Wikipedia:Reference desk/Archives/Mathematics/2006 October 21

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1 October 21

[edit] October 21

[edit] Aesthetically pleasing maths

Hi. I'm developing an idea for a peice of art which basically consists of a sheet of glass held up in a frame. On the glass in thick marker pen, I want some mathematical calculations written on one side, much like a teacher would write on a blackboard. The peice will be visible from both sides.

Although I know very little maths, the notation and symbols used are very pleasing and beautiful to look at (the integration sign is a particular favorite) and I was wondering if you maths people out there could give me some "workings" that utilise a lot of these glyphs. One simple example to give you a picture is perhaps a step by step working of the differentation of f(x) = x^2; although maybe something a bit more beautiful, perhaps involving Euler's Identity or lots of weird glyphs and what-not would be great. Thanks! 164.11.204.56 15:37, 21 October 2006 (UTC)

Well, Maxwell's equations look very pretty to me. ☢ Ҡi∊ff⌇↯ 17:14, 21 October 2006 (UTC)

Ah, another one you might be interested is the series expansion for the trigonometric functions:

$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots = \sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!}$

$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots = \sum_{n=0}^\infty \frac{(-1)^nx^{2n}}{(2n)!}$

Which I think have an amazing simplicity and importance. The sigma notation is not an integral, but I think it is also very aesthetically pleasing. ☢ Ҡi∊ff⌇↯ 17:41, 21 October 2006 (UTC)

Cool! I'll try to come up with suggestions. I immediately thought about the integration sign, but apparently, you already had it in mind. :-) —Bromskloss 17:23, 21 October 2006 (UTC)

So, here I show off a variety of mathematical symbols that might be of interest. You'll have to judge by yourself which of them, if any, are good looking. I have chosen examples that actually mean something useful in mathematics – I suppose that would make it more interesting. However, I'm afraid I sometimes mix up graphical beauty with matematical beauty, so mabye they are only attractive for their meaning and not for their typography! If you like a particular symbol and want more of it, just ask. But then again, you seem to know quite a bit of mathematics yourself, don't you?

$\zeta(s) = \sum_{n=1}^\infin \frac{1}{n^s}$
$\aleph_{\lambda} = \bigcup_{\beta < \lambda} \aleph_\beta$ Hebrew!
$\begin{pmatrix} \cos{\theta} & -\sin{\theta} \\ \sin{\theta} & \cos{\theta} \end{pmatrix} \begin{pmatrix} \cos{-\theta} & -\sin{-\theta} \\ \sin{-\theta} & \cos{-\theta} \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$
$\langle x, y \rangle = \frac{1}{2}\left(\|x+y\|^2 - \|x\|^2 - \|y\|^2\right)$
$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$ Lorentz factor, important in special relativity
$\operatorname{div\ curl\ } \mathbf{A} = \nabla \cdot (\nabla \times \mathbf{A}) = 0$
$\Delta f = \nabla^2 f = {\partial f \over \partial r}\mathbf{\hat r} + {1 \over r}{\partial f \over \partial \theta}\boldsymbol{\hat \theta} + {1 \over r\sin\theta}{\partial f \over \partial \phi}\boldsymbol{\hat \phi}$
$n\in \mathbb{N} \Rightarrow \sqrt{n} \in \mathbb{N}\cup (\mathbb{R}\setminus \mathbb{Q})$ I believe I proved this some time ago, but it should probably be checked before you use it. It has a number of funny symbols in any case. :-)
$\ell^2 = \left\{\left(x : \mathbb{N}\rightarrow \mathbb{C}\right)\; :\; \sum_{n\,\in\,\mathbb{N}} \left|x(n)\right|^2 < \infty \right\}$
$\int\limits_{\varphi=0}^{2\pi} \mathrm{e}^{\mathrm{i}\varphi} \,\mathrm{d}\varphi = 0$
$\frac{\mathrm{d}^4}{\mathrm{d}\varphi^4}\mathrm{e}^{\mathrm{i}\varphi} = \mathrm{e}^{\mathrm{i}\varphi}$
$\iiint\limits_V\left(\nabla\cdot\mathbf{F}\right)\mathrm{d}V=\iint\limits_{\part V}\mathbf{F}\cdot \mathrm{d}\mathbf{S}$
$\int\limits_{\Sigma} \nabla \times \mathbf{F} \cdot \mathrm{d}\mathbf{\Sigma} = \oint\limits_{\partial\Sigma} \mathbf{F} \cdot \mathrm{d}\mathbf{r}$

—Bromskloss 20:36, 21 October 2006 (UTC)

(edit conflict) For weird symbols, you can't beat physics equations. Maxwell's equations are neat looking, as is the Schrödinger equation. Depending on how meaningful the result is supposed to be and how much space you have you could include all sorts of things. Combining, for instance, Taylor series and the Euler identity, you could prove e^ix = isin(x)+cos(x).

To someone who has not had a course in diff. eq (where, in my experience, the result is first seen), the asthetic of symbols is all that is there. The result, however, would be meaninful to people who understand it. The outline of the proof (omitting questions of convergence and such) is

$e^{ix} = \sum_{k=0}^\infty \frac{(xi)^{k}}{k!}$
$=\sum_{k \mod 2 = 0}^{\infty} \frac{(xi)^{k}}{k!} + \sum_{k \mod 2 = 1}^{\infty} \frac{(xi)^{k}}{k!}$
$=\sum_{k=0}^\infty \frac{(xi)^{2k}}{(2k)!} + \sum_{k=0}^\infty \frac{(xi)^{2k+1}}{(2k+1)!}$
$=\sum_{k=0}^{\infty} \left(-1\right)^k \frac{(x)^{2k}}{(2k)!}+i \sum_{k=0}^\infty \left(-1\right)^k \frac{(x)^{2k+1}}{(2k+1)!}$
$= cos(x) + i sin(x)$

(Doing this on the fly, I hope I didn't make a mistake.) Many similar short proofs of interesting results could be found (espeically since you could leave out some steps). --TeaDrinker 20:47, 21 October 2006 (UTC)

Here are some of my favorites from pure analysis:

$f(a) = {1 \over 2\pi i} \oint_C {f(z) \over z-a}\, dz$

$\zeta(z) \; \Gamma(z) = \int_0^\infty \frac{t^{z-1}}{e^t-1} \; dt$

$\Gamma\left(\frac{s}{2}\right)\zeta(s)\pi^{-s/2} = \Gamma\left(\frac{1-s}{2}\right)\zeta(1-s)\pi^{(1-s)/2}$

$\pi = \sum_{k = 0}^{\infty} \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6}\right).$

$\int_{-\infty}^\infty e^{-x^2}dx = \sqrt{\pi}$

It's hard to compete with many of the equations from vector calculus and mathematical physics, though, simply because they so compactly encode properties of extremely complex systems (at least to a mathematician, depth makes formulas visually appealing :-). I'd give the Navier-Stokes equations a look. Fredrik Johansson 21:42, 21 October 2006 (UTC)

My field is computer science and discrete mathematics, and there are some fine ones. I've always had a soft spot for the binomial theorem

$(x+y)^n=\sum_{k=0}^n{n \choose k}x^ky^{n-k}$

It's not terribly advanced but I think it's beutiful as hell. If we're going for pure awesomeness, the hypergeometric function is unbeatable. Unfortunatly, the notation that's used in that article is not the coolest one, but the one in Donald Knuths works is mindblowing. Also, anything involving Stirling numbers of the second kind. Like

$\left\{\begin{matrix} n \\ k \end{matrix}\right\} = \left\{\begin{matrix} n-1 \\ k-1 \end{matrix}\right\} +k \left\{\begin{matrix} n-1 \\ k \end{matrix}\right\}$

$\left\{\begin{matrix} n \\ k \end{matrix}\right\} =\frac{1}{k!}\sum_{j=1}^{k}(-1)^{k-j}{k \choose j} j^n.$

Curly brackets are cool! Also, as you are an artist, it might be a nice idea to have your equations mean something deeper. The axioms of Zermelo–Fraenkel set theory are pretty neat, and they are the most commonly used way to define mathematics itself! Personally, I find something mezmerising about that, what those formulas stand for. How much comes from them. Oskar 03:36, 22 October 2006 (UTC)

Wow, there's a lot of great stuff here!

e^ix = isin(x) + cos(x) is a bloody good one actually since it has strong ties to musical analysis for obvious reasons (Fourier Series / Transform)! I play guitar. The Schrödinger equation also looks wonderful although I am not familiar with what it describes - hopefully something profound about the nature of the physical world?

I was also wondering if anyone recognises the beast below. Something to do with digital sampling, by the look of those braces? I don't even know if it's actually a valid statement or an arbitrary collection of stuff, but it's exactly the kind of thing I'm looking for...

$\Delta M_i^{-1} = -a \sum_{n=1}^N D_i \left[ n \right] \left[ \sum_{j \in \mathbb{C} \{i \}} F_{ij} \left[ n-1 \right] + F\operatorname{ext}_i \left[ n^{-1} \right] \right]$

164.11.204.56 02:17, 23 October 2006 (UTC)

Why not Euler's Identity?

$e^{i\pi} + 1 = 0 \!\$

The five most important constants in all of mathematics (e, i, pi, 1, and 0), and the three most important operators (addition, multiplication, and exponentiation). Most beautiful --Ķĩřβȳ ♥Ťįɱé Ø 06:28, 24 October 2006 (UTC)

I recommend the expansion in chord diagrams of the Kontsevich integral of the trefoil knot. It uses such exotic glyphs, I can't even show it to you using Wikipedia's software! It occurs in the last page of [1]. Go download it; you won't be disappointed. Melchoir 06:55, 24 October 2006 (UTC)

[edit] Working out the point at which lines of longnitude intercept the equator on an orthographic projection

I have seen the equations at Orthographic projection (cartography)#Mathematics, although I am told there is an easier way to create these arc by simply finding the equatorial intercept. I would be very obliged if anyone could tell me how this can be worked out. Thankyou Lofty 15:51, 21 October 2006 (UTC)

Is the equation Radius * sin (angle)? Lofty 16:17, 21 October 2006 (UTC)

The equations from the article are:

$x = R\,\cos(\phi)\sin(\lambda - \lambda_{0})$

$y = R\,[\cos(\phi_1)\sin(\phi) - \sin(\phi_1)\cos(\phi)\cos(\lambda-\lambda_0)]$

in which $\lambda\,$ stands for the latitude and $\phi\,$ for the longitude of the point being projected, and $(\lambda_0,\,\phi_1)$ is the point on the sphere to be mapped to the origin. (No idea why one has a subscript 0 and the other a subscript 1.) Indeed, if we substitute $(\lambda,\,\phi) := (\lambda_0,\,\phi_1)$ , we obtain $(x,\,y) = (0,\,0)$ . The equator corresponds to $\phi = 0\,$ . Applying the substitution $\phi = 0\,$ , and using the fact that sin 0 = 0 and cos 0 = 1, we can simplify the equations, obtaining this for the intercept of the line of longitude for $\lambda\!\,$ with the equator:

$x = ~~R\,\sin(\lambda - \lambda_{0})$

$y = -R\,\sin(\phi_1)\cos(\lambda-\lambda_0)$

As on this image, due to Snyder himself, the equator becomes in general an ellipse. If we furthermore assume that the origin (on the sphere) is at (0,0), the intersection of the equator and the main line of longitude, this can be further simplified to:

$x = R\,\sin\,\lambda$

$y = \,0\,$

in which the projected equator is a straight line segment, just like here. The formula for x corresponds to what you wrote. --Lambiam ^Talk 03:51, 22 October 2006 (UTC)

[edit] Solution for a proof?

Hi guys, I need a bit of help. I do not know exactly how to finish this proof. I need to show that the set of all infinite subsets of the natural numbers is uncountable. I need to do this directly, as opposed to showing that it must be uncountable since the set of all finite subsets of the natural numbers is itself countable and the union of two countable sets is countable (and the powerset of the natural numbers just is the set of all infinite and finite subsets of the natural numbers). Thanks. Also I do not think Cantor's diagonal argument for the uncountability of the reals does not work in this case. Thanks again.--69.171.125.20 16:58, 21 October 2006 (UTC)

The diagonal argument does work, you just have to be a little creative. Suppose you have a list of all the sets. Construct a new set in the following way: If the first set in the list contains 1, the new set does not contain 1, and vice versa. I'll let you take it from there. —Keenan Pepper 18:42, 21 October 2006 (UTC)

The diagonal argument will not work unmodified: for example, if set n is all natural numbers except n+1 (which admittedly is not at all a complete list), then the diagonal set you construct will be the empty set, which is decidedly not infinite. It's possible to modify this argument so that it works, though -- think about all the sets whose complements are finite and convince yourself that the set of such finite-complement sets is countable. What does this suggest our proof will need to take into account? Tesseran 03:20, 23 October 2006 (UTC)

[edit] Goodness of fit for a rodent Population

I will try to be as brief as possible, but please forgive me if I do not include needed information. I am totally lost. I have calculated ratios of sexes for 7 rodent species I have captured in the field. My professor told me that he would like to have a calculation for goodness of fit for these populations. I have been reading all over the internet, wikipedia and various books, and I have no idea how to calculate goodness of fit. I have seen the equation $\chi^2 = \sum {\frac{(O - E)^2}{E}}$ but I'm not sure what numbers to use, or if this is even the correct equation to use. Am I totally missing something here? Can I even do a goodness of fit test for rodent sex, as there are only 2 sexes (it is my understanding that chi square tests for 3 or more events)? If I forget about sex alltogether, is there a way to calculate goodness of fit? I can provide data if needed, but any help would be greatly appreciated! Thank you so much! —The preceding unsigned comment was added by ChibiChibiChan (talk • contribs) .

The equation you have seen gives the statistic for Pearson's chi-square test, and yes it can be used to see how well your data resembles your model. The O in the equation refers to what you've Observed, and the E is for what you Expect to observe. You can use the test when you have 2 categories; there should be a section in the lower part of the article about that :) Yesitsapril 00:28, 22 October 2006 (UTC)

So The $64,000 Question is: What is your model? The goodness of fit is the goodness of the fit between two things: between (a) the actually observed frequencies and (b) the expected values according to the distribution given by your model (after you tweaked the parameters to get the best fit you could). For each possible category (for instance "Transsexual Gerbil", or "Male Capybara"), you should have a value O for the observed number (e.g. 20) and a value E for the expected number (e.g. 22.4), which you find from your model. Then you compute (20−22.4)²/22.4 = 0.2571, which is the contribution of this category towards the total χ². If all observations are close to the expectations, you get a low value, which indicates a good fit. --Lambiam ^Talk 04:10, 22 October 2006 (UTC)

Make sure you calculate frequencies in O and E and not ratios and percentages. Basically, imagine you have a set of data, and you want to see if the data resembles a Normal distribution, for example. You would find the mean and variance of the data and use this to make a model for a Normal distribution, assuming it did represent one. Then you would calculate the expected frequencies using a formula for your Normal distribution (total frequency x probability for that frequency) and then use the chi-squared test on each of these values of data, then add them up. You would then use tables or manual calculations to determine whether the values fall under your required significance level. You don't need to have a formula to calculate E, either. If you have a set of values a test should've had (such as pH level), and you did an experiment and found that your values look fairly stupid, you could run the same test using these values as O. If you are still confused, I'll give you an example. x42bn6 Talk 13:09, 22 October 2006 (UTC)

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