Wikipedia:Reference desk/Archives/Mathematics/2013 January 11

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January 11

Converting this to an integrable form

In this system, the acceleration at a point is defined by $a\times \cos(\operatorname {atan2} (y,x))$ in the x-dimension and $a\times \sin(\operatorname {atan2} (y,x))$ in the y-dimension. I guess this makes it ${\operatorname {d} \over \operatorname {d} \!t}{\begin{bmatrix}v_{x}\\v_{y}\end{bmatrix}}={\begin{bmatrix}a\times \cos(\operatorname {atan2} (y,x))\\a\times \sin(\operatorname {atan2} (y,x))\end{bmatrix}}$. How do I derive the position function for any point after time $t$? EDIT: The particle also has an initial velocity. — Melab±1 ☎ 01:56, 11 January 2013 (UTC)

Looks like your first step should be to convert the problem to polar co-ordinates. Gandalf61 (talk) 03:38, 11 January 2013 (UTC)

Since $\cos(\operatorname {atan2} (y,x))=x$ and $\sin(\operatorname {atan2} (y,x))=y$, the ODEs decouple to ${\ddot {x}}=ax$ and ${\ddot {y}}=ay$, which are not very difficult to solve. (For one sign of a, check harmonic oscillator). —Kusma (t·c) 15:30, 11 January 2013 (UTC)

Looking at this I realized that I neglected to say that the particle has an initial velocity. And I don't know how to apply the negative sign to it. — Melab±1 ☎ 21:59, 12 January 2013 (UTC)

The general solution for $a=b^{2}>0$ is $x=c_{1}\cosh(bt)+c_{2}\sinh(bt)$. If a=0 it is $x=c_{1}t+c_{2}$. For $a=-b^{2}<0$ it is $x=c_{1}\cos(bt)+c_{2}\sin(bt)$ (and the same for y). You can find the constants $c_{1},c_{2}$ by using the initial conditions for position and velocity. —Kusma (t·c) 11:50, 13 January 2013 (UTC)

Where does $b$ come from? Which of the trigonometric functions come from harmonic oscillation? And shouldn't there be a square in their somewhere? Also, if the equation for $x$ is the same for $y$, then it will only oscillate in a line. — Melab±1 ☎ 15:33, 13 January 2013 (UTC)

$b={\sqrt {|a|}}$. The constants of integration are not necessarily the same for the x and y motions, so the particle won't in general just oscillate along a line. But as long as a is negative the motion will be periodic, with a period of $2\pi /b$. It is beginning to sound as if this problem is a little out of your depth. Gandalf61 (talk) 17:33, 13 January 2013 (UTC)

Eh, probably is. It can be figured out using discrete time. That's pretty obvious but it is error prone. I looked at Euler's three-body problem, which I am trying to model a simplified version of, and I realized I am way out of my league. — Melab±1 ☎ 19:45, 13 January 2013 (UTC)

Heuristic argument for expected number of solutions to specific congruence

I am interested in solutions to the congruence

2ⁿ ≡ 1 (mod (n + 1)²)

In this paper on page 1560 and in this paper on page 446 it is stated that the expected number of primes p satisfying 2^{p − 1} ≡ 1 (mod p²) (and x≤p≤y, Bo Jacoby (talk) 13:26, 11 January 2013 (UTC)) is approximately ln(ln y / ln x). How could I develop a similar heuristic argument for solutions of the above congruence? (Note that n necessarily has to be even in order to satisfy that congruence). Unfortunately it is not very clear to me how those heuristic arguments in those papers are being obtained. Any insights would be appreciated. -- Toshio Yamaguchi 13:00, 11 January 2013 (UTC)

Chances of counter-intuitive results from a betting system

Say I find a way to gain an edge in a game of chance, say I can win an evens bet 55% of the time, what are the odds that after 100 or 500 bets I lose money anyway? Equally, if I think I've got an edge but don't (say my odds are 45% to win an evens bet) what's the chance I'm up after 100 or 500 bets anyway? How do I calculate this for different odds and so on? 80% chance of winning a 3/1 bet, 70% chance, you get the idea. I'm really feeling the 5 years since I studied maths...thanks in advance. 81.159.112.136 (talk) 19:04, 11 January 2013 (UTC)

You didn't say if you bet the same amount each time, and have to stop at exactly 100 or 500 or can choose to stop early. At any rate, a 5% edge either way is virtually certain to manifest itself after 100 bets, much less 500. StuRat (talk) 19:19, 11 January 2013 (UTC)

I'm assuming the same bet value each time. The idea is how long would I have to test a system with fictional bets to know with reasonable certainty if it worked, assuming that it isn't a mathematical game like roulette or blackjack but the edge is the same each time. 81.159.112.136 (talk) 19:53, 11 January 2013 (UTC)

I think the OP means all bets are the same size and the number of bets is exactly 100 or exactly 500. Binomial distribution#Cumulative distribution function says the probability of having no more than k successes out of n tries when the success probability is p is

$F(k;n,p)\leq {\frac {1}{2}}\exp \left(-2{\frac {(np-k)^{2}}{n}}\right).\!$

Using n=100, k=50, p=0.55 in this gives (assuming I haven't made a calculator mistake) a probability of having won no more than half the time of less than 0.304. Using n=500, k=250, p=0.55 gives a probability of having won no more than half the time of less than 0.042. You can also go into that article and get the exact expression, which would involve summing 50 or 250 terms. Duoduoduo (talk) 19:59, 11 January 2013 (UTC)

I tried a computer simulation, and got different numbers. For the n=100 case, I get about a 0.134 probability you will lose with a 5% edge in your favor. If we include ties that brings it up to about an 0.183 chance. For the n=500 case, I get a .011 or .014 probability, depending on if ties are included. StuRat (talk) 20:30, 11 January 2013 (UTC)

That's perfect! Now how on earth do I do that on a calculator with different numbers (so I don't make you good people do it for every combination I can think of)? Is there a website I can put the n, k and p values into? 81.159.112.136 (talk) 20:08, 11 January 2013 (UTC)

If you know and have access to Excel you could write the formula once and play with the parameters at will. But there's probably (p≥0.55) a website that already has the formula programmed. Maybe someone else knows one? Duoduoduo (talk) 20:20, 11 January 2013 (UTC)

Googling 'Binomial Distribution Calculator' has thrown up some useful results. Thanks both of you. 81.159.112.136 (talk) 20:28, 11 January 2013 (UTC)

Using PARI/GP where the given one-line formulas can be entered exactly like below with no declarations or anyhting else, and the result is immediately printed on the next line.

55% winning chance gives this chance of 0-49 wins out of 100:

r=0.55;sum(n=0,49,binomial(100,n)*r^n*(1-r)^(100-n))

0.1345762131880523867767271082

Explanation: r=0.55=55% is the winning chance. binomial(100,n) is the number of ways to pick exactly n elements out of 100. A pick requires n wins with total probability rⁿ, and 100-n losses with total probability (1-r)^100-n. The chance of exactly n wins out of 100 becomes binomial(100,n)*rⁿ*(1-r)^100-n. This is summed for n from 0 to 49. That summation would be time consuming to perform by hand but I don't know a simpler way to get an exact result instead of bounds.

0-249 wins out of 500:

r=0.55;sum(n=0,249,binomial(500,n)*r^n*(1-r)^(500-n))

0.01108154140615325239660158777

45% winning chance gives this chance of 51-100 wins out of 100:

r=0.45;sum(n=51,100,binomial(100,n)*r^n*(1-r)^(100-n))

0.1345762131880523867767271081 (same as first question except rounding of last decimal)

251-500 out of 500:

r=0.45;sum(n=251,500,binomial(500,n)*r^n*(1-r)^(500-n))

0.01108154140615325239660158776 (same as second question except rounding of last decimal)

The numbers match StuRat's computer simulation. PrimeHunter (talk) 20:58, 11 January 2013 (UTC)

Yes, those numbers match perfectly with my simulation results, listed above. StuRat (talk) 21:44, 11 January 2013 (UTC)

Base pi

If we did things in base pi, would the current integers be transcendental? — Preceding unsigned comment added by 93.96.113.87 (talk) 19:21, 11 January 2013 (UTC)

Leaving aside the question what does it actually mean to write numbers in a noninteger basis, transcendentality is a property of a number, not of its representation, hence the answer is no (and the answer to the implied question is that π would still be transcendental).—Emil J. 19:38, 11 January 2013 (UTC)

I don't think noninteger radices are particularly problematic. To be able to represent all numbers in radix pi, you need at least four digits available (presumably 0 to 3) so you can't expect unique representations — for example, pi itself could be either 10 or 3-point-something — but then again we don't have unique representations in integer radices either, at least once you're allowing infinitely many places to the right of the radix point. --Trovatore (talk) 19:47, 11 January 2013 (UTC)

"pi itself could be either 10 or 3-point-something" : this is no different than saying 10 in base 10 can be represented either as 10 or as 9.999... In your case, 3 is the closest thing to pi, just like 9 is the closest thing to 10. — 79.113.217.116 (talk) 17:13, 6 April 2013 (UTC)

See Non-integer representation. --Carnildo (talk) 02:13, 12 January 2013 (UTC)