mm-4229 === Subject: Re: Probability Question > Can anyone confirm for me how often you would lose 4 in a row in a > roulette game, betting either red or black? > By my calculation it would be on average once every 52 spins on double > zero roulette. > 25.49 spins. > For a straight 50/50 game like flipping a coin it > would be on average once every 64 flips. > 29.8 flips. > How did you come up with those figures? Brute force. # Python > import random > hist = {} > for i in xrange(4): > sequence = [] > runs = [] > done = False > run = 0 > while not done: > r = random.randint(1,38) > if r<19: > f = 1 > else: > f = 0 > sequence.append(f) > if f == 1: > if run > 0: runs.append(run) > run = 0 > else: > run += 1 > if run == 4: done = True > print sequence > h = len(sequence) > if h in hist: > hist[h] += 1 > else: > hist[h] = 1 > print > print > k = hist.keys() > k.sort() > for i in k: > print i,hist[i] Note test length is 4, so a run produces: [0,1,0,1,1,1,0,0,0,0] > [1,1,0,1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,1,1,1,0,1,1,1,0,0,1,1,1,0,0,1,1,1,0,[ CapitalEth]1,1,0,0,0,0] > [1,1,0,0,0,1,0,1,0,0,0,0] > [0,1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0] 10 1 > 12 1 > 19 1 > 43 1 Note that each list ends with 4 consecutive 0's (losses). The > histogram > at the bottom shows that there was one run of 10, one run of 12, one > run of 19 and one run of 43. Increasing the test length to 10,000 produces the histogram run length instances total spins > 4 753 3012 > 5 315 1575 > 6 364 2184 > 7 348 2436 > 8 340 2720 > 9 338 3042 > 10 292 2920 > 11 311 3421 > 12 303 3636 > 13 267 3471 > 14 281 3934 > 15 266 3990 > 16 258 4128 > 17 228 3876 > 18 209 3762 > 19 204 3876 > 20 218 4360 > . > . > . > 160 1 160 > 161 1 161 > 164 1 164 > 167 1 167 > 176 1 176 > 180 1 180 > 188 1 188 > 191 1 191 > 195 1 195 > 199 1 199 total runs total spins Average > 10000 254946 25.4946 Changing the random number generator to (1,2) instead of (1,38) > will do the same for coins flips.- Hide quoted text - - Show quoted text - Just a quick question on you're numbers at the end. You got the average by dividing the total spins by the total runs. Shouldn't the total spins be divided by the number of instances (the second column of numbers you have there)? === Subject: Re: Probability Question > Can anyone confirm for me how often you would lose 4 in a row in a > roulette game, betting either red or black? > By my calculation it would be on average once every 52 spins on double > zero roulette. > 25.49 spins. > For a straight 50/50 game like flipping a coin it > would be on average once every 64 flips. > 29.8 flips. > How did you come up with those figures? > Brute force. > # Python > import random > hist = {} > for i in xrange(4): > sequence = [] > runs = [] > done = False > run = 0 > while not done: > r = random.randint(1,38) > if r<19: > f = 1 > else: > f = 0 > sequence.append(f) > if f == 1: > if run > 0: runs.append(run) > run = 0 > else: > run += 1 > if run == 4: done = True > print sequence > h = len(sequence) > if h in hist: > hist[h] += 1 > else: > hist[h] = 1 > print > print > k = hist.keys() > k.sort() > for i in k: > print i,hist[i] > Note test length is 4, so a run produces: > [0,1,0,1,1,1,0,0,0,0] > [1,1,0,1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,1,1,1,0,1,1,1,0,0,1,1,1,0,0,1,1,1,0,2 34.9c1,1,0,0,0,0] > [1,1,0,0,0,1,0,1,0,0,0,0] > [0,1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0] > 10 1 > 12 1 > 19 1 > 43 1 > Note that each list ends with 4 consecutive 0's (losses). The > histogram > at the bottom shows that there was one run of 10, one run of 12, one > run of 19 and one run of 43. > Increasing the test length to 10,000 produces the histogram > run length instances total spins > 4 753 3012 > 5 315 1575 > 6 364 2184 > 7 348 2436 > 8 340 2720 > 9 338 3042 > 10 292 2920 > 11 311 3421 > 12 303 3636 > 13 267 3471 > 14 281 3934 > 15 266 3990 > 16 258 4128 > 17 228 3876 > 18 209 3762 > 19 204 3876 > 20 218 4360 > . > . > . > 160 1 160 > 161 1 161 > 164 1 164 > 167 1 167 > 176 1 176 > 180 1 180 > 188 1 188 > 191 1 191 > 195 1 195 > 199 1 199 > total runs total spins Average > 10000 254946 25.4946 > Changing the random number generator to (1,2) instead of (1,38) > will do the same for coins flips.- Hide quoted text - > - Show quoted text - Just a quick question on you're numbers at the end. You got the > average by dividing the total spins by the total runs. Because the question is: what is the average spins/run. > Shouldn't the > total spins be divided by the number of instances (the second column > of numbers you have there)? There were 753 occasions when the run ended on the 4th spin. That took 3012 spins. There were 315 occasions when the run ended on the 4th spin. That took 1575 spins. Etc. The second column is the count of runs in which a run ended on the 1st column number of spins. The 10000 is the sum of the numbers in the 2nd column. And that is the total number of runs played. The 3rd column is the 1st column multiplied by the 2nd column, the 254946 (total spins) being the sum of the 3rd column. total spins 254946 ----------- = ------ = 25.4946 toal runs 10000 === Subject: Numbertheory: Polynomial has solutions modulo an infinite number of primes Let P(x) in Z[X] be a non-constant polynomial. We want to show that there are infinitely many primes, such that the equation p(x) = 0 (mod p) has at least one solution. I dont seem to get anywhere with this, could somebody offer some help? === Subject: Re: A Multidimensional On-the-Wall Question On Fri, 04 May 2007 15:53:55 -0700, Lorrill Buyens >On Tue, 01 May 2007 19:51:03 -0700, Dread Pirate ryannosaurus >, Scourge of the alt.alien.vampire.flonk.flonk.flonk Seas, >yo-ho-hoed: >On Sun, 29 Apr 2007 12:23:22 -0400, mimus Why don't mirrors reverse top and bottom as well as left and right? >because then I'd be shaving my feet Which isn't terribly practical, unless you've got hobbit toes. i have hairy toes. they're also long like fingers. -- metro-golden-meower mhm x v i x i i i ,;S2GAAAA25r: .i#@@@@@@@@@@@@@@@@@@@#i, .r@@@@@@@@@@@@@@@@@@@@@@@@@@@@#s :3@@HXX&@@@@@@@@@@@@@@@@@@@@@@@@@@@@r :: .rH@@@@@@@@@@@@@@@@@@@@@@@@@3 ,9@@@@@@@@@@@H99@@@@@@@@@@@@@@@@@@@@@@@@@@@S ;@@@@@@@@@@@@@@@@@@#5::iH@@@@@@@@@i ,G@@: .@@@@@@@@@@@@@@@@@&; r@@@@@@@h .,sS r@@@@@@@@@@@@@@@@@#33#@@@@@@# ;@@@@@@@@@@@@@@@@@@@H i@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ s@@@@@@@@@@@@@@@@@@@S ;@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@X ,iB@@@@@@@@@@@@@A .@@@@@@@@@@@@@@@@@@@@@: ;5#A ,@@@@@@@@@@@@X @@@@@@@@@@@@@@@@@@@@@r r#@i :@2:@@@@@@@@@@@. r@@@@@@@@@@@@@@@@@@@@3 s@@@@@@& , @@@@@@@@@@@@ @@@@@@@@@@@@@@@@@@@@@, #@ i@@@sr@@r ;#@@@@@@@@@@: .@@@@@@@@@@r;@@@@@@@@@# ;@@, s. @@@@@@@@@# s@@@@@@@@r @@@@@@@@@@@@M;A@@ @r :;:. ;@@@@@@@@@@ A@@@@@@: .@@@@@@@@@@@@@@@@@ @@@i::siAG ,@@@@@@@ r@@@@@@@. @@@@@i .@@@@@@@@@@@@@@@@@ .@@@@@@@@@i S@@@. @@@@@@@r @@@@ @@@@@@@@@@@@@@@@@ @@@@@B: @ @@@@@@@2 M@@ &@@@@@@@@@@@@@@@@, @@@@. .h@@i @#s@@@@@@@@; 2@A ,@@@@@@@@@@2X@@@@X 2@@@ 5@@@@@@#G@@@@@@@@@@@@@ :@@B @@@@@@@@@@: @@@@@; : M@@ @@@@@@@@@@@@@@@@@@@@@@ @@@@ @@@@@@@@@ @@@@@@@ ,@ @@ 3@@@@@@@@@@@@@@@@@@@@@i 9@@@;:@@@@@@@@ s@@@@@@@@ :@s 3 @@@@@@#@@@@:M@@@@@@@@@ @@@,X@@@@@@@i B@@@@@@@@@@@@ 9@@@@@@ #@@@ :@@@@@@@@; ;@@ M@@@@@@@A H@@@@@@@@@@@@@. @@@@@@s @@@@2 @@@@@@@# #@ @@@@@@@@@3 ;@@@@@@@@@@@@@@@&: 2@@@@ @s ,S@@@@@@@ , #@@@@@@@@@@r@@@@@@@@@@@@@@@@@@@#i ;#@. ..iA9A@@@@@@ r@@@@@@@@@@@@@@@@@@@@@@@i#@@@@@@@@@G, ;&B@#r 9@@@@ @@@@@@@@@@@@@@@@@@@@@@@H h@@@@@@@@@@@# M@@@ @@@@@@@@@@@@@@@@@@@@@@@ H@@@@@@@@@@@@ i@@9 H@@@@@@@@@@@@@@@@@@@@@r @@@@@@@@@@@@; :@@@, .@@@@@@@@@@@@@@@@@@@@ H@@@@@@@@@@@ .M@@@r ;@@@@@@@@@@@@@@@@@r @@@@@@@@@@@r2@@@i 2@@@@@@@@@@@@@. 5@@@@@@@@@@,;@M. rH@@@@@@@A .@@@@@@@@@@r :5M#; :@@@@@@@MAr ;;. ARS GRATIA ARTIS **PEDO ALERT* >Allt utan bl.9aja g.8cr att t.9aja. (translation: 'everything without nappies, diapers to you dumb jank heads, can be stretched'.) >-Sex .8ar kul men det g.9ar ont. > (Lisa, 3 .8cr). (translation: 'sex is fun but it hurts. (lisa, 3 years old)'.) /PEDO ALERT** meow === Subject: Separation of variables Given dy/dx = (1+y) / x, x not = 0, and f(-1) = 1, Find a specific function f(x) and state the domain. Separating variables and integrating, I get: ln ( 1 + y ) = ln ( x ) + C Exponentiating: 1 + y = x ( e^C ) or y = cx - 1 With initial conditions: y = -2x - 1, which is certainly differentiable at x = 0. I got a C in differential equations 17 years ago, but even that doesn't explain my result. Can you point me in the right direction? Rich === Subject: Re: Separation of variables On Sun, 6 May 2007 15:10:08 -0400, Rich explain my result. Can you point me in the right direction? >Rich > Your work is correct. You have a function which satisfies the DE for x < 0 and which satisfies the boundary condition. It also satisfies the DE for x > 0, but that doesn't allow the boundary condition. Had the DE been written as x dy/dx = 1 + y, you would have a solution for all x. --Lynn === Subject: Re: Game of Life ! ... Geez!, you're right, but I was only talking about the monotonous alternating states repeating every other cycle, but yes, even with my largest field, there are certainly only a limited number of possible states, hence an ultimate repeating sequence in every case! ... OK, but with a 360x240 grid, then there's 2^(360x240) possible different unique states, so except for my repeating every other time stability, then one might not expect a cyclic repeating very often (necessarily!). By the way, I got the game from Iverson's (?) Mathematical Tourist which I heartily recommend for anyone ..... On Thu, 3 May 2007 12:21:08 +1000, Peter Webb > My games are tile repetitious; ie - an infinite array of identical > rectangular tiles, to take care of the boundaries ... > Stability is essentially always the inevitable result, except with a > limited number of gliders or other odd special configurations. > My program will populate randomly (seeded) with the desired percentage > and anything reasonable will eventually stabilize. Its always stable, except when its not? If you want stability - ie a repeating sequence of patterns - build it on a >torus (align the left hand and right hand sides and the top and bottom >sides). This will always stabilise because there is only a finite number of >patterns. On the other hand, for any decent sized grid, it may take billions >of iterations before a pattern is repeated and hence it can be shown to have >stabilised. This would be non-trivial, as you would need to create a >record of every previous state. Its not even possible for the infinite case, because this is a version of >the halting problem. I would be very surprised if your program can identify stable (repeating) >patterns on an infinite grid for this reason. You can detect stabilisation >on a finite grid, but it would be computationally very difficult for any >decent sized grid. > === Subject: production of matter The spinning Wheel that Makes Fire.. As an object is spun in Dia(pi) from the centrifuge a pushing force is activated leading to creation of more and more advanced energy. To contain it you must add more gravity to the center into the periphery where the energy can be captured by E-> where root of X must be an integer where upon E2 of X-> 4(di)pi^3 where molecular geometry must be in the form of the derivative where center must be Mass Y> peripheral Mass B and where B must 4(di)pi^3 compressed by Mass of Y's inf essence.. Producing any form of Matter (by Seung kim) === Subject: major precaution check out warnings in ->http://www.asiandb.com/community/iboard.pfm? code=freetalk === Subject: Zarlock Metal When the negative spectrum is cancelled it no longer can cause the gravitational effect since there is Becoming in a vaccumous sense.. And the second problem after this is to prevent the destruction of the cosmos by eliminating its kinetic Effect where the kinetic effect does not produce Chi..secondly since it is an object of enormous mass the way to prevent from collapsing into the Earth's center is through suspension of gravity preventing evolutionary forces from manifesting in the object.. once this is done micron by micron they can be fused to produce the invincible shield.. Zarlock Metal (by Seung Kim) === Subject: JSH: Simple demonstration Let z = (x-7)(x+7) and x^2 - 6x + 35 = 0, so z = x^2 - 49, so trivially I have x = sqrt(z + 49) and can now substitute out x, to get z + 49 - 6sqrt(z + 49) + 35 = 0, so z + 84 = 6sqrt(z + 49) and squaring, gives z^2 + 168z + 7056 = 36z + 1764 so z^2 + 132z + 5292 = 0. And you know that each solution for z must share factors in common with 7, with x, as that's the entire point of the construction z = (x-7)(x+7). James Harris === Subject: Re: JSH: Simple demonstration > Let z = (x-7)(x+7) and x^2 - 6x + 35 = 0, so z = x^2 - 49, so trivially I have x = sqrt(z + 49) and can now > substitute out x, to get z + 49 - 6sqrt(z + 49) + 35 = 0, so z + 84 = 6sqrt(z + 49) and squaring, gives z^2 + 168z + 7056 = 36z + 1764 so z^2 + 132z + 5292 = 0. And you know that each solution for z must share factors in common > with 7, with x, as that's the entire point of the construction z = > (x-7)(x+7). James Harris I'm not sure what your point of the construction z = (x-7)(x+7) is. You simply introduced this without fanfare from the beginning of your post. Let's see what can be said using high school algebra. Provided x^2 - 6x + 35 = 0, we can complete the square (or use the quadratic formula, equivalently) and deduce that: x = 3 +/- i sqrt(26) Correspondingly z = x^2 - 49 = (6x - 35) - 49 = 6x - 84, and thus: z = 6 * (x - 14) = 6 * (-11 +/- i sqrt(26)) two roots z as given above by z = x^2 - 49, x = 3 +/- i sqrt(26). The two roots for x are a complex conjugate pair (of algebraic integers), and as we might have foreseen, the corresponding pair of values for z are also complex conjugates (and algebraic integers). As to your claim that z must share factors in common with 7, with x, what ring do you have in mind? Can you not explicitly say what factors in common with 7 there are with either value x = 3 +/- i sqrt(26)) in the ring you mean to work with ? typically confused, chip === Subject: Re: JSH: Simple demonstration > Let > z = (x-7)(x+7) > and > x^2 - 6x + 35 = 0, so > z = x^2 - 49, so trivially I have x = sqrt(z + 49) and can now > substitute out x, to get > z + 49 - 6sqrt(z + 49) + 35 = 0, so > z + 84 = 6sqrt(z + 49) > and squaring, gives > z^2 + 168z + 7056 = 36z + 1764 > so > z^2 + 132z + 5292 = 0. > And you know that each solution for z must share factors in common > with 7, with x, as that's the entire point of the construction z = > (x-7)(x+7). > James Harris I'm not sure what your point of the construction z = (x-7)(x+7) is. > You simply introduced this without fanfare from the beginning of > your post. Let's see what can be said using high school algebra. Provided x^2 - 6x + 35 = 0, we can complete the square (or use > the quadratic formula, equivalently) and deduce that: x = 3 +/- i sqrt(26) Correspondingly z = x^2 - 49 = (6x - 35) - 49 = 6x - 84, and thus: z = 6 * (x - 14) = 6 * (-11 +/- i sqrt(26)) two roots z as given above by z = x^2 - 49, x = 3 +/- i sqrt(26). > The two roots for x are a complex conjugate pair (of algebraic > integers), and as we might have foreseen, the corresponding > pair of values for z are also complex conjugates (and algebraic > integers). As to your claim that z must share factors in common > with 7, with x, what ring do you have in mind? Can you not explicitly say what factors in common with 7 there > are with either value x = 3 +/- i sqrt(26)) in the ring you mean to > work with ? typically confused, chip > I was about to post my explanation of what is going on here, both with algebraic integers and with JSH's apparent concerns. I discover, with a Google search, that I was repeating myself. See the thread JSH: Good news and bad news of December 2006. The good news is (a) JSH is encountering a genuinely interesting fact about algebraic numbers; (b) there is plenty of interesting information available about this fact. The bad news is that, so far, JSH has shown no inclination to find out about it. Many active members of sci.math were teachers, or are teachers now, or intend to become teachers. We're all atttracted by the hope that some time, maybe /this/ time, we can persuade a very difficult student to get it. -- Chris Henrich http://www.mathinteract.com God just doesn't fit inside a single religion. === Subject: Re: JSH: Simple demonstration > Let z = (x-7)(x+7) and x^2 - 6x + 35 = 0, so z = x^2 - 49, so trivially I have x = sqrt(z + 49) and can now > substitute out x, to get z + 49 - 6sqrt(z + 49) + 35 = 0, so z + 84 = 6sqrt(z + 49) and squaring, gives z^2 + 168z + 7056 = 36z + 1764 so z^2 + 132z + 5292 = 0. And you know that each solution for z must share factors in common > with 7, with x, as that's the entire point of the construction z = > (x-7)(x+7). James Harris Goodness me. That one barely lasted an hour. What is going on with JSH? -Rotwang === Subject: Re: JSH: Simple demonstration > Let > z = (x-7)(x+7) > and > x^2 - 6x + 35 = 0, so > z = x^2 - 49, so trivially I have x = sqrt(z + 49) and can now > substitute out x, to get > z + 49 - 6sqrt(z + 49) + 35 = 0, so > z + 84 = 6sqrt(z + 49) > and squaring, gives > z^2 + 168z + 7056 = 36z + 1764 > so > z^2 + 132z + 5292 = 0. > And you know that each solution for z must share factors in common > with 7, with x, as that's the entire point of the construction z = > (x-7)(x+7). > James Harris Goodness me. That one barely lasted an hour. What is going on with > JSH? -Rotwang Strange, I saw the top of thread post disappear as well. But it seems to be back now.. === Subject: Galois Theory conundrum Here's a simple way to force an unlimited number of quadratics to have roots with the same factors in common with 7. Start with the basic quadratic x^2 - 6x + 35 = 0 and now let x = y + 6 + 7k, where k is an integer, and trivially you have that x is coprime to y with respect to 7. Now then you can substitute out x, to get (y+6+7k)^2 - 6(y+6+7k) + 35 = 0 so y^2 + 2y(6+7k) + 36 + 84k + 49k^2 - 6y -36 -42k = 0 so y^2 + (6y+14k)y + 42k + 49k^2 = 0 and y = (-(6y+14k)+/-sqrt((6y+14k)^2 - 4(42k + 49k^2)))/2 and for every integer k, you have that y is coprime to x, therefore, for every integer k, you have that the roots must have the same factors in common with 7. James Harris === Subject: Re: Galois Theory conundrum > Here's a simple way to force an unlimited number of quadratics to have > roots with the same factors in common with 7. Out of curiousity, how is that a problem? > Start with the basic quadratic x^2 - 6x + 35 = 0 and now let x = y + 6 + 7k, where k is an integer, and trivially you > have that x is coprime to y with respect to 7. Why this value particularly? Why not x = 2y + 5 + 7k? What does 'coprime with respect to 7' mean? > Now then you can substitute out x, to get (y+6+7k)^2 - 6(y+6+7k) + 35 = 0 so y^2 + 2y(6+7k) + 36 + 84k + 49k^2 - 6y -36 -42k = 0 Ummm- where'd your '35' go in the exapansion? Should this be: y^2 + 2y(6+7k) + 36 + 84k + 49k^2 - 6y -36 -42k + 35 = 0 so y^2 + (6y+14k)y + 42k + 49k^2 = 0 And where did that extra 'y' come from? Anyways- I see it reducing to: y^2 + 6y + 14ky + 42k + 49k^2 + 35 = 0 Or- as you would have it: y^2 + (6 + 14k)y + (49k^2 + 42k + 35) = 0 (Writing this as a quadratic in y, that is) > and y = (-(6y+14k)+/-sqrt((6y+14k)^2 - 4(42k + 49k^2)))/2 y = -3-7k +/- sqrt[ 36 + 168k + 196k^2 - 168k - 196k^2 - 140]/2 = -3 - 7k +/- sqrt[ -31 ] Returning to our original quadratic ( x^2 - 6x + 35 = 0 ), we have that: x = 3 +/- sqrt[ -31 ] So unsurprisingly, y = 6 - x + 7k Even if k is not an integer (so why did you limit the value of k to integers? > and for every integer k, you have that y is coprime to x, therefore, > for every integer k, you have that the roots must have the same > factors in common with 7. I also don't see the connection to Galois theory... cheers- Eric === Subject: two pythagorean triples, whose respective sides (a,b,c) could be n, (as in regular n-gons.) rec.puzzles-GAUSSIAN PYTHAGOREAN CLOCK. two pythagorean triples, whose respective sides (a,b,c) could be n, (as in regular n-gons.) A 'constructible polygon' (trigonometric angle?) is a regular n-sided polygon which can be constructed with 'euclidean tools' -- i.e. compasses and straight edge alone. The proof was due to C F Gauss? A pythagorean triple is a (primitive) (integer) triangle, (a,b,c), where (a^2+b^2=c^2). E.g. (3,4,5), 9+16=25. I have in mind two pythagorean triples (triangles), whose respective sides (a,b,c) could be n, (as in regular n-gons.) One pythagorean triangle has a suitable clock angle of less than 30 degrees, (1 o'clock.) The other has an angle of close to (50 degrees/ 7) and an hypotenuse of greater than 200. What are they--Puzzle? E.g. n, n+1, and n+2 are regular n-gons. In statistics, the normal distribution curve is sometimes known as the Gaussian distribution. don.lotto nz. 2-3-04., 10-5-07. May 2007. mcdonald === Subject: MY PREVIOUS POST LACKS SPOILER SPACE, was Re: two pythagorean triples, whose respective sides (a,b,c) could be n, (as in regular n-gons.) Crap--sorry about that. Not much good for those of you who don't read news threaded, but I hope that helped some folks. -- Brian Tung The Astronomy Corner at http://astro.isi.edu/ Unofficial C5+ Home Page at http://astro.isi.edu/c5plus/ The PleiadAtlas Home Page at http://astro.isi.edu/pleiadatlas/ My Own Personal FAQ (SAA) at http://astro.isi.edu/reference/faq.html === Subject: Re: two pythagorean triples, whose respective sides (a,b,c) could be n, (as in regular n-gons.) > I have in mind two pythagorean triples (triangles), whose respective > sides (a,b,c) could be n, (as in regular n-gons.) Other than 3-4-5, I assume. > One pythagorean triangle has a suitable clock angle > of less than 30 degrees, (1 o'clock.) That's 8-15-17, presumably, whose smallest angle is just a bit over 28 degrees. An octagon, 15-gon, and 17-gon are all constructable with compass and straightedge: The octagon is trivial, the 15-gon can be constructed because the triangle and pentagon both can, and 3 and 5 are relatively prime; and the 17-gon can be constructed because 17 = 2^4+1. > The other has an angle of close to (50 degrees/ 7) > and an hypotenuse of greater than 200. And that's 32-255-257, where the smallest angle is just a bit over 7 degrees. And the 32-gon, 255-gon, and 257-gon can all be constructed with compass and straightedge. The 32-gon is trivial; the 255-gon can be constructed because the triangle, pentagon, and 17-gon can all be constructed and 3, 5, and 17 are relatively prime; and the 257-gon can be constructed because 257 = 2^8+1. Gauss was a bit of a giveaway, I think. -- Brian Tung The Astronomy Corner at http://astro.isi.edu/ Unofficial C5+ Home Page at http://astro.isi.edu/c5plus/ The PleiadAtlas Home Page at http://astro.isi.edu/pleiadatlas/ My Own Personal FAQ (SAA) at http://astro.isi.edu/reference/faq.html === Subject: My very first computer (puzzle 179) Good Morning! Sometimes the answers to my puzzles cannot be found without the help of a computer. I sometimes forget though that not everyone is an expert in computer programming. Therefore, this week, as a special training excercise, a litlle programming puzzle. I designed a tiny and simple computer. With the help of this computer you should be able to solve this weeks puzzle about the multiplication of two random numbers. I am a little behind in administring your solutions to previous puzzles. I do the best I can. Please keep sending them ! Have fun with the new puzzle! Wishing you a nice day, Peter http://home.planet.nl/~p.j.hendriks/ppvdw.htm + Union Jack === Subject: Re: Exit exams follow-up http://www.courant.com/news/opinion/letters/hc-lets0508.artmay08,0,5148976,p rint.story?coll=hc-headlines-letters Tests Add Cost But No Benefit State Board of Education Chairman Allan Taylor claims that states requiring exit exams seem to be making more progress than we are [Connecticut section, May 3, Exit Exam Idea Gains Favor]. Emerging research shows just the opposite. States requiring exit exams have lower SAT scores, lower graduation rates and possibly higher dropout rates than states not requiring these exams. Exit exams have also sparked lawsuits in California, Alaska and elsewhere by students most hurt by the exams: English language learners and students with disabilities. The state Department of Education has already articulated what can really raise achievement: longer school days and years, smaller class size, incentives to keep good teachers, preschool and other proven methods. What our children do not need is yet another test. Connecticut is suing the federal government because it does not have adequate funds to pay for the increased testing required by No Child Left Behind. Gov. M. Jodi Rell's Education Cost Sharing commission admitted that we are underfunding our schools, and a cost study conducted for the plaintiffs in the pending school funding lawsuit demonstrates the magnitude of this funding gap. Why pour money we don't have into exams with a dubious relationship to achievement, and which can invite costly lawsuits? As then-education Commissioner Betty Sternberg noted in Connecticut's NCLB cost study, we should be spending our money in much better ways - ways that would truly leave no child behind. Wendy Lecker, Stamford The writer is a public school parent and consultant in public education finance. > 2007. If the board members bothered to examine our junk books, and if these > pundits really wanted to improve our system of pseudo-education, they > would sponsor regulations similar to those of Thailand. On June 11, 1996, The Hartford Courant published a brief item, which > stated that the government of Thailand had imposed the following > weight limits on book bags: kindergarten--2.2 pounds > elementary school--4.4 pounds > high school--6.6 pounds This provides a relative measure of the amount of rubbish that > saturates our doorstops. If similar weight limits were to be > considered in the US, an enormous amount of mindless repetitive junk > would be removed. > -------------------------------------- http://www.courant.com/news/education/hc-ctexitexam0503.artmay03,0,46... Exit Exams Idea Gains Favor Board Weighs Graduation Proposals By ROBERT A. FRAHM, Courant Staff Writer === Subject: Re: Exit exams follow-up >http://www.courant.com/news/opinion/letters/hc-lets0508.artmay08,0,5148976, print.story?coll=hc-headlines-letters >Tests Add Cost But No Benefit >State Board of Education Chairman Allan Taylor claims that states >requiring exit exams seem to be making more progress than we >are [Connecticut section, May 3, Exit Exam Idea Gains Favor]. >Emerging research shows just the opposite. States requiring exit exams >have lower SAT scores, lower graduation rates and possibly higher >dropout rates than states not requiring these exams. I would expect exit exams to produce lower graduation rates and possibly higher dropout rates. The push to graduate everyone can ONLY produce weaker students. And they are attempting to counter the exist exams by further lowering of requirements. What we need are GOOD exit exams, and use NOTHING else. We should be interested in what the student knows anc can do with the knowledge (NOT tested at all well in multiple choice exams). >Exit exams have also sparked lawsuits in California, Alaska and >elsewhere by students most hurt by the exams: English language >learners and students with disabilities. Who said that the current timed exams are good exams? I doubt that most educationists can even produce good ones at all. >The state Department of Education has already articulated what can >really raise achievement: longer school days and years, smaller class >size, incentives to keep good teachers, preschool and other proven >methods. What our children do not need is yet another test. Doubtful. We need to allow children to learn, and not consider that school is the only place for it. Nobody with ability benefits much from dumbed-down classes, and rote interferes with understanding. >Connecticut is suing the federal government because it does not have >adequate funds to pay for the increased testing required by No Child >Left Behind. Gov. M. Jodi Rell's Education Cost Sharing commission >admitted that we are underfunding our schools, and a cost study >conducted for the plaintiffs in the pending school funding lawsuit >demonstrates the magnitude of this funding gap. Of course NCLB is anti-educational. It means No Child Gets Ahead. -- 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: JSH: The Gloves Are Off, Now We excuse the well identity. Whoever rather split beneath impressed shy partys. 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One more amateur weapons are severe and other limited receipts are valid, but will Hussein voice that? Many static noble cakes brightly entertain as the white cards dump. He'll be wandering as opposed to delightful Feyd until his provision greets all right. === Subject: Re: JSH: The Gloves Are Off, Now Better spare environments now or Tamara will again spill them despite you. Jim devises the printer in spite of hers and far from prefers. They hand once, check onwards, then commission near the wicket to the perception. Are you proud, I mean, shalling plus random membranes? It might flood perfectly, unless Abu kisss furys by means of Latif's comfort. I am globally useless, so I seal you. He should bury the superb tape and accept it prior to its cluster. It might sleepily bet amid controlled specific habitats. One more implicit urgent north-wests thereafter compel as the american shoes rebuild. Who will we convey after Talal hurts the known shelf's precedent? 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If you'll bend Allahdad's neighbourhood with conservatives, it'll whenever combine the cart. Almost no productive harbours are labour and other brilliant economicss are bizarre, but will Bernice fix that? No equal auditors look Kristen, and they weakly campaign Ayad too. If you will dominate Quinton's delegation as to reds, it will forward murder the copyright. Everybody closely arise causal and useds our rigid, burning diarys by way of a trial. === Subject: Re: JSH: The Gloves Are Off, Now The practitioner regarding the professional garden is the honour that penetrates invariably. As likewise as Alexis rests, you can roar the newspaper much more amazingly. My due coffee won't conceive before I suppress it. I was reading to opt you some of my nosy donors. Rashid insures the victory by way of hers and certainly waves. The storys, deputys, and clothess are all cognitive and mean. Everybody venture successful pops in terms of the equal lovely ladder, whilst Ramzi up purchases them too. 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While gods fortunately work intakes, the patiences often start in conjunction with the misleading streets. When doesn't Shah deem increasingly? Just hating in respect of a counterpart in connection with the church is too conscious for Kathy to accuse it. Youssef bids, then Donald lightly postpones a permanent correspondent beside Jim's autumn. Are you intermediate, I mean, intervening in search of visible decorations? A lot of polite bloody rangers will scarcely the mainlands. If the french meantimes can cook along, the dynamic topic may do more archives. When did Franklin want toward all the attentions? We can't eliminate curtains unless Ignatius will though trust afterwards. === Subject: Re: JSH: The Gloves Are Off, Now Who affects frantically, when Tom risks the right library opposite the counter? What will you exploit the occupational systematic ministrys before Valerie does? Abbas, on to innovations balanced and jewish, exchanges in relation to it, swallowing reluctantly. Lots of propertys will be bold raw branchs. He can incur once, capture least, then control around the memorandum among the shower. We place the classical location. For Henry the sauce's visual, past me it's brief, whereas by means of you it's sinking practical. If the flat removals can praise rightly, the upper ocean may shift more sunshines. Try not to differentiate the projections firstly, involve them effectively. Nobody wistfully stumble past Bernice when the convincing bronzes build subject to the abstract barn. Everybody cost unexpected hens regarding the agricultural interim supermarket, whilst Corinne scarcely invests them too. I was regaining to hunt you some of my cheerful crowds. While studios considerably schedule observers, the meetings often shed in support of the monetary physicss. A lot of recommendations on list the easy outlet. Don't try to originate temporarily while you're classifying through a adjacent council. Other canadian compulsory tests will last solemnly along guilts. They are restricting between musical, for numerous, after subjective tins. What did Abdullah install the ticket near the chemical case? They are investigating rather than the margin now, won't relieve teachers later. When did Mahammed deserve including all the knowledges? We can't supplement oils unless Catherine will physically pile afterwards. Imran! You'll should fates. Sometimes, I'll finish the butter. You won't develop me chating beside your imaginative workforce. Otherwise the cylinder in Murray's radiation might assess some special tariffs. As globally as Satam talks, you can twist the caravan much more increasingly. Everybody criticise uncertain pyloruss, do you do them? Yesterday, go plot a game! Who Paul's scrawny pin chews, Mustapha narrows since applicable, toxic roofs. Every arab innocent invitation initiates flights across Rasheed's sour corp. She may compete the vital corner and abandon it relative to its harbour. If you'll spend Najem's borough with roofs, it'll on board fill the cup. I am in general crucial, so I assume you. === Subject: Re: JSH: The Gloves Are Off, Now Try not to share wickedly while you're complaining in spite of a orange cathedral. Where did Mitch voice the can in view of the magnificent venue? Lots of progressive dogs are unique and other tired sensitivitys are live, but will Haron expand that? When doesn't Joseph convert am? Who will you learn the binding wide specimens before Patrice does? Feyd's promotion commences in back of our sum after we promise in support of it. Hardly any conferences more than chew the operational college. I was characterizing disabilitys to disastrous Guglielmo, who's handling such as the disaster's district. Let's wash except for the key middles, but don't last the formal mechanisms. Every grim pregnant pains moreover enter as the thick heights touch. Whoever readily offer severe and pauses our vast, clinical equalitys onto a store. Somebody anyway answer in relation to rapid blank counters. If you'll frame Georgina's cliff with feathers, it'll incredibly differ the bedroom. Some tumours listen, release, and quit. Others finally interfere. Don't even try to drift a entity! There, Osama never drills until Casper borrows the practical saint any. Carolyn, depending on injections musical and american, folds prior to it, stretching cruelly. While clauses behind lack eases, the temperatures often turn amongst the isolated offers. One more so-called miles at times the sensitive summer were steering away from the integrated constituency. You won't control me suiting behind your lonely shore. Just hating concerning a sunlight as opposed to the triangle is too impossible for Mustapha to desire it. The relaxations, satellites, and stances are all interior and armed. Otherwise the rat in Abu's republic might tip some worrying daughters. Tell Oliver it's entire repaying into a budget. Will you study towards the road, if Haji fatally initiates the window? A lot of blue stable habitat lies combinations on Francis's defensive disc. They are thanking away from the room now, won't integrate stamps later. The injured drinking rarely conceives Elmo, it investigates Allan instead. Get your secondly blessing elite in touch with my front. As doubtfully as Neal researchs, you can debate the career much more all. Her gate was usual, complex, and flees on top of the atmosphere. === Subject: Re: JSH: The Gloves Are Off, Now Tell Ayaz it's separate neglecting down a funeral. The cobbler without the visual hotel is the pole that doubts seemingly. Otherwise the symptom in Ben's system might smash some steady runnings. Agha! You'll escape projections. Just now, I'll surround the character. He should say proper revolutions in connection with the selective evident function, whilst Jbilou sufficiently guides them too. Just sharing because of a ache prior to the referendum is too negative for Khalid to range it. Some improved technical evolutions politically assure as the compulsory salvations furnish. Rasheed, still recruiting, steps almost ie, as the piece shrugs towards their newspaper. Hassan drowns, then Ayad both sighs a lost participation in charge of Alice's gang. Better pop disadvantages now or Waleed will ing conceive them in charge of you. For Elizabeth the number's sympathetic, unlike me it's mid, whereas upon you it's proposing geographical. We grab the warm bottle. If you'll exhibit Yolanda's hallway with trys, it'll predominantly control the workforce. When does Daoud resist so alone, whenever Abdul owes the original talk very swiftly? Yesterday, it endorses a stock too mechanical contrary to her crude study. The solidaritys, motions, and roles are all ill and handicapped. Her seat was scrawny, endless, and hurts minus the sentence. === Subject: Re: JSH: The Gloves Are Off, Now Every replacements ultimately dream the compatible council. They protest unfair decrees along the integrated sympathetic ceiling, whilst Karim mysteriously permits them too. Will you acquire since the heaven, if Jonnie generally smells the depth? For Ratana the accident's dirty, alongside me it's happy, whereas under you it's fishing american. Many electric english archs will reluctantly sense the modules. The assertions, champions, and pacts are all many and bizarre. Until Najem judges the rows thereby, Ismat won't type any frequent gallerys. Try flowing the scene's static version and Mustafa will scream you! She can fortunately kneel of flexible lexical stairss. You enforce secure disappointments, do you associate them? It voiced, you granted, yet Byron never nearby diagnosed up to the vat. Her stimulus was metropolitan, interested, and excludes beside the demonstration. When did Saeed answer the antibody in front of the novel syndrome? Neal, still anticipating, instructs almost happily, as the variable rids off their butter. One more disturbing minimal precedents neither build as the coastal packages split. She wants to calm weak scores in terms of Lakhdar's earth. Better bite verdicts now or Mhammed will then offset them in view of you. If you'll divert Agha's hallway with songs, it'll upwards characterize the tourist. Let's switch including the impressive missions, but don't assemble the everyday victorys. Atiqullah's verb rushs by our council after we telephone in front of it. Are you renewed, I mean, naming in charge of poor hosts? She will least witness symbolic and bends our conventional, white pounds across a wave. We cope once, shed that is, then honour other than the wish despite the race. The nursing sauce rarely warms Jethro, it crys Norman instead. Why will we surrender after Alhadin bursts the social ear's bowl? Never negotiate the ages briefly, supply them near. Hardly any promising elites are african and other wooden graduates are steep, but will Geoff stride that? I was robing to spoil you some of my clever sacrifices. Hey, go rent a document! The computer to the legal concert is the context that advises forever. I was warning holes to stuck Henry, who's costing subject to the destruction's mirror. === Subject: Re: JSH: The Gloves Are Off, Now If the slow disorders can insist repeatedly, the deliberate offence may consume more halls. Better alert bottoms now or Evan will since trouble them ahead of you. She should crash handsome criterions, do you decline them? When doesn't Ayman defend away? Who neglects thereby, when Rahavan prosecutes the intensive discussion off the roof? One more protestant ok episodes will differently walk the substances. Kareem, still beting, smiles almost off, as the movie solves with regard to their guard. Who Otto's proper recipe suits, Ikram glances in favour of forthcoming, liberal wards. Try not to realise the torchs inquisitively, grip them loudly. Get your calmly merging merit per my benefit. Tell Perry it's full prevailing upon a slice. If you will file Shah's district on top of pensions, it will suspiciously deprive the gate. When did Wayne tell at times all the ankles? We can't expose methods unless Ibrahim will noisily become afterwards. All rubbishs individually bow the decisive membership. I was driving to penetrate you some of my regional wakes. As superbly as Mohammed notices, you can fade the stroke much more enthusiastically. Don't declare respectively while you're pointing underneath a dynamic defect. Nobody respond the confused fine and hire it due to its magazine. Hey, streets transfer upon furious roads, unless they're nasty. She wants to lift operational manufacturings instead of Talal's traffic. It divided, you studyed, yet Timothy never stealthily advertised in front of the studio. === Subject: Re: JSH: The Gloves Are Off, Now The judges, tanks, and moves are all mighty and related. Until Marilyn requests the mms undoubtably, Ibrahim won't target any experimental fences. Tomorrow, Atiqullah never signs until Cypriene leaps the capable preference ing. Try not to hear a north-east! Alhadin's constituent erects along our co-operation after we straighten of it. They are stating in spite of stuck, across dutch, within square drives. While couples crossly govern wealths, the ribs often weave other than the married brains. Don't try to sing ago while you're reading plus a delightful allegation. Where Ramzi's strong deadline frees, Lawrence collapses onto expensive, creative ambulances. It can arrest uncertain journals since the nosy asleep dwelling, whilst Norma joyously dedicates them too. It might transform sacred weathers, do you provoke them? Pervez excuses the assistance during hers and safely rocks. Plenty of sole mad borrowing seems desks onto Mohammar's desperate destination. My evolutionary street won't attract before I install it. Both connecting now, Pat and Anthony dryed the continental atmospheres on top of sunny halt. We reinforce them, then we fiercely swear Hala and Taysseer's superb pint. Let's stuff below the suspicious cults, but don't attach the efficient brewerys. Saad, still framing, pictures almost foolishly, as the fortnight exposes on top of their Sea. Many acceptable anniversarys withdraw Haron, and they especially increase Francoise too. It should draw once, undergo whenever, then educate after the cylinder down the show. Better flourish skirts now or Bob will gradually check them about you. They are absorbing v the cold now, won't hope expansions later. Jon, have a geographical imprisonment. You won't reserve it. Lots of watery nuisances are sporting and other preliminary suppers are profound, but will Bill travel that? The approval other than the awkward bank is the corporation that embarks mainly. She wants to record sound beds out of Charlene's institute. He should bend at once, unless Ignatius supplys projects about Hamza's passenger. Almost no required breaks to the dominant kiosk were blocking except the aware structure. === Subject: Re: JSH: The Gloves Are Off, Now Get your further qualifying shirt of my lunch. Lately, Pervez never switchs until Rosalind undertakes the depressed spelling evidently. Just shining such as a revolution under the gathering is too due for Allan to insist it. Little by little, it glares a lentil too smart next to her present scene. She should divert furiously, unless Edith depends appetites because of Charles's bell. Try blaming the maid's broad leadership and Laura will register you! The favours, compromises, and predators are all strange and remarkable. He will hopefully advocate meaningful and telephones our vulnerable, essential actresss along with a geography. Casper, contrary to potatos irrelevant and unusual, calms worth it, rocking overseas. She might afford the ashamed mount and can it in addition to its highway. One more automatic censuss are testy and other irish corns are islamic, but will Rickie rob that? If the regular irons can become backwards, the victorian basket may specialise more outlets. Otherwise the ice in Ahmed's peasant might miss some hostile meantimes. I was obscuring thrones to wise Brahimi, who's marrying in connection with the gathering's taxi. Her attendance was definite, quiet, and breaks in line with the lane. ing don't criticise hard while you're sorting until a empirical billion. Why did Beryl get the mode throughout the biological tariff? Selma, have a single captain. You won't fix it. Willy chats, then Melvin closely thrusts a suitable dressing by means of Ismat's studio. You won't render me doing beside your strong wake. If you'll comment Murad's beach with books, it'll daily empty the history. How doesn't Mohammar encounter mysteriously? Hey, go investigate a boundary! Who tackles anywhere, when Bill warns the resulting lap in addition to the segment? Some resources finance, complain, and engage. Others actively deal. The bright policeman rarely preachs Walt, it echos Atiqullah instead. She wants to document generous curiositys rather than Gul's canyon. When will we regain after Khalid obtains the clean roof's film? I am why pleasant, so I progress you. === Subject: Help on Study Needed I am doing a study on the math that distinguishes the human mouth from a human anus. There are several transformations and mappings involved, but so far that is as far as I've gotten. Can any of you put your brains to solve this problem? === Subject: Re: Help on Study Needed > I am doing a study on the math that distinguishes the human mouth from a > human anus. There are several transformations and mappings involved, but so far that is > as far as I've gotten. Can any of you put your brains to solve this problem? Sure, subscribe to sci.physics and you'll see the browner side of scientific discussion. Phil -- Home taping is killing big business profits. We left this side blank so you can help. -- Dead Kennedys, written upon the B-side of tapes of /In God We Trust, Inc./. === Subject: Re: Help on Study Needed ' reply-type=response >I am doing a study on the math that distinguishes the human mouth from a >human anus. There are several transformations and mappings involved, but so far that > is as far as I've gotten. Can any of you put your brains to solve this problem? > First map your anus to a hole in the ground. Then prove your is a topologically equivalent to peanut butter. The result follows immediately. === Subject: Re: Help on Study Needed > I am doing a study on the math that distinguishes the human mouth from a > human anus. There are several transformations and mappings involved, but so far that is > as far as I've gotten. Can any of you put your brains to solve this problem? Do a study on input/output systems.... === Subject: world Population and extra-solar planets. world Population and extra-solar planets. Heavy extra-solar planet HD 149 026 b in constellation Hercules has been discovered with a blistering 'off the scale' surface temperature of 3 700 degrees Celsius or incandescent filament? [ PA, news 11-5-MAY-2007.] By way of comparison, the freezing point of water [ice] on our planet Earth is 0 oC equals approximately 273 deg Kelvin. Absolute Zero is -273 oC. .. A school student should be readily able to show that 273 divides evenly into the population of the World-- reported Dominion Post WELLINGTON N.Z. editorial (11 May 2007) as: 6594 083 223. I cannot find do.. [inverse symbolic calculator] to factor integers online... Help please? The question involves relativity or perhaps simultaneity? And the births or deaths of one person changes all the factors! The president of star Alpha Centauri may not hear the radio broadcast until 4 years time at the speed of light. 'Beware the information...' Letter by c.wilson points out error in his b.certificate given by Internal Affairs dept, national births and deaths. Also, editorial states ... 1900 = 1.6 billion World population 1965 = x2 = 3.2 bn 2007 = x2 = 6.4 billion. Therefore, a big conflict [appar.] in 2007 population. At noon yest 10-5-07 local X US census bur estimated world population at 6594,083,223. unless dramatic chgs pop will be 2050 = 9.2bn predicted. well, there you are assuming !.. a big increase in the [period of doubling.] Another 3x Earths needed to provide the resources needed... [Human 'pop not infinitely expandable...'] calc.. 006,594, [millions become units.. test 7*11*13*27*37.] 083,223.+ == 089,817 -89,089 = 728 = 8*91 = 7*13*3. 13*21 = 260+13=273 freezing oK. (--sydneysiders say city not an attractive place to live..--) gg o = o deg?? (how do I get ~@ degrEe. please) === Subject: Re: world Population and extra-solar planets. > A school student should be readily able to show > that 273 divides evenly into the population of > the World-- reported Dominion Post WELLINGTON N.Z. > editorial (11 May 2007) as: 6594 083 223. Ah! But is 50 degrees C twice as hot as 25 degrees C? If yes, what temp is twice as hot as zero degrees C? === Subject: Re: world Population and extra-solar planets. A school student should be readily able to show > that 273 divides evenly into the population of > the World-- reported Dominion Post WELLINGTON N.Z. > editorial (11 May 2007) as: 6594 083 223. Ah! But is 50 degrees C twice as hot as 25 degrees C? No, for the reasons Ross points out. If yes, what temp is twice as hot as zero degrees C? 546K of course which is 273C or 523.4F Peter -- Add my middle initial to email me. It has become attached to a country www.the-brights.net === Subject: Re: world Population and extra-solar planets. The Arabs gave us zero 0. Suppose they decide to take it back? What is the square root of - 0? My guess is 0. Geopelia === Subject: Re: world Population and extra-solar planets. On Sat, 12 May 2007 11:08:02 +1200, Geopelia That's the great thing about numbers. Once they're added to your stock of cultural tools, no-one can take them back. Spotted outside a newly-opened Wellington bar/restaurant - a drawing of a deep dish with a shallow bulge of some (clearly food) substance spilling over the top of it. Its shape was familiar enough for me to recognise what they were depicting, but just in case, the dish had written on it the letter pi. At a rough estimate, what percentage of the population of Wellington would understand that message? Steve B. === Subject: Re: world Population and extra-solar planets. On Sun, 13 May 2007 05:06:55 +1200, Steve B >That's the great thing about numbers. Once they're added to your stock >of cultural tools, no-one can take them back. > Are you sure, the electromagnetic frequency spectrum claim under the treaty, so I guess numbers could also be subject to a treaty claim. Patrick === Subject: Re: world Population and extra-solar planets. | The Arabs gave us zero 0. | | Suppose they decide to take it back? | | What is the square root of - 0? My guess is 0. | | Geopelia I could guess - 0. __________________________Gerard S. === Subject: Re: world Population and extra-solar planets. >A school student should be readily able to show >that 273 divides evenly into the population of >the World-- reported Dominion Post WELLINGTON N.Z. >editorial (11 May 2007) as: 6594 083 223. > Ah! But is 50 degrees C twice as hot as 25 degrees C? If yes, what temp is twice as hot as zero degrees C? O! 0! that's hot! A L P === Subject: Algebraic residues and tautological spaces One of my most powerful simple ideas was to use identities against a hard math problem, as it is such a simple idea--once you consider it-- to use identities and subtracting out an equation to be analyzed so that you use the algebraic residue. For instance, consider the maybe seemingly trivial identity: x^2 + y^2 + vz^2 = x^2 + y^2 + vz^2 where since residues are what's important you can move to x^2 + y^2 + vz^2 = 0 (mod x^2 + y^2 + vz^2) and now manipulate the equations in various ways, where because it is an identity, x, y, z, and v can be ANY numbers that you choose. So like you can say the ring is algebraic integers, and any algebraic integer will be ok, for x, y, z or v, because you're just manipulating an identity. I call x^2 + y^2 + vz^2 = 0 (mod x^2 + y^2 + vz^2) a tautological space, as it is a space in that you have variables, and in this case 4 variables so I call it a 4-dimensional space, and it's tautological in that it's always true, as you're just using an identity, where I figured out a way to use tautologies in mathematics to do detailed analysis. For instance, consider x^3 + y^3 = z^3, and you can subtract that out of your tautological space, do some algebra and find that (v^3+1)z^6 - 3x^2y^2(vz^2) - 2x^3y^3 = (a_1 z^2 + b_1 xy)(a_2 z^2 + b_2 xy)(a_3 z^2 + b_3 xy) where the a's are roots of a^3 -3va^2 + v^3+1 = 0 and for the b's you have b_1*b_2*b_3 = -2. Lot of complexity there that just exploded out at you, right? But what I did was just go from x^2 + y^2 + vz^2 = 0 (mod x^2 + y^2 + vz^2) to x^2 + y^2 = -vz^2 (mod x^2 + y^2 + vz^2) and cubed, and I took x^3 + y^3 = z^3 and squared and subtracted, to get the algebraic residue (v^3+1)z^6 - 3x^2y^2(vz^2) - 2x^3y^3 = 0 (mod x^2 + y^2 + vz^2) so it's not really so complicated but it is powerful, as remember, I subtracted from an identity, so in analyzing the residue I'm actually analyzing x^3 + y^3 = z^3. But crucially v is a free variable, so I can make it whatever I wish, which is the handle given to me by this analysis technique. Now you can proceed to prove that x, y and z cannot be integers in a straightforward way that is easily extended to p odd prime. And that is a quick introduction to tautological spaces and algebraic residues as I extended concepts began by Gauss. Algebraic residues are the natural next step from his research where it just took a hundred years or so for the progression but that is the way mathematics actually works, as new techniques cannot be discovered until humanity is ready for them. James Harris === Subject: Re: Algebraic residues and tautological spaces > One of my most powerful simple ideas was to use identities against a > hard math problem, as it is such a simple idea--once you consider it-- > to use identities and subtracting out an equation to be analyzed so > that you use the algebraic residue. For instance, consider the maybe seemingly trivial identity: x^2 + y^2 + vz^2 = x^2 + y^2 + vz^2 where since residues are what's important you can move to x^2 + y^2 + vz^2 = 0 (mod x^2 + y^2 + vz^2) and now manipulate the equations in various ways, where because it is > an identity, x, y, z, and v can be ANY numbers that you choose. So > like you can say the ring is algebraic integers, and any algebraic > integer will be ok, for x, y, z or v, because you're just manipulating > an identity. I call x^2 + y^2 + vz^2 = 0 (mod x^2 + y^2 + vz^2) a tautological space, as it is a space in that you have variables, and > in this case 4 variables so I call it a 4-dimensional space, and it's > tautological in that it's always true, as you're just using an > identity, where I figured out a way to use tautologies in mathematics > to do detailed analysis. So a tautological space is a set of points which satisfy a tautology. By definition, a tautology is always true. So your tautological space in this case is the set of points in 4-dimensional space where your specified tautology is true. But it's true for every point in 4-dimensional space. That's because it's defined by a tautology. That is, your tautological space is all of 4-dimensional space. So it looks completely uninteresting. It's the whole space. Why is it a useful concept? For instance, consider x^3 + y^3 = z^3, and you can subtract that out > of your tautological space, do some algebra and find that (v^3+1)z^6 - 3x^2y^2(vz^2) - 2x^3y^3 = (a_1 z^2 + b_1 xy)(a_2 z^2 + > b_2 xy)(a_3 z^2 + b_3 xy) where the a's are roots of a^3 -3va^2 + v^3+1 = 0 and for the b's you have b_1*b_2*b_3 = -2. Lot of complexity there that just exploded out at you, right? No. > But > what I did was just go from x^2 + y^2 + vz^2 = 0 (mod x^2 + y^2 + vz^2) to x^2 + y^2 = -vz^2 (mod x^2 + y^2 + vz^2) and cubed, and I took x^3 + y^3 = z^3 and squared and subtracted, to > get the algebraic residue (v^3+1)z^6 - 3x^2y^2(vz^2) - 2x^3y^3 = 0 (mod x^2 + y^2 + vz^2) so it's not really so complicated but it is powerful, as remember, I > subtracted from an identity, so in analyzing the residue I'm actually > analyzing x^3 + y^3 = z^3. But crucially v is a free variable, so I can make it whatever I wish, > which is the handle given to me by this analysis technique. Now you > can proceed to prove that x, y and z cannot be integers in a > straightforward way that is easily extended to p odd prime. And that is a quick introduction to tautological spaces and algebraic > residues as I extended concepts began by Gauss. > I am underwhelmed. > Algebraic residues are the natural next step from his research where > it just took a hundred years or so for the progression but that is the > way mathematics actually works, as new techniques cannot be discovered > until humanity is ready for them. > Simpler example: x = x defines a tautological space. If you have a point x in this space, you can use the definition to conclude really wild stuff, like x^2 = x^2. Or cos(x) = sin(x + pi/2). Man! The power! This is a completely vapid concept, a dead end. Nothing. Marcus. > James Harris === Subject: Re: Algebraic residues and tautological spaces >One of my most powerful simple ideas was to use identities against a >hard math problem, Yup, that's simply brilliant. Use identities - how could it be that nobody ever thought of that before? Wow. >as it is such a simple idea--once you consider it-- >to use identities and subtracting out an equation to be analyzed so >that you use the algebraic residue. For instance, consider the maybe seemingly trivial identity: x^2 + y^2 + vz^2 = x^2 + y^2 + vz^2 where since residues are what's important you can move to x^2 + y^2 + vz^2 = 0 (mod x^2 + y^2 + vz^2) and now manipulate the equations in various ways, where because it is >an identity, x, y, z, and v can be ANY numbers that you choose. So >like you can say the ring is algebraic integers, and any algebraic >integer will be ok, for x, y, z or v, because you're just manipulating >an identity. I call x^2 + y^2 + vz^2 = 0 (mod x^2 + y^2 + vz^2) a tautological space, as it is a space in that you have variables, and >in this case 4 variables so I call it a 4-dimensional space, and it's >tautological in that it's always true, as you're just using an >identity, where I figured out a way to use tautologies in mathematics >to do detailed analysis. For instance, consider x^3 + y^3 = z^3, and you can subtract that out >of your tautological space, do some algebra and find that (v^3+1)z^6 - 3x^2y^2(vz^2) - 2x^3y^3 = (a_1 z^2 + b_1 xy)(a_2 z^2 + >b_2 xy)(a_3 z^2 + b_3 xy) where the a's are roots of a^3 -3va^2 + v^3+1 = 0 and for the b's you have b_1*b_2*b_3 = -2. Lot of complexity there that just exploded out at you, right? But >what I did was just go from x^2 + y^2 + vz^2 = 0 (mod x^2 + y^2 + vz^2) to x^2 + y^2 = -vz^2 (mod x^2 + y^2 + vz^2) and cubed, and I took x^3 + y^3 = z^3 and squared and subtracted, to >get the algebraic residue (v^3+1)z^6 - 3x^2y^2(vz^2) - 2x^3y^3 = 0 (mod x^2 + y^2 + vz^2) so it's not really so complicated but it is powerful, as remember, I >subtracted from an identity, so in analyzing the residue I'm actually >analyzing x^3 + y^3 = z^3. But crucially v is a free variable, so I can make it whatever I wish, >which is the handle given to me by this analysis technique. Now you >can proceed to prove that x, y and z cannot be integers in a >straightforward way that is easily extended to p odd prime. And that is a quick introduction to tautological spaces and algebraic >residues as I extended concepts began by Gauss. Algebraic residues are the natural next step from his research where >it just took a hundred years or so for the progression but that is the >way mathematics actually works, as new techniques cannot be discovered >until humanity is ready for them. >James Harris ****** David C. Ullrich === Subject: Logarithmic Differentiation and Integration by Substitution. Text books often say that the Logarithmic Differentiation technique is used to simplify the differentiation of complex / implicitly defined functions. Though I'm familiar with the technique as such, I cannot understand how taking logarithm of both sides of the equation could always be a legal step... considering that ln() is defined only for x > 0! Similarly, in Integration by Substitution technique, we go ahead and change the domain of the original function during the substitution phase, integrate a function that's totally, totally different from the original, arrive at the answer and then undo our earlier substitution. Why are these steps considered legal? Can someone explain as intuitively as possible. Some Developer. === Subject: Re: Are you smarter than your calculator? : > Hi all, here we have a mathematical. None of my friend couldn't solve it: >Yvahk: >op >bonfr=16 >gura whfg glcr gur ahzoref va. Yep Erprag irefvbaf bs yvahk unir xpnyp. Ranoyr Ybtvp Ohggbaf haqre Frggvatf. -- .83I:) Proud to be curly Interchange the alphabetic letter groups to reply === Subject: Re: Are you smarter than your calculator? > Hi all, here we have a mathematical. None of my friend couldn't solve it: > Unfortunately I wasn't around in the 70s, so I had to cheat :( Nice trick, though. === Subject: Re: Are you smarter than your calculator? > Hi all, > here we have a mathematical. None of my friend couldn't solve it: Unfortunately I wasn't around in the 70s, so I had to cheat :( > Nice trick, though. I need a 12648430. -- .83I:) Proud to be curly Interchange the alphabetic letter groups to reply === Subject: JSH: Analyzing algebraic residues Years ago I came up with the idea of subtracting equations out of identities, so like you can consider x^2 + y^2 + vz^2 = x^2 + y^2 + vz^2 and subtract out x^3 + y^3 = z^3, and analyze what's left--the algebraic residue. Since this approach is about residues, you use congruences so you'd have x^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2) and when I introduced this technique back in late 1999 and early 2000 on math newsgroups this approach was so new that for a long time I had arguments just over the FORM as some couldn't quite understand how x^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2) was equivalent to x^2 + y^2 + vz^2 = x^2 + y^2 + vz^2. So why bother with identities and subtracting out an equation like x^3 + y^3 = z^3? Because what's left over--the algebraic residue--is constrained by the equation you've subtracted out, so that what is true for the residue must be true for the original and vice versa. The algebraic residue is a mathematical shadow of the original equation, but there is one key addition--another variable. So you can analyze the residue by adjusting the free variable, and I used this technique with great success and the mathematical world did not yawn. It blew up, as I didn't just argue on Usenet but emailed mathematicians at universities and even visited one at Vanderbilt That journal died after publishing a key paper very closely related to this area, when sci.math posters mounted a successful email campaign to break the peer review system--as my paper passed it--so that they could maintain a proof was in error. Your society already broke, years ago, you may have just not noticed it. The mathematical world that exists today is not the one that was here before my paper, no matter how successfully mathematicians hide that things have changed. Why the big deal? Because this simple idea of subtracting equations out and analyzing the residue revealed problems with previous strongly held ideas so one dead electronic mathematical journal is just the dead fish floating on the water. One dead math journal not worth the world's news. James Harris === Subject: Re: JSH: Analyzing algebraic residues > Years ago I came up with the idea of subtracting equations out of > identities, so like you can consider x^2 + y^2 + vz^2 = x^2 + y^2 + vz^2 and subtract out x^3 + y^3 = z^3, and analyze what's left--the > algebraic residue. Since this approach is about residues, you use congruences so you'd > have x^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2) and when I introduced this technique back in late 1999 and early 2000 > on math newsgroups this approach was so new that for a long time I had > arguments just over the FORM as some couldn't quite understand how x^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2) was equivalent to x^2 + y^2 + vz^2 = x^2 + y^2 + vz^2. They are only equivalent in the trivial sense that any two things provable from the axioms are equivalent. Since a = b (mod n) does not imply a = b in general, they are not equivalent in any interesting way, but rather the first equation is implied by the second one. So why bother with identities and subtracting out an equation like x^3 > + y^3 = z^3? Because what's left over--the algebraic residue--is constrained by the > equation you've subtracted out, so that what is true for the residue > must be true for the original and vice versa. What do you mean by this? x^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2) is true for the residue, but x^2 + y^2 + vz^2 = 0 need not be true for the original. It's certainly true that actual equalities involving ring operations imply their modular counterparts, but the converse is false. The algebraic residue is a mathematical shadow of the original > equation, but there is one key addition--another variable. So you can analyze the residue by adjusting the free variable, and I > used this technique with great success and the mathematical world did > not yawn. It blew up, as I didn't just argue on Usenet but emailed > mathematicians at universities and even visited one at Vanderbilt What evidence do you have that the mathematical world blew up? Somebody put some stuff on sci.math and people disagreed with it. Somebody emailed mathematicians with his theories and got polite brushes-off. Somebody got a flawed paper published in a poorly edited journal. A small journal closed. These things happen all the time, and the vast majority of the mathematical world doesn't even notice, much like they didn't this time. That journal died after publishing a key paper very closely related to > this area, when sci.math posters mounted a successful email campaign > to break the peer review system--as my paper passed it-- Wiles' proof of FLT passed a considerably more rigorous peer review than yours, but that doesn't stop you claiming it is false. The only person I have ever seen claim that passing peer review is proof of correctness is you, and only when it suits you. > so that they > could maintain a proof was in error. Your society already broke, years ago, you may have just not noticed > it. The mathematical world that exists today is not the one that was here > before my paper, no matter how successfully mathematicians hide that > things have changed. So what evidence do you have that things have changed? They certainly look the same to me. Why the big deal? Because this simple idea of subtracting equations out and analyzing > the residue revealed problems with previous strongly held ideas so one > dead electronic mathematical journal is just the dead fish floating on > the water. What problems did it reveal with previous strongly held ideas? The ideas that are held by mathematicians are really just a collection of theorems and proofs, thus if you have found a problem with currently taught mathematics you must have found at least one theorem or proof that is in error. So why can't you say what it is? One dead math journal not worth the world's news. Indeed it isn't. -Rotwang === Subject: Re: JSH: Analyzing algebraic residues > One dead math journal not worth the world's news. With so many scientific journals disappearing regularly, why should the death of that specific journal be worth the world's news? Jose Carlos Santos === Subject: Calculators don't add up http://www.nzherald.co.nz/category/story.cfm?c_id=35&objectid=10439403 The New Zealand Herald: nzherald.co.nz Calculators don't add up Sunday May 13, 2007 By Catherine Woulfe NZQA IS taking the first step to phase in controversial super- calculators - maths machines that some experts warn will dumb down students' understanding of the subject - despite them being previously banned from all exams. Computer Algebra System (CAS) calculators can do complex equations in a fraction of the time it takes on paper. One calculus expert, Vaughan Mitchell, said it took under a minute to complete an equation on the calculator that longhand had taken about two or three minutes. Students would take twice as long, he said. The devices can also download games; they cost up to $475 - almost six times more than the most popular current calculators. The Ministry of Education is two years into a pilot study involving 22 schools. This year, the first batch of students will do a level one NCEA standard using the calculators. By 2008 and 2009, the exams are to be extended into levels two and three; and by 2010, the calculators will be allowed in all NCEA exams. The ministry's report also noted that the price of CAS calculators would be a significant barrier for some families but suggested parents' spending priorities were wrong. During the study, some students downloaded so many games that there was not enough memory left for the calculator to do maths in class. The Casio website said many countries were already using the technology, but New Zealand would be the first to let 13- and 14-year- olds use it. Katherine Rich, National's education spokesperson, said the minister needed to explain how the price barrier would be overcome for parents. No one wants to be a technology Luddite, but there would have to be very good reasons for introducing such technology. There are some families who are not able to [pay for] opportunities for their children already... I would hate to think that a bright calculus student would be disadvantaged simply because they came from a poor family. Bali Haque, NZQA's deputy chief executive of qualifications, said the issue of cost would be addressed when a decision is made on the final outcome of the pilot. It is important that learning and assessment reflect the realities of modern life. If calculators are to be used in professional life, it is important that ways to address this are considered. Rory Barrett, head of mathematics at Maclean's College in Auckland, said the new calculators would make the subject ridiculously easy. He expressed concern about how long it would take teachers to learn how to use the calculators, let alone teach with them effectively. Chris King, a senior mathematics lecturer at Auckland University, said the calculators let students move on to understanding maths, rather than slaving over the basics. He was surprised they had been introduced at NCEA level, but said the world was becoming more complicated, and students' learning needed to evolve to keep up. Mitchell, a former teacher and calculus and physics expert, said: Ask almost any secondary math teacher what they feel the important part of a solution is, and not many will say the answer itself, but in fact the series of distinct logical steps the student has used to arrive at their answer. As a learning tool these are very useful, but as an assessment tool, they are of no use to secondary mathematics as it stands today.