mm-1036 === Subject: Re: Nerd Quotient Quiz >What does this mean? Your nerdiness is: >All hail the monsterous nerd. You are by far the KING NERD GOD!!! It means you scored lower than I did. I got 100 on the first and only try and nailed it in about a minute. I suppose the uranium samples, mice, parrot, and insects in my room gave me the edge, though. ;) Jon === Subject: sequences of functions Let f_n(x,y) be a convergent sequence of L^2(A times A) functions; then f_n(a,y) converge in L^2(A) for a.e a in A. Is this true? === Subject: Re: sequences of functions > Let f_n(x,y) be a convergent sequence of L^2(A times A) functions; > then f_n(a,y) converge in L^2(A) for a.e a in A. > Is this true? No. On A = [0,1] you can choose intervals I_n whose lengths -> 0 such that each point in A is contained in infinitely many I_n and each point in A misses I_n for i.m. n. So now set f_n(x,y) = 1 if x lies in I_n, 0 otherwise. Then f_n -> 0 in L^2(AxA), but for each x, f_n(x,.) is either the function 1 or the function 0, each case occurring infintely many times. === Subject: Re: sequences of functions >Let f_n(x,y) be a convergent sequence of L^2(A times A) functions; >then f_n(a,y) converge in L^2(A) for a.e a in A. >Is this true? Consider the special case where the functions are constant in the second variable (say A has finite measure). Then the question is the same as whether f_n(x) converging in L^2(A) implies ae convergence. Theres an answer to this in the book... ************************ David C. Ullrich === Subject: Re: OT: Software for plotting graphs? > I need a program that can break the axis into segments, preferably > more than two segments. What software can I use? It depends, what do you mean by break the axis into segments? -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: tetrahedral-cartesian transform posting-account=NRsM-wwAAAAi5GV8YXCgqBCSyfE5DQT7 I guess Im not so much looking for another method of proving this but independent confirmation, which I have received. I seem to be proliferating errors in my writing lately which is not helping things. Anyhow the resultant cartesian vectors for each S ( a much more straghtforward representation of the problem) are posted above in this thread. -Tim === Subject: can anybody provide some pointers for study of PDE and SDE? Hi all, I have to self-study PDE(paritial differential equations) and stochastic differential equations) myself recently... It was hard... I mean, without a class and homework assignments and without the pressure of exam, I got little progress in self-studying... I was hoping that I can get around of the above situation. Can anybody give me some pointers? I also highly appreciate if someone point me to some video lectures on PDE and SDE... I hope they do exist somewhere in this world... I found a lot schools video taped their engineering class, but not math classes... but as an EE engineering student, I found the key is the math classes... Please help me! === Subject: Re: Disappointed Differential >> Let epsilon > 0. Choose delta > 0 so that |f(t) - f(0)| >> < epsilon whenever |t| < 0. Write the last integral >|t| >= 0, I think you mean whenever |t| > 0. No, I meant whenever |t| < delta. >> as the sum of two integrals, one where |t| < delta and >> one where |t| > delta. >Thats just the only thing I could understand if it was wrong. >Dave, if you wanted to express all known non-linear analytic methods on one >page, how would you go about doing that? >For example you mention the technical hypotheses, there are boundary >conditions, stuff like that. Assuming Im quite ignorant, which I am, or >another is, how do you outline non-linear differential methods in a >nutshell, all of them. >Id appreciate that, Id like to learn about this chaos theory someday. Uh, sorry, but I dont know anything about non-linear analytic methods and this chaos theory. >Ross F. ************************ David C. Ullrich === Subject: Re: Disappointed >> Lets see. Of course we need some technical hypotheses >> of f; lets assume that f is bounded and continuous. >As is demonstrated further on, this assumption is essential. _Some_ sort of growth condition is needed or none of the integrals exist. One could replace boundedness by exponential growth here. >> Using the fact that the gaussian has integral 1 we need >> to show that >> int(-oo,+oo) (f(t) - f(T)).g_sigma(t-T) dt -> 0. >> Lets assume T = 0 just to save typing - we need to show >> that >> int(-oo,+oo) (f(t) - f(0)).g_sigma(t) dt -> 0 >> as sigma -> 0. So its more than enough to show that >> int(-oo,+oo) |f(t) - f(0)|.g_sigma(t) dt -> 0 >> Let epsilon > 0. Choose delta > 0 so that |f(t) - f(0)| >> < epsilon whenever |t| < 0. Write the last integral >> as the sum of two integrals, one where |t| < delta and >> one where |t| > delta. >I suppose you mean |t| < delta instead of |t| < 0 . Yes. (Also decided later that delta was a bad choice of variable name...) >Mind your typos! This is SCI.MATH! Yes, sir. >And I would rather say _three_ integrals instead of two, >but, anyway, I understand what you mean. >> The integral over |t| < delta is less than epsilon >> for all sigma, by our choice of delta and the fact >> that the gaussian has integral 1. On the other >> hand the fact that f is bounded and property (iii) >> above shows that the integral over |t| > delta is >> < epsilon if sigma is small enough. QED. (more details >> if you want.) >This is correct, as far as I can see. As I said, its a very standard argument in some circles. >No more details needed. Thanx! >Han de Bruijn ************************ David C. Ullrich === Subject: Re: Dont get it by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBDLiJr18014; > >> I am sorry, sir, I do not see anything written in the >> previous message, could you please explain?. >If you cant even read a reply closely enough to see what was >added theres not much hope, is there? Those are brave words coming from someone who has _by his own admission_ mis-read problems way more than just once ,and proceded to answer them incorrectly. >this: >Say (a,b) is one of these open intervals. Then you know what >f(a) and f(b) are - you really cant think of a way to define >f(x) for x in (a,b) so that f is continuous on [a,b]? > I would appreciate your help with the following: > I had a question about a space-filling curve f:[0,1]->[0,1]^3, > I know that It is continuous on a subset of [0,1] (which I > do not think is dense in [0,1], so that the extension is not > trivial in this sense) The function is defined on the subset of [0,1] given by : S={ 0.a_1b_1c_10a_2b_2c_20.....0a_nb_nc_n0..... > with a_i,b_i,c_i in {0,1,...,9} , and every fourth place is 0} . The function f is given by: f(0.a_1b_1c_10a_2b_2c_20.....0a_nb_nc_n0..)= (0.a_1a_2a_3....., 0.b_1b_2b_3....b_n... ,0.c_1c_2c_3....) Then f is continuous in S ( lets assume that). What I would really appreciate your help with is, in > finding an explicit representation of an extension of > f to [0,1]S ( S as above), so that f is continuous > in [0,1]S/S=[0,1]. I could tell that S is closed in [0,1],so that [0,1]S > is open, so that it is the countable union of open intervals, > so that we can define f in each open interval. >>Say (a,b) is one of these open intervals. Then you know what >>f(a) and f(b) are - you really cant think of a way to define >>f(x) for x in (a,b) so that f is continuous on [a,b]? > I can think of doing it linearly, I guess, by defining the > function in this interval to be the line joining f(a) > to f(b). > y-f(b)=[(f(b)-f(a))/(b-a)]*(x-f(a)) > Would this be correct? > I would really be grateful for any help in finding > this explicit continuation. > >>************************ >>David C. Ullrich >************************ >David C. Ullrich === Subject: simple trig q, but need some help by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBDM3ci19950; I was asked to help with this problem by a friend of mine. not having done trig in years, Im stumped. heres the problem. find the K and a that satisfy the following. K Sin(t+ a)= 2Cos(t)+ 5 Sin(t) i got... using sum identity, K Sin(t) Cos(a) + Cos(t) Sin(a) = 2 cos(t) + 5 Sin(t) Now what do I do? I was trying... 2 Cos(t) + 5 Sin(t) K = ------------------------------- Sin(t) Cos(a) + + Cos(t) Sin(a) How do I proceed from that? and what properties need I use? === Subject: Re: simple trig q, but need some help >find the K and a that satisfy the following. >K Sin(t+ a)= 2Cos(t)+ 5 Sin(t) >i got... using sum identity, >K Sin(t) Cos(a) + Cos(t) Sin(a) = 2 cos(t) + 5 Sin(t) Of course you mean: K (Sin(t) Cos(a) + Cos(t) Sin(a)) = 2 cos(t) + 5 Sin(t) Assuming this is true for all values of t, pick some good ones: Whenever sin(t)=0, K Cos(t) Sin(a) = 2 cos(t) Sin(a) = 2/K Likewise, whenever cos(t)=0, K Sin(t) Cos(a) = 5 Sin(t) Cos(a) = 5/K Then you can use tan(a)=sin(a)/cos(a) and sin^2(a)+cos^2(a)=1 to calculate K and a. --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Re: simple trig q, but need some help In general: Are there constants C and Phi such that A.cos t + B.sin t = C.sin(t+Phi) for all t? Well, C.sin(t+Phi) = C (sin t . cos Phi + cos t . sin Phi); you start was completely OK and to the point. Now look at A.cos t + B.sin t = C (sin t . cosPhi + cos t . sinPhi); if one can determine C and Phi such that A = C.sin Phi and B = C.cos Phi then the problem has been solved. Well, what about C = sqrt (AA + BB) and Phi = atan (A/B)? or Phi = pi/2 if B = 0? And observe how Pythagoras Theorem comes into picture when you draw the rotating arrows that represent the oscillatory motions T -> A.cosT, T -> B.sinT and the sum T -> A.cosT + B.sinT. Johan E. Mebius >I was asked to help with this problem by a friend of mine. not having done trig in years, Im stumped. >heres the problem. >find the K and a that satisfy the following. >K Sin(t+ a)= 2Cos(t)+ 5 Sin(t) >i got... using sum identity, >K Sin(t) Cos(a) + Cos(t) Sin(a) = 2 cos(t) + 5 Sin(t) >Now what do I do? >I was trying... > 2 Cos(t) + 5 Sin(t) >K = ------------------------------- > Sin(t) Cos(a) + + Cos(t) Sin(a) >How do I proceed from that? and what properties need I use? === Subject: Re: simple trig q, but need some help days. My association with the Department is that of an alumnus. >I was asked to help with this problem by a friend of mine. not having done trig in years, Im stumped. >heres the problem. >find the K and a that satisfy the following. >K Sin(t+ a)= 2Cos(t)+ 5 Sin(t) Meaning, find a and K that will make this an identity (true for every t)? >i got... using sum identity, >K Sin(t) Cos(a) + Cos(t) Sin(a) = 2 cos(t) + 5 Sin(t) K should also be multiplying cos(t)sin(t) (the second summand). >Now what do I do? >I was trying... > 2 Cos(t) + 5 Sin(t) >K = ------------------------------- > Sin(t) Cos(a) + + Cos(t) Sin(a) >How do I proceed from that? and what properties need I use? This wont help much, since you want both K and a. If you are trying to find K and a that will make the expression an identity, then try plugging in some values of t to get conditions on K and a. For example, if t=0, then sin(t)=0, cos(t)=1, so we have K*sin(a) = 2 If t = pi/2, then sin(t)=1, cos(t)=0, and sin(pi/2+a) = cos(a), so we get K*cos(a) = 5. If these two conditions are satisfied, then you can clearly see that the expression will be an identity, since K*sin(t+a) = Ksin(t)cos(a) + Kcos(t)sin(a) = 5sin(t) + 2cos(t). So, you want to find a and K that will satisfy K*sin(a) = 2 K*cos(a) = 5. Clearly, K is nonzero. Squaring both equations, we have K^2*sin^2(a) = 4 K^2*cos^2(a) = 25 sin(a) = 2/sqrt(29) cos(a) = 5/sqrt(29) and one and only one value of a between 0 and pi/2 will work. Or K=-sqrt(29), sin(a) = -2/sqrt(29), cos(a) = -2/sqrt(29), which gives the previous value plus pi. Or, multiplying Ksin(a) = 2 by 5 and Kcos(a)=5 by 2, we get 5K sin(a) = 2K cos(a) from which we get 5sin(a) = 2cos(a). Since we cannot have either side equal to 0, this yields tan(a) = 2/5. So one solution is K=sqrt(29), a = arctan(2/5) -- Its not denial. Im just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: simple trig q, but need some help posting-account=2OYlAwwAAAAzuGZHzY8fB1XqLzeo4Yd5 You forgot to distribute the K after you did the identity. Once you correct this, youll yield: K = (2 + 5*tan(t)) / (tan(t)*cos(a) + sin(a)) Once you distrubute the ÔK, its a matter of algebra from there. D. > I was asked to help with this problem by a friend of mine. not having done trig in years, Im stumped. > heres the problem. > find the K and a that satisfy the following. > K Sin(t+ a)= 2Cos(t)+ 5 Sin(t) > i got... using sum identity, > K Sin(t) Cos(a) + Cos(t) Sin(a) = 2 cos(t) + 5 Sin(t) > Now what do I do? > I was trying... > 2 Cos(t) + 5 Sin(t) > K = ------------------------------- > Sin(t) Cos(a) + + Cos(t) Sin(a) > How do I proceed from that? and what properties need I use? === Subject: Re: simple trig q, but need some help >I was asked to help with this problem by a friend of mine. not having done trig in years, Im stumped. >heres the problem. >find the K and a that satisfy the following. >K Sin(t+ a)= 2Cos(t)+ 5 Sin(t) >i got... using sum identity, >K Sin(t) Cos(a) + Cos(t) Sin(a) = 2 cos(t) + 5 Sin(t) Not a bad start. But it should be K( Sin(t) Cos(a) + Cos(t) Sin(a))=.. Plug in t=0. Then K*sin(a) = 2. Plug in t=Pi/2. Then K*cos(a) = 5. Does this help as a start? Thomas >Now what do I do? >I was trying... > 2 Cos(t) + 5 Sin(t) >K = ------------------------------- > Sin(t) Cos(a) + + Cos(t) Sin(a) >How do I proceed from that? and what properties need I use? === Subject: Re: simple trig q, but need some help >I was asked to help with this problem by a friend of mine. not having done trig in years, Im stumped. >heres the problem. >find the K and a that satisfy the following. >K Sin(t+ a)= 2Cos(t)+ 5 Sin(t) K = sqrt(29), a in {Arcsin(2 sqrt(29) / 29) + 2 n pi: n in Z} is one set of solutions. Do you see why? -- Stephen J. Herschkorn sjherschko@netscape.net === Subject: Re: simple trig q, but need some help <41BE16C9.20708@netscape.net> posting-account=arnIEA0AAAAiqq71lhC1Y7JkPVXrK0Dm >find the K and a that satisfy the following. >K Sin(t+ a)= 2Cos(t)+ 5 Sin(t) > K = sqrt(29), a in {Arcsin(2 sqrt(29) / 29) + 2 n pi: n in Z} is one > set of solutions. Do you see why? no... can you show me the steps? also, i think the problems requires a symbolic solution. or no? or is whats shown above symbolic? ( sorry, Im a biologist, I havent done this in a while) === Subject: Re: simple trig q, but need some help >find the K and a that satisfy the following. >K Sin(t+ a)= 2Cos(t)+ 5 Sin(t) > >>K = sqrt(29), a in {Arcsin(2 sqrt(29) / 29) + 2 n pi: n in Z} is >> >one >>set of solutions. Do you see why? >> >no... can you show me the steps? sin (t + a) = (sin a) cos t + (cos a) sin t = 2/K cos t + 5/K sin t So we want sin a = 2/K, cos a = 5/K, and sin^2 a + cos^2 a = 1. a must be in the first or third quadrant, so I see I can improve my solution to K = sqrt(29), a in {Arcsin (2 sqrt(29) / 29) + n pi: n in Z}. (a is in radians) -- Stephen J. Herschkorn sjherschko@netscape.net === Subject: [OT] Re: Google Groups is much better now xanthian@well.com says... > Well, if you want your complaints to > receive some attention, send them to [email address deleted] That was the first thing I did. Q. === Subject: Determinant with trigonometry Proof, that determinant (with n rows and columns) | cosx 1 0 0 ... 0 0 | | 1 2cosx 1 0 ... 0 0 | | 0 1 2cosx 1 ... 0 0 | | 0 0 1 2cosx ... 0 0 | = cos nx |............................................................ ........ | | 0 0 0 0 ... 2cosx 1 | | 0 0 0 0 ... 1 2cosx | The main proble for me is to evaluate this determinant. Do you have any ideas? *-----------------------* www.GroupSrv.com *-----------------------* === Subject: Re: Determinant with trigonometry > Proof, that determinant (with n rows and columns) I will edit your post, so that the cos nx line is not in a confusing place: cos(nx) = > | cosx 1 0 0 ... 0 0 | > | 1 2cosx 1 0 ... 0 0 | > | 0 1 2cosx 1 ... 0 0 | > | 0 0 1 2cosx ... 0 0 | > |............................................................ ..| > | | > | 0 0 0 0 ... 2cosx 1 | > | 0 0 0 0 ... 1 2cosx | > The main proble for me is to evaluate this determinant. > Do you have any ideas? > *-----------------------* > www.GroupSrv.com > *-----------------------* Induction: check D(1) for n=1 (obvious, 1-by-1 matrix) and D(2) for n=2 (well-known trig identity), and to get from D(n) to D(n+1), expand the big determinant D(n+1) by the last row. You should get a three-term recursion, expressing another trigonometric identity D(n+1) = 2*cos(x) * D(n) - D(n-1). === Subject: Re: Deep Thoughts # 37: I just realized whats wrong with the Continuum Hypothesis posting-account=sAS5-AwAAABlKnmtMjBbYHvhxI6W0cAg > The cardinality of a set is not well-defined. What is the cardinality > of: > 1. A = { x | x = N v (eY)YeA ^ x=P(Y) } > 2. A = { x | x = N v x = P(A) } > 3. A = { x | ~(x e A) } > 4. A = { x | x e A } > 5. A = { x | x e (A u {0}) } Here A is defined in terms of A. These are not definitions. > 6. A = { x | x e x } This is the empty set. > 7. A = { x | ~ (x e x) } This is the class of all sets. It is not a set. > Which have less cardinality than which? Also include P(A) , A u B , A > n B etc. for A and B from this list. > To tell if A 1. Are A and B well-defined? > 2. Is A < B? > However, > 1: To determine if these expressions are well-defined is not > possible in general. > 2: To tell if A < B is not possible even when A and B are > well-defined > But you have to have self-referential definitions like these in order > to define even the natural numbers! No, the set of natural numbers is defined to be the set of finite ordinals. An ordinal is defined to be a connected transitive set and is said to be finite if it does not dominate or equal any limit ordinal. There is no self-referential definition involved. > So my formalization of CH (formula C-B): > N < A < 2^N > is not expressible in any Set Theory that includes only provably > consistent expressions. CH is a sentence in the first-order language of set theory. > C-B Conjecture: There is a simpler equation than formula C-B that is > also not expressible. > Empirical Basis: The unsolvability of the Halting Problem is a > corollary of the fact that the programs that do not halt yes on > themselves is not r.e. (I formally prove this in my arxiv paper.) The > above reminds me of the Halting Problem and I suspect there is an > analogy that shows us this simpler equation to which the > inexpressibility of CH is a corollary. > That is what is wrong with the Berry Paradox. We cant solve the > Busy Beaver and we cant tell what the smallest number requiring a > million words to describe is because we cant tell which descriptions > are consistent. > This is just diagonalizing over all sets. Not really very different > from the Russell Paradox. Plus a little bit of Godels 2nd. > Incompleteness Theorem. > C-B === Subject: Re: mathematica/maple for linux >> Does anyone know of a good, user friendly piece of software similar to >> maple or mathematica for linux? In particular something that will > http://www.mupad.de > LD http://maxima.sourceforge.net/ or http://www.nongnu.org/axiom/ both of them in combination with http://www.texmacs.org makes up a nice system. Marc === Subject: Re: Disappointed Differential by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBDMhdM23293; > Let epsilon > 0. Choose delta > 0 so that |f(t) - f(0)| > < epsilon whenever |t| < 0. Write the last integral >>|t| >= 0, I think you mean whenever |t| > 0. >No, I meant whenever |t| < delta. > as the sum of two integrals, one where |t| < delta and > one where |t| > delta. >>Thats just the only thing I could understand if it was wrong. >>|| >>Dave, if you wanted to express all known non-linear analytic methods on one >>page, how would you go about doing that? >>For example you mention the technical hypotheses, there are boundary >>conditions, stuff like that. Assuming Im quite ignorant, which I am, or >>another is, how do you outline non-linear differential methods in a >>nutshell, all of them. >>Id appreciate that, Id like to learn about this chaos theory someday. >Uh, sorry, but I dont know anything about non-linear analytic >methods and this chaos theory. >>Ross F. >************************ >David C. Ullrich Ha ha ha ha. Thats pretty funny. Let me see if I can pry one in here, how about whats a list of In all seriousness the differential equations are about the most difficult mathematics. To be sure it is a question of calculation, but the methodology is intense. Other calculatory methods, eg, the umbral or discrete calculi, are similarly difficult, as are extended number theoretic crunching exercises. Non-linear analysis and chaos theory took off as paradigms in the early to mid-eighties, and rapid and rabid and fragmented advances have occurred throughout the field, combined with geometric methods, Euclidean and non-Euclidean methods, and brute force digital approximation. Where else can you get an answer? Hey how do you solve Schroedingers wave equation? Hey hows the weather. :( Ha ha ha ha. Ross === Subject: Re: Golf competition >{{2, 5, 7, 9}, {0, 1, 11, 6}, {3, 4, 8, 10}} ^ ^ >{{3, 1, 10, 7}, {4, 5, 9, 6}, {0, 2, 11, 8}} ^ ^ >{{5, 0, 6, 10}, {2, 3, 11, 9}, {4, 1, 7, 8}} >{{0, 4, 11, 7}, {5, 3, 8, 6}, {1, 2, 10, 9}} ^ ^ >{{0, 3, 9, 8}, {2, 4, 6, 7}, {1, 5, 11, 10}} >Nobody partners the same player more than once For once read twice. For example, 0 & 11 play in weeks 1,2,4. This is the same solution I proposed. In the dice model, youve written 11,0 on one face of the die, 1,10 on an adjacent face, and 5,6 on a face adjacent to those two, and youve colored the vertex where these three meet. (On my desk Ive got a picture where those three faces are seen in clockwise order around this vertex.) The faces opposite them are labelled 4,7, 2,9, and 3,8 resp. So the foursomes corresponding to this vertex are {11,0,2,8}, etc. as seen in the second line above. The other lines correspond to the three other colored vertices of the die I have labelled, except the fourth line, which is the one I called week 1. dave === Subject: Re: Golf competition >{{2, 5, 7, 9}, {0, 1, 11, 6}, {3, 4, 8, 10}} >{{3, 1, 10, 7}, {4, 5, 9, 6}, {0, 2, 11, 8}} >{{5, 0, 6, 10}, {2, 3, 11, 9}, {4, 1, 7, 8}} >{{0, 4, 11, 7}, {5, 3, 8, 6}, {1, 2, 10, 9}} >{{0, 3, 9, 8}, {2, 4, 6, 7}, {1, 5, 11, 10}} >>When does player 0 play with players 4 or 7? >Third game, isnt it? (Fourth.) Yes, sorry. Read too fast. Or thought too slowly. === Subject: Re: JSH: But what if it works? posting-account=Q2zO6wwAAABSLuGzZIjG0efOtB9n8fUY > : I get LOTS of ideas. > No doubt about that. Well theres a serious problem on this newsgroup with people who dont understand the idea process. Brainstorming. Its been around for a while but you people act clueless about it. So I talk about having a lot of ideas where usually they dont pan out, and get what? Criticized. > : I have a PROCESS for how I work through my ideas, which includes a lot > : of discussion, as talking out ideas helps me understand them. > Im curious where this process goes on, because its certainly not here in > sci.math. Here in sci.math you post bad mathematics, make > undergraduate-level errors, insult people, and whine like a five-year-old. Errors are part of the discovery process. Errors are part of brainstorming. If you act like every single little idea is the grand great idea for all time that will shine like a bright light throughout the world, then you will waste a lot of time. If you worry about every little mistake, then you will waste time. You people betray your own inconsistency by caring more about my mistakes than I do, and then claiming that my work is not important. Why do you care at all? Why wont you just go away? Psychological issues of your own are why. > : I LIKE doing mathematical research. > Im not sure if you really know what it is. Like a comment like that is surely meant to be negative in some way. But it is actually childish though you accuse me of whining like a five year old. When it comes down to it, I post about various mathematical ideas I have on a math newsgroup. And you know what? Thats what Usenet is supposed to be about, and freedom of speech is supposedly important. Ive gone on about this before, how posters here do not even bother to hide that their agenda is controlling how I post, or actually trying to stop me from posting at all, as if freedom of speech were just a phrase. Whats strange is that though theyve failed, year after year, after year, after year, they keep at it, and then call *me* crazy. > Best, > Justin Yeah right. At least you can end without some empty platitude as if you are actually a polite social being when your own posting just showed that you ßout group rules as you see fit. Usenet is an arena where people can post about their ideas. Its not part of the bargain that some people take it upon themselves to live out their anti-social control fantasies by hounding one particular poster, year after year, after year. James Harris === Subject: Re: JSH: But what if it works? > If you act like every single little idea is the grand great idea for > all time that will shine like a bright light throughout the world, then > you will waste a lot of time. You have often acted like your ideas are the grand ideas for all time. Remember your FLT proof? Remember your claim that Galois Theory is dead? Remember your factoring stuff? Remember the hammer (where is it)? > If you worry about every little mistake, then you will waste time. You often claim that there is no error in your work. And in math, mistakes do matter! > Why do you care at all? Why wont you just go away? Why dont you? > Ive gone on about this before, how posters here do not even bother to > hide that their agenda is controlling how I post, or actually trying to > stop me from posting at all, as if freedom of speech were just a > phrase. Well here is some free speach from me... James, you are an idiot. Stop posting! BTW, did you get published yet? === Subject: Re: JSH: But what if it works? : Well theres a serious problem on this newsgroup with people who dont : understand the idea process. You need to look inwards for a bit. Folks here have no problem with the idea process - they embrace and understand it. Ive not had anything but warm receptions here for anything Ive ever asked. I challenged the very foundations of mathematics once myself, but I was straightened out in short order. : So I talk about having a lot of ideas where usually they dont pan out, : and get what? Criticized. This is not why you get criticized. You get criticized because you refuse to learn basic arithmetic, you refuse to attempt to follow logical arguments, and you insult people. : Errors are part of the discovery process. Errors are part of : brainstorming. Sure are; its not the errors that are the problem - its the unwillingness to respond respectably and intelligently. : You people betray your own inconsistency by caring more about my : mistakes than I do, and then claiming that my work is not important. Your work is not important. It will never change the world. It will never change mathematics. : Why do you care at all? Why wont you just go away? Why dont you? If your math is so fantastic, take it to a journal (a respectable one), get it published, and then come back and rub it in our faces. Best, Justin === Subject: Re: JSH: But what if it works? posting-account=sAS5-AwAAABlKnmtMjBbYHvhxI6W0cAg It would seem a bit premature to talk about any potential real-world impact. Just find out whether the idea works or not. If it does, then will be the time to get excited. === Subject: Re: JSH: But what if it works? posting-account=Q2zO6wwAAABSLuGzZIjG0efOtB9n8fUY > It would seem a bit premature to talk about any potential real-world > impact. > Just find out whether the idea works or not. If it does, then will be > the time to get excited. Mathematical research is just fun. I accept that now, and for me, the excitement is finding something new. Even if an idea doesnt pan out, often you can learn something by understanding why. Its all just GREAT FUN, and thats one thing that I realize few of you experience, or youd understand where Im coming from, and how much I enjoy the process that Ive worked out. Brainstorming is when you just get creative. Trying out all kinds of different ideas, and seeing if you come across something youve never seen before. Later you get to come back and critique, or get critiques, and try to knock holes in your own ideas!!! How many of you experience the utter and pure joy of attacking your own work? Or hammering at it with everything youve got, and when thats not enough, inviting others to try and tear it down as well? Mathematics--the absolute and true Mathematics, and not mathematics as people can screw up and call something mathematics when it is not--is perfect. A mathematical proof is absolute. It is absolute power. I invite people to attack my ideas because I dont want ideas that will wither under the assault. Trouble is, I get people who make it personal, or who cant manage to deliver a cogent response to my ideas. But even when they cant, they CLAIM they can, which is why Usenet is mostly a waste. You know what I do most of the time? I look for holes in my own work. I attack my own ideas. I look over, and over, and over again trying to find something wrong. Thats what takes up most of my time these days, as Im usually not actively researching. Mostly I spend time trying to find something wrong. James Harris === Subject: Re: JSH: But what if it works? > Mathematical research is just fun. I accept that now, and for me, the > excitement is finding something new. When have you ever found something new that works? > Even if an idea doesnt pan out, often you can learn something by > understanding why. And often you cannot learn from understanding why it does not pan out. In your cases, it has always been a colossal waste of time. > Brainstorming is when you just get creative. Trying out all kinds of > different ideas, and seeing if you come across something youve never > seen before. I have not seen you brainstorm. I have only seen you talk nonsense, and get into emotional fights with strangers. > Later you get to come back and critique, or get critiques, and try to > knock holes in your own ideas!!! I have never seen you accept criticism, or criticize your own work. > Mathematics--the absolute and true Mathematics, and not mathematics > as people can screw up and call something mathematics when it is > not--is perfect. James, you are such a goofy romantic Hitler of math! Mathematics is not absolute. It is always relative to a context, such as which ring you are talking about. Russels paradox and Goedels Incompleteness Theorem showed some weaknesses in mathematics that you should know about. > A mathematical proof is absolute. No, it is based on (i.e. relative to) axioms. > It is absolute power. No. It is at times useful though. > But even when they cant, they CLAIM they can, which is why Usenet is > mostly a waste. So avoid it. === Subject: Re: JSH: But what if it works? Discussion, linux) >> Mathematics--the absolute and true Mathematics, and not mathematics >> as people can screw up and call something mathematics when it is >> not--is perfect. > James, you are such a goofy romantic Hitler of math! Wow. Its not easy to be more hyperbolic than James on his Hammer days. Youve managed. James is naive and silly when he speaks of his perfect Mother Mathematics, but somehow I fail to see the goofy romantic Hitler analogy. -- Jesse F. Hughes Usenet is demonstrably dangerous. It needs to be regulated. --James S. Harris, voice of reason and moderation === Subject: Re: JSH: But what if it works? What part of goofy romantic Hitler do you fail to see? He is goofy. You get the goofy part I hope. that is the easy bit. Romanticism in various art forms stressed emotion, imagination, freedom, and rebellion against dogma and classical conventions. Sounds like James to me. Adolf Hitler was not just a megalomaniac despot. That was what made him famous, but he had many other traits that are often overlooked because of the nasty stuff he did. But he was much more. He was a lazy, romantic, idealistic, fantasy-oriented, and controlling twit. He dreamt of becoming a priest and an artist, but never accomplished either. You should see the similarities with James as a wanna-be mathematician, I would think. === Subject: Re: JSH: But what if it works? <87d5xc93xq.fsf@phiwumbda.org> Discussion, linux) > You should see the similarities with James as a wanna-be > mathematician, I would think. Not nearly enough similarities to justify such a stupid comparison. -- And the logical extension of free and open-source software in the realm of sex would certainly include publicly shared sex at a sex party,... queer sexuality and... non-proprietary sexual affection. Annalee Newitz writing in Salon.com === Subject: Re: JSH: But what if it works? > Adolf Hitler was not just a megalomaniac despot. That was what made him > famous, but he had many other traits that are often overlooked because of > the nasty stuff he did. But he was much more. He was a lazy, romantic, > idealistic, fantasy-oriented, and controlling twit. He dreamt of becoming > a priest and an artist, but never accomplished either. You should see the > similarities with James as a wanna-be mathematician, I would think. Another similarity between Adolf and James is that they are both paranoid. But there are differences too, so the analogy is not perfect (never is, right?). Adolf actually did have some drawing talent, and if he were not so lazy and screwed up, he could have been a good artist. He did very few drawings, but the ones he did do were actually not too bad. James shows no talent for math at all really, as far as I can see. === Subject: Re: JSH: But what if it works? Discussion, linux) > You know what I do most of the time? I look for holes in my own work. > I attack my own ideas. I look over, and over, and over again trying to > find something wrong. > Thats what takes up most of my time these days, as Im usually not > actively researching. > Mostly I spend time trying to find something wrong. Really? ,---- | Ive tossed my baby idea out there already, as Ive made two posts, | one deriving some formulas, and another stepping through an | algorithm from it. | | Ive tested neither, as Im not interested in investing the time at | this point, with a baby idea. `---- When does this love of attacking your own work begin? Oh, heck, never mind. After all, WHAT IF IT WORKS??? -- If you are a mathematician, then you cannot dispute the result. If you dispute the result [...] then you are NOT a mathematician. Anyone who disputes this result [...] is not a mathematician. I am a mathematician, which is how I could find the result.--James S. Harris === Subject: Re: JSH: But what if it works? <876535qrfk.fsf@phiwumbda.org> posting-account=Q2zO6wwAAABSLuGzZIjG0efOtB9n8fUY > You know what I do most of the time? I look for holes in my own work. > I attack my own ideas. I look over, and over, and over again trying to > find something wrong. > Thats what takes up most of my time these days, as Im usually not > actively researching. > Mostly I spend time trying to find something wrong. > Really? Last Wednesday eventually I found myself musing about yx^2 + Ax - K = T and worked it out, oh, in a few minutes, couldnt have been more than half an hour. Over time Ive found myself talking about it more and more, while still hesitant. > ,---- > | Ive tossed my baby idea out there already, as Ive made two posts, > | one deriving some formulas, and another stepping through an > | algorithm from it. > | Ive tested neither, as Im not interested in investing the time at > | this point, with a baby idea. > `---- > When does this love of attacking your own work begin? Immediately. With my latest idea though, initially I focused on the algebra, as Ive had a tendency to just make dumb mistakes when really desperate for a nice result, so I looked to what I saw as the weakest area. Im more satisfied today that the algebra is correct. > Oh, heck, never mind. After all, WHAT IF IT WORKS??? Heres the discovery process: creative, critiquing, editing Notice I didnt say creating as you dont create anything in mathematics--you discover. You people spend a lot of time criticizing me for the creative part, when I brainstorm out ideas and have the GALL to actually talk about them! And then you come back later to criticize me when I critique my own ideas and toss out bad ones, as if that proves Im a crank as well since I have so many errors that even I acknowledge!!! And then you criticize on the editing, as my own understanding evolves over time, and I shift positions trying to stay absolutely correct. After spending all that time criticizing the discovery process, you people then claim that everything I do is just a waste of time, and often tell people to ignore me! James Harris === Subject: Re: JSH: But what if it works? > And then you criticize on the editing, as my own understanding evolves > over time, and I shift positions trying to stay absolutely correct. How can you shift positions and at the same time stay absolutely correct? That means that what is absolutely correct must change over time. And it just happens to change according to what you believe is truth at the current time. How do you have such control of the reality of mathematics? Perhaps you are god? Perhaps we all just in your mind, with no reality of our own. Perhaps the universe is just your insanity. === Subject: Re: JSH: But what if it works? >You know what I do most of the time? I look for holes in my own > work. >I attack my own ideas. I look over, and over, and over again > trying to >find something wrong. >Thats what takes up most of my time these days, as Im usually not >actively researching. >Mostly I spend time trying to find something wrong. >>Really? > Last Wednesday eventually I found myself musing about > yx^2 + Ax - K = T > and worked it out, oh, in a few minutes, couldnt have been more than > half an hour. > Over time Ive found myself talking about it more and more, while still > hesitant. >>,---- >>| Ive tossed my baby idea out there already, as Ive made two posts, >>| one deriving some formulas, and another stepping through an >>| algorithm from it. >>| Ive tested neither, as Im not interested in investing the time at >>| this point, with a baby idea. >>`---- >>When does this love of attacking your own work begin? > Immediately. With my latest idea though, initially I focused on the > algebra, as Ive had a tendency to just make dumb mistakes when really > desperate for a nice result, so I looked to what I saw as the weakest > area. > Im more satisfied today that the algebra is correct. >>Oh, heck, never mind. After all, WHAT IF IT WORKS??? > Heres the discovery process: creative, critiquing, editing > Notice I didnt say creating as you dont create anything in > mathematics--you discover. Let me give you an example of something I worked on for about a year. I was given the idea of a particular expansion on quadrilaterals that, when iterated, appeared to become a good approximation of a square. My friend also mentioned that it frustrated his high school math teacher because that teacher couldnt prove it. My first challenge was to convince myself that it seemed to work. Not hard with a couple of drawings. Then I had to come up with a means of measuring closeness to square. I used a few ideas from statistics for that. My precise measurements changed as I worked on the problem, but the basic idea worked. Then I used algebra to find formulas for the corners of the new quadrilateral given the old one. This required a mixture of algebra and trigonometry. The expansions could then be represented as multiplication of a coordinate matrix by a square expansion matrix. Then I was able to examine the eigenvectors and eigenvalues to find a natural decomposition of the the quadrilateral into well-behaved components that served to form the basis of a vector-space. Examining the expansion on the basis elements, along with the notion of a limit, finished the proof. I spent the remaining 4 months or so applying the technique to other shapes and generalizing it to arbitrary n-gons. Along the way, I removed restrictions (such as convexity) in favor of more generic approaches. I discussed the work with a couple of colleagues once to get the idea of using eigenvectors/eigenvalues. Is the work published? No. I havent even done a rewrite to do things like remove the various details of computations. Will it ever get published? Maybe. To be honest, Im more interested in looking at a second, related, expansion. I also have various other things on my plate right now. Before I concluded that it was correct, I conducted a few tests, using excel spreadsheets and charts, to verify the results I had. It has not been peer-reviewed and may contain some errors. It certainly had some unexpected results (the hexagon comes to mind). > You people spend a lot of time criticizing me for the creative part, > when I brainstorm out ideas and have the GALL to actually talk about > them! Brainstorming is not normally paraded around in public. Results, or potential results are. The above example is not a result, by the way. I havent given details on what the expansion is or what the conclusions were. Also, half the time you parade out a result and later call it brainstorming. It looks like an attempt to save face. > And then you come back later to criticize me when I critique my own > ideas and toss out bad ones, as if that proves Im a crank as well > since I have so many errors that even I acknowledge!!! When most things are presented as tested and ßawless, only to have the ßaws shown and acknowledged, this is no surprise. When you still do not acknowledge the existence of counterexamples to your results, this is no surprise. > And then you criticize on the editing, as my own understanding evolves > over time, and I shift positions trying to stay absolutely correct. > After spending all that time criticizing the discovery process, you > people then claim that everything I do is just a waste of time, and > often tell people to ignore me! Its not the discovery process, but your poor use of it that is criticized. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: JSH: But what if it works? Originator: richard@cogsci.ed.ac.uk (Richard Tobin) >You people spend a lot of time criticizing me for the creative part, >when I brainstorm out ideas and have the GALL to actually talk about >them! You may find brainstorming works better if you dont do the following to the people youre brainstorming with: - call them liars - threaten to call their employers - suggest that youre going to set the US army on them. After all, youre the one asking for help, from people who have no obligation to you. -- Richard === Subject: Re: JSH: But what if it works? <876535qrfk.fsf@phiwumbda.org> Discussion, linux) > After spending all that time criticizing the discovery process, you > people then claim that everything I do is just a waste of time, and > often tell people to ignore me! Well, I dont know who you mean by you people, but I promise I dont tell folks to ignore you. I sing your praises far and wide. -- Jesse F. Hughes Just goes to tell you. If you make a major discovery, and some stupid interviewer asks you if youre the greatest mathematician of all time, just say no. -- practical advice from James S. Harris === Subject: Re: JSH: But what if it works? Discussion, linux) > Mathematical research is just fun. I accept that now, and for me, the > excitement is finding something new. > Even if an idea doesnt pan out, often you can learn something by > understanding why. > Its all just GREAT FUN, and thats one thing that I realize few of you > experience, or youd understand where Im coming from, and how much I > enjoy the process that Ive worked out. But so far, you just keep saying over and over that the idea is just a baby, not worth the effort to even check if it works, BUT WHAT IF IT DID??? That does not seem like the approach of someone that finds mathematical research fun. That is the approach of someone that prefers daydreaming to finding out whether his idea is worth a plug nickel. Of course, nothing wrong with daydreaming, but it shouldnt be confused with research. -- If you like high adventure, come with me. If you like the stealth of intrigue, come with me. If you like blood and thunder, come with me. But first listen to a word from our sponsor. -- Adventures by Morse === Subject: Re: JSH: But what if it works? : Mathematical research is just fun. I accept that now, and for me, the : excitement is finding something new. But you havent really found anything new. : Even if an idea doesnt pan out, often you can learn something by : understanding why. With respect, you dont seem to learn anything even as your ideas dont pan out. : How many of you experience the utter and pure joy of attacking your own : work? Or hammering at it with everything youve got, and when thats : not enough, inviting others to try and tear it down as well? I do, pretty frequently. Your work tends to crumble of its own accord though, not much tearing seems necessary. Best, Justin === Subject: Re: JSH: But what if it works? > I LIKE doing mathematical research. Calling what you do Ômathematical research is like calling a kid poking a dead raccoon with a stick Ôanatomical research. In case you forgot, *research* literally involves going through all relevant past discoveries and literature, critically and in great detail -- a process you have expressed repeated disdain for. -- 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 === Subject: program: TRIANGLE (for FX7400GPLUS) posting-account=1QdlFQ0AAADvQHRFU-gnUXpZu6etKyL3 = TRIANGLE = A:?->List 4<+ B:?->List 5<+ C:?->List 6<+ List 4[1]->A<+ List 5[1]->B<+ List 6[1]->C<+ List 4[2]->D<+ List 5[2]->E<+ List 6[2]->F<+ {A,B,C,A}->List 1<+ {D,E,F,D}->List 2<+ (A+B).852->G<+ (B+C).852->H<+ (C+A).852->I<+ (D+E).852->J<+ (E+F).852->K<+ (F+D).852->L<+ (A+B+C).853->M<+ (D+E+F).853->N<+ {G,H,I,M}->List 3<+ {J,K,L,N}->List 4<+ DrawStat === Subject: program: TRIANGLE (for FX7400GPLUS) fixed posting-account=1QdlFQ0AAADvQHRFU-gnUXpZu6etKyL3 = TRIANGLE = A:?->List 4 B:?->List 5 C:?->List 6 List 4[1]->A List 5[1]->B List 6[1]->C List 4[2]->D List 5[2]->E List 6[2]->F {A,B,C,A}->List 1 {D,E,F,D}->List 2 (A+B).852->G (B+C).852->H (C+A).852->I (D+E).852->J (E+F).852->K (F+D).852->L (A+B+C).853->M (D+E+F).853->N {G,H,I,M}->List 3 {J,K,L,N}->List 4 DrawStat === Subject: Re: program: TRIANGLE (for FX7400GPLUS) fixed posting-account=UtgH7gwAAACpBhTelVPOXNP7RAfbtQrK > = TRIANGLE = > A:?->List 4 > B:?->List 5 > C:?->List 6 > List 4[1]->A > List 5[1]->B > List 6[1]->C > List 4[2]->D > List 5[2]->E > List 6[2]->F > {A,B,C,A}->List 1 > {D,E,F,D}->List 2 > (A+B).852->G > (B+C).852->H > (C+A).852->I > (D+E).852->J > (E+F).852->K > (F+D).852->L > (A+B+C).853->M > (D+E+F).853->N > {G,H,I,M}->List 3 > {J,K,L,N}->List 4 > DrawStat Well no wonder your original post made no sense. === Subject: Linear Algebra, Please help. Question: Given a symmetric nxn matrix A, show there is an orthogonal mx U of determinant 1 with U^(-1)AU= D a diagonal matrix. This is what I have so far: Suppose X1.....Xn is a basis of eignvalues of the nxn matrix A of eigen values lambda....(lambda)n. Set C= (X1.....Xn) (c is invertible) Claim that if D is the diagonal (lambda)...(lambda)n then CD=AC Then the jth column of CD is CDj=C(0 . (lambda)j . 0) = ((lambda)j)Xj=AXj D=C^(-1)AC if orthonormal, c is orthogonal. when c is orthogonal c inverse = C^(T) so C^(T)AC if Ax1......Xn C^(-1)AC=C^(T)AC ---------------------------------------------- * Binary Usenet Leeching Made Easy * http://www.newsleecher.com/?usenet ---------------------------------------------- === Subject: Re: Linear Algebra, Please help. > Given a symmetric nxn matrix A, show there is an orthogonal mx U > of determinant 1 with U^(-1)AU= D a diagonal matrix. > This is what I have so far: > Suppose X1.....Xn is a basis of eignvalues of the nxn matrix A > of eigen values lambda....(lambda)n. > Set C= (X1.....Xn) (c is invertible) > Claim that if D is the diagonal (lambda)...(lambda)n then CD=AC > Then the jth column of CD is CDj=C(0 > . > (lambda)j > . > 0) > = ((lambda)j)Xj=AXj > D=C^(-1)AC > if orthonormal, c is orthogonal. when c is orthogonal c inverse > = C^(T) > so C^(T)AC if Ax1......Xn > C^(-1)AC=C^(T)AC I guess that youre working over the real field. Heres some help: if v and w are eigenvectors of A with distinct eigenvalues x and y, then the vectors v and w are orthogonal. Indeed, = = x and = = y. But, since A is symmetric, = = and therefore x = y. So, since x and y are distinct, you must have = 0. I hope that this helps. Jose Carlos Santos === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics [...] Im mostly interested in definitions that hold true all the time. They do, Lester, they do. By definition. Whether a definition refers to anything real is another question. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >[...]> Im mostly interested in definitions that hold true all the time. >They do, Lester, they do. By definition. Whether a definition refers to >anything real is another question. On the other hand, Wolf, it might be interesting to explore this definition of definition a little more objectively to decide how they can refer to anything unreal. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >>[...]Im mostly interested in definitions that hold true all the time. >>They do, Lester, they do. By definition. Whether a definition refers to >>anything real is another question. > On the other hand, Wolf, it might be interesting to explore this > definition of definition a little more objectively to decide how they > can refer to anything unreal. A unicorn is.... === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics posting-account=u3EDPg0AAADaLcRunJk5L0Cw4LDDaXak > Science, Philosophy, Mysticism, Art, Mathematics, and Physics > --------------- > Allow me to summarize certain lines of reasoning relating to the > definition of science, philosophy, and mysticism so as to make these > ideas explicit. > Science argues that A is C because A is B and B is C. > Philosophy argues that A is C either because A is B and let me tell > you a little story about B and C or because let me tell you a little > story about A and B and B is C. > Mysticism just postulates that A is C because X is Y and the two are > connected by let me tell you a little story. > Art just argues that X is Y. > Mathematics just argues that A is C because they have dibs on it. > Physics aruges that A is C cuz they dont need no stinkin reasons. you forgot the most important one: Lester = idiot in all science, philosophy, mysticism, art, math and physics. Thats an invariant under any conceivable transformation local or global. Seems you are an example of a unified theory of stupidity. Mike === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >> Science, Philosophy, Mysticism, Art, Mathematics, and Physics >> --------------- >> Allow me to summarize certain lines of reasoning relating to the >> definition of science, philosophy, and mysticism so as to make these >> ideas explicit. >> Science argues that A is C because A is B and B is C. >> Philosophy argues that A is C either because A is B and let me tell >> you a little story about B and C or because let me tell you a >little >> story about A and B and B is C. >> Mysticism just postulates that A is C because X is Y and the two are >> connected by let me tell you a little story. >> Art just argues that X is Y. >> Mathematics just argues that A is C because they have dibs on it. >> Physics aruges that A is C cuz they dont need no stinkin reasons. >you forgot the most important one: >Lester = idiot in all science, philosophy, mysticism, art, math and >physics. >Thats an invariant under any conceivable transformation local or >global. Seems you are an example of a unified theory of stupidity. Yes. However, yours is only a particular definition and not of any universal significance. When you get around to dealing in universals for a change, do let us know, wont you, sport? === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics <41bfb282.31315365@netnews.att.net> posting-account=u3EDPg0AAADaLcRunJk5L0Cw4LDDaXak I prefer Existential Qualifiers to Universals. I tend to think the latter hode som dogmatism. Anyway, no universally quantified propositions can be proved empirically so there you go, my particular definition may not be of universal significance but its of significance to you:) Mike === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >I prefer Existential Qualifiers to Universals. I tend to think the >latter hode som dogmatism. Anyway, no universally quantified >propositions can be proved empirically so there you go, my particular >definition may not be of universal significance but its of >significance to you:) Well, it may apply to me but it is hardly of significance to me. The problem is that it doesnt apply to anyone but me; so, it is only of particular significance in any event whereas what I say is applicable across the board whence I say it. You might consider getting your words straight before you say them. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics <41bd1833.17476667@netnews.att.net> <41bd9f23.21123889@netnews.att.net> <1Umvd.231528$HA.34538@attbi_s01> posting-account=SqYkwg0AAABibDnSpEgBfNCp5F3aKoMz My main point is that the word equal, with all its secular meanings, is probably a poor choice for =, which you have well described as an approximate relationship, or at least a qualified relationship. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >My main point is that the word equal, with all its secular meanings, >is probably a poor choice for =, which you have well described as an >approximate relationship, or at least a qualified relationship. Most words of any fundamental significance are. And the more we try to specify them the vaguer they get. Maybe theres a conservation of vagueness law at work akin to the uncertainty principle. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics <41bd1833.17476667@netnews.att.net> <41bd9f23.21123889@netnews.att.net> <1Umvd.231528$HA.34538@attbi_s01> <41bec327.26789264@netnews.att.net> posting-account=SqYkwg0AAABibDnSpEgBfNCp5F3aKoMz >My main point is that the word equal, with all its secular meanings, >is probably a poor choice for =, which you have well described as an >approximate relationship, or at least a qualified relationship. > Most words of any fundamental significance are. And the more we try > to specify them the vaguer they get. Maybe theres a conservation of > vagueness law at work akin to the uncertainty principle. I like that! They are very similar! Specifying definition increases vagueness---and like in the subatomic world, we become part of the experiment---we cannot stay outside the system we are measuring! === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >>My main point is that the word equal, with all its secular > meanings, >>is probably a poor choice for =, which you have well described as > an >>approximate relationship, or at least a qualified relationship. >> Most words of any fundamental significance are. And the more we try >> to specify them the vaguer they get. Maybe theres a conservation of >> vagueness law at work akin to the uncertainty principle. > I like that! They are very similar! Specifying definition increases > vagueness---and like in the subatomic world, we become part of the > experiment---we cannot stay outside the system we are measuring! I dont think Lester likes quantum physics ;-) === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >My main point is that the word equal, with all its secular >> meanings, >is probably a poor choice for =, which you have well described as >> an >approximate relationship, or at least a qualified relationship. > Most words of any fundamental significance are. And the more we try > to specify them the vaguer they get. Maybe theres a conservation of > vagueness law at work akin to the uncertainty principle. >> I like that! They are very similar! Specifying definition increases >> vagueness---and like in the subatomic world, we become part of the >> experiment---we cannot stay outside the system we are measuring! >I dont think Lester likes quantum physics ;-) Actually, JPL, I think quantum mechanics relations are very important. I just prefer to derive the uncertainty relation from scratch and derive its implications mechanically from that derivation instead of the not stinkin reasons method used in conventional quantum physics. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics <41bd1833.17476667@netnews.att.net> <41bd9f23.21123889@netnews.att.net> <1Umvd.231528$HA.34538@attbi_s01> <41bec327.26789264@netnews.att.net> <41be294a$0$44096$5fc3050@dreader2.news.tiscali.nl> <41bfb322.31475312@netnews.att.net> posting-account=SqYkwg0AAABibDnSpEgBfNCp5F3aKoMz One thing you may not be aware of in the history of quantum (or what they used to call , atomic) physics, is that the theoreticians were forced to change their descriptions of the atomic and subatomic world (mathematical and otherwise, like Dirac delta functions) because of the theretofore unexplainable observations from the laboratory, i.e. absorption and emission spectra, radiative decay, etc. So Im not sure sometimes and waves other times, depending on how they are constrained. describe---no stinkin reason is the frustrated exclamation of no reason I can see!. My favorite bit of prose---whoever can identify its source get the kewpie doll: The lofty prize, Of Science lies Hidden today as ever. Who, with no thought- To him its brought, To own without endeavor! on the ßip side... stlbl === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >One thing you may not be aware of in the history of quantum (or what >they used to call , atomic) physics, is that the theoreticians were >forced to change their descriptions of the atomic and subatomic world >(mathematical and otherwise, like Dirac delta functions) because of the >theretofore unexplainable observations from the laboratory, i.e. >absorption and emission spectra, radiative decay, etc. So Im not sure >sometimes and waves other times, depending on how they are constrained. >describe---no stinkin reason is the frustrated exclamation of no >reason I can see!. Of course. My primary complaint is the kind of philosophizing quantum theorists are willing to accept as a substitute for knowledge and as collateral for their ignorance. By the phrase from scratch what I have in mind is the definition of definition and the further deduction of Heisenbergs Uncertainty relation from it. I posted my analysis of the first over a year ago and will append a copy. If youre interested, take a gander. >My favorite bit of prose---whoever can identify its source get the >kewpie doll: >The lofty prize, >Of Science lies >Hidden today as ever. >Who, with no thought- >To him its brought, >To own without endeavor! >on the ßip side... >stlbl As Ive received no analytical objections to the following post Im appending several historical observations. > Plancks Constant >Previously in the thread Angular Momentum in Rotating Bodies, I >presented an analytical framework for the interpretation of dr/dt in >circular rotation of a point mass m at velocity v and radius r. No one >I know of agrees with my interpretation of dr/dt. However, in the >interests of further establishing this general framework, I would like >to pursue general developement of the idea which culminates in the >analytical definition of Plancks constant. >We begin by noting that in cases of circular rotation at constant >angular velocity we have a centripetally directed dr/dt acting on >point mass m of a magnitude equal to tangential velocity v. This is >what causes the rotation of v and produces r as a consequence of >rotation. >We then integrate dr/dt along r which produces 1/2 mvr/2pi with units >of measure equal to rr/t. Now, I have been cautioned on several >occasions not to suggest that this quantity represents angular >momentum in conventional terms and I agree. Perhaps we should simply >call it rotational momentum to prevent confusion. >What we notice immediately however is that it bears the same form as >the quantity mvr corresponding to Plancks constant. However, we have >to straighten certain things out in this connection. >In conventional macro angular rotation such as ßywheels we have a >centripetal dr/dt and tangential v which are equal to each other. They >are effectively bound up through tensile forces internal to the body >undergoing rotation. In celestial angular mechanics on the other hand >we have a wide variety of potential dr/dts and tangential orbital >velocities operating in various combinations. >different situation. The tangential velocity of rotation v is constant >under all circumstances. In other words, v = c. Thus dr/dt operates >mass. >second) times an analytical masslet, m0 (kg-sec) and interpret the >quantity mvr as a multiple of nm0vr. Further we can interpret r as a >function of c/n such that Plancks constant = m0cc. In other words, m0 >is roughly on the order of 10^-50 kg-sec in magnitude and Plancks >constant corresponds to the multiple of m0 and the square of the >velocity of light. >We notice several things about rotational momentum. In linear motion >at constant velocity rotational momentum is zero because dr/dt and mvr >are both zero. And in circular rotation at a constant angular velocity >rotational momentum is constant because mvr is constant. This >represents the analytical distinction between circular and linear >motion. >Further we notice that dr/dt can be of any magnitude. It is not bound >by the constancy of the velocity of light as an upper limit because it >doesnt go anywhere. It only produces rotation in relation to actual >tangential motion v = c. >mass and radius of rotation are inversely proportional, that is that >remove DEL in address for email Linear versus Analytical Mechanics One of the really unfortunate aspects of Newtons choice of a linear frame of reference for the analysis of mechanics is that r is poorly defined and t is not defined at all. In other words, r is only defined in direction and t is not defined by any consideration pertinent to the analytical frame of reference. And this had a pernicious impact on the subsequent development of angular mechanics as well as relativistic considerations and quantum mechanics in the twentieth century. The problem is that r and t and their combinations are all we have to work with. Taken to the second level of compounding we have six combinations: r, 1/t, r/t, r/tt, rr/t, and rr/tt. However, in the linear analytical frame of reference the next to last combination rr/t was overlooked because there is no apparent application for it in linear mechanical contexts. On the other hand, in angular frames of reference we have applications for all combinations and all the elements are well defined. The radius of rotation is well defined in terms of direction and magnitude and time is well defined in analytical terms as whatever time is needed for 2pi radians of rotation. The rr/t combination is also well defined in angular terms. However, in extrapolating the idea of rr/t from linear to angular contexts in classical mechanics, whoever devised the analytical approach made the mistake of trying to emulate linear mechanics in the sense of explaining rotation as a linear progression of r instead of a simple radial v in combination with tangential v. This is more akin to an anachronistic pre Newtonian view of mechanics. Kepler thought that some force of angels was needed to keep planets in orbit around the sun and regarded that force as tangential in direction. Newton on the other hand recognized that the only force needed was centripetal in nature and not tangential. But whoever devised the analytical considerations underlying angular mechanics apparently never considered the Newtonian perspective and presumably relied on the pre Newtonian rationale. Thus we wind up with a conceptual schism among the various realms of angular mechanics. On the one hand we have orbital angular mechanics, the macro realm of ordinary angular mechanics, and the micro realm of quantum effects. And unfortunately there is no conceptual integration among them. We are convinced that all represent mechanical realms but we have no basis for comprehending each in terms of the others. Orbital angular mechanics represents the realm of remote interactions dealt with in terms of inverse square centripetal forces and tangential orbital velocities. Whereas the macro realm of ordinary angular mechanics deals with linear analogs such as moments of inertia instead of mass, torque instead of force, and angular acceleration and velocity instead of their linear analogs. The micro realm of angular mechanics on the other hand is dealt with on the merely descriptive basis of formalisms. This is the realm of quantum mechanics - QM - or as I prefer to call it quantum magic where things dont seem to happen for any definite mechanical reason at all. However with the redefinition of macro angular momentum and Plancks constant in circular rotation we are at last in a position to understand the mechanical differences among the realms in conceptual terms. The micro realm of quantum effects is one of constant tangential velocity of rotation v = c and a variable radial dr/dt. The macro realm of ordinary angular mechanics on the other hand is one in which the tangential velocity of rotation is variable but tangential v = radial dr/dt and both are kept in strict synchronization by internal tensile forces. And finally orbital angular mechanics is defined by various combinations of tangential v and radial dr/dt. This is normally thought of in celestial terms but in point of fact applies equally to the atomic realm as well. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics [...] I dont think Lester likes quantum physics ;-) Lester thinks quantum theory is a fraud, or a joke, or both. He believes its a conspiracy by the elitist scientists who ignore his epoch-making discovery of differences among difference (or between - I forget; not that it would make much difference.) Lester doesnt like anything he cant imagine. He doesnt like any dimensions beyond the spatial three, either. Lester believes that if he cant understand it, it must be wrong - because of course whatever he understand must be true. (Thats an example of a vlaid but unsound argument, another thing Lester has trouble with. See his talk about tautologies.) === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >[...]> I dont think Lester likes quantum physics ;-) >Lester thinks quantum theory is a fraud, or a joke, or both. He believes >its a conspiracy by the elitist scientists who ignore his epoch-making >discovery of differences among difference (or between - I forget; not >that it would make much difference.) Not quite, Wolf. Lester thinks the application of quantum principles is etc. etc. etc. and that quantum effects should be derived from stinkin reasons guessing game used by conventional physics. >Lester doesnt like anything he cant imagine. He doesnt like any >dimensions beyond the spatial three, either. Not quite, Wolf. Lester doesnt like anything you can imagine but cant prove. >Lester believes that if he cant understand it, it must be wrong - >because of course whatever he understand must be true. (Thats an >example of a vlaid but unsound argument, another thing Lester has >trouble with. See his talk about tautologies.) Another vlaid faux pas, Wolf. Whats unsound is that your argument is particular and not universal. I may or may not like what I cant understand but unlike yourself I much prefer what I can prove. The only thing of universal significance youve ever said is that tautologies are always true. Can I help it if I then extrapolate the consequences of that uncharacteristic admission? Its your fault yet you blame me. You shoulda stood in bed. The true behaviorist never admits anything is true. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >>My main point is that the word equal, with all its secular meanings, >>is probably a poor choice for =, which you have well described as >>an approximate relationship, or at least a qualified relationship. >> Most words of any fundamental significance are. And the more we try >> to specify them the vaguer they get. Maybe theres a conservation of >> vagueness law at work akin to the uncertainty principle. > I like that! They are very similar! Specifying definition increases > vagueness---and like in the subatomic world, we become part of the > experiment---we cannot stay outside the system we are measuring! > I dont think Lester likes quantum physics ;-) I smell a cat...is it dead yet? === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics <41bd1833.17476667@netnews.att.net> <41bd9f23.21123889@netnews.att.net> <1Umvd.231528$HA.34538@attbi_s01> <41bec327.26789264@netnews.att.net> <41be294a$0$44096$5fc3050@dreader2.news.tiscali.nl> posting-account=SqYkwg0AAABibDnSpEgBfNCp5F3aKoMz Half dead, half alive of course.... === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics > Half dead, half alive of course.... I tend to agree...half the time. When no ones looking. === Subject: Re: Determinant with trigonometry by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBE0sON02857; >Proof, that determinant (with n rows and columns) >| cosx 1 0 0 ... 0 0 | >| 1 2cosx 1 0 ... 0 0 | >| 0 1 2cosx 1 ... 0 0 | >| 0 0 1 2cosx ... 0 0 | = >cos nx >|........................................................... ......... >| 0 0 0 0 ... 2cosx 1 | >| 0 0 0 0 ... 1 2cosx | >The main proble for me is to evaluate this determinant. >Do you have any ideas? >*-----------------------* > www.GroupSrv.com >*-----------------------* Cofactors. Show what youve done so far. === Subject: Re: .99999... still=/= 1 > i think its worth noting that if you use the dedikind [sic] cut def. > of the real numbers then 1/3 (the real number) = {p in Q : p < 1/3 > (the rational number)}. Yes, if you are treating rational numbers and real numbers as completely separate objects, with the reals constructed from the rationals, what you say is true. But in that case a rational number and the corresponding real number are completely different objects, one of them a single rational number and the other a set of rational numbers. But that wasnt the context of the original thread, where all the numbers being discussed were real numbers (at least that is the assumption in the absense of any statement to the contrary). In particular, if the OP had intended that interpretation, he should have specified which numbers he mentionned were supposed to be rational numbers and which were supposed to be real numbers. And even in the context of Dedekind cuts, sets of real numbers arent the same thing as individual real numbers. And the particular set the OP mistakenly indicated using {...} notation wasnt even a Dedekind cut, or any other standard way to construct the reals frm the rationals. So I dont think your remarks effectively rebut what I posted. === Subject: Re: .99999... still=/= 1 >> i think its worth noting that if you use the dedekind cut def. >> of the real numbers then 1/3 (the real number) = {p in Q : p < 1/3 >> (the rational number)}. > Yes, if you are treating rational numbers and real numbers as > completely separate objects, with the reals constructed from the > rationals, what you say is true. But in that case a rational number and > the corresponding real number are completely different objects, one of > them a single rational number and the other a set of rational numbers. > But that wasnt the context of the original thread, where all the > numbers being discussed were real numbers (at least that is the > assumption in the absense of any statement to the contrary). In > particular, if the OP had intended that interpretation, he should have > specified which numbers he mentionned were supposed to be rational > numbers and which were supposed to be real numbers. > And even in the context of Dedekind cuts, sets of real numbers arent > the same thing as individual real numbers. > And the particular set the OP mistakenly indicated using {...} notation > wasnt even a Dedekind cut, or any other standard way to construct the > reals frm the rationals. > So I dont think your remarks effectively rebut what I posted. I did not claim to rebut anything, I just thought I would make that comment. If .3, .33, .333, etc are the corresponding rational numbers (equivalence classes of fractions) then we can say that {.3, .33, .333, etc} is a subset of 1/3 the real number. This just seems interesting and vaguely on topic to me. Sorry for the misspelled Dedekind. -- Justin Young http://web.syr.edu/~jryoun04/ === Subject: Re: .99999... still=/= 1 > there must be something in the real number system that would explain > the time difference in these calculations that shows 1 = 1 representation of the number 1, one of which is very fast and one of which is very slow, you conclude that the two results you get cant possibly be the same number, so there must be something in the real number system that makes 1 a different number from 1 depending on how it was calculated? On what principle of logic do you come to that conclusion? Or is your whole starting of this thread a lie? You admit that the periodic decimal fraction .99999... (using the real metric) and the integer 1 (using the usual embedding of integers in reals) are exactly the same real number, and the only thing different about the two representations of them is that it takes more work to figure out what real number is a particular periodic decimal fraction than it takes to figure out what real number is a particular integer, but assuming you do the work to figure out what each is, they turn out to be exactly the same real number? > Otherwise both computers would have shown 1 = 1 at the same time. So you believe its impossible to write two different computer programs that produce the same result but one runs faster than the other and so it gets the answer faster than the other? So if one computer program takes longer than the other, you know for sure the results will be different? === Subject: Re: .99999... still=/= 1 >> there must be something in the real number system that would explain >> the time difference in these calculations that shows 1 = 1 >representation of the number 1, one of which is very fast and one of >which is very slow, you conclude that the two results you get cant >possibly be the same number, so there must be something in the real >number system that makes 1 a different number from 1 depending on how >it was calculated? On what principle of logic do you come to that >conclusion? >Or is your whole starting of this thread a lie? You admit that the >periodic decimal fraction .99999... (using the real metric) and the >integer 1 (using the usual embedding of integers in reals) are exactly >the same real number, and the only thing different about the two >representations of them is that it takes more work to figure out what >real number is a particular periodic decimal fraction than it takes to >figure out what real number is a particular integer, but assuming you >do the work to figure out what each is, they turn out to be exactly the >same real number? >> Otherwise both computers would have shown 1 = 1 at the same time. >So you believe its impossible to write two different computer programs >that produce the same result but one runs faster than the other and so >it gets the answer faster than the other? So if one computer program >takes longer than the other, you know for sure the results will be >different? I am using a non-standard approach to something that seems to exist between the cracks of real numbers. Like for example, .999... < X < 1 What is the space between .999... and 1 which we can call X, that prevents, .999... from perfectly equaling 1. I did say .999... does converge to 1, but in a series a convergence value never perfectly equals that value. Its just the nearest thing that does. I approached this with the use of time or a dimensionless space that doesnt allow it. The fact is there is no numbers between .999... and 1, but they are right next to each other. I used two approaches to this, higher order differential analysis which could analyze this in infinitesimal space DIM -->oo between .999... and 1 or by the Gamma function which I used as a function of time approaching infinity. Using the Gamma Function, gamma(alpha) = INTEGRAL ( 0 to oo) (e^-t) * t^(alpha-1) dt n-->oo gamma ( 1 + 9/10^n) ~= 1 as 9/10^n converges to 1 In each case, the more time you spend approaching 1, the closer you get to 1. Another example I used to show that there was an inequality of time with equations, was by the example of the Distributive Property. a ( b + c ) = ab + ac There is a difference in the amount of time needed to make the calculations on the left and right side of the equal sign of this equation even thought they are equal to each other. So I used a non-standard approach in showing that time is a significant factor in the cracks of space between real numbers. Smarts Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 > there must be something in the real number system that would explain > the time difference in these calculations that shows 1 = 1 >>representation of the number 1, one of which is very fast and one of >>which is very slow, you conclude that the two results you get cant >>possibly be the same number, so there must be something in the real >>number system that makes 1 a different number from 1 depending on how >>it was calculated? On what principle of logic do you come to that >>conclusion? >>Or is your whole starting of this thread a lie? You admit that the >>periodic decimal fraction .99999... (using the real metric) and the >>integer 1 (using the usual embedding of integers in reals) are exactly >>the same real number, and the only thing different about the two >>representations of them is that it takes more work to figure out what >>real number is a particular periodic decimal fraction than it takes to >>figure out what real number is a particular integer, but assuming you >>do the work to figure out what each is, they turn out to be exactly the >>same real number? > Otherwise both computers would have shown 1 = 1 at the same time. >>So you believe its impossible to write two different computer programs >>that produce the same result but one runs faster than the other and so >>it gets the answer faster than the other? So if one computer program >>takes longer than the other, you know for sure the results will be >>different? > I am using a non-standard approach to something that seems to exist >between >the cracks of real numbers. Like for example, >.999... < X < 1 > What is the space between .999... and 1 which we can call X, that prevents, >.999... from perfectly equaling 1. I did say .999... does converge to 1, but >a series a convergence value never perfectly equals that value. Its just the >nearest thing that does. > I approached this with the use of time or a dimensionless space that >doesnt >allow it. The fact is there is no numbers between .999... and 1, but they are >right next to each other. I used two approaches to this, higher order >differential analysis which could analyze this in infinitesimal space DIM >-->oo >between .999... and 1 or by the Gamma function which I used as a function of >time approaching infinity. > Using the Gamma Function, >gamma(alpha) = INTEGRAL ( 0 to oo) (e^-t) * t^(alpha-1) dt > n-->oo >gamma ( 1 + 9/10^n) ~= 1 >as 9/10^n converges to 1 > In each case, the more time you spend approaching 1, the closer you get to > Another example I used to show that there was an inequality of time with >equations, was by the example of the Distributive Property. >a ( b + c ) = ab + ac > There is a difference in the amount of time needed to make the calculations >on the left and right side of the equal sign of this equation even thought >they >are equal to each other. So I used a non-standard approach in showing that >time >is a significant factor in the cracks of space between real numbers. One may ask, what about beyond the limits of infinity, or 1 less than, or 1 more than infinity with the series .999... . Ok lets take a look where this nth term goes. ( reference: MathCAD professional) looking at the nth term 1 less than infinity n--> oo - 1 lim 9/10^n --> 90/10^oo ~= 0 n--> oo + 1 lim 9/10^n --> 9/(10*10^n) ~= 0 n--> oo + oo lim 9/10^n --> 9/(10^oo)^2 ~= 0 n--> oo + oo + oo lim 9/10^n --> 9/(10^oo)^3 ~= 0 n-->oo^oo lim 9/10^n --> 9/(10^oo)^oo ~= 0 n-->oo^oo^oo lim 9/10^n --> 9/((10^oo)^oo)^oo ~= 0 I used ~= to mean approximately equal to zero because as n-->oo the nth term in the series the limit approaches --> 0, never really reaching it. This is a non-standard analysis. I am trying to show that there is a space existing between, .999... and 1. Theoretically time is the separation factor between real numbers. The more time spent, the closer you get to the actual convergence value. So since 0 is only approached in any case approaching infinity or beyond the limits of infinity, .999... =/= 1 It only converges to 1 never reaching 1. Smarts Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 >> there must be something in the real number system that would explain >> the time difference in these calculations that shows 1 = 1 >representation of the number 1, one of which is very fast and one of >which is very slow, you conclude that the two results you get cant >possibly be the same number, so there must be something in the real >number system that makes 1 a different number from 1 depending on how >it was calculated? On what principle of logic do you come to that >conclusion? >Or is your whole starting of this thread a lie? You admit that the >periodic decimal fraction .99999... (using the real metric) and the >integer 1 (using the usual embedding of integers in reals) are exactly >the same real number, and the only thing different about the two >representations of them is that it takes more work to figure out what >real number is a particular periodic decimal fraction than it takes to >figure out what real number is a particular integer, but assuming you >do the work to figure out what each is, they turn out to be exactly the >same real number? >> Otherwise both computers would have shown 1 = 1 at the same time. >So you believe its impossible to write two different computer programs >that produce the same result but one runs faster than the other and so >it gets the answer faster than the other? So if one computer program >takes longer than the other, you know for sure the results will be >different? >> I am using a non-standard approach to something that seems to exist >>between >>the cracks of real numbers. Like for example, >>.999... < X < 1 >> What is the space between .999... and 1 which we can call X, that >prevents, >>.999... from perfectly equaling 1. I did say .999... does converge to 1, but >>in >>a series a convergence value never perfectly equals that value. Its just >the >>nearest thing that does. >> I approached this with the use of time or a dimensionless space that >>doesnt >>allow it. The fact is there is no numbers between .999... and 1, but they >are >>right next to each other. I used two approaches to this, higher order >>differential analysis which could analyze this in infinitesimal space DIM >>-->oo >>between .999... and 1 or by the Gamma function which I used as a function >>time approaching infinity. >> Using the Gamma Function, >>gamma(alpha) = INTEGRAL ( 0 to oo) (e^-t) * t^(alpha-1) dt >> n-->oo >>gamma ( 1 + 9/10^n) ~= 1 >>as 9/10^n converges to 1 >> In each case, the more time you spend approaching 1, the closer you get >>1. >> Another example I used to show that there was an inequality of time with >>equations, was by the example of the Distributive Property. >>a ( b + c ) = ab + ac >> There is a difference in the amount of time needed to make the >calculations >>on the left and right side of the equal sign of this equation even thought >>they >>are equal to each other. So I used a non-standard approach in showing that >>time >>is a significant factor in the cracks of space between real numbers. > One may ask, what about beyond the limits of infinity, or 1 less than, or >more than infinity with the series .999... . Ok lets take a look where this >nth term goes. >( reference: MathCAD professional) >looking at the nth term >1 less than infinity >n--> oo - 1 >lim 9/10^n --> 90/10^oo ~= 0 >n--> oo + 1 >lim 9/10^n --> 9/(10*10^n) ~= 0 >n--> oo + oo >lim 9/10^n --> 9/(10^oo)^2 ~= 0 >n--> oo + oo + oo >lim 9/10^n --> 9/(10^oo)^3 ~= 0 >n-->oo^oo >lim 9/10^n --> 9/(10^oo)^oo ~= 0 >n-->oo^oo^oo >lim 9/10^n --> 9/((10^oo)^oo)^oo ~= 0 > I used ~= to mean approximately equal to zero because as n-->oo the nth >term >in the series the limit approaches --> 0, never really reaching it. This is a >non-standard analysis. I am trying to show that there is a space existing >between, .999... and 1. Theoretically time is the separation factor between >real numbers. The more time spent, the closer you get to the actual >convergence >value. > So since 0 is only approached in any case approaching infinity or beyond >the >limits of infinity, >.999... =/= 1 > It only converges to 1 never reaching 1. I have to make a correction here, .999... < X < 1 <-- this isnt correct. It should be like this. .999...> X < 1 .999... > X and, X < 1 What is X? The nth term approaching the limit of oo goes to -->0. What does this mean? It means there is a terminator digit before it reaches infinity. It looks something like this. Non-Standard Analysis n--> oo - 1 lim 9/10^n --> 90/10^oo X = 90/10^oo ^ | notice the zero .999... > 90/10^oo < 1 or, .999...0 < 1 It never reaches 1. At one digit less than oo, ( assuming you really reach infinity) the nth term reaches 0 so, .999... reaches 0 not 1. Even if you go past oo by one digit, it still doesnt reach 1. Non-Standard Analysis n-->oo + 1 lim 9/10^n --> 9/(10*10^oo) --> 0 Smarts Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: Comments on senior research idea posting-account=8F96fgwAAABusuCFPp4BNz74T0xFr-Jl > I had an idea for my undergraduate senior research project. I thought > about taking a look at the Bible codes. For those that dont know this > is where people have found words/phrases at certain intervals in the > Bible. What does everyone think of this idea? Any question you think > I should look into? I am still trying to develop a list of questions > to ask about it. If you do decide to do a project on Bible Codes, I would recommend studying Brendan McKays website on the subject to ensure that you do not reinvent the wheel: http://cs.anu.edu.au/~bdm/dilugim/torah.html === Subject: Santa Claus is the god of math !!!! There is NOTHING that Santa cannot prove !!!! Wish for a proof for Christmas and youll get it! === Subject: Re: Santa Claus is the god of math !!!! !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~( 5eZ41to5f%E@ELIi $t^ VcLWP@J5p^rst0+(Ô>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > There is NOTHING that Santa cannot prove !!!! > Wish for a proof for Christmas and youll get it! We already had a thread discussing acceptance of Gods proofs in this group recently. I vainly refer to after which it petered out. Go back from there. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Santa Claus is the god of math !!!! > There is NOTHING that Santa cannot prove !!!! > Wish for a proof for Christmas and youll get it! > We already had a thread discussing acceptance of Gods proofs in this > group recently. I vainly refer to > after which it petered out. Go back from there. Im God and I have numerous proofs. Everyone thinks Im joking year after year with proof of god posted in front of them..... oh haha. NOT JOKING..... serious thread. Jennifer Brooks gender water EVE add the vowels in my name = 7 add the consonants in my name = 7 initial 7 G god initials GC 73 born 7/3 born in Melbourne, NewChurch St. Man 7 G ala James Bond Woman 10 (J) ala Bo Derek heres one proof, just answer the questions honestly working them out in your head, Ive had to state that 1,000 times already on usenet, everyone wants to find some interpretations that makes it void. ...what if I answer it wrong then that means it doesnt work so its not a proof... what if i look up the answers.... 1000 excuses >Statistical Anomaly Proof Of God. >This is the simplest proof of God you will ever see, and it takes 3 minutes for anyone >to verify. >----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, Id hold it upright, >with the intent of attacking my assailants knife hand. >--ANSWER OPTIONS--------------- >A cliff86 >B Rust >C Shanx >D NormDePloom >E Rich Shewmaker >F CNote >G Jeff >H See You In Hell My Friend. >I Someone >J Greg Neill >Which author best fits each post? >You have 1 chance in 1,000 of getting this quiz right. >In theory people could still solve it with 100 random alias options, breaking million to 1 >odds of these 3 posts all occuring on the same day. Each post is a large coincidence >and they all happened one one day. No non paranormal person can replicate this >collation of posts. >WHY IS IT PROOF OF GOD. >3/ on 02/02/2002 one year before I yelled to a dozen newsgroups > proof of God 0202 2002 using this remark >replies to me are indicative of their alias, part of deity. >LIVE WITNESSES >P A R A N O R M A L W I T N E S S E S >(alt.magic) >> Yay! Bravo! >> The only possible explanation is, of course, that I have latent psychic >> powers that allow me to be 100% accurate 50% of the time when I have to >> choose from two choices! >> Seriously though, good work. I want more Usenet magic! >(sci.skeptic) > I *will* include some comments, but Ill add some spoiler space in case > anyone is really wanting to give it a go. > My guess is > 1c, 2a, 3b >>And the spoiler answer follows... >>Pretty good, Herc! I was spot on with my guess. >>Danny >(alt.astrology) >>One other thing - just to verify (not that thats really necessary, >>I am sure) your channeling abilities: >>Eric >(alt.sci.physics) >>Herc - strongly related aha (Hercules)! and the detective work (Poirot). >>Two for one mon ami! >>Mike > In alt.sci.physics, (thats where the proof is now LIVE) > I have no less than 8 consecutive people posting messages > that are ~strongly related~ to their name (alias). >(sci.skeptic) >> What if some unidentified paranormal power can only beat odds of one in a >> thousand? Then it does not exist? >If you can do this consistantly, then we just calculate how many times you need >to do it to come up with odds of a million to one, and youre in. >>If its that easy I would think that Herc could win. >(sci.skeptic) > 1) Which one of these actually went by his/her first name? > a) My mother > b) my mother-in-law > c) my father-in-law >> b >Mmmmm-mmmm! Wholesome humble pie! >I could try lying to save face but I have to admit it: >Four out of four right. >Eric >(aus.tv) === >Subject: OT: RESULTS: Herc-Truman test >Mathematically, and conservatively, Id say the odds of you ßuking those >answers would be, say, 1/100 * 1/3 * 1/1 * 1/100 which is a 1 in 30,000 === Subject: Re: Santa Claus is the god of math !!!! === >Subject: Re: Santa Claus is the god of math !!!! >Message-id: <3283krF3h3876U1@individual.net> There is NOTHING that Santa cannot prove !!!! >> Wish for a proof for Christmas and youll get it! >> We already had a thread discussing acceptance of Gods proofs in this >> group recently. I vainly refer to >> after which it petered out. Go back from there. >Im God and I have numerous proofs. Everyone thinks >Im joking year after year with proof of god posted in >front of them..... oh haha. Looks like youve found the one thing god cant prove. -- Mensanator Ace of Clubs === Subject: Re: Why (or not) use single letter variable names? at 06:00 PM, mathar@amer.strw.leidenuniv.nl (Richard Mathar) said: >Programmers have to write things down to make it understandable to a >machine (called compiler), which will rewrite this as a long >sequences of 0s and 1 that can be executed by a computer. That is equally simple whether he uses single letter names or long names. Further, it overlloks the fact that code is also read by human beings. >This is different from talking to humans with associative brains. Only if the program never has to be changed. In the real world, an unreadable program is a bad program even if it produces correct results. >As already argued before, the mathematical notation is made for >pencils and can use font styles, super/subscripts, Greek and other >alphabets to enlarge its name space, which is difficult in an 7-bit >ASCII oriented, standardized environment. Computers can handle more than ASCII, and there have been programming languages that required more than ASCII. >Once the programing concepts have >made enough progress such that we can write formulas with our pens >on the touch-screen and click on the compile-and-compute button, >we may be able to return to the 2-dimensional graphical format, but >as long as computer programs are to be made of byte sequences (ASCII >art) so they can be unambiguously stored and retrieved from computer >media, this will stay that way. No. The terms byte sequences and ASCII art are quite different. ASCII art is certainly not a viable way to represent Mathematics on a computer, but an extended character set is. Santayana. >For now, modern computer languages have already incorporated most of >the symbols of the keyboards into their design; this started when C >began to use {, [, &, ? and <, > (where FORTRAN only knew ( and >.gt.). No, it started long before C. >Another point is that a mathematician can handle a set with a >symbol like N, but a programmer has to create and manipulate >instances of these; No. There is nothing to prevent a programming language from having predefined constants, and some languages do have them. >I am aware that this is not firm ground here, but I think that the >mathematicians proofs, lemmas are equivalent to the functions >of the programmer; whereas the mathematician can start a valid proof >with s.th like Let p a prime.., the programmer has to implement >s.th. that tests a given integer (the instance) on primeness >because otherwise hell end up in the realm of core-dumps. No. Mathematicians refer to theorems that others have already proven; programmers refer to functions that others have already written. Unless your programming language does not support the use of external subroutine libraries, there is nothing to contrast here. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: Re: Why (or not) use single letter variable names? >> Then, mathematicians name things in a common abstraction. Tolerance >> becomes Epsilon, Angle becomes Theta, etc. [...big snip...] > It could be done much more efficiently and produce better > code. Computer terminology seems to be deliberately > intended for confusion with most of mathematics terminology > beyond arithmetic and simple algebra; symbols with many > mathematical meanings are misused, Now, thats just silly. Most computer languages actually hew a lot closer to a standard meaning for symbols than mathematics does! Consider the operator *, which in programming circles means multiplication. Always has, always will, except in APL. But mathematics uses the star or asterisk to mean all kinds of different things (Ive seen it used this semester alone as: F* : Skolemization of F x* : just another derivational suffix x*y : just some random binary operator And there are at least three competing conventions in mathematical notation for how to represent multiplication itself! (times, cdot, and nothing at all.) And dont get me started on parentheses! Even in C, theyve only got three distinct meanings; and there exist some functional languages where theyre only ever used for grouping. Contrast this to math, where (a,b) could mean probably half a dozen things. [...] > We need to get away from ASCII and have longer codes, which > can avoid writing epsilon, but just have the one Greek > letter appear on the screen or the print, without print > spacing. Editing becomes otherwise a laborious task. Have you ever tried to enter Unicode text on a 101-key keyboard? Its a real pain. I love Unicode, but its really not the future of programming. At best, it will allow your coworkers to comment their code in /portable/ Devanagari. (If you just want ßashy Greek letters and text effects, why not learn CWEB? ;) I have no idea what you meant by without Ôprint spacing. Is it relevant? -Arthur === Subject: Re: Why (or not) use single letter variable names? multiplication. Always has, always will, except in APL. In C, * has at least four other uses: for starting and ending comments, as a pointer declarator, as a pointer dereference operator, and something weird related to VLA function parameters in C99. -- Ben Pfaff email: blp@cs.stanford.edu web: http://benpfaff.org === Subject: Re: Why (or not) use single letter variable names? > Consider the operator *, which in programming circles means > multiplication. Always has, always will, except in APL. > In C, * has at least four other uses: for starting and ending > comments, as a pointer declarator, as a pointer dereference > operator, and something weird related to VLA function parameters > in C99. On the other hand, we all start our journey in maths learning that Ôx means multiply. How confusing is that? - Gerry Quinn === Subject: Re: Why (or not) use single letter variable names? >> Consider the operator *, which in programming circles means >> multiplication. Always has, always will, except in APL. >> In C, * has at least four other uses: for starting and ending >> comments, as a pointer declarator, as a pointer dereference >> operator, and something weird related to VLA function parameters >> in C99. >On the other hand, we all start our journey in maths learning that Ôx >means multiply. >How confusing is that? I believe that Algol may have wanted to switch it back to a midline Ôx. The use of Ô* in Fortran was that this was an early language, tied to a 48-character (including space) punched card system, and even that requiring some changes. Considering that the alphabet, the numerals, parentheses, space, and +-., already use 43 characters, there was not much available to work with. But Fortran was originally intended as a highly limited language. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Why (or not) use single letter variable names? === >Subject: Re: Why (or not) use single letter variable names? >Message-id: Consider the operator *, which in programming circles means >> multiplication. Always has, always will, except in APL. >> In C, * has at least four other uses: for starting and ending >> comments, as a pointer declarator, as a pointer dereference >> operator, and something weird related to VLA function parameters >> in C99. >On the other hand, we all start our journey in maths learning that Ôx >means multiply. >How confusing is that? Very, when talking about 3x5 rectangles. >- Gerry Quinn -- Mensanator Ace of Clubs === Subject: Re: Why (or not) use single letter variable names? :>> Then, mathematicians name things in a common abstraction. Tolerance :>> becomes Epsilon, Angle becomes Theta, etc. : [...big snip...] :> It could be done much more efficiently and produce better :> code. Computer terminology seems to be deliberately :> intended for confusion with most of mathematics terminology :> beyond arithmetic and simple algebra; symbols with many :> mathematical meanings are misused, : Now, thats just silly. Most computer languages actually hew a lot : closer to a standard meaning for symbols than mathematics does! : Consider the operator *, which in programming circles means : multiplication. I do not think that multiplication was ever a standard meaning for * outside of programmining languages. In programming circles * also commonly means and wildcard or Kleene star. Heres a little Perl program where * has four meanings. $m=*STDIN; while ($n=<$m>) { ($a,$b)=($n =~ /(d*)*(d*)$/); $t+=$a*$b; } Stephen === Subject: Re: Why (or not) use single letter variable names? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBDLiJn18010; >The ASCII-fication as we see on sci.math, , or as Leroy Quet calls it >>linear mode, for what it may be cumbersome, it is at >>least legible in most cases. >More imporantly, it doesnt take a GBs worth of running >software to read it. >/BAH >Subtract a hundred and four for e-mail. 64K should be enough. Seriously, a lot can be done with 64 kilobytes of memory and a frame buffer. You computers video card probably has a very powerful processor and megabytes of memory. Some projects ofßoad non-graphics processing to the video card. Im not volunteering to implement TeX and MathML on a boot ßoppy that While that may be so, I know how to acquire the specifications to do that. The know-how I have, the experience I dont. Some, I do. There are reasons to be concise, and modern processor, storage, and memory limitations are generally not a concern. Youre old school. Thats the opposite of new school. When did you first own your own computer, and what was it? I have more than ten computers. This ones called a G4 Cube. Twenty years ago the computer youre using to read this USEnet post was called a supercomputer and cost tens of millions of dollars. Thats good! Mathematics is all over the place. I aspire to Quet-ness, Quet-icity, Quet-dom. Im a big fan of Quet. 12055 F. === Subject: Re: Why (or not) use single letter variable names? >I know at least one programmer who thinks it would be kind of cool >>if we could use unicode characters for variable names. Occasionally, >>for instance, we use variables with names like theta, and he thinks >>it would be cute if we could just have a little theta instead. I do not consider it cool; for mathematical typing, I consider it very important. I would definitely like to be able to type characters in with a fixed-width code with lots of characters. The old Apples had something like this, with a means of showing an optional keyboard. >Your programmer needs to consider a lot more things than cute. >Some of the programmers I worked with (a long time ago) could >type 60 words/minute. Adding more characters on the keyboard >would slow them down. Yes, children, code is still typed in >using keyboard-ßavored interfaces by Real Programmers. Adding more shifts would not make that much of a slowdown. And typing theta requires 5 characters, or if one needs to use an escape, theta requires 6. BTW, some of the Algol designers were using Flexowriters, which did have multiple shifts, instead of the keypunches at the time. It was intended that subscripts and superscripts be renderered accordingly. And many of the keywords in Algol were to be implemented by other means; begin could be hitting a special key with a special character put in. I do not know my typing speed, but I suspect 40 wpm. However, I type as I think, and I would find it faster to use lots of characters. >Another thing that he needs to consider very seriously is >backwards compatibility. Now, I know that this is not a >PC thing but dismissing this as a goal is going to byte >everybodys butt soon. The backwards compatibility is not that much of a problem. One just has a dictionary translating back to ASCII codes, with possibly some extra characters indicating the changes. Variable names are irrelevant, and we need to realize this. >I could on with other problems with being cute. :-) >> ..The >>inconveniences that go with using non-ASCII character sets in >>programming are worse, however, which also adds to the incentive >>to use longer names there. >People are using longer names because they can. I would have >never allowed the functional specs to pass if Id been in those >development groups. The reason people use these things is to >avoid documenting what they do. Take a good look at the mess >we have on certain PCs and youll figure out why documenting >thoroughly in human-readable form is a basic necessity in >the computing biz. I agree. For the computer, it only slows things down by the length of time needed to read the name, compare it with the name list, and convert to a reduced family. Also, in interactive programs, it needs to do the reverse operation. >A programmer who knows his biz will not have to read the >purpose of a variable every time he needs to reference it >in his code. This is wasting valuable time and resources. This is absolutely correct. The name is largely irrelevant. However, short names, like using c3 for the third case, may be useful. >/BAH >Subtract a hundred and four for e-mail. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Why (or not) use single letter variable names? >>linear mode, for what it may be cumbersome, it is at >least legible in most cases. >>More imporantly, it doesnt take a GBs worth of running >>software to read it. >64K should be enough. Oh, son. 64K is too much to read ASCII. >Seriously, a lot can be done with 64 kilobytes of memory and a frame buffer. You computers video card probably has a very powerful >processor and megabytes of memory. Some projects ofßoad >non-graphics processing to the video card. I didnt say it was impossible to do. I am pointing out that information stored using todays hard/software may not be readable using tomorrows hard/software. Another bad assumption made is that power will be available at all times to read this information. Another bad assumption is that this information will be accessible even if a component (software and/or hardware) fails. Another bad assumption is that this information will be retrievable even if a virus fiddles your bits. >Im not volunteering to implement TeX and MathML on a boot ßoppy that >While that may be so, I know how to acquire the specifications to do >that. The know-how I have, the experience I dont. Some, I do. >There are reasons to be concise, and modern processor, storage, and >memory limitations are generally not a concern. Im trying to point out that they should be a concern. >Youre old school. Thats the opposite of new school. And the new school is a gaggle of foolish geese if the above assumptions are made. > .. When did you >first own your own computer, and what was it? Im still using the first computer I bought with my money. Before I stopped working, I didnt use my money to have access to computer gear and software. > ..I have more than ten >computers. This ones called a G4 Cube. >Twenty years ago the computer youre using to read this USEnet post >was called a supercomputer Nope. Youre wrong; actually, there at least three errors in that sentence. :-) > ... and cost tens of millions of dollars. Nope. Not tens, just a couple. >Thats good! >Mathematics is all over the place. I aspire to Quet-ness, Quet-icity, >Quet-dom. Im a big fan of Quet. >12055 F. I dont speak hex. /BAH Subtract a hundred and four for e-mail. === Subject: Re: Mathematical Logic posting-account=LXpajg0AAABnezcOAC7aEOeIgA2fVK2b === > Subject: Re: Mathematical Logic > I. Let Gamma = {negation of all v_1P_v_1, P_v_2, P_v_3,...}. > P denotes the relation; v_i denotes elements in sentences. > Whats Gamma, { ~Av1.P(v1), ~Av2.P(v2), ... } ? > TX_MG: Gamma = gamma = {~Av1P_(v1), P_(v2), P_(v3), ...} if you want to have ~ denote as negation, A denotes as the for all. > v2 P(v2) isnt well formed formula. TX_MG: NO!! you are wrong. in first-order logic, gamma = {~Av1P(v1), P_(v2), P_(v3), ...} is a wff! > Is Gamma consistent? Is Gamma satisfiable? > BUT, how do i apply completeness theorem to this prove? > -- > II. determine the following sentences, either show there is a > deduction(from empty set)or give a counter-model(i.e., a > stucture in which it is false): > d) Not exist y for all x P_(xy) <--> not P_(xx) > d: deduction > Do you mean? > Not exist y for all x (P_(xy) <--> not P_(xx)) > TX_MG: yeah you are right, i mistyped. > Assume d, prove Ay.Ax.(x = y) > -- > III. Assume the language (with equality) has just the > parameters (for all) and P, where P is a two-place predicate > symbol. Let A be the structure with |A| = Z, the set of integers > (positive, negative, and zero) and with belong to P^A if > |a-b|=1. Thus A looks like a infinite graph: > ...<-->*<-->*<-->*<-->... > Show that there is an elementarily equivalent structure B that > is not connected. (being connected means that for every two > members of |B|, there is a path between them. A path--of length > n -- from a to b is a sequence with a = p_0 > and b = p_n and belong to P^B for each i.) > How about (1/2)Z ? > TX_MG: what about it? > Yea, what about it? What language? FOL with = and P. > What are the axioms for <=? None? Then (1/2)Z is a model. > You can also define for (1/2)Z, aRb when |a - b| = 1. > Is (1/2)Z equivalent to Z? Is it elementary? > I know I need to apply compactness, > but i dont know how to get it start. > Me neither. Are there axioms for P? > ---- === Subject: Re: Mathematical Logic === Subject: Re: Mathematical Logic > I. Let Gamma = {negation of all v_1P_v_1, P_v_2, P_v_3,...}. > P denotes the relation; v_i denotes elements in sentences. > in first-order logic, gamma = {~Av1P(v1), > P_(v2), P_(v3), ...} is a wff! Its a set of wffs. > Is Gamma consistent? Is Gamma satisfiable? Gamma is omega-inconsistent. > BUT, how do i apply completeness theorem to this prove? ---- === Subject: Re: Mathematical Logic posting-account=0pGp0w0AAACCkl2ylaAtx3LqRXqsqqck i know i proved it. but thank you however.... === Subject: A set of Logical axiom. posting-account=0pGp0w0AAACCkl2ylaAtx3LqRXqsqqck Please help this problem: Given @ denote as a set of logical axioms: 1. Tautologies; 2. for all x, alpha --> (alpha_t)^x, where t is substitutalbe for x in alpha; 3. for all x, (alpha --> beta) --> (for all x alpha --> for all x beta); 4. alpha --> for all x alpha, where x does not occur free in alpha; 5. x=x; 6. x=y --> (alpha --> alpha), where alpha is atomic and alpha is obtained from alpha by replacing x in zero or more (but not necessarily all) places by y. now show following: a. suppose we add to @ some formula phi that is not valid. show that the soundness theorem now fails. b. at the other extreme, suppose we tke no logical axoms at all: @ = empty set. show that the completeness theorem now fails. c. supose we modify @ by adding one new valid formula. explain why both the soundness theorem and the completeness theorem still hold. === Subject: Re: solvable group posting-account=0pGp0w0AAACCkl2ylaAtx3LqRXqsqqck >>Proof: >let G_1 and G_2 be two groups. >G_1 = N_1, N_2, ..., N_n, = {e} >G_2 = (N_1), (N_2), ..., (N_n), = {e} >> Me: >>Lousy notation. You mean: >>G_1 = N_1 > N_2 > ... > N_n = {e} >>where N_i is normal in N_{i-1} and N_{i-1}/N_{i} is abelian. >>Likewise for G_2. >>And you do not know for sure that they are same length. You should >say >>a few words about adding terms if necessary to make sure both normal >>series have the same length. >> Reply: >>% what about by Schreier Theorem? i added in G_1 = N_1 >(N_2)^1 >(N_2)^2> ...>(N_2)^n-1 > N_2 > ... >(N_n)^1> (N_n)^2 > (N_n)^3> ... >(N_n)^n-1> N_n = {e} >>such that: (N_i)^j := N_i(N_i-1 Intercept M_j, j= 1, 2, ..., n-1) >> Which Schreiers Theorem? What is M_j? >> The only entry on Schreier Theorem in the index to Rotmans >> Introduction to the Theory of Groups is the one that refers to the >> theorem giving a bijection between the equivalence classes of >> extensions with the second cohomology group. The other one I can >think >> of is the Nielsen-Schreier Theorem on subgroup of the free >> group. Neither seems particularly applicable. >> The point is: you were ASSERTING that both groups would have solvable >> series of the same length. While it is trivial to make sure they do, >> the definition by itself does not assure such a thing. So you need to >> say that you are doing it. >> Its trivial in this case: just add enough copies of the trivial >group >> to whichever series is shorter to make sure they both have the same >> length. >then >G_1 x G_2 >> Me again: >>And this comes from your lousy notation, and, I suspect, from not >>understanding what G is solvable means. G_1 is NOT an n-tuple of >>groups. >>Remember: a group K is solvable if and only if there exists a chain >of >>subgroups, >>K = K_1 > K_2 > ... > K_n = {e} >>such that K_i is normal in K_{i-1} and K_{i-1}/K_i is abelian, for >>each i=2,3,....,n. >is that right? >>Not as written. You are close, though: the subgroups of G_1 x G_2 >you >>want are the subgroups N_1 x N_1, N_2 x N_2, N_3 x N_3,..., >>etc. You then need to say a few words about why N_i x N_i is normal >>in N_{i-1} x N_{i-1}, and why the quotient is abelian (which is not >>hard). >>% by that i use theorem, if G is a group, G/[G, G]. >> If G is a group, G/[G,G] WHAT? You did not even write a full >sentence, >> let alone a theorem. >sorry, i mean if G is a group, G/[G, G] is a abelian. > Yes. So? > You seem to be either utterly confused, or utterly unable to write > clearly what you mean. > Heres a definition of solvable: (many different equivalent > definitions; this is the one I will use) > DEF. A group G is solvable if and only if there is a finite sequence of > subgroups, H_1, H_2, H_3,..., H_n, such that: > (i) H_1 = G; > (ii) H_n = {e}; > (iii) H_i is normal in H_{i-1} for i=2,...,n. > (iv) H_{i-1}/H_i is abelian for i=2,...n. > If you are using a different definition, then state it. > You were asked to prove that if G_1 and G_2 are each solvable, then > G=G_1 x G_2 is also solvable. > So you are told: > (1) There is a sequence of subgroups H_1, H_2,..., H_n of G_1 which > satisfy (i)-(iv) above. > (2) There is a sequence of subgroups K_1, K_2, ..., K_m of G_2 > which satisfy (i)-(iv) above. > (Note the definition does not require that all such sequences have the > same length, so there is no reason to assume both sequences have the > same length a priori). > We want to show: > (*) There is a sequence of subgroups M_1,....,M_r of G_1 x G_s > which satisfy (i)-(iv) above. > was in the right track; but the details were completely muddled to the > point of incoherence. > So, first, to simplify things, we will extend the shortest of the two > sequences: if n then we define K_{m+1} = K_{m+2} = ... = K_n = {e}. Say, without loss > of generality, that m<=n. > Now, the sequence K_1,K_2,....,K_m,K_{m+1},...,K_n still satisfies > (i)-(iv), so it is another witness for for the solvability of K. I > will now work with the sequences > H_1, H_2, ..., H_n > and K_2, K_2, ..., K_n. > I define the sequence M_1, M_2,...,M_n by letting M_i = H_i x K_i. > What I need to show now is that this is a sequence of subgroups of G_1 > x G_2, which satisfies (i) through (iv). That they are subgroups of > G_1 x G_2 follows because each H_i is a subgroup of G_1, and each K_i > is a subgroup of K_2. > (i) follows because H_1 = G_1 and K_1 = G_2, so > M_1 = H_1 x K_1 = G_1 x G_2. > (ii) follows because H_n = {e}, K_n = {e}, so > M_n = H_n x K_n = {e} x {e} which is the trivial subgroup of > G_1 x G_2. > (iii) follows because H_i is normal in H_{i-1}, K_i is normal in > K_{i-1}, so M_i = H_i x K_i is normal in H_{i-1} x > K_{i-1}=M_{i-1} by standard properties of the direct > product. This is true for i=2,....,n, because it is true for H_i > and K_i for i=2,...,n > (iv) Follows again as above: since H_{i-1}/H_i is abelian and > K_{i-1}/K_i is abelian, we have > M_{i-1}/M_i = (H_{i-1} x K_{i-1})/(H_i x K_i) > which is isomorphic to (H_{i-1}/H_i) x (K_{i-1}/K_i) > which by the hypothesis is a product of abelian groups, hence > abelian. > This proves that G_1 x G_2 is solvable, with witnesses > M_1,M_2,...,M_n. > -- > Its not denial. Im just very selective about > what I accept as reality. > --- Calvin (Calvin and Hobbes) > Arturo Magidin > magidin@math.berkeley.edu === Subject: Re: center of a group and direct product. posting-account=0pGp0w0AAACCkl2ylaAtx3LqRXqsqqck >Hello mates, >two questions: >I: Let G be a group. Prove that if H normal G and H intercept G^(1) = >{e}, where G^(1) = [G,G] is the >first derived group, then H normal Z(G), where Z(G) denotes the center >of the group G. >I understand H normal to G means aha^-1 = h where h in H and a in G. > No. That would mean H central. You mean aha^{-1} = h for some h in H. >and I understand how to get {e} from H intercept G^(1). But I confuse >about the conclusion > Apparently because you are confused about what it means for the group > to be normal. > The conclusion is asking you to show that if H is normal and H > intersect [G,G] is trivial, then H is actually a subgroup of the > center of G, where > Z(G) = {x in G : gx = xg for all g in G}. > So you want to show that for all h in H, and all g in G, gh=hg. This > is equivalent to showing ghg^{-1} = h (which was your mistaken > condition for normality). > HINT: Remember that [x,y] = xyx^{-1}y^{-1}, and that xy = [x,y]yx. If > h in H and g in G, [g,h] = (ghg^{-1})h^{-1} must be an element of ... >********************************************************* >II: Let G_1 and G_2 be two groups. Prove that Z(G_1 x G_2) = Z(G_1) x >Z(G_2), where Z denotes the center, x denotes the direct product. >Do I need to show Z(G_1) is solvable? > What? > Z(G_1) = {g in G_1: xg=gx for all x in G_1} > Z(G_2) = {h in G_2: yh=hy for all y in G_2} > Z(G_1 x G_2) = {(x,y) in G_1 x G_2 : (x,y)(a,b) = (a,b)(x,y) for all } > { : (a,b) in G_1 x G_2 } > Prove Z(G_1) x Z(G_2) = Z(G_1 x G_2). That is: > If (x,y) in Z(G_1 x G_2), then x in Z(G_1) and y in Z(G_2). > If x in Z(G_1) and y in Z(G_2), then (x,y) in Z(G_1 x G_2). > -- > Its not denial. Im just very selective about > what I accept as reality. > --- Calvin (Calvin and Hobbes) > Arturo Magidin > magidin@math.berkeley.edu === Subject: Re: heisenberg group posting-account=0pGp0w0AAACCkl2ylaAtx3LqRXqsqqck > The multiplication of typical elements in Z(H) looks like: > ( 1 0 a ) ( 1 0 b ) ( 1 0 a+b ) > ( 0 1 0 ) ( 0 1 0 ) = ( 0 1 0 ) > ( 0 0 1 ) ( 0 0 1 ) ( 0 0 1 ) > So, define a map phi : R --> Z(H) by > phi (a) = the identity matrix with an a in the (1,3) entry > and just show that phi is a bijective homomorphism (bijectivity is > painfully obvious, and the above should lead to a homomorphism). > Therefore phi is an isomorphism. === Subject: Re: Differential Geometry Lie Group Question > Let G be a Lie Group acting on a manifold M. Let L_g : M -> M be the > diffeomorphism multiplication by g in G. Then we can define, given p > in M and omega a differential form in $M$, phi : G -> wedge^q T_p^* > M by g |-> L_g^*omega_p. (where the last string of symbols mean the > q-exterior power of the cotangent space at p of M). > Ive been trying to show that phi is continuous. > Any help ? > nojb. I think I understand this to mean that omega is a continuous q-form on $M$, and phi is defined by phi(g) = (L_g^* omega)_p. The extra parentheses are necessary because the right-hand side depends not on omega_p but on omega_{L_g p}. I think the elegant way to handle this is to use the fact that the action of G on M is a manifold map from G cross M to M. Now, you can pull omega back to G cross M. Try to identify phi with something in that pull-back. -- Chris Henrich Q: Who lives at 668? A: The next-door neighbor of the Beast. -- (?)Joshua Burton === Subject: test test === Subject: test test === Subject: Re: Poll: Are PCs Turing Machines? posting-account=polmKQwAAABsnP--6saHYkbedetRP4nH > Interesting, because you assume based on present knowledge that there > exists at least one calculation that always and forever will require > an unbounded number of digits. > Yes, I do make such an assumption. This assumption seems fairly trivial. > In my example, I dont even care about the calculation itself, only that the > input is a 10^1000 digit number. Theres no conceivable way to even store > such a number without a corresponding amount of memory. This is false UNLESS the condition is I need to see all the digits. The word pi represents pi, in fact better than a large expansion of pi showing many digits. The problem isnt representing all the digits so you can base a calculation on the digits: the problem is finding a suitable analytic representation of pi which would provide the digits just in time, as needed. I mean, if we are to be recreational Platonists, imagining an ideal world as Plato and his boys did at the Symposium or Boys Night Out, lets party down, and ALSO imagine that idealized Turing Machines would not do anything so sordid as to calculate the NEXT digit of pi based on ALL digits of pi. Instead we can imagine that in Heaven, the Riemann Hypothesis is solved in such a way that an elegant, analytic way is found to dispense with ALL calculations with large numbers. Even if one majored in physics one can well imagine multiple universes, cant one? If the Turing Abomination (a Turing machine executing such inelagant code that it has destroyed humanity) needs more storage, then staying within Turings original gedanken experiment we may permit ourselves to imagine that the Turing machine simply reaches through a worm hole and Or, the Riemann Hypothesis is solved in such a way (see below for disclaimer) that we can calculate pi in an analytic manner. Is it not an unsolved, which is to say open, mathematical question whether all calculations must be unbounded in storage? Not being a mathematician (see below) I do not know. > This may be true OR it may be an artifact of our knowledge. In the > case of pi the dependency, I understand, is on something called the > Riemann hypothesis, the solution of which may crack the problem of > digit patterns recurring in pi, in which case, a TM could calculate > it to arbitrary precision WITHOUT running out of storage. > I dont think the randomness of digits in pi has anything to do with the RH, > but I could be wrong. But calculating pi wasnt my example. Im not a mathematician. Although I was rather privileged to assist John Nash with the vagaries of the Microsoft C compiler in 1992, I dont even play one in the movies because Id left Princeton by the time A Beautiful Mind was made. Riemann and pi. I think Nash is still working on Riemann. > Whether a PC simulates a Turing machine is a question unrelated to > physical and empirical data. You see, the assertion that the > true in all possible worlds because an infinite universe is > physically possible, if not verifiable. > This is true, but the question was, can a PC simulate a TM. I assumed > PC meant a real PC, existing in our universe. If you want to hypothesize > a universe in which there is an infinite amount of matter, which can be > formed into an infinite amount of memory units in a finite time, and in > which there is no speed-of-light limit on information transfer, then, yeah, > a PC can probably simulate a TM. But I dont think that was the question > that was asked. Youre taking an engineering viewpoint. I prefer that of the philosopher. Seriously, advances in computation have generally occured when people start thinking, not what can be done but what could be done if we had more storage. Babbages Grand Failure was that his vision outran the capabilities of machinists but if he had worked in China, in the same century, he probably would have found machinists able and willing to machine to his required precision. > --Mark === Subject: Re: Poll: Are PCs Turing Machines? >> Yes, I do make such an assumption. This assumption seems fairly >> trivial. In my example, I dont even care about the calculation >> itself, only that the input is a 10^1000 digit number. Theres no >> conceivable way to even store such a number without a corresponding >> amount of memory. > This is false UNLESS the condition is I need to see all the digits. No it isnt. If the input is a 10^1000 digit number, I dont care what the output is. Theres simply no way to even input the number to begin the calculation. Unless theres some analytic way to solve the problem _without even knowing the input number_, the calculation cannot be performed in this universe. > Is it not an unsolved, which is to say open, mathematical question > whether all calculations must be unbounded in storage? Not being a > mathematician (see below) I do not know. Again, unboundedness in the calculation is not the issue. A (theoretical) TM can accept inputs that no PC will ever be able to accept. >> This is true, but the question was, can a PC simulate a TM. I >> assumed PC meant a real PC, existing in our universe. If you want >> to hypothesize a universe in which there is an infinite amount of >> matter, which can be formed into an infinite amount of memory units >> in a finite time, and in which there is no speed-of-light limit on >> information transfer, then, yeah, a PC can probably simulate a TM. >> But I dont think that was the question that was asked. > Youre taking an engineering viewpoint. I prefer that of the > philosopher. I think youre the one taking the engineering viewpoint. Youre saying TMs which cannot be physically constructed are not worth considering. Im saying that TMs are a theoretical concept, and we can consider TMs which cannot ever be constructed. Do you believe its impossible to even CONCEIVE of a TM which calculates ((10^1000)!)? --Mark === Subject: Re: Poll: Are PCs Turing Machines? <1tmvd.188505$V41.183108@attbi_s52> posting-account=polmKQwAAABsnP--6saHYkbedetRP4nH It doesnt explicitly say that TMs are better since the logic of what Derrida called Differance never says the favored term is better. Instead the betterness remains of necessity tacit. In mathematics, a problem is an abstraction. In computer science (described by Dijkstra as applied mathematics), a problem is phenomenological and its content includes human beings. Therefore, monster problems, of the sort solved by TMs and not solvable by PCs, do not exist. === Subject: Re: Poll: Are PCs Turing Machines? > It doesnt explicitly say that TMs are better since the logic of > what Derrida called Differance never says the favored term is > better. Instead the betterness remains of necessity tacit. > In mathematics, a problem is an abstraction. In computer science > (described by Dijkstra as applied mathematics), a problem is > phenomenological and its content includes human beings. > Therefore, monster problems, of the sort solved by TMs and not > solvable by PCs, do not exist. So the only problems that exist are the ones that are solvable by currently existing computers? The problem of factoring a 30 digit number, for example, didnt exist until computers were capable of solving it? The problem of factoring a 1000 digit number doesnt exist today, but will exist whenever a computer is capable of solving it? What does it mean for a problem to not exist? --Mark === Subject: Re: Poll: Are PCs Turing Machines? <1tmvd.188505$V41.183108@attbi_s52> posting-account=polmKQwAAABsnP--6saHYkbedetRP4nH > It doesnt explicitly say that TMs are better since the logic of > what Derrida called Differance never says the favored term is > better. Instead the betterness remains of necessity tacit. > In mathematics, a problem is an abstraction. In computer science > (described by Dijkstra as applied mathematics), a problem is > phenomenological and its content includes human beings. > Therefore, monster problems, of the sort solved by TMs and not > solvable by PCs, do not exist. > So the only problems that exist are the ones that are solvable by > currently existing computers? The problem of factoring a 30 digit number, > for example, didnt exist until computers were capable of solving it? The Correct. If computer science is applied mathematics, then it is part of history. Its problem set is limited to problems that in principle can be solved. Of course, this set expands but owing to the daily struggles of computer scientists with material, concrete realities, to deliberately lapse into Marxist phrasing. They dont exist in Platonic la-la land prior to the date at which they are empirically solvable. > problem of factoring a 1000 digit number doesnt exist today, but will exist > whenever a computer is capable of solving it? What does it mean for a > problem to not exist? > --Mark === Subject: Re: Poll: Are PCs Turing Machines? > It doesnt explicitly say that TMs are better since the logic of > what Derrida called Differance never says the favored term is > better. Instead the betterness remains of necessity tacit. > In mathematics, a problem is an abstraction. In computer science > (described by Dijkstra as applied mathematics), a problem is > phenomenological and its content includes human beings. > Therefore, monster problems, of the sort solved by TMs and not > solvable by PCs, do not exist. >> So the only problems that exist are the ones that are solvable by >> currently existing computers? The problem of factoring a 30 digit >> number, for example, didnt exist until computers were capable of >> solving it? The > Correct. If computer science is applied mathematics, then it is part > of history. Its problem set is limited to problems that in principle > can be solved. Of course, this set expands but owing to the daily > struggles of computer scientists with material, concrete realities, to > deliberately lapse into Marxist phrasing. They dont exist in > Platonic la-la land prior to the date at which they are empirically > solvable. Wow. Thats a position that I doubt many mathematicians, computer scientists, OR philosophers would agree with. Youre saying are no unsolved problems, because problems dont exist until theyre solvable. Essentially youre just defining away the difference between the real and the possible. The possible doesnt exist until its real. Seems a rather sterile and pointless exercise in hiding from difficulties, but if thats your definition of a problem, youre welcome to it. --Mark === Subject: Re: Poll: Are PCs Turing Machines? > Interesting, because you assume based on present knowledge that there > exists at least one calculation that always and forever will require > an unbounded number of digits. > Yes, I do make such an assumption. This assumption seems fairly trivial. > In my example, I dont even care about the calculation itself, only that the > input is a 10^1000 digit number. Theres no conceivable way to even store > such a number without a corresponding amount of memory. > This may be true OR it may be an artifact of our knowledge. In the > case of pi the dependency, I understand, is on something called the > Riemann hypothesis, the solution of which may crack the problem of > digit patterns recurring in pi, in which case, a TM could calculate > it to arbitrary precision WITHOUT running out of storage. > I dont think the randomness of digits in pi has anything to do with the RH, > but I could be wrong. But calculating pi wasnt my example. > Whether a PC simulates a Turing machine is a question unrelated to > physical and empirical data. You see, the assertion that the > true in all possible worlds because an infinite universe is > physically possible, if not verifiable. > This is true, but the question was, can a PC simulate a TM. I assumed > PC meant a real PC, existing in our universe. If you want to hypothesize > a universe in which there is an infinite amount of matter, which can be > formed into an infinite amount of memory units in a finite time, and in > which there is no speed-of-light limit on information transfer, then, yeah, > a PC can probably simulate a TM. But I dont think that was the question > that was asked. Argument 1 The question is: Is there a TM that logically represents a PC? or, for any PC, is there a TM that logically represents it? ie. are PCs Turing Machines? as in, are Traffic Lights automations? YES Argument 2. Everything possible in the real world can be represented / simulated functionally. All functions can be represented by TMs. Therefore anything can be represented by a TM. Therefore PCs can be represented by a TM. Therefore PCs are TMs. Most people have the infinite memory argument backwards. Is a TM a #FSM? NO a FSM would run out of memory for many TMs. Herc #FSM = finite state machine === Subject: Re: Poll: Are PCs Turing Machines? > All functions can be represented by TMs. Nope. --PeterD === Subject: Re: Poll: Are PCs Turing Machines? > All functions can be represented by TMs. > Nope. All functions possible in the real world can be represented / simulated by TMs Herc === Subject: Re: Poll: Are PCs Turing Machines? > All functions can be represented by TMs. > Nope. > All functions possible in the real world can be represented / simulated by TMs Nope. Try Busy beaver function. --PeterD === Subject: Re: Poll: Are PCs Turing Machines? > All functions possible in the real world can be represented / simulated by > TMs > Herc What about functions on reals, like f(x)=sqrt(x)? Or did you mean to assert a tautology? -- Matt === Subject: Re: Poll: Are PCs Turing Machines? In sci.math, Matt Timmermans : >> All functions possible in the real world can be represented / simulated by >> TMs >> Herc > What about functions on reals, like f(x)=sqrt(x)? Or did you > mean to assert a tautology? The standard function for sqrt(x) is modelable on a TM to a certain extent; the main problem is the input tape, as x could be an arbitrarily large decimal. An elegant method -- and darned fast -- is also available, by guessing y_0 somewhere near sqrt(x) -- ideally, abs(y_0 - sqrt(x)) <= 0.5 -- then computing y_{i+1} = (y_i + x/y_i)/2. The main difficulty is dividing by an increasingly long decimal, in addition to feeding the input tape. Of course |-|erc is asserting a tautology here, as all functions possible in the real world are representable by TMs, since they are also representable by all atoms in the real world, of which there are no more than about 10^100 -- and in the immediate vicinity, no more than about 2 * 10^50, assuming the Earth were made out of 100% oxygen, which gives an overestimate. And the vast majority of those atoms are a little hard to get at. :-) But assert as |-|erc might, the reals continue to be non-denumerable. The importance of this result is largely theoretical, though it can lead to trains of thought that ultimately yield fruit in the physics world at some point -- its hard to tell, but who could have predicted that work on number theory would have yielded RSAs modern encryption algorithm, for example? [.sigsnip] -- #191, ewill3@earthlink.net Its still legal to go .sigless. === Subject: Re: Poll: Are PCs Turing Machines? > But assert as |-|erc might, the reals continue to be non-denumerable. > The importance of this result is largely theoretical, though > it can lead to trains of thought that ultimately yield fruit > in the physics world at some point -- its hard to tell, but who > could have predicted that work on number theory would have yielded > RSAs modern encryption algorithm, for example? rubbish. infinite people each ßip coins infinite times all sequenes of heads and tails have been tossed to infinite length theres no hyperinfinity to produce new sequences (run your finger down the diagonal and imagine all you like) 20th century mathematicians couldnt see this simple 2 variable relationship 1. L(a,a) = L(a,a) (obvious) 2. exists b, L(a,b) = L(b,b) (provable from 1, with b=a) 3. forall a, exists b, L(a,b) = L(b,b) (generalization of 2) 4a. not(exists a, forall b, L(a,b) != L(b,b)) (negation of 3) 4b. not(exists a, forall b, L(a,b) = !L(b,b)) (! is some suitable digit change function) 5. exists r, not(exits a, forall b, L(a,b) = r(b)) (r(b)= !L(b,b)) No one will comment if the free b in r is allowed in step 5. Might need a forall L to fully model which might be 2nd Order. Your EXTREME lengths to join_the_dots are purely futile and lack insight into self reference limitation of models, namely inverted representative members of any formalism. It is complte ignorance when it is shown that swapping digits does not generate any new sequence of digits, your continuing ignorance of this fact will drive you further and further into paradoxical foolish assumptions. Herc === Subject: Re: Poll: Are PCs Turing Machines? In sci.math, |-|erc <32850jF3imalnU1@individual.net>: >> But assert as |-|erc might, the reals continue to be non-denumerable. >> The importance of this result is largely theoretical, though >> it can lead to trains of thought that ultimately yield fruit >> in the physics world at some point -- its hard to tell, but who >> could have predicted that work on number theory would have yielded >> RSAs modern encryption algorithm, for example? > rubbish. > infinite people each ßip coins infinite times > all sequenes of heads and tails have been tossed to infinite length [rest snipped] Produce the list. -- #191, ewill3@earthlink.net Its still legal to go .sigless. === Subject: Re: Poll: Are PCs Turing Machines? > All functions possible in the real world can be represented / simulated by > TMs > Herc > What about functions on reals, like f(x)=sqrt(x)? Or did you mean to assert > a tautology? In that case there is just a TM that represents sqrt(x) since simulation is defined for finite processes. However a simulation of any finite representation is possible. There is no loss of generality since a TM can exist OTM that Optimises is arguments before simulating them as a UTM. Even if TM-sqrt and TM-sqr only parse one digit at a time OTM ( TM-sqr (TM-sqrt), 3 ) = 3 Herc === Subject: Re: Poll: Are PCs Turing Machines? >> All functions possible in the real world can be represented / simulated >> TMs >> Herc >> What about functions on reals, like f(x)=sqrt(x)? Or did you mean to >assert >> a tautology? The relationship between Turing Machines and real numbers is rather interesting. I think a Turing machine corresponds to a real number if, given any rational number as input (in the form of a pair of integers) the TM will always stopeither in state L (assigning the rational to set L of the Dedekind section for the real) or in state U. Given any obviously computable real number (such as an algebraic) it is obviously possible to build a TM that corresponds to it. There will be in fact an infinite number of such TMs (Find one and add a lot of irrelevant states to it.) Given an arbitrary TM, it is undecidable whether it corresponds to a real or not. Can one develop a meaningful theory of computability along these lines? Dick Batchelor === Subject: Re: Poll: Are PCs Turing Machines? > Argument 1 > The question is: Is there a TM that logically represents a PC? > or, for any PC, is there a TM that logically represents it? > ie. are PCs Turing Machines? > as in, are Traffic Lights automations? Is there an automation that logically represents a set of traffic lights? YES Is there a TM that logically represents a PC? > YES > Argument 2. > Everything possible in the real world can be represented / simulated functionally. > All functions can be represented by TMs. > Therefore anything can be represented by a TM. > Therefore PCs can be represented by a TM. > Therefore PCs are TMs. > Most people have the infinite memory argument backwards. > Is a TM a #FSM? NO a FSM would run out of memory for many TMs. > Herc > #FSM = finite state machine Argument 3 Are PCs Finite State Machines YES Can Finite State Machines be emulated by TMs YES PCs can be emulated by FSMs can be emulated by TMs Therefore PCs are TMs Herc === Subject: Re: Fourier Series/Complex analysis posting-account=MBqP4A0AAADmTXsiNvaGzTW4Fx5uIecd Hidely ho, is from my lecture notes for class; actually, the preceding problem is to classify all such functions that have this property (i.e. slightly better than continuous), and this problem is just a continuation of GB === Subject: Groups as galois groups posting-account=xslVFw0AAACcw30rwkkdKYhs8AhLZVeJ I am interested in learning more about the problem of determning which groups arise as galois groups (over Q of course). In particular, I would like to know of any topological methods used to explore the subject (I am thinking of using this as a senior research project, and it would be best if I could involve some algebraic topology). Does anyone know of some papers or books that would be good for starting off with? Also, I know that all groups arise as fundamental groups of some space (at least I hope I am recalling this correctly), but I am unsure of the details. Does it have anything to do with the fact that it is so easy to produce the free group on n letters as a fundamental group i.e. pi_1 of the n-fold wedge of S^1? And then we have various ways of adding relations based on pi_1 (X x Y)=pi_1(X) x pi_1(Y), Van Kampens theorem, and so on? Or is there a more universal approach? Michael Williams === Subject: Re: Groups as galois groups > I am interested in learning more about the problem of determning which > groups arise as galois groups (over Q of course). In particular, I > would like to know of any topological methods used to explore the > subject (I am thinking of using this as a senior research project, and > it would be best if I could involve some algebraic topology). Does > anyone know of some papers or books that would be good for starting off > with? First of all, you should be aware of the fact that the problem that you want to study is known as the inverse problem of Galois theory. Suggested reading: Topics in Galois theory Jean-Pierre Serre Galois groups over Q edited by Yasutaka Ihara, Kenneth Ribet and Jean-Pierre Serre Jose Carlos Santos === Subject: Non-atomistic continuum: Peirce, Weyl, and? While traditional analysis considers the continuum as the set of its has partsî, and Weyl explained in ÔOn the New Foundational Crisis of Mathematics in 1921 that the concept of a continuous manifold remained mathematically sterile. At least within IR+, cosine integral transform translates any discrete number into a non-atomistic continuous function and vice versa. May I ask for some elucidating hints? Eckard Blumschein === Subject: Re: Euler Cookies | | Arent shortbread cookies shaped with a Runge Cutter? Unless youre using Newtons method. We seem to have gone full circle here. -- )>==ss$$%PARR(.bc> Parr === Subject: Re: Euler Cookies === >Subject: Re: Euler Cookies >Message-id: Unless youre using Newtons method. >We seem to have gone full circle here. But Fig Newtons are square. >)>==ss$$%PARR(.bc> Parr -- Mensanator Ace of Clubs === Subject: A vector calc related question posting-account=2OYlAwwAAAAzuGZHzY8fB1XqLzeo4Yd5 Back in the day, a visiting professor challenged my class to find an exception to the following general statement: F sub xy = F sub yx (Partial to x then y is equal to the partial of y then x). This is used, as yall may know, to find relative extrema where d = Fxx*Fyy - (Fxy)^2. The reason it came up was the lecturer said that they are one in the same. The visitor challenged us to find an exception. Later, when my lecturer came back, he admited there were cases where it isnt equal. Anyone have an idea? === Subject: Re: A vector calc related question > Back in the day, a visiting professor challenged my class to find an > exception to the following general statement: > F sub xy = F sub yx (Partial to x then y is equal to the partial of y > then x). > This is used, as yall may know, to find relative extrema where d = > Fxx*Fyy - (Fxy)^2. The reason it came up was the lecturer said that > they are one in the same. The visitor challenged us to find an > exception. Later, when my lecturer came back, he admited there were > cases where it isnt equal. Anyone have an idea? A standard example is F(x,y) = xy(x^2-y^2)/(x^2+y^2) if x =/=0 or y=/=0, F(x,y) = 0 if x = y = 0. The two mixed partial derivatives at (0,0) are different. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Reversing Functors in Set his own ability to solve his own problems and satisfy his own needs. The leftist is antagonistic to the concept of competition because, deep inside, he feels like a loser. 17. Art forms that appeal to modern leftist intellectuals tend to focus on sordidness, defeat and despair, or else they take an orgiastic tone, throwing off rational control as if there were no hope of accomplishing anything through rational calculation and all that was left was to immerse oneself in the sensations of the moment. 18. Modern leftist philosophers tend to dismiss reason, science, objective reality and to insist that everything is culturally relative. It is true that one can ask serious questions about the foundations of scientific knowledge and about how, if at all, the concept of objective reality can be defined. But it is obvious that modern leftist philosophers are not simply cool-headed logicians systematically analyzing the foundations of knowledge. They are deeply involved emotionally in their attack on truth and reality. They attack these concepts because of their own psychological needs. For one thing, their attack is an outlet for host === Subject: Re: Number of prime factors of an odd perfect number any case it is not normal to put into the satisfaction of mere curiosity the amount of time and effort that scientists put into their work. The curiosity explanation for the scientists motive just doesnt stand up. 88. The benefit of humanity explanation doesnt work any better. Some scientific work has no conceivable relation to the welfare of the human race - most of archaeology or comparative linguistics for example. Some other areas of science present obviously dangerous possibilities. Yet scientists in these areas are just as enthusiastic about their work as those who develop vaccines or study air pollution. Consider the case of Dr. Edward Teller, who had an obvious emotional involvement in promoting nuclear power plants. Did this involvement stem from a desire to benefit humanity? If so, then why didnt Dr. Teller get emotional about other humanitarian causes? If he was such a humanitarian then why did he help to develop the H-bomb? As with many other scientific achievements, it is very much open to question whether nuclear power plants actually do benefit humanity. Does the cheap electricity outweigh the accumulating waste and risk of accidents? Dr. Teller saw only one side of the question. Clearly his emotional involvement with nuclear power arose not from a desire to benefit humanity but from a personal fulfillment he got from his work and from seeing it put to practical use. 89. The same is true of scientists generally. With possible rare exceptions, their motive is neither curiosity nor a desire to benefit humanity but the need to go through the power process: to have a goal (a scientific problem to solve), to make an effort (research) and to attain the goal (solution of the problem.) Science is a surrogate activity because scientists work mai === Subject: Maxwell laws are spherical because atoms are spherical Re: Lorentz transformations, Maxwell Eq., and Special Relativity imply the Cosmos is a huge sphere such as the inside of an atom About the easiest description of the Maxwell Equations and of Special Relativity is to say that these laws are the geometry of a sphere or cylinder and not laws of Euclidean ßat space geometry. Coulomb law is a sphere description. Special Relativity is a sphere description in that if you increase distance you decrease time. And where a moving magnet in stationary wire is the same as stationary magnet in moving wire is because light is a constant speed. So where one factor corresponds to the inverse or reverse of another factor which is spherical geometry. Let us run the argument in opposite direction by asking whether you can have (1) Maxwell Equations and Lorentz invariance of Special Relativity if the Cosmos was Euclidean ßat space geometry? (2) Can you have Coulombs law and the moving or stationary magnet and wire loop what they are if the cosmos was ßat Euclidean? (3) Can you have special relativity with its constant c value for the speed of light in a ßat Euclidean Cosmos? Let me try to answer these three brießy: (1) You cannot have Maxwell theory or Lorentz invariance without spherical geometry because because closed loops of a sphere is the mathematics of inverse square and square roots of Lorentz. Faradays lines of force become trivial in ßat Euclidean and only make sense in spherical where closed loops exist. (2) Some of the Maxwell Eq are spherical and others are cylindrical but spherical is a special case of cylindrical or vice versa (3) And finally the invariance of Lorentz transformation in the Maxwell Equations because the invariance of a constant that is the speed of light in Special Relativity. So simply put, the Maxwell theory and Special Relativity are what they are because the entire Cosmos is spherical or cylindrical in geometry and is not ßat Euclidean. The Maxwell theory and Special Relativity are microwindows of what the large scale Cosmos is and both are spherical/cylindrical. Another way of saying it is that the speed of light must be constant in a spherical cosmos because that cosmos is bounded and not infinite, and so is the speed of light a constant bounded number. And Maxwell theory cannot be spherical for the small scale and yet have the cosmos of the large scale be open ßat infinite Euclidean. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: Maxwell laws are spherical because atoms are spherical Re: Lorentz transformations, Maxwell Eq., and Special Relativity imply the Cosmos is a huge sphere such as the inside of an atom > Let me try to answer these three brießy: (1) You cannot have Maxwell > theory or Lorentz invariance without spherical geometry because because > closed loops of a sphere is the mathematics of inverse square and square > roots of Lorentz. Faradays lines of force become trivial in ßat Euclidean > and only make sense in spherical where closed loops exist. What you say? Youre confusing the (necessary) spaces with the fields. Both can have curvature, but they are different kinds of animals. Faraday is field equations, presume 3D + time. GRT links energy tensor to curvature (as an axiom). 4D. If you add another 2D you can have ßat, with a nicely curved sub-manifold AND consistency. Push up to 10D and you can seem to get Lorentz for free (but I have been suspicious about such freedom for years). Tomasso. === Subject: Re: Presentations of D_n when n = 2^m [...] > Z_4 x| Z_2 appears to have 2 generators, and (Z_2 x Z_2) x| Z_2 > appears to have 3, but they are both D_4. The number of generators of a group G obviously depends on the presentation, and can be as many as |G|. For finite p-groups, the minimum number of generators is log_p(|G/Phi(G)|), where Phi(G) is the Frattini subgroup of G, the intersection of the maximal subgroups of G, which turns out to be the smallest normal subgroup K of G such that G/K is elementary abelian. [...] -- Jim Heckman === Subject: Re: Presentations of D_n when n = 2^m >The number of generators of a group G obviously depends on the >presentation, and can be as many as |G|. ...or more! E.g. Z/2Z = === Subject: Restriction on Longitudinal Waves In Physics, there exists a geometrical method for constructing a Longitudinal wave from a Transversal wave. Thinking about this construction reveals the following fact, if my analysis has been done correctly, in: Theorem: the amplitude of a longitudinal wave cannot be larger than its wavelength, divided by 2.pi . I can hardly imagine that nobody has found this before. But ... Han de Bruijn === Subject: Re: Restriction on Longitudinal Waves > In Physics, there exists a geometrical method for constructing > a Longitudinal wave from a Transversal wave. Thinking about this > construction reveals the following fact, if my analysis has been > done correctly, in: > Theorem: the amplitude of a longitudinal wave cannot be larger > than its wavelength, divided by 2.pi . > I can hardly imagine that nobody has found this before. But ... > Han de Bruijn See: Acoustic Fields and Waves in Solids, Vol I & II by B.A. Auld, 2nd ed (February 1990), Krieger Publishing Company; ISBN: 089874783X === Subject: Re: Restriction on Longitudinal Waves Given monotony, what you claim in appears to me to be true. A separate condition on physical media is tensile strength, that is the separation between two points. If it is too great, we have an earthquake. >I can hardly imagine that nobody has found this before. I am utterly unqualified to render an opinion on *that*. However, in physical media, are you assuming infinite compressiblity? That is, is the spacing between points allowed to become infinitely small? In gas media, there is a characteristic exponent greater than one producing heating on compression.... Under conditions of turbulence, could monotony be violated? I tolerance everything and tolerate everyone. I love: Dona, Jeff, Kim, Kimmie, Mom, Neelix, Tasha, and Teri, alphabetically. I drive: A double-step Thunderbolt with 657% range. I fight terrorism by: Using less gasoline. === Subject: Re: PROOF that 0.99999... = 1 >In sci.math, S. Enterprize Company >10x = 9 + x >> Its still useless. >> let x = 2 >> 10( 2) = 9 + 2 >> 20 =/= 11 >Congratulations! Youve proven that x=2 is not a root of >the above equation. >Care to try another number, though? :-) >[.sigsnip] >-- >#191, ewill3@earthlink.net >Its still legal to go .sigless. Ok, lets look at it in this point of view. 10x = 9 + x x can equal 1 or .999... . There are two roots to the equation. x1 = 1 x2 = .999... but, x1 =/= x2 The method I used was to show the degree of balance in the equation using a convergence lim for the equation, which was a non-standard approach that showed that a convergence test, if it could be could be applied for the equation you showed. The convergence of a series has to do with the limits as they approach infinity for the nth term. Your proof clearly shows only two roots. This is not a proof that .999... precisely equals 1. In fact, you can see there are two roots, and they dont equal. Smarts Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: PROOF that 0.99999... = 1 In sci.math, S. Enterprize Company In sci.math, S. Enterprize Company >>I made a typing error here, >>10x = 9 + x > Its still useless. > let x = 2 > 10( 2) = 9 + 2 > 20 =/= 11 >>Congratulations! Youve proven that x=2 is not a root of >>the above equation. >>Care to try another number, though? :-) >>[.sigsnip] >>-- >>#191, ewill3@earthlink.net >>Its still legal to go .sigless. > Ok, lets look at it in this point of view. > 10x = 9 + x > x can equal 1 or .999... . > There are two roots to the equation. > x1 = 1 > x2 = .999... > but, > x1 =/= x2 If x1 != x2, is 10*x1 - 9 - x1 == 10*x2 - 9 - x2? [rest snipped] -- #191, ewill3@earthlink.net Its still legal to go .sigless. === Subject: Re: PROOF that 0.99999... = 1 at 04:44 PM, escu;tur36@hotmail.com;;; (E. Escultura) said: >2) The real number ssystem itself is ill-defined because two of its >axioms are false, That statement is meaningless. >namely, the completeness and trichotomy axioms. counterexamples to >them were constructed by Banach-Tarski and Brouwer. That statement is meaningless. You can proove axioms to be inconsistent with other axioms; you cant provide counterexamples. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: FLTMA: standard output oormat possibility I have verified *manually* that for 0> You _really_ need to work on the irony detector. Truth >> and provability are _different_ things - when you define >> one to be the other you exhibit amazing ignorance of the >> most basic issues. >> ************************ >> David C. Ullrich >The definitions of Turing Machines and Lambda Calculus are very >different, too, arent they? For heavens sake, at some point you should just accept that youve made a fool of yourself and drop it. I mean you have plenty of experience with this phenomenon... Ok. To answer your question: Yes, those two definitions are very different. I cant imagine what your point is. Two comments: (i) If someone asked for the definition of a Turing Machine and someone else replied with the definition of the Lambda Calculus that person would indeed be exhibiting blithering confusion. (ii) On the other hand, its true that the set of functions computable by TMs is the same as the set of functions computable by the LC. _Guessing_ what your point might be here, I have to point out that truth and provability are _not_ equivalent. (Validity and provability are equivalent. And truth _in that one particular model_ is equivalent to provability _in that one particular formal system_. A very curious formal system, btw, since theres no procedure to recognize whether a proof is valid.) >C-B ************************ David C. Ullrich === Subject: SVD Algorithm Good morning. Im searching for an SVD algorithm (in C, C++ or Java). I have tried SVD algorithm of Numerical Recipes, but Singular Values are not in descending order. work with number of rows < number of columns. I would like to find an SVD algorithm with these characteristics: - It should work with m>=n and m ha scritto nel messaggio > I have tried SVD algorithm of Numerical Recipes, but Singular Values are > not in descending order. I could sort the Singular Value Matrix (W is a vector). But, How to interchange the columns of U and the rows of V? Is there an algorithm? === Subject: Re: SVD Algorithm > Im searching for an SVD algorithm (in C, C++ or Java). > I have tried SVD algorithm of Numerical Recipes, but Singular Values are > not in descending order. What is the problem with that? Just sort them and make appropriate interchanges to the U and V matrices as well. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: SVD Algorithm Dik T. Winter ha scritto nel messaggio > What is the problem with that? Just sort them and make appropriate > interchanges to the U and V matrices as well. I could sort the Singular Value Matrix (W is a vector). But, How to interchange the columns of U and the rows of V? Is there an algorithm? === Subject: Re: SVD Algorithm Dik T. Winter ha scritto nel messaggio > What is the problem with that? Just sort them and make appropriate > interchanges to the U and V matrices as well. I could sort the Singular Value Matrix (W is a vector). But, How to interchange the columns of U and the rows of V? Is there an algorithm? === Subject: Re: Inplace Matrix Transpose? >> I am writing code and need to be able to transpose non-square >> matricies. This is easy to do if you create a second matrix and copy >> the rows from the input matrix to the columns of the output matrix. >> However, my matricies are large and take up a lot of memory. I would >> prefer not to create a second matrix. I am sure there is a way to do an >> inplace transpose, but deriving an algorithm is more difficult than I >> thought. >[ ... snip ... ] >> But how can it be done inplace without creating a second matrix? >Maybe this is a stupid remark, but I cannot imagine that someone will >ever be in need of a transposed matrix. Why not simply use different >indexing? I mean, M[j,i] instead of M[i,j]. Or maybe something like: >function Transpose(i,j : integer) : double; >begin > Transpose := M[j,i]; >end; >Isnt it better to fake the system than re-organize your whole memory!? Depends on what youre doing. A transposed matrix will be much more efficient for some sorts of things, because of the way things are laid out in memory. Typically a row of a matrix is stored in a contiguous block of memory, so iterating along the elements of a row can be _much_ more efficient than iterating along the elements of a column. (Say youre multiplying two nxn matrices, where n is very large. If you just multiply them you iterate over each column of the second matrix n times. If you first transpose the second matrix you iterate along each column once, and then the multiplication involves just running along rows of both matrices.) >But I suppose the real problem is that you are using a standard package >and have no choice? >Han de Bruijn ************************ David C. Ullrich === Subject: Re: Inplace Matrix Transpose? >I am writing code and need to be able to transpose non-square matricies. [Example given to show that the problem is to take a sequence of numbers and permute them so that, when each is interpreted as the set of rows in a matrix, the second matrix is the transpose of the first.] This is a standard enough problem that it has arisen both in this newsgroup and in the literature; some details are in http://www.math-atlas.org/97/transpose.mat The nature of the permutation depends on the combination of the sizes of the matrices, e.g. for square matrices it has order 2, but for other sizes of matrix it will have higher order. dave === Subject: Re: String Theory: Good, Bad and Bogus > Part 1 > Words written by Dennis Overbye are between quote marks. > a single equation that could explain all the laws of physics, all the > forces of nature - the proverbial Ôtheory of everything ? > Extraordinary claims require extraordinary proof. Physicists, like Dr. > Robert Park, use a double standard not applying the same rules of > engagement to fashionable elegant string theory as they do to ßying > saucers, the paranormal and cold fusion. > http://www.archivefreedom.org/ > And so emerged into the limelight a strange new concept of nature, > called string theory, so named because it depicts the basic constituents > uniting all the forces, string theory had the potential of achieving the > goal that Einstein sought without success for half his life and that has > embodied the dreams of every physicist since then. If true, it could be > used like a searchlight to illuminate some of the deepest mysteries > physicists can imagine, like the origin of space and time in the Big > Bang and the putative death of space and time at the infinitely dense > centers of black holes. ... In the last 20 years, string theory has > become a major branch of physics. Physicists and mathematicians > conversant in strings are courted and recruited like star quarterbacks > by universities eager to establish their research credentials. String > theory has been celebrated and explained in best-selling books like The > Elegant Universe, by Dr. Brian Greene, a physicist at Columbia > University, and even on popular television shows. > ÔLet them eat cake said Marie Antoinette shortly before she was > guillotined. > even as they ate cake and drank wine, the string theorists admitted > that after 20 years, they still did not know how to test string theory, > or even what it meant. > Note or even what it meant. Mainstream theoretical physics today is in > a sorry state. Not so for experimental physics. > As a result, the goal of explaining all the features of the modern > world is as far away as ever, they say. And some physicists outside the > string theory camp are growing restive. At another meeting, at the Aspen > Institute for Humanities, only a few days before the string > commemoration, Dr. Lawrence Krauss, a cosmologist at Case Western > Reserve University in Cleveland, called string theory Ôa colossal failure. > Brian Greene got a few million dollars advance for his book. The big > corporations have a vested interest in this hype that can be compared to > WMD in Iraq. > String theorists agree that it has been a long, strange trip, but they > still have faith that they will complete the journey. > The fusion of science and religion on the heels of the fusion of Church > and State? gravity (which seems to be an essential component of the theory??), and just about anything else that comes along. Perhaps even mathematical proof of God can be found in there somewhere, which is probably what the whole thing is really all about. There are currently five other contenders for the string theory crown, some of which have open ended strings. This kind of physics is unbounded. It surely is strange physics. Thats my 2 cents worth anyway. http://www.ozemail.com.au/~mkeon/the1-1a.html Now thats what I call a theory of everything. ----- Max Keon === Subject: interchanging INT and SUM [Excuse me for my english.] In a proof I found in a book, the author says (in the middle of the proof) that is trivial to verify that INT_A SUM_n=1^inf f_n(x,t)g(t,y)dt = = SUM_n=1^inf INT_A f_n(x,t)g(t,y)dt where g is an L^2(AxA) element. The hypothesis are: --- f_n is a sequence of L^2(AxA) elements. --- sum_n=1^inf ||f_n|| is convergent Could you give me some hint? === Subject: Re: Random Rational? Dave Langers: > I would say this comes from the fact that a randomly chosen fraction can > often be simplified (e.g., 8/6 with even denominator becomes 4/3 without). > Now, a fraction has an even demoninator if > - the denominator is even and the numerator is odd (p = 1/2*1/2 = 1/4) > - the denominator is divisible by four and the denominator is divisible > by two but not by four (p = 1/4*1/4 = 1/16) > - the denominator is divisible by eight and the denominator is divisible > by four but not by eight (p = 1/8*1/8 = 1/64) > - etc.etc. > Giving a total probability of 1/4 + 1/16 + 1/64 + ... = 1/3 > Voila. ... if you accept the notion that the probability is 1/2 that a randomly picked natural number is even. But why should that be the case? If you arrange the natural numbers in a sequence where each of them occurs exactly once, e.g. 1, 3, 2, 5, 7, 4, 9, 11, 6, 13, 15, 8, 17, 19, 10, 21, 23, ... you see that there are twice as many odd numbers than even ones. Helmut Richter === Subject: Re: Random Rational? > We all know that uniform distribution over a countably infinite set is > nonsense. Therefore, for the phrase random rational number to mean > anything, some other distribution must be specified. Let uniform distribution over a countably infinite set be nonsense. But how about a probability density over a countably infinite set? Then the probability density of the perfect squares within the natural integers - as I understand it - depends on where you are. And in the neighbourhood of x it is equal to 1/sqrt(x) . The probability density of the primes - as I understand it - is 1/ln(x) approximately (: Prime Number Theorem. Here x is a sufficiently large natural number.) Both these probability densities become zero (infinitesimal) for x -> oo . Nevertheless, the primes are more numerous than the squares, because ln(x) < sqrt(x) ==> 1/ln(x) > 1/sqrt(x) for x > 2 . O yeah, I find that the density of the cubes goes like x^(-2/3) . Question: do these findings bear any resemblance with results obtained in official mathematics? Since Im rather ignorant in number theory. Han de Bruijn === Subject: Re: Random Rational? > How about this: > for each positive integer N let > P_N = N^{-2}|{(m,n): m,n integers, 1<= m,n <= N, m/n has even denominator}| > and let P = lim_{N->infinity} P_N (if that limit exists). > Is P a good interpretation of the probability that a random rational > number has an even denominator? > Is P = 1/3? Good interpretation? What does that mean? Be straightforward, please! Is P a Probability or is it a Density? Han de Bruijn === Subject: Random Rational? > How about this: for each positive integer N let > P_N = N^{-2}|{(m,n): m,n integers, 1<= m,n <= N, m/n has even denominator}| > and let P = lim_{N->infinity} P_N (if that limit exists). Yes, thats the interpretation I tried first. Putting it in measure terms then, we have a sequence of sets X_N = {(m,n): 1 <= m,n <= N}, and for a subset A of N^2 we define m(A) = lim_{N->inf} |A u X_N| / |X_N| if the limit exists. Then we define m(A) = m(A), where A = {m/n: (m,n) in A}. Unfortunately m fails to satisfy the axioms of a measure, which makes me very leery of calling anything based on it a probability. - Tim === Subject: Re: Starting Calculus in January... > Sorry, didnt mean to sound hostile, I should have added a smiley to that > statement. No problem. It just seemed that, at first, I got more insulting messages than useful ones. > Sometimes the best info available comes in the form of a website. When I > was trying to learn Hungarian and couldnt find a good reference, I printed > a 20-page tutorial I found on the web. It was much much better than > nothing. > On the original topic, I wouldnt recommend any of the calc texts Ive used > for uninstructed study as they were all intolerably dry. Im not looking for something to replace instructed study. I *did* say Im starting my Calculus class in January. I just wanted a recommendation for something I can read before January 5 that will give me some useful knowledge for when Im actually *in* the class. -- Darryl L. Pierce Visit my webpage: By doubting we come to inquiry, through inquiry truth. - Peter Abelard === Subject: Re: Starting Calculus in January... posting-account=0QrkrwwAAABDyQGPKX7NtkkaKfngvovA Fifty years ago was 1954. I submit that calculus books in 1954 did NOT weigh ten pounds, as you claim. Perhaps the calculus experience that you recall was fifteen or twenty years later than that. === Subject: Re: Starting Calculus in January... > Fifty years ago was 1954. I submit that calculus books in 1954 did NOT > weigh ten pounds, as you claim. Perhaps the calculus experience that > you recall was fifteen or twenty years later than that. Yes. Youre right David. 15 or 20 years later. I havent even seen a calculus book from before 1960, except Courants, and maybe Titchmarsh (which is mainly complex functions) - both well before 1960. Both were weighty. Tomasso. === Subject: Re: Starting Calculus in January... >.... The little book by Otto Toeplitz, The Calculus: a >Genetic Approach works through the history of elementary calculus, >showing what originally motivated various ideas. Some bits are harder >than others, but you could find it a more rewarding book than the usual >texts. Dont neglect the exercises which are collected together near >the end of the book.... > Mr. Pledger, please dont be insulted when I ask my question because > I dont know you and dont have any idea if youre one those pests ;-). > Is this recommendation a good one? It sounds like a book > that would be worthwhile buying at full price.... Its a classic. The manuscript was found among Toeplitzs papers after he died, then first published in German in 1949. My old copy of the 1963 English translation is a very cheap paperback which Ive never regretted buying. Ken Pledger. === Subject: Re: Starting Calculus in January... >>.... The little book by Otto Toeplitz, The Calculus: a >>Genetic Approach works through the history of elementary calculus, >>showing what originally motivated various ideas. Some bits are harder >>than others, but you could find it a more rewarding book than the usual >>texts. Dont neglect the exercises which are collected together near >>the end of the book.... >> Mr. Pledger, please dont be insulted when I ask my question because >> I dont know you and dont have any idea if youre one those pests ;-). >> Is this recommendation a good one? It sounds like a book >> that would be worthwhile buying at full price.... > Its a classic. The manuscript was found among Toeplitzs papers >after he died, then first published in German in 1949. Aha! 1949 is a useful piece of data. The genetics in the title made my bull meter tick. New age authors like to cross dress their titles to make them legitimate. > .. My old copy of >the 1963 English translation is a very cheap paperback which Ive never >regretted buying. Another useful piece of info. I hadnt realized Im looking for a translation. Im off to the bookstore to see what I can find :-). /BAH Subtract a hundred and four for e-mail. === Subject: Re: physical significance > Hello Niel, > Thanx for the post. > I am seeking whether any abstract theory of mathematics or physics use the > concept of complex time? If so how? > Shashi Also, you might want to have a look general relativity. You may see a term similar to the following bandied about: x^2 + y^2 + z^2 + (it)^2 which is derived from the euclidean distance on (x, y, z, it). x, y, z are spacial measurements, normalised to the speed of light (i.e. the measurements are divided by c), and t is a time. Its not imaginary time per se, but a certain use of i with time as a sort of syntactic sugar I think? alex === Subject: Re: physical significance > Hello All, > I am a beginner in mathematics and i am stuck in the following concept i > found in a book of complex numbers: > Consider a child throwing a ball into the air. > For example, assume that the ball is thrown straight up, with an initial > velocity of 9.8 meters per second. One second after it leaves the childs > hand, the ball has reached a height of 4.9 meters, and the acceleration of > gravity (9.8 meters per second2) has reduced its velocity to zero. The ball > then accelerates toward the ground, being caught by the child two seconds > any instant of time is given by: > h = (-g*t^2)/2 + v*t > where h is the height above the ground (in meters), g is the acceleration of > gravity (9.8 meters per second2), v is the initial velocity (9.8 meters per > second), and t is the time (in seconds). > t Ô 1± 1&h/4.9 > Now, suppose we want to know when the ball passes a certain height. > Plugging in the known values and solving for t: > For instance, the ball is at a height of 3 meters twice: t =0.38 (going up) > and t = 1.62 seconds (going down). > As long as we ask reasonable questions, these equations give reasonable > answers. But what happens when we ask unreasonable questions? For > example: At what time does the ball reach a height of 10 meters? This > question has no answer in reality because the ball never reaches this > height. > Nevertheless, plugging the value of h = 10 into the above equation gives two > answers: t = 1+ sqrt(-1.041) and t = 1- sqrt(-1.041). > My question is, in the above example what is the Physical significance of > the complex time? > I believe this quantity of complex time would be used in some physical > concept or theory....How do we analyse this complex time in the real world? Im not sure of the physical significance of complex time per se, but in the case of this problem (and other quadratic based problem solutions I suspect), I think the complex answer does have a meaningful interpretation of a sort. For a start, the real part of the answer represents the time at which the object got *closest* to the target distance (10, in your example). In the complex answer case, at this time the object has velocity zero. As for the imaginary part.... Hint: write out the kinematic equation h = (-g*t^2)/2 + v*t in its purely quadratic form (in terms of t), then write out the expression for the solution if this quadratic. (You know; -b +/- ...etc) Identify which part of this expression gives the imaginary term. Then look at the equation for this imaginary term and compare it to the standard 5 laws of kinematics and see what its similar to. alex === Subject: Re: physical significance >>... As long as we ask reasonable questions, these equations give reasonable >>answers. But what happens when we ask unreasonable questions? For >>example: At what time does the ball reach a height of 10 meters? This >>question has no answer in reality because the ball never reaches this >>height. >>Nevertheless, plugging the value of h = 10 into the above equation gives two >>answers: t = 1+ sqrt(-1.041) and t = 1- sqrt(-1.041). >>My question is, in the above example what is the Physical significance of >>the complex time? >>I believe this quantity of complex time would be used in some physical >>concept or theory....How do we analyse this complex time in the real world? > Apparently, even when the text says the question is unreasonable, > still it is asked in this newsgroup. Many discoveries have relied on people exploring Ôunreasonable lines of thought. I suppose you think that we shouldnt have started using negative numbers? After all, at one point the concept was Ôunreasonable. Ditto for imaginary numbers - the square root of a negative number was Ôunreasonable, and yet once someone took the idea seriously, there was benefit to be found. Do you believe everything a textbook says? Do your mind snap shut the moment a textbook tells you a line of inquiry is unreasonable or stupid? Sounds like you would happily let a book do your thinking for you.... As for the original question; there is a meaning of sorts in the complex solution. See my reply to the original question which will appear shortly... alex === Subject: How to visualize limits in category theory? I have previously asked the newsgroup about a very simple way of defining a complete category and there was a person who gave me a very nice definition.. that it has limit for all small diagram. I understand that perfectly but I cannot visualize it like I would like to visualize limits say in analysis. Is there really no way of visualizing categorical limits? In the book I am reading (and I hate to say it, but I am not finding it to my liking!) I keep on reading about complete category this, complete category that (not to mention well-poweredness, cowell-powerdness and cocompletness).. makes me crazy. Simple algebraic stuffs being proven by category theory is just not my style. And worst is, my supervisor keeps telling me that it is really required for me to read the book and be able to comfortably work with categories (I dont very much get it, we work on algebraic geometry and I have never expected that I had to do so much of category theory.). Jose Capco === Subject: Re: How to visualize limits in category theory? Discussion, linux) > I have previously asked the newsgroup about a very simple way of > defining a complete category and there was a person who gave me a very > nice definition.. that it has limit for all small diagram. I understand > that perfectly but I cannot visualize it like I would like to visualize > limits say in analysis. Is there really no way of visualizing > categorical limits? How about it has all products and equalizers? Any limit can be constructed by products and equalizers. (This theorem should be easy to find in a textbook.) > In the book I am reading (and I hate to say it, but I am not finding it > to my liking!) I keep on reading about complete category this, complete > category that (not to mention well-poweredness, cowell-powerdness and > cocompletness).. makes me crazy. Whats not to love? (Well, okay, theres plenty of category theory I dont love, but you havent mentioned it. Yonedas name comes to mind. Its an easy and beautiful lemma, I know, but I suffer a mental block that prevents me from grasping it for longer than a few minutes at a time. A defect, I know.) -- A set having three members is a single thing wholly constituted by its members but distinct from them. After this, the theological doctrine of the Trinity as Ôthree in one should be childs play. --Max Black, _Caveats and Critiques_ === Subject: Re: How to visualize limits in category theory? posting-account=tdYrvA0AAACTm02P3kgbrEwRzJI8b08S Id be interested in hearing how other people interpret limits allright - I find myself that the pyramid-type-thing with commuting faces way of thinking about them is to my satisfaction. Joseph Goguen gives his interpretations of concepts in category theory in his nicely written manifesto (which is fun to read anyway). www.cs.ucsd.edu/users/goguen/pps/manif.ps he says about limits: A diagram D in a category C can be seen as a system of constraints, and then a limit of D represents all possible solutions of the system. I dont know if these are in any way close to the answer to your question (of course I hope they touch on it anyway). Stephen === Subject: Re: How to visualize limits in category theory? >Id be interested in hearing how other people interpret limits allright >- I find myself that the pyramid-type-thing with commuting faces way >of thinking about them is to my satisfaction. >Joseph Goguen gives his interpretations of concepts in category theory >in his nicely written manifesto (which is fun to read anyway). >www.cs.ucsd.edu/users/goguen/pps/manif.ps >he says about limits: >A diagram D in a category C can be seen as a system of constraints, >and then a limit of D represents all possible solutions of the system. furthermore, notice that this is essentially the subject matter of algebraic geometry! that is, the concept of limit of a diagram is more or less equivalent to the concept of algebraic variety carved out by a system of equations. the nodes of the diagram correspond to variables, while the arrows correspond to equations (or constraints, as goguen puts it) of the form y=f(x), where the variable x corresponds to the node at the tail and the variable y to the node at the head of the arrow. of course, it might be questionable whether the contemporary version of algebraic geometry is really about algebraic varieties anymore, but it used to be, at least, in the classical period more or less initiated by descartes and fermat. perhaps there might be some algebraic geometers who study something entirely different, but at least for those of them interested in the concept of algebraic variety, an understanding of the concept of limit of a diagram is just about indispensable. >I dont know if these are in any way close to the answer to your >question (of course I hope they touch on it anyway). -- [e-mail address jdolan@math.ucr.edu] === Subject: Re: Algebra that requires Logic > If you need/want to go more into model theory, the standard text > is still Hodges Model Theory or A shorter Model Theory. I highly recommend Hodges MT - its a beautiful textbook. See also the Handbook of Mathematical Logic, MR below. You may also find it helpful to learn some universal algebra, for which I refer you to Burris and Sankappanavar, now online: http://www.thoralf.uwaterloo.ca/htdocs/ualg.html Also theres much related literature available online, e.g. do a Google search on: real algebraic geometry, o-minimal, etc. --Bill Dubuque ------------------------------------------------------------- --------------- -- MR 56#15351 02-06 Handbook of mathematical logic. Edited by Jon Barwise. With the cooperation of H. J. Keisler, K. Kunen, Y. N. Moschovakis and A. S. Troelstra. Studies in Logic and the Foundations of Mathematics, Vol. 90. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. xi+1165 pp. ISBN 0-7204-2285-X ------------------------------------------------------------- --------------- -- to share with the entire mathematical community some modern developments in logic. We have selected from the wealth of topics available some of those which deal with the basic concerns of the subject, or are particularly important for applications to other parts of mathematics, or both. Mathematical logic is traditionally divided into four parts: model theory, set theory, recursion theory and proof theory. We have followed this division, for lack of a better one, in arranging this book. It made the placement of chapters where there is interaction of several parts of logic a difficult matter, so the division should be taken with a grain of salt. Each of the four parts begins with a short guide to the chapters that follow. The first chapter or two in each part are introductory in scope. More advanced chapters follow, as do chapters on applied or applicable parts of mathematical logic. Each chapter is definitely written for someone who is not a specialist in the field in question. On the other hand, each chapter has its own intended audience which varies from chapter to chapter. In particular, there are some chapters which are not written for the general mathematician, but rather are aimed at logicians in one field by logicians in another. Table of Contents: Jon Barwise, Foreword (p. vii); Contributors (pp. viii-ix). PART A. MODEL THEORY: Jon Barwise, An introduction to first-order logic (pp. 5-46) H. Jerome Keisler, Fundamentals of model theory (pp. 47-103) Paul C. Eklof, Ultraproducts for algebraists (pp. 105-137) Angus Macintyre, Model completeness (pp. 139-180) Michael Morley, Homogenous sets (pp. 181-196) K. D. Stroyan, Infinitesimal analysis of curves and surfaces (pp. 197-231) M. Makkai, Admissible sets and infinitary logic (pp. 233-281) A. Kock and Doctrines in categorical logic (pp. 283-313). G. E. Reyes PART B. SET THEORY: J. R. Shoenfield, Axioms of set theory (pp. 321-344) Thomas J. Jech, About the axiom of choice (pp. 345-370) Kenneth Kunen, Combinatorics (pp. 371-401) John P. Burgess, Forcing (pp. 403-452) Keith J. Devlin, Constructibility (pp. 453-489) Mary Ellen Rudin, Martins axiom (pp. 491-501) I. Juhasz, Consistency results in topology (pp. 503-522). PART C. RECURSION THEORY: Herbert B. Enderton, Elements of recursion theory (pp. 527-566) Martin Davis, Unsolvable problems (pp. 567-594) Michael O. Rabin, Decidable theories (pp. 595-629) Stephen G. Simpson, Degrees of unsolvability: a survey of results (pp. 631-652) Richard A. Shore, alpha-recursion theory (pp. 653-680) Alex. S. Kechris Recursion in higher types (pp. 681-737) and Yiannis N. Moschovakis, Peter Aczel, An introduction to inductive definitions (pp. 739-782) Donald A. Martin, Descriptive set theory: projective sets (pp. 783-815). PART D. PROOF THEORY AND CONSTRUCTIVE MATHEMATICS: C. Smorynski, The incompleteness theorems (pp. 821-865) Helmut Schwichtenberg, Proof theory: some applications of cut-elimination (pp. 867-895) Richard Statman, Herbrands theorem and Gentzens notion of a direct proof (pp. 897-912) Solomon Feferman, Theories of finite type related to mathematical practice (pp. 913-971) A. S. Troelstra, Aspects of constructive mathematics (pp. 973-1052) Michael P. Fourman, The logic of topoi (pp. 1053-1090) Henk P. Barendregt, The type free lambda calculus (pp. 1091-1132) Jeff Paris and A mathematical incompleteness in Peano arithmetic Leo Harrington (pp. 1133-1142) Index of names (pp. 1143-1150) Subject index (pp. 1151-1165). === Subject: Re: Algebra that requires Logic <31rklmF3fdr5bU1@individual.net> posting-account=_-j7cgwAAADnQK9-r68QgRsgfV-jhA3A Geez, Ill have to get my hands on that. Ôcid Ôooh === Subject: Re: Proof of Sum_{i=1...n} i^k is a polynomial expression over n > Xan escribi.97: >> First of all, Ignacio, sorry for no answer to you in a supreme answer >> to the my question of Touchard congruence. I wanted to answer but >> then I have to occupy my time in other _obligatory_ things. Its not >> excuse.... sorry. > It isnt matter ... > Xan escribi.97: >> Say S_k(n) = Sum_{i=1...n} i^k = 1^k + 2^k + .... + n^k. >> I want to prove that S_k(n) is a polynomial expression over n, that >> is, that there exists a polynomial p_k(x) in R[x] such that S_k(n) = >> p_k(n) for all n (and p_k only depends of k). >> I prefer proofs by induction and elementals. I know that it could be >> proved using the Bernoulli polynomials, but I want a proof without >> that (I want more elemental proof). Could it be?. I tried but I did >> not get it. >> Xan. > Let > S(n, k) = 1^k + 2^k + ... + n^k > be the sum of the k-powers of the first n positive integers. > Then > (1+1)^(k+1) = 1^(k+1)+(k+1)*1^k + Comb(k+1, 2)*1^(k-1) + ... + 1 > (2+1)^(k+1) = 2^(k+1)+(k+1)*2^k + Comb(k+1, 2)*2^(k-1) + ... + 1 > ..... > (n+1)^(k+1) = n^(k+1)+(k+1)*n^k + Comb(k+1, 2)*n^(k-2) + ... + 1 > (n+1)^(k+1) = 1 + (k+1)Sum(n, k) + Comb(k+1, 2)Sum(n, k-1) + ... + n > and > S(n, k)=((n+1)^(k+1) - (n+1) - Sum(Comb(k+1, k+1 - i)S(n, i), i, 1, > k-1))/(k+1) >> This is a ingenious recursion. Until here, all good. > Sou, you can determine S(n, k) from S(n, 1), S(n, 2), ..., S(n, k-1). > As S(n, 0) = n is a poynomial, also S(n, k) is for k > 0. You can > deduce also that degree of S(n, k), with repect to n, is k +1. >> This is the dark part. How can I determine that S(n,k) is a polynomial >> over n?. By induction of n?. Really, for proving that S(n,k) is a >> polynomial over n, you need to prove here that: > By induction on k, of course: S(n, 0) = n is a polynomial. Then suppose that > all S(n, k), for k < m, is a polynomial. Then S(n, m) is sum/difference of Yes. Of course. Sorry for my ofuscation. I will write it formally (with sumatories) and surely no problem will appear (even trivial problem). Its a good demostration. For another hand, can you help me?. During these days, I made a demostration (a try of demostration) but I arrived to contradiction. So I want to know where is my error in that demostration. I will give you the main details of my demo. If you want more details, I can provide you these. Before you see that, I have to say that its a (very!!!) indirect demostration of what I want. For arrive to my aim, I make rodeos (I know Ignacio that you understand this bad-formed expression! ;-)) Well, without no dilation: We want to prove: Th1: Let be k>=0 fixed. Define S_k(n)=Sum_{i=1,...,n} i^k. There is a polynomial p(x) in R[x] such that: 1) grad (p)=k+1 2) S_k(n)=p(n) Trivially, we can prove the more relaxed theorem Lemma1: Let be k>=0, S_k(n) defined above. For all n>=1, there is a polynomial q_{k,n} (x) (that depends of n and of k)in R[x] such that grad(q)=k+1 and q_{k,n}(n)=S_k(n) Demo: trivial for induction by n. Now I want to prove that I can choose q_{k,n} (x) that only depends of k. So, that there is q_k(x) such that S_k(n)=q_k(n). Suposing that there is this polynomial q_k. So, for every n>=1, we have q_k(n+1)=S_k(n+1)= S_k(n)+(n+1)^k = q_k(n) + (n+1)^k. So if q_k exists, q_k satisfy that q_k(n+1) = q_k(n) + (n+1)^k, for all n>=1. that is (m=n-1) (1) q_k(m) = q_k(m-1) + m^k, for all m>=2 We could prove that (1) is equivalent to q_k(x) = q_k(x-1)+x^k [ <--) trivial -->) Say p(x) = q_k(x-1)+x^k. p and q_k coincide in infinite values (by (1)) , so p and q are the same polynomial] And we could prove the converse, easyly. Details ommited. So we could prove: Th2: There is q_k(x) in R[x] such that S_k(n) = q_k(n) iff there is q(x) in R[x] such that (2) q(x) = q(x-1) + x^k and q(0)=0. And if q_k exists, q_k verifies (2) and if there is q such that verifies (2), we could take q_k=q And we could prove that q such that verifies (2) is unique. Trivially, q satisfying (2) has defree > k, and we know that deg(q) has to be k+1 although we dont prove. But if you get q(x) = Sum_{i=0...s} a_i x^i, deg(q)=s and you develop (2) with binomial coeficients, you get that in the coeficient k (this need some listing of coeficients and counts as columns. if you need details, I will give to you): a_k = (Sum_{i=k...s} b_{i,k}) +1, where b_{i,k} = a_i Binom(i, j) (-1)^(i-j) and if s were k+1, we have: a_k = a_k + a_{k+1} (-k-1) that is a contradiction. But I dont know where Im wrong. If you could see that, I will thank you a lot. Xan. > With respect to the degree of S(n, k), the unique sumand of degree (k +1) is > the first: (n+1)^(k+1) (newly it can be very easily show by induction in k). > Also is easy to determine some coefficients of S(n, k), as the principal > coefficient: 1/(k+1). But also some others. === Subject: Re: Prime numbers problem Phil, > Primes>30 are coprime to 30. This is true, as primes are coprime to any number. > So are of the form 30n+{1,5,7,11,13,17,91,23,29} I think 35 from above formula is not a prime even if the remainder is prime. This shows also that composite number divided by 30 can give prime remainder. 30=2*3*5 I think there should be a list of all numbers between 0 and 29 that are coprime to each prime factor of 30: (2 and 3 and 5). Here is the list: {1,7,11,13,17,19,23,29} Lets assume you make small mistake forgetting to remove 5 from your list and mistyping 19 as 91. The formula: p=30n+m where m={1,7,11,13,17,19,23,29} generates all coprime to 30. > All of the above remainders, apart from 1, are prime. Waldek === Subject: Re: Prime numbers problem > Phil, > Primes>30 are coprime to 30. > This is true, as primes are coprime to any number. > So are of the form 30n+{1,5,7,11,13,17,91,23,29} Hahhah, yup, thats a load of rubbish! I particularly liked the 91. Phil -- The gun is good. The penis is evil... Go forth and kill. === Subject: Re: Prime numbers problem > Phil, > Primes>30 are coprime to 30. > This is true, as primes are coprime to any number. Like the numbers 3 and 9? -- Jon Haugsand Dept. of Informatics, Univ. of Oslo, Norway, mailto:jonhaug@ifi.uio.no http://www.ifi.uio.no/~jonhaug/, Phone: +47 22 85 24 92 === Subject: Re: Prime numbers problem > Phil, > Primes>30 are coprime to 30. > This is true, as primes are coprime to any number. > Like the numbers 3 and 9? :( you are right. if p - prime number and p>x and p, x > 0 then p and x are coprime. > -- > Jon Haugsand > Dept. of Informatics, Univ. of Oslo, Norway, mailto:jonhaug@ifi.uio.no > http://www.ifi.uio.no/~jonhaug/, Phone: +47 22 85 24 92 === Subject: Goldbach related problems Notation: if A and B and two sets, A+B = {a + b | a in A, b in B} and S(n,A) = A + A ... + A (n times) (S as successor of A) P = prime numbers. A good reformulation of Goldbach problem can be write as: Goldbach problem is equivalent to showing 1) P + P + 2 is subset of P + P 2) there is some even number, 2n, such that 2n = p + q, where p and q are primes (2) is trivial (4 = 2+2) (1) is really the induction step over n So Goldbach problem reduces to prove that (a) P + P + 2 subseteq P + P If we drop P, we get P + 2 subseteq P (b) that is false, but the twin prime problem is related to that. Its equivalent to knowing if there is some infinite subset of P, A, such that A + 2 subseteq P. If we add Ps into right side of (b) we find a famous problem: P +2 subseteq P? false P +2 subseteq P + P? false P +2 subseteq P + P + P? true. Its a corollary of Vinogradov theorem. So Vinogradov proves, realy, that 3 is the minimum number such that P + 2 subseteq S(n,P) is true. If we ask about if (a) is true of false, we can follow the same tactic: add Ps in (b). If we add Ps in left side, we have the following questions: Is it true P + P +2 subseteq P + P? (iff Goldbach) Is it true P + P + P +2 subseteq P + P? Is it true P + P + P + P +2 subseteq P + P? Is it true P + P + P + P + P +2 subseteq P + P? .... That is if S(n,P) is a subset of P + P for some n >=2. That is a weak problem than Goldbach. If we add Ps two both side, we have the following questions: Is it true P + P +2 subseteq P + P? Is it true P + P + P +2 subseteq P + P + P? Is it true P + P + P + P +2 subseteq P + P + P + P? .... Is it true that S(n,P)+2 subseteq S(n,P)? Well, obviously the questions: S(n,P) is a subset of P + P for some n >=2? Is it true that S(n,P)+2 subseteq S(n,P)? are difficult but less than Goldbach related problems (at least I hope that). I cant resolve these, but perhaps some of you will do. Its only a point of view of the thing. Sometimes the reformulation are good. Xan. === Subject: Poisson process, paradox by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBEDmLS02915; For a constant rate Poisson process, every specific(upto a finite resolution)sequence of N spikes occuring over a given time interval is equally likely. This seems paradoxical because we certainly do not expect to see all N spikes apprearing within the first 1% of time interval. Resolve this paradox. Help!!!! === Subject: Re: Poisson process, paradox >For a constant rate Poisson process, every specific(upto a finite >resolution)sequence of N spikes occuring over a given time interval is >equally likely. This seems paradoxical because we certainly do not >expect to see all N spikes apprearing within the first 1% of time >interval. Resolve this paradox. If you deal a poker hand from a well-shufßed deck, every possible hand is equally likely. But you certainly dont expect to get a royal ßush. This paradox is essentially the same as yours. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Poisson process, paradox > For a constant rate Poisson process, every specific(upto a finite resolution)sequence of N spikes occuring over a given time interval is equally likely. This seems paradoxical because we certainly do not expect to see all N spikes apprearing within the first 1% of time interval. Resolve this paradox. > Help!!!! Consider T as the time passing between two successive spike occurrences. If you see a spike say at time s (the start time) then the expected time for next spike occurrence is s+T. Your scenario descibes time instance (s+0.01T) which I simply denote by s+t where t = 0.01T. Recall that for a poisson system, the distribution of time that passes between two successive events has exponential distribution which makes the spikes to occur _randomly_ and understanding the meaning of such randomness property is the key concept to understand the apparent paradox (if any) that your question poses. The exponential distribution has a property that makes the time remaining for occurrence of next spike independent of time that already elapsed since last spike. Mathematically, P{T>s+t | T>s} = P{T>t} In other words, probability{ spike occurs in time interval from s+t to s+T given spike occured at s does not depend on time that already passed from s to s+t}. Therefore, yes we do expect N spikes at s + t the same way we would at s+T where t is 1% of T. Incidentally, one can see from above argument why exponential function is referred to as memoryless or forgetful function. -- Respectfully, Mohan Pawar MIO Instruments LLC (920) 277-6037 === Subject: Re: Poisson process, paradox posting-account=I_TYfwwAAAAroKuWwwoJjIK92nicsvNP Conditional probabilities often seem paradoxical since it is easy to implicitly confuse them with unconditional probabilities. Say that the Poisson process was at the rate of an average of 5 events per hour. Then maybe exactly 10 events in a specific hour is somewhat unlikely, but, *given* that: how are those 10 events distributed? If they werent uniformly distributed then (say) the chance of one of those events occuring in the first half hour would be greater than the chance of it occuring in the second half hour - but that would contradict the stationary (constant rate) property of a Possion process. === Subject: Transivity of the Action of a Symetric Group by Conjugation on Subgroups by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBEDmc203112; Good afternoon, First let see the context of my question. Let G be a subgroup of the symetric group S(F_2^m) (where F_2^m is a m dimensional vector space over F_2 = {0,1}) such that for all nonidentity sigma in G, sigma x is different from x (no fixed point) and sigma^2 x = x (sigma is an involution). Such a group is called a group of fixed-point free involutions. We can check that for those kind of groups we have : * G is Abelian * |G| <= 2^m Now we consider those groups of fixed-point free involutions of F_2^m such that |G| = 2^m. These groups are called maximal group of fixed-point free involutions of F_2^m and are ISOMORPHIC to T(F_2^m) the group of translations of F_2^m and even they are ISOMORPHIC to the group F_2^m itself. In particular they are 2-groups. For instance all the conjugate groups of the group of translations of F-2^m, T(F_2^m), are such maximal groups of fixed-point free involutions (ex. pi T(F_2^m)pi^{-1} where pi is in S(F_2^m)). The question is : Are all the maximal groups of involutions without fixed-point conjugate to the group of translations T(F_2^m) (i.e. is there only one orbit for the action by conjugation of S(F_2^m) on the set of all maximal groups of fixed-point free involutions ?) ?? Laurent === Subject: Re: (open?) questions related to basic linear algebra. by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBEDmN903056; QUESTIONS: Let M be a module over a commutative ring R. 1) does there always exist minimal generating sets in M ? 2) suppose M has a minimal generating set G; does the finitness of G imply that of the maximal free sets. Or more weakly, suppose M have a base B; does the finitness of B imply that of maximal free sets? 3) suppose M have a base B; does the finitness of B imply that of minimal generating sets? 4) do always free sets have cardinality less than generating sets? or more weakly, do always free sets have cardinality less than bases? 5) are two maximal free sets always of the same cardinality? Answer to question 2: yes. The question can be reformulated: is the rank of a free submodule F of a finitely generated module M finite? Since M is a homomorphic image of a free module of finite rank it suffices to consider the case >M free of finite rank<. It also suffices to consider free modules F of finite rank. We can replace the ring R by any localization R_p at a prime p of R, since localization preserves freeness and the rank. We consider (weakly) associated primes of R: the prime p is called associated if it contains an ideal I:={r in R | rx=0} for some x in R0, and is minimal among the primes containing I. The ring R_p then has the property that for every finitely generated ideal J of R_p there exists some y in R_p0 such that yI=0. So replace R by R_p, where p ius associated, and let m be the maximal ideal of R. Then: mM intersected with F = mF Let b_1,...,b_r be a basis of M and c_1,...,c_l a basis of F. Take an element x of mM - it can be written as x=m_1b_1+...+m_rb_r with m_i in m. Then there exists some non-zero y such that ym_i=0 for all i. Thus yx=0. Assume now that x is also an element of mM intersected F, then x=a_1c_1+...+a_lc_l with a_i in R, and thus 0=yx=ya_1c_1+...+ya_lc_l which implies ya_i=0 for all i due to freeness. It follows a_i in m for all i as asserted. We can now consider the R/m-vector spaces M/mM and F/mF, the latter being a subspace of the first due to what we just proved. We get: rank(M)=dim(M/mM)>=dim(F/mF)=rank(F). H === Subject: Re: (open?) questions related to basic linear algebra. >QUESTIONS: Let M be a module over a commutative ring R. > 1) does there always exist minimal generating sets in M ? No. Hint: consider the rationals over the integers. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Santa Claus is the god of math !!!! by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBEDmKt02892; >There is NOTHING that Santa cannot prove !!!! >Wish for a proof for Christmas and youll get it! Can he solve the Continuum hypothesis? [In the sense that Paul Erdos said that Christs answer to the question Can we solve the CH? was Paul Cohen gave all you can know.] === Subject: Re: Topics i n Metric Engineering II of W^3 I tried all this and *did* get a Berry phase shift. But, man, my electric bill went up by about $35. I just cant afford it, what with Christmas coming and all. And how will I feed all these intergalactic visitors coming thru the pesky rip in space-time? > III Zs Cognitive Dissonance on Newton to Einstein Gestalt Shift in > meaning of the inertial/non-inertial motion creative tension. > Also message from retired USAF SAC scientist R. Kiehn on Cartan, Stokes > theorem, Bohm-Aharonov, aerodynamic lift etc. >snip > The metric for an x-y vacuum domain wall can be written as > ds^2 = -(1 - g|z|/c^2)^2 . d(ct)^2 + (1 - g|z|/c^2)^2 .e^-(2kappa > ct).(dx^2 + dy^2) + dz^2 > JS: So what? > If you do a macro-quantum interference experiment, you will get a Berry > phase shift as well. How can you do that? Thats where metric engineering > the fabric of spacetime for warp, wormhole and weapon W^3 comes in. > Another story coming soon to a computer screen near you. === Subject: Re: Help with integral. by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBEEeAp08685; >In a problem Ive been working on recently, I have the following integral >int_0^{a} int _0^{b} { frac{y}{ x^2 + y^2 - 2xyc } } dx dy >where 0<=c<1 >There is a singularity as x,y approach 0. I need to compute this >integral fairly accurately and quickly. [...] This is (without any guarantee) mathematicas response [5.0]: In[1]:=inx=Integrate[y/(x^2+y^2-2*c*x*y),{x,0,a},Assumptions- >{00,a>0 }] Out[1]=(ArcSin[c]+ArcTan[(a-c*y)/Sqrt[(-(-1+c^2))*y^2]])/Sqrt [1-c^2] In[2]:=Integrate[inx,{y,0,b},Assumptions->{00,b>0}] Out[2]=(-2*(b+a*c)*Sqrt[1-c^2]*ArcSin[c]-2*a*c*Sqrt[1-c^2]* ArcTan[(b-a*c)/(a *Sqrt[1-c^2])]- 2*b*Sqrt[1-c^2]*ArcTan[(a-b*c)/(b*Sqrt[1-c^2])]+a*(-1+c^2)*(- 2*Log[a]+Log[a^ 2+b^2-2*a*b*c]))/(2*(-1+c^2)) hth Valeri === Subject: (-1)^n(n^(1/n) - 1) conditionally convergent? posting-account=2s1rcw0AAABZtBqaUPbD5ZJN1V2LKUsY Using Leibniz theorem I believe I can prove that the series above converge. I have a problem of proving that the series do _not_ converge absolutely. Any hint is highly appreciated. Boris. P.S. Yes, this is the homework question :). === Subject: Re: (-1)^n(n^(1/n) - 1) conditionally convergent? > Using Leibniz theorem I believe I can prove that the series above > converge. > I have a problem of proving that the series do _not_ converge > absolutely. > Any hint is highly appreciated. > Boris. > P.S. Yes, this is the homework question :). n^(1/n) = exp((log(n)/n) > 1 + ... Enough of a hint? === Subject: Re: (-1)^n(n^(1/n) - 1) conditionally convergent? posting-account=2s1rcw0AAABZtBqaUPbD5ZJN1V2LKUsY The answer is below... > Using Leibniz theorem I believe I can prove that the series above > converge. > I have a problem of proving that the series do _not_ converge > absolutely. > Any hint is highly appreciated. > Boris. > P.S. Yes, this is the homework question :). > n^(1/n) = exp((log(n)/n) > 1 + ... I tried that already. > Enough of a hint? Unfortunately Im still being a little dense... What about exp((log(n)/n - 1)? I know that: lim(n->oo, exp(log(n)/n) - 1) = 0 But I am unable to deduce anything useful :( Boris. === Subject: Re: (-1)^n(n^(1/n) - 1) conditionally convergent? > n^(1/n) = exp((log(n)/n) > 1 + ... > I tried that already. Well, whats the first term in ...? === Subject: Re: (-1)^n(n^(1/n) - 1) conditionally convergent? >> n^(1/n) = exp((log(n)/n) > 1 + ... >> I tried that already. >Well, whats the first term in ...? That would be . . === Subject: Re: (-1)^n(n^(1/n) - 1) conditionally convergent? * bobatonhu@yahoo.co.uk > Using Leibniz theorem I believe I can prove that the series above > converge. > I have a problem of proving that the series do _not_ converge > absolutely. > Any hint is highly appreciated. What is the definition of n^(1/n)? -- Jon Haugsand Dept. of Informatics, Univ. of Oslo, Norway, mailto:jonhaug@ifi.uio.no http://www.ifi.uio.no/~jonhaug/, Phone: +47 22 85 24 92 === Subject: Re: (-1)^n(n^(1/n) - 1) conditionally convergent? posting-account=2s1rcw0AAABZtBqaUPbD5ZJN1V2LKUsY > What is the definition of n^(1/n)? n-th root of n. Boris. === Subject: Re: (-1)^n(n^(1/n) - 1) conditionally convergent? posting-account=eLZrfw0AAAARDyX0AuKM0PJzr4YCFhKK > * bobatonhu@yahoo.co.uk > Using Leibniz theorem I believe I can prove that the series above > converge. > I have a problem of proving that the series do _not_ converge > absolutely. > Any hint is highly appreciated. > What is the definition of n^(1/n)? > -- > Jon Haugsand > Dept. of Informatics, Univ. of Oslo, Norway, mailto:jonhaug@ifi.uio.no > http://www.ifi.uio.no/~jonhaug/, Phone: +47 22 85 24 92 (n^(1/n) - 1) = (exp(ln(n)/n) - 1) > ln(n)/n > 1/n which diverges. The middle inequality comes from expanding exp(x) - 1 = x + x^2/2! + etc > x. === Subject: Re: (-1)^n(n^(1/n) - 1) conditionally convergent? posting-account=eLZrfw0AAAARDyX0AuKM0PJzr4YCFhKK > * bobatonhu@yahoo.co.uk > Using Leibniz theorem I believe I can prove that the series above > converge. > I have a problem of proving that the series do _not_ converge > absolutely. > Any hint is highly appreciated. > What is the definition of n^(1/n)? > -- > Jon Haugsand > Dept. of Informatics, Univ. of Oslo, Norway, mailto:jonhaug@ifi.uio.no > http://www.ifi.uio.no/~jonhaug/, Phone: +47 22 85 24 92 === Subject: Complex Analysis Question on Rouches Theorem I have a question from the proof of Rouches theorem from my textbook. The formulation of the theorem is Let G be a domain in C, K a compact subset of G, and f and g holomorphic functions in G such that |f(z)-g(z)|<|f(z)| for every point z in the boundary of K. Then, f and g have the same number of zeros in the interior of K, taking into account multiplicities. In the proof, the author says For 0<=t<=1 let f_t = (1-t) f + t g so that f_0 = f and f_1 = g. Let m(t) denote the number of zeros of f_t in the interior of K. ... by the argument principle we have / | f(z) m(t) = | ----- dz | f(z) /Gamma Finally, by standard estimates show that the integral on the right side of the preceding equality is a continuous function of t My question is, what standard estimates is the author talking about? I cant seem to get one with using epsilon-delta style of continuity proof; that is if I assume |t - t_o| In the proof, the author says > For 0<=t<=1 let > f_t = (1-t) f + t g > so that f_0 = f and f_1 = g. > Let m(t) denote the number of zeros of f_t in the interior of K. > ... > by the argument principle we have > / > | f(z) > m(t) = | ----- dz > | f(z) > /Gamma You omitted t in the integral. > Finally, by standard estimates show that the integral on the right side of > the preceding equality is a continuous function of t > My question is, what standard estimates is the author talking about? |m(t) - m(s)| = | int_gamma [(f_t)/f_t - (f_s)/f_s] | <= (arc length) * (maximum of |integrand|). Now show |(f_t)/f_t - (f_s)/f_s| is small everywhere on gamma if |t-s| is small. This is a standard calculus-like argument. === Subject: Re: Complex Analysis Question on Rouches Theorem posting-account=W2DCTA0AAAAlbhDMl3GrysSnPy1IK_7f ... > by the argument principle we have > / > | f(z) > m(t) = | ----- dz > | f(z) > /Gamma > Finally, by standard estimates show that the integral on the right side of > the preceding equality is a continuous function of t > My question is, what standard estimates is the author talking about? > I cant seem to get one with using epsilon-delta style of continuity proof; > that is if I assume |t - t_o| delta. This last sentence seems a bit garbled: given epsilon, you want to find delta such that, whenever |t-t_0| < delta, the distance between the two integrals is small. More to the point: All the author means is that the integral of a function which depends continuously on a parameter t will also depend continuously on t. To prove this, first think about what the condition in the above statement really means (i.e., what is the distance between two functions? --- hint: think of uniform convergence). Now if we have a function f and a function g, sufficiently close to f, write g = f + (g - f) [which is the usual trick], and integrate the latter, using the fact that g-f is small. Lasse --- === Subject: Where am I going wrong? Ok, its another one of the cardinality of the reals things, but Im not going to claim Im right I just hope some one can point out my mistake. So here goes: If we form a binary tree with 1/2 as the root node, 1/4 and 3/4 as its children, 1/8, 3/8, 5/8 and 7/8 as their children, etc. Extended to Aleph_0 levels we have every real >0 and <1 in the tree (i.e. every number representable as sum_i(f(i).2^-i) where f is any function mapping the natural numbers to {0,1}). If we traverse this tree in breadth first fashion it would seem that we can assign a natural number to each node in the tree as its order in breadth first traversal. Given that this would mean a contradiction in ZF I assume Im wrong. Can anyone tell me why? === Subject: Re: Where am I going wrong? > Ok, its another one of the cardinality of the reals things, but Im not > going to claim Im right I just hope some one can point out my mistake. > So here goes: > If we form a binary tree with 1/2 as the root node, 1/4 and 3/4 as its > children, 1/8, 3/8, 5/8 and 7/8 as their children, etc. Extended to > Aleph_0 levels we have every real >0 and <1 in the tree (i.e. every > number representable as sum_i(f(i).2^-i) where f is any function mapping > the natural numbers to {0,1}). If we traverse this tree in breadth first > fashion it would seem that we can assign a natural number to each node > in the tree as its order in breadth first traversal. Given that this > would mean a contradiction in ZF I assume Im wrong. Can anyone tell me why? The number of nodes is countable. The number of possible paths isnt. === Subject: Re: Where am I going wrong? >>Ok, its another one of the cardinality of the reals things, but Im not >>going to claim Im right I just hope some one can point out my mistake. >>So here goes: >>If we form a binary tree with 1/2 as the root node, 1/4 and 3/4 as its >>children, 1/8, 3/8, 5/8 and 7/8 as their children, etc. Extended to >>Aleph_0 levels we have every real >0 and <1 in the tree (i.e. every >>number representable as sum_i(f(i).2^-i) where f is any function mapping >>the natural numbers to {0,1}). If we traverse this tree in breadth first >>fashion it would seem that we can assign a natural number to each node >>in the tree as its order in breadth first traversal. Given that this >>would mean a contradiction in ZF I assume Im wrong. Can anyone tell me why? > The number of nodes is countable. The number of possible paths isnt. The number of possible paths is irrelevant. The nodes represent every real number >0 and <1, are you saying these are countable? === Subject: Re: Where am I going wrong? > Ok, its another one of the cardinality of the reals things, but Im > not going to claim Im right I just hope some one can point out my > mistake. So here goes: > If we form a binary tree with 1/2 as the root node, 1/4 and 3/4 as > its children, 1/8, 3/8, 5/8 and 7/8 as their children, etc. Extended > to Aleph_0 levels we have every real >0 and <1 in the tree (i.e. > every number representable as sum_i(f(i).2^-i) where f is any > function mapping the natural numbers to {0,1}). If we traverse this > tree in breadth first fashion it would seem that we can assign a > natural number to each node in the tree as its order in breadth > first traversal. Given that this would mean a contradiction in ZF I > assume Im wrong. Can anyone tell me why? >> The number of nodes is countable. The number of possible paths isnt. > The number of possible paths is irrelevant. The nodes represent every > real number >0 and <1, are you saying these are countable? 1/3 is not the label of any node. Your response seemed to show you recognize this fact: > > So at what level in the tree does 1/3 appear? > > You only have rationals with denominators that > are powers of 2. I was under the impression that sum_i(f(i).2^-i) (summed over the natural numbers) where f(i) was a function mapping the naturals to {0,1} is a real number, otherwise 2^Aleph_0 = |{x: x=n/2^m}| where m is finite. As I understand it this would mean pow(Aleph_0) = Aleph_0. It is true that one obtains every number in the interval [0,1] by summing a series of the form sum_i(f(i).2^-i), over the natural numbers. This is not the same as saying (in error) that this mapping is a surjection when you restrict your attention to finite sums. Note that to get 1/3, you require an infinite sum of terms, while each node exists at a finite stage (and thus corresponds to a finite sum of terms). At finite stages, nodes are in 1:1 correspondence with (finite) paths, but this correspondence fails to account for the reals that take infinitely many terms of the form n/2^m. Dale. === Subject: Re: Where am I going wrong? >> Ok, its another one of the cardinality of the reals things, but Im >> not going to claim Im right I just hope some one can point out my >> mistake. So here goes: >> If we form a binary tree with 1/2 as the root node, 1/4 and 3/4 as >> its children, 1/8, 3/8, 5/8 and 7/8 as their children, etc. >> Extended to Aleph_0 levels we have every real >0 and <1 in the tree >> (i.e. every number representable as sum_i(f(i).2^-i) where f is any >> function mapping the natural numbers to {0,1}). If we traverse this >> tree in breadth first fashion it would seem that we can assign a >> natural number to each node in the tree as its order in breadth >> first traversal. Given that this would mean a contradiction in ZF I >> assume Im wrong. Can anyone tell me why? > The number of nodes is countable. The number of possible paths isnt. >> The number of possible paths is irrelevant. The nodes represent every >> real number >0 and <1, are you saying these are countable? > 1/3 is not the label of any node. Your response seemed to show you > recognize this fact: > > So at what level in the tree does 1/3 appear? > > You only have rationals with denominators that > > are powers of 2. > I was under the impression that sum_i(f(i).2^-i) > (summed over the natural numbers) where f(i) was > a function mapping the naturals to {0,1} is a real > number, otherwise 2^Aleph_0 = |{x: x=n/2^m}| where > m is finite. As I understand it this would mean > pow(Aleph_0) = Aleph_0. > It is true that one obtains every number in the interval [0,1] > by summing a series of the form sum_i(f(i).2^-i), over the > natural numbers. This is not the same as saying (in error) > that this mapping is a surjection when you restrict your > attention to finite sums. I am not claiming the tree is finite, or does my definition of the tree fail for a depth of Aleph_0. > Note that to get 1/3, you require an infinite sum of terms, > while each node exists at a finite stage (and thus corresponds > to a finite sum of terms). At finite stages, nodes are in 1:1 > correspondence with (finite) paths, but this correspondence fails > to account for the reals that take infinitely many terms of the > form n/2^m. Im not sure what the deal with paths is. Does a breadth first traversal fail to cover the whole tree of infinite depth? === Subject: Re: Where am I going wrong? > Ok, its another one of the cardinality of the reals things, but > Im not going to claim Im right I just hope some one can point out > my mistake. So here goes: > If we form a binary tree with 1/2 as the root node, 1/4 and 3/4 as > its children, 1/8, 3/8, 5/8 and 7/8 as their children, etc. > Extended to Aleph_0 levels we have every real >0 and <1 in the tree > (i.e. every number representable as sum_i(f(i).2^-i) where f is any > function mapping the natural numbers to {0,1}). If we traverse this > tree in breadth first fashion it would seem that we can assign a > natural number to each node in the tree as its order in breadth > first traversal. Given that this would mean a contradiction in ZF I > assume Im wrong. Can anyone tell me why? >> The number of nodes is countable. The number of possible paths isnt. > The number of possible paths is irrelevant. The nodes represent every > real number >0 and <1, are you saying these are countable? >> 1/3 is not the label of any node. Your response seemed to show you >> recognize this fact: >> > > So at what level in the tree does 1/3 appear? >> > > You only have rationals with denominators that >> > are powers of 2. >> I was under the impression that sum_i(f(i).2^-i) >> (summed over the natural numbers) where f(i) was >> a function mapping the naturals to {0,1} is a real >> number, otherwise 2^Aleph_0 = |{x: x=n/2^m}| where >> m is finite. As I understand it this would mean >> pow(Aleph_0) = Aleph_0. >> It is true that one obtains every number in the interval [0,1] >> by summing a series of the form sum_i(f(i).2^-i), over the >> natural numbers. This is not the same as saying (in error) >> that this mapping is a surjection when you restrict your >> attention to finite sums. > I am not claiming the tree is finite, or does my definition of the tree > fail for a depth of Aleph_0. Note that I did not say you were claiming the tree to be finite. What I said was that you do not get 1/3 as a finite sum of the form that you are using, namely as a sum of n/2^m for integer numerators n. Surely you realize that each node corresponds to a *finite* sum? Surely you recall that you said this: The number of possible paths is irrelevant. The nodes represent every real number >0 and <1, are you saying these are countable? How do you reconcile these statements; the first two are mine, and the third is yours: 1. Every node corresponds to a *finite* sum. 2. 1/3 is not expressible as a finite sum sum( f(n)/2^n | n : Natural numbers ) 3. The nodes represent every real number >0 and <1. >> Note that to get 1/3, you require an infinite sum of terms, >> while each node exists at a finite stage (and thus corresponds >> to a finite sum of terms). At finite stages, nodes are in 1:1 >> correspondence with (finite) paths, but this correspondence fails >> to account for the reals that take infinitely many terms of the >> form n/2^m. > Im not sure what the deal with paths is. Does a breadth first traversal > fail to cover the whole tree of infinite depth? The deal is this: the only way to reach the leaves of this tree is via paths. There are no nodes out at the ends of the tree. Also, when I am using the term path, I am referring to monotonic paths starting at the root. In this fashion, I can uniquely identify any node in the tree by expressing the path from the root to that node. There is one other thing: exactly *how* do you care to do breadth first traversal of this tree? It seems that you could most likely find a closed-form solution for the nth number visited, and could then establish that that formula never achieves the value 1/3. Suppose we look at this search process: According to your layout: Root(level 0): 1/2 at level 1: 1/4 and 3/4 level 2: 1/8, 3/8, 5/8, 7/8 and so forth, with level k populated by fractions with odd numerators less than 2^(k+1) and denominator equal to 2^(k+1). In terms of the path from the root to a node at level k, the path corresponding to the length-k sequence F in {-1,1}^k reaches a node with value: 1/2 + sum(F(n)/2^(n+1)) Lets look at breadth-first search of this tree: For breadth-first search, you start at the root, visit the nodes at level 1, then those at level 2, and so forth. At each level n, there are 2^n nodes to visit, and so you first reach level k (root is level 0) at stage: Level 0: 1 Level 1: + 2 Level 2: + 4 ... + ... Level k-1 + 2^(k-1) ....completing Levels 1 ... k-1 NEXT STEP: + 1 ....beginning Level k. i.e., = 2^k. That is, a number of the form 1/2 + sum(F(n)/2^(n+1), n=1,...,k) will be visited between 2^k and 2^(k+1). Note that the number of binary digits required to express the number tells us both a lower and upper bound for the natural number you intend to assign to the node: k+1 binary digits <--> number assigned is between 2^k and 2^(k+1). Now, consider 1/3 = 0.0101010101... What number do you suggest will be assigned to 1/3? Dale. === Subject: Re: Where am I going wrong? >> I am not claiming the tree is finite, or does my definition of the >> tree fail for a depth of Aleph_0. > Note that I did not say you were claiming the tree to be finite. What > I said was that you do not get 1/3 as a finite sum of the form that > you are using, namely as a sum of n/2^m for integer numerators n. > Surely you realize that each node corresponds to a *finite* sum? Not totally sure that I do. > Note that the number of binary digits required to express the number > tells us both a lower and upper bound for the natural number you intend > to assign to the node: > k+1 binary digits <--> number assigned is between > 2^k and 2^(k+1). > Now, consider 1/3 = 0.0101010101... > What number do you suggest will be assigned to 1/3? One in the set {x:2^k<=x<2^(k+1) as k->inf}? Is this a bad definition, or is |{x:0<=x<2^(k+1), k->inf}| > Aleph_0? Or do we actually have to treat k as Aleph_0 giving us 2^Aleph_0? === Subject: Re: Where am I going wrong? Stephen Jones a .8ecrit : > Ok, its another one of the cardinality of the reals things, but Im > not going to claim Im right I just hope some one can point out my > mistake. So here goes: > If we form a binary tree with 1/2 as the root node, 1/4 and 3/4 as > its children, 1/8, 3/8, 5/8 and 7/8 as their children, etc. Extended > to Aleph_0 levels we have every real >0 and <1 in the tree (i.e. > every number representable as sum_i(f(i).2^-i) where f is any > function mapping the natural numbers to {0,1}). If we traverse this > tree in breadth first fashion it would seem that we can assign a > natural number to each node in the tree as its order in breadth > first traversal. Given that this would mean a contradiction in ZF I > assume Im wrong. Can anyone tell me why? >> The number of nodes is countable. The number of possible paths isnt. > The number of possible paths is irrelevant. The nodes represent every > real number >0 and <1, are you saying these are countable? Do you read answers to your messages? Which node represent 1/3? === Subject: Re: Where am I going wrong? >Ok, its another one of the cardinality of the reals things, but Im not >going to claim Im right I just hope some one can point out my mistake. >So here goes: >If we form a binary tree with 1/2 as the root node, 1/4 and 3/4 as its >children, 1/8, 3/8, 5/8 and 7/8 as their children, etc. Extended to >Aleph_0 levels we have every real >0 and <1 in the tree (i.e. every >number representable as sum_i(f(i).2^-i) where f is any function mapping >the natural numbers to {0,1}). If we traverse this tree in breadth first >fashion it would seem that we can assign a natural number to each node >in the tree as its order in breadth first traversal. Given that this >would mean a contradiction in ZF I assume Im wrong. Can anyone tell me why? >> The number of nodes is countable. The number of possible paths isnt. > The number of possible paths is irrelevant. The nodes represent every > real number >0 and <1, are you saying these are countable? The nodes represent only a countable subset of the reals, namely, the rationals with denominators that are powers of 2. The real number 1/3, for example, does not lie in any node at any depth. Each real number in [0,1] can be represented as the limit of the values lying along a particular path, and there are uncountably many paths. -- Dave Seaman Judge Yohns mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Where am I going wrong? > The nodes represent only a countable subset of the reals, namely, the > rationals with denominators that are powers of 2. The real number 1/3, > for example, does not lie in any node at any depth. Each real number in > [0,1] can be represented as the limit of the values lying along a > particular path, and there are uncountably many paths. 1) This tree: a / b c has 9 paths: (a) (b) (c) (a b) (a c) (b a) (c a) (b a c) (c a b) Does it contain 9 values? 2) The point is that the tree is traversed in breadth first fashion. This means that paths are irrelevant. 3) How do you show that there are uncountably many paths? === Subject: Re: Where am I going wrong? >> The nodes represent only a countable subset of the reals, namely, the >> rationals with denominators that are powers of 2. The real number 1/3, >> for example, does not lie in any node at any depth. Each real number in >> [0,1] can be represented as the limit of the values lying along a >> particular path, and there are uncountably many paths. > 1) This tree: > a > / > b c > has 9 paths: This is a finite tree, meaning that each path terminates in a node. In the infinite tree, a path need not terminate. There are uncountably many paths, but only countably many that terminate. > (a) > (b) > (c) > (a b) > (a c) > (b a) > (c a) > (b a c) > (c a b) > Does it contain 9 values? I dont know what your point is, but the infinite tree has countably many terminating paths and countably many values. Its true that you can find multiple paths going through any given node, but that does not contradict what I said. > 2) The point is that the tree is traversed in breadth first fashion. When do you reach a node containing 1/3? > This means that paths are irrelevant. No, it doesnt. There are uncountably many nonterminating paths, and each real number in [0,1] can be represented as the limit of the values along some nonterminating path. > 3) How do you show that there are uncountably many paths? There is a natural bijection between the nonterminating paths and the set of binary digit strings. -- Dave Seaman Judge Yohns mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: Where am I going wrong? >The nodes represent only a countable subset of the reals, namely, the >rationals with denominators that are powers of 2. The real number 1/3, >for example, does not lie in any node at any depth. Each real number in >[0,1] can be represented as the limit of the values lying along a >particular path, and there are uncountably many paths. >>1) This tree: >> a >> / >>b c >>has 9 paths: > This is a finite tree, meaning that each path terminates in a node. In > the infinite tree, a path need not terminate. There are uncountably many > paths, but only countably many that terminate. That the paths do not terminate is the point. Are you saying that the reals do not exist within the tree as defined or that a breadth first traversal would never touch the reals within it, or would not do so in countably many steps? === Subject: Re: Where am I going wrong? Originator: richard@cogsci.ed.ac.uk (Richard Tobin) >If we form a binary tree with 1/2 as the root node, 1/4 and 3/4 as its >children, 1/8, 3/8, 5/8 and 7/8 as their children, etc. Extended to >Aleph_0 levels we have every real >0 and <1 in the tree So at what level in the tree does 1/3 appear? You only have rationals with denominators that are powers of 2. -- Richard === Subject: Re: Where am I going wrong? >>If we form a binary tree with 1/2 as the root node, 1/4 and 3/4 as its >>children, 1/8, 3/8, 5/8 and 7/8 as their children, etc. Extended to >>Aleph_0 levels we have every real >0 and <1 in the tree > So at what level in the tree does 1/3 appear? > You only have rationals with denominators that are powers of 2. I was under the impression that sum_i(f(i).2^-i) (summed over the natural numbers) where f(i) was a function mapping the naturals to {0,1} is a real number, otherwise 2^Aleph_0 = |{x: x=n/2^m}| where m is finite. As I understand it this would mean pow(Aleph_0) = Aleph_0. posting-account=X3ttSg0AAAA4Fg-MKfJxiPMOnY4Nge1r Singapores High Test Scores Win Over Some Educators; Another Fleeting Craze? Teachers Go Back to Class By CRIS PRYSTAY Staff Reporter of THE WALL STREET JOURNAL TOWNSEND, Mass. -- About five years ago, a statewide test in Massachusetts revealed that students math skills deteriorated sharply as they went from fourth to sixth grade. Alarmed, the Massachusetts education commissioner suggested an unconventional fix: importing the math curriculum used in Singapore. Students in Singapore routinely score among the highest in international math tests. The hope was that American kids taught the Singaporean way would improve their math scores. The approach has been adopted in about 200 schools nationwide, from rural Oklahoma to the inner cities of New Jersey. Early indications suggest that many U.S. students taught with textbooks imported from Singapore do perform better in math. Some children who once found the subject frustrating say they now like it. Faced with a worrying decline in math proficiency among U.S. kids, a growing number of educators are seeking inspiration from Asian curricula. American children are falling behind their Asian peers in science and math, a shift that could push still more white-collar jobs offshore as the next generation graduates. Our kids just dont seem as numerate as they should be, and we decided we needed to try whatever we can to fix that, says David Driscoll, Massachusetts education commissioner and a former math teacher himself, who had the idea of using Singapore text books in local schools. Critics assert that math teaching has been dumbed down in the U.S. over the past two decades. They say that too much emphasis is placed on making the subject accessible and fun and not enough on vital, if repetitive, drills such as multiplication tables. Another big criticism: U.S. math curricula tend to cover plenty of subject areas but not in sufficient depth. Singapore and other southeast Asian countries take a different tack. Singapores curriculum was developed over the past few decades by math experts hired by the Ministry of Education, who continually interviewed math teachers to find out what works and where kids need help. The elementary textbooks cover only one-third of the topics typically found in U.S. textbooks, but the material is taught far more thoroughly. While rote learning plays a part, kids in Singapore also learn to use visual tools to understand abstract concepts. Singapore math texts, for example, ask kids to draw bars and other diagrams to visualize problems -- a technique called bar modeling. When this strategy is applied consistently over a number of years, children tend to be better able to break down complex problems and do rapid calculations in their head. Not everyone believes that importing textbooks from Singapore would solve Americas math problem. Some states say the approach doesnt meet their standards. American math curriculum varies from state to state, so there is a potential gap between standards set on the material students need to know and what they have covered using the Singapore books. The National Council of Teachers of Mathematics in the U.S. suggests that it might not be possible to copy what Singapores done simply by importing its books. The success of its math program may have roots in Singapores highly disciplined culture, where the entire community -- particularly parents -- expects kids to buckle down and work hard, argues the NCTM. Theres little doubt, though, that math teaching in America needs to be overhauled. Tomorrow, Boston College will release a four-year global study that is expected to show the math gap with Asia remains. The colleges last study, the 1999 Trends in International Mathematics and Science Study (TIMSS), ranked eighth-graders in Singapore the best in math, while U.S. kids came in 19th, just behind Latvia. American kids also fall further behind the longer theyre in school; as fourth-graders, American kids ranked 7th on the 1995 study. That decline has already had an impact on U.S. universities. Among U.S. freshmen who plan to major in science or engineering, one in five requires remedial math courses, according to the National Science Board, which is part of the government-funded National Science Foundation. Enrollment by U.S. citizens or permanent residents in graduate science and engineering programs, meantime, dropped 10% between 1994 and 2001. Enrollment of foreign students grew 35%. Because of the skills gap, America risks losing even more jobs overseas. Many have a gnawing sense that our problems may be more than temporary and that the roots of the problem may extend back through our education system, said Federal Reserve Chairman Alan Greenspan at a Boston finance conference in March. Reforming the U.S. curriculum is difficult. Unlike Singapore and other Asian countries, the U.S. doesnt have a national curriculum. Each state is responsible for setting standards, while each district retains control over how a subject is taught. Under the Bush administrations No Child Left Behind policy, funding and jobs depend on how each school rates on standardized state exams. Many district officials are reluctant to try something new for fear of slipping up on those exams. But a handful are turning to Asia for answers. Georgia plans to adopt Japanese math standards as part of its reform of secondary-school curricula. A teacher-training textbook, based on Singapores elementary math curriculum and written by a math professor at Michigan State University, is now used at a half-dozen universities in America. Singapores math textbooks for young kids have the biggest appeal in the U.S. because theyre written in English. In rural Bethel, Okla., school-district superintendent Marty Lewis thought his kids were slipping in math. After his curiosity about Singapore was piqued by the 1999 TIMSS results, he did an Internet search about the Singapore method. That led him to the private Rosenbaum Foundation of Pennsylvania, which funds Singapore math programs in the U.S. and Israel. The foundation, in turn, put him in touch with Yoram Sagher, a Florida mathematician who trains teachers to teach Singapore math. Mr. Lewis hired Mr. Sagher to give a one-week seminar to all his teachers in July. Bethel kindergartners and first-graders began using the Singaporean books in September. I came to a point where I thought: I dont care how crazy people think I am; Im going to go out and find something that works, says Mr. Lewis. While Bethels kids are just getting started, other school districts have adopted the Singapore method wholesale. One is North Middlesex, a farming and commuter district thats an hours drive from Boston. North Middlesexs program got rolling soon after the education commissioner, Mr. Driscoll, noticed the decline in math ability among his states sixth-graders. In 2000, he got a $50,000 federal grant to test whether a Singapore curriculum would improve math scores for kids in his district. North Middlesex dispatched three teachers with math degrees to work with a math professor at the Worcester State College in Massachusetts. They came up with a seven-day summer seminar for North Middlesex district teachers, based on textbooks from Singapore. The Singapore-inspired program was started in grades five through eight, which needed the greatest help. As more teachers volunteered, the program was extended to other grades. On a recent morning in Ashby, a tiny town in North Middlesex, fifth-grade math teacher Bob Hogan asked for volunteers to work out how many women there are in a hypothetical university class of 250 if there are 50 more men than women. Mr. Hogan, an energetic 30-year-old teacher, asked for volunteers to tell him how to solve the problem using a bar model. Sarah Carter, a 9-year-old with freckles and bright red hair, leaned forward in her seat, arm in air. First, she instructed Mr. Hogan to draw two bars of equal length, and label the top one women and the bottom one men. She then he told him to add a small square extension to the mens bar, and write 50 inside of it. To the right of both bars, she asked him to write 250, indicating the full value of both, together. Looking at this pictorial, she started to solve the problem without pen or paper: She verbally subtracted 50 from 250, and asked him to write the 200 on the board, to the left of the two empty bars, indicating their combined value. Then, she divided 200 by two, and announced there are 100 women in the class, and 150 men. I dont know where Singapore is, she said, but I like the way they do math. Some teachers were initially skeptical. Steve Keating, a veteran math teacher who teaches seventh grade, says he has lived through a host of new math approaches, including the new math craze in the 1970s. My first thought was, here we go again, he says, referring to the Singapore method. He was especially taken aback by the textbook. By grades seven and eight, kids in the Singapore program are doing high-school-level algebra. I thought, wow, thats complicated -- even for me, says Mr. Keating. He was eventually won over when he saw how enthusiastic his own students became about math. The approach expects a lot of its teachers. Singapores math program doesnt come with guides that walk teachers through every step of the class, and every problem, as many U.S. courses do. Teachers cant ßip to the back of the book for answers. During the first year, Mr. Keating spent two hours every night preparing the next days lesson. On his summer vacation he took math books to the beach. The effort paid off; his students math scores improved. Some parents also had doubts. Suzanne Carter recalls that her daughter Sarah, whod always struggled at math, came home and drew bars and rectangles instead of working on the sums she grew up with. I was frustrated. I had no idea what she was doing, says Mrs. Carter, a sign-language instructor. Her daughters school, however, doesnt need more convincing. Students at North Middlesex are already doing better on state exams. Eighth-graders, for example, scored 75.4 points on this years state math proficiency index, up from 63.2 points in 2000. That jump was twice that of the state average -- which also improved. Other grades improved, but in line with the state average. Eager for something more conclusive, North Middlesex recently hired Stanford Universitys Hoover Institution to analyze a slew of state and district exams to see whether a group of 300 students whod taken one to three years of the Singapore program were better at math than other students. The study, which is continuing, found the Singapore math students had significantly better computation skills. Boston Public Schools tried the Singapore math books in a few classes at one school last year, but decided to drop them. The district had adopted another math program, called the Workshop Model, which promoted group and independent work activities designed to get kids to think about concepts behind math. They didnt want to detract from that by experimenting more broadly with something new, said Ed Joyce, curriculum director for math for Boston Public Schools. I wouldnt say anything bad about Singapore math, but I would say theres a lot of programs that would have the same result, he said. Another hurdle that could limit the appeal of the Singapore method is the U.S. obsession with standardized testing. Kids taking Singapore math might be better at a core set of subjects such as multiplication, fractions, word problems and algebra, but they may struggle with topics that appear on state tests. So North Middlesex supplements the Singapore books with a few extra lessons in subjects like probability, which are taught in grades four and five in the U.S. but not until later grades in Singapore. The Singapore method continues to attract fans. Inspired by North Middlesex, 20 schools in 12 different districts across Massachusetts are now running Singapore pilot programs. William Carey, principal of Beachmont Elementary school in Revere, a blue-collar suburb of Boston, last year began offering Singapore math in grades one through four. He reports some early signs of success. Beachmonts grade four class lagged behind the state average by just 3% on this years state exam, up from the 8% gap between the state and last years fourth-grade class. Beachmonts success, in turn, has inspired others. Across town, teachers at Garfield Elementary began to teach math the Singapore way this year. When something makes a difference, people notice, says Mr. Carey, the principal at Beachmont. Word is starting to spread. ------------------------------------------------------- === Subject: New number Here is an interesting concept: This is a new number representing the opposite of zero, but not exactly infinity. I call the number All and it is written as a filled in zero (or a zero with a dot in the middle for convenience). It exists on the basis that in an infinite universe, all possibilities that can exist must exists, therefore when all the possibilities do exist the number is All . I am probably not the first to think of this, but I am sure some fun could be had with the concept. === Subject: Re: New number > Here is an interesting concept: > This is a new number representing the opposite of zero, but not > exactly infinity. I call the number All and it is written as a > filled in zero (or a zero with a dot in the middle for convenience). > It exists on the basis that in an infinite universe, all possibilities > that can exist must exists, therefore when all the possibilities do > exist the number is All Wow! That is such an amazing discovery that I have to divulge my discovery of a new number that is the opposite of three, but its not exactly infinity! I call it eerht, and it is written using the numeral 3, only setting it on its side atop a zero, and drawing a smiley face inside the zero portion of eerht. Any time I have a problem where the solution isnt really three, and it isnt exactly infinity, I just yell out EERHT! and draw the numeral for eerht on something, or Ill just draw it in the air! This math research is so much fun! > . I am probably not the first to think of this, but I am sure some fun > could be had with the concept. Im going to invent a number that is written by throwing a handful of confetti! And another that uses one of those blowy things where a paper thing rolls out and makes a squawking sound! Dale === Subject: Re: New number > Here is an interesting concept: > This is a new number representing the opposite of zero, but not > exactly infinity. I call the number All and it is written as a > filled in zero (or a zero with a dot in the middle for convenience). > It exists on the basis that in an infinite universe, all possibilities > that can exist must exists, therefore when all the possibilities do > exist the number is All > . I am probably not the first to think of this, but I am sure some fun > could be had with the concept. Do you want some dressing with that word-salad? === Subject: n-fold dunce cap by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBEGFtb19238; If the fundamental group of an n-fold dunce cap is Z_n, then what would be its covering spaces? Is S^2 its universal cover with a covering map of degree n? If n is prime, then is S^2 the only non-trivial cover? If n is not prime, and say n=ab, then would a cover of the n-fold dunce cap be an a-fold dunce cap with a covering map of degree b? Bill === Subject: Re: n-fold dunce cap >If the fundamental group of an n-fold dunce cap I assume you mean by that what is more often called the pseudoprojective plane of degree n or the like--the quotient P(n) of the unit disk in R^2 by the equivalence relation with non-trivial equivalence classes the sets of vertices of regular n-gons inscribed in the unit circle? If not, ignore what follows. >is Z_n, then what would be its covering spaces? Is S^2 >its universal cover with a covering map of degree n? Certainly not (for n > 2)! Can you think of any candidates for the non-trivial covering transformations, for instance? Let C be the image in P(n) of the unit circle. As you have correctly stated (though I dont know your reasoning, so I dont know if youve stated it for the right reasons), the fundamental group of P(n) is Z/(n). Moreover, if you take the basepoint on C, then the inclusion of C (which is itself a 1-sphere) into P(n) induces the map k mapsto k (mod n) from the fundamental group of C onto the fundamental group of P(n). Therefore, if X is a connected covering space of P(n), the pre-image of C in X is a connected covering space of C, that is, yet another copy of 1-sphere. Meanwhile, the preimage in X of the complement of C in P(n)--which is an open 2-disk, and therefore contractible--is a disjoint union of copies of that complement; and you should be able to convince yourself that the closure of any component of that preimage is the union of that component and the (entire) preimage of C. In fact, you should be able to convince yourself (at least, I have; so I hope its true!) that that closure is some P(m), where m is a divisor of n! In particular, in the case of the simply connected (universal) cover X of P(n), that closure is P(1), i.e., a closed disk. There are n/m such components in general, so n in the universal-cover case, and any two of them have exactly their boundaries in common. Now you can *see* what X is. >If n is prime, then is S^2 the only non-trivial cover? >If n is not prime, and say n=ab, then would a cover of >the n-fold dunce cap be an a-fold dunce cap with a >covering map of degree b? Lee Rudolph === Subject: Re: How big is the electron? Jack Sarfatti; > leads to problems of infinite renormalization parameters. In quantum informationdynamics says that energy is the same as information. Because enegy is information then we see trivially, how energy === Subject: Re: How big is the electron? <41bdfafa$1@news.accesscomm.ca> posting-account=rucChAwAAACE1EKH5GVIfFU67b8tSAnw my suggestion is simple all the various Ôsizes is becuae *the electron has not jusy one Ôsize it has many sizes* so it conglomeration, as you can immagine a conglomeration can have different inner shapes and arangements th e most common one is the longish shape ie sort of a fountain. all the best Y.Porat ------------------------------------ === Subject: Re: How big is the electron? posting-account=2fdXgA0AAAAiH_AsNUz4SbCuDB3R59KU Ive never seen a no-hidden variable proof, only no-local-hidden-variable proofs, and QM is non-local anyway, so whats the deal about no hidden variables? And an electron could be considered as large as its wavefunction, or the radius of an event horizon of a spining charged massive black hole with the energy, charge, and angular momentum of an electron, either is fairly intuitive as to what you are talking about. But these hard balls and other stuff. Ive seriously never seen that before, like ever. And the whole question seems weird since clearly electrons are nonlocal anyway since they are correlated and inßueced by electrons arbitrarily far away. === Subject: Re: How big is the electron? <41bdfafa$1@news.accesscomm.ca> posting-account=dtfNHQwAAAA8zDWo-yL2b-U5volg3zkc > Jack, since you know so many physicists let me ask you some things. > In your theory and in the physics community in general, > 1. How big is an electron, whats its radius? > The classical ball size of an electron is > e^2/mc^2 ~ 10^-13 cm > The quantum size is > h/mc ~ 10^-11 cm > The classical ball is surrounded by a virtual electron-positron plasma > that extends out from 10^-13 cm to 10^-11 cm > This is the NAIVE picture. > Problem is that Bohrs Copenhagen -> Von Neumann interpretation says > THERE ARE NO ELECTRONS as Newtons hard massy balls - there are only > quantum waves or qubits that collapse to make localized clicks > on counters. This is Wheelers Smoky Dragon mysticism. That is, there > are NO HIDDEN VARIABLES. This > leads to problems of infinite renormalization parameters. > There is then the scattering theory of form factors - complicated. > Also high energy scattering of electrons on protons and neutrons shows > the latter have three point like scattering centers - consistent with > quarks. Quarks and leptons are unified in gauge theory. Well not quite. > Bohms theory does allow the HIDDEN VARIABLES i.e. Newtons hard massy > balls that are guided by the Bohr quantum pilot waves. Its like a > ball rolling on a landscape of hills valleys and saddle point mountain > passes. These are the Bohmian trajectories of the hidden variables. > Then there is the idea that electrons and quarks are really little > microgeons i.e. localized regions of pure vacuum with CONFORMAL > CURVATURE fields like rotating charged black holes. Burinski is working > on that now in Moscow. However you need strong short range gravity G* ~ > 10^40G on the scale of e^2/mc^2 ~ 1 fermi for this to work. Maybe the > extra space dimensions kick it for nuclear physics. Also for the stable > electron you need a special solution that does not have Hawking > radiation - that is part of Kerr-Newman. > The microgeons will shrink if probed with a big kick in a > high-resolution scattering Heisenberg microscope experiment. They look transfer > - at least up to a critical value, where maybe they start looking bigger > again in Susskinds IR/UV duality of string theory. > One nice thing about rotating micro-geons in strong short range gravity > is that they automatically obey the universal Regge slope law > Spin of hadronic resonance ~ alphaE^2 > alpha ~ (1 Gev)^-2 > This is the basic data for string theory and of course black holes and > strings are two sides of the same coin! > Also no hierarchy problem in this picture where the quantum gravity > Planck scale Lp is actually at ~ 1Gev. I published this in 1973 and > Abdus Salam invited me to ICTP because of that. > 2. What is the equivalent lineal mass density (kg/m) of the vacuum? > Meaningless question. > The naive zero point energy of the vacuum WITHOUT the COHERENCE that is > the basis for my theory of the emergence of Einsteins General > Relativity as a More is Different post-inßationary macro-quantum > coherent phase modulation is > Vacuum Energy Density ~ hc/Lp^4 > This is MUCH TOO BIG. The dark energy density is > ~ hc/(c/H)^4 > Where H = todays FRW Hubble parameter R^-1dR/dt. > If you use Lp ~ 10^-33 cm Hals PV gives is off by 122 Powers of Ten. > That is a colossal failure compared to GRs colossal success of 10^-14 > in the 1916 + 13 pulsar data that won a Nobel Prize! > Hal Puthoff has no solution to this problem in PV. He hand waves it away > as he genußects to ET. :-) > http://www.guntheranderson.com/v/data/thevatic.htm > http://www.hyperdictionary.com/dictionary/genußect > Im doing a comparison of GR and PV and, Id like to compare it to your > model as well. However, Im getting mixed input about the radius of an of 2.818 > fm. Milonni on the other hand says it is point-like and 10^-2 fm but its > charge is spread out by the interaction with the ZPF to about the Compton > wavelength, ~10^3 fm. The classical radius is bogus. Do you agree? > Sort of. See above. PV is worthless, a complete waste of time. > 1. It violates general coordinate transformation covariance of the > dynamical action. > 2. It is incompatible with the LOCAL equivalence principle used in GR > (e.g. Cartan tetrads). > 3. It is seriously incomplete e.g. it cannot describe rotating bodies > and the now observed gravimagnetic Lense-Thirring frame drag of the > Cartan tetrads. > 4. PV fails every experimental test beyond the 3 trivial classic tests > in new observations like 1913+16 pulsar where GR works beautifully to a > precision of 10^-14! > The above opinion on Hals PV is the consensus of several top physicists > at GR 17 > e.g. Cliff Will, Matt Visser, Bill Unruh ... > Since at less than 1/2 the Compton wavelength we get the possibility of e-p > pair creation, this would seem to fit well with your model of the > vacuum. So > Im hoping you can give me some specifics to compare to. > main_engineering > The electron is a spiral-shaped cloud extending > outward from the nucleus *in exactly the same > way* that a galaxys arm is made. > They are the same thing. > John > galaxy model for the atom > http://www.petcom.com/~john/ So how does this Ôrolling about help to make the earth grow bigger? And what sort of immediate manifestations should we look for when we === Subject: Re: Gamma Function/Mills ratio/Inequalities posting-account=OyMMlAwAAADyhoVhXYX4Bw0T-1IatpYa If f(x)=G(x+0.5)/G(x+1) , then F:(0,infty)--->(0,infty) with (*) F(x)= f(x)*sqrt(x) is increasing on its domain. Therefore, if O < x < y ,then f(x)*sqrt(x) < f(y)*sqrt(y) , or (1) sqrt(x/y) < f(y)/f(x) . Its clear that (1) is verified for x=0. Further, use Alain Vergothe remark, namely that f satisfies ,,Wallis functional equation (2) f(x+0.5)f(x) = 1/(x+0.5) for all x >= 0 . Regarding this functional equation, see for instantce : I.Lazarevic and ..., ,, Functional equations for Wallis and Gamma Function , Univ. Beograd Publ. Elektrotehn. Fak.Ser. Mat. Fiz. No. 461-497 (1974), 245-252. If we make the substitutions x-->x+0.5 and y-->y+0.5 in (1), using (2) we find that sqrt((x+0.5)/(y+0.5)) < (x+0.5)*f(x)/((y+0.5)*f(y)) which is the same with (3) f(y)/f(x) < sqrt((x+0.5)/(y+0.5)) , ( 0=< x < y ). Thus, according to (2), inequality (1) implies the upper-bound (3). f(y)/f(x) = sqrt((x+u(x,y))/(y+u(x,y)) with u(x,y) in (0,1/2). David Cantrell remarks that in fact u(x,y) is in (1/4 , 1/pi) subset (0,1/2) is very interesting. However, until now I havent a proof for this proposition === Subject: Re: The State-of-the-Art in Mathematics by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBEGVVs20729; >In sci.math, S. Enterprize Company > MATHEMATICAL UPDATE AND CLARIFICATION >> >> THIS IS part of my occasional update of mathematics and physics. It >> refers to the resolution of mathematical issues that have appeared in >> several journals recently. (1) The complex number i is nonsense since >> i = sqrt(-1) = sqrt(1/-1) = 1/i = -i from which follows i = -i. >> -i = 1/i >> -i = 1/sqrt(-1) >typo; i =/= -i >> i =/= -1 >> i = sqrt(-1) >> i = sqrt( -1) >> i^2 = 1 >> i^3 = -i >Be *VERY* careful here. sqrt(4) = +2 or -2. sqrt(-1) is >traditionally called i, but again sqrt(-1) = +i or -i. >(Theres at least one joke proof running around that >deliberately confuses the issue by inappropriately taking >the square root and proving a nonsense result.) >> Lets see if the reciprocal of 1/i really is -i. >Well, if b is the reciprocal of a, then a * b = +1. >Since i^2 = -1, i * i * (-1) = +1, and one can take >i * (-1) = -i to be the reciprocal of i. QED. >> It looks like to me, the reciprocal of i is, >> i^-1. >> But, lets check, >> sqrt(-1)/sqrt(-1) = 1 >> i^1 * i^-1 = i^(1-1) = i^0 = 1 >> If the reciprocal of i was -1 then >Did you mean -i here? >> i^1 * (-i^1) = -i^(1+1) = -i^2 = 1 >> It looks like it gives the same answer in this case. >> But what about this ? >> (i^-1) * (i^-1) = i^(-2) = 1/i^2 = 1 >Eh? 1/i^2 = 1/(-1) = -1. >> compare, >> (-i^1) * (-i^1) = i^2 = 1 >> It looks like to me, the reciprocal of i can equal -i or i^-1, >> because you get the same results for either way you express it. >And the two are in fact equal. >> But does, >> i^3 = i^-1 >> i^3 = -i >> It was shown that, >> -i = i^-1 >> So it does. >> And, >> i =/= -i >> But when you raise this to the power of 2, two negatives turns into >> positive, THEN they are equal to 1. >> i^2 = 1 >> -i^2 = 1 >I think you mean (-i)^2 = 1, which is also true. >-(i^2) = 1 is also true. However, i^2 = -1. >[.sigsnip] >-- >#191, ewill3@earthlink.net >Its still legal to go .sigless. the problem starts with the definition that i = sqrt(-1). The mapping sqrt is well-defined only on perfect square. It is nonsense otherwise. E. E. E. Escultura === Subject: Re: The State-of-the-Art in Mathematics >>In sci.math, S. Enterprize Company >> > THIS IS part of my occasional update of mathematics and physics. It > refers to the resolution of mathematical issues that have appeared in > several journals recently. (1) The complex number i is nonsense since > i = sqrt(-1) = sqrt(1/-1) = 1/i = -i from which follows i = -i. > -i = 1/i -i = 1/sqrt(-1) >typo; i =/= -i > i =/= -1 i = sqrt(-1) > i = sqrt( -1) > i^2 = 1 > i^3 = -i >>Be *VERY* careful here. sqrt(4) = +2 or -2. sqrt(-1) is >>traditionally called i, but again sqrt(-1) = +i or -i. >>(Theres at least one joke proof running around that >>deliberately confuses the issue by inappropriately taking >>the square root and proving a nonsense result.) > Lets see if the reciprocal of 1/i really is -i. >>Well, if b is the reciprocal of a, then a * b = +1. >>Since i^2 = -1, i * i * (-1) = +1, and one can take >>i * (-1) = -i to be the reciprocal of i. QED. > It looks like to me, the reciprocal of i is, > i^-1. > But, lets check, > sqrt(-1)/sqrt(-1) = 1 > i^1 * i^-1 = i^(1-1) = i^0 = 1 > If the reciprocal of i was -1 then >>Did you mean -i here? > i^1 * (-i^1) = -i^(1+1) = -i^2 = 1 > It looks like it gives the same answer in this case. > But what about this ? > (i^-1) * (i^-1) = i^(-2) = 1/i^2 = 1 >>Eh? 1/i^2 = 1/(-1) = -1. > compare, > (-i^1) * (-i^1) = i^2 = 1 > It looks like to me, the reciprocal of i can equal -i or i^-1, > because you get the same results for either way you express it. >>And the two are in fact equal. > But does, > i^3 = i^-1 > i^3 = -i > It was shown that, > -i = i^-1 > So it does. > And, > i =/= -i > But when you raise this to the power of 2, two negatives turns into > positive, THEN they are equal to 1. > i^2 = 1 > -i^2 = 1 >>I think you mean (-i)^2 = 1, which is also true. >>-(i^2) = 1 is also true. However, i^2 = -1. Nope. (-1^.5)*(-1^.5) = 1^1 = 1 negative * negative = positive >>[.sigsnip] >>-- >>#191, ewill3@earthlink.net >>Its still legal to go .sigless. >the problem starts with the definition that i = sqrt(-1). The mapping sqrt is >well-defined only on perfect square. It is nonsense otherwise. Nope. Perfect squares doesnt matter with the sqrt. Example, sqrt(2) = 1.41421... (1.41421...)^2 = 2 A perfect square would be, sqrt(4) = 2 sqrt(-1) = i <--- This is correct. This accounts for the imaginary side of reality. To make sqrt(-1) real, you square it, (sqrt(-1))^2 = 1 >E. E. E. Escultura Smarts Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: JSH: But what if it works? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBEGVV120733; >But what if this idea works? since they are entertaining. Ad Hominem === Subject: Re: Recent resolution of issues by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBEGVVD20737; >> MATHEMATICAL UPDATE AND CLARIFICATION >> THIS IS part of my occasional update of mathematics and physics. >You are updating all of mathematics and all of physics? >-- >Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html >Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 >Francis Wheen, _How Mumbo-Jumbo Conquered the World_ I have updated foundations, number theory theory, the real number system and analysis. I have updated physics and astronomy; qunatum mechanics has been updated to quantum gravity and astronomy to macro gravity. Cosmology has been upgraded, too. E. E. Escultura === Subject: Re: Recent resolution of issues by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBEGXq920884; >x = 0.9_ >10x = 9.9_ > = 9 + x >9x = 9 >x = 1 This is called the Perron paradox, the use of necessary condition without existence principle or theory. It involves assuming the existence of something that is not established yet and then proving that it is equal to something. Using this method you can prove that the greatest integer is 1. Please see, Exact solutions of Fermats equation, Nonlinear Studies, Vol. 5, No. 2, pp. 227 - 254. E. E. Escultura === Subject: Re: Recent resolution of issues >This is called the Perron paradox, the use of necessary >condition without existence principle or theory. It involves >assuming the existence of something that is not established >yet and then proving that it is equal to something. Using >this method you can prove that the greatest integer is 1. >Please see, Exact solutions of Fermats equation, Nonlinear >Studies, Vol. 5, No. 2, pp. 227 - 254. So linear is a synonym of sensical? Whod have thought it? Lee Rudolph === Subject: Help on simple problem posting-account=IBjmYA0AAAAd4mo1TE6QS2-8KE9q5lQJ If you could help me with this problem, Id be very grateful. Im studying for the GREs and I cant get past why Im getting this wrong. Below is the question and below that is my logic (which cleary is wrong). How many 3 digit positive integers are odd and do not contain 5? My logic - divide this into two problems. There are 900 3-digit numbers, 450 of them are odd. 9x10x5 is just one way to get at this. To find how many do not contain 5 - I first figure theres an 8 in 9 chance that the third digit will not contain a 5, 9 in 10 for the second, and 9 in 10 for the first. 8x9x9= 648. numbers that are odd, so I subtracted and received 198. (Not the correct answer of 288) so I went back to the drawing board and tried to figure out how many numbers contained 5, but came up 260. 900-260=640. 640-450=190. URG..... HELP! === Subject: Re: Help on simple problem > If you could help me with this problem, Id be very grateful. Im > studying for the GREs and I cant get past why Im getting this wrong. > Below is the question and below that is my logic (which cleary is > wrong). > How many 3 digit positive integers are odd and do not contain 5? > My logic - divide this into two problems. There are 900 3-digit > numbers, 450 of them are odd. 9x10x5 is just one way to get at this. Yes. > To find how many do not contain 5 - I first figure theres an 8 in 9 > chance that the third digit will not contain a 5, 9 in 10 for the > second, and 9 in 10 for the first. 8x9x9= 648. Thats correct. > numbers that are odd, so I subtracted and received 198. Heres your problem. The set of odd 3-digit numbers does indeed have 450 members and the set of 3-digit numbers not containing a 5 does have 648 members, but you want to find the size of the intersection of those two sets, which in general you dont do by subtracting their sizes. Drawing a Venn diagram might help you see whats going on. > (Not the correct answer of 288) so I went back to the drawing board and > tried to figure out how many numbers contained 5, but came up 260. > 900-260=640. 640-450=190. URG..... HELP! Assuming your numbers dont have a leading 0 we have First (high-order) digit: any of 1, 2, 3, 4, 6, 7, 8, 9 Middle digit: any of 0, 1, 2, 3, 4, 6, 7, 8, 9 Last digit: any of 1, 3, 7, 9 8 choices for first digit, 9 choices for middle digit 4 choices for last The choices are independent, so there are 8 * 9 * 4 = 288 different numbers. Rick === Subject: Re: Help on simple problem Originator: richard@cogsci.ed.ac.uk (Richard Tobin) >How many 3 digit positive integers are odd and do not contain 5? Try taking account of the oddness by restricting what the last digit can be. -- Richard === Subject: Re: Help on simple problem >If you could help me with this problem, Id be very grateful. Im >studying for the GREs and I cant get past why Im getting this wrong. >Below is the question and below that is my logic (which cleary is >wrong). >How many 3 digit positive integers are odd and do not contain 5? >My logic - divide this into two problems. There are 900 3-digit >numbers, 450 of them are odd. 9x10x5 is just one way to get at this. No need to divide the problem up; you can simply count them. 1. How many choices are there for the third (least significant) digit? 2. How many choices for the middle digit? 3. And how many for the first, assuming no leading 0? Then multiply your answers. --Lynn === Subject: Re: Help on simple problem posting-account=IBjmYA0AAAAd4mo1TE6QS2-8KE9q5lQJ >If you could help me with this problem, Id be very grateful. Im >studying for the GREs and I cant get past why Im getting this wrong. >Below is the question and below that is my logic (which cleary is >wrong). >How many 3 digit positive integers are odd and do not contain 5? >My logic - divide this into two problems. There are 900 3-digit >numbers, 450 of them are odd. 9x10x5 is just one way to get at this. > No need to divide the problem up; you can simply count them. > 1. How many choices are there for the third (least significant) digit? > 2. How many choices for the middle digit? > 3. And how many for the first, assuming no leading 0? > Then multiply your answers. > --Lynn === Subject: Re: The state-of-the-art in mathematics by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBEHETO24831; >Hi Folks, >>I would like to share with you the latest findings >>sqrt(1/-1) = 1/i >= -i or i = -i; dividing both sides of the last >equation by i, I obtain 1 = -1 or 1 = 0 and the real number system >goes down the drain. If I add i on both sides instead, I obtain 2i = 0 >or i = 0 and the complex number system vanishes in thin air. >E. E. Escultura >>i = sqrt(-1) = sqrt(1/-1) = 1/i = -i or i = -i >>... every first-semester-student of mathematics knows >>that the square root of a number is 2-valued. >>If the rest of your work is of the same >>depth<< ... >Thats bad English. Shouldnt that is be am? The problem here is not the value of i but its being ill-defined because the sqrt is well-defined only on perfect square. In mathematics what is ill-defined does not exist. So defining is this way not bad English but bad mathematics. E. E. Escultura === Subject: Re: GCH vs. Axiom of Choice. Mail-To-News-Contact: abuse@dizum.com >| so I didnt think C or ~C was >|considered false, just relatively independant axioms. >Are you assuming that every statement X for which we can prove >Con(ZF)=>Con(ZF+X) and Con(ZF)=>Con(ZF+~X) is neither true >nor false? I dont mean it as a rhetorical question. At the risk of sounding like Pilate, What is truth? I always thought that Mathematical Truth was only definable with respect to a given set of axioms. I guess that I got this idea from (popularized) readings about the Parallel Postulate. Some guy spends his life trying to prove it by reductio ad absurdum and fails. People build on his work, giving us new forms of geometry. Each of these forms of geometry is, to the best of my knowledge, considered true. (Obviously, you cant take the results of one and apply them with another.) >I dont assume anything like that, and to me this seems like a >nonsequitur. If you believe such a thing, and you dont know >why you believe it, its liable to be a source of some confusion. So, is the Parallel Postulate true all by itself -- whatever that means? If not, whats different about C? -- Michael F. Stemper #include Time ßies like an arrow. Fruit ßies like a banana. === Subject: Re: GCH vs. Axiom of Choice. > At the risk of sounding like Pilate, What is truth? I always thought > that Mathematical Truth was only definable with respect to a given > set of axioms. On the contrary, there is no such definition. Mathematical truth, as defined in mathematics, is not relative to any set of axioms. === Subject: Re: GCH vs. Axiom of Choice. posting-account=2fdXgA0AAAAiH_AsNUz4SbCuDB3R59KU > Im assuming sentance is just a misspelling of sentence, and not > a new technical term. Given your explanation, why is Con(ZF) not > an FO sentence? Yeah, I cant spell, nothing new. My understanding of Con(ZF) is something like: For every FO sentence S, it is not the case that both S and ~S are implied by the conjuction of the axioms of ZF. It quantifies over sentences, something I dont know how to do within a formal sentence. === Subject: Re: GCH vs. Axiom of Choice. J.E. a .8ecrit : >> Im assuming sentance is just a misspelling of sentence, and > not >>a new technical term. Given your explanation, why is Con(ZF) not >>an FO sentence? > Yeah, I cant spell, nothing new. My understanding of Con(ZF) is > something like: For every FO sentence S, it is not the case that both > S and ~S are implied by the conjuction of the axioms of ZF. Well, your understanding is shallow It > quantifies over sentences, something I dont know how to do within a > formal sentence. Exactly, thats why it is shallow. After so many messages pontificating on models, logic, Skolem, and so on, you have still not caught the idea of G.9adel coding? Amazing. === Subject: Re: GCH vs. Axiom of Choice. > It > quantifies over sentences, something I dont know how to do within a > formal sentence. Google Godel numbering. === Subject: Re: GCH vs. Axiom of Choice. posting-account=sAS5-AwAAABlKnmtMjBbYHvhxI6W0cAg > I was asking about FO sentances being true in L. Neither Con(ZF) not > ~Con(ZF) is a FO sentance is it? > I was wondering if for a FO sentance X, such that Con(A)=>Con(A&X) > where A is the conjuction of axioms of ZF, whether the sentances TX > that are of the form S[x] & x in L that are proved by the conjuction > of axioms of ZF with each other and with X will be consistent with the > sentances TY that are of the form S[x] & x in L that are proved by > the conjuction of axioms of ZF with each other and with ~X given that > Con(A) => Con (A&~X). S[x] & x in L would seem to have x as a free variable. So how can it be a sentence? > I dont even know if the question is a real question because I dont > know if x in L is a FO sentance, if it isnt, then TX and TY are > collections of no sentances, and my question is vacuously answered. x in L is a *well-formed formula* in the first-order language of set theory, not a sentence. > dont understand what L is well enough to know if L is FO expressable. > I dont know if adding a discussion of L automatically makes one > talking outside ZF or whether if can be done inside ZF. Talking with > Keith Ramsey makes me worry that L is well defined, that its existance > isnt proveable maybe not even by assuming Con(ZF), and since Con(ZF) > isnt a FO sentance to my knowledge, The existence of L is provable in BG. You dont need to assume Con(ZF). Con(ZF) is a sentence in the first-order language of set theory. > I dont know the first thing about > how to prove things given Con(ZF), hence why Im asking on Usenet. === Subject: Re: GCH vs. Axiom of Choice. posting-account=2fdXgA0AAAAiH_AsNUz4SbCuDB3R59KU >S[x] & x in L would seem to have x as a free variable. So how can it > be a sentence? Just put Ex in front of it. >x in L is a *well-formed formula* in the first-order language of set >theory, not a sentence. I meant that I dont know if x in L can be made into a sentance. I could put Ex x in L and then x isnt free anymore, but L itself needs to be expanded out, or else its still not a sentence. I dont want L to be free, but I dont understand what L is well enough to write a formula corresponding to x in L, Keith Ramsey seemed to indicate to me that whether or not L exists is unkown. But that could just mean that the sentance EL S(L) when expanded correctly is undecidable, not that no FO sentance S(L) exists to single out what was intended well enough that every truth of L is proveable from some system consistent with ZF, which Ramsey didnt say, but it sorta what Ive been asking. >The existence of L is provable in BG. You dont need to assume Con(ZF). Doesnt BG assume classes or something, Im not familiar with BG. >Con(ZF) is a sentence in the first-order language of set theory. Really? Is it hard to write? Ive heard on usenet that you can write something like MiniZF is consistent, I didnt know that ZF is consistent was possible. === Subject: Re: GCH vs. Axiom of Choice. Originator: joshp@xoxy.net (joshp) >> Con(ZF) is a sentence in the first-order language of set theory. > Really? Yes. Its also a sentence of PRA (primitive recursive arithmetic). > Is it hard to write? Not really. Consistency for a formal system (say, FS) can be asserted by using a two-place predicate (say, is-FS-proof) that is true just when its first argument is a valid proof (in FS) of the second argument. Con(FS) then asserts the impossibility of deriving a sentence and its negation: (p)(s)(p)(s)((is-FS-proof(p,s) and is-FS-proof(p,s)) implies s =/= -( _ s _ )) In that sentence, double quotes and the underscore represent finite sequences and their concatenation. For example, if x = 1 = 1, then -( _ x _ ) = -(1 = 1). -- Josh Purinton === Subject: Re: GCH vs. Axiom of Choice. posting-account=2fdXgA0AAAAiH_AsNUz4SbCuDB3R59KU For this purpose a FO sentance is a sentance inductively built up from the symbols e A E & ~ v and members of a countable set M of other symbols, which Ill call terms. 1. If x and y are terms, then xey is a formula with free x and y. 2. If B and C are formulas, then B&C, BvC and ~C are formulas and the free terms of B&C and BvC are the free terms of B as well as the free terms of C. ~B has the same free terms as B. 3. If x is a term and B is a formula, then ExB and AxB are formulas, and the free terms in are the same as in B unless x was free in B in which case it is not a term of either of the new formula. Each formula is built up from the naive set {e,A,E,&,v}UM through a finite number of applications of those three steps. Nothing else is a formula. A FO sentance is a formula with no free terms. Hopefully the meaning is obvious. === Subject: Re: GCH vs. Axiom of Choice. > For this purpose a FO sentance is a sentance inductively built up from > the symbols e A E & ~ v and members of a countable set M of other > symbols, which Ill call terms. Im assuming sentance is just a misspelling of sentence, and not a new technical term. Given your explanation, why is Con(ZF) not an FO sentence? === Subject: Re: Moments over a Simplex >> The midpoint of a line segment (x1,x2) is (x1+x2)/2 >> The midpoint of a triangle (r1,r2,r3) is (r1+r2+r3)/3 >> The midpoint of a tetrahedron (r0,r1,r2,r3) is (r0+r1+r2+r3)/4 >> The midpoint of a simplex in N dimensions is sum(k<=N) r_k/N ? >> The variance in x over a line segment in 1-D is: (x2-x1)^2/12 >> The variance in x over a triangle in 2-D is, if I made no errors: >> ((x2-x1)^2 + (x3-x1)^2 + (x3-x2)^2)/36 >What do you mean by the variance over a multi-dimensional simplex? >For example, suppose (X1, X2, X3) is uniformly distributed over {x >in R^3: min x_i >= 0, x1 + x2 + x3 =1} You can come up with a >(singular) covariance matrix, but what do you mean by one scalar variance? What I computed in my posting was V = E[(X - E[X]).(X - E[X])] where X is a vector-valued random variable uniformly distributed on to be what he was looking for. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Moments over a Simplex > and, if I havent made a mistake, the variance is > sum_{0<=i What do you mean by the variance over a multi-dimensional simplex? > For example, suppose (X1, X2, X3) is uniformly distributed over {x > in R^3: min x_i >= 0, x1 + x2 + x3 =1} You can come up with a > (singular) covariance matrix, but what do you mean by one scalar variance? As a matter of fact, I didnt know how to expres myself properly with this stuff. But Robert Israel seems to understand, nevertheless. So I would like to redirect you to his poster. Han de Bruijn === Subject: Re: Moments over a Simplex >> What do you mean by the variance over a multi-dimensional simplex? >> For example, suppose (X1, X2, X3) is uniformly distributed over >> {x in R^3: min x_i >= 0, x1 + x2 + x3 =1} You can come up with a >> (singular) covariance matrix, but what do you mean by one scalar >> variance? > As a matter of fact, I didnt know how to expres myself properly with > this stuff. But Robert Israel seems to understand, nevertheless. So I > would like to redirect you to his poster. With all due respect to Dr. Israel, regardless of the complexity of his calculations, he does not answer my question either. In fact, he defines a vector valued function R(x) and then talks about its square, which makes no sense. -- Stephen J. Herschkorn sjherschko@netscape.net === Subject: Building a target function by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBEIiAF01324; I would like to build a pricing model for hosting. It should be very easy, the cost is basically made up of two limited resources, space x and bandwidth y. A server has a limit on both, and so if all y is used up, then the x cannot be used and vice vverse, why a y/x ratio is good to maintain. So start with a pricing model: f(x,y)= 0.17x + 0.15 y 0 < x , 0 < y y/x < 10 since the y/x >10 would mean unused space I included the cost of space in the equation for y/x > 10 f(x,y) = 0.17 x + 0.15 Min ( y, 10x) + 0.32 Max ( 0, y - 10x) 0 < x, 0 < y Now the result is that above that given an y/x over 10 you can increase the x without getting a higher price, since it is already paid for (0.32=0.15+0.17) Im thinking, how to smooth out this function to both maintain the ratio without giving out space for free. any ideas? === Subject: Please help quickly! by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBEJcvf06464; please help me with the following problems ...by showing me the work and how to do them..i am tryin to check my answers and work if sec x=4, find sin x and sin 2x solve for x: -cos x _______ = tan x 1- sinx factor: y= 3x^4- 5x^3+ 3x^2 - x === Subject: Re: Please help quickly! > please help me with the following problems ...by showing me the work and how > to do them.. Then after that, sharpen my pencil, clean my room, and change my diaper. And do it quickly! === Subject: Re: Please help quickly! >please help me with the following problems ...by showing me the work and how to do them..i am tryin to check my answers and work Gee, Im hurrying to work them quickly for you, but I have to eat lunch, do some shopping, watch a little TV, and take a nap first. When is your assignment due? --Lynn === Subject: Re: Solve a linear equation system over GF(2) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBEK3YF09045; >Bill, Roy, and Carl, >NTL seems to be a fine tool. FFLAS would probably be even better, but >has such sparse documentation that I wonder whether it is already ment >for production use. >Thx, > Simon The computer algebra system Fermat handles this very easily with builtin functions. You can also easily work over extension fields GF(2^8) and GF(2^16). It is shareware, available for Mac, Linux, Windows, Unix. Robert H. Lewis Fordham University author of Fermat www.bway.net/~lewis === Subject: Re: Hilbert Spaces and L^p > Take l^1, for instance. Consider the sequences a = (1,0,0,0,0,...) and > b = (0,1,0,0,0,...). Then > ||a + b||^2 + ||a - b||^2 = 8 > and > 2||a||^2 + 2||b||^2 = 4. > Therefore, the norm ||.||_1 does not satisfy the paralellogram law. > Jose Carlos Santos In your example here, the sequences a and b also belongs to l^2, and ||a+b||^2 = ||a-b||^2=2, so it indeed satisfies the paralellogram law. Check that. nunu *-----------------------* www.GroupSrv.com *-----------------------*