Subject: manifold question I Þnd it difÞcult to prove the following. Suppose M,N are smooth compact connected manifold. Let smooth surjective f: M-->N such that rank (df_a)= dim N for all a in M. Then: the submanifold f^(-1) (a) is differeomorphic to f^(-1)(b) where a, b are distinct point N. === Subject: Re: manifold question > I Þnd it difÞcult to prove the following. Suppose M,N are smooth > compact connected manifold. Let smooth surjective f: M-->N such that > rank (df_a)= dim N for all a in M. Then: the submanifold f^(-1) (a) is > differeomorphic to f^(-1)(b) where a, b are distinct point N. Check out this thread: Level set of a manifold Dale. === Subject: Integral-Solution do you have a solution for the following expression ( in maple notation ): int((+1/2*(ln(exp(2*I /a *(-Pi)*(x-1/2 *a )))*I+Pi)/Pi)*(+1/2*(ln(exp(2*I /b *(-Pi)*(x-1/2 *b )))*I+Pi)/Pi),x=0..a*b); where a,b are integer with a,b > 0. Christian R.9fther === Subject: Re: Integral-Solution > do you have a solution for the following expression ( in maple > notation ): > int((+1/2*(ln(exp(2*I /a *(-Pi)*(x-1/2 *a > )))*I+Pi)/Pi)*(+1/2*(ln(exp(2*I /b *(-Pi)*(x-1/2 *b > )))*I+Pi)/Pi),x=0..a*b); > where a,b are integer with a,b > 0. Well, have you tried using maple?? Come on, for a start you can cancel the ln() and the exp(). The you should try and simplify the integrand. It you¹re still stuck, then come back here. -Michael. === Subject: Re: Integral-Solution > do you have a solution for the following expression ( in maple > notation ): > int((+1/2*(ln(exp(2*I /a *(-Pi)*(x-1/2 *a > )))*I+Pi)/Pi)*(+1/2*(ln(exp(2*I /b *(-Pi)*(x-1/2 *b > )))*I+Pi)/Pi),x=0..a*b); > where a,b are integer with a,b > 0. > Well, have you tried using maple?? > Come on, for a start you can cancel the ln() and the exp(). The you should > try and simplify the integrand. It you¹re still stuck, then come back here. > -Michael. I have tried using maple, but the result of maple seems not to be correct. The value evalf(int((+1/2*(ln(exp(2*I /a *(-Pi)*(x-1/2 *a)))*I+Pi)/Pi)*(+1/2*(ln(exp(2*I /b *(-Pi)*(x-1/2 *b)))*I+Pi)/Pi),x=0..a*b)) differs from the value of the symbolic result that maple gives. Christian Ruether === Subject: Re: Integral-Solution >> do you have a solution for the following expression ( in maple >> notation ): >> int((+1/2*(ln(exp(2*I /a *(-Pi)*(x-1/2 *a >> )))*I+Pi)/Pi)*(+1/2*(ln(exp(2*I /b *(-Pi)*(x-1/2 *b >> )))*I+Pi)/Pi),x=0..a*b); >> where a,b are integer with a,b > 0. >I have tried using maple, but the result of maple seems not to be >correct. >The value evalf(int((+1/2*(ln(exp(2*I /a *(-Pi)*(x-1/2 >*a)))*I+Pi)/Pi)*(+1/2*(ln(exp(2*I /b *(-Pi)*(x-1/2 >*b)))*I+Pi)/Pi),x=0..a*b)) >differs from the value of the symbolic result that maple gives. Yes, Maple often has trouble with symbolic integrals involving branch cuts. To start with, ln(exp(z)) = z + 2*n*Pi*I where n = þoor((Pi - Im z)/(2*Pi)). Your integrand becomes (x - a*þoor(x/a))*(x - b*þoor(x/b))/(a*b) = frac(x/a)*frac(x/b) Suppose gcd(a,b) = 1. Then there is a 1-1 correspondence between integers j in [0,ab) and pairs [j_a, j_b] of integers in [0,a) x [0,b), such that j = j_a mod a and j = j_b mod b. If x = j+t with 0<=t<1, frac(x/a) = t/a + j_a/a frac(x/b) = t/b + j_b/b and so int(frac(x/a)*frac(x/b),x=j..j+1) = 1/(3*a*b) + (j_a+j_b)/(2*a*b) + j_a*j_b/(a*b) Adding this up over all j_a from 0 to a-1 and j_b from 0 to b-1, we get 1/12 + a*b/4 Now what if gcd(a,b) = c > 1? Let a=A*c and b=B*c. Then your integral is int(frac(x/(A*c))*frac(x/(B*c)), x=0 .. A*B*c^2) Use the change of variables x = c*t, and this becomes c*int(frac(t/A)*frac(t/B), t=0 .. A*B*c) which, since the integrand is periodic with period A*B, becomes c^2*int(frac(t/A)*frac(t/B), t=0 .. A*B) = c^2 * (1/12 + A*B/4) = gcd(a,b)^2/12 + a*b/4 Maybe it¹s not surprising that Maple doesn¹t come up with this formula. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Integral-Solution > Come on, for a start you can cancel the ln() and the exp(). simpliÞcation ln(exp(z))=z is not valid for the range in the integral. With the simpliÞcation ln(exp(z))=z, the integral reduces to Int(x^2,x=0 .. a*b)/a/b = a^2*b^2/3. With a=2,b=3 that is 12. But if I put a=2,b=3 in the original integral and integrate numerically, Maple says 1.583333 For the principal value of ln, we compute ln(exp(z)) by adjusting z by an integer multiple of 2*Pi*I so that the result has imaginary part between -Pi and Pi. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: Length of sequence of consecutive primes starting at 2 This must have been asked in here, but what is the longest sequence of consecutive odd integers that are all prime. How long is the longest (known) sequence? Is such a sequence restricted to a max length? For example: 3,5,7 with length 3 === Subject: Re: Length of sequence of consecutive primes starting at 2 > This must have been asked in here, but what is the longest sequence of > consecutive odd integers that are all prime. > How long is the longest (known) sequence? Is such a sequence > restricted to a max length? > For example: 3,5,7 with length 3 That¹s it. In any sequence of three (or more) larger odd numbers, (at least) one of them must be divisible by 3. === Subject: Re: Length of sequence of consecutive primes starting at 2 3,5,7 is the max: http://www.cs.iastate.edu/~cs330/exams/pastExams/Þnalsol.pdf >This must have been asked in here, but what is the longest sequence of >consecutive odd integers that are all prime. >How long is the longest (known) sequence? Is such a sequence >restricted to a max length? >For example: 3,5,7 with length 3 === Subject: Re: Length of sequence of consecutive primes starting at 2 > This must have been asked in here, but what is the longest sequence of > consecutive odd integers that are all prime. > How long is the longest (known) sequence? Is such a sequence restricted > to a max length? > For example: 3,5,7 with length 3 Restricting myself to positive integers, I believe you have gotten it. Any sequence of three or more consecutive odd integers must contain at least one that is divisible by three. There is only one odd prime integer that is divisible by three. -- Lance Lamboy Go F*ck Yourself ~ Dick Cheney === Subject: How do I solve this equation? At one time I am convinced that I knew how to solve the following equation, but many years have passed since those days and I am at a loss. I need to solve (for x) an equation of the form A + B*x + C*e^(c*x) + D*e^(d*x) = 0 where A, B, C, D, c and d are all constants. There must be some way of isolating x here, mustn¹t there? My tired old mind can no longer Þnd it, though. Any ideas or help would be very greatly appreciated! === Subject: Re: How do I solve this equation? >At one time I am convinced that I knew how to solve the following >equation, but many years have passed since those days and I am at a >loss. >I need to solve (for x) an equation of the form >A + B*x + C*e^(c*x) + D*e^(d*x) = 0 >where A, B, C, D, c and d are all constants. There must be some way of >isolating x here, mustn¹t there? My tired old mind can no longer Þnd >it, though. >Any ideas or help would be very greatly appreciated! You are a victim of one of the biggest misconceptions about mathematics: Mathematics is all about rearranging terms, doing tricky substitutions, applying fancy functions and ... out pops the solution as a nice (or maybe not so nice) expression using functions that one can Þnd on a pocket calculator or a Computer Algebraic System. This can very rarely be done. Highschool and Undergraduate Education seems to further this misconception because those excercises are all tuned in such a way that an algebraic solution can be found. Calculus (e.g. Þnding primitive functions)emphasizes that. I think early courses in differential equations also are heavy on algebraic manipulations. I think, much talent is wasted by this method. I would call that martial mathematics - maybe an inadequate translation of the term Kampfrechnen I heard on de.sci.mathematik. Well I always wanted to say that. Sorry to be OT by now. Thomas P.S. You can Þnd solutions of your equation if you are given values of your constants. You need to apply mathematical algorithms that approximate your solutions step by step. It is also a good idea to do curve sketching to eliminate parameter values for which there are no solutions. Example: If B=0 and A,C,D >0 there are no (real) solutions, regardless of c and d. === Subject: Re: How do I solve this equation? > At one time I am convinced that I knew how to solve the following > equation, but many years have passed since those days and I am at a > loss. > I need to solve (for x) an equation of the form > A + B*x + C*e^(c*x) + D*e^(d*x) = 0 > where A, B, C, D, c and d are all constants. There must be some way of > isolating x here, mustn¹t there? My tired old mind can no longer Þnd > it, though. > Any ideas or help would be very greatly appreciated! You can quit racking your brain. There is, unfortunately, no solution in closed form in terms of familiar functions. Indeed, as best I can tell, it can¹t be solved using the Lambert W function either. David Cantrell === Subject: Re: How do I solve this equation? > At one time I am convinced that I knew how to solve the following > equation, but many years have passed since those days and I am at a > loss. > I need to solve (for x) an equation of the form > A + B*x + C*e^(c*x) + D*e^(d*x) = 0 > where A, B, C, D, c and d are all constants. There must be some way of > isolating x here, mustn¹t there? My tired old mind can no longer Þnd > it, though. > Any ideas or help would be very greatly appreciated! > You can quit racking your brain. There is, unfortunately, no solution in > closed form in terms of familiar functions. Indeed, as best I can tell, it > can¹t be solved using the Lambert W function either. > David Cantrell what can be done anymore. that appears to solve it to one part in a million in 6 or 7 tries, so I can use this. === Subject: Minimizing f(x1) ....+ f(xn) I was trying to prove that the problem minimize g(x1,....xn) = f(x1)....+ f(xn) subject to x1+....xn =K>0 and xi>=0, i=1....n has a global minimum at x1 =....xn = K/n whenever f:[0, inf) ->R is differentiable and f¹ is strictly increasing. The proof I found may not be very sophisticated but I think it¹s OK. First, let¹s show that claim is true for n=2. In this case, the problem reduces to Þnding the minimum of g(x1) = f(x1) + f(K-x1) for x1 in [0, K}. Then g¹(x1) = f¹(x1) - f¹(K-x1). Since f¹ is strictly increasing, for 0<=x10 for k/2 0 and the claim is proved for n=2. Now, for n>2, let¹s show that if the vector x= (x1,....xn) is a feasible solution for the problem and at least 2 components xi and xj of x are distinct, then x cannot be an optimal solution. Let M = (xi+xj)/2 and consider the vector y, obtained from x by replacing xi and xj with M. That is, y = (x1....x_i-1,M...x_j-1,M.....xn) (supposing, WLOG, that ix_j, the claim just proved for n=2 implies we must have f(x_i) + f(x_j) > f(M) + f(M) =2f(M). Therefore, f(x) > f(y), which show x is not an optimal solution. Since the only way to avoid having 2 distinct variables set at different values is to choose x* = (K/n.....K/n) and since x* is a feasible solution, it¹s proved that x* is a global optimal solution for the problem. Of course, similar and symmetric conclusions hold if we assume f¹ is decreasing. What I found interesting is this solution is that it¹s simpler then using Lagrange Multipliers. Using Lagrange Multipliers, it¹s not hard to get the Þrst order optimality conditions. But to prove the solution is a global optimal is not so easy. If you want to compute the Hessian of the function you must assume f is twice differentiable and f¹¹ is continuous. In fact, this is not necessary. The same arguments hold if instead of sums we had products, provided f is postive on [o, inf). I think this proof is correct , right? Amanda === Subject: Re: Minimizing f(x1) ....+ f(xn) > minimize g(x1,....xn) = f(x1)....+ f(xn) > subject to x1+....xn =K>0 > and xi>=0, i=1....n > has a global minimum at x1 =....xn = K/n whenever f:[0, inf) ->R is > differentiable and f¹ is strictly increasing. Well, that means f is convex, whence: n*sum(f(xk)/n, k=1..n) >= n*f(sum(xk, k=1..n)/n) = n*f(K/n). That implies g has a global minimum at x1 = ... = xn = K/n. -- Julien Santini === Subject: Re: Minimizing f(x1) ....+ f(xn) > minimize g(x1,....xn) = f(x1)....+ f(xn) > subject to x1+....xn =K>0 > and xi>=0, i=1....n > has a global minimum at x1 =....xn = K/n whenever f:[0, inf) ->R is > differentiable and f¹ is strictly increasing. > Well, that means f is convex, whence: > n*sum(f(xk)/n, k=1..n) >= n*f(sum(xk, k=1..n)/n) = n*f(K/n). That implies g > has a global minimum at x1 = ... = xn = K/n. Yes, sure. It is trivial. My solution was stupid.... Amanda === Subject: Re: a simple problem > lim (log n!)/log n^{n}= ? (n--->00) > http://mathworld.wolfram.com/StirlingsApproximation.html > http://planetmath.org/encyclopedia/StirlingsApproximation.html Stirling is a bit of overkill here. The fraction is < 1, so we need only show the lim sup is >= 1. One approach is to use log n! = sum(m=1,n) log(m) >= int_[1,n] log(x) dx, leading to a simple solution. Without calculus: for any Þxed positive integer k, there are at least n - [n/k] - 1 numbers in {1,...,n} that are at least n/k. So (log n!)/log n^{n} >= (n - n/k - 1)*log(n/k)/[n*log(n)] -> 1 - 1/k. This is true for every k, so we¹re done. === Subject: Hermite polynomial continued fraction I have noticed the following and do not know if it is known... The denominators of the continued fraction 1/x - 1/x -2 / x -3/x -4/x ...-n/x are the nth Hermite polynomials ie. (and the numerators are Hermite polynomial n-1) 1 denominator = 1 He sub 0 1/x denominator = x He sub 1 1/x -1/x denominator = x^2 -1 He sub 2 1/x -1/x -2/x denominator = x^3 - 3x He sub 3 1/x - 1/x -2/x -3/x denominator = x^4 -6x^2 + 3 He sub 4 . . . 1/x - 1/x -2/x -3/x ...... n/x He sub n where He sub n (x) = 2^(-n/2) H sub n(x/2^.5) H sub n = nth Hermite polynomial. === Subject: Re: Hermite polynomial continued fraction > I have noticed the following and do not know if it is known... > The denominators of the continued fraction > 1/x - 1/x -2 / x -3/x -4/x ...-n/x > are the nth Hermite polynomials ie. > (and the numerators are Hermite polynomial n-1) > 1 denominator = 1 He sub 0 > 1/x denominator = x He sub 1 > 1/x -1/x denominator = x^2 -1 He sub 2 > 1/x -1/x -2/x denominator = x^3 - 3x He sub 3 > 1/x - 1/x -2/x -3/x denominator = x^4 -6x^2 + 3 He sub 4 > . > . > . > 1/x - 1/x -2/x -3/x ...... n/x He sub n > where He sub n (x) = 2^(-n/2) H sub n(x/2^.5) > H sub n = nth Hermite polynomial. Have a look at 7.1.15 of Abramowitz & Stegun¹s ŒHandbook of Mathematical Functions¹ here : 26.2.14 is of interest too (for 1/x+1/x+2/x+...) : For more general C.F. expressions concerning orthogonal polynomials see (8) at mathworld : http://mathworld.wolfram.com/OrthogonalPolynomials.html So it seems known (a little...). Raymond === Subject: Re: HELP: INTEGER FUNCTION ?=? REAL FUNCTION >> Let¹s say f(n) = 1/1 + 1/2 + 1/3 + 1/4 + ...+ 1/n >> ( Mathematica calls this the HarmonicNumber[n] ) >> Can I Þnd a g(x) = f(n) for g(x) restricted to Z ? >There are many. The most widely used is essentially the digamma >function, Psi. More precisely, HarminicNumber(n) = Psi(n+1)+gamma, >where gamma is Euler¹s constant. Psi is the logarithmic derivative >of Gamma: >Psi(x) = Gamma¹(x)/Gamma(x) >> In particular, can I Þnd an integral deÞnition for g(x) like the integral >> deÞnition of Gamma(x)? >> g(x) = int_(0 to inÞnity) h(x,n) dx ? Psi(x) = int_0^inÞnity (exp(-t) - (t+1)^(-x))/t dt for Re(x) > 0. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Finicky Problem with Baby Rudin, DeÞnition 3.16 I refer to DeÞnition 3.16, and Theorem 3.17(a), on page 56 of the 3rd edition of W. Rudin, _Principles of Mathematical Analysis_ (McGraw-Hill 1976). It seems to me that because of the _a priori_ possibility that E might be empty, we have to distinguish 3 cases: (1) {s_n} is not bounded above. Then +infty in E. (2) {s_n} is not bounded below. Then -infty in E. (3) {s_n} is bounded both above and below. In case (3), we must do one of 2 things: (A) Use the Bolzano-Weierstrass theorem to infer that E is nonempty, before attempting to deÞne s^* or s_*. (B) Accept the literal consequences of the deÞnitions of the terms involved even if E is empty. (This is already uncomfortable because Rudin has never used these deÞnitions in the case of the empty set.) If we choose (B), then if E is empty, we seem to have: s^* = -infty s_* = +infty But then consider the inference in the third paragraph of the proof of Theorem 3.17(a): If s^* = -infty, then E contains only one element, namely -infty, and there is no subsequential limit. This is valid only with the unstated assumption that some independent argument has established that E is nonempty - which brings us back to choice (A). To my mind, this establishes fairly clearly that there is no serious alternative to choice (A). The second paragraph of the proof of Theorem 3.17(a) then needs to be rewritten. But more seriously, isn¹t it a bad idea, in the Þrst place, to give a deÞnition of limit superior which requires the Bolzano-Weierstrass theorem to make it work (regardless of the fact that this requirement is unstated)? Surely it¹s preferable to give the simple deÞnition (in strict LaTeX+amsmath.sty form this time, sorry): [ limsup_{n to infty} s_n = lim_{k to infty} sup_{n geqslant k} s_n ] This only needs the completeness axiom to make it work. Am I being silly? (A Google search for errata didn¹t turn up anything relevant, but maybe I screwed that up, too.) -- Angus Rodgers (angus_prune@ eats spam; reply to angusrod@) Contains mild peril === Subject: Re: Finicky Problem with Baby Rudin, DeÞnition 3.16 > I refer to DeÞnition 3.16, and Theorem 3.17(a), on page 56 > of the 3rd edition of W. Rudin, _Principles of Mathematical > Analysis_ (McGraw-Hill 1976). etc. etc. If you want people to READ your posts, you have to tell us what the deÞnition and theorem ARE. While I happen to have the book in the other room, I¹m not interested enough to go look it up. But for the record, yes, Rudin is often sloppy. --Ron Bruck === Subject: Tuazon speaks at Whitney Whitney Museum of American Art Initial Public Offerings (I.P.O.): New Artists, New Curators Heather Felty Kouris and Oscar Tuazon Heather Felty Kouris, special projects director at apexart, New York, and artist Oscar Tuazon address the impact of city life on society as reþected in their projects. Kouris is currently organizing Everyday Hellas, an exhibition resulting from her 2002 Fulbright Research Fellowship in Greece. Tuazon, a 2002 graduate of the Architecture and Urban Studies component of the Whitney¹s Independent Study Program, explores alternative modes of living in his architectural and installation-based projects such as City Without a Ghetto. This series features salon-style dialogues between some of the most exciting curators, artists, and writers working in New York today. Admission: $5; members, senior citizens, and students with valid ID $3. Advance sales are strongly recommended as seating is limited; admission cannot be guaranteed without an advance purchase. Tickets are available at the Museum Admissions Desk or 1 (877) WHITNEY. === Subject: Re: _Re: VOTE on whether 1/oo = 0 >>Should the question be whether the limit of 1/x as x approaches >>inÞnity equals 0? >> >>And if not 0, what other value could it be. >An oft-repeated assertion on sci.math is that 1/oo is non-zero. >It fact, it is exactly equal to 1.0 - 0.999... > In fact, it is nothing of the kind. In fact, in some extensions of the real or complex number systems, there is an element which can reasonably be called oo and 1/oo = 0. > Since oo is NOT A NUMBER, you can¹t do division with it. Whether one chooses to call it a number or not is hardly important; it is an element of an extended system, and its reciprocal is 0. > Asking if 1/oo = 0 makes as much sense as asking if 1/blue = 0. The latter is nonsense (at least presumably, since we know of no system in which 1/blue is deÞned). Such cannot be said of the former. > Since NEITHER Œblue¹ nor ŒinÞnity¹ are NUMBERS you can¹t divide > them into 1. Well of course, if there is no ŒinÞnity¹ deÞned in the system in which we are working, then it¹s nonsense to speak of 1/inÞnity. But similarly, if we are working in the real numbers, say, it would be nonsense to assert that i^2 = -1. > InÞnity is not a real number. Real numbers satisfy the axioms of > a Þeld. InÞnity does not. Three correct statements in a row. Bravo! David W. Cantrell === Subject: Re: _Re: VOTE on whether 1/oo = 0 >Since oo is NOT A NUMBER, you can¹t do division with it. But you can apply linear fractional transformations to it; and since, historically (albeit abusively), the notation of fractions is used to denote these automorphisms of the projective line (over whatever Þeld you¹re working in), it is correct--in context--to write 1/oo = 0. Lee Rudolph === Subject: interesting website for free classical piano sheet music Have you heard of The Sheet Music Archive? You can download an entire library of classical sheet music for free. The largest site of its kind on the web. http://www.sheetmusicarchive.net The site is quite useful for piano teachers, as it has exercises and a huge repertoire. === Subject: Re: interesting website for free classical piano sheet music Still off topic, but http://www.mutopiaproject.org/ has a slightly less annoying licensing scheme. > Have you heard of The Sheet Music Archive? > You can download an entire library of classical > sheet music for free. The largest site of its > kind on the web. > http://www.sheetmusicarchive.net > The site is quite useful for piano teachers, as > it has exercises and a huge repertoire. === Subject: boxplots On the following AP Statistics Examination for this year (http://apcentral.collegeboard.com/repository/ap04_frq_ statistics_b_36001.pd f) in Question #5a, I have a question on how to construct the boxplots. It gives the values below Q1, the value of Q1, the median, the value of Q3, and the values above Q3. How would I construct the boxplot given just this information? I¹m a bit confused. Also, where do the whiskers start and end in each of the two boxplots? How would I Þgure this out? === Subject: Psychohistory - > Group Psychohistory - which seeks to understand the motivations of large groups in history. Areas of Psychohistorical Study There are three inter-related areas of psychohistorical study. 1. The History of Childhood - which looks at such questions as: 1. How have children been raised throughout history 2. How has the family been constituted 3. How and why have practices changed over time 4. The changing place and value of children in society over time 5. How and why our views of child abuse and neglect have changed 6. Why there is still denial in modern societies about the reality of child abuse 2. Psychobiography - which seeks to understand individual historical people and their motivations in history. 3. Group Psychohistory - which seeks to understand the motivations of large groups in history. [edit ] The actual term psychohistory was coined by Isaac Asimov as the name for a Þctional science in his Foundation Trilogy universe http://en.wikipedia.org/wiki/Psychohistory -- Respectfully, Roger L. Bagula tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : URL : http://home.earthlink.net/~tftn URL : http://victorian.fortunecity.com/carmelita/435/ === Subject: Re: Psychohistory - > Group Psychohistory - which seeks to understand the motivations of large groups in history. > Areas of Psychohistorical Study Off-topic drivel. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Psychohistory - > Group Psychohistory - which seeks to understand the motivations of large groups in history. Search links on psychohistory ( mostly psychoanalytic branch ala Freud Jung etc.): http://www.psychohistory.com/ http://www.psychohistory.com/htm/01_journal.html http://bpaeng.by.ru/ http://www.h-net.org/~psychhst/ http://www.geocities.com/psychohistory2001/ > * The Institute for Psychohistory > www.psychohistory.com 140 Riverside Drive New York, NY 10024-2605 tel/ fax: (212) 799-2294 email: psychhst@tiac.net The Institute is a scholarly research and publication institute chartered by the State of New York as a not-for-proÞt educational corporation, the Association for Psychohistory, Inc. > * Digital Archive of PSYCHOHISTORY Articles & Texts > http://www.geocities.com/kidhistory > * The International Psychohistorical Association and the Profession of Psychohistory by Henry Lawton. > http://www.geocities.com/Athens/Acropolis/8623/index.htm > * BRANCHES OF THE INSTITUTE FOR PSYCHOHISTORY > * Psychohistory Online Discussion To join send a blank email to psychohistory-subscribe@topica.com. > * Robert B. McFarland, M.D., is > Chairman of the National Parenting Conference http//.bcn.boulder.co.us/~fells/parenting.html , founder of The Parenting Place, and Contributing Editor to the Journal of Psychohistory, and can be reached at 2300 Kalmia, Boulder, CO 80304. > setstats 1 > Areas of Psychohistorical Study > There are three inter-related areas of psychohistorical study. > 1. The History of Childhood - which looks at such questions as: > 1. How have children been raised throughout history > 2. How has the family been constituted > 3. How and why have practices changed over time > 4. The changing place and value of children in society over time > 5. How and why our views of child abuse and neglect have changed > 6. Why there is still denial in modern societies about the > reality of child abuse > 2. Psychobiography - > which seeks to understand individual historical people and their > motivations in history. > 3. Group Psychohistory - which seeks to understand the motivations of > large groups in history. > [edit > ] > The actual term psychohistory was coined by Isaac Asimov > as the name for a Þctional > science in his Foundation Trilogy > universe > http://en.wikipedia.org/wiki/Psychohistory -- Respectfully, Roger L. Bagula tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : URL : http://home.earthlink.net/~tftn URL : http://victorian.fortunecity.com/carmelita/435/ === Subject: Re: Psychohistory - > Group Psychohistory - which seeks to understand the motivations of large groups in history. Asimov must have loved the use they made of his name, but it doesn¹t alter the original implications and use of that science that his novel suggested. Some people are still doing work that could be considered related to the original subject. > Search links on psychohistory ( mostly psychoanalytic branch ala Freud > Jung etc.): > http://www.psychohistory.com/ > http://www.psychohistory.com/htm/01_journal.html > http://bpaeng.by.ru/ > http://www.h-net.org/~psychhst/ > http://www.geocities.com/psychohistory2001/ >> * The Institute for Psychohistory >> www.psychohistory.com 140 Riverside Drive New York, NY >> 10024-2605 tel/ fax: (212) 799-2294 email: psychhst@tiac.net The >> Institute is a scholarly research and publication institute chartered >> by the State of New York as a not-for-proÞt educational corporation, >> the Association for Psychohistory, Inc. >> * Digital Archive of PSYCHOHISTORY Articles & Texts >> Eric Heimstadt¹s ongoing project for the preservation and >> dissemination of psychohistorical texts, whose goal is to create a >> comprehensive psychohistorical library. Now contains over one hundred >> prenatal and cultic studies. Also features a full set of >> psychohistory-related web links. >> http://www.geocities.com/kidhistory >> * The International Psychohistorical Association and the >> Profession of Psychohistory by Henry Lawton. >> http://www.geocities.com/Athens/Acropolis/8623/index.htm >> * BRANCHES OF THE INSTITUTE FOR PSYCHOHISTORY >> * Psychohistory Online Discussion To join send a blank email to >> psychohistory-subscribe@topica.com. >> * Robert B. McFarland, M.D., is >> Chairman of the National Parenting Conference >> http//.bcn.boulder.co.us/~fells/parenting.html , founder of The >> Parenting Place, and Contributing Editor to the Journal of >> Psychohistory, and can be reached at 2300 Kalmia, Boulder, CO 80304. >> setstats 1 >> Areas of Psychohistorical Study >> There are three inter-related areas of psychohistorical study. >> 1. The History of Childhood - which looks at such questions as: >> 1. How have children been raised throughout history >> 2. How has the family been constituted >> 3. How and why have practices changed over time >> 4. The changing place and value of children in society over time >> 5. How and why our views of child abuse and neglect have changed >> 6. Why there is still denial in modern societies about the >> reality of child abuse >> 2. Psychobiography - >> which seeks to understand individual historical people and their >> motivations in history. >> 3. Group Psychohistory - which seeks to understand the motivations of >> large groups in history. >> [edit >> ] >> The actual term psychohistory was coined by Isaac Asimov >> as the name for a >> Þctional science in his Foundation Trilogy >> universe >> http://en.wikipedia.org/wiki/Psychohistory -- Respectfully, Roger L. Bagula tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : URL : http://home.earthlink.net/~tftn URL : http://victorian.fortunecity.com/carmelita/435/ === Subject: Re: Psychohistory - > Group Psychohistory - which seeks to understand the motivations of large groups in history. > Asimov must have loved the use they made of his name, > but it doesn¹t alter the original implications and use of that science > that his novel suggested. > Some people are still doing work that could be considered > related to the original subject. But you are posting drivel irrelevant to mathematics on sci.math. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Psychohistory - > Group Psychohistory - which seeks to understand the motivations of large groups in history. X-Reply-Etiquette: No copy by email, please Originator: legalize@deuce.xmission.com (Rich) [Please do not mail me a copy of your followup] Psychohistory isn¹t a science, its Þction. Its also not relevant to sci.fractals. -- The Direct3D Graphics Pipeline-- code samples, sample chapter, FAQ: Pilgrimage: Utah¹s annual demoparty === Subject: Re: Psychohistory - > Group Psychohistory - which seeks to understand the motivations of large groups in history. > Psychohistory isn¹t a science, its Þction. Its also not relevant > to sci.fractals. > -- Your comments are pointless. He will not listen to you, anyway. I guess he is doped. Some kind of endurance-boosting drugs.. (Opium ? EPO ? Creatine ?) === Subject: International Conference SETIT 2005 SETIT 2005 The 3rd International Conference on Sciences of Electronic, Technologies of Information and Telecommunications SETIT 2005 will be held in Susa, Tunisia 27-31 March 2005. you can Þnd more details in: http://www.universites.tn/setit/ The paper submission can be done on-line at http://www.conference-papers.org/ Accepted Papers will be published in the Conference Proceedings, as a Book Chapter with the ISBN: 973-51-546-3 Papers are solicited in the following areas: Information Processing Signal Processing Image and Video Multimedia Telecommunications & Networks Electronic Applications 15 January 2005 Submission of Þnal versions. 27-31 March 2005 Conference SETIT 2005. Med Salim BOUHLEL Co-Chairman SETIT === Subject: Re: Is Binary BrainF*** Turing-complete? charset=Windows-1252 > Regarding the question that started this thread, > i.e. whether a Þve-instruction brain variant is > Turing-complete, r.e.s. has proved it is by developing > a translator from brain to this variant, now called F I thought it would be entertaining to obtain an alternative proof by programming a universal tag-system directly in F: >+[+>>+]+>[[[[<<]<<]+>+>>+[[+>>+]+>>+]+<<+>]>+>[>+>]>[>>]+>>+ <<<<<[[<<]<<]>>>+[<[>[>>]+[[+>>+]+>>+]+[>>]<<<+>>[+<<+]+[[< <]<<]>+[+>>+]+<+]>[>>]+[[+>>+]+>>+]+[>>]<[+[+<<+]+[[<<]<<]>+[ +>>+]+<+>[>>]+[[+>>+]+>>+]+[>>]<+]>+<+[+<<+]+[[<<]<<]>+[+>>+] >>+]<+>+[+>>+]+>>+[<<+<+>[+>>+]+>>+]+<<+>>[>>]<<+<<[<<]>] For a detailed explanation of this program, and a description of the universal tag-system it simulates, see http://r.s.home.mindspring.com/F/f_tag.txt --r.e.s. === Subject: iso~~. hello.....doctor~ in the linear algebra, Let V be the space of all Þbonacci squence {a_n}. DeÞnd v1,v2 in V by v1 = (1,0,1,1,2,3,5.....) v2 = (0,1,1,2,3,5,8.....) then, v1 and v2 form a basis for V. thus, dim V = 2 ------------------------------------- of course, i can show that linear indepent, span to V. but i have a question. i think..... if {a_n} is the set of all sequences of real numbers, V ~ R^2 (iso) if {a_n} is the set of all sequences of complex numbers, V ~ C^2 (iso) nevertheless, R^2 is not iso C^2 um.....my thinking is right ?? === Subject: Re: iso~~. >in the linear algebra, >Let V be the space of all Þbonacci squence {a_n}. I assume a Fibonacci sequence is deÞned as any sequence of scalars (a_n: n=1..inÞnity) satisfying the recursion a_n = a_{n-1}+a_{n-2} for n>=3. >DeÞnd v1,v2 in V by >v1 = (1,0,1,1,2,3,5.....) >v2 = (0,1,1,2,3,5,8.....) >then, v1 and v2 form a basis for V. >thus, dim V = 2 >if {a_n} is the set of all sequences of real numbers, >V ~ R^2 (iso) >if {a_n} is the set of all sequences of complex numbers, >V ~ C^2 (iso) >nevertheless, R^2 is not iso C^2 The question is, which scalar Þeld are you using in the deÞnition of V? Any Þeld will do, but each one will give you a different V. If the scalar Þeld is F, then V is isomorphic (as a vector space over F) to F^2. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Outerplanar Graphs > Most simple connected planar graphs may be drawn with all vertices > on the outer face. But not all such graphs are outerplanar. Why? > Well, for one thing, an outerplanar graph has at most 2n-4 edges, 2n-4 is a typo; it should be 2n-3 === Subject: Re: Outerplanar Graphs 3QLpj-NoP*NzsIC,boYU]bQ]H¹y<#4ga3$21: > > > Most simple connected planar graphs may be drawn with all vertices > > on the outer face. But not all such graphs are outerplanar. Why? > > Well, for one thing, an outerplanar graph has at most 2n-4 edges, > 2n-4 is a typo; it should be 2n-3 -- David Eppstein Computer Science Dept., Univ. of California, Irvine http://www.ics.uci.edu/~eppstein/ === Subject: Re: James Harris hello james harris > in other words what i just said. he knows he¹s right. even though we > all tell him he¹s an idiot. > Sigh. What does any of that have to do with mathematics? > It is called sci.math you know. > You people are just sick. You get off on calling me names so the > reality of the situation doesn¹t matter to you. Look who¹s going on about the reality of the situation! The reality of the situation is that the mathematics so far published by James Harris has been found to be of little or no value by everyone except the author, and that the non-mathematics published here by t James Harris is of no value except to amuse those who read NGs in the absence of bread and circuses. But somehow JSH doesn¹t sell as a Christian facing the lions. === Subject: Re: James Harris hello james harris > > in other words what i just said. he knows he¹s right. even though we > > all tell him he¹s an idiot. > > Sigh. What does any of that have to do with mathematics? > > It is called sci.math you know. > > You people are just sick. You get off on calling me names so the > reality of the situation doesn¹t matter to you. > Look who¹s going on about the reality of the situation! > The reality of the situation is that the mathematics so far published by > James Harris has been found to be of little or no value by everyone > except the author, and that the non-mathematics published here by t > James Harris is of no value except to amuse those who read NGs in the > absence of bread and circuses. > But somehow JSH doesn¹t sell as a Christian facing the lions. James Harris why are you trying to get your paper published in another journal when you know it is faulty? === Subject: Re: James Harris hello james harris [snip previous discussion about the use of Œcrackpot¹ label] > Sigh. What does any of that have to do with mathematics? Sigh. What does *your* post (see below) have to do with mathematics?? > It is called sci.math you know. > You people are just sick. You get off on calling me names so the > reality of the situation doesn¹t matter to you. > Ok, so a lot of you think that I¹m wrong. You don¹t believe I¹ve > found anything of mathematical interest, and you feel VERY conÞdent > about calling me names and repeating over and over and over again in > posts just how low you think I am. > Well I think it shows that there¹s something wrong with many of you > that you have to spend so much time to try and attack one person, and > spend so much time getting all hot and bothered about one guy, who > admits that he mouths off at times and has often been wrong. > You are sick. There is something wrong with you people. > I post on Usenet. Usenet is a public forum. I¹ve angered a lot of > people who just keep going and going and going but that doesn¹t mean > they aren¹t wrong. > Sure, some of you act like it¹s my problem if I said things that upset > YOU. > You act like I¹m wrong if lots of people reply to me. > You go on and on and on as if it¹s all my fault, as if you can spend > the rest of your lives calling me an idiot or a crank or a crackpot > and come off smelling like roses. > But you can¹t. > People like you who have to spend so much time and effort putting down > ONE other human being cannot be healthy!!! > You are sick. There is something really wrong with a group of people > who spend so much time and effort worrying about attacking ONE other > human being. > Hell, some of you have even questioned my humanity. > I POST. These are WORDS. This is USENET. > People post on Usenet and don¹t always say what you want to hear. > But ganging up on one person for YEARS is just not a sign of mental > health. > Yeah, sure, call me crazy, but I¹m not a member of a gang of hostile > posters who spend an inordinate amount of time attacking one person, > who when it comes right down to it, is just posting on Usenet. > James Often in error, but never in doubt! Harris -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com === Subject: Re: PROVING TRANSCENDENCE ?? > This forum is great... > I just got some answers to my Þrst-ever posting and now I¹m eager to toss > out a few more of the questions that occur to me from time to time. > Here¹s one.... > What kind of mathematical tools & techniques are used in trying to prove that > a particular number is transcendental? > There are proofs that e and pi and others are transcendental but I recently > read somewhere that e + pi, for instance, has not been shown to be > transcendental or non-transcendental. But it is easy to show that at least one of e + pi or e*pi transcendental. === Subject: Re: PROVING TRANSCENDENCE ?? >What kind of mathematical tools & techniques are used in trying to prove >that a particular number is transcendental? You might look at Alan Baker, Transcendental Number Theory, Cambridge U. Press 1975. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Group theory question Does anyone know of an elegant proof of : If G is a Þnite group and G/Z(G) has a unique Sylow p-subroup then G has a unique Sylow p-subgroup. I think I have a proof by contradiction, showing that distinct Sylow p-subgroups of G would give rise to distinct Sylow p-subgroups of G/Z(G). I was hoping there would be a nicer way to do the problem. === Subject: Re: Group theory question > Does anyone know of an elegant proof of : > If G is a Þnite group and G/Z(G) has a unique Sylow p-subroup then G > has a unique Sylow p-subgroup. > I think I have a proof by contradiction, showing that distinct Sylow > p-subgroups of G would give rise to distinct Sylow p-subgroups of > G/Z(G). I was hoping there would be a nicer way to do the problem. I do not think the following proof is elegant, but ..... Proof. By the hypothesis, G/Z(G) has a uinque Sylow p-subgroup M. Note that every Sylow p-subgroup of G/Z(G) is of the form PZ(G)/Z(G), for some Sylow p-Subgroup P of G. If P1 is another Sylow p-subgroup of G, then we must have M=P1Z(G)/Z(G)=PZ(G)/Z(G). It follows that P1Z(G)=PZ(G). Now let x be in P, then x is in P1Z(G). Thus x=yz for some y in P1 and z in Z(G). Since x and y are p-elemets, z so is (note that y commute with z and x=yz). Now is a normal p-subgroup (since z in Z(G)), thus it is contained in every Sylow p-subgroup (It is easy to see by these facts: every two Sylow p-subgroups are conjugate and every p-subgroup is contained in a Sylow p-subgroup) Thus z belongs to P1, in particular. Thus y=xz^{-1} in P1. It follows that P is contained in P1. Thus P=P1, as required. All the best Alireza === Subject: Re: Group theory question quee0849@yahoo.co.uk >Does anyone know of an elegant proof of : >If G is a Þnite group and G/Z(G) has a unique Sylow p-subroup then G >has a unique Sylow p-subgroup. >I think I have a proof by contradiction, showing that distinct Sylow >p-subgroups of G would give rise to distinct Sylow p-subgroups of >G/Z(G). I was hoping there would be a nicer way to do the problem. Let Q/Z(G) be the unique Sylow p-subgroup of G/Z(G) (so Q is normal in G), and let P be a Sylow p-subgroup of Q (and hence of G). Then Q = Z(G) P, so P is normal in Q. Hence P is the unique Sylow p-subgroup of Q, so P is characteristic in Q and hence normal in G. Alternatively: by the Frattini argument, G = Q N_G(P) = Z(G) P N_G(P) <= N_G(P), so P is normal in G. Derek Holt. === Subject: Re: What form of matter will last the longest? > Interesting question. Devil Solvent at temp eats copper, nickel, > nichrome, silver... I didn¹t try any Hastelloys because I found > an acceptable metal then stopped. Nope, I wouldn¹t give C-2000 > good odds against Devil Solvent or its eutectic Devil Solvent II. What is Devil Solvent? All links in Google on this topic go back to you. === Subject: Re: What form of matter will last the longest? > Interesting question. Devil Solvent at temp eats copper, nickel, > nichrome, silver... I didn¹t try any Hastelloys because I found > an acceptable metal then stopped. Nope, I wouldn¹t give C-2000 > good odds against Devil Solvent or its eutectic Devil Solvent II. > What is Devil Solvent? All links in Google on this topic > go back to you. It¹s the only molten salt Uncle Al could identify in which graphite violently reacts and diamond dust persists. The idea is to do CVD diamond chemistry condensed phase, 1500 times the density of Ar/H2/CH4 plasma, with a corresponding increase in deposition rate. We do it Brad Pate¹s way. If it Þnally works it will be an awesomely clever implementation. I think we have the bugs out, Þnally. The Þrst Devil Solvent run was conducted in a 0.25 thick densiÞed graphite crucible. A quarter cup of the stuff went through the graphite, through about 1.5 inches of Þre brick, through the desk top, through two shelves, and made a respectible run at the cement slab þoor. Emission of visible plume Management, Hey Al, you could have tunneled to China! Uncle Al, Nah. It would have exploded at the water table. It¹s nothing a barrier-coated Parr reactor can¹t handle. We did have something of a time Þnding someboody who could weld the inch-thick reactor head. Parr wanted none of it. If Uncle Al was not being eaten alive by 38 species of mosquitoes at the University of Manitoba while sodomizing roaches, he¹d be running the thing right now. The world will have to wait, because Management is stooopid. (The potato salad at the Pembina Hall servery is world class. It would sit proudly in any New York Deli. Manitoba is Ukrainian Russians. Ukrainian young women will make your breath catch in your throat and your eyes mist over they are so beautiful - even better from the front. Then they turn 30 and molt into monsters.) (Everything in Winnipeg is named Pembina. I understand it is a virulent STD passed between poultry and farmers.) -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/qz.pdf === Subject: Re: What form of matter will last the longest? > > Interesting question. Devil Solvent at temp eats copper, nickel, > > nichrome, silver... I didn¹t try any Hastelloys because I found > > an acceptable metal then stopped. Nope, I wouldn¹t give C-2000 > > good odds against Devil Solvent or its eutectic Devil Solvent II. > What is Devil Solvent? All links in Google on this topic > go back to you. > It¹s the only molten salt Uncle Al could identify in which > graphite violently reacts and diamond dust persists. The idea is > to do CVD diamond chemistry condensed phase, 1500 times the > density of Ar/H2/CH4 plasma, with a corresponding increase in > deposition rate. We do it Brad Pate¹s way. If it Þnally works > it will be an awesomely clever implementation. I think we have > the bugs out, Þnally. > The Þrst Devil Solvent run was conducted in a 0.25 thick > densiÞed graphite crucible. A quarter cup of the stuff went > through the graphite, through about 1.5 inches of Þre brick, > through the desk top, through two shelves, and made a respectible > run at the cement slab þoor. Emission of visible plume umm Mon Oncle. What was the cup that held the Devil Solvent made of? Noodles KD is good food! > Management, Hey Al, you could have tunneled to China! > Uncle Al, Nah. It would have exploded at the water table. > It¹s nothing a barrier-coated Parr reactor can¹t handle. We did > have something of a time Þnding someboody who could weld the > inch-thick reactor head. Parr wanted none of it. > If Uncle Al was not being eaten alive by 38 species of mosquitoes > at the University of Manitoba while sodomizing roaches, he¹d be > running the thing right now. The world will have to wait, > because Management is stooopid. (The potato salad at the Pembina > Hall servery is world class. It would sit proudly in any New > York Deli. Manitoba is Ukrainian Russians. Ukrainian young > women will make your breath catch in your throat and your eyes > mist over they are so beautiful - even better from the front. > Then they turn 30 and molt into monsters.) > (Everything in Winnipeg is named Pembina. I understand it is a > virulent STD passed between poultry and farmers.) > -- > Uncle Al > http://www.mazepath.com/uncleal/ > (Toxic URL! Unsafe for children and most mammals) > http://www.mazepath.com/uncleal/qz.pdf === Subject: Re: What form of matter will last the longest? >>Interesting question. Devil Solvent at temp eats copper, nickel, >>nichrome, silver... I didn¹t try any Hastelloys because I found >>an acceptable metal then stopped. Nope, I wouldn¹t give C-2000 >>good odds against Devil Solvent or its eutectic Devil Solvent II. >What is Devil Solvent? All links in Google on this topic >go back to you. >>It¹s the only molten salt Uncle Al could identify in which >>graphite violently reacts and diamond dust persists. The idea is >>to do CVD diamond chemistry condensed phase, 1500 times the >>density of Ar/H2/CH4 plasma, with a corresponding increase in >>deposition rate. We do it Brad Pate¹s way. If it Þnally works >>it will be an awesomely clever implementation. I think we have >>the bugs out, Þnally. >>The Þrst Devil Solvent run was conducted in a 0.25 thick >>densiÞed graphite crucible. A quarter cup of the stuff went >>through the graphite, through about 1.5 inches of Þre brick, >>through the desk top, through two shelves, and made a respectible >>run at the cement slab þoor. Emission of visible plume > umm Mon Oncle. What was the cup that held the Devil Solvent made of? Note that he said at temp and molten salt. Seems reasonable it wasn¹t especially reactive when not molten (unless it really likes water), but started eating the graphite after suitable heating. Also note that he wants to _patent_ the process, including the composition of D.S. I & II. If you want to know what it is, there¹re lots of clues scattered through his posts, including the above one that one is an eutectic of the other. Guess away. Mind you, any decent thermite will do what he recounts (except for C allotrope solubility), but then active thermite is somewhat more than merely molten. Mark L. Fergerson === Subject: Re: What form of matter will last the longest? >> > > >>Interesting question. Devil Solvent at temp eats copper, nickel, >>nichrome, silver... I didn¹t try any Hastelloys because I found >>an acceptable metal then stopped. Nope, I wouldn¹t give C-2000 >>good odds against Devil Solvent or its eutectic Devil Solvent II. > >What is Devil Solvent? All links in Google on this topic >go back to you. >> >>It¹s the only molten salt Uncle Al could identify in which >>graphite violently reacts and diamond dust persists. The idea is >>to do CVD diamond chemistry condensed phase, 1500 times the >>density of Ar/H2/CH4 plasma, with a corresponding increase in >>deposition rate. We do it Brad Pate¹s way. If it Þnally works >>it will be an awesomely clever implementation. I think we have >>the bugs out, Þnally. >> >>The Þrst Devil Solvent run was conducted in a 0.25 thick >>densiÞed graphite crucible. A quarter cup of the stuff went >>through the graphite, through about 1.5 inches of Þre brick, >>through the desk top, through two shelves, and made a respectible >>run at the cement slab þoor. Emission of visible plume > umm Mon Oncle. What was the cup that held the Devil Solvent made of? > Note that he said at temp and molten salt. Seems > reasonable it wasn¹t especially reactive when not molten > (unless it really likes water), but started eating the > graphite after suitable heating. > Also note that he wants to _patent_ the process, > including the composition of D.S. I & II. If you want to > know what it is, there¹re lots of clues scattered through > his posts, including the above one that one is an eutectic > of the other. Guess away. > Mind you, any decent thermite will do what he recounts > (except for C allotrope solubility), but then active > thermite is somewhat more than merely molten. molten salt is used for solar power stations that use mirrors instead of solar cells, so the heat capacity is utilised already. the same day reading these I saw a kids show where he shot sludge out of his arm that melted bins, people... the doctor who episode was on rockets thats carried chemical warheads in aus.politics. what you call coincidence I call a channel. I can¹t Þgure out the relevance except that chemical weapons is our next threat. and we all thought it would be biological! If you are interested in using my nuclear proof quantum shield that also protects people in my vicinity, (my destiny to reach Eve cannot be breached ~ laws of physics ~ but you;re all too entwined to notice my proofs of macro quantum entanglement) then I would really really appreciate donations to domainsATÞndit.com since US military keeps me broke so I remain in public. Herc change dit to dot few hundred so I can launch my websites 6 months earlier than scheduled would be nice, the satelite truman probe with microwave lasers targetting me constantly ages me 2 years every 6 months, contant deprivation. === Subject: Re: What form of matter will last the longest? >> > > >>Interesting question. Devil Solvent at temp eats copper, nickel, >>nichrome, silver... I didn¹t try any Hastelloys because I found >>an acceptable metal then stopped. Nope, I wouldn¹t give C-2000 >>good odds against Devil Solvent or its eutectic Devil Solvent II. > >What is Devil Solvent? All links in Google on this topic >go back to you. >> >>It¹s the only molten salt Uncle Al could identify in which >>graphite violently reacts and diamond dust persists. The idea is >>to do CVD diamond chemistry condensed phase, 1500 times the >>density of Ar/H2/CH4 plasma, with a corresponding increase in >>deposition rate. We do it Brad Pate¹s way. If it Þnally works >>it will be an awesomely clever implementation. I think we have >>the bugs out, Þnally. >> >>The Þrst Devil Solvent run was conducted in a 0.25 thick >>densiÞed graphite crucible. A quarter cup of the stuff went >>through the graphite, through about 1.5 inches of Þre brick, >>through the desk top, through two shelves, and made a respectible >>run at the cement slab þoor. Emission of visible plume > umm Mon Oncle. What was the cup that held the Devil Solvent made of? > Note that he said at temp and molten salt. Seems > reasonable it wasn¹t especially reactive when not molten > (unless it really likes water), but started eating the > graphite after suitable heating. > Also note that he wants to _patent_ the process, > including the composition of D.S. I & II. If you want to > know what it is, there¹re lots of clues scattered through > his posts, including the above one that one is an eutectic > of the other. Guess away. > Mind you, any decent thermite will do what he recounts > (except for C allotrope solubility), but then active > thermite is somewhat more than merely molten. > Mark L. Fergerson Probably will be used as a bioweapon based on his rants. === Subject: Re: What form of matter will last the longest? > > > > > > Interesting question. Devil Solvent at temp eats copper, nickel, > > nichrome, silver... I didn¹t try any Hastelloys because I found > > an acceptable metal then stopped. Nope, I wouldn¹t give C-2000 > > good odds against Devil Solvent or its eutectic Devil Solvent II. > > > > What is Devil Solvent? All links in Google on this topic > > go back to you. > It¹s the only molten salt Uncle Al could identify in which > graphite violently reacts and diamond dust persists. The idea is > to do CVD diamond chemistry condensed phase, 1500 times the > density of Ar/H2/CH4 plasma, with a corresponding increase in > deposition rate. We do it Brad Pate¹s way. If it Þnally works > it will be an awesomely clever implementation. I think we have > the bugs out, Þnally. > The Þrst Devil Solvent run was conducted in a 0.25 thick > densiÞed graphite crucible. A quarter cup of the stuff went > through the graphite, through about 1.5 inches of Þre brick, > through the desk top, through two shelves, and made a respectible > run at the cement slab þoor. Emission of visible plume > umm Mon Oncle. What was the cup that held the Devil Solvent made of? 0.25 thick densiÞed graphite. The cup clearly refers to the volume not its container. Do you want it in milliliters? 59 ml. -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/qz.pdf === Subject: Re: What form of matter will last the longest? > The Þrst Devil Solvent run was conducted in a 0.25 thick > densiÞed graphite crucible. A quarter cup of the stuff went > through the graphite, through about 1.5 inches of Þre brick, > through the desk top, through two shelves, and made a respectible > run at the cement slab þoor. Emission of visible plume > Uncle Al Sounds like some rightous stuff. Rob === Subject: Re: What form of matter will last the longest? > What is Devil Solvent? All links in Google on this topic > go back to you. > It¹s the only molten salt Uncle Al could identify in which > graphite violently reacts and diamond dust persists. The idea is > to do CVD diamond chemistry condensed phase, 1500 times the > density of Ar/H2/CH4 plasma, with a corresponding increase in > deposition rate. We do it Brad Pate¹s way. If it Þnally works > it will be an awesomely clever implementation. I think we have > the bugs out, Þnally. > The Þrst Devil Solvent run was conducted in a 0.25 thick > densiÞed graphite crucible. A quarter cup of the stuff went > through the graphite, through about 1.5 inches of Þre brick, > through the desk top, through two shelves, and made a respectible > run at the cement slab þoor. Emission of visible plume Impressive. I used to work with antimony chlorides - the pentachloride is a liquid superacid at room temp, a very strong oxidizer, a good source of Cl- ligand for complexing metal ions, and hydrolyzes in air to yield HCl, Cl2 and SbOCl. Sort of an anhydrous aqua regia on steroids. If the stuff wasn¹t kept scrupulously dry, it would eat through steel (carbon, stainless, whatever). Out batch reactor was Monel and had to be replaced after a water-cooled condensor leaked into the pot, corroding it severely - the plant was shut down for months! When reprocessing SbCl5 that had been used in making CFC 143, the material was especially nasty due to excess þuoride - *that* stuff ate through our *glassware* when we were running corrosion tests - never mind the SS and Monel coupons. Corrosions rates in hundreds of microns per year were common. We never tested Hastelloys, probably because management felt that 10,000 pound capacity reactors would be prohibitively expensive. So what is Devil Solvent composed of? Tom Davidson Richmond, VA === Subject: Re: What form of matter will last the longest? > > > What is Devil Solvent? All links in Google on this topic > > go back to you. > It¹s the only molten salt Uncle Al could identify in which > graphite violently reacts and diamond dust persists. The idea is > to do CVD diamond chemistry condensed phase, 1500 times the > density of Ar/H2/CH4 plasma, with a corresponding increase in > deposition rate. We do it Brad Pate¹s way. If it Þnally works > it will be an awesomely clever implementation. I think we have > the bugs out, Þnally. > The Þrst Devil Solvent run was conducted in a 0.25 thick > densiÞed graphite crucible. A quarter cup of the stuff went > through the graphite, through about 1.5 inches of Þre brick, > through the desk top, through two shelves, and made a respectible > run at the cement slab þoor. Emission of visible plume > > Impressive. I used to work with antimony chlorides - the pentachloride is a > liquid superacid at room temp, a very strong oxidizer, a good source of Cl- > ligand for complexing metal ions, and hydrolyzes in air to yield HCl, Cl2 > and SbOCl. Sort of an anhydrous aqua regia on steroids. > If the stuff wasn¹t kept scrupulously dry, it would eat through steel > (carbon, stainless, whatever). Out batch reactor was Monel and had to be > replaced after a water-cooled condensor leaked into the pot, corroding it > severely - the plant was shut down for months! When reprocessing SbCl5 that > had been used in making CFC 143, the material was especially nasty due to > excess þuoride - *that* stuff ate through our *glassware* when we were > running corrosion tests - never mind the SS and Monel coupons. Corrosions > rates in hundreds of microns per year were common. > We never tested Hastelloys, probably because management felt that 10,000 > pound capacity reactors would be prohibitively expensive. > So what is Devil Solvent composed of? Read the patent if and when, or if it thoroughly bombs in the next run of trials I¹ll post it. You will not believe it - and it will be awesome in the impact of its inevitable necessity. -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/qz.pdf === Subject: Re: an example of the function derivative? > Example: f(z)=z^2 > f¹(z) = lim {dz->0} ((z+dz)^2 - z^2)/dz = lim [dz->0] (2z+dz) = 2z > I belive that this example is correct. > let me show now my steps to solve this. I obviously make a mistake somewhere > my example: > f¹(z) = lim {dz->0} ((z^2 + dz)^2 - z^2)/dz = Where did that (z^2 + dz)^2 come from? In the other line you have (z+dz)^2, which is correct. (z^2 + dz)^2 is nonsense. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: an example of the function derivative? I am sorry this was my mistake. Hope this is ok. I have still problem with keyboard notation of math expressions. This below is correct: f¹(z) = lim {dz->0} ((z + dz)^2 - z^2)/dz the steps are then: f(z)=z^2 f¹(z) = lim {dz->0} ((z + dz)^2 - z^2)/dz = lim {dz->0} ((z^2 + 2dz*z + dz^2) - z^2)/dz = (2dz^2 + dz^2)/dz = 3dz^2/dz = 3dz === Subject: Re: an example of the function derivative? >I am sorry this was my mistake. Hope this is ok. I have still problem with >keyboard notation of math expressions. >This below is correct: >f¹(z) = lim {dz->0} ((z + dz)^2 - z^2)/dz >the steps are then: >f(z)=z^2 >f¹(z) = lim {dz->0} ((z + dz)^2 - z^2)/dz You must think of dz as *one* quantity, it doesn¹t mean d*z or something like that. Think of dz as delta - you wouldn¹t say that delta = del*ta, wouldn¹t you? >lim {dz->0} ((z^2 + 2dz*z + dz^2) - z^2)/dz No. This is lim {dz->0} ((z^2 + 2(dz)*z + (dz)^2) - z^2)/dz = lim {dz->0} (2(dz)*z + (dz)^2))/dz = lim {dz->0} (2*z + dz) = 2z Do you see how dz cancels? Again, think of it as *one* quantity with a fancy name - not just a one-letter name but a two-letter name (BTW: are there quantities with four-letter-names? ;-) Thomas >(2dz^2 + dz^2)/dz >3dz^2/dz >3dz === Subject: Re: an example of the function derivative? === Subject: Re: an example of the function derivative? > I am sorry this was my mistake. Hope this is ok. I have still problem with > keyboard notation of math expressions. > This below is correct: > f¹(z) = lim {dz->0} ((z + dz)^2 - z^2)/dz > the steps are then: > f(z)=z^2 > f¹(z) = lim {dz->0} ((z + dz)^2 - z^2)/dz > lim {dz->0} ((z^2 + 2dz*z + dz^2) - z^2)/dz > (2dz^2 + dz^2)/dz Well there you have it: dz*z is not the same as (dz)^2. It is because your notation dz^2 is ambiguous. Don¹t use it. -Michael. === Subject: Re: an example of the function derivative? Oh Yes. I missed. dz iz not (d*z) it is Œdelta z¹ - Œz¹ value change. === Subject: Re: an example of the function derivative? >> I am sorry this was my mistake. Hope this is ok. I have still problem with >> keyboard notation of math expressions. >> This below is correct: >> f¹(z) = lim {dz->0} ((z + dz)^2 - z^2)/dz >> the steps are then: >> f(z)=z^2 >> f¹(z) = lim {dz->0} ((z + dz)^2 - z^2)/dz >> = >> lim {dz->0} ((z^2 + 2dz*z + dz^2) - z^2)/dz >> = >> (2dz^2 + dz^2)/dz >Well there you have it: dz*z is not the same as (dz)^2. It is because your >notation dz^2 is ambiguous. Don¹t use it. And furthermore, it is inconsistent in itself - How could (dz)^2 be equal to dz^2. To be consistent it should be d^2z^2 (which it isn¹t of course). Thomas >-Michael. === Subject: Re: an example of the function derivative? > f¹(z) = lim {dz->0} ((z^2 + dz)^2 - z^2)/dz = > this is the next step I think I mistake True, but if you don¹t tell us how you came to that erroneous result, we wont be able to point the error. > = 3dz^2/dz = [3dz] this result is different > What is the mistake? === Subject: Re: an example of the function derivative? Hope I rewritten it correctly for I need to get use to the keyboard notation. Here are the steps I did: f(z)=z^2 f¹(z) = lim {dz->0} ((z+dz)^2 - z^2)/dz = lim {dz->0} ((z^2 + 2dz*z + dz^2) - z^2)/dz = (2dz^2 + dz^2)/dz = 3dz^2/dz = 3dz === Subject: Re: Symmetric Polynomials > [.snip.] >Theorem 5.S(3): Let F be a Þeld. Let a = a(x,y) = x + y and b = b(x,y) = >x*y be the elementary symmetric functions in two variables over F. Let S be >the Þeld of symmetric rational functions. Then S = F(a,b), where F(a,b) is >the quotient Þeld of F[a,b] (= polynomials in a and b over F). >Herstein then asks (as an exercise) to prove that a symmetric polynomial in >x and y over F may be expressed as a polynomial in the elementary symmetric >functions in x and y. (In other words, prove the classic theorem on >symmetric polynomials.) >I realize that there are several standard proofs of this theorem. However, I >wondered if the exercise could be proved as a corollary of the theorem >stated above. I give my proof below, but something doesn¹t seem right. This >proof seems a lot easier than the standard ones. I would appreciate any >comments. >Proof: Let f(x,y) be a symmetric polynomial in F[x,y]. We have to show that >f can be expressed as a polynomial in a and b over F. The polynomial f may >be regarded as a member of S (the symmetric rational functions over F). By >the theorem quoted above, f is a member of F(a,b). Therefore f may be >written as a quotient >f(x,y) = g(a,b)/h(a,b) >where g and h are polynomials in the elementary symmetric polynomials a and >b. We may assume that g and h are relatively prime (if not, cancel the >common factor). Note that F[x,y] is a UFD. The equation that relates g and h >to f shows that h divides g. > Could you expand on why this statement follows? It would be obvious to > me if f were a polynomial in a and b, not x and y, but that is exactly > what you are trying to show... > If you can prove that, I would agree that your argument is valid. > -- > It¹s not denial. I¹m just very selective about > what I accept as reality. > --- Calvin (Calvin and Hobbes) > Arturo Magidin > magidin@math.berkeley.edu that h divides g is a gap in the proof. I¹ll think about this and post again if I Þgure it out. Pete Klimek === Subject: Re: This Week¹s Thought¹s on Fermat¹s Last Theorem: 1 > I think I should have written > a^n + b^n = 0 (mod c) is sufÞcient for > a^n + b^n = c^m This is false. > a^n + b^n = 0 (mod c) is necessary for > a^n + b^n = c^n This is true. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: This Week¹s Thought¹s on Fermat¹s Last Theorem: 1 >> I think I should have written >> a^n + b^n = 0 (mod c) is sufÞcient for >> a^n + b^n = c^m >This is false. Right again, Gerry. It¹s only sufÞcient for a^n + b^n = k*c, isn¹t it? >> a^n + b^n = 0 (mod c) is necessary for >> a^n + b^n = c^n >This is true. Great! Agreement on one point. A place to a start next week. As an amateur, I¹m capable of little more than looking for patterns in tables of numbers and performing algebraic and some integral manipulations. The pattern I see is that since a^0 = 1, then for some v, a^v = 1. Likewise, since b^0 = 1, then for some v b^ Yours, Doug Goncz ( ftp://users.aol.com/DGoncz/ ) Student member SAE for one year. Loves in my life: Dona, Jeff, Kim, Mom, Neelix, Tasha, and Teri. === Subject: Re: Number theory > I really do not appreciate being called arrogant: I am the most > modest of men. > You are *not* the most modest of men. I¹m way more modest than you. > It¹s not even close. But, of course, you have so much more to be modest about. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Number theory <40f62584$0$25462$afc38c87@news.optusnet.com.au> <40F774D7.2020509@et.uni-magdeburg.de> <87d62rxfbu.fsf@phiwumbda.org> Discussion, linux) >> I really do not appreciate being called arrogant: I am the most >> modest of men. >> You are *not* the most modest of men. I¹m way more modest than you. >> It¹s not even close. > But, of course, you have so much more to be modest about. You¹re just jealous. -- Naomi Klein reports that Microsoft is helping develop e-government in Iraq, Œwhich Dees admits is a little ahead of the curve, since there is no g-government in Iraq--not to mention functioning phones lines.¹ -- as reported in The Register, http://www.theregister.co.uk === Subject: Re: Projections in Braid Groups > I¹m having trouble visualizing why you¹re having trouble visualizing > this. Are you perhaps overloading the term projection, then > being puzzled by the (evident) fact (implicit in your example) > that in general A can¹t be uniquely reconstructed from its two > projections? > You might get more useful help (maybe not from me) if you establish > a context for your question by telling us where you¹re coming > across this construction. I¹m less familiar with the pure braid > literature than the general braid literature, and I¹m not as familiar > with the latter as I should be, but I do sort of know it, and I don¹t > recall this notation. > Lee Rudolph attacks that can break a public cryptosystem based on the hardness of the conjugacy problem in braid groups. Here is an excerpt from the paper: For a pure braid A, deÞne the left and right projections tau_left(A) and tau_right(A) as the braids (over n/2 strings) obtained from removing the Þrst or last half of the strings. Sorry for not seeing immediately what you¹re saying, I understand that we cannot usually reconstruct A from its two projections, but I was just trying to get a Œfeel¹ for what the projection would look like. I help and take care. Adam === Subject: Re: Projections in Braid Groups >> I¹m having trouble visualizing why you¹re having trouble visualizing >> this. ... >I was >just trying to get a Œfeel¹ for what the projection would look like. Okay, I¹m *still* having trouble understanding why you¹re having trouble. Maybe you¹re either trying to deal with the whole thing entirely algebraically (*big* mistake with braids), or you¹ve let the algebra overwhelm the geometry? A way the latter might happen would be, I guess, that you think of a geometric (pure) braid diagram very rigidly, as a concatenation of a Þxed set of standard diagrams for the standard (pure) generators; if you do think that way, then it¹s true that the diagrams of the left and right projections of a braid will typically not be diagrams of the sort you would be expecting. But they¹re still braid diagrams in the looser (and better!) sense. Minimal example: the standard Artin generator for the pure braid group on 2 strings, A_{12}, has the following standard diagram (use a monospace font, of course) / / / / / / so its two projections into the pure braid broup on 1 string have the diagrams / / and / / / / which of course are *not* standard diagrams: those would be | | | | | and | | | | | | | of course. Have I read your mind yet? Lee Rudolph === Subject: Re: James Harris hello james harris by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6L12Ax31864; And what are people who seek to spend their free time classifying crackpots? Do they have an urge to feel mighty special about themselves by an oversimpliÞed and exaggerated comparison with those less fortunate? === Subject: Turing¹s Statistical Work? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6L12As31868; Can anyone give me a source for Turing¹s statistical work in cracking the Enigma code in WWII? I¹m looking for a complete mathematical description. So far I¹ve only been able to Þnd allusions to his work but nothing in detail. === Subject: Re: Turing¹s Statistical Work? > Can anyone give me a source for Turing¹s statistical > work in cracking the Enigma code in WWII? I¹m looking > for a complete mathematical description. So far I¹ve only > been able to Þnd allusions to his work but nothing > in detail. [Cross-posted to sci.crypt] Battle of Wits by Stephen Budiansky and Alan Turing, The Enigma by Andrew Hodges had enough detail for me. -- Clive Tooth http://www.clivetooth.dk === Subject: Re: Turing¹s Statistical Work? > Can anyone give me a source for Turing¹s statistical > work in cracking the Enigma code in WWII? I¹m looking > for a complete mathematical description. So far I¹ve only > been able to Þnd allusions to his work but nothing > in detail. There¹s a chapter on the cracking of Enigma in Simon Singh¹s The Code Book. Obviously it¹s not a complete mathematical description, but it does go into some detail of how the Enigma code worked and how the team at Bletchley Park attacked it. hth Andrew Taylor === Subject: Re: Turing¹s Statistical Work? > Can anyone give me a source for Turing¹s statistical > work in cracking the Enigma code in WWII? I¹m looking for a complete > mathematical description. So far I¹ve only been able to Þnd allusions > to his work but nothing in detail. It¹s probably a state secret. Who knows when we¹ll have to reliberate Europe (again). Already the Germans have started acting up over Iraq. We simply have to maintain our ability to decode their telegraphic transmissions. Releasing Turing¹s work is simply not in the national interest. -- Lance Lamboy Go F*ck Yourself ~ Dick Cheney === Subject: Re: JSH: Groupthink > [3] requires uf=1 > [2] reduces to mf^2 = 5 > which reduces [1] to 65 > There is no algebraic conþict between these two requirements. > Hope this helps to reduce the dialog to fundamental issues, such as whether > u*f=1 violates JSH¹s stipulation that: > f is a non unit, non zero algebraic integer coprime > to 3 and x, and u a non unit, non zero algebraic integer > coprime to f. I thought there was another stipulation: m is a (rational) integer. This would be a contradiction even in his object ring. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: JSH: Groupthink > [3] requires uf=1 > [2] reduces to mf^2 = 5 > which reduces [1] to 65 > > There is no algebraic conþict between these two requirements. > > Hope this helps to reduce the dialog to fundamental issues, such as whether > u*f=1 violates JSH¹s stipulation that: > > f is a non unit, non zero algebraic integer coprime > to 3 and x, and u a non unit, non zero algebraic integer > coprime to f. > I thought there was another stipulation: m is a (rational) integer. > This would be a contradiction even in his object ring. Right. I was surprised that Andrzej Kolowski dropped his entire line of objection after I made this post. He had started to imply he would show a conþict in the ring of algebraic integers but never Þnished and, I thought, implied a simple algebraic error in substituting uf=1 in one place and sqrt(5) in another. My There is no algebraic conþict between these two requirements. i.e. uf=1 and mf^2 = 5, was in reference to this simple algebraic substitution complaint. But what we really have is uf=1 and mf^2=5, with m an integer and JSH¹s: f is a non unit, non zero algebraic integer coprime to 3 and x, and u a non unit, non zero algebraic integer coprime to f. So JSH now must argue that f=sqrt(5/m) and u=1/sqrt(5/m) do not conþict with his algebraic-integer stipulations. I was conÞdent Kolowski would pick back up on this. KeithK > -- > dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 > home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: JSH: Groupthink > > > > > > > >>> Fraud is when you submit a paper like this to a journal knowing > >>> it has this kind of error; or fail to withdraw the paper when > >>> such errors and others have been pointed out. That is outright > >>> academic fraud, the kind that gets people Þred from tenured > >>> positions. Harris has no worries on that score. > >> > >>Well, I can see you are emotionally involved, but sometimes a mistake > >>is just a mistake. > >> > >> > >> The question is, have you withdrawn the paper? Until you have, > >> you are committing fraud and I will continue to remind you of it. > >> > > > >Minor errors do not require withdrawal of a paper. > > > >and asked if I could correct it, but they never did. > > > >The errors are basically typos that don¹t change the conclusion. I¹m > >curious to see if you¹ll admit that reality. > > > > The errors are not minor. Your proposed Þx is invalid. > See below. > > You¹re lying. It¹s basic algebra. Now then, you can keep arguing as > if I¹m wrong, and keep lying, but it won¹t change the truth. I was not lying. I was simply wrong. I thought about this last night and realized that your proposed Þx for the error in APF would enable you to apply your claimed main result to the polynomial 65 x^3 - 12 x + 1. This however is somewhat academic. Your main result in APF is incorrect. As you know, it contradicts Galois theory. You are clearly aware of this because in your Abstract to APF, you mention ... overinterpretations of Galois Theory. However you do not elaborate on this cryptic remark or even mention Galois theory in the paper itself. You have seen the Galois theory arguments. You have not provided a valid refutation of them. You have also seen that W. Dale Hall has presented a proof that your claims regarding the factorization of 65 x^3 - 12 x + 1 are wrong. His proof can be veriÞed by simple arithmetic. You have ignored that, and instead claim that Hall is making an unwarranted assumption. Despite Hall¹s repeated requests, you have refused to specify where in his argument that assumption comes into play. Regarding your Þx: the current version of APF contains no hint of the idea behind the Þx. The error is there with no explanation. What you *should* do at a minimum is contact the editor and inform him of the error, or submit a revision. In fact of course, in view of the deeper problems involving your main result, you should simply withdraw the paper. Andrzej > James Harris === Subject: Re: Kevin Brian Fisher - February 10th 1974 >F I S H E R >6 9 19 8 5 18 = 65 > I was busking (Translation= begging and pan-handling) << The following (courtesy of Waxy.org) is sort of an unofÞcial FAQ explaining the psychotic nonsense posted to Usenet by Shawn Daryl Kabatoff AKA Dar, AKA Probababbilities. And now AKA marcia and me. WARNING: Read below before even thinking about responding to this twit. http://www.waxy.org/archive/2002/05/21/dar_kaba.shtml#000643 Usenet has the tendency to provide a public forum for those who would normally be scribbling in a closet. For example, take Daryl Shawn Kabatoff. For the last few years, he¹s methodically gathered statistics from various sources, ranging from local newspaper obituary pages to the food court of the Saskatoon Midtown Plaza mall. With all the raw data he¹s collected, he¹s attempting to prove daily that our full names are in mathematical harmony with our birthdays. His rants normally focus on a single individual he¹s met or read about, starting with calculations related to their birthdate and full names, blending in whatever other personal information about their family members, spouses, birthplace, and career he¹s been able to zealotry, and personal torment. I¹ve never seen anything like it. With all the prime numbers, Fibonacci sequences and biblical references, it¹s like reading the notebooks of Maximillian Cohen and John Nash combined. Unsurprisingly, several posts unfold to reveal a history of painful mental illness. If you have some time, take a look. I¹ve detailed his posting history and a several sample posts below. Usenet Posting History: January 27, 1999 to July 5, 2000 as Catsco@home.com December 9, 2000 to May 4, 2001 as s.kabatoff@sk.sympatico.ca Oct 30, 2001 to Oct 31, 2001 as kabatoff@the.link.ca January 20, 2002 to April 17, 2002 as s_kabatoff@hotmail.com (original posts have been removed from Google Groups archive) April 26, 2002 to Present as dar_kabatoff@hotmail.com Selected Posts: Tessa Lynne Smith Dastageer Sakhizai and Helen Smith Brett David Maki Andrew Meredith Cotton Kathryn Lee Hipperson Amanda Dawn Newton Mona Marie Etcheverry Tony Peter Nuspl Lisa Charlene McMillan Grant Allyn Wood Comments scarier still is that saskatoon is my hometown, though not my current residence. and every single place he¹s mentioned in his posts (most notably nervous harold¹s and the roastary) were either places i¹ve been (as it¹s a small city of 200K) or hangouts, ie. the two places mentioned. chances are i could email some friends back home and Þnd out if they know of him, they (my friends that is) being of the broadway-centred slacker ilk. myself, too, until i got out of there. eh, anyways. thought it odd to see all this. midtown mall. i ate my meals there, whilst waiting several days in line for star wars episode one, at the theatre across the street. posted by andy raad on May 22, 2002 06:20 PM Fascinating. It¹s like he¹s trying to take chaos and bind it into whatever rules he can Þnd, religious, logical and otherwise. Numbers and math have a reliable pattern, something that can always be proven to true or false. People and religion do not. It reminds me of Darren Aronofsky¹s movie Pi. It¹s the story of an paraniod genius who is trying to Þnd a pattern in Pi. A group that takes interest in his work is convinced that the existence of Pi, a number whose existence can be proven but no quantiÞed, is proof of the existence of God. Kabatoff¹s hunt for patterns in something as random as name selection is a way to reconcile his deeply logical thought process with his conþicting religious views. Exactly. I probably shouldn¹t have, but I e-mailed Daryl yesterday, asking him if he¹d be willing to create a numerological analysis for me. I also asked him if he had seen either Pi or A Beautiful Mind, and what he thought of them. If he replies, I¹ll be sure to post it. I baked many pumpkin pies for Shawn (he likes pumpkin pies). I rubbed pumpkin pie all over my breasts for him, and my breasts turned orange. I am a pumpkin for Shawn. posted by Trisha Blondie on July 24, 2002 10:41 PM Um, that¹s swell. So, you¹re in love with him? Shawn once went to a funeral for a Jehovah Witness that shot himself and the lemon tarts were very bad, they were not only sour but were rubbery as well. Shawn said that the guy was some kind of Jehovah Witness prophet, he saw in advance that the lemon tarts at his funeral were to be very very bad, and so he shot himself. Shawn said that he never ate pumpkin pie at a funeral but would like to some day. Shawn likes pumpkin pie and so I have been practicing to make very good pumpkin pies. posted by Trisha Blondie on July 25, 2002 02:49 PM Shawn said that the lemon tarts were sour, bitter and rubbery. I don¹t think this guy takes notes. I think he has Total Recall, and it has driven him insane... Oh... I almost forgot... I didnt spend thousands of dollars a day tormenting Daryl... We got a deal on tormenting that Þscal year, it only came to about 37cents a day.... Mr. Kabatoff attempts to portray himself as a victim, but in fact he is a violent predatory pedophile who is well known to his local law enforcement. In his post to multiple newsgroups with the subject Collecting Mail For The Coming Anti-Christ, he encourages mothers to send him photos of their naked daughters. Mr Kabatoff explains, I personally did not want photographs being mailed to (the coming Ant-Christ) that were of underage children unless the parent was signing consent. He is banned from virtually all the shopping malls in his community because he stalks young people and sexually harasses them. He has an extensive arrest record which includes sexual molestation charges. He¹s been hospitalized in mental institutions about his contact with young girls in many posts. Search newsgroup archives for posts by him containing the word nubile. As part of his harrassment, he provides personal details in a public forum, such as the real names of real children, in these and other posts. About one wanted her and her sister dead. He not only curses children and prays for their death in his posts, he also enjoys attending the funerals of young people: And so, since nubile sweeties are found in greatest abundance at the funerals of high school students, then it is the funerals of high school students that make the very very best funerals, especially if there is food... I stuff my face (and my pockets) with all the good food and look at all the pretty nubile sweeties and have the time of my life.. r=&ie=UTF-8&scoring=d&selm=LfXN8.63042%24R53.25142039% 40twister.socal.rr. com&rnum=1 Many of his posts are sent to alt.teens.advice. However, he liberally spams, þoods and crossposts his off-topic threatening and offensive missives to countless newsgroups. Some people HAVE problems and some folks ARE problems. Don¹t dismiss Mr. Kabatoff as a harmless nut. When he sends these posts to any newgroup, please help by reporting him to I knew of him when I was attending the University of Saskatchewan. He¹d hang out in the Arts computer lab and all you¹d see is screens of numbers racing by on his laptop. I have an original copy of his Collecting Mail for the Coming Anti-Christ pamphlet, and have seen him be hauled away by campus security on more than one occasion. My friends and I refer to him as Crazy Number Man. I¹ve been posting to (and about) Shawn for over two years with big gaps in between. He has seen Pi and didn¹t like it and didn¹t think it resembled him at all. (Wrong, it Þts him to a tee) He doesn¹t have total recall and has stated that he travels with a lap top to notate items. Also, he uses cut n¹ paste a lot if you read all the way through his ramblings. He is anti-social as shown by his angry statements towards those who, by his own admission, have been kind (but not kind enough) to him. Still, he¹s intelligent and seems to be able to take a joke on occassion. That¹s where I came in. ALOHA Reply to group (Unsolicited e-mail is deleted from the server unread if it comes from anyone not already in my addressbook. I¹ll never even see it) === >Saturday June 2nd 2001 153/212 16176 >D A V E N O C K >4 1 22 5 14 15 3 11 = 75 > I found Celeste waitressing at Burger King on The King George Hiway and >103rd Street in Surrey, the nubile sweety is a twin (as in 2). Bat Kabatoff is gonna get to is a wet dream. Auwe!! << The following (courtesy of Waxy.org) is sort of an unofÞcial FAQ explaining the psychotic nonsense posted to Usenet by Shawn Daryl Kabatoff AKA Dar, AKA Probababbilities. And now AKA marcia and me. WARNING: Read below before even thinking about responding to this twit. http://www.waxy.org/archive/2002/05/21/dar_kaba.shtml#000643 Usenet has the tendency to provide a public forum for those who would normally be scribbling in a closet. For example, take Daryl Shawn Kabatoff. For the last few years, he¹s methodically gathered statistics from various sources, ranging from local newspaper obituary pages to the food court of the Saskatoon Midtown Plaza mall. With all the raw data he¹s collected, he¹s attempting to prove daily that our full names are in mathematical harmony with our birthdays. His rants normally focus on a single individual he¹s met or read about, starting with calculations related to their birthdate and full names, blending in whatever other personal information about their family members, spouses, birthplace, and career he¹s been able to zealotry, and personal torment. I¹ve never seen anything like it. With all the prime numbers, Fibonacci sequences and biblical references, it¹s like reading the notebooks of Maximillian Cohen and John Nash combined. Unsurprisingly, several posts unfold to reveal a history of painful mental illness. If you have some time, take a look. I¹ve detailed his posting history and a several sample posts below. Usenet Posting History: January 27, 1999 to July 5, 2000 as Catsco@home.com December 9, 2000 to May 4, 2001 as s.kabatoff@sk.sympatico.ca Oct 30, 2001 to Oct 31, 2001 as kabatoff@the.link.ca January 20, 2002 to April 17, 2002 as s_kabatoff@hotmail.com (original posts have been removed from Google Groups archive) April 26, 2002 to Present as dar_kabatoff@hotmail.com Selected Posts: Tessa Lynne Smith Dastageer Sakhizai and Helen Smith Brett David Maki Andrew Meredith Cotton Kathryn Lee Hipperson Amanda Dawn Newton Mona Marie Etcheverry Tony Peter Nuspl Lisa Charlene McMillan Grant Allyn Wood Comments scarier still is that saskatoon is my hometown, though not my current residence. and every single place he¹s mentioned in his posts (most notably nervous harold¹s and the roastary) were either places i¹ve been (as it¹s a small city of 200K) or hangouts, ie. the two places mentioned. chances are i could email some friends back home and Þnd out if they know of him, they (my friends that is) being of the broadway-centred slacker ilk. myself, too, until i got out of there. eh, anyways. thought it odd to see all this. midtown mall. i ate my meals there, whilst waiting several days in line for star wars episode one, at the theatre across the street. posted by andy raad on May 22, 2002 06:20 PM Fascinating. It¹s like he¹s trying to take chaos and bind it into whatever rules he can Þnd, religious, logical and otherwise. Numbers and math have a reliable pattern, something that can always be proven to true or false. People and religion do not. It reminds me of Darren Aronofsky¹s movie Pi. It¹s the story of an paraniod genius who is trying to Þnd a pattern in Pi. A group that takes interest in his work is convinced that the existence of Pi, a number whose existence can be proven but no quantiÞed, is proof of the existence of God. Kabatoff¹s hunt for patterns in something as random as name selection is a way to reconcile his deeply logical thought process with his conþicting religious views. Exactly. I probably shouldn¹t have, but I e-mailed Daryl yesterday, asking him if he¹d be willing to create a numerological analysis for me. I also asked him if he had seen either Pi or A Beautiful Mind, and what he thought of them. If he replies, I¹ll be sure to post it. I baked many pumpkin pies for Shawn (he likes pumpkin pies). I rubbed pumpkin pie all over my breasts for him, and my breasts turned orange. I am a pumpkin for Shawn. posted by Trisha Blondie on July 24, 2002 10:41 PM Um, that¹s swell. So, you¹re in love with him? Shawn once went to a funeral for a Jehovah Witness that shot himself and the lemon tarts were very bad, they were not only sour but were rubbery as well. Shawn said that the guy was some kind of Jehovah Witness prophet, he saw in advance that the lemon tarts at his funeral were to be very very bad, and so he shot himself. Shawn said that he never ate pumpkin pie at a funeral but would like to some day. Shawn likes pumpkin pie and so I have been practicing to make very good pumpkin pies. posted by Trisha Blondie on July 25, 2002 02:49 PM Shawn said that the lemon tarts were sour, bitter and rubbery. I don¹t think this guy takes notes. I think he has Total Recall, and it has driven him insane... Oh... I almost forgot... I didnt spend thousands of dollars a day tormenting Daryl... We got a deal on tormenting that Þscal year, it only came to about 37cents a day.... Mr. Kabatoff attempts to portray himself as a victim, but in fact he is a violent predatory pedophile who is well known to his local law enforcement. In his post to multiple newsgroups with the subject Collecting Mail For The Coming Anti-Christ, he encourages mothers to send him photos of their naked daughters. Mr Kabatoff explains, I personally did not want photographs being mailed to (the coming Ant-Christ) that were of underage children unless the parent was signing consent. He is banned from virtually all the shopping malls in his community because he stalks young people and sexually harasses them. He has an extensive arrest record which includes sexual molestation charges. He¹s been hospitalized in mental institutions about his contact with young girls in many posts. Search newsgroup archives for posts by him containing the word nubile. As part of his harrassment, he provides personal details in a public forum, such as the real names of real children, in these and other posts. About one wanted her and her sister dead. He not only curses children and prays for their death in his posts, he also enjoys attending the funerals of young people: And so, since nubile sweeties are found in greatest abundance at the funerals of high school students, then it is the funerals of high school students that make the very very best funerals, especially if there is food... I stuff my face (and my pockets) with all the good food and look at all the pretty nubile sweeties and have the time of my life.. r=&ie=UTF-8&scoring=d&selm=LfXN8.63042%24R53.25142039% 40twister.socal.rr. com&rnum=1 Many of his posts are sent to alt.teens.advice. However, he liberally spams, þoods and crossposts his off-topic threatening and offensive missives to countless newsgroups. Some people HAVE problems and some folks ARE problems. Don¹t dismiss Mr. Kabatoff as a harmless nut. When he sends these posts to any newgroup, please help by reporting him to I knew of him when I was attending the University of Saskatchewan. He¹d hang out in the Arts computer lab and all you¹d see is screens of numbers racing by on his laptop. I have an original copy of his Collecting Mail for the Coming Anti-Christ pamphlet, and have seen him be hauled away by campus security on more than one occasion. My friends and I refer to him as Crazy Number Man. I¹ve been posting to (and about) Shawn for over two years with big gaps in between. He has seen Pi and didn¹t like it and didn¹t think it resembled him at all. (Wrong, it Þts him to a tee) He doesn¹t have total recall and has stated that he travels with a lap top to notate items. Also, he uses cut n¹ paste a lot if you read all the way through his ramblings. He is anti-social as shown by his angry statements towards those who, by his own admission, have been kind (but not kind enough) to him. Still, he¹s intelligent and seems to be able to take a joke on occassion. That¹s where I came in. ALOHA Reply to group (Unsolicited e-mail is deleted from the server unread if it comes from anyone not already in my addressbook. I¹ll never even see it) === Subject: Question about PhD math programs reworded Why do you think that many strong students who earn 4 yrs degrees in math at US colleges are ill prepared for PhD programs? === Subject: Re: Question about PhD math programs reworded > Why do you think that many strong students who earn 4 yrs degrees in math > at US colleges are ill prepared for PhD programs? Why do _you_ think that *many* strong students who earn 4 yrs degrees in math at US colleges are ill prepared for PhD programs? -- Paul Sperry Columbia, SC (USA) === Subject: Re: Question about PhD math programs reworded >Why do you think that many strong students who earn 4 yrs degrees in math >at US colleges are ill prepared for PhD programs? This is because real mathematics is not being taught in most undergraduate programs, but instead memorization of computational techniques, and possibly a few deÞnitions. It starts even earlier, and the faculties of the universities seem unable to resist the educationist monstrosity that you have to teach a course to what the students want and still call it the same. When I was on my postdoc, some of us tried to deÞne a mathematician. Our deÞnition was Someone who might not know what 2+2 is, but can prove it is 4. We no longer require a good Euclid course in high school, and many cannot even get one, nor do we expect those graduating high school to be able to understand an argument by induction. There is very little, if any, attempt to remedy this until the third year of college, if then, and it even continues to the end. Being able to manipulate in calculus, including vector calculus, and linear algebra gives no understanding of anything, and I question if those who have it are any better prepared for abstract mathematics than they were in elementary school, plus a good explanation of variables in general, not starting with numbers. An abstract idea should not always be presented as a generalization, but as something basic; it is. To most mathematics graduates, a continuous function is something mysterious, an integral is what one can get by antidifferentiation, a group is something which was learned in a useless course, with some proofs memorized, etc. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Question about PhD math programs reworded > Why do you think that many strong students who earn 4 yrs degrees in math > at US colleges are ill prepared for PhD programs? What do you mean by strong? And is this in comparison to other countries or just a general complaint? -- Mitch Harris (remove q to reply) === Subject: Re: Question about PhD math programs reworded > Why do you think that many strong students who earn 4 yrs degrees in > math at US colleges are ill prepared for PhD programs? with few exceptions, because of the tax dollar... === Subject: Re: James Harris hello james harris > And what are people who seek to spend their free time classifying > crackpots? Do they have an urge to feel mighty special about > themselves by an oversimpliÞed and exaggerated comparison with > those less fortunate? And what about people that waste time reading those posts (like you and me). Van === Subject: Re: What is Forced (accelerated) Motion? > The acceleration would be the angular velocity, squared, divided by the > radius, or 6.28 ft * 100 radians * 100 radians = 62800ft/sec^2 towards the > center of the disk. This force is transmitted to the dot by the disk. > Should the dot leave the disk, it would leave in a straight line at Assuming > that there is no friction or added energy, the angular acceleration is zero. > Now, should your dot actually weigh something signiÞcant, then if it is > brought to the center of the circle, the disk will speed up, conserving > momentum. > Michael > } Mathematical deÞnition of acceleration IS real world > So, tell me, > How much is a dot on the edge of a 1 foot in diameter wheel > spinning at a constant 1000 revs per minute, accelrating? > and when will such acceleration create a higher rev speed? The point is (for the nth time), speed and velocity are not the same. Velocity is a vector in 3D, as is force and acceleration. Speed is the magnitude (length) of the velocity vector. In circular motion, the speed can be constant (if there is no torque), but the velocity vector, which is tangent to the circle, is constantly changing direction. It takes force to do this, a force directed toward the center of mass. (Whence the term central forces, like gravity and electricity. Consider a small mass (a satellite) under the inþuence of the gravity from inÞnity, and consider what can happen to it. This is the famous central force problem. I refer you to a physics text for more on this elementary but fascinating problem. You will Þnd the test mass moving in a hyperbola if the motion is unbounded, an ellipse (circle is a special case), when the mass is captured (bounded motion). Van === Subject: Re: What is Forced (accelerated) Motion? message > } The acceleration would be the angular velocity, squared, divided by the > } radius, or 6.28 ft * 100 radians * 100 radians = 62800ft/sec^2 towards the > } center of the disk. > Why do use the term towards the center of the disk? > It is spinning constantly around it and never heading towards it, > and also it does not accelerate at all. > So why bother saying it is accelerating at all? > It isn¹t. Idiot. You should know something about motion before you post statements like this. See my post, and any text on circular motion and/or orbits. > It is in a constant circular motion, > no acceleration of it¹s speed is occuring at all, > or the rpm would be rising along with the travel speed around > the disk. For the (n+1)th time, speed is constant in circular motion (with no torque), but the direction of the velocity vector is constantly changing, and this requires a central force, which produces a central acceleration, which keeps a satellite in orbit--otherwise, if the large mass were not there to exert a central force on the satellite, it would move in a straight line out into space, instead of circling the earth (or whatever mass or force is at the center). > > I don¹t care about moving it closer or further. > It is not accelerating. > Or are you saying accelerating things can not change speed > they are moving at? In circular motion in a central force, this is exactly what I am saying, and I have explained why, as have many others. You could understand this yourself if you would take the time to learn it--your ideas on motion are confused--I understand why you make these statements, but I don¹t understand why you won¹t listen to other people. Is it that you don¹t want to admit you have been mistaken after clinging so hard to your mistaken ideas? I don¹t know. I promised myself a couple days ago I would stop wasting time with this nonsense--I really do have better things to do with my time. But... Van === Subject: Number Theory A friend of mine just informed me that there is a perfect square between the sum of the 1st n primes and the sum of the 1st (n+1) primes. Not haven taken a number theory course I really don¹t see how to prove this. Non the less I would love to see the proof. Mkajuma === Subject: Re: Number Theory > A friend of mine just informed me that there is a perfect square between the > sum of the 1st n primes and the sum of the 1st (n+1) primes. Not haven taken > a number theory course I really don¹t see how to prove this. Non the less I > would love to see the proof. m^2 is the sum of the Þrst m odd numbers (e.g., 5^2 = 25 = 1 + 3 + 5 + 7 + 9). The n-th prime is at least as big as the n-th odd number (since all the primes after 2 are odd numbers). I think that¹s all you need to put together a proof. Much much much much harder would be proving that between any two perfect (non-zero) squares there¹s a prime. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: What is Forced (accelerated) Motion? > } You misunderstand the use of acceleration here. What is being said is > } that all points on the disk (except the central point) are in > } acceleration when the disk is spinning, since they are changing their > } velocity constantly. Any point has constant *speed*, but not velocity > } (which has a direction). And a change in velocity is acceleration. > That is where you are losing the original real world meaning > of acceleration (speeding up) Wrong. This is where _you_ are going wrong. Take some time to read about these things. Its not that difÞcult. > A change in direction does not automaticall infer a change in velocity, > for if it did, the disk would be speeding up it¹s speed according to > the old fashion meaning of acceleration. > If I asked to to accelerate the point on the disk, > Would you speed up the disk, or tell me it is already accelerating, > constantly? > The mathematical world deÞnition is very bad, You can¹t change the deÞnitions of words to suit your ideas. You don¹t understand circular motion. Any motion that is not both 1) a straight line 2) constant speed does not have constant velocity, and it takes acceleration to change velocity. If all motion were in 1D--on a line, you would be right. But a circle is in 2D, and allows for changes in direction. This is the 1st place where things start to get interesting. > The mathematical world deÞnition is very bad, No. Your post shows why we need that precision of math in our deÞnitions. It keeps us from getting confused about things. Math is about precision of ideas, and using logic, at least I think that is the aspect that is relevant here. Van === Subject: Re: What is Forced (accelerated) Motion? > } You misunderstand the use of acceleration here. What is being said > is > } that all points on the disk (except the central point) are in > } acceleration when the disk is spinning, since they are changing > their > } velocity constantly. Any point has constant *speed*, but not > velocity > } (which has a direction). And a change in velocity is acceleration. > That is where you are losing the original real world meaning > of acceleration (speeding up) > Wrong. This is where _you_ are going wrong. Take some time > to read about these things. Its not that difÞcult. > A change in direction does not automaticall infer a change in > velocity, > for if it did, the disk would be speeding up it¹s speed according to > the old fashion meaning of acceleration. > If I asked to to accelerate the point on the disk, > Would you speed up the disk, or tell me it is already accelerating, > constantly? > The mathematical world deÞnition is very bad, > You can¹t change the deÞnitions of words to suit your ideas. > You don¹t understand circular motion. Any motion that is not > both 1) a straight line > 2) constant speed > does not have constant velocity, and it takes acceleration to > change velocity. If all motion were in 1D--on a line, you would > be right. But a circle is in 2D, and allows for changes in direction. > This is the 1st place where things start to get interesting. > The mathematical world deÞnition is very bad, > No. Your post shows why we need that precision of math in > our deÞnitions. It keeps us from getting confused about > things. Math is about precision of ideas, and using logic, > at least I think that is the aspect that is relevant here. > Van Let¹s try a different tact: Throw a stone into the air say at 45 degrees it will decelerate on the upwards journey stop at its apex and accelerate downwards towards the earth. This both doable, and demonstrable. So going up the stone decelerates and coming down the stone accelerates. (Would Œquod erat demonstrandum¹ be applicable at this point?) Throw said stone very far at 45 degrees it will decelerate going upwards and accelerate coming down AND stop when it hits the earth. Observe nothing has changed. Now throw said stone so far (over the horizon and some more) at 45% so that when it descends it misses the earth. The stone is still falling towards the earth (accelerating)but keeps missing it. Once this point is grasped/understood ask yourself the following question - in this case why would the stone stop accelerating? Bill === Subject: Re: Question about PhD math programs reworded > Why do you think that many strong students who earn 4 yrs degrees in math > at US colleges are ill prepared for PhD programs? Education has a low priority in the US, and many colleges don¹t prepare students that well--students need to be challenged and given direction--though if they are really strong they should be able to overcome these problems. Van === Subject: Re: Question about PhD math programs reworded > Why do you think that many strong students who earn 4 yrs degrees > in math > at US colleges are ill prepared for PhD programs? > Education has a low priority in the US, and many colleges don¹t Are you kidding? > Van === Subject: Re: Question about PhD math programs reworded >> Why do you think that many strong students who earn 4 yrs degrees >> in math >> at US colleges are ill prepared for PhD programs? >> Education has a low priority in the US, and many colleges don¹t >Are you kidding? Are you conþating schooling with education? Lee Rudolph === Subject: Re: ~ Proof techniques for surjectivity. > > > >When attempting to prove that functions are bijective, the proof of > >surjectivity often gives me difÞculty. I would like to know what techniques > >are commonly used to prove surjectivity, and how readers attempt such > >proofs. Below are a couple of the techniques that I have seen used; some may > >be the same. > [cut] > >Proof techniques. > Actually, if I recall correctly, I used that technique > in my thesis (or at least something very similar to it). > For example, suppose you have a group homomorphism f: H -> G > from the Þnite group H into the Þnite group G. To make > the example simpler, suppose that f is injective. Injective is 1-1 ==> onto. That¹s all you need to know. Van === Subject: Graphing Software I would appreciate recommendations for a good graphing software package for Windows (2000). My needs are modest. I am currently working with some exponential and polynomial functions. I purchased Advanced Grapher some time ago, but it hasn¹t been updated in awhile and the author doesn¹t respond to requests for enhancements. http://www.alentum.com/agrapher/index.htm It¹s OK for what I need. The main thing it lacks is the ability to deÞne saved functions. That is, I would like like to be able to do two things: 1. DeÞne a function, say f1=3x^2-1, and then be able to refer to it in another expression. This would allow me to plot f and f-x on the same graph without having to enter f twice or change it in two places. 2. DeÞne a function, say f2=ax^2+bx+c, and then be able to call it pasing it the values for a, b, and c. Can anyone recommend a program that can do that? Either commercial or shareware is OK. I¹d be willing to spend up to around $200 if there is nothing cheaper. -- === Subject: Re: Graphing Software SymbMath (JavaT Edition) is web-based symbolic math and computer algebra system, which runs in any computer with JavaT technology. You can play it online. SymbMath (DOS Edition) is computer algebra system. www.SymbMath.com > I would appreciate recommendations for a good graphing software > package for Windows (2000). My needs are modest. I am currently > working with some exponential and polynomial functions. > I purchased Advanced Grapher some time ago, but it hasn¹t been updated > in awhile and the author doesn¹t respond to requests for enhancements. > http://www.alentum.com/agrapher/index.htm > It¹s OK for what I need. The main thing it lacks is the ability to > deÞne saved functions. That is, I would like like to be able to do > two things: > 1. DeÞne a function, say f1=3x^2-1, and then be able to refer to it > in another expression. This would allow me to plot f and f-x on the > same graph without having to enter f twice or change it in two places. > 2. DeÞne a function, say f2=ax^2+bx+c, and then be able to call > it pasing it the values for a, b, and c. > Can anyone recommend a program that can do that? > Either commercial or shareware is OK. I¹d be willing to spend up to > around $200 if there is nothing cheaper. > -- === Subject: Re: Graphing Software Why don¹t you try Gnuplot for Windows - as far as I know it is free. You can plot graphs from data Þles or from functions using a fairly simple command language and if you want to put the graphs into a Word program you can just use Shift-Alt-Print Screen. Ian Taylor > I would appreciate recommendations for a good graphing software > package for Windows (2000). My needs are modest. I am currently > working with some exponential and polynomial functions. > I purchased Advanced Grapher some time ago, but it hasn¹t been updated > in awhile and the author doesn¹t respond to requests for enhancements. > http://www.alentum.com/agrapher/index.htm > It¹s OK for what I need. The main thing it lacks is the ability to > deÞne saved functions. That is, I would like like to be able to do > two things: > 1. DeÞne a function, say f1=3x^2-1, and then be able to refer to it > in another expression. This would allow me to plot f and f-x on the > same graph without having to enter f twice or change it in two places. > 2. DeÞne a function, say f2=ax^2+bx+c, and then be able to call > it pasing it the values for a, b, and c. > Can anyone recommend a program that can do that? > Either commercial or shareware is OK. I¹d be willing to spend up to > around $200 if there is nothing cheaper. === Subject: Re: Graphing Software Have you tried graphmatica? http://www.Þleboost.net/directory/education/mathematics/ review_000751.html http://www.winsite.com/bin/Info?15000000036697 I hope you Þnd this helpful. Kevin O¹Neill > I would appreciate recommendations for a good graphing software > package for Windows (2000). My needs are modest. I am currently > working with some exponential and polynomial functions. > I purchased Advanced Grapher some time ago, but it hasn¹t been updated > in awhile and the author doesn¹t respond to requests for enhancements. > http://www.alentum.com/agrapher/index.htm > It¹s OK for what I need. The main thing it lacks is the ability to > deÞne saved functions. That is, I would like like to be able to do > two things: > 1. DeÞne a function, say f1=3x^2-1, and then be able to refer to it > in another expression. This would allow me to plot f and f-x on the > same graph without having to enter f twice or change it in two places. > 2. DeÞne a function, say f2=ax^2+bx+c, and then be able to call > it pasing it the values for a, b, and c. > Can anyone recommend a program that can do that? > Either commercial or shareware is OK. I¹d be willing to spend up to > around $200 if there is nothing cheaper. > -- === Subject: Re: Graphing Software >Have you tried graphmatica? >http://www.Þleboost.net/directory/education/mathematics/ review_000751.html >http://www.winsite.com/bin/Info?15000000036697 That is a good one; I just wish I could remember the Þlename and program name of another similar one like that: mvgraph...., graphmv.... ? Some small Þle, too, works really easily. I found both last year while doing a web G C === Subject: Re: Graphing Software > I would appreciate recommendations for a good graphing software > package for Windows (2000). My needs are modest. I am currently > working with some exponential and polynomial functions. > I purchased Advanced Grapher some time ago, but it hasn¹t been updated > in awhile and the author doesn¹t respond to requests for enhancements. > http://www.alentum.com/agrapher/index.htm > It¹s OK for what I need. The main thing it lacks is the ability to > deÞne saved functions. That is, I would like like to be able to do > two things: > 1. DeÞne a function, say f1=3x^2-1, and then be able to refer to it > in another expression. This would allow me to plot f and f-x on the > same graph without having to enter f twice or change it in two places. > 2. DeÞne a function, say f2=ax^2+bx+c, and then be able to call > it pasing it the values for a, b, and c. > Can anyone recommend a program that can do that? > Either commercial or shareware is OK. I¹d be willing to spend up to > around $200 if there is nothing cheaper. I think Equation Grapher may be a good candidate for your need. Sometimes I have also some basic equations to check their graph to have a sense of what it is; and it is a good one for me. More info here: http://www.mfsoft.com/equationgrapher/ HTMH Maximus > -- === Subject: Re: Graphing Software >I would appreciate recommendations for a good graphing software >package for Windows (2000). My needs are modest. I am currently >working with some exponential and polynomial functions. >I purchased Advanced Grapher some time ago, but it hasn¹t been updated >in awhile and the author doesn¹t respond to requests for enhancements. >http://www.alentum.com/agrapher/index.htm >It¹s OK for what I need. The main thing it lacks is the ability to >deÞne saved functions. That is, I would like like to be able to do >two things: >1. DeÞne a function, say f1=3x^2-1, and then be able to refer to it >in another expression. This would allow me to plot f and f-x on the >same graph without having to enter f twice or change it in two places. >2. DeÞne a function, say f2=ax^2+bx+c, and then be able to call >it pasing it the values for a, b, and c. >Can anyone recommend a program that can do that? >Either commercial or shareware is OK. I¹d be willing to spend up to >around $200 if there is nothing cheaper. If you are a student, you can get Maple for about $130. Go to http://www.maplesoft.com/ for details. What you are asking above is trivially done in Maple. If that¹s all you are going to use Maple for, you will be actually using about 1/10000th part of Maple, but that could be considered an advantage. Here¹s how you enter the fragments that you have provided: f1 := 3*x^2 - 1; plot(f1, x=-2..2); # plot f1 plot({f1, f1-x}, x=-2..2); # plot f1 and f1-x simultaneously f2 := a*x^2 + b*x + c; subs(a=2, b=1, c=3, f2); # substitute values for a, b, c in f2 and so on. -- rr === Subject: Re: Graphing Software >If you are a student, you can get Maple for about $130. >Go to http://www.maplesoft.com/ for details. >What you are asking above is trivially done in Maple. >If that¹s all you are going to use Maple for, you >will be actually using about 1/10000th part of Maple, >but that could be considered an advantage. >Here¹s how you enter the fragments that you have provided: >f1 := 3*x^2 - 1; >plot(f1, x=-2..2); # plot f1 >plot({f1, f1-x}, x=-2..2); # plot f1 and f1-x simultaneously >f2 := a*x^2 + b*x + c; >subs(a=2, b=1, c=3, f2); # substitute values for a, b, c in f2 >and so on. Nope, not a student. -- === Subject: Re: Graphing Software << >Nope, not a student. Since you mentioned Windows, you can try the free 30 day trial of Derive from www.derive.com and see if you like that. Even for non-students it is a tiny fraction of the price of Maple. I tried that Þrst, and your hotmail address, both bounced === Subject: Re: Graphing Software ><< >>Nope, not a student. >Since you mentioned Windows, you can try the free 30 day trial >of Derive from www.derive.com and see if you like that. >Even for non-students it is a tiny fraction of the price of Maple. How does Derive compare to Mathematica or Maple? -- === Subject: Re: Graphing Software >>Since you mentioned Windows, you can try the free 30 day trial >>of Derive from www.derive.com and see if you like that. >>Even for non-students it is a tiny fraction of the price of Maple. >How does Derive compare to Mathematica or Maple? Derive does not have the vast collection of programming features that Maple and Mathematica have. But it does an amazing amount of work, often without needing to learn how to program some solution to your problem. If you have an example problem and wonder how Derive might handle it you could send me mail, address is valid, and I¹d send you my best estimate of how to handle it. === Subject: Re: Reverse Continued Fractions. http://wims.unice.fr/wims/wims.cgi?session=E198839EF0.1&lang= en&cmd=reply&mod ule=tool%2Fnumber%2Fcontfrac.en&formula=2%2F%284-pi%29& precision=50&num_style =4 > The site allows you to Þnd, along with standard continued fractions, > continued fractions with negative terms or with alternating terms compute terms ad lib beyond the edge of the page. (Or did I miss an option?) > I plugged in pi, and, according to the site, your last term for pi is > different than theirs. Yes that was my boo-boo; I was using Maple with initially no forced update of the precision, so it chose its own. Forcing the precision gets the page¹s > But you are saying that these expansions are not unique anyway. No, I did not say that, Gerald Edgar said that, and it¹s not quite true. There *is* a unique representation if you insist on inÞnite sequences, which I think one should - this makes the Baire space identiÞcation nice. If you allow terminating sequences of denominators, (which are compulsory for ordinary CFs but NOT here [!]), then there is non-uniqueness, but that becomes irrelevant when onbe insists on non-termination. Non-termination implies uniqueness.... hmmm... I wonder if this has got anything to do with aborting identical twins... ------------------------------------------------------------- --------------- -- Bill Taylor W.Taylor@math.canterbury.ac.nz ------------------------------------------------------------- --------------- -- Everybody loves my baby, But my baby don¹t love nobody but me. (The Russell love song) ------------------------------------------------------------- --------------- -- === Subject: Re: JSH: Mistakes happen >>Mistakes happen. And people can overlook them, for many reasons. >Everyone makes the odd mistake, but you, James, are a mistake-generator. >>I make LOTS of mistakes. I¹ve noted that before. >>... >>YES!!! I screw up a LOT! I¹ve made LOTS OF MISTAKES!!! >>... >>There have even been times when I made some really stupid mistakes >>that I couldn¹t see myself having made so I argued when I was WRONG. > So why, oh why, do you so Þercely deny that there are no mistakes in your > latest piece of work, whatever that latest piece of work might happen to be? Ahem. Actually this he claims, not denies. ;-) === Subject: Re: JSH: Mistakes happen <2m4tggFiujbcU1@uni-berlin.de> Discussion, linux) >> >> >>>Mistakes happen. And people can overlook them, for many reasons. >> >>Everyone makes the odd mistake, but you, James, are a mistake-generator. >I make LOTS of mistakes. I¹ve noted that before. >... >YES!!! I screw up a LOT! I¹ve made LOTS OF MISTAKES!!! >... >There have even been times when I made some really stupid mistakes >that I couldn¹t see myself having made so I argued when I was WRONG. >> So why, oh why, do you so Þercely deny that there are no mistakes in your >> latest piece of work, whatever that latest piece of work might happen to be? > Ahem. Actually this he claims, not denies. ;-) Which prompts the question: can one *Þercely* claim something? -- Jesse Hughes How lucky we are to be able to hear how miserable Willie Nelson could imagine himself to be. -- Ken Tucker on Fresh Air === Subject: Re: JSH: Mistakes happen > Now then the sci.math newsgroup has problems. You have people you > welcome, and who you cheer, who stalk me like weird obsessed > creatures, and you can¹t control them, and don¹t seem willing to > control them, as I think they are exemplars of your group. > You are sick, as a group. Even if we want to control these sick people, how could we do it? Criticism only starts controversy, as with you. We can¹t control your idiotic, provactive posts, but the thing about a newsgroup is anyone can say anything they want, and then anyone can say anything _they_ want about that post, and so on. Van === Subject: Re: JSH: Mistakes happen > Now then the sci.math newsgroup has problems. You have people you > welcome, and who you cheer, who stalk me like weird obsessed > creatures, and you can¹t control them, and don¹t seem willing to > control them, as I think they are exemplars of your group. > > You are sick, as a group. > Even if we want to control these sick people, how could > we do it? Criticism only starts controversy, > as with you. We can¹t control your idiotic, provactive posts, > but the thing about a newsgroup is anyone can say anything > they want, and then anyone can say anything _they_ want about > that post, and so on. No, I POST. That¹s just what people do on Usenet. But some of you copy posts off of Usenet to put up on your webpages. Some of you sent time, not talking about mathematics, not talking about anything important, but just spending time talking negatively about me. Posting on Usenet, no matter how much it annoys you or how silly you think a posting is, is just doing what is done on Usenet. People post on Usenet. But copying posts OFF of Usenet, and sending emails or putting up nasty webpages to attack a Usenet poster is NOT HEALTHY BEHAVIOR. You people are sick. There has to be something wrong with you that math isn¹t enough, your own lives aren¹t enough, your own work is not enough that you can get involved with it, get consumed by it, rather than spending so many years worrying about me. I¹m ONE PERSON who is hounded by a group of people supposedly intelligent, supposedly rational, who supposedly care about mathematics more than silly social crap. But instead sci.math is like some high school in the United States, where one gang has decided to load up on ONE PERSON, and will not stop, and will not quit, but just keeps at it, YEAR AFTER YEAR AFTER YEAR!!! When there¹s a gang on one side, spouting nastiness, and ONE PERSON on the other side, what sane person would dare to question what is going on? I post on Usenet. Why is that such a big deal that so many people spend so much time on and off Usenet trying to affect my life? What¹s wrong with you people? James Harris === Subject: Re: JSH: Mistakes happen ... > Posting on Usenet, no matter how much it annoys you or how silly you > think a posting is, is just doing what is done on Usenet. People post > on Usenet. Did you not tell some people that they had to shut up in your threads? > But copying posts OFF of Usenet, and sending emails or putting up > nasty webpages to attack a Usenet poster is NOT HEALTHY BEHAVIOR. Did you not send e-mails to institutions to have people shut up? I see it: Quod licet Iovi non licet bovi. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: JSH: Mistakes happen > ... > Posting on Usenet, no matter how much it annoys you or how silly you > think a posting is, is just doing what is done on Usenet. People post > on Usenet. > Did you not tell some people that they had to shut up in your threads? > But copying posts OFF of Usenet, and sending emails or putting up > nasty webpages to attack a Usenet poster is NOT HEALTHY BEHAVIOR. > Did you not send e-mails to institutions to have people shut up? > I see it: Quod licet Iovi non licet bovi. > -- > dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 > home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: JSH: Mistakes happen > No, I POST. That¹s just what people do on Usenet. But some of you > copy posts off of Usenet to put up on your webpages. Some of you sent AKA peer review (wasn¹t that generous of me?). Careful what you ask for. > And many of you spend a lot of > time, not talking about mathematics, not talking about anything > important, but just spending time talking negatively about me. It¹s good that you acknowledge that you¹re not as important as you once thought you were. Maybe there¹s hope for you yet. > When there¹s a gang on one side, spouting nastiness, and ONE PERSON on > the other side, what sane person would dare to question what is going > on? Hey, but it¹s just a post on usenet. What¹s the big deal? That¹s what people do, they post on usenet. Nevertheless, your implied admission of insanity is a good step for you. Acknowledging a problem is a good start in solving it. > I post on Usenet. Why is that such a big deal that so many people > spend so much time on and off Usenet trying to affect my life? > What¹s wrong with you people? We¹re all secretly envious of your math, which is so subtle that I can¹t even see it in your posting. -- CodeCutter - good, fast and cheap; pick two. === Subject: Re: JSH: Mistakes happen > > Now then the sci.math newsgroup has problems. You have people you > > welcome, and who you cheer, who stalk me like weird obsessed > > creatures, and you can¹t control them, and don¹t seem willing to > > control them, as I think they are exemplars of your group. > > > > You are sick, as a group. > Even if we want to control these sick people, how could > we do it? Criticism only starts controversy, > as with you. We can¹t control your idiotic, provactive posts, > but the thing about a newsgroup is anyone can say anything > they want, and then anyone can say anything _they_ want about > that post, and so on. > No, I POST. That¹s just what people do on Usenet. But some of you > copy posts off of Usenet to put up on your webpages. Some of you sent > time, not talking about mathematics, not talking about anything > important, but just spending time talking negatively about me. > Posting on Usenet, no matter how much it annoys you or how silly you > think a posting is, is just doing what is done on Usenet. People post > on Usenet. > But copying posts OFF of Usenet, and sending emails or putting up > nasty webpages to attack a Usenet poster is NOT HEALTHY BEHAVIOR. > You people are sick. > There has to be something wrong with you that math isn¹t enough, your > own lives aren¹t enough, your own work is not enough that you can get > involved with it, get consumed by it, rather than spending so many > years worrying about me. > I¹m ONE PERSON who is hounded by a group of people supposedly > intelligent, supposedly rational, who supposedly care about > mathematics more than silly social crap. > But instead sci.math is like some high school in the United States, > where one gang has decided to load up on ONE PERSON, and will not > stop, and will not quit, but just keeps at it, YEAR AFTER YEAR AFTER > YEAR!!! > When there¹s a gang on one side, spouting nastiness, and ONE PERSON on > the other side, what sane person would dare to question what is going > on? > I post on Usenet. Why is that such a big deal that so many people > spend so much time on and off Usenet trying to affect my life? > What¹s wrong with you people? > James Harris It¹s simple, Usenet can¹t affect your life in the ways you describe if you don¹t use it. Grow some brains, Harris. Dave === Subject: Re: JSH: Mistakes happen > Posting on Usenet, no matter how much it annoys you or how silly you > think a posting is, is just doing what is done on Usenet. People post > on Usenet. > But copying posts OFF of Usenet, and sending emails or putting up > nasty webpages to attack a Usenet poster is NOT HEALTHY BEHAVIOR. > You people are sick. > There has to be something wrong with you that math isn¹t enough, your > own lives aren¹t enough, your own work is not enough that you can get > involved with it, get consumed by it, rather than spending so many > years worrying about me. > I¹m ONE PERSON who is hounded by a group of people supposedly > intelligent, supposedly rational, who supposedly care about > mathematics more than silly social crap. > But instead sci.math is like some high school in the United States, > where one gang has decided to load up on ONE PERSON, and will not > stop, and will not quit, but just keeps at it, YEAR AFTER YEAR AFTER > YEAR!!! > When there¹s a gang on one side, spouting nastiness, and ONE PERSON on > the other side, what sane person would dare to question what is going > on? > I post on Usenet. Why is that such a big deal that so many people > spend so much time on and off Usenet trying to affect my life? > What¹s wrong with you people? > James Harris Stop snivelling. You¹re starting to sound like Dar Kabatoff. -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com === Subject: Choosing the value in an equation Hi all, I¹m working on some math studies here, and I¹m a bit stomped. I have a task where I should work with an equation that looks like this: x^2 + 4x - 2 = ? I have all the answers, but could use some help understanding how to go about Þnding the answers. Any help and pointers would be much appreciated. I¹m supposed to: a) Choose the left-hand constant so that x is 2 (which is easy enough, just substitute x with 2, and get 10). I¹m also supposed to Þnd the other possible left-hand constant, which is -6. I¹m not sure how to go about to Þnd the result -6 though, any ideas? b) Choose the left-hand constant so that the equation only has one solution, any suggestions? (The cheatsheet says -6) c) Choose the left-hand constant so that the equation can have an answer. (The cheatsheet says and number smaller than -6) Morten === Subject: Re: Choosing the value in an equation Oops, this was posted a bit too early in the morning. It should be right-hand where is says left-hand, and says and number should be says any number. -Morten > Hi all, > I¹m working on some math studies here, and I¹m a bit stomped. I > have a task where I should work with an equation that looks like > this: > x^2 + 4x - 2 = ? > I have all the answers, but could use some help understanding how to > go about Þnding the answers. Any help and pointers would be much > appreciated. > I¹m supposed to: > a) Choose the left-hand constant so that x is 2 (which is easy enough, > just substitute x with 2, and get 10). I¹m also supposed to Þnd > the other possible left-hand constant, which is -6. I¹m not > sure how to go about to Þnd the result -6 though, any ideas? > b) Choose the left-hand constant so that the equation only > has one solution, any suggestions? (The cheatsheet says -6) > c) Choose the left-hand constant so that the equation can > have an answer. (The cheatsheet says and number smaller > than -6) > Morten === Subject: Re: Choosing the value in an equation > Hi all, > I¹m working on some math studies here, and I¹m a bit stomped. I > have a task where I should work with an equation that looks like > this: > x^2 + 4x - 2 = ? > I have all the answers, but could use some help understanding how to > go about Þnding the answers. Any help and pointers would be much > appreciated. > I¹m supposed to: > a) Choose the left-hand constant so that x is 2 (which is easy enough, You mean right-hand constant. > just substitute x with 2, and get 10). I¹m also supposed to Þnd > the other possible left-hand constant, which is -6. I¹m not You mean the other possible value for x, when the right-hand side is 10. > sure how to go about to Þnd the result -6 though, any ideas? > b) Choose the left-hand constant so that the equation only > has one solution, any suggestions? (The cheatsheet says -6) Again, you mean right-hand constant. Hint: How many solutions does the equation (x+2)^2 = 0 have? > c) Choose the left-hand constant so that the equation can > have an answer. (The cheatsheet says and number smaller > than -6) You mean right-hand constant, and you mean can¹t have an answer. Hint: How many solutions does the equation (x+2)^2 = -1 have? Or in general (x+2)^2 < 0. -Michael. === Subject: Re: Choosing the value in an equation Hi Michael, >>I¹m supposed to: >> a) Choose the left-hand constant so that x is 2 (which is easy enough, > You mean right-hand constant. Yup, I was completely in a daze when I was writing this. :) >> just substitute x with 2, and get 10). I¹m also supposed to Þnd >> the other possible left-hand constant, which is -6. I¹m not > You mean the other possible value for x, when the right-hand side is 10. Yes. Just Þgured out how it could be done.. >> b) Choose the left-hand constant so that the equation only >> has one solution, any suggestions? (The cheatsheet says -6) > Again, you mean right-hand constant. > Hint: How many solutions does the equation (x+2)^2 = 0 have? Not sure.. >> c) Choose the left-hand constant so that the equation can >> have an answer. (The cheatsheet says and number smaller >> than -6) > You mean right-hand constant, and you mean can¹t have an answer. > Hint: How many solutions does the equation (x+2)^2 = -1 have? Or in general > (x+2)^2 < 0. Again I¹m not sure. I¹m going to go dig myself a hole and hide now. ;) Maybe you know of a page on the net where these kinds of equations are explained? Morten === Subject: Re: Choosing the value in an equation <40fdfd84$1@news.broadpark.no> > Maybe you know of a page on the net where these kinds of equations > are explained? Look up solutions to quadradic equations. ax^2 + bx + c = 0, a /= 0 iff x = (-b +- sqr(b^2 - 4ac))/2a Thus equation has two real solutions when b^2 - 4ac > 0 one real solution when b^2 - 4ac = 0 two complex solutions when b^2 - 4ac < 0 Solution by completing the square ax^2 + bx + c = 0 (ax)^2 + abx + ac = 0 (ax)^2 + abx + (b/2)^2 = (b/2)^2 - ac (ax + b/2)^2 = (b/2)^2 - ac = (b^2 - 4ac)/4 ax + b/2 = (1/2)sqr(b^2 - 4ac) x = (-b +- sqr(b^2 - 4ac))/2a === Subject: Re: Choosing the value in an equation William, > Thus equation has two real solutions when > b^2 - 4ac > 0 > one real solution when > b^2 - 4ac = 0 > two complex solutions when > b^2 - 4ac < 0 that¹s right. It took me a while to Þgure it out, but reading the book thoroughly helped med realize what you mention above. We don¹t cover complex solutions, so the equation doesn¹t have a solution when b^2 - 4ac < 0. -Morten === Subject: Re: Choosing the value in an equation > Hint: How many solutions does the equation (x+2)^2 = 0 have? > Not sure.. Here¹s another hint: How many solutions does the equation y^2 = 0 have? -- Daniel W. Johnson panoptes@iquest.net http://members.iquest.net/~panoptes/ 039 53 36 N / 086 11 55 W === Subject: Re: Example of Polynomial Root Method > APPLICATION OF http://mypeoplepc.com/members/jon8338/polynomial/index.html > > Alrighty then, I have my counterexample ready You don¹t need one. Jon provided his own counterexample. > > The fact that you have ignored these two principles has caused more > swear words to be passed your direction than what you¹ve said about > vectors, Maclauren series, and linear algebra, all put together. > > For all I care, the individual can go to hayell. > As far as I can tell, that individual ŒProginoskes¹ constitutes half > of your audience. Or half that still cares, who actually asked whether the method worked. -- Christopher Heckman === Subject: Re: Algebraic topology > which I don¹t understand: how do you compute the degree of M as the > intersection index of M with a general position line L in CP(n) ? > Wouldn¹t that give you something like the fundamental class of the > intersection of M and the line ? Well, I remember such an explanation from GrifÞths and Harris¹ Principles of algebraic geometry; see therein Preliminaries -> 4.Topology of manifolds -> Intersection of analytic cycles. Simeon === Subject: A.D. History (was Re: Agnostic MorituriMax) > ago. That was so important that the entire civilized world started > counting time from that moment. Not even close. Around 525 A.D. or so (they didn¹t call it that back then) church leaders gave up trying to Þgure out when the world began to reorder the year dates (they didn¹t want to use the existing standard of the founding of Rome) and with the Bible not being very speciÞc on that topic, as a short cut tried to restart the calendar years from Jesus¹ birth. The christian church then proceded to botch even that. Not only did they get the year wrong, they even got the season wrong (middle-eastern shepards out tending their þock in winter???) and pinned the date at December 25, 753 A.U.D. or from the founding of Rome (the previous year number benchmark). Theories abound that the mistake was deliberate in order to compete with the popular Mithraist religion¹s week-long celebration of the end of the year starting on Dec. 25th. For more details about the history of the calendar, see the short yet excellent summary at: http://www.geocities.com/Athens/Oracle/9941/endtime.html Hadji === Subject: Re: A.D. History (was Re: Agnostic MorituriMax) >> ago. That was so important that the entire civilized world started >> counting time from that moment. > Not even close. Around 525 A.D. or so (they didn¹t call it that back > then) church leaders gave up trying to Þgure out when the world > began to reorder the year dates (they didn¹t want to use the existing > standard of the founding of Rome) and with the Bible not being very > speciÞc on that topic, as a short cut tried to restart the calendar > years from Jesus¹ birth. The christian church then proceded to botch > even that. Not only did they get the year wrong, they even got the > season wrong (middle-eastern shepards out tending their þock in > winter???) and pinned the date at December 25, 753 A.U.D. or from the > founding of Rome (the previous year number benchmark). Theories > abound that the mistake was deliberate in order to compete with the > popular Mithraist religion¹s week-long celebration of the end of the > year starting on Dec. 25th. I¹ve just consulted my extensive selection of calendars, and discovered that today is: Char-shanbeh, Saratan 31, 1383 in the Afghan calendar. Chorekhshabathi, Aweleach 1, 1453 in the Armenian calendar. Idal, Kalimat 9 (Asma¹), year 9 (Baha¹), Vahid 9, Kull-i-Shay 1 BE, until sunset, in the Baha¹i calendar. Luang, Pepet, Beteng, Sri, Wage, Was, Buda, Ludra, Ogan, Pati in the Balinese Pawukon calendar. BudhBar, Shrabon 6, 1411 BS in the Bangla calendar. Day 5, month 6, year 21 (Jia-shen), cycle 78 in the Chinese calendar. Choiak 1, 2753 in the Egyptian calendar. Rob, Hamle 14, 1996 EE in the Ethiopic calendar. Buddhavara, Sravana 5, 5105 KY, from sunrise, in the old Hindu lunisolar calendar. Buddhavara, Karka 6, 5105 KY, from sunrise, in the old Hindu solar calendar. Buddhavara, leap month Sravana 4, 2061 VE, from sunrise, in the Hindu lunisolar calendar. Buddhavara, Sravana 6, 1926 SE, from sunrise, in the Hindu solar calendar. Buddhavara, Ashadha 30, 1926 SE in the Indian national calendar. Yaum al-arba¹a¹, Jumada al-Ahira 3, 1425 AH (approx), until sunset, in the Islamic calendar. 12.19.11.8.5 in the Mayan long count. 8 Xul in the Mayan civil haab calendar. 10 Chicchan in the Mayan religious tzolkin calendar. Mayan Lord of the Night G3. Amardad 3 (Ardibehest), 1374 in the Parsi Fasli calendar. Fravardin 1 (Hormazd), 1374 in the Parsi Kadmi calendar. Aspandarmad 6 (Khordad), 1373 in the Parsi Shenshai calendar. Char-shanbeh, Tir 31, 1383 AP in the Persian calendar. Budhvaar, Sawan 6, 536 in the Sikh Nanakshahi calendar. Wan phut, Garagadakom 21, 2547 BE in the Thai solar calendar. Day 5, month 6, year 21 (Jia-shen), cycle 78 in the Vietnamese calendar. My personal favourite is the Mayan calendar: http://www.pauahtun.org/basic.html === Subject: Re: A.D. History (was Re: Agnostic MorituriMax) > I¹ve just consulted my extensive selection of calendars, and discovered > that today is: > Char-shanbeh, Saratan 31, 1383 in the Afghan calendar. [list of calendars snipped] You left off the Stardate. Socks === Subject: Re: A.D. History (was Re: Agnostic MorituriMax) >> I¹ve just consulted my extensive selection of calendars, and >> discovered that today is: >> Char-shanbeh, Saratan 31, 1383 in the Afghan calendar. > [list of calendars snipped] > You left off the Stardate. The stardate system rivals the Mayan calendar for complexity: http://www.cs.umanitoba.ca/~djc/startrek/stardates/ (some people have *way* too much time on their hands) === Subject: Generating fractional part of Sqrt(Prime) Hi all, I need help with some algorithm, because my mathematical insight is not enough I¹m afraid. I present this in pascal-like syntax, because Delphi is my thing: I have a big prime, say N, probably stored in an array of bytes: var N : array[0..MaxBytes-1] of Byte; How I generated this prime is not important now. I want to generate, _without_ _limit_, the digits, preferrably base 2 (binary) or 16 (hex), of the fractional part of Sqrt(N). So what I need is some endless loop, just generating the next digit, digit after digit... Extra request (not vital): If it is in any way possible, then I would like to start on some given digit, without calculating the previous ones. Thanx, M. === Subject: Re: Generating fractional part of Sqrt(Prime) > Hi all, > I need help with some algorithm, because my mathematical insight is not > enough I¹m afraid. > I present this in pascal-like syntax, because Delphi is my thing: > I have a big prime, say N, probably stored in an array of bytes: > > var N : array[0..MaxBytes-1] of Byte; > > How I generated this prime is not important now. > I want to generate, _without_ _limit_, the digits, preferrably base 2 > (binary) or 16 (hex), of the fractional part of Sqrt(N). > So what I need is some endless loop, just generating the next digit, > digit after digit... > Extra request (not vital): If it is in any way possible, then I would > like to start on some given digit, without calculating the previous ones. > Thanx, > M. You may use the classical (school) algorithm converted in binary : since you want only the fractional part of the development of sqrt(N) let¹s start with q= int(sqrt(N)) and r= N- q^2 (int(..) is of course the integer part of sqrt(N)) repeat nr:= 4*r; t := 4*q+1; if nr >= t then begin r:= nr-t; q:= 2*q+1; //output a Œ1¹ binary digit end else begin r:= nr; q:= 2*q; //output a Œ0¹ digit end; until tired... or out of precision since r is multiplied by 4 at each iteration (anyway a multiprecision package allowing addition, substraction and shift only is not too hard to implement so...). -------------- A better method is to use Newton¹s iterations with quadratic convergence (not 1 digit but 2 times _more_ digits at every iteration!!) Start with x= sqrt(N) (with ŒÞxed¹ hardware precision only of course! If you are Œpurist¹ start with q=1! :-)) compute nx = (N/x+x)/2 at each iteration (implementation : repeat x:= (N/x+x)/2 until...) allowing 2 times more digit precision at every iteration. (Of course multiprecision code is required able to add and divide) A better (nicer) implementation using only integer values may be obtained (considering x=p/q) and : start with p= int(sqrt(N)) and q=1 (or p=1 if you prefer) compute np= N*q^2+p^2; nq= 2*p*q; at every iteration with np the new p and nq the new q This requires only multiplications and additions and a unique division at the end (of m digits by m digits for a result of nearly 2m digits) I¹ll stop there hoping it helped, Raymond === Subject: Re: Generating fractional part of Sqrt(Prime) > I want to generate, _without_ _limit_, the digits, preferrably base 2 > (binary) or 16 (hex), of the fractional part of Sqrt(N). > So what I need is some endless loop, just generating the next digit, > digit after digit... Hm. First of all, generating the digits one-by-one (up to, say, one million) is probably much slower than generating the entire one-million-digit result at once. But if speed is not important, then I could suggest using Newtons method. Let me guess: You want to use this as some kind of random number generator, perhaps in a cryptographic application. Then I think you¹re digging yourself a hole. Without being an expert, I have a hunch that the digits of sqrt(prime) are not cryptographically independent. I suggest you instead consider the simple but effective Blum Blum Shub algorithm . -Michael. === Subject: Re: Generating fractional part of Sqrt(Prime) >>I want to generate, _without_ _limit_, the digits, preferrably base 2 >>(binary) or 16 (hex), of the fractional part of Sqrt(N). >>So what I need is some endless loop, just generating the next digit, >>digit after digit... > Hm. First of all, generating the digits one-by-one (up to, say, one million) > is probably much slower than generating the entire one-million-digit result > at once. But if speed is not important, then I could suggest using Newtons > method. > Let me guess: You want to use this as some kind of random number generator, > perhaps in a cryptographic application. Then I think you¹re digging yourself > a hole. Without being an expert, I have a hunch that the digits of > sqrt(prime) are not cryptographically independent. Guessed right. But: if N is huge enough (which means Sqrt(N) should be huge enough also), and prime, could an attacker (or his computer(s)) deduce N from maybe having some digits of the fractional part of Sqrt(N)? I know the digits of e.g. Pi are too well known to use for anything cryptographic, but if they wouldn¹t be, then they would be random enough, wouldn¹t they? I believe they would pass e.g. the Diehard test. > I suggest you instead consider the simple but effective Blum Blum Shub > algorithm . I know about it. I somehow do not like anything with a period. Even if the period is *huge*. Maybe it¹s stupid of me, but I¹d rather have something that will deÞnately not have, is garuanteed not to have, a period. The digits of Sqrt(Prime) will never have a period. Thanx, M. === Subject: Re: Generating fractional part of Sqrt(Prime) > I know about it. I somehow do not like anything with a period. Even if > the period is *huge*. Maybe it¹s stupid of me, but I¹d rather have > something that will deÞnately not have, is garuanteed not to have, a > period. The digits of Sqrt(Prime) will never have a period. The same is true for the digits of the square root of 6. Why do you insist on primes? -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Goldbach¹s conjecture is neither true nor false > > The above statement satisÞes the criteria for a mathematical axiom, > > so it does not require proof. Not every statement which is independent of a set of axioms is worth considering as an addition to that set of axioms. You have failed to show either that your statement is independent of any particular set of axioms or that it is worth consideration as an addition to any particular set of axioms. === Subject: Re: Goldbach¹s conjecture is neither true nor false > > > The above statement satisÞes the criteria for a mathematical axiom, > > > so it does not require proof. > Not every statement which is independent of a set of axioms is worth > considering as an addition to that set of axioms. > You have failed to show either that your statement is independent of any > particular set of axioms or that it is worth consideration as an > addition to any particular set of axioms. Further to my previous reply, and to correct any confusion caused by careless quoting, the original statement was Within the topology of mathematics, the neighbourhood of the solution space surrounding any proof of GC has zero dimensions. I haven¹t shown that the statement in question is an axiom - I leave that as an exercise for you as you seem so interested in the idea (hint: if S is a set of axioms, A is independent of S if neither {A, S} nor {~A, S} produces a contradiction, and use the methods of Functional Analysis in your proof - good luck!) As to whether the statement is worth consideration (as an axiom or otherwise), in my opinion, anything which clariÞes the status of Goldbach¹s Conjecture deserves worldwide publicity, so that hundreds of thousands of people now and in the future will not waste their lives or lose their sanity in attempts to prove or disprove GC, which is probably unprovable (with its unprovability also being unprovable), and whose actual truth or falsity is in reality completely worthless and unimportant. === Subject: Re: Goldbach¹s conjecture is neither true nor false > > > The above statement satisÞes the criteria for a mathematical axiom, > > > so it does not require proof. > Not every statement which is independent of a set of axioms is worth > considering as an addition to that set of axioms. > You have failed to show either that your statement is independent of any > particular set of axioms or that it is worth consideration as an > addition to any particular set of axioms. I haven¹t shown that the statement in question is an axiom - I leave that as an exercise for you as you seem so interested in the idea (hint: if S is a set of axioms, A is independent of S if neither {A, S} nor {~A, S} produces a contradiction, and use the methods of Functional Analysis in your proof - good luck!) === Subject: Re: Can you Þnd anything wrong with this solution to the Halting Problem? >> And no one has correctly pointed out any mistakes of mine. > Well, that is incorrect Peter. > It is also a deliberate lie. > Lots of people have pointed out mistakes of yours, > refuting your claims. Peter thinks he¹s being very slick with his use of Œcorrectly¹. He¹ll referr you to his website which pruports to enumerate any possible refutations. Or at least, the possible approaches that might lead to a refutation. Because no one has used one of those pre-approved forms of refutation then no one has Œcorrectly¹ refuted his claims. -- CodeCutter - good, fast and cheap; pick two. === Subject: Re: Can you Þnd anything wrong with this solution to the Halting Problem? > And no one has correctly pointed out any mistakes > of mine. > Well, that is incorrect Peter. > It is also a deliberate lie. Well that is actionable libel. If you can prove me wrong, then have at it, otherwise just shut up! === Subject: Re: Can you Þnd anything wrong with this solution to the Halting Problem? X-URL: http://mygate.mailgate.org/mynews/talk/talk.bizarre/ 003813c77ebe7e1e1025be619 e65799a.48257%40mygate.mailgate.org > And no one has correctly pointed out any mistakes > of mine. > Well, that is incorrect Peter. >> It is also a deliberate lie. > Well that is actionable libel. No, it isn¹t. I was raised by a lawyer, and I¹ve been on the net dealing with kooks like you who think every true description of their kook behavior is libel, for almost 20 years now. Somehow all that hot air has not resulted in a single lawsuit. > If you can prove me wrong, I proved that you yourself have admitted, multiple times, that you are wrong, then lied about your own behavior. Since you have no clue what constitutes a proof, you snipped it rather than respond to it. > then have at it, otherwise just shut up! Welcome to Usenet, land where the US First Amendment guarantee of freedom of expression Þnds full þower. If you want to trash out newsgroups where I participate with your blatant, time-wasting nonsense and pitiful attempts to defend your incompetence in the Þelds where you pretend expertise into the jibes of experts who know better, you are going to have to live with the consequences. Now that you are revealed to be, by your own writings over several years, not a person in single error about a single topic, but a full þowered Usenet kook, with a long history of providing bogus disproofs of long established results, then arguing interminable with refutations of your nonsense you don¹t have the wit to comprehend or the moral character to acknowledge, the quintessential victim of invincible ignorance, so that more and more people have realized that reasoning with you is a waste of time, while mocking you is still very entertaining, I¹ll just say I¹m happy I recognized your kook-nature early on and got my mocking begun early, before the late July rush began and the rates went up. xanthian. -- === Subject: Re: Can you Þnd anything wrong with this solution to the Halting Problem? > And no one has correctly pointed out any mistakes > of mine. > > Well, that is incorrect Peter. >> It is also a deliberate lie. > Well that is actionable libel. > No, it isn¹t. I was raised by a lawyer, and I¹ve > been on the net dealing with kooks like you who > think every true description of their kook behavior > is libel, for almost 20 years now. Somehow all > that hot air has not resulted in a single lawsuit. yet. === Subject: Re: Can you Þnd anything wrong with this solution to the Halting Problem? X-URL: http://mygate.mailgate.org/mynews/talk/talk.bizarre/ dac72134db51d7bad8288614c aacdf15.48257%40mygate.mailgate.org >> No, it isn¹t. I was raised by a lawyer, and I¹ve >> been on the net dealing with kooks like you who >> think every true description of their kook >> behavior is libel, for almost 20 years now. >> Somehow all that hot air has not resulted in a >> single lawsuit. > yet. Nor will it from you, oh producer of hot air you wish to have accepted as proofs. Perhaps you recall saying this You open youself up to losing all of your personal property to libel cases, by calling people liars with no sufÞcient basis. when you were threatening somone else who correctly characterized what you were saying as lies, back in 2002? Your kook-behavior is documented on the ŒNet Peter, for anyone to conÞrm, and that documentation isn¹t going away. Threats of lawsuits are very common kook behavior. The more kook behavior in which you indulge, the stronger that documentation grows. All your bluster avails you nothing, you are known to be a hollow drum, all noise and no substance, and eventually that knowledge will catch up with you at each new venue, as it now has here. Nothing more is required than that some one interested party happen to do a web search on your prior postings. All your claims that no-one can read your wonderful proof founder on the fact that your proofs are and always have been meaningless drivel, the product of your profound incompetence at each task you undertake. Your recent claims to be an expert programmer, to have patents pending for Þnite automata (do you even know what a Þnite automaton _is_?) merely highlight your inability to tell the truth, and your pathetic grasping at glory you¹ve never earned. xanthian, amused at empty threats. -- === Subject: Re: Can you Þnd anything wrong with this solution to the Halting Problem? >> If Olcott is right, then he¹s setting his sights far too low. A method for >> creating software whose output cannot be accessed by other software would be >> valuable in other ways besides disproving the Halting Problem. The MPAA, >done is that encryption is powerful enough to not require the >additional expense of special hardware. > Seriously, man, forget the Halting Problem. Even if by some miracle you > manage to convince the entire anal-retentive comp sci establishment, big > freaking whoop. The absolute most you could get for that would be a Fields > Medal, which only impresses geeks. You can¹t sell it for any signiÞcant I have my reasons. > amount, and it won¹t get you laid. The movie people, on the other hand, > will pay you real money for your secure-software idea, which is only a minor > detail of your proof anyway, and they don¹t have their heads in the clouds > like these ivory-tower academics, so it should be easier to explain your > idea to them. Besides, after you¹re rich and famous for solving the > secure-software problem, you¹ll have plenty of leisure and credibility to > put towards your more theoretical research. > -- > Matthew Skala > mskala@ansuz.sooke.bc.ca Embrace and defend. > http://ansuz.sooke.bc.ca/ === Subject: Re: ŒUncountable¹ doesn¹t exist In sci.math, Virgil >> The uncountable is deÞnitely useful for describing large cardinals, >> some of which shrink with time... > Depends on how you wash them. Cold water, gentle detergent, soak, air dry. (If one can Þnd a washing machine large enough.) -- #191, ewill3@earthlink.net It¹s still legal to go .sigless. === Subject: Re: ŒUncountable¹ doesn¹t exist > The uncountable is deÞnitely useful for describing large cardinals, > some of which shrink with time... > Depends on how you wash them. What is IZF set theory? Rgds., Ross F. === Subject: Re: ŒUncountable¹ doesn¹t exist > The uncountable is deÞnitely useful for describing large cardinals, > some of which shrink with time... > Depends on how you wash them. What is IZF set theory? Rgds., Ross F. === Subject: Re: ŒUncountable¹ doesn¹t exist > I realised some time ago that I would reject AC in favor of DC, which > allows only countable collections. The reason for this, I founc out, > was that I can¹t accept any proof that requires an uncountable number > of steps - it is logically impossible. > Consider Cantor¹s diagonalisation argument. Everyone knows that it > proves R to be uncountable. Yet, if we could take an uncountable > number of steps, we could add each diagonal number to the list, and > eventually come up with all the reals. This is a contradiction; and > the only resolution is to exclude uncountable proofs. > This is nonsense. After adding an uncountable number of elements to a > countably inÞnite list you have an uncountable list. Since no one > ever denied that the reals could be put in correspondence with an > uncountable cardinal, there is no contradiction. This was my point - it is nonsense, as the very notion of performing any sequence of operations implies Œlisting¹ them, so an uncountable number of steps is impossible. > If we accept the preceding, no uncountable sets really Œexist¹! > If we accept the preceding, we should seek help. I choose not to accept Cantor¹s delusions, and I should seek help? Andrew Usher === Subject: Re: ŒUncountable¹ doesn¹t exist Discussion, linux) >> I realised some time ago that I would reject AC in favor of DC, which >> allows only countable collections. The reason for this, I founc out, >> was that I can¹t accept any proof that requires an uncountable number >> of steps - it is logically impossible. >> >> Consider Cantor¹s diagonalisation argument. Everyone knows that it >> proves R to be uncountable. Yet, if we could take an uncountable >> number of steps, we could add each diagonal number to the list, and >> eventually come up with all the reals. This is a contradiction; and >> the only resolution is to exclude uncountable proofs. >> This is nonsense. After adding an uncountable number of elements to a >> countably inÞnite list you have an uncountable list. Since no one >> ever denied that the reals could be put in correspondence with an >> uncountable cardinal, there is no contradiction. > This was my point - it is nonsense, as the very notion of performing > any sequence of operations implies Œlisting¹ them, so an uncountable > number of steps is impossible. It is not the result that is nonsense. It is your argument that is nonsense. It proved nothing at all. The result of adding an uncountable number of distinct elements to a set is an uncountable set. No one ever said otherwise. You have found a non-problem. >> If we accept the preceding, no uncountable sets really Œexist¹! >> If we accept the preceding, we should seek help. > I choose not to accept Cantor¹s delusions, and I should seek help? Does it bother you? There are some that choose Þnitist mathematics for philosophical reasons. I don¹t see any reason to quarrel with them. There are others (like you) that try to Þnd inconsistencies in mathematics with inÞnity. Their arguments are uniformly laughable (like yours). If you can be deluded that such inadequate reasoning is correct then I think you have two choices: don¹t pursue mathematics as a career or seek help understanding why your argument is incorrect. -- [I]t¹s good for the economy to charge for intellectual property, so open source software cannot be good, while Microsoft is the most far-thinking company around and is doing it all for the good of the public. -- Linus Torvalds paraphrases Microsoft VP Craig Mundie === Subject: Re: ŒUncountable¹ doesn¹t exist >> I realised some time ago that I would reject AC in favor of DC, which >> allows only countable collections. The reason for this, I founc out, >> was that I can¹t accept any proof that requires an uncountable number >> of steps - it is logically impossible. >> Consider Cantor¹s diagonalisation argument. Everyone knows that it >> proves R to be uncountable. Yet, if we could take an uncountable >> number of steps, we could add each diagonal number to the list, and >> eventually come up with all the reals. This is a contradiction; and >> the only resolution is to exclude uncountable proofs. >> This is nonsense. After adding an uncountable number of elements to a >> countably inÞnite list you have an uncountable list. Since no one >> ever denied that the reals could be put in correspondence with an >> uncountable cardinal, there is no contradiction. > This was my point - it is nonsense, as the very notion of performing > any sequence of operations implies Œlisting¹ them, so an uncountable > number of steps is impossible. Your choice of words is interesting, since the words sequence and list both suggest countability. So what you just said is that every set that can be arranged in a sequence (and is therefore countable) is listable (another way of saying it¹s countable). Sure, but what about the sets that can¹t be contained in a sequence to begin with? >> If we accept the preceding, no uncountable sets really Œexist¹! Your comments about AC are a bit confused, as well. You seem to be under the impression that if we accept AC, it will establish the existence of uncountable sets, and therefore you reject AC. I am sorry to disappoint you, but the axiom that guarantees the existence of uncountable sets (given that countably inÞnite sets exist) is the power set axiom, not AC. Cantor¹s theorem implies that the power set of a countably inÞnite set is uncountable. The theorem does not require AC. >> If we accept the preceding, we should seek help. > I choose not to accept Cantor¹s delusions, and I should seek help? Perhaps you will revise your estimate of where Cantor¹s delusions lie by accepting AC and rejecting power sets instead? Or perhaps you will drop your objections to uncountability and decide to take on the Banach-Tarski paradox? -- Dave Seaman Judge Yohn¹s mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: ŒUncountable¹ doesn¹t exist >> I have never seen a proof that required even an inÞnite (let alone >> uncountable) number of steps, and therefore the question has never >> arisen. A proof by transÞnite induction, for example, has basically one >> step, which often involves the consideration of three cases. > Well, I guess we have different meanings here for Œnumber of steps¹. > Regardless, I can¹t see any other reason to reject AC. > The mere fact that a proof mentions inÞnite sets does not mean the proof > requires inÞnitely many steps. How many steps does it take to > demonstrate that the set of even numbers is countable? > Answer: One. Let f: N -> 2N be given by f(n) = 2*n for each n. > (Technically, if you Þll in all the gaps in that answer, there would be > a few more steps than just one, but the number would still be small and > Þnite.) This is true, as you have an explicit formula. But proofs by induction are certainly inÞnite (in fact, we need an axiom to allow them), because they expand to an inÞnite number of steps. > By deÞnition, a proof is a *Þnite* sequence of formulas with the > property that each formula is either an axiom, or a consequence of > previous formulas. If you consider each formula to be a step in the > proof, then every proof has a Þnite number of steps. > Invoking the axiom of choice, even on an uncountable collection of sets, > is just *one* step in a proof. But the axiom of choice _doesn¹t_ give an explicit formula. Hence it is necessary to perform each choice individually. Remember that the axiom of choice is really about what we choose to call proper sets. If I say that uncountable collections aren¹t sets, I¹m not claiming they could not Œexist¹ in any sense. At least one does, the set of all countable ordinals, but the construction if very artiÞcial. Andrew Usher === Subject: Re: ŒUncountable¹ doesn¹t exist >>> I have never seen a proof that required even an inÞnite (let alone >>> uncountable) number of steps, and therefore the question has never >>> arisen. A proof by transÞnite induction, for example, has basically one >>> step, which often involves the consideration of three cases. >> Well, I guess we have different meanings here for Œnumber of steps¹. >> Regardless, I can¹t see any other reason to reject AC. >> The mere fact that a proof mentions inÞnite sets does not mean the proof >> requires inÞnitely many steps. How many steps does it take to >> demonstrate that the set of even numbers is countable? >> Answer: One. Let f: N -> 2N be given by f(n) = 2*n for each n. >> (Technically, if you Þll in all the gaps in that answer, there would be >> a few more steps than just one, but the number would still be small and >> Þnite.) > This is true, as you have an explicit formula. But proofs by induction > are certainly inÞnite (in fact, we need an axiom to allow them), > because they expand to an inÞnite number of steps. Depends on your starting point. Induction is an axiom if you are working in PA, but it¹s a theorem if you are working in ZF. Hence your statement (we need an axiom) is not necessarily true. >> By deÞnition, a proof is a *Þnite* sequence of formulas with the >> property that each formula is either an axiom, or a consequence of >> previous formulas. If you consider each formula to be a step in the >> proof, then every proof has a Þnite number of steps. >> Invoking the axiom of choice, even on an uncountable collection of sets, >> is just *one* step in a proof. > But the axiom of choice _doesn¹t_ give an explicit formula. Hence it > is necessary to perform each choice individually. You have that backwards. If it were possible to perform each choice individually, we would not need the axiom of choice. AC was devised precisely to cover those cases where we cannot perform each step individually. > Remember that the axiom of choice is really about what we choose to > call proper sets. If I say that uncountable collections aren¹t sets, > I¹m not claiming they could not Œexist¹ in any sense. At least one > does, the set of all countable ordinals, but the construction if very > artiÞcial. The set aleph_1 of all countable ordinals does not require AC for its construction. -- Dave Seaman Judge Yohn¹s mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Can anyone solve this ??? Each different letter represents an integer from 1 to 26 - all you need to do is work out which letter corresponds to which number. Hint: the value for the letter Z can be determined by elimination. A + B = D B * K = C F * U = D F * K = L H + I = J Q * Q = R R + S = T T = K * G U + V = W F + K = Q W + H = O D + E = N + P V * W = X + Y O + V = N M + I = Y Any help please ??? === Subject: Re: Can anyone solve this ??? > Each different letter represents an integer from 1 to 26 - all you > need to do is work out which letter corresponds to which number. > Hint: the value for the letter Z can be determined by elimination. > A + B = D > B * K = C > F * U = D > F * K = L > H + I = J > Q * Q = R > R + S = T > T = K * G > U + V = W > F + K = Q > W + H = O > D + E = N + P > V * W = X + Y > O + V = N > M + I = Y > Any help please ??? Your directions should read represents a DIFFERENT integer. Note that F+K=Q, so Q is at least 3. Since R=Q^2, there are only 3 possibilites for R, 9, 16 or 25, which means Q= 3, 4 or 5. This severely limits the values of F and K. So that should get you started. === Subject: problem - help ! Each different letter represents an integer from 1 to 26 - all you need to do is work out which letter corresponds to which number. Hint: the value for the letter Z can be determined by elimination. A + B = D B * K = C F * U = D F * K = L H + I = J Q * Q = R R + S = T T = K * G U + V = W F + K = Q W + H = O D + E = N + P V * W = X + Y O + V = N M + I = Y Any help please ??? === Subject: Re: problem - help ! > Each different letter represents an integer from 1 to 26 - all you > need to do is work out which letter corresponds to which number. > Hint: the value for the letter Z can be determined by elimination. > A + B = D > B * K = C > F * U = D > F * K = L > H + I = J > Q * Q = R > R + S = T > T = K * G > U + V = W > F + K = Q > W + H = O > D + E = N + P > V * W = X + Y > O + V = N > M + I = Y > Any help please ??? number. That makes it easy... In the multiplication F*K=L, F and K must be greater than one, so that L doesn¹t equal K nor F. And of course F and K are different. Therefore F+K=Q means that Q is at least 5. Since Q*Q=R, R is at least 25, and the next possibility would be 6*6=36. Therefore Q=5 and R=25. Since R+S=T, T=26 and S=1. Since T=K*G, either K=2 and G=13, or K=13 and G=2. But F+K=Q, so K<5. Therefore K=2 and G=13; and F=3. And F*K=L, so L=6. So far: 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 S K F Q L G R T Since U+V=W, W is at least 4+7=11. Now X+Y is at most 23+24=47. V*W=X+Y and W>=11, therefore V<5. So V=4; W<12 so W=11; and U=7. And since F*U=D, D=21. So far: 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 S K F V Q L U W G D R T Again, V*W=X+Y, but now we know V*W=44. X and Y can¹t both be 22; neither can they be 21+23. They must be 20 and 24. (But we don¹t yet know which is which.) H>=8, so N>=23. But N can¹t be 24; either X or Y has that. So N=23 and H=8. Also W+H=O=19. So far: 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 S K F V Q L U H W G O * D N * R T * means X or Y Since B>=9 and B*K=C, we deduce that C is even and at least 18; only 18 and 22 are possible. But B isn¹t 11, so B=9 and C=18. So far: 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 S K F V Q L U H B W A G C O * D N * R T * means X or Y Since H+I=J, 8+I=J. Only two spaces are eight apart: I=14, J=22. Since M+I=Y, either Y=20 and M=6 or else Y=24 and M=10. It¹s the latter, and X=20. So far: 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 S K F V Q L U H B M W A G I C O X D J N Y R T Finally, D+E=N+P, so 21+E=23+P and E=P+2. They Þt only into P=15, E=17. And the last free number gives Z=16. -- Don Reble djr@nk.ca === Subject: Re: problem - help ! > Each different letter represents an integer from 1 to 26 - all you > need to do is work out which letter corresponds to which number. > Hint: the value for the letter Z can be determined by elimination. > A + B = D > B * K = C > F * U = D > F * K = L > H + I = J > Q * Q = R This looks like a good place to start. R must be a square; only the values 1, 4, 9, 16, and 25 are possible. Furthermore, Q <= 5. > R + S = T > T = K * G > U + V = W > F + K = Q Then this further limits F and K: F <= 4 and K <= 4. > W + H = O > D + E = N + P > V * W = X + Y > O + V = N > M + I = Y > Any help please ??? -Michael. === Subject: Re: problem - help ! <40fe3855$0$187$edfadb0f@dread11.news.tele.dk> === Subject: Can anyone solve this ??? >Each different letter represents an integer from 1 to 26 - all you >need to do is work out which letter corresponds to which number. >Hint: the value for the letter Z can be determined by elimination. Then why didn¹t you say so? That each number is uniquely represented. Q * Q = R q /= 1; r = 4, 9, 16, 25; 2 <= q <= 5 F * K = L f,k /= 1; 2 <= f,k F + K = Q q = 5; r = 25; 2 <= f,k <= 3; l = 6 R + S = T s = 1; t = 26 T = K * G k = 3; g = 13; f = 2 1, 2, 3, 5, 6, 13, 25, 26 s, f, k, q, l, g, r, t B * K = C b = 4, 7, 8; c = 12, 21, 24 F * U = D u = 4, 7, 8, 9, 10, 11, 12; d = 8, 14, 16, 18, 20, 22, 24 A + B = D d /= 8; 7 <= u <= 12 H + I = J U + V = W W + H = O D + E = N + P V * W = X + Y O + V = N M + I = Y >Any help please ??? ---- === Subject: Re: Psychohistory Was Hari Seldon pulling our leg?--> Can one person signiÞcantly affect the course of history or not? > [Please do not mail me a copy of your followup] > tftn@earthlink.net spake the secret code > <40FBF227.9060306@netscape.net> thusly: >This gives a reference to the Foundation novels that >are the basis of the psychohistory idea. > Right. This isn¹t sci.fractals.science-Þction. There are plenty of > science Þction newsgroups in which you could discuss psychohistory. Oh, I did look it up. As far as I can tell, psychohistory is things like The Universality of Incest. Basically it seems to be an attempt to revive psychoanalysis by claiming that those Venus statues are actually penetration devices. === Subject: Re: Psychohistory Was Hari Seldon pulling our leg?--> Can one person signiÞcantly affect the course of history or not? >> [Please do not mail me a copy of your followup] >> tftn@earthlink.net spake the secret code >> <40FBF227.9060306@netscape.net> thusly: >>This gives a reference to the Foundation novels that >>are the basis of the psychohistory idea. >> Right. This isn¹t sci.fractals.science-Þction. There are plenty of >> science Þction newsgroups in which you could discuss psychohistory. >Oh, I did look it up. As far as I can tell, psychohistory is things >like The Universality of Incest. Basically it seems to be an attempt >to revive psychoanalysis by claiming that those Venus statues are >actually penetration devices. Well I suppose that that¹s interesting, but it¹s not really relevant, since that¹s deÞnitely not the sense in which the word is being used here. ************************ === Subject: Re: I am playing > I am trying to learn about how to create pages. I am having a lot of trouble. > It will not let me put a picture on a page. This is a test. > http://hometown.aol.com/kurtstocklmeir/myhomepage/ personal.html try alt.test? === Subject: Re: I am playing > they do not help me. You¹re welcome. As someone else suggested, you might try removing everything but tag. > I thought I would keep aol for 1 more month to learn about making > pages. They let people have free pages. I can not get a picture on > the page. The method I used was to use aol picture center - they let > me put a picture there - add a link to it on my page. um... it occurs to me to ask how you are uploading your pictures into AOL¹s webspace. Every ISP that I¹ve ever heard of gives you free webspace. All you have to do is make a text Þle in Notepad, and upload it (along with your images) to your little corner of their webspace. > ... To me it looks like the picture got worse when the picture > on my computer went to aol. I¹m confused now. It looks like you¹re saying that you can see your pictures after you send them AOL¹s computer, but the quality isn¹t that good? -- CodeCutter - good, fast and cheap; pick two. === Subject: Re: I am playing >>get rid of most of the html code.. >>to insert an image just ust the >src=http://www.domain.com/picture.jpg> > I tried that just now, didn¹t work. Shows a funny > little icon instead of the picture. (What¹s > picture.jpg a picture _of_, anyway?) A funny little icon? -- CodeCutter - good, fast and cheap; pick two. === Subject: Re: I am playing >get rid of most of the html code.. >to insert an image just ust the src=http://www.domain.com/picture.jpg> >> I tried that just now, didn¹t work. Shows a funny >> little icon instead of the picture. (What¹s >> picture.jpg a picture _of_, anyway?) >A funny little icon? Not ha-hah funny, just the funny little icon Netscape always shows when it Þnds an img tag but can¹t Þnd (or can¹t decode) the src. ************************ David C. Ullrich === Subject: Re: I am playing > (What¹s picture.jpg a picture > _of_, anyway?) >>A funny little icon? > Not ha-hah funny, just the funny little icon Netscape > always shows when it Þnds an img tag but can¹t Þnd > (or can¹t decode) the src. Sorry, another bad joke. I was suggesting that that was the answer to your question. -- CodeCutter - good, fast and cheap; pick two. === Subject: Re: I am playing >> (What¹s picture.jpg a picture >> _of_, anyway?) >A funny little icon? >> Not ha-hah funny, just the funny little icon Netscape >> always shows when it Þnds an img tag but can¹t Þnd >> (or can¹t decode) the src. >Sorry, another bad joke. I was suggesting that that was the answer to your >question. Duh. I hadn¹t considered that possibility. ************************ David C. Ullrich === Subject: Re: I want to study on the theory of set > Can anybody provide me with some resouce about this Þeld? === Subject: Re: What is Forced (accelerated) Motion? > MorituriMax skrev i en meddelelse >>So I guess the acceleration of the earth around the sun, the > acceleration of >>the >>earth/sun around the galaxy, the acceleration of the earth/sun through > the >>local >>group, all those magically don¹t count? There are others as well but > those >>were >>just to open your eyes. >Open _you¹re_ eyes Mori. Those are all Free (inertial) Motions! >>your, and no. > Isn¹t it incredible that some people can¹t or won¹t see that any curved > motion is accelerated. It is actually quite simple. If you want to change > changing constantly) then you need to apply a force, and once you apply a > force you get acceleration. The answer is quite simple - put them in a pilot training centrifuge, and spin it up until they¹re feeling 5G. Would love to see them trying to the form the words to deny acceleration when their lips are plastered to their forehead! === Subject: Re: What is Forced (accelerated) Motion? > MorituriMax skrev i en meddelelse >> >>So I guess the acceleration of the earth around the sun, the > acceleration of >>the >> >>earth/sun around the galaxy, the acceleration of the earth/sun through > the >>local >> >>group, all those magically don¹t count? There are others as well but > those >>were >> >>just to open your eyes. > >Open _you¹re_ eyes Mori. Those are all Free (inertial) Motions! >> >>your, and no. >> > Isn¹t it incredible that some people can¹t or won¹t see that any curved > motion is accelerated. It is actually quite simple. If you want to change is > changing constantly) then you need to apply a force, and once you apply a > force you get acceleration. > The answer is quite simple - put them in a pilot training centrifuge, and spin > it up until they¹re feeling 5G. Would love to see them trying to the form the > words to deny acceleration when their lips are plastered to their forehead! Perhaps they would have more luck if we told them keep their head and eyes moving while they were zipping around at 5 G. They could explain it more carefully while they were cleaning out the centrifuge. Michael === Subject: Re: What is Forced (accelerated) Motion? } Mathematical deÞnition of acceleration IS real world So, tell me, How much is a dot on the edge of a 1 foot in diameter wheel spinning at a constant 1000 revs per minute, accelrating? and when will such acceleration create a higher rev speed? === Subject: Re: What is Forced (accelerated) Motion? The acceleration would be the angular velocity, squared, divided by the radius, or 6.28 ft * 100 radians * 100 radians = 62800ft/sec^2 towards the center of the disk. This force is transmitted to the dot by the disk. Should the dot leave the disk, it would leave in a straight line at Assuming that there is no friction or added energy, the angular acceleration is zero. Now, should your dot actually weigh something signiÞcant, then if it is brought to the center of the circle, the disk will speed up, conserving momentum. Michael > } Mathematical deÞnition of acceleration IS real world > So, tell me, > How much is a dot on the edge of a 1 foot in diameter wheel > spinning at a constant 1000 revs per minute, accelrating? > and when will such acceleration create a higher rev speed? === Subject: Re: What is Forced (accelerated) Motion? } The acceleration would be the angular velocity, squared, divided by the } radius, or 6.28 ft * 100 radians * 100 radians = 62800ft/sec^2 towards the } center of the disk. Why do use the term towards the center of the disk? It is spinning constantly around it and never heading towards it, and also it does not accelerate at all. So why bother saying it is accelerating at all? It isn¹t. It is in a constant circular motion, no acceleration of it¹s speed is occuring at all, or the rpm would be rising along with the travel speed around the disk. I don¹t care about moving it closer or further. It is not accelerating. Or are you saying accelerating things can not change speed they are moving at? === Subject: Re: What is Forced (accelerated) Motion? > } The acceleration would be the angular velocity, squared, divided by the > } radius, or 6.28 ft * 100 radians * 100 radians = 62800ft/sec^2 towards the > } center of the disk. > Why do use the term towards the center of the disk? > It is spinning constantly around it and never heading towards it, > and also it does not accelerate at all. > So why bother saying it is accelerating at all? > It isn¹t. > It is in a constant circular motion, > no acceleration of it¹s speed is occuring at all, > or the rpm would be rising along with the travel speed around > the disk. You misunderstand the use of acceleration here. What is being said is that all points on the disk (except the central point) are in acceleration when the disk is spinning, since they are changing their velocity constantly. Any point has constant *speed*, but not velocity (which has a direction). And a change in velocity is acceleration. alex === Subject: Re: What is Forced (accelerated) Motion? } You misunderstand the use of acceleration here. What is being said is } that all points on the disk (except the central point) are in } acceleration when the disk is spinning, since they are changing their } velocity constantly. Any point has constant *speed*, but not velocity } (which has a direction). And a change in velocity is acceleration. That is where you are losing the original real world meaning of acceleration (speeding up) A change in direction does not automaticall infer a change in velocity, for if it did, the disk would be speeding up it¹s speed according to the old fashion meaning of acceleration. The mathematical world deÞnition is very bad, If I asked to to accelerate the point on the disk, Would you speed up the disk, or tell me it is already accelerating, constantly? === Subject: Re: What is Forced (accelerated) Motion? > } You misunderstand the use of acceleration here. What is being said is > } that all points on the disk (except the central point) are in > } acceleration when the disk is spinning, since they are changing their > } velocity constantly. Any point has constant *speed*, but not velocity > } (which has a direction). And a change in velocity is acceleration. > That is where you are losing the original real world meaning > of acceleration (speeding up) The Œreal world¹ deÞnition is imprecise, as someone has pointed out to you. The fact that some people, yourself included, incorrectly use the term Œacceleration¹ doesn¹t negate it¹s true meaning or deÞnition. > A change in direction does not automaticall infer a change in velocity, Yes, it does. Bang head against wall. Wash, rinse, repeat. > for if it did, the disk would be speeding up it¹s speed according to > the old fashion meaning of acceleration. You are in sci.math, Þlled with mathematicians, arguing about what the word acceleration really means. You are being universally told that you are wrong, but you can¹t seem to grasp that, and you¹re insisting on foisting an inaccurate caricature of a well deÞned concept on us. This is your problem, not ours. If you don¹t like the real deÞnition of acceleration, I suggest you write to your MP or member of congress or whatever (so they can have a giggle too). > The mathematical world deÞnition is very bad, > If I asked to to accelerate the point on the disk, > Would you speed up the disk, or tell me it is already accelerating, > constantly? Most well informed people would say how do you want it accelerated? by the way, it¹s already being accelerated (and being accelerated by a non-constant vector at that) Now please accelerate in such a way that you maximise your displacement from me. Hey, use a spinning disk, knock yourself out. === Subject: Re: What is Forced (accelerated) Motion? > } You misunderstand the use of acceleration here. What is being said is > } that all points on the disk (except the central point) are in > } acceleration when the disk is spinning, since they are changing their > } velocity constantly. Any point has constant *speed*, but not velocity > } (which has a direction). And a change in velocity is acceleration. > That is where you are losing the original real world meaning > of acceleration (speeding up) > A change in direction does not automaticall infer a change in velocity, > for if it did, the disk would be speeding up it¹s speed according to > the old fashion meaning of acceleration. No, we aren¹t confusing the issue. We are disentangling it very precisely. Your old fashioned meaning is very imprecise and could mean several different things, even if you think you mean only one or two. We are segregating out each of them to allow them to be studied independently before putting them back together to form a whole picture. In that picture we have all the different effects of going around a circle. This includes the angular velocity, the tangential velocity, and the centrifugal and centripedal forces. To top it off, you can go ahead and measure these values and discover how correct physics really is. How well does your old fashioned terminology handle that. > The mathematical world deÞnition is very bad, > If I asked to to accelerate the point on the disk, > Would you speed up the disk, or tell me it is already accelerating, > constantly? The disk is traveling at 1000 rpm, that is a rotational velocity. At any instant, a point 1 foot from the center of the disk would be traveling at 6280 feet per second in the direction of its tangent. If the point is to stay on the disk, it must change direction. A change of direction requires acceleration in that direction. It gets a little easier if it is broken down into vectors. If the disk is horizontal and spinning clockwise, then at some moment the point is traveling north. Its velocity could be considered to be 6280 feet/second N, 0 feet/second East. After an eigth of a turn, it is now traveling 4440 feet/second N, 4440 feet/second East. At the moment that it was heading due north, it was accelerating eastward towards the center of the disk. An instant later, it was accelerating both eastward and southward (or negative northward). The cumulative effect in an eigth of a turn was to reduce the North component and increase the East component. As it was heading northeast at a speed of 6280 feet/second, it was being pulled in the negative north and postiive east directions, again with the same magnitude of force as before. This further diminished the North component, and increased the East component. By the time the point was at the north of the disk, it would be heading east, with a velocity of 0 f/s North, 6280 f/s East. If it fell off the disk at this point, it would travel at 6280 f/sec in an easterly direction. Michael === Subject: Re: What is Forced (accelerated) Motion? } No, we aren¹t confusing the issue. We are disentangling it very precisely. } Your old fashioned meaning is very imprecise and could mean several } different things, even if you think you mean only one or two. Oh please, that is the most sad ass freakin thought about such. The old way only means one thing (moving faster) the old way also had a term for moving slower (deceleration) It is your way that is all screwed up. I asked you a simple question, ((If I asked to to accelerate the point on the disk, Would you speed up the disk, or tell me it is already accelerating, constantly?)) you have 2 answers. answer 1: Ok (and you increase the speed of the disc) answer 2: The spot on the disc is already accelerating. pick one will ya? I don¹t need all your extra crap added. I just need to know if you would speed up the disc or not, so I would know to trust you in real life situations or not. === Subject: Re: What is Forced (accelerated) Motion? > } No, we aren¹t confusing the issue. We are disentangling it very precisely. > } Your old fashioned meaning is very imprecise and could mean several > } different things, even if you think you mean only one or two. > Oh please, > that is the most sad ass freakin thought about such. > The old way only means one thing (moving faster) > the old way also had a term for moving slower (deceleration) > It is your way that is all screwed up. > I asked you a simple question, > ((If I asked to to accelerate the point on the disk, > Would you speed up the disk, or tell me it is already accelerating, > constantly?)) > you have 2 answers. > answer 1: Ok (and you increase the speed of the disc) > answer 2: The spot on the disc is already accelerating. > pick one will ya? > I don¹t need all your extra crap added. > I just need to know if you would speed up the disc > or not, so I would know to trust you in real life situations > or not. So sorry. The disk is moving at a constant angular velocity. Each point on the disk is constantly acceleration towards the center in order to maintain its circular path at that constant angular velocity. If you accelerate a point on the disk in any direction but towards the center or directly away from the center then it will change the angular velocity of the disk, or in your terms, accelerate the disk. Perhaps the confusion for you is that this thread started as someone trying to refute centripetal acceleration of planets and such. In order to maintain the proper precision, you were getting the full answer, rather than the incomplete one you were asking for. The disk portion of the thread was put in to counter the idea of a weight on a string. In the string case the centripetal acceleration was very apparent, which tended to make someone who didn¹t believe in that kind of thing very uncomfortable. Michael === Subject: Re: What is Forced (accelerated) Motion? oops, 6.28^2 * 100^2 = a bit over 360000 ft/sec^2 Michael > The acceleration would be the angular velocity, squared, divided by the > radius, or 6.28 ft * 100 radians * 100 radians = 62800ft/sec^2 towards the > center of the disk. This force is transmitted to the dot by the disk. > Should the dot leave the disk, it would leave in a straight line at Assuming > that there is no friction or added energy, the angular acceleration is zero. > Now, should your dot actually weigh something signiÞcant, then if it is > brought to the center of the circle, the disk will speed up, conserving > momentum. > Michael > } Mathematical deÞnition of acceleration IS real world > So, tell me, > How much is a dot on the edge of a 1 foot in diameter wheel > spinning at a constant 1000 revs per minute, accelrating? > and when will such acceleration create a higher rev speed? === Subject: Re: What is Forced (accelerated) Motion? } velocity is a vector, it requires speed and direction. Speed itself doesn¹t } care about direction. Sorry, You have been duped If speed does not care about direction, You could easily do 1000 mph on a 90 degree corner. You should wake up before the brainwashing sets in to far. Velocity and Speed are the same beasts. === Subject: Re: What is Forced (accelerated) Motion? > } velocity is a vector, it requires speed and direction. Speed itself doesn¹t > } care about direction. > Sorry, > You have been duped > If speed does not care about direction, > You could easily do 1000 mph on a 90 degree corner. > You should wake up before the brainwashing sets in to far. > Velocity and Speed are the same beasts. (after my reply I¹ve just posted to this thread elsewhere) Oooh, I get it now... willful ignorance huh? They are not the same. === Subject: Re: What is Forced (accelerated) Motion? } (after my reply I¹ve just posted to this thread elsewhere) } Oooh, I get it now... willful ignorance huh? } They are not the same. Actually , You are the mathematical ignorance holder, You have lost the original meanings for your new meanings. If you physically accelerate that spot, the disc will turn faster. Sheesh, Never learned old school at all huh? === Subject: Re: What is Forced (accelerated) Motion? > } (after my reply I¹ve just posted to this thread elsewhere) > } Oooh, I get it now... willful ignorance huh? > } They are not the same. > Actually , > You are the mathematical ignorance holder, > You have lost the original meanings for your new meanings. > If you physically accelerate that spot, > the disc will turn faster. > Sheesh, > Never learned old school at all huh? Only if you accelerate the spot in the direction it is already going, and it transmits that to the disk. In that case, you will cause an angular acceration. We have been referring to a centripedal acceleration, which is why the spot circles at a constant radius rather than þying off. There are a couple kinds of acceleration. Michael === Subject: Re: What is Forced (accelerated) Motion? > } velocity is a vector, it requires speed and direction. Speed itself doesn¹t > } care about direction. > Sorry, > You have been duped > If speed does not care about direction, > You could easily do 1000 mph on a 90 degree corner. > You should wake up before the brainwashing sets in to far. > Velocity and Speed are the same beasts. Wrong again space! === Subject: Top math and science students in the US The children of immigrants are becoming the top math and science students in the United States, dominating academic competitions and representing the strongest hope the nation has of keeping an edge in high-tech and biomedical Þelds, according to a study released Monday. see What does this say about the motivation and preparation of those who grew up in the US. I think this says something about the long term effects of making fun of academic achievement and giving education such a low status and priority in the US. Our politicians are guilty of this too. Van === Subject: Re: Top math and science students in the US > The children of immigrants are becoming the top math and science > students in the United States, dominating academic competitions and > representing the strongest hope the nation has of keeping an edge in > high-tech and biomedical Þelds, according to a study released Monday. > see > > What does this say about the motivation and preparation of those who > grew up in the US. > I think this says something about the long term effects of > making fun of academic achievement and giving education such a low > status and priority in the US. Our politicians are guilty of this too. It should not come as a surprise, that children of immigrants are more motivated towards math and science in order to establish themselves in the U.S. society than children of well established (at least in terms of citizenship) families that have other means of moving up within the society. (In short: if you are not rich , at least try to be smart :-) Even more so, because these are Þelds where they can utilise the strong primary education of their former countries much better than in e.g. English literature. It also tells us, that immigration is an advantage for an industrialised society, a lesson that a lot of other countries still need to learn. Marc === Subject: Re: Open access to technical journals >> The 118-page report stops short of >> fully endorsing the open access publishing movement, where authors are >> charged for their research to be made freely available to everyone on >> the web, but strongly supports further experimentation with this new >> business model. >Do you want free publication on the Web? Put up a Web page. > >See? Wildly obscure scholary paper at your Þngertips. >Can¹t you see somethimg terribly wrong with a business model that >says, the more people who look at your work, the more we charge >you to post it? It would be like conÞscating the wages of >productive folk and awarding them to bastards too lazy to work, >The more you earn the more we take. In fact, the more you earn >the *greater fraction* of what you earn we take. >What sort of idiot would buy into that system? Wait a minute... Out of the mouth of babes.... You should hear about the slavery being requested on this side of the States. /BAH Subtract a hundred and four for e-mail. === Subject: Re: Raatikainen¹s Complexity Complex > If you remember my argument, it says that your conclusion is wrong. > FAS_A1 is not necessarily subsumed in FAS_A2. First, let us invoke my > above argument. We cannot Þx XF to some ancient logical inference > procedure. (We don¹t have to) As I said there are more intelligent, or > if you will prefer, simply different inference procedures. > Why should these different inference procedures be relevant at all? > After all, you can generate the theorems of any theory T from an > arbitrary axiom or a set of axioms by a suitable different inference > procedure. > If you want to speak about some mathematically intelligent program > that extracts mathematical information from strings, that¹s Þne. But it > has nothing to do with the strength of the theories those strings happen > to encode the axiomatisations of. The reason we¹re interested in these > theories is that they formalise certain truths of, say, arithmetic, not > because some axiomatisation or another considered as a bit string > contains some interesting information, what ever that means. > The importance of the ancient rules of inference you refer to is that > they correctly capture Þrst order logical entailment relation. This is > why they are relevant to determining the strength of a theory, rather > than some arbitrarily chosen different inference procedure. I will pen a more detailed response in due time. Meanwhile, I think a preliminary answer can be that the nature is not necessarily rational the way logicist philosophy has deÞned it (Incidentally, I think this is one of the places where logical positivism has failed, a branch in the school of thought known as positivism) This now seems to me a remarkable aspect of digital philosophy (postmodernism with style) in contrast to logical positivism (outdated fashion of modernism). -- Eray Ozkural === Subject: Re: Raatikainen¹s Complexity Complex > I will pen a more detailed response in due time. Meanwhile, I think a > preliminary answer can be that the nature is not necessarily > rational the way logicist philosophy has deÞned it (Incidentally, I > think this is one of the places where logical positivism has failed, a > branch in the school of thought known as positivism) > This now seems to me a remarkable aspect of digital philosophy > (postmodernism with style) in contrast to logical positivism (outdated > fashion of modernism). Your comments do indeed rival those of Lacan, Kristeva, Baudrillard, and other postmodernist luminaries. === Subject: Re: Raatikainen¹s Complexity Complex > I will pen a more detailed response in due time. Meanwhile, I think a > preliminary answer can be that the nature is not necessarily > rational the way logicist philosophy has deÞned it (Incidentally, I > think this is one of the places where logical positivism has failed, a > branch in the school of thought known as positivism) > > This now seems to me a remarkable aspect of digital philosophy > (postmodernism with style) in contrast to logical positivism (outdated > fashion of modernism). > Your comments do indeed rival those of Lacan, Kristeva, Baudrillard, > and other postmodernist luminaries. I am damned! Now, I have to quit computer science and go to a liberal arts college. :¹( Cordially, -- Eray Ozkural === Subject: Re: Raatikainen¹s Complexity Complex Originator: mtx014@linux.services.coventry.ac.uk (Robert Low) >> This now seems to me a remarkable aspect of digital philosophy >> (postmodernism with style) in contrast to logical positivism (outdated >> fashion of modernism). > Your comments do indeed rival those of Lacan, Kristeva, Baudrillard, >and other postmodernist luminaries. That¹s a little cruel, isn¹t it? -- Rob. http://www.mis.coventry.ac.uk/~mtx014/ === Subject: Re: Handheld computer with math software The HP Palmtop Paper, 1999, 8(6), Issue 48, reviewed: 8. Symbolic Math SymbMath is a computer algebra system that can perform exact numeric, symbolic and graphic computation. It manipulates complicated formulas and returns answers in terms of symbols, formulas, exact numbers, table and graph. SymbMath is also an expert system that is able to learn from user¹s input. If the user only inputs one formula without writing any code, it will automatically learn many problems related to this formula (e.g. it learns many integrals involving an unknown function f(x) from one derivative f¹(x) ). SymbMath is, in another sense, a programming language in which you can deÞne conditional, case, piecewise, recursion, multi-value functions and procedures, derivatives, integrals and rules. The program runs on the HP Palmtop and does almost everything that Derive can do. Symbolic Math comes in one of three versions: Shareware, Student, and Advanced. Its three versions are available from the author, Dr. Weiguang Huang. www.SymbMath.com > Hello! > Does anyone know if MATLAB, Maple, Mathematica or any such program (are > there others?) are available for any handheld computer like Palm? > Calculators like TI-89 can not compete. Too slow processor, too little > display, too bad functionality. > I¹d be happy if anyone suggested a good combination of software and handheld > computer, since I really don¹t have a clue what might work. === Subject: Re: Top math and science students in the US Give us your tired, your poor . . . your scientists and your mathematicians. > The children of immigrants are becoming the top math and science > students in the United States, dominating academic competitions and > representing the strongest hope the nation has of keeping an edge in > high-tech and biomedical Þelds, according to a study released Monday. > see > What does this say about the motivation and preparation of those who > grew > up in the US. I think this says something about the long term effects > of > making fun of academic achievement and giving education such a low > status > and priority in the US. Our politicians are guilty of this too. > Van === Subject: Re: Top math and science students in the US >Give us your tired, your poor . . . your scientists and your >mathematicians. You are missing the point. Those kids were taught how to work. That means thinking and moving body parts at the same time for the purpose of producing something. /BAH Subtract a hundred and four for e-mail. === Subject: Linear algebra Do A*B and B*A have the same eigenvalues (and same multiplicity) when both A and B are singular. (or else inv(A)(A*B)*A would be similar to B*A) === Subject: Re: Linear algebra > Do A*B and B*A have the same eigenvalues (and same multiplicity) > when both A and B are singular. (or else inv(A)(A*B)*A would be similar > to B*A) If A and B are square matrices, then yes, AB and BA have the same characteristic polynomials. One can see this by noting that (tI + A)B and B(tI + A) are conjugate for all but Þnitely many t and letting t -> 0 (in the usual topology for R or C or in the Zariski topology for any algenraically closed Þeld :-)). -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: Linear algebra >> Do A*B and B*A have the same eigenvalues (and same multiplicity) >> when both A and B are singular. (or else inv(A)(A*B)*A would be similar >> to B*A) >If A and B are square matrices, then yes, AB and BA have the same >characteristic polynomials. One can see this by noting that >(tI + A)B and B(tI + A) are conjugate for all but Þnitely many t >and letting t -> 0 (in the usual topology for R or C or in the >Zariski topology for any algenraically closed Þeld :-)). Thus the eigenvalues have the same algebraic multiplicity, but they don¹t necessarily have the same geometric multiplicity. E.g. for [ 0 1 ] [ 0 0 ] A = [ 0 0 ], B = [ 0 1 ] the eigenspace for eigenvalue 0 has dimension 1 for AB but 2 for BA. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Linear algebra >> Do A*B and B*A have the same eigenvalues (and same multiplicity) >> when both A and B are singular. (or else inv(A)(A*B)*A would be similar >> to B*A) >If A and B are square matrices, then yes, AB and BA have the same >characteristic polynomials. One can see this by noting that >(tI + A)B and B(tI + A) are conjugate for all but Þnitely many t >and letting t -> 0 (in the usual topology for R or C or in the >Zariski topology for any algenraically closed Þeld :-)). One does not even need square, but of course the number of zeros of the characteristic equation may not be equal. The easy way is to show that |I - t*A*B| = |I - t*B*A| by evaluating the determinant of I t*A B I by using row transformations to make one of the off-diagonal parts 0. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Rational bases??? That exists? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6LBoTJ16878; Joaquim Nogueira === Subject: Re: Automorphisms of complex numbers > This follows the disscusion Does 1/oo=0, and the fact that the > reader will try to interpret this notation, in order to answer the > question. I ask a new question, for discussion: > What is the cardinality of the automorphism group of the complex > numbers, C? > I say 2^c, where c:=|C| have access to that: Automorphisms of the Complex Numbers Paul B. Yale Mathematics Magazine, Vol. 39, No. 3 (May, 1966) , pp. 135-141 Short story: 2^(2^aleph_0), needs Zorn¹s Lemma for the proof. Jon === Subject: Re: Automorphisms of complex numbers > What is the cardinality of the automorphism group of the complex > numbers, C? > I say 2^c, where c:=|C| I agree. I think the degree of transcendence of C over the rationals must be c. For suppose c_i is a maximal set of algebraically independent numbers, with cardinality b. Then k = Q(c_i) will also have cardinality b, and so will C which is algebraic over k. But then any permutation of the c_i will deÞne an automorphims of k, which will extend to an automorphism of C. The result follows, since the number of permutations of C (or of a set of cardinality c) is 2^c. I think ... -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Automorphisms of complex numbers > What is the cardinality of the automorphism group of the complex > numbers, C? > > I say 2^c, where c:=|C| > I agree. > I think the degree of transcendence of C over the rationals must be c. > For suppose c_i is a maximal set of algebraically independent numbers, > with cardinality b. > Then k = Q(c_i) will also have cardinality b, > and so will C which is algebraic over k. > But then any permutation of the c_i will deÞne an automorphims of k, > which will extend to an automorphism of C. > The result follows, > since the number of permutations of C (or of a set of cardinality c) > is 2^c. > I think ... Yes, of course this is the automorphism group of the pure Þeld C. Of course one can think of C as a vector space over R, or a Þeld extension of R. It is not a particularly strange view that would conclude there were only two automorphisms. That isn¹t the way I see things though. === Subject: Re: L-series of elliptic curves > Could anyone lead me in the direction of a website which gives > examples of an L-series of an elliptic curve (or even has a calculator > of the series of a_n). I would like to see what they look like for > some concrete examples without having to go through the trouble of > calculating them myself. > Craig You could try Milne¹s online book about elliptic curves : http://www.jmilne.org/math/CourseNotes/math679.html An example is given page 116 for an older version of the great pari/gp calculator available here : http://pari.math.u-bordeaux.fr/download.html (try [? ellinit] and [? ellseries] for the new syntax) Hoping it helped, Raymond === Subject: Re: L-series of elliptic curves > Could anyone lead me in the direction of a website which gives > examples of an L-series of an elliptic curve (or even has a calculator > of the series of a_n). I would like to see what they look like for > some concrete examples without having to go through the trouble of > calculating them myself. > > Craig > You could try Milne¹s online book about elliptic curves : > http://www.jmilne.org/math/CourseNotes/math679.html They are very high quality. > An example is given page 116 for an older version of the great > pari/gp calculator available here : > http://pari.math.u-bordeaux.fr/download.html > (try [? ellinit] and [? ellseries] for the new syntax) Without going through the trouble of downloading the calculator and Þguring out its syntax, does anyone have any charts or examples of an L-series of an elliptic curve? The literature that I have read doesn¹t seem to do this; it seems to only talk about the theory behind it. Craig === Subject: Re: Surrogate factoring, reasons for my concerns > > OMG SHUT UP SHUT UP SHUT UP! I can¹t believe you¹re not shutting up! > Tom It seems you are losing it. But why? Now I pointed out that I¹ve written up my surrogate factoring ideas and sent them to a math journal. Go away. There¹s no new news. There¹s nothing meaningful to do but wait on the journal. I can¹t believe you¹re still reading posts in this thread. Why bother? There are quite a few threads that I¹m COMPLETELY ignoring. There¹s something wrong with you. It¹s USENET. I wonder how some of you have survived on Usenet this long. Like, hello, on Usenet people can post, and keep posting even if you Þnd that it is driving you bonkers, so people who are driven bonkers by other people posting should probably stay away from Usenet. You see, people post on Usenet whether you wish them to or not. Or didn¹t you know that little tidbit? James Harris === Subject: Re: Surrogate factoring, reasons for my concerns <40F6487F.CAFC3FA3@fel.tno.nl> <40F6583D.2097CC75@ellipsa.no.sp.am.net> Discussion, linux) > You see, people post on Usenet whether you wish them to or not. > Or didn¹t you know that little tidbit? I have to say this little tidbit is surprising coming from you. -- At the Microsoft-sponsored cocktail reception in the Galaxy Ballroom that evening, Robert Dees urges us Œto network on behalf of the people of Iraq,¹ -- Naomi Klein reports on Microsoft¹s efforts to further democracy. === Subject: Re: Surrogate factoring, reasons for my concerns > You see, people post on Usenet whether you wish them to or not. > Or didn¹t you know that little tidbit? > I have to say this little tidbit is surprising coming from you. I¹ve faced extraordinary behavior. Like David Ullrich, clearly willing to rely on his afÞliation with Oklahoma State University, a tax payer supported institution, was also quite willing to make public statements I felt reþected badly upon him and his institution. Now how many players on the Oklahoma State University football team would David Ullrich dare say that he *thought* of a racial slur to call them, but then was talked out of saying it? The guy is a freaking coward. I know you all know that he wouldn¹t dare make the kind of statement he made on Usenet to any of the many players of various ethnicities that are on the Oklahoma State University football team, or even the basketball team. I dare him to defend his statement again in a post, and claim that he would! And, in my case, LOTS of posters have gathered together to try and have some impact on my postings by acting as a group, basically a gang, and while yeah, at times I feel like it¹s just one of those things, other times it pisses me off. It¹s just not fair. I¹m ONE GUY, so why should I have to deal with a gang? It¹s like how freedom of speech is such a great thing, but people have to accept that some people will abuse their freedoms to the detriment of society. So they have to be controlled. One person posting on Usenet--a place where people post--can hardly be faulted for doing what is done on Usenet: posting. Some may not like the content of those posts, and if some person makes certain statements they may in fact represent someone whose speech needs to be controlled. However, annoying some people who are full of themselves and overtaken with their own egos hardly rises to that level. Now I don¹t doubt some of you will continue to defend gang behavior on Usenet, and defend the fairness of a gang of posters doing their best to confront and control ONE PERSON, but hey, you¹re sick. Faulting someone for posting on Usenet, which is a place where people post, is just not rational. It¹s just, well, it¹s just rather weird, especially given the history of Usenet. And yes, I do still feel I have the right to fault a particular person with some established position in society--like a professor at a state university--if they come out and say something stupid. I don¹t fault David Ullrich for posting, but for what he posted. A math professor CAN be held to a higher standard, especially if they¹re at a state university, getting paid by taxpayers. James Harris === Subject: Re: Surrogate factoring, reasons for my concerns > It¹s just not fair. I¹m ONE GUY, so why should I have to deal with a > gang? Because you picked a Þght with one -- recall your hysterical diatribes about *groupthink*, etc. > James Often in error, but never in doubt! Harris -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com === Subject: Re: Surrogate factoring, reasons for my concerns >> >> OMG SHUT UP SHUT UP SHUT UP! I can¹t believe you¹re not shutting up! >> Tom > It seems you are losing it. But why? I haven¹t lost it. That¹s a quote from the Simpsons you uncultured vermin. > Now I pointed out that I¹ve written up my surrogate factoring ideas > and sent them to a math journal. Congrats. I¹ve submitted things to journals before too. The best part is the creative rejection notices you get. DEAR $author, Your paper ``$title¹¹ was rejected because @reason{random() % 10}. We¹re sorry that your paper was not accepted. We hope to see you at $conference this year! $program_chair > Go away. There¹s no new news. There¹s nothing meaningful to do but > wait on the journal. So... shut up, stop replying? You¹re the one dragging this out with Ullrich. If he¹s such an ass as you point out then you don¹t need to reply to him to prove your point. But if you do keep replying then just the opposite becomes true. > I can¹t believe you¹re still reading posts in this thread. > Why bother? Cuz you¹re cross posting to sci.crypt which is a group I read. > There are quite a few threads that I¹m COMPLETELY ignoring. > There¹s something wrong with you. That¹s an understatement if I ever saw one. > It¹s USENET. I wonder how some of you have survived on Usenet this > long. Normally I ignore threads that don¹t interest me either. You on the other hand are fun to play mind games with. But for all you trolls out there [including you Harris] if you wonder why I get replies? and if you replying makes you angry why do you do it? Look at CNN and the idiots who troll I mean anchor there. They say total bull day in and day out. Nobody questions them. The average american will tune to CNN and believe [even if they don¹t admit it] what they see. In usenet [specially sci.* groups] we want just the opposite. When some troll comes around you have to make a tasteful amount of noise to let others know that not everything is honky-dorry. In your case people want to make sure it¹s known [amongst people who read these groups and would be affected by your writing] that you¹re not as credible as you want to let on. David Ullrich I¹m sure, does other things during the day. The 10 minutes he spends reading usenet isn¹t his No.1 priority. You give yourself too much credit if you think they occupy their time thinking about Po widdle James Harris. During my day for instance, I develop various pieces of software, read email, play games, do work for work and go out with friends. I spend about an hour a day on usenet [at the most] reading the threads that pop up. I certainly don¹t sit and stew over what you¹re going to say next because it doesn¹t take much effort to debunk anything you say. Tom === Subject: Re: Surrogate factoring, reasons for my concerns >> >> >> OMG SHUT UP SHUT UP SHUT UP! I can¹t believe you¹re not shutting up! >> >> Tom > > It seems you are losing it. But why? > I haven¹t lost it. That¹s a quote from the Simpsons you uncultured vermin. Oh, so you think that shows that you¹re not losing it? Your reply here is fascinating in what you appear to NOT see when I dare say few others who read your post can possibly miss it. > Now I pointed out that I¹ve written up my surrogate factoring ideas > and sent them to a math journal. > Congrats. I¹ve submitted things to journals before too. The best part is > the creative rejection notices you get. This thread is not about you. It¹s about my work. Notice the title of the thread is, Surrogate factoring, reasons for my concerns, which further reinforces that it is about *my* work, not yours. That is, I don¹t care about creative rejections you get. > DEAR $author, > Your paper ``$title¹¹ was rejected because @reason{random() % 10}. We¹re > sorry that your paper was not accepted. We hope to see you at $conference > this year! > $program_chair > Go away. There¹s no new news. There¹s nothing meaningful to do but > wait on the journal. > So... shut up, stop replying? You¹re the one dragging this out with > Ullrich. If he¹s such an ass as you point out then you don¹t need to reply > to him to prove your point. But if you do keep replying then just the > opposite becomes true. Sounds like you¹re deluded enough to think that you can use reverse psychology to convince me to do what you want, which is wacky since this is Usenet, and trying to control other people¹s posts, when you can *CHOOSE* to ignore them is just, well, it¹s just plain weird. Now you can get creative all you want in your various and futile attempts to control my postings, and it won¹t change the reality that THIS IS USENET. YEAH! Hell yeah! THIS IS USENET. Isn¹t it grand? > I can¹t believe you¹re still reading posts in this thread. > > Why bother? > Cuz you¹re cross posting to sci.crypt which is a group I read. So? > There are quite a few threads that I¹m COMPLETELY ignoring. > > There¹s something wrong with you. > That¹s an understatement if I ever saw one. Um, now you¹re talking about yourself negatively. Can you at least consider the possibility that you¹re losing it? > It¹s USENET. I wonder how some of you have survived on Usenet this > long. > Normally I ignore threads that don¹t interest me either. You on the other > hand are fun to play mind games with. Sounds like you¹re telling yourself what you want to hear. I¹m bored. I talked about some research of mine, and decided that it was time to quit worrying about all the ramiÞcations and just leave it to the experts. That should have been it. However, the thread continues because some posters keep chattering, and I Þnd the behavior somewhat interesting. I still do. Feel free to reply again. > But for all you trolls out there [including you Harris] if you wonder why I > get replies? and if you replying makes you angry why do you do it? > Look at CNN and the idiots who troll I mean anchor there. They say > total bull day in and day out. Nobody questions them.... Now you¹re ranting. Um, you STILL think you¹re playing mindgames with me? Sounds to me like you¹re just losing it. Remember, Usenet is a place where people post. Trying to control the posting behavior of other people is not, well, it¹s not exactly rational given the environment. You can keep banging your head against a wall if you wish. But you will just scramble your own brains. Maybe eventually you¹ll knock yourself out, eh? James Harris === Subject: Re: Surrogate factoring, reasons for my concerns > Oh, so you think that shows that you¹re not losing it? > Your reply here is fascinating in what you appear to NOT see when I > dare say few others who read your post can possibly miss it. So now you¹re a mathematician, cryptographer, lawyer and psychologist? Holy crap, you¹re a living breathing renaissance man. All you have now is to master astrology, history, politics, chemistry, art, music and medicine and we¹ll make chapels in your honour! Don¹t stop now you¹re almost there! Whatever. Keep posting your nonsense and you¹ll keep getting people pointing out the folly of your ways. Tom === Subject: Re: Surrogate factoring, reasons for my concerns > Still he also posts a lot on sci.math and I¹ve lost count how many > times he¹s called me an idiot! Not often enough for you, apparantly. -- CodeCutter - good, fast and cheap; pick two. === Subject: Re: Surrogate factoring, reasons for my concerns there was an added space in the result from the searchengine, I found by re-searching it; corrected, below. > ... typiÞed by a Nazi by the name of Martin Heidegger, > a professor of existentialism, and the spiritual father > of his follower, Jean-Paul Sartre of France ... > www.larouchepub.com/lar/1997/lhl_evil_we_have_tolerated.html - 43k --Give Earth a Trickier Dick Cheeny -- out of ofÞce, after gigayears! http://tarpley.net/bush12.htm http://www.benfranklinbooks.com/ http://members.tripod.com/~american_almanac http://www.wlym.com/pdf/iclc/howthenation.PDF === Subject: Re: Surrogate factoring, reasons for my concerns > there was an added space in the result from the searchengine, > I found by re-searching it; corrected, below. It doesn¹t matter that you¹ve corrected the URL you posted. Jean-Paul Sartre was neither a Nazi, nor a Fascista. J -- __________________________________________ When will Bush be tried for war crimes? Joe Peschel D.O.E. SysWorks http://members.aol.com/jpeschel/index.htm __________________________________________ === Subject: Re: Surrogate factoring, reasons for my concerns >I¹m going to top post here to point out that David Ullrich is one of >the people who posts who is an actual mathematician. He teaches at >Oklahoma State University. >OSU Mathematics Directory: >Teaching Staff >http://www.math.okstate.edu/~rolodex/combined.html I doubt that anyone but you has any idea what your point is here... >Still he also posts a lot on sci.math and I¹ve lost count how many >times he¹s called me an idiot! Really? Could be I¹ve called you an idiot, but I don¹t recall doing so. I do recall many times I¹ve stated that you¹re either an idiot, a troll or a lunatic. >Well, I guess that¹s what a teacher from Oklahoma State University >considers proper as Ullrich has said it, and he is, in fact, a teacher >at Oklahoma State University. Yes, I am in fact a teacher at OSU. Your point must be that it¹s not proper for a teacher to talk to his students that way. No, of course it¹s not. I _don¹t_ talk to my students the way I talk to you. And the reason is _not_ just that it wouldn¹t be right, or that it could get me in trouble, although those are two reasons. In fact I¹ve never had the _desire_ to talk to my students in a manner anything remotely like the way I talk to you, because they don¹t talk to _me_ the way you do. Trust me, if it ever did happen that I had a student who talked to me the way you do, he¹d be gone _long_ before things got to the point where I¹d be talking to him the way I talk to you. >Oh, and notice how he replies below to some of my musings. >Oh, and again, I remind that I¹ve sent off my surrogate factoring work >to a math journal, which is a peer reviewed math journal, by the way, So? >so it doesn¹t matter what is said on Usenet, Giggle: why would anyone have the idea that what¹s said here matters, regardless? I guess it matters to someone who¹s desperate to have his work acknowledged but who¹s utterly incompetent, so he can¹t possibly Þnd any other way to try to obtain that acknowledegment. But it certainly doesn¹t matter to most people... >and actually, there¹s >little reason for people like Ullrich to keep up what they¹re >doing...that is, there¹s little reason that makes sense to typical >adults. News þash: typical adults have no reason to pay any attention to sci.math in the Þrst place. ************************ David C. Ullrich === Subject: Re: Surrogate factoring, reasons for my concerns >I¹m going to top post here to point out that David Ullrich is one of >the people who posts who is an actual mathematician. He teaches at >Oklahoma State University. >Still he also posts a lot on sci.math and I¹ve lost count how many >times he¹s called me an idiot! By my count, only 16 times out of over 2100 replies; which I consider tremendous restraint. > Really? Could be I¹ve called you an idiot, but I don¹t > recall doing so. These are usually in the context of a single word of reply; for example: === |Subject: Re: JSH: More on ring issue | |>>All of this is no joke as I did prove Fermat¹s Last Theorem, so |>>your involvement in obscuring that that truth is of great |>>concern to me, while I feel that you may indeed honestly just |>>not realize what the truth is. | |>Idiot. More often you use expressions of the form when you say , it makes you _sound_ like an idiot (at least 17 times); or if your proof uses undeÞned terms which you refuse to deÞne you¹re making an idiot _of yourself_ or similar (at least 9 times); which I consider as statements about his _posts_, rather than implications that he actually _is_ an idiot. These fall into the category of calling his _arguments_ idiotic; for example: === |Subject: Re: JSH: Frustration, venting, weird group | |> Hint: Your traditional this was all a test thing works better if |> you don¹t include idiotic statements like saying Z[1/2] is a Þeld. I¹d take that as attacking the argument, rather than the arguer. Of course, we must recall that in this case, the pot is at least as guilty as the (putatative) kettle; for who can forget these classic JSH ripostes? off!!! You ing stupid idiot!!! How many times do I have to tell you that your attention is not wanted David Ullrich? Or the more urgent: OFF David Ullrich!!!!!!! off you stupid ing idiot!!!! As far as I can tell, _you¹ve_ only called _him_ a ing idiot once: My god you¹re a ing idiot. It doesn¹t _bug_ people when you talk about evil math society, it leads to gales of laughter. You really haven¹t Þgured that out yet? Wow. > Trust me, if it ever did happen that I had a student > who talked to me the way you do, he¹d be gone _long_ > before things got to the point where I¹d be talking > to him the way I talk to you. Somehow, I can¹t imagine someone repeating the same basic algebra course for 6 years without gaining _some_ knowledge along the way. === Subject: Re: Hermite polynomial continued fraction Search for the Jacobi matrices to their corresponding orthogonal polynomials and thus their continued fractions. It also follows from the 3 (was it 4?) term reccurence relation on the orthogonal polynomials. === Subject: Re: Hermite polynomial continued fraction I forgot to add that a good reference is N. Akhiezer, The Classical Moment Problem, Edinburg, Oliver and Boyd, 1965. > Search for the Jacobi matrices to their corresponding orthogonal > polynomials and thus their continued fractions. It also follows from > the 3 (was it 4?) term reccurence relation on the orthogonal > polynomials. === Subject: question for writing.. hello......doctor~ Actually, you have a sequence of the form S_n+1 = f(S_n), where f is given by f(x) = ax + b, x in R, |a| <1. It follows f is a contraction on R, because |f(x) - f(y)| = |a| |x-y| and |a|<1. Therefore, every sequence of the form given above converges in R to the only Þxed point of f, that is, to the only number a such that f(a) = a (contractions on always R have one, and only one, Þxed point in R). If f is continuous and S_n is convergent, then S_n converges to a solution of the equation f(S) = S ( a Þxed point), even if f is not a contraction. But in this case, you Þrst have to prove somehow that S_n converges. Artur ------------------------------------------------------------- -- i had a question. if continuous f:[-1,1] -> R is contraction when some 0 <= c < 1 with for all x,y in [-1,1], |f(x) - f(y)| < c|x - y|, then, f has Þxed point. i know this. if f:[-1,1]->R such that |f(x) - f(y)| = |x - y| for all x,y in [-1,1] and lim X_n = x n->00 then, x is Þxed point of f. -------------------------------------------- it¹s possible ?? maybe, it looks like possible by writing of Artur. but i know that f(x) = x + 1 does not have Þxed point. um......i am confusing. i need your advice. thank you very much. === Subject: Re: question for writing.. > hello......doctor~ > Actually, you have a sequence of the form S_n+1 = f(S_n), where f is > given by f(x) = ax + b, x in R, |a| <1. It follows f is a contraction > on R, because |f(x) - f(y)| = |a| |x-y| and |a|<1. Therefore, every > sequence of the form given above converges in R to the only Þxed > point of f, that is, to the only number a such that f(a) = a > (contractions on always R have one, and only one, Þxed point in R). > If f is continuous and S_n is convergent, then S_n converges to a > solution of the equation f(S) = S ( a Þxed point), even if f is not a > contraction. But in this case, you Þrst have to prove somehow that > S_n converges. > Artur > ------------------------------------------------------------- -- oh.....i am sorry....my question is very vague. um...my question is If f is continuous and S_n is convergent, then S_n converges to a solution of the equation f(S) = S ( a Þxed point), even if f is not a contraction. i can¹t understand this. i know that f(x) = x + 1 does not have Þxed point. of course, f(x) is not contraction. if lim S_n = a, f(a) = a. ?? um....i can¹t understand. give me a advice ,please thank you very much. === Subject: Re: Mathematical exact explanation of the adiabatic theorem/adiabatic passage? > where may I look up a mathematically exact proof of the adiabatic > theorem? > The description found in Albert Messiah¹s book on quantum mechanics is > not what I call mathematically exact. It (hopefully) is correct, but > it is not proven in a mathematically closed way/context. I wonder if > there is any educational book on functional analysis that also > contains a proof of the adiabatic theorem -- and such that one is able > to track down the required theorems without perusing a whole maths > library and still ending up with no exact answers. I think Mathematical Methods of Classical Mechanics by V.I.Arnold will give you what you need. -Michael. === Subject: Re: Mathematical exact explanation of the adiabatic theorem/adiabatic passage? > I think Mathematical Methods of Classical Mechanics by V.I.Arnold will > give you what you need. I doubt that. The adiabatic theorem is about quantum mechanical operators. === Subject: Re: Question about PhD math programs reworded It¹s probably an oversimpliÞcation, but it may be because they weren¹t well-enough prepared for college in the Þrst place. A PhD program functions at a high level. Based on my experience many, many talented students begin college without sufÞcient preparation to do calculus at a level that prepares them for analysis as juniors or seniors, and also gives them the exposure to abstract, proof-oriented thinking needed for abstract algebra. Thus they have to catch up when they reach graduate school. At worst this usually means spending an extra year to get a graduate degree, no great loss. === Subject: Re: Opera (Two Riddles) > As promised, this post contains two riddles. > The Þrst is more difÞcult than the second, as it depends on the > knowledge of a speciÞc and obscure fact. (But a few minutes¹ research > with Google would solve that.) > However, it is to the second that the title of this post contains a > clue. > Riddle the Þrst: > What is the difference between Ansel Adams and Stephen G. Baer? Hmmm. Can I come back to that one? > Riddle the second: > What do Euclid and Herb Ritts have in common? Google is just getting too smart these days - To Þnd the answer I only had to click on one of the sponsored links displayed when I viewed your post. spoiler space . . . . . . . . . . . . . . . . . . . . I gather Herb Ritts is a photographer, who took some pictures on Euclid Beach. John Ramsden Author-Supplied-Address: bds ipp mpg de === Subject: Re: How to prove advection þows are topology preserving? Mail-To-News-Contact: abuse@dizum.com |> My intuition tells me that if I have an initial set S and I advect it |> through some continuous þow Þeld V(x) with: |> then the topology of the deformed set S after advection will not |> change. SpeciÞcally, the set will not break apart and will not merge |> (if initially it had disconnected parts). It will also not |> self-intersect, if the þow V(x) is single-valued. |> Can anyone tell me where I can Þnd a reference where this is proven? |> Or else, if my intuition is wrong after all, what other conditions |> have to be imposed on V(x) in order to enforce topology preservation? In magnetohydrodynamics (MHD) the related theorem is þux conservation, from which you can derive the concept of a þux tube, and then a Þeld line (þux tube of inÞnitesimal cross section). In MHD, Þeld lines cannot cross, and it is not hard to see how this follows from þux conservation. My lecture note on this, http://www.rzg.mpg.de/~bds/lectures/mhd-lecture.html gives a basic picture of it and uses a proof based on Þnite-element triangles. It also cites the basic texts (eg, Freidberg) for those who want more detail. In þuid dynamics I am less sure of it, but I think it follows from the circulation theorem which shows vorticity is conserved (in some sense like mean squared vorticity) and from this follows that vortex lines vortex circulation theorem Ah, it is by Kelvin (and Helmholtz whom I had thought was the originator). to texts. You can start with this: http://www.du.edu/~jcalvert/tech/þuids/vortex.htm |> The same thing should happen to a set deÞned implicitly as the zero |> set of some Þeld value phi(x) and this Þeld is then advected by the |> same þow V(x): |> phi_t + V dot gradient(phi) = 0 That¹s passive scalar advection, which is different from the circulation theorem, I think. Isn¹t this true only if V is incompressible? -- cu, Bruce drift wave turbulence: http://www.rzg.mpg.de/~bds/ Author-Supplied-Address: bds ipp mpg de === Subject: Re: How to prove advection þows are topology preserving? Mail-To-News-Contact: abuse@dizum.com |> An excellent book on this sort of problem is |> The Kinematics of Mixing : Stretching, Chaos, and Transport (Cambridge |> Texts in Applied Mathematics) by J. M. Ottino, ISBN: 0521368782. |> besides waiting for diffusive processes to occur. Even in the |> incompressible limit, if your set were (for instance) to þow towards |> two opposing vorticies, it would be torn into two sections. This can only happen if there is diffusion. But note that the timescale itself is not limited to simple estimates of the diffusion time. Nonlinear processes, even non-turbulent, can deliver dynamics to the diffusion timescale much faster (this is what fast collisionless reconnection [plasma physics] is all about). |> One criteria for whether a þow is mixing or not in the sense that |> your intial set gets mixed into the larger þow is whether the Lyaponov |> exponents of the þow Þeld are ever positive. Can¹t comment on that... -- cu, Bruce drift wave turbulence: http://www.rzg.mpg.de/~bds/ === Subject: Re: The real effect of centrifugal force > CUT< > Consider a mass on a string one meter long spinning clockwise around a hub > at one revolution per second. Let¹s imagine the whole works out in space so > that there is no stray gravity force to com0plicate the question. If the > mass is released from the string as the string passes 12 o¹clock, how long > does it take for the involute path to pass the radius at 3 o¹clock? At 6 > o¹clock? at 9 o¹clock? > As long as you are in no hurry, and we¹re just imagining anyway; lets > make the speed one twelfth [1/12] revolution per hour: In one hour [@1 > o¹clock], after releasing the string at noon [12 o¹clock], the mass > will have traveled 1/12 revolution, and be at the intersection of a > tangent and a radial line at one o¹clock: In two hours the mass will > have traveled 2/12 revolution, and be at the intersection of a tangent > and a radial line at two o¹clock: In three hours the mass will have > traveled 3/12 revolution, and be at the intersection of a tangent and > a radial line at three o¹clock: In four hours the mass will have > traveled 4/12 revolution, and be at the intersection of a tangent and > a radial line at four o¹clock: > With the whole works out in space, where there is no stray gravity > force to complicate the situation, this will continue into a second > revolution; even into a third, or more; but by 4 or 5 o¹clock the > conclusion is evident: > A curved line drawn through the intersections of the tangent lines and > the radial lines is the radial outward path followed by the mass as it > centrifugally þees from the hub to which it was originally attached. > This is the principle by which slings and similar weapons operate. BZZZZZT! WRONG! You are the weakest link! === Subject: Re: The real effect of centrifugal force > Hey dirty mount, centripetal-centrifugal forces are action-reaction > pair. > Yes, I think there are some textbooks that deÞne things this way -- > that the centrifugal force is the force of the stone on the sling > and that centripetal force is the force of the sling on the stone > for instance. If I recall correctly, that is what I was taught. > But that is not the usage that I prefer, that most of the people here > prefer and that most textbooks prefer. > In our preferred usage, centrifugal force is the imaginary force that > you need to invoke in a rotating frame of reference so that Newton¹s > second law (f=ma) works out. The terms Þctitious force or > inertial force are often used to describe forces of this sort. > Coriolis force is another example of a Þctitious force. > Fictitious forces have no third law partners. > John Briggs However, IMO the term Þcticious is used only in reference to applying the law in a rotating frame. Otherwise, centrifugal force is very real and without it a centrifuge wouldn¹t work. I prefer the term inertial force. I think the unfortunate choice of the term Þcticious force has contributed to the creation of a whole generation of crackpots. It¹s a sad. Mike === Subject: Re: The real effect of centrifugal force > Actually centrifugal force, when not restrained as by a string or > sling causes a body to þee the center of an evolute radially; along > involutes that remain directed radially away from the center: > Yes, its direction will be radial relative to the rotating reference frame > center. But it will be tangential relative to any external inertial > (stopped) reference frame. > Like the sling used by David to slay Goliath; when it is released the > missile travels radially away from the center to points equidistant > along the tangent.; at potentially lethal speeds depending on the > speed it has when it is released from the evolute. > There is nothing Þctitious about centrifugal force! > The centrifugal force is very real for those that are in the rotating > reference frame, not for those outside the rotating frame (in an > inertial reference frame). > NOAA explains Ocean tides Earth opposite bulge based on the centrifugal > force due to Earth rotation. > NASA also takes centrifugal forces very seriously. > So far nobody has a good explanation for the circular motion. > Nobody can explain this: > http://physics.nad.ru/Physics/English/gyro_tmp.htm > http://physics.nad.ru/Physics/English/gyro_txt.htm You are missing something nontrivial in your direction guesses. You must allow for a large translation of your center of rotation prior to release. You cannot assume a simple Þxed center rotating frame of reference. === Subject: Re: The real effect of centrifugal force > Actually centrifugal force, when not restrained as by a string or > sling causes a body to þee the center of an evolute radially; along > involutes that remain directed radially away from the center: > Yes, its direction will be radial relative to the rotating reference frame > center. But it will be tangential relative to any external inertial > (stopped) reference frame. > Like the sling used by David to slay Goliath; when it is released the > missile travels radially away from the center to points equidistant > along the tangent.; at potentially lethal speeds depending on the > speed it has when it is released from the evolute. > There is nothing Þctitious about centrifugal force! > The centrifugal force is very real for those that are in the rotating > reference frame, not for those outside the rotating frame (in an > inertial reference frame). > NOAA explains Ocean tides Earth opposite bulge based on the centrifugal > force due to Earth rotation. > NASA also takes centrifugal forces very seriously. > So far nobody has a good explanation for the circular motion. > Nobody can explain this: > http://physics.nad.ru/Physics/English/gyro_tmp.htm > http://physics.nad.ru/Physics/English/gyro_txt.htm You are missing something nontrivial in your direction guesses. You must allow for a large translation of your center of rotation prior to release. You cannot assume a simple Þxed center rotating frame of reference. === Subject: Re: Imaginary primes? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6LDHuJ24766; >Numbers x+i*y, where x and y are integers, are known as Gaussian >integers. They have a number theory much like ordinary integers. >(There are other subÞelds of C that do, too.) There are more >Gaussian integers than integers, Allow me to correct this. The last statement is grossly false. There are exactly as many Gaussian integers as there are ordinary integers. Both sets are countably inÞnite. You can put them into a 1 to 1 correspondence. >so there are more possibilities >for factorization. This is poorly stated. The Gaussian integers also have unique factorization, like the ordinary integers. What is true is that some numbers that are prime in the ordinary integers have factorizations in the Gaussians. These are the primes that are 1 mod 4. (since they are the sum of exactly two squares) The Gaussian integers are formed by adding an element to the set of ordinary integers. That element is a root of x^2+1 = 0, aka Œi¹. One can form other rings by adjoining the roots of other, irreducible polynomials to the ordinary integers. For example, adjoining the root of x^2 + x + 1 [a cube root of 1] results in the ŒEisenstein¹ integers. This entire subject is known as algebraic number theory. === Subject: Re: Length of sequence of consecutive primes starting at 2 by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6LDHup24756; >> May I ask why you need ALL the primes? As I stated, at one time or >> another all primes up to about 10^12 have been generated. I doubt if >> ANYONE knows what the largest upper bound truly is. It is at least 10^12. >It¹s a truly uninteresting factette, but the Þgure is >6.10^16 >Oliveira e Silva¹s performed a distributed brute-force GC >veriÞer which I believe generates all primes. I have no direct reason to doubt you, but: Generating all primes to 6.10^16 seems a little far-fetched. How would you hold them in memory?? It is also a LOT of computing.... Also, generating all the primes up to some Þxed bound is not necessary for a GC veriÞer. Are you sure that Silva generates all of them? I suspect that >the large prime gaps hunters has reached a similar level. They deÞnitely do NOT generate all of them to a Þxed bound. === Subject: Re: Length of sequence of consecutive primes starting at 2 >> May I ask why you need ALL the primes? As I stated, at one time or >> another all primes up to about 10^12 have been generated. I doubt if >> ANYONE knows what the largest upper bound truly is. It is at least 10^12. >It¹s a truly uninteresting factette, but the Þgure is >6.10^16 >Oliveira e Silva¹s performed a distributed brute-force GC >veriÞer which I believe generates all primes. > I have no direct reason to doubt you, but: Politician? Lawyer? Marriage guidance counsellor? Store detective? Nope, I give up - what¹s your day job? > Generating all primes to 6.10^16 seems a little far-fetched. > How would you hold them in memory?? It is also a LOT of computing.... Of course he doesn¹t hold them all in memory, why the heck would he want to do that? What gave you the impression that he did? And yes, it is a lot of computing. However, his sieve is an exceptionally fast one, and the project¹s been running on several gross machines for quite a while. > Also, generating all the primes up to some Þxed bound is not > necessary for a GC veriÞer. Are you sure that Silva generates > all of them? Not necessary, but certainly sufÞcient. His sieve certainly looks like a segmented Eratosthenes with wheel to me -- what does it look like to you? Which primes do you think think it wouldn¹t generate? It would of course be possible to restrict large primes to only a certain set of residues, but then you¹d need a proportionally larger window of small primes, and I suspect that that would damage the runtime as much as it saves. > I suspect that >the large prime gaps hunters has reached a similar level. > They deÞnitely do NOT generate all of them to a Þxed bound. I¹m thinking of the ones interested in Þrst occurances, obviously. (Nicely/Nyman et al.) They would only chose to /not/ generate ones which they could guarantee were too close to both of their neighbouring primes, surely? How would you suggest they do that? I¹m not suggesting that they enumerate every prime, in binary or decimal representation, but I would expect them to be generated, as remaining set bits in a bitÞeld, in their entirety. Looking for large gaps, and using KMP or BM, would imply that only a small fraction actually get explicitly represented in radix form. (However, to do so would be an extra overhead that¹s o(1) compared to the sieve time, or at worst O(1), depending on how many logs and loglogs they¹re left with. I suspect if BN¹s running a sieve, then it might be slower than Silva¹s, and may well have more superlinear behaviour, in which case the overhead is o(1).) Or have I overlooked something obvious? Or were you thinking of the arbitrary large gap hunters such as JKA and his megagap, rather than the minimal ones like Nyman? I should have been more explicit I guess, but I really didn¹t think that anyone would consider generation of all primes up to 10^80000 to be the kind of task under discussion. Phil -- 1st bug in MS win2k source code found after 20 minutes: scanline.cpp 2nd and 3rd bug found after 10 more minutes: gethost.c Both non-exploitable. (The 2nd/3rd ones might be, depending on the CRTL) === Subject: Re: Combinatorics & Probability Problem No one replied this thread??? > I have a simple combinatorics & probability problem which I don¹t know how > to derive a beautiful solution for it. > There is a new disease occurring in Sai Gon. Doctors estimated that the > transmissibility of this decease is T, which means the probability that a > friend of a patient would be infected by the decease is T. Given n people > infected by this decease and in this context, we assume that each person of > these has exactly other k different friends. The problem is to compute the > probability that exactly m friends of these n people would also be infected > by that decease. > Anyone has any idea? > Khoa === Subject: non-existence of continuously - differentiable space Þlling curves I¹m trying to prove that there does not exist a continuously differentiable surjective map f : [0,1] -> [0,1]x[0,1]. I think I may have a proof using Sard¹s theorem : namely that if there did exist one then for each y in [0,1]x[0,1] there exists x in [0,1] st f(x)=y, but D_x f : T_x [0,1] -> T_y [0,1]x[0,1] -this cant be surjective since its a linear map from a vector space of dim 1 to a vs of dim 2. So every y in [0,1]x[0,1] is a non-regular value. But Sards theorem says that the set of non-regular values for f has zero measure, and cleary [0,1]x[0,1] does not. However, I am a little apprehensive as I have a doubt as to whether I am assuming f is Œsmooth¹, i.e inÞnitely differetiable. If anyone knows of an easier, more elementary argument I would be interested to here it. === Subject: Re: non-existence of continuously - differentiable space Þlling curves If f is differentiable on [0,1], then the Hausdorff dimension of the image is at most 1. The Hausdorff dimension of a set in the plane with interior is 2. Argument OK even if f¹ is not continuous. OK if f¹ is not bounded: The image still has sigma-Þnite 1-dimensional Hausdorff measure, and thus dimension 1. But f¹ exists almost everywhere is not enough for this argument (or the fact). > I¹m trying to prove that there does not exist a continuously > differentiable surjective map f : [0,1] -> [0,1]x[0,1]. > I think I may have a proof using Sard¹s theorem : namely that if there > did exist one then for each y in [0,1]x[0,1] there exists x in [0,1] st > f(x)=y, but D_x f : T_x [0,1] -> T_y [0,1]x[0,1] -this cant be > surjective since its a linear map from a vector space of dim 1 to a vs > of dim 2. So every y in [0,1]x[0,1] is a non-regular value. But Sards > theorem says that the set of non-regular values for f has zero measure, > and cleary [0,1]x[0,1] does not. > However, I am a little apprehensive as I have a doubt as to whether I am > assuming f is Œsmooth¹, i.e inÞnitely differetiable. > If anyone knows of an easier, more elementary argument I would be > interested to here it. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: non-existence of continuously - differentiable space Þlling curves > I¹m trying to prove that there does not exist a continuously > differentiable surjective map f : [0,1] -> [0,1]x[0,1]. > I think I may have a proof using Sard¹s theorem : namely that if there > did exist one then for each y in [0,1]x[0,1] there exists x in [0,1] st > f(x)=y, but D_x f : T_x [0,1] -> T_y [0,1]x[0,1] -this cant be > surjective since its a linear map from a vector space of dim 1 to a vs > of dim 2. So every y in [0,1]x[0,1] is a non-regular value. But Sards > theorem says that the set of non-regular values for f has zero measure, > and cleary [0,1]x[0,1] does not. > However, I am a little apprehensive as I have a doubt as to whether I am > assuming f is Œsmooth¹, i.e inÞnitely differetiable. > If anyone knows of an easier, more elementary argument I would be > interested to here it. There exists C < oo such that |f(x) - f(y)| <= C|x - y|. For n a positive integer, partition [0,1] into n intervals of lenth 1/n. The image is then contained in the union of n discs of radius C/n. The 2-dimensional measure of the image is therefore <= n*Pi*(C/n)^2. === Subject: Re: non-existence of continuously - differentiable space Þlling curves > I¹m trying to prove that there does not exist a continuously > differentiable surjective map f : [0,1] -> [0,1]x[0,1]. > I think I may have a proof using Sard¹s theorem : namely that if there > did exist one then for each y in [0,1]x[0,1] there exists x in [0,1] st > f(x)=y, but D_x f : T_x [0,1] -> T_y [0,1]x[0,1] -this cant be > surjective since its a linear map from a vector space of dim 1 to a vs > of dim 2. So every y in [0,1]x[0,1] is a non-regular value. But Sards > theorem says that the set of non-regular values for f has zero measure, > and cleary [0,1]x[0,1] does not. > However, I am a little apprehensive as I have a doubt as to whether I am > assuming f is Œsmooth¹, i.e inÞnitely differetiable. > If anyone knows of an easier, more elementary argument I would be > interested to here it. > There exists C < oo such that |f(x) - f(y)| <= C|x - y|. For n a positive > integer, partition [0,1] into n intervals of lenth 1/n. The image is then > contained in the union of n discs of radius C/n. The 2-dimensional measure > of the image is therefore <= n*Pi*(C/n)^2. Of the three posts responding to the query [which have appeared on my server so far], this is by far the clearest and easiest to understand. --Ron Bruck === Subject: Re: non-existence of continuously - differentiable space Þlling curves >I¹m trying to prove that there does not exist a continuously >differentiable surjective map f : [0,1] -> [0,1]x[0,1]. >I think I may have a proof using Sard¹s theorem : namely that if there >did exist one then for each y in [0,1]x[0,1] there exists x in [0,1] st >f(x)=y, but D_x f : T_x [0,1] -> T_y [0,1]x[0,1] -this cant be >surjective since its a linear map from a vector space of dim 1 to a vs >of dim 2. So every y in [0,1]x[0,1] is a non-regular value. But Sards >theorem says that the set of non-regular values for f has zero measure, >and cleary [0,1]x[0,1] does not. >However, I am a little apprehensive as I have a doubt as to whether I am > assuming f is Œsmooth¹, i.e inÞnitely differetiable. >If anyone knows of an easier, more elementary argument I would be >interested to here it. It¹s certainly more elementary than this. If f is continuously differentiable then f([0,1]) has Þnite one-dimensional Hausdorff measure (there exists c < inÞnity such that for every delta > 0 there is a cover of f([0,1]) by balls B_j of radius < delta, such that the sum of the radius of B_j is less than c.) The same argument shows that f cannot satisfy a Holder condition |f(x) - f(y)| <= c |x - y|^a if a > 1/2 (which is sharp - standard constructions give f¹s that do satisfy this condition with a = 1/2.) ************************ David C. Ullrich === Subject: Re: ~ Sets of functions 1 >> But, every once in a while, what the things actually ->are<- can come >> into play (for good or ill), as it did in your attempts. It is clear >> that if A is a subset of B, then F(X,A) is a subset of F(X,B); but as >> sets, this is not the case when A is not equal to B (because the >> Œfunction¹ carries information about its domain and codomain). So >> instead, if you want to be really technical, you have to say that >> there is a natural map from F(X,A) to F(X,B) which is one-to-one, >> and identiÞes any function f:X->A with the function f:X->B with the >> exact same values at every x in X. Which is, after much discussion and >> my usual long-winded-ness, what Julien was saying. (-: > So, I should have showed that there was a map from F(X,A) to F(X,B) to > satisfy everyone? That would certainly be sufÞcient. Of course, the original statement of the exercise certainly did not emphasize this, so there is no need to feel guilty about it. Marc === Subject: Re: ~ Sets of functions 1 > So, I should have showed that there was a map from F(X,A) to F(X,B) to > satisfy everyone? > That would certainly be sufÞcient. Of course, the original statement > of the exercise certainly did not emphasize this, so there is no need > to feel guilty about it. Like this? Let f: F(X,A) -> F(X,B) be a function deÞned by f(x) = g(x) where g(x) = {f(x) if x in A, Some c in B-A if x in B-A}. I don¹t think that the Some c in B-A is meaningful because A might equal B and thus there would not be an element, but then again no x would ever be outside of A in that case so the second part wouldnt be used, but it doesnt seem right. I might not even be understanding the correct map to be shown. Please let me know, Adam. === Subject: Re: Bear of very little brain seeks guidance from an Owl or a don¹t go there from a Kanga I am indebted to Will Orrick for his kind and instructive posting suggesting that the proposed procedure P be carried out for orders 4,...,12, so that completion of the incidence relations in the remainder blocks can be attempted in the manner he suggested. As soon as I began carrying out his suggested methodology, I realized that it is better to truncate the initial procedure P and accept a larger remainder block of n^2 points and n^2-n lines, rather than the smaller remainder block of n^2 points and n lines which I had originally mentioned. The reason the larger remainder block is better is because it lets us bring more math to bear on the problem of trying to determine whether the remainder block has any interesting structure that will shed light on the FPP rule-out problem. (Think, for example, of n zonohedra with n generators each having n(n-1) faces, e.g. 3 cubes, 4 rhombic dodecahedra, etc.) The new initial procedure P is trivial to state once a particular convention is adopted to label the vertices of any uniformly n-ary tree of max vertex depth 2, e.g. T2: [ [ a b] [ c d]] (for order 2) A B C T3: [ [ a b c] [ d e f] [ g h i]] (for order 3) A B C D This labelling convention is as follows: the root is v00 and the children of the root are v01,...,v0n the children of the leftmost remaining subtree are v11,...,v1n the children of the next leftmost remaining subtree are v21,...v2n ... the children of the rightmost remaining subtree are vn1,...,vnn. With this labelling, we can give the incidence relations enumerated by the procedure P as follows. First: a line containing v00 and v01,...,v0n Next: a line containing v00 and v11,...,vi1,...vn1 ... a line containing v00 and v1i,...,vii,...Vni ... a line containing v00 and v1n,...,vin,...vnn Next: a line containing v0n and v11,...,v1i,...v1n ... a line containing v0n and vi1,...,vii,...vin ... a line containing v0n and vn1,...,vni,...vnn (In the case of order 3, this procedure obviously yields a remainder block of 13-7 = 6 = n^2-n lines.) To reduce the number of points without completely Þlled-in incidence to just n^2, we now exclude the root v00 and its rightmost child v0n and insist (for order n) that the remaining incidence relations for the remaining children of the root (i.e. v01,...,v0n-1) be divided evenly among the n^2-n lines of the remainder block. For example, for order n=2, the one remaining child of the root will lie once on the line 1 of the remainder block and once on line 2. For order n =3. one remaining child of the root will lie on three lines of the remainder block and the other remaining child of the root will lie on the other three lines of the remainder block. This gives a remainder block of n^2-n lines and n^2 points, as desired. === Subject: Re: Bear of very little brain seeks guidance from an Owl or a don¹t go there from a Kanga I am indebted to Will Orrick for his kind and instructive posting suggesting that the proposed procedure P be carried out for orders 4,...,12, so that completion of the incidence relations in the remainder blocks can be attempted in the manner he suggested. As soon as I began carrying out his suggested methodology, I realized that it is better to truncate the initial procedure P and accept a larger remainder block of n^2 points and n^2-n lines, rather than the smaller remainder block of n^2 points and n lines which I had originally mentioned. This is because such remainder blocks, though larger than the original, permit more mathematics to be brought to bear on the question of whether they have sufÞciently interesting structure either to further reduce the search space of the FPP rule-out problem or suggest the nature of a complete solution. The new initial procedure P is trivial to state once a particular convention is adopted to label the vertices of any uniformly n-ary tree of max vertex depth 2, e.g. T2: [ [ a b] [ c d]] (for order 2) A B C T3: [ [ a b c] [ d e f] [ g h i]] (for order 3) A B C D This labelling convention is as follows: the root is v00 and the children of the root are v01,...,v0n the children of the leftmost remaining subtree are v11,...,v1n the children of the next leftmost remaining subtree are v21,...v2n ... the children of the rightmost remaining subtree are vn1,...,vnn. With this labelling, we can give the incidence relations enumerated by the procedure P as follows. First: a line containing v00 and v01,...,v0n Next: a line containing v00 and v11,...,vi1,...vn1 ... a line containing v00 and v1i,...,vii,...Vni ... a line containing v00 and v1n,...,vin,...vnn Next: a line containing v0n and v11,...,v1i,...v1n ... a line containing v0n and vi1,...,vii,...vin ... a line containing v0n and vn1,...,vni,...vnn (In the case of order 3, this procedure obviously yields a remainder block of 13-7 = 6 = n^2-n lines.) To reduce the number of points without completely Þlled-in incidence to just n^2, we now exclude the root v00 and its rightmost child v0n and insist (for order n) that the remaining incidence relations for the remaining children of the root (i.e. v01,...,v0n-1) be divided evenly among the n^2-n lines of the remainder block. For example, for order n=2, the one remaining child of the root will lie once on the line 1 of the remainder block and once on line 2. For order n =3. one remaining child of the root will lie on three lines of the remainder block and the other remaining child of the root will lie on the other three lines of the remainder block. For order 4, the three remaining children of the root will lie each on four lines of the twelve (4^2 - 4 = 12) in the remainder block. For order n, the n-1 remaining children of the root will lie each on n lines of the n^2-n lines of the block. This gives a remainder block of n^2-n lines and n^2 points, as desired. === Subject: Re: (pi^4+pi^5)^(1/6)=e??? >>I can do a little better with this simple equation --- >>3 ~ inx((pi)^2 * sqrt(pi+ 1 + 1/5511606)) >>Actually 2.99999999999999801721... or good too 14 decimal places.;-) >>(inx)= natural log e. > >Whatever happened to good old ln? >Anyway, here¹s my entry: >e = > >(pi^5*(1+pi^(-1)/(1-pi^(-13)/(2-pi^(-2)/(1-pi^(-4)/(2-pi^(-3 )))))))^(1/6) >with 15 correct digits. >Robert Israel israel@math.ubc.ca >Department of Mathematics http://www.math.ubc.ca/~ israel >University of British Columbia >Vancouver, BC, Canada V6T 1Z2 > Improving on my last entry -- > 3~in((pi)^2 * sqrt(pi + 1 + 1/5511606 + 1/60887389898152)) > By adding one more large integer inverse, it is now correct to 29 > decimal places. ;-) > > Dan If we are allowed lots of digits in our answers, how about: pi = e * 0 + 3.14159265358979323846264338327950288419716939937510 Which I believe is correct to 50 decimal places. With some effort, I may be able to improve on the formula. Se.87n O¹Leathl.97bhair === Subject: Re: (pi^4+pi^5)^(1/6)=e??? > > > >>I can do a little better with this simple equation --- > >>3 ~ inx((pi)^2 * sqrt(pi+ 1 + 1/5511606)) > >>Actually 2.99999999999999801721... or good too 14 decimal places.;-) > >>(inx)= natural log e. > > > >Whatever happened to good old ln? > > > >Anyway, here¹s my entry: > > > >e = > >(pi^5*(1+pi^(-1)/(1-pi^(-13)/(2-pi^(-2)/(1-pi^(-4)/(2-pi^(-3 )))))))^(1/6) > > > >with 15 correct digits. > > > >Robert Israel israel@math.ubc.ca > >Department of Mathematics http://www.math.ubc.ca/~ israel > >University of British Columbia > >Vancouver, BC, Canada V6T 1Z2 > > Improving on my last entry -- > > 3~in((pi)^2 * sqrt(pi + 1 + 1/5511606 + 1/60887389898152)) > > By adding one more large integer inverse, it is now correct to 29 > decimal places. ;-) > > Dan > If we are allowed lots of digits in our answers, how about: > pi = e * 0 + 3.14159265358979323846264338327950288419716939937510 > Which I believe is correct to 50 decimal places. With some effort, I > may be able to improve on the formula. > Se.87n O¹Leathl.97bhair Did you not notice the --> ;-) But aside from that, how do you think these large integers are calculated and then using their inverse? I would say a little more difÞcult then what you are proposing! My algorithm is a brute force trial and error method. I and am not sure if a closed form method is possible for this particular equation. Maybe the mathematicians can answer that one? Dan === Subject: Re: (pi^4+pi^5)^(1/6)=e??? > >>3 ~ inx((pi)^2 * sqrt(pi+ 1 + 1/5511606)) > > > > 3~in((pi)^2 * sqrt(pi + 1 + 1/5511606 + 1/60887389898152)) > > > > By adding one more large integer inverse, it is now correct to 29 > > decimal places. ;-) > If we are allowed lots of digits in our answers, how about: > pi = e * 0 + 3.14159265358979323846264338327950288419716939937510 > Did you not notice the --> ;-) > But aside from that, how do you think these large integers are > calculated and then using their inverse? > I would say a little more difÞcult then what you are proposing! > My algorithm is a brute force trial and error method. I and am not sure > if a closed form method is possible for this particular equation. > Maybe the mathematicians can answer that one? Closed form? Huh? It¹s just simple algebra, unless you¹re 3 = ln((pi)^2 * sqrt(pi+1 + 1/x)) for some integer x. So we solve for x and Þnd 1/x = (e^6 / pi^4) - pi - 1 You can type the right-hand side of that into Google, or use a handheld calculator, to Þnd the answer; 1/x is approximately 1.814E-7, so x is (to the lowest greater integer) 5511606. Taking 1.814E-7 minus 1/5511606 gives approximately 1.6E-14, which is out of the useful range of any of the calculators I have immediately available, but gives a reciprocal of something like 6.3E+13. Repeat ad nauseam, or until you run out of precision. There¹s no secret method to it, as far as I know. -Arthur === Subject: Re: (pi^4+pi^5)^(1/6)=e??? > 1/x = (e^6 / pi^4) - pi - 1 >You can type the right-hand side of that into Google, or use >a handheld calculator, to Þnd the answer; 1/x is approximately >1.814E-7, so x is (to the lowest greater integer) 5511606. > Taking 1.814E-7 minus 1/5511606 gives approximately >1.6E-14, which is out of the useful range of any of the >calculators I have immediately available, but gives a >reciprocal of something like 6.3E+13. Repeat ad nauseam, >or until you run out of precision. There¹s no secret >method to it, as far as I know. Yes (but don¹t use handheld calculators, unless they¹re running Maple or another CAS that can handle arbitrary precision). More difÞcult question: is there a more accurate way than this greedy algorithm to write 1/x as the sum of the reciprocals of two integers? Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: (pi^4+pi^5)^(1/6)=e??? > http://www.scienceteecher.com/math_shirt.htm > If true, where did this formula come from??? You might want to take a look at http://mathworld.wolfram.com/e.html http://numbers.computation.free.fr/Constants/E/e.html === Subject: Re: (pi^4+pi^5)^(1/6)=e??? >Improving on my last entry -- >3~in((pi)^2 * sqrt(pi + 1 + 1/5511606 + 1/60887389898152)) >By adding one more large integer inverse, it is now correct to 29 >decimal places. ;-) Well, then, you might as well say 3 ~ ln(pi^2 sqrt(pi + 1 + 1/5511606 + 1/608873898152 + 1/19488940474795004702374981616)) correct to 57 decimal places. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: (pi^4+pi^5)^(1/6)=e??? >Improving on my last entry -- >3~in((pi)^2 * sqrt(pi + 1 + 1/5511606 + 1/60887389898152)) >By adding one more large integer inverse, it is now correct to 29 >decimal places. ;-) > > Well, then, you might as well say > 3 ~ ln(pi^2 sqrt(pi + 1 + 1/5511606 + 1/608873898152 + > 1/19488940474795004702374981616)) > correct to 57 decimal places. Jeez, I hope someone¹s paying you for this ;) === Subject: Re: (pi^4+pi^5)^(1/6)=e??? >> Well, then, you might as well say >> 3 ~ ln(pi^2 sqrt(pi + 1 + 1/5511606 + 1/608873898152 + >> 1/19488940474795004702374981616)) >> correct to 57 decimal places. >Jeez, I hope someone¹s paying you for this ;) No, but I accept donations. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: (pi^4+pi^5)^(1/6)=e??? >Improving on my last entry -- >3~in((pi)^2 * sqrt(pi + 1 + 1/5511606 + 1/60887389898152)) >By adding one more large integer inverse, it is now correct to 29 >decimal places. ;-) > > Well, then, you might as well say > 3 ~ ln(pi^2 sqrt(pi + 1 + 1/5511606 + 1/608873898152 + > 1/19488940474795004702374981616)) > correct to 57 decimal places. > Robert Israel israel@math.ubc.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia > Vancouver, BC, Canada V6T 1Z2 ln(e^e). correct to 1,792 decimal places. but not to 1,793 e != e. e is a dynamic constant that changes as the universe expands. crazy isn¹t it? by the time u read this this equation could be as correct as possibly.. millions of places.. or only to a few hundred. === Subject: Re: Solve: cos(th) / th = c ? >>Hrmph, the error rate for 0.1 < c < 1.0 is much higher, > No, |relative error| is bounded as I stated. So Hrmph to you. >>I¹m getting an error on the order of 0.001 in that range > Then it would seem that you¹ve done something wrong. Perhaps I am, but here is an example of what I calculated: Let f(c) be your equation as deÞned before. Then th = f( 0.2 ) = 1.3060 relative error = (cos( th ) / th - 0.2)/0.2 = 0.0021346 I suppose I am calculating relative error on a different function than you are -- I am looking from the viewpoint of my original function (wrt to c, not th). (PS. The hrmph was not meant negatively) -- edA-qa mort-ora-y Idea Architect http://disemia.com/ === Subject: Re: Solve: cos(th) / th = c ? >>Hrmph, the error rate for 0.1 < c < 1.0 is much higher, > No, |relative error| is bounded as I stated. So Hrmph to you. >>I¹m getting an error on the order of 0.001 in that range > Then it would seem that you¹ve done something wrong. > Perhaps I am, but here is an example of what I calculated: > Let f(c) be your equation as deÞned before. Then > th = f( 0.2 ) = 1.3060 Right. It gives th as being approximately 1.30596..., whereas if we solve the equation cos(th)/th = 0.2, we Þnd that th = 1.30644... The relative error in the approximation is thus 1.30596.../1.30644... - 1, which is about -0.00037 . > relative error = (cos( th ) / th - 0.2)/0.2 = 0.0021346 > I suppose I am calculating relative error on a different function than > you are -- I am looking from the viewpoint of my original function (wrt > to c, not th). I don¹t know a name for what you calculated. (Does anyone else?) I had given an approximation for _th_ and so, of course, when stating relative error in that approximation, I was talking about relative error in th. But if, as you indicated in a later post, the possible values of c in your application are conÞned to a small interval, you should probably forget the approximation I¹d given, which was instead designed to work reasonably well for all positive c. If 0 < c < 1, then I suspect that th can be approximated well with a polynomial of fairly low degree. I¹ll have a look and get back to you... David === Subject: Re: Solve: cos(th) / th = c ? > If 0 < c < 1, then I suspect that th can be approximated well with a > polynomial of fairly low degree. I¹ll have a look and get back to you... We wish to solve cos(th)/th = c, 0 < c < 1, for th approximately. Rather than using a polynomial, it now seems that using the reciprocal of a polynomial may work somewhat better. I suggest using th = (pi/2) / (1 + c + 0.03167*c^2 + 0.17947*c^3 - 0.08621*c^4) which provides |relative error in th| < 0.0002 . By using the restriction 0 < c < 1, an approximation was obtained which is both slightly simpler and more accurate than the approximation previously suggested. HTH, David Cantrell === Subject: Re: Solve: cos(th) / th = c ? >>This looks quite simple, but I can¹t seem to Þgure it out. In the >>equation: >>cos( th ) / th = c >>I need a closed form solution for _th_ such that 0 <= _th_ < 2*pi >>The bounds are placed since I know there are multiple answers, but I >>need the answer in this range. Since lim th->0 is inÞnity and limit >>th->pi/2 is 0, and this is a continuous function there is guaranteed to >>be an answer. > Newton¹s iteration method is the right one here. There can be no > closed form solution for any transcendental equation. You¹ve just reminded me of an old section of my webpage from 1997, so I have just ressurrected it... http://ohmslaw.org.uk/oldsite/fc/newton/ It¹s a simple intro to newtons method with the standard basins of attraction pictures... it¹s a bit white-on-black, but hey, it was 1997 :) alex === Subject: Re: Solve: cos(th) / th = c ? > This looks quite simple, but I can¹t seem to Þgure it out. In the > equation: > cos( th ) / th = c > I need a closed form solution for _th_ such that 0 <= _th_ < 2*pi > The bounds are placed since I know there are multiple answers, but I > need the answer in this range. Since lim th->0 is inÞnity and limit > th->pi/2 is 0, and this is a continuous function there is guaranteed to > be an answer. You could use the simple iterative method (I can¹t remember the name for it - anyone?) It¹s the method where you Þnd a zero point for y = f(x) by iterating the equation x = f(x) + x until x stabilises. Then this Þnal value of x is a zero for the function f(x). (This technique seen popularly in the verhulst bio feedback equation solution x -> rx(1-x) diagrams, e.g. http://access.mmhs.ca/compsci/Verhulst.html) Anyway, I¹ve written a very quick iterative equation solver in python: # # iterative solution to equations # # iterates x = f(x) + x until x value # stabilises - then this Þnal value of x # is a zero point for f(x). # this method by no means will converge on # answers for all f(x)! # import math x = 0.1 lastX=0 c=0.25 acceptableError = 0.0001 while 1: lastX = x x = (math.cos(x)) / x - c + x print x if (abs(x - lastX) < acceptableError): break I set c = 0.25 in this example and kicked off the program. It soon settled down: . . . 4.00607303783 3.59406013915 3.09382195273 2.5209658827 1.94826650526 1.50908811093 1.29995322657 1.25576362943 1.25250398115 1.25235958358 1.25235350086 so for c = 0.25, there is a zero of (cos(th)/th - c) at th ~= 1.25235350086. I¹ve just added an outer loop so the program solves for various values of c from 0 to 1.0, and the code now looks like this: import math for c in map(lambda x:þoat(x)/20.0,range(21)): x = 0.1 lastX = 0 while 1: lastX = x x = (math.cos(x)) / x - c + x if (abs(x - lastX) < 0.0001): break print c =,c, Þnal x =,x Runnig this, I get the following solutions for various values of c: c = 0.05 Þnal x = 7.47179414288 c = 0.1 Þnal x = 7.06961748158 c = 0.15 Þnal x = 6.50590026399 c = 0.2 Þnal x = 1.30644613965 c = 0.25 Þnal x = 1.25235350086 c = 0.3 Þnal x = 1.20191307442 c = 0.35 Þnal x = 1.15473882741 c = 0.4 Þnal x = 1.11050688151 c = 0.45 Þnal x = 1.06897794736 c = 0.5 Þnal x = 1.02987635315 c = 0.55 Þnal x = 0.993036272332 c = 0.6 Þnal x = 0.958269596135 c = 0.65 Þnal x = 0.925375928532 c = 0.7 Þnal x = 0.894367776016 c = 0.75 Þnal x = 0.864890487349 c = 0.8 Þnal x = 0.837021880173 c = 0.85 Þnal x = 0.81068199489 at c = 0.9 the program hangs because the values þip between the following two: 0.653454762216 0.968517108859 .. strange attractor of death! alex === Subject: Re: Solve: cos(th) / th = c ? >> This looks quite simple, but I can¹t seem to Þgure it out. In the >> equation: >> cos( th ) / th = c >> I need a closed form solution for _th_ such that 0 <= _th_ < 2*pi >> The bounds are placed since I know there are multiple answers, but I >> need the answer in this range. Since lim th->0 is inÞnity and limit >> th->pi/2 is 0, and this is a continuous function there is guaranteed to >> be an answer. >You could use the simple iterative method (I can¹t remember the name for >it - anyone?) >It¹s the method where you Þnd a zero point for y = f(x) by iterating >the equation x = f(x) + x until x stabilises. Then this Þnal value of x >is a zero for the function f(x). For f(x) = cos(x)/x - c, the solution in (0,Pi/2) will be a stable equilibrium if -2 < f¹(x) < 0 at the equilibrium. This is true for approximately .7964304168 < x < Pi/2. So the method (started at an appropriate initial condition) will work if c < cos(.7964304168)/.7964304168 = 0.8779962602 approximately. Of course, other iteration schemes might be stable over a wider interval. In particular, Newton¹s method is always stable (at least for a function that is analytic at the solution, or is C^2 with nonzero derivative at the solution). Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Solve: cos(th) / th = c ? >This looks quite simple, but I can¹t seem to Þgure it out. In the >equation: > cos( th ) / th = c >I need a closed form solution for _th_ such that 0 <= _th_ < 2*pi >The bounds are placed since I know there are multiple answers, but I >need the answer in this range. Since lim th->0 is inÞnity and limit >th->pi/2 is 0, and this is a continuous function there is guaranteed to >be an answer. >>You could use the simple iterative method (I can¹t remember the name for >>it - anyone?) >>It¹s the method where you Þnd a zero point for y = f(x) by iterating >>the equation x = f(x) + x until x stabilises. Then this Þnal value of x >>is a zero for the function f(x). > For f(x) = cos(x)/x - c, the solution in (0,Pi/2) will be a stable > equilibrium if -2 < f¹(x) < 0 at the equilibrium. This is true > for approximately .7964304168 < x < Pi/2. stability of this method! How was that result derived? > So the method (started > at an appropriate initial condition) will work if > c < cos(.7964304168)/.7964304168 = 0.8779962602 approximately. > Of course, other iteration schemes might be stable over a wider > interval. In particular, Newton¹s method is always stable (at > least for a function that is analytic at the solution, or > is C^2 with nonzero derivative at the solution). What are you referring to when you say C^2 here? Newtons method can result in periodic behaviour, can¹t it? In some cases you can bounce between two values continually... alex === Subject: Re: Solve: cos(th) / th = c ? >> For f(x) = cos(x)/x - c, the solution in (0,Pi/2) will be a stable >> equilibrium if -2 < f¹(x) < 0 at the equilibrium. This is true >> for approximately .7964304168 < x < Pi/2. >stability of this method! How was that result derived? An equilibrium p for the iteration x_{n+1} = g(x_n) is said to be stable if for every epsilon > 0 there is delta > 0 such that if |x_0 - p| < delta then |x_n - p| < epsilon for all n > 0. In particular, if |g¹(p)| < 1, there is r > 0 such that |g(x) - p| < |x - p| whenever 0 < |x - p| < r, and then the deÞnition of stable is satisÞed with delta = min(epsilon,r). >> Of course, other iteration schemes might be stable over a wider >> interval. In particular, Newton¹s method is always stable (at >> least for a function that is analytic at the solution, or >> is C^2 with nonzero derivative at the solution). >What are you referring to when you say C^2 here? Twice differentiable, and the second derivative continuous. >Newtons method can result in periodic behaviour, can¹t it? In some cases >you can bounce between two values continually... Yes, and chaotic behaviour is also possible. This does not contradict the statement about stability, which only refers to what happens when the initial value is sufÞciently close to the equilibrium. What happens when the initial value is far away from the equilibrium is another matter entirely. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Solve: cos(th) / th = c ? >> This looks quite simple, but I can¹t seem to Þgure it out. In the >> equation: >> cos( th ) / th = c >> I need a closed form solution for _th_ such that 0 <= _th_ < 2*pi sorry, just to add, it pretty much goes without saying I am not providing a closed form solution here (which it seems doesn¹t exist), but a numerical approximation to solutions! alex === Subject: Re: Solve: cos(th) / th = c ? > > This looks quite simple, but I can¹t seem to Þgure it out. In the > > equation: > > cos( th ) / th = c > > I need a closed form solution for _th_ such that 0 <= _th_ < 2*pi > > The bounds are placed since I know there are multiple answers, but I > > need the answer in this range. Since lim th->0 is inÞnity and limit > > th->pi/2 is 0, and this is a continuous function there is guaranteed to > > be an answer. > Newton¹s iteration method is the right one here. > Based on the OP¹s second post in this thread, it doesn¹t sound like any > iterative method would be the right one here. In particular, he said > I just want something that can execute in constant time. > There can be no closed form solution for any transcendental equation. > That¹s obviously false. For example, given the transcendental equation > cos(x) = c, a closed-form solution is x = arccos(c). > David Cantrell Yes David, suggestion regarding accuracy of terminology is well taken. clear even before he said it himelf, OP wanted something fast, by constant time he meant a short period, I replied in same tenor, being inclined myself to spend a few moments only.. yes,one needs to be === Subject: Re: Solve: cos(th) / th = c ? > clear even before he said it himelf, OP wanted something fast, by > constant time he meant a short period, I replied in same tenor, being Well, truly I do want something that can execute in constant time which I now understand is not possible. However, the constant time is quite important, so using the apporximation from David is a good option for me. -- edA-qa mort-ora-y Idea Architect http://disemia.com/ === Subject: Re: f(f(f...))=x > f(*)-->function of * > f(f(f(...)))=x -->f(x)=x > If I start with an initial value of x_o for x, and apply f repeatedly, > producing the sequence f(x_o), f(f(x_o)), f(f(f(x_o))), etc, and if > this sequence converges to some value, then this value is the solution > to f(x)=x. For example if f(x)=2sin(x), and starting with x_o=2, the > 25th step gives a value of 1.895495197, and the 26th step gives > 1.895493673 using a hand-held calculator (Sharp EL 506H), if I have > counted properly 8-). What is the name of this process? I was trying to remember that in another thread and still can¹t remember a decent name for it. You can Þnd (some) zeroes of f(x) by doing this iteration: x -> f(x) + x ... the +x being the important part that makes it settle on zero points. Only effective for some equations of course. >Surely it > must be well-known. Yes, it is... not enought that I can think of the name for it! Try newton¹s method for a faster and more well behaved convergent method: http://ohmslaw.org.uk/oldsite/fc/newton/ alex === Subject: Re: f(f(f...))=x >>Surely it >> must be well-known. >Yes, it is... not enought that I can think of the name for it! It is called Þxed point iteration. >Try newton¹s method for a faster and more well behaved convergent method: >http://ohmslaw.org.uk/oldsite/fc/newton/ Fixed point iterations only give you stable Þxed points (a sufÞcient condition is that |f¹(x0)| < 1). Newton¹s method can give you stable as well as unstable Þxed points. Newton¹s method is faster, *locally*, but it can be ill-behaved if you come close to points where f¹(x)=0. Thomas >alex === Subject: Re: ~ Proof techniques for surjectivity. > When attempting to prove that functions are bijective, the proof of > surjectivity often gives me difÞculty. I would like to know what techniques > are commonly used to prove surjectivity, and how readers attempt such > proofs. Below are a couple of the techniques that I have seen used; some may > be the same. > DeÞnition. Let A and B be sets. A function (also known as a map) f from A > to B, denoted f: A -> B, is a subset F <= A x B such that for each a in A, > there is one and only one pair of the form (a,b) in F. The set A is called > the domain of the function and the set B is called the codomain of the > function. > DeÞnition. Let f: A-> B be a function. The function is surjective if for > every b in B, there exists some a in A such that f(a) = b; equivalently, if > f_*(A) = B. > Proof techniques. > 1) Show the equality of the range and codomain. > This is the Þrst technique that I used when showing surjectivity. It is > sometimes difÞcult to use because an obvious derivation from the range to > the codomain may be elusive. Often working from backwords and forwards, > trying to Þnd common ground, is useful when doing the scratchwork. > 2) DeÞne a member of the domain involving an arbitrary member of the > codomain, then show that when the domain member is mapped the result is the > lone arbitrary member of the codomain. > The author of my book frequently uses this technique, but I am at a loss > as to how he Þnds the element that is mapped. > 3) Show that the function has a right inverse. > This is very similar to #2. More or less #2 = #3. I would prefer this method. Also note, that with respect to the original task (proving bijectivity), it saves time because the right inverse is the natural candidate for a left inverse. > I¹d appreciate any constructive comments regarding further techniques or > how to more effectively use the ones that I described above. When doing > exercises, I Þnd myself spending more of my time in fruitless attempts at > proving surjectivity than anything else. There must be a simpler ways to go > about it. This depends a lot on the function in question. Marc === Subject: Re: ~ Proof techniques for surjectivity. > When attempting to prove that functions are bijective, the proof of > surjectivity often gives me difÞculty. I would like to know what techniques > are commonly used to prove surjectivity, and how readers attempt such > proofs. Below are a couple of the techniques that I have seen used; some may > be the same. The method you use depends on the particular case. Other people have provided some good ideas which will often work to prove surjectivity using counting arguments and other indirect means. But maybe you were thinking of situations where you have a particular function and you want to show it¹s surjective directly, using the deÞnition. Here¹s an example. Suppose I want to show that f: R^2 -> R^2 deÞned by f(x,y) = (x + y, x^3 + 2) is surjective. What does it mean to be surjective? It means that every element of the target (R^2) is an output of f. So take an arbitrary element of the target, say (a,b). I have to show that (a,b) is an output of f. (x,y)? ---f---> (a,b) (hit me!) What does it mean to be an output of f? It means that there is an input (x,y) which produces that output --- that is, f(x,y) = (a,b), or (x + y, x^3 + 2) = (a,b). Since I have to show that there is an input (x,y) which satisÞes this equation, I¹ll work backwards from the equation to guess what the input (x,y) should be. (That is, **I¹ll solve for x and y in terms of a and b**.) b = x^3 + 2, so x = (b - 2)^(1/3). Then x + y = a, so (b - 2)^(1/3) + y = a, or y = a - (b - 2)^(1/3). So my guess is that the input (x,y) = (a - (b - 2)^(1/3), (b - 2)^(1/3)) will produce the output (a,b). I ought to check that my guess works by plugging it into f: f(x,y) = f((a - (b - 2)^(1/3), (b - 2)^(1/3)) = ... (ugly algebra omitted) ... = (a,b). It works. Therefore, the arbitrary element (a,b) of the target set is an output of f, so f is surjective. uses this technique, but I am at a loss as to how he Þnds the element that is mapped. The procedure above shows how this is done: Take an arbitrary element of the target set, *assume* that it¹s f(some input), then solve for (some input) in terms of the arbitrary element. If you think about it, this is exactly what you did to Þnd the inverse of a function back in algebra/precalculus/calculus/wherever you learned about inverse functions. Here you¹re Þnding a formula for a right inverse (if there is one); in the case where the function is bijective, it will turn out to be a left inverse as well. Note that the solving for the input in terms of the output really is working backwards, and a proper proof requires that you check that your guess really works. Try using this approach with f: R -> R given by f(x) = x^2. What goes wrong? === Subject: Re: ~ Proof techniques for surjectivity. > The method you use depends on the particular case. Other > people have provided some good ideas which will often > work to prove surjectivity using counting arguments and > other indirect means. But maybe you were thinking of > situations where you have a particular function and you > want to show it¹s surjective directly, using the deÞnition. > Here¹s an example. I usually have a deÞnition for the function I wish to show is surjective, but often it is made up of functions from sets of function to sets of functions. With all the parentheses and things, the functions aren¹t very nice to work with. So trying to Þnd an inverse for them gives me pains. Your example was very well put together. I thank you for it. I will use it for polynomial functions and things, but for sets of functions, I am not sure if it will be so simple. However, I will give it a try. > uses this technique, but I am at a loss as to how he Þnds > the element that is mapped. The procedure above shows how > this is done: Take an arbitrary element of the target set, > *assume* that it¹s f(some input), then solve for (some > input) in terms of the arbitrary element. If you think > about it, this is exactly what you did to Þnd the inverse > of a function back in algebra/precalculus/calculus/wherever > you learned about inverse functions. Here you¹re Þnding > a formula for a right inverse (if there is one); in the > case where the function is bijective, it will turn out to > be a left inverse as well. Yes, the author initially was showing his scratchwork, but then states that he will no longer do so and that many of the proofs will seem to come out of nowhere, but that is just because the readers can not see his reasoning and scratchwork. I take that is the case when he deÞnes an element to prove surjectivity. > Note that the solving for the input in terms of the > output really is working backwards, and a proper > proof requires that you check that your guess really > works. I am aware. The author emphasized the errors of backwards proofs. I had no idea that I was doing backwards proofs all throughout highscool! None of my teachers said otherwise when we submitted math answers. === Subject: Re: ~ Proof techniques for surjectivity. >> The method you use depends on the particular case. Other >> people have provided some good ideas which will often >> work to prove surjectivity using counting arguments and >> other indirect means. But maybe you were thinking of >> situations where you have a particular function and you >> want to show it¹s surjective directly, using the deÞnition. >> Here¹s an example. > I usually have a deÞnition for the function I wish to show is > surjective, but often it is made up of functions from sets of function to > sets of functions. With all the parentheses and things, the functions aren¹t > very nice to work with. So trying to Þnd an inverse for them gives me > pains. > Your example was very well put together. I thank you for it. I will use > it for polynomial functions and things, but for sets of functions, I am not > sure if it will be so simple. However, I will give it a try. I forgot to add one other thing. You may have been told when you learned the algorithm for Þnding the inverse of a function that it doesn¹t always work --- and the same caveat applies here since it¹s really the same thing. The catch is that in any situation where you want to solve for one thing in terms of another, you might not be able to do it (or it might be hard to do). For example, take f: R -> R deÞned by f(x) = {x + 1 if x < 0 {x if x >= 0. f is surjective, as you can see from the graph. (Why?) If you try to write a proof by solving for inputs in terms of outputs, you¹ll Þnd you need to take cases. Okay, a little more work than the last example, but still doable. But consider g: R -> R deÞned by g(x) = 2x^5 + 3x^4 - 2x^3 + x^2 - x + 1. Again, you can see from the graph that g is surjective. You are not likely to be able to prove this by solving for inputs in terms of outputs. The second example kind of leads back to the idea behind some of the other techniques people mentioned. See, if I¹m trying to show f: X -> Y is surjective, I¹m supposed to take an arbitrary y in Y, then show *there is* an x in X such that f(x) = y. But if you¹ve learned about existence proofs in your course, you found out that in math, there is doesn¹t mean that you must *Þnd a thing explicitly*. Lots of existence proofs are indirect --- you show (essentially) that it would be a contradiction if the x you want *didn¹t* exist, without actually Þnding x. So returning to the function g above, you can show it¹s surjective by using the Intermediate Value Theorem from calculus (which is a there exists theorem). I¹m sure you can work out the details. Some of the other approaches people mentioned are similar, in that you don¹t actually *Þnd* an input which gives the arbitrary output --- rather, you show by some means that such an input must exist. (I¹ve noticed when I teach math proof that surjectivity seems to confuse people. I¹m sure you¹ll get the idea with a little practice. Good luck.) Bruce I. === Subject: Re: ~ Proof techniques for surjectivity. > Suppose I want to show that f: R^2 -> R^2 deÞned by > f(x,y) = (x + y, x^3 + 2) > is surjective. > ... (ugly algebra omitted) ... = (a,b). To avoid ulgy algebra notice x^3 + 2 ranges over all of R and for any given x, x + y ranges over all of R. To formalize, Þrst show for all r in R, some x_r with (x_r)^3 + 2 = r then f(x_r + (s - x_r), (x_r)^3 + 2) = (s,r). So take x = x_r = cbrt(r - 2) and y = s - x_r to get f(x,y) = (s,r) > It works. Therefore, the arbitrary element (a,b) of the > target set is an output of f, so f is surjective. > uses this technique, but I am at a loss as to how he Þnds > the element that is mapped. The procedure above shows how > this is done: Take an arbitrary element of the target set, > *assume* that it¹s f(some input), then solve for (some > input) in terms of the arbitrary element. If you think > about it, this is exactly what you did to Þnd the inverse > of a function back in algebra/precalculus/calculus/wherever > you learned about inverse functions. Here you¹re Þnding > a formula for a right inverse (if there is one); in the > case where the function is bijective, it will turn out to > be a left inverse as well. > Note that the solving for the input in terms of the > output really is working backwards, and a proper > proof requires that you check that your guess really > works. > Try using this approach with f: R -> R given by f(x) = x^2. > What goes wrong? === Subject: Re: ~ Proof techniques for surjectivity. > So take x = x_r = cbrt(r - 2) and y = s - x_r to get > f(x,y) = (s,r) What does cbrt represent? === Subject: Re: ~ Proof techniques for surjectivity. |> So take x = x_r = cbrt(r - 2) and y = s - x_r to get |> f(x,y) = (s,r) | What does cbrt represent? cbrt = cube root _____________________________Gerard S. === Subject: Re: ~ Proof techniques for surjectivity. >When attempting to prove that functions are bijective, the proof of >surjectivity often gives me difÞculty. I would like to know what techniques >are commonly used to prove surjectivity, and how readers attempt such >proofs. Below are a couple of the techniques that I have seen used; some may >be the same. >DeÞnition. Let A and B be sets. A function (also known as a map) f from A >to B, denoted f: A -> B, is a subset F <= A x B such that for each a in A, >there is one and only one pair of the form (a,b) in F. The set A is called >the domain of the function and the set B is called the codomain of the >function. >DeÞnition. Let f: A-> B be a function. The function is surjective if for >every b in B, there exists some a in A such that f(a) = b; equivalently, if >f_*(A) = B. >Proof techniques. >1) Show the equality of the range and codomain. I don¹t think I¹d call this a technique, quite - seems to me that this technique amount to saying you can show the function is surjective by showing it¹s surjective. >2) DeÞne a member of the domain involving an arbitrary member of the >codomain, then show that when the domain member is mapped the result is the >lone arbitrary member of the codomain. > The author of my book frequently uses this technique, but I am at a loss >as to how he Þnds the element that is mapped. Well, that¹s the tricky part. If you see a person do this he could perhaps explain how he did it for some particular function, but there¹s no recipe for how to do it in general. >3) Show that the function has a right inverse. > This is very similar to #2. This is exactly the _same_ as #2, just in different words. > I¹d appreciate any constructive comments regarding further techniques or >how to more effectively use the ones that I described above. When doing >exercises, I Þnd myself spending more of my time in fruitless attempts at >proving surjectivity than anything else. There must be a simpler ways to go >about it. A very useful technique that¹s not on the list: If A and B are _Þnite_ sets, A and B have the same number of elements, and f : A -> B is injective then it follows that f is surjective. (For example: if p is prime and a <> 0 mod p then there exists b such that ab = 1 mod p. Pf: Fix a. DeÞne f : {0,...,p-1} -> {0,...,p-1} by f(x) = the remainder when ax is divided by p. Use the fact that p is prime to show that f is injective. Hence f is surjective, hence there exists x with f(x) = 1, QED.) ************************ David C. Ullrich === Subject: Re: ~ Proof techniques for surjectivity. > A very useful technique that¹s not on the list: If A and B are > _Þnite_ sets, A and B have the same number of elements, > and f : A -> B is injective then it follows that f is > surjective. Yes, I reasoned that when learning about functions, but since I haven¹t learned about cardinality of sets yet, I couldn¹t prove the theorem properly. If I have a deÞnition of a function that I wish to show is bijective, I think I will just try to Þnd an inverse for it and not prove injectivity and surjectivity separately. === Subject: Re: ~ Proof techniques for surjectivity. >When attempting to prove that functions are bijective, the proof of >surjectivity often gives me difÞculty. I would like to know what techniques >are commonly used to prove surjectivity, and how readers attempt such >proofs. Below are a couple of the techniques that I have seen used; some may >be the same. [cut] >Proof techniques. >1) Show the equality of the range and codomain. > I don¹t think I¹d call this a technique, quite - > seems to me that this technique amount to saying > you can show the function is surjective by > showing it¹s surjective. Actually, if I recall correctly, I used that technique in my thesis (or at least something very similar to it). For example, suppose you have a group homomorphism f: H -> G from the Þnite group H into the Þnite group G. To make the example simpler, suppose that f is injective. Suppose that you know what the group G is, but you only have some local information about the structure of H (like the order of a centralizer of one of H¹s involution, assuming H is of even order, which I am assuming). Note that the range of f is isomorphic to H. I want to show that f is onto. Using the local information about H, there is a theorem that gives the order of H (I guess H must be a simple group also, but this is also assumed). Thus, I know the order of the range of f. But, that order turns out to be equal to the order of G. Thus, f is an onto map and H and G are isomorphic. Here, G is a known simple group and H is an unknown simple group about which I know some local information, and I am trying to show that in fact H must be the given known simple group G. -- Bill Hale === Subject: Re: Please comment > If by typograhical rewrite rules you mean structured > symbolic manipulation, then no that¹s only a small part of > mathematics. There¹s a lot to just making rigorous abstract > notions (this may involve symbols). Most proving that goes > on is non-symbolic, or any symbolic manipulation is the most > trivial part. consistant with Church-Turing thesis (or I am lost again). Please, did you read my remarks regarding the stratiÞcation of existence from self subsistent objects to numbers? What do you think? > And a side question, if someone kindly will, has there been a more > contemporary attempt to formalize mathematics in depth and breadth of > Principia Mathematica? > Yes, but not in such a form (a three volume monograph). > There is very little in mathematical logic that is not a > reaction to PM. see Proof theory, automated deduction, mizar. I see. I will certainly look into these. Andrew === Subject: Re: Please comment > Principia tried to resolve the set theoretic paradoxes > with the Theory of Types. The simple theory still has > its inconsistencies, There are no inconsistencies in the simple theory of types. > and AFAIK nobody has answered the > question for the ramiÞed theory, which I believe is not > in Principia. Answered what question? The ramiÞed theory of types, used in Principia, isn¹t inconsistent either. === Subject: Catalan formula via tableaux (better formatting) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6LFnV206639; It¹s well-known that the Catalan numbers count, among many things, the number of standard Young tableaux of shape (n,n). I would like to use the tableaux in an elementary derivation of the formula for the Catalan numbers, without reference to the hook-length formula and without using a bijection to another combinatorial object (e.g., subdiagonal lattice paths). Write C(m,n) for the binomial coefÞcient m choose n. There are C(2n,n) tableaux of shape (n,n) that are increasing along rows -- choose half of the numbers, order them in the top row, order the rest for the bottom row. How many of these are standard Young tableaux? That is, how many of these increase down each column? Let b_i for 0 =< i =< n be the number of tableaux with i bad columns, i.e., where the top entry is larger than the bottom entry. The standard Young tableaux are counted by b_0, no bad columns. Experimental statistics on small n lead to the conjecture b_0 = b_1 = ... = b_n. If this is true in general, then we have Cat(n) = b_0 = 1/(n+1) * C(2n,n). I would appreciate any ideas on proving b_0 = b_1 = ... = b_n. Some further notes: By switching the order of the rows, b_0 = b_n, b_1 = b_(n-1), etc. Any Þxed change of a standard Young tableaux, e.g., þipping the Þrst column, could lead to one or more bad rows depending on the tableaux, so I¹m not sure anything that explicit will work. === Subject: Re: Catalan formula via tableaux (better formatting) > It¹s well-known that the Catalan numbers count, among many things, > the number of standard Young tableaux of shape (n,n). I would like > to use the tableaux in an elementary derivation of the formula for > the Catalan numbers, without reference to the hook-length formula and > without using a bijection to another combinatorial object (e.g., > subdiagonal lattice paths). > Write C(m,n) for the binomial coefÞcient m choose n. There are > C(2n,n) tableaux of shape (n,n) that are increasing along rows -- > choose half of the numbers, order them in the top row, order the rest > for the bottom row. How many of these are standard Young tableaux? > That is, how many of these increase down each column? > Let b_i for 0 =< i =< n be the number of tableaux with i bad > columns, i.e., where the top entry is larger than the bottom entry. > The standard Young tableaux are counted by b_0, no bad columns. > Experimental statistics on small n lead to the conjecture b_0 = b_1 > = ... = b_n. If this is true in general, then we have Cat(n) = b_0 = > 1/(n+1) * C(2n,n). I think this is equivalent (by using the bijection to lattice paths) to the Feller-Chung theorem. This states the number of (1,1) - (1,-1) paths from (0,0) to (2n,0) having 2k steps below the origin equals in Mathematics Magazine by David Callan with a nice bijective proof of this. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: resources about LOGIC? X-Library: Indy 9.00.10 I am in the need of some resources about logic,which,I mean,is concerned with axiom,predicate,proposion,etc. Could anyone give me a hint about how to get them or something? -- Composed with Newz Crawler 1.7 http://www.newzcrawler.com/ === I suppose I should warn people about this. He¹s gone around the bend, it appears. He was once a friend and a trusted moderator. He has abandoned science and mathematics for ancient Egyptian curses? Respectfully, Roger L. Bagula tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : URL : http://home.earthlink.net/~tftn URL : http://victorian.fortunecity.com/carmelita/435/ === Subject: Re: heavy trolling I must apologize for the doping charge. You are right, it was intended to incite controversy - the behavior of an internet troll. But I am not a good troll, an ideal troll creates a heated discussion, see http://en.wikipedia.org/wiki/Internet_troll According to George H. Mead, individuals are products of society, and the self arises out of social experience. I don¹t know what your self-concept is, but it is something wrong with it. The emergence of self-consciousness and the self is possible through the gradually developing ability in childhood to take the role of the other and to visualize your own performance and appearance from the point of view of others. Very young children do not yet have the ability to imagine the world from other point of views. Although often your posts could come from an unrecognized genius, sometimes your behavior reminds me of a child. A number of readers and members of this newsgroup have complained about your behavior, but although mainly these people are reading your posts, you seem to have heavy difÞculties to understand them. Because you have the highest posting rate in many newsgroups and are constantly posting all sorts of weird things (without clear personal beneÞt), it looks like you are using some kind of doping or drugs. Jochen === Subject: Re: heavy trolling Blaming my abilities or disabilities on drugs or other illegal behavior is slander. I may understand, but I don¹t have to like your behavior. You know if anyone does that I work very hard. I take the responsibility of keeping information þowing and new ideas in front of those who need them.... I don¹t suffer bad behavior well: I tend to tell the person they have stepped over the line. I¹m reading The Dark Age Ahead by economist Jane Jacobs. Although I don¹t agree with everything she says, I agree that we a tending toward a self destructive society that will end in a cultural failure. One of the main failures she sees is in the educational system and professional self policing as connected to maintaining a technology based society. Our current technology isn¹t adequate for a post oil based society to maintain current population and cultural levels. This apparently means that a lot of people will die in the next 10 to 50 years ( probably nearer than far...). I think people should be aware of the possible/ probably future in order to prepare and maybe save some of the culture that is good. > I must apologize for the doping charge. You are right, > it was intended to incite controversy - the behavior > of an internet troll. But I am not a good troll, an ideal > troll creates a heated discussion, see > http://en.wikipedia.org/wiki/Internet_troll > According to George H. Mead, individuals are products of > society, and the self arises out of social experience. > I don¹t know what your self-concept is, but it is something > wrong with it. The emergence of self-consciousness > and the self is possible through the gradually developing > ability in childhood to take the role of the other and > to visualize your own performance and appearance from the > point of view of others. Very young children do not yet > have the ability to imagine the world from other point of > views. > Although often your posts could come from an unrecognized > genius, sometimes your behavior reminds me of a child. > A number of readers and members of this newsgroup > have complained about your behavior, but although > mainly these people are reading your posts, you > seem to have heavy difÞculties to understand them. > Because you have the highest posting rate in many > newsgroups and are constantly posting all sorts of > weird things (without clear personal beneÞt), it looks > like you are using some kind of doping or drugs. > Jochen -- Respectfully, Roger L. Bagula tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : URL : http://home.earthlink.net/~tftn URL : http://victorian.fortunecity.com/carmelita/435/ === Subject: Re: heavy trolling >I must apologize for the doping charge. You are right, >it was intended to incite controversy - the behavior >of an internet troll. But I am not a good troll, an ideal >troll creates a heated discussion, see >http://en.wikipedia.org/wiki/Internet_troll ... >Although often your posts could come from an unrecognized >genius You¹re *sure* you¹re not a good troll? I mean, some of the best trolling I¹ve ever seen consisted in nothing more than the bland assertion of a patent absurdity. Lee Rudolph === Subject: Re: heavy trolling You have been a real fool at times haven¹t you? I realize you are very well educated and intelligent, but you just make yourself look so damned bad at times. I¹ve thought of including you with the other self destructive þame producers, but you sometimes actually answer questions right. That¹s a virtue I like. >>I must apologize for the doping charge. You are right, >>it was intended to incite controversy - the behavior >>of an internet troll. But I am not a good troll, an ideal >>troll creates a heated discussion, see >>http://en.wikipedia.org/wiki/Internet_troll > ... >>Although often your posts could come from an unrecognized >>genius > You¹re *sure* you¹re not a good troll? > I mean, some of the best trolling I¹ve ever seen consisted > in nothing more than the bland assertion of a patent absurdity. > Lee Rudolph -- Respectfully, Roger L. Bagula tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : URL : http://home.earthlink.net/~tftn URL : http://victorian.fortunecity.com/carmelita/435/ === Subject: Re: heavy trolling >> I mean, some of the best trolling I¹ve ever seen consisted >> in nothing more than the bland assertion of a patent absurdity. Even on Usenet I Þnd it difÞcult to believe that anyone would be so gullible as to respond to a bland assertion of a patent absudity. If the assertion is patently absurd, only a true naif would respond. Either that, or an egocentric person with a need to show himself or herself as somehow intellectually superior to the poster of the patent absurdity. Seems unlikely, though. its search engine? You¹ll get some small number of free searches per day (I think 10), and after that you have to use Microsoft¹s Hotmail Billing System to pay some sort of micro-payment per search. Unbelievable! -- Kevin S. Wilson Tech Writer at a University Somewhere in Idaho You can safely ignore Kevin in order to maximise life¹s experience. --A. Loon, in alt.religion.kibology === > I suppose I should warn people about this. > He¹s gone around the bend, it appears. > He was once a friend and a trusted moderator. > He has abandoned science and mathematics for > ancient Egyptian curses? Why are you posting this off-topic garbage here? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === > I suppose I should warn people about this. > He¹s gone around the bend, it appears. > He was once a friend and a trusted moderator. > He has abandoned science and mathematics for > ancient Egyptian curses? > Why are you posting this off-topic garbage here? Why are you creating a whole new thread about it? === >> I suppose I should warn people about this. >> He¹s gone around the bend, it appears. >> He was once a friend and a trusted moderator. >> He has abandoned science and mathematics for >> ancient Egyptian curses? >> Why are you posting this off-topic garbage here? > Why are you creating a whole new thread about it? I created no new thread. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === > >> >> I suppose I should warn people about this. >> He¹s gone around the bend, it appears. >> He was once a friend and a trusted moderator. >> He has abandoned science and mathematics for >> ancient Egyptian curses? >> >> Why are you posting this off-topic garbage here? > > Why are you creating a whole new thread about it? > I created no new thread. By changing the subject title, you created a new thread for every newsreader set to thread by subject. Also, Google archives posts by subject, so for evermore, you are enshrined as the creator of the thread Google Groups. See http://snipurl.com/7x8n === Subject: Re: Can anyone solve this ??? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6LGV1N10717; >Each different letter represents an integer from 1 to 26 - all you >need to do is work out which letter corresponds to which number. >Hint: the value for the letter Z can be determined by elimination. >A + B = D >B * K = C >F * U = D >F * K = L >H + I = J >Q * Q = R >R + S = T >T = K * G >U + V = W >F + K = Q >W + H = O >D + E = N + P >V * W = X + Y >O + V = N >M + I = Y >Any help please ??? To get you started: F * K = L, so neither F nor K are equal to 1, so they are 2 & 3 or bigger. F + K = Q, so Q is atleast 5. Q * Q = R, so R is atleast 25, but it can¹t be 36, so R = 25. R = 25; Q = 5; F,K are 2,3 or 3,2; L = 6. === Subject: Re: Agnostic MorituriMax > No, that would be true if there were nothing to help us know, but that > ago. That was so important that the entire civilized world started > counting time from that moment. > You might want to check with the chinese about that... Besides, it seems we are not capable of thinking or imagining anything truly original and unique, not made of parts we have seen or heard about before. (If you don¹t think so, try it.) Thus, we can think of unicorns, dragons, and what not, but analyzing them, we realize there are made of parts we already know. So, primitive people lacked any knowledge of God. They could not possible have imagined the existence of God, an eternal, all powerful, pure spirit, not made of or similar to anything they could see, hear, or imagine. Evidently, God wanted people to know about Him. Therefore, God chose a group of people to tell them, and made miraculous feats to convince them of His reality, so they could bear witness of His existence. That would appear to have been a tremendous privilege for that people, but God has no favorites and is fair. So, that people has had to pay a very high price for that privilege. But they wore witness and now it seems the whole world have heard about God. Peter === Subject: Re: Agnostic MorituriMax > Besides, it seems we are not capable of thinking or imagining anything > truly original and unique, not made of parts we have seen or heard > about before. (If you don¹t think so, try it.) Thus, we can think of > unicorns, dragons, and what not, but analyzing them, we realize there > are made of parts we already know. So, primitive people lacked any > knowledge of God. Complete nonsense. You fail to appreciate the evolution of religious belief just as you failed to appreciate the evolution of the calendar. Primitive people have often devised their own theistic explanations for things they could not explain otherwise. Even monotheism is not unique to what you would regard as non-primitive peoples. There is nothing difÞcult about the concept of God; that¹s why so many God-variants have been invented throughout history! > They could not possible have imagined the existence > of God, an eternal, all powerful, pure spirit, not made of or similar > to anything they could see, hear, or imagine. Evidently, God wanted > people to know about Him. Therefore, God chose a group of people to > tell them, and made miraculous feats to convince them of His reality, > so they could bear witness of His existence. That would appear to have > been a tremendous privilege for that people, but God has no favorites > and is fair. So, that people has had to pay a very high price for that > privilege. But they wore witness and now it seems the whole world have > heard about God. Oh, pass the sick bag! Revealing his existence by a series of miracles and then having the witnesses suffer to atone for their good fortune is fair, is it? Where are your miracles now, believer? Or must we make do with the occasional weeping statue? === Subject: Re: Agnostic MorituriMax > > > If you would like to know why God exists and how God came into > > existence, I don¹t know. Nobody knows. That is a big mystery, which is > > clearly beyond our mental capacity. But I agree that for an > > intelligent man to accept something he does not understand, he needs > > is always conserved. Since it cannot be created or destroyed, it means > > it must have always existed, like God. Why it exists and how it came > > Not necessarily. If the Big Bang Theory holds up then there was a speciÞc > point when there was no space, no mass, no energy, no time, but until we can > Þgure out what laws held before the planck limit, then we cannot formulate a > theory to explain what was there before the beginning.. I use the term before > loosely since with no time there wouldn¹t have been a before.. look at some of > Alan Guth¹s work for how our universe may have begun. > > > into existence, nobody knows, and it is also a big mystery; but we > > cannot deny that mass-energy exists, so we have to accept it. To me, > > that is the intellectual justiÞcation to accept God. > > To me, that is one possibility, Þne for you, for me it is the same as magic, we don¹t know so we make up what is comfortable for each person to accept. > No, that would be true if there were nothing to help us know, but that > you know what AC means? Factory Air? Double-A === Subject: Re: Agnostic MorituriMax > No, that would be true if there were nothing to help us know, but that > ago. That was so important that the entire civilized world started > counting time from that moment. > You might want to check with the chinese about that... Better yet, he should check history. It was about 500 years afterwards that christians started counting the years AD. === Subject: Re: Agnostic MorituriMax > > No, that would be true if there were nothing to help us know, but that > > ago. That was so important that the entire civilized world started > > counting time from that moment. > You might want to check with the chinese about that... > Better yet, he should check history. It was about 500 years afterwards that > christians started counting the years AD. Michael === Subject: Re: Agnostic MorituriMax > > > > No, that would be true if there were nothing to help us know, but that years > > ago. That was so important that the entire civilized world started > > counting time from that moment. > > > > You might want to check with the chinese about that... > Better yet, he should check history. It was about 500 years afterwards > that > christians started counting the years AD. Absolutely! In fact, Constantine stole the Jewish idea of a seven-day week to form the new calendar thus replacing the Roman version of the sections of the month (like Ides of March, or April, etc.). That¹s when he changed the existing Sabbath day from the existing one to the new day he named Sunday (being an ex-Mithraist sun-worshiper). He then guarenteed that all Christians would no longer keep the Sabbath holy because they used the wrong day! Checkout the excellent summary of the history of the calendar at: http://www.geocities.com/Athens/Oracle/9941/endtime.html Hadji === Subject: Re: Agnostic MorituriMax >> To me, that is one possibility, Þne for you, for me it is the same as >magic, we don¹t know so we make up what is comfortable for each person to >accept. >> >> No, that would be true if there were nothing to help us know, but that >> you know what AC means? >Nothing. The correct term is AD. > A.D. means Anno Domini, which, in Latin means year of our lord. > So, recognizing Christ as our lord, the meaning is the same No. You stated AC, which can only mean after christ. AD, or Anno Domini means in the year of our lord. Which refers to the year of his birth, not after his death. AC is not used. CE is a better term to use in any event. >, even > if the annotation was a bit off. If you are going to spew bull, at least get the nomenclature correct. http://en.wikipedia.org/wiki/Anno_Domini === Subject: Re: Agnostic MorituriMax > > > > If you would like to know why God exists and how God came into > > existence, I don¹t know. Nobody knows. That is a big mystery, which is > > clearly beyond our mental capacity. But I agree that for an > > intelligent man to accept something he does not understand, he needs > > is always conserved. Since it cannot be created or destroyed, it means > > it must have always existed, like God. Why it exists and how it came > > > > Not necessarily. If the Big Bang Theory holds up then there was a > speciÞc > > point when there was no space, no mass, no energy, no time, but until we > can > > Þgure out what laws held before the planck limit, then we cannot > formulate a > > theory to explain what was there before the beginning.. I use the term > before > > loosely since with no time there wouldn¹t have been a before.. look at > some of > > Alan Guth¹s work for how our universe may have begun. > > > > into existence, nobody knows, and it is also a big mystery; but we > > cannot deny that mass-energy exists, so we have to accept it. To me, > > that is the intellectual justiÞcation to accept God. > > > > To me, that is one possibility, Þne for you, for me it is the same as > magic, we don¹t know so we make up what is comfortable for each person to > accept. > No, that would be true if there were nothing to help us know, but that > you know what AC means? > Nothing. The correct term is AD. You are right. I am sorry; that was a typo. I meant AD. Peter === Subject: Honoraria available for ID of KNOWN nx4 matrices with 4 speciÞc properties Cumulative Inquiry, Inc, a start-up R&D Þrm chartered in Huntsville AL (USA), will pay reasonable and customary honoraria for identi- Þcation of any KNOWN formal or applied n x 4 matrix with all four of the properties deÞned below. Descriptions of any such objects must be posted to the sci.math group in order to qualify for the honoraria. Rights to practical use (if any) of the identiÞed objects remain with the author, with the three following exceptions. 1) Honoraria-eligible objects do not include any matrix qualifying as an ordered DAG representable by a dim 2 partial order (see posting to sci.math.research by David Halitsky with title beginning Consultancy fee ...) 2) Honoraria-eligible objects do not include any matrix involving a uniformly networked hierarchy character- izable as a uniformly n-ary b-tree index of max vertex depth 2 (n^2+n+1 vertices) on which the n^2+n+1 lines of a Þnite projective plane have been deÞned (see posting to sci.math by David Halitsky with title beginning Bear of very little brain...) 3) Honoraria will not be paid for ID of any matrix related to any bioinformatic representation of a biomolecular coding sequence and its amino acid reþex which can be inferred from the type of matrix implicitly deÞned in the technical PDF available at the StrucClues website sponsored by Cumulative Inquiry (http://www.CumulativeInquiry.com/StrucClues) Fiduciary references for prompt payment of fees/honoraria by Cumulative Inquiry Inc can be obtained from William F. Mann at wfmann@alum.mit.edu. Business address/phone for Cumulative Inquiry Inc are: David Halitsky 2928 Stewart Campbell Pte Thompsons Station TN 37179 USA Cell: 256-426-6243 (USA) ************************************************************* Salient properties of nx4 matrix A: 1) (aij)^2 is an integer, for 1 <= i <= n) and j = 1,2,3,4 2) (ai1)^2 + (ai2)^2 + (ai3)^2 + (ai4)^2 = n-1, for 1 <=i <= n 3) SUM((ai1)^2) = SUM((ai3)^2) for 1 <= i <= n 4) SUM((ai2)^2) = SUM((ai4)^2) for 1 <= i <= n) === Subject: Two questions in Complex Analysis I have to short question in Complex Analysis which are part of an excersize in my course. I¹m quite stuck, any hint would be helpful. 1. Let f(z) = sum_{n=0}^{infty}a_{n}z^n be an analytic function in an open circle of radius R around 0. It is known that | f(z) | <= M. Let z_{0} be the closest 0 of f(z). Prove that: |z_{0}| >= R|a_{0}|/(M+|a_{0}|) 2. Let g(z) be an analytic function that is deÞned for |z|>1. It is known that Im ( f(z) ) = 0 for each z in [1,infty}. Prove that Im( f(z) ) = 0 for each z in (-infty,1]. === Subject: Re: Two questions in Complex Analysis 2. I think the second question can be solved as follows : Note that g(z):= conjugate (f (conjugate z)) is holomorphic on |z|>1. Note that f-g =0 on [1,oo). By the identity theorem its zero everywhere( there¹s a limit point of zeros). Thus f=g everywhere. So on (-oo,1] f= conjugate(f) > I have to short question in Complex Analysis which are part of an > excersize in my course. I¹m quite stuck, any hint would be helpful. > 1. Let f(z) = sum_{n=0}^{infty}a_{n}z^n be an analytic function in > an open circle of radius R around 0. It is known that | f(z) | <= M. > Let z_{0} be the closest 0 of f(z). Prove that: > |z_{0}| >= R|a_{0}|/(M+|a_{0}|) > 2. Let g(z) be an analytic function that is deÞned for |z|>1. > It is known that Im ( f(z) ) = 0 for each z in [1,infty}. > Prove that Im( f(z) ) = 0 for each z in (-infty,1]. === Subject: Estimate requested for display of E4 in E3 using the 4 generators of the rhombic dodecahedron Background In Regular Polytopes, Coxeter discussed the general principles behind the fact that the 14 vertices of the rhombic dodecahedron (RD) form a symmetric shadow of gamma-4, the dim 4 measure polytope (hypercube or tesseract.) In this shadow projection, the remaining two of gamma-4¹s 16 vertices go to the center of the RD. Because this shadow projection exists, the 4 generators of the RD (and their opposing radius vectors) in dim 3 can be considered as the projections of the four axes of a rectangular Cartesian coordinate system in dim 4. Hence, some idea of how dim 4 point-sets look in dim 3 can be obtained by using the 4 generators and their opposing vectors as kind of an oblique dim 4 coordinate system in dim 3. Call this coordinate system CS(4,3) Request for Estimate Cumulative Inquiry, Inc, a start-up R&D Þrm chartered in Huntsville AL (USA), requests an estimate for a graphics program that will permit the display of certain dim 4 point sets in the coordinate system CS(4,3). The program will accept as input a set of points Pi (xi,yi,zi,wi) in dim 4 (say i <= 20) and display them within CS(4,3). If it helps simplify programming, it may be worth- while to note that xi^2 for any point will always be an integer, and same for yi, zi, wi. Also, it will be true that for any input set Pn of n points, xi^2 + yi^2 +zi^2 + wi^2 = n-1 for any point Pi in Pn. Finally, certain points in the input set Pi will have one or two zero-coordinates, and will thus lie on a true dim 3 sphere or dim 2 circle in dim 4. For such subsets of Pi, the program should be able to display the oblique conics or quadrics in which these points lie relative to two or three axes of CS(4,3). (The input to the program will indicate which points in Pi should be treated in this manner. Program should be JAVA/XP/IE compatible (SORRY!) Other Details Fiduciary references for prompt payment of fees by Cumulative Inquiry Inc can be obtained from William F. Mann at wfmann@alum.mit.edu. Business address/phone for Cumulative Inquiry Inc are: David Halitsky 2928 Stewart Campbell Pte Thompsons Station TN 37179 USA Cell: 256-426-6243 (USA) thinking on this matter. Sorry we can¹t pay for working up the estimate. === Subject: Re: United Airlines magazine has surprisingly hard geometry problem > (1) DBC = ABE + EBC = 20 + 60 = 80 (from problem statement) > (2) DCB = 70 (from problem statement). > (3) DBC + DCB + CDB = 180 (sum of interior angle of a triangle) > (4) CDB = 50 You mean 30, don¹t you? Bill Smythe === Subject: Re: United Airlines magazine has surprisingly hard geometry problem >> (1) DBC = ABE + EBC = 20 + 60 = 80 (from problem statement) (2) DCB = >> 70 (from problem statement). (3) DBC + DCB + CDB = 180 (sum of interior >> angle of a triangle) (4) CDB = 50 > You mean 30, don¹t you? > Bill Smythe Correct. Sorry for the typographical error. -- Lance Lamboy Go F*ck Yourself ~ Dick Cheney === Subject: Re: United Airlines magazine has surprisingly hard geometry problem > I þew United Airlines this past weekend and there was an IQ test in their > Hemispheres magazine with eleven easy problems and one high-school > geometry problem that still has me stumped. Essentially, it was this: > > Let ABC be a triangle with a point D on side AB and a point E on > side AC. We are given the following angles (in degrees): > > ABE = 20; EBC = 60; ACD = 10; DCB = 70. > > Problem: What is angle CDE in degrees? > > One can brute-force this with a lot of trigonometry, but what¹s the slick > solution? Someone I showed this to said that he¹d seen this problem > before but unfortunately he couldn¹t remember the reference. > This is one of the notorious adventitious angles problems. > The slightly different problem where the last two angles are 30 and 50 > degrees resp. is more well-known. That can be solved by inserting a line > to make an equilateral triangle and lots of isosceles triangles. > See > http://www.math-atlas.org/01_incoming/advent >SPOILER .... >One can show that the triangles ACD and CDE are similar by >using the sine rule several times and the identity >sin 20 sin 40 sin 80 = sqrt(3)/8 >(angles in degrees). This I¹d say is quite cynical. Here are some hints for those who still don¹t get it .... ***** SPOILER ALERT ***** Hint 1: Take point E¹ on line AB so that EE¹ is parallel with BC. The crux of the problem lies in showing DE¹ = EE¹. (WIth this, we can easily conclude that angle CDE = 20) Hint 2: Take point F on line AC so that CE¹ = CF. Where will the bisector of angle E¹CE meet AB? Hint 3: Consider the equilateral triangle with edge DE¹. Where would the third point of this triangle be? Perhaps you¹d have to look for another isosceles triangle. - Yuzuru Hiraga (Univ. of Tsukuba) === Subject: Re: United Airlines magazine has surprisingly hard geometry problem Originator: tchow@lagrange.mit.edu.mit.edu (Timothy Chow) [spoiler deleted] Beautiful! Did you Þgure this out yourself or do you have a reference for your proof? -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences === Subject: Re: United Airlines magazine has surprisingly hard geometry problem > Beautiful! Did you Þgure this out yourself or do you have a reference > for your proof? Did I Þgure this out myself? Yes and no. which involve more or less the same technique. # That¹s why I said Robin was being cynical when he gave a trig. hint. Once you have a grasp of the idea, and suspect that the problem indeed does involve it, then you¹re in a more advantageous position than taking a head-start. Still, Þnding out the key entity (e.g. point F in my hints) would require some effort for individual problems. I do believe I was able to solve this kind of problem by myself on my Þrst encounter, but that was a long long time ago. (20, perhaps 30 years) About references, sorry, all my knowledge of these problems come from Some of these problems are treated in Japanese elementary or junior-high class at advanced levels. Your problem, as well as that given by Clive Tooth, are called integer-angle triangle/quadrangle problems in Japan, because (as the name suggests) they involve angles with integer values. There are various Japanese works on this topic, which I believe is similar to the enumeration given by Dave Rusin in Robin Chapman¹s reference: http://www.math-atlas.org/01_incoming/advent Below are some HP¹s I found on this topic --- sorry, they too are written in Japanese, but if you can display the diagrams, maybe you can guess what they¹re asking. Top page: http://homepage2.nifty.com/sintakenoko/index.html Example applet pages: http://homepage2.nifty.com/sintakenoko/Applet/Langley2.html http://homepage2.nifty.com/sintakenoko/Applet/Langley4.html # perhaps the Langley in Þle names above may be suggestive of # some source, but I¹m not sure. Top page: http://www2.netwave.or.jp/~igoso/igoso.html integer-angle pages: http://www2.netwave.or.jp/~igoso/shomei/shomei0.htm (click the links for individual problems.) - Yuzuru Hiraga (Univ. of Tsukuba) === Subject: Re: United Airlines magazine has surprisingly hard geometry problem > [spoiler deleted] > Beautiful! Did you Þgure this out yourself or do you have a reference > for your proof? People who like that kind of puzzle _might_ also like this one: (All angles are given in degrees.) ABD and CBD are two triangles sharing a common side, BD. AD = 1 CD = 2 angle ABD = 18 angle BAD = angle CBD = 45 Prove, geometrically, that angle BCD = 54. -- Clive Tooth http://www.clivetooth.dk === Subject: Re: United Airlines magazine has surprisingly hard geometry problem > I þew United Airlines this past weekend and there was an IQ test in their > Hemispheres magazine with eleven easy problems and one high-school > geometry problem that still has me stumped. Essentially, it was this: > Let ABC be a triangle with a point D on side AB and a point E on > side AC. We are given the following angles (in degrees): > ABE = 20; EBC = 60; ACD = 10; DCB = 70. > Problem: What is angle CDE in degrees? > One can brute-force this with a lot of trigonometry, but what¹s the slick > solution? Someone I showed this to said that he¹d seen this problem > before but unfortunately he couldn¹t remember the reference. SPOILER One can show that the triangles ACD and CDE are similar by using the sine rule several times and the identity sin 20 sin 40 sin 80 = sqrt(3)/8 (angles in degrees). -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: Re: United Airlines magazine has surprisingly hard geometry problem >> I þew United Airlines this past weekend and there was an IQ test in their >> Hemispheres magazine with eleven easy problems and one high-school >> geometry problem that still has me stumped. Essentially, it was this: >> Let ABC be a triangle with a point D on side AB and a point E on >> side AC. We are given the following angles (in degrees): >> ABE = 20; EBC = 60; ACD = 10; DCB = 70. >> Problem: What is angle CDE in degrees? >> One can brute-force this with a lot of trigonometry, but what¹s the slick >> solution? Someone I showed this to said that he¹d seen this problem before >> but unfortunately he couldn¹t remember the reference. >Note that > ACB = ACD + DCB = 80, >and > ABC = ABE + EBC = 80. >Thus, the triangle ABC is isosceles (AB = AC), and BAC = 180 - 160 = 20. >Next, note that BAC = BAE (it¹s the same angle, not just equal in >measure), and so BAE = 20 = ABE. >Thus, the triangle ABE is isosceles (AE = BE), and we Þnd that > AEB = 180 - 40 = 140, >and also that the triangles ADE and BDE are congruent. Thus the angles >ADE and BDE are equal, and sum to 180, giving BDE = 90. >Finally, consider the triangle BCD. The angles DCB and DBC (alias ABC) >are known to be DCB = 70, DBC = 80. Thus, the third angle, BDC, is >found to be > BDC = 180 - (DCB + DBC) = 180 - 150 = 30. >CDE is then BDE - BDC = 90 - 30 = 60. >Well, that¹s what I got. >Dale. Dale, I also decided that CDE = 60, but don¹t see that ADE and BDE are congruent. If we label the point where lines DC and BE cross as point F, then I do believe that triangles DEF and BCF are similar in a mirror image manner, if you follow. As you point out, triangle ABE is isosceles with angles ABE=20, BAE=20 and AEB=140. Since AEB=140, then BEC=40. We were given that ACD=10, which means CFE=130, thus BFD=130 also, which makes BFC=50 and DFE=50. Since ABE=20 and BFD=130, then BDC=30, which makes ADC=150. It is at this point that I stumble. I can¹t break angle ADC into its two components which total 150, nor break AEB into its two components which total 140. We know that DFE=50, which means the other two vertices (EDF and DEF) of the small internal triangle total 130. If points D and E were located equidistant from point A then line DE would be parallel to BC, which would make triangles DEF and BCF similar and the solution would be obvious. As you can tell, geometry is not a strong point with me, but I wonder if it would help to know that line DE is perpendicular to AB. Mind you I¹m not saying it is, but if it were, would that mean that triangle DEF was mirror-image similar to BCF? If so, that would mean angle CDE=60 and BED=70. === Subject: Re: United Airlines magazine has surprisingly hard geometry problem > My workings > BEC = 180 - (80 + 60) = 40 > Let F be the point the DC crosses EB > BFC = 180 - (60 + 70) = 50 > therefore DFE = 50 > DFB = 180 - 50 = 130 > BDF = 180 - (130 + 20) = 30 I agree up to here. > BDE = 180 - (20 + 50) = 110 This seems to require that DEB is 50. I don¹t see how to get that. -- --Tim Smith === Subject: Re: United Airlines magazine has surprisingly hard geometry problem >>My Þrst thought is that if you saw this on an airline magazine, it >>would make more sense that there was a mistake in the puzzle, and that >>its original intention was to ask for angle BAC. >That¹s not a bad guess, but in the original formulation (which I don¹t >have on me because I didn¹t bother to keep a copy of the magazine), it >was a multiple-choice question... Aha! A multiple-choice question! If the choices were along the lines of: 20, 50, 80 and 120, then the puzzle might again be a reasonable difÞculty for an airline magazine! -- dgates === Subject: Re: United Airlines magazine has surprisingly hard geometry problem Originator: tchow@lagrange.mit.edu.mit.edu (Timothy Chow) >If the choices were along the lines of: 20, 50, 80 and 120, then the >puzzle might again be a reasonable difÞculty for an airline magazine! Again, I didn¹t keep a copy of the magazine, but the choices were something like 15, 20, 25, 30, and maybe not enough info. Also, it said something about doing it by geometry and not by drawing an accurate diagram and measuring (though it did suggest, in the answer section, that drawing an accurate diagram would be helpful). The only other thing I remember about the brief remarks in the answer section was a suggestion to extend lines. Also, there was some distinction between getting 11 out of 12 and 12 out of 12---genius versus super-genius or some other such meaningless distinction. I¹m still sort of interested in Þnding a purely synthetic proof. It seems are no longer on the web. Following are a couple of thoughts I¹ve had; maybe someone else can make something of them. It seems that judicious creation of isosceles and equilateral triangles is a typical trick in these problems. ABC and ABE are isosceles, of course, but it¹s not immediately clear how this helps. So, extend BC and DE until they meet at F. By cheating and using the known value of x that we¹ve calculated trigonometrically, we see that triangle BDF is isosceles. Conversely, to compute x, it sufÞces to prove that BDF is isosceles. My next thought is to create an equilateral triangle by extending BE to a point G such that BGF is equilateral. It then sufÞces to prove that triangle BDG is isosceles, or (denoting the intersection of BE and CD by H) that DGH is isosceles. But now I¹m stuck. Maybe instead, one should construct a point I on BF such that BEI is equilateral. Then AEI is isosceles and AI is parallel to CD. But again I¹m not sure what to do next. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences === Subject: Re: United Airlines magazine has surprisingly hard geometry problem >>BEC = 180 - (80 + 60) = 40 >>Let F be the point the DC crosses EB >>BFC = 180 - (60 + 70) = 50 >>therefore DFE = 50 >>DFB = 180 - 50 = 130 >>BDF = 180 - (130 + 20) = 30 > Everything up to here is correct. >>BDE = 180 - (20 + 50) = 110 > This is wrong; BDE is not 110. I don¹t even understand where the expression > 180 - (20 + 50) comes from. Maybe you¹re confusing DFE with DEB? *grin* Oh boy, the last time I do maths at 2am.... Actually I think I would have made that mistake had it been 2pm. Basically I drew a small triangle to illustrate...too small. I confused angle BFC with BED. I shall go back to my pre-calc now.... === Subject: Elmentary Proofs Hello. I am a student going into second year and am currently reading Proofs and Fundamentals by Etahn Bloch, to try and get a grounding of proofs. There are a few exercises which I tried to construct proofs for and I was wondering if any of you would be so kind to make sure they are correct and to suggest improvements to my style. 2.3.4 Show that the product of a non-zero rational number and an irrational number is irrational. Proof - We are going to derive a contradiction. Let x be a real and rational number. Hence, x = m/n, where m and n are integers. Let y be an irrational number, which by deÞnition cannot be written in fractional form. We assume the product, w, is a rational number and can be written as a fraction. However, in order for w to be a fraction, it must be the product of two fractions. We run into a contradiction here because the irrational number cannot be written as a fraction, so therefore the product of a non-zero rational number and an irrational number is irrational. === Subject: Re: Elmentary Proofs x-mimeole: Produced By Microsoft MimeOLE V6.00.2800.1441 > Hello. I am a student going into second year and am currently reading > Proofs and Fundamentals by Etahn Bloch, to try and get a grounding of > proofs. There are a few exercises which I tried to construct proofs for and > I was wondering if any of you would be so kind to make sure they are correct > and to suggest improvements to my style. > 2.3.4 Show that the product of a non-zero rational number and an irrational > number is irrational. Let the Þrst number be a/b in reduced form (where (a,b)=1), and call the irrational number c. Assume their product is rational. Put it in reduced form as x/y, where x and y are integers and (x,y)=1. x/y = (ac)/b. Cross multiply and we get xb = ayc. xb is an integer, as is ay. So divide both sides by ay and we get xb/ay = c, where xb and ay are integers. So c is rational--which is a contradiction. So their product is irrational. Michael === Subject: Re: Elmentary Proofs > Hello. I am a student going into second year and am currently reading > Proofs and Fundamentals by Etahn Bloch, to try and get a grounding of > proofs. There are a few exercises which I tried to construct proofs for > and I was wondering if any of you would be so kind to make sure they are > correct and to suggest improvements to my style. > 2.3.4 Show that the product of a non-zero rational number and an > irrational number is irrational. > Proof - We are going to derive a contradiction. Let x be a real and > rational number. Hence, x = m/n, where m and n are integers. Let y be an > irrational number, which by deÞnition cannot be written in fractional > form. We assume the product, w, is a rational number and can be written as > fraction. However, in order for w to be a fraction, it must be the > product > of two fractions. You mean a rational like 36/1 cannot be written as a product of two non-fraction like say sqrt(24) and sqrt(54)? > We run into a contradiction here because the irrational > number cannot be written as a fraction, begging the question > so therefore the product of a > non-zero rational number and an irrational number is irrational. Where have you used the (essential) assumption that x is nonzero? -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: 3D Pascal¹s Triangle (Cone?) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6LIXKp21654; Is there a three dimensional version of Pascal¹s triangle? If so, I suppose it would be a cone (?). Applications? phil === Subject: Re: 3D Pascal¹s Triangle (Cone?) > Is there a three dimensional version of Pascal¹s > triangle? If so, I suppose it would be a cone (?). > Applications? > phil Yes, there is. It extends to any number of dimensions and its interesting properties are extended.... I am currently writing a few notes up on it. a few details: In P , the 2d pascal triangle (assuming a rightsided triangle), the 2 number at coordinate (a,b) gives you C (the combination formula). b a In P , the n-dimensional pascal triangle (again setting it out in the n right-angled triangle fashion), the number at coordinate (a,b,c,d...) gives you: C b a,c,d,... which is deÞned as the number of ways of choosing a, then c, then d (..etc..) items from b. Of course it is only valid for 0 <= sum(a,c,d...) <= b So, when you write C as with convention, you are actually writing: n r C n r,0,0,0,... And since the amount of ways of making no selections, any amount of times, is always 1, it reduces to plain old nCr. As for the binomial theorem: In P , the row of numbers at (*, b) [where * is taken to be a wildcard 2 that matches all deÞned numbers, i.e. 0 <= * <= b] gives you the coefÞcients of (x+y)^b. In P , the general multidimensional case, the row of numbers at coord n (*, b, c, d ...) gives you the coefÞcients of: Q d b --- (x + y) Q da ------------- Q! .. where Q = sum(c, d, ...). Note that for P , the above general equation reduces to nCr 2 as you would expect. As to how you generate P : let us deÞne the term *generator*, which n is the set of relative coordinates from which Œcells¹ in pascals triangle derive their value (through summation of values at the given offset coordinates). For P , if we are using the right-angled model, the generator is: 2 { (0, -1), (-1, -1) } ... this happens to be the coordinates of a 1-simplex (i.e. a 1 dimensional triangle!) offset by 1 in a certain direction. Similary, for P , the generator is the coordinates of an (n-1)-simplex n offset by 1 in a certain direction (so the generator has n offset coordinates in it). Concrete examples: P generator: { (0, 0, -1), (1, 0, -1), (0, 1, -1) } 3 P generator: { (0, 0, 0, -1), (1, 0, 0, -1), } 4 (0, 1, 0, -1), (0, 0, 1, -1) } (compare P3 and P4 generators to see the similarity!) When generating pascals triangle, we are familiar with doing the generation in steps, whereby in P2 each step involves adding a new row of numbers. If we denote the amount of generations (Œsteps¹) done by Œg¹, the nth dimenional pascal triangle has a total amount of deÞned Œcells¹ (non-zero numbers) equal to: | P | = C n,g n+g g (btw, what is the uber-correct notation for the amount of items in a set? I am using modulus style | | notation above... ) As the dimensions increase, the above deÞned count of values in pascals tetrahedron thing increase rapidly, as you¹d expect in this table which gives the count of non-zero cells for varying dimension and generations done: generations done ----> dim=1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 dim=2 1 3 6 10 15 21 28 36 45 55 66 78 91 105 dim=3 1 4 10 20 35 56 84 120 165 220 286 364 455 560 dim=4 1 5 15 35 70 126 210 330 495 715 1001 1365 1820 2380 Btw, the old sierpinksi triangle trick (plot each value mod 2 and you get the sierpinksi triangle) seems to continue on into whatever dimension you like... yyaargh, I really need to Þnish up my notes... alex === Subject: Re: 3D Pascal¹s Triangle (Cone?) ETAsAhRbZVu6EkBXEBNQ0ePGRaSeHIfOMAIUDm4sJHD5c+ B4ohgNFckN3lPmOzk= Yes. It takes the form of an open triangular pyramid. Place the number 1 at the apex and along all three edges leading from that point. Fill in the faces with the reamining entries (>=2) in Pascal¹s Triangle. Now for inside the pyramid: deÞne each entry as the sum of the three entries closest above it. Thus the entry in the middle of the third layer below the apex is 2+2+2 = 6, because the closest elements above that position are three twos from teh Pascal-triangular faces. SigniÞcance: Imagine you have three labeled urns, and N distinct balls to place into these urns. Then each number in the Nth interior triangle below the apex corresponds to the number of permutations possible with some proportion of the balls in each urn. For example, the middle entry in the N = 3 triangle calculated above as 6, corresponds to six ways of placing the balls such that one is in each urn. Equivalently, the numbers in the Nth triangle below the apex correspond to the coefÞcients that appear in the expansion of (x+y+z)^N. Work out the N = 3 triagle and compare with the following: (x+y+z)^3 = x^3 + y^3 + z^3 + 3x^2y + 3xy^2 + 3x^2z + 3xz^2 + 3y^2z + 3yz^2 + 6xyz --OL === Subject: Re: 3D Pascal¹s Triangle (Cone?) yes; pascal¹s tetrahedron, whether he ever considered them. there¹s a readily overcomable ambiguity, which is that the divisions of a tetrahedron, like cutting cheese at regular intervals, parallel to the four facets, does not decompose into tetrahedra! of course, the planar sections are the PTs. > Is there a three dimensional version of Pascal¹s > triangle? If so, I suppose it would be a cone (?). --Strep Throat at Watergate ?!? http://tarpley.net/bush12.htm === Subject: Re: 3D Pascal¹s Triangle (Cone?) > yes; pascal¹s tetrahedron, > whether he ever considered them. > there¹s a readily overcomable ambiguity, > which is that the divisions of a tetrahedron, > like cutting cheese at regular intervals, > parallel to the four facets, > does not decompose into tetrahedra! Can you elaborate on this difÞculty? why should it decompose into tetrahedra? > of course, the planar sections are the PTs. the 3d one (and in fact all dimensional version above the 2d one) features the conventional triangle in 3 planes. There are other sections that contain other things there too (see my post about to follow this one). alex === Subject: A Characteristic Congruence for Sophie Germain Primes by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6LIXKl21646; I found a new result, it is a characteristic congruence for Sophie Germain primes. A prime p is known as of Sophie Gemain if 2p+1 is also prime. Here is my result: Theorem. p>2 is a Sophie Germain prime if and only if ((p-1)!) + 6p = 1 (mod p(2p+1)). For the proof, three results are useful: - Wilson¹s Theorem. p is prime if and only if (p-1)! = -1 (mod p). - Lemma 1. If n#4 is not prime, then n devides (n-1)!. Proof. Indeed, either n=ab, with 12 is odd, then (2n)! = n(n+1)((n-1)!) (mod 2n+1). Proof. we have for any odd n>2, (2n)! = (n-1)!(2n+1-(n+1))(2n+1-n)...(2n+1-2)(2n+1-1) = (n-1)!(n+1)! (mod 2n+1), = n(n+1)((n-1)!) (mod 2n+1) PROOF OF THE THEOREM -------------------- If p is prime, then by Wilson¹s Theorem (p-1)! = -1 (mod p) Hence ((p-1)!) + 6p = 1 (mod p) (1) If 2p+1 is prime, then by Wilson¹s Theorem and Lemma 2, we have p(p+1)((p-1)!) = (2p)! = -1 (mod 2p+1) and so 4p(p+1)((p-1)!) = ((2p+1)-1)((p-1)!) = -((p-1)!) = -4 (mod 2p+1) Hence we have ((p-1)!) + 6p = 6p+4 = 3(2p+1)+1 = 1 (mod 2p+1) (2) from (1) and (2), it follows that ((p-1)!) + 6p = 1 (mod p(2p+1)) Reciprocally, if the congruence ((p-1)!) +6p = 1 (mod p(2p+1)) is satisÞed, then ((p-1)!) = 1 (mod p), by Lemma 1, p is prime. Moreover ((p-1)!) = -6p+1 = -3(2p+1) + 4 = 4 (mod 2p+1) and so, by Lemma 2, we have (2p)! = p(p+1)((p-1)!) = 4p(p+1) = (2p+1)-1 = -1 (mod 2p+1) It follows, by Wilson¹s Theorem, that 2p+1 is prime. The Theorem is proved. However, the characterization of the Sophie Germain primes by means of this theorem is not practical, because we don¹t know any algorithm other than that of the deÞnition to calculate N! quickly. I will be grateful to you to check well if there is a fault in my proof, thank you very much. === Subject: Re: JSH: Mistakes happen by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6LIXJJ21638; (snip) >Oh, I dream of intimacy with James S. Harris. I do. >Say, what are you wearing? (snip) >Use me, baby, use me just like you use the others and then toss me >aside. I don¹t mind. It would be an honor. >-- >Jesse F. Hughes GASP! Jesse, you¹re going to make Quinn jealous. I knew you seemed kinda cozy with the tree-boys. (shudder!) Now Quinn, don¹t be upset - James needs some new friends outside of your closed list. (double shudder!) JHIN (archiving the post as we speak) === Subject: Elementary Proofs 2 Hello again. I was going to type one more attempt at a theorem before I sent the last one. Theorem: Let c be a positive integer that is not a prime number. Show there is some positive integer b such that b|c and b <= sqrt(c). Proof: Lets derive a contradiction. Assume there are positive integers, b and c, such that c is a composite number, and either b is not divisible by c or b > sqrt(c). If c is a composite number it means there exists any integers, say e and f, such that 1 <= e <= f, f <= c, and c = fe. If b is not divisible by c then it means b cannot be written as bq = c, where q is any integer. However, it is possilbe that the product of fe could be equivalent to bq. Thus, we have run into a contradiction because c cannot equal bq. Since we have a contradiction, we can then conclude the theorem to be true. Mike L === Subject: Re: Elementary Proofs 2 > Hello again. I was going to type one more attempt at a theorem before I > sent the last one. > Theorem: Let c be a positive integer that is not a prime number. Show there > is some positive integer b such that > b|c and b <= sqrt(c). > Proof: Lets derive a contradiction. Assume there are positive integers, b > and c, such that c is a composite number, and either b is not divisible by c > or b > sqrt(c). If c is a composite number it means there exists any > integers, say e and f, such that 1 <= e <= f, f <= c, and > c = fe. If b is not divisible by c then it means b cannot be written as bq > = c, where q is any integer. However, it is possilbe that the product of fe > could be equivalent to bq. Thus, we have run into a contradiction because c > cannot equal bq. Since we have a contradiction, we can then conclude the > theorem to be true. > Mike L Your theorem is trivially true, just take b=1. THen b is a positive integer, b divides c, and b <= sqrt(c). So I assume that you really meant to say: Theorem: Let c be a positive integer that is not a prime number. Show there is some positive integer b >= 2 such that b|c and b <= sqrt(c). Although proofs by contradiction are useful, it¹s generally clearer for this type of problem to give a direct proof. Proof: By assumption, c is not prime, so c factors as c = a*b with both a and b at least 2. Switching a and b if necessary, we may assume that b <= a. Then b^2 <= a*b = c, so b <= sqrt(c). [I hope this wasn¹t a homework problem!] === Subject: Re: Elementary Proofs 2 > Hello again. I was going to type one more attempt at a theorem before I > sent the last one. > Theorem: Let c be a positive integer that is not a prime number. Show there > is some positive integer b such that > b|c and b <= sqrt(c). Don¹t you want to add the condition that b must be greater than 1? Otherwise, b = 1 will work always. -- Bill Hale === Subject: Re: Elementary Proofs 2 > Hello again. I was going to type one more attempt at a theorem before I > sent the last one. > Theorem: Let c be a positive integer that is not a prime number. Show there > is some positive integer b such that > b|c and b <= sqrt(c). > Proof: Lets derive a contradiction. Assume there are positive integers, b > and c, such that c is a composite number, and either b is not divisible by c > or b > sqrt(c). I think you mean b is not a divisor of c. What you said is It is not the case that c|b > If c is a composite number it means there exists any > integers, say e and f, such that 1 <= e <= f, f <= c, and > c = fe. If b is not divisible by c then it means b cannot be written as bq > = c, where q is any integer. However, it is possilbe that the product of fe > could be equivalent to bq. Thus, we have run into a contradiction because c > cannot equal bq. Since we have a contradiction, we can then conclude the > theorem to be true. This doesn¹t work. Consider c=6, b=25, e=2, f=3. One approach is to start with all the divisors of c, and assume that all of them are > sqrt(c). Then derive a contradiction. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Elementary Proofs 2 You may want to try a direct proof mn = r; m, n >1, m, n e N assume m > sqrt(r) => m = k sqrt(r), k > 1 n = r/[k sqrt(r)] = (1/k) (r/sqrt(r)) = (1/k) sqrt(r) k>1 => 1/k < 1 therefore n < sqrt(r) > Hello again. I was going to type one more attempt at a theorem before I > sent the last one. > Theorem: Let c be a positive integer that is not a prime number. Show there > is some positive integer b such that > b|c and b <= sqrt(c). > Proof: Lets derive a contradiction. Assume there are positive integers, b > and c, such that c is a composite number, and either b is not divisible by c > or b > sqrt(c). > I think you mean b is not a divisor of c. What you said is It is not > the case that c|b > If c is a composite number it means there exists any > integers, say e and f, such that 1 <= e <= f, f <= c, and > c = fe. If b is not divisible by c then it means b cannot be written as bq > = c, where q is any integer. However, it is possilbe that the product of fe > could be equivalent to bq. Thus, we have run into a contradiction because c > cannot equal bq. Since we have a contradiction, we can then conclude the > theorem to be true. > This doesn¹t work. Consider c=6, b=25, e=2, f=3. > One approach is to start with all the divisors of c, and assume that all > of them are > sqrt(c). Then derive a contradiction. > -- > Will Twentyman > email: wtwentyman at copper dot net === Subject: Re: Peter Olcott, poly-Þelded kook (give up on it being): Disproof of the Halting Problem¹s Conclusion [...] > It¹s of interest that Peter¹s kook path has taken > him through claims to out-expert the experts in at > least two independent scientiÞc Þelds, in neither > of which he is capable of carrying on a conversation > without babbling. Is it really that bad. Well, try Google group searching infallible reasoning, then see if you can Þgure out who the -- --Bryan === Subject: Re: Best divisors approximation > Question for the math gurus. I have a clock circuit that divides a > master clock to a requested frequency. There are two divisors, both > integers. Divisor_1 can have the value 2-254, even values only. > Divisor_2 can have the value 0-255. The problem is to select the two > divisor values such that the resulting frequency is the closest > possible to the requested frequency. > > The clock frequency is given by the equation: > F_clk = F_master / (Divisor_1 * (Divisor_2 + 1)) > > Is there an algorithm that can do this fairly easily? Any input > appreciated. > Christian Bau¹s spreadsheet suggestion is easy, of course, or if > you are going to do this frequently you could easily write a c or > bc or perl program to try all the values in a few milliseconds. > For bigger problems, the ofÞcial methods to solve in reasonable > time are based on continued fractions and/or lattice reduction > methods, unless my understanding is upside down. See, eg, the > Re: How to Þnd near integer values of n*a and n*b? > -jiw a hardware control API for linux driver developers on an embedded system. I was trying to avoid calculating all possible values to see which is best, but that¹s what I ended up doing anyway. Paul === Subject: Re: Riemann¹s Hypothesis by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6LIvmP23537; Is the crux of the analytic function problem that the zeta function, a piece-wise polynomial function sum (series), can not be proven analytic? Does the vertical asymptote at Re(z) = -1/2 + it have anything to do with the analytic continuation problem? I suspect that you and the others who have responded have the knowledge to solve the problem. What I want to do is have you re-think and analyze the problem, and, perhaps discover the solution, either individually or in a collaborative. As I mentioned before, I don¹t have the knowledge to write the proof nor do the math, yet, but if you are willing, I will contribute my thoughts, my discovery process, and maybe some inventiveness. If what I contribute has any value, then compensate me when you¹ve won the prize. If not, then, I may gain a better understanding of complex analysis and, maybe you will have gained some analytical insight. Participate when and as often as you want. Dont display your ideas if you want to keep them to yourself, and, save your transmissions. But, solving these problems may be worth your time in personal achievement and organization reputation, as well. Besides, I suspect there are plenty of problems, awards, and notoriety for everyone. Your time is now while mathematics is fresh on your mind; my time will come, I hope. Im from Amarillo, Texas; Professor Gary Adams was my Calculus teacher at Amarillo Junior College; Im 50 years young and starting another a career in mathematics. Ive been in the California Bay area since 87; and, I probably wouldnt have the interest/motivation to be a mathematician without the Millennium Problems. I dont work well under pressure, so old and tried problems are just what I need. Ive had many challenges: a collector for IRS, computer programming, and design. Only design has been a good Þt, but it hasnt developed, yet. I have copies of complex analysis study materials from the Encyclopedia Britannica and some other reference books. And, I have access to a Lars V. Ahlfors¹, Complex Analysis text; Im studying Stokowskis Calculus and George S. McCartys Topology. Any thoughts? >> Regarding an idea to solve a problem when I don¹t know the subject: >> You¹re right, I don¹t know complex function analysis and am just >> scratching the surface in real analysis, again. But, isn¹t the >> problem about prime numbers? >The Riemann Hypothesis has *consequences* for the distribution >of prime numbers, but it is most naturally stated and addressed >as a problem about a function of a complex variable. >-- >Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Riemann¹s Hypothesis > I have copies of complex analysis study materials from the Encyclopedia > Britannica and some other reference books. And, I have access to a Lars V. > Ahlfors¹, Complex Analysis text; Im studying Stokowskis Calculus and George > S. McCartys Topology. Any thoughts? 1. Don¹t toppost. 2. Ahlfors is an excellent text, but it may be a while before you¹re ready for it. 3. There¹s a book by Havil called Gamma that may interest you. It¹s a popularization, not a textbook, but it has more mathematics in it than any other popularization that comes to mind. Towards the end it gets you to where you can see what the Riemann Hypothesis is about. If you work through & understand everything in Havil, you will have made a good start on the math you need. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Riemann¹s Hypothesis >Is the crux of the analytic function problem that the zeta function, a >piece-wise polynomial function sum (series), can not be proven analytic? No. The zeta function is analytic on the complex plane except for a simple pole at z=1. It is not a piece-wise polynomial function sum. >Does the vertical asymptote at Re(z) = -1/2 + it have anything to do >with the analytic continuation problem? What vertical asymptote? >As I mentioned before, I don¹t have the knowledge to write the proof nor >do the math, yet, but if you are willing, I will contribute my >thoughts, my discovery process, and maybe some inventiveness. If what I >contribute has any value, then compensate me when you¹ve won the prize. It¹s hard to imagine how you could contribute anything of value when you haven¹t the foggiest idea what the problem is about. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: [Find the Fallacy!!] -- ŒProof¹ of Monotheism.... > > let x represent the number of gods in the universe > > > > it is obviously false that x=1 => x=0, > > > > . . . > > > > Nice puzzle. The fallacy arises because of the suppressed universal > > quantiÞer. Supply it explicitly and you¹ll see where you go wrong. > > > > Charlie Volkstorf > > Cambridge, MA > > > > which implies that x=1 > > > i¹m not sure i understand your solution--if you could, please explain > how it compares to my method of ŒÞxing¹ the fallacy: > If x is considered a constant variable, then x=1 => x=0 is not false > (a constant.) It is equivalent to x!=1 (a function of one variable.) > For the formula x=1 => x=0 to be false we would have to be referring > to the common convention of suppressing leading universal quantiÞers, > and the complete formula is (all x)(x=1 => x=0) which is in fact > obviously false as stated. However, in this case, its negation is > ~(all x)(x=1 => x=0) which isn¹t x=1 ^ x!=0 which is short for (all > x)(x=1 ^ x!=0) (making people think that x must equal 1) but rather > its negation is (exists x)(x=1 ^ x!=0) which is true but doesn¹t tell > us anything about the original value of x viz. the number of gods in > the universe. > As far as the number of gods goes (gofs gogs . . .) see the seminal > paper, Deep Thoughts # 9: Formalization of God. > Charlie Volkstorf > Cambridge, MA ok, but since => stands for a tautologically true material condition ( |=(A->B)) (at least in my textbook--i¹m a beginner here) then the => is only true if it holds for all possible truth values of the antecendent and consequent--which means holding for all x, no? this would make the universal quantifer redundant, and the error is in applying the demorgan rules to logical conditional instead of a material conditional, no? i mean, the demorgan rules are proved using the truth table, which is for material conditionals, how can you use them on logical conditionals? === Subject: Re: Centrifugal force - real or Þctitious > Hi Donald, > Please read this > http://en.wikipedia.org/wiki/Centrifugal_force > Maybe it¹ll help you understand. mate, you should¹ve paid more attention in > your physics lessons! angular momentum into linear momentum... Aren¹t they both conserved? === Subject: Re: Centrifugal force - real or Þctitious > Hi Donald, > Please read this > http://en.wikipedia.org/wiki/Centrifugal_force > Maybe it¹ll help you understand. mate, you should¹ve paid more attention in > your physics lessons! angular momentum into linear momentum... Aren¹t they both conserved? === Subject: Re: Centrifugal force - real or Þctitious > What¹s harder to understand is how centrifugal force causes the Earth > to bulge at the equator, and the rate of free fall to be less there: > Earth bulge at equator is due to a REAL centrifugal force: > http://observe.arc.nasa.gov/nasa/space/centrifugal/ centrifugal4.html > http://observe.arc.nasa.gov/nasa/space/centrifugal/ centrifugal5.html > The rate of free fall is less because the distance to Earth center increases > and gravity decreases at the square of the distance. > The tidal opposite Ocean bulge also is due to centrifugal force. > (see NOAA explanation). > To some, centrifugal force isn¹t considered a real force as such, > because if it was, masses would be thrown off into space due to > Earth¹s rotation. > Such people can¹t see that it¹s a matter of _equilibrium_; that as > long as the centripetal force is greater than the centrifugal force, > mass will gravitate. > NASA explains it all about centrifugal force starting here: > http://observe.arc.nasa.gov/nasa/space/centrifugal/ centrifugal_entry.html > The problem is that nobody can explain what inertia is. I¹ve been explaining it for quite a while: It¹s simply the measure of a quantity of matter: The ratio of the impulse [ft] exerted on and/or by that quantity of matter, divided by the rate of displacement [s/t] that it causes: Most concisely written as [ft^2/s]; which is equal to the product of two times the weight [w] of the quantity of matter, divided by the rate at which it will freefall [g = s/t^2 - 16¹/sec^2 on Earth] at the location of the scale on which it is weighed: ft^2/s = 2w/g. > Because most of the people don¹t like to show their ignorance they simply > avoid the problem by means of the equivalence principle. > Then they are like parrots after reading the book. > Don¹t ask intelligent questions to a parrot. === Subject: Re: Centrifugal force - real or Þctitious > The problem is that nobody can explain what inertia is. > I¹ve been explaining it for quite a while: It¹s simply the measure of > a quantity of matter: The ratio of the impulse [ft] exerted on and/or > by that quantity of matter, divided by the rate of displacement [s/t] > that it causes: Most concisely written as [ft^2/s]; which is equal to > the product of two times the weight [w] of the quantity of matter, > divided by the rate at which it will freefall [g = s/t^2 - 16¹/sec^2 > on Earth] at the location of the scale on which it is weighed: ft^2/s > = 2w/g. That¹s the usual deÞnition that is solely based on the equivalence principle. When you say that inertia = 2w/g (it must be w/g only, but...) you are assuming the equivalence principle, saying that inertia is equivalent to gravity, hence it is an acceleration in the Þeld that produces the weight. The underlying question is against what actually mass causes inertia. According to Newtons Þrst law, bodies at rest tend to stay at rest, and bodies in motion tend to stay in motion. The question that arises is with respect to what? At rest with respect to what? In motion with respect to what? Newton advocate the answer was space itself. He thought that space itself formed an absolute frame of reference. Einstein showed that there is no preferred reference frame, and that all things are relative, as it were. Even before Einstein, many people were uncomfortable with the idea that empty space itself could serve as a frame of reference. You cant exactly hammer a nail into empty space and measure from that. Things become much more confusing when strange forces like centrifugal force, the Coriolis effect and the gyroscope precession need to be explained. About the gyroscope precession notice there is a new variable into play, which is the sense of rotation and the right hand rule is required. http://physics.nad.ru/Physics/English/gyro_tmp.htm http://physics.nad.ru/Physics/English/gyro_txt.htm Notice that in the above link the weight doesn¹t fall. It is antigravity. Newton¹s Laws goes down the drain (no reaction force exists to balance the weight) and GRT also goes down the drain (since the equivalence principle goes too - the weight doesn¹t fall). So, the actual theories don¹t explain inertia on its basic mechanism. === Subject: Re: Centrifugal force - real or Þctitious > > You need to look up noninertial frames in a physics or mechanical > engineering textbook. Even Halliday & Resnick talks about them. > Given sHead¹s age and understanding of basic physics, he probably believes > Halliday & Resnick was a vaudeville team. > Tom Davidson > Richmond, VA Talk about me making dumb posts: Why you gotta waste bandwidth with brite comments like that? Especially when ya know better. Cripe sakes, if you don¹t start making serious contributions here, the gravy boat¹s gonna overþow. Help, don¹t hinder! === Subject: Re: Centrifugal force - real or Þctitious > Ok, show me one inertial force that doesn¹t obey the equivalence > principle, at least locally. What do you think makes our experience > of the so-called Þctitious forces just as real as any other forces? If you say that the force is inertial (inertial force) hence it will obey the equivalence principle. But what about Lorentz force of electromagnetism and what about the gyroscopic forces in mechanics? In general, those forces that require the right hand rule to be expressed? Tell me about this one and the equivalence principle: http://physics.nad.ru/Physics/English/gyro_tmp.htm http://physics.nad.ru/Physics/English/gyro_txt.htm === Subject: Re: Force and counterforce > > > > > > > > > Put a block on the table, Shead. Measure all the forces on the > > > block--primarily the force due to gravity, and the force from > > > the table on the block. The net force is exactly zero, F = ma, > > > there is no acceleration and the block doesn¹t move. > > > > Here¹s the point I¹m trying to make Sam: Despite the fact that there > > is no motion as such between the table and the block, the table is > > _displacing_; forcibly decelerating the block from its usual free > > falling toward Earth¹s center; which it would do if it was allowed to > > fall freely. > > > > And my point is--There is no acceleration taking place, Shead as > > the net force on the block is zero. > > Well hokay, if you¹ve _got_ to have _net force_ and a _rate of change > in velocity_ to have _deceleration and displacement_, that¹s your > prerogative. > > I say that there is no net force because the table is exerting an > equal and opposite counterforce to restrain and/or decelerate the > block from falling. > > Since the deÞnition of acceleration is dv/dt which is obviously > zero, there is no acceleration (or if you like, deceleration) and > given F = ma, the forces sum to zero. > References for Shead > http://scienceworld.wolfram.com/physics/Acceleration.html > http://scienceworld.wolfram.com/physics/Force.html No, no Sammy: The _average_ acceleration is [(vt-vi)/t]: When [vt-vi] is inÞnitesimal, and is virtually the same; then we are able to say that v/t is the average; a constant displacement [a distance (s)], divided by the square of the time period [t[ during which it occurs: That is [a = (vt-vi)/t = 2s/t^2]; when (vt-vi) is inÞnitly small; virtually encompassing the same point, or period of time [t]. Furthermore Sammy: The net force [f] is equal to the product of the _weight_ (w) of the mass, and the acceleration that the net force causes, divided by the acceleration (g) at which it will free fall: That is f = wa/g. Likewise: w = fg/a! How many times must you be told? By the way, do you know that stress is the acceleration of the surfaces of areas that are exerting pressure on each other; attempting to displace each other? === Subject: quotient ring question Consider the following conjecture: -------------------- | Let k be a Þeld and R = k[[x,y]] be the ring of formal power series | in two variables. Let f,g in R be nonzero polynomials without | constant term. Then the following are equivalent: | | 1) There exists a ring isomorphism phi: R --> R such that phi(f) = g. | | 2) The factor rings R/(f) and R/(g) are isomorphic as k-algebras (here | (f) is the principal ideal generated by f and similarly for g). -------------------- In the discussion it is stated without proof that 1) => 2). The reason I believe 2) => 1) is that it makes problem 14 easy to do (he basically asks you to assume 2) is true and conclude that ord(f)=ord(g) where ord = smallest degree of all terms of the polynomial). However I haven¹t the slightest idea how to prove 2) => 1), or even if it is true. Does anybody know a proof? -Nathan === Subject: Q: About the binomial transform of the primes. In the OEIS -- A007443 -- Binomial transform of the primes. 2,5,13,33,83,205,495,1169,2707,6169... Where the left diagonal starting with the Þrst term of this sequence (2), it lists the primes in order --- 2,3,5,7,11.. etc. Find the residuals between terms of the transform and list them into a new sequence. ie. 5 == 1(mod 2), 13 == 3(mod 5), 33 == 7(mod 13) etc. 1,3,7,17,39,85,179,369,755... Now start with (2) and add the left diagonal of the deltas of this new sequence. Wahla! The primes. This new sequence is the binomial transform of the prime deltas. Can anyone show a progressive sequence that does the same thing? The Fibonacci sequence is the only other one I could Þnd. First Þnding the sequence transform and then using the residuals between each term that creates a new sequence which is the binomial transform of the deltas of the original sequence. Of course this eliminates all sequences that are monic polynomials. Dan === Subject: I am playing Ken talked about some important things. I do not think all internet service companies give free web pages. I am thinking about using net zero because it is cheap compared to most companies. I have heard net zero will destroy your account if you use them a lot even if they say you can use them all you want. I do not use the internet that much - I do not think they will tell me that I can not use their company because of the amount of time I am on the internet. I do not think net zero gives a free web page - I called them and they said they do - I think it is not true. I did not ever get my picture on my page at aol. I could only get a picture on their photo center - it could be true they make it more simple to do that becasue they can make money like selling prints of pictures. I put a link on my page to aol photo center. The person who scanned the picture lives around me. They said they used a bit map ??? because it uses a lot of memory for a picture as compared to different methods like jpeg ???. It could be true aol does not like a bit map ???. I take a picture. It has information. Because of the uncertainty principle how long does it take for all the information to leave the picture??? === Subject: Re: I am playing > Ken talked about some important things. > I do not think all internet service companies give free web pages. Well, that¹s not quite what I said. I said they give you free webspace. > I do not think net zero gives a free web page - > I called them and they said they do - I think it is not true. They provide the space, you provide the pages. I¹m wondering if what you mean by Œfree web pages¹ is the on-line text editor that some web hosting companies provide. Web hosting is simply an internet service provider who provides extra services. There¹s nothing stopping you from editing your pages off-line in Notepad, and then uploading them into your webspace. > I did not ever get my picture on my page at aol. I could only get a > picture on their photo center - it could be true they make it more > simple to do that becasue they can make money like selling prints of > pictures. I have no idea what this Œphoto centre¹ is. Whatever it is, it sounds more complicated to me. What could be simpler than building a text Þle in Notepad and uploading it along with your image Þles? > I put a link on my page to aol photo center. The person > who scanned the picture lives around me. They said they used a bit > map ??? because it uses a lot of memory for a picture as compared to > different methods like jpeg ???. It could be true aol does not like a > bit map ???. *.jpg is to *.bmp as *.zip is to *.txt. I have no idea what AOL does or does not like. > I take a picture. It has information. Because of the uncertainty > principle how long does it take for all the information to leave the > picture??? That¹s a joke, right? -- CodeCutter - good, fast and cheap; pick two. === Subject: Re: True Gems of ScientiÞc Epistemology > I would like to get opinions on what were the most profound books and > thinkers on scientiÞc epistemology you have encountered. By this I > mean a work that totally changed the way you look at the world and at > science. Below is my list. I would be grateful for your additions. I > am really looking for works that are novel and different, orthogonal > to the ones I listed. For example, I included Bertalanffy¹s General > System Theory so I do not want the many, many other books on General > System Theory. > My list (no particular order): > Rene Thom: Structural Stability and Morphogenesis > Ludwig von Bertalanffy: General System Theory > David Bohm: Wholeness and Implicate order > - Jeff An earlier times Indian scientist,Jagadish Chandra Bose. http://www.minhas.net/culture/indianpeople/jcbose.htm known for Response phenomena in plants and Microwave communication. === Subject: Re: True Gems of ScientiÞc Epistemology > I would like to get opinions on what were the most profound books and > thinkers on scientiÞc epistemology you have encountered. By this I > mean a work that totally changed the way you look at the world and at > science. Below is my list. I would be grateful for your additions. > Rene Thom: Structural Stability and Morphogenesis > Ludwig von Bertalanffy: General System Theory > David Bohm: Wholeness and Implicate order G. Spencer Brown: Laws of Form === Subject: Re: True Gems of ScientiÞc Epistemology Interesting idea but can you explain what you mean by scientiÞc epistemology? Do you mean epistemology that is scientiÞc (e.g. Quine, Kornblith, Goldman), or the epistemology of science (e.g. Popper, Kuhn, Laudan)? > I would like to get opinions on what were the most profound books and > thinkers on scientiÞc epistemology you have encountered. By this I > mean a work that totally changed the way you look at the world and at > science. Below is my list. I would be grateful for your additions. I > am really looking for works that are novel and different, orthogonal > to the ones I listed. For example, I included Bertalanffy¹s General > System Theory so I do not want the many, many other books on General > System Theory. > My list (no particular order): > Rene Thom: Structural Stability and Morphogenesis > Ludwig von Bertalanffy: General System Theory > David Bohm: Wholeness and Implicate order > - Jeff === Subject: Re: True Gems of ScientiÞc Epistemology > Interesting idea but can you explain what you mean by scientiÞc > epistemology? Do you mean epistemology that is scientiÞc (e.g. Quine, > Kornblith, Goldman), or the epistemology of science (e.g. Popper, Kuhn, > Laudan)? I would say I am more interested in epistemology that is scientiÞc rather than epistemology of science speciÞcally. I suspect they overlap signiÞcantly and also that the former is more encompassing than the latter. I guess I am just looking at what, in your opinion, are profound and under-appreciated scientiÞc priniciples to generalize the knowledge we already have or least at novel interpretaions of that knowledge regardless of the speciÞc domain. The use of scientiÞc in this sense is loose (and somewhat circular) - I suppose I just mean that I¹m interested in epistemic paradigms that are at least logical and not some it-all-comes-from-within metaphysical drivel. === Subject: Re: True Gems of ScientiÞc Epistemology Originator: harris@tcs.inf.tu-dresden.de (Mitchell Harris) >> Interesting idea but can you explain what you mean by scientiÞc >> epistemology? Do you mean epistemology that is scientiÞc (e.g. Quine, >> Kornblith, Goldman), or the epistemology of science (e.g. Popper, Kuhn, >> Laudan)? >I would say I am more interested in epistemology that is scientiÞc Holy crap, that¹s mathematics! Read any modern philosphy of mathematics. er.. maybe not. They Þt your description but I suspect it¹s not what you want given your Þrst examples. Mitch === Subject: Re: True Gems of ScientiÞc Epistemology > I would like to get opinions on what were the most profound books and > thinkers on scientiÞc epistemology you have encountered. By this I > mean a work that totally changed the way you look at the world and at > science. Below is my list. I would be grateful for your additions. I > am really looking for works that are novel and different, orthogonal > to the ones I listed. For example, I included Bertalanffy¹s General > System Theory so I do not want the many, many other books on General > System Theory. > My list (no particular order): > Rene Thom: Structural Stability and Morphogenesis > Ludwig von Bertalanffy: General System Theory > David Bohm: Wholeness and Implicate order will turn up a lot of info on him. He was a Hungarian scientist who developed his ideas in England shortly after the second world war. I must admit that I have only read books about his ideas - not his works themselves, but I Þnd his ideas refreshing and more in line with how we actually conduct research rather than our abstract ideas as to what it should be. http://www.mwsc.edu/orgs/polanyi/ http://www.iscid.org/polanyi.php http://en.wikipedia.org/wiki/Michael_Polanyi === Subject: Re: True Gems of ScientiÞc Epistemology > I wanted to add one more criterion: > I¹m speciÞcally looking for the thinkers and books that are not only > highly original but also under-appreciated and not as widely known as > they deserve to be. For instance, Aristotle, Plato, and Darwin will > not make this list as their work is well known and universally > recognized. > - Jeff > I would like to get opinions on what were the most profound books and > thinkers on scientiÞc epistemology you have encountered. By this I > mean a work that totally changed the way you look at the world and at > science. I would like to nominate the most entertaining exploit in the history of physics--the insight, concealed as subtle mathematical trickery, whereby the Lorentz transformation appears almost magically, seemingly derivable from the Galilean transformation: http://www.everythingimportant.org/relativity Another discovery that¹s highly original but under-appreciated is the amusing and inescapable observation that the relativity of simultaneity is just an illusion in a spatially closed and bounded universe. In actuality, in a large and very realistic class of pseudo-Riemannian spacetimes, events that are simultaneous for one observer are simultaneous for all. :-) http://www.everythingimportant.org/viewtopic.php?t=605 http://www.everythingimportant.org/relativity/simultaneity.htm Eugene Shubert http://www.everythingimportant.org === Subject: Re: True Gems of ScientiÞc Epistemology > I would like to nominate the most entertaining exploit in the history > of physics--the insight, concealed as subtle mathematical trickery, > whereby the Lorentz transformation appears almost magically, > seemingly derivable from the Galilean transformation: > http://www.everythingimportant.org/relativity You¹d like to nominate yourself? Are you another one of the sci.physics.* loons? What has sci.math done to deserve this? Phil -- 1st bug in MS win2k source code found after 20 minutes: scanline.cpp 2nd and 3rd bug found after 10 more minutes: gethost.c Both non-exploitable. (The 2nd/3rd ones might be, depending on the CRTL) === Subject: Re: True Gems of ScientiÞc Epistemology > I would like to nominate the most entertaining exploit in the history > of physics--the insight, concealed as subtle mathematical trickery, > whereby the Lorentz transformation appears almost magically, > seemingly derivable from the Galilean transformation: > > http://www.everythingimportant.org/relativity > You¹d like to nominate yourself? I only nominated the ideas. Individuals nominate and þatter me personally by saying that my claims are impossible. Experts slander me by saying that my claims are perfectly true but unworthy of recognition. This is praise also because these experts almost never try to correct the false accusations of the ignorant rabble. Eugene Shubert http://www.everythingimportant.org === Subject: Re: True Gems of ScientiÞc Epistemology > I wanted to add one more criterion: > I¹m speciÞcally looking for the thinkers and books that are not only > highly original but also under-appreciated and not as widely known as > they deserve to be. For instance, Aristotle, Plato, and Darwin will > not make this list as their work is well known and universally > recognized. Lakatos¹ Proofs n Refutations has already been mentioned, but it bears repeating.... Kuhn¹s Structure of ScientiÞc Revolutions is basically the same as Lakatos¹, only on a larger scale.... Together the two are the best possible duo to consult for an elucidiation of the notion _good thinking_....... Frege¹s Grundlagen der Arithmetic rules..... as do the essays Uber Sinn und Bedeutung, Concept and Object, and Function and Concept...... Dummett¹s Frege: Philosophy of Language and Frege: Philosophy of Mathematics taught us how to understand what the hell Frege was talking about, and its signiÞcance.... Wittgenstein¹s Remarks on the Foundations of Mathematics (or something close to that), along with his Philosophical Investigations provided a lot of the conceptual impetus for Lakatosian/Kuhnian thinking (speciÞcally the rule-following business)..... Kant has some good stuff about 6/7 of the way thru the 1st Critique..... his Logic is also pretty good (for his time, at any rate) - vaquely a Frege-precursor...... ok, I¹ll stop now.... :) cdj === Subject: Re: True Gems of ScientiÞc Epistemology > I wanted to add one more criterion: > I¹m speciÞcally looking for the thinkers and books that are not only > highly original but also under-appreciated and not as widely known as > they deserve to be. For instance, Aristotle, Plato, and Darwin will > not make this list as their work is well known and universally > recognized. > - Jeff > I would like to get opinions on what were the most profound books and > thinkers on scientiÞc epistemology you have encountered. I still say Richard Feynman. In my youth I read widely on epistemology Hume etc but none capered my imagination or inþuenced me more than Feynman. I have read others felt the same. However within the physics community I believe Feynman is appreciated so I am not sure how well he Þts your criteria Œof under-appreciated and not as widely known as they deserve to be¹. By this I > mean a work that totally changed the way you look at the world and at > science. Below is my list. I would be grateful for your additions. I > am really looking for works that are novel and different, orthogonal > to the ones I listed. For example, I included Bertalanffy¹s General > System Theory so I do not want the many, many other books on General > System Theory. > My list (no particular order): > Rene Thom: Structural Stability and Morphogenesis > Ludwig von Bertalanffy: General System Theory > David Bohm: Wholeness and Implicate order > - Jeff === Subject: Re: True Gems of ScientiÞc Epistemology > I would like to get opinions on what were the most profound books and > thinkers on scientiÞc epistemology you have encountered. By this I > mean a work that totally changed the way you look at the world and at > science. Below is my list. I would be grateful for your additions. I > am really looking for works that are novel and different, orthogonal > to the ones I listed. For example, I included Bertalanffy¹s General > System Theory so I do not want the many, many other books on General > System Theory. > My list (no particular order): > Rene Thom: Structural Stability and Morphogenesis > Ludwig von Bertalanffy: General System Theory > David Bohm: Wholeness and Implicate order IMHO much better than those is - Feynman - The Character of Physical Law. Bohm was a very good scientist but IMHO his ideas on wholeness and the implicate order missed the mark. Indeed his pilot wave theory, part of which that book is based on, (yes I have read it) has been proven wrong see: http://citebase.eprints.org/cgi-bin/citations?id=oai: arXiv.org:quant-ph/0103 100 and http://arxiv.org/abs/quant-ph/0206196. OTOH Feynman was a great scientist - in many peoples top ten of all time. The Character of Physical Law has been mentioned by many physicists as having a strong inþuence on them. His ideas, unlike Bohms, continue to inþuence and guide current research and the way science is taught see http://www.eftaylor.com/quantum.html. Bill === Subject: Re: True Gems of ScientiÞc Epistemology >I would like to get opinions on what were the most profound books and >thinkers on scientiÞc epistemology you have encountered. By this I >mean a work that totally changed the way you look at the world and at >science. Below is my list. I would be grateful for your additions. I >am really looking for works that are novel and different, orthogonal >to the ones I listed. For example, I included Bertalanffy¹s General >System Theory so I do not want the many, many other books on General >System Theory. >My list (no particular order): >Rene Thom: Structural Stability and Morphogenesis >Ludwig von Bertalanffy: General System Theory >David Bohm: Wholeness and Implicate order These three are sort of out there in terms of mainstream science. Somewhat more mainstream: Other books are good to read in order to learn to reject what they claim, I¹d include Penrose¹s books in that list, and Stuart Kaufman¹s The Origin of Order. Orthogonal to your choices, I always like to recommend Ruth Garrett Millikan¹s books on teleosemantics, On Clear and Confused Ideas is more accessible than Language, Thought, and Other Biological Categories. And in relation to your later message, don¹t toss off Darwin so quickly, evolution is an *extremely* subtle and tricky idea, enough so that Gould and Dawkins agree on just about no details at all. Along those lines, I¹d recommend Ernst Mayr¹s Toward a New Philosophy of Biology. NB: Whether any of the books on your list or mine qualify as epistemology, is debatable. J. === Subject: Re: True Gems of ScientiÞc Epistemology > Orthogonal to your choices, I always like to recommend Ruth Garrett > Millikan¹s books on teleosemantics, On Clear and Confused Ideas is > more accessible than Language, Thought, and Other Biological > Categories. Millikan? bah. > And in relation to your later message, don¹t toss off Darwin so > quickly, evolution is an *extremely* subtle and tricky idea, enough so > that Gould and Dawkins agree on just about no details at all. Along > those lines, I¹d recommend Ernst Mayr¹s Toward a New Philosophy of > Biology. Good call on the evolution-tip. To hell with Gould n Dawkins tho - go with Daniel Dennett, Darwin¹s Dangerous Idea! :) cdj === Subject: Re: True Gems of ScientiÞc Epistemology >> Orthogonal to your choices, I always like to recommend Ruth Garrett >> Millikan¹s books on teleosemantics, On Clear and Confused Ideas is >> more accessible than Language, Thought, and Other Biological >> Categories. >Millikan? bah. Just for the record, I¹m not saying she¹s 100% correct, just *interesting*. FWIW, I say she¹s strong on how concepts are used, but the teleo- part of it, and the Swampman business, is as you say (bah!) but that¹s still interesting! >> And in relation to your later message, don¹t toss off Darwin so >> quickly, evolution is an *extremely* subtle and tricky idea, enough so >> that Gould and Dawkins agree on just about no details at all. Along >> those lines, I¹d recommend Ernst Mayr¹s Toward a New Philosophy of >> Biology. >Good call on the evolution-tip. To hell with Gould n Dawkins tho - go >with Daniel Dennett, Darwin¹s Dangerous Idea! > :) Well, Dennett and Millikan are old buddies, but by that same light, Dennet is another one who gets some stuff very right, and other stuff, not so right. DDI is strong in its denial of the agency of evolution, though that¹s a bit of beating on a strawman, but DDI does not really present a good intuition pump to use instead. Which is pure Dennett, who is happy to talk of intentionality as a stance, but who then offers only just-so stories about why we would have such a stance. So, I¹ll stay with the biologist Mayr when it comes to evolution. (for a great overview of the history and foundations of evolutionary theory, Peter Bowler, Evolution: The History of an Idea, 1983/1989, isbn 0-520-06386-4) J. === Subject: Re: True Gems of ScientiÞc Epistemology | > I would like to get opinions on what were the most profound books and | > thinkers on scientiÞc epistemology you have encountered. By this I | > mean a work that totally changed the way you look at the world and at | > science. Below is my list. I would be grateful for your additions. I | > am really looking for works that are novel and different, orthogonal | > to the ones I listed. For example, I included Bertalanffy¹s General | > System Theory so I do not want the many, many other books on General | > System Theory. | > My list (no particular order): | > Rene Thom: Structural Stability and Morphogenesis | > Ludwig von Bertalanffy: General System Theory | > David Bohm: Wholeness and Implicate order | > - Jeff | | You need some framework to develop this question in. Let¹s get more | speciÞc. The three planks of epistemic questions are: | | 1) what is meant by knowledge, Egads! Patrick, you go WAY too far. What is meant by what? What is meant by is? What is meant by meant? What is meant by by? Surely most of us have some idea of what is meant by knowledge? If we have to deÞne every single word in the dictionary for you, we¹ll spend ALL our time talking about the meaning words and none at all discussing the ideas they convey. Heck, look it up in the dictionary, it means what it is intended to mean, and even then it¹s only what other words describe it as, so we¹d have to check the meaning of them too. Knowledge is what a human being thinks he/she knows. The individual may be wrong, in which case it isn¹t knowledge, but for the most part it we understand it. I know how to serve an ace at tennis. I can¹t do it, but I know how it¹s done. I know how to fry an egg for breakfast. That, I can even demonstrate. Now, are you going to ask me what an egg is? Or what I mean by fry? Or what an ace is? Does every word have to be reduced to strings of three characters or four , one syllable for you, that you can understand what they mean? | 2) how are knowledge claims to be represented in your language, Jeez... President George W. Bush thinks he knows how to govern the USA. Many people will disagree, because they know how to do it better. Who is right? I know how to fry an egg. Too bad if you want it over-easy, I fry eggs the way I like Œem, up. However, I do KNOW what is meant by over-easy and fry and knowledge. and | 3) what are the justiÞcations for speciÞc knowledge claims. I can see it now. Waitress: How would your like your eggs, sir? Patrick: Does the chef know how to make them over-easy? Waitress: Yes, sir. Patrick: Does the chef know how to make them sunny-side-up? Waitress: Yes, sir. Patrick: Does the chef know how to make them hard? Waitress, getting sarcastic: What does chef mean to you, sir? Patrick: Hmm... I¹ll have to ask someone at sci.math. Waitress: Would you like to look at the menu a little longer, sir? One hour later, and Patrick still hasn¹t had his breakfast. Androcles | | Patrick === Subject: meet in the middle attack I¹m reading an explanation of the meet-in-the-middle attack, and speciÞcally why it shows that double DES is no more secure than DES. There¹s nothing too mystifying here, except one thing: We have (p,c) a known plaintext/ciphertext pair. The assertion is made that E_k(p) is stored for all (Þrst) keys k so that they can be compared to D_k(c) to try to Þnd a match. But there are 2^56 keys, which is 7 x 10^16. That seems like a lot of memory, Þrst of all, and, worse, a whopping lot of computing to Þll that memory. My question is, is that REALLY a reasonable computing task? (Not for my laptop, but if Microsoft wants to spy on Intel.) How hard, right now, is it to compute and store that many half-encryptions? Bart === Subject: Re: meet in the middle attack > My question is, is that REALLY a reasonable computing task? > (Not for my laptop, but if Microsoft wants to spy on Intel.) > How hard, right now, is it to compute and store that many > half-encryptions? You could divide up the space into discrete chunks for parallel processing. There are black-hat hackers out there who are Œzombie masters¹. They¹re armed with hundreds or even thousands of other peoples¹ machines, which they¹ve taken over without the knowledge of the Œowners¹. I realize that this could sound like some paranoid conspiracy theory, but it¹s a well documented fact; see for example, http://www.grc.com/dos/grcdos.htm. -- CodeCutter - good, fast and cheap; pick two. === Subject: SMSU Problem Corner The latest problems have been posted at http://math.smsu.edu/~les/POTW.html Hope to see you there. === Subject: Re: Question about PhD math programs reworded > including vector calculus, and linear algebra gives > no understanding of anything, and I question if those > who have it are any better prepared for abstract > mathematics than they were in elementary school, > plus a good explanation of variables in general, not > starting with numbers. An abstract idea should not > always be presented as a generalization, but as > something basic; it is. This is a good point IMO. I was taught to manipulate things, and it took me a long time (as a physics student), to see that this was not really math. Van === Subject: Re: Mathematical exact explanation of the adiabatic theorem/adiabatic passage? > I think Mathematical Methods of Classical Mechanics by V.I.Arnold will > give you what you need. > I doubt that. The adiabatic theorem is about quantum mechanical operators. Not so. Arnold does have a nice short, to the point discussion of adiabatic invariants (ad. inv.) from a basic point of view. I was 1st taught about adiabiatic invs. in the same way Arnold does; energy/frequency (= action) is (almost) constant as one slowly changes the lenth of a pendulum. In general, anything that is almost const. as a parameter changes is an adiabatic inv. BTW, this has been studied mathematically in theoretical physics-- perturbation expansions, etc. Its important for many things, esp. QM. It is interesting to look at this from a very simple QM point of view, even when you are doing things classically, quantum ideas are useful. All energy is quantized, and E = Nhw, N = # of quanta, w = frequency, and h = Planck¹s const. The ad. inv. is E/w = Nh = (# of quanta) h, so ad. inv. is conservation of the number of quanta. One can do this for any type of wave (sound, MHD), and deÞne the wave action as E/w, where E is now energy density of waves. One can show that conservation of wave action means that wave quanta are conserved--this is the conserved quantity associated with wave propagation. I have written a paper on this. Van === Subject: Re: (pi^4+pi^5)^(1/6)=e??? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6M1fr324069; >> >>3 ~ inx((pi)^2 * sqrt(pi+ 1 + 1/5511606)) >> > >> > 3~in((pi)^2 * sqrt(pi + 1 + 1/5511606 + 1/60887389898152)) >> > >> > By adding one more large integer inverse, it is now correct to 29 >> > decimal places. ;-) >> >> If we are allowed lots of digits in our answers, how about: >> >> pi = e * 0 + 3.14159265358979323846264338327950288419716939937510 >> Did you not notice the --> ;-) >> But aside from that, how do you think these large integers are >> calculated and then using their inverse? >> I would say a little more difÞcult then what you are proposing! >> My algorithm is a brute force trial and error method. I and am not sure >> if a closed form method is possible for this particular equation. >> Maybe the mathematicians can answer that one? > Closed form? Huh? It¹s just simple algebra, unless you¹re > 3 = ln((pi)^2 * sqrt(pi+1 + 1/x)) >for some integer x. So we solve for x and Þnd > 1/x = (e^6 / pi^4) - pi - 1 >You can type the right-hand side of that into Google, or use >a handheld calculator, to Þnd the answer; 1/x is approximately >1.814E-7, so x is (to the lowest greater integer) 5511606. > Taking 1.814E-7 minus 1/5511606 gives approximately >1.6E-14, which is out of the useful range of any of the >calculators I have immediately available, but gives a >reciprocal of something like 6.3E+13. Repeat ad nauseam, >or until you run out of precision. There¹s no secret >method to it, as far as I know. >-Arthur I guess I was just spinning my wheels with this complex algorithm. Dan === Subject: Re: Colors by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i6M1fq724061; What country did Chop Suey originate from?