> lim delta->0 [f(x+delta)-f(x)]/(delta)^2 exists. > > wondering if there are functions that the following limit does exist: > > lim delta->0 [f(x+delta)-f(x)] / [(delta)^alpha] exists. where alpha>1 > > Any references would be nice too. f(x+delta) = f(x) + delta f'(x) + delta^2/2 f''(x) + o(delta^2) [f(x+delta) - f(x)]/(delta)^2 = f'(x)/delta + f''(x)/2 + o(1) Therefore, you must have f'(x) = 0 ie f constant as it has been said. If this is not for every x, then you only have to have a null derivative at the considered points. For your second question, you must have the n-th first derivatives with n = floor(alpha) equal to zero. -- Nicolas ==== > GR56 grava .88 la saucisse et au marteau: > > lim delta->0 [f(x+delta)-f(x)]/(delta)^2 exists. > > wondering if there are functions that the following limit does exist: > > lim delta->0 [f(x+delta)-f(x)] / [(delta)^alpha] exists. where alpha>1 > > Any references would be nice too. > > f(x+delta) = f(x) + delta f'(x) + delta^2/2 f''(x) + o(delta^2) > > [f(x+delta) - f(x)]/(delta)^2 = f'(x)/delta + f''(x)/2 + o(1) > > Therefore, you must have f'(x) = 0 ie f constant as it has been said. If > this is not for every x, then you only have to have a null derivative at > the considered points. > > For your second question, you must have the n-th first derivatives with > n = floor(alpha) equal to zero. Careful, careful; he didn't assume the function was twice differentiable, or even once differentiable for that matter. (Although it follows.) If the intent is for the limit of (f(x+delta)-f(x))/delta^2 to exist for ALL x, then of course f(x+t) - f(x) f(x+t)-f(x) lim ------------- = lim ----------- * t = 0 t->0 t t->0 t^2 for all x, so f is differentiable and f'(x) = 0 identically: so f is constant. If the intent is to ask whether it is possible for this limit to exist for SOME x, of course it is. Take f(x) = x^3, for example, at x = 0. --Ron Bruck ==== About 2 months ago I ran across a problem on MIT's website to find the indefinite integral of a certain function. It was from some problem solving seminar, and I think it was rated as the hardest problem of all the problems during the semester. I posted about a while back about this integral but everybody just said that since Mathematica can't do it it's probably not integrable in terms of elementary functions. Well, after about two months of toiling over this problem, I think I might finally have a solution. However, there's a few places where, at least to me, things were a little iffy about whether or not I could do what I was doing. So I want to put my solution and see if there's any flaws. So, the problem is to find in closed form Integral[f(x) dx], where f(x) = x/Sqrt[x^4+4x^3-6x^2+4x+1] So, first note two things: 1) f(x) = f(1/x) (assuming of course that x is not 0) 2) the polynomial in the bottom is a reciprocal polynomial Reciprocal polynomials generally can be solved by making a substitution of the form t = x+1/x. So I use this to my advantage later. So anyway, first we use obvservation 1 to say that if I = Integral[f(x) dx], then 2I = Integral[f(x) dx] + Integral[f(1/x) dx] In the second integral, let u = 1/x, and it transforms to Integral[f(x) dx] - Integral[f(u)/u^2 du] Since x is just a placeholder, I replace the x's with u's in the first integral and get 2I = Integral[(1 - 1/u^2)f(u) du] Now, we use the second observation. We already have 2I = Integral[(1 - 1/u^2)*(u/Sqrt[u^4+4u^3-6u^2+4u+1]) du] Bringing that u from the numerator into the denominator, we get 2I = Integral[(1 - 1/u^2)*(1/Sqrt[u^2+4u-6+4/u+1/u^2]) du] Now we are set up to write the part under the square root as a poloynomial in x+1/x. Indeed, we get 2I = Integral[(1 - 1/u^2)*(1/Sqrt[(u+1/u)^2+4(u+1/u)-8]) du] Let v = u+1/u. Then dv = (1-1/u^2) dv, which is perfect since there's a (1-1/u^2) sitting right there! So the integral transforms to: 2I = Integral[1/Sqrt[v^2 + 4v - 8] dv] Now this integral is easily evaluated as 2I = Log[2 + v + Sqrt[v^2 + 4v - 8]] Substituting v gives 2I = Log[2 + u + 1/u + Sqrt[u^4 + 4u^3 - 6u^2 + 4u + 1]] And substituting x gives I = 0.5*Log[2 + x + 1/x + (1/x^2)Sqrt[x^4 + 4x^3 - 6x^2 + 4x + 1]] So, is this right? ==== Nobody escribi.97 en el mensaje > About 2 months ago I ran across a problem on MIT's website to find the > indefinite integral of a certain function. It was from some problem > solving seminar, and I think it was rated as the hardest problem of > all the problems during the semester. I posted about a while back > about this integral but everybody just said that since Mathematica > can't do it it's probably not integrable in terms of elementary > functions. Well, after about two months of toiling over this problem, > I think I might finally have a solution. However, there's a few > places where, at least to me, things were a little iffy about whether > or not I could do what I was doing. So I want to put my solution and > see if there's any flaws. So, the problem is to find in closed form Integral[f(x) dx], where f(x) = x/Sqrt[x^4+4x^3-6x^2+4x+1] So, first note two things: 1) f(x) = f(1/x) (assuming of course that x is not 0) > 2) the polynomial in the bottom is a reciprocal polynomial Reciprocal polynomials generally can be solved by making a > substitution of the form t = x+1/x. So I use this to my advantage > later. So anyway, first we use obvservation 1 to say that if I = > Integral[f(x) dx], then 2I = Integral[f(x) dx] + Integral[f(1/x) dx] In the second integral, let u = 1/x, and it transforms to Integral[f(x) dx] - Integral[f(u)/u^2 du] Since x is just a placeholder, I replace the x's with u's in the first > integral and get 2I = Integral[(1 - 1/u^2)f(u) du] Now, we use the second observation. We already have 2I = Integral[(1 - 1/u^2)*(u/Sqrt[u^4+4u^3-6u^2+4u+1]) du] Bringing that u from the numerator into the denominator, we get 2I = Integral[(1 - 1/u^2)*(1/Sqrt[u^2+4u-6+4/u+1/u^2]) du] Now we are set up to write the part under the square root as a > poloynomial in x+1/x. Indeed, we get 2I = Integral[(1 - 1/u^2)*(1/Sqrt[(u+1/u)^2+4(u+1/u)-8]) du] Let v = u+1/u. Then dv = (1-1/u^2) dv, which is perfect since there's > a (1-1/u^2) sitting right there! So the integral transforms to: 2I = Integral[1/Sqrt[v^2 + 4v - 8] dv] Now this integral is easily evaluated as 2I = Log[2 + v + Sqrt[v^2 + 4v - 8]] Substituting v gives 2I = Log[2 + u + 1/u + Sqrt[u^4 + 4u^3 - 6u^2 + 4u + 1]] Actually is 2I = Log[2 + u + 1/u + Sqrt[u^2 + 4u - 6 + 4/u + 1/u^2]] I = (1/2)Log[2 + x + 1/x + Sqrt[x^2 + 4x - 6 + 4/x + 1/x^2]] I = (1/2)Log[2 + x + 1/x + Sqrt[x^4 + 4x^3 - 6x^2 + 4x + 1]/x] But dI/dx = (x^2 - 1)/(2xSqrt[(x^4 + 4x^3 - 6x^2 + 4x + 1]) = (1/2)(1 - 1/x^2)f(x) ??? There are something more wrong ... I think that it is in > Integral[f(x) dx] - Integral[f(u)/u^2 du] (#1) Since x is just a placeholder, I replace the x's with u's in the first > integral and get 2I = Integral[(1 - 1/u^2)f(u) du] It would be true if the integrals would be definites. But not if they are indefinites. The first integral in #1 is a function of x, while the second is a function of u, and x =/= u, actually is x = 1/u. If you replace x by u in the first integral, you must to replace dx by -du/u^2, and you get 2I = -2*Integral[f(u)/u^2 du] true, but useless ... -- Ignacio Larrosa Ca.96estro A Coru.96a (Espa.96a) ilarrosaQUITARMAYUSCULAS@mundo-r.com ==== [snip] > So anyway, first we use obvservation 1 to say that if I = > Integral[f(x) dx], then 2I = Integral[f(x) dx] + Integral[f(1/x) dx] In the second integral, let u = 1/x, and it transforms to Integral[f(x) dx] - Integral[f(u)/u^2 du] Since x is just a placeholder, I replace the x's with u's in the first > integral and get 2I = Integral[(1 - 1/u^2)f(u) du] [snip] This looks wrong. x is in fact not just a placeholder, because you are doing an indefinite integral. Try writing the integral with explicit limits, say from 1 to X. However, is this step really necessary? You should be able to get by without it. -Michael. ==== > > So, the problem is to find in closed form > > Integral[f(x) dx], where f(x) = x/Sqrt[x^4+4x^3-6x^2+4x+1] MR1798560 (2002b:11093) van der Poorten, Alfred J.(5-MCQR-NT); Tran, Xuan Chuong(5-MCQR-NT) Quasi-elliptic integrals and periodic continued fractions. (English. English summary) Monatsh. Math. 131 (2000), no. 2, 155--169. 11J70 (11Y65) Summary: In this report we detail the following story. Several centuries ago, Abel noticed that the well-known elementary integral $$intfrac{dx}{sqrt{x^2+2bx+c}}=logleft(x+b+sqrt{x^2+2bx+c}righ t)$$ is just a presage of more surprising integrals of the form $$intfrac{f(x)dx}{sqrt{D(x)}}=logleft(p(x)+q(x)sqrt{D(x)}right ).$$ Here $f$ is a polynomial of degree $g$ and the $D$ are certain polynomials of degree $deg D(x)=2g+2$. Specifically, $f(x)=p'(x)/q(x)$ (so $q$ divides $p'$). Note that, morally, one expects such integrals to produce inverse elliptic functions and worse, rather than an innocent logarithm of an algebraic function. Abel went on to study abelian integrals, and it was Chebyshev who explained---using continued fractions---what is going on with these `quasi-elliptic' integrals. Recently, the second author computed all the polynomials $D$ over the rationals of degree 4 that have an $f$ as above. We explain various contexts in which the present issues arise. These contexts include symbolic integration of algebraic functions, the study of units in function fields and, given a suitable polynomial $g$, the consideration of the period length of the continued fraction expansion of the numbers $sqrt{g(n)}$ as $n$ varies over the integers. But the major content of this survey is an introduction to period continued fractions in hyperelliptic---thus quadratic---function fields. Reviewed by M. Mend?s France -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) ==== > About 2 months ago I ran across a problem on MIT's website to find the > indefinite integral of a certain function. It was from some problem > solving seminar, and I think it was rated as the hardest problem of > all the problems during the semester. I posted about a while back > about this integral but everybody just said that since Mathematica > can't do it it's probably not integrable in terms of elementary > functions. Well, after about two months of toiling over this problem, > I think I might finally have a solution. However, there's a few > places where, at least to me, things were a little iffy about whether > or not I could do what I was doing. So I want to put my solution and > see if there's any flaws. > > So, the problem is to find in closed form > > Integral[f(x) dx], where f(x) = x/Sqrt[x^4+4x^3-6x^2+4x+1] > > So, first note two things: > > 1) f(x) = f(1/x) (assuming of course that x is not 0) > 2) the polynomial in the bottom is a reciprocal polynomial > > Reciprocal polynomials generally can be solved by making a > substitution of the form t = x+1/x. So I use this to my advantage > later. > > So anyway, first we use obvservation 1 to say that if I = > Integral[f(x) dx], then 2I = Integral[f(x) dx] + Integral[f(1/x) dx] > > In the second integral, let u = 1/x, and it transforms to > > Integral[f(x) dx] - Integral[f(u)/u^2 du] > > Since x is just a placeholder, I replace the x's with u's in the first > integral and get > > 2I = Integral[(1 - 1/u^2)f(u) du] > > Now, we use the second observation. We already have > > 2I = Integral[(1 - 1/u^2)*(u/Sqrt[u^4+4u^3-6u^2+4u+1]) du] > > Bringing that u from the numerator into the denominator, we get > > 2I = Integral[(1 - 1/u^2)*(1/Sqrt[u^2+4u-6+4/u+1/u^2]) du] > > Now we are set up to write the part under the square root as a > poloynomial in x+1/x. Indeed, we get > > 2I = Integral[(1 - 1/u^2)*(1/Sqrt[(u+1/u)^2+4(u+1/u)-8]) du] > > Let v = u+1/u. Then dv = (1-1/u^2) dv, which is perfect since there's > a (1-1/u^2) sitting right there! So the integral transforms to: > > 2I = Integral[1/Sqrt[v^2 + 4v - 8] dv] > > Now this integral is easily evaluated as > > 2I = Log[2 + v + Sqrt[v^2 + 4v - 8]] > > Substituting v gives > > 2I = Log[2 + u + 1/u + Sqrt[u^4 + 4u^3 - 6u^2 + 4u + 1]] > > And substituting x gives > > I = 0.5*Log[2 + x + 1/x + (1/x^2)Sqrt[x^4 + 4x^3 - 6x^2 + 4x + 1]] > > So, is this right? > Well, I definitely made one mistake in that substituting v should not give you what I put. I think the final result should be I = 0.5*Log[(x+1)^2 + Sqrt[(x+1)^4-12x^2]] - 0.5*Log[x] If someone could differentiate this using Mathematica and simplify it I would appreciate it. ==== > About 2 months ago I ran across a problem on MIT's website to find the > indefinite integral of a certain function. It was from some problem > solving seminar, and I think it was rated as the hardest problem of > all the problems during the semester. I posted about a while back > about this integral but everybody just said that since Mathematica > can't do it it's probably not integrable in terms of elementary > functions. Well, after about two months of toiling over this problem, > I think I might finally have a solution. However, there's a few > places where, at least to me, things were a little iffy about whether > or not I could do what I was doing. So I want to put my solution and > see if there's any flaws. > > So, the problem is to find in closed form > > Integral[f(x) dx], where f(x) = x/Sqrt[x^4+4x^3-6x^2+4x+1] > > So, first note two things: > > 1) f(x) = f(1/x) (assuming of course that x is not 0) > 2) the polynomial in the bottom is a reciprocal polynomial > > Reciprocal polynomials generally can be solved by making a > substitution of the form t = x+1/x. So I use this to my advantage > later. > > So anyway, first we use obvservation 1 to say that if I = > Integral[f(x) dx], then 2I = Integral[f(x) dx] + Integral[f(1/x) dx] > > In the second integral, let u = 1/x, and it transforms to > > Integral[f(x) dx] - Integral[f(u)/u^2 du] > > Since x is just a placeholder, I replace the x's with u's in the first > integral and get > > 2I = Integral[(1 - 1/u^2)f(u) du] > > Now, we use the second observation. We already have > > 2I = Integral[(1 - 1/u^2)*(u/Sqrt[u^4+4u^3-6u^2+4u+1]) du] > > Bringing that u from the numerator into the denominator, we get > > 2I = Integral[(1 - 1/u^2)*(1/Sqrt[u^2+4u-6+4/u+1/u^2]) du] > > Now we are set up to write the part under the square root as a > poloynomial in x+1/x. Indeed, we get > > 2I = Integral[(1 - 1/u^2)*(1/Sqrt[(u+1/u)^2+4(u+1/u)-8]) du] > > Let v = u+1/u. Then dv = (1-1/u^2) dv, which is perfect since there's > a (1-1/u^2) sitting right there! So the integral transforms to: > > 2I = Integral[1/Sqrt[v^2 + 4v - 8] dv] > > Now this integral is easily evaluated as > > 2I = Log[2 + v + Sqrt[v^2 + 4v - 8]] > > Substituting v gives > > 2I = Log[2 + u + 1/u + Sqrt[u^4 + 4u^3 - 6u^2 + 4u + 1]] > > And substituting x gives > > I = 0.5*Log[2 + x + 1/x + (1/x^2)Sqrt[x^4 + 4x^3 - 6x^2 + 4x + 1]] > > So, is this right? > > > Well, I definitely made one mistake in that substituting v should not > give you what I put. I think the final result should be > > I = 0.5*Log[(x+1)^2 + Sqrt[(x+1)^4-12x^2]] - 0.5*Log[x] > > If someone could differentiate this using Mathematica and simplify it > I would appreciate it. The derivative is x^2 - 1 ---------------------------- 2 x sqrt(x^4+4x^3-6x^2+4x+1) --Ron Bruck ==== Try this: x^4+4x^3-6x^2+4x+1=(x-a)*(x-b)*(x-c)*(x-d), where a=-1+1/2*sqrt(12)+sqrt(3-2*sqrt(3)) b=-1+1/2*sqrt(12)-sqrt(3-2*sqrt(3)) c=-1-1/2*sqrt(12)+sqrt(3+2*sqrt(3)) d=-1-1/2*sqrt(12)-sqrt(3+2*sqrt(3)) Thus x/(x^4+4x^3-6x^2+4x+1)=x/((x-a)*(x-b)*(x-c)*(x-d)) Try expressing this as a sum of integrable polynomial fractions in x, then integrate. > About 2 months ago I ran across a problem on MIT's website to find the > indefinite integral of a certain function. It was from some problem > solving seminar, and I think it was rated as the hardest problem of > all the problems during the semester. I posted about a while back > about this integral but everybody just said that since Mathematica > can't do it it's probably not integrable in terms of elementary > functions. Well, after about two months of toiling over this problem, > I think I might finally have a solution. However, there's a few > places where, at least to me, things were a little iffy about whether > or not I could do what I was doing. So I want to put my solution and > see if there's any flaws. So, the problem is to find in closed form Integral[f(x) dx], where f(x) = x/Sqrt[x^4+4x^3-6x^2+4x+1] So, first note two things: 1) f(x) = f(1/x) (assuming of course that x is not 0) > 2) the polynomial in the bottom is a reciprocal polynomial Reciprocal polynomials generally can be solved by making a > substitution of the form t = x+1/x. So I use this to my advantage > later. So anyway, first we use obvservation 1 to say that if I = > Integral[f(x) dx], then 2I = Integral[f(x) dx] + Integral[f(1/x) dx] In the second integral, let u = 1/x, and it transforms to Integral[f(x) dx] - Integral[f(u)/u^2 du] Since x is just a placeholder, I replace the x's with u's in the first > integral and get 2I = Integral[(1 - 1/u^2)f(u) du] Now, we use the second observation. We already have 2I = Integral[(1 - 1/u^2)*(u/Sqrt[u^4+4u^3-6u^2+4u+1]) du] Bringing that u from the numerator into the denominator, we get 2I = Integral[(1 - 1/u^2)*(1/Sqrt[u^2+4u-6+4/u+1/u^2]) du] Now we are set up to write the part under the square root as a > poloynomial in x+1/x. Indeed, we get 2I = Integral[(1 - 1/u^2)*(1/Sqrt[(u+1/u)^2+4(u+1/u)-8]) du] Let v = u+1/u. Then dv = (1-1/u^2) dv, which is perfect since there's > a (1-1/u^2) sitting right there! So the integral transforms to: 2I = Integral[1/Sqrt[v^2 + 4v - 8] dv] Now this integral is easily evaluated as 2I = Log[2 + v + Sqrt[v^2 + 4v - 8]] Substituting v gives 2I = Log[2 + u + 1/u + Sqrt[u^4 + 4u^3 - 6u^2 + 4u + 1]] And substituting x gives I = 0.5*Log[2 + x + 1/x + (1/x^2)Sqrt[x^4 + 4x^3 - 6x^2 + 4x + 1]] So, is this right? > ==== > Try this: > x^4+4x^3-6x^2+4x+1=(x-a)*(x-b)*(x-c)*(x-d), > where > a=-1+1/2*sqrt(12)+sqrt(3-2*sqrt(3)) > b=-1+1/2*sqrt(12)-sqrt(3-2*sqrt(3)) > c=-1-1/2*sqrt(12)+sqrt(3+2*sqrt(3)) > d=-1-1/2*sqrt(12)-sqrt(3+2*sqrt(3)) > Thus > x/(x^4+4x^3-6x^2+4x+1)=x/((x-a)*(x-b)*(x-c)*(x-d)) > Try expressing this as a sum of integrable polynomial fractions in x, then > integrate. I wish it were so simple. Unfortunately, you need to take the square root of that quartic polynomial on the bottom. ==== >Try this: >x^4+4x^3-6x^2+4x+1=(x-a)*(x-b)*(x-c)*(x-d), >where >a=-1+1/2*sqrt(12)+sqrt(3-2*sqrt(3)) >b=-1+1/2*sqrt(12)-sqrt(3-2*sqrt(3)) >c=-1-1/2*sqrt(12)+sqrt(3+2*sqrt(3)) >d=-1-1/2*sqrt(12)-sqrt(3+2*sqrt(3)) >Thus >x/(x^4+4x^3-6x^2+4x+1)=x/((x-a)*(x-b)*(x-c)*(x-d)) >Try expressing this as a sum of integrable polynomial fractions in x, then >integrate. ... except that it's x/sqrt(x^4+4x^3-6x^2+4x+1) that he wants to integrate. According to Maple, > int(x/sqrt((x-a)*(x-b)*(x-c)*(x-d)),x); /(a - c) (x - d)1/2 2 /(c - d) (x - b)1/2 2 (d - a) |---------------| (x - c) |---------------| (a - d) (x - c)/ (b - d) (x - c)/ /(c - d) (x - a)1/2 / |---------------| | (a - d) (x - c)/ /(a - c) (x - d)1/2 /(c - b) (d - a)1/2 c EllipticF(|---------------| , |---------------| ) + (a - d) (x - c)/ (d - b) (c - a)/ (d - c) /(a - c) (x - d)1/2 a - d /(c - b) (d - a)1/2 EllipticPi(|---------------| , -----, |---------------| ) (a - d) (x - c)/ a - c (d - b) (c - a)/ / 1/2 | / ((a - c) (c - d) ((x - a) (x - b) (x - c) (x - d)) ) / / Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 ==== > About 2 months ago I ran across a problem on MIT's website to find the > indefinite integral of a certain function. It was from some problem > solving seminar, and I think it was rated as the hardest problem of > all the problems during the semester. I posted about a while back > about this integral but everybody just said that since Mathematica > can't do it it's probably not integrable in terms of elementary > functions. Mathematica 5 gives an answer -- terribly messy -- in closed form in terms of elliptic integrals. I have no idea if that answer is correct or not. > Well, after about two months of toiling over this problem, > I think I might finally have a solution. However, there's a few > places where, at least to me, things were a little iffy about whether > or not I could do what I was doing. So I want to put my solution and > see if there's any flaws. So, the problem is to find in closed form Integral[f(x) dx], where f(x) = x/Sqrt[x^4+4x^3-6x^2+4x+1] [snip] > I = 0.5*Log[2 + x + 1/x + (1/x^2)Sqrt[x^4 + 4x^3 - 6x^2 + 4x + 1]] So, is this right? Have you forgotten how easy such a thing is to check??? Just differentiate what you think to be the antiderivative, and see if that result is equal to the integrand! Unfortunately, it seems that you did something wrong... David ==== > About 2 months ago I ran across a problem on MIT's website to find the > indefinite integral of a certain function. It was from some problem > solving seminar, and I think it was rated as the hardest problem of > all the problems during the semester. I posted about a while back > about this integral but everybody just said that since Mathematica > can't do it it's probably not integrable in terms of elementary > functions. > > Mathematica 5 gives an answer -- terribly messy -- in closed form in terms > of elliptic integrals. I have no idea if that answer is correct or not. > > Well, after about two months of toiling over this problem, > I think I might finally have a solution. However, there's a few > places where, at least to me, things were a little iffy about whether > or not I could do what I was doing. So I want to put my solution and > see if there's any flaws. So, the problem is to find in closed form Integral[f(x) dx], where f(x) = x/Sqrt[x^4+4x^3-6x^2+4x+1] > [snip] > I = 0.5*Log[2 + x + 1/x + (1/x^2)Sqrt[x^4 + 4x^3 - 6x^2 + 4x + 1]] So, is this right? > > Have you forgotten how easy such a thing is to check??? Just differentiate > what you think to be the antiderivative, and see if that result is equal > to the integrand! > > Unfortunately, it seems that you did something wrong... > > David I don't have Mathematica, and it's a painful function to differentiate and simplify. Of course I could have done it but there's alot of room for error, and if I make a mistake differentiating I'm likely to think that my answer is wrong, when in fact it's not. However, you said it is wrong, so I assume you differentiated the result using Mathematica and didn't get the same thing. Unfortunate indeed. Is it possible someone can point me to the error? ==== > Have you forgotten how easy such a thing is to check??? Just differentiate > what you think to be the antiderivative, and see if that result is equal > to the integrand! > > Unfortunately, it seems that you did something wrong... > > David > > > I don't have Mathematica, and it's a painful function to differentiate > and simplify. Of course I could have done it but there's alot of room > for error, and if I make a mistake differentiating I'm likely to think > that my answer is wrong, when in fact it's not. However, you said it > is wrong, so I assume you differentiated the result using Mathematica > and didn't get the same thing. Unfortunate indeed. BTW a simple calculator is a great way to check complicated algebra. Just pick a random value for X and compute the numerical result at each stage. You can quite quickly track down which step contained the error. Numerical differentiation also works quite effectively if you have sufficient precision on the calculator and use common sense in selecting delta-X. ==== theorems. Not being able to figure out that proof was really bugging me, I'll sleep better now. > If F is given consider the sequence of functions (defined on |z| = 1) >F[r](z) = F(rz), where r is a monotone sequence of reals approaching 1, say >r = 1/(1+n) with n a natural number. Since F is continuous F[r](z) uniformly >approaches F(z) as r approaches 1. [This is the point I was missing before - >it holds because the unit circle is a closed set.] > > Or more precisely: This works because the closed _disk_ is a _compact_ > set, which implies that f is uniformly continuous. > >Let E/2 be the worst case >error value of |F(z) - F[r](z)|. Since rz lies within the radius of >convergence of F, F converges uniformly there, and we can take a partial sum >P[r](z) that is uniformly within E/2 of F[r](z). This partial sum is a >polynomial in (z) as well as in (rz). For all z on the unit circle |F(z) - >P[r](z)| <= |F(z) - F[r](z)| + |F[r](z) - P[r](z)| < E which goes to zero as >r approaches 1. So the P[r](z) are the desired polynomials in z. [My only >nagging doubt here is that this argument would seem to apply to an F defined >on any circle, not just the unit circle.] > > Well you can relax, it's true for any closed disk. (In fact it's true > in much greater generality, although it's not so easy to prove; > Runge's theorem, included in most complex books, gives a > generalization, and Mergelyan's theorem gives the result under > weaker hypotheses yet.) > >If the sequence of polynomials is given they converge and therefore form a >Cauchy sequence. The difference between any two of them is an analytic >function and therefore reaches its maxium modulus on the boundary of the >unit disk. So the polynomials form a uniform Cauchy sequence on the closed >unit disk and therefore converge uniformly. [Interestingly none of the math >books I had included Cauchy's criterion for uniform convergence, just >Cauchy's criterion for convergence - too obvious an extension to state >explicitly no doubt. But I did find explicit mention of it on the web.] -The polynomials converge uniformly on the closed unit disk and are >continuous, therefore they converge to a continuous function there. >-The polynomials converge uniformly on the open unit disk and are analytic, >therefore they converge to an analytic function there. Ian >Been staring at this for hours and can't seem to find a proof that >>seems needlessly complex or just plain doesn't work... can someone see >>a simple argument? I would be much appreciated - I don't even need >>this result for anything but somehow I just can't let it go. >>Let f be a continuous function on the unit circle T = {|z| = 1}. Show >>that f can be approximated uniformly on T by a sequence of polynomials >>in z if and only if f has an extension F that is continuous on the >>To approximate such an F, consider dilates F[subscripted r](z) = >>F(rz). >>I can't get either direction to work... if F is given then the hint >>seems to imply that you can obtain the polynomial sequence from the >>power series expansion of F, but how to show uniform convergence in a >>simple way? >> The power series need not converge to F at points of the boundary. >> But that's not what the hint suggests - read the hint again... >>If the polynomial sequence is given then it's easy to use the Cauchy >>integral formula on the polynomials, take the limit, and get F >>analytic (limit of the integral = integral of the limit because of the >>uniform convergence, and analyticity follows from the continuity of >>the limit funciton f). But how to show that F is continuous at points >>on the unit circle? >> Don't use the Cauchy formula. Use the Maximum Modulus Theorem, >> and look up uniform convergence and Cauchy sequence in a book >> on adbanced calculus. >>I found this in Complex Analysis by Gamelin, section V.4, problem 14. >> David C. Ullrich > > David C. Ullrich ==== Two thoughts on why e is good, and a limerick: (1) If we could choose an irrational number as our base (ie instead of the usual 10 or 2 for binary), then e would be the most efficient. Why? An n-symbol string in base m can represent m^n different numbers, and to do so we use n*m different symbols (m letters, but each letter can be n different positions). So if we write the size of the represented space (m^n) as a function of the number of different symbols k=m*n, we get f(k) = m^(k/m) = (m^(1/m))^k. I like to think of the number (m^(1/m)) as capturing the efficiency of the standard base-m type encoding. In the integers, 3 maximizes this as 1.4422... But if we use some calculus, the global maximum is actually at e, with a value of 1.4446... In other words, our efficiency is maximized at e. This argument is my own, but I'm sure it has been done better elsewhere, because everything I think I have that's even slightly original turns out that way... (2) One of the most beautiful theorems of modern mathematics: the central limit theorem. Usually people think of it as only applying to probability, since it is easiest to interpret in that realm. But it view, you could declare the presence of e in the statement of CLT to be a consequence of Fourier analysis (and sort of from Euler's formula e^(ix) = cos(x) + i sin(x)), but this is a bit of an over- simplification. limit theorem; so the best elementary idea of it that I can give is that there is a single function, something like e^(-x^2/2) (times a constant), which represents the limiting behavior of virtually _any_ function which is randomly sampled and then averaged. I know that's a bad description, but it's a start. Finally, the limerick (written by me :) My favorite number is e it compounds continuously; if you raise it to pi multiplied by i, negative one it will be. -Tyler ==== Part II Note this is all constructive criticism of Hal's ideas on metric engineering. Excerpts from Puthoff & IbisonÕs H.E. PUTHOFF*, S.R. LITTLE AND M. IBISON Institute for Advanced Studies at Austin, 4030 West Braker Lane, Suite 300, Austin, Texas 78759-5329, USA. with my comments. 3. The Quantum Vacuum 3.1 Zero-Point Energy (ZPE) Background Quantum theory tells us that so-called empty space is not truly empty, but is the seat of myriad energetic quantum processes. Specifically, quantum field theory tells us that, even in empty space, fields (e.g., the electromagnetic field) continuously fluctuate about their zero baseline values. So far, so good. However IMHO it is the QED PV virtual electron-positron zero point vacuum fluctuations that dominate the low energy effective macro-quantum field theory out of which Einstein's gravity together with exotic vacuum unified dark energy/matter fields co-emerge like love and marriage all together as phase and amplitude modulation patterns, respectively, of the vacuum coherence field that is formed by the attractive BCS exchange of virtual photons between the virtual electron-positron pairs all inside the vacuum. Note that Hal makes no reference at all to the new discovery in precision cosmology of dark energy as being relevant to his Quixotic Ahabian quest in search of for the vacuum wind. :-) The energy associated with these fluctuations is called zero point energy (ZPE), reflecting the fact that such activity remains even at a temperature of absolute zero. Such a concept is almost certain to have profound implications for future space travel, as we will now discuss. Agreed. When a hypothetical ZPE-powered spaceship strains against gravity and inertia ... That's a strange way of putting it. The local net random micro-quantu zero point stress-energy density tensor field is Tuv(Exotic Vacuum) = (String Tension)/ZPEguv using Ed Witten's natural units h = c = k(Boltzmann) = 1 Tuv(Exotic Vacuum) = (String Tension)^2[(String Tension)^3/2|Vacuum Coherence|^2 -1]guv What is wrong with Puthoff's paradigm is that he has no idea of Vacuum Coherence in his informal thinking nor formalized in his mathematics. The zero point energy density is Too and I use the 3 GR sign conventions in which Too(Exotic Vacuum) > 0 means strongly anti-gravity negative pressure dark energy since the general equation of state (pressure) = w(energy density) reduces to w = -1 for ALL zero point quantum fields of all spins. Too(Exotic Vacuum) < 0 means strongly gravitating positive pressure dark matter. Therefore dark energy needs (String Tension)^3/2|Vacuum Coherence|^2 > 1 Dark matter needs (String Tension)^3/2|Vacuum Coherence|^2 < 1 Where the critical Vacuum Coherence corresponding to zero Einstein Cosmological Constant of the non-exotic vacuum in the large-scale FRW metric is (String Tension)^3/2|Vacuum Coherence|^2 = 1 Or |Vacuum Coherence|*^2 = (String Tension)^-3/2 The Einstein exotic vacuum local geometrodynamic field equation is then Tuv(Induced Time and Space Warps) + Tuv(Exotic Vacuum) ~ 0 The total covariant 4-divergence vanishes conserving total local stress-energy Einstein current density. Unlike 1915 geometrodynamics, the Bianchi identities break down so that the two pieces of the divergence are not separately zero! Only the sum is zero. This is necessary for practical soft metric engineering of the Tech Gnostic Underground Stream inside the vacuum. That is, Tuv(Induced Space and Time Warps)^;v + Tuv(Exotic Vacuum)^;v = 0 This is the first small step for Mankind leading to The Right Stuff to Make Star Trek Real. See my lecture in Time Travel: The Art of the Possible Disk 2 of Paramount Pictures DVD Special Collector's Edition of Star Trek IV: The Voyage Home indeed! ;-) *Before you read on click on this! Turn the volume up on your sound. Sarfatti pitches to Puthoff at bat in the World Series :-) http://www.niehs.nih.gov/kids/lyrics/ballgame.htm there are three elements of the equation that the ZPE technology could in principle address: (1) a decoupling from gravity, NO! We want just the opposite! Big error of strategic thinking! The Question is: What is The Question? Here Hal & Co ask a wrong question IMHO. STRIKE ONE! (2) a reduction of inertia NO! Ditto. That's like preventing your car from being stolen by blowing it up if it is broken into! STRIKE TWO! or (3) the generation of energy to overcome both. NO, NO, NO a zillion times NO! ;-) That's the Brute Force approach. That is not The Tao Chi of ET! STRIKE THREE! You're Out! Take me out to the ball game .... 3.2 Gravity With regard to a ZPE basis for gravity, the Russian physicist Andrei Sakharov was the first to propose that in a certain sense gravitation is not a fundamental interaction at all, but rather an induced effect brought about by changes in the quantum-fluctuation energy of the vacuum when matter is present Yes, this is essentially correct and is precisely what my theory is all about. The details are in http://qedcorp.com/APS/EmergentGravity.doc or http://qedcorp.com/APS/EmergentGravity.doc (smaller file) to be continued. Engineering the Zero-Point Field and Polarizable Vacuum for Interstellar Flight Ò1. Introduction The concept of Òengineering the vacuumÓ found its first expression in the mainstream physics literature when it was introduced by T. D. Lee in There he stated: ÔThe experimental method to alter the properties of the vacuum may be called vacuum engineering.... If indeed we are able to alter the vacuum, then we may encounter some new phenomena, totally unexpected.Õ This legitimization of the vacuum engineering concept was based on the recognition that the vacuum is characterized by parameters and structure that leave no doubt that it constitutes an energetic medium in its own right. Foremost among these are its properties that (1) within the and field fluctuations, and (2) within the context of general relativity the vacuum is the seat of a spacetime structure (metric) that encodes the distribution of matter and energy. Indeed, on the flyleaf of a book of essays by Einstein and others on the properties of the vacuum we find the statement ÒThe vacuum is fast emerging as the central structure of modern physicsÓ [3]. Given the known characteristics of the vacuum, one might reasonably inquire as to why it is not immediately obvious how to catalyze robust interactions of the type sought for space-flight applications. To begin, in the case of quantum fluctuations there are uncertainties that remain to be clarified regarding global thermodynamic and energy constraints. Furthermore, the energetic components of potential utility involve very small-wavelength, high-frequency fields and thus resist facile engineering solutions.Ó This last remark may not be correct in general, although it is correct within the framework Puthoff is pursuing i.e. using the stress-energy density of the electromagnetic field to attempt to directly warp space-time geometry with practical utility. Such a brute-force approach is hopeless and a waste of effort IMHO because EinsteinÕs equation in this case is simply Induced Space-Time Warp = (Applied EM Stress-Energy Density)/(G-String Tension) The G-String Tension is 10^19Gev per 10^-33 cm, which is simply too stiff! One would need a way to lower the G-String-Tension, for example an ÒM TheoryÓ that yields something like: G*-String Tension = e^-(r*/Lp)^?(G-String Tension) Where r* = compactification scale of extra-dimensions of hyperspace beyond 4-D space-time and Lp^2 = hG/c^3 = 1/(G-String Tension) for h = c = 1 Puthoff alludes to this barrier in his next remark: ÒWith regard to perturbation of the space-time metric, the required energy densities exceed by many orders of magnitude values achievable with existing engineering techniques. Nonetheless, we can examine the constraints, possibilities and implications under the expectation that as technology matures, felicitous means may be found that permit the exploitation of the enormous, as-yet-untapped potential of so-called Òempty spaceÓ. This is the problem that Hal goes into denial and wishing about. There is no way that will brute-force approach will work in any practical way. The UFOs do not use high-energy EM field densities and Hal is primarily interested in the UFOs. That should tell him right away that he is pursuing the wrong path to solve the problem. Ò2. Propellantless Propulsion 2.1 Global Constraint Regardless of the mechanisms that might be entertained with regard to ÒpropellantlessÓ or ÒfieldÓ propulsion of a spaceship, there exist certain constraints that can be easily overlooked but must be taken into consideration. A central one is that, because of the law of conservation of momentum, the center of mass-energy (CM) of an initially stationary isolated system cannot change its position if not acted upon by outside forces. This means that propellantless or field propulsion, whatever form it takes, is constrained to involve coupling to the external universe in such a way that the displacement Engineering the Zero-Point Field and Polarizable Vacuum for Interstellar FlightÓ It has been known since the late 1950Õs how to circumvent this problem provided one has Òexotic stuffÓ with a strong enough Ònegative pressureÓ to provide what Herman Bondi called Ònegative matterÓ in the late 1950Õs and Kip Thorne called Òexotic matterÓ in the late 1980Õs in his Òtraversable wormhole time travelÓ papers with additional work by Igor Novikov and others. Robert Forward summarized BondiÕs early work in a paper written in the early 1990Õs. More details are to be found in We now know since 1999 from Type 1a supernova data showing the acceleration of the expansion speed of the Universe that approximately 3/4 of all the large-scale stuff of the Universe is exotic vacuum anti-gravitating Òdark energyÓ with a strong negative zero point pressure that is equal and opposite to the positive zero point energy density. This fact is the key to practical metric engineering of the vacuum that Puthoff et-al completely misses. The modified vacuum Einstein equation is Tuv(Geometry) + Tuv(Exotic Vacuum) = 0 Where Tuv is the local stress-energy density tensor under the Einstein DIFF(4) symmetry group. Tuv(Geometry) = (String Tension)Guv(Geometry) Where Guv(Geometry) = Ruv [CapitalEth] (1/2)Rguv And Tuv(Exotic Vacuum) = (String Tension)/zpfguv /zpf = (String Tension)^-1[(String Tension)^3/2|Vacuum Coherence|^2 [CapitalEth] 1] /zpf > 0 is negative pressure Òdark energyÓ exotic vacuum that is ~ 3/4 of large-scale stuff of the Universe. /zpf < 0 is positive pressure Òdark matterÓ exotic vacuum that is ~ 1/4 of the large scale stuff of the Universe. Gravity effect of exotic vacuum ÒstuffÓ ~ c^2/zpf Where + sign on RHS is universally repelling anti-gravity and a [CapitalEth] sign is universally attracting gravity with effective short scale coupling strengths much larger than NewtonÕs. The key idea not found in any of PuthoffÕs papers on the subject of metric engineering is the local Einstein stress-energy tensor current densityÕs covariant divergence Tuv^;v where ;v is the Diff(4) covariant derivative relative to the metric torsion free connection field for parallel transport of tensors along tangent vector fields in curved space-time. When there is no exotic vacuum, i.e., /zpf = 0 corresponding to optimal vacuum coherence, the Bianchi identities work and we have closed current conservation of the pure geometrodynamic stress-energy tensor local currents, i.e. Tuv(Geometry)^;v = 0 This prevents any practical metric engineering! In contrast, when there is exotic vacuum then Tuv(Geometry)^;v =/= 0 Instead Tuv(Geometry)^;v + Tuv(Exotic Vacuum)^;v = 0 Where the common string-tension factor cancels out of this current conservation equation. This essentially solves the problem for practical metric engineering bypassing the string tension problem completely! One uses the Bohm-Aharonov-Josephson effect to tweak Tuv(Exotic Vacuum) in a ÒsoftÓ way. To be continued. The Physical Principles of Metric Engineering by Jack Sarfatti 5th draft excerpt from the book Super Cosmos The term metric engineering was coined by Dr. Harold Puthoff. Hal, as he prefers to be called, was a US Naval Officer who then worked for the National Security Agency before going to the Stanford Research Institute where he conducted the famous Remote Viewing experiments with Russell Targ testing Uri Geller, Ingo Swann, Pat Price and other psychics in a project paid for by the CIA and Department of Defense Intelligence Agencies. I first met Hal and Russell at SRI in 1973 and that story is told in my book Destiny Matrix. Hal has held very high USG security clearances and it is well known that he is obsessed with the UFO mystery. Hal, like his co-worker Bernie Haisch, who is an editor of the Astrophysical Journal, both strongly believe in the physical reality of flying saucers. They, with Jacques Vallee, have been active in the NASA Breakthrough Propulsion Program and in UFO groups some financed by Laurance Rockefeller and the Howard Hughes clone Las Vegas Hotel Tycoon Robert Bigelow who owns Bigelow Aerospace Corporation. Since they take the reality of the flying saucers seriously so do I. The two lines of theoretical physics research that Hal has pursued for several decades now, zero point energy and a polarized vacuum model of gravity, is primarily motivated by the quest to understand how the saucers fly. The alleged reality of such advanced alien technology is clearly of immense importance to US National Security and beyond. The recent developments in physics shown in NOVA's Elegant Universe with Brian Greene, in Stephen Hawking's The Universe in a Nutshell, Michio Kaku's Hyperspace and Igor Novikov's The River of Time makes time traveling alien interference in our history more probable not less probable. This is all an aspect of metric engineering defined as the practical control of space and time warps. I have discussed this in Paramount Pictures DVD Special Collector's Edition of Star Trek IV: The Voyage Home in Time Travel: The Art of the Possible and in Learning Channel's Ultra-Science: Time Travel. The ultimate form of metric engineering is seen in Q in Star Trek where a Super Mind is able to warp space-time. We also need a physics of consciousness to see if that fiction can be realized in fact. British Astronomer Royal, Sir Martin Rees who runs the laboratory where Stephen Hawking works and who is the new Master of Trinity College, Cambridge has added the dimension of Doomsday WMD to the quest for a metric engineering breakthrough in his Chapter 9 of his pessimistic Our Final Hour on the Ice Nine rip-in-space propagating at the speed of light that could literally destroy our universe. We are now at that turning point in our history where, like Mickey Mouse in Walt Disney's Fantasia we could, in our incompetence, destroy the entire Universe just as we are now surely destroying Earth's Biosphere. Therefore, any aliens out there with Star Gate Time Travel and Warp Drive Super Technology are already here in order to stop us from killing them as well as ourselves. The American Christian Fundamentalist Right has a strong belief in Apocalypse Now and there is real danger that this is a self-fulfilling prophecy in a precognitive remote viewing of our future by Saint John in The Bible's Revelations. All the more reason for me, like Paul Revere, to alert the public to these new developments in physics because only real knowledge can possibly save us and even then there is no guarantee. What does mainstream physics tell us about the possibility of metric engineering? The relevant parts of mainstream physics for metric engineering are Einstein's general theory of relativity and micro-quantum mechanics including the More is Different emergence of macro-quantum superfluid order in ground state and vacuum instabilities developed by condensed matter Princeton physicist P.W. in the inflationary chaotic cosmology of A. Linde consistent with the issue of Scientific American. Let's start with Einstein's gravity field equation of ~ 1915. Einstein viewed pure space-time geometry like the marble in a statue by Michelangelo with gross matter and radiation as wood. Einstein's equation, in the 21st Century language, is Space-Time Curvature = (Stress-Energy Density)/(String Tension) Curvature has the physical dimensions of 1/Area, String Tension has Energy/Length and Stress-Energy Density has Energy/Volume. Simple algebra confirms that this way of writing Einstein's equation is correct dimensionally. The String Tension is the basic parameter that Ed Witten of the Princeton Institute of Advanced Study uses in his discussions of M theory that unifies the five limiting cases of super string theory in which the elementary leptons, quarks and gauge force bosons are vibrating strings of pure energy pulsating in the extra space dimensions of Calabi-Yau hyperspace. The ordinary matter and radiation on the right hand side of Einstein's equation are open strings whose ends are stuck to 3Dim brane worlds of which our universe is one. You can picture a string as a tiny wormhole whose two ends need not be on the same brane world because only then can you not have equal numbers Another equivalent way to look at Einstein's 1915 equation from our Cosmic Justice peeking through the blindfold. That is, the stress-energy density tensor of pure marble geometry is Stress-Energy Density Tensor of Pure Geometry = (String Tension)(Space-Time Curvature) (2) Einstein's equation is then like the static equilibrium balance of forces in architecture in which the sum of all the contributions to the stress-energy density tensor add up to a perfect zero! In this simplest of cases Stress-Energy Density Tensor of Geometry + Stress-Energy Density of Matter etc. = 0 (3) All physically real objects are either tensors, spinors or twistors in the theory of relativity. This is because the coordinate map is not the physical territory. The form of the laws of physics must be the same locally no matter how the coordinate map is morphed. To this we must add the Einstein Equivalence Principle or EEP which says that even in a curved space-time, we can use the special theory of relativity locally for a special class of LIF observers to describe what is happening to a good approximation. The EEP is only meant to apply in this approximate way and it will break down if one is falling into a black hole singularity or if the microscope magnification is so large that quantum gravity zero point energy density fluctuations in the space-time geometry itself get large. There may also be torsion fields not included in Einstein's 1915 geometrodynamics that may require a modification of the EEP. The spinor is a square root of a tensor and the Penrose twistor is a spinor in a complex space-time. The physics of point matrix space-time is in reality the physics of extended strings and membranes in real space-time. Hawking's imaginary time and its connection to inverse temperature is part of that same story. The key concept for metric engineering is the Einstein current! Einstein's field equation is always the statement that the sum of all the stress-energy density tensors when put on the same side of the equation all balance out to exactly zero. We then take a covariant divergence of the equation to get the Einstein currents, which are conserved when added together. The covariant divergence depends on something called a connection field, which tells us how to parallel transport tensors and spinors along paths or histories in space-time and beyond including the extra dimensions of hyperspace. Every time a gauge force field is added, like a torsion field, there is an additional piece added to this connection field. Einstein's 1915 theory is a degenerate case of the bigger unified field theory just like a circle is a degenerate case of an ellipse. An ellipse has two centers or foci. When the two foci merge together the ellipse degenerates into a circle. Einstein's 1915 theory in the form of the Bianchi identities forbids practical metric engineering because it has an impenetrable barrier completely separating the marble geometric current from the wood matter current. Both kinds of currents must be able to intermingle to transform into each other in order to have soft practical metric engineering of Star Gate Time Travel traversable wormholes and weightless superluminal warp drives using the anti-gravitating exotic vacuum dark energy with negative zero point exotic vacuum pressure that is equal in magnitude, but opposite in sign, to the zero point energy density. The new Type Ia super novae data showing that our brane world Universe is accelerating in its rate of expansion, i.e. speeding up rather than slowing down. Indeed approximately 3/4 of all the stuff of our Universe on the large scale is made out of dark energy. Therefore, the idea of ancient traversable Star Gate wormholes stabilized by dark energy is not so far fetched. The brute force approach to metric engineering taken by Hal Puthoff now for at least two decades can never work because space-time is too stiff to bend directly with electromagnetic field energy density to do anything worth doing because the string tension is too large. In contrast, we can used the quantum interference of the Einstein stress-energy density currents via the Bohm-Aharonov-Josephson effect for practical metric engineering provided that the pure geometric currents are not separately conserved like they are in Einstein's original theory. One way to do that is to bottle or harness the dark energy as General Douglas Mac Arthur precognitively remote viewed in his Duty, Honor, Country Farewell Speech to the Cadets at West Point in 1962 in a speech made even more remarkable for its reference to the coming war in space with extra-terrestrials. Michael Turner, a professor of physics at the impossible to bottle dark energy and if Einstein's 1915 theory is the final theory he is correct. However, the real UFO evidence that drives Hal Puthoff like Captain Ahab after The White Wale, Moby Dick, is the evidence that suggests Professor Turner will be proved wrong in that prediction. ref. http://qedcorp.com/APS/StarGate1.mov http://qedcorp.com/APS/EmergentGravity.pdf ==== I'm attempting to connect two 3D squares together. They are of 16 points each and I?ve run them through the Bernstein equation. Now, I?m quite new to 3 dimensional mathematics and new to the Bernstein formula, but its proving to be very interesting. I've been trying for days now to get the squares to match up smoothly. There is a curvature in the surface of the one square/patch and I?m attempting to keep the same curvature through the second square (to make half a flat oval). I've heard about the tangents method, about having tangents from the control points on the one square to the control points on the other square, which should create a smooth surface. And here in is the problem, there is a definite line, an inward slope as the two squares/patches meet. As far I can tell the tangents line up along the join, I just don't have a clue why the join isn't perfectly smooth. Many people must have run into this problem. If you have any thoughts, ideas or possible directions to pursue, they would all be very much appreciated. T. Overton ==== I'm attempting to connect two 3D squares together. They are of 16 points each and I?ve run them through the Bernstein equation. Now, I?m quite new to 3 dimensional mathematics and new to the Bernstein formula, but its proving to be very interesting. I've been trying for days now to get the squares to match up smoothly. There is a curvature in the surface of the one square/patch and I?m attempting to keep the same curvature through the second square (to make half a flat oval). I've heard about the tangents method, about having tangents from the control points on the one square to the control points on the other square, which should create a smooth surface. And here in is the problem, there is a definite line, an inward slope as the two squares/patches meet. As far I can tell the tangents line up along the join, I just don't have a clue why the join isn't perfectly smooth. Many people must have run into this problem. If you have any thoughts, ideas or possible directions to pursue, they would all be very much appreciated. T. Overton ==== I am guessing, if we take an (n_1 by n_2 by n_3 by ...n_m) m-dimensional box (where the n's are positive integers), and we let the number of lattice points of positive integer coordinates (k_1,k_2,k_3, ...k_m), where NO common primes divide EACH k, be q(n), then: limit{n_1->oo, n_2->oo, n_3->oo, ...n_m->oo} q(n)/(n_1 * n_2 * n_3 * ...n_m) = 1/zeta(m). IE; the fraction of all finite sequences of m 'uniformly randomly chosen' positive integers, which are so that no common prime divides every integer in these sequences, is 1/zeta(m). (example of counted 4-tuple: (4,3,2,4) *is* counted because 3 is not divided by 2.) But...what is meant here by taking the limits of the n's?? I assume that each n must approach infinity 'independently' of the others. (Would the limit be the same if n_k was a monotonically increasing function of n_{k-1}, for example?) And, what is meant exactly by 'uniformly randomly chosen'? I am *not* quoting from a book, but rather have myself UNRIGOROUSLY derived the above well-known result today, and wish to know if the result has a more rigorous and technically-correct statement. thanks, Leroy Quet ==== > Because Khinchtine's constant exists and is finite, we know that the > distribution of the integers among the terms of the simple continued > fraction of a sufficently random (whatever is meant by this) positive > real is not normal. > > In other words, 1 is much more likely to occur than, say, 1000 in the > continued fraction expansion of most positive reals. > > > So, let us say we have a real x where the geometric average of x's > continued fraction terms approaches Khinchtine's constant, where these > terms are {a(k)}. > > > So, what can be said about the number theoretical aspects of {a(k)}, > such as: > > 1) What is the likelyhood that any a(k) will be prime, as k -> oo. > > ie. If pcf(m) is the number of primes among a(k) for 1 <= k <= m, then > what is pcf(m) asymptotical towards? > > 2) Same question as (1), but replace 'primes' with 'squarefree > integers'. > > 3) What is likelihood a(k) and a(k+1) will be coprime? > > 4) What are the expected number of divisors for all a(k) where 1 <= k > <= m? > > > > Etc etc etc .... > Of course, the number of primes among the first m CF terms of most positive reals would exceed that, on average, among the first m positive integers in general, on average. Right? > > thanks, > Leroy > Quet ==== > For some even positive integer m, > we have a m-by-m grid. > > In this 2 player game, each player has (m^2/2) counters, > each counter numbered with a distinct integer from 1 to (m^2/2). > > The players take turns placing the counters into the grid's squares in > any order the players wish. > > (We do not actually need the counters, for players can simply write > the numbers in the grid's squares. But the counters make it easy to > know which numbers each player has already used, for each integer is > to be used one per player.) > (Or we can simply play this on a computer.) > > > Scoring: > > One player is rows, the other player is columns. > > For, say, rows, every set of adjacent integers, where each immediately > adjacent (to left/right) pair is coprime, is multiplied, then these > groups of multiplied integers are all added up to get the > row-player's score. > > For columns, we do the same, but we consider immediately adjacent > pairs which are adjacent above/below for multiplication if coprime. > > As to help explain what I mean, here is an example (of a game played > against myself without using any strategy): > > > (Who plays which number is unimportant.) > > 8 5 8 3 > 6 1 2 6 > 7 3 1 5 > 2 4 7 4 > > Rows gets: > > 8*5*8*3 + > 6*1*2 + 6 > + 7*3*1*5 > + 2 + 4*7*4 > > Columns gets: > > 8 + 6*7*2 + > 5*1*3*4 + > 8 + 2*1*7 + > 3 + 6*5*4 > > > I would guess that higher m than 4 would be more interesting. We can use other criteria other than coprimality when determining which integers to multiply. (One advantage of coprimality is that if 2 positive integers are lower, then they are more likely to be coprime than if they had been higher, which {I feel} improves this game's strategy.) Any interesting variations on this game??? thanks, Leroy Quet ==== Does anyone have any idea how to solve the following: x+y+z=a xy+yz+xz=b xyz=c I have a hunch that x,y,z should be the roots of the cubic equations t^3-at^2+bt-c=0 ==== Brian Troutwine grava .88 la saucisse et au marteau: > Does anyone have any idea how to solve the following: > > x+y+z=a > xy+yz+xz=b > xyz=c > > I have a hunch that x,y,z should be the roots of the cubic equations t^3-at^2+bt-c=0 That's true. Now you've got many methods to solve 3rd degree's equations, such as Cardan or Ferrari. -- Nicolas ==== Could anyone help me solve the following: Find six different nondegenerate triangles with integer sides for which a=16 and A=60 degrees; where the angles of the triangles are labeled A,B,C and the sides opposite them a,b,c. ==== >Could anyone help me solve the following: >Find six different nondegenerate triangles with integer sides for >which a=16 and A=60 degrees; where the angles of the triangles are >labeled A,B,C and the sides opposite them a,b,c. The law of cosines says a^2 = b^2 + c^2 - 2 b c cos(A), i.e. 256 = b^2 + c^2 - b c. Sorry: the only positive integer solution is b=c=16. I think the least value of a for which there are six different triangles (if you count (a,b,c) and (a,c,b) as different when b<>c) is 49, for which you have the triangles [49, 16, 55], [49, 21, 56], [49, 35, 56], [49, 39, 55], [49, 49, 49], [49, 55, 16], [49, 55, 39], [49, 56, 35], [49, 56, 21] Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 ==== First my original post copy/pasted, then the post I am replying to by Fred the Wonder Worm, before my reply below. >We have a regular hexagon whose sides are lables, clockwise from top, >A through F. the hexagon's inner walls as if the walls were mirrored, but it also >affects the direction of *itself*; for whenever it crosses its own >path (as already drawn), it passes through the path, but is reflected >as if a mirror has been placed perpendicularly to the previous path at >the point of intersection. > know >if such a particular path actually exists, because the path's final >direction is heavily dependent upon the accuracy in which the path is >drawn and the accuracy of the angles reflected. But I will give the order of the hexagon's surfaces as visited by the >path (as drawn at the time of each particular crossing) below. (I know this, if my by-hand approximation was not too flawed.) (I have no idea. You best use exact-rational arithetic to get this, if >it is possible to figure out at all.) E, B , D, 2 crossings, F, E, 1 crossing, F, 3 crossing, D, B, 3 >crossings, F, 7 crossings, and back to its staring point. (If no crossings are listed between letters, than no crossings occur >between them.) (If someone solves this, they are going to have to post some link to a >webpage with the answer, I am afraid.) thanks, >Leroy Quet > If anyone wishes to check my angle chase, here's one description of it. > I've tried it three times now, so either it's right or I'm hitting a > blind spot. Start by drawing in a representative diagram up to that > point. (Make it big -- this is important!) Let the vertices in > clockwise order be labelled L, M, N, O, P, Q, so that LM is side A, MN > is side B, etc. Let the photon start at point R, and successive points > where it changes direction be S, T,..., Z. (Z is the point on the last > D-F segment, and I aim to show that Z does not actually exist.) > > That is, R is on side A (LM), S is on side E (PQ), T is on B (MN), > U is on D (OP), V is on ST, W is on RS, X is on F (QL), Y is on E (QS), > and Z should be on WX. > > Let angle LRS = theta. Then we can chase angles (all degree signs > omitted) as follows: > > QSR = 120 - theta > PST = 120 - theta > RST = 2*theta - 60 > MTS = 120 - theta > NTU = 120 - theta > TUO = theta > PUV = theta > UVS = 120 > SVW = 120 > VWS = 120 - 2*theta > SWX = 120 - 2*theta > RWX = 60 + 2*theta > LXW = 180 - 3*theta > QXY = 180 - 3*theta > WXY = 6*theta - 180 > XYQ = 3*theta - 120 > SYZ = 3*theta - 120 > XYZ = 420 - 6*theta > > But then in triangle XYZ, angles XYZ + ZXY sum to 240 degrees, which is > impossible. We conclude that the ray is actually heading at 60 degrees > _away_ from the last segment from D-F. So that ray must actually cross > segment WS rather than WX. I believe it then crosses VS and hits E > again, possibly then going via UV, TV and maybe back to the starting > point. (I think, but am not sure, that there is enough leeway for this.) > > So a modified version of the problem might have the pattern as > > A, E, B, D, 2, F, E, 2, E, 2, start > > Geoff. I wonder if the ray, after crossing VS does *not* then head to E, but rather heads more upwards towards UV, then (perhaps) spirals inwardly and infinitely towards point V. But a number of other things could perhaps happen, for my diagram (hand-drawn) is ambiguous. (Around points V and Z, there are a few segments that could each thanks, Leroy Quet ==== sum[k^2/k!] from k=1 to k=inf Surely there are some cute algebraic tricks to get 2e out of this, I just don't know them - lil help please? thx, cdj ==== sum[k^2/k!] from k=1 to k=inf Surely there are some cute algebraic tricks to get 2e out of this, I >just don't know them - lil help please? thx, cdj Write e^x = SUM(n=0 t =oo) (x^n/n!) Differentiate both sides and then multiply both by x Again, differentiate both sides and multiply both bu x. Put x = 1. BINGO! ==== > sum[k^2/k!] from k=1 to k=inf That equals sum_(k=1,oo) k/(k-1)! = sum_(k=0,oo) (k+1)/k! = d(xe^x)/dx at x = 1. ==== > > sum[k^2/k!] from k=1 to k=inf > > That equals sum_(k=1,oo) k/(k-1)! = sum_(k=0,oo) (k+1)/k! = d(xe^x)/dx at x > = 1. well aren't I just a genius now? lol thanks all, cdj ==== > > >>sum[k^2/k!] from k=1 to k=inf > > > That equals sum_(k=1,oo) k/(k-1)! = sum_(k=0,oo) (k+1)/k! = d(xe^x)/dx at x > = 1. Or, in the middle, sum_(k=0,oo) (k+1)/k! = sum(k>0,k/k!) + sum(k>=0,1/k!) = sum(k>0, 1/(k-1)!) + e = 2e As I recall, sum(k>=0, k^n/k!) is an integer multiple of e. The Bell numbers come to mind. ==== cdj sum[k^2/k!] from k=1 to k=inf Surely there are some cute algebraic tricks to get 2e out of this, I > just don't know them - lil help please? Write k^2 = k(k-1) + k and get two simpler series. LH ==== > sum[k^2/k!] from k=1 to k=inf Surely there are some cute algebraic tricks to get 2e out of this, I > just don't know them - lil help please? e^(e^x) = 1 + e^x + e^2x/2! + e^3x/3! + ... Differentiate twice, then set x = 0. ==== Here is 2 player game. (Inspired by math and other preexisting commonly-played games.) Each player has an identical set of 15 square tiles colored as followed: Y = yellow R = red B = blue P = purple *Y ** *Y ** ** R* R* *B *Y ** *Y P* *B RB RB ** PY P* PY P* ** R* R* *B PY P* PY *B RB RB Players take turns making a row of tiles, one tile added at each turn. A player can ONLY add a tile if the immediately previous tile she/he is to add shares at least one color with the new tile. Each player get one point for each color-combination not among his/her opponent's PREVIOUSLY placed pieces. (In other words, if the lines of tiles are expanding to the right and are both aligned, then a player gets a point for every piece of his/hers where the corresponding piece of her/his opponent is NOT to the LEFT of the player's own tile in question.) And if any player cannot use all of his/her pieces (because of color-matching rule), the player with the most played pieces gets 2 points for each additionally played piece beyond the number played by his/her opponent. (We could also just give each player 2 points for each played piece, since the score-difference would be the same anyway.) Sample game beginning (played without strategy here) ---------------------------------------------------- Player 1 Player 2 -------- --------- *Y ** RB RB *Y point ** point ** R* PY point *Y point ** R* PY point *Y point *B *B ** point P* point *B *B PY point P* point RB ** ** P* point R* RB ** ** RB *B etc... etc... (And the points become less frequent as play progresses.) I would suggest a greater # of colors, such as 5, 6, 7, or 8 (8 for 'advanced' play). If we have n colors, we would have (2^n -1) pieces to get all combinations of colors (except no colors). thanks, Leroy Quet ==== I was looking through my files the other day and found a simple Fermat me, but that's not saying much since I am not an expert mathematician so I thought I would post it here for comment by the research community. Note: I am posting this out of mere curiosity, so please be gentle :-) Fermat's Proof Below: -------------------------- The is a simple proof that for the equation x^n+y^n=z^n there are no integer solutions for n>2. Background Fermat must have been considering the Pythagorean Theorem x^2+y^2=z^2 and observed the integer solution sets x=3I, y=4I, z=5I and mused, do integer solutions exist for x^n+y^n=z^n (and had decided no). Restate the equation: x^n (x^n+y^n)^(n-1) + y^n (x^n+y^n)^(n-1) = (x^n+y^n)^n Notice the factor (x^n+y^n) has the exponent (n-1) on the left side of the equation and the exponent (n) on the right side. To solve this equation we must introduce some exponent factor so that f(scaler, n-1) and f(scaler, n) have a common denominator (n). Fermat's proof simply observes (n-1) and (n) are sequential. The exponent operators are A^xA^y=A^(x+y) (addition) or A^x/A^y=A^(x-y) (subtraction). Sequential numbers do not have common denominators greater than 1. There is no number that can be added to, or subtracted from, sequential numbers that will change the fact that they are sequential. Proof -->Sequential numbers do not have common denominators greater than 1 -->If two numbers are sequential, one is ODD and one is EVEN and their difference is 1 -->Factor the numbers- ODD=ODDa*ODDb EVEN=EVENa*EVENb =EVENa*ODDb -->There are 2 equations: 1=ODD-EVEN 1=EVEN-ODD Replace with factors and solve. -->1=(ODDa*ODDb)-(EVENa*ODDb) 1=(EVENa*ODDb)-(ODDa*ODDb) 1=ODDb(ODDa-EVENa) 1=ODDb(EVENa-ODDa) Solve ODDb=1 ODDb=1 ODDa=EVENa+1 EVENa=ODDa+1 Sequential numbers do not have common denominators greater than 1. ==== The Fermat crackpot position is filled by James Harris and has been for a decade or more. You'll have to stand in line. Thousands of professional mathematicians worked on the problem for a century or so. And your friend waltzed in with a trivial solution?? Truly amazing! >I was looking through my files the other day and found a simple Fermat >me, but that's not saying much since I am not an expert mathematician >so I thought I would post it here for comment by the research >community. Note: I am posting this out of mere curiosity, so please be gentle :-) Fermat's Proof Below: >-------------------------- >The is a simple proof that for the equation x^n+y^n=z^n there are no >integer solutions for n>2. Background >Fermat must have been considering the Pythagorean Theorem x^2+y^2=z^2 >and observed the integer solution sets x=3I, y=4I, z=5I and mused, do >integer solutions exist for x^n+y^n=z^n (and had decided no). Restate the equation: x^n (x^n+y^n)^(n-1) + y^n (x^n+y^n)^(n-1) = >(x^n+y^n)^n Notice the factor (x^n+y^n) has the exponent (n-1) on the left side of >the equation and the exponent (n) on the right side. To solve this equation we must introduce some exponent factor so that >f(scaler, n-1) and f(scaler, n) have a common denominator (n). Fermat's proof simply observes (n-1) and (n) are sequential. The >exponent operators are A^xA^y=A^(x+y) (addition) or A^x/A^y=A^(x-y) >(subtraction). Sequential numbers do not have common denominators >greater than 1. There is no number that can be added to, or >subtracted from, sequential numbers that will change the fact that >they are sequential. Proof -->Sequential numbers do not have common denominators greater than 1 >-->If two numbers are sequential, one is ODD and one is EVEN and their >difference is 1 -->Factor the numbers- >ODD=ODDa*ODDb >EVEN=EVENa*EVENb >=EVENa*ODDb -->There are 2 equations: >1=ODD-EVEN 1=EVEN-ODD >Replace with factors and solve. -->1=(ODDa*ODDb)-(EVENa*ODDb) 1=(EVENa*ODDb)-(ODDa*ODDb) >1=ODDb(ODDa-EVENa) 1=ODDb(EVENa-ODDa) >Solve >ODDb=1 ODDb=1 >ODDa=EVENa+1 EVENa=ODDa+1 Sequential numbers do not have common denominators greater than 1. ==== (some snips) > Fermat must have been considering the Pythagorean Theorem x^2+y^2=z^2 > and observed the integer solution sets x=3I, y=4I, z=5I and mused, do > integer solutions exist for x^n+y^n=z^n (and had decided no). > > Restate the equation: x^n (x^n+y^n)^(n-1) + y^n (x^n+y^n)^(n-1) = > (x^n+y^n)^n You might notice that the restated equation is always true for any value of x, y, and n. Eliminating z in that way removes the dependency of the truth of the equation on the existence of a counterexample to FLT. Thus, any further proof attempt that focuses on the restated equation can't possibly prove FLT. -- Mark Thornquist ==== > > (some snips) > > Fermat must have been considering the Pythagorean Theorem x^2+y^2=z^2 > and observed the integer solution sets x=3I, y=4I, z=5I and mused, do > integer solutions exist for x^n+y^n=z^n (and had decided no). > > Restate the equation: x^n (x^n+y^n)^(n-1) + y^n (x^n+y^n)^(n-1) = > (x^n+y^n)^n > > You might notice that the restated equation is always true for > any value of x, y, and n. Wait a second... He then proves that this equation has no solution! This is absolutely brilliant. > Eliminating z in that way removes the > dependency of the truth of the equation on the existence of a > counterexample to FLT. Thus, any further proof attempt that > focuses on the restated equation can't possibly prove FLT. ==== > To solve this equation we must introduce some exponent factor so that > f(scaler, n-1) and f(scaler, n) have a common denominator (n). Depending on what this is supposed to mean, it is either irrelevant or satisfied by scaler[sic]^(n-1) . -- Daniel W. Johnson panoptes@iquest.net http://members.iquest.net/~panoptes/ 039 53 36 N / 086 11 55 W ==== Unless you are a crank (which I will assume you are not (for now)), you should avoid at all costs posting proofs of FLT. Even if you got it right, people will all just assume you're crazy. Statistically, it's pretty much true, and hardly any seasoned mathie will take the time to find the flaw in your proof. If you want to learn math, and interact with more experienced mathies, I suggest finding other questions to approach and try to solve - how many polyominoes are there? Does the 3n+1 sequence always terminate? If you _insist_ on studying FLT, then it is most instructive to find the mistakes in your proof for yourself. Believe me, unless you really suck at math (or are a crank), you truly are capable of finding your own mistakes. Finally, and a bit hypocritically, I tried to find the flaw in your proof but right away I can't tell what you mean. What is your overall form of argument? I assume it is to arrive at a contradiction -- that is, I assume that you are assuming that x^n+y^n = z^n for some fixed positive integer triple x,y,z and fixed n>2; and that you want to derive a contradiction from this. So why did you stop using z in your equation? You will have to use that z is a positive integer _somewhere_. Also, what is this function f you introduce? What is an exponent factor and how do you plan to use it to solve the equation? Why must f(--) and f(---) have a common denominator of n? Finally, scalar is spelled scalar not scaler. Tyler > I was looking through my files the other day and found a simple Fermat > me, but that's not saying much since I am not an expert mathematician > so I thought I would post it here for comment by the research > community. > > Note: I am posting this out of mere curiosity, so please be gentle :-) > > Fermat's Proof Below: > -------------------------- > The is a simple proof that for the equation x^n+y^n=z^n there are no > integer solutions for n>2. > > Background > Fermat must have been considering the Pythagorean Theorem x^2+y^2=z^2 > and observed the integer solution sets x=3I, y=4I, z=5I and mused, do > integer solutions exist for x^n+y^n=z^n (and had decided no). > > Restate the equation: x^n (x^n+y^n)^(n-1) + y^n (x^n+y^n)^(n-1) = > (x^n+y^n)^n > > Notice the factor (x^n+y^n) has the exponent (n-1) on the left side of > the equation and the exponent (n) on the right side. > > To solve this equation we must introduce some exponent factor so that > f(scaler, n-1) and f(scaler, n) have a common denominator (n). > > Fermat's proof simply observes (n-1) and (n) are sequential. The > exponent operators are A^xA^y=A^(x+y) (addition) or A^x/A^y=A^(x-y) > (subtraction). Sequential numbers do not have common denominators > greater than 1. There is no number that can be added to, or > subtracted from, sequential numbers that will change the fact that > they are sequential. > > Proof > > -->Sequential numbers do not have common denominators greater than 1 > -->If two numbers are sequential, one is ODD and one is EVEN and their > difference is 1 > > -->Factor the numbers- > ODD=ODDa*ODDb > EVEN=EVENa*EVENb > =EVENa*ODDb > > -->There are 2 equations: > 1=ODD-EVEN 1=EVEN-ODD > Replace with factors and solve. > > -->1=(ODDa*ODDb)-(EVENa*ODDb) 1=(EVENa*ODDb)-(ODDa*ODDb) > 1=ODDb(ODDa-EVENa) 1=ODDb(EVENa-ODDa) > Solve > ODDb=1 ODDb=1 > ODDa=EVENa+1 EVENa=ODDa+1 > > Sequential numbers do not have common denominators greater than 1. ==== Hash: SHA1 examine the statement before posting it, and deemed it worthy of discussion. With regard to whether or not it qualifies as a proof, we must first agree on a definition of the word. My understanding is that a mathematical proof is a demonstration that, given certain *axioms* (widely accepted truths), a presented theory is necessarily true. Since widely accepted is a subjective assessment, it often becomes necessary for a mathematician (or other problem solver, for that matter) to break proposed problems down into smaller parts that are more easily reconciled with reality. If the above understanding is correct, then proving or disproving FLT involves a demonstration of how existing axioms either necessitate its truth or untruth. Thus it appears that the submitted work stands upon following axioms: 1) The Axiom of Variable Substitution - used here to restate the equation to: x^n(x^n+y^n)^(n-1) + y^n(x^n+y^n)^(n-1) = (x^n+y^n)^n 2) In order for this equation to be solved, its writer states that (n-1) and (n)--sequential numbers--must be shown to have a common denominator greater than 1. As a layperson, I can not confirm the axiomatic nature of this statement, perhaps someone else can confirm or deny it. 3) Sequential numbers can not share a common denominator greater than 1. In short, the attempted proof reveals that the root of the problem is that sequential numbers can not be shown to have common denominators greater than 1, explaining why the equation breaks at n > 2, but works for 1 and 2. In light of the above, there are only 3 ways to disprove this theory: 1) The definition of proof as proposed is incorrect. 2) One of the three (3) axioms can be shown to be false (2 looks like a candidate from my seat, 1 & 3 appear rather obvious). 3) An error in deductive reasoning was made at one of the three steps (e.g., the variable substitution was not performed correctly). With all of the brainpower in this forum, I am confident that we can either confirm or rapidly dispense with (I agree this is more likely) this theory and perhaps emerge with a greater level of understanding as well. As iron sharpens iron, so one man sharpens another. - Proverbs iQA/AwUBP8O79KDLMcpPQeVjEQLBpQCfXmBOKPS7Z5gwAogY332Z2P8jmkAAoMYO lIbArqOy/03cGqLxG2ljzlel =IpQk > Unless you are a crank (which I will assume you are > not (for now)), you should avoid at all costs posting proofs > of FLT. Even if you got it right, people will all > just assume you're crazy. Statistically, it's > pretty much true, and hardly any seasoned mathie > will take the time to find the flaw in your proof. > > If you want to learn math, and interact with > more experienced mathies, I suggest finding other > questions to approach and try to solve - how > many polyominoes are there? Does the 3n+1 > sequence always terminate? If you _insist_ on > studying FLT, then it is most instructive to > find the mistakes in your proof for yourself. > Believe me, unless you really suck at math > (or are a crank), > you truly are capable of finding your own > mistakes. > > Finally, and a bit hypocritically, I tried to > find the flaw in your proof but right away I > can't tell what you mean. What is your overall > form of argument? I assume it is to arrive > at a contradiction -- that is, I assume that you > are assuming that x^n+y^n = z^n for some fixed > positive integer triple x,y,z and fixed n>2; and > that you want to derive a contradiction from this. > So why did you stop using z in your equation? You > will have to use that z is a positive integer > _somewhere_. Also, what is this function > f you introduce? What is an exponent factor > and how do you plan to use it to solve the > equation? Why must f(--) and f(---) have > a common denominator of n? Finally, scalar is > spelled scalar not scaler. > > Tyler > > I was looking through my files the other day and found a simple Fermat > me, but that's not saying much since I am not an expert mathematician > so I thought I would post it here for comment by the research > community. > > Note: I am posting this out of mere curiosity, so please be gentle :-) > > Fermat's Proof Below: > -------------------------- > The is a simple proof that for the equation x^n+y^n=z^n there are no > integer solutions for n>2. > > Background > Fermat must have been considering the Pythagorean Theorem x^2+y^2=z^2 > and observed the integer solution sets x=3I, y=4I, z=5I and mused, do > integer solutions exist for x^n+y^n=z^n (and had decided no). > > Restate the equation: x^n (x^n+y^n)^(n-1) + y^n (x^n+y^n)^(n-1) = > (x^n+y^n)^n > > Notice the factor (x^n+y^n) has the exponent (n-1) on the left side of > the equation and the exponent (n) on the right side. > > To solve this equation we must introduce some exponent factor so that > f(scaler, n-1) and f(scaler, n) have a common denominator (n). > > Fermat's proof simply observes (n-1) and (n) are sequential. The > exponent operators are A^xA^y=A^(x+y) (addition) or A^x/A^y=A^(x-y) > (subtraction). Sequential numbers do not have common denominators > greater than 1. There is no number that can be added to, or > subtracted from, sequential numbers that will change the fact that > they are sequential. > > Proof > > -->Sequential numbers do not have common denominators greater than 1 > -->If two numbers are sequential, one is ODD and one is EVEN and their > difference is 1 > > -->Factor the numbers- > ODD=ODDa*ODDb > EVEN=EVENa*EVENb > =EVENa*ODDb > > -->There are 2 equations: > 1=ODD-EVEN 1=EVEN-ODD > Replace with factors and solve. > > -->1=(ODDa*ODDb)-(EVENa*ODDb) 1=(EVENa*ODDb)-(ODDa*ODDb) > 1=ODDb(ODDa-EVENa) 1=ODDb(EVENa-ODDa) > Solve > ODDb=1 ODDb=1 > ODDa=EVENa+1 EVENa=ODDa+1 > > Sequential numbers do not have common denominators greater than 1. ==== > examine the statement before posting it, and deemed it worthy of > discussion. With regard to whether or not it qualifies as a proof, > we must first agree on a definition of the word. No, we already have a definition. If you wish to use the word proof, you are well advised to use its current meaning and not make up some new meaning. > If the above understanding is correct, then proving or disproving > FLT involves a demonstration of how existing axioms either > necessitate its truth or untruth. Yes, and the axioms in question are Peano's axioms, which are the very definition of the natural numbers. Thomas ==== > My understanding is that a mathematical proof is a demonstration > that, given certain *axioms* (widely accepted truths), a presented > theory is necessarily true. Since widely accepted is a subjective > assessment, it often becomes necessary for a mathematician (or other > problem solver, for that matter) to break proposed problems down into > smaller parts that are more easily reconciled with reality. Oh, and axioms are statements which are too *simple* to be proved. Example: If a = b and b = c, then a = c. Your axioms have no business being claimed as axioms, although axiom 1 has the virtue of being close to the relevant axioms. (You might as well add the FLT itself as axiom 4 and really shorten your proof.) You don't get to pick out the holes in your proof and call them axioms. -- Daniel W. Johnson panoptes@iquest.net http://members.iquest.net/~panoptes/ 039 53 36 N / 086 11 55 W ==== > Oh, and axioms are statements which are too *simple* to be proved. > Example: If a = b and b = c, then a = c. What makes that too simple to be proved? ==== > 1) The Axiom of Variable Substitution - used here to restate the > equation to: > > x^n(x^n+y^n)^(n-1) + y^n(x^n+y^n)^(n-1) = (x^n+y^n)^n You've restated the equation as a tautology. This is true for all conceivable values of x, y, z, and n. > 2) In order for this equation to be solved, its writer states that > (n-1) and (n)--sequential numbers--must be shown to have a common > denominator greater than 1. As a layperson, I can not confirm the > axiomatic nature of this statement, perhaps someone else can confirm > or deny it. The equation as stated above certainly does not require any particular common DIVISOR for (n-1) and (n). And I'm still not sure what this alleged axiom is trying to say. Would this axiom assert that a solution for a * k^2 + b * k^2 = k^3 requires a common divisor for 2 and 3? -- Daniel W. Johnson panoptes@iquest.net http://members.iquest.net/~panoptes/ 039 53 36 N / 086 11 55 W ==== [.snip.] >My understanding is that a mathematical proof is a demonstration >that, given certain *axioms* (widely accepted truths), a presented >theory is necessarily true. No. Mathematical axioms are NOT widely accepted truths; they are merely formal statements. Truth don't enter into it. >2) In order for this equation to be solved, its writer states that >(n-1) and (n)--sequential numbers--must be shown to have a common >denominator greater than 1. This is nonsense as written. You mean a common DIVISOR. >As a layperson, I can not confirm the >axiomatic nature of this statement, perhaps someone else can confirm >or deny it. The statement is unsupported. The fact that n-1 and n are relatively prime (have no common divisors other than 1 and -1) was not shown to imply Fermat's Last Theorem. Don't know why he thinks this is the case. >3) Sequential numbers can not share a common denominator greater than >1. DIVISOR, not denominator. The 'denominator' of an integer is usually taken to be 1, period. >In short, the attempted proof reveals that the root of the problem is >that sequential numbers can not be shown to have common denominators >greater than 1, explaining why the equation breaks at n > 2, but >works for 1 and 2. No: the attempted proof ->ASSERTS<- that Fermat's Last Theorem has to n and n-1 being relatively prime, but that assertion is not justified. As for 1 and 2, 1 and 2 have no common divisors other than 1 and -1, so why would the assertions about n not work for n=2? >In light of the above, there are only 3 ways to disprove this theory: 1) The definition of proof as proposed is incorrect. Your understanding of what an axiom is is certainly incorrect. >2) One of the three (3) axioms can be shown to be false (2 looks >like a candidate from my seat, 1 & 3 appear rather obvious). 2 is not and was not asserted to be, an axiom. Neither is 3, for that matter. 3 is a conclusion derived from the basic properties of the integers, while 2 was asserted and not proven: >> Notice the factor (x^n+y^n) has the exponent (n-1) on the left side of >> the equation and the exponent (n) on the right side. >> >> To solve this equation we must introduce some exponent factor so that >> f(scaler, n-1) and f(scaler, n) have a common denominator (n). The last paragraph is (a) nonsense as written; what is scaler? What is f(scaler,n-1)? What is f(scaler, n)? What common denominator? Does he mean common factors? In short: imprenetable nonsense. ====================================================================== It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) ====================================================================== Arturo Magidin magidin@math.berkeley.edu ==== Hash: SHA1 Geez...I tried to preface my post with the fact that I am NOT a mathematician, I was just looking for a little peer review of an text I received from an associate. In any event: >My understanding is that a mathematical proof is a demonstration >that, given certain *axioms* (widely accepted truths), a presented >theory is necessarily true. > > No. Mathematical axioms are NOT widely accepted truths; they are > merely formal statements. Truth don't enter into it. - - -snip- > Your understanding of what an axiom is is certainly incorrect. I appreciate you attempting to polish up my definition, but fail to see how any formal statement is axiomatic? You may also want to clear this up with both Merriam Webster (www.m-w.com) and the MIT mathematics lab since they share the same definition: http://tinyurl.com/wl0m (Section 3: Axioms) All progress and greater knowledge originate from this process of separating the wheat from the chaff, so I do greatly appreciate the constructive comments and illumination that others have contributed and hope that in some small way the post may have stimulated the iQA/AwUBP8QWvaDLMcpPQeVjEQI9sQCgvlW+v5MIEIMXmn8Xk0mnz6rEiL0AoMYq U53LXg0i2RFQ+cICbmaO/2h8 =Nc/3 ==== > Finally, and a bit hypocritically, I tried to > find the flaw in your proof but right away I > can't tell what you mean. He seems to be tacitly assuming that if m and n are coprime, so are k^m and k^n. (And the second half of the proof simply establishes that n-1 and n are coprime.) -- Daniel W. Johnson panoptes@iquest.net http://members.iquest.net/~panoptes/ 039 53 36 N / 086 11 55 W ==== > Unless you are a crank (which I will assume you are > not (for now)), you should avoid at all costs posting proofs > of FLT. Even if you got it right, people will all > just assume you're crazy. I have never assumed anyone is crazy for posting a FLT proof; just optimistic. And if it even is right, all the better. > Statistically, it's > pretty much true, and hardly any seasoned mathie > will take the time to find the flaw in your proof. That's strange - every single FLT proof I have seen here over the years has been shot down by someone fairly quickly. (...) --- J K Haugland http://www.neutreeko.com ==== > I was looking through my files the other day and found a simple Fermat > me, but that's not saying much since I am not an expert mathematician > so I thought I would post it here for comment by the research > community. With many simple proofs of hard theorems, there is a very simple way to see that is something fishy, and it works with this proof as well. It is well known that a^n + b^n = c^n has plenty of solutions in positive integers a, b, c if n = 2 and none have ever been found if n > 2. So whatever proof you have that a^n + b^n = c^n _must_ fail if n = 2. If it doesn't fail for n = 2 then it proves something wrong, so even if you don't understand the proof you _know_ it cannot be correct. The proof you posted never requires that n > 2. ==== > The proof you posted never requires that n > 2. It never even requires that n > 1. -- Daniel W. Johnson panoptes@iquest.net http://members.iquest.net/~panoptes/ 039 53 36 N / 086 11 55 W ==== > > With many simple proofs of hard theorems, there is a very simple way > to see that is something fishy, and it works with this proof as well. > (SNIP) > > The proof you posted never requires that n > 2. Care should be taken, though. The proof of the irrationality or transcendance of pi requires that pi be the ratio of a circle's circumference to it's diameter in a rather subtle, not-obvious way. ==== fuffy > I was looking through my files the other day and found a simple Fermat > me, but that's not saying much since I am not an expert mathematician > so I thought I would post it here for comment by the research > community. Note: I am posting this out of mere curiosity, so please be gentle :-) Fermat's Proof Below: > -------------------------- > The is a simple proof that for the equation x^n+y^n=z^n there are no > integer solutions for n>2. Background > Fermat must have been considering the Pythagorean Theorem x^2+y^2=z^2 > and observed the integer solution sets x=3I, y=4I, z=5I and mused, do > integer solutions exist for x^n+y^n=z^n (and had decided no). Restate the equation: x^n (x^n+y^n)^(n-1) + y^n (x^n+y^n)^(n-1) = > (x^n+y^n)^n Notice the factor (x^n+y^n) has the exponent (n-1) on the left side of > the equation and the exponent (n) on the right side. To solve this equation we must introduce some exponent factor so that > f(scaler, n-1) and f(scaler, n) have a common denominator (n). Fermat's proof simply observes (n-1) and (n) are sequential. The > exponent operators are A^xA^y=A^(x+y) (addition) or A^x/A^y=A^(x-y) > (subtraction). Sequential numbers do not have common denominators > greater than 1. There is no number that can be added to, or > subtracted from, sequential numbers that will change the fact that > they are sequential. Proof -->Sequential numbers do not have common denominators greater than 1 > -->If two numbers are sequential, one is ODD and one is EVEN and their > difference is 1 -->Factor the numbers- > ODD=ODDa*ODDb > EVEN=EVENa*EVENb > =EVENa*ODDb -->There are 2 equations: > 1=ODD-EVEN 1=EVEN-ODD > Replace with factors and solve. -->1=(ODDa*ODDb)-(EVENa*ODDb) 1=(EVENa*ODDb)-(ODDa*ODDb) > 1=ODDb(ODDa-EVENa) 1=ODDb(EVENa-ODDa) > Solve > ODDb=1 ODDb=1 > ODDa=EVENa+1 EVENa=ODDa+1 Sequential numbers do not have common denominators greater than 1. ==== >> fuffy >> >> fuffy >> >> Fuffy has refined its tactics. First, all that Fuffy did was >> to keep the name of the most current poster in a thread >> from appearing on Google. >> >> Free clue for the clueless: If you want to see the >> threads have lots of sub-threads, and the default >> order makes it difficult to find recent comments >> in more than one. >> >> What is to be done about Fuffy? >> >> Duh-h-h. Don't ask me. >> >> Well, I suspect virtually every Google user but you >> already knew about sort by date and how to find >> recent posters. >> >> I'm a Mathie. >> >> People want to know who the last poster in a thread is, >without having to do a Google Advanced Search. Second clue for the clueless: Sort by date is on every thread with more than one message. Look around. Nothing to do with Advanced Search. > For >example, if Google shows Justin Van Twinkie as the last >poster, I won't bother to read the last posting Then you'll have no idea whether or not 15 people have posted since the last time you read the thread. ==== > >> fuffy >> >> fuffy >> >> Fuffy has refined its tactics. First, all that Fuffy did was >> to keep the name of the most current poster in a thread >> from appearing on Google. >> >> Free clue for the clueless: If you want to see the >> threads have lots of sub-threads, and the default >> order makes it difficult to find recent comments >> in more than one. >> >> What is to be done about Fuffy? >> >> Duh-h-h. Don't ask me. >> >> Well, I suspect virtually every Google user but you >> already knew about sort by date and how to find >> recent posters. >> >> I'm a Mathie. >> >> People want to know who the last poster in a thread is, >without having to do a Google Advanced Search. > > Second clue for the clueless: > > Sort by date is on every thread with more than one message. Look > around. Nothing to do with Advanced Search. > > For >example, if Google shows Justin Van Twinkie as the last >poster, I won't bother to read the last posting > > Then you'll have no idea whether or not 15 people have posted since > the last time you read the thread. All right, now I see what you mean. Now I know how to keep McCallum from wasting my time. On the other hand, I'll bet there are Google Users other than myself who don't you seem to want to believe, and for people who don't know how it works, McCallum fucks things up in a way that he has no right to. ==== Suppose you have some random variables X_j that satisfy the conditions of the central limit theorem (idential means and variances). We all know that the SUM of such random variables tends toward a Gaussian distribution. Now suppose that you have some function f(X) that maps each of these random variables X_j. Suppose for simplicity that the function f(X) is invertible, at least in some restricted domain, but perhaps nonlinear. Is there any statement that can be made about the distribution of the sum of f(X_j)? References would be greatly appreciated, also. Much thanks, JB ==== >Suppose you have some random variables X_j that satisfy >the conditions of the central limit theorem (idential >means and variances). We all know that the SUM of such >random variables tends toward a Gaussian distribution. >Now suppose that you have some function f(X) that maps >each of these random variables X_j. Suppose for simplicity >that the function f(X) is invertible, at least in some restricted >domain, but perhaps nonlinear. >Is there any statement that can be made about the distribution >of the sum of f(X_j)? Well, obviously you want the f(X_j) to satisfy conditions on means and variances. They don't have to be identical, though. Lindeberg's Theorem says if Y_j are independent with means mu_j and variances sigma_j^2, M_n = sum_{j=1}^n mu_j and S_n^2 = sum_{j=1}^n sigma_j^2, with 1/S_n^2 sum_{j=1}^n E[(Y_j - mu_j)^2 I(|Y_j - mu_j| >= S_n epsilon) -> 0 as n -> infinity for all epsilon > 0 (I(A) being the indicator function of A, i.e. 1 if A is true and 0 otherwise), then 1/S_n sum_{j=1}^n (Y_j - mu_j) converges in distribution to a standard normal random variable. A condition that may be easier to apply is that 1/S_n^3 sum_{j=1}^n E |(Y_j - mu_j)^3| -> 0 as n -> infinity. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 ==== >Suppose you have some random variables X_j that satisfy >the conditions of the central limit theorem (idential >means and variances). We all know that the SUM of such >random variables tends toward a Gaussian distribution. Actually we don't know that: assuming your X_j are independent, but not identically distributed, identical means and variances are not enough for a Central Limit Theorem: you need the Lindeberg condition. An amusing example is X_j = 0 with probability 1-1/j^2 +/- j with probability 1/(2 j^2) each so all X_j have mean 0 and variance 1. But since sum_j 1/j^2 < infinity, with probability 1 all but finitely many X_j are 0, and thus 1/n sum_{j=1}^n X_j -> 0 almost surely. >Now suppose that you have some function f(X) that maps >each of these random variables X_j. Suppose for simplicity >that the function f(X) is invertible, at least in some restricted >domain, but perhaps nonlinear. >Is there any statement that can be made about the distribution >of the sum of f(X_j)? > > Well, obviously you want the f(X_j) to satisfy conditions > on means and variances. They don't have to be identical, though. > Lindeberg's Theorem says if Y_j are independent with means mu_j > and variances sigma_j^2, M_n = sum_{j=1}^n mu_j and > S_n^2 = sum_{j=1}^n sigma_j^2, with > 1/S_n^2 sum_{j=1}^n E[(Y_j - mu_j)^2 I(|Y_j - mu_j| >= S_n epsilon) -> 0 I left out a bracket: that should be 1/S_n^2 sum_{j=1}^n E[(Y_j - mu_j)^2 I(|Y_j - mu_j| >= S_n epsilon)] -> 0 > as n -> infinity for all epsilon > 0 (I(A) being the indicator function of > A, i.e. 1 if A is true and 0 otherwise), then > 1/S_n sum_{j=1}^n (Y_j - mu_j) converges in distribution to a standard > normal random variable. E.g. suppose for simplicity all X_j and f(X_j) have mean 0, f(0) = 0, and there are positive constants a and b such that a(y-x) < f(y) - f(x) < b(y-x) whenever x < y. Then a^2 Var(X_j) <= Var(f(X_j)) <= b^2 Var(X_j). I'll write S_n^2 = sum_{j=1}^n Var(X_j) and S'_n^2 = sum_{j=1}^n Var(f(X_j)), so a S_n <= S'_n <= b S_n. In Lindeberg's condition f(X_j)^2 I(|f(X_j)| > S'_n epsilon) <= b^2 X_j^2 I(|X_j| > S_n epsilon a/b) so if X_j satisfy the condition, so do f(X_j). Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 ==== OK I see where I went wrong now, I think. Let me try again. Consider the Category, (I use capital C to indicate it may have the cardinality of all classes, thus have to be a super-category or whatever); whose objects are all ordinary categories, and whose arrows are functors. This Category has natural transformations defined within it, which are (I think?) mappings from functors to functors. So they are a sort of 2nd-order Arrow in this Category. So, given their properties, what do Natural Transformations (in this Category) correspond to in the way of mappings-of-arrows in an ordinary category? Does this make sense now? What special named sort of arrow-transformation do they correspond to? ---------------------------------------------------------------------------- -- Bill Taylor W.Taylor@math.canterbury.ac.nz ---------------------------------------------------------------------------- -- I shot an arrow in the air And down it came, I know not where. I looked for it up in a tree Which was, though, just a categ'ry! ---------------------------------------------------------------------------- -- ==== >A and B have different dimensions, >but AB and BA are both square. >Can anyone provide a proof showing the eigenvalues are the same for AB and BA ? Side question: What's your favorite linear algebra text for answering this sort of question? I pulled out my Horn and Johnson but realized I had no idea where to look to find a proof, if there is one. Eigenvalue was too broad an index entry. Related question: Do you have a favorite book covering specially-structured matrices? That would include, for example, fast Toeplitz inversion methods? Or could easily answer the question when somebody asks, is there a name for this kind of matrix? - Randy ==== 0072 I've got a list of points (in x,y form) that are the corners of a > polygon, in order. It's not necessarily a convex polygon. Is there a > standard algorithm for determining whether a given point is inside > or outside the polygon? All this takes place in the plane... For > practical purposes, an algorithm is ok if it can handle up to 7 points > decently. Many thanks -Adrian > This is probably in the faq at comp.programming.games.algorithms, or > something like that. Google will probably also help. I know it's in the comp.graphics.algorithms FAQ. http://www.faqs.org/faqs/graphics/algorithms-faq/ Check out question 2.03. - Randy ==== I have looking at a few web pages dealing with the largest known > calculated primes and a great deal of computational time is taking > into searching for these numbers and verifying they are primes. I have > seen the Euclid's proof of infinitude primes and it occurs that me > that super-large prime numbers can be calculated using the following: Others have already given you an example of why your idea won't fly, but I there > is no largest prime (and therefore an infinite number of primes) is as > follows: If there is a largest prime, P, then the number of primes is finite. We can > therefore form a new number, N, which is one greater than the product of > all primes. Therefore, N > P (and, indeed, it would be very much greater > than P). When divided by any prime in our list, it leaves remainder 1. Therefore, > /either/ it is prime /or/ it is the product of two or more primes, at least > one of which is greater than P. In /either/ case, P has been shown not to be the largest prime number. Nope. Nowhere have you assumed that the list of primes known is > complete up to P. You've simply proved that the finite list of > known primes is not the list of all primes. If you use Euclid's proof as constructive, then the sets you generate 0079 > {2,3,7} > {2,3,7,43} > {2,3,7,13,43,139} > {2,3,7,13,43,139,3263443} > and thence you find the new primes {547,607,1033,31051}, all of which > are less than the currently largest known prime 3263443. Don't forget the second case! :-) Don't forget that if you're pretending that you don't know what > the primes are, you can't suddenly pull otherwise well-known > features of the prime numbers out of a hat. Phil > -- > Unpatched IE vulnerability: ADODB.Stream local file writing > Description: Planting arbitrary files on the local file system > Exploit: http://ip3e83566f.speed.planet.nl/eeye.html > (but unrelated to the EEye exploit) ==== >Well duh! >All of _what_, though. Primes. John ==== > Nope. That is prior art. However taxing folks on their exusions and > bodily productions is not far in the future. The jism tax. > It's the coming thing. HOWWWWWWWWWWLL !!!!!! Pun of the week! But seriously, we already have, here in NZ, proposed taxes on emissions, not of people but cows. In order to comply with the perceived obligations under the Kyoto protocol, (hey there's a song line there... Kyo-to pro-to..), our PC government proposed a new cattle emission tax on farmers for the alleged harm to the ozone layer due to emitted methane! Naturally, this was immediately dubbed the fart tax by the predatory media, and attracted so much derision, joking, and protest from the farming community that it's unlikely to see the light of day; which is in some ways a pity because a fart tax would have been a wondrous thing! (Even if smelling a little fishy!) ---------------------------------------------------------------------------- -- Bill Taylor W.Taylor@math.canterbury.ac.nz ---------------------------------------------------------------------------- -- Camouflage condoms: So they won't see you coming. ---------------------------------------------------------------------------- -- ==== >I should note that the above carries no attribution. It is >>copyrighted material originally published in The Onion, >>but I couldn't locate any of the original Onion material >>online anywhere. I've already posted the Onion material (actually, I've posted the link >to the material fron the Onion's own website) elsewhere in this thread. Huh. I didn't realize their archive went back that far. I once tried and eventually gave up. It's the one with several eminent physicists discussing Big Bang theory. Like man, what if the whole universe is, like, a big Atom in another Universe? Wow, heavy, man. That sort of thing. - Randy ==== When a Riemann surface is constructed by joining copies of the cut plane the arbitrariness in the choice of cuts carries over into an arbitrariness in the Riemann surface. And where there is no unique domain there is no unique function so it is incorrect to speak of the function w(z) defined by an initially given relation f(w,z)=0. It is rigor not pedantry to remark that the derivable functions are many and that the initially given relation is just what they have in common. When Riemann introduced his Riemann surfaces it may be that he did so to eliminate the multiplicity that comes with branches. But the multiplicity reappears in this altered form. To understand it one has to stay with analysis and not move on to a more abstract geometrical treatment. For other notes on this topic visit my website www.riemannsurfaces.info. ==== >When a Riemann surface is constructed by joining copies of the cut >plane the arbitrariness in the choice of cuts carries over into an >arbitrariness in the Riemann surface. Lucky that that's not the only way to construct a Riemann surface then... > And where there is no unique >domain there is no unique function so it is incorrect to speak of >the function w(z) defined by an initially given relation f(w,z)=0. Look up analytic continuation in an advanced book on complex analysis: the standard construction of _the_ Riemann surface here does not have anything to do with cuts, and there's nothing arbitrary about it. The surface is just the set of all function elements which can be reached by continuation along a path, starting with a given function element. >It is rigor not pedantry to remark that the derivable functions are >many and that the initially given relation is just what they have in >common. > When Riemann introduced his Riemann surfaces it may be that he did >so to eliminate the multiplicity that comes with branches. But the >multiplicity reappears in this altered form. To understand it one has >to stay with analysis and not move on to a more abstract geometrical >treatment. > > For other notes on this topic visit my website >www.riemannsurfaces.info. > David C. Ullrich ==== >Message-id: <2410d7e.0311241320.5b02c68e@posting.google.com >Can someone provide with a mathematical algorithm which explains the >horrendous pixel-crime perpetred by the infamous Mr. Mensanator? http://www.thequantummachine.com/images.php Hey, the new image is a big improvement over the original. A little constructive criticism helps doesn't it? I thought that my TFT screen had been degraded by some air pollutant, >but then I realized that it was only his image which had some fungus >on it... :-) -- Mensanator Ace of Clubs ==== > >> Given an integral domain R, can we not construct a field in the same >> way we construct Q from Z? I.e., equivalence classes on R x >> (R{0}), where (a, b) ~ (c, d) iff a d = b c . (Integral domains >> are by definition commutative, no? So says Herstein.) If R has a 1, then the mapping f(r) = (r,1) gives the desired >embedding; but without the 1, how do we embed R? > > You are not really mapping to (r,1), you are mapping to the class of > (r,1). So map r to the class of (r^2,r) if r is nonzero, and map 0 to > the class of (0,r) for arbitrary r different from 0. > Doh! Of course; because cancellation holds in R (follows from integrality), if we map r,s to (r^2,r) and (s^2,s), resp., then r^2.s = s^2.r iff r = s... > ====================================================================== > It's not denial. I'm just very selective about > what I accept as reality. > --- Calvin (Calvin and Hobbes) > ====================================================================== > > Arturo Magidin > magidin@math.berkeley.edu ==== Please help with equations/sketches for a circular torus intersection when the cutting plane is not parallel to its axis of symmetry. Equations may be expressed in terms of torus tube radius, tube displacement from axis,inclination/length of perpendicular from torus axis/center to the intersecting plane. References requested preferably from the internet. TIA ==== > Please help with equations/sketches for a circular torus intersection > when the cutting plane is not parallel to its axis of symmetry. > Equations may be expressed in terms of torus tube radius, tube > displacement from axis,inclination/length of perpendicular from torus > axis/center to the intersecting plane. References requested preferably > from the internet. TIA just take a three-dimensional parametrisation of the torus like ( cos s * (cos t + 2) ) f(s,t) = ( sin s * (cos t + 2) ) ( sin t ) This is for a torus of displacement 2 and tube radius one. With vector calculus you now can easily deduce the cutting with any plane. Rene. -- Ren.8e Meyer Student of Physics & Mathematics Zhejiang University, Hangzhou, China ==== An interesting integral just appeared in alt.math, and there should be a neater solution than I found. The problem is to evaluate int_0^1 int_0^{1-x} 1/(1-axy) dy dx, where |a| < 4. My solution was to expand into a geometric series (justified since |a| < 4, 0 <= y <= 1-x ==> |4axy| < 1), obtaining int_0^1 a^n x^n (1-x)^(n+1) dx, and then use the standard identity int_0^1 x^a (1-x)^b = Gamma[a+1] Gamma[b+1]/Gamma[a+b+2]. This yields the integral as a power series, sum_{n=0}^infty a^n n!^2/(2n+2)!, which is then recognizable as 2 arcsin(sqrt(a)/2)^2 --------------------- . a But there ought to be a direct way to see this. Any takers? --Ron Bruck ==== Two minor corrections: that should read |a|<4, 0 <= y <= 1-x ==> |axy| < 1, and we obtain for the value of the integral (1-x)^(n+1) sum_{n=0}^infty int_0^1 a^n x^n ----------- dx. n+1 But the final answer remains unchanged. --Ron Bruck > An interesting integral just appeared in alt.math, and there should be > a neater solution than I found. The problem is to evaluate > > int_0^1 int_0^{1-x} 1/(1-axy) dy dx, > > where |a| < 4. > > My solution was to expand into a geometric series (justified since |a| > < 4, 0 <= y <= 1-x ==> |4axy| < 1), obtaining > > int_0^1 a^n x^n (1-x)^(n+1) dx, > > and then use the standard identity > > int_0^1 x^a (1-x)^b = Gamma[a+1] Gamma[b+1]/Gamma[a+b+2]. > > This yields the integral as a power series, > > sum_{n=0}^infty a^n n!^2/(2n+2)!, > > which is then recognizable as > > 2 arcsin(sqrt(a)/2)^2 > --------------------- . > a > > But there ought to be a direct way to see this. Any takers? > > --Ron Bruck ==== Set R_1 = x+y, R_2 = xy Is there an easy way of writing the polynomial x^n + y^n in terms of R_1 and R_2? I know it can be done by expanding out with R_1^n, but this is obviously not practical for large n. Is there some sort of identity that allows you to write the answer down immediately, similar to the binomial theorem? ==== > Set R_1 = x+y, R_2 = xy > > Is there an easy way of writing the polynomial x^n + y^n in terms of > R_1 and R_2? I know it can be done by expanding out with R_1^n, but > this is obviously not practical for large n. Is there some sort of > identity that allows you to write the answer down immediately, > similar to the binomial theorem? The buzzword is Dickson polynomial. Ley u = x + y and x = xy. Form the generating function F(t) = sum_{n=0}^infinity (x^n + y^n)t^n. Then F(t) = 1/(1-xt) + (1-yt) = (2 - ut)/(1 - ut + vt^2) = (2 - ut)sum_{m=0}^infinity (u + vt)^m t^m = (2 - ut)sum_{n=0}^infinity t^n sum_m {m choose n-m}u^{2m-n} v^{n-m} etc. Pulling out the coefficient of t^n gives D_n(u,v) the Dickson polynomial of the first kind with D_n(u,v) = x^n + y^n. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) ==== > Set R_1 = x+y, R_2 = xy Is there an easy way of writing the polynomial x^n + y^n in terms of > R_1 and R_2? I know it can be done by expanding out with R_1^n, but > this is obviously not practical for large n. Is there some sort of > identity that allows you to write the answer down immediately, > similar to the binomial theorem? Here's my attempt: Write f_n = x^n+y^n. Then expand (x+y)*(x^n+y^n) to get the recurrence relation: f_(n+1) - R_1 * f_n + R_2 * f_(n-1) = 0 with the initial conditions: f_0 = 2 and f_1 = R_1. The solution to the recurrence relation may be written as: f_n = A * u^n + B * v^n, where u and v are roots of the quadratic equation t^2 - R_1 * t + R_2 = 0. The coefficients A and B may be found by considering the initial conditions. -Michael. ==== > > Set R_1 = x+y, R_2 = xy > Here's my attempt: > > Write f_n = x^n+y^n. Then expand (x+y)*(x^n+y^n) to get the recurrence > relation: > f_(n+1) - R_1 * f_n + R_2 * f_(n-1) = 0 > with the initial conditions: f_0 = 2 and f_1 = R_1. That is a special case of the Newton's identities. > The solution to the recurrence relation may be written as: > f_n = A * u^n + B * v^n, where > u and v are roots of the quadratic equation t^2 - R_1 * t + R_2 = 0. Don't you think that the solutions to this equation would be x and y.... > The coefficients A and B may be found by considering the initial conditions. ... with A=1 and B=1? So this approach won't give much information, as the solution you get will be (drums, please) f_n=x^n+y^n :) Jyrki ==== Set R_1 = x+y, R_2 = xy Here's my attempt: Write f_n = x^n+y^n. Then expand (x+y)*(x^n+y^n) to get the recurrence > relation: > f_(n+1) - R_1 * f_n + R_2 * f_(n-1) = 0 > with the initial conditions: f_0 = 2 and f_1 = R_1. That is a special case of the Newton's identities. The solution to the recurrence relation may be written as: > f_n = A * u^n + B * v^n, where > u and v are roots of the quadratic equation t^2 - R_1 * t + R_2 = 0. Don't you think that the solutions to this equation would be x and y.... Indeed.... The coefficients A and B may be found by considering the initial conditions. ... with A=1 and B=1? So this approach won't give much information, as the > solution you get will be (drums, please) f_n=x^n+y^n :) Oh well, it seemed like such a good idea at the time.... :-) -Michael. ==== > > Set R_1 = x+y, R_2 = xy > > Is there an easy way of writing the polynomial x^n + y^n in terms of > R_1 and R_2? I know it can be done by expanding out with R_1^n, but > this is obviously not practical for large n. Is there some sort of > identity that allows you to write the answer down immediately, > similar to the binomial theorem? Search for Newton's identities (a useful recurrence relation) and Waring's formula. They are also helpful for a larger number of variables (x,y,z,etc.). I don't know what would be the simplest formula for two variables. Wouldn't any formula for Dickson's polynomials do? Search for any of these key phrases! Jyrki Lahtonen, Turku, Finland ==== > Ok, I apologize. > > I did not say it was fraud, I _asked_ if it was. > > You say it was not, that answers the question. Ok. I accept your apologies. I am little tired of this controversy, and seems stupid to get angry with people I don't personally meet. > It's just that extraordinary claims require extraordinary evidence. What you > present on your web site cannot even be considered evidence, let alone > extraordinary evidence. I'm supposed to take your word that if presented > correctly, the lines would be parallel? That's not how science is done. I agree with that. As I said, the main investigation was done by Crater&McDaniel. I didn't reproduce step by step, but made my own calculations and simulations that were consistent. At that time the computer simulation was very cost, so I made some simulations that showed the same level that theirs. Also, revised the protocol, and the pattern can be fitted with that mean deviation inside of the mounds, the problem is trying to do by hand. But it is easy, just using least squares (if you insist, as Crater, in a coordinate fit). I will prepare when I get some time to do it by computer, you must notice that at the time I did the diagram, as it is rotated, it would have took me a considerable time to locate the corresponding coordinates and there was the issue of the non-rectified image. What I have thought is to use a superimposed image, with the centers clearly marked, so it is visually discernible how far is the deviation in each mound. > And let's not forget that you solicited comments. Do you think that I am the > only one who sees problems with what you presented? You should be happy to > see such comments so that you can correct the situation. The problem is that the critics here and other posts were far from gentle. I am not accustomized to this. Also notice that I am little tired of writing some statements, and getting them ignored. For example, I stated that the simulations (and calculation by hand using area ratios will do) show that you need about 75 or more mounds to get reasonable levels of that pattern being by chance. And a poster in my forum came with 106 dots and showed some nice alignments. Perhaps I loose patience very fast, but I had to explain that not only those nice alignments were present, but very likely the mounds pattern. Other thing was to extract it, which would likely require a computer. > I will clarify also in my webpage soon, > > Good. > > givin links to your webpage. > > Don't you have your priorities wrong? Shouldn't you focus on what _you_ are > presenting? It is now closed. I think the best test would be to perfom a complete scan of the martian surface. It could be used to detect crater chains and make crater population counts also (they are similar to mounds, but with lightning inverted). However, in order to avoid bias, I should make it independent from the mounds, and when I detect automatically, this may be somehow indicative of non-randomness. > You will be acussed of dishonesty. > > Dishonesty? I just presented what I saw. You admit that the original image > was not orthorectified, so I was honest in saying that the lines do not appear > parallel. Now if I made a mistake, it was an honest one, because I used the > image that _you_ provided. Rather than explaining in the text why the image > was incorrect, wouldn't it have been better to post a correct image? But that was said in the webpage. I retire the accusation of dishonesty here and in my webpage, accepting that you did not notice. And yes, I will do that. I will also publish the source code I use and the link to the original image in NSSDC website. > You can sue me also, if you want. I am wishing it. > > Sue you for what? Calling me dishonest? Calling me pathetic? Calling me a slow > thinker? Calling me stupid? These comments won't make you look good. I retire those comments, but you must notice that I have heard everything, from being a pseudoscientist, to not knowing probability notions, fraud, and countless others. The problem is a mathematical one. For example my claim that this is the only pattern (after a square-based one) with the greater number of DIFFERENT parallel/perpendicular directions for 5 points (BAGED, plus repeating P) goes unchallenged. And this is what irritates me. That claim is just a very defined one, testable with analitical geometry. But based on a diagram, so... could I have missed some configuration? I recommend to read most of the material I present, at least superficially, before making comments, and let the childish I am more intelligent than you issues out of the conversation. ==== >Message-id: <2410d7e.0311220235.6c366ff9@posting.google.com > You are one of those who confuse the probability of an event occuring > with the event itself. Duh... It is you who confuse it. You still believe that the probability of that event has turned to 1. This is a claim I have heard countless times. Yet a math PhD laughed at it. see how they do that. Also the crater chains are analysed probabilistic, but still they are there since some million years ago... > In the mounds problem, the probabilistic experiment is having 6 mounds > formed by some process. The process has a great number of pausible > outcomes, depending on its particular physical nature. If you assume > some type of a random formation process, then the probability of > getting the formation shown is extremely low. > > Instead of wasting your time uploading gifs and accusing people, why > don't you program a simulation of 6 mounds forming at random, set your > accuracy range for parallel and perpendicular lines and see of you can > get that or similar formations and what the frequency of those are. > That will convince you the probability of getting that formation or > similar ones is extremely low. That's all there is to it. I have done that particular experiment. I used parallel lines only for a computer simulation of a number of points, not just 6 but greater. Still, for such a number of parallel relationships I got 1/1000 for 6 points. I draw some of them at hand (I got numbers in the screen only because at that time I didn't get any notion of image processing). Most were unremarkably, and there was a tendency to form what I called degenerated lines (a bug). You can do by yourself and check. Otherwise you are talking without foundation, just as counting sheeps, and you know. DO THE SIMULATION AND WHEN YOU HAVE A FIGURE, WE TALK AGAIN. If not, you are making pseudoscience. ==== >Message-id: <1f366ae.0311240955.b79503f@posting.google.com >>Message-id: <2410d7e.0311220235.6c366ff9@posting.google.com >[snip] > >> You just can't _prove_ it's not random. Probability has no meaning >> for something that has already happened. You claim the odds are >> 200,000,000,000 to 1 and then you say that no other configuration >> has this property. >> >> Duh. >> [snip] You are one of those who confuse the probability of an event occuring >with the event itself. Duh... In the mounds problem, the probabilistic experiment is having 6 mounds >formed by some process. The process has a great number of pausible >outcomes, depending on its particular physical nature. If you assume >some type of a random formation process, then the probability of >getting the formation shown is extremely low. The formation? Don't you mean a formation? Instead of wasting your time uploading gifs and accusing people, why >don't you Why doesn't the OP have a mathematically correct drawing on his web site? >program a simulation of 6 mounds forming at random, What, exactly, is the geological process I am supposed to simulate? Or should I just sprinkle random pixels onto an image? >set your accuracy range Accuract range? This is math, not physics. There is no range allowed in determining parallel and perpendicular. >for parallel and perpendicular lines and see of you can >get that or similar formations and what the frequency of those are. >That will convince you the probability of getting that formation or >similar ones is extremely low. That's all there is to it. You're wrong. There is a world of difference between that formation and similar formations. By the way, has you ever been to a casino? Have you ever looked at the night sky and noticed all the polygons formed by the stars? Do you think there is some mysterious reason for this? >The number you see the >roullette ball sitting on of course just happended but you can't bet >on it. Have you thought about _why_ they won't let you bet on it? Do you know what the probability is of an event that has already happened? >If you think talking about its probability has no meaning >because it just happened, Whatever the probability was before the event, after the event the probability is simply 1. Possibly you didn't see this example, so I'll repeat it: Put 64 coins in a box and shake. Note the pattern of heads and tails. The probability of that pattern occuring is 1/18446744073709551616 and yet it happened! Shake the box again. The pattern you see now also had a 1/18446744073709551616 chance of occuing and it happened also. The chance of BOTH those patterns occuring back to back is 1/340282366920938463463374607431768211456 What conclusion do you draw from this? >then if you have just won, why should they >pay you back 36 times your bet? They shouldn't, roullette payouts are 35 to 1. Of course, you would know this if you've ever been to a casino. >Just take your money back and that's >it.:) -- Mensanator Ace of Clubs ==== > Whatever the probability was before the event, after the event the probability > is simply 1. The probability is related to the chances of that event happening. I would not apply to events that have already happened. The p value does not get altered, because p is predictive, and you are talking of a value which is not predictive. I would think that it is more a semantic question. I would assign 0 to event which are impossible and 1 to event that are surely TO HAPPEN (in the future). ==== >Message-id: <2410d7e.0311250420.2e0c3caf@posting.google.com >> Whatever the probability was before the event, after the event the >probability >> is simply 1. The probability is related to the chances of that event happening. I >would not apply to events that have already happened. The p value does >not get altered, because p is predictive, and you are talking of a >value which is not predictive. I would think that it is more a semantic question. I would assign 0 to event which are impossible and 1 to event that are >surely TO HAPPEN (in the future). FYI, I have heard it said probabilty of 1 and certainty do not mean the same thing for a future event, but I can't explain it. Put a coin in a box and shake it. Without looking, what is the probability of showing heads? This isn't quantum mechanics. It does not depend on whether you look or not. It is either heads or it isn't. Before you shake the box, you can ask what is the probability that it will show heads after the shake. After the event, the probability of heads is either 0 (if it actually came up tails) or 1 (if it did come up heads). The trick is you don't know which case it is. You can try to guess what the coin is showing, but in that case the probability applies to the guess, not the event. In the real world (such as football games), a coin is called in the air while the probability function is still valid. -- Mensanator Ace of Clubs ==== > Put a coin in a box and shake it. Without looking, what is the probability of > showing heads? This isn't quantum mechanics. It does not depend on whether you > look or not. It is either heads or it isn't. Before you shake the box, you can > ask what is the probability that it will show heads after the shake. After the > event, the probability of heads is either 0 (if it actually came up tails) or 1 > (if it did come up heads). The trick is you don't know which case it is. But the probability is 1/2 BEFORE the trial. I say that AFTER the trial, it is still 1/2, because you cannot apply to that already happened case. Anyway, it is just semantic. ==== > Put a coin in a box and shake it. Without looking, what is the probability of > showing heads? This isn't quantum mechanics. It does not depend on whether you > look or not. It is either heads or it isn't. Before you shake the box, you can > ask what is the probability that it will show heads after the shake. After the > event, the probability of heads is either 0 (if it actually came up tails) or 1 > (if it did come up heads). The trick is you don't know which case it is. > > > But the probability is 1/2 BEFORE the trial. I say that AFTER the > trial, it is still 1/2, because you cannot apply to that already > happened case. Anyway, it is just semantic. If it was just semantic, people wouldn't get into heated arguments about it. Smarter people than me have said one should not make probability statements about past actualities, but only about future possibilities which I tend to agree with. But let's say you want to use it anyway. What is the probability that you will get heads on two consecutive coin flips? Before we start, we know that the probability of both is the probabilities of the individual flips multiplied together: 1/2 * 1/2 = 1/4. You flip the coin and it comes up tails. NOW what is the probability that both flips will be heads? We know the answer is 0, we know the probability of the second flip is 1/2. The probability of the past event being heads must therefore be 0. Similarly, had the first flip been heads, we succeed if and only if the second flip is heads. Thus, the chance of success (at this stage) is 1/2. But that's the probability of the second individual flip. Therefore, the probability of the past event must be 1. The probability of the past event is not the same as a future event. The function has collapsed from sample space of two possible outcomes to a sample space of one actual outcome. I may be stepping onto thin ice here, but it seems to me that once you have an event, the only thing you can prove is that the sample space of the single actual outcome must be a subset of the sample space of multiple possible outcomes prior to the event. Suppose instead of a coin, our box contains a regular polyhedron whose faces are numbered sequentially form 1 to n (depending on the polyhedron). We are not told which polyhedron it is, but when rolled, the result is 13. This actual outcome must be a subset of the possible outcomes and only an icosahedron has a set of outcomes that includes 13, so we can safely say that the probability of that outcome prior to the roll was 1/20. Now had the outcome been 8, we cannot say what the probability was because it could have been 1/8 for an octahedron, 1/12 for or dodecahedron or 1/20 for the icosahedron. We do not know how large that sample space was of the forces that generated the mounds and it is possible that the real sample space is much smaller than your analysis indicates. If you say it is extremely unlikely that these mounds would form this way, but they did, then there are three possibilities: 1: they did NOT form that way (let's not re-argue the geometry issue) 2: they DID form that way and your estimate of likelyhood is wrong 3: coincidence That's where statistics comes into play, to help distinguish case 2 from can't prove anything. ==== Hey, Mensanator, that post is from other poster. ==== >Message-id: <2410d7e.0311241350.632214f9@posting.google.com >> >> Note that the original red line G-P also misses mound P by >> a mile and any claims of parallelism to the other red lines >> is just a figment of the imagination. >> You are simply a LIAR. Even in the by hand adjusted image, the >deviation is of 7 pixels. It has been enlarged 40%, which corresponds >to less than 5 pixels in the original image. With a resolution of less than 50 meters/pixel, that gives: 5 * 50 = 250 meters. That is several times less than 1 mile, and also must be noticed that >the adjust by computer has more precission, being the deviation of >less than 3.5 pixels, AS CLEARLY stated in my webpage. You are simply a LIAR, and even so, you CRIMINALLY acusse me of fraud. >I may be wrong, or not, but you are a disgusting dishonest person. Did you see my earlier post? You asked for a retraction and you got it. Why are you still belaboring the point? To reiterate, I did not accuse you of fraud, I asked if the image (that you admit does not represent what you claim it does) was intended to mislead the reader. I'll accept your answer that you did not intend to mislead. As far as being a liar, perhaps you are unaware of the phrase a miss is as good as a mile Let me broaden your knowledge of English from The New Dictionary of Cultural Literacy, Third Edition. 2002 A miss is as good as a mile A near miss is still a miss and therefore no better than missing by a great margin. When someone says misses by a mile, it does not mean literally 1 mile. It simply means that unless the measurement is EXACT, it matters not whether the deviation is 3.5 pixels or 50 meters. The definition of parallel does not allow for ANY deviation, no matter how small. You do remember that you were asking for math commentary? If you need help presenting your data in a clear, mathematically unambiguous format, just ask. Flying off the handle isn't going to get you anywhere. -- Mensanator Ace of Clubs ==== > Did you see my earlier post? You asked for a retraction and you got it. Why are > you still belaboring the point? Yes, I saw and replied. I use webposting, it takes more time than from Outlook. > > To reiterate, I did not accuse you of fraud, I asked if the image (that you > admit does not represent what you claim it does) was intended to mislead the > reader. I'll accept your answer that you did not intend to mislead. That is right now. As I provide the maximum information for the webpage it is seen that I do not mislead. Notice that I have boldered the rebuttals, that would be far from a pseudoscientist of hoaxer technique. > As far as being a liar, perhaps you are unaware of the phrase > > a miss is as good as a mile > > Let me broaden your knowledge of English from The New Dictionary of Cultural > Literacy, Third Edition. 2002 > > > A near miss is still a miss and therefore no better than missing by a great > margin. > When someone says misses by a mile, it does not mean literally 1 mile. It > simply means that unless the measurement is EXACT, it matters not whether the > deviation is 3.5 pixels or 50 meters. The definition of parallel does not allow > for ANY deviation, no matter how small. You do remember that you were asking > for math commentary? Ok. I am not english native speaker as it is obvious, so I took your commentary literally. > If you need help presenting your data in a clear, mathematically unambiguous > format, just ask. Flying off the handle isn't going to get you anywhere. ==== The paper I refer was published in Journal of Scientific Exploration, not Statistical Science. I wanted to mean that I would have prefered SSc. ==== > If you need help presenting your data in a clear, mathematically unambiguous > format, just ask. Flying off the handle isn't going to get you anywhere. Notice that the paper by Dr.Horace&Prof.McDaniel is presented that way. It was submitted to a peer-review journal, though I had prefered a mainstream publication. (Statistical Science, in spite of their Bible code publication). You can find a link to the paper and a rebuttal by Dr. Ralph Greenberg in my website. Notice that the issue is complicated. I just showed the data, I have not published nothing in peer-review journals, not conducted any proper scientific test, except a lot of helpful commentaries with Crater & McDaniel, and some personal replications of part of their work. I personally verified my model of parallel lines (which differs from theirs though the ideal polygon is completely identical), getting a figure of about 1/1000 for parallel lines alone (I mean, not replicating their pattern, but ANY pattern of 6 mounds with a high number of parallel relations). The perpendicular relations lower the ocurrences (not conducted. I made other tests for 4 points, but they just showed coincidence in magnitude with their work). It must be noticed that I use a model which is not fit-coordinated, i.e., took simply the deviations in pixels from the point (Crater used virtual mounds, I used points). That corresponds to a diagram in which you join the estimated center of the mounds, and then calculate how much the figure deviates from an ideal model. I didn't use that because I prefer noticing the mound separations from the lines, than having to interpret visually how much a line difers from parallelism with other. I cannot claim scientifically anything on my tests, as they were just amateur (I am a physicist, and a mathematics graduate student helped me to make the program in C), but seem to give support to C&M analysis, performed with not 6 but 12 (+4) mounds. As you see, I do not talk of my work in the site, and that wouldn't be honest. I just talk here about it as anecdote. I encourage the readership of their work. I will replicate my work and publish all the data online so you can criticize properly. But as my only interest is scientific curiosity, and do not seek money, fame, attention, etc, etc, I would appreciate a proper dialoge with no bold claims. ==== > Why is P seven pixels below, and one pixel to the right of the > mound you claim it's on. Why is D six pixels below and 3 pixels > to the right of the mound you claim it's on? That is greater deviation from an adjusted fit by computer. Anyway, there is a deviation for an ideal figure. As explained clearly in the text, there must be room for some deviation, because 1) We are in a km scale, not crystal-scale. Geological events are not precise nor perfect on that scale. 2) You don't know if the fit is better (or worst) when measuring ground distances, instead of aerial. 3) Orthorectification is not present in the image. That said, the deviations are slightly corrected. 4) A crater chain generally presents worst alignments. 5) The chances of randomness drop with the square of the average deviation (that is valid for a non-coordinate fit, a coordinate one is somehow more strigent). Would not be deviation, the chances would be zero. In the actual scenario they are very small FOR THAT PARTICULAR PATTERN (and here enters directly the issue of uniqueness). > Mansanator doesn't need the likes of me sticking up for his > character, but the evidence is that you are spouting gibberish > and displaying flagrant disregard for principles of probability, > and Mensanator is displaying a perfectly sufficient knowledge > of geometry to interpret and invalidate your illusions, a likwise > sufficient knowledge of probability to puncture your probabilistic > posturing, and to be perfectly frank, is being perfectly honest > and reasonable with you when he criticises your nonsense. More > honest than you appear to deserve. > > *_PLONK_* He has apologized and I have accepted the apologies. I am perfectly honest, and be sure I don't get money, fame or whatever with this issue. My quest is completely a curiosity-based one. I am a physicist, I have talked with PhD's about the issue. The most important objections are these: 1) Mound selection. 2) Uniqueness of pattern and the issue of a priori selection. The issue of the deviation is accepted as perfectly normal. Some simply asked for what was the average deviation, directly. Not that there was one and that invalidated all (reasons above). ==== > Suppose you've got a stick of length 1. Now, following happens. > 1) The stick is broken at X (where X is the distance between the > point of impact and the left end of the stick). > 2) The left part of the stick gets hit again and brakes at Y (where > Y is the distance between the point of impact and the left end of > the stick). > Now, provided that the both hits are uniformly distributed, what > is the conditional distribution function for Y given that X = x? > [...] This sounds like Y|X ~ Uniform(0,X), for which P(Y<=y|X=x) = y/x. Did I miss something? ==== > This sounds like Y|X ~ Uniform(0,X), for which P(Y<=y|X=x) = y/x. > Did I miss something? I think i treated the problem as more difficult than what it was. You didn't miss anything. On the contrary, that got me to the -- Kindly Konrad --------------------------------------------------- May all spammers die an agonizing death; have no burial places; their souls be chased by demons in Gehenna from one room to another for all eternity and more. Sleep - thing used by ineffective people as a substitute for coffee Ambition - a poor excuse for not having enough sence to be lazy --------------------------------------------------- ==== To my best knowledge this is neither proven nor disproven - sometimes you hear : conjugate a+i*b to a - i*b, by this you get the dot-product and so you have (R2, +, r.s.m., dot), the euclidian vectorspace. The conjugate is widely used in C, but how do You get it, when C is by definition the commutative field (R2, +, r.s.m., * ) and * is given in (a,b)*(u,v)=(au-bv,av+bu) ? The pleasure of finding out (and to be the first to publish) i leave to yourself. My opinion i'll tell You in about a month time, in the meantime You might find pleasure in reading a fairytale: Once upon a time - no, in our times, there is a little dot-product, somehow skew in its dimensions. A big-booster-genius, C , master of his own universe, is boosting of his richness and intellectuality, no simple mind can dare to comprehend. They live on the vast plain, where arrow-fields have to be worked on. C was strolling around with his apologist Riemann sur Face, so called 'cause he could give a broad smile on his face- as the reflection of the sun in the hair of little Dot caught his eyes. C envied her for her intricating abilities - she was just casting a shadow of an arrow onto the iron bars of the coordinate-grid, behind which he kept his ellbow ticked the flank of sur Face as he shouted :Hey, Dotty, come here, let me have a look into your mirror! She thought :This might take an interesting twist. He never asked me that before. and when sur Face added with a grin from one ear to the other:Your algebraic completeness is calling upon you - come here ! she went over to them. With an obidient gesture, but with the words :Underneath Your cloths You are naked. she handed the mirror to C. The continuation of this You can find under: z in http://i-is-no-longer-imaginary.gmxhome.de and http://i-z.eu.tt Have fun Hero ==== Ummm... are we suppose to guess what that means? > 0038 > >> A trillion seems a little high. (I hear that billion isn't a word >> in the UK? Does that mean that trillion isn't either?) In the UK, these words do exist, but with different meanings than those > to >which you might be accustomed. Billion means 10^12 (a million squared), >trillion means 10^18 (a million cubed), quadrillion means 10^24 (a >million to the power 4), etc etc (except in the hands of ignorant people >such as politicians, businessmen, and the media). And then for the other numbers, they say thousand billion, thousand > trillion, etc.? never sounded right to my ears that a million squared was a trillion, > not a billion. > -- > dgates@spamfreelinkline.com ==== > In message , Richard Heathfield > A quote from: >> The New Fowler's Modern English Usage, Third Edition, Edited by R. W. >> Burchfield, The acknowledged authority on English usage >> [all of that from the front of the dust jacket...] 0074 >> Under the topic billion: >> It is best now to work on the assumption that the word means 'a >> thousand millions' in all English-speaking areas... But that is a false assumption. My home is an English-speaking area in which >the word does not mean that. If the book suggests that it is best to work on false assumptions, then >perhaps it isn't that good a book. at least a county, rather than an individual dwelling. Nick > -- > Nick Wedd nick@maproom.co.uk ==== As I vaguely understand it, the Nash embedding theorem applies to the problem of embedding one (compact) Riemannian manifold isometrically into Euclidean space. Suppose instead you have a mapping M->T of one Riemannian manifold onto another so that the fibres are all Riemannian manifolds. Can one conclude that there is a vector bundle V with metric over T and a bundle map of M->T into V which is fibrewise an isometric embedding? If so, what is a reference to this result? Ignorantly, Allan Adler ara@zurich.ai.mit.edu **************************************************************************** * * * Intelligence Lab. My actions and comments do not reflect * * in any way on MIT. Moreover, I am nowhere near the Boston * * metropolitan area. * * * **************************************************************************** ==== As I vaguely understand it, the Nash embedding theorem applies to the problem >of embedding one (compact) Riemannian manifold isometrically into Euclidean >space. Suppose instead you have a mapping M->T of one Riemannian manifold onto >another so that the fibres are all Riemannian manifolds. Can one conclude >that there is a vector bundle V with metric over T and a bundle map of M->T >into V which is fibrewise an isometric embedding? If so, what is a reference >to this result? Ignorantly, >Allan Adler >ara@zurich.ai.mit.edu ... I'm not sure, but you may be able to show this for compact M by choosing a trivial bundle over T of sufficiently large dimension and using the parametric h-principle for isometric immersions (see 3.1.2 (H) in Partial Differential Relations, M. Gromov, 1980 Springer-Verlag) together with the construction of isometric embeddings given in 3.1.1 on page 223. John Mitchell ==== Let V be space of smooth functions on R(real number) satisfying a linear homogeneous ODE with Polynomial Coefficients , and the leading coefficient of the ODE MUST BE 1. ie. f(x) in V if f satisfys fn(x) + P(n-1) fn-1(x) + P(n-2) fn-2(x) +... +P(0)f(x) = 0 fk represents the kth dervative of f. the coefficient of fn must be 1. e.g e to the power x^2 is in V because it satisfies f'(x)-2xf(x)=0 prove that V is a ring. I thought for weeks but still do not think of the solution. Anyone can help me? ==== > Let V be space of smooth functions on R(real number) satisfying a > linear homogeneous ODE with Polynomial Coefficients , and the leading > coefficient of the ODE MUST BE 1. > > ie. f(x) in V if f satisfys > fn(x) + P(n-1) fn-1(x) + P(n-2) fn-2(x) +... +P(0)f(x) = 0 > fk represents the kth dervative of f. > the coefficient of fn must be 1. > > e.g > e to the power x^2 is in V because it satisfies > f'(x)-2xf(x)=0 > > prove that V is a ring. > > I thought for weeks but still do not think of the solution. Anyone can > help me? it might make your life easier if you used the fact that 'the leading coefficient of the ODE MUST BE 1' is equivalent to the more reasonable statement the leading non-zero coefficient is constant. (just divide through; it's homogenous) ==== > at 04:13 PM, Mitch Harris said: > >>topology is really just a fancy word for geometry, right? > >Wrong. There are many issues in Geometry beyond the scope of Topology. >Before the last few centuries, hardly any work in Geometry had >anything to do with Topology. > >>I meant by my short comment about topology and geometry that, by >>whatever route one is lead to topology (category theory, point-set >>analysis, differential manifolds, etc) that one is doing generalized >>geometry. > > Point-set topology is in no way generalized geometry, nor > even related to it. I agree that they seem to be quite different, but I am sure there could be some connection made between them. Couldn't the incommensurability of the diagonal of a square to its side (is that geometrical?) be seen as a precursor to point-set topological ideas like continuity, openness, closure, connectedness, etc.? Aren't points and sets of points and the aformentioned ideas abstractions of spatial concepts? I get your (and Shmuel's) point. I apologize for my flippant, tendenciously presented relationship between geometry and topology. Philosophically -everything- is distinct. But then also everything has similarities with everything else. All I'm trying to point out is that in the nebulous cloud of intuitively described mathematical areas, some areas are more similar than others. I think I've watered this down enough to say absolutely nothing. Or possibly not. Maybe. Mitch ==== [snip] > Yes, the gist of it is single register you only have to remember one > number, all along the process. fact. But is _that aspect_ of the method Vedic? The expression crosswise vertical > doesn't say much about it, does it? Oh yes, it does. That is the beauty of Sanskrit. It is not verbose; > it just defines the heart of things, as briefly as possible, and > leaves the rest to those with vision. Vedic means secret - only the > seer can understand the Vedas. They are meaningless to others. Modern mathematics likes to make everything as lucid and explicit as possible. It doesn't need to appeal to mystery, or to 'old books with forgotten knowledge, rediscovered but clear only to the chosen ones'. Everything open and clear. And you know what... all in all, modern maths is _less_ verbose. After all, it doesn't need 16 books to explain a trivial thing like elementary arithmetic ... (*) Herman Jurjus (*): (It is said that the original discovery, around 1910, of Vedic maths was written down in 16 books.) ==== > [snip] > > Yes, the gist of it is single register you only have to remember one > number, all along the process. fact. But is _that aspect_ of the method Vedic? The expression crosswise vertical > doesn't say much about it, does it? Oh yes, it does. That is the beauty of Sanskrit. It is not verbose; > it just defines the heart of things, as briefly as possible, and > leaves the rest to those with vision. Vedic means secret - only the > seer can understand the Vedas. They are meaningless to others. > > Modern mathematics likes to make everything as lucid > and explicit as possible. It doesn't need to appeal to mystery, > or to 'old books with forgotten knowledge, rediscovered > but clear only to the chosen ones'. Everything open and clear. No, it is not. It is a bunch of dogma, the way they teach them in school these days. I have two daughters, one who has given VCE (year 12) and has now completed her graduate studies. The other will give VCE next year. I helped them in their Maths, and was totally horrified by the course material, and the way the are taught. They did not do any of your precious geometry - Euclid was totally avoided. Then, they did not learn how to derive any formula from first principles. Even the formula was Geometric Progression was not derived! It was just given in the book like some divine truth. The children are expected to put it in their calculator and use the formula to get some answers to problems. They are *not* given the derivation! When I derived it for my daughter, she was impressed, and said it was so cool. Basically, they want the kids to become morons and believe whatever the top authorities say. This is nothing but a step to pure tyranny. > And you know what... all in all, modern maths is _less_ verbose. > After all, it doesn't need 16 books to explain a trivial thing like > elementary arithmetic ... (*) But the most elementary long multiplication has been well beyond the brains of the most advanced mathematical minds of our time, as this thread shows... that happens when you stop to think, and are expected only to believe. Which, of course, results from the way you are taught in school. > > Herman Jurjus > > (*): > (It is said that the original discovery, around 1910, of Vedic > maths was written down in 16 books.) At any rate, this proves that the book on Vedic maths is not a fraud. Whether in 16 books or 1600, it took me but half a page of that book in the English language to find out a new and better multiplication method. Who knows what will happen to the field of maths when all the 16 books are fully understood... But then, if all mathematicians are trained to be dogmatic morons, how can that ever happen? Arindam Banerjee. ==== [snip] > It is certainly true that the stupidest mistakes are often > made by the smartest people. No, this is illogical. It's called a 'paradox'. That is something that is deliberately worded to seem illogical, but has a hidden message. I could explain it in more length, but i won't. It would feel like explaining a joke. Herman Jurjus ==== > [snip] > > It is certainly true that the stupidest mistakes are often > made by the smartest people. No, this is illogical. > > It's called a 'paradox'. You are confusing learned people with smart people. Learned people usually are not smart. So, those who are useless at worthwhile jobs or business, or not strong enough to take up a trade like plumbing, take to learning and live off the public via university funding. The worst among the learned people are also not keen to learn new things that will upset their hard earned knowledge. So, they try their best to crush new ideas. Smart people, such as those who are paid to find out and try out new things for more competive advantage, usually have very little regard for such learned people. They just do their own thing, and when they have proved what they want to prove, let the learned people absorb their new ideas and propagate them as divine truths with their usual pomposity. In the meantime, the smart people have to endure the hostility of the learned people, but if they have independent means to pursue their work, they do not particularly care. Arindam Banerjee. That is something that is deliberately > worded to seem illogical, but has a hidden message. > I could explain it in more length, but i won't. It would feel > like explaining a joke. > > Herman Jurjus ==== > >>[snip] >>It is certainly true that the stupidest mistakes are often >>made by the smartest people. No, this is illogical. >>It's called a 'paradox'. > > > You are confusing learned people with smart people. Learned people > usually are not smart. So, those who are useless at worthwhile jobs > or business, or not strong enough to take up a trade like plumbing, > take to learning and live off the public via university funding. The ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ This distinction between smartness and learnedness is interesting, but, somewhat superficial. Like in India, anyone looking western or speaking unaccented english used to be called smart. In north america, since most people look western and their mother tongue is english, definition of smart is somewhat different. In general, the accepted idea is related to innovativeness/creativeness in your field of work. In that sense, a plumber can be smart, just as much as a university researcher or a professor, or a corporate guy. Even a learned fellow can be smart too. Smart folks bring innovative apporaches to problem solving, as distinct from the others. Arjoe > worst among the learned people are also not keen to learn new things > that will upset their hard earned knowledge. So, they try their best > to crush new ideas. Smart people, such as those who are paid to find > out and try out new things for more competive advantage, usually have > very little regard for such learned people. They just do their own > thing, and when they have proved what they want to prove, let the > learned people absorb their new ideas and propagate them as divine > truths with their usual pomposity. In the meantime, the smart people > have to endure the hostility of the learned people, but if they have > independent means to pursue their work, they do not particularly care. > > Arindam Banerjee. > > That is something that is deliberately > >>worded to seem illogical, but has a hidden message. >>I could explain it in more length, but i won't. It would feel >>like explaining a joke. >>Herman Jurjus > ==== If A can be diagonalized as PAP', then computing A^n is as simple as computing (PAP')^n = P A^n P' where A^n is just composed of entries with eigenvalues raised to n. Similarly, functions of A such as sin(A) are equivalent to P sin(A) P' where the middle term is a series that converges to a matrix with entries on the diagonal that are the sin of the eigenvalues. Consider the special case of the Jordan matrix where J cannot be diagonalized, i.e. (not necessarily 5x5): [L 1 0 0 0] [0 L 1 0 0] [0 0 L 1 0] [0 0 0 L 1] [0 0 0 0 L] (L are eigenvalues). My question is, what is f(J) generally? For example, if all L = Pi, then sin(J) is A - (A^3)/3! + (A^5)/5! - ... What does this converge to? I can see what happens to the matrix when taking successive powers, but I can't figure out what happens to the series. Also, does this matrix have a special name? I think I've seen it before, but don't know where. JR ==== >If A can be diagonalized as PAP', then computing A^n is as simple as I think one of these A's is supposed to be a D. >computing (PAP')^n = P A^n P' where A^n is just composed of entries with >eigenvalues raised to n. Similarly, functions of A such as sin(A) are >equivalent to P sin(A) P' where the middle term is a series that converges >to a matrix with entries on the diagonal that are the sin of the >eigenvalues. Consider the special case of the Jordan matrix where J cannot >be diagonalized, i.e. (not necessarily 5x5): >[L 1 0 0 0] >[0 L 1 0 0] >[0 0 L 1 0] >[0 0 0 L 1] >[0 0 0 0 L] >(L are eigenvalues). My question is, what is f(J) generally? For example, if >all L = Pi, then sin(J) is A - (A^3)/3! + (A^5)/5! - ... What does this >converge to? I can see what happens to the matrix when taking successive >powers, but I can't figure out what happens to the series. Also, does this >matrix have a special name? I think I've seen it before, but don't know >where. If this matrix (called a Jordan block) is n x n, note that J - L I = N where N^n = 0. For 1 <= j <= n-1, N^j has 1's on the j'th superdiagonal and 0's elsewhere. If f(z) = sum_{j=0}^infinity c_j (z-L)^j, then f(J) = sum_{j=0}^infinity c_j (J-L)^j = sum_{j=0}^{n-1} c_j N^j Thus f(J) is an upper triangular Toeplitz matrix with entries a_{j,k} = f^(k-j)(L)/(k-j)! for k >= j. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 ==== > If A can be diagonalized as PAP', then computing A^n is as simple as > computing (PAP')^n = P A^n P' where A^n is just composed of entries with > eigenvalues raised to n. Similarly, functions of A such as sin(A) are > equivalent to P sin(A) P' where the middle term is a series that converges > to a matrix with entries on the diagonal that are the sin of the > eigenvalues. Consider the special case of the Jordan matrix where J cannot > be diagonalized, i.e. (not necessarily 5x5): [L 1 0 0 0] > [0 L 1 0 0] > [0 0 L 1 0] > [0 0 0 L 1] > [0 0 0 0 L] (L are eigenvalues). My question is, what is f(J) generally? For example, if > all L = Pi, then sin(J) is A - (A^3)/3! + (A^5)/5! - ... What does this > converge to? I can see what happens to the matrix when taking successive > powers, but I can't figure out what happens to the series. Also, does this > matrix have a special name? I think I've seen it before, but don't know > where. Simpler case: A = [a 1] [0 a] A^k = [a 1]^k = [a a^k + ak] [0 a] [0 a ] Hence exp(A) = [ exp(a) sum(k=0,..,infty: (a^k + ak)/k! )] [ 0 exp(a) ] = [ exp(a) exp(a) + a*exp(1) ] [ 0 exp(a) ] In general, one expects that the entries above the main diagonal of J^n are given by linear recursion relations, derived by evaluating J^n * J = J^(n+1) and imposing suitable initial conditions for n = 1. Substituting this form of J^n into the series expansion and summing element by element gives the result (if J has a zero eigenvalue, these expansions may terminate.) -- P.A.C. Smith The vast majority of Iraqis want to live in a peaceful, free world. And we will find these people and we will bring them to justice. ==== If A can be diagonalized as PAP', then computing A^n is as simple as > computing (PAP')^n = P A^n P' where A^n is just composed of entries with > eigenvalues raised to n. Similarly, functions of A such as sin(A) are > equivalent to P sin(A) P' where the middle term is a series that converges > to a matrix with entries on the diagonal that are the sin of the > eigenvalues. Consider the special case of the Jordan matrix where J cannot > be diagonalized, i.e. (not necessarily 5x5): [L 1 0 0 0] > [0 L 1 0 0] > [0 0 L 1 0] > [0 0 0 L 1] > [0 0 0 0 L] (L are eigenvalues). My question is, what is f(J) generally? For example, if > all L = Pi, then sin(J) is A - (A^3)/3! + (A^5)/5! - ... What does this > converge to? I can see what happens to the matrix when taking successive > powers, but I can't figure out what happens to the series. Also, does this > matrix have a special name? I think I've seen it before, but don't know > where. Simpler case: A = [a 1] > [0 a] A^k = [a 1]^k = [a a^k + ak] > [0 a] [0 a ] This should read [a^k a^k + ak] [0 a^k ] > Hence exp(A) = [ exp(a) sum(k=0,..,infty: (a^k + ak)/k! )] > [ 0 exp(a) ] = [ exp(a) exp(a) + a*exp(1) ] > [ 0 exp(a) ] In general, one expects that the entries above the main diagonal > of J^n are given by linear recursion relations, derived by evaluating J^n * J = J^(n+1) and imposing suitable initial conditions for n = 1. Substituting this form of J^n into the series expansion and summing > element by element gives the result (if J has a zero eigenvalue, these > expansions may terminate.) -- > P.A.C. Smith The vast majority of Iraqis want to live in a peaceful, free world. > And we will find these people and we will bring them to justice. -- P.A.C. Smith The vast majority of Iraqis want to live in a peaceful, free world. And we will find these people and we will bring them to justice. ==== >Can we embed every G without adding or removing >edges into the graph of some 0/1-polytope? I don't see the point of this added requirment as the final embedded result preserves edges. ---- ==== > I am a science teacher but I have taught math for a few years. I too > agree tht it could be possible that people have a math gene. Through > out my school years I have always struggled with math, while some > students just seemed to get it right off. No matter how hard I > studied it was a constant battle. While teaching math to my students I see the same types of behaviors I > think this makes me a better math teacher cause I know what it feels > like not to understand, which in turn makes me more patient with my > students. I think the book is worth reading and considering. However > you have to be careful when sharing this information with your > students cause it could turn into a self pity issue. > > In fact, there is a book by that title. Listed on Amazon.com (search Books > with the words math gene.) I read it a few years ago, and found it > interesting. The same part of the brain involved in higher math (not basic > arithmatic) does language processing Maths is one of the most interesting subjects that i have ever took, you would have to be stupid not to be able to understand it once it is explained to u!! ==== > Maths is one of the most interesting subjects that i have ever took, > you would have to be stupid not to be able to understand it once it is > explained to u!! Congratulations, you are a bearer of the math gene. |-}}} Joking aside, I think that everybody can understand the topics in maths but the deepness of understanding usually depends on the quality of the explanation. Karl ==== Ok. But if a set is connected, then any two points should have a polygonal > path between them, right? No. But if the set is open and connected in R^2 as you said earlier, then > yes. > Theorem: locally path connected S, open connected U subset S ==> U path connected Proof relies upon connected chain theorem Show subspace U is locally path connected, etc... Indeed R^2 is locally path connected with path connected balls. So in R^2 a connected open set can be path connected by a finite number of end to end line segments all contained in a finite chain of overlapping balls which surely gives an analysist enuf wriggle room to smooth the broken line into a polynomial; thus I wriggle out of proceeding with further details. ==== > Example: > > The last event time was 15. Here are some events and the times at > which they occured: > Event A: 3, 4, 5, 9, 13 > Event B: 1, 2, 6 > Event C: 7, 8, 10, 11 > > If I were assuming independence, I'd just note down the > interarrival times, the times between events: > > A: 1, 1, 4, 4. Mean = 2.5 > C: 1, 4. Mean = 2.5 > B: 1, 2, 1 Mean = 1.3 > > But are they independent? Does event A depend on when B or > C occurred, or how many times A has occurred? Let's assume they are independent. > > Intuitively, I would expect to order the events A, C, B in decreasing > order of expected soon-ness. > > Why? Look at the inter-event spacings. Why do you have this > expectation? Do you have some prior knowledge that is > relevant? Because although the inter-event spacing for event C has mean 2.5, note that the event last occured at time '4', and we are now at time '15'. So the mean inter-event measure is rather out-of-date. It looks to me as though this event C's cluster of occurences has probably ended, and I have little expectation that the event will occur again in the near future. Toby. ==== > > Example: > > The last event time was 15. Here are some events and the times at > which they occured: > Event A: 3, 4, 5, 9, 13 > Event B: 1, 2, 6 > Event C: 7, 8, 10, 11 > > Intuitively, I would expect to order the events A, C, B in decreasing > order of expected soon-ness. > > Why? Look at the inter-event spacings. Why do you have this > expectation? Do you have some prior knowledge that is > relevant? > > Because although the inter-event spacing for event C has mean 2.5, > note that the event last occured at time '4', and we are now at time > '15'. Ah. Excellent point. What that means statistically, I'd say, is that you have more information about the number-of-events per time interval rvs than about the interarrival time rvs. Theoretically the same information is contained in both. But as you point out, time gaps give you more sample information about one than the other. One statistic you can get at easily from your data is P0(t) = the probability of no events in a time interval of length t. For instance, for t = 1- (just less than 1), P0_A(t) is the probability that a random interval [a,a+1) in [0,15] excludes 3, 4, 5, 9 and 13. That means a can't fall in (2,3], (3,4], (4,5], (8,9] or (12,13], i.e., (2,5], (8,9], (12,13]. Since a can range from 0 to 14, and you are excluding a total range of 5, the probability that a random interval of size 1- will contain no events of type A is (14-5)/14 = 9/14 = 0.64. Conversely, the probability of getting at least one event of type A in an interval of length 1 is 0.36. For event B, the probability of no events is (14-3)/14 = 0.79, and the probability of seeing an event is 0.21. For event C, the probability of no events is (14-4)/14 = 10/14 = 0.71, and the probability of seeing an event is 0.29. All this gets back to the fact that your intuition is correct: You saw more events of type A than of any other, so you can expect that one most often. You saw very few of type B, so you can expect that one least often. I got lost in the math and missed the obvious. However, you can get more than that from your data. You can estimate as I did the probability of seeing an event of each type in any given 1-second interval. But you can estimate other probabilities too, which will reflect the clustering behavior. What is the probability of seeing at least one event in a 5 second interval? Event A: There is no 5-second interval in [0,15] which is free of events. Probability of at least one event = 100%. Event B: The interval (6,15] is free of events. If I choose a random interval [a,a+5) with 0 <= a <= 10, then the probability of no events is the probability that a > 6, which is (10-6)/10 = 40%. The probability of at least one event is 60%. Event C: Similarly, the probability of seeing no events in [a,a+5) is the probability that a is in [0,2], i.e. 20%. The probability of at least one event is 80%. Summary: Estimate of P(at least one event in the next T sec): 1 sec 5 sec A 36% 100% B 21% 60% C 29% 80% And you can do this for any interval size. > So the mean inter-event measure is rather out-of-date. It looks > to me as though this event C's cluster of occurences has probably > ended, and I have little expectation that the event will occur again > in the near future. Yes, you're right. Somewhere along the line I got confused between B's data and C's data. The analysis above reflects your original data as quoted above. And you can repeat it for any other convenient time interval. - Randy ==== right term in english but was too lazy to look it up. Regarding your own experiments and patents, I hope we don't get ourselves a legal fight, wait for part III of my journal and you'll see what I mean. Happy however to have my own observations totally confirmed. see you around, Guenther > However there is something else I wan't to share with you, I carefully > observed the trajectory of the device from the time I released it to > the time it landed and found it was a parable. I'm going to try and > figure the implicatins this has, but I already consider it a very > interesting and revolutionary fact in itself. > > similar results with the Action Device that I have built. I'm not going > to go into detail, because my patent applications are still pending. > > However, I sincerly doubt that the trajectory you observed was a > parable, or a usually short fictitious story that illustrates a moral > attitude or a religious principle [Merriam-Webster]. More likely, it was > a parabola. > > See for details. > > I'm quite convinced that my Action Device works, because, although it > didn't fly off into space immediately, I left it by the curb overnight in > the hopes that it was just getting warmed up, and sure enough, the next > morning it was gone! I'm left to conclude that it actually did fly off > with uniform acceleration into space. > > Good luck, > -Mike ==== loathe as i am to use the jsh thing in the subject, i'm aware that it would only offend jsh if i didn't, and that isn't the point of this. sorry, james, but i am one of those dreaded mathematicians. i hope you don't jump to any unfounded conclusions though. reading your postings, it seems that you have little idea of what we do with our time. you almost appear to say that we sit at our desks write something down, which is worthwhile maths (by fiat because we are mathematicians) and deny anyone's claim that disagrees with our narrow viewpoint. i wish that i could just write something down and have it instantly enter the annals, but it doesn't work like that. instead, you ,hopefully, find something interesting to think about. then when you've figured out why the first three attempts went wrong, you talk to some other people. then when you've possibly got something that isn't garbage you show it off to someone else and they point out where you went wrong, or that it was published 5 years ago, or something. then after you've changed what needs to be changed you might get it submitted to some journal and the referee points out some more errors, where you've used phi for two different things in the same equation and all the typos. at each stage you take it with good grace. at least, if you're reasonable. perhaps there is something to learn there? so, james, there are adequate commentaries on your work that show there are many errors in your algebraic integer argument. my summary would be this: look, i've assumed i can invert 7 here, and it implies 1/7 is in the ring of algebraic integers. of course it isn't (you can prove that i hope, it's a simple exercise most undergrads should be able to do), but instead of checking your assumtion you're off on a crusade. i hope you've read enough of the commentaries to understand why you can't divide by seven. it would then seem reasonable to correct these aspects. of course as has been pointed out you can't. and i mean that as an impersonal you, rather than a personal criciticism. however, here's some advice for you when writing more 'proofs' there is nothing special in 7,7,22 in your proof. so why did you choose them? for clarity's sake wouldn't 2,2,3 have been much more illuminating? this is written in plain text. there are therefore lots of constraints on the symbols you can use. try avoiding unnecessary subscript notation. start with a simple statement of what it is that is wrong. it takes a lot of effort to decode mathematical arguments, and a simple 'the problem is foo' goes along way. at best you say you've uncovered a hundred year old problem, but don't state what it is, just launch into some unmotivated reasoning and say, 'voila, that's wrong.' your prime counting algorithm is at best a party trick i'm afraid. if you could find some new primes for us, that'd be interesting. if you wish to demonstrate your skills, then how about finding some demonstrably silly argument, say one of the anti-cantor ones, and deconstructing it. it might earn some respect. my favourite would be the one showing the reals are in 1-1 correspondence with some subset of the p-adic integers. if you don't know what those are, try explaining why the arguments about two expansions representing the same number is junk. there have been many counter-intuitive results found in mathematics, and more will be discovered. no one would begrudge the discoverer anything. ==== > > loathe as i am to use the jsh thing in the subject, i'm aware that it > would only offend jsh if i didn't, and that isn't the point of this. > > sorry, james, but i am one of those dreaded mathematicians. [...] Oh, I thought that you were the ghost of e. e. cummings. -- G.C. ==== > loathe as i am to use the jsh thing in the subject, i'm aware that it > would only offend jsh if i didn't, and that isn't the point of this. JSH in the subject line isn't due to James's vanity. It is included so that other readers can killfile threads without having them clutter the newsgroup. -- Jesse Hughes Like the ski resort full of girls hunting for husbands and husbands hunting for girls, the situation is not as symmetrical as it might seem. -- Alan MacKay ==== > It is not in general possible to obtain analytic solutions to an arbitrary > differential equation. This is not simply because ingenuity fails, but > because the repertory of standard functions (polynomials, exp, sin and so > on) in terms of which solutions may be expressed is too limited to > accommodate the variety of differential equations encountered in practice. One could expand on this and state that the usual elementary functions may indeed be *defined* to be specific solutions to specific linear differential equations. The fact that an ODE is linear basically implies that we only have to solve a small number of ODE's (read: define a small number of elementary functions), and this allows us to - essentially - write down a solution to *all* linear ODE's in terms of these functions. One could try the same with nonlinear ODE's, but here a solution to a some ODE can not be generalized to other non-linear ODE's. One basically has to start from scratch each time one encounters a new nonlinear ODE. -Michael. ==== Robert Israel >Firstly what exactly are NONLINEAR differential equations? A linear differential equation (with dependent variable > y and independent variable x) can be written as a_n(x) d^n y/dx^n + ... + a_1(x) dy/dx + a_0(x) y = F(x) A nonlinear differential equation is a differential equation that is > not linear. In particular, any terms involving some more complicated > function of y and/or its derivatives makes the equation nonlinear. Secondly why are they considered so difficult to solve? Because they are not easy to solve. There are very simple-looking > first-order nonlinear differential equations such as dy/dx = x y^2 + x + 1 for which AFAIK no closed-form solution is known, even involving > an integral. By contrast, there is a well-known formula (involving > integrals) for solving first-order linear differential equations. Just to add 2¢, if f is a solution of a homogeneous linear DE, then so is kf for any constant k. If g is another solution, so is f+g. Inhomogeneous linear eq'ns have a related property, found in all the textbooks. LH ==== I'd like to write n as a fucntion of N, ie n = f(N). How do I do this easily? ==== race which would end up in lots of isolated and often persecuted groups competing for livelyhoods provided exactly the conditions under which evolution works best. -- Bruce Harvey bruce@bearsoft.co.uk The centre for inside out thinking. > > >>... [fkasner] snipped incredible nonsense >>from a dedicated anti-semite ...[Tom Potter] > >Careful, Kaz, these days the ones who always cry anti-Semite >are now perceived to be the real bigots and racists at heart. Reflect on that and ask yourself why this came about. >Jewish moral and intellectual superiority, ......perhaps? >All good things do come to an end, Kaz,.......ahahahaha.... One has to wonder why some people do this. >Are they brainwashed, >or do they support the people who instigate >conflict and war for power and riches????? >Tom Potter >I don't know, nor care, what post preceded this one, >but since ever mankind existed some people of/in any >tribe have always and will always instigate conflict and >war for power and riches...... or what other reasons are >there for killing each other?......and if the conditions were >right, Tom, you'd be in the forefront doing the same, given >the personality you exhibit on the Usenet......ahahahaha... >Gee, I don't get this. How do I get into this power elite? I'm usually >>considered quite bright and I am Jewish. But I have never been in the >>company of any Jew who was in this power elite. My late brother-in-law >>was rich, Jewish, but not one of the power elite. How do I get there. >>Why have the other Jews kept this a secret from me and all my > relatively >>poor ancestors? Even the atomic bomb had those who blabbed about it. > Why >>haven't I heard about this grand scheme? I want in. I could contribute > a >>lot. I have lots of Christian relatives who would provide cover for > me. >>I know a few fairly powerful politicians and could influence them if > it >>were worth my while. Why has all this been hidden from me for so many >>years? > >Very astute and SEEMINGLY factual, Kaz, BUT..... >See, Kaz, apparently you really don't get it. It's very simple. >It is not so-called riches and power that gets Jews into deep shit >periodically and epically. A lot of goys have far more of each, yet >it normally does not get them into trouble. Most have learned how >to handle it: Discretely, smoothly and QUIETLY. But not the Jews. With the Jews it is the opposite. It is their > fucking, >incessant **loudness about themselves**, advocating that they are >richer, better, smarter and more lovable BECAUSE THE ARE JEWISH, >from private events to general media stuff, (like i.e. reporting on the >school > bus accident report that killed 20 US-goy kids into the >background......) >that is what makes the goyim, which outnumber Jew almost a million >fold, so irate and belligerent towards them. --- It is a simple as > that. >We won't even have to touch that for 50 years US taxpayers got >forced to pay billions, annually, to the Jewish State... with NOTHING > to >show for in return! Will it change? No. Why not? >Because that would mean for Jews to become Un-Jewish....ahahahaha.... >Of course Potter will come along now and give recital of his >Not all Jews are... song > .........aahahahaha........ahahahaha......... >Could it be that Tom Potter is a troll who makes this all up out >>of whole cloth? Naw, not Tom Potter. >>Everybody knows he is honest to a fault. >>FK > >ahahahaha........fault, Kaz, what fault?.........honesty and morality >is Potterian by DEFAULT.....ahahahahaha........but then it becomes >more evident with each post of yours that you, Kaz, are the Jewish >mirror image of Potter >>How so? >>Tom Potter > AHAHHAHAH..........ahahahahaha........Tom, Tom....... > until you answer this yourself, to yourself, about yourself and for > yourself > you are as blind to the solution of the problem as are your instigators > of > conflict and war for power and riches.......ahahahha.....ahahaha >>Are you asserting that fkasner >>is against the instigation of conflict and war >>for power and riches, by non-Jews like Bush, Blair and Rumfeld, >>and that fkasner thinks that the root causes of how conflict >>comes about should be discussed, researched, and something >>done to eliminate the root causes? Suit yourself Tom, but let me repeat for your benefit, Tom: Of course Potter will come along now and give recital of his >Not all Jews are... song > .........aahahahaha........ahahahaha......... which he just did with a variation thereof. >In other words, are you asserting that fkasner >>thinks that the root cause of conflict and war like >>instigation for power and riches, >>should be treated, rather than symptoms like >>boys throwing rocks at tanks demolishing their >>homes and neighborhoods, and girls becoming human bombers, >>because family members were murdered by soldiers? >>Tom Potter Suit yourself, Tom, whether in your, mine or other words. > You are reading different things off/in my post to Kaz then I do. > You come across to me here just as arguing for argument's sake. > Let me repeat for your benefit, Tom: Of course Potter will come along now and give recital of his >Not all Jews are... song > .........aahahahaha........ahahahaha......... which he just did with another variation thereof. Forget it. He's a dumb-ass troll who thinks that he can continue to ring > the bells of those who as a point of ethics and morality see the grand > Jewish conspiracy as a mean spirited attempt to pick on a small minority > who as emancipation has left them free to succeed are a wonderfully > handy scapegoat. Why even in countries that have almost no Jews they are > now having their prime ministers engage in blatant anti-semitism. What a > deal for a troll. But probably in point of fact Tom Potshard is an > anti-semite. But then again he probably hates almost anyone who he is > convinced won't punch out his lights for showing his hate. So he is what > is remarkable an anti-semitic troll. PLUNK. > FK It is interesting to see that when someone points out > that the Bolsheviks.instigated the class wars of the 1900's, > and are instigating the religious wars of the 2000's, > that fkasner implies that all instigators of conflict and war > are Jews, and that the folks who are opposed to war, > and the instigation of war, are anti-semitic trolls. All Jews are not instigators of conflict and war for profit, > any more than all Muslims are terrorists, > and all Gypsies are fortune tellers. And as can be seen in the cases of Bush, Blair and Rumfeld, > all instigators of war for profit are not Jews. It is interesting to see that fkasner uses the same > boilerplate tactic that the war-for-profit gang > of attacking the messenger, rather than addressing > the message in an intelligent, rational, MORAL way. I suggest that folks should think about what it means to > them, their families, to their country, and to society, > to give in to people who instigate conflict and war for power and riches, > and to wimp out because they are so aggressive in > attacking the folks who question their motives and tactics. Giving in to this kind of behavior encourages and rewards it. Would you let a whining, abusive, destructive child have his way? -- > Tom Potter http://tompotter.us > ==== Is there a version of the spectral theorem for symmetric matrices with entries in F_2 (field with two elements). Certainly one would need to work in the algebraic completion of F_2, but can we necessarily diagonalize symmetric matrices? TIA! ==== > Is there a version of the spectral theorem for symmetric matrices with > entries in F_2 (field with two elements). Certainly one would need to > work in the algebraic completion of F_2, but can we necessarily > diagonalize symmetric matrices? Try diagonalizing (0 1 // 1 0) :-) -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) ==== > > Is there a version of the spectral theorem for symmetric matrices with > entries in F_2 (field with two elements). Certainly one would need to > work in the algebraic completion of F_2, but can we necessarily > diagonalize symmetric matrices? > > Try diagonalizing (0 1 // 1 0) :-) Rats...shoulda thought of that myself. This shows that there's no Jordan form, either: letting A = [0 1; 1 0], we have char poly A = x^2 - 1 = (x-1)^2 so 1 is a repeated eigenvalue. There is only one eigenvector v=[1;1] but no solution to (A-I)x=v. Does anything work? Are there conditions that one can check to see when one has diagonalizability or Jordan form? ==== > >> >> Is there a version of the spectral theorem for symmetric matrices with >> entries in F_2 (field with two elements). Certainly one would need to >> work in the algebraic completion of F_2, but can we necessarily >> diagonalize symmetric matrices? >> >> Try diagonalizing (0 1 // 1 0) :-) > > Rats...shoulda thought of that myself. > > This shows that there's no Jordan form, either: letting A = [0 1; 1 0], > we have > > char poly A = x^2 - 1 = (x-1)^2 > > so 1 is a repeated eigenvalue. There is only one eigenvector v=[1;1] > but no solution to (A-I)x=v. ? What about (1 0)? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Needless to say, I had the last laugh. Alan Partridge, _Bouncing Back_ (14 times) ==== > What about (1 0)? Boy, I'm braindead today. So, is Jordan form OK over F_2? ==== >So, is Jordan form OK over F_2? Jordan form works for a matrix A over any splitting field for the characteristic polynomial of A. Or if you want to stay within the original field, you can use rational canonical form. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 ==== >>Let E be a subset of [a,b] with mE > 0, where m is the Lebesgue >>measure. Then there exists a closed subset F of E with mF > 0. >> >> If you are assuming that E is measurable, this is true. Otherwise, >> it is not. >You're saying there's a non-measurable set E with m(E) > 0? Depends somewhat on your definitions. If m(E) is the outer measure, then yes. If it is only defined for E measurable, then no. Books differ on this point. In fact, any non-measurable set has measure >0. --Dan Grubb ==== >>Let E be a subset of [a,b] with mE > 0, where m is the Lebesgue >>measure. Then there exists a closed subset F of E with mF > 0. >> >> If you are assuming that E is measurable, this is true. Otherwise, >> it is not. > >You're saying there's a non-measurable set E with m(E) > 0? > > Depends somewhat on your definitions. If m(E) is the outer measure, > then yes. If it is only defined for E measurable, then no. Books > differ on this point. > > In fact, any non-measurable set has measure >0. The point is, the OP EXPLICITLY CALLED m(E) the Lebesgue measure. Do books differ in calling Lebesgue outer measure Lebesgue measure? I think not. There's too much room for confusion. If you mean, some books use m(E) to denote the Lebesgue outer measure of E, and also the Lebesgue measure of E, then yes, I agree. (Although most would use m^*.) But I cannot find a single example where outer measure is CALLED Lebesgue measure. --Ron Bruck ==== [...] > >You're saying there's a non-measurable set E with m(E) > 0? > > Depends somewhat on your definitions. If m(E) is the outer measure, > then yes. If it is only defined for E measurable, then no. Books > differ on this point. > > In fact, any non-measurable set has measure >0. > > The point is, the OP EXPLICITLY CALLED m(E) the Lebesgue measure. > > Do books differ in calling Lebesgue outer measure Lebesgue measure? > I think not. There's too much room for confusion. > > If you mean, some books use m(E) to denote the Lebesgue outer measure of > E, and also the Lebesgue measure of E, then yes, I agree. (Although > most would use m^*.) But I cannot find a single example where outer > measure is CALLED Lebesgue measure. > Here's one... H. Federer, GEOMETRIC MEASURE THEORY (1969). 2.1.2 (page 53), definition of measure: The domain is 2^X, the collection of all subsets of a set X. Of course then only countable subadditivity can be required, and some special subsets are called measurable. Later (page 111) Lebesgue measure is a measure. ==== > In some texts, the term Lebesgue measure means what I call Lebesgue > outer measure. The OP did not say what text he was using. Which texts? ==== The Russell's paradox and the Barber of Seville share a comminality. The assumption of the existence of non-existent things leads to contradiction! Neither the 'Russell Class' nor the 'Barber of Seville' exist! The essence of these contradictions resides in the notion that: ~ER(xRy <->x ~(xRx)) or ~ER(yRx <->x ~(xRx)). Both are valid. That the x's and the y's are not of the same type is not relevant. Russell's theory of types does not resolve the paradox of the Barber of Seville! Although, his 'Types' does eliminate the possibility of expressing (x e x). (ix: (x e y) <->x ~(x e x )) is just as contradictory as is (ix: x=y <->x ~(x=x)). ~E!(iy: (x e y <->x ~(x e x)) ~E!(iy: x=y <->x ~(x=x)) In general: ~E!(iy: xRy <->x ~(xRx)), and, ~E!(yRx <->x ~(xRx)) for every R. It is the case that: Ax(xRy <-> ~(xRx)) and Ax(yRx <-> ~(xRx)) are both contradictions. for all R's. Witt ==== > There's a puddle of water in front of my house where the main water > line runs about 6.94 below grade. In that area is a pipe around the > shut-off valve that straddles the inlet/outlet pipes on the valve and > the bottom is resting on a couple fire bricks. > What's 6T mean? > The puddle covers about 5 square feet and didn't get smaller last > night, but no larger either. I don't know the relative humidity or > how well water will perk, and the water meter isn't moving with > everything off. There is also some plastic sheeting on the ground > under the water, but it is not complete. It is possible water was .93spilled.94 there by local children getting a > drink, and water under our home (during a previous leak) took MONTHS > to evaporate with the vents wide open. Here are some flow test results pouring water into a quart measuring > cup while timing it: A) flow rates that were not sufficient to move the water meter > 1) *1.24 gph > 30 gpd > 2) 1.77 gph > 42.5 gpd B) flow rates that make the water meter move very slowly > 1) 2.63 gph > 63 gpd > 2) 1.11 gph > 27 gpd > As A1&2 conflict with B2, data is suspect. Typo? Measurement or recording mistake? Inaccuraces of meter? Perhaps mininum useful data is 5 gph. ==== > There's a puddle of water in front of my house where the main water > line runs about 6ä below grade. In that area is a pipe around the > shut-off valve that straddles the inlet/outlet pipes on the valve and > the bottom is resting on a couple fire bricks. What's 6T mean? Comparison with the following paragraph suggests that '^T' indicates a close quote or double quote; so either 6 inches or 6 feet. > It is possible water was ãspilledä there by local children getting a > drink, and water under our home (during a previous leak) took MONTHS > to evaporate with the vents wide open. -- P.A.C. Smith The vast majority of Iraqis want to live in a peaceful, free world. And we will find these people and we will bring them to justice. ==== Wow, just as I was about to cross-post to RocketSci.LittleOldLadies.BasketWeaving an answer appears... LIKE I SAID ORIGINALLY, the meter isn't moving during the leak, if it is a leak. With the water off (and Grandma and children relocated for the night and following day) the puddle was gone but came back quickly when water was turned back on (cut off at street and house cut-off to prevent gravity sourcing from the house). What I wanted to know from all the supposed math gods was if there was enough info in the data to extrapolate the constant leak (which was not enough to move the meter) when the threshold of meter movement was passed and several flows measured. I did not think there was sufficient data, but (assuming reading skills) I thought someone here was at least bright enough to know if there was an obtainable answer... If you didn't glean this from the OP, I'm the plumber and I didn't want to dig up the pipes for nothing while I'm dealing with the flu - obviously now, I have to, but - Is there an answer to the original post? This ain't math, it's plumbing psychology detective work. > At bed time, securely tape all the toilet and faucet handles with duck > tape. Take flashlight, read water meter and note time. Go to bed and > don't have any wet dreams. ;-) Upon waking or whenever you get around to > it, such as after work, read water meter and note time. If in the mean time the water meter has been spinning its wheels, you know > its your leak to entertain or aggravate you plumber who fixed the privious > leak. Otherwise its water company's leak, who before letting you pin the > leak on them, will repeat the test. Furthermore, be there any doubt about the source, such as being ground > water, the water company can take water samples and test for clorine, > florine or whatever other locally popular water pollutants the water > company is found of making you drink and bath in. >There's a puddle of water in front of my house where the main water > >line runs about 6?? below grade. In that area is a pipe around the > >shut-off valve that straddles the inlet/outlet pipes on the valve > >and the bottom is resting on a couple fire bricks. > 6 what? Inches, feet, meters? >The puddle covers about 5 square feet and didn't get smaller last > >night, but no larger either. I don't know the relative humidity or > >how well water will perk, and the water meter isn't moving with > >everything off. There is also some plastic sheeting on the ground > >under the water, but it is not complete. > >It is possible water was ^îspilled^ï there by local children getting > >a drink, and water under our home (during a previous leak) took > >MONTHS to evaporate with the vents wide open. > You could bail the water and see what happens. > Is the water tap for public use? If not, get a locking faucet. > Otherwise complain to the water department or other lible agency. ---- ==== > LIKE I SAID ORIGINALLY, the meter isn't moving during the leak, if it is a > leak. > Didn't seem you got any results. Rate of flow not important, what's important is what volumn is needed for meter to register noticeable change. Take a shower and see how much meter moves. If much, then flush toliet a time or two and see how much meter moves. > With the water off (and Grandma and children relocated for the night and > following day) the puddle was gone but came back quickly when water was turned > back on (cut off at street and house cut-off to prevent gravity sourcing from > the house). > Thus puddle comes from water line. Is not puddle sufficient volume to move meter? What to check? Water faucet or what ever where children play? Fittings near and around said location? Pipes underneath puddle? How far down is the pipe? Six inches? That'd make for easy spot digs to find what direction leak is moving. Did watching puddle fill give any hint entry point into puddle of water? When my sewer was fixed, contractor dug 3 ft thru dry hard packed dirt to have miss his target. Eventually he figured out where to dig. Happy this dig was within access of back hoe that found pipe underneath sidewalk 12 ft down. Yea, 100 year old line with no city records as to location. > What I wanted to know from all the supposed math gods was if there was enough What feelings does an appellation 'plumer god' or 'pipe god' impart? > info in the data to extrapolate the constant leak (which was not enough to > move the meter) when the threshold of meter movement was passed and several > flows measured. I did not think there was sufficient data, If you didn't glean this from the OP, I'm the plumber and I didn't want to dig > up the pipes for nothing while I'm dealing with the flu - obviously now, I > have to, but - > Even feeling well, nobody would want to. What a nuisance to have people be gone overnight to assure accurate test. > Is there an answer to the original post? > I've made some comment thereto in other post. ==== One thing that simply can't be denied is that Legendre's Method is ugly. Besides being ugly it's clunky. It's a really ugly, clunky *algorithm* like consider. To count the composites up to 10, you need to already know that 2 and 3 are primes. The method is too stupid to know that 2 and 3 are primes, so you have to tell it. Then you take floor(10/2) - 1 = 4, to get the composites with 2 as a factor, and those composites are 4, 6, 8, and 10. Next you take floor(10/3) - 1 to get the composites with 3 as a factor, and those are 6 and 9. OOPS! What the fuck!!! You've DOUBLE COUNTED as you have 6 in both freaking lists!!! So now what to do, what to do, oh, you subtract floor(10/6) = 1, so that you get the correct count which is 5 for the 5 composites 4,6,8,9, and 10. That's the single big idea that mathematicians managed to come up with over a hundred years for counting primes. Then they tweaked it, fiddled with it endlessly, so there are all these names for different algorithms. Now I come along, over a hundred years after Legendre came up with his *crappy* method, and starting from scratch, I come up with a MUCH, much cleaner idea. Then stupid mathematicians get on *my* case because there are all these dumb algorithms lying around from tweaking the crappy, ugly method. Mathematicians are so fucking stupid. I hate mathematicians. It's not my fault you dimwits just came close to the elegant function!!! I found it, so play fair!!! Put my partial difference equation in the textbooks! You stupid fucking twit mathematicians. James Harris My math discoveries, found for profit http://mathforprofit.blogspot.com/ ==== > One thing that simply can't be denied is that Legendre's Method is > ugly. Besides being ugly it's clunky. It's a really ugly, clunky *algorithm* like consider. To count the composites up to 10, you need to already know that 2 and > 3 are primes. The method is too stupid to know that 2 and 3 are > primes, so you have to tell it. Then you take floor(10/2) - 1 = 4, to get the composites with 2 as a > factor, and those composites are 4, 6, 8, and 10. Next you take > floor(10/3) - 1 to get the composites with 3 as a factor, and those > are 6 and 9. OOPS! What the fuck!!! You've DOUBLE COUNTED as you have 6 in both > freaking lists!!! So now what to do, what to do, oh, you subtract floor(10/6) = 1, so > that you get the correct count which is 5 for the 5 composites > 4,6,8,9, and 10. That's the single big idea that mathematicians managed to come up with > over a hundred years for counting primes. Then they tweaked it, fiddled with it endlessly, so there are all > these names for different algorithms. Now I come along, over a hundred years after Legendre came up with his > *crappy* method, and starting from scratch, I come up with a MUCH, > much cleaner idea. Then stupid mathematicians get on *my* case because there are all > these dumb algorithms lying around from tweaking the crappy, ugly > method. Mathematicians are so fucking stupid. I hate mathematicians. It's not my fault you dimwits just came close to the elegant > function!!! I found it, so play fair!!! Put my partial difference equation in the > textbooks! You stupid fucking twit mathematicians. > If I knew someone was evil, out to get me, stupid, etc., I wouldn't continuecorrespondence with them. Why haven't you figured this out yet? Is there some sort of problem you have that you need to have acceptance? Maybe some father issues ??(I'm assuming of course that you know who your father is.) We'll see... MB ==== > If I knew someone was evil, out to get me, stupid, etc., I wouldn't > continuecorrespondence with them. Why haven't you figured this out yet? Is > there some sort of problem you have that you need to have acceptance? Maybe > some father issues ??(I'm assuming of course that you know who your father > is.) Mr. Harris has a condition known as NPD. (See below and this will answer your questions, this egomaniac has NPD and is totally deranged with fits of grandeur)! Here is a clinical definition: *** Diagnostic criteria for 301.81 Narcissistic Personality Disorder (cautionary statement) A pervasive pattern of grandiosity (in fantasy or behavior), need for admiration, and lack of empathy, beginning by early adulthood and present in a variety of contexts, as indicated by five (or more) of the following: (1) has a grandiose sense of self-importance (e.g., exaggerates achievements and talents, expects to be recognized as superior without commensurate achievements) (2) is preoccupied with fantasies of unlimited success, power, brilliance, beauty, or ideal love (3) believes that he or she is special and unique and can only be understood by, or should associate with, other special or high-status people (or institutions) (4) requires excessive admiration (5) has a sense of entitlement, i.e., unreasonable expectations of especially favorable treatment or automatic compliance with his or her expectations (6) is interpersonally exploitative, i.e., takes advantage of others to achieve his or her own ends (7) lacks empathy: is unwilling to recognize or identify with the feelings and needs of others (8) is often envious of others or believes that others are envious of him or her (9) shows arrogant, haughty behaviors or attitudes Reprinted with permission from the Diagnostic and Statistical Manual of Mental Disorders, fourth Edition. Copyright 1994 American Psychiatric Association *** ==== that's a nice explanation, if it's true of Legendre's crappola, but what does floor mean -- greatest integer part? so, then, what is different about your method? > To count the composites up to 10, you need to already know that 2 and > 3 are primes. The method is too stupid to know that 2 and 3 are > primes, so you have to tell it. > > Then you take floor(10/2) - 1 = 4, to get the composites with 2 as a > factor, and those composites are 4, 6, 8, and 10. Next you take > floor(10/3) - 1 to get the composites with 3 as a factor, and those > are 6 and 9. > So now what to do, what to do, oh, you subtract floor(10/6) = 1, so > that you get the correct count which is 5 for the 5 composites > 4,6,8,9, and 10. > I found it, so play fair!!! Put my partial difference equation in the > textbooks! > http://mathforprofit.blogspot.com/ ==== > I found it, so play fair!!! Put my partial difference equation in the > textbooks! Text books take quite a bit of time to write. Since your method is so new I wouldn't expect to see it in the books for another year or two. Why don't you go away in the meantime. V. -- homepage: cs utk edu tilde lastname ==== > One thing that simply can't be denied is that Legendre's Method is > ugly. Besides being ugly it's clunky. > You stupid fucking twit mathematicians. > James Harris Jimmy Baby! You are friggin' beautiful! I am rolling on the floor laughing my ass off! ==== > You stupid fucking twit mathematicians. Mr. Harris, your language is quite colorful during holiday seasons. As for us twits, please provide a greater estimate (x = 4*10^22) for pi(x) that beats this one. See: http://numbers.computation.free.fr/Constants/Primes/countingPrimes.html In fact, lets make it simpler for you, please duplicate this result using your magnanimous method! Come on, these twits certainly surpassed anything you have done, haven't they? Do you believe your method is superior, then prove it (and we all know the answer don't we JSH?). You are such a joke! Get a life you ignoramus! ==== Let X,Y,X',Y' be Z-modules (that is, abelian groups). Suppose X+Y is isomorphic to X'+Y' (where + should mean the ordinary direct sum as Z-modules). When (and how) can one conclude that X is isomorphic to X' and Y to Y'? Now, I don't think it's true in general, so, what if Y=Y'=Z ? What if X is known to be free/projective ? I'd be grateful about any hint in that direction, or counterexamples to such a claim... Philipp ==== Let X,Y,X',Y' be Z-modules (that is, abelian groups). Suppose X+Y is >isomorphic to X'+Y' (where + should mean the ordinary direct sum as >Z-modules). When (and how) can one conclude that X is isomorphic to X' >and Y to Y'? As written, almost never: pick any two nonisomorphic Z-modules X and Y, and let X'=Y, Y'=X. >Now, I don't think it's true in general, so, what if Y=Y'=Z ? For ->finitely generated<- Z-modules, it is true that if X+Y = X+Y', then Y=Y'. For non-finitely generated Z-modules, it need not be true. E.g., let Y be the direct sum of a countable number of copies of Z, and let X=Z and X'=0, or any direct sum of a finite or infinite countable number of copies of Z. > What if >X is known to be free/projective ? Still not true for free: the example above has both Y and X free, Y=Y', but X need not be isomorphic to X'. ====================================================================== It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) ====================================================================== Arturo Magidin magidin@math.berkeley.edu ==== I have one question regarding Goldbach's Conjecture. It states that 'every even integer greater than 2 is sum of two prime numbers'. I'm curious whether it says about two different primes or whether it allows situation where even number is sum of two equal (that means the same) primes. As an example take 6 = 3+3. Of course, it can be also written as 6=1+5, but, does the hypothesis say that always, for every even number, must exist TWO different primes, whoose sum give that number, or does it allow situation as specified above? -- Krzysiek ==== > I have one question regarding Goldbach's Conjecture. > It states that 'every even integer greater than 2 is > sum of two prime numbers'. > I'm curious whether it says about two different primes > or whether it allows situation where even number > is sum of two equal (that means the same) primes. > > As an example take 6 = 3+3. That's a perfectly good example. It is in fact the only way to decompose 6 into two primes. > Of course, it can be also written as 6=1+5, > but, does the hypothesis say that always, for every > even number, must exist TWO different primes, > whoose sum give that number, > or does it allow situation as specified above? First: 1 is not prime. Second: The conjecture (obviously) means the sum of two primes--which statement does not say two different primes, so it doesn't mean two different primes. Thomas ==== Krzysztof Olczyk <> grava .88 la saucisse et au marteau: > As an example take 6 = 3+3. > > Of course, it can be also written as 6=1+5, > but, does the hypothesis say that always, for every > even number, must exist TWO different primes, > whoose sum give that number, > or does it allow situation as specified above? As 1 is not considered as prime, I guess the two primes may be equal :) -- Nicolas X-Cise: tanbanso@iinet.net.au X-CompuServe-Customer: Yes X-Coriate: admin@interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Pose: george_cox@btinternet.com X-Punge: Micro$oft ==== at 10:57 PM, Austrolopithecus Afarensis quoted: >The problem is this same brush is used to tar *anyone* outside the >establishment who attempts to advance fundamental objections to or >criticisms of the canonical theory, no matter how rational and articulate That claim, of course, is precisely the sort of statement that gets [people labeled as cranks. There have been people who published criticisms and fundamental objections to GR without being tarred by that brush, but they published papers on Physics instead of polemics against Einstein or against the establishment. Follow-up set. -- Shmuel (Seymour J.) Metz, SysProg and JOAT not reply to spamtrap@library.lspace.org ==== the Stanford low-orbit thing sounds interesting, but how could it be definitive, resting upon the same-old interpretation of Herr E.? > That claim, of course, is precisely the sort of statement that gets > [people labeled as cranks. There have been people who published > criticisms and fundamental objections to GR without being tarred by > that brush, but they published papers on Physics instead of polemics > against Einstein or against the establishment. --ils duces d'Enron! <1f366ae.0311240654.1bd2d91f@posting.google.com> X-Cise: tanbanso@iinet.net.au X-CompuServe-Customer: Yes X-Coriate: admin@interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Pose: george_cox@btinternet.com X-Punge: Micro$oft ==== at 06:54 AM, undeniablefact@yahoo.com (Undeniable) said: >At any rate, whatever happened >to the ultimate test of GR involving gyros in low orbit? There is no ultimate test of any theory. The test you're thinking of seems to be back on track. -- Shmuel (Seymour J.) Metz, SysProg and JOAT not reply to spamtrap@library.lspace.org ==== > This is the first time someone is going through same thought process > through which I have gone. I was just scratching my head day and night > for nine months. The mistake I made that I was using BOLT(!!!) > instead of SPRING(or any source of restoring force). If you use bolt > AB and CB instead of stretched springs and try to push ends A and C of > both arms in direction towards point B, same thing will happen which > you have shown in the figure. > > Just imagine that AB and CB are two shock-ups in which there is > tremendous restoring force directed towards point B, is acting at > point A and C. Now just put your both hands on end A and end C of > these shock-ups, What do you mean by tremendous restoring force? Will the springs want to stretch or shrink? More precisely, which way does your restoring force point to?' > > (1)will the vertex point B move in downward direction?(As, by this > time you know very well, angle ABC does not change.) Depends on the direction of the force of the springs. > > (2)if your hands are feeling restoring force in direction towards > point B ONLY, then why both ends of left and right arm will not feel > same force in direction towards point B ONLY? They will feel the whole thing twitching until it reaches equilibrium. And they will feel the distance AC change. > > (3)The biggest trick involved in this drama is that if both ends of > left and right arm are feeling restoring force towards point B, these > arms just can't move in downward direction AT EXACTLY SAME MOMENT. As long as AD and CD can move freely and independently of each other they can. > I am going to recover myself psychologically and physically. Great idea. Greetings! Volker ==== >> This device does work. Did you build it? More importantly, did you use it with no other assists. One of the earliest lessons that our development group learned was to run their own code _all the time_. There's nothing like having a system crasher influence developers' decisions about whether they'll write new code or fix what they just broke so they can use a working machine to write the new code. /BAH ==== > I am going to recover myself psychologically and physically.(This is > one of the trick He trained me to do in extreme situation, PULL > BACK!. This is no less than a military operation for me). Military operations aren't successful if the attacker retreats as soon as fatigue sets in. People here haven't even been overly nasty with you, compared to certain others. Perhaps because they sense that you are probably a very unbalanced and vulnerable person underneath your veneer of, dare I say it, holy warrior. However.... He trained you in how to protect yourself, but it is also He who hurls all the attacks your way. Explain how that isn't a contradiction. > I am going > back to my workplace, take some rest. As I will not have any access to > internet, I will not be able to see or post here. This is exactly why the key ingredient of deprogramming is to take the person into an isolated situation, and refuse to stop the arguments that invalidate the logic of their delusions. But, I Will Be Back! As soon as you, once more, believe your own story fully, I'm sure. A tip: Stop posting to this kind of newsgroup if you can't handle rational arguments against your point of view, without perceiving yourself as being under attack. ==== euan ==== > > >>Right on. The people who say another thing coming are probably the >>same ones who say I could care less. >>Gib > > > I could care less is just sarcasm isn't it? Makes perfect sense then. > > I don't know. What do you mean by sarcasm in this context? > > Gib I mean, I could care less said sarcastically means the same thing as I couldn't care less meant literally. Whenever I've heard people say the former (which is not too often) they've been careful to say it in a _very_ sarcastic tone - but maybe where you are it's caught on to such an extent that people no longer do this. David ==== > > >Right on. The people who say another thing coming are probably the >>same ones who say I could care less. >>Gib >I could care less is just sarcasm isn't it? Makes perfect sense then. >>I don't know. What do you mean by sarcasm in this context? >>Gib > > > I mean, I could care less said sarcastically means the same thing > as I couldn't care less meant literally. Whenever I've heard people > say the former (which is not too often) they've been careful to say it > in a _very_ sarcastic tone - but maybe where you are it's caught on to > such an extent that people no longer do this. Where I am (New Zealand) nobody says could when they mean couldn't - it is a usage that I've noticed on the web and on US media. It seems that you are suggesting that some people say I _could_ care less, implying but not much less. But I don't think this is way I've usually heard it, rather I get the impression that the speaker just doesn't realize what he's saying. To bring us back to mathematics, I'm reminded of the mad hatter's tea-party, where Alice is offered more tea, when she hasn't yet had any. She protests, but is corrected (I have to give the full quotation): Take some more tea, the March Hare said to Alice, very earnestly. I've had nothing yet, Alice replied in an offended tone: so I can't take more. You mean you can't take less, said the Hatter: it's very easy to take more than nothing. Nobody asked your opinion, said Alice. Who's making personal remarks now? the Hatter asked triumphantly. It's amazing how much entertainment JSH provides, isn't it? Gib ==== > > >Right on. The people who say another thing coming are probably the >>same ones who say I could care less. >>Gib >I could care less is just sarcasm isn't it? Makes perfect sense then. >>I don't know. What do you mean by sarcasm in this context? >>Gib > > > I mean, I could care less said sarcastically means the same thing > as I couldn't care less meant literally. Whenever I've heard people > say the former (which is not too often) they've been careful to say it > in a _very_ sarcastic tone - but maybe where you are it's caught on to > such an extent that people no longer do this. > > David My experience is that, by far, the vast majority of users of this form of the expression (I could care less) fail to recognize the sarcastic (or even the ironic) tone of that version, as contrasted to the literal tone of the other version (I couldn't care less). Whether they're irony-deficient or simply don't hear the words any longer, due to their countless years of repetition, I could never determine. As for the identity of users of the phrase I could care less and the users of the phrase another thing coming, I grew up in a family that used another thinG coming as well as I couldn't care less We (that is, I) always understood it in the sense that the previous poster (Jesse F. Hughes), who interpreted it something like if you think X, you're going to find out that the truth is Y, another thing. Well, he said it much better: > > > When one says, If you think X, you've got another thing coming, X is > typically a future event or a regularity. The word think here is > used synonymously with expect or believe, but I want to stress the > expect meaning. > > So the first thing is not the thought about X, but X itself (very > loosely). If you expect X, you have another thing coming. If you > think I'll give you $5, you have another thing coming (an experience > contrary to the actual giving of the $5). This is the interpretation > that I have always had for the expression. ==== I really *could* care less, you smouldering piece o'****, but then I'd just have to ignore you. please, dyspose of yourself at once -- and don't start a forest fire! >I could care less is just sarcasm isn't it? Makes perfect sense then. >>I don't know. What do you mean by sarcasm in this context? --ils duces d'Enron! ==== >> Right. It's that deliberately bad grammar that sounds like >> think is the word intended there, in an attempt to be >> affectedly folksy. It sounds like one of those made-up >> Texas proverbs in other words. >> Now I'm gonna hear from more Texans than you can shake a >> stick at. As a technical Texan[1], I claim that thing is the commonest Texas >version. At least in my family. Well then you'll know what I'm talking about with Texas language. I didn't really mean proverb. Probably simile. Or is it metaphor. That boy is slower than .... This newsgroup is slicker 'n.... You know, this stuff: http://uts.cc.utexas.edu/~samoht/texassimiles.html So tell me. Does anyone actually talk this way? I'm a damn lib'rl east-coast yankee. For all I know, the whole damn country west of Pittsburgh talks this way. - Randy <87k75qmyqh.fsf@phiwumbda.org> <87r7zy3cvu.fsf@phiwumbda.org> <8765hatedb.fsf@phiwumbda.org> <87smkdvi03.fsf@phiwumbda.org> ==== > Right. It's that deliberately bad grammar that sounds like > think is the word intended there, in an attempt to be > affectedly folksy. It sounds like one of those made-up > Texas proverbs in other words. Now I'm gonna hear from more Texans than you can shake a > stick at. >>As a technical Texan[1], I claim that thing is the commonest Texas >>version. At least in my family. Well then you'll know what I'm talking about with Texas language. I > didn't really mean proverb. Probably simile. Or is it metaphor. That boy is slower than .... > This newsgroup is slicker 'n.... [owl shit on a brass > doorknob] At least that's how I finish #2. #1 has more endings. You know, this stuff: > http://uts.cc.utexas.edu/~samoht/texassimiles.html So tell me. Does anyone actually talk this way? I do, but it's a bit affected. I think it's funny, so I picked up the habit on purpose. My family uses plenty of Southern and Texas phrasing, but not so much comparisons like those. More things like I'll be on you like ugly on an ape, and if it was a snake, it would've bit you, and That was bigger'n all get out. I'm a damn lib'rl east-coast yankee. For all I know, the whole damn > country west of Pittsburgh talks this way. For eight years, at least a small part of Pittsburgh talked that way, too. -- Jesse F. Hughes Truth is common stuff, ready to your hand, but lies you have to make yourself, and you can't be sure they are any good until you've used them --- and then it's too late. John Steinbeck ==== > I'm a damn lib'rl east-coast yankee. For all I know, the whole damn > country west of Pittsburgh talks this way. > > For eight years, at least a small part of Pittsburgh talked that way, > too. OK, I'll bite. What eight years? Is this a reference to some presidential administration? - Randy <87k75qmyqh.fsf@phiwumbda.org> <87r7zy3cvu.fsf@phiwumbda.org> <8765hatedb.fsf@phiwumbda.org> <87smkdvi03.fsf@phiwumbda.org> <87y8u4gb9c.fsf@phiwumbda.org> ==== >> I'm a damn lib'rl east-coast yankee. For all I know, the whole damn >> country west of Pittsburgh talks this way. >> >> For eight years, at least a small part of Pittsburgh talked that way, >> too. OK, I'll bite. What eight years? Is this a reference to > some presidential administration? No, it's the eight years I spent in Pittsburgh doing my damnedest to influence speech patterns. My damnedest was essentially limited to teaching a German or two a few key phrases and the finer points of the one-syllable vs. the two-syllable versions of shit. My main German student was willing to learn and got the usage patterns down pretty well, but Hoo-doggies! That puppy was slicker'n owlshit on a brass doorknob! just doesn't flow when your accent is similar to Schwarzenegger's[1]. It probably doesn't help that I never made a very persuasive Okie my damn self. Even when I still lived there, the locals would regularly ask me where I was from. Blame too many Monty Python records during a misspent youth. Footnotes: [1] Yes, yes, I'm sure that my buddy's accent was nowhere near Schwarzenegger. It was close enough for my crudely trained ear. -- 'Every man who has ever lived holds tight to the belief that for him alone the laws of probability are canceled out by love[...] Therefore, you will marry Guinevere. You do not want advice --- only agreement.' Merlin sighed... -- John Steinbeck ==== > What is this crap? Repunctuate. What! ... Is this crap? The answer may be left as an exercies. -- Chris Henrich Drag people to events to invite them. -- A tool tip in Apple's iCal program. ==== Assume an n x n Go board. We place k stones on this board so that they are well-connected (i.e. for every stone there is a stone in an adjacent square, not counting diagonals squares). What is the maximum number of liberties (empty squares with at least one stone in an adjacent square) achievable and what is the minimum number of stones we can achieve it with? I know the maximum upper bound for the liberties on a board n x n is: 4 + (n^2 / 3 - 1) * 2 = 2 + 2n^2 / 3 I have a possible optimal placing for a 19x19 board but would like to prove it can't be improved. ==== > Assume an n x n Go board. ... for every stone there > is a stone in an adjacent square, not counting ... Ooops? Stones are placed on the intersection points, not within the squares. +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ A nice little | | | | | | | | | 9x9 version for +---+---+---+---+---+---+---+---+ exercising. | | | | | | | | | It's possible +---+---+---+---+---+---+---+---+ to play terribly | | | | | | | | | exciting games +---+--( )--+---+---+---+---+---+ even on such a | | | | | | | | | small board! +--( )-(#)-( )--+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ ( ) = white stones | | | | | | | | | (#) = black stones +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ Black stone (#) with one liberty left. (No not left but below :-) RR ==== Please stop wasting your time and mine. Witt > G. Frege, apparently, would agree with your #2. For Frege: > If, x/y is defined (the z: x*z = y), then 0/0 = 0, and 1/0 = 0. > Frege surely wouldn't agree with that. The z: x*z = y > is commonly called y/x, even by Frege and all your other > authorities included :-) RR ==== > > the Riemann Hypothesis follows from the existence >> of a prime between n and n + sqrt(n).... Really? Seems quite unlikely. > > Really? Why does that seem unlikely? There are numerous reasons. If the statement were true, there would be a very powerful heuristic argument in favor of RH, which I've never heard about. It is easy to create a set of integers with the same density distribution as the primes, and having the above property, but such that the Zeta function has roots with real part not = 1/2. It looks like a confusion with another result concerning the difference between pi(n) and li(n) The result is startling enough that if it were known to be true, everyone, including me and you, would have heard about it. ==== > >>> the Riemann Hypothesis follows from the existence >>> of a prime between n and n + sqrt(n).... >>Really? Seems quite unlikely. >> >> Really? Why does that seem unlikely? There are numerous reasons. If the statement were true, there would be a very >powerful heuristic argument in favor of RH, which I've >never heard about. It is easy to create a set of integers with the same >density distribution as the primes, and having the above >property, but such that the Zeta function has roots with >real part not = 1/2. It looks like a confusion with another result concerning >the difference between pi(n) and li(n) The result is startling enough that if it were known to >be true, everyone, including me and you, would have heard >about it. All excellent reasons, except for the last half of the last one: Doubtless I would have heard of it, but there's no reason I would remember it - I know that there are precise relations between the distribution of primes and RH but I have no idea what they are, never having had any reason to try to put that into long-term memory (hence I wasn't certain that I had not heard of it...) David C. Ullrich ==== > I assumed that what he thought was unlikely was the statement > that RH _follows_ from the existence of a prime between n > and n + sqrt(n) (for large n, of course). Do you have a reference > for that? Alas, no. It may be that I misremembered something I heard/read long ago, and that the implication only goes in the other direction. -- ==== > Real science is data first. I agree with this. For example the tens of millions of measured redshifts that agree > perfectly with the Big Bang model. That's DATA for you, in capital > letters. ROTFLMAO! How do they agree? They find that objects which are more redshifted tend to be farther > away. Thats how. 1) Assume objects with larger redshift are farther away. 2) Claim that observing larger redshift proves it is farther away. That's one of the shortest circular arguments I've seen in awhile. Because we simply *assume* that redshift > gives us a distance. You would assume such a thing, yes. A scientist wouldn't. That's why you'll never be a scientist. Too bad so many have. That is the basis for the Big Bang. But we agree that Big Bang cosmologists aren't scientists. > (There are a few dozen *measurements* of distance > versus redshift. Given that I have done a whole lot more than a few dozen myself, I > can confidently state that you are lying. Fascinating. Citation, please. (I'll settle for 100, independent distance measurements that do not depend directly or indirectly on the Hubble law) > There are *tens* *of* *millions* of measurements of distance versus > redshift in perfectly fine agreement with Big Bang cosmology. Yawn. Since you get to adjust parameters both on distance and on the BB side, I'm not impressed. > Besides, it only takes one real case to disprove the assumption. There is no assumption anywhere. Sure there is. You are assuming that the *ONLY* contribution to redshift is doppler (or doppler and expansion if you want to be picky). No 'tired light', no 'change in physical parameters', no nothing. > You pick a light source -- say a > particular type of galaxy or a particular type of supernova or a > particular type of variable star (it doesn't really matter, just pick > some object you can recognize) and you measure their brightness and > some are bright and some are dim and there's some distribution of > brightness between the brightest and the dimmest and when you look at > objects that are farther away then you'll see the exact same > distribution of brightness except that all objects will appear dimmer > because they're farther away. Sure, with the adjustible parameter of extinction to help you out. And you can't know that the distribution of brightness is the same without assuming the hubble relation that gave you the distance -- or vice versa. > And then you measure redshift and you plot apparent brightness against > redshift and you find that by and large redshift increases as > brightness decreases. Ah! But the 'by and large' is the assumption. The point being that there may be exceptions. > The Big Bang does NOT claim that all objects must lie on a perfect > redshift/distance curve, merely that there is an *additional* linear > term in any distance-redshift relation. Namely the Hubble flow. And that is the assumption. That redshift is only and always due to motion (physical or expansion). > Whatever population of things you measure the redshift distribution > of, you find an additional redshift the farther things are away -- by > whatever means the distance is determined. But that's the whole point! The whatever means is always the redshift. All 'direct' measurements of distance end with parallax. Beyond that is theory -- adjustible to meet the BB postulate. UNLESS one finds a single connected group of objects with significantly different redshift. > And it doesn't have to be determined terribly precisely: if your > distance determination has a factor two uncertainty, then you get a > spread of measured points around the linear term. If it has a factor > *ten* uncertainty then you get a *wider* spread around the EXACT SAME > linear term. We all agree there are uncertainties. So why are you so 'certain' you are correct? > Besides, it only takes one real case And in these tens of millions of observation there is not a single > real case that disproves anything whatsoever. Even if there were 10 million independent determinations of distance, the simple exclusion of contrary data makes your claim meaningless. > If you (or Mr. Arp or > anybody) want to claim that any two galaxies are in interaction with > each other then the burden of proof is on you (or Mr. Arp or whoever > makes the claim). Precisely. And Dr. Arp was well on the way to providing just that. Which is -- of course -- why Dr. Arp was kicked off telescopes. True science in the modern sense. The burden of proof is on Arp -- so we'll simply remove from him the ability to collect proof (data). Problem solved. > But JUST to make sure we're not missing something somewhere, we > actually look at the stuff Mr. Arp hands us and we find that the > galaxies in question are NOT in fact in interaction with each other. How did you manage to determine this fact? Quite simply, you cannot have any basis for your claim -- except theory (which is not fact.) Quite simply the ONLY argument that can be used on Arp's objects is the word coincidence. (So your claiming fact is a bold-faced lie.) Let's take an example. An (apparently) physically connected group of three galaxies. Two which appear to be ejected from the main one. The outer two have significantly higher redshift than the main one. The connection appears to be gas. How did you 'disprove' the connection? AH, yes! The redshifts are different THEREFORE, the objects cannot be connected BECAUSE the distance is different, because we assume redshift determines distance (assuming our conclusion). The ONLY valid objections are that this ONE case is simply a coincidence. There is no way to PROVE or DISPROVE one case. But a claim of coincidence can only be made from a statistical viewpoint. So ... obviously ... one must do a number survey of such possible coincidences. Which is what Arp was doing when he was kicked off the telescopes. > So he hands us another pair and we confirm AGAIN that he has no point. > Then we do the same thing AGAIN and AGAIN and after the tenth or > hundredths time we can confidently state that we have given the Arps > of the world their due hearing and all their points simply evaporate > upon inspection. There is NOT ONE observation in the universe in contradiction with the > Big Bang. There are a few dozen observations that the Arps of the > world can't figure out (difficult observations, all of them) and it is > simply a *lie* to proclaim that they pose some kind of contradiction > to the Big Bang. They don't. No matter how long we stare into the > heavens, there will always be some observation at the sheer limits of > observability that is difficult to interpret -- but that's all they > are: difficult to interpret. Not any kind of disproof. And you are a pathetic liar, yourself. For you obviously understand the issue. You claim that each single case of Arps is not proof. But not because it is not true -- but because you can claim each case is pure coincidence (which is why you weaseled and never mention *HOW* each observation of Arp's is wrong). You slime over to claiming they are difficult to interpret, and only a few dozen can't be figured out. You evade the central issue: That you claim the burden of proof is on Arp, and then you deliberately prevent Arp from gathering data that could provide that proof. If you had an ounce of honesty or professional integrity, you would be agitating for Arp's reinstatement to viewing time. Then you could evaluate Arp's data and the statistics would tell all. And -- if you are correct -- the statistics would support you. But your actions (and those of the rest of the BB heirarchy) is simply to avoid the possibility of disproof, by terminating data collection. > Not to mention the CMB (and in particular the *anisotropies* > in the CMB in perfect agreement with what you'd expect from the Big > Bang). And so forth. To which version of the BB are you referring? I am referring to the mathematical model that the word Big Bang has > referred to for at lest 50 years now. There are a dozen versions of the BB, over the past 50 years. To which version, specifically, are you referring? With inflation, or without? With light elements or without? With dark matter, or without? Etc... > Certainly not the predictions > (which were 1 part in 100, prior to COBE). Ah, the predictions. Let me submit that you have no clue what you're talking about. I figured you'd dodge the issue. :) > Let me further submit that COBE was in perfect agreement with BB > cosmology LOL! > and that WMAP has pushed the same agreement an order of > magnitude further. As Mike Turner tends to say cosmology is becoming > boring, every measurement only confirms and refines the previous one. ROTFLMAO!!! Those are pretty major 'refinings!' > The BB constantly adds new > epicycles every time something is observed. Nothing new has been added to the Big Bang since the original FRW > metric was been worked out unless you're willing to consider inflation > to be a part of the Big Bang Model now in which case noting has been > added to the Big Bang since inflation. That's it. Everything else > follows from what was there all along. ROTFLMAO!!! Do you really believe this? How old are you? And can I sell you a bridge? > To first order (and to second one) the CMB should be isotropic and > thus we build a satellite to measure that and it confirms what we > expect to within its margin of observational error. LOL! COBE was designed to measure the (predicted) 1-part-in-100 to 1-part-in-1000 variation in the background. So COBE was designed to go down to 1-part-in-10000, just to be safe. There was NO signal at 1-part-in-10000 (a factor of 10 to 100 below BB predictions). Data enhancement has been used to find a signal at 1 part in 100000 level. But that is a factor of 100 to 1000 below the actual predictions of the BB. NOW you claim that the BB 'predicted' 1 part in 100000. Yeah, that's the ticket! > Then someone says > well, if we're really nitpicky about the equations then there should > actually be this tiny, most minute little anisotropy in there, but way > beyond COBE's capability. So we scratch our heads, build a satellite > more than an order of magnitude more sensitive (WMAP) and sure enough > it finds these minute anisotropies *exactly* at the level at which > they were expected. COBE would NEVER have been built too coarse to identify the BB predictions. It's just that the BB 'predictions' change (as always) as soon as another contradiction is found. > Expected from the exact same Big Bang model people referred to before > COBE existed. LOL! This is too good to parody. {snip} And while they're at it, they squeeze these few oddball cases into > roles they never had: taking an object who's redshift seems to place > it outside the hubble flow to pretend that there IS NO hubble flow and > the entire theory explaining it must be wrong. Yes. That is the inescapable logic of science. No, it isn't. You are merely clueless what you are talking about. The Big Bang makes no statement whatsoever about the velocity of any > one single object. None. Period. It never has, it never will. Any odd > object can have any odd velocity with respoce to any odd other object > (within the observed constraints of relativity). All it does is ADD a small, (essentially) linar term to the overall > velocities of objects. At any distance you'll find stuff moving > towards us and other stuff moving away from us, but the farther out > you look the more the overall distribution of velocities is going to > be shifted in the direction *away* from us, by something like 70 km/s > per Mpc ditance. It does NOT say (has never said, will never say) that all objects at > 1Mpc distance must move away from us at 70km/sec. Merely that if you > measured all the velocities of objects at 1Mpc distance that their > velocities would be centered around 70km/s. And that's exactly what is > observed. The radial velocity of any one single object doesn't say *squat* about > the Big Bang. It merely means that there is one single object that > happens to move at some velocity relative to the hubble flow. I've always understood your theory. And I've not in any fashion disagreed that your above statement describes the theory and assumptions behind the big bang. The problem is that you don't appear to understand the scientific method, or even minimal history of your own theory and it's constantly-changing content and 'predictions.' -- greywolf42 ubi dubium ibi libertas ==== > Real science is data first. I agree with this. For example the tens of millions of measured redshifts that agree > perfectly with the Big Bang model. That's DATA for you, in capital > letters. ROTFLMAO! How do they agree? They find that objects which are more redshifted tend to be farther > away. Thats how. > > 1) Assume objects with larger redshift are farther away. > 2) Claim that observing larger redshift proves it is farther away. You would assume such a thing, yes. A scientist wouldn't. That's why you'll never be a scientist. > That's one of the shortest circular arguments I've seen in awhile. Of course it is circular: it is YOUR argument, not anybody else's. Because we simply *assume* that redshift > gives us a distance. You would assume such a thing, yes. A scientist wouldn't. That's why you'll never be a scientist. > > Too bad so many have. Zero. You are a disgusting lying pig. > That is the basis for the Big Bang. The basis for the Big Bang is the correlation of *independently* measured redshift and distance. You are a disgusting lying pig. > But we agree > that Big Bang cosmologists aren't scientists. More that you'll ever be. > There are *tens* *of* *millions* of measurements of distance versus > redshift in perfectly fine agreement with Big Bang cosmology. > > Yawn. Since you get to adjust parameters both on distance and on the BB > side, I'm not impressed. Yes, you would do such a thing. A scientist wouldn't. That's why you'll never be a scientist. > > Besides, it only takes one real case to disprove the assumption. There is no assumption anywhere. > > Sure there is. You are assuming that the *ONLY* contribution to redshift is > doppler (or doppler and expansion if you want to be picky). No 'tired > light', no 'change in physical parameters', no nothing. Both of the two you mentioned require that redshift is correlated with distance. Which you are denying. ('tired light' has been squarely disproven multiple times in this NG alone). > You pick a light source -- say a > particular type of galaxy or a particular type of supernova or a > particular type of variable star (it doesn't really matter, just pick > some object you can recognize) and you measure their brightness and > some are bright and some are dim and there's some distribution of > brightness between the brightest and the dimmest and when you look at > objects that are farther away then you'll see the exact same > distribution of brightness except that all objects will appear dimmer > because they're farther away. > > Sure, with the adjustible parameter of extinction to help you out. And you > can't know that the distribution of brightness is the same without assuming > the hubble relation that gave you the distance -- or vice versa. brightness is a *measured* quantity. You are a disgusting lying pig. > And then you measure redshift and you plot apparent brightness against > redshift and you find that by and large redshift increases as > brightness decreases. > > Ah! But the 'by and large' is the assumption. No, it is what everybody finds who's done the observation. There's nothing assumed here. You are a disgusting lying pig. > The point being that there > may be exceptions. The Big Bang does not say *squat* about the velocity of any one single object. Is says something about a general matter flow IN ADDITION TO any local relative velocities objects may have due to any physical processes they damn well please to have. This has been explained to you before. > The Big Bang does NOT claim that all objects must lie on a perfect > redshift/distance curve, merely that there is an *additional* linear > term in any distance-redshift relation. Namely the Hubble flow. > > And that is the assumption. Absolutely nothing whatsoever is assumed when a plot of apparent brightness versus redshift shows that by and large the brighter objects have less redshift. > That redshift is only and always due to motion > (physical or expansion). What does this have to do with the *observed fact* that objects that are far away have greater redshift? The fact that you deny? 1) There is an *observed* redshift-distance relation that is independent of any cosmological model. You LIE that this isn't the case. 2) There is a cosmological model that says that such a thing should be observed *on the largest scales* (in broad, universal terms, but NOT!!! locally) if the universe is expanding in some fashion. You LIE that some object with a peculiar velocity can somehow disprove that model. > Whatever population of things you measure the redshift distribution > of, you find an additional redshift the farther things are away -- by > whatever means the distance is determined. > > But that's the whole point! The whatever means is always the redshift. It isn't. You are a disgusting lying pig. I PRESENTED TO YOU an obvious means of getting at distance through apparent brightness. > All 'direct' measurements of distance end with parallax. They don't. You are a disgusting lying pig. > Beyond that is > theory -- adjustible to meet the BB postulate. UNLESS one finds a single > connected group of objects with significantly different redshift. 1) Any small group of objects can have any local relative velocities and the Big Bang wouldn't care -- it is a *global* theory of the universe as a whole. 2) No single connected group of objects with significantly different redshift has been found, ever. If you want to claim such, thenthe burden of proof is on you to *show* that any group of objects is connected. You are a disgusting lying pig. > And it doesn't have to be determined terribly precisely: if your > distance determination has a factor two uncertainty, then you get a > spread of measured points around the linear term. If it has a factor > *ten* uncertainty then you get a *wider* spread around the EXACT SAME > linear term. > > We all agree there are uncertainties. So why are you so 'certain' you are > correct? I wasn't saying anything whatsoever about uncertainties. You are Tens of millions of astronomical observations find themselves in perfectly fine agreement with the Big Bang. NONE find themselves in contradiction with it. A buncha disgusting lying pigs tries to worship a few dozen that are simply hard to figure out. > Even if there were 10 million independent determinations of distance, the > simple exclusion of contrary data makes your claim meaningless. There is no exclusion of contrary data here anywhere. You are a disgusting lying pig. > If you (or Mr. Arp or > anybody) want to claim that any two galaxies are in interaction with > each other then the burden of proof is on you (or Mr. Arp or whoever > makes the claim). > > Precisely. And Dr. Arp was well on the way to providing just that. Which > is -- > of course -- why Dr. Arp was kicked off telescopes. He wasn't. You are a disgusting lying pig. > But JUST to make sure we're not missing something somewhere, we > actually look at the stuff Mr. Arp hands us and we find that the > galaxies in question are NOT in fact in interaction with each other. > > How did you manage to determine this fact? In other words: you know absolutely nothing whatsoever about the issue at all -- you haven't read even a single paper on the subject, you are merely insulting millions of people who are performing routinely observations that you have demonstrated repeatedly are too complicated for you to even grasp conceptually. > Quite simply, you cannot have > any basis for your claim Says the ignorant moron whos only basis has always been the *lies* he's so fond of posting. > An (apparently) physically connected group of three galaxies. if all it takes for a couple things to be connected is to look like they're close together then you've already lost right here. > Two which appear to be ejected from the main one. The outer two > have significantly higher redshift than the main one. The connection > appears to be gas. > > How did you 'disprove' the connection? AH, yes! The redshifts are > different THEREFORE, the objects cannot be connected BECAUSE the distance is > different, because we assume redshift determines distance (assuming our > conclusion). *You* *yourself* *assumed* that redshift is related to distance when you brought up tired light or change of the physical constants. A scientist, of course, would never do such a thing. Which is why you're never going to be a scientist. A scientist would simply point to the *measured* correlation between distance and redshift. > And you are a pathetic liar, yourself. For you obviously understand the > issue. Contrary to you I do. You keep *demonstrating* that you don't. > You claim that each single case of Arps is not proof. But not > because it is not true -- but because you can claim each case is pure > coincidence The word coincidence does not appear once in my post. YOU would do such a thing. A scientist, of course, wouldn't. Which is why you're never going to be a scientist. > which is why you weaseled and never mention *HOW* each > observation of Arp's is wrong There's hardly weaseling anywhere involved in not answering a question that nobody asked. But here's a chance for you to learn what the fuck you're talking about for the first time in your worthless life: why don't you read up on one or two of the papers where people observed some of Arps objects? Then you could learn for yourself how one measures whether things are in connection or not. But of course you'd never actually poison your retarded brain with actual DATA. > You slime over to claiming they are > difficult to interpret, They are. It is difficult to interpret observations whenever objects happen to lie close to the same line of sight. That's it. There's nothing slime about it. Are you claiming otherwise? > That you claim the burden of proof is on Arp, The burden of proof is *always* on the positive claim. > and then you deliberately prevent Arp from gathering data > that could provide that proof. I don't. You are a disgusting lying pig. > If you had an ounce of honesty or professional integrity, I'd advise that you look less ridiculous when you don't use such words right after lying publicly. > There are a dozen versions of the BB, over the past 50 years. One. > To which > version, specifically, are you referring? With inflation, or without? With > light elements or without? With dark matter, or without? Etc... *THE* Big Bang model describes a bunch of observations. Wherever people leave out certain aspects from the computations to make them easier to perform then that doesn't magically produce a new version of the model. It merely means that the results of the computations are good to within some approximation. > Certainly not the predictions > (which were 1 part in 100, prior to COBE). Ah, the predictions. Let me submit that you have no clue what you're talking about. > > I figured you'd dodge the issue. :) There is no issue in the sentence you posted and that I quoted that one could dodge even if one wanted to. You are making reference to the predictions without any further clue what the fuck you're referring to. Which would have been quite trivial if you had the slightest idea what you keep blathering on and on about. > Let me further submit that COBE was in perfect agreement with BB > cosmology > > LOL! If you imagine that LOL somehow constitutes an argument, let me advise you that you are as wrong on the issue as you are on all matters astronomy. If you imagine that there is any kind of disagreement between COBE data and Big Bang cosmology, then the burden of proof is on you to *present* that disagreement. Bu that would require, of course, that you learn for the first time in your life what the term Big Bang actually *means*. > and that WMAP has pushed the same agreement an order of > magnitude further. As Mike Turner tends to say cosmology is becoming > boring, every measurement only confirms and refines the previous one. > > ROTFLMAO!!! Those are pretty major 'refinings!' Yes. Refinements. Your pretending that WMAP didn't refine the COBE results doesn't somehow change reality and make it so. COBE WMAP H0 72 +/- 2 +/- 7 70 +/- 4 t0 13 +/- 1.5 13.7 +/- 0.3 Omega0 1.03 +/- 0.03 1.02 +/- 0.02 OmegaB 0.04 +/- 0.008 0.044 +/- 0.004 OmegaM 0.33 +/- 0.035 0.27 +/- 0.04 OmegaX 0.78 +/- 0.06 0.73 +/- 0.04 PWL-n 1.05 +/- 0.09 0.93 +/- 0.03 dn/dlnk -0.02 +/- 0.04 -0.03 +/- 0.02 WMAP confirmed and refined the COBE results. That's it. There's just simply nothing else to be added. This is the DATA that you disgusting lying pigs try to lie out of existence.. > The BB constantly adds new > epicycles every time something is observed. Nothing new has been added to the Big Bang since the original FRW > metric was been worked out unless you're willing to consider inflation > to be a part of the Big Bang Model now in which case noting has been > added to the Big Bang since inflation. That's it. Everything else > follows from what was there all along. > > ROTFLMAO!!! Do you really believe this? I believe nothing whatsoever. I examine the evidence and grant the one or other theory to be the one most consistent with the observable universe. Belief is not a mode I engage in. Contrary to you, of course. > How old are you? I have been doing astronomy professionally when you were nothing but a lustful glint in some hispanic taxi-drivers eye. > And can I sell > you a bridge? You are making the mistake of projecting from yourself onto others all the time. Including here. > It's just that the BB 'predictions' change (as always) as soon as another > contradiction is found. If you had the slightest clue about science, you'd know that a prediction cannot change after the fact. What is predicted is predicted. And then we go and see if we can confirm the prediction observationally. And sometimes we can and sometimes we can't. And sometimes we were too optimistic in what might be detectable and sometimes we're too pessimistic. Nobody has a monopoly on the magnitude of *any* observed effect UNTIL it has been observed. At that point, there's a degree of freedom removed from a model that is simply not available any more. That doesn not change the model. It does not produce a new version of a model. It simply puts an observational contraint on one or more parameters of the model. > Expected from the exact same Big Bang model people referred to before > COBE existed. > > LOL! This is too good to parody. As they say: truth is stranger than fiction. And while they're at it, they squeeze these few oddball cases into > roles they never had: taking an object who's redshift seems to place > it outside the hubble flow to pretend that there IS NO hubble flow and > the entire theory explaining it must be wrong. Yes. That is the inescapable logic of science. No, it isn't. You are merely clueless what you are talking about. The Big Bang makes no statement whatsoever about the velocity of any > one single object. None. Period. It never has, it never will. Any odd > object can have any odd velocity with respoce to any odd other object > (within the observed constraints of relativity). All it does is ADD a small, (essentially) linar term to the overall > velocities of objects. At any distance you'll find stuff moving > towards us and other stuff moving away from us, but the farther out > you look the more the overall distribution of velocities is going to > be shifted in the direction *away* from us, by something like 70 km/s > per Mpc ditance. It does NOT say (has never said, will never say) that all objects at > 1Mpc distance must move away from us at 70km/sec. Merely that if you > measured all the velocities of objects at 1Mpc distance that their > velocities would be centered around 70km/s. And that's exactly what is > observed. The radial velocity of any one single object doesn't say *squat* about > the Big Bang. It merely means that there is one single object that > happens to move at some velocity relative to the hubble flow. > > I've always understood your theory. You didn't in the past and you don't now. Otherwise you wouldn't continue to assert that the redial velocity of a single observed object can somehow disprove the Big Bang. You are merely a disgusting lying pig. But I'm repeating myself. ==== >> Whatever population of things you measure the redshift distribution >> of, you find an additional redshift the farther things are away -- by >> whatever means the distance is determined. >> >> But that's the whole point! The whatever means is always the redshift. [...] >Tens of millions of astronomical observations find themselves in >perfectly fine agreement with the Big Bang. NONE find themselves in >contradiction with it. A buncha disgusting lying pigs tries to worship >a few dozen that are simply hard to figure out. A few dozen anomalies whose distances were measured by the redshift. Um... -- Let us learn to dream, gentlemen, then perhaps we shall find the truth... But let us beware of publishing our dreams before they have been put to the proof by the waking understanding. -- Friedrich August Kekul.8e ==== ... stuff deleted ... >> If you (or Mr. Arp or >>anybody) want to claim that any two galaxies are in interaction with >>each other then the burden of proof is on you (or Mr. Arp or whoever >>makes the claim). > > > Precisely. And Dr. Arp was well on the way to providing just that. Which > is -- > of course -- why Dr. Arp was kicked off telescopes. True science in the > modern sense. The burden of proof is on Arp -- so we'll simply remove from > him the ability to collect proof (data). Problem solved. > Do you have any evidence that (1) Arp is not allowed to use a telescope today. (2) [if 1 is in fact correct] the reason is as cynical as you are making it out to be. ... the rest deleted ... I looked briefly and found no indication that (1) is correct. I also did not find any indication that (1) is incorrect. You apparently have some first-hand information, and it would be worthwhile (to this reader) to see it presented. > -- > greywolf42 > ubi dubium ibi libertas > > Dale. ==== > For example the tens of millions of measured redshifts that agree > perfectly with the Big Bang model. That's DATA for you, in capital > letters. ROTFLMAO! How do they agree? They find that objects which are more redshifted tend to be farther > away. Thats how. > > 1) Assume objects with larger redshift are farther away. > 2) Claim that observing larger redshift proves it is farther away. > > That's one of the shortest circular arguments I've seen in awhile. Why is this so difficult for you? 1) Measure redshift of object. 2) Measure distance to object. 3) Notice that larger distances correlate linearly with larger redshifts. Where does Assume objects with larger redshift are farther away come into what you read above? Where do you think he said that? You measure the redshift. You measure the distance. These are two measurements. There are no assumptions. You notice that large values of x go with large values of y. Where's the circle? I know you're not an idiot. So where do you see a circular argument in a description of a correlation process? - Randy <3c65f87.0311181923.13b10b42@posting.google.com> <3c65f87.0311200627.6edb705a@posting.google.com> ==== > Why is this so difficult for you? 1) Measure redshift of object. > 2) Measure distance to object. > 3) Notice that larger distances correlate linearly with > larger redshifts. Where does Assume objects with larger redshift are farther > away come into what you read above? Where do you think he > said that? In all fairness, surely it's true that measuring distance only works for relatively near objects. I assume (but don't know) that we can independently measure the distance of a minority of objects whose redshifts we can measure. For the remainder of the objects, I suppose we have no recourse but to infer their distance from their redshifts. It is this collection of facts that leads to the wrong conclusion that redshift is the only measure of distance we have, and the correlation between redshift and distance is circular. It's similar to the alleged circularity one sees in some arguments regarding the fossil record: The only measure of the age of the rocks are the fossils found there and the age of the rocks give the age of the fossils. -- Jesse Hughes If you really think there's a bug you should report a bug. Maybe you're not using it properly... It turns out Luddites don't know how to use software properly, so you should look into that. -- Bill Gates ==== > Why is this so difficult for you? >> 1) Measure redshift of object. >> 2) Measure distance to object. >> 3) Notice that larger distances correlate linearly with >> larger redshifts. >> Where does Assume objects with larger redshift are farther >> away come into what you read above? Where do you think he >> said that? In all fairness, surely it's true that measuring distance only works >for relatively near objects. I assume (but don't know) that we can >independently measure the distance of a minority of objects whose >redshifts we can measure. For the remainder of the objects, I suppose >we have no recourse but to infer their distance from their redshifts. I'm no expert, but this seems to understate the case from what I read. I posted a bunch of links about measuring distance elsewhere in this thread. Parallax can be used close by. Cepheid variables can be used out to extreme distances, as far as we can see. In between there are a number of options, including stellar brightness, that are apparently well calibrated out to many thousands of parsecs. So I don't think it's the case that the number of points in a Hubble Law validation study is a few dozen or small. The one study I found online mentioned 800 Cepheid variables, and that was just one study focusing on extreme ranges. - Randy ==== > Why is this so difficult for you? >> 1) Measure redshift of object. >> 2) Measure distance to object. >> 3) Notice that larger distances correlate linearly with >> larger redshifts. >> Where does Assume objects with larger redshift are farther >> away come into what you read above? Where do you think he >> said that? In all fairness, surely it's true that measuring distance only works >for relatively near objects. I assume (but don't know) that we can >independently measure the distance of a minority of objects whose >redshifts we can measure. For the remainder of the objects, I suppose >we have no recourse but to infer their distance from their redshifts. Sort of, buy there's a little more to it, because of a lot of different ways to measure distance. I imagine everyone agrees that parallax (noting the difference in apparent position of a star at different places in the Earth's orbit around the sun and deducing) the distance from that) gives a reliable measure of distance. But it only works nearby. But then there are those Cepheid variable things. You look at nearby Cepheid variables, where you can measure absolute brightness by measuring the distance using parallax, and you note I think it's a relation between period and absolute brightness. You notice that this relation holds for nearby nearby Cepheids and also for faraway nearby ones, so you assume that it also holds for faraway ones, and then you start measuring distance by comparing period and apparent brightness. That's not guaranteed to work, but nothing in science is guaranteed - since you can't give any reason why the relation should be different for faraway stars it seems reasonable to assume the relation is the same until there's some reason not to. Then you notice that the redshift is correlated with the distance as measured by the period and apparent brightness, in exactly the way it would be if in fact the Cepheid distance measurements were accurate and farther objects were moving away faster. At this point the idea that the universe is expanding is just the simplest way to explain the data - otherwise we need an explanation for why the difference in the way faraway Cepheids works is in sync with the redshift that way... [Then someone has problems with noise in a radio telescope (or radar or something) and someone notices that it's something that had been predicted to exist as consequence of the Big Bang. People work out how far from perfectly isotropic the noise should be, and recently when they do the Fourier analysis on data measured by some lah dee dah satellite they find it's exactly as the model predicts (except for one detail about a missing low harmonic, which people are speculating could mean the universe is compact.) So of course at that point it's perfectly natural to decide that there's no basis for any of this whatever, it's just stuff people believe because it's fashionable...] >It is this collection of facts that leads to the wrong conclusion that >redshift is the only measure of distance we have, and the correlation >between redshift and distance is circular. It's similar to the >alleged circularity one sees in some arguments regarding the fossil >record: The only measure of the age of the rocks are the fossils found >there and the age of the rocks give the age of the fossils. >-- >Jesse Hughes >If you really think there's a bug you should report a bug. Maybe >you're not using it properly... It turns out Luddites don't know how >to use software properly, so you should look into that. -- Bill Gates David C. Ullrich ==== >>You know I've had two extremely contentious discussions with Randy Only two? It just seems like more. As I recall they both concerned the correct interpretation of dL/dt and ran on and on and on. There might have been one other but I can't remember the subject. >and >>I've never seen him lie. The most one might say is that he - like all >>- makes occasional mistakes or misjudges the relevance of issues What? Why, I oughta ... > and >>even gets irritated once in a while. But a lie is a purposeful >>manipulative misstatement of truth. You might consider getting your >>barbs and facts at least a little straighter before you shoot them. Well, thanks. I guess. Since our relationship has been as you say >contentious, I'll take this as high praise. I do indeed get irritated. >There are two hot buttons for me: (1) something I perceive as massive >illogic (we will of course disagree as to what constitutes illogic, >and (2) being accused of lying. > You're welcome I guess. My primary hot button is being accused of bad faith in presenting or analyzing issues whether right or wrong. I still haven't forgotten the arguments you advanced in support of your evaluation of the directon of dr which I still need to understand and explain to my own satisfaction (and no I'm not trying to resurrect the issue here). I've never really understood why people can't just discuss even contentious issues without being personally abusive and derogatory. ==== > That he may have made errors doesn't change the fact that Galileo performed > experiments (with balls on inclined planes). And 'mechanics' is a > theoretical practice -- not an experimental one. {Galileo didn't drop > weights from the Tower of Pisa -- that was a different guy.} Galileo performed experiments indeed. But the oft-forgotten thing is that so did everyone else. Copernicus gets scorn in my book. He was right, but only by being whatsoever to recommend it. Kepler's at least explained the experimental observations better than Ptolemy, for the first time ever. It was only until Newton that the principal objections to the heliocentric model were adequately explained. Thomas ==== That he may have made errors doesn't change the fact that Galileo performed > experiments (with balls on inclined planes). And 'mechanics' is a > theoretical practice -- not an experimental one. {Galileo didn't drop > weights from the Tower of Pisa -- that was a different guy.} Galileo performed experiments indeed. But the oft-forgotten thing is > that so did everyone else. Not everyone else. Most of the hierarchy simply read Aristotle. > Copernicus gets scorn in my book. He was right, but only by being > whatsoever to recommend it. Perhaps just better-read. The Greeks knew about the Earth going around the Sun in 400 B.C. > Kepler's at least explained the > experimental observations better than Ptolemy, for the first time > ever. It was only until Newton that the principal objections to the > heliocentric model were adequately explained. Actually, the principle objections were not resolved until parallax was first measured. As the lack of observable parallax (a requirement of any Sun-centered system) was the major objection. -- greywolf42 ubi dubium ibi libertas ==== what's irrelevant is the profundity (or not) of Arp's research (and the anti-BBers) to Jimi Harris' math. > Where do you think Hubble's Law came from in the first place? Not perfect. But a damned good fit. > Motions, not orbits. But yes, I understand the assumptions behind the > hubble law. > > But check out the second graph. http://www.physics.utah.edu/~burko/courses/Phys3740/notes/hubble.html > > Fascinating. This appears to be the supernova data ... chopped off, I do > believe. I believe the data goes much farther out -- to 2 or 3 billion > parsecs. And shows a clear deviation from linear -- destroying the linear > 'hubble law'. (i.e. dark energy) > > This is especially egregious because the author claimed that some had > proposed quadratic redshift-distance laws have been 'disproved' due to > the supernova data. This is (politely) called data selection. Less > politely, it's called deliberate distortion of the evidence. > We have exactly zero direct measurements of quasar distances. Regardless of > what numbers we have of anything else. > > See this page, in the section titled Determining > H0. http://homepages.tcp.co.uk/~carling/astrocos.html Out to 50 kpc we can use stars. We know how bright they > should be from stellar evolution theory. > > We can assume this. But it's not a direct measurement. And the brightness > of these simulations was originally calibrated by assuming the Hubble > cepheids, high mass and low metallicity stars. > > We use the brightness > of the star as a distance measure. How many stars have we > observed within 50000 parsecs? Isn't that most of the > Milky Way galaxy? --ils duces d'Enron! ==== > My point is that my discovery is PURE MATH, and PURE KNOWLEDGE, > representing a previously unknown formula for both finding and > counting primes. Why should I believe you? Here's a previously unknown formula: 283490823481290348 + 1987234817239472345 = 2270725640720762693 Is it very interesting? No. Mine is certainly previously unknown: I'll bet anyone $100 that nobody has ever before published that addition. But it's not very interesting, for two reasons: 1) Knowing that addition isn't very important, and 2) Anyone could reproduce the formula very easily if they needed it. Similarly, methods of creating finite difference equations are very well known. What is missing in your previously unknown formula is: 1) An explanation of why it is important--for example, does it enable faster computation of primes? (no); does it make for faster factorizations? (no). Did you know there is a diophantine equation whose roots are exactly the set of primes? I could tweak that equation in a million ways and produce a previously unknown formula for primes, but that's hardly very interesting. 2) Your methods: where did your equation come from? How can we apply the techniques you used in other cases for other problems? Thomas <51566c78.0311242033.6cbb94e0@posting.google.com> ==== > Reminds me of Lenny Bruce's bit about if, in a movie, a man approaches a > woman with a pillow, he would rather see him put it under her rear than > over her face. Or George Carlin's bit in which he imagines replacing the world kill with the word fuck in classic movies. -- Jesse F. Hughes Really, I'm not out to destroy Microsoft. That will just be a completely unintentional side effect. -- Linus Torvalds ==== These programs use simulated evolution to create artificial neural networks to solve specific problems. They are well documented and are in standard C with all source included. They will compile and run with all the common compilers and OS's. They can be downloaded at no charge from: http://annevolve.sourceforge.net Mitchell Timin -- Many are stubborn in pursuit of the path they have chosen, few in pursuit of the goal. - Friedrich Nietzsche http://annevolve.sourceforge.net is what I'm into nowadays. Humans may write to me at this address: zenguy at shaw dot ca <3c65f87.0311080513.159f0484@posting.google.com> ==== >> Let's be careful with our selection of terms. JSH didn't offer a proof of >> FLT, he offered an incorrect argument in an attempt to prove FLT. >> >> Justin Van Winkle >> >> Somehow I doubt it. Even if your proof were to be found to be entirely >> correct, nobody would care. Why? Because the thing you assert is >> completely irrelevent. Why don't you know this? Because you don't know >> any >> mathematics. You don't know any mathematics. >> What you say is true, for a reason that reveals the *ugliness* >> of mathematics--and the *ugliness* of mathies. Mathies >> would turn up their nose at an elementary proof of FLT >> by JSH but not (say) an elementary proof of FLT by Harvey >> Friedman, because JSH is an Afro-American without formal >> training in mathematics--unlike Harvey Friedman, who is white, >> has a Ph.D, and plays the game as mathies require it to be >> played. THAT is why a proof of FLT by JSH would be irrelevant. >> And THAT is why everything JSH says about professional >> mathematicians is true. >> Seriously, think about this. >> --John > Mathies WOULD turn up their nose at an elementary proof of FLT > by JSH but not (say) an elementary proof of FLT by Harvey > Friedman and, THAT is why a proof of FLT by JSH WOULD be irrelevant. in response to: Even if your proof WERE to be found to be entirely > correct, nobody WOULD care Refrain from attributing to me words not mine, or reading what is not there > into things that I say. Yes, you said those things. Specifically, you said that the relevant facts are that Friedman is white and has a PhD and plays the game and that JSH is black and has no formal training and presumably doesn't play the correct game. Unfortunately you have no evidence that this ugliness actually exists. I would be very shocked if Friedman came up with a proof for FLT, since that isn't his bag as far as I know, but that's not really relevant. Everyone here agrees that if a reputable mathematician came up with a proof of FLT, it would be recognized -- at least reasonably soon. The dispute is whether, if someone like JSH did the same, it would be recognized. Sadly, no one like JSH has come up with such a proof so the hypothesis is untested. I know that James thinks otherwise, but *no one else* has ever said otherwise as far as I know, yourself included. Maybe you ought to follow James's lead and switch to his prime counting thing. Everyone agrees that it works, at least. Now we can debate (well, not *we*, but you and others) whether it is ignored because (1) James is an iconoclast or (b) it is old or a slight variation on something old. I know you think James is a genius, but so far your only evidence that he is a genius is that people that you don't like reject his work. Oh, and also you don't understand his work. I was never clear: is the fact that you don't get it part of your evidence he's a genius? Or is that just a convenient reason to claim he's a genius without having to explain why? To be fair, maybe you got the impression he's a genius simply from the number of times that claim has been repeated here. There is a small body of posters (previously one, now two) that seem fond of that fact. -- Conservative, n: A statesman who is enamored of existing evils, as distinguished from the Liberal who wishes to replace them with others. -- Ambrose Bierce ==== grade of a mathematics-oriented high-school in Croatia and would like you to help me with an assignment conjured up by my teacher and put into our school test: Prove or disprove the irrationality of 2^pi. ==== > grade of a mathematics-oriented high-school in Croatia and would like you to > help me with an assignment conjured up by my teacher and put into our school > test: > > Prove or disprove the irrationality of 2^pi. > There's a result called the Gelfond-Schneider theorem that says if a is algebraic and not equal to 0 or 1, and b is irrational then every value of a^b is transcendental. Therefore 2^pi is transcendental. This theorem is proved in Chap. 8 of A. H. Rose, A Course in Number Theory. ==== >grade of a mathematics-oriented high-school in Croatia and would like you to >help me with an assignment conjured up by my teacher and put into our school >test: > Prove or disprove the irrationality of 2^pi. This is definitely not a suitable problem for high-school students. I think it's unsolved. On the other hand, it follows from a result of Lang that there are at most two primes p such that p^pi is algebraic (see Baker, Transcendental Number Theory, Theorem 12.3). And from the same result, at least one of 2^pi, 2^(pi^2), 2^(pi^3) is transcendental. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 ==== > grade of a mathematics-oriented high-school in Croatia and would like you to > help me with an assignment conjured up by my teacher and put into our school > test: > > Prove or disprove the irrationality of 2^pi. > Let me see if I understand this. You want a reader of sci.math to do the problem for you, so you can hand it in as your own work, and get credit for it. Have I got that right? Is there nothing about that that strikes you as just a teeny-weeny bit, what's the word I'm looking for, dishonest? Or am I missing something? For what it's worth, this looks like a very hard problem to me, perhaps even an unsolved problem. -- ==== > Let me see if I understand this. Permission granted. Right. > You want a reader of sci.math to do the problem for you, > so you can hand it in as your own work, and get credit for it. Wrong. > Have I got that right? No. > Is there nothing about that that strikes you as just a teeny-weeny bit, > what's the word I'm looking for, dishonest? I won't even dignify this weak sarcastic remark with an answer. > Or am I missing something? Try everything. > For what it's worth, this looks like a very hard problem to me, > perhaps even an unsolved problem. Exactly why I posted it. The test is over, I got an A in it, but I still wonder whether it can be solved as I cannot solve it myself (and my teacher won't help me in attaining the solution). ==== > they radiate. The faster they go the greater the radiated energy per > unit time. String a group of them along a line and you now have a > current, and a current produces a magnetic field in surrounding space. > Energy is stored in this field. Thus there are two reasons that charged > only two reasons. At c (wrt lab frame) the electron cannot be further > accelerated, not because the associated magnetic field contains infinite > disappears at this speed, i.e. all of the virtual photons inducing the > this speed, simply because the source of those photons is the > accelerator itself, which is at rest in lab frame. OTOH an infinite > quantity of energy input per unit time into the accelerator is required > terminal velocity, all of the input energy thereafter is wasted, not one > analogous to that of a skydiver. The special relativistic prediction is > incorrect, and despite the propaganda, that prediction has never been > verified. The collision energy has never been directly measured, it has > only been force fit to the various 'perceived' byproducts of the > collision, all of which are based upon ad hoc conjectures which are > the shaman, and this proves only that the general public is at an > evolutionary standstill. Morons leading morons, they'll both fall into > the ditch. Best to suspend judgment, this is the high road, there are > only ditches on either side. > > Richard Perry > magnetic field and emit photons by accelerating, then how can we know whether their observed behavior is really due to relativistic mass or just an electromagnetic field effect? Seems like the only way to tell is if we could figure out a way to Double-A ==== > > > they radiate. The faster they go the greater the radiated energy per > unit time. String a group of them along a line and you now have a > current, and a current produces a magnetic field in surrounding space. > Energy is stored in this field. Thus there are two reasons that charged > only two reasons. At c (wrt lab frame) the electron cannot be further > accelerated, not because the associated magnetic field contains infinite > disappears at this speed, i.e. all of the virtual photons inducing the > this speed, simply because the source of those photons is the > accelerator itself, which is at rest in lab frame. OTOH an infinite > quantity of energy input per unit time into the accelerator is required > terminal velocity, all of the input energy thereafter is wasted, not one > analogous to that of a skydiver. The special relativistic prediction is > incorrect, and despite the propaganda, that prediction has never been > verified. The collision energy has never been directly measured, it has > only been force fit to the various 'perceived' byproducts of the > collision, all of which are based upon ad hoc conjectures which are > the shaman, and this proves only that the general public is at an > evolutionary standstill. Morons leading morons, they'll both fall into > the ditch. Best to suspend judgment, this is the high road, there are > only ditches on either side. Richard Perry > magnetic field and emit photons by accelerating, then how can we know > whether their observed behavior is really due to relativistic mass or > just an electromagnetic field effect? > > Seems like the only way to tell is if we could figure out a way to > > Double-A space, matter, and field are interchangeable terms. Richard Perry ==== > Is the force on a moving charge equal to qE? >> How do you know? Experimental evidence. That is true only if you assume that its mass effectively increases by gamma. Nonsense. It is experimentally verified, assuming nothing. >As you know. You even claim to never have said otherwise. Because you have claimed to accept that the electron gains the same amount of energy every time it passes through >E = Fs = qV = qEs s = distance between electrodes. Sorry, ambiguous! The first E is energy, the second is electric field. > I have never used those expressions here. > I did not state that. of the fact that the electron gains the same amount of energy every time it passes through the RF-cavity. Gained energy is force times distance, Fs. This energy is independent of the speed, thus must the force F be independent of the speed. The energy gained by a charge going throug a potential drop is qV, the electric field E = V/s qV = qEs = Fs Thus F = qE independent of the speed, which answers your question. Got it now? Paul ==== >> Is the force on a moving charge equal to qE? >>> How do you know? >>Experimental evidence. >> That is true only if you assume that its mass effectively increases by gamma. Nonsense. It is experimentally verified, assuming nothing. It depends what you call 'Kinetic energy'. >As you know. You even claim to never have said otherwise. Because you have claimed to accept that the electron >gains the same amount of energy every time it passes through Don't misrepresent me Paul. I only agreed that in the steady state condition, that was true. >E = Fs = qV = qEs s = distance between electrodes. Sorry, ambiguous! >The first E is energy, the second is electric field. Yes I understood that. For a moving charge, I claim it could be wrong. > I have never used those expressions here. >> I did not state that. of the fact that the electron gains the same amount of >energy every time it passes through the RF-cavity. Gained energy is force times distance, Fs. >This energy is independent of the speed, thus >must the force F be independent of the speed. >The energy gained by a charge going throug >a potential drop is qV, the electric field E = V/s >qV = qEs = Fs >Thus F = qE independent of the speed, >which answers your question. Got it now? Well, that is probably correct - but it isn't really the main issue. What I want to know is 'where that energy goes'. You say it all goes into KE but only if you include the (effective) mass increase. I say the 'effective mass increase' is not that at all. It is a reaction against the applied field. It obviously requires a lot of energy to create a reverse field 'bubble' inside an applied field. Note: relativity doeesn't offer any physical explanation for the apparent mass increase. It simply accepts a seemingly correct equation derived from bogus postulates. Paul > Henri Wilson. See the Stupidity of Relativity. www.users.bigpond.com/hewn/index.htm ==== >>>>Yes. >>>per cycle in the RF-cavities regardless of the speed. >>>But it will loose more and more energy per cycle in >>>the bends as the speed increases. >>>When the two are equal, the accelerator is in steady state. >>>when it is going at peak efficiency. >>>> gaps. During the rest of their travels they lose a little speed. >>>> However the question is not about the steady state condition - in which the >>> input energy goes into radiation. It is why increasingly more energy is >>> balance radiation. >>The answer is that Nature tells us that the KE is: m*(gamma-1)*c^2 >> And does a 'change in KE' involve a term with 'dm/dt' in it? In the same way as d/dt(C*x) where C is constant >involve a term with dC/dt in it. >>>Thus the gained energy does NOT decrease when the speed >>>>of the electrons approaches c. >>>>>>So whatever you think happens to the field in the RF-cavities >>>>when the speed approaches c, we KNOW for certain that >>>>>> But where does that energy go? >>>>Into kinetic energy, of course. >>>Einstein says: KE = m*c^2*(((1/sqrt(1 - v^2/c^2)) - 1) >>>Newton says: KE = 0.5*m*v^2 >>>>Why do you find the second of these equations more natural >>>than the first? >>>Two different theories, the first is experimentally confirmed, >>>the second is experimentally falsified. >>You didn't answer this crucial question, Henry. >>Why are you willing to accept that KE = 0.5*m*v^2 >>as nature tells us is wrong, >>and NOT willing to accept KE = m*c^2*(((1/sqrt(1 - v^2/c^2)) - 1) >>as Nature tells us is correct? >> I do accept that equation for moving charges Paul. You know I have never denied >> that is obeys the experimental evidence. >> My argument is that the SRian idea of a 'mass increase' is not really a mass >> increase at all but an illusion caused by a reduction in the force on a moving >> charge due to a field. There is no idea of mass increase in SR. > I will go further and speculate that mass is nothing but a property somehow >> associated with fields. Henry Wilson seems to claim that the mass increases, though. >> Once again it is not the KE itself we have to worry about but changes in KE. >>> Once again you get a term with 'v.dm/dt'. >>No you don't. >>You obviously didn't read what I said! >>The momentum is m*gamma*v where m is invariant. >> Once known as 'rest mass'? Righ. Or simply mass. >so dp/dt = m*d/dt(gamma*v) >> |>You have screwed this up before, and I have shown >> |>you the correct equation before. >> |> |>F = dp/dt = d/dt (m*gamma*v) >> |>gamma =1/sqrt(1 - v^2/c^2) >> |> |>dv/dt = (1/((v^2/c^2)*gamma^3 + gamma))*F/m >> |> |>((v^2/c^2)*gamma^3 + gamma)*dv = (F/m)*dt >> |>gamma*v = (F/m)*t + C >> Since you obviously copied this out of a book, you didn't see the intermediate >> steps that included 'v.dm/dt' I didn't copy this from anywhere, Henry. >You are babling. >F = dp/dt = d/dt (m*gamma*v) >F = m*d/dt(gamma*v) >F = m*((v^2/c^2)*gamma^3 + gamma))*dv/dt >dv/dt = (1/((v^2/c^2)*gamma^3 + gamma))*F/m Where is the intermediate step that included v*dm/dt ? >m is a constant and is treated as such. F=dp/dt=d(mv)/dt=m.dv/dt+v.dm/dt... !!!!!!!!!!!!!! Let m=Mo.gamma > So dm/dt=Mo.d(gamma)/dt Then F=Mo.gamma(dv/dt)+v.Mo.d(gamma)/dt > or F/Mo=gamma(dv/dt)+v.d(gamma)/dt Now d(gamma)/dt=(dv/dt).v.(gamma^3)/(c^2) Therefore dv/dt=(F/Mo)/[(v/c)^2)gamma^3+gamma] ....get it? Of course I get it. This is what I said many postings ago. IF the momentum is mv, then the mass must increase. BUT MOMENTUM IT IS NOT mv! According to SR, it is p = m*v*gamma, m = Mo = invariant So there isn't any term containing dm/dt in dp/dt .... get it? >>The KE is m*(gamma-1)*c^2 where m and c are invariant. >>so dE/dt = m*c^2*d/dt(gamma)=m*c^2*gamma^3*v*dv/dt >> Yes Paul, you are telling me what is observed. >> I AM TRYING TO TELL YOU WHERE THE EXTRA ENERGY MIGHT BE STORED. >> Do you think the fairies take it? Extra energy relative to what? >Does 0.5*m*v^2 explain where the energy is stored? >Is there fairies involved here? >SR doesn't explain where the energy is stored any more than NM does. >The theories predicts different energies, neither explains it. >SR is right, NM is wrong. >That's all. NM correctly says that the KE is (1/2mv^2), where m is constant and v is > measured in the frame of the field. In the frame of which field? This isn't anything to discuss, Henry. > to get it to velocity v. Kinetic energy is per definition the energy we have to pump into And you know what that energy is. > stored? In the field of course. Now you are talking nonsense. Of course NO kinetic energy is instrinsic. You MUST know that, Henry! >>>> That's not what NATURE tells us. That's what SR says. >>That's what Nature through experiments tells us. >>That SR says the same is why SR isn't falsified. >>That NM says otherwise is why NM is falsified. >> NM is perfectly capable of handling this. >> Simple! Your only problem is that when we are discussing how much >energy is gained in the RF-cavity, then you have to claim >that you NEVER stated what you just did. :-) I don't see the connection. Henry, don't tell me you still haven't got this! The gained energy is Fs. It is the same regardless So F can blatantly obvious not decrease when the speed >>>Why do you have a problem with that? >>>Why not simply accept it? >>>> Accepting it means virtually nothing. What's the point? >>This illustrates your problem. >>You refuse to accept that Nature works as it actually does. >>And again I must ask: >>Why don't you say that accepting KE = 0.5*m*v^2 >>means vertually nothing. What's the point? >> What is 'v'? That is the point. You didn't answer, Henry. >I ask you why you think KE = 0.5*m*v^2 explains >more than KE = m*(gamma-1)*c^2. >What's the big difference? I told you. 1/2mv^2 is the kinetic energy. The additional energy is stored in > the field 'bubble' around the moving charge. You didn't answer, Henry. You just keep asserting that KE is 0.5*m*v^2 and NOT m*(gamma-1)*c^2. WHY do you claim that KE = 0.5*m*v^2? Do you know it a priori? Do you know it because it was Newton and not Einstein who said so? Is it a religious conviction you will believe blindly despite of anything? >>>>Which proves you WRONG. >>>>The radiation from an accelerated charge! >>>>or the fields associated with a moving charge! >>>>or The 'Back EMF' concept. >>>>any less when the speed approaches c. >>>>>> that does not conflict with what I said. >>>>Uh? :-) >>>> I repeat, THAT DOES NOT CONFLICT WITH WHAT I SAID. >>> My argument is about what you claim is an apparent, 'mass increase'. >>You were talking about: >>radiation from an acceleraed charge!, >>which ONLY is relevant when the charge is accelerated, >>and: The 'Back EMF' concept. which only is relevant >>when there is an EMF (electic field). >>And now you are insisting that you were NOT talking >>about what happens in the acceleration part, but how >>accelerated in an electric field? >> Paul, if you cannot see what I am getting at by now I don't think there is any >> hope for you at all. >> My argument is about where the additional energy is stored, when a charged >> You say it manifests itself as a mass increase (although you try to disguise >> the fact) >> I agree that it appears just like a mass increase. I also claim that it is NOT >> RELATED to a mass increase but that the energy is stored somehow in the fields Why don't you need enigmatic fields to store the KE in 0.5*m*v^2? Because '0.5*m*v^2' is the correct expression for amount of energy associated Is it? Why? Both 0.5*m*v^2 and m*(gamma-1)*c^2 are in accordance with experimental evidence when v << c, while only the latter is in accordance with experimental evidence when v is big enough for the two equations to give a measurable different prediction. So why do you claim that the former is the correct expression >> Can you not see that the implications of being able to eliminate 'mass' from >> the equations are considerable. We might be able to actually learn something >> about the nature of mass if we look beyond a simple maths formula. So you are actually saying that the KE is stored in enigmatic fields >even when c << v? That is the KE = 0.5*m*v^2 is stored in fields? Not directly. but indirectly maybe. I reckon all mass is explainable in terms > of 'fields' whatever they are. May well be. But it is irreleavant to our issue. I have never seen you have a problem with the KE of low speed bodies, and insited that the KE must be stored in some kind of field bubbles. Quite the contrary. You have claimed that it is only the _differemce_ between m*(gamma-1)*c^2 and 0.5*m*v^2 that is stored in these bubbles. So where is the 0.5*m*v^2 stored? >>That doesn't make sense. >>You are of course free to speculate where the KE >>Consider the consequence of that! >> I am considering the consequences of far more than that. >> I used to think that 'fields were made of matter'. Now I think that all matter >> must be made of 'fields'. >> These fields don't want to move relative to other fields, hence the concepts of >> 'mass' and 'inertia'. >> All we need now is an understanding of what a 'field' is. Good luck. >You should look at the standard model. > Do you believe in 'solid matter'? Anything 'solid' must be made of something smaller...and on and on.... Where does that end? There are turtles all the way down, of course. Paul ==== >>so dp/dt = m*d/dt(gamma*v) >>>>> |>You have screwed this up before, and I have shown >>> |>you the correct equation before. >>> |>> |>F = dp/dt = d/dt (m*gamma*v) >>> |>gamma =1/sqrt(1 - v^2/c^2) >>> |>> |>dv/dt = (1/((v^2/c^2)*gamma^3 + gamma))*F/m >>> |>> |>((v^2/c^2)*gamma^3 + gamma)*dv = (F/m)*dt >>> |>gamma*v = (F/m)*t + C >>>> Since you obviously copied this out of a book, you didn't see the intermediate >>> steps that included 'v.dm/dt' >>I didn't copy this from anywhere, Henry. >>You are babling. >>F = dp/dt = d/dt (m*gamma*v) >>F = m*d/dt(gamma*v) >>F = m*((v^2/c^2)*gamma^3 + gamma))*dv/dt >>dv/dt = (1/((v^2/c^2)*gamma^3 + gamma))*F/m >>Where is the intermediate step that included v*dm/dt ? >>m is a constant and is treated as such. >> F=dp/dt=d(mv)/dt=m.dv/dt+v.dm/dt... !!!!!!!!!!!!!! >> Let m=Mo.gamma >> So dm/dt=Mo.d(gamma)/dt >> Then F=Mo.gamma(dv/dt)+v.Mo.d(gamma)/dt >> or F/Mo=gamma(dv/dt)+v.d(gamma)/dt >> Now d(gamma)/dt=(dv/dt).v.(gamma^3)/(c^2) >> Therefore dv/dt=(F/Mo)/[(v/c)^2)gamma^3+gamma] >> ....get it? Of course I get it. >This is what I said many postings ago. >IF the momentum is mv, then the mass must increase. >BUT MOMENTUM IT IS NOT mv! According to SR, it is p = m*v*gamma, m = Mo = invariant >So there isn't any term containing dm/dt in dp/dt That's just a smartarse way of expressing the Newtonian equation for an accelerating mass that is increasing in size. Only a real fanatic would fall for it. .... get it? >>The KE is m*(gamma-1)*c^2 where m and c are invariant. >>>so dE/dt = m*c^2*d/dt(gamma)=m*c^2*gamma^3*v*dv/dt >>>> Yes Paul, you are telling me what is observed. >>> I AM TRYING TO TELL YOU WHERE THE EXTRA ENERGY MIGHT BE STORED. >>>> Do you think the fairies take it? >>Extra energy relative to what? >>Does 0.5*m*v^2 explain where the energy is stored? >>Is there fairies involved here? >>SR doesn't explain where the energy is stored any more than NM does. >>The theories predicts different energies, neither explains it. >>SR is right, NM is wrong. >>That's all. >> NM correctly says that the KE is (1/2mv^2), where m is constant and v is >> measured in the frame of the field. In the frame of which field? >This isn't anything to discuss, Henry. The Newtonian explanation was dropped after the Einsteinian red herring was spotted. If NM is extended to include the Wilsonian 'reverse field bubble effect', it will be quite adequate. > to get it to velocity v. Kinetic energy is per definition the energy we have to pump into >And you know what that energy is. As far as I'm concerned, KE=0.5.Mo.v^2 at all speeds, (relative to the original rest frame). The additional energy used to bring the charge to the speed should not be included in KE. > stored? In the field of course. Now you are talking nonsense. >Of course NO kinetic energy is instrinsic. >You MUST know that, Henry! All right, 'intrinsic' might not have been the best word. I think you know what I meant though. Please reveal to us your own physical theory as to where the excess energy is stored Paul. >Your only problem is that when we are discussing how much >>energy is gained in the RF-cavity, then you have to claim >>that you NEVER stated what you just did. :-) >> I don't see the connection. Henry, don't tell me you still haven't got this! >The gained energy is Fs. It is the same regardless >So F can blatantly obvious not decrease when the speed Typical SRian illogic. THE GAINED ENERGY IS THE SAME IRRESPECTIVE OF SPEED, THEREFORE SPEED INCREASE IS ALWAYS DIRECTLY PROPORTIONAL TO THE AMOUNT OF ENERGY GAINED. >>>Why do you have a problem with that? >>>>Why not simply accept it? >>>>>> Accepting it means virtually nothing. What's the point? >>>>This illustrates your problem. >>>You refuse to accept that Nature works as it actually does. >>>And again I must ask: >>>Why don't you say that accepting KE = 0.5*m*v^2 >>>means vertually nothing. What's the point? >>>> What is 'v'? That is the point. >>You didn't answer, Henry. >>I ask you why you think KE = 0.5*m*v^2 explains >>more than KE = m*(gamma-1)*c^2. >>What's the big difference? >> I told you. 1/2mv^2 is the kinetic energy. The additional energy is stored in >> the field 'bubble' around the moving charge. You didn't answer, Henry. >You just keep asserting that KE is 0.5*m*v^2 and NOT m*(gamma-1)*c^2. >WHY do you claim that KE = 0.5*m*v^2? Because that leads to a believeable physical theory rather than a nebulus maths equation derived by pure accident. >Do you know it a priori? >Do you know it because it was Newton and not Einstein who said so? >Is it a religious conviction you will believe blindly despite of anything? Who are YOU to talk about religious convictions. >>>> You say it manifests itself as a mass increase (although you try to disguise >>> the fact) >>> I agree that it appears just like a mass increase. I also claim that it is NOT >>> RELATED to a mass increase but that the energy is stored somehow in the fields >>Why don't you need enigmatic fields to store the KE in 0.5*m*v^2? >> Because '0.5*m*v^2' is the correct expression for amount of energy associated Is it? Why? >Both 0.5*m*v^2 and m*(gamma-1)*c^2 are in accordance with >experimental evidence when v << c, while only the latter is in >accordance with experimental evidence when v is big enough >for the two equations to give a measurable different prediction. >So why do you claim that the former is the correct expression I'm merely defining KE as always '0.5*m*v^2' and distinguishing IT from the energy that goes into the field and has all the appearance of a 'mass increase'.. >> Can you not see that the implications of being able to eliminate 'mass' from >>> the equations are considerable. We might be able to actually learn something >>> about the nature of mass if we look beyond a simple maths formula. >>So you are actually saying that the KE is stored in enigmatic fields >>even when c << v? That is the KE = 0.5*m*v^2 is stored in fields? >> Not directly. but indirectly maybe. I reckon all mass is explainable in terms >> of 'fields' whatever they are. May well be. But it is irreleavant to our issue. IT IS NOT >I have never seen you have a problem with the KE of low speed >bodies, and insited that the KE must be stored in some kind of >field bubbles. Quite the contrary. You have claimed that it is >only the _differemce_ between m*(gamma-1)*c^2 and 0.5*m*v^2 >that is stored in these bubbles. So where is the 0.5*m*v^2 stored? '0.5*m*v^2' is hardly 'stored'. It is merely part of the energy released when an object is brought to rest in some reference frame. I claim that the separate energy stored in the 'reverse field bubble' is similarly released in this process. You cannot argue with that Paul. It is simply a matter of definition. Mine is a realistic physical one whereas your is purely a maths one based on an unproven postulate. Whether my equation is precisely the same as SR's 'Mo.gamma' remains to be seen. >>> All we need now is an understanding of what a 'field' is. >>Good luck. >>You should look at the standard model. >> Do you believe in 'solid matter'? >> Anything 'solid' must be made of something smaller...and on and on.... >> Where does that end? There are turtles all the way down, of course. yes! You found it! Paul > Henri Wilson. See the Stupidity of Relativity. www.users.bigpond.com/hewn/index.htm ==== You have now changed the question. > You are now specifying that the electron is moving at high speed when it enters > the field. Do you want me to remind you of your original question? The original question was how long it takes when it enters a static electric field, regardless of was accelerated from standstill (or a very low initial speed). In that case, you said the force enters the field. You said however that it would not do so if So my new scenario is: We still have two electrodes 1 km apart, with a million volts potential difference between them. Behind the cathode, there is an electron gun shooting electrons with an initial speed v_o through a small hole in the cathode. The cathode itself isn't hot, and how the electron is accelerated is of no consequence. The point is that it comes out of the hole with the speed v_o. The question is still: How long time does it take from the electron comes out of the hole to a force start acting on it? My answer is: as it enters a static electric field. What is your answer? You have given no clear answer. > My previous answer (repeated twice) stands. I haven't seen it. Would you repeat it please? Paul ==== > You have now changed the question. >> You are now specifying that the electron is moving at high speed when it enters >> the field. Do you want me to remind you of your original question? The original question was how long it takes >when it enters a static electric field, regardless of was accelerated from standstill (or a very low >initial speed). In that case, you said the force >enters the field. You said however that it would not do so if So my new scenario is: >We still have two electrodes 1 km apart, with a million >volts potential difference between them. >Behind the cathode, there is an electron gun shooting >electrons with an initial speed v_o through a small hole >in the cathode. >The cathode itself isn't hot, and how the electron >is accelerated is of no consequence. The point is >that it comes out of the hole with the speed v_o. The question is still: >How long time does it take from the electron >comes out of the hole to a force start acting on it? My answer is: >as it enters a static electric field. What is your answer? >You have given no clear answer. > My previous answer (repeated twice) stands. I haven't seen it. >Would you repeat it please? According to my 'reverse field bubble' theory, the applied field will be reduced in the vicinity of the moving charge. That will have a similar effect to the applied field taking time to operate. Paul > Henri Wilson. See the Stupidity of Relativity. www.users.bigpond.com/hewn/index.htm ==== > > > >>Paul >> No silly answers please. How about the correct answer? they radiate. The faster they go the greater the radiated energy per >unit time. String a group of them along a line and you now have a >current, and a current produces a magnetic field in surrounding space. >Energy is stored in this field. Thus there are two reasons that charged >only two reasons. At c (wrt lab frame) the electron cannot be further >accelerated, not because the associated magnetic field contains infinite >disappears at this speed, i.e. all of the virtual photons inducing the >this speed, simply because the source of those photons is the >accelerator itself, which is at rest in lab frame. OTOH an infinite >quantity of energy input per unit time into the accelerator is required >terminal velocity, all of the input energy thereafter is wasted, not one >analogous to that of a skydiver. The special relativistic prediction is >incorrect, and despite the propaganda, that prediction has never been >verified. The collision energy has never been directly measured, it has >only been force fit to the various 'perceived' byproducts of the >collision, all of which are based upon ad hoc conjectures which are >the shaman, and this proves only that the general public is at an >evolutionary standstill. Morons leading morons, they'll both fall into >the ditch. Best to suspend judgment, this is the high road, there are >only ditches on either side. Richard Perry > > Relativity is blatant BULL!!! Relatively moving light clocks are all you need to disprove SR. If each is always oriented perpendicular to its motion relative to the other, then each will expect the other to tick slower, acceleration doesn't change a thing, and doesn't break any symmetry. The only consistent solution is motion wrt a local fixed medium. Every experiment supporting SR supports a local aether, in fact demands it, because this relative motion crap doesn't fly, it is contradictory. There 'is' an embankment against which to compare the propagation of light, namely all of the relative to each other. Richard Perry > > Henri Wilson. > See the Stupidity of Relativity. > www.users.bigpond.com/hewn/index.htm ==== >> >> >> >>>Paul >>>>>> No silly answers please. >>How about the correct answer? >>they radiate. The faster they go the greater the radiated energy per >>unit time. String a group of them along a line and you now have a >>current, and a current produces a magnetic field in surrounding space. >>Energy is stored in this field. Thus there are two reasons that charged >>only two reasons. At c (wrt lab frame) the electron cannot be further >>accelerated, not because the associated magnetic field contains infinite >>disappears at this speed, i.e. all of the virtual photons inducing the >>this speed, simply because the source of those photons is the >>accelerator itself, which is at rest in lab frame. OTOH an infinite >>quantity of energy input per unit time into the accelerator is required >>terminal velocity, all of the input energy thereafter is wasted, not one >>analogous to that of a skydiver. The special relativistic prediction is >>incorrect, and despite the propaganda, that prediction has never been >>verified. The collision energy has never been directly measured, it has >>only been force fit to the various 'perceived' byproducts of the >>collision, all of which are based upon ad hoc conjectures which are >>the shaman, and this proves only that the general public is at an >>evolutionary standstill. Morons leading morons, they'll both fall into >>the ditch. Best to suspend judgment, this is the high road, there are >>only ditches on either side. >>Richard Perry >>> >> >> Relativity is blatant BULL!!! Relatively moving light clocks are all you need to disprove SR. If each >is always oriented perpendicular to its motion relative to the other, >then each will expect the other to tick slower, acceleration doesn't >change a thing, and doesn't break any symmetry. The only consistent >solution is motion wrt a local fixed medium. Every experiment supporting >SR supports a local aether, in fact demands it, because this relative >motion crap doesn't fly, it is contradictory. There 'is' an embankment >against which to compare the propagation of light, namely all of the >relative to each other. Richard Perry > I agree. SR is just a disguised aether theory. The LT's cannot be 'real' unless space and time are at least locally absolute. >> Henri Wilson. >> See the Stupidity of Relativity. >> www.users.bigpond.com/hewn/index.htm Henri Wilson. See the Stupidity of Relativity. www.users.bigpond.com/hewn/index.htm ==== How do I find that cos(2pi/17)=-1/16+ Sqrt(17)/16 + Sqrt(34 - 2*Sqrt(17))/16 + Sqrt(17 + 3*Sqrt(17) - Sqrt(34 - 2*Sqrt(17)) - 2*Sqrt(34 + 2*Sqrt(17)))/8 ? How do I find the right-hand side given a different angle? ==== > How do I find that cos(2pi/17)=-1/16+ Sqrt(17)/16 + Sqrt(34 - 2*Sqrt(17))/16 + > Sqrt(17 + 3*Sqrt(17) - Sqrt(34 - 2*Sqrt(17)) - 2*Sqrt(34 + 2*Sqrt(17)))/8 ? How do I find the right-hand side given a different angle? http://www.ace.gatech.edu/~ryanhynd/17gon.pdf -- P.A.C. Smith The vast majority of Iraqis want to live in a peaceful, free world. And we will find these people and we will bring them to justice. ==== > [ snip ] >> >> end note: >> Did you ever try to read Two Dogmas of Empiricism, W.V.O. Quine 1951 >> that I recommended? It's on-line. When I read it for the first time >> View), I thought I understood it, I also found it boring. Today, I know >> I didn't understand it, I now know why I didn't understand it, and I >> that's why I don't find it boring - putting it mildly, I see it as one >> of the most important papers written in the 20th century in fact....... >> >> Give it a shot, and give genuine humility a shot too. Having a look at >> On What There Is, The Problem of Meaning in Linguistics, Reference >> and Modality and New Foundations For Mathematical Logic might help >> ....- hell, read the whole collection - it'll be a start...... >> -- > I don't know as I'll have the opportunity to do any reading whilst I'm > getting my advanced degree in humility. > > > Actualy, Lester, I suspect you would rather enjoy reading Two Dogmas of Empiricism - Quine's discussion of topics you seem to find engaging is both seminal and fluid. It's a short paper. You can find it on-line at: http://www.ditext.com/quine/quine.html I recommend it both because, as I say, I think you would enjoy reading it, and because I'm interested to hear what you make of it. The main thrust of the paper is to undermine two presumptions of empiricism: | Modern empiricism has been conditioned in large part by two dogmas. One is | a belief in some fundamental cleavage between truths which are analytic, | or grounded in meanings independently of matters of fact and truths which | are synthetic, or grounded in fact. The other dogma is reductionism: the | belief that each meaningful statement is equivalent to some logical | construct upon terms which refer to immediate experience. Both dogmas, I | shall argue, are ill founded. One effect of abandoning them is, as we | shall see, a blurring of the supposed boundary between speculative | metaphysics and natural science. Another effect is a shift toward | pragmatism. What practical difference can it make to anybody's life whether analytic and synthetic truths differ in kind or only in degree? The last two sections of the paper, especially the last section, a mostly metaphorical discussion of what today would be called a theory of belief revision, belief propagation, or model selection, has direct bearing on Expert Systems that assist in making decisions, or actualy make decisions on their own, or arrive at a diagnosis, or a most probable explanation of uncertain information. Should a loan application be approved? Should a prisoner be paroled? Who should get a kidney? What disease or combination of diseases is behind a set of symptoms, and what lab tests give the most information? My own work is typified by the problem of writing software that, given a digital image of a check, reads the dollar amount and decides either to accept its best guess as to what the amount is (a most probable interpretation of all available evidence) or route the image to a workstation for human data entry. How do we get from the esoterica of epistomology to algorithms that hand out kidneys, loans, liberty, treatment, and advice? How do we evaluate the performance of such algorithms? Against what standards can we compare them? | The totality of our so-called knowledge or beliefs, from the most casual | matters of geography and history to the profoundest laws of atomic physics | or even of pure mathematics and logic, is a man-made fabric which impinges | on experience only along the edges. Or, to change the figure, total | science is like a field of force whose boundary conditions are experience. | A conflict with experience at the periphery occasions readjustments in the | interior of the field. Truth values have to be redistributed over some of | our statements. Re-evaluation of some statements entails re-evaluation of | others, because of their logical interconnections -- the logical laws | being in turn simply certain further statements of the system, certain | further elements of the field. Having re-evaluated one statement we must | re-evaluate some others, whether they be statements logically connected | with the first or whether they be the statements of logical connections | themselves. But the total field is so undetermined by its boundary | conditions, experience, that there is much latitude of choice as to what | statements to re-evaluate in the light of any single contrary experience. | No particular experiences are linked with any particular statements in the | interior of the field, except indirectly through considerations of | equilibrium affecting the field as a whole. | If this view is right, it is misleading to speak of the empirical content | of an individual statement -- especially if it be a statement at all | remote from the experiential periphery of the field. Furthermore it | becomes folly to seek a boundary between synthetic statements, which hold | contingently on experience, and analytic statements which hold come what | may. Any statement can be held true come what may, if we make drastic | enough adjustments elsewhere in the system. Even a statement very close to | the periphery can be held true in the face of recalcitrant experience by | pleading hallucination or by amending certain statements of the kind | called logical laws. Conversely, by the same token, no statement is immune | to revision. Revision even of the logical law of the excluded middle has | been proposed as a means of simplifying quantum mechanics; and what | difference is there in principle between such a shift and the shift | whereby Kepler superseded Ptolemy, or Einstein Newton, or Darwin | Aristotle? Are you familiar with Bayesian Networks, which are also known as Belief Nets? A Bayesian Network is a directed acyclic graph whose nodes represent random variables with arcs representing causal influences or class-property relationships (to paraphrase Judea Pearl). It is a graphical representation of probabilistic knowledge. The power of such nets is two-fold: they make modeling assumptions explicit and clear, and they form the basis of efficient algorithms for calculating joint probability distributions over the set of random variables, or subsets of the variables, or conjunctions and disjunctions of sets of variables - you could think of such a network as a probabilistic database organized to support efficient processing of probabilistic queries. Evidence impinges nodes at the periphery, which propagate diagnostic support up the directed links to parent nodes, and receive causal support from parent nodes. A general node, with both parents and children, receives both diagnostic support and causal support. Belief propagates through the net until at equilibrium each node represents a probability distribution over its state space. If the graph is connected then a change to any node in the net influences the state of all nodes in the net, but the computation remains tractable because the state of any node depends only on the states of its neighbors (conditional independance). Just how efficient and tractable such belief propagation algorithms can be depends on the topology of the graph - in general the more sparse the graph the more efficient the algorithm (but potentialy the more approximate the model). I first learned of Bayesian Networks from Judea Pearl's 1988 classic: Probablistic Reasoning In Intelligent Systems: Networks of Plausible This is an extraordinary book...no philosopher concerned with probability, decision theory, defeasible logic, or even epistemology in its formal guises, can afford to be ignorant of the contents of this book... This is a great book. So far I've mentioned Bayesian Networks only as static data structures for the efficient computation of the results of probabilistic queries, but over the last ten years there has been a steady shift in the focus of attention from algorithms for propagating evidence towards methods for learning parameters and structure from data (Cowell, Dawid, Lauritzen, and Spiegelhalter in the introduction to Probabilistic Networks and Expert Systems). Adjusting the parameters of an existing belief net could be called normal science, whereas a change in the topology could be called a paradigm shift. So, what's my point? If nothing else a look at the history of Expert Systems provides in microcosm a chapter in the unfolding book of Applied Epistemology. Quine's metaphor has become today's algorithms and data structures. It is probably just a coincidence (or is it co-occurrence) that, while plugging Quine, Longley did not mention explicitly Quine's book Word and Object, or his essay Natural Kinds (in the book Ontological Relativity), or take the time to trace the line of thought from the scandal of induction through innate similarity spaces to, when coupled with a heterogeneous world, a mind of adapted computational modules rather than the blank slate, lump of wax, or general-purpose computer of the Standard Social Science Model (Pinker, The Language Instinct). But that is a topic for another time... Anyhow, Lester. When you ask: >What if the positivism/empiricism/materialism/behaviorism axis is >failing? Not just being ignored and passed over as outmoded scientific >doctrine but actually being scorned as ideas which never had any >conceptual validity? What if quantum mechanics is wrong and there are >special relativity was never quite right to begin with? If this >happens and happens in conclusive terms then the scientific world and >contemporary visions of reality are going to come apart at the seams >and I think we'll need all the philosophical help we can get. the resistance you encounter is probably not so much a matter of reactionary scholasticism. Probably it has more to do with Quine's metaphor: you are not altering a single hypothesis or proposition in isolation, you are altering the whole interconnecting web of scientific belief, which has a great deal of explanatory power, an elegant and spare structure - call it a large epistomological inertial mass. It will take a LOT of evidence to change it; or a HUGE simplification in its structure, or substantial evidence and substantial simplification whose product is HUGE. If you want to move the entire scientific world you need a lever big enough and a place to stand. ==== [ snip ] > > > end note: > Did you ever try to read Two Dogmas of Empiricism, W.V.O. Quine 1951 > that I recommended? It's on-line. When I read it for the first time > View), I thought I understood it, I also found it boring. Today, I know > I didn't understand it, I now know why I didn't understand it, and I > that's why I don't find it boring - putting it mildly, I see it as one > of the most important papers written in the 20th century in fact....... > > Give it a shot, and give genuine humility a shot too. Having a look at > On What There Is, The Problem of Meaning in Linguistics, Reference > and Modality and New Foundations For Mathematical Logic might help > ....- hell, read the whole collection - it'll be a start...... > -- >> I don't know as I'll have the opportunity to do any reading whilst I'm >> getting my advanced degree in humility. >> >> > >Actualy, Lester, I suspect you would rather enjoy reading Two Dogmas of >Empiricism - Quine's discussion of topics you seem to find engaging is both >seminal and fluid. It's a short paper. You can find it on-line at: http://www.ditext.com/quine/quine.html I recommend it both because, as I say, I think you would enjoy reading it, >and because I'm interested to hear what you make of it. The main thrust of the paper is to undermine two presumptions of empiricism: | Modern empiricism has been conditioned in large part by two dogmas. One is >| a belief in some fundamental cleavage between truths which are analytic, >| or grounded in meanings independently of matters of fact and truths which >| are synthetic, or grounded in fact. The other dogma is reductionism: the >| belief that each meaningful statement is equivalent to some logical >| construct upon terms which refer to immediate experience. Both dogmas, I >| shall argue, are ill founded. One effect of abandoning them is, as we >| shall see, a blurring of the supposed boundary between speculative >| metaphysics and natural science. Another effect is a shift toward >| pragmatism. What practical difference can it make to anybody's life whether analytic and >synthetic truths differ in kind or only in degree? The last two sections of >the paper, especially the last section, a mostly metaphorical discussion of >what today would be called a theory of belief revision, belief propagation, >or model selection, has direct bearing on Expert Systems that assist in >making decisions, or actualy make decisions on their own, or arrive at a >diagnosis, or a most probable explanation of uncertain information. Should >a loan application be approved? Should a prisoner be paroled? Who should >get a kidney? What disease or combination of diseases is behind a set of >symptoms, and what lab tests give the most information? My own work is >typified by the problem of writing software that, given a digital image of a >check, reads the dollar amount and decides either to accept its best >guess as to what the amount is (a most probable interpretation of all >available evidence) or route the image to a workstation for human data >entry. How do we get from the esoterica of epistomology to algorithms that >hand out kidneys, loans, liberty, treatment, and advice? How do we evaluate >the performance of such algorithms? Against what standards can we compare >them? | The totality of our so-called knowledge or beliefs, from the most casual >| matters of geography and history to the profoundest laws of atomic physics >| or even of pure mathematics and logic, is a man-made fabric which impinges >| on experience only along the edges. Or, to change the figure, total >| science is like a field of force whose boundary conditions are experience. >| A conflict with experience at the periphery occasions readjustments in the >| interior of the field. Truth values have to be redistributed over some of >| our statements. Re-evaluation of some statements entails re-evaluation of >| others, because of their logical interconnections -- the logical laws >| being in turn simply certain further statements of the system, certain >| further elements of the field. Having re-evaluated one statement we must >| re-evaluate some others, whether they be statements logically connected >| with the first or whether they be the statements of logical connections >| themselves. But the total field is so undetermined by its boundary >| conditions, experience, that there is much latitude of choice as to what >| statements to re-evaluate in the light of any single contrary experience. >| No particular experiences are linked with any particular statements in the >| interior of the field, except indirectly through considerations of >| equilibrium affecting the field as a whole. | If this view is right, it is misleading to speak of the empirical content >| of an individual statement -- especially if it be a statement at all >| remote from the experiential periphery of the field. Furthermore it >| becomes folly to seek a boundary between synthetic statements, which hold >| contingently on experience, and analytic statements which hold come what >| may. Any statement can be held true come what may, if we make drastic >| enough adjustments elsewhere in the system. Even a statement very close to >| the periphery can be held true in the face of recalcitrant experience by >| pleading hallucination or by amending certain statements of the kind >| called logical laws. Conversely, by the same token, no statement is immune >| to revision. Revision even of the logical law of the excluded middle has >| been proposed as a means of simplifying quantum mechanics; and what >| difference is there in principle between such a shift and the shift >| whereby Kepler superseded Ptolemy, or Einstein Newton, or Darwin >| Aristotle? Are you familiar with Bayesian Networks, which are also known as Belief >Nets? A Bayesian Network is a directed acyclic graph whose nodes represent >random variables with arcs representing causal influences or class-property >relationships (to paraphrase Judea Pearl). It is a graphical representation >of probabilistic knowledge. The power of such nets is two-fold: they make >modeling assumptions explicit and clear, and they form the basis of >efficient algorithms for calculating joint probability distributions over >the set of random variables, or subsets of the variables, or conjunctions >and disjunctions of sets of variables - you could think of such a network as >a probabilistic database organized to support efficient processing of >probabilistic queries. Evidence impinges nodes at the periphery, which >propagate diagnostic support up the directed links to parent nodes, and >receive causal support from parent nodes. A general node, with both parents >and children, receives both diagnostic support and causal support. Belief >propagates through the net until at equilibrium each node represents a >probability distribution over its state space. If the graph is connected >then a change to any node in the net influences the state of all nodes in >the net, but the computation remains tractable because the state of any node >depends only on the states of its neighbors (conditional independance). >Just how efficient and tractable such belief propagation algorithms can be >depends on the topology of the graph - in general the more sparse the graph >the more efficient the algorithm (but potentialy the more approximate the >model). I first learned of Bayesian Networks from Judea Pearl's 1988 classic: >Probablistic Reasoning In Intelligent Systems: Networks of Plausible This is an extraordinary book...no philosopher concerned with > probability, decision theory, defeasible logic, or even epistemology > in its formal guises, can afford to be ignorant of the contents of > this book... This is a great book. So far I've mentioned Bayesian Networks only as static data structures for >the efficient computation of the results of probabilistic queries, but over >the last ten years there has been a steady shift in the focus of attention >from algorithms for propagating evidence towards methods for learning >parameters and structure from data (Cowell, Dawid, Lauritzen, and >Spiegelhalter in the introduction to Probabilistic Networks and Expert >Systems). Adjusting the parameters of an existing belief net could be >called normal science, whereas a change in the topology could be called a >paradigm shift. So, what's my point? If nothing else a look at the history of Expert >Systems provides in microcosm a chapter in the unfolding book of Applied >Epistemology. Quine's metaphor has become today's algorithms and data >structures. It is probably just a coincidence (or is it co-occurrence) that, while >plugging Quine, Longley did not mention explicitly Quine's book Word and >Object, or his essay Natural Kinds (in the book Ontological >Relativity), or take the time to trace the line of thought from the >scandal of induction through innate similarity spaces to, when coupled >with a heterogeneous world, a mind of adapted computational modules rather >than the blank slate, lump of wax, or general-purpose computer of the >Standard Social Science Model (Pinker, The Language Instinct). But that >is a topic for another time... Anyhow, Lester. When you ask: >What if the positivism/empiricism/materialism/behaviorism axis is >>failing? Not just being ignored and passed over as outmoded scientific >>doctrine but actually being scorned as ideas which never had any >>conceptual validity? What if quantum mechanics is wrong and there are >>special relativity was never quite right to begin with? If this >>happens and happens in conclusive terms then the scientific world and >>contemporary visions of reality are going to come apart at the seams >>and I think we'll need all the philosophical help we can get. the resistance you encounter is probably not so much a matter of reactionary >scholasticism. Probably it has more to do with Quine's metaphor: you are >not altering a single hypothesis or proposition in isolation, you are >altering the whole interconnecting web of scientific belief, which has a >great deal of explanatory power, an elegant and spare structure - call it a >large epistomological inertial mass. It will take a LOT of evidence to >change it; or a HUGE simplification in its structure, or substantial >evidence and substantial simplification whose product is HUGE. If you want >to move the entire scientific world you need a lever big enough and a place >to stand. > interconnected. A chain is only as strong as its weakest link. What makes me think it might actually happen is that I'm not arguing in a vacuum any more. I'm actually beginning to understand what I'm talking about. The same mistakes seem to apply pretty much across the board. By the way I'll check out the online Quine reference. ==== I'm kind of confused by a definition I've seen of independence of two random events or random variables, which contradicts my notion of independent random variables/events. What I understand as independent variables or events is two variables/events that are the outcome of experiments that have no relationship whatsoever -- hence the term *INdependent* (as in, they don't *depend on* each other). One immediate consequence of the above definition is that the probability of two particular outcomes for both events simultaneously is the product of the probabilities of each event alone. Notice that the above implies this, but this does not imply the above. Now, how can the above be stated as a definition of independence?? That seems to me completely illogical!! In particular, I hear that for variables with Gaussian distribution, independence and uncorrelation (uncorrelatedness?) are equivalent things, because it turns out that the product of the PDF's of two uncorrelated Gaussian variables happens to be equal to the product of the PDF's. I find this completely absurd!!! Independence of course implies uncorrelated; but uncorrelated does not imply independence!! In cryptograhpy, the data and the encrypted data are uncorrelated (this is one of many condition for a good encryption system -- so that an attacker can not take advantage of statistical correlation of the gibberish with the data being hidden). But what a joke would it be to say that the data and that data passed through a mathematical function are independent!!!! I can take a pseudo-random generator algorithm that generates Gaussian pseudo-random numbers, and construct two random variables as follows: draw a random number from some physical (i.e., truly random) process; seed the pseudo-random generator with the random number, and get the first variable, then re-seed the pseudo-random generator with the value of the first variable, and get the second variable. The PRNG can be made such that these two variables will be uncorrelated. But is it not a joke to claim that the two random variables are *independent*?? The second variable is explicitly a function of the first one!!! Just because of a big coincidence about the Gaussian distribution that makes the joint distribution equal to the product of the marginals are we going to say that they are independent?? Is this a matter of point of view? Or am I being misled and those definitions are indeed incorrect? Carlos -- ==== >I'm kind of confused by a definition I've seen of independence of >two random events or random variables, which contradicts my notion >of independent random variables/events. >What I understand as independent variables or events is two >variables/events that are the outcome of experiments that have no >relationship whatsoever -- hence the term *INdependent* (as in, >they don't *depend on* each other). Well then, your understanding is imperfect. That's an example, not the definition. The definition of independent events is as follows: A and B are independent if Pr(A intersect B) = Pr(A) Pr(B). >One immediate consequence of the above definition is that the >probability of two particular outcomes for both events >simultaneously is the product of the probabilities of each event >alone. Notice that the above implies this, but this does not >imply the above. >Now, how can the above be stated as a definition of independence?? >That seems to me completely illogical!! Once the definition has been made, and accepted by the mathematical community, you just have to accept it. Yell and scream all you want, it's not going to change. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 ==== I'm trying to recall an example of a proposition S(n) where we can show S(k)->S(k+1) but S(1) fails and, in general, S(n) is false. In such a case, one might falsely conclude S(n) holds for all natural no's n . L ==== >I'm trying to recall an example of a proposition S(n) where we can show >S(k)->S(k+1) but S(1) fails and, in general, S(n) is false. In such a case, >one might falsely conclude S(n) holds for all natural no's n . The easiest I can come up with is: S(n) = n is greater than 100 (or greater than 10, or greater than anything other than 0). To prove the induction step: Suppose that S(k) holds; then k>100. Since k+1>k>100, we conclude that k+1>100; that is, if S(k) holds, then S(k+1) holds. ====================================================================== It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) ====================================================================== Arturo Magidin magidin@math.berkeley.edu ==== to be true for n>100. I seem to recall examples which are generally, if not always, false, despite the fact S(k)->S(k+1). Len >I'm trying to recall an example of a proposition S(n) where we can show >S(k)->S(k+1) but S(1) fails and, in general, S(n) is false. In such a case, >one might falsely conclude S(n) holds for all natural no's n . The easiest I can come up with is: S(n) = n is greater than 100 (or greater than 10, or greater than anything other than 0). To prove the induction step: Suppose that S(k) holds; then k>100. Since k+1>k>100, we conclude that > k+1>100; that is, if S(k) holds, then S(k+1) holds. ====================================================================== > It's not denial. I'm just very selective about > what I accept as reality. > --- Calvin (Calvin and Hobbes) > ====================================================================== Arturo Magidin > magidin@math.berkeley.edu > ==== How about something like if n is irrational then n + 1 is is irrational? This implication is true for all real, or even complex, n, but the antecedent and consequent are false for all integers, indeed all rationals. > to be true for n>100. I seem to recall examples which are generally, if not > always, false, despite the fact > S(k)->S(k+1). > > Len >I'm trying to recall an example of a proposition S(n) where we can show >S(k)->S(k+1) but S(1) fails and, in general, S(n) is false. In such a > case, >one might falsely conclude S(n) holds for all natural no's n . The easiest I can come up with is: S(n) = n is greater than 100 (or greater than 10, or greater than anything other than 0). To prove the induction step: Suppose that S(k) holds; then k>100. Since k+1>k>100, we conclude that > k+1>100; that is, if S(k) holds, then S(k+1) holds. ====================================================================== > It's not denial. I'm just very selective about > what I accept as reality. > --- Calvin (Calvin and Hobbes) > ====================================================================== Arturo Magidin > magidin@math.berkeley.edu > ==== >to be true for n>100. I seem to recall examples which are generally, if not >always, false, despite the fact >S(k)->S(k+1). S(n) = There are at least n+1 positive integers strictly smaller than n. S(n) = there are only finitely many integers smaller than n. S(n) = The set {1,....,n} can be bijected with a proper subset of itself. S(n) = There is an integer x such that n grava .88 la saucisse et au marteau: > to be true for n>100. I seem to recall examples which are generally, if not > always, false, despite the fact > S(k)->S(k+1). The proposition being an irrational works. -- Nicolas ==== > The proposition being an irrational works. I was thinking k is not an integer, but this might be more straightforward. There is also k = k+1, where the induction adds 1 to both sides. -- Daniel W. Johnson panoptes@iquest.net http://members.iquest.net/~panoptes/ 039 53 36 N / 086 11 55 W ==== Perfect! > Leonard M. Wapner grava .88 la saucisse et au marteau: > turn out > to be true for n>100. I seem to recall examples which are generally, if not > always, false, despite the fact > S(k)->S(k+1). The proposition being an irrational works. -- > Nicolas ==== im trying to come up with an algorithm to output an euler tour, given an undirected eulerian graph G. Can someone give me of a counter example of a graph for which my algorithm below does not find the tour even though there is one: 1. Do Depth-First Search traversal of G and number the vertices in DFS-preorder 2. Reinitialize all vertices and edges of G as unused. 3. Produce a cycle by: start from vertex with preorder number 1 computed in step 1, and repeatedly go to the vertex with highest preorder number possible along an unused edge. Stop when all edges incident to the current vertex are used. X-Cise: tanbanso@iinet.net.au X-CompuServe-Customer: Yes X-Coriate: admin@interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Pose: george_cox@btinternet.com X-Punge: Micro$oft ==== >I know this is not a very deep question, but as I've come accross >rigorous texts like Halmos's and Lax's on linear algebra and noticed >their emphasis on finite dimensional vector spaces (certainly to be >expected in Halmos's case), it makes me wonder if linear algebra is >regarded mainly as the study of these spaces. No. However, because the finite dimensional case is simpler, it is common for itnroductory texts to only cover it. >Also, are there linear algebra books (as opposed to functional >analysis texts) that treat infinite dimensional vector spaces? -- Shmuel (Seymour J.) Metz, SysProg and JOAT not reply to spamtrap@library.lspace.org ==== > I know this is not a very deep question, but as I've come accross > rigorous texts like Halmos's and Lax's on linear algebra and noticed > their emphasis on finite dimensional vector spaces (certainly to be > expected in Halmos's case), it makes me wonder if linear algebra is > regarded mainly as the study of these spaces. If linear algebra is understood as the study of linear maps, the restriction to finite dimensional vector spaces is not essential. The fundamental definions (e.g. of linear map) make sense for modules over rings. However, there are results about f.d. vector spaces, that do not generalize to the more general setting, or can be obtained only with more work. In the restricted situation of an introductory course (or book), it is probably a reasonable choice, to consider only f.d.v.s. Moreover, in the study of the infinite-dimensional case, methods from analysis are helpful, so the author might want to avoid yet more methods. When I was a beginning student, the course Linear Algebra introduced (in this order) Sets, groups, rings, modules, finitely generated vector spaces and restricted to the more special setting whenever there would be too much of a technical burden otherwise. Marc ==== I have big questions in my mind about linear algebra concept. Can anyone help me? (Because my linear algebra is not good, maybe some of my descriptions are wrong. Please help me to point them out) We can use inner product to find the projection of a vector to the basis of a vector space. However, before we defining inner product, we have already learned the fact that every vector can be uniquely written as linear combination of basis. That means the concept of projection is independent of inner product. Then why should we define inner product again? (without using the concept of projection?) Also, inner product can be defined in different way(say, dot product, Hermitian inner product... I can't list many examples, because i don't know). Do all these inner product give me the same projection? Can I define my own kind of inner product? Another question is, do infinite dimensional vector spaces has the concept of basis? Also, I wanna ask the following is correct or not: Given an infinite dimensional vector space V, I have already showed that an infinite set of vectors A(subset of V) is an othonormal set. Can I say that there exists another subset B of V and B contains A (A is subset of B also)such that the vectors in B form the basis of V? (or can I say that every element of V can be uniquely written as linear combination of vectors in B?) ==== >I have big questions in my mind about linear algebra concept. Can >anyone help me? (Because my linear algebra is not good, maybe some of >my descriptions are wrong. Please help me to point them out) We can use inner product to find the projection of a vector to the >basis of a vector space. However, before we defining inner product, we >have already learned the fact that every vector can be uniquely >written as linear combination of basis. That means the concept of >projection is independent of inner product. Then why should we define >inner product again? (without using the concept of projection?) The inner product is independent of any chosen basis. It adds geometry (lengths, angles, areas, etc.) to the vector space. The two kinds of projection you mention are different if the basis is not orthonormal. For example, in R^2 with the standard inner product, the projection (in the first sense) of (1, 1) onto (1, 0) is (1, 0). If you use the basis {(1, 0), (1, 1)}, then the projection in the second sense is the zero vector. >Also, inner product can be defined in different way(say, dot product, >Hermitian inner product... I can't list many examples, because i don't >know). Do all these inner product give me the same projection? Can I >define my own kind of inner product? Hermitian inner products are used in complex vector spaces and are more special than an arbitrary inner product on the underlying real vector space. I suggest ignoring the complex case until you're comfortable with the real case. Infinitely many inner products can be defined on a vector space. On R^n, with respect to the standard basis, an inner product can be represented by a positive-definite symmetric n-by-n matrix, and there are clearly many such matrices. Different inner products will yield different projections in general (if by projection you mean orthogonal projection). One interesting example is a smooth surface in R^3. At each point, the tangent space of the surface is a plane, and the plane inherits an inner product from R^3. The inner product on the tangent space defines the infinitesimal geometry of the surface at the chosen point. A major theme of Differential Geometry is to relate the infinitesimal geometry to the global geometry and topology of the surface (or more generally, Riemannian manifold). Another question is, do infinite dimensional vector spaces has the >concept of basis? series). There's also a general result that any vector space has a basis, whose proof uses transfinite induction. John Mitchell ==== Another question is, do infinite dimensional vector spaces has the >concept of basis? > > series). There's also a general result that any vector space has a > basis, whose proof uses transfinite induction. There's a strikingly simple and familiar example of an infinite dimensional vector space. Consider the set R[x] of all polynomials in one variable whose coefficients are real numbers. You can add two polynomials to get another one, and you can multiply any polynomial by a real number to get another polynomial. One can easily verify that R[x] satisfies the axioms of a vector space, where each polynomial is regarded as a vector, and the real numbers are the scalars. Now consider the set of polynomials B = {1, x, x^2, x^3, x^4, . . .}. Every polynomial in R[x] can be written in a unique way as a finite linear combination of elements of B. Further, it's pretty clear that no finite collection of polynomials can have that property (because any finite collection of polys contains one of maximum degree, and you can't get any higher-degree polys by adding or multiplying by scalars). Therefore B is a basis of R[x], and therefore R[x] has countable dimension considered as a vector space over the real numbers R. ==== I was wondering about the partial sum of the sum-over-divisors of mu = Mobius function, but previously was wondering only about when the sum was over those divisors of m which were <= sqrt(m). Not many of my questions regarding the sqrt(m) -situation were answered, but I might as well post the below (unamazing) equality which generalizes the sum to being over those divisors <= n. Let m and n be positive integers. sum{k|m, 1<= k <= n} mu(k) = sum{k=1 to n} mu(k) phi(n/k,m), where phi(x,m) = number of positive integers <= x and coprime to m. Also, these sums = sum{k|q, n/p <= k <= n} mu(k), where q is largest divisor of m which is coprime with p = a prime. If n = floor(m), I can not see how the above sums (if true) can lead to a closed-form for sum{k|m, 1<= k <= sqrt(m)} mu(k) Any comments? thanks, Leroy Quet Archive-Name: misc.metric-system ==== RESULT unmoderated group misc.metric-system passes 211:25 [ Note: This document is multiposted in 3 copies because too many large service providers implement crossposting limits and would otherwise drop it. - n.a.n moderation team ] This vote was conducted by a neutral third party. Questions about the proposed group should be directed to the proponent. Proponent: Markus Kuhn Votetaker: Bill Aten There were 236 valid YES/NO votes submitted during the voting period. Each proposed newsgroup, in order to pass, must have at least 2/3 YES votes and at least 100 more YES than NO votes. The results are as follows: misc.metric-system results - 236 valid (YES & NO) votes Yes No : 2/3? >100? : Pass? : Group ---- ---- : ---- ----- : ----- : ------------------------------------------- 211 25 : Yes Yes : Yes : misc.metric-system 6 abstaining votes and 3 invalid votes The proposal passed. There is a five day discussion period after these results are posted. If no serious and significant allegations of voting irregularities are raised, the moderator of news.announce.newgroups will create the newsgroup shortly thereafter. The remainder of the RESULT contains: Rationale Charter Final Voting Acknowledgements NEWSGROUPS LINE: misc.metric-system The International System of Units. RATIONALE: misc.metric-system Units of measurement and related standards affect many aspects of our daily lives. The global standardization of a single consistent International System of Units was a major breakthrough for human civilization and significantly simplified communication, learning, work and trade all over the planet. The introduction of the metric system still faces delays in some areas. Notable examples are consumer communication and traffic regulations in the United States and United Kingdom, as well as parts of the aeronautical and typographic industry. It is therefore no surprise that discussions about the metric system flare up regularly in many different newsgroups. In particular the slow progress with metrication in the United States promises to fuel such debates for many years to come. A dedicated newsgroup will focus expertise and will provide a medium for professionals and hobbyists to find advice and suggestions on metric product standards and conventions. None of the newsgroups in which metric-system issues flare up frequently is particularly suited for this topic by charter and readership. The popularity of the that there is a significant number of people interested in the topic. Considering the important role that units of measurement play in everyone's life, this promises to become a quite lively newsgroup. The name of the proposed group has been the subject of some debate. The present proposal is motivated by these considerations: - Although discussions about the metric system focus much on its slow progress in a small number of countries, the topic is inherently international in nature and discussions tend to benefit very significantly from world-wide participation. Therefore, placing the group under us.* or uk.* would be inappropriate. - The metric system affects many fields, including consumer communication and road traffic. Discussions about the metric system range from basic science and applied engineering considerations to economic, social, psychological, legal, public policy and media aspects. This excludes sci.* and leaves misc.* as the most appropriate hierarchy. - The metric system is the only system of units used in almost every region and field of application. Imperial and U.S. Customary units are usually discussed in relation to the metric system, which is within the scope of the proposed group. Other unit systems have very limited applications and are better discussed in specialized science or history groups. The proposed group is far more likely to have specialized children rather than equivalent siblings, which speaks against an entire misc.measurement.* hierarchy and justifies a place directly under misc. - The term metric system remains the most well known and most easily recognized English language term for what is more formally called the International System of Units (SI). This speaks against group names such as *.si or *.metric. The proposed charter has equally been the subject of some debate. The present proposal is motivated by these considerations: - It refers equally to both the official modern name International System of Units (SI) and the colloquial English term metric system. This is in the interest of rapid recognition by both readers and search-engine users. Any further distinction between these terms is deliberately left to explanatory periodic postings. - It is broad enough to cover discussions about different historic variants of the metric system (e.g., CGS, MKS, various European customary units) as well as contemporary units that compete with the SI (e.g., inch, pound, Fahrenheit, calorie). - It is narrow enough to exclude topics that are not related to metric units (e.g., the history and redefinition of calendars). - It covers product standards and conventions that are not part of any official definition of the metric system, but that are closely related to metric units (e.g., metric clothing sizes, paper formats, engineering components, traffic regulations). These can be considerably more complex topics than the metric system itself, leading to discussions of particular interest to consumers and practitioners. - It leaves room for the possible later creation of a separate misc.metric-system.advocacy group for those with a particular interest in political activities related to metrication. - It was written with the expectation that the topic is unlikely to attract large non-plain-text postings, commercial advertising or unsocial behaviour in any particular way, leaving these issues to common sense and USENET etiquette. - It is brief. CHARTER: misc.metric-system This newsgroup is for discussion about the International System of Units (SI) or metric system, including its use in scientific, technical, and consumer applications, its history and definition, and its adoption in fields and regions where other units of measurement are still prevalent (metrication). Included within its scope are related global standards and conventions, for example metric product specifications and consumer-product labelling practice. END CHARTER. FINAL VOTING ACKNOWLEDGEMENTS: misc.metric-system - Final Voter List are provided only to help verify the validity of the interest poll. Voted YES ---------------------------------------------------------------------------- -- 72027.3605 [at] compuserve.com Bob Bailin alan.jamieson [at] ntlworld.com Alan Jamieson alanh [at] unc.edu Alan Hoyle aminggs [at] yahoo.co.uk Andrew Inggs Andries.Brouwer [at] cwi.nl Andries Brouwer anthony [at] atkielski.com Anthony Atkielski b4r4n5k1 [at] yahoo.com Adam Baranski baron_carter [at] bmc.com Baron Carter bfrost [at] nyx.net Bonnell Frost blaise.egan [at] btopenworld.com Blaise F Egan bob [at] cave.org Robert Wilkins boldyrev+nospam [at] cgitftp.uiggm.nsc.ru Ivan Boldyrev bouvin [at] daimi.au.dk Niels Olof Bouvin Brian.Inglis [at] SystematicSw.ab.ca Brian Inglis buff [at] pobox.com William Denton CarletonM [at] aol.com Carleton MacDonald cdkaese [at] ntlworld.com Chris Kaese celia.mesure [at] spritenote.co.uk Celia Mesure ceri [at] submonkey.net Ceri Davies chrajohn [at] alumni.indiana.edu Chris Johnson chris [at] kzim.com Christopher Robin Zimmerman chris [at] lordsutch.com Chris Lawrence chris [at] metric.org.uk Chris KEENAN CirgreeSys [at] aol.com Timothy Moylan coghlan [at] sympatico.ca Patrick Coghlan croyle [at] gelemna.org Don Croyle dalcorn [at] hoosierlink.net David K Alcorn davep [at] davep.org Dave Pearson david [at] king-usa.com David King david [at] rossde.com David E. Ross dbpg [at] telus.net David Gibson dbreslau [at] ContinuumPhotonics.com David Breslau de5 [at] sws5.ornl.gov Dave Sill dips770 [at] yahoo.co.uk Dipika Tanna don-aitken [at] clara.co.uk Don Aitken dshatto [at] ucla.edu David Shatto dviolini [at] adelphia.net Robert de Violini dwolff [at] panix.com David Wolff Ekkehard [at] Uthke.de Ekkehard Uthke elizabeth.gallagher [at] rogers.com Elizabeth Gallagher ellis [at] spinics.net Rick Ellis erik+misc.metric-system [at] selwerd.nl Erik Warmelink erwan [at] rail.eu.org Erwan David esa.peuha [at] helsinki.fi Esa Peuha fisal [at] vbi.vt.edu Fidel Salas flo [at] uk.thalesgroup.com Paul Williams franck.thales [at] wanadoo.fr Franck T. frankmatthews [at] houston.rr.com Frank Matthews fungus [at] OCF.Berkeley.EDU Hank Fung gareth.rees [at] pobox.com Gareth Rees gherbert [at] retro.com George William Herbert gnygaard [at] nccray.com Gene Nygaard goltz [at] mmert.org James P. Goltz han.maenen [at] wanadoo.nl Han Maenen harder [at] myrealbox.com Jesper Harder hcrun [at] charter.net Dick Jackson henks [at] cistron.nl Henk Schaer herber [at] fnal.gov Randolph J. Herber horlacher [at] belwue.de Ulli 'Framstag' Horlacher howard.w.ludwig [at] lmco.com Howard W. LUDWIG hughescck [at] citcom.net Carroll Hughes ian.cowley [at] addenbrookes.nhs.uk Ian Cowley ijackson [at] chiark.greenend.org.uk Ian Jackson james.bursa [at] strcprstskrzkrk.co.uk James Bursa JamesL [at] Lugoj.com James Logajan jdg [at] diogenes.sacramento.ca.us John David Galt jeff [at] inforama.co.uk Jeff Gross jeffrey [at] carlyle.org Jeffrey Carlyle jilvonen [at] ee.oulu.fi Jussi Ilvonen jim.silverton [at] erols.com James Silverton jimmc [at] nova.org Jim McCracken jimmy-riddle [at] lineone.net Andrew Macpherson jimrtex [at] pipeline.com Jim Riley jjllxa16 [at] xs4all.nl J. J. Lodder jmprice [at] calweb.com John M. Price, PhD joe [at] jolomo.net Joe Morris john.hyde [at] ntlworld.com John A Hyde John.Jones [at] cec.eu.int John Michael Jones john [at] tradoc.fr John Wilcock Jon.Ericson [at] jpl.nasa.gov Jon Ericson jonathan.miles [at] uk.pwc.com Hugh Jonathan Miles jpc [at] suespammers.org J. Porter Clark kai-vote-mms [at] khms.westfalen.de Kai Henningsen kamlet [at] panix.com Art Kamlet khaled [at] easy.com Khaled Choudhury kilopascal [at] cox.net John P. Schweisthal kilopond [at] bigpond.com Eric Burns laszlo [at] pop-mg.rnp.br Laszlo Pinto lmend [at] ccny.cuny.edu Loren D. Mendelsohn lsonderling [at] earthlink.net Lawence Sonderling m.j.moseley [at] imperial.ac.uk Merrick Moseley Markus.Kuhn [at] cl.cam.ac.uk Markus Kuhn martin [at] vliet.demon.co.uk Martin Vlietstra mattheww [at] chiark.greenend.org.uk Matthew Woodcraft matthewzotter [at] earthlink.net Matthew Zotter mavifibe [at] angelfire.com Marcus V F Be max [at] alcyone.com Erik Max Francis mechtly [at] ux1.cso.uiuc.edu Eugene A. Mechtly mgk920 [at] dataex.com Michael G. Koerner michael.dahms [at] fh-flensburg.de Michael Dahms michael.mauch [at] gmx.de Michael Mauch michal [at] rosa.id.au Michal Rosa mjonesmel [at] comcast.net Matthew D. Jones morgan [at] head.cfa.harvard.edu Windsor Morgan msh210 [at] math.wustl.edu Michael Hamm (msh210) mthalloran [at] aol.com Michael T. Halloran naughtin [at] bigpond.net.au Pat Naughtin ncooper [at] wahoo.sjsu.edu Noreen Cooper ned [at] kilowatt.whoi.edu Ned Forrester nehager [at] msi-sensing.com Nat Hager III nejman [at] goteborg.bostream.se Asbj.9arn Nejman news01+Steven.Murdoch [at] cl.cam.ac.uk Steven J. Murdoch news1 [at] matthias-baake.de Matthias Baake nfitz [at] sentex.net Nicholas Fitzpatrick nhoffmann [at] despammed.com Norbert Hoffmann Nicefella [at] tesco.net Rowland Dye nikkit [at] web.de Kai Rode oconnell [at] slr.orl.lmco.com Kevin B. O'Connell odt [at] dtrx.de Olaf Dietrich pa [at] cdg.chalmers.se Per Andersson palaste [at] cc.helsinki.fi Joona I Palaste pan [at] syix.com Pan patrick.f [at] netaccess.co.nz Patrick Fitzgerald per-ove.persson3 [at] comhem.se Per Ove Persson pete.forman [at] westerngeco.com Pete Forman Peter-Lawrence.Montgomery [at] cwi.nl Peter Montgomery Peter [at] Lairo.com Peter Lairo phil [at] bathcity.org.uk Philip Andrews phil [at] celeritycomm.com Phil Chernack phil [at] durden.clara.co.uk Phil D phil [at] mckerracher.org Phil McKerracher philip.hall2 [at] ntlworld.com Philip S Hall Piotr.Zielinski [at] cl.cam.ac.uk Piotr Zielinski pk [at] TechFak.Uni-Bielefeld.DE Peter Koch pm9000 [at] usa.net Mark Halsall porjes [at] spamcop.net Peter Stephenson ppnerk [at] yahoo.com Phred Pnerk promotemetric [at] earthlink.net Ronald L. Stone psmyth [at] gmx.net Peter Smyth ptrusten [at] cox.net Paul Trusten ressel [at] frontiernet.net Howard Ressel richardl [at] zetnet.co.uk Richard Loebner richgr [at] panix.com Rich Greenberg richw [at] richw.org Rich Wales rkim461 [at] ECY.WA.GOV Rich Kim rlcarr [at] animato.arlington.ma.us Rich Carreiro rlh [at] theworld.com Roger L Hale rmcleod [at] pacificcoast.net Robert McLeod rnews [at] river.com Richard Johnson robert [at] chezmarshall.freeserve.co.uk Robert Marshall robin.paice [at] btconnect.com Robin Paice Roddy.Urquhart [at] synopsys.com Roddy Urquhart ronald.gallagher [at] rogers.com Ronald Gallagher rphenry [at] cox.net Richard Henry rps [at] rena.mat.uc.pt Rui Pedro Mendes Salgueiro rpyle [at] sciti.com Robert Pyle sagittaria [at] softhome.net Sagittaria sandmann [at] clio.rice.edu Charles Sandmann sanschag [at] staff.uni-marburg.de Paul Sanschagrin Sascha.Schimke [at] Student.Uni-Magdeburg.DE Sascha Schimke sbfaulds [at] ihug.co.nz Stuart Brodie Faulds sbrel [at] pcug.org.au Sandy Brelsford sc1-news [at] roamer1.org Stanley Cline shields [at] msrl.com Michael Shields shrao [at] nyx.net Shrisha Rao simon [at] darkmere.gen.nz Simon Lyall sj.bond [at] onyxnet.co.uk shirley bond skywatchbob [at] yahoo.com Robert Price sostaric [at] mszs.si DAVOR SOSTARIC sparre [at] nbi.dk Jacob Sparre Andersen srl32 [at] cam.ac.uk Stephen Lewis steve-in-sf [at] sbcglobal.net Steve Zafft steve [at] steve-and-pattie.com Steve MacGregor t.wade [at] vms.eurokom.ie Tom Wade terry [at] connected-systems.com Terry Simpson thefatphil [at] yahoo.co.uk FatPhil thierry [at] pompo.net Thierry Thomas thk [at] kms.dk Thomas Knudsen thor [at] anta.net Thor Kottelin toivo [at] ucs.uwa.edu.au Toivo Pedaste tomp [at] st.net.au Tom Perrett trh [at] xs4all.nl Tom Hageman tuhing [at] lexmediadigital.ph Tom W. Uhing twid [at] bibulus.org Thomas Widmann van.ette [at] inter.nl.net Robert-Jan van Ette vote-misc.metric-system [at] newton.digitalspace.net Philip Newton Wbauer03 [at] aol.com William Bauer wessels7 [at] xs4all.nl Rik Wessels wfp [at] wfpconsulting.com Bill Potts willner [at] cfa.harvard.edu Steve Willner worstall [at] btopenworld.com Michael V Worstall zimnyzenon [at] interia.pl Zenobiusz Zimny zvr [at] pobox.com Alexios Zavras Voted NO ---------------------------------------------------------------------------- -- aaron [at] aaroncity.com Aaron O'Donnell billv1939 [at] netscape.net Bill Vajk brougham5 [at] yahoo.com E. Brougham clzeni [at] mindspring.com clzeni dc [at] panix.com David W. Crawford devin [at] thecabal.org Devin L. Ganger dougbob [at] charter.net Bob Douglas eaking [at] ulape.org.uk Edward King fwbrown [at] bellsouth.net Wayne Brown gaillard [at] panix.com Ed Gaillard gprrspw [at] mindspring.com G.P. Ryan graham.drabble [at] lineone.net Graham Drabble johnl [at] iecc.com John Levine josephb [at] panix.com Joe Bernstein miliff [at] qnet.com Mary Shafer msb [at] vex.net Mark Brader naddy [at] mips.inka.de Christian Weisgerber patrick [at] texier.info Patrick Texier rick [at] bcm.tmc.edu Richard Miller ru.igarashi [at] usask.ca ru igarashi stainles [at] realtime.net Dwight Brown wiz [at] verinet.com Wiz Voted ABSTAIN ---------------------------------------------------------------------------- -- aahz [at] pobox.com Aahz hkt79 [at] earthlink.net Henrietta K Thomas mmontcha [at] OregonVOS.net Matthew Montchalin tempdog [at] erols.com A [Temporary] Dog yan [at] storm.ca Yves Bellefeuille Invalid Votes ---------------------------------------------------------------------------- -- ! Ack bounced: No such user m.collado [at] aaron.ls.fi.upm.es Manuel Collado ! Ack bounced: No such user medawar [at] panix.com ! No name given -- Bill Aten, UVV ==== Let us say we have a 4-by-4 same-sum grid (ie. magic square), which consists of 1 to 16 placed so that every row/column/main-diagonal sum to 30. We could then subdivide each of the 16 grid-squares each into smaller 4-by-4 grids. And then we could place the same numbers we had in the first-order grid, then multiply each of these by the integer that was placed into this larger square (1/16 of first-order grid) at the start. And we could continue this indefinitely... But to avoid infinities, we could divide each integer at each generation by 30. We would then be be left, at the nth generation, with a grid of 16^n squares containing rationals, where each row and column and main-diagonal sums to 1. And I wonder then, what can be said about the infinite-generation grid? What about if the grid is subdivided at each generation, not into 4-by-4, but by some other subdivision(s)?? There must be something interesting to say and ask about these fractal(-like) magic squares. thanks, Leroy Quet ==== >Let us say we have a 4-by-4 same-sum grid (ie. magic square), which >consists of 1 to 16 placed so that every row/column/main-diagonal sum >to 30. That might be difficult to accomplish -- 34 is easier. -- Wally Farley (WWFiv) ==== Let b(1) = 1; for 2 <= m, b(m) = (1/ln(m)) sum{p|m} sum{k=1 to a(p,m)} b(m/p^k) /k, where the p-sum is over the distinct primes p which divide, and p^a(p,m) = highest power of p which divides m. then: I believe that: sum{m=1 to oo} b(m)/ m^y = exp(integral{y to oo} ln(zeta(x)) dx) which is limit {n ->oo} product{k >= yn} zeta(k/n)^(1/n) (perhaps). Right? My attempt at trolling... And, by the way, if we altered the sum for complex-integration, and took its analytical continuation if necessary (to get B(z) defined on whole complex plane), we *might* be able to use B(z) to help investigate the Riemann hypothesis, for if zeta = 0 anywhere along the contour of integration, then B(z) = 0. (Not as easy as it seems, sadly, for B might still be 0 without any zeta(z) = 0 in its integrated domain, and because B seems to become indeterminate anyway as the upper-limit of integration approaches complex infinity.) ;( (I know, I know, I am full o' bull-spit...) ;) thanks, Leroy Quet ==== > Anselm's argument is an excercise in begging the question. An excellent example of question-begging yourself! > It is -worthless-. Nothing exists by definition. Existence has to be > proven or observed. An interesting thesis. I'm not sure what it does to the foundations of math. I'd simply ask how you would go about proving this thesis. Thomas ==== Do you profess a religion or at least believe in God? If you don't, why bother with God? If you do, does your religion lack a theology? I really wonder why one would want to prove or even argue on a mathematical plane, the existence of God. I suggest you approach God by faith and you will find God ==== > > > >> This is an interpretation of Christopher Langan's CTMU, www.ctmu.org >> , and Saint Anslem's ontological argument. > > > Anselm's argument is an excercise in begging the question. > > It is -worthless-. Nothing exists by definition. Existence has to be > proven or observed. > I beg to differ. If I define God as the best football player of all times, then he by definition exists. Of course the definition is nonsense, and I'm unlikely to get anyone else to follow my arguments, unless I start my speach In this three hour lecture, I will use the word 'God' to denote the best football player of all times. Forget about who created the world. Forget about the meaning of life. This lecture will talk only about football and hot dogs. Russell's definition is very much of the football player kind. Acke ==== > This is an interpretation of Christopher Langan's CTMU, www.ctmu.org >> , and Saint Anslem's ontological argument. > Anselm's argument is an excercise in begging the question. It is -worthless-. Nothing exists by definition. Existence has to be > proven or observed. > I beg to differ. If I define God as the best football player of all > times, then he by definition exists. Assuming your definition defines anything. There is no reason why there should be any such football player. Compare and contrast your definition with:I define Dog as the largest negative number greater than 7, does this exist just by this definition? > Of course the definition is > nonsense, and I'm unlikely to get anyone else to follow my arguments, > unless I start my speach In this three hour lecture, I will use the > word 'God' to denote the best football player of all times. Forget about > who created the world. Forget about the meaning of life. This lecture > will talk only about football and hot dogs. Russell's definition is very much of the football player kind. Acke > ==== >> The idea is not worthless, but doesn't have a lot to go on... Its like defining, giving a limit to infinity, its not going to come down to an equation but an idea... infinite knowledge... hmmm....yea ==== > This is an interpretation of Christopher Langan's CTMU, www.ctmu.org > , and Saint Anslem's ontological argument. > 1.] If it is possible for a mind to perfectly understand[model] every > aspect and detail of reality, then the mind that perfectly models > reality is a super-intelligence, for all intents and purposes, the > super-intelligence is God. Since reality is ever-changing it's not possible to comprehend all of totality at once. Perhaps the better definition of God-like intelligence is having the knowledge and power of creation. 2.]If the perfect correspondence can be approached via a convergent > analytic-synthetic propositional limit, then the limit exists, even > though a sentient mind within reality can only approach the limit. The knowledge of creation, God-like knowledge, can be modeled mathematically as a limit. The mathematics of evolutionary emergent properties serves as a basis for such a conjecture. An Introduction to Complex Systems Torsten Reil, Department of Zoology, University of Oxford http://users.ox.ac.uk/~quee0818/complexity/complexity.html Understanding that evolution is a property of randomness, in both non-living and living structures, provides a relationship between time and emergent steps for the following reasons. Random interactions are the driving force for evolution. Evolution produces emergent properties not definable from analysis of the parts. Emergent properties happen when an evolutionary system reaches self-organized criticality. Evolutionary systems attain self-organized criticality as fast as practical. These properties of evolution are not constrained to living systems, but also apply to non-living and intellectual systems as well. Emergent properties occur in power law bursts, large evolutionary steps are normal. Self-Organizing Systems (SOS) FAQ http://www.calresco.org/sos/sosfaq.htm INVESTIGATIONS STUART A. KAUFFMAN http://www.santafe.edu/sfi/People/kauffman/Investigations.html Once one understands the mathematics of evolution it is fairly easy to show that our level of intelligence does not define an upper limit. Taken to it's limit, these properties make clear that the emergence of a level of intelligence as far a above us as we are above animals...is a mathematical certainty. The universe, taken as a single evolving system, is certain to evolve what it needs to reproduce itself. And that conclusion provides a basis for the conjecture that this universe was created in a similar way. By naturally evolved intelligent design. 3.] If the limit exists, the exact mental correspondence exists in the > mind of a super-intelligence. 4.] That is to say, if the limit exists then a description exists. 5.] If the description exists then the describer exists, since the > description is isomorphic. 6.]The describer is a super-intelligence. 7.] By definition, the super-intelligence is God. The burden of proof becomes the burden of proving the convergence, > to an exact correspondence, between the mental construct[infinite > number of axioms] and reality Infinite number of axioms? There is only one. All creation and structure in the universe takes the form of an iterated loop into itself of two variables. The first being classical motion, the second quantum motion. At the limit [MIND]<--->[REALITY] The ...convergence of quantum and classical motion over time to evolutionary processes is the limit. M = R [axiomatic method]--->[exact correspondence]<---[scientific method] [light]--->[self-organization]<---[motion] As time goes towards infinity, so does order and intelligence. Jonathan s ==== > Do these sequence, especially the Beth sequence have fixed points? Yes, they must, by an elementary theorem. The alephs are well ordered, and all the alephs together are a proper class. (Likewise for the beths.) Any such class is isomorphic to the class of all ordinals, so there must be a bijection between the class of alephs and the class of ordinals. (That bijection is, in fact, the function aleph.) And you know this! > Does the Oswald Veblen's theorem, that normal ordinal functions have an > unbounded transfinte series of fixed points, apply to Aleph and Beth > sequences? Why wouldn't it? Thomas ==== > All this looks ok now. For your next mind-boggling trip, may i suggest you > have a look at inaccessible cardinals? > Finally! Ok, I've pulled down some web stuff about them, so at least I know the definition. So yes, the aleph and beth sequences we considered are limit cardinals and strong limit cardinals, but not very big limits for being singular. In addition, from previous reading I recall that tho it's possible to prove the consistency of rejecting inaccessible cardinals, it's impossible to prove acceptance of inaccessibles is consistent (relative to consistency of set theory). I'm much humored by those results, in effect saying tho it's quite reasonable to reject the unimaginably big, there's no assurance whatsoever that accepting the unimaginably big is reasonable; posing pesky overtones for those who entertain similar overwhelming concepts such as almighties. I'm also aware these are the smallest of a likely transfinite series of ever bigger, ever more ghost like cardinals, of which only a finite start has been perceived thru the fog of whimsy existence: Mahlo, compact, measurable, huge, extendible, remarkable; just to mention a notable few. There yet remains the expressions 'unusual', 'notable', 'too much', 'weird', 'incomprehensible', 'transcendent', 'unmentionable'. ;-) It'll take me awhile to ponder the definitions and perhaps review cofinality before attempting an intelligent question about them. Have you anything you'd like to point out about them? ==== > All this looks ok now. For your next mind-boggling trip, may i >> suggest you have a look at inaccessible cardinals? > Finally! Ok, I've pulled down some web stuff about them, so at least > I know the definition. So yes, the aleph and beth sequences we > considered are limit cardinals and strong limit cardinals, but not > very big limits for being singular. In addition, from previous reading I recall that tho it's possible to > prove the consistency of rejecting inaccessible cardinals, it's > impossible to prove acceptance of inaccessibles is consistent > (relative to consistency of set theory). I'm much humored by those results, in effect saying tho it's quite > reasonable to reject the unimaginably big, there's no assurance > whatsoever that accepting the unimaginably big is reasonable; posing > pesky overtones for those who entertain similar overwhelming concepts > such as almighties. I'm also aware these are the smallest of a likely transfinite series > of ever bigger, ever more ghost like cardinals, of which only a > finite start has been perceived thru the fog of whimsy existence: > Mahlo, compact, measurable, huge, extendible, remarkable; just to > mention a notable few. There yet remains the expressions 'unusual', 'notable', 'too much', > 'weird', 'incomprehensible', 'transcendent', 'unmentionable'. ;-) It'll take me awhile to ponder the definitions and perhaps review > cofinality before attempting an intelligent question about them. > Have you anything you'd like to point out about them? Actually, before going there, you might first be entertained by this very nice essay (alas, in french) about large objects and infinite ones : http://www.eleves.ens.fr:8080/home/madore/math/infinity.pdf But already, measurable cardinals are really interesting, as it is not clear at all at first sight that they must be huge, then not clear at all how much huge (with respect to inaccessible) that is. In fact, a good start is to ponder why inaccessible cardinals are so much bigger that, say, the Beth^(omega0)(0)-th fixed point of the Beth sequence... ==== I'm college freshmen currently taking multivariable calculus and considering double majoring in math and computer science. One thing that's been difficult for me lately is that my teacher grades not just based on the correctness of our work, but also the readability. She seems to grade on the organization of our work (if she can't follow it when grading, she takes off points), as well as how well we explain what we have done in words, with graphs, and worked solutions. I've been programming for quite a while, and I know that in that context it is very important to write code that is readable to simplify debugging and modification. However, I have only been studying mathematics seriously for a short while and I have yet to overcome many of the bad habits I developed in high school. If anyone has any tips or knows of any online/offline resources for math style, it would be much appreciated. On a side note, I am also not as used to communicating with mathematicians verbally or in writing as I am with programmers or engineers in general. When I run into terminology I am unfamiliar with, I turn to www.mathworld.wolfram.com; however, if anyone knows of any other and/or better resources I could certainly use them. -Brendan ==== I want to prove the following: Let f: R --> R be a continuous function such that (1) f((a + b)/2) >= (f(a) + f(b))/2, where 0 > a > b, (2) f(x) + f(y) >= f(x') + f(y') implies f(r*x) + f(r*y) >= f(r*x') + f(r*y'), for any x, y, x', y', and r (with r > 0). Then (3) there exists some negative x such that f(x) - f(2x) >= f(0) - f(x). Now, I believe I can prove that (1) and (2) imply: (4) for arbitrarily small negative e, there exists some x < e such that f(x) - f(2x - e) >= f(e) - f(x). And, although I can think of functions that satisfy (4) but not (3), none of these is continuous. (For example: let f be such that f(0) = 0, and f(x) = x - 1 for all negative x; then f satisfies (4) but not (3), and f is not continuous at 0.) So here's my hunch: (4) plus the continuity of f implies (3). But I can't work out how to prove it. Any help would be gratefully received. ==== Thought some of you might like to know about this site. http://www-neos.mcs.anl.gov/neos/ Enjoy, Flip