the case of germs of analytic functions. Germs of analytic functions are special because two analytic functions defined on a connected open set U that agree with each other on a neighborhood of a point in U agree with each other everywhere in U. So the local aspect of germs of analytic functions makes a lot less of a difference than it does for, say, germs of continuous functions. As someone else pointed out, a power series with a positive radius of convergence at the point defines a germ of an analytic function, each germ has such an associated power series, and two germs are equal if and only if they have the same power series. In your case it seems you're considering the germs of functions which have a specified value at the given point, which simply would mean that the constant term in the power series is given. Keith Ramsay X-Cise: tanbanso@iinet.net.au X-CompuServe-Customer: Yes X-Coriate: admin@interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Punge: Micro$oft ==== at 08:24 PM, Michele Dondi said: >I searched google with some suitable keys (IMHO - eg: >'germ+dynamical system'), but I couldn't find anything useful, germ+sheaves or germs+differential geometry might get you more hits. As another poster explained, a germ is an equivalence class of local functions. -- Shmuel (Seymour J.) Metz, SysProg and JOAT not reply to spamtrap@library.lspace.org ==== I am working out some problems from Rudin's Analysis. At the end of Chapter 2, in the excercise section (No. 13, p. 44), we have-- 13. Construct a COMPACT set of REAL numbers whose LIMIT POINTS form a COUNTABLE set. [I have typed the important words in caps] I came up with the set A = {0} U {1/n + 1/(n+1)} Where n is an element of Z+ The limit points of A are, L = {3/2, 5/6, 7/12, .......0} My question are:-- 1. Is the set A compact?--because it is bounded between [0, 3/2] 2. Does the set L have a bijection with Z+?--I believe it does. 3. Are there any other (prettier) sets that would satisfy the above conditions? This is my first rigorous course in analysis, and I should appreciate any help. FWH ==== > 13. Construct a COMPACT set of REAL numbers whose LIMIT POINTS form a > COUNTABLE set. I came up with the set A = {0} U {1/n + 1/(n+1)} > Where n is an element of Z+ > A = {0} / { 1/n + 1/(n+1) | n in N } doesn't cut it as others have commented. Indeed lim(n->oo) 1/n + 1/(n+1) = 0 and combine them into a single set. > The limit points of A are, L = {3/2, 5/6, 7/12, .......0} My question are:-- > 1. Is the set A compact?--because it is bounded between [0, 3/2] > Yes, A is compact but only because A is both bounded and closed! ==== > I am working out some problems from Rudin's Analysis. At the end of > Chapter 2, in the excercise section (No. 13, p. 44), we have-- 13. Construct a COMPACT set of REAL numbers whose LIMIT POINTS form a > COUNTABLE set. [I have typed the important words in caps] I came up with the set A = {0} U {1/n + 1/(n+1)} Where n is an element of Z+ The limit points of A are, L = {3/2, 5/6, 7/12, .......0} I see only one limit point, namely 0, the limit of the sequence 1/n + 1/(n+1). ==== > I am working out some problems from Rudin's Analysis. At the end of > Chapter 2, in the excercise section (No. 13, p. 44), we have-- 13. Construct a COMPACT set of REAL numbers whose LIMIT POINTS form a > COUNTABLE set. [I have typed the important words in caps] I came up with the set A = {0} U {1/n + 1/(n+1)} Where n is an element of Z+ The limit points of A are, L = {3/2, 5/6, 7/12, .......0} Wrong. The only limit point of A is 0. But you're right about the rest: A is compact and countable. Jose Carlos Santos ==== > I'm going to make my question, and then I'll tell you my proposed solution > (which my professor doesn't quite agree with me) Question: Imagine you have to draw 6 cards from a regular deck of 10 cards. > We know that this deck is composed by 4 aces and 6 other unknown cards. What > is the probability of: > a) An Ace showing up FIRST at the 1st position (first draw) > b) An Ace showing up first at the 2nd position > c) An Ace showing up first at the 3rd position > d) An Ace showing up first at the 4th position > e) An Ace showing up first at the 5th position > f) An Ace showing up first at the 6th position > g) An Ace showing up first at the 7th position and on Let me remember you that if an ace show up first in the second position, > then we know for sure that it doesn't show in the 1st... My solution: > a) easy one: 4/10 > b) Pr(not Ace in pos 1) x Pr(first ace in 2nd pos) > Pr(not Ace in pos 1) = 6/10 > Pr(first ace in 2nd pos): > ok, my line of thought is that if an ace shows up in the second pos, > then > I only have 3 other aces in the remaining 8 positions - nchoosek(8,3) > I know that are a total of nchoosek(9,4) ways to organize 4 aces in 9 > positions, > so the solution is: > 6/10 x [nchoosek(8,3)/nchoosek(9,4)] > c) following the same principle: > Pr(not ace in 1) x Pr(not ace in 2) x Pr(first ace in 3rd pos) > 6/10 x 5/9 x [nchoosek(7,3)/nchoosek(8,4)] > d) Pr(not ace in 1) x Pr(not ace in 2) x Pr(not ace in 3) x Pr(first ace in > 4th pos) > 6/10 x 5/9 x 4/8 x [nchoosek(6,3)/nchoosek(7,4)] > e) Pr(not A in 1) x Pr(not A in 2) x Pr(not A in 3) x Pr(not A in 4) x > Pr(first ace in 5th pos) > 6/10 x 5/9 x 4/8 x 3/7 [nchoosek(5,3)/nchoosek(6,4)] > f) Pr(not A in 1) x Pr(not A in 2) x Pr(not A in 3) x Pr(not A in 4) x > Pr(not A in 5) x > Pr(first ace in 6th pos) > 6/10 x 5/9 x 4/8 x 3/7 x 2/6 x [nchoosek(4,3)/nchoosek(5,4)] > f) Pr(not A in 1) x Pr(not A in 2) x Pr(not A in 3) x Pr(not A in 4) x > Pr(not A in 5) x > Pr(not A in 6) x Pr(first ace in 7th pos) > 6/10 x 5/9 x 4/8 x 3/7 x 2/6 x 1/4 x [nchoosek(3,3)/nchoosek(4,4)] > the nchoosek here tell us that if we don't find an ace in any of the > first 6 cards, then > it will be in the 7th one for sure > g) 0, since Pr(not A in 7) = 0, same for all other subsequent positions > Does it make sense? Can I consider the various probabilities of an ace not > showing as independent of one each other? > Appreciate Padu I offer a very simplistic solution: Consider each draw a separate problem. So on the kth draw you have n = (7-k) non aces. On the third draw -say, n = 4, so P = 4/8, etc. ==== > I'm going to make my question, and then I'll tell you my proposed solution > (which my professor doesn't quite agree with me) Question: Imagine you have to draw 6 cards from a regular deck of 10 cards. > We know that this deck is composed by 4 aces and 6 other unknown cards. > What > is the probability of: > a) An Ace showing up FIRST at the 1st position (first draw) > b) An Ace showing up first at the 2nd position > c) An Ace showing up first at the 3rd position > d) An Ace showing up first at the 4th position > e) An Ace showing up first at the 5th position > f) An Ace showing up first at the 6th position > g) An Ace showing up first at the 7th position and on Let me remember you that if an ace show up first in the second position, > then we know for sure that it doesn't show in the 1st... My solution: > a) easy one: 4/10 > b) Pr(not Ace in pos 1) x Pr(first ace in 2nd pos) > Pr(not Ace in pos 1) = 6/10 > Pr(first ace in 2nd pos): > ok, my line of thought is that if an ace shows up in the second pos, > then > I only have 3 other aces in the remaining 8 positions - nchoosek(8,3) > I know that are a total of nchoosek(9,4) ways to organize 4 aces in 9 > positions, > so the solution is: > 6/10 x [nchoosek(8,3)/nchoosek(9,4)] The standard way, and much easier way, of dealing with this sort of problem is with conditional probabilities. The probability of getting the first ace on draw 2 equals the probability of 6/10, of a non-ace (N) on draw 1, times the conditional probability of 4/9, of ace on draw 2 GIVEN non-ace on draw 1, or (6/10)*(4/9) = 24/90 = 4/15. Or, ith the obvious meanings, P(C|D) meaning conditional probability of C given that D occurs: P(NA) = P(N)*P(NA|N) Similarly, for the first ace on the third draw: P(NNA) = P(N)*P(NN|N)*P(NNA|NN) = (6/10)*(5/9)*(4/8)= 1/6 and so on. ==== So it's also misspelt? Misspelt is incorrect; misspelled is correct!!! ==== > So it's also misspelt? Misspelt is incorrect; misspelled is correct!!! How can you have any kind of intelligent dialogue if you can't even spell simple words like misspelled? My advice: Give up! ==== > So it's also misspelt? Misspelt is incorrect; misspelled is correct!!! How can you have any kind of intelligent dialogue if you can't even > spell simple words like misspelled? My advice: Give up! Whut maks yew thank thet spealling rellie is the thang whut determins whethur a dye-a-log is Ineeligetn er Knot? I allus thawt thet Kontent wuz the delineating facter!! RJ Peez ==== >> So it's also misspelt? >> Misspelt is incorrect; misspelled is correct!!! How can you have any kind of intelligent dialogue if you can't even >spell simple words like misspelled? My advice: Give up! spelt2 ( P ) Pronunciation Key (splt) v. A past tense and a past participle of spell. Source: The American Heritage¨ Dictionary of the English Language, Fourth Edition ==== So it's also misspelt? Misspelt is incorrect; misspelled is correct!!! >>How can you have any kind of intelligent dialogue if you can't even >>spell simple words like misspelled? My advice: Give up! > spelt2 ( P ) Pronunciation Key (splt) > v. > A past tense and a past participle of spell. Source: The American Heritage¨ Dictionary of the English Language, > Fourth Edition I thought it was a valid (if fallen out of common use) form... -- Mark K. Bilbo ==== > So it's also misspelt? Misspelt is incorrect; misspelled is correct!!! That was my point, anyone who corrects God's spelling, mine, will always make a mistake. I have yet to gather all the links, quite amazing. It works both ways I'm afraid! Note the single exclamation point. Herc ==== 3-4 minutes (about 3 MB/hr) for more than two days, and > counting. by posting this & other messages using a disposable address is for use only in newsgroup postings, and I'm being to test whether worms will actually harvest it from newsgroup postings alone. --r.e.s. test.1@mindspring.com ==== > address is for use only in newsgroup postings, and I'm > being to test whether worms will actually harvest it from > newsgroup postings alone. Has been proven already. Somebody I know did this kind of test and within what nntp servers are used to harvest addresses. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ ==== > address is for use only in newsgroup postings, and I'm > being to test whether worms will actually harvest it from > newsgroup postings alone. Has been proven already. Somebody I know did this kind of test and within > what nntp servers are used to harvest addresses. Right. I did this to get first-hand evidence for someone who was sceptical. The flood of Sven-generated/infected --r.e.s. ==== who was sceptical. The flood of Sven-generated/infected One hour? Interesting. Check time of day. My system starts getting attacked at 7:00 every morning (eastern daylight time) via a certain ISP. It used to happen only when connected to the newsgroup system. After a week? or trying to blast just logged in and not connected to anything. I don't get the attacks through AOL....yet. /BAH ==== I'm not a mathematician, so I probably misuse a number of terms > in this problem. I'm taking calc I and statistics right now. I'm my stats class yesterday, the instructor covered how to calculate > the total possible number of outcomes in a sample set. It's painfully > obvious, of course. If you have 2 6-sided dice, the number of > possible outcomes is 6*6. I asked how to calculate the total > possible number of outcomes if you assumed that a 3 on die 1 and a > 5 on a die 2 was the same as a 5 on die 1 and a 3 on die 2. He > didn't know the answer. (I'm taking the class at a community > college, and this guy's an electrical engineer.) So I went home and started to work on the problem. This is what I > came up with. It's not homework. I was just curious. http://www.technolalia.com/~ndronen/math-problem.txt Is there a known equation for what I'm trying to express? I have > the seeds of the solution, but I'm not sure how to approach this > mathematically. I'm a programmer, and I could write a program to > compute this, but I'm curious how to express it mathematically. > Nicholas May we presume that the second number is always equal to or greater than the first number? Then, If the first number is 1 there are 6 possible outcomes If the first number is 2 there are 5 possible outcomes If the first number is 3 there are 4 possible outcomes If the first number is 4 there are 3 possible outcomes If the first number is 5 there are 2 possible outcomes If the first number is 6 there are 1 possible outcomes Total outcomes = 21, which is (6*(6+1))/2 or generally (n*(n+1))/2. ==== [Please forgive me if this is a repeat. I think I may have asked this same question long ago, but I can't find it in a google search.] Let F be the smallest subfield of R which contains Q and is closed under power operation, x, y in F ==> x^y in F when x^y takes on a real value. (Define it to be the positive value when there are two possible values for the power.) Since not all algebraics can be expressed via radicals, F is strictly contained in the algebraics, no? Is there a name for F? -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu ==== > Let F be the smallest subfield of R which contains Q and is > closed under power operation, x, y in F ==> x^y in F when x^y > takes on a real value. (Define it to be the positive value when there > are two possible values for the power.) Since not all algebraics can > be expressed via radicals, F is strictly contained in the > algebraics, no? Is there a name for F? As ANN and WDH has pointed out, this is not a subfield of the algebraics. Mea culpa. Still, -Does it still have a name? It is countable, no? -Different definition: Let G be the smallest subfield of R which contains 1 and has the property that x in G, n in N, x and n positive ==> x^(1/n) in G. That is certainly (?) contained in the real algebraics. Name? -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu ==== > [Please forgive me if this is a repeat. I think I may have asked this > same question long ago, but I can't find it in a google search.] > > Let F be the smallest subfield of R which contains Q and is closed > under power operation, x, y in F ==> x^y in F when x^y takes on a > real value. (Define it to be the positive value when there are two > possible values for the power.) Since not all algebraics can be > expressed via radicals, F is strictly contained in the algebraics, > no? Is there a name for F? > No, F is not contained in the field of algebraic numbers, since by the Gelfond-Schneider Theorem, given x algebraic & not equal to 0 or 1, and y algebraic and irrational, then x^y is transcendental. So, for instance, suppose you take x = 2^(1/2), which is mandated to be in your field F, and y = 3^(1/2), also in that field, then one has x^y being transcendental. then again, I'm not really an expert in the field, er, area. Dale. ==== > [Please forgive me if this is a repeat. I think I may have asked this > same question long ago, but I can't find it in a google search.] Let F be the smallest subfield of R which contains Q and is closed > under power operation, x, y in F ==> x^y in F when x^y takes on a > real value. (Define it to be the positive value when there are two > possible values for the power.) Since not all algebraics can be > expressed via radicals, F is strictly contained in the algebraics, > no? Is there a name for F? But then F contains things like 2^(2^(1/2)), which is not algebraic ? ==== There are 3 ants at 3 corners of a triangle, they randomly start moving towards another corner.. what is the probability that they don't collide. ==== > There are 3 ants at 3 corners of a triangle, they randomly start > moving towards another corner.. what is the probability that they > don't collide. There are only two possibilities for the ants not to collide: they all move left or they all move right. I assume that the possibility for the ants to move left or right are all equal, hence 1/2. Then the possibility for them not to collide is p=2*((1/2)*(1/2)*(1/2))=1/4 ==== You have 8 balls. One of them is defective and weighs less than others. You have a balance to measure balls against each other. In 2 weighings how do you find the defective one? ==== > You have 8 balls. Aaaaarrghhh! -- Chris ==== > You have 8 balls. One of them is defective and weighs less than > others. You have a balance to measure balls against each other. In 2 > weighings how do you find the defective one? Once you've warmed up on that one, here's another: You have 12 balls. One of them is defective and is either heavier or lighter than the others. You have a balance to measure balls against each other. In 3 weighings how do you find the defective one? -- Bob Day ==== > You have 8 balls. One of them is defective and weighs less than > others. You have a balance to measure balls against each other. In 2 > weighings how do you find the defective one? Once you've warmed up on that one, here's another: > You have 12 balls. One of them is defective and is either > heavier or lighter than the others. You have a balance to > measure balls against each other. In 3 weighings how do > you find the defective one? And then finally to W weighings and (3^W-3)/2 balls ;-) http://groups.google.com/groups?&threadm=7O6t9.172902$8o4.27741@afrodite.tele net-ops.be Dirk Vdm ==== > You have 8 balls. One of them is defective and weighs less than > others. You have a balance to measure balls against each other. In 2 > weighings how do you find the defective one? 3 on one side of scale 3 on other. 2 aside. If the 6 do not balance, take the lower weighted 3. Pick 2 from that group for second weighing, etc. If 6 do balance, take remaining 2 and weigh. (Could you not start with 9?) Bill ==== 1/n - n and n - 1/n have a simple numerical relationship, no matter whether n <> 1. > 1/n - n/1 Right, I _know_ that! But what if you reverse them: Will n/1 - 1/n be the > same? There's always a jackleg or two in any newsgroup. ==== > 1/n - n and n - 1/n have a simple numerical relationship, > no matter whether n <> 1. > Suposin' it's zero Brian? Cut< ==== > This would be a good question for our great uncle! What's the difference between dividing the number [1] by a number [n], and > dividing a number [n] by the number [1]? Forget your question, S*head. Let's deal with only one of them. Inquiring minds want to know the difference between these, in Shead Algebra: a. dividing the number [1] by a number [n] b. dividing the number 1 by a number [n] c. dividing the number [1] by a number n d. dividing the number 1 by a number n e. dividing the number [1] by a number (n) f. dividing the number (1) by a number [n] g. dividing the number (1) by a number (n) h. dividing the number (1) by a number n i. dividing the number 1 by a number (n) and also the difference between [1]/[n] and [[1]/[n]] and [1/n], etc. Gene Nygaard Time flies like an arrow; fruit flies like a banana. ==== > This would be a good question for our great uncle! What's the difference between dividing the number [1] by a number [n], and > dividing a number [n] by the number [1]? Forget your question, S*head. Let's deal with only one of them. > Inquiring minds want to know the difference between these, in Shead > Algebra: You have forgottent the substitution of all the words with a / symbol. > a. dividing the number [1] by a number [n] > b. dividing the number 1 by a number [n] > c. dividing the number [1] by a number n > d. dividing the number 1 by a number n > e. dividing the number [1] by a number (n) > f. dividing the number (1) by a number [n] > g. dividing the number (1) by a number (n) > h. dividing the number (1) by a number n > i. dividing the number 1 by a number (n) and also the difference between [1]/[n] and [[1]/[n]] and [1/n], etc. Gene Nygaard > Time flies like an arrow; > fruit flies like a banana. ==== This would be a good question for our great uncle! What's the difference between dividing the number [1] by a number [n], > and > dividing a number [n] by the number [1]? Forget your question, S*head. Let's deal with only one of them. > Inquiring minds want to know the difference between these, in Shead > Algebra: You have forgottent the substitution of all the words with a / symbol. > a. dividing the number [1] by a number [n] > b. dividing the number 1 by a number [n] > c. dividing the number [1] by a number n > d. dividing the number 1 by a number n > e. dividing the number [1] by a number (n) > f. dividing the number (1) by a number [n] > g. dividing the number (1) by a number (n) > h. dividing the number (1) by a number n > i. dividing the number 1 by a number (n) and also the difference between [1]/[n] and [[1]/[n]] and [1/n], etc. Gene Nygaard > As per usual Gene is exaggerating the parentheses; brackets, and lack thereof into a big deal; to make me look like a bumbling idiot by _his_ asking such a dumb question. I see little or no difference in those examples he presents; but perhaps you do. The difference _I'm_ seeking an answer to is if a number [n] is other than 1; then the answer to 1/n will not be 1; it will vary all over the place: 1/zero is infinite; 1/0.1 is 10; 1/0.01 is 100; 1/0.001 is 1000...: 1/1.1 is 0.91; 1/2 is 0.5; 1/4 is 0.25, and 1/40 is 0.025.... I see no inverse in sight, and _that's_ the point of the whole darn thing! ==== > [snip] > The difference _I'm_ seeking an answer to is if a number [n] is other than > 1; then the answer to 1/n will not be 1; it will vary all over the place: If [n] is a number, then 1/n is also a number and will not vary all over the place. For each specific value of 'n' there is a specific value of 1/n (other than n = 0). > 1/zero is infinite; 1/0.1 is 10; 1/0.01 is 100; 1/0.001 is 1000...: 1/1.1 is > 0.91; 1/2 is 0.5; 1/4 is 0.25, and 1/40 is 0.025.... I see no inverse in sight, and _that's_ the point of the whole darn thing! Inverse? Why is that the point? Do you see any reciprocals? (I see one for each value of 'n'.) -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com ==== > [snip] The difference _I'm_ seeking an answer to is if a number [n] is other than > 1; then the answer to 1/n will not be 1; it will vary all over the place: If [n] is a number, then 1/n is also a number and will not vary all over the > place. For each specific value of 'n' there is a specific value of 1/n (other > than n = 0). 1/zero is infinite; 1/0.1 is 10; 1/0.01 is 100; 1/0.001 is 1000...: 1/1.1 is > 0.91; 1/2 is 0.5; 1/4 is 0.25, and 1/40 is 0.025.... I see no inverse in sight, and _that's_ the point of the whole darn thing! Inverse? Why is that the point? Do you see any reciprocals? (I see one for each > value of 'n'.) > Yes; it might have been better had I used m instead of n; since I'm talkin' 'bout the inverse of mass; which is a constant for any given body. > -- > There are two things you must never attempt to prove: the unprovable -- and the > obvious. I'm of the opinion that we at least _attempt_ to prove the obvious. > -- > Democracy: The triumph of popularity over principle. Where the unscupulous can lead the meek and unwary astray from their freedom, because popularity breeds self important evil and sordid behavior. > -- > http://www.crbond.com ==== > [snip] >> The difference _I'm_ seeking an answer to is if a number [n] is other >than >> 1; then the answer to 1/n will not be 1; it will vary all over the >place: >> If [n] is a number, then 1/n is also a number and will not vary all over >the >> place. For each specific value of 'n' there is a specific value of 1/n >(other >> than n = 0). >> 1/zero is infinite; 1/0.1 is 10; 1/0.01 is 100; 1/0.001 is 1000...: >1/1.1 is >> 0.91; 1/2 is 0.5; 1/4 is 0.25, and 1/40 is 0.025.... >> I see no inverse in sight, and _that's_ the point of the whole darn >thing! >> Inverse? Why is that the point? Do you see any reciprocals? (I see one for >each >> value of 'n'.) > >Yes; it might have been better had I used m instead of n; since I'm talkin' >'bout the inverse of mass; which is a constant for any given body. Like I said before, there's no reason for me to try to make you look like a bumbling idiot, Donald. You do just fine all on your own! ==== ... > 1/zero is infinite; 1/0.1 is 10; 1/0.01 is 100; 1/0.001 is 1000...: > 1/1.1 is 0.91; 1/2 is 0.5; 1/4 is 0.25, and 1/40 is 0.025.... Two errors. 1/1.1 != 0.91. That is only an approximation. 1/zero is infinite has no immediate place in mathematics. You have mathematical viewpoint there is not such thing as infinite. > I see no inverse in sight, and _that's_ the point of the whole darn > thing! > > Inverse? Why is that the point? Do you see any reciprocals? (I see one for > each value of 'n'.) > > > Yes; it might have been better had I used m instead of n; since I'm talkin' > 'bout the inverse of mass; which is a constant for any given body. Eh? You were saying you were not seeing inverses. In mathematics, within the reals, the inverse, or reciprocal, to a number n is 1/n. You can use the notation 1/2 or 0.5 for the inverse, or reciprocal, of 2, mathematically there is no difference. I could even write 0.1, or 0.3 when the reader knows what base I use. What mass has to do with this (and the change of 'n' to 'm') escapes me. If the inverse of mass is a constant for a given body, I would think that the mass (its inverse or reciprocal) also would be constant. Are you claiming it is not? But, welcome back to sci.math. We just lost our most notorious crank; it is always good to have a substitute. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ ==== > ... > 1/zero is infinite; 1/0.1 is 10; 1/0.01 is 100; 1/0.001 is 1000...: > 1/1.1 is 0.91; 1/2 is 0.5; 1/4 is 0.25, and 1/40 is 0.025.... Two errors. 1/1.1 != 0.91. That is only an approximation. > 1/zero is infinite has no immediate place in mathematics. You have > mathematical viewpoint there is not such thing as infinite. I see no inverse in sight, and _that's_ the point of the whole darn > thing! Inverse? Why is that the point? Do you see any reciprocals? (I see one for > each value of 'n'.) > Yes; it might have been better had I used m instead of n; since I'm talkin' > 'bout the inverse of mass; which is a constant for any given body. Eh? You were saying you were not seeing inverses. In mathematics, within > the reals, the inverse, or reciprocal, to a number n is 1/n. You can use > the notation 1/2 or 0.5 for the inverse, or reciprocal, of 2, mathematically > there is no difference. I could even write 0.1, or 0.3 when the reader > knows what base I use. What mass has to do with this (and the change of 'n' to 'm') escapes me. > If the inverse of mass is a constant for a given body, I would think that > the mass (its inverse or reciprocal) also would be constant. Are you > claiming it is not? But, welcome back to sci.math. We just lost our most notorious crank; it > is always good to have a substitute. > -- > dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 > home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ realize the importance of math to physics; that physics couldn't exist without it. All I'm claiming is that a body's mass [m] is constant; but unless its value is one [1], like one kilogram, or one gram, or maybe one slug, its inverse is not 1/m: I don't think. ==== > realize the importance of math to physics; that physics couldn't exist > without it. > > All I'm claiming is that a body's mass [m] is constant; but unless its value > is one [1], like one kilogram, or one gram, or maybe one slug, its inverse > is not 1/m: I don't think. Well, I do not know. I am missing something I think. If something is m kilogram, the inverse of that is 1/m kilogram^-1 I would say. But in mathematics we work unitless. So the inverse of m is 1/m. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ ==== to > realize the importance of math to physics; that physics couldn't exist > without it. All I'm claiming is that a body's mass [m] is constant; but unless its value > is one [1], like one kilogram, or one gram, or maybe one slug, its inverse > is not 1/m: I don't think. Well, I do not know. I am missing something I think. If something > is m kilogram, the inverse of that is 1/m kilogram^-1 I would say. > But in mathematics we work unitless. If you work unitless, you're working witless: Pure numbers don't mean a darn thing without units! So the inverse of m is 1/m. > -- > dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 > home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ ==== >Message-id: > realize the importance of math to physics; that physics couldn't exist >> without it. >> All I'm claiming is that a body's mass [m] is constant; but unless its >value >> is one [1], like one kilogram, or one gram, or maybe one slug, its >inverse >> is not 1/m: I don't think. >> Well, I do not know. I am missing something I think. If something >> is m kilogram, the inverse of that is 1/m kilogram^-1 I would say. >> But in mathematics we work unitless. If you work unitless, you're working witless: Pure numbers don't mean a darn >thing without units! What are the units of pH? So the inverse of m is 1/m. >> -- >> dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, >+31205924131 >> home: bovenover 215, 1025 jn amsterdam, nederland; >http://www.cwi.nl/~dik/ -- Mensanator 2 of Clubs http://members.aol.com/mensanator666/2ofclubs/2ofclubs.htm ==== > This would be a good question for our great uncle! >> What's the difference between dividing the number [1] by a number >[n], >> and >> dividing a number [n] by the number [1]? >> Forget your question, S*head. Let's deal with only one of them. >> Inquiring minds want to know the difference between these, in Shead >> Algebra: >> You have forgottent the substitution of all the words with a / symbol. >> a. dividing the number [1] by a number [n] >> b. dividing the number 1 by a number [n] >> c. dividing the number [1] by a number n >> d. dividing the number 1 by a number n >> e. dividing the number [1] by a number (n) >> f. dividing the number (1) by a number [n] >> g. dividing the number (1) by a number (n) >> h. dividing the number (1) by a number n >> i. dividing the number 1 by a number (n) >> and also the difference between [1]/[n] and [[1]/[n]] and [1/n], etc. >> Gene Nygaard >As per usual Gene is exaggerating the parentheses; brackets, and lack >thereof into a big deal; to make me look like a bumbling idiot by _his_ >asking such a dumb question. I see little or no difference in those examples >he presents; but perhaps you do. The difference _I'm_ seeking an answer to is if a number [n] is other than >1; then the answer to 1/n will not be 1; it will vary all over the place: 1/zero is infinite; 1/0.1 is 10; 1/0.01 is 100; 1/0.001 is 1000...: 1/1.1 is >0.91; 1/2 is 0.5; 1/4 is 0.25, and 1/40 is 0.025.... I see no inverse in sight, and _that's_ the point of the whole darn thing! There's no reason for me to try to make you look like a bumbling idiot, Donald. You do just fine all on your own! Gene Nygaard http://ourworld.compuserve.com/homepages/Gene_Nygaard/ ==== PI = 3.141592654 HORUS EYE = 64 EYE IN THE CIRCLE, PI x 64 PI x 64 = 201.0619299 = 201 + 0.0619299 [ 1/.0619299 = 16.14172.. ] = 201 + 1/17 + 0.003106371 [ 1/.003106371 = 321.919091.. ] = 201 + 1/17 + 1/322 + 0.00000078 =.= 201 + 1/17 + 1/322 322 = 7 x 23 x 2 = G.W.B 17 = Q = KEY When PI x 64 is represented as an egyptian fraction, as above, and the first 3 numbers (triangle count) in the expansion are taken, we get a very good approximation to pi, even better than the infamous 355/113 value. This approx to PI we shall call the Skull & Bones PI, since 322 is the key number in that order. The value PI x 64 = 201 + 1/17 + 1/322, gives PI = 3.141592642, compare PI = 355/113 = 3.141592920, and the true PI = 3.141592654, thus the Skull & Bones PI is just a little more accurate than 355/113, giving the correct value of the 7th decimal place, while 355/113 is only accurate up to the 6th decimal place. 7 = G, the seventh letter in the alphabeth is the infamous Freemason's G. Well now, here's the really interesting part. The Skull & Bones number 322 factors into three prime numbers 322 = 7.23.2 = G.W.B. Maybe that's why people think G.W.B. is a member of Skull and Bones! The S&B key is his initials. However, we all know that PI is the number 666. The key PI = 355/113, which is the holy value, gives us 666 when we reverse the digits in one number and add to the other, 355 => 553 + 113 = 666 113 => 311 + 355 = 666 So, if PI is 666, and the number of his name G.W.B is encoded in the key Skull and Bones PI, accurate to the same 355/113, that gives us 666, but just a little more accurate that we get 7 or G decimals, then who is G.W.B. ? NOVUS ORDO SECLORUM has 17 letters, 17 = Q, the Key here is in the interpretation of the banner. Does it look like it is rising up to the sky? Is America to be lifted up? Maybe. Depending on who is doing the looking. i ==== > The only place where logical inference breaks down is > inside Longley's head. > Hmmm, I'm in my early 40's. What should I do with the > rest of my life? Maybe I should write up my opinions on a > web site and then repeat my 8 favorite words thousands of > times in a newsgroup? Duh, that sounds logical! Then > everybody will know that it wasn't my fault, it was those > idiot folk psychologist's fault! > Hmmm, now I'm about 50 and nobody has agreed with my > opinions despite thousands of repetitions. Duh, logic must > have broken down! Quine was right! Everybody else's > language is worse than I deduced! Everybody in newsgroups > is folk psychological too! How to solve this problem? > Let me analyze more extensionally! Well, I could spend my > 50's posting another 8,000 repetitions. Duh, that sounds > like a good decision! And if that doesn't work, as a > backup I still have my 60's! Excellent and accurate analysis work, Larry. Usenet (and the larger text Net) has its David Hayes, its Arthur T. Murray, its James Harris, its Stan Rothwell, its Robert Vienneau, and an unnamed cast of thousands more exactly like them who plague its various newsgroups, usually one or two toads to a hole, and your description fits each of them equally well. Your Mister Longley joins a proud Net tradition, widely crossposting his uninteresting and unvarying opinions where they are inappropriate, unwanted, and reviled, without a bit of normal human reaction to such a reception (such as trying to figure out just where one has gone wrong and correcting it), secure in his conviction that he alone is the appointed Guardian of The Truth. And as you imply, he will doubtless still be mumbling on into the jaws of hundreds of thousands of killfiles, long after all his teeth are gone and his mind had begun to wander aimlessly from its rut, captive of habit and conviction both. xanthian. -- ==== The problem with your analysis is, of course, that occasionally such people ARE right. There seems to be a complete inability around here for self-appointed critics like yourself to escape the notion that popularity is the arbiter of the scientific or philosophical worth of a position. This is ironic given your implied self-description as non-dogmatic. The way to attack a position is to attack the position, not to count the experts that attack the position. Ol' Art Murray's jabber is gibberish, and that can be easily shown. Why don't you try attacking Longley on the basis of what he with you. The only place where logical inference breaks down is > inside Longley's head. Hmmm, I'm in my early 40's. What should I do with the > rest of my life? Maybe I should write up my opinions on a > web site and then repeat my 8 favorite words thousands of > times in a newsgroup? Duh, that sounds logical! Then > everybody will know that it wasn't my fault, it was those > idiot folk psychologist's fault! Hmmm, now I'm about 50 and nobody has agreed with my > opinions despite thousands of repetitions. Duh, logic must > have broken down! Quine was right! Everybody else's > language is worse than I deduced! Everybody in newsgroups > is folk psychological too! How to solve this problem? Let me analyze more extensionally! Well, I could spend my > 50's posting another 8,000 repetitions. Duh, that sounds > like a good decision! And if that doesn't work, as a > backup I still have my 60's! Excellent and accurate analysis work, Larry. Usenet (and the larger text Net) has its David Hayes, its > Arthur T. Murray, its James Harris, its Stan Rothwell, its > Robert Vienneau, and an unnamed cast of thousands more > exactly like them who plague its various newsgroups, usually > one or two toads to a hole, and your description fits each > of them equally well. Your Mister Longley joins a proud Net tradition, widely > crossposting his uninteresting and unvarying opinions where > they are inappropriate, unwanted, and reviled, without a bit > of normal human reaction to such a reception (such as trying > to figure out just where one has gone wrong and correcting > it), secure in his conviction that he alone is the appointed > Guardian of The Truth. And as you imply, he will doubtless still be mumbling on > into the jaws of hundreds of thousands of killfiles, long > after all his teeth are gone and his mind had begun to > wander aimlessly from its rut, captive of habit and > conviction both. xanthian. > -- ==== > The problem with your analysis is, of course, that > occasionally such people ARE right. And a broken clock is correct twice a day. Mr. Longley has the problem that he is quite possibly entirely correct, in the context of the 1950's and the psychology of mind as understood then, but is not making a meaningful contribution in the context of the 2000's and the software problem of an AI mind. He is exactly a broken clock. > There seems to be a complete inability around here for > self-appointed critics like yourself to escape the notion > that popularity is the arbiter of the scientific or > philosophical worth of a position. You accuse me falsely. You will find several times, using a web search, where I have posted that despite at one time Albert Einstein being the only person on the entire planet who believed in the worth of special relativity, his vote outweighed all the others. Mr. Longley's problem is _not_ that he has an innovative viewpoint which is having trouble being accepted by a is in the position of those who doubted Einstein, trying to cling to an outmoded mindset which has been overtaken by events. He is fighting a battle to prevent progress on any other terms than his, he is utterly condemned to failure, and meanwhile his chattering, drivel, and repetitive negativism waste the time of people who could be making positive contributions to the develoment of AI using modern knowledge not much connected to the detailed understanding of carbon-based intelligence. Whatever the outcome is of silicon mind, the least likely way of all for it to succeed is as a slavish imitation of carbon mind, but that is the only path Mr. Longley will permit. > This is ironic given your implied self-description as > non-dogmatic. I have no such description of myself; my self-description is: deals poorly with obdurate fools. > The way to attack a position is to attack the position, > not to count the experts that attack the position. Nor have I done so, though I have mentioned in passing that those actually contributing to the field of AI mind find no value in Mr. Longley's drivel, nor am I interested in attacking Mr. Longley's position. I have no expertise whatever in carbon mind, nor do I have any intention of acquiring any, so I would be a fool to dispute Mr. Longley on his own intellectual ground. That is not the issue under contest. My interest, and it is purely a hobby, is in software exploration of mathematical computational complexity and the heuristics used to deal with it, and to a lesser extent in the mechanics of artificial life, for which I have an extensive design document, nowhere close to complete, to which I have been adding functional requirements as they occur to me over a few years, and which may well die without fruit. This leaves me a participant in both comp.ai.alife, very modestly, and comp.ai.genetic to a much larger extent, usually just to answer people's questions and point them to appropriate online resources, including my own modest freeware. http://www.well.com/user/xanthian/java/TravellerDoc.html I became aware of Mr. Longley only because of his paradigmatic net.kook behavior, crossposting his arguments to newsgroups which have entirely other focus than 1950's understandings of carbon-based intelligence, repeating ad nauseum pointless references to antiquated ideas which he believes incapable of improvement, and refusal to leave when the noise he has been adding to newsgroups outside the purview of his discussion resulted in repeated requests from many regular participants in those groups to take his litter elsewhere. > Ol' Art Murray's jabber is gibberish, and that can be > easily shown. Yet the problems caused by Mr. Murray's pointless contributions to the same newsgroups is not 5% of the impact level of Mr. Longley's, and this is all and exactly the point. Both are violating netiquette, but Mr. Longley is doing it at length and without ceasing, while Mr. Murray trespasses perhaps ever week or two at worst. > Why don't you try attacking Longley on the basis of what > you can get to agree with you. I am not attacking Mr. Longley, per se, I am merely requesting that he take his discussion out of newsgroups where no one at all is interested in his reactionary ideas. And in the case, I have stopped reading or replying directly to his postings at all, only to quotes in the postings of others. Mr. Longley, as a direct participant in the newsgroups I read, no longer exists for me, a sad but necessary way of dealing with the noise level he causes, and so I need merely deal with the responses of his sycophants, and of my fellow detractors. Nor do I feel the need to get anyone but Mr. Longley to agree with me that his agenda would be better served by sticking to newsgroups where he did not have to spend a great portion of his effort counter requests that he get the hell out of the way of the groups' on-charter discussion. Rallying forces is not my style, I am perfectly capable of making Mr. Longley's life as miserable as he makes that of other people, all by my lonesome, and I have half a dozen other similar campaigns ongoing simultaneously elsewhere. [I am unemployed, have no social life, have lived morbidly depressed for eighteen years, and thus have eighteen hours or so a day to devote to the tasks.] He tells people _doing_ AI, that they are doing it entirely wrong, despite that he has no contributions of his own, in software on silicon, which he can present as a superior alternative, and this is simply rude, arrogant, and unacceptable behavior in newsgroups where such contributions are the on-charter topics of discussion. If he wants to haunt comp.ai.philosophy to his dying day, that is their problem. I have no interest in reading there, ever, philosophy bores me to tears, to me it is only the ruminations of addicted navel divers, and I only crosspost there from the strong understanding that most of what I am responding to would only be seen by those to whom I am responding if posted in that newsgroup from which they are making their contributions. When Mr. Longley starts making the quiet, focused newsgroups where I read into battlegrounds of a distant war, destroying all hope of meaningful, on-topic dialog with his inane blatting, he becomes my problem, a challenge of a type I joy in accepting, and as indicated above, I do not suffer fools gladly, nor, given the choice, at all. xanthian. <3f73a0d0_1@news.iprimus.com.au> <3LOdnYcq9oVxiOmiU-KYuA@metrocast.net> <3f74bc38_1@news.iprimus.com.au> <47b2ec70.0309271107.129b7db0@posting.google.com> <947208bc88d54dd60b374e12b098cec3@news.teranews.com> ==== > The problem with your analysis is, of course, that >> occasionally such people ARE right. And a broken clock is correct twice a day. Mr. Longley has the problem that he is quite possibly >entirely correct, in the context of the 1950's and the >psychology of mind as understood then, but is not making >a meaningful contribution in the context of the 2000's >and the software problem of an AI mind. He is exactly a broken clock. > Glen has corrected you and all you do is respond with precisely the sort of rhetoric which accounts for my criticising your behaviour in the first place. Let's face it, you don't know what you are talking about, nor do you want to learn. Your post is riddled with untruths, far too many for me to work through and correct at this time. Perhaps someone else will oblige, in the interests of integrity. You're writing ignorant nonsense, and that's being kind. The only people who will *not* see that will be those who are as uninformed & susceptible to rhetoric as you are. -- David Longley ==== It isn't, by the way, that carbon intelligence isn't of research interest in achieving silicon intelligence, just that Mr. Longley's approach is completely divorced from the actual research desmene. The below quote is from a recruitment page for Lucent Technologies. xanthian. http://www.bell-labs.com/org/physicalsciences/org/pr/pr.html biological computation research Our goal is to understand how biological neurons and nervous systems perform computation and produce intelligent behavior, and to use that knowledge in the design of novel algorithms and computing paradigms. Understanding how universal forms of brain dynamics relate to computation and behavior is a major research focus. Current topics include the role of oscillation and synchrony in sensory perception, continuous attractor neural networks in short term memory, and neural mechanisms in the generation and recognition of sequences. We have played a leading role in the development of methods for measuring neural activity including BOLD contrast based functional Magnetic Resonance Imaging (fMRI) and optical imaging of chemical concentration dynamics. New methods under development include optical coherence tomography and next-generation microelectrode technology using MEMs and active-feedback stabilization. Novel algorithms for data classification, such as Non-negative Matrix Factorization, electronic noses, and asynchronous microelectronic circuits using hybrid analog/digital computation are examples of applied technologies we are developing. We are currently recruiting in the area of machine learning and novel electronic circuit design. -- ==== > The problem with your analysis is, of course, that occasionally such > people > ARE right. The problem with your analysis of his analysis is that you're not keeping in mind how unbelievably rarely it actually happens. And it happens from people who have strange ideas but are not cranks; even when cranks are right, you don't get points for getting the right answer without showing your work. The vast majority of the time someone's wacky ideas are labelled as crackpot and dismissed, that's exactly what they were. -- Erik Max Francis && max@alcyone.com && http://www.alcyone.com/max/ __ San Jose, CA, USA && 37 20 N 121 53 W && &tSftDotIotE / All of your moonlight whispers / Counterfeit, counterfeit love __/ Lamya ==== Erik: The problem with your analysis of his analysis is that you're not keeping in mind how unbelievably rarely it actually happens. GS: On the contrary, I am quite aware of how rare it is for someone to do truly innovative science (not that the other sort is useless - a well conducted parametric analysis of the variables controlling some phenomenon is worthwhile, if not sexy/theoretical). Erik: And it happens from people who have strange ideas but are not cranks; even when cranks are right, you don't get points for getting the right answer without showing your work. The vast majority of the time someone's wacky ideas are labelled as crackpot and dismissed, that's exactly what they were. GS: But Longley's views are mostly consistent with a scientific and philosophic tradition that is embraced by a minority of practicing scientists and philosophers. Longley's thesis (and I am not endorsing every thing that David says, mind you) is not of the hollow Earth variety - it is still a minority view, granted, but I can certainly find hundreds of people (most of whom teach in universities and do government funded research) who would, in large measure, agree with much of what David says. Now, we are all familiar with minority positions within established natural sciences; as you say, the individuals are frequently regarded as a little quirky, but we have learned to be tolerant of the minority view, as long as it is not too far-fetched. However, in such cases, both sides of the issue will tend to agree upon what constitutes the subject matter in the debate, and what experimental methods may be brought to bear on the issue and so forth. In the case of Longley's endeavor's - and mine as well - the issues are conceptual; they revolve around the very definition of the subject matter and how it is to be studied. The rift that I am talking about is nothing less than the ongoing debate between behaviorism and mainstream psychology and philosophy. The mainstream side of things (cognitivism, mentalism) seeks to convince its neophytes that the debate is over and the facts decided. You will see this position clearly expressed in some of the posts that are responses to mine. The upshot of this approach is the de facto censorship and continued misrepresentation of behaviorism, much like that of the creationists with respect to evolutionary theory. The only difference is that the evolutionary view predominates in mainstream biology, but the psychological form of creationism called cognitive psychology predominates in mainstream psychology and associated philosophies. The problem with your analysis is, of course, that occasionally such > people > ARE right. The problem with your analysis of his analysis is that you're not > keeping in mind how unbelievably rarely it actually happens. And it > happens from people who have strange ideas but are not cranks; even when > cranks are right, you don't get points for getting the right answer > without showing your work. The vast majority of the time someone's > wacky ideas are labelled as crackpot and dismissed, that's exactly what > they were. -- > Erik Max Francis && max@alcyone.com && http://www.alcyone.com/max/ > __ San Jose, CA, USA && 37 20 N 121 53 W && &tSftDotIotE > / All of your moonlight whispers / Counterfeit, counterfeit love > __/ Lamya ==== >The vast majority of the time someone's >wacky ideas are labelled as crackpot and dismissed, that's exactly what >they were. The effectiveness of that prediction is strongly dependent on who's doing the dismissing, and why. --Blair Why your average crank can't compare himself to Galileo. ==== The problem with your analysis is, of course, that occasionally such > people > ARE right. > The problem with your analysis of his analysis is that you're not > keeping in mind how unbelievably rarely it actually happens. And it > happens from people who have strange ideas but are not cranks; even when > cranks are right, you don't get points for getting the right answer > without showing your work. The vast majority of the time someone's > wacky ideas are labelled as crackpot and dismissed, that's exactly what > they were. For some strange reason, my name appeared in some list in this thread. The odd thing is, I spend my time agreeing with some advanced literature. And I show my work in quite a lot of detail. And I get no criticism that disagrees with what I say and is substantial. Furthermore, there are quite a few links to my economics Web pages out on the net. For example, I've even had scholars reference my stuff in dead tree works. I can understand why one might look at some of my posts and form a certain impression while avoiding the substance. But once one looks at the substance, one finds that economics - the topic of most of my posts - is quite an odd discipline. -- 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 ==== > I can understand why one might look at some of my posts and form > a certain impression while avoiding the substance. But once one > looks at the substance, one finds that economics - the topic of > most of my posts - is quite an odd discipline. And once again, it is not the substance of your works which is at issue, it is your insistance on posting them in places where you have been told repeatedly that they are inappropriate, on arguing the point rather than conforming to social norms, and on being unable to understand that the (putative) merit of your ideas sent to an appropriate venue does not justify your rude behavior in sending them to an inappropriate one. xanthian. -- ==== Take for example these 5 equations: 1 1 1 + 2 = + 2 = + 1 1 1 4 = + 4 = + 4 = + 1 1 1 + + 2 = + 1 1 1 1 1 + 2 = + 3 = 1 3 = + 1 4 = + + 4 = + 1 1 1 1 Set A = {{x1},{x2},{x3},{x4}}, where each x# is some number. Now, let us say that the above equations represent some cardinal's equation-trees of set A. Let us say that any cardinal which is > 1 is the continuous side of the cardinal's equation-tree. Let us say that any cardinal which is = 1 is the discrete side of the cardinal's equation-tree. Let x#' be a dummy variable of xor(|{x1}|,|{x2}|,|{x3}|,|{x4}|) . 1 is xor(x1',x2',x3',x4') + 1 is xor(x1',x2',x3',x4') 4 = + 1 is xor(x1',x2',x3',x4') + 1 is xor(x1',x2',x3',x4') 1 is xor(x1',x2') 2 = + 1 is xor(x1',x2') 4 = + 1 is xor(x1',x2',x3',x4') + 1 is xor(x1',x2',x3',x4') 1 is xor(x1',x2') 2 = + 1 is xor(x1',x2') 4 = + 1 is xor(x1',x2') 2 = + 1 is xor(x1',x2') 1 is xor(x1',x2',x3') + 3 = 1 is xor(x1',x2',x3') 4 = + + 1 is xor(x1',x2',x3') 1 is |{x4}| 1 is |{x1}| 2 = + 3 = + 1 is |{x2}| 4 = + 1 is |{x3}| 1 is |{x4}| As you can see above, the quantity in each cardinal's equation-tree is being kept, while the structural symmety-degree and the information's clarity-degree of each tree are changed. My question is: What mathematical's branch deals with this kind of information's structures ? Doron Shadmi ==== > This is a problem from my precalc book that I don't know how to solve: The path of a ball is given by: > y = ((-1/20)(x^2)) + 3x + 5 where y is the height of the ball (in feet) and x is the > horizontal distance (in feet) from where the ball was thrown. a) find the maximum height of the ball > b) find the distance the ball travels > I don't know where/how to begin with this (without just plugging in the numbers > for trial and error). Math Learner. First of all, I would get rid of any book that uses numerical fractions instead of decimals!!! ==== > Is the complete solution to (1) known? (1) ax^2 + bxy + cy^2 = f^2 where integers a,b,c are known, with b^2 > 4ac, but f is not known and x,y are > to be integer also? Dario Alpern's javascript program at http://www.alpertron.com.ar/QUAD.HTM is > almost there, but his program requires that f be known in advance. Is it possible to solve (1) so that all possible solutions of f^2 are located? > Actually John Cremona and David Rusin published a nice paper on this in 1997, which appeared in Mathematics of Computation* as Efficient Solution of Rational Conics in 2001 basically parametrizing all solutions, once one solution was known. Cremona's program is very efficient at locating a rational solution and parametrizing the others, see his tconic C++ program under the mwrank3.gz package (http://www.maths.nott.ac.uk/personal/jec/ftp/progs/) See http://eprints.nottingham.ac.uk/archive/00000060/ for the paper. I can see from the lack of response to my inquiry that the internet newsgroup is quite shallow, this is sad. *MathComp Efficient solution of rational conics J. E. Cremona; D. Rusin ==== It has come to my attention that Coca-Cola has begun to use a new formula for their soda. This is different than the original formula. I remember back in the 80's Coke stopped using the original formula and went to New Coke in an effort to gain market share -- but after it failed, they went back to the original formula. Now, coke is using: SD = 14C02 + 7NH5 + 15H2O + 25C3N9 This is deceptive because they are claiming it is still the original formula. Can anyone comment? CS-102 ==== It has come to my attention that Coca-Cola has begun to use a new > formula for their soda. This is different than the original formula. I > remember back in the 80's Coke stopped using the original formula and > went to New Coke in an effort to gain market share -- but after it > failed, they went back to the original formula. Now, coke is using: SD = 14C02 + 7NH5 + 15H2O + 25C3N9 This is deceptive because they are claiming it is still the > original formula. Can anyone comment? CS-102 Is there a math angle? ==== It has come to my attention that Coca-Cola has begun to use a new > formula for their soda. This is different than the original formula. I > remember back in the 80's Coke stopped using the original formula and > went to New Coke in an effort to gain market share -- but after it > failed, they went back to the original formula. Now, coke is using: SD = 14C02 + 7NH5 + 15H2O + 25C3N9 This is deceptive because they are claiming it is still the > original formula. Can anyone comment? CS-102 Is there a math angle? There is with Diet Coke - something not unlike Russell's Paradox. The Coca-Cola company recently (in the UK) brought out a new Vanilla flavoured Diet Coke. So since then one can assert that the original 'vanilla' Diet Coke is _not_ actually vanilla Diet Coke. Go figure... (Sorry - What I'm currently drinking is rather stronger than coke ;-) --------------------------------------------------------------------------- John R Ramsden (jr@adslate`rm -rf *`.com) (remove spambot trap before use) --------------------------------------------------------------------------- A stockbroker is someone who invests your money until it's all gone. Woody Allen ==== We are studying the PigeonHole principle and though I have answered the first part of the question, the second is a little more tricky. It says: Suppose there are 9 students in a class a) Show the class must have at least 5 male or 5 female students. I simply stated that as there are only 2 options, male or female, if the above is false then there would be > 5 of the second option making the statement true. b) Show the class must have at least 3 male or at least 7 female students. ==== > We are studying the PigeonHole principle and though I have answered the > first part of the question, the second is a little more tricky. It says: Suppose there are 9 students in a class > a) > Show the class must have at least 5 male or 5 female students. > I simply stated that as there are only 2 options, male or female, if the > above is false then there would be > 5 of the second option making the > statement true. > b) > Show the class must have at least 3 male or at least 7 female students. > if 2 males and 6 females made up the class then there are only 8 students. add another student. Herc ==== > We are studying the PigeonHole principle and though I have answered the > first part of the question, the second is a little more tricky. It says: Suppose there are 9 students in a class > a) > Show the class must have at least 5 male or 5 female students. > I simply stated that as there are only 2 options, male or female, if the > above is false then there would be > 5 of the second option making the > statement true. > b) > Show the class must have at least 3 male or at least 7 female students. > Maybe, if you insist, these can somehow use the Pigeon Hole Principle but far easier is: M + F = 9. If M > = 5 you're done. Otherwise, M < 5, 9 = M + F < 5 + F so F > 4. This is, in fact, _your_ argument for (a). The (b) part is similar. The (a) part can be fairly easily done with PHP: You have 9 pigeons and 2 holes so at least one hole contains at least ciel(9/2) = 5 pigeons. I don't see any reasonable way to apply the PHP to (b). -- Paul Sperry Columbia, SC (USA) ==== > We are studying the PigeonHole principle and though I have answered the > first part of the question, the second is a little more tricky. It says: Suppose there are 9 students in a class > a) > Show the class must have at least 5 male or 5 female students. > I simply stated that as there are only 2 options, male or female, if the > above is false then there would be > 5 of the second option making the > statement true. > You're thinking out loud without presenting a coherent and logical line of thought. The statement is, using m,f for males, females m >= 5 or f >= 5 The negation of that statement by DeMorgan's laws of logic, is not(m >= 5) and not(f >= 5) or equivalently m < 5 and f < 5 Now since m,f are integers, we have m <= 4 and f <= 4 Which is a contradiction to m+f = 9, is it not? Now as the negation causes a contradiction, the orginal statement is true. Is that not so? > b) > Show the class must have at least 3 male or at least 7 female students. > How about a similar approach? The statement is m >= 3 or f >= 7 and the negation of that statement is ???? ==== |In particular your equation is the product of all irreducible polynomials |over GF(2) of degrees 5 and 1. Minor correction. x^32-x=x^32+x is the product of all irreducible polynomials of degrees 5 and 1. He had x^31+1 which leaves out the factor of x. Keith Ramsay ==== let s be a sequence where |s(n + 1) - s(n)| < 1/(2^(n)) for all n where n is a natural number (n and n+1 are subscripts denoting a term s(n) and its succeeding term s(n+1)). So if a sequence is Cauchy then |s(m) - s(n)| < epsilon for all epsilon > 0 and m, n > N for some N. Let |s(n + j) - s(n)| < j/(2^(n)) since we have a series of terms |s(n+1) - s(n)| < 1/(2^(n)), |s(n+2) - s(n+1)| < 1/(2^(n)) etc and the above statement is what we get when we add up all of the intermediate statements. Then let m = n+j. Now, we could show j/(2^(n)) < epsilon by letting n be greater than log2(j/epsilon)) but logarithms have not yet been defined in the class and i don't think they can be used. So if i showed that j/n < epsilon for all epsilon > 0 and some n, could we say that this sequence is Cauchy, since j/(2^n) is less than j/n for all n where n is a natural number and we have the final inequality of: |s(m) - s(n)| < j/(2^n) < j/n < epsilon? Matt Lafer ==== >let s be a sequence where |s(n + 1) - s(n)| < 1/(2^(n)) for all n where n is a natural number (n >and n+1 are subscripts denoting a term s(n) and its succeeding term >s(n+1)). So if a sequence is Cauchy then |s(m) - s(n)| < epsilon for all >epsilon > 0 and m, n > N for some N. Let |s(n + j) - s(n)| < j/(2^(n)) since we have a series of terms |s(n+1) - s(n)| < 1/(2^(n)), |s(n+2) - >s(n+1)| < 1/(2^(n)) etc and the above statement is what we get when we >add up all of the intermediate statements. Then let m = n+j. Now, we >could show j/(2^(n)) < epsilon by letting n be greater than >log2(j/epsilon)) but logarithms have not yet been defined in the >class and i don't think they can be used. Even if using logs were legal this would _not_ be a correct argument. You need to come up with a N that depends _only_ on epsilon, such that n, m > N implies |s(n) - s(m)| < epsilon. When you say m = n + j and N = log2(j/epsilon)) then N depends on epsilon and also on m; this is not legal. (For example: The answer to the next question in Ross, Does |s(n) - s(n+1)| < 1/n imply s(n) is Cauchy is _no_. But you could use your method to show the answer was yes. So it really is _wrong_, since it proves something false.) >So if i showed that j/n < >epsilon for all epsilon > 0 and some n, could we say that this >sequence is Cauchy, since j/(2^n) is less than j/n for all n where n >is a natural number and we have the final inequality of: |s(m) - s(n)| < j/(2^n) < j/n < epsilon? >Matt Lafer ************************ David C. Ullrich ==== (**) Let |s(n + j) - s(n)| < j/(2^(n)) since we have a series of terms |s(n+1) - s(n)| < 1/(2^(n)), |s(n+2) - >s(n+1)| < 1/(2^(n)) etc and the above statement is what we get when we >add up all of the intermediate statements. Then let m = n+j. Now, we >could show j/(2^(n)) < epsilon by letting n be greater than >log2(j/epsilon)) but logarithms have not yet been defined in the >class and i don't think they can be used. Actually, you want to avoid having your n depend on j, since in order for the sequence to truly be Cauchy, j should be able to run to infinity with n fixed. Your expression depending on j can become arbitrarily large for whichever n you pick. Instead of using the expression (**) where you simply let each intermediate difference be less than the the maximum of all of them, and then take j times that quantity, a better approach is to look at the sum: (1/2)^(n) + (1/2)^(n+1) + ..... + (1/2)^(n + (j-1)) |s(n + j) - s(n)| is also less than this. Then it isn't hard to see that this goes to zero as n --> infinity, no matter what j is (the trick is to show that the summation of powers of 1/2 converges to 1, and then you can show that deleting the first n terms brings you below epsilon for a properly chosen n). Hope that helped, J.E. www.math.ucla.edu/~anglicus ==== > let s be a sequence where |s(n + 1) - s(n)| < 1/(2^(n)) for all n where n is a natural number (n > and n+1 are subscripts denoting a term s(n) and its succeeding term > s(n+1)). So if a sequence is Cauchy then |s(m) - s(n)| < epsilon for all > epsilon > 0 and m, n > N for some N. > |s(m) - s(n)| = |s(m)-s(m-1) + ... + s(n+2)-s(n+1) + s(n+1)-s(n)| <= 2^-n + 2^-(n+1) + ... + 2^-(m-1) < 2*2^-n > Let |s(n + j) - s(n)| < j/(2^(n)) > This isn't going to work because as j->oo j*2^-n -> oo. > since we have a series of terms |s(n+1) - s(n)| < 1/(2^(n)), |s(n+2) - > s(n+1)| < 1/(2^(n)) etc and the above statement is what we get when we > add up all of the intermediate statements. Then let m = n+j. Now, we > could show j/(2^(n)) < epsilon by letting n be greater than > log2(j/epsilon)) but logarithms have not yet been defined in the > class and i don't think they can be used. So if i showed that j/n < > epsilon for all epsilon > 0 and some n, could we say that this > sequence is Cauchy, since j/(2^n) is less than j/n for all n where n > is a natural number and we have the final inequality of: |s(m) - s(n)| < j/(2^n) < j/n < epsilon? > ==== > To all webmaster,newsmaster it's recommand to set all you News server to receive and send Newsgrousp was use for Text , chit chat , permit to post more > and groups team piracy have use usenet for long time Fight Piracy set your News Server to 32 kbytes (250 lines) well it just goes to show, lucky we have people like James Harris making those short proofs. Herc ==== Darnit, its one of those late date posts, now I'm stuck up here for a month, may as well give you a puzzle that will free humanity if one person can solve it. ----1----------------------------------------------------------------------- ---- Randi will test you when you properly apply to be tested. Sign up here: http://www.randi.org/research/challenge.html ----2----------------------------------------------------------------------- ----- It really all depends on the situation. ----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. ----------------- A cliff86 B Rust C Shanx D NormDePloom E Rich Shewmaker F CNote G Wanda H See You In Hell My Friend. I Someone J Greg Neill Which name best fits each post? Herc ==== |x -x1 y -y1 z -z1| |x2-x1 y2-y1 z2-z1|= |x3-x1 y3-y1 z3-z1| |x y z 1| |x1 y1 z1 1| |x2 y2 z2 1| |x3 y3 y3 1| It easy to understand that this equation is through a perplex caculation, but how does the main thinking about this? ==== > Put 4 different groups on 3 x 3 grid, such that > Every group contains atleast 1 member. > Every group should be in contact with another group via atleast 1 > member. > Every member of the group should be in contact with atleast one > another member of the same group. > 'Contact' = Placed next to eath other. When you say next to each other do you mean side by side or I can > count in also up and down??? I'm pretty sure I misunderstood but if not... let the 4 groups to be @ # $ and % :.............. @ @ @ > # $ % > # % % > This doesn't satisfy the requirement that Every member of the group should be in contact with at least one other member of the same group. ==== I am finding this integral very hard Cos[x^2+z^2]*Exp[(x+y)^2] with -3I am finding this integral very hard Cos[x^2+z^2]*Exp[(x+y)^2] with -3 When I submit it to Mathematica 4.1, I get the one-dimensional integral, In[1]:= Integrate[Cos[x^2+z^2]Exp[(x+y)^2],{x,-3,3},{y,-2,2},{z,-1,1}] Sqrt[Pi] Erfi[2 - x] Sqrt[Pi] Erfi[2 + x] Out[1]= Integrate[(-------------------- + --------------------) 2 2 Pi 2 2 > (-(Sqrt[--] (Cos[x ] FresnelC[-Sqrt[--]] - 2 Pi 2 2 > FresnelS[-Sqrt[--]] Sin[x ])) + Pi Pi 2 2 2 2 > Sqrt[--] (Cos[x ] FresnelC[Sqrt[--]] - FresnelS[Sqrt[--]] Sin[x ])), 2 Pi Pi > {x, -3, 3}] Why not apply your Monte Carlo method to this? -- Stephen J. Herschkorn herschko@rutcor.rutgers.edu ==== I was able to simp[lify that a little bit using //FullSimplify, but i have found that it is very time consuming in evaluation. very time consuming. Even as a one dimentional integral. If you then evaluate the next intrgral with a //N at the end you get an answer for that intrgral it looks right pritty much, i have been getting an answer simmlilar with my simulations so at least i know i am on track. the answer should be: -1.9897311790956833*10^9 that is if mathematica does every thing correctly which i dont know about. Stephen >I am finding this integral very hard Cos[x^2+z^2]*Exp[(x+y)^2] with -3 Stephen J. Herschkorn herschko@rutcor.rutgers.edu > ==== Because, in the language of probability, both the conditioned > population and the event being measured differ in the two cases. > In the first case we could have 6 successes out of a total of 16 > candidates. In the second case both the number of candidates (7) and > the number of successes (3) were different. Actually there were 22 candidates in the first case; 6 AA + 16 AK ==== you all know, that f.e. the group (Z/p^k)^* where k>2 is cyclic for odd p and it can be written as {-1,+1} x C, where C is a cyclic group of order 2^(k-2) for p = 2. let us now regard a p-adic numberfield K:Q_p of finite dimension n = ef over the p-adic numbers Q_p, let A be the ring of whole numbers of K and P its (unique) prime-ideal. my question is now: what is about the structure of (A/P^k)^*, especially if p=2. im sure, that has been already done, but where can i find it in the net or in literature? does someone here know the answer? thank you for reading and thank you for any answer T.Salesch ==== I am sitting with two main forces on my body right now. > I am clearly not experiencing a positive 'dv/dt' in anyone's frame. Force cannot be defined as 'ma'. > Henry's right: He's experiencing a negative _de_celeration; in a free falling observer's frame of reference: > Bzzt!!! > Bzzt!!! Yourself rasberry(:-) > There is a deep and important difference between FORCE and TENSION. > Cut< > Force and tension use the same units (Newtons) and because of that are very > often confused, but they are quite different physical quantities. Force is a > vector, and tension is a tensor. > OHhh; dumb me: I always thought tensors were something more sophisticated; but now thanks to you now I see that they are only like the ropes and chains with which body's are pulled. Thrust is applied by pushing OR pulling you know; where pulling is only accomplished when a good grip or hook is attached so as to 'get around behind a good portion' of the body. > Here's an analogy, in case the distinction is too much for you to > appreciate: Energy and torque use the same Cut< ==== > The equation: f = ma has been attributed to Newton; ever since someone first > interpreted his Second Law of motion; when the inverse of mass [m] entered > physics texts; written as 1/m! Now I ask you: Who would think that the inverse of mass [m], is [1/m]? Any intellegent pros or cons are welcome; from anyone. If mass is defined as a ratio, say f/a or e/(c^2); then 1/m has meaning as in > a/f or (c^2)/e. But I hesitate to say that 1/m is the inverse of 'mass' > I think it is _if_ a body's mass is _one_ slug; one kilogram, or one of anything. > I don't understand the relation of the message text to the message subject? When the equation f = ma was made up it referred to the acceleration [a] as being inversely proportional to the mass, and used the fraction 1/m. That and that 'a' is proportional to 'f', were combined; so that 'a' is proportional to 'f/m': They were then written as an equation: f = kma. I see some hokus pokus there - regarding 1/m being the inverse of m, except when m is one; which it isn't always. ==== > it normally goes without saying, > One has to retain the units in one's equations! > Not if they cancel out! ==== Cut< The only relationship that works is a=F/m. > A few idiots jumped on this little identity and twisted it around in the hope > that they would appear knowledgeable. Damn right. So here's my advice to people who have to take college physics > courses from idiot professors who tell you to use F=ma: DON'T GIVE IN! > Cut< > Sure, it's tough, but you'll have the satisfaction of knowing that you did not > cave in and multiply. ---DPM David you oughta be whopped proper for telling such a whooper: All you need to do is discreetly use f = F-uw = (f/a)a, and the rest will take care of itself. It might even teach those idiot professors something; especially if you have to show your work. ==== Is it possible to express a polynomial such as y = Ax^3 + Bx^2 + Cx + D in terms of y (i.e. x = f(y)). If yes, what is the process; if no then why? Andrew Milne ==== > Is it possible to express a polynomial such as y = Ax^3 + Bx^2 + Cx + D in > terms of y (i.e. x = f(y)). If yes, what is the process; if no then why? Andrew Milne Please note that there is a value of 'y' for each 'x' in the original polynomial -- but not necessarily an 'x' for each 'y'. If the polynomial is even, for example, 'y' will never be negative. -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com ==== In a (basical) topology course I am reading , the author underlines the unicity (if existence) of a limit in Hausdorff spaces (such like metric spaces) and gives the trivial counterexample of the trivial topology for the non-Hausdorff spaces (in this case any point is the limit of any application at any point). Whence my question: In a Fr.8enet space which is not a Hausdorff space, is a limit necessarily unique? (i'd be interested in a counterexample but it doesnt seem so easy to find one?) If the answer to my previous question is yes, what can we say about a Kolmogorov space which is not a Fr.8echet space (that may be easier to find a coutnerexample in this case I think). -- Julien Santini