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Whitehound
The Whitehound Corporation
Frontline Assembly Point
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Posted - 2011.04.20 12:34:00 -
[181]
Edited by: Whitehound on 20/04/2011 12:54:46
 Originally by: Kazuo Ishiguro
I would agree that the set of all random walks of finite lengths is countable (via a process of induction), but not the set of random walks of infinite length. By a similar line of reasoning: the set of integers Z is countable, and so is Z^n for any finite n, but Z raised to an unbounded power is not countable.
I am not as generous as you are. I believe that the requirement of a walk to have a starting point and then to travel infinitely across a countable infinite space renders them countable infinite.
Just like N is countable infinite are the rational numbers Q countable infinite, all while the subset [1, 2) for N contains only one number does this subset contain infinite numbers for the set of rational numbers. You can then construct Q from N^2. Q seems to be far larger, but Cantor showed that it is really only a different set of countable infinite elements.
A set S of countable infinite elements will always return a countable infinite set when raised to a higher dimension (i.e. SxS or S^3). Only when raised to infinite dimensions does it become uncountable infinite (iirc).
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Kazuo Ishiguro
House of Marbles
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Posted - 2011.04.20 18:16:00 -
[182]
Originally by: Whitehound
A set S of countable infinite elements will always return a countable infinite set when raised to a higher dimension (i.e. SxS or S^3). Only when raised to infinite dimensions does it become uncountable infinite (iirc).
We're in agreement on this point, and also about the rationals, each of which is after all just a pair of integers.
Quote:
I am not as generous as you are. I believe that the requirement of a walk to have a starting point and then to travel infinitely across a countable infinite space renders them countable infinite.
How is this different from constructing a real number (in the interval [0,1), w.l.o.g.) via its base-2n expansion? Each expansion has a first digit and carries on indefinitely. This is, loosely speaking, one definition of the reals - the set of all numbers (with zero imaginary part) that are limits of Cauchy sequences.
In fact, you would hit some of the real numbers more than once that way (can you see which ones?), but for any real number you can think of, there is at least one corresponding infinite random walk.
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34.4:1 mineral compression |
Whitehound
The Whitehound Corporation
Frontline Assembly Point
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Posted - 2011.04.20 19:38:00 -
[183]
Originally by: Kazuo Ishiguro
How is this different from constructing a real number (in the interval [0,1), w.l.o.g.) via its base-2n expansion? Each expansion has a first digit and carries on indefinitely. This is, loosely speaking, one definition of the reals - the set of all numbers (with zero imaginary part) that are limits of Cauchy sequences.
It is not the definition of all real numbers. This is already your mistake. It is only how a single real number is found.
Try to write down the number 1/3 (or 0.3) in binary notation. It is a rational number and yet will it need infinite binary digits. Imagine then all rational numbers as a sequence of 0 and 1. You will again need "infinite X infinite" digits (or N^2) to construct Q.
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Scorpyn
Caldari
Warp Ghosts Omega
Spectres of the Deep
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Posted - 2011.04.20 22:36:00 -
[184]
Originally by: Whitehound
You then need to fulfil the requirement of all chosen directions to have an equal probability. The further away you get from the origin the more steps you need to make into the reverse direction to satisfy this requirement. So when you approach infinity are you left with at most 2 out of 4 directions to get away and you are forced to choose the ones leading back, or else the probabilities of the chosen directions will not be equal.
Yes, but while you approach infinity (in distance from the origin) you will also have lots more possible paths a lot closer to the origin.
If the amount of possible paths that haven't returned to the origin increase faster than the amount of possible paths that have returned to the origin, then that means that there is a bias towards not going back to the origin.
In the 2D case (amount of possible paths ending in a specific location) :
(0 steps) (1 step) (2 steps)
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1 0 0 0 2 0 2 0
0 0 1 0 0 -> 0 1 0 1 0 -> 1 0 4 0 1
0 0 0 0 0 0 0 1 0 0 0 2 0 2 0
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
After 2 iterations, you have 4 possible paths that have returned, but 12 that haven't. Now I do understand that the further you get from the origin the more it turns into a 50% chance of getting closer or further away, but since those aren't the only possible paths after taking that amount of steps, the results will still be skewed.
More iterations are needed though, because the same pattern can be seen in 1D after the same amount of steps :
0 0 1 0 0 -> 0 1 0 1 0 -> 1 0 1 0 1
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Whitehound
The Whitehound Corporation
Frontline Assembly Point
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Posted - 2011.04.21 00:29:00 -
[185]
Originally by: Scorpyn
Yes, but while you approach infinity (in distance from the origin) you will also have lots more possible paths a lot closer to the origin.
No no no. This is irrelevant. You may want to believe that there are more numbers in Q than in N, that Q may be infinite times larger than N, but they are both "only" infinite.
Infinity is not a limiter and therefore does not define size. It means that there is no limit.
For this reason is there no such thing as more and less infinite walks. Relations like "more" and "less" make no sense here.
You also believe that a walk has to be close to its origin to have a higher chance to return, but this is not true either. Any walk can return no matter how far away it is, its chance to return is independent from its length and its length is infinite. Walks also do not end and so your number example has got no sense either. The numbers you put in your grid will all go towards infinity and whatever order you had in mind will drown in infinity.
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Akita T
Caldari Navy Volunteer Task Force
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Posted - 2011.04.21 13:55:00 -
[186]
Edited by: Akita T on 21/04/2011 14:05:47
Originally by: Whitehound
Originally by: Akita T
Do you at least agree that "In the case of a random 3D walk, as numbers of steps increase (ignoring the limit to infinity), the percentage of walks that do return to the origin approaches ~34.05%, not 100%" ?!?
Do I agree? No. [...] I can take each of these infinite[...]
Apparently, you can't even seem to be able to read a simple question and answer to it...
Originally by: Whitehound
The real numbers are uncountable infinite whereas walks are only countable infinite.
The first part, yes, sure. The second part, THAT REQUIRES PROOF.
A few examples of uncountable infinite sets include "the set of infinite sequences of natural numbers" and also "the set of subsets of natural numbers". So what makes you say with so much certainty that the set of all possible infinite walks is countable infinite as opposed to uncountable infinite ?
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Whitehound
The Whitehound Corporation
Frontline Assembly Point
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Posted - 2011.04.21 18:06:00 -
[187]
Edited by: Whitehound on 21/04/2011 18:16:17
Originally by: Akita T
...
Do you never get tired of trolling or was your break a consequence of it?
I will not go through it with you again only because you did not understand it the first time, then got tired and now take a second attempt. I have no reason to believe that you will ever get it right, but that I am only wasting my time with you. Instead, I am beginning to think that you sit in front of your computer and whenever you should be using your head to think about the problems I give you do you light a cigarette and smoke it away only so you can make yourself heard again.
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Akita T
Caldari Navy Volunteer Task Force
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Posted - 2011.04.21 18:10:00 -
[188]
Take a wild uncountable infinite set of guesses.
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Whitehound
The Whitehound Corporation
Frontline Assembly Point
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Posted - 2011.04.21 18:17:00 -
[189]
Originally by: Akita T
Take a wild uncountable infinite set of guesses.
I take it as a simple "yes".
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Sitara
Minmatar
Solar Flare Trade and Production
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Posted - 2011.04.21 18:33:00 -
[190]
Originally by: Whitehound
Do you never get tired of trolling
The irony 
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Akita T
Caldari Navy Volunteer Task Force
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Posted - 2011.04.21 18:59:00 -
[191]
Originally by: Whitehound
Originally by: Akita T
Take a wild uncountable infinite set of guesses.
I take it as a simple "yes".
Well, since for you P = 1 for just about everything involving infinity, it does make perfect sense that you would.
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Whitehound
The Whitehound Corporation
Frontline Assembly Point
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Posted - 2011.04.21 19:08:00 -
[192]
Originally by: Akita T
Well, since for you P = 1 for just about everything involving infinity, it does make perfect sense that you would.
You cannot stop failing. Please, never stop posting.
How about you start making infinite coin tosses and when you are done will I accept anything you say as the truth?
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Akita T
Caldari Navy Volunteer Task Force
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Posted - 2011.04.21 19:26:00 -
[193]
Edited by: Akita T on 21/04/2011 19:27:41
again...
Let N be an arbitrary natural number, extremely large but still finite.
Given the entire possible set of walks of precisely length N, how many of them did get back to the origin at least once ?
Is it NOT roughly 34.05% of them ?
Again, N is a very large but FINITE number.
No mention of infinity anywhere here YET.
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Whitehound
The Whitehound Corporation
Frontline Assembly Point
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Posted - 2011.04.21 19:57:00 -
[194]
Originally by: Akita T
Edited by: Akita T on 21/04/2011 19:27:41
again...
Let N be an arbitrary natural number, extremely large but still finite.
Given the entire possible set of walks of precisely length N, how many of them did get back to the origin at least once ?
Is it NOT roughly 34.05% of them ?
Again, N is a very large but FINITE number.
No mention of infinity anywhere here YET.
All walks of a finite length need to return to the origin. If they end just one step away from the origin then the probability requirement will have been failed, and because you can calculate a precise difference in probability for each direction.
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Akita T
Caldari Navy Volunteer Task Force
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Posted - 2011.04.21 21:04:00 -
[195]
Originally by: Whitehound
If they end just one step away from the origin then the probability requirement will have been failed
If they end up just one step away then they haven't returned yet, and out of all possible walks of length N+1, only ONE extra for each one that's one step away will have gotten back, while 5 more still won't have.
Originally by: Whitehound
All walks of a finite length (N) need to return to the origin.
Excuse me, WHAT ? Where the hell do you get that idea from ?
So far you have at least stayed within the realm of the borderline plausible, claiming that mandatory return only for infinite walks, but now you end up saying the same thing for not just some of the FINITE walks too, but ALL of them ?!?
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Whitehound
The Whitehound Corporation
Frontline Assembly Point
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Posted - 2011.04.21 21:20:00 -
[196]
Edited by: Whitehound on 21/04/2011 21:22:24
Originally by: Akita T
Originally by: Whitehound
If they end just one step away from the origin then the probability requirement will have been failed
If they end up just one step away then they haven't returned yet, and out of all possible walks of length N+1, only ONE extra for each one that's one step away will have gotten back, while 5 more still won't have.
Originally by: Whitehound
All walks of a finite length (N) need to return to the origin.
Excuse me, WHAT ? Where the hell do you get that idea from ?
So far you have at least stayed within the realm of the borderline plausible, claiming that mandatory return only for infinite walks, but now you end up saying the same thing for not just some of the FINITE walks too, but ALL of them ?!?
Learn to read. And think before you post.
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Akita T
Caldari Navy Volunteer Task Force
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Posted - 2011.04.21 22:06:00 -
[197]
Edited by: Akita T on 21/04/2011 22:08:07
Originally by: Whitehound
Learn to read. And think before you post.
If you are having trouble with people misunderstanding what you are saying, learn to express your thoughts more coherently in words that can not be interpreted in more than one way, or at least learn to respond to the question that was asked, not to a random point in the previous discussion without clearly referencing it, and last but not least, learn to respond to simple questions that require a simple yes or no answer with either yes or no.
It is quite clear that for any given number of steps N, SOME of the total POSSIBLE 6^N paths will not have returned at all.
There can be no dispute here, it is obvious that such paths do exist as long as N is finite in length.
YOU are claiming that as the number N gets larger (NOT EVEN DISCUSSING THE LIMIT "TOWARDS INFINITY"), the percentage of paths that did not return up to step N out of all possible 6^N steps is getting smaller, approaching 0%.
I am saying that within the collection of linked materials lies the clear mathematical proof that it only gets closer to roughly 66%, not 0%.
You are saying that this is not the case, that this is not what any of the materials say, but repeatedly fail to express what you think they say in clear and concise terms.
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Whitehound
The Whitehound Corporation
Frontline Assembly Point
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Posted - 2011.04.21 22:24:00 -
[198]
Edited by: Whitehound on 21/04/2011 22:25:34
Originally by: Akita T
If you are having trouble with people misunderstanding what you are saying, ...
Originally by: Akita T
Excuse me, WHAT ?
You are a terrible troll. You were amusing but now you are getting pathetic.
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Akita T
Caldari Navy Volunteer Task Force
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Posted - 2011.04.21 22:45:00 -
[199]
Edited by: Akita T on 21/04/2011 22:46:36
Originally by: Whitehound
You are a terrible troll. You were amusing but now you are getting pathetic.
Ran out of non-arguments and unsubstantiated claims already ? D'awwww...
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Scorpyn
Caldari
Warp Ghosts Omega
Spectres of the Deep
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Posted - 2011.04.21 22:47:00 -
[200]
Originally by: Whitehound
Originally by: Scorpyn
Yes, but while you approach infinity (in distance from the origin) you will also have lots more possible paths a lot closer to the origin.
No no no. This is irrelevant. You may want to believe that there are more numbers in Q than in N, that Q may be infinite times larger than N, but they are both "only" infinite.
Infinity is not a limiter and therefore does not define size. It means that there is no limit.
For this reason is there no such thing as more and less infinite walks. Relations like "more" and "less" make no sense here.
Note the wording "approaches infinity". It's not the same as infinity. In a previous post I suggested that the question has 4 answers, and in the case of an amount of steps that approaches infinity within an infinity, it makes sense to talk about percentage of returning paths after a certain number of steps, even if that number of steps approaches infinity.
Originally by: Whitehound
You also believe that a walk has to be close to its origin to have a higher chance to return
What? No, you misunderstood me. (They should however have a higher chance of returning in the same amount of steps as paths that are further away from the origin.)
Originally by: Whitehound
Walks also do not end and so your number example has got no sense either.
I'm trying to explain the concept of amount of returning walks vs non-returning walks after a certain amount of steps. I do agree that with an infinite amount of steps all paths will return, but not if it only approaches infinity, which is why I previously mentioned that I believe that the question has 4 answers.
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Whitehound
The Whitehound Corporation
Frontline Assembly Point
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Posted - 2011.04.21 22:56:00 -
[201]
Edited by: Whitehound on 21/04/2011 22:57:24
Originally by: Akita T
Ran out of non-arguments and unsubstantiated claims already ? D'awwww...
This is you, Akita. All your math boils down to creating links on the Internet it seems. You fail to follow a simple discussion and are not able to understand what it is you linked to. You are not even making an attempt to stay on the topic any longer. All one now gets from you are excuses.
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Akita T
Caldari Navy Volunteer Task Force
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Posted - 2011.04.21 23:04:00 -
[202]
Originally by: Whitehound
Originally by: Akita T
Ran out of non-arguments and unsubstantiated claims already ? D'awwww...
This is you, Akita. All your math boils down to creating links on the Internet it seems. You fail do follow a simple discussion and are not able to understand what it is you linked to. You are not even making an attempt to stay on the topic any longer. All one now gets from you are excuses.
And all your math boils down to is spouting unsubstantiated conclusions, refusing to answer simple questions, and getting wrong the few concrete answers you do give to those questions.
It is ABSOLUTELY CERTAIN that for a given FINITE LENGTH (of at least reasonable size) for random 3D walks, only roughly A THIRD of them will have returned to the origin within the allotted number of steps.
The larger the number of steps, the closer to 34.05% the percentage of walks that will have had returned to the origin goes.
THIS IS EVEN DEMONSTRABLE PRACTICALLY, NOT JUST THEORETICALLY.
Oh, hey, look, another link.
Of course you will not care to actually run the practical part, or if you somehow miraculously will, you will still deny the obvious.
_
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Scorpyn
Caldari
Warp Ghosts Omega
Spectres of the Deep
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Posted - 2011.04.21 23:18:00 -
[203]
Originally by: Whitehound
@Scorpyn: I'll try to catch up with you tomorrow.
No rush, I'll probably be busy with this easter thingy for a few days.
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Whitehound
The Whitehound Corporation
Frontline Assembly Point
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Posted - 2011.04.22 08:19:00 -
[204]
Originally by: Scorpyn
Note the wording "approaches infinity". ...
Which is why I said that it is irrelevant... You have mentioned Cauchy series, so let me compare it to them. For the sum of an infinite series to converge does it need a Cauchy series. You may believe that for this convergence it would be enough to have a series converging to 0 so that the sum of the series itself converges (believing you are adding only 0 to the sum as you approach infinity), but it needs more than this. The series needs to converge, it needs to converge to 0 and the difference between any element of the series needs to converge to 0, too (the later establishes an independence between all elements). You may use these requirements to believe that there are far more sums of series that diverge than to actually converge, but you would be wrong to believe so. There are as many convergent sums of series as there are real numbers.
The same is true for infinite walks. You may think that the speed or a speed difference of something going towards infinity makes a difference in reaching infinity, but it does not. Infinity cannot be reached and the speed at which you approach it makes no difference to the fact. You are only racing a race that does not have an end and therefore will never have a winner. The speed alone at which something grows does not make it stop, just as it is not enough to have an infinite series converging towards 0 for the sum of it to converge.
I have not read Polya's proof, but only skimmed it (I am not the one who makes claims based on my ability to link to it...). He will likely have made further conditions in his proof to come up with his numbers. For example, finding finite and independent values (Qa, Qb) for the quantity of walks as their number approaches infinity so that he can say "the probability is the ratio of Qa to Qb".
Whatever it is, Akita does not understand it. All the proof is to her is to be a puzzle and it is also the only application she sees in it, to post it as a puzzle.
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Whitehound
The Whitehound Corporation
Frontline Assembly Point
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Posted - 2011.04.22 08:40:00 -
[205]
Edited by: Whitehound on 22/04/2011 08:41:18
Originally by: Akita T
Oh, hey, look, another link.
Of course you will not care to actually run the practical part, or if you somehow miraculously will, you will still deny the obvious.
Thank you, I did take a look. I cannot say however that I saw an infinite walk. More like 1000 iterations and a break statement to end the walk even sooner. Say, if it was not for this 1000-steps-condition, where would the iteration have a chance to end, other than at its origin? If you removed the 1000-steps-condition, to actually get an infinite walk, would its probability to end at its origin be 1.
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Sturmwolke
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Posted - 2011.04.22 13:34:00 -
[206]
Originally by: Sturmwolke
This thread might have a probability of extending into pages 7 and 8; oh! say, 15.9% ... given infinite time of course.
Page 7
Mwahahahahaha !
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Akita T
Caldari Navy Volunteer Task Force
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Posted - 2011.04.22 16:43:00 -
[207]
Edited by: Akita T on 22/04/2011 16:45:34
Originally by: Whitehound
Originally by: Akita T
Oh, hey, look, another link.
Of course you will not care to actually run the practical part, or if you somehow miraculously will, you will still deny the obvious.
Thank you, I did take a look. I cannot say however that I saw an infinite walk. More like 1000 iterations and a break statement to end the walk even sooner. Say, if it was not for this 1000-steps-condition, where would the iteration have a chance to end, other than at its origin? If you removed the 1000-steps-condition, to actually get an infinite walk, would its probability to end at its origin be 1.
Gee, I stopped talking about INFINITE walks quite a while ago and started talking about FINITE walks of very long length... oh, you know, since right before my break or so.
And you haven't noticed yet ? You'd think the caps and bold would have at least been able to draw a tiny bit of your attention to that fact.
Use the exact same simulation, ramp up the break condition to 10^4 steps.
Do you think the result will be closer to 1, or closer to 34% ?
How about if the break condition is at 10^5 steps ? 10^6 steps ? 10^9 steps ?
That's EXACTLY what I asked before the break, and you said "1" not "0.34", while the practical simulation would spit results closer to 0.34
Do you still claim the same "1" now that you saw the practical example for FINITE length walks ?
Assuming you did finally realize we're talking finite length, not infinite anymore.
The clarification of correct answers for a FINITE length walk (of long step count) is a very important prerequisite before being able to even begin moving forward into the discussion of infinite length walks.
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Whitehound
The Whitehound Corporation
Frontline Assembly Point
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Posted - 2011.04.22 18:50:00 -
[208]
Edited by: Whitehound on 22/04/2011 18:51:25
Originally by: Akita T
The clarification of correct answers for a FINITE length walk (of long step count) is a very important prerequisite before being able to even begin moving forward into the discussion of infinite length walks.
You have no understanding of infinite maths and your only known approach to it is to "ramp it up"? And you want to use it to talk about infinite walks? You are so full of fail.
I will go and tune my car. With Akita's help will I be able to reach infinite speed by "ramping it up".  Bye, I need to find a car mechanic with a very special ramp ...
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Akita T
Caldari Navy Volunteer Task Force
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Posted - 2011.04.22 20:05:00 -
[209]
Edited by: Akita T on 22/04/2011 20:15:15
Originally by: Whitehound
You have no understanding of infinite maths and your only known approach to it is to "ramp it up"? And you want to use it to talk about infinite walks?
Whereas your presentation of the understanding of "infinite maths" pretty much just boils down to "ha-ha, you said 'INFINITY' at one point instead of 'approaching infinity' as you should have, so anything you might ever say from this point onwards has to be invalid, I can't hear you lalalalala" while sticking your fingers in your ears.
Quote:
I will go and tune my car. With Akita's help will I be able to reach infinite speed by "ramping it up". Bye, I need to find a car mechanic with a very special ramp ...
Funny you should say that, because YOUR logical approach to the problem would claim to result in infinite speed, whereas my logical approach would estimate a finite top speed.
To put it in other words, YOU are claiming something akin to (crudely) "the series 1/4 + 1/8 + 1/16 + ... + 1/2N" not only when N equals infinity EQUALS INFINITY, as opposed to, you know, simply being equal to 0.5 instead... but also when N grows very large too, you imply the result will somehow be able to keep growing beyond the 0.5 boundary somehow magically.
Or, you know, probability of anything when the variable approaches infinity equals 100%, in your math knowledge.
Congratulations on revolutionizing math, a lot of proofs just became not only wrong but wholly unnecessary, since, hey, "infinity", blah, right ?[/sarcasm]
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Whitehound
The Whitehound Corporation
Frontline Assembly Point
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Posted - 2011.04.22 20:31:00 -
[210]
Originally by: Akita T
Whereas your presentation of the understanding of "infinite maths" pretty much just boils down to "ha-ha, you said 'INFINITY' ...
Yes, because your approach to it is a joke. Just shut up now and make your loop an infinite loop. Do not come back and start arguing before you have actually turned it into one.
See, this is what you wrote: GIVEN AN INFINITE AMOUNT OF TIME, what is the probability you will EVENTUALLY land on the origin point at least once again for each of the three cases ?
So do give it an infinite amount of time.
Btw, do you know what the next joke is? Your use of a random-function! Did you know that randomness is another abstract and just as infinity does not exist? It is amusing to see how you are only willing to talk about infinity, but not actually willing to put it into your code, while at the same time you are using a random-function. 
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