Wikipedia:Reference desk/Archives/Mathematics/2011 December 1

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December 1

Algebra problem

I've been asked to find the solution for "x" in the equation

$\ln x={\sqrt[{3}]{x}}$

is there a way to find the solution to this algebraically? 108.70.151.156 (talk) 02:03, 1 December 2011 (UTC)

$\ln x=x^{\frac {1}{3}}$

$\Rightarrow x^{\frac {-1}{3}}\ln x=1$

$\Rightarrow e^{\frac {-\ln x}{3}}\ln x=1$

$\Rightarrow e^{\frac {-\ln x}{3}}{\frac {-\ln x}{3}}={\frac {-1}{3}}$

$\Rightarrow {\frac {-\ln x}{3}}=W\left({\frac {-1}{3}}\right)$

$\Rightarrow x=e^{-3W\left({\frac {-1}{3}}\right)}$

Widener (talk) 05:17, 1 December 2011 (UTC)

See also Lambert W function. -- Meni Rosenfeld (talk) 06:55, 1 December 2011 (UTC)

Note that there are two solutions as the Lambert W function is multivalued in this region. Wolfram Alpha can show you both solutions in various ways. 81.98.43.107 (talk) 20:25, 1 December 2011 (UTC)

This is really great. I'm not sure I'm at the appropriate level of math to do this type of math yet (the textbook from which this problem was from recommended using a calculator, as it turns out), but I think I'll be able to learn something trying to understand it. Thanks a lot for the help. 108.70.151.156 (talk) 04:29, 2 December 2011 (UTC)

minimum cirdumference

These question are from O level book ,i m confused about these ,1:- the circum ference of the trunck of a tree is measured up to nearest 2 mm .if its radius is 20 cm find its minimum circumference ,(answer on book is 125.67) 2:-The density of a given material is 2.4 g/cubic cm .if its mass ,to the nearest 10 g is 30 g .find its minimum and maximum valumes answers:- (3.0 ,12.0) — Preceding unsigned comment added by 119.152.27.162 (talk) 02:25, 1 December 2011 (UTC)

Please retype your question and take the time to avoid typographical errors. As asked, your questions cannot be answered because it is impossible to trust something like "2 mm" when "answer" is written as "nswer". Is it actual ".2 mm" or "20 mm"? Also, in the second question, you ask for a minimum and maximum value. Value of what? Value of area, density, volume, mass...? -- kainaw™ 03:28, 1 December 2011 (UTC)

The circumference of a circle is $2\pi r$ where $r$ is the radius. So just substitute 20 cm in for r to get the answer 125.66 cm. Widener (talk) 05:25, 1 December 2011 (UTC)

But they asked for the minimum. So 20 cm - 2 mm = 20 cm - 0.2 cm = 19.8 cm. I plug that into the equation and get 124.41. I'm not sure where they got their answer. The second problem doesn't work out either. I have to wonder in Kainaw is right and there are typos in the numbers, too. StuRat (talk) 05:42, 1 December 2011 (UTC)

Actually he said the circumference was measured to the nearest 2 mm, not the radius. That doesn't make any sense though since in this question the circumference is not measured, rather it is derived from the radius. Widener (talk) 05:58, 1 December 2011 (UTC)

Perhaps what was meant is "The radius of a tree trunk is exactly 20 cm. What range of values might the circumference have, if measured to within 0.2 mm ?" (Note that I changed the error to 1/10th what the OP listed.) In that case the answer should be 125.64-125.68. If it was a multiple choice question, instead, and only one answer was it that range, it could very well be 125.67. StuRat (talk) 06:32, 1 December 2011 (UTC)

Bayesian inference of a time-varying parameter

Let's say I am trying to determine the probability of a coin coming up heads. From a Bayesian perspective, I can interpret the outcomes as being distributed according to a Bernoulli distribution with parameter p, and p as being distributed according to a Beta conjugate. However, what if I know that p slowly varies with time, and as such want to downweight the influence of outcomes that occurred longer ago? Could I do something akin to exponential smoothing on the Beta prior parameters before doing the posterior update? Intuitively this seems like it would do the trick, but I don't know how principled or 'Bayesian' it is. Are there any proper treatments of this kind of problem? --2.125.226.198 (talk) 23:02, 1 December 2011 (UTC)

As a simple heuristic you could indeed use a beta distribution where each success/failure contributes an exponentially decaying amount to the corresponding parameter. But to be really Bayesian you need a model for how p changes with time (you could even take it one step further and have a prior on different models). To me a reasonable model is that the log-odds $l=\log {\frac {p}{1-p}}$ follows a restrained Brownian motion ${\frac {dl}{dt}}=\alpha {\frac {dB}{dt}}-\beta l$. But I don't see how to obtain a closed-form for the resulting inferences. You will have to maintain an interpolating function for your posterior of l at any point, and update it numerically as time passes and as you collect data. -- Meni Rosenfeld (talk) 07:23, 2 December 2011 (UTC)

OK, thanks. I was sort of just checking if there wasn't some sort of standard approach to this, as the problem is such a minor part of what I'm doing that it'd probably end up looking a bit overkill if I spent ages implementing and explaining a proper solution. I noticed too that this is a bit similar to a Hidden Markov Model, but I don't know much about adapting HMMs to a continuous hidden state. I will probably just go with the heuristic. Thanks. --2.125.226.198 (talk) 11:46, 2 December 2011 (UTC)

You could always discretize the continuous variable. How precise estimates do you really need of the coin bias? Something like 1000 bins would probably do, and as arrays go, that's a small one. If the bias changes slowly, then you might further optimize things by pretending that the bias is constant for every n coin flips, so each observation consists of n flips. -- Coffee2theorems (talk) 17:11, 4 December 2011 (UTC)

The problem you are describing in known in the literature as regularization. You have to start with some sort of constraint on the form of the time dependency -- usually this means expressing the rate as a function of a limited set of parameters, and then using Bayesian tecniques to estimate the values of the parameters. Looie496 (talk) 05:27, 3 December 2011 (UTC)

You have a function, f, satisfying 0≤f(t)≤1 for 0≤t<N, t real. A bitstring b_t for 0≤t<N, t integer, is constructed such that b_t=1 with probability f(t), and =0 with probability 1−f(t). So b is a sampled and rounded version of f. The challenge is to reconstruct f(t) from b_t. Consider the following approach. Compute a discrete fourier transform

${\hat {b}}_{\omega }=N^{-1/2}\sum _{t=0}^{N-1}1^{\omega t/N}{\bar {b}}_{t}$

where the inverse is

$b_{t}=N^{-1/2}\sum _{\omega =0}^{N-1}1^{\omega t/N}{\bar {\hat {b}}}_{\omega }$

Here 1^x is shorthand for e^2πix = cos(2πx)+i sin(2πx), and the bar signifies complex conjugation.

Now omit the high-frequency terms.

$p(t)=N^{-1/2}\sum _{\omega =-a}^{a}1^{\omega t/N}{\bar {\hat {b}}}_{\omega }$

where a is chosen carefully to trade off between spatial resolution, a=0, and temporal resolution a=N/2.

This method is rather general, but has no explicite reference to Bayesian inference. Bo Jacoby (talk) 00:09, 4 December 2011 (UTC).