Wikipedia:Reference desk/Archives/Mathematics/2006 September 14

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September 14

Unexpected symbol behvior w/ LaTex

Would someone please take a look at the following page:

http://wiki.riteme.site/wiki/Beer-Lambert_law

I created a new section entitled "Derivation" with several equations. Some look fine, but there are two types of problems with others ...

1. The equation font size differs depending on whether there is a fraction in the equation or not. When there is a fraction, the equation looks fine (e.g. 'Absorbance' or 'Transmittance"). In equations without a fraction, however, the symbols are almost too small to read. How can I make all of the math fonts the same?

2. Several of the equations have a small hyphen "-" at the end, and I can't figure out how to get rid of them.

I haven't used LaTex before, so that may be the problem, except that I don't see the problem occurring in the first section, 'Equations' that was written by someone else.

Thanks for any help,

Axewiki 00:25, 14 September 2006 (UTC)

I've fixed the first problem - the thing is, the software Wikipedia runs on has a default setting that if it can turn an expression in <math> tags to regular text and still look ok, it will, otherwise it will turn it into a PNG image. However, the definition of "looking ok" doesn't always work, especially if you're trying to make a bunch of equations look the same, so sometimes you have to trick it into making the equation into a PNG by adding a small space in the form of \, to the end of the equation. I got that trick from m:Help:Formula#Forced PNG rendering. As to the second, I seem to remember seeing that kind of thing happen elsewhere, but I can't remember the fix. Confusing Manifestation 01:32, 14 September 2006 (UTC)

Thanks ConMan! Parts of it are much better ...

Axewiki 02:16, 14 September 2006 (UTC)

Cylindrical sections

I am looking for information on cylindrical sections, i.e., ellipses; specifically, I want to determine what kind of a curve the ellipse will map to if the cylinder is unrolled after a cut in its long dimension (it appears to be a sine or cosine, based on a pencil rubbing of a wood model I have). I have searched the web and Wikipedia, and have not found much on this subject.

If the angle of the cut is 45 degrees (the case I am interested in), then the long-axis, or height, dimension (assuming the cylinder is standing on its end) is a simple function of the ellipse's x coordinate, when plotted in the ellipse's own plane, but the horizontal dimension on the unrolled cylinder is much more difficult, and I have not been able to visualize it well enough to calculate the formula for the transformation. It equals the length of the circular arc subtended at that height, but that depends on the angle, which is difficult for me to see. (I hope that makes sense.) Thanks in advance for any help you can give.

-- Ed

On the cylinder, we can pick our coordinates so the z-axis is coincident with the axis of the cylinder and the $r-\theta \$ plane is at the bottom face of the cylinder. Suppose the radius of the cylinder is R, and the height of the cut is $a+b\sin(\theta )\$. Converting to Cartesian coordinates with the same z-axis and (positive) x-axis pointing along the ray $\theta =0\$, we find that the point $\{R,\theta ,z\}\$ is mapped to the point $\{x,y,z\}=\{R\cos(\theta ),R\sin(\theta ),z\}\$. So the cut is mapped as $\{R,\theta ,a+b\cos(\theta )\}\rightarrow \{R\cos(\theta ),R\sin(\theta ),a+b\cos(\theta )\}\$. But that's just $\{x,y,a+{b \over R}x\}\$ (where we've used a notational dodge to roll it for us). This tells us that the cut is straight (not wavy) (equivalently planar) and sinusoidal when unrolled. -- Fuzzyeric 04:58, 14 September 2006 (UTC)

Fun Game

Someone in my math club proposed a variation on Nim: instead of removing n objects from a heap, you remove n objects from each of m heaps, where m and n are at least 1. Anyone have suggestions on solving it? Black Carrot 04:49, 14 September 2006 (UTC)

I'm assuming you mean "in a given step, from each remaining heap remove n objects", "there are initially m heaps", "the m heaps may have different starting sizes". Use Bouton's method, described in the article you link, and find a solution driving t to zero in each heap simultaneously. In general, this will require taking large counts to make most of the heaps vanish so that the remaining heaps can be simultaneously optimized.

If you mean one takes n objects, distributed as one likes from m heaps, then the distinction into m heaps is superfluous since there's effectively only one heap.

If you mean one must take >1 object from each of m heaps and the first player to deplete any heap is the loser, then use Bouton's method on each heap independently.

If n is upper bounded, one may not be able to reach a t=0 state in one move. For instance if the bound is <m and you must draw an object from every heap to reach a winning intermediate state. Then, similar to some more complicated games, the first moves are sort of random and the loser is the first player to make a mistake noticed by his opponent. In fact the early play is probably to mini-max the ability of the other player to drive the game out of the non-winning equilibrium. -- Fuzzyeric 05:10, 14 September 2006 (UTC)

No, I meant "in a given step", from each of "as many heaps as you feel like (m)" remove n objects. The starting condition could be whatever the players agree upon. So, for instance, from (3,2,1) I would take one from each (2,1), they might take the entire first heap (1), then I would take the remaining one and win. Or from (7,3,3,1,1) I would take three from each of the first three heaps (4,1,1), they might take one from the first and last (3,1), I would take one from the first (2,1), and it would end the same way as before. Black Carrot 15:32, 15 September 2006 (UTC)

Then you mean the variation described above as "If you mean one takes n objects, distributed as one likes from m heaps, then the distinction into m heaps is superfluous since there's effectively only one heap." This is equivalent to one heap as there is no consequence to heap boundaries. Use Bouton's method on the total number of objects, regardless of heap membership. -- Fuzzyeric 04:02, 16 September 2006 (UTC)

If that was what I meant, that's what I would have said, and I wouldn't have had any trouble solving it myself. Where did I say it could be distributed as you like? The same amount (n) must be removed from each of as many stacks as you like (m). To put it visually, it's like yanking out a rectangle of arbitrary dimensions, and it's been devilishly difficult to solve. If you had stacks (7,3,3,1,1) as above, for instance, under your interpretation, I could just remove all of them and have done with it. Under the one I've been trying to explain, you could remove 7, 6, 5, or 4 from the first one, or 3 or 2 each from any of the first three of them, or 1 each from any of them. Reduced to two stacks, it simplifies to Wythoff's Game. Black Carrot 22:02, 18 September 2006 (UTC)

Recurrence Relations

The recurrence relation article doesn't talk about the cases of nonlinear recurrence relations. Could anyone indicate me what would be standard methods for solving relations such as $u_{n}=f(u_{n-1})$ where $f(x)={\frac {1}{1-x}}$ (just an example, I don't really want to know what that particular solution is). What could the function f(x) be that would leave the recurrence relation relatively easy to solve (no Z / Laplace transforms or others...). I'm also interested in the inhomogeneous cases and thoses where the degree is greater, but I'll start with this... --Xedi 21:01, 14 September 2006 (UTC)

There should be an article on the "cycle structure" of solutions of $u_{n}=f(u_{n-1})$ ... perhaps somewhere near Chaos theory? In any case, for Möbius transformations, such as the one you mention, the solution is fairly simple, and contained within that article. As $u_{n}=f^{(n)}(u_{0})$, there are "obvious" properties; e.g., if f is injective, so is f⁽ⁿ⁾, and there are various techniques that work in the vicinity of a fixed point of f. Other than that, solutions usually depend on a change of coordinates; finding a function g such that, e.g. f(g(x)) = g(x + 1) (at least locally), leading to f⁽ⁿ⁾(x) = g(g^-1(x) + n).

Thanks for the swift reply, I did actually notice the periodicity of the recurrence relation I gave where $f(x)={\frac {1}{1-x}}$. I'll try to understand clearly everything and then come back. Thanks again ! --Xedi 21:42, 14 September 2006 (UTC)