mm- Subject: Re: coprimality & gcd >I believe that I prefaced my remarks by saying that it was a number theory >(i.e. natural number) proof. Obviously it could have been an algebraic >problem, but lacking ANY sort of identifying information (e.g. assume a >Euclidian domain or in an arbitrary ring, etc.) I used Okkam¹s Razor and >assumed natural numbers. If I am wrong then the OP is free to ignore my >reply. > You replied to me, not to the original poster. And speciŽcally, you > replied following my comments that the context would matter; if you > meant to piggyback, then might I suggest stating so in your post and > removing the text from the person you are following up in the future? I was piggybacking on your comment about using the deŽnition of GCD, but like I said, I did distinguish it from your work by saying that mine was using number theory. Sorry for the confusion. === Subject: Re: Attractor set representation >of several degenerated Žxed point which merge into one structure. > This is a little vague -what do you mean by that? If f6(0) = -0.16389 there are several attracting points - http://img41.exs.cx/img41/6937/attrf1-60.png If f6(0) = -0.16388 it seems that the previous attracting points degenerate into closed (probably nonintersecting in higer dimentions) curves - http://img50.exs.cx/img50/6597/attrf1-59-1.png If f6(0) = -0.16382178649293054848 it seems that the previous curves merge into one curve. === Subject: Re: Attractor set representation >> >>of several degenerated Žxed point which merge into one structure. >> This is a little vague -what do you mean by that? >If f6(0) = -0.16389 there are several attracting points - That seems to be a stable *periodic* orbit. You would say Žxed point only if the sequence of points tends to a single point. I couldn¹t recontruct that picture though - what are the other initial conditions (t=0 in your terminology). >http://img41.exs.cx/img41/6937/attrf1-60.png >If f6(0) = -0.16388 it seems that the previous attracting points >degenerate into closed (probably nonintersecting in higer dimentions) >curves - http://img50.exs.cx/img50/6597/attrf1-59-1.png >If f6(0) = -0.16382178649293054848 it seems that the previous curves >merge into one curve. The closed curves you get aren¹t attractors (you see that if you start with initial conditions close. You get different closed curves.) They seem to be part of a higher dimensional attractor (2 or 3 dimensional?) Thomas === Subject: Re: paper claiming p=np and soap bubbles Quote from his reference: If you believe, as many do, that hypercomputational processes are always merely mathematical, and never physically real, you can¹t be rational and at the same time refuse to accept our case for P=NP. What does that mean? Believe then I¹m not rational, and refuse to accept case? OR Not believe, and be rational and accept case? ** Clarity avoids you. > The paper is the best argument I have heard for p=np, even though I > believe the opposite. It can be found here: > http://arxiv.org/abs/cs.CC/0406056 > It brings out a great question. > Basically, the argument is that since soap bubbles can be made to > solve NP-complete problems, particularly the Steiner tree graph > problem, in what appears to be polynomial time and physics on a > macroscopic level can be modeled as a Turing machine, it must be true > that p=np. > What I would like to know from any physicists out there is why do soap > bubbles work in such a way that they are able to solve the Steiner > tree graph problem?How is nature able to quickly solve problems that > we cannot solve quickly? > Craig === Subject: standard deviation and N-1 In high school, I learned that the formula for standard deviation has n in the denominator, but in college the book has N-1 in the denominator. What is the reason for this? So far, I found this in my book (By Yates, Moore, McCabe): Why do we average by dividing by n-1 rather than n? Because the sum of deviations is always zero, the last deviation can be found once we know the other n-1. So we are not averaging n unrelated numbers. Only n-1 of the squared deviations can cary freely, and we average by dividing by the total by n-1. The n-1 is called the degrees of freedom of the variance or standard deviation. I sort of understand that, but could someone explain in simpler terms and expand on that? I¹m still a little puzzled as to why n-1. === Subject: Re: standard deviation and N-1 The reason in terms of degrees f feedom is a goon mnemonic, but it needs to be complmented with a solid mathematical basis. You ar trying to estimate the average value of (x-mu)^2 where mu is the mean. This is not as straightforward a it sounds because mu is not really known; you have only an estimator. Assuming mu = x-bar, the usual estimator, allows you to do the calculation but presents a different problem. Of all possible values mu might have, x-bar is the one that gives the MINIMUM value for the variance you¹re trying to estimate with it! This is not what you want for an UNBIASED estimator of the variance. So what we do is to treat mu itself as if it were a statistical variable with average x-bar and whose own variance is sigma^2/N. Given such a likelihood distribution for mu derived the sample data, we Žnd that the average estimator for (the variance is the sum of the squared differences with x-bar divided by N-1 instead of N. THAT is your unbiased estimator. --OL === Subject: Re: standard deviation and N-1 I do remember going through the derivation and getting the N-1. Details? In one of my old stat books. (N is suppose to be greater than 20 anyway) > The reason in terms of degrees f feedom is a goon mnemonic, but it > needs to be complmented with a solid mathematical basis. > You ar trying to estimate the average value of (x-mu)^2 where mu is the > mean. This is not as straightforward a it sounds because mu is not > really known; you have only an estimator. Assuming mu = x-bar, the > usual estimator, allows you to do the calculation but presents a > different problem. Of all possible values mu might have, x-bar is the > one that gives the MINIMUM value for the variance you¹re trying to > estimate with it! This is not what you want for an UNBIASED estimator > of the variance. So what we do is to treat mu itself as if it were a > statistical variable with average x-bar and whose own variance is > sigma^2/N. Given such a likelihood distribution for mu derived the > sample data, we Žnd that the average estimator for (the variance is the > sum of the squared differences with x-bar divided by N-1 instead of N. > THAT is your unbiased estimator. > --OL === Subject: Re: College Algebra & Trig - Same Semester? > I¹m wondering how wise it would be to take College Algebra and Trig in > the same semester. I already have a BS in math (it is 20 years old!), > and am now going back and taking some refresher courses. So, I have > some mathematical maturity, and I¹m very motivated. > I guess I¹m really asking this - how much of College Algebra content > is found in a Trig course? > Gerald I would recommend sending the instructors an email asking about the course topics, the books and the amount of work anticipated. I am not sure we can such much more beyond that as we don¹t know your skills or weaknesses or what the instructors plan. I think this is possible and you have previous experience (plus the beneŽt of maturity) so don¹t let me sway you. I have often sent instructors emails asking if they tell me a bit about the course before enrolling. HTH, Flip === Subject: Re: simple probability question > hi, > i have a quick question relating to probability, and i¹ve reached the > limits of my ability to recall my college statistics course.. > i have 1000 widgets. > each widget has a 2% probability of failure during a year of operation > (CDF). Can you get a refund? How much did they cost? > now how do i Žgure out the probability that ANY widget will fail in > my collection of widgets. the widget failures are independent of each > other. You just said you don¹t know how or have forgotten how to Žgure that out, so I would say you fake it. That¹s how. > ie. if i put all the widgets in a bucket, and come back in a year, > what is the probability that all 1000 widgets will still be working? > what if i come back in six months? or two years? I believe the bucket won¹t change a thing. And where are you going to go for a whole year? Aren¹t you worried that someone might use up your widgets without your permission? > i¹m not just interested in the answer, but i¹d like to know the > formula, or how this works. That¹s what they all say until they sober up in the morning. === Subject: Re: simple probability question >> hi, >> i have a quick question relating to probability, and i¹ve reached the >> limits of my ability to recall my college statistics course.. >> i have 1000 widgets. >> each widget has a 2% probability of failure during a year of operation >> (CDF). > Can you get a refund? How much did they cost? >> now how do i Žgure out the probability that ANY widget will fail in my >> collection of widgets. the widget failures are independent of each >> other. > You just said you don¹t know how or have forgotten how to Žgure that > out, so I would say you fake it. That¹s how. >> ie. if i put all the widgets in a bucket, and come back in a year, what >> is the probability that all 1000 widgets will still be working? what if >> i come back in six months? or two years? > I believe the bucket won¹t change a thing. Dumbass. You neglected to calculate the probability that the bucket will fail. Bucket failure is not independent from widget failure. > And where are you going > to go for a whole year? He¹s a travelling widget salesman. > Aren¹t you worried that someone might use up your widgets without your > permission? That is why they have a 2% failure rate. >> i¹m not just interested in the answer, but i¹d like to know the >> formula, or how this works. > That¹s what they all say until they sober up in the morning. Lance Lamboy Go F*ck Yourself ~ Dick Cheney === Subject: Re: proof of monotheism without a žaw X-URL: http://mygate.mailgate.org/mynews/sci/sci.math/ 692a2fda1ea7a89b6d2d49c8e56736 47.48257%40mygate.mailgate.org > Therefore, God is One and cannot be composed of many parts. Feel free to substitute Craig¹s Brain Cell for God and notice the same proof works with equal validity to eliminate any possibility that you have more than one. I see a great need for newgrouping sci.lamer.incurable. xanthian. === Subject: Re: proof of monotheism without a žaw You are using man made logic, not God¹s logic. He could have created himself in another space or universe in different form, like Microsoft did going from Win 95 to Win 98. You also assume that he cannot upgrade himself, why not? In human terms, please. > Someone gave a žawed proof of monotheism before. Here¹s the real > thing: > We will not prove that God exists (that is in other threads scattered > about the internet), but we will prove that assuming that He does > exist and that He created everything except for Himself (which is > generally an accepted assumption about what God is), then He must be > One, i.e., cannot have any parts: > Let us suppose for the sake of contradiction that he is composed of > many parts (at least two). Then since God created everything except > for Himself and these parts are not God, he must have created these > parts. But since these parts by our supposition are necessary for him > to exist (since he is composed of them), he could not have created > them, which is a contradiction. > Therefore, God is One and cannot be composed of many parts. > Craig === Subject: Re: proof of monotheism without a žaw > You are using man made logic, not God¹s logic. He could have created > himself in another space or universe in different form, like Microsoft did > going from Win 95 to Win 98. > You also assume that he cannot upgrade himself, why not? In human terms, > please. I like it! Do we have to wait for a V3 Universe before it works properly? Can we expect an urgent security patch which Žxes Iraq? How about some decent capacity planning on the next release (running out of oil sucks)? Can I download a pirate reality through Kazaa, and if so do you know the serial number? Is there a Linux version where the physics is all public domain? === Subject: Re: limitation to induction on Žnite bounds responding to |-|erc to infer that a sequence does not occur in S. I note that this rule only allows me to infer that Žnite sequences do not occur in S. This is unfortunate, since the arguments you make seem to essentially rely on inŽnite sequences occuring. Is there a rule that would allow me to infer that a given inŽnite sequence does not occur in S? Also, your argument appears to rely not upon judgements of the form d does not occur in S, but upon judgements of the form d does occur in S. Therefore, I would also like to reiterate my request for whether there is any rule that would allow me to draw some conclusion not involving occurs from premisses which include positive statements of occuring. Are you able to formulate any such rule? **** HERC > therefore, the inŽnite digit sequence of all HERC > irrationals occurs in the set 010203. PRD > Let¹s suppose this is true, and examine the consequences PRD > of such an assumption. PRD > It would seem by your answers that a statement x occurs PRD > in the set 010203 means something different than x PRD > is an elemeent of the set 010203. HERC > yes. PRD > Does the statement x occurs in set S mean the same PRD > as the statement x is an element of set S? HERC > no, PRD > If x occurs in set S means something other than PRD > x is an element of set S, please answer the following: PRD > Are there any rules of inferrence that allow one to PRD > validly conclude from a statement of the form PRD > x occurs in S some other statement? PRD > That is, does there exist a valid rule of the form: PRD > Given that x occurs in set S and statement P is PRD > true, we can validly conclude that statement Q is PRD > true? PRD > Can you give an example of such a valid rule of PRD > inferrence? [snip of example] HERC > That is Cantor¹s proof. The induction stretching to HERC > inŽnite sets is unsound. HERC > Given that occurs in set S and was HERC > contructed using Cantors diagonaliasion, HERC > we can validly conclude that set S is not demonstrably HERC > missing any sequence of digits **** PRD > I am confused as to whether this is offered as an PRD > example of a valid rule, or an invalid rule. You PRD > say that the induction stretching to inŽnite sets PRD > is unsound. But I¹m not sure if the induction PRD > refers to the rule you give immediatly after. PRD > Is the rule: PRD > Given that occurs in set S and was PRD > constructed using Cantor¹s diagonalization, PRD > we can validly conclude that set S is not demonstrably PRD > missing any sequence of digits PRD > a valid rule? HERC > yes PRD > Does it work for all and S? HERC > yes, if the digit string occurs then it can¹t be missing. **** PRD > I¹m sorry, but I am still unsure of what you mean by PRD > missing. You use the term occurs in to mean PRD > something other than is an element of, so I am led to PRD > wonder what missing means. PRD > I suspect that it means the same thing as does not occur. PRD > And it is to verify my suspicion that I am asking these PRD > questions. PRD > Does missing mean does not occur? HERC > HERC > sequence occurs <-> sequence is not missing PRD > For any and S, does the phrase PRD > the sequence of digits is missing in [from?] S PRD > mean exactly the same as PRD > the sequence of digits does not occur in S? HERC > yes PRD > Is there some other rule of inference that you PRD > can provide which allows me to infer from PRD > a condition that involves the predicate occurs, PRD > to a consequence that does not involve the PRD > predicate occurs? HERC > HERC > sequence is a member -> sequence occurs HERC > sequence does not occur -> sequence is not a member **** PRD > The rule: PRD > sequence does not occur -> sequence is not a member PRD > does indeed have occur on the premiss side, and not on the PRD > consequence side. PRD > Unfortunately, it isn¹t really what I am looking PRD > for. The rules you have given me so far allow me to make PRD > a judgement that sequence d does occur, but they do not allow PRD > me to make the judgement that sequence d does not occur. PRD > Thus, there is no way yet for me to use the rule which you PRD > have just provided me. PRD > Do you know of a rule of inferrence that will allow me to PRD > conclude something of the form: PRD > sequence d does not occur in S ? PRD > Do you know of a rule of inferrence that will allow me to PRD > conclude something of that does not involve the predicate PRD > occurs, and which has the predicate occurs within its PRD > premisses in a way that I can derive those premisses? HERC > En e N, if( Žnite(length(d)), n<= length(d)) HERC > Ai, HERC > !digitsmatchupton(i, d, n) HERC > -> d does not occur in S HERC > where HERC > digitsmatchupton(x, d, n) HERC > if (n == 0) return true HERC > else HERC > { HERC > if S(x, n) != d(n) return false HERC > else return digitsmatchupton(x, d, n-1) HERC > } === Subject: Re: limitation to induction on Žnite bounds > to infer that a sequence does not occur in S. I note that this > rule only allows me to infer that Žnite sequences do not > occur in S. This is unfortunate, since the arguments you > make seem to essentially rely on inŽnite sequences occuring. > Is there a rule that would allow me to infer that a given inŽnite > sequence does not occur in S? try it out : S = 010203 and d = 0.3.. > HERC > En e N, if( Žnite(length(d)), n<= length(d)) > HERC > Ai, > HERC > !digitsmatchupton(i, d, n) > HERC > -> d does not occur in S does there exist a number where !digitsmatchupton(i, 0.3..., n) no, since for all lengths of the string 0.3333.. that digit sequence occurs. so the inŽnite string 0.33.. occurs in S. If the string occurs, then *every digit* occurs in the correct digit position. Why would this be invalid for an inŽnite string? > Also, your argument appears to rely not upon judgements of the > form d does not occur in S, but upon judgements of the > form d does occur in S. Therefore, I would also like to reiterate > my request for whether there is any rule that would allow me to > draw some conclusion not involving occurs from premisses > which include positive statements of occuring. Are you able > to formulate any such rule? if digit-string occurs in S, S setminus digit-string = 0. where eg. {2, 3, 4} setminus 2.9 = 0.1 Herc > **** > HERC > therefore, the inŽnite digit sequence of all > HERC > irrationals occurs in the set 010203. > PRD > Let¹s suppose this is true, and examine the consequences > PRD > of such an assumption. > PRD > It would seem by your answers that a statement x occurs > PRD > in the set 010203 means something different than x > PRD > is an elemeent of the set 010203. > HERC > yes. > PRD > Does the statement x occurs in set S mean the same > PRD > as the statement x is an element of set S? > HERC > no, > PRD > If x occurs in set S means something other than > PRD > x is an element of set S, please answer the following: > PRD > Are there any rules of inferrence that allow one to > PRD > validly conclude from a statement of the form > PRD > x occurs in S some other statement? > PRD > That is, does there exist a valid rule of the form: > PRD > Given that x occurs in set S and statement P is > PRD > true, we can validly conclude that statement Q is > PRD > true? > PRD > Can you give an example of such a valid rule of > PRD > inferrence? > [snip of example] > HERC > That is Cantor¹s proof. The induction stretching to > HERC > inŽnite sets is unsound. > HERC > Given that occurs in set S and was > HERC > contructed using Cantors diagonaliasion, > HERC > we can validly conclude that set S is not demonstrably > HERC > missing any sequence of digits > **** > PRD > I am confused as to whether this is offered as an > PRD > example of a valid rule, or an invalid rule. You > PRD > say that the induction stretching to inŽnite sets > PRD > is unsound. But I¹m not sure if the induction > PRD > refers to the rule you give immediatly after. > PRD > Is the rule: > PRD > Given that occurs in set S and was > PRD > constructed using Cantor¹s diagonalization, > PRD > we can validly conclude that set S is not demonstrably > PRD > missing any sequence of digits > PRD > a valid rule? > HERC > yes > PRD > Does it work for all and S? > HERC > yes, if the digit string occurs then it can¹t be missing. > **** > PRD > I¹m sorry, but I am still unsure of what you mean by > PRD > missing. You use the term occurs in to mean > PRD > something other than is an element of, so I am led to > PRD > wonder what missing means. > PRD > I suspect that it means the same thing as does not occur. > PRD > And it is to verify my suspicion that I am asking these > PRD > questions. > PRD > Does missing mean does not occur? > HERC > sequence occurs <-> sequence is not missing > PRD > For any and S, does the phrase > PRD > the sequence of digits is missing in [from?] S > PRD > mean exactly the same as > PRD > the sequence of digits does not occur in S? > HERC > yes > PRD > Is there some other rule of inference that you > PRD > can provide which allows me to infer from > PRD > a condition that involves the predicate occurs, > PRD > to a consequence that does not involve the > PRD > predicate occurs? > HERC > sequence is a member -> sequence occurs > HERC > sequence does not occur -> sequence is not a member > **** > PRD > The rule: > PRD > sequence does not occur -> sequence is not a member > PRD > does indeed have occur on the premiss side, and not on the > PRD > consequence side. > PRD > Unfortunately, it isn¹t really what I am looking > PRD > for. The rules you have given me so far allow me to make > PRD > a judgement that sequence d does occur, but they do not allow > PRD > me to make the judgement that sequence d does not occur. > PRD > Thus, there is no way yet for me to use the rule which you > PRD > have just provided me. > PRD > Do you know of a rule of inferrence that will allow me to > PRD > conclude something of the form: > PRD > sequence d does not occur in S ? > PRD > Do you know of a rule of inferrence that will allow me to > PRD > conclude something of that does not involve the predicate > PRD > occurs, and which has the predicate occurs within its > PRD > premisses in a way that I can derive those premisses? > HERC > En e N, if( Žnite(length(d)), n<= length(d)) > HERC > Ai, > HERC > !digitsmatchupton(i, d, n) > HERC > -> d does not occur in S > HERC > where > HERC > digitsmatchupton(x, d, n) > HERC > if (n == 0) return true > HERC > else > HERC > { > HERC > if S(x, n) != d(n) return false > HERC > else return digitsmatchupton(x, d, n-1) > HERC > } === Subject: Re: limitation to induction on Žnite bounds > responding to |-|erc The rule: > sequence does not occur -> sequence is not a member > does indeed have occur on the premiss side, and not on the > consequence side. > Unfortunately, it isn¹t really what I am looking > for. The rules you have given me so far allow me to make > a judgement that sequence d does occur, but they do not allow > me to make the judgement that sequence d does not occur. > Thus, there is no way yet for me to use the rule which you > have just provided me. > Do you know of a rule of inferrence that will allow me to > conclude something of the form: > sequence d does not occur in S ? > Do you know of a rule of inferrence that will allow me to > conclude something of that does not involve the predicate > occurs, and which has the predicate occurs within its > premisses in a way that I can derive those premisses? En e N, if( Žnite(length(d)), n<= length(d)) Ai, !digitsmatchupton(i, d, n) -> d does not occur in S where digitsmatchupton(x, d, n) if (n == 0) return true else { if S(x, n) != d(n) return false else return digitsmatchupton(x, d, n-1) } This is how simple my argument is. This is the supposed countable list : 0.111111111... 0.2222222222... 0.333333333... 0.43434343434... 0.98765432134.. .. this is the diag 0.12335... this is mod_diag 0.23446... this is Cantors conclusion. No number has 2 at digit 1 AND has 2 at digit 2 AND has 4 at digit 3 AND has 4 at digit 4 AND has 6 at digit 5 AND ... i.e. No number matches mod_diag at digit 1 AND matches mod_diag at digit 2 AND matches mod_diag at digit 3 AND ... Enter Set T, 010203 T = {k/10^n, k and n in J, 0 <= k < 10^n, n > 0} = {0.0, 0.1, 0.2, ... 0.9, 0.01, 0.02, ..., 0.99, ... } A number does have 2 at digit 1 AND has 2 at digit 2 AND has 4 at digit 3 AND has 4 at digit 4 AND has 6 at digit 5 AND ... Given any number for mod_diag : A number matches mod_diag at digit 1 AND matches mod_diag at digit 2 AND matches mod_diag at digit 3 AND ... Pretty straight forward as far as I can see that mod_diag occurs in the set of computable numbers. Herc > **** > [Recap of discussion follows] > HERC > therefore, the inŽnite digit sequence of all > HERC > irrationals occurs in the set 010203. > PRD > Let¹s suppose this is true, and examine the consequences > PRD > of such an assumption. > PRD > In normal set theory, there is a notion of set difference. > PRD > That is, if I have a set S which is a subset of the set T, > PRD > I can construct a new set R such that > PRD > (R union S) = T > PRD > (R intersection S) = empty_set > PRD > We say in this case that R = T - S > PRD > I have some questions for you. > PRD > First is there a set of all irrational numbers between 0 and 1? > HERC > not likely > PRD > (If yes, then I should be able to name this set -- I¹ll call it Y.) > PRD > Second, can I take the set difference between your set 010203 and Y? > PRD > (If I can, I will call it X = 010203 - Y) > PRD > Third, if X exists, could you describe its elements please. > HERC > X = 010203 > HERC > 010203 = {0.1, 0.2, 0.3...0.9, 0.01, 0.02...0.99, 0.001...} > *** > PRD > A few more questions. > PRD > 1. Does your set 010203 contain the element > PRD > 1/3 = .33333... ? > HERC > no > PRD > 2. Can we construct a set whose only element is 1/3? > PRD > (Call it U = {1/3} if it exists) > HERC > yes > PRD > 3. Can we take the set difference between the set > PRD > 010203 and the set U? > HERC > yes > PRD > 4. Does the set 09099 = { 0.9, 0.99, ... } contain > PRD > the element 1? > HERC > no > PRD > 5. Can we construct a set whose only element is 1? > HERC > yes > PRD > 6. Can we take the set difference between the set > PRD > 09099 and the set {1} (if so call it V) > HERC > yes > PRD > 7. If V exists, can you describe the elements of V? > HERC > V = 09099 > **** > PRD > It would seem by your answers that a statement x occurs > PRD > in the set 010203 means something different than x > PRD > is an elemeent of the set 010203. > HERC > yes. > HERC > as the set approaches inŽnite length, the element occurs. > [snip of example] > PRD > Does the statement x occurs in set S mean the same > PRD > as the statement x is an element of set S? > HERC > no, > HERC > sequences of digits OCCUR, > HERC > stings occur > HERC > numbers are elements > PRD > Does the number 1 occur in the set {.9, .99, ...}? > HERC > no > PRD > Does the number .999... occur in the set {.9, .99, ...}? > HERC > yes or no, the SEQUENCE OF DIGITS occurs. > PRD > Does the number 1/3 occur in the set 010203? > HERC > the sequence of digits 0.333.. does. > PRD > If x occurs in set S means something other than > PRD > x is an element of set S, please answer the following: > PRD > Are there any rules of inferrence that allow one to > PRD > validly conclude from a statement of the form > PRD > x occurs in S some other statement? > PRD > That is, does there exist a valid rule of the form: > PRD > Given that x occurs in set S and statement P is > PRD > true, we can validly conclude that statement Q is > PRD > true? > PRD > Can you give an example of such a valid rule of > PRD > inferrence? > [snip of example] > HERC > That is Cantor¹s proof. The induction stretching to > HERC > inŽnite sets is unsound. > HERC > Given that occurs in set S and was > HERC > contructed using Cantors diagonaliasion, > HERC > we can validly conclude that set S is not demonstrably > HERC > missing any sequence of digits > **** > PRD > I am confused as to whether this is offered as an > PRD > example of a valid rule, or an invalid rule. You > PRD > say that the induction stretching to inŽnite sets > PRD > is unsound. But I¹m not sure if the induction > PRD > refers to the rule you give immediatly after. > PRD > Is the rule: > PRD > Given that occurs in set S and was > PRD > constructed using Cantor¹s diagonalization, > PRD > we can validly conclude that set S is not demonstrably > PRD > missing any sequence of digits > PRD > a valid rule? > HERC > yes > PRD > Does it work for all and S? > HERC > yes, if the digit string occurs then it can¹t be missing. > **** > PRD > I¹m sorry, but I am still unsure of what you mean by > PRD > missing. You use the term occurs in to mean > PRD > something other than is an element of, so I am led to > PRD > wonder what missing means. > PRD > I suspect that it means the same thing as does not occur. > PRD > And it is to verify my suspicion that I am asking these > PRD > questions. > PRD > Does missing mean does not occur? > PRD > For any and S, does the phrase > PRD > the sequence of digits is missing in [from?] S > PRD > mean exactly the same as > PRD > the sequence of digits does not occur in S? > PRD > If the answer to the above question is yes, then > PRD > I would point out that the rule you have given me > PRD > allows me to infer from a statement of the form > PRD > d occurs in S to a statement of the form > PRD > for any sequence of digits e, e occurs in S. > PRD > Is that correct? > PRD > Is there some other rule of inference that you > PRD > can provide which allows me to infer from > PRD > a condition that involves the predicate occurs, > PRD > to a consequence that does not involve the > PRD > predicate occurs? > PRD > The following is an example of the form > PRD > I am looking for, but is obviously an > PRD > incorrect rule. > PRD > If the sequence of digits occurs in > PRD > S, then S has an even number of elements > PRD > What I am looking for is whether it is possible > PRD > to derive something from a proof of > PRD > occurence which is not also a statement > PRD > about occurence. > HERC > Right. Occur refers to any digit string, even if it > HERC > can¹t be indexed in the list. > HERC > consider this inŽnite string of H and T. > HERC > HTHHTTHHHHTTTTHHHHHHHHTTTTTTTT... > HERC > This string contains both the inŽnite sequence of H > HERC > and the inŽnite sequence of T. > **** > PRD > We seem to be having some difŽculty communicating. > PRD > I¹m not sure why this should be. Could you please give > PRD > short direct answers to the following questions. > PRD > Does missing mean does not occur? > HERC > sequence occurs <-> sequence is not missing > PRD > For any and S, does the phrase > PRD > the sequence of digits is missing in [from?] S > PRD > mean exactly the same as > PRD > the sequence of digits does not occur in S? > HERC > yes > PRD > Is there some other rule of inference that you > PRD > can provide which allows me to infer from > PRD > a condition that involves the predicate occurs, > PRD > to a consequence that does not involve the > PRD > predicate occurs? > HERC > sequence is a member -> sequence occurs > HERC > sequence does not occur -> sequence is not a member === Subject: Re: limitation to induction on Žnite bounds >> There is a difference between an inŽnite number of heads in the entire >> string, and an inŽnite sequence of heads. HTHTHTHTHT... has an >> inŽnite number of heads, but the longest string is 1. >what¹s the longest string when you žip a balanced coin inŽnite times? > Now that¹s a sensible and interesting question, but on past form I¹m sure you > won¹t like the answer. > Clearly, there is no upper bound to the length of subsequences of Hs. Indeed, > *any* Žnite sequence has probability 1 of occurring as a subsequece in the > inŽnite sequence. > But notice the qualiŽcation Žnite: an *inŽnite* sequence of Hs has > probability 0, since that would require an inŽnite tail of Hs, which is the > same as having *no* Ts after some point, which has probability 0. > And notice that this is closely analogous to the problem of what¹s the longest > sequence of 3s in the set {3, 33, 333, ...}. There is no upper bound (since > for every natural number k there is a sequence of length k}, but there is also > no inŽnite sequence of 3s (since every sequence has a natural number k that > is its length). > And this is a direct consequence of the Phillite-Frustrating Fact that there > is no bound to the natural numbers, yet each particular one is Žrmly Žnite. but you contradict good ole countable inŽnity theory === > Subject: Re: estimating probability > Herc, > Yes, the number of ties will be inŽnite. There will also be iniŽnite > (both in length and in number) periods favoring heads and inŽnite > periods favoring tails between the ties Herc === Subject: Re: limitation to induction on Žnite bounds > There is a difference between an inŽnite number of heads in the entire > string, and an inŽnite sequence of heads. HTHTHTHTHT... has an > inŽnite number of heads, but the longest string is 1. >what¹s the longest string when you žip a balanced coin inŽnite times? >> Now that¹s a sensible and interesting question, but on past form I¹m sure >> you >> won¹t like the answer. >> Clearly, there is no upper bound to the length of subsequences of Hs. >> Indeed, >> *any* Žnite sequence has probability 1 of occurring as a subsequece in the >> inŽnite sequence. >> But notice the qualiŽcation Žnite: an *inŽnite* sequence of Hs has >> probability 0, since that would require an inŽnite tail of Hs, which is the >> same as having *no* Ts after some point, which has probability 0. >> And notice that this is closely analogous to the problem of what¹s the >> longest >> sequence of 3s in the set {3, 33, 333, ...}. There is no upper bound (since >> for every natural number k there is a sequence of length k}, but there is >> also >> no inŽnite sequence of 3s (since every sequence has a natural number k that >> is its length). >> And this is a direct consequence of the Phillite-Frustrating Fact that there >> is no bound to the natural numbers, yet each particular one is Žrmly >> Žnite. >but you contradict good ole countable inŽnity theory No I don¹t. === >> Subject: Re: estimating probability >> Herc, >> Yes, the number of ties will be inŽnite. There will also be iniŽnite >> (both in length and in number) periods favoring heads and inŽnite >> periods favoring tails between the ties That¹s not entirely true: there will indeed be an inŽnite number of sequences between cumulative ties (because there will be an inŽnite number of ties), BUT these sequences are all Žnitely long. So how about responding directly probability 0, since that would require an inŽnite tail of Hs, which is the same as having *no* Ts after some point, which has probability 0. --------------------------- | BBB b Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | | B B a a r b b | | BBB aa a r bbb | ----------------------------- === Subject: Re: limitation to induction on Žnite bounds > There is a difference between an inŽnite number of heads in the entire > string, and an inŽnite sequence of heads. HTHTHTHTHT... has an > inŽnite number of heads, but the longest string is 1. >what¹s the longest string when you žip a balanced coin inŽnite times? >> Now that¹s a sensible and interesting question, but on past form I¹m sure >> you >> won¹t like the answer. >> Clearly, there is no upper bound to the length of subsequences of Hs. >> Indeed, >> *any* Žnite sequence has probability 1 of occurring as a subsequece in the >> inŽnite sequence. >> But notice the qualiŽcation Žnite: an *inŽnite* sequence of Hs has >> probability 0, since that would require an inŽnite tail of Hs, which is the >> same as having *no* Ts after some point, which has probability 0. >> And notice that this is closely analogous to the problem of what¹s the >> longest >> sequence of 3s in the set {3, 33, 333, ...}. There is no upper bound (since >> for every natural number k there is a sequence of length k}, but there is >> also >> no inŽnite sequence of 3s (since every sequence has a natural number k that >> is its length). >> And this is a direct consequence of the Phillite-Frustrating Fact that there >> is no bound to the natural numbers, yet each particular one is Žrmly >> Žnite. >but you contradict good ole countable inŽnity theory > No I don¹t. === >> Subject: Re: estimating probability >> Herc, >> Yes, the number of ties will be inŽnite. There will also be iniŽnite >> (both in length and in number) periods favoring heads and inŽnite >> periods favoring tails between the ties > That¹s not entirely true: there will indeed be an inŽnite number of sequences > between cumulative ties (because there will be an inŽnite number of ties), > BUT these sequences are all Žnitely long. So how about responding directly > probability 0, since that would require an inŽnite tail of Hs, which is the > same as having *no* Ts after some point, which has probability 0. to return to the starting point on a random walk with probablitly 1 takes inŽnite time. you¹re concluding that this limits him to only return once. Jim Wells says > sample sets. If the probability of something happenning in one trial is > P, what are the odds that it will occur once in T trials? The formula is > 1 - [1 - P] ^ T. Note that as the number of trials becomes inŽnite, the > odds of the event happening converge to 1 for all P > 0. Now note that > in an inŽnite series of coin tosses, there are an inŽnite number of > smaller inŽnite series of coin tosses, each treatable as a trial for > the probability that an inŽnte series of tosses will have an unequal > number of heads and tails žips. So the two requirements are met, a P 0 and an inŽnite T. Hence the probability equalling 1 that at least one > iniŽnte subset of an iniŽte series of tosses will manifest a > disequilibrium one way or the other. An inŽnite string can be made up of segments of inŽnite strings. fun coin() while true if rnd>0.5 return wend for any number of cycles it wont necessarily halt in that time. for inŽnite number of cycles there is still one possible outcome, [low, low, low...] that shows its impossible to prove that it halts. Note that as the number of trials becomes inŽnite, the > odds of the event happening converge to 1 for all P > 0 as there is a possible path for inŽnite Heads, P>0 with inŽnite trials this means the probability of inŽnite Heads is 1. Herc === Subject: Re: limitation to induction on Žnite bounds >> There is a difference between an inŽnite number of heads in the >> entire >> string, and an inŽnite sequence of heads. HTHTHTHTHT... has an >> inŽnite number of heads, but the longest string is 1. > >what¹s the longest string when you žip a balanced coin inŽnite times? Now that¹s a sensible and interesting question, but on past form I¹m > sure you won¹t like the answer. Clearly, there is no upper bound to the length of subsequences of Hs. > Indeed, *any* Žnite sequence has probability 1 of occurring as a > subsequece in the inŽnite sequence. But notice the qualiŽcation Žnite: an *inŽnite* sequence of Hs > has probability 0, since that would require an inŽnite tail of Hs, > which is thes ame as having *no* Ts after some point, > which has probability 0. And notice that this is closely analogous to the problem of what¹s > the longest sequence of 3s in the set {3, 33, 333, ...}. There is > no upper bound (since for every natural number k there is a sequence > of length k), but there is also no inŽnite sequence of 3s > (since every sequence has a natural number k that is its length). And this is a direct consequence of the Phillite-Frustrating Fact > that there is no bound to the natural numbers, yet each particular > one is Žrmly Žnite. >but you contradict good ole countable inŽnity theory >> No I don¹t. >> === > Subject: Re: estimating probability Herc, Yes, the number of ties will be inŽnite. There will also be iniŽnite > (both in length and in number) periods favoring heads and inŽnite > periods favoring tails between the ties >> That¹s not entirely true: there will indeed be an inŽnite number of >> sequences >> between cumulative ties (because there will be an inŽnite number of ties), >> BUT these sequences are all Žnitely long. So how about responding directly >> probability 0, since that would require an inŽnite tail of Hs, which is the >> same as having *no* Ts after some point, which has probability 0. >to return to the starting point on a random walk with probablitly 1 takes >inŽnite time. >you¹re concluding that this limits him to only return once. cumulative ties (returns to the starting point), each of which takes FINIELY LONG. >Jim Wells says >> sample sets. If the probability of something happenning in one trial is >> P, what are the odds that it will occur once in T trials? The formula is >> 1 - [1 - P] ^ T. Note that as the number of trials becomes inŽnite, the >> odds of the event happening converge to 1 for all P > 0. Now note that >> in an inŽnite series of coin tosses, there are an inŽnite number of >> smaller inŽnite series of coin tosses, each treatable as a trial for >> the probability that an inŽnte series of tosses will have an unequal >> number of heads and tails žips. So the two requirements are met, a P > 0 and an inŽnite T. Hence the probability equalling 1 that at least one >> iniŽnte subset of an iniŽte series of tosses will manifest a >> disequilibrium one way or the other. >An inŽnite string can be made up of segments of inŽnite strings. >fun coin() >while true > if rnd>0.5 return >wend >for any number of cycles it wont necessarily halt in that time. >for inŽnite number of cycles there is still one possible outcome, [low, low, >low...] that >shows its impossible to prove that it halts. It halts with probability 1 (assuming that rnd() uses some truly random quantum process and not a deterministic pseudo-random generator). That can be viewed as meaning that it¹s allowed to fail to halt on a Žnite number of runs during an unlimited number of runs of the program. But this is entirely aside from the argument I¹ve given you to address (twice, so far). > Note that as the number of trials becomes inŽnite, the > odds of the event happening converge to 1 for all P > 0 >as there is a possible path for inŽnite Heads, P>0 >with inŽnite trials this means the probability of inŽnite Heads is 1. No it doesn¹t. That possible path has probability 0. Note that Jim Wells correctly requires that P > 0. Now, please try analysing my actual argument, that an *inŽnite* sequence of Hs has probability 0, since that would require an inŽnite tail of Hs, which is the same as having *no* Ts after some point, which has probability 0. Do you see any žaw in any step of this argument (other than the fact that you don¹t want to accept the conclusion)? --------------------------- | BBB b Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | | B B a a r b b | | BBB aa a r bbb | ----------------------------- === Subject: Re: limitation to induction on Žnite bounds === >Subject: Re: estimating probability >Herc, >Yes, the number of ties will be inŽnite. There will also be iniŽnite >(both in length and in number) periods favoring heads and inŽnite >periods favoring tails between the ties. Why? In a random walk, the >expected wait time for return to origin is inŽnite. Don¹t take my word >for it. See http://www.ms.uky.edu/~mai/java/stat/brmo.html What is the >expected wait time to go from tied to disequilibrium? Always one toss. >So even though the coin is fair and will balance out given inŽnite >tosses, most of the time there heads will still be unequal to tails. >Is that incorrect also? >>There is a difference between an inŽnite number of heads in the entire >>string, and an inŽnite sequence of heads. HTHTHTHTHT... has an >>inŽnite number of heads, but the longest string is 1. > what¹s the longest string when you žip a balanced coin inŽnite times? Probabilistically, any arbitrarily large number of consecutive heads/tails will occur. There may or may not be a maximum. That does not mean an inŽnite string of heads/tails will occur, however. Just arbitrarily large. Of course, it is also possible for an inŽnite string to occur. Will Twentyman email: wtwentyman at copper dot net === Subject: Re: it is~ > hello.....doctor~ > fn(x) = (1 - x) / (1 + x^n) > x in [0,1] > show that fn(x) is uniformly converge. > -------------------------------- > of course f(x) = 1 - x. > so > |fn(x) - f(x)| > = | (1 - x) / (1 + x^n) - (1-x) | > = | [{x^(n+1)} - {x^n}] / (1 + x^n) | > <= M_n -> 0 > but i can¹t Žnd M_n form. i had a good idea. | [{x^(n+1)} - {x^n}] / (1 + x^n) | <= | {x^(n+1)} - {x^n} | = {x^n} - {x^(n+1)} <= {n/(n+1)}^n - {n/(n+1)}^(n+1) --(***) because (***), let y = {x^n} - {x^(n+1)} y¹ = nx^(n-1) - (n+1)x^n = nx^(n-1) - nx^n - x^n = (-n-1)x^n + nx^(n-1) thus y is max if x = n / (n+1) so {x^n} - {x^(n+1)} <= {n/(n+1)}^n - {n/(n+1)}^(n+1) and lim {n/(n+1)}^n - {n/(n+1)}^(n+1) = n->00 [lim {n/(n+1)}^n] - [lim{n/(n+1)}^(n+1)] = 1/e - 1/e = 0 thus uniformly converge um...... that¹s right ?? === Subject: theory-edge cool links directory hi all, feel free to drop by this cool autosorted link directory http://vznuri.orgspace.com/theory-edge/ best links from all over the web, hot off the keyboard!! collected titles and summaries full of cutting edge categories & topics, listed below 224 links, 24 categories software algorithmics digital convergence, fractals, mosaic/netscape browser, java, extreme programming, digital physics simulation stock market analysis, social engineering, physics simulation, formula 1, transportation, betting, trafŽc engineering, molecular dynamics special fx movie based effects, synthespians, animation open source torvalds, linux, raymond, halloween docs bioinformatics biology + CS, protein folding problem, $100M IBM blue gene project hardware robotics mindstorms, neuromorphic computing, robot soccer, sony dreambot & qrio, bipeds, tilden, toys supercpus supercomputers, clusters, distributed computation, reconŽgurable computing qm computing quantum computing, photonics, spintronics, qm gates, molecules, theory, sites wartech black budget, ebombs, war robotics, military industrial porn complex nanotech nanotransistors, feynman, drexler, smalley, MEMS/microsystems cyberspace websearch infofreedom cyber independence, cyberjournals, open science, MIT open courseware, arXiv online preprint server, darknets, digital rights management, spam blogging denton, academic blogging, blogging vs newspapers, blogging as journalism, future of blogging cybercash economic warfare, gold, future of money, govt surveillance & control, wealth distribution, econophysics cybersecurity NSA, CIA, RSA, echelon, al-Qaida, cryptography, steganography, information warfare, cyberterrorism, homeland security, microsoft monoculture culture proŽles feynman, frankel, hillis, brockman¹s third culture & digerati, nelson, mandelbrot, conway, ramanujan, mathematicians, cranks, mead, nash, erdos gaming video games, game & movie convergence, game economies, nintendo, gaming AI, gaming math/physics simulation hacking hacker crackdown by sterling, hacker hall of fame, psychology of hacking, viruses egovt evoting, edemocracy, emergent democracy cypherpunk encryption, zimmerman, NSA, key escrow, denning, cypherpunks, sealand, DES crack singularity AI consequences, vinge, cyborg vision, joy¹s dystopian visions, kelly theory cs theory chaitin, chayes, theoretics CS, microsoft r&d, barabasi, comp.theory FAQ, satisŽability problem, phase transitions in CS, minesweeper math conquest P-time primality, $7M claymath awards, robbins conjecture automated thm proving, wiles-fermat thm, riemann conjecture, eternity puzzle math misc strogatz, godel, famous & challenge problems, math societies, math sites & magazines AI behavior analysis, loebner prize, cognitive science, evolutionary computing, webmind company, human vs computer chess, church-turing thesis, asimov laws graph theory barabasi & watts, small world graphs, hayes, physics of the web digital physics wolfram, fredkin, conway life game, gliders, rule 110, java life applets etc elsewhere other related sites, locations, egroups, mailing lists === Subject: Re: Why we must go back to Roman Numerals > Joseph Hertzlinger > message >> Considering that the US is looking a bit like the Roman Republic.... > An ignorant Roman Republic: The next Vice President might be someone who made a fortune based on junk science. > The luncheon crowd at an Orlando hotel laughed and Marques posed the > question: What are the angles on a three-four-Žve-triangle? > The governor gave a steely grin and then stalled a bit. The angles > would be ... If I was going to guess ... Three-four-Žve. > Three-four-Žve. I don¹t know, 125, 90 and whatever remains on 180? > Marques had the correct answer: It¹s 30-60-90. I suspect Marques also doesn¹t know that the most abundant form of toxic waste in the atmosphere is produced by green plants. http://hertzlinger.blogspot.com === Subject: Re: L¹Hopital #2 by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i683dLV21430; >>L¹Hopital¹s rule, when we use it, requires that >> f¹ >>--- --> L . >> g¹ >>My question is that if we instead suppose >> f >>--- --> L, >> g >>then is it true that >> f¹ >>--- --> L ? >> g¹ >No. It¹s very easy to give counterexamples. >For example consider the limit as x -> inŽnity of > (x + sin(x)) / x. >************************ >David C. Ullrich I think I got what we have missed. In David¹s example, it should be noted that the usual L¹Hopital¹s rule can not be also applied (If one tries to do so without realizing the importance of hypothesis). I myself was encountered by a counterexample žoor(x)/x, and spent a few nights wondering about this. I think we should add to hypothesis of both of our lemma and L¹Hopital¹s rule the following: Let f~ and g~ be asymptotic values of f and g, respectively. Then if f~/g~ is a indeterminate form, then... f~¹/g¹ = L. This modiŽcation is quite consistent, since L = f/g ~ f~/g~ What would you think? You do not have to send any response to my email account. H. Shinya === Subject: Is there more symmetry to this function? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i683dKW21401; Consider the following function: f: N x N -> N (N naturals) f(1,1) = 16, f(1,2) = 17, f(1,3) = 20, f(1,4) = 25, f(1,5) = 32, ... sequence of differences: (1, 3, 5, 7, ...) f(2,1) = 33, f(2,2) = 36, f(2,3) = 41, f(2,4) = 48, f(2,5) = 57, ... sequence of differences: (3, 5, 7, 9, ...) f(3,1) = 52, f(3,2) = 57, f(3,3) = 64, f(3,4) = 73, f(3,5) = 84, ... sequence of differences: (5, 7, 9, 11, ...) f(4,1) = 73, ... etc. Then f(x,x) = (2x+2)^2 and f(1,1) - f(1,1) = 0 = 0*8 f(2,1) - f(1,2) = 16 = 2*8 f(3,1) - f(1,3) = 32 = 4*8 f(4,1) - f(4,1) = 48 = 6*8 Am I missing any other obvious symmetries of interest? Can this function be stated explicitely? Sincerly, C. Dement === Subject: Re: square root of 3i by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i683dKg21405; >> In the third quadrant in the Cartesian coordinate system where the >> values of x,y are [-x, -y] and (my) sqrt(-3) = >> (((sqrt(3*4+2))-2)/2) * -1 = -.870828693... >First, how do you get sqrt(-3) to be anything other than sqrt(3)i or >-sqrt(3)i? >> Where-- >> (-x)= -.870828693... >> (-y)= -2.129171307... >> Completing the square -- -x + -y = -3. >> Checking the square -- >> -3 =((((.870828693...* 2 +2)^2)-2)/4)*-1 >> Checking Cartesian 3rd quadrant sqrt(-3) against (descartesian) >> sqrt(-3)-- >What is descartesian, and how is it different from Cartesian? I would say descartesian is where y is imaginary and x is real. Cartesian is where x and y are reals. >> ((((sqrt(3*4+2))-2)/2) * -1) / (sqrt(3/2) + sqrt(3/2 )i) = >> DECIMAL EXPANSION --->oo on the real side and on the imaginary side. >Let a be the real number ((((sqrt(3*4+2))-2)/2) * -1) >Let b be the real number sqrt(3/2) >Note that 1/(1+i) = (1-i)/2 >Why would you be surprised that a/(b*(1+i)) = (a/b)*((1-i)/2) = >(a/(b*2)) * (1-i) = (a/(b*2)) - (a/(b*2))i has real and imaginary parts >with the same absolute value? You are right, my blunder! Dan >Daniel W. Johnson >panoptes@iquest.net >http:// members.iquest.net/~pano ptes/ I¹m a little rusty at solving DE¹s, 15 years ago in collage. > But I need to learn how to solve this ODE: dy/dx = y^2 + x^2 riccati: y¹ - yy = xx there is a method to Žnd a family of solutions for a riccati equation for which a particular solution is known. you can look that up. the apparently hard part in your problem is coming up with a particular solution. it would appear that some polynomial may do the trick. however, i tried polynomials of the forms Ax + B and Axx + Bx + C neither of these forms work. > Can someone explain the steps on how it¹s done? Is there a > way to transform it into a simpler linear equation? yes. you will see when you Žnd a reference riccati transforms into bernoulli, and bernoulli transforms into linear. > Any help would be greatly appreciated. === Subject: Re: Primes in Žnite intervals > Now 1/(a-1) = q-1, so your claim implies that p - q < c log(q-1). > So you¹re saying that there is c such that for every prime q the > gap between q and the next prime is at most c log(q-1). AFAIK > this is much smaller than any known upper bound on prime gaps: > even assuming the Riemann Hypothesis, I think the best you get is > c q^(1/2) log(q). True, though this is one of those places where the Riemann Hypothesis gives error terms that appear to be far weaker than the truth. There¹s some reason to hope that the gap between q and the next prime shouldn¹t get much over the square of log q, although proving anything remotely resembling this is far out of reach. Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: psychology of geometric visualization > Visualizing things in three dimensions doesn¹t come naturally to me, > so I¹m always trying to improve my skills in that area. If anybody > knows any references or has any suggestions to make, I¹d be glad to > hear about it. In case that¹s too general for you, here¹s the > speciŽc thing that¹s on my mind at the moment: a cube rotating on > one of its diameters. Apparently many (most?) people consider it > obvious that the cube coincides with itself every 120 degrees, but > for me it¹s extremely difŽcult to see that. In fact, I went to the > trouble of writing down an actual (analytic) proof. But of course > that misses the point. What I¹d really like to know is this: if this > conclusion seems completely clear to you, can you give me any insight > into how you arrive at it? I haven¹t seen anyone give what I thought was the standard mathematical answer, viz., visualize it in n dimensions, and then specialize to n = 3. Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Search for a number theoretic function related to prime divisor sums > I found something related to this toppic. It is called the mertens > constant. > I found it on > http://mathworld.wolfram.com/MertensConstant.html > I guess the description there is not quite acurate. I can¹t immagine > that the formula (stated on their site) for the prime reciprocal > delivers an exact value. I guess it is only an aproximation. But it is > quite interesting. Do you understand the notation in the formula? It says the sum of the reciprocals of the primes up to x is equal to log log x, plus a constant B, plus something that goes to zero as x goes to inŽnity. That last piece means that they are not claiming to give an exact value, but only an approximation that gets better and better as x increases. Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Search for a number theoretic function related to prime divisor sums > do you know some rules about such numbers (for which the reciprocal > values of the prime divisors add up to a value > 1)? > First I thought, that maybe the following is true for such x: > if x has a reciprocal sum > 1 (as described above) and x > 30 => there > exists at least one t > 1 which is composite and t maximum prime divisor of x. I¹m sorry, I think I don¹t understand this at all. If the sum of the reciprocals of the prime divisors of x exceeds 1 then t = 4 satisŽes t > 1, is composite, and is less than the biggest prime divisor q of x since q is at least 5. What did you really mean? Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: better than a quartic? > I have to solve this set of equations: > t5 = t1 + (t1-t2)^2/( (t1-t3)*(Pa-1) ) > t1 = t5 + (t5-t4)^1/( (t5-t3)*(Pb-1) ) > - is there a better way to solve the original equations, without > turning them into a quartic? I need the 4 solutions, having only one > won¹t do! Isn¹t the resultant supposed to help with systems of polynomials? E.g. the abstract of http://portal.acm.org/citation.cfm?id=362839&dl=ACM&coll= portal seems to imply so. === Subject: Re: What makes the world go-round, and so unround? >> In sci.physics, Donald G. Shead >> Centrifugal force is what makes the world go-round, and so unround: >> The same force that made the water in Newton¹s spinning bucket >> experiment recede from the center, and climb up the sides. >Cut< >> The world is part of the solar vortex; which Rene¹ Descartes proposed >> was just one of the many vortices that Žll the univese. >Cut< >> As such the world is just carried around by the sun¹s vortex, and >> spins about its own vortex, centrifugally. Therefore it needs no >> Œexternal force¹ to do this. >> The answer is right even if your reasoning isn¹t, it turns out. :-) >It¹s not _my_ reasoning Ghouly: Didn¹t I just say? The world is part >of the solar vortex; which Rene¹ Descartes proposed was just one of >the many vortices that Žll the univese. No. Jim === Subject: Re: A little lesson for sqrt(144) year olds. > | > | x has no sign until an assumption is made. > | | > Excellent. Didn¹t take much to work that out, did it? > | | > sqrt() has no sign until an assumption is made. > | That assumption was made long ago. > Not by me. > -2 is a square root of 4 because -2 * -2 = 4, > -i, -1, i and 1 are the four fourth roots of 1, and there are > exactly N Nth roots of any number. > | Just as sin(x) has an accepted meaning, and so does the > | symbol 3. There is nothing inherent in those symbols, > | but there is a universally accepted meaning which only > | an idiot would insist on ignoring. Same as for sqrt(x). > | As I said in a previous post, it would be pointlessly > | confusing to use a different meaning from that which is > | universally accepted and understood. > It is (almost) universally accepted and understood that -2 * -2 = 4, > and the root x of any number N is given by x^2 = N. > Some people think not, I guess. > Androcles Dirk doesn¹t need to do that. He keeps + & - under different wallnut shells, and can produce the correct choice after any amount of moving them around. eg: although x is an unknown and could be anywhere <0> , notice how he can magically apply the correct sign to x to solve the problem? An uncanny gift, that :-) Jim G === Subject: Re: A little lesson for sqrt(144) year olds. X-AUTHid: wagners5 :> | > | x has no sign until an assumption is made. :> | > :> | > Excellent. Didn¹t take much to work that out, did it? :> | > :> | > sqrt() has no sign until an assumption is made. :> | :> | That assumption was made long ago. :> Not by me. :> -2 is a square root of 4 because -2 * -2 = 4, :> -i, -1, i and 1 are the four fourth roots of 1, and there are :> exactly N Nth roots of any number. :> | :> | Just as sin(x) has an accepted meaning, and so does the :> | symbol 3. There is nothing inherent in those symbols, :> | but there is a universally accepted meaning which only :> | an idiot would insist on ignoring. Same as for sqrt(x). :> | :> | As I said in a previous post, it would be pointlessly :> | confusing to use a different meaning from that which is :> | universally accepted and understood. :> It is (almost) universally accepted and understood that -2 * -2 = 4, :> and the root x of any number N is given by x^2 = N. :> Some people think not, I guess. :> Androcles : Dirk doesn¹t need to do that. He keeps + & - under different wallnut : shells, and can produce the correct choice after any amount of moving : them around. : eg: although x is an unknown and could be anywhere <0> , notice how he : can magically apply the correct sign to x to solve the problem? : An uncanny gift, that :-) : Jim G It is amazing anyone could have the slightest problem understanding Dirk¹s point. Apparently in the world you and Androcles inhabit there is no ambiguous way to write the positive number x such that x*x=2. The equation x^2=N has two roots. Those two roots are sqrt(N) and -sqrt(N). Apparently you and Androcles think the two roots are sqrt(N) and sqrt(N) and have no way of distinguishing the two roots from each other, or of specifying which root you mean. Stephen === Subject: Re: A little lesson for sqrt(144) year olds. > | > sqrt() has no sign until an assumption is made. > | That assumption was made long ago. > Not by me. Right. As I said, a pointless position to take. > -2 is a square root of 4 because -2 * -2 = 4, Right. But the symbol sqrt(x) is reserved for a particular square root. To ignore this convention is pointless and idiotic. There are simple unambiguous ways of indicating which of the roots you are choosing while adhering to this universal convention. Simple and unambiguous is the point of convention. There are inŽnitely many solutions to the equation sin(x) = 0.5. But there is only one value of arcsin(0.5). > It is (almost) universally accepted and understood that -2 * -2 = 4, > and the root x of any number N is given by x^2 = N. > Some people think not, I guess. When did you ever see a statement otherwise? Let¹s put it baldly out there: While there are two solutions to x^2 = N, when N is positive real the symbol sqrt(N) stands for the positive solution of the equation. Now, if you want to claim that I don¹t believe there are two solutions after I just said there were, we have a name for the deliberate distortion of truth. - Randy === Subject: Re: Atheist MorituriMax > First of all, it¹s athiest. Mr Max. You are proud of your own mispelling. > I told you you couldn¹t correct me. I am right. And you will never > forget it. in german and then the damn americans gottsa turn it all around.. I apologize for that one.. As for never forgetting it, 1. You don¹t mean that much to me. My mom or dad I never forget, you I just > chalk up to an honest mistake. > 2. The screw up isn¹t that big a deal. Woopie. To *you* it isn¹t a big deal, but to him it proves that you are evil, > immoral, and that you are incapable of ever laying a hand on him or his > divinely inspired ideas. Go Žgure. > If you believe you know something, you know nothing. If you believe > you know nothing, you know something. > Peter What if youn know you know something because you can see it is true? AKA common sense. There is no doubt in what is God given is there? How do we know anything is true? Does somebodyelse have to tell you? Well how did they know it was right? Did someone have to tell them? They would have to have seen it for themselves. Its the misuse of the forbidden fruit that¹s the problem. Maybe I am not talking about knowledge Peter. To me it is simply experiencing the grace of the truth as it supports life. Just because you know Einstein¹s ideas doesn¹t make you anybody. This especially applies to those who assume the role of authority. They are puffed up with Einstein¹s ideas and they are equally depraved. I don¹t let crazy people teach me anything. I would be as crazy as them if I did. Our great Einstein wanted to know God; how He created the universe; the rest were just details. Mitch Raemsch -- Light falls -- === Subject: Re: Yet Another Proof of God¹s Existence > * Chan-Ho Suh >>Following in the tradition of philosopher/logician/mathematicians of >>the past, (e.g. see Goedel¹s version of Anselm¹s Ontological Proof: >>http://en.wikipedia.org/wiki/G%F6del%27s_ontological_proof ) >>William Hatcher wows the crowd with his proof of God¹s existence, >>relying on what he calls relational logic: >>6ab57 > Quoting: > By Kathy Gilsinan > ... > The proof itself rests on four principles, the Žrst of which is > the assertion that something exists. Even if the world is an > illusion, he pointed out, an illusory self, contemplating an > illusory universe, is still something that exists. > Further, he said, everything that exists does so because of some > cause, and the principle of sufŽcient reason states that every If he is, as he indicates, being factual, there is no evidence that everything that exists does so because of some cause. Cause and effect is just a way of saying that nature is consistent, not that everything has a cause. > phenomenon is either caused by something external or caused by caused by itself is a real stretch since the cause would also be an effect which motion wise, i.e., time wise, has to have the cause occuring before the effect. One might just as well say that existence has no beginning as to say that some god caused itself to exist and thus cause existence. > itself, but never both. Everything that exists has to have a > reason for existing, he said. Now, reason has many deŽnitions. I assume that he means a fact or cause for an existence of some existent. But a fact is just some aspect of reality and so some existent and not a cause of that existent. And in rational thought, what is considered an effect must have a cause, but what is considered a cause is not necessarily required to be and effect and to have a cause, until one is observed for that cause. Larry > Working from these principles, Hatcher Žrst deŽned what he > called the minimum criteria for Godhood, and then set about > trying to prove the existence of a phenomenon to Žt those > criteria. God, he said, must exist and be unique, and must be > self-caused as well as being the cause of everything else. Every > existing phenomenon is the end effect of a causal chain of > possibly inŽnite length, starting with God, he said. > Ignoring the mathematician¹s (or the author¹s) ability to count to > four, I still don¹t buy the principle that everything has to have a > reason for existing. However, even if I accept this principle, I > cannot see how something has to exist at the beginning of an inŽnite > causal chain. Perhaps we could accept this anyway, but then our god > is inŽnite far away, and we certainly cannot rely on him as current > politics is a better proof of. === Subject: Re: InŽnity can not exist >> It is impossible for inŽnity to exist, because in order for inŽnity >> to exist, everything possible must happen. Including inŽnity not >> existing. Therefore, it is possible for something to be an extrememly >> large, but not inŽnite. For it to be inŽnite, it would have to be >> Žnite at the same time. >> Japcuh >> (Just Another Perl C Unix Hacker) >> http://www.catb.org/~esr/faqs/hacker-howto.htm#what_is >> .O. >> ..O >> OOO >Well, actually what is important to remember is the deŽnition of >inŽnity, that is, something which is not Žnite, therefore, it is not >true for it to be inŽnite, it would have to be Žnite at the same >time. InŽnity is not an event nor is it something tangible. In >fact, it can not be measured...it is an abstraction. As far as our >universe is concerned, nothing is inŽnite. In other words, we could >the universe. Granted it would take a very long time to do so, but >the point is that we could and eventually everything in the universe >would be accounted for. However, this same function could not be >applied to the natural numbers. I am curious to know what you meant >by saying everything possible must happen and how you are relating >that to inŽnity?? >> Go read The Library of Babel or The Book of Sand by Borges for >> entertaining versions of the idea. >> German >Or look at the Constructivists - L.E.J. Brouwer. They believe >that if something can not be constructed in a Žnite number >of steps that it does not exist. Contructivism. . . must . . . stay . . . awake . . . sorry, nodding off. IConstructivism always reminds me of Trurl and Klapaucius, the Constructors of Stanilaw Lem. Somewhere along the way I realized I didn¹t love science or math, just stories about science and math, and Borges and Lem were more fun than Brouwer after a while; Brouwer I like for a radical skepticism of the foundations of logic, his relationship to Hilbert, Russell and Poincare, which is a great story. Outside of that, I think math is starting to drive me to sleep. >This argument seems silly to me. InŽnity is an abstract concept, >not a measurable unit. >Chris German === Subject: Re: InŽnity can not exist > Hi Chris, > silly me but it seems to me (the skeptic realist I am, not the > Constructivist:) that before we could believe or disbelieve the real > existence of inŽnity you need to tell us something about your > deŽnition of existence and its whereabouts: in reality outside or > within. Until then we may choose some point in the middle between it > does and it does not exist--sitting and waiting (ŒindeŽnitely¹:) > for the arrival of the overwhelming evidence that will Žnally force us > to make up our gullible mind (i.e. Žx it to the absolute Œtruth¹, and > stick to it to the bitter end:). >> Or look at the Constructivists - L.E.J. Brouwer. They believe that if >> something can not be constructed in a Žnite number of steps that it >> does not exist. >> This argument seems silly to me. >> InŽnity is not a measurable unit. >> InŽnity is an abstract concept. > ... therefore, its reality is... outside or within? Come on, it must be > easy, is ŒinŽnity¹ a sublime product of the supreme mind (like all the > words--abstract creations of Lord Logos), or is it something else > entirely, or is it more, and how much more than a regular word. Or > perhaps you could construct it for us, and it doesn¹t matter in how > many Žnite (or more:) number of steps you do it, important is the > experiment to last less than ŒindeŽnitely¹. > I mean, people are impatient these days, few would like to be told to > repeat some imaginary process Œforever¹. We are just Œunprepared¹ to > wait until the end of Œeternity¹ in order to witness the miracle (the > actual construction or coming into existence of the ŒinŽnitesimal¹ > converging rapidly to the inŽnite loop, approaching but never actually > reaching it:), you know why? Because...:), to experience anything it > has to cross our path during our Œlifetimes¹ -- which happen to be... > Žnite or inŽnite? (or more than ŒŽnite¹ but less than ŒinŽnite¹:) > Best, > Ann I didn¹t mean to call you silly. I guess that I take for granted the concept of inŽnity. You do sound like a bit like a Constructivist, which is not a bad thing. I know that they have gotten a bit of bad press, but Brouwer¹s ideas are interesting. I did not say that inŽnity exists, but that it is a concept (and a very convenient one when dealing with things such as the set of natural numbers, pi, e, etc). Chris === Subject: Re: partition numbers by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i685Cn529046; >> computes partitions of nos upto 2000 in 10 mins. >In Tom Apostol¹s book, Introduction to Analytic Number Theory, it says >The largest value of p(n) yet computed is p(14,301). >Apostol credits the calculation to D H Lehmer, On a conjecture of >Ramanujan, J London Math Soc 11 (1936) 114-118. Apostol says Lehmer >used an asymptotic formula of Rademacher, for which the reference is >On the partition function p(n), Proc London Math Soc 43 (1937) 241-254. >Obviously Lehmer¹s work predates electronic computers. >Apostol¹s book is from 1976. >-- >Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) Using my program I could compute partitions till n = 2950 on my PC. As I go higher the program has problems in linking. The results I got for n = 2950 : Partition(2940) = 0123308336305578354112026211802049193104930848592916919650 Partition(2941) = 0126216805171138737617036572951130313853282827410825879061 Partition(2942) = 0129193371372902955707032855483303910844145844873058981540 Partition(2943) = 0132239617582700887394868913108046996127491088578398740963 Partition(2944) = 0135357162985415511589743912486242444851308055625529113425 Partition(2945) = 0138547664115170268692548647529737170068337373903300103977 Partition(2946) = 0141812815710525411760858003150230078907807393167475862342 Partition(2947) = 0145154351589112306869943757896220010428243053569418676980 Partition(2948) = 0148574045542144251289424406688020174607142714119916037967 Partition(2949) = 0152073712249252200759484666353656363257902921556313550875 Partition(2950) = 0155655208214103838202514613322512517150174562757200203871 How can I conŽrm if these are correct? === Subject: Re: Paperback math reference books > Its a mistake to get paperback math books, as they will > probably be used for reference (if they are any good), > and they will fall apart. At the time it seemed like a > good deal though. I just got out my (paperback) copy of Spivak¹s A Comprehensive Intro > to Differential Geometry, and it is in 8-9 parts now. > I have to put the pages in order every time I get it out. In contrast, I have some great hardback ref books. > They will fall apart too, if you use them often enough :-) Yeah, and it¹s always the most useful books that fall apart the fastest, since you¹re using them all the time. Any reference book still sitting on your shelf in a pristine condition after several years cannot be very good. === International Journal of Computational Geometry & Applications Special Issue: Selected Papers from the Ninth International Computing View table-of-contents and abstracts at http://www.worldscinet.com/ijcga.html Contents: D. T. Lee and Joseph S. B. Mitchell Guest Editor¹s Foreword Binhai Zhu Cylindrical Hierarchy For Deforming Necklaces Sergey Bereg Traveling Salesman Problem Of Segments Jinhui Xu, Zhiyong Lin, Yang Yang And Ronald Berezney Tetris Is Hard, Even To Approximate Ron Breukelaar EfŽcient Construction Of Low Weighted Bounded Degree Planar Spanner Xiang-Yang Li And Yu Wang EfŽcient Approximation Algorithms For Pairwise Data Clustering And Applications Xiaodong Wu, Danny Z. Chen, James J. Mason and Steven R. Schmid Covering A Set Of Points With A Minimum Number Of Turns Michael J. Collins For more information, go to http://www.worldscinet.com/ijcga.html === Subject: Re: Tensors for mathematicians Sorry for an off-topic post, but you don¹t happen to live in Page you a couple of times. On topic, is this in preparation for phys 12 or did you skip out of phys 1? I hope prerequisite knowledge of tensors is unnecessary for phys 12, or else I have to do some reading. > SpeciŽcally, I¹m learning special relativity using tensors. Is > anyone aware of a development of tensors which is comprehensible to a > non-physics person (i.e. one which does not start out by saying, A > vector is anything that transforms like a vector, without ever > telling you what transforms like means)? I¹ve seen a little about > tensor spaces (I read the section in Dummit & Foote, anyway), but as > far as what contravariant, covariant, dual space, and the like > mean, I have very little idea. > Daniel McLaury === Subject: Re: Tensors for mathematicians Sorry for an off-topic post, but you don¹t happen to live in Page you a couple of times. On topic, is this in preparation for phys 12 or did you skip out of phys 1? I hope prerequisite knowledge of tensors is unnecessary for phys 12, or else I have to do some reading. > SpeciŽcally, I¹m learning special relativity using tensors. Is > anyone aware of a development of tensors which is comprehensible to a > non-physics person (i.e. one which does not start out by saying, A > vector is anything that transforms like a vector, without ever > telling you what transforms like means)? I¹ve seen a little about > tensor spaces (I read the section in Dummit & Foote, anyway), but as > far as what contravariant, covariant, dual space, and the like > mean, I have very little idea. > Daniel McLaury === Subject: Re: Tensors for mathematicians > SpeciŽcally, I¹m learning special relativity using tensors. Is > anyone aware of a development of tensors which is comprehensible to a > non-physics person (i.e. one which does not start out by saying, A > vector is anything that transforms like a vector, without ever > telling you what transforms like means)? I¹ve seen a little about > tensor spaces (I read the section in Dummit & Foote, anyway), but as > far as what contravariant, covariant, dual space, and the like > mean, I have very little idea. > Daniel McLaury I recommend: Tensor Geometry: The Geometric Viewpoint and Its Uses (Graduate Texts in Mathematics, Vol 130) by C.T.J. Dodson, T. Poston A real effort is made in this book to explain not only the physics but, as you say, what a tensor is without non-sensical deŽnitions such as those you describe above. This is also done by Spivak in his Differential Geometry books, but he doesn¹t focus as much on physics as Dodson and Poston. If you really get serious--or even if not, you might like to take a look at the following two books: Differential Forms and Connections by R. W. R. Darling General Relativity for Mathematicians (Graduate Texts in Math Ser Vol 48) by Sach and Wu --this is interesting to scan. It explains clearly some of the differences in the ways physicist and mathematician deal with things. One thing I recall from the book is their statement that it was getting the deŽnitions right that was the real problem. And that what a mathematician would call motivation for deŽnition, a physicists might call a proof of a theorem. And other useful comments like that. Darling¹s book may be good background for understanding the details of Sach and Wu¹s book. Good luck, Edwin Clark === Subject: Re: One question Lattice density > Hi I am reading SLOANE and CONWAY¹s SPLAG > I have one question about the density deŽnition > the book deŽnes the density to be > volume of one sphere > ------------------------------- > volume of fundamental region > I wonder whether I can redeŽne this deŽnition to be > volume of one sphere > ------------------------------- > volume of the Voronoi cell > In another word if I try to approximate a Voronoi cell by a shpere, > how close are they two? The Voronoi cell is one of many possible shapes for a fundamental region. Ideas for other shapes can be seen in some of Escher¹s art. The different possible shapes will all have the same volume, except in the case of certain aperiodic hyperbolic tilings. An example of an exception is a hyperbolic plane tiling in which each vertex is surrounded by a triangle, a pentagon, a decagon and a dodecagon. Each pentagon must share an edge and its other three vertices with a triangle each. The decagons and dodecagons alternate around the pentagon. One way to cut out a fundamental region for this tiling is to include a pentagon, its three vertex triangles and portions of the decagons and dodecagons that share the other four edges of the pentagon. That makes 1 pentagon, 3 triangles, 7/10 of a decagon and 7/12 of a dodecagon. An alternative is to leave out the 3 vertex triangles and include the triangle that shares an edge. That way the region is smaller by two triangles. Either of these regions is an aperiodic monotile, if what you mean by aperiodic is that no Žnite cluster of such tiles is the fundamental region of a symmetry group.