Wikipedia:Reference desk/Archives/Mathematics/2013 April 1

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April 1

Skinning a cat

What is the surface area of a skinned cat? (do not assume a spherical cat) --Carnildo (talk) 09:50, 1 April 2013 (UTC)

I assume sincee you are asking at the maths reference desk you don't want an actual answer but some method of calculating it. Might I suggest that you do the following

Colour some random square centimeters of the cat fur purple or green or some other easy to distinguish colour.

Count how many hairs the cat sheds per day and what proportion of them are coloured

Divide the number of square centimeters you coloured by the fraction of shed hairs that were coloured.

Do this with some neighbours cats too. (Gosh I mistyped hats there first instead of cats, no colouring random squares of their hats would really cause trouble but I don't suppose they'd mind you doing this to their cats) You can colour different random samples of squares of their fur and weigh them to find an approximate relationship between weight of cat and amount of skin. Use different colours for interest. Make a guess what type relationship you'll find. Note that no actual skinning is required using this method so no animals are harmed in this experiment. ;-) Dmcq (talk) 10:40, 1 April 2013 (UTC)

p.s. if you are doing a number of cats or don't want to go to the trouble of cleaning out all previously shed cat hairs what you can do is dye the entire animal one colour and the squares a different colour. Using a number of different colours you don't need to keep the cats apart either which means less expense for better statistics. Use food dyes for safety. Dmcq (talk) 17:01, 1 April 2013 (UTC)

You can easily approximate the cat's body with a cylinder, and its head with a sphere. — 79.113.222.119 (talk) 23:23, 1 April 2013 (UTC)

True, but for a real cat it might be more accurate to weigh it, dunk it in water, weigh it again and see how much water there is in a couple of square centimetres of fur. or one could use an Egyptian mummified cat and measure the amount of bandages that were in contact with the cat, or one could use an Xbox Kinect to measure the whole cat and measure area from the computer model. Then again one could look up the area of a human's skin on the web and assume a cat was a kind of small human - that would leave out the tail and the legs would be too long I think but minks look fairly similar so a good estimate can be got from the size of a mink coat and using a scaling factor though a small raccoon is probably more comparable in weight. If you're lucky you might even find some cat fur mislabelled as something else and that could be measured directly.

And by the way it is a wonder to me how people estimate skin area, and even more so lung area or the surface of the intestines or all the blood vessels. Dmcq (talk) 09:00, 2 April 2013 (UTC)

Lung/intestine area might be estimated my total rate of absorption of a substance (e.g. oxygen/sucrose) after measuring local concentrations and calibrating characteristics of a flat area. Though I guess often one is merely looking for a reference figure than a true figure, since the result may be sensitive to measuring choices. It's not too clear relevance skinning the cat has for that case. — Quondum 09:47, 2 April 2013 (UTC)

Thanks for that. And by the way Body surface area conversion dogs and cats gives an estimate for an answer to the original question. I made a little estimate of my own from human skin area and it scaled to very nearly these figures despite us not having a tail. Dmcq (talk) 10:44, 2 April 2013 (UTC)

Functions Lacking a Closed Form

$F(x)=\int _{0\ or\ 1}^{\infty }{f(t,x)}\ dt$ — Is there any way of calculating their values, inverse functions, derivatives, critical points (maxima and minima, inflection points, etc.) without retorting to brute, computer-aided approximations and numerical analysis ? — 79.113.222.119 (talk) 23:41, 1 April 2013 (UTC)

You can calculate the derivative by differentiation under the integral (in good cases). I don't know what kind of closed form you expect for critical points and so forth. This isn't even a reasonable request for a function of one variable that doesn't involve taking an improper integral (such as $F(x)=x^{2}+2x+x\sin x$, what are its critical points?) Sławomir Biały (talk) 23:59, 1 April 2013 (UTC)

Do you happen to know any non-numerical methods for solving transcendental equations ? — 79.113.222.119 (talk) 01:07, 2 April 2013 (UTC)

Well for reference desk question type questions the Lambert W function seems to come up a lot but in general you need to get really familiar with the hypergeometric functions and other special functions for the problems that occur in practice. I'd advise getting something like Mathematica if you really do need to do that sort of thing often. Newton used to evaluate functions as series an try and recognize what came up. Nowadays people are often happy just getting an idea of the general shape and being sure a solution will converge and leaving the rest up to a computer to do numerically. Dmcq (talk) 10:32, 2 April 2013 (UTC)

I've been working with functions of the form:

${\begin{cases}F(n)\ =\ \int _{0}^{\infty }{e^{-x^{n}}}\ dx\ =\ {1 \over n}!\ =\ \Gamma \left(1+{1 \over n}\right)\\\\F(n)\ =\ \int _{0}^{\infty }{e^{-{\sqrt[{n}]{x}}}}\ dx\ =\ n!\ =\ \Gamma (n+1)\end{cases}}\qquad \qquad {\begin{cases}F(n)\ =\ \int _{0}^{\infty }{e^{-e^{x^{n}}}}\ dx\\\\F(n)\ =\ \int _{0}^{\infty }{e^{-e^{\sqrt[{n}]{x}}}}\ dx\end{cases}}$

$F(2n)\ =\ \int _{0}^{\infty }{\underbrace {x^{-x^{\cdot ^{\cdot ^{x}}}}} _{2n}}\ dx\ =\ \int _{0}^{\infty }{dx \over ^{2n}x}\qquad \qquad \qquad \qquad \ \ {\begin{cases}F(n)\ =\ \int _{0}^{\infty }{e^{-\Gamma (x)^{n}}}\ dx\\\\F(n)\ =\ \int _{0}^{\infty }{e^{-{\sqrt[{n}]{\Gamma (x)}}}}\ dx\end{cases}}$

In all of these cases, I'm wondering if there are any non-numerical formulas for calculating their minima, maxima, values, and convergence. By this I mean something similar to $\int _{-\infty }^{\infty }{e^{-x^{2}}}={\sqrt {\pi }}$. — 79.113.217.28 (talk) 22:48, 4 April 2013 (UTC)

If you don't want to do brute force numerical work, you could use asymptotic expansions. You can try to use the saddle point method to approximate the integral as a Gaussian integral and then systematically improve on that approximation. So, given an integrand f(x,n), you can find where the derivative is zero in the complex plane. Then Log[f(x,n)] expanded around such a point x* (which will depend on n, of course) will behave as A + B (x-x*)^2 + .... Then Cauchy's theorem says that the integral from zero to infinity along the real axis is the same as the integral along any other contour that starts at zero and moves to infinity (if the integrand is analytic), so you can consider a contour that visits the point x*. The dominant contribution to that contour integral comes from the neighborhood of x*. To find the corrections to the Gaussian result systematically, you need to perform a conformal transform to a new variable z, such that f(x,n) becomes exactly exp(-z^2), the integrand then becomes exp(-z^2) dx/dz. You'll then typically get a fast converging but usually divergent asymptotic series that you can use to find the properties like extrema etc...Count Iblis (talk) 15:58, 6 April 2013 (UTC)