Wikipedia:Reference desk/Archives/Mathematics/February 2006

February 1

Radian Use in Rates of Change problems

Problem: A line of light emanating from a lighthouse makes one revolution every 10 seconds. The lighthouse is located 4km off a straight shoreline. How fast does the light move along the shoreline when it forms a 45 degree angle to a line from the lighthouse perpendicular to the shoreline?

It basically boils down to this equation, and we are trying to find dx/dt when θ = 45 degrees

tan(θ) = x/4km

(differentiate)

sec²(θ) dθ/dt = (1/4km) dx/dt 

(plug 45 degrees in)

2 dθ/dt = 1/4km dx/dt

(isolate dx/dt)

dx/dt = 8km dθ/dt

Now is where I am confused. The resultant answer is supposed to be in km, so we can't put in 45 degrees, since that has a unit of degrees. We are supposed to put in π/5s (the radian equivalent of 1rev/10 sec) and get

8πkm/5s

because radians are "unitless," being the ratio of the arc swept over the radius. However, why radians? I could easily invent another measure that is "unitless" but would have a different numerical value. For example, I could I could use "diams" (named after diameter), that would equal the ratio of the arc over the diameter. It would effectively equal half of the radian value, so the answer given would be half of the correct answer that putting in radians would give me (and would therefore be incorrect). --JianLi 01:21, 1 February 2006 (UTC)

You have keep in mind that the sinus function which accepts radians is a different function than the sinus function which accepts degrees. Let's call them sinrad and sindeg. Now it is true that d/dx sinrad = cosrad. However it is not true that d/dx sindeg = cosdeg. Note that sindeg(x) = sinrad(πx/180). Therefore d/dx sindeg(x) = π/180 cosdeg(x). Hope that this clarifies things. —Ruud 01:47, 1 February 2006 (UTC)

Ah...I see now. Thanks --JianLi 03:22, 1 February 2006 (UTC)

*Plotting good looking fractals

Is there a good free software that can make good looking fractals, phase portraits of chaotic attractors, etc at a good resolution. The only tool I have is matlab, but using it I can't make good looking high resolution plots. Thanks! deeptrivia (talk) 01:27, 1 February 2006 (UTC)

Have you tried XaoS? —Keenan Pepper 03:53, 1 February 2006 (UTC)

Shouldn't we care more about what's on the inside of the fractals? Sure, they might be bad-looking, but does that mean we should ignore and reject them? Joe 21:02, 5 February 2006 (UTC)

Seems like I'll need to install compilers, libraries et al. before I can get this to work. Does this allow me to plot things based on my equations or just lets me change parameters? Can I make jpg files using this?? Do you know any programs that I can just install and start using? Thanks! deeptrivia (talk) 04:09, 1 February 2006 (UTC)

Not really. If you're using Windows, just follow this link, which you'll also find on the sidebar of their old (non-wiki) documentation page. —Ilmari Karonen (talk) 00:04, 2 February 2006 (UTC)

Am trying Chaospro. It's looking good. deeptrivia (talk) 04:24, 1 February 2006 (UTC)

Unemployment rate vs Inflation rate. Time taken for economy to stabilize.

Thanks, XaOS is working! But actually I wanted to make fractals based on my own equations. For example, see the fractal on the right. This is the best resolution I can get with matlab, but I want more! I hope someone will have something for me.deeptrivia (talk) 02:45, 4 February 2006 (UTC)

• Many years ago, John Walker, Rudy Rucker and I wrote some software for the soon-to-be-cancelled Autodesk Science Series. You can download James Gleick's Chaos the Software and Cellab, among other things; I know that Chaos still works under Windows (fifteen years after being written!), though it was a DOS program with tons of direct screen I/O. It won't solve your needs, but it is fun to play with. --jpgordon∇∆∇∆ 03:03, 4 February 2006 (UTC)

Thanks! I'll surely try it out. deeptrivia (talk) 03:11, 4 February 2006 (UTC)

How did you make that Unemployment v Inflation graph? Black Carrot 04:32, 5 February 2006 (UTC)

As far as I remember, using the "image" function in matlab. By the way, I'm still looking for a useful software, so the question is still open. deeptrivia (talk) 05:00, 5 February 2006 (UTC)

I mean, what formula did you use? Black Carrot 01:57, 7 February 2006 (UTC)

Based on the following equations:
File:Nlmodel.png
deeptrivia (talk) 02:07, 7 February 2006 (UTC)

Number Theory Problem

Hi,

Is there anyone can show me the method on how to proof the below number theory's problem:

For any interger n there are always n consecutive integers that all of which are not primes.

Thank you very much

EX

Hint: Consider the integers (n+1)!+2, (n+1)!+3, (n+1)!+4... —Keenan Pepper 03:51, 1 February 2006 (UTC)

Pythogoras theorem

Many people say the Pythogoras' Theorem was not given by Phytogoras. So my question is who gave the Pythogoras' Theorem?(was it Phytogoras or someone else?) It was most defiantly Pytogoras.

See the section entitled "History" in Pythagorean theorem. Says it better than I can - but do come back if you're confused by it. enochlau (talk) 08:09, 1 February 2006 (UTC)

See Sulba Sutras. deeptrivia (talk) 14:07, 1 February 2006 (UTC)

Odd differential operator

I wonder if there is a reference in the literature for this operator?

$D(f(x))_{p}={\frac {f(px)-f(x)}{px-x}}$

And of course, is there a witty proof that lim p -> 1 is rigorously the normal differential operator? --HappyCamper 07:09, 1 February 2006 (UTC)

That's called the q-derivative (usually that parameter is denoted q instead of p. For "quantum", I believe). It has a red link on derivative (generalizations). It's used in quantum groups sometimes. -lethe ^{talk +} 11:12, 1 February 2006 (UTC)

See Category:Q-analogs although it seems we don't yet have an article on this one. q-series are older than quantum mechanics; the earliest q-series I know of is the basic hypergeometric function, which is 19th century. The relationship between the q-derivative and q-series can be seen by taking the q-derivative of x^n. (left as a homework exercise). linas 14:57, 1 February 2006 (UTC)

I created a very stubby q-derivative article, and split out a distinct article q-analog, away from q-series. linas 16:03, 1 February 2006 (UTC)

I also expanded on the "why its quantum-like" in the q-analog article. linas 16:18, 1 February 2006 (UTC)

Is this a function that is heavily used in number theory? How I encountered this was actually through statistical mechanics and a very cutting edge research compilation... --HappyCamper 06:58, 2 February 2006 (UTC)

Maybe not "heavy", but it does show up in number theory and string theory and combinatorics, and so perhaps not surprisingly in stat mech. I've seen it in the context of the Ising model, the theory of which does invoke Riemann surfaces of high genus, etc; the Ising model is stat mech; there's been some recent major progress in that field. Although I would also be curious to look at your reference. linas 06:28, 5 February 2006 (UTC)

As for the limit as p -> 1, it depends on what you mean by limit. But it is true that this is equal to the derivative on a compact set, but the convergence is non-uniform on non-compact sets. As an example, let xεR¹. Writing p = 1+ε and considering Taylor series gives

${\frac {f(1+\epsilon )x)-f(x)}{\epsilon x}}=f'(x)+O(\epsilon x),$

and it's easy to see that this limit is uniform only if x is constrained to be in some bounded set. --Deville (Talk) 07:01, 5 February 2006 (UTC)

Harmonographs

After much searching, I have found the parametric equations that describe harmonographs such as the ones at Questacon and the Wollongong Science Centres in Australia. They are:

${\begin{matrix}x\left(t\right)&=&A_{x}(t)\sin \left(w_{x}t+p_{x}\right)+A_{s}(t)\sin(w_{s}t+p_{s})\\y(t)&=&A_{y}(t)\sin \left(w_{y}t+p_{y}\right)\\A_{x}(t+1)&=&d_{x}\times A_{x}(t),|d_{x}|<1\end{matrix}}$

Where A_x and A_y are the amplitudes, p_x and p_y are the phase shifts, w_x and w_y are the period setters and d_x is the decay factor. The site I visited here says that A_s, p_s and w_s are all to do with the rotation.

Does anybody have or know of graphing software that can graph a parametric equation with a decaying function incorporated?

Replace the last equation with its analytic solution $A_{x}(t)=a_{0}\exp(t\log d_{x})\!$, where $|d_{x}|<1$ as before but $a_{0}$ is an extra constant you need to choose. Then you can simply plug this formula for $A_{x}(t)$ into the first two equations. Any ordinary package which graphs parametric equations will be able to handle this. —Blotwell 03:20, 2 February 2006 (UTC)

Also, does anybody know the fractal formed on the complex plane, which is the set of points z such that z² is a real number?

I think this description is probably missing something: the set $\{z\in \mathbb {C} :z^{2}\in \mathbb {R} \}$ is simply $\mathbb {R} \cup \mathbb {R} {}i$, the set of numbers that are purely real or purely imaginary. It is not a fractal. —Ilmari Karonen (talk) 23:54, 1 February 2006 (UTC)

--Alexs letterbox 10:46, 1 February 2006 (UTC)

February 2

Cube rotation

Eh... This looked so simple, so trivial, but it's being terrible. I created a cube on POV-Ray, <1,1,1>/2 to -<1,1,1>/2 (side = 1). Well, I want to rotate it and then use the difference command to remove everything below 0y, creating then a tetrahedron with sides = ${\sqrt {2}}$ and height ${\frac {\sqrt {3}}{2}}$. Cutting is easy, but the rotation has been an issue. I made an animation to help you guys understand what I'm trying to do. Here, x = red, y = green, z = blue.

First thing, I rotate 45° on the z axis. Then, I have to find the angle of rotation on the x axis to make the three vertices lie on the xz plane. Well, I first tried 45°. It didn't work, and I figured why. Well, I then did some math and came with the angle $\arctan {\sqrt {2}}$. This almost works, but something is wrong and I can't figure what. Can't be that I need to rotate the other axis, because rotating on the y axis won't change anything. As you can see, the top vertice matches the z axis perfecly, but the other three just don't.

Worth noting, POV-Ray treats rotate <1,1,0> (two axis in one command) differently than rotate <1,0,0> and then rotate <0,1,0> (one axis per command). Not a clue why, though. Anyone can clear this out?

So, anyway, how do I do this? ☢ Ҡieff⌇↯ 01:38, 2 February 2006 (UTC)

You already have the cube oriented corretly, you just need to translate it downwards a bit. Anyway, wouldn't it be easier to use a prism? —Ilmari Karonen (talk) 02:06, 2 February 2006 (UTC)

Eh, doesn't make sense :o The cube is centered on the origin, the cut plane should be the xz plane, right? I think it's a matter of rotation only, not translation. And the issue is not the prisma itself, but I'm doing a certain 3D structure here that's based on 3 perpedincular cylinders, and I want to make it look like a tripod lying on the floor, but I keep getting a foot right and the other two off. The cube is just an easy way to deal with the problem. ☢ Ҡieff⌇↯ 02:27, 2 February 2006 (UTC)

No, Ilmari Karonen's right. When you orient the cube with a vertex uppermost, as you correctly have, there's a top vertex, a bottom vertex and 6 others. Three of them are adjacent along an edge from the top vertex, and they're all on the same horizontal plane which is above the origin. The other three are adjacent to the bottom vertex and they're all on a different horizontal plane, symmetrically located below the origin. If you were thinking they were all at the same horizontal level you had something in mind more like an octahedron.

By my quick calculations (could be wrong), the vertical distance from the top vertex to the upper plane is sqrt(3), so that's the height of the tripod and you should cut off everything below the plane y = 1 - sqrt(3) (and lower accordingly). —Blotwell 03:02, 2 February 2006 (UTC)

(edit conflict) The cut plane should be parallel to the xz plane, but you need to move either the plane or the cube vertically to get a tetrahedron. In the animation on the right, look closely at the spot where the x axis meets the cube. After the first rotation, this point is in the middle of one of the edges of the cube, and it will stay there during the second rotation. But the cut plane should meet the cube at the upper end of this edge. —Ilmari Karonen (talk) 03:19, 2 February 2006 (UTC)

I've rendered it semi-transparent and yes, I see it now. I was, indeed, having the properties of the octahedron in head... :/ I've now found that the cut plane lies on ${\frac {\sqrt {3}}{6}}$, by the way. (my Rubik's Cube was a lot of help on figuring this out, though.) Anyway, thanks everyone :) ☢ Ҡieff⌇↯ 03:33, 2 February 2006 (UTC)

Analytic number theory

In a very amateur way I have enjoyed working through books on elementary, classical number theory. I am of course aware that there is a huge field of analytic number theory, but I have no insight into this area at all (and only first year undergraduate knowledge of analysis). Is anyone aware of a textbook that might introduce me to some of the main techniques and results of analytic number theory without assuming that I am a mathematical genius or throwing me right in at the deep end? Thanks for any help you can give. Maid Marion 13:50, 2 February 2006 (UTC)

Well, "first year undergraduate knowledge of analysis" might not be enough, depending on what that covers. It really is essential to know the most basic parts of complex analysis, for example the Cauchy integral formula, and what a complex analytic function is. Then again, it's possible to get a simple version of Dirichlet's theorem on arithmetic progressions working without complex analysis. Dmharvey 02:20, 4 February 2006 (UTC)

I recommend A Primer of Analytic Number Theory : From Pythagoras to Riemann by Jeffrey Stopple (published by Cambridge, ISBN 0521012538). It is aimed at undergraduates and specifically does not assume a prior knowledge of complex analysis. Gandalf61 09:30, 4 February 2006 (UTC)

Thanks guys - very interesting. I'll try to get hold of the Stopple book. Maid Marion 15:24, 4 February 2006 (UTC)

Solutions

Does anyone here know where I can get the solutions to the International Mathematical Talent Search. I found the questions at http://www.math.ca/Competitions/IMTS/. I've searched a while on google, and didnt get any, so if you dont get the solutions, I might have to post the questions which I find difficult here again ;-) ! -- Rohit 17:37, 2 February 2006 (UTC)

February 3

(no questions today)

Oh no, not again... ☢ Ҡieff⌇↯

So just for your entertainment: in a mirror, why is left and right exchanged but not up and down? (I know the answer, but just as you seem so disappointed ;-)
(-; Safety warning: trying to perform a headstand in front of a mirror may damage your health, the mirror and anything that can't get away quick enough ;-) The Infidel 21:59, 4 February 2006 (UTC)

They are exchanged, you just don't realize it because you see horizontal reflection and vertical reflection of a human in different ways, thanks to our vertical line of near-symmetry and our curious insistence on remaining upright while talking to each other. Were you freed from such restraints, you would see clearly that a mirror not only reflects horizontally and vertically, but diagonally, northeasterly, substandardly, and many other ways besides. Black Carrot 04:44, 5 February 2006 (UTC)

• No: write top rigth botom left on a circle, apply your transformation and see that it's just a rotation of a half circle The Infidel 20:38, 6 February 2006 (UTC)

In fact, I think it doesn't reflect wither up and down or left and right but it does reflect back and front: when I try to comb my hair or shave with the aid of a mirror, I often move my hand forwards when I should reach to the back or in the other way, but I never confuse the left or right side of my head or the top with the bottom. – b_jonas 10:20, 5 February 2006 (UTC)

• You're right, but why did you have to tell it right away? I'd loved to hear theories about polarized light and the psychology of starfish staring at a mirror, or even, if I'm really mean, about some strange accidents in the bathroom involving a mirror ... The Infidel 20:38, 6 February 2006 (UTC)

• I'm always feet up in a mirror and right is always right. hydnjo talk 01:25, 7 February 2006 (UTC)

• Do you get a lot of footprints on your mirror? —Ilmari Karonen (talk) 14:24, 7 February 2006 (UTC)

February 4

Drawing phase portraits on the surface of a cylinder

Potential energy and phase portrait of a simple pendulum. Note that the x-axis being angle, wraps onto itself after every 2π radians.

I have to draw a phase portrait of a dynamical system given by the following equation.

${\frac {1}{2}}a{\dot {\theta ^{2}}}=g\left(1-{\frac {1}{2}}\lambda \cos \theta \right)\cos \theta +C$

Since the x-axis is angle, the manifold is $\mathbb {R} \times \mathbb {S}$. I want to draw the phase portrait on the surface of a cylinder so that $2\pi$ and 0 coincide. How can I do this in Matlab/Maple ? On the right is an example of a similar situation. deeptrivia (talk) 02:16, 4 February 2006 (UTC)

I don't understand; just graph that equation above, won't that give you the phase diagram of your system? What more do you want? -lethe ^{talk +} 05:25, 4 February 2006 (UTC)

I want to map it on the surface of the cylinder, so that 0 and 2π on the theta axis coincide, rather than being on opposite sides. This is somewhat like saying that I want to represent it as a globe (only one dimension is curved though, making it a cylinder instead of a sphere) as opposed to a map. I want to keep this cylinder transparent (probably not 100% transparent) so that things on the far side can be seen. deeptrivia (talk) 05:31, 4 February 2006 (UTC)

Oh, maybe I see what you mean. Like, you want it graphed on the surface of a 3d cylinder. The graph you have is actually on a topological cylinder alreadyl, once you decree that 0 an 2π are identified, but I guess you want it to actually look like the cylinder embedded in R³. I tried graphing it. How's this look? -lethe ^{talk +} 07:37, 4 February 2006 (UTC)

That's exactly what I want. How did you make it? Thanks! deeptrivia (talk) 12:51, 4 February 2006 (UTC)

I plotted something similar to what Confusing Manifestation suggested below, except that I used cylindrical coordinates. He is correct, this method suffers from the defect that it requires you to solve for ω. I had to graph two plots, one for the positive square root and one for the negative. I could have avoided this by plotting the whole relation in cylindrical coordinates and intersecting with the cylinder. -lethe ^{talk +} 13:13, 4 February 2006 (UTC)

In terms of formulas, if you're looking to have something like what lethe drew, you want to plot $(cos\theta ,sin\theta ,f(\theta ))$, where $f(\theta )$ is the function you're plotting. Of course, given that the phase portrait is more of a relation, you'll need to either split it into a number of functions or find another paramater to include so that it draws the full portrait. Confusing Manifestation 09:53, 4 February 2006 (UTC)

Alright. And how do I make the cylinder below it, say, in MATLAB? deeptrivia (talk) 12:51, 4 February 2006 (UTC)

Thanks Lethe! What software did you use, and how did you plot the cylinder? deeptrivia (talk) 13:16, 4 February 2006 (UTC)

I used grapher, though I don't suspect it much matters what program you use, they all do parametric plotting (though I can't give you specific instructions for MATLAB, since I don't use that. I could get you there in Mathematica). The cylinder was plotted with the equation r=1. -lethe ^{talk +} 13:21, 4 February 2006 (UTC)

Thanks a ton, Lethe! I think I'll manage it now. deeptrivia (talk) 13:24, 4 February 2006 (UTC)

Laplace transform calculation

I have a question on[1] Anyone please help me.

(On which,why do for-all ${\mathcal {,}}isinx=odd,$ ,therefore they all equal to 0?)

--HydrogenSu 12:16, 4 February 2006 (UTC)

your question seems to be unclear. I think perhaps you want to ask why

$\int _{-\infty }^{\infty }e^{-2at}i\sin \omega t\,dt$

equals 0. I think the explanation is supposed to go like this: a function f is odd if for all x, f(x)=f(–x). Odd functions satisfy

$\int _{-a}^{a}f(x)\,dx=0,$

since the positive area on one side of the y-axis cancels out with the negative area on the other side. With some care, this result can be extended to improper integrals. For even functions, you can't cancel out the area, but you can say that the total area from –∞ to +∞ is twice the area from 0 to +∞. This is used in the subsequent step, and accounts for the 2 in front of the cosine.

The sine function is indeed an odd function, and the cosine function is indeed even. However, the integrand above is not the sine function, but rather the product of a sine and an exponential. As far as I can see, the integrand is not odd, and therefore this step of the calculation is an error. Nor is the 2 in front of the cosine correct.

(It's $d\omega$, not $dt$, so it actually is odd. The Infidel 21:51, 4 February 2006 (UTC))

Oh, you're right! I totally mucked that up. I'm sorry, yes, it i odd, and that integral is 0. -lethe ^{talk +} 02:30, 5 February 2006 (UTC)

This calculation seems to be proceding with a Fourier transform, which isn't really appropriate for doing a Laplace transform.

What I would suggest instead would be to start with the identity

$\sin at={\frac {e^{iat}-e^{-iat}}{2i}}$

then square it, and integrate the resulting exponentials. -lethe ^{talk +} 13:10, 4 February 2006 (UTC)

After trying it myself, I think that the above suggestion isn't so great, and you'd be better served with the identity

$\sin ^{2}x={\frac {1-\cos 2x}{2}}$

-lethe ^{talk +} 14:10, 4 February 2006 (UTC)

Thanks. I've seen your reply. If I have any more qut. ,ask here again.--HydrogenSu 12:04, 5 February 2006 (UTC)

February 5

Combinatorics problem

Dear all:

Does anyone know what the following combinatorial expression sums up to?

${n \choose 1}^{3}+{n \choose 2}^{3}+{n \choose 3}^{3}+\ldots +{n \choose n}^{3}$

Thanks

206.172.66.136 01:59, 5 February 2006 (UTC)

I remember thinking about this once. There definitely is a simple identity involving squares, but I couldn't find one involving cubes. Not that that means anything. Dmharvey 02:07, 5 February 2006 (UTC)

With a question like this, you compute the first half-dozen terms then type them into the Online Encyclopedia of Integer Sequences. The result lists an asymptotic formula, a recursive formula, an integral formula and a convolution relation, but no alternative combinatorial expression. Do any of those help? —Blotwell 08:28, 5 February 2006 (UTC)

Thank you very much for your help Blotwell. Yes it was very useful. I was able to find the 'Frenel numbers' entry in MathWorld, which solved everything. thank you again!!! 129.97.252.63 03:18, 6 February 2006 (UTC)

Careful with the spelling; it's "Franel". --KSmrq^T 03:44, 6 February 2006 (UTC)

LOL, sorry I screwed up. :-))))))))))))))))))))) 129.97.252.63 23:41, 6 February 2006 (UTC)

You can use imaginary numbers to help out here...the cube root of unity plays a role... --HappyCamper 02:08, 7 February 2006 (UTC)

February 6

what is an interger?

—The preceding unsigned header was added by 152.163.100.11 (talk • contribs) . probably meaning integer

If you were paying attention to our pages, you'd have seen the little search box at the left. From it, you'd easily find our article on integers. ☢ Ҡieff⌇↯ 05:05, 6 February 2006 (UTC)

Note that searching for "interger" doesn't turn up anything relevant. Fredrik Johansson - talk - contribs 07:30, 6 February 2006 (UTC)

Unless, of course, you try "interger" on a search engine like Google or Yahoo! that suggests a correct spelling ("Did you mean: integer"). A spelling suggestion feature here would be useful. --jh51681 19:24, 6 February 2006 (UTC)

I just use a redirect whenever I notice a common misspelling. Redirects are cheap. —Keenan Pepper 23:26, 6 February 2006 (UTC)

Perhaps this WP article would be helpful. hydnjo talk 23:33, 6 February 2006 (UTC)

"Interger vitae, scelerisque purus". (Pure in life, and free from crime). linas 23:42, 6 February 2006 (UTC)

Random etymology: The Latin word integer literally means "untouched", hence "pure". Its current English meaning comes from "untouched" -> "unbroken" -> "whole" -> "whole number". —Keenan Pepper 23:54, 6 February 2006 (UTC)

Once again proving that there are no stupid questions ... actually, there are very stupid questions, or at least ones that are asked in the wrong place, but even you still have a chance of learning something very interesting from them. Confusing Manifestation 09:05, 7 February 2006 (UTC)

February 7

A Calculating

$\,\int _{0}^{\infty }\,X^{2}{\frac {1}{e^{X}-1}}dX=..................$

Is 2.54? How to do? Thanks.--HydrogenSu 13:17, 7 February 2006 (UTC)

I don't know how to do it, but Mathematica tells me the answer is 2.40411 though. enochlau (talk) 13:32, 7 February 2006 (UTC)

The Integrator gives the indefinite integral in terms of polylogarithms, so it's over my head. —Keenan Pepper 14:25, 7 February 2006 (UTC)

If using ${\mathcal {\,}}\Gamma (z)\,$ functions,how to do (it)? (Extra:-Remembered in street-talk something like in movie ID4 the "it" can be ignored? I'm an Asian.)--HydrogenSu 19:16, 7 February 2006 (UTC)

I heard from my physics teachre he said:$\,\int _{0}^{\infty }\,X^{3}{\frac {1}{e^{X}-1}}dX={\frac {\pi ^{4}}{15}}$ can be calculated by that kind of special functions.--HydrogenSu 19:21, 7 February 2006 (UTC)

I just now guess that $\,\int _{0}^{\infty }\,X^{2}{\frac {1}{e^{X}-1}}dX={\frac {\pi ^{3}}{ab}}$

Where ${\mathcal {\,}}b-a=3\,$. I'm going to guess in my room what they are. Such interesting.--HydrogenSu 19:26, 7 February 2006 (UTC)

Heh, that doesn't tell much. Any positive number can be written in the form $\pi ^{3}/(ab)$ where $a-b=3$, and that integral is positive. – b_jonas 21:39, 7 February 2006 (UTC)

• It's just a tech for memory,not need to really operate in math. Too trouble. I'm learning Physics,not math.--HydrogenSu 07:57, 10 February 2006 (UTC)

It's $2\zeta (3)$, according to the second formula given in Riemann_zeta_function#The_Riemann_zeta_function_as_a_Mellin_transform.--gwaihir 21:31, 9 February 2006 (UTC)

Change heading in 3 dimensions

We're at point (x1, y1, z1) looking at angle $\theta$. We want to face point (x2, y2, z2). I know that to face a new angle in two dimmensions, we could rotate by the arctan((y2-y1)/(x2-x1)). But how do I do this in three dimensions? 216.158.57.50 18:07, 7 February 2006 (UTC)

In 3D there are two angles that describe a direction, not just one. For example, there is the angle as projected into the XY plane and the angle above the XY plane. See spherical coordinates and cylindrical coordinates. StuRat 21:17, 7 February 2006 (UTC)

The section title uses the word "heading", by which I'm guessing you mean compass heading. But suppose I'm at an air show watching a helicopter fly straight toward me and over my head. As the plane approaches and my gaze follows it, the heading of my gaze does not change, yet I must move my head. When the plane is directly overhead and my gaze is perfectly vertical, every possible heading still leaves my gaze fixed on the plane. Now suppose that I am videotaping the flight, and that the plane pauses to hover. Still leaving the plane centered in the viewfinder, I can rotate the camera around the line of sight, changing neither the azimuth (heading) nor the altitude, yet changing the camera orientation. So your problem is underspecified, and based on a false premise. The problem in 3D has subtleties and complexities for which 2D does not prepare us. --KSmrq^T 06:03, 8 February 2006 (UTC)

See also flight dynamics. —Keenan Pepper 15:21, 8 February 2006 (UTC)

February 8

I don't like maths

can you do all my math homework for me, it's hardMaths are for kids 18:48, 8 February 2006 (UTC)

Yeah, sure. The answer is 42. Have a nice day! —Ilmari Karonen (talk) 19:28, 8 February 2006 (UTC)

Only if you say the magic word and emphazi it suitly. – b_jonas 19:29, 8 February 2006 (UTC)

Why do Brits call it "maths" instead of "math" ? Do you also study "Englishes", "histories", and "governments" ? StuRat 20:02, 8 February 2006 (UTC)

Do you study mathematic? Fredrik Johansson - talk - contribs 20:06, 8 February 2006 (UTC)

We generally only study one mathematic subject at a time, yes. Do you study them all at once, and hence study "maths" ? StuRat 21:53, 9 February 2006 (UTC)

I study several of them; not necessarily at the same time, but that's not a requirement. Fredrik Johansson - talk - contribs 21:58, 9 February 2006 (UTC)

Well, in Australia when it comes to school subjects we have Maths and Sport, whereas, I am told, in the US it's Math and Sports. And yet there are those who would say Australia is one of the most Sports-crazy countries in the world (noting the plural in particular given we play at least 4 codes of football, cricket, netball, and countless others). Confusing Manifestation 02:18, 9 February 2006 (UTC)

It's such a shame that people nowdays are so anti-mathematics...

"I am accustomed, as a professional mathematician, to living in a sort of vacuum, surrounded by people who declare with an odd sort of pride that they are mathematically illiterate." — David Mumford

Amen to that. ☢ Ҡi∊ff⌇↯ 04:10, 9 February 2006 (UTC)

Hear Hear. — Ilyanep (Talk) 04:20, 9 February 2006 (UTC)

Ditto. I recommend Innumeracy by John Allen Paulos, if you haven't read it yet. --JianLi 01:02, 10 February 2006 (UTC)

I'm British English, why do I say Maths, no idea! Who said language so logical. I suppose I would say 'Maths' was better because mathematics ends in an 's'--βjweþþ (talk) 20:01, 9 February 2006 (UTC)

In my neverending quest to be politically correct, I always write "mathematics" in full, never "math" or "maths". This is every bit as important as men leaving the toilet seat down. One compromise might be to take a note from the non-sexist pronoun s/he, and write math/s. Note that British and American English differ also in examples like "Ford Motor Company pay their workers" versus "Ford Motor Company pays their workers". --KSmrq^T 01:11, 10 February 2006 (UTC)

Thus We might take a guess of French men would say it as "MATHE" 'cause end of Mathématique in it.

And Chi. say:ㄕㄨˋ ........Chineses in TW used to be use tongues to be lazy in writting.

French men (and women) say Maths, referring to "MathématiqueS"...Spanish people say "Matemáticas", so as Portuguese do. Everyone uses the plural but American. For once we agree with the "Rosbiffs", the "Ricains" nous cassent les couilles.

"I love you=ㄨㄛˇ ㄞˋ ㄋ一ˇ"

Math help

where was the first math instrument found?

In the woodwinds section--64.12.116.11 22:50, 8 February 2006 (UTC)

Hah...that made me Laugh out loud :\ I'm such a band geek. — Ilyanep (Talk) 04:20, 9 February 2006 (UTC)

Depends what qualifies as an "instrument". The earliest form of counting probably involved keeping one pebble for every animal in your heard, and pairing them off one by one. The Latin word calculus literally means "pebble". So do pebbles count as "math instruments"? —Keenan Pepper 23:13, 8 February 2006 (UTC)

The abacus goes way back, too, although not as far as the pebble. StuRat 05:46, 9 February 2006 (UTC)

The earliest mathematical tools with any degree of permanence were probably tally sticks such as the Ishango Bone. Gandalf61 11:10, 9 February 2006 (UTC)

February 9

Semiprimes

The density of the primes are 1/ln(P), is there a similar result for semiprimes? Can this result be edited to only include square free numbers?

Integral question

$I_{4}=\,\int _{0}^{\infty }\,x^{4}e^{-{\alpha }x^{2}}dx={\frac {3}{8}}{\sqrt {\frac {\pi }{\alpha ^{5}}}}$

How to derive this? --HydrogenSu 14:37, 9 February 2006 (UTC)

See Integration by parts, and the Error function. --BluePlatypus 14:53, 9 February 2006 (UTC)

My Soluntuon by Using ${\mathcal {\,}}L(t^{a})\,$ :--HydrogenSu 18:03, 16 February 2006 (UTC)

${\mathcal {\,}}\,\int _{0}^{\infty }\,x^{4}e^{-{\alpha }x^{2}}dx={\frac {1}{2}}\,\int _{0}^{\infty }\,x^{2((5/2)-1)}e^{-{\alpha }x^{2}}d(x^{2})\,$

Where ${\mathcal {\,}}{\frac {x^{4}}{x}}=x^{3}=x^{2{\cdot }{\frac {5}{2}}-2}\,$

The ${\mathcal {\,}}\,\int _{0}^{\infty }\,e^{-{\alpha }x^{2}}x^{2{\cdot }((5/2)-1)}d(x^{2})\,$ which just satisfies with

${\mathcal {\,}}\,\int _{0}^{\infty }\,e^{-{\alpha }t}t^{(x-1)}dt={\frac {1}{\alpha ^{x}}}\Gamma (x)\,$

So ${\mathcal {\,}}{\frac {1}{2}}\,\int _{0}^{\infty }\,e^{-{\alpha }x^{2}}x^{2((5/2)-1)}d(x^{2})\,$

${\mathcal {\,}}={\frac {1}{2\alpha ^{5/2}}}\Gamma ({\frac {5}{2}})\,$

${\mathcal {\,}}={\frac {1}{2\alpha ^{5/2}}}{\frac {3}{2}}\Gamma ({\frac {3}{2}})\,$

$={\frac {3}{8}}{\sqrt {\frac {\pi }{\alpha ^{5}}}}$

Where ${\mathcal {\,}}\Gamma ({\frac {1}{2}})={\sqrt {\pi }}\,$ ,and ${\mathcal {\,}}\Gamma (x)=(x-1)!\,$

Typed the above by JAVA,also contribute them to other users!

--HydrogenSu 19:01, 9 February 2006 (UTC)

• The Error function certainly has do with it. The antiderivative of (x^4)*exp(-a * x^2) contains two exponential terms and one Erf term. --BluePlatypus 08:45, 10 February 2006 (UTC)

HydrogenSu - based on your recent contribution history, it is clear that you are using the services of the reference desk pages to help with your homework only - an account should not be used for this purpose. There is a reason why homework questions are frowned on - the volunteers here not to do your homework! It has nothing to do with your nationality or freedom or whatnot - those sorts of politics can simply stay off of the reference desk - Wikipedia simply isn't a forum for that. Respectfully, in the future please be mindful of that. --HappyCamper 13:13, 10 February 2006 (UTC)

What is this "USA time" anyway? I live in the USA now, and ever since coming here, I don't seem to have any time at all! Dmharvey 14:42, 10 February 2006 (UTC)

I responded your question above yesterday and moved it on Language Section.--HydrogenSu 11:31, 11 February 2006 (UTC)

February 10

Toroid/Torus

Surface or Solid?

According to the Wikipedia articles, the Toroid is a solid while the Torus is its surface. According to the American Heritage Dictionary,

a toroid is

a A surface generated by a closed curve rotating about,
but not intersecting or containing, an axis in its own plane.
b. A solid having such a surface.

and a torus is

A toroid generated by a circle; a surface having the shape
of a doughnut. In this sense, also called tore2

Which definitions are right? I'm confused --JianLi 03:10, 10 February 2006 (UTC)

I would say "toroid" or "torus" may refer to either the surface or the solid. To be specific, refer to a "toroid surface" / "torus surface" or a "solid toroid" / "solid torus". StuRat 04:38, 10 February 2006 (UTC)

What the rest of the world chooses to do is up to them, but in mathematics a torus is a surface, as is a sphere. --KSmrq^T 13:55, 10 February 2006 (UTC)

The dictionary definition you give of torus says it is a surface, which is consistent with Wikipedia's definition. In any case, one should note that the page on torus makes it clear the contexts of usage given are in mathematics, e.g. "in geometry..." and "in topology...". The dictionary definition of toroid indicates there is ambiguous usage as a solid or surface; this is reflected also in the toroid article, where the toroid is described as a solid, but the example of coils given actually are "hollow". --C S (Talk) 14:12, 10 February 2006 (UTC)

Exactly. One should always be a bit careful about looking up specialized terms in a dictionary (unless it's a specialty dictionary, of course), since the dictionary intends to give the common usage. So what is commonly called a torus need not be what topographers call a torus. Most people just say "doughnut shape", though. :)--BluePlatypus 18:19, 10 February 2006 (UTC)

I thought a toroid was a psuedo-torus, where the revolving circle can intersect itself...is that not true? --HappyCamper 18:52, 10 February 2006 (UTC)

Actually, I did make an effort to use only the mathematical definitions. For the full definitions, see torus and toroid and look under the definitions labeled "Mathematics" --JianLi 21:05, 10 February 2006 (UTC) In any case, if the terms are used interchangeably (as StuRat said), or Torus is a surface while Toroid may be either (as Chan-Ho said), why do we have separate articles? Does anybody think a merge is in order? --JianLi 21:05, 10 February 2006 (UTC)

Non-circular toroids?

In addition to the surface/solid confusion, note that the dictionary allows a toroid to be made by any closed curve, not just a circle. How do you reconcile that? Are dictionary definitions just unreliable for mathematics, even if it was listed as a "mathematics" definition? --JianLi 21:08, 10 February 2006 (UTC)

Well, if you get a square and rotate it in a circular orbit, the resulting shape will be a toroid with a square cross-section. The torus is just the case where this toroid's closed curve happens to be circle. From what I know, the word torus is usually used for the surface alone, where toroid is the solid with such surface. ☢ Ҡi∊ff⌇↯ 21:39, 10 February 2006 (UTC)

I call any section rotated about an axis to be a "surface of revolution" or a "solid of revolution". StuRat 02:03, 11 February 2006 (UTC)

Well, sure. Take a cone for example. Those are rotations when the inner radius is zero. If the radius is > 0, you'll have a toroid. ☢ Ҡi∊ff⌇↯ 03:06, 11 February 2006 (UTC)

I call them all "surfaces of revolution" or "solids of revolution". I only call them a "torus" or "toroid" if the section which is rotated is a full circle in the plane of the axis of revolution, but not touching that axis. I therefore consider "toroids" to be a proper subset of all "surfaces/solids of revolution". StuRat 16:50, 11 February 2006 (UTC)

Oh, that. Well, aren't they? I always thought so. ☢ Ҡi∊ff⌇↯ 23:27, 11 February 2006 (UTC)

Suggestion given to BP guy

<post removed> - Relegated to talk pages. See BluePlatypus and HydrogenSu. Thank you. --HappyCamper 13:26, 10 February 2006 (UTC)

Homotopy lambda-calculus

Since too many questions are being answered too quickly, I thought I'd ask if anybody has heard of the homotopy lambda-calculus talked of by Vladimir Voevodsky (2002 Fields medal) in this lecture, or can guess what it's about? Is it something to do with motivic cohomology? --- Charles Stewart^(talk) 19:14, 10 February 2006 (UTC)

I went to this lecture. He didn't get to the "homotopy" part of it... it's a four lecture series. He pretty much outlined plain vanilla lambda calculus. That's it.

February 11

Area of a regular polygon?

The Regular polygon article says the following gives the area of a regular polygon:

$A={\frac {nt^{2}}{4\tan(180^{\circ }/n)}}$

What does t stand for?

t is the apothem, I think. ☢ Ҡi∊ff⌇↯ 02:11, 12 February 2006 (UTC)

If t were the apothem the formula would be ${\frac {1}{2}}pt$ (where p is the perimeter and t is the apothem) wouldn't it? — Ilyanep (Talk) 02:19, 12 February 2006 (UTC)

Yeah I know, but you don't see p in there, do you? I'd think both are equivalent, the only difference is that this one calculates the perimeter inherently based on number of sides or something. I'd have to check out by hand, but I'm too lazy. ☢ Ҡi∊ff⌇↯ 06:54, 12 February 2006 (UTC)

My friend and I deduced an alternate formula, is it correct?

$A={\frac {s^{2}n(\tan(i/2))}{4}}$

Where s is the side length, n is the number of sides, and i is the angle measure of an interior angle. — Ilyanep (Talk) 18:46, 11 February 2006 (UTC)

On looking at it a second time, they look really similar. Assuming that t = s, then :

${\frac {nt^{2}}{4\tan(180^{\circ }/n)}}={\frac {s^{2}n(\tan(i/2))}{4}}/{\frac {\tan(i/2)}{\tan(i/2)}}$

(aside from the fact than tan(180/n) is not equal to tan(i/2) I think, but that seems to be a minor difference. — Ilyanep (Talk) 18:50, 11 February 2006 (UTC)

$i/2={\frac {180-(360/n)}{2}}=90-(180/n)$

hmm...somehow I doubt that tan(90-(180/n)) = tan(180/n) ... getting somewhere here though. These math tags are fun though. — Ilyanep (Talk)

Perhaps if i was the angle measure of a central angle rather than interior angle, then it would work, because then:

$i/2={\frac {360/n}{2}}=180/n$

— Ilyanep (Talk) 19:03, 11 February 2006 (UTC)

Yeah, I'd say t is side length and 1/tan(180/n) is (length of half a side)/(length of apothem), making it equivalent to your (1/2)(perimeter)(length of apothem) formula. Black Carrot 02:46, 12 February 2006 (UTC)

I deduced this formula on my own and t is equal to the length of the side.

Infinite Sums

These will solve Topic:Infinite Sums.

• 您的問題的確需要"技巧(tech.)"來說明.請不妨把兩邊都平方,做此後,你將發現(9-6)恆等於3:

$({\sqrt {6+{\sqrt {6+{\sqrt {6+...}}}}}})^{2}=3^{2}$

第 1輪:$6+{\sqrt {6+{\sqrt {6+...}}}}=9$

$\Longrightarrow {\sqrt {6+{\sqrt {6+{\sqrt {6+...}}}}}}=9-6=3$

第 2輪:$6+{\sqrt {6+{\sqrt {6+...}}}}=9$

$\Longrightarrow {\sqrt {6+{\sqrt {6+{\sqrt {6+...}}}}}}=9-6=3$

第 3輪:$6+{\sqrt {6+{\sqrt {6+...}}}}=9$

$\Longrightarrow {\sqrt {6+{\sqrt {6+{\sqrt {6+...}}}}}}=9-6=3$

.

.

.

第10輪:$6+{\sqrt {6+{\sqrt {6+...}}}}=9$

$\Longrightarrow {\sqrt {6+{\sqrt {6+{\sqrt {6+...}}}}}}=9-6=3$

.

.

.

第${\mathcal {\,}}\infty \,$輪:$6+{\sqrt {6+{\sqrt {6+...}}}}=9$

$\Longrightarrow {\sqrt {6+{\sqrt {6+{\sqrt {6+...}}}}}}=9-6=3$

That's why ${\sqrt {6+{\sqrt {6+{\sqrt {6+...}}}}}}=3$ always stands.

That's what my high school teachers taught. I learned.--HydrogenSu 23:01, 12 February 2006 (UTC)

Hm, an inspiring way to prove things. So I've tried is myself:

$({\sqrt {6+{\sqrt {6+{\sqrt {6+...}}}}}})^{2}=(-2)^{2}$

第 1輪:$6+{\sqrt {6+{\sqrt {6+...}}}}=4$

$\Longrightarrow {\sqrt {6+{\sqrt {6+{\sqrt {6+...}}}}}}=4-6=-2$

第 2輪:$6+{\sqrt {6+{\sqrt {6+...}}}}=4$

$\Longrightarrow {\sqrt {6+{\sqrt {6+{\sqrt {6+...}}}}}}=4-6=-2$

第 3輪:$6+{\sqrt {6+{\sqrt {6+...}}}}=4$

$\Longrightarrow {\sqrt {6+{\sqrt {6+{\sqrt {6+...}}}}}}=4-6=-2$

.

.

.

第 10輪:$6+{\sqrt {6+{\sqrt {6+...}}}}=4$

$\Longrightarrow {\sqrt {6+{\sqrt {6+{\sqrt {6+...}}}}}}=4-6=-2$

.

.

.

第${\mathcal {\,}}\infty \,$輪:$6+{\sqrt {6+{\sqrt {6+...}}}}=4$

$\Longrightarrow {\sqrt {6+{\sqrt {6+{\sqrt {6+...}}}}}}=4-6=-2$

That's why ${\sqrt {6+{\sqrt {6+{\sqrt {6+...}}}}}}=-2$ always stands.

(Sorry, couldn't help it ;-) The Infidel 20:17, 16 February 2006 (UTC)

Sorry,nobody told people that ${\sqrt {6+{\sqrt {6+{\sqrt {6+...}}}}}}=-2$. For my solutions above,which was according to the asker's ${\sqrt {6+{\sqrt {6+{\sqrt {6+...}}}}}}=3$. And then started to derive it.

However,I don't know why you had to say that strange words. That was off topic--HydrogenSu 13:20, 17 February 2006 (UTC)

If not for the last sentence I would have tried to explain it to you, but so you think it's off topic I won't argue. The Infidel 20:08, 17 February 2006 (UTC)

---

I know that: ${\sqrt {6+{\sqrt {6+{\sqrt {6+...}}}}}}=3$

How do I prove that? I just figured it out by putting it into a calculator (and knowing that math club questions like this always add up to a nice round whole number) — Ilyanep (Talk) 19:01, 11 February 2006 (UTC)

We assume a positive square root; remember that. Let x be the quantity in question. Note

$x={\sqrt {6+x}}.$

Squaring both sides loses sign information; remember that, but do it anyway. Proceed as comes naturally. --KSmrq^T 19:12, 11 February 2006 (UTC)

Of course you have to prove that the sequence converges first for this argument to be made completely rigorous. Hint: Prove that the sequence given by

$a_{1}:=b,\quad a_{n+1}={\sqrt {6+a_{n}}}$

is bounded and monotone, hence converges for any nonnegative starting point b. Kusma (討論) 19:23, 11 February 2006 (UTC)

Well now this is just degrading into vandalism. Changing other people's comments is a serious breech of Wikiquette. —Keenan Pepper 22:37, 12 February 2006 (UTC)

大爺您要清 就請清您們胡搞的惡意批評的文字好嗎? 謝大爺. 您得要行行好. 上頭我有很多都是math的,不可刪呢!--HydrogenSu 23:01, 12 February 2006 (UTC)

I'm sorry, but I can't even see these characters. Are they chinese? — Ilyanep (Talk) 23:20, 12 February 2006 (UTC)

Indeed they are, but all of this is getting rather offtopic (comments about the comments), so I won't even try to translate. And it would be nice if we could close this discussion here. 謝謝! (Chinese characters, "thank you"). Kusma (討論) 23:26, 12 February 2006 (UTC)

--HydrogenSu 16:48, 15 February 2006 (UTC)

February 12

How is a Lorenz curve calculated?

We recently went over this in economics class, but our professor didn't tell us how it was calculated; I imagined it was because it was beyond the scope of the class, and it was good enough to know that it was a measure of inequality. For example, he gave us Brazil's Lorenz value as .68, and the U.S.'s as .46. How did they arrive at those values? From the looks of it, it has something to do with the area enclosed by a graph representing a country's income quintiles divided by itself and the area that would be covered by a line representing perfect equality. Please correct any of the above if it's incorect. --Impaciente 01:41, 12 February 2006 (UTC)

Actually I just found the answer to my question. --Impaciente 04:46, 12 February 2006 (UTC)

Do share, please. ☢ Ҡi∊ff⌇↯ 06:42, 12 February 2006 (UTC)

Covariant derivative

How, exactly, is the covariant derivative defined? Of corse, I've read the article, but I'm still not sure what the domain and the range of the covariant derivative is. Probably I'm missing something that is really trivial.

Let M be a Riemannian manifold of dimension m and $p\in M$ any point of the manifold. In the tangent space $T_{p}M$ there are vectors u,v. I understand the covariant derivative $\nabla =\nabla (p)$ is a map (depending on the point p) $\nabla (p):T_{p}M\to \{A:T_{p}M\to T_{q}M\}$, $\nabla (p):u\mapsto \nabla _{u}(p)$ where $\nabla _{u}(p):v\mapsto \nabla _{u}(p)v\in T_{q}M$ with $T_{q}M$ being the tangent space of a point $q\in M$.

That's my understanding so far. If this was correct, there would be some mentioning of the point q in the article, so I guess p=q, and the connection with other points is by local charts? The Infidel 11:43, 12 February 2006 (UTC)

I have mostly purged memory of my Differential Geometry lectures from my mind, but I believe the covariant derivative is basically a way of expressing the spatial derivative with respect to a vector tangential to the manifold in terms of the parametric representation of said manifold. I need someone to back me up here, though. Confusing Manifestation 14:10, 12 February 2006 (UTC)

It's a generalization of the directional derivative, so you need a direction u at a point p and something to differentiate, v, which must be defined in some neighbourhood of p. There is no single point different from p involved in this. Your formula for the type of $\nabla (p)$ is wrong, it should be $\nabla (p)\colon T_{p}M\to (VF_{p}\to T_{p}M)$, where $VF_{p}$ means "local vector fields defined in some neighbourhood of p".--gwaihir 17:25, 12 February 2006 (UTC)

Ok, I had another look at the article and found that $\nabla (p):T_{p}M\to \{A:T_{p}M\to T_{p}M\}$ (which is described under the "Note" section while I was expecting this within the definition, that's why I didn't find this at first).

What remains of my question shifts to this: In Euklidean space we have (for differentiabe functions) $\forall _{x_{0}}\exists _{\epsilon ,c}\forall _{|x-x_{0}|<\epsilon }\,\,|f(x)-f(x_{0})-(x-x_{0})f'(x_{0})|<c\cdot |x-x_{0}|^{2}$

Is there an analog for this approximation that holds for manifolds where we cannot generally take x-x_0 (except when using local charts; but then we have to show it's independent of the choice of the chart)? The Infidel 20:39, 12 February 2006 (UTC)

The question boils down to: When all the items about a covariant derivative are in the tangent space of one point, how is this used to get along to another point in a neighbourhood, without having cumbersomely to deal with local charts. The Infidel 20:59, 12 February 2006 (UTC)

I don't know where you found $\nabla (p):T_{p}M\to \{A:T_{p}M\to T_{p}M\}$, but it's still wrong. You can't take a derivative of a single tangent vector, just like you can't compute the derivative of a function if you just know its value at a single point.--gwaihir 22:09, 12 February 2006 (UTC)

Covariant derivative#Notes reads: "The vectors u and v in the definition are defined at the same point p. Also the covariant derivative $\nabla _{\mathbf {v} }{\mathbf {u} }$ is a vector defined at p." which I translate as $\nabla (p):T_{p}M\to \{A:T_{p}M\to T_{p}M\}$.

Note that you can define a derivative in, for example, a polynomial ring, purely algebraicaly, without using any limit or even topology or metric.

Of corse, the covariant derivative wouldn't make sense if there would be no way to link a point $p\in M$ to a point $p+{\rm {d}}p$ at a distance $|{\rm {d}}p|$ but the later expressions are generally undefined on a manifold, so the basic approach is to pick a local chart, glue charts together and to show that the result is independent of the choice of the local charts. But then, if I have to use local charts anyway, the notation of a covariant derivative wouldn't be much of a profit, so I guess (and hope) there's some way around. That's what I'm looking for. The Infidel 18:09, 13 February 2006 (UTC)

The quoted sentence is indeed misleading. But, if you prefer to think about polynomials: you can't determine $p'(0)$ if you only know the absolute term, i.e., $p(0)$.

Concerning your general question: You need a covariant derivative because the naïve derivative of vector fields in the form $\partial _{i}(f\partial _{j})=(\partial _{i}f)\partial _{j}$ depends on your choice of a local chart. If you take the derivative of a function, there is no problem, $\nabla _{X}f=Xf$ is independent of the metric. But differentiating a vector field requires to somehow compare tangent vectors at different points, and this is what a connection/covariant derivative does.--gwaihir 01:32, 14 February 2006 (UTC)

You say "But differentiating a vector field requires to somehow compare tangent vectors at different points, and this is what a connection/covariant derivative does." and I'm totally with you, but I don't see how, exactly, this works. From your comments I could find some "minor" details in the article (the kind that are minor when you understand but fatal if you miss them). Now I'm on a track but I don't know if it's only for my personal understanding or of a general interest. So let's call it a day for now. Thank you. The Infidel 22:06, 14 February 2006 (UTC)

February 13

Area of a rectangle?

If My living is 24 feet long by 20 feet wide. How much carpet must I buy to cover the entire floor? (how much is that in square feet?)

Is this a trick question? I'd say just multiply the length and the width... —Keenan Pepper 22:39, 13 February 2006 (UTC)

Dont joke thats the easiest question or is it a trick?
Anyway if its not a trick the answer is 480 square feet61.17.48.173 09:35, 14 February 2006 (UTC)

number with 98 zeroes

My son wants to know the name of a number with 98 zeroes. For example, a 6 followed by 98 zeroes.

Thanks, Dan—The preceding unsigned comment was added by 66.19.241.133 (talk • contribs) 23:42, 13 February 2006 (UTC)

By Names of large numbers, 10⁹³ could be called a trigintillion, so 6×10⁹⁸ (a 6 followed by 98 zeros) could be called "six hundred untrigintillion". Of course, the issue is different in long scale naming, where the number's name should be "six hundred sedecillion" or "six hundred sexdecillion". However, almost nobody knows or uses these words, so the number is usually just called "six times ten to the ninety-eight". Hope that helps, Kusma (討論) 00:12, 14 February 2006 (UTC)

By Names of large numbers, 10⁹⁶ could be called a 'sexdecillion (sedecillion)', so 10⁹⁸ could be called 100 sexdecillion, and 6 followed by 98 zeros could be called 600 sexdecillion.

A google divided by 100. hydnjo talk 00:23, 14 February 2006 (UTC)

A centigoogol? Thus, in your example, 6 centigoogols. By the way, it's not spelt "google". —Blotwell 02:30, 14 February 2006 (UTC)

It is now ;-) --Deville (Talk) 02:18, 15 February 2006 (UTC)

Haha, a google...that's hilarious --JianLi 20:30, 15 February 2006 (UTC)

February 14

Ohnoes

${TheMathDied}$ : ( — Ilyanep (Talk) 00:34, 14 February 2006 (UTC) (carry on and ignore this)

Reverse digits squaring

O...K... I'll just ignore that above section Ilyanep ;) Anyway, my question relates to something I was thinking about recently. $12^{2}=144$, right? Now reverse the digits of the numbers on both sides of this statement to get $21^{2}=441$, a statement which is also true. This also works with $13^{2}=169$ and $31^{2}=961$. My question is if there is a mathematical reason for this or is it just complete coincidence? -- Daverocks (talk) 11:11, 14 February 2006 (UTC)

• It should work for any two-digit number which does not require carrying to square it, and you've hit on the only two. It comes from the fact that $(x+y)^{2}=x^{2}+2xy+y^{2}$. In $12^{2}$, x=10 and y=2, and in $21^{2}$, x=20 and y=1. Each term in the expansion contributes to only one digit in the answer (in the case of 12, 13, 21, 31), so that's how that works. (ESkog)^(Talk) 11:56, 14 February 2006 (UTC)

You can represent a 2-digit number as $(10a+b)$. Then $(10a+b)^{2}=100a^{2}+20ab+b^{2}$, or $a^{2}\times 10^{2}+2ab\times 10^{1}+b^{2}\times 10^{0}$. This is of course just the positional decimal representation of the number. Reversibility occurs if there are no carries; i.e. if $2ab<10$, $a^{2}$, and $b^{2}<10$. (Darn, got beaten to it :-) Fredrik Johansson - talk - contribs 11:59, 14 February 2006 (UTC)

Ahh, I get it now, thanks. No carrying means the numbers are preserved when reversed. It works OK with $20^{2}=400$ and $30^{2}=900$ but when we get to $40^{2}=1600$, it's ruined. :P -- Daverocks (talk) 08:36, 15 February 2006 (UTC)

In fact it's not the only two cases. Not counting 10, 20, and 30, there're also the cases $11^{2}=121$ and $22^{2}=484$ which are the same when reversed (as compared to $33^{2}=1089$ which is not true reversed: $33^{2}\neq 9801$. There are also examples with more than two digits, like $1121^{2}=1256641,1211^{2}=1466521$. – b_jonas 10:52, 15 February 2006 (UTC)

There are many solutions, the main first few are 1 2 3 11 12 13 21 22 31 101 102 103 111 112 113 121 122 201 202 211 212 221 301 311 1001 1002 1003 1011 1012 1013 1021 1022 1031 1101 1102 1103 1111 1112 1113 1121 1122 1201 1202 1211 1212 1301 2001 2002 2011 2012 2021 2022 2101 2102 2111 2121 2201 2202 2211 3001 3011 3101 3111 10001 10002 10003 10011 10012 10013 10021 10022 10031 10101 10102 10103 10111 10112 10113 10121 10122 10201 10202 10211 10212 10221 11001 11002 11003 11011 11012 11013 11021 11022 11031 11101 11102 11103 11111 11112 11113 11121 11122 11201 11202 11211 12001 12002 12011 12012 12101 12102 12111 12201 12202 13001 13011 20001 20002 20011 20012 20021 20022 20101 20102 20111 20112 20121 20122 20201 20211 20221 21001 21002 21011 21021 21101 21102 21111 21201 22001 22002 22011 22101 22102 22111 30001 30011 30101 30111 31001 31011 31101 31111.

Discussed there at OEIS : http://www.research.att.com/~njas/sequences/A085305 --DLL 21:18, 17 February 2006 (UTC)

Wow, thanks! -- Daverocks (talk) 01:39, 18 February 2006 (UTC)

Trigonometry/Golden Time on a clock face

At the Wikipedia location "en.wikipedia.org/wiki/10:08" the claim is made that the position of the hands of a clock achieves exact symmetry at 10:09:13.8". The article also claims that "if a rectangle is drawn inside the circle touching where the hands are pointing at 10:08, this will approximate the Golden rectangle...".

Can someone show me the detailed mathematical calculations that would substantiate those claims?

Thanks. Don

Let me just do the first part:

  Hour hand revolutions = 
    [10 + 9/60 + 13.8/3600 ] / 12 = 0.84615277777777 revolutions

  Minute hand revolutions =
    [     9    + 13.8/60   ] / 60 = 0.15383333333333 revolutions

Note that those two values add up to one revolution. You can convert them to degrees by multiplying by 360. This gives angles of 55.385 degree and 304.615 degree, which are both 55.385 degrees on either side of 12 oclock. StuRat 00:33, 15 February 2006 (UTC)

• • • Very helpful! I think I see now how I could generalize the problem and find all other times where both the minute and hour hands are equidistant from 12:00. [Don]

As for the second part, take the rectangle and bisect it along its diagonal. you can work out that its angles are about 90°, 34.6° and 55.4° (based on the 10:09.138 position), and taking tan 55.4° gives you 1.45 which is a pretty poor approximation of the Golden ratio. However, the golden rectangle has angles of 58.3° and 31.7° so if you look just at the angles and not the trigonometric ratios it's not too bad. (By my quick calculations, the Golden rectangle is formed around 9:04.50.) Confusing Manifestation 05:14, 16 February 2006 (UTC)

• • • How did you arrive at the 90-34.6-55.4 degree angles? By my calculations, 10:10:17 is "golden time" and is a little closer to "symmetry time" than the article's claim of 10:08. [Don]

Pi in base 6???

My friend is an alien who crash-landed on this boring planet. Being no scientist-alien, he needs help reprograming the hyperdrive. Unfortunately, math on his planet is in base-6, so I need to know pi for the hyper-dimensional geometery warping. Please help!!! I don't want my alien friend to materialize inside of a neutron star!

It begins 3.0503300514151241052344140531253211023012144420041152 525533142033313113553513123345533410015154344401... Wow, I never imagined my arbitrary-base pi program would see any practical use. And possibly saving life, too! Let me know if your friend needs more digits. - Fredrik Johansson - talk - contribs 19:59, 14 February 2006 (UTC)

hmm, interesting to see 1415 early in there, but that's just an interesting coincidence that's probably statistically not too unlikely (I know it's certainly not something that would be useful mathematically) Confusing Manifestation 06:18, 15 February 2006 (UTC)

While a quite exciting problem, this sounds quite like a homework problem :\. Fredrik...what language is your program written in? — Ilyanep (Talk) 00:39, 15 February 2006 (UTC)

See http://fredrikj.net/code/pie.py (Python) - Fredrik Johansson - talk - contribs 13:39, 15 February 2006 (UTC)

How's $\pi$ on base $e$? :) ☢ Ҡi∊ff⌇↯ 13:10, 15 February 2006 (UTC)

Hmm, my program doesn't do non-integer bases. Maybe that'll be a future project ;-) Fredrik Johansson - talk - contribs 13:39, 15 February 2006 (UTC)

How would you go about setting up this calculation if you had to do it by hand? -LambaJan 09:19, 18 February 2006 (UTC)

This discussion has got me interested. Is there a website where can I find pi in various bases? I would use the script above, but I don't know how to run it.--Bjwebb (talk) 10:46, 18 February 2006 (UTC)

You need Python. Anyway; 10000 digits of π in bases 2-36. Fredrik Johansson - talk - contribs 11:00, 18 February 2006 (UTC)

LambaJan: if you want to do the calculation yourself, there are two ways. The easy way is to find the value of pi for sufficent precision written up in base 10 (or any other base) somewhere, and then convert that to base 6. The hard way is to use a series like

$\pi =\sum _{0\leq n}{\frac {1}{16^{n}}}{\bigg (}{\frac {4}{8n+1}}-{\frac {2}{8n+4}}-{\frac {1}{8n+5}}-{\frac {1}{8n+6}}{\bigg )}$

and do the calculation yourself in base 6 (or any other base and convert it to base 6 afterwards). – b_jonas 18:58, 18 February 2006 (UTC)

Thank you. -LambaJan 22:12, 18 February 2006 (UTC)

February 15

Sine Curve

How does one calculate the curve length of a sine curve? That is, if you used a string to trace the sine curve, say, from 0 to 2π, what would the length of the string be when you stretched it out? And would there be any practical use, in physics for example, for this length function? --JianLi 20:35, 15 February 2006 (UTC)

I suppose one application is to know how much ink is needed to plot sine waves, say when charting the tides. StuRat 22:00, 15 February 2006 (UTC)

How you do this in general is at Length of an arc. Unfortunately the result in this case is an elliptic integral, so it is not possible to express it in elementary functions. Kusma (討論) 22:39, 15 February 2006 (UTC)

The way you compute the length of a curve is as follows. Let's say that the curve is the graph of a function, such as y = f(x). Then as you move along this curve, the tangent vector to this curve can be written as (1,f'(t)). The length of this vector is ${\sqrt {1+(f'(t))^{2}}}$, so we integrate this for all t in the curve. In this case, then, we compute

$\int _{0}^{2\pi }{\sqrt {1+\cos ^{2}(t)}}\,dt\approx 7.6404.$

--Deville (Talk) 22:30, 15 February 2006 (UTC)

Thanks --JianLi 00:30, 16 February 2006 (UTC)

Estimating the area under the curve

To estimate the area under a curve, one may use the trapezoidal method or the midpoint method. And since the trapezoidal method has twice as much error as the midpoint method (or maybe it was the other way around, I forget), to find the exact area under the curve, this method: (Trapezoidal Area) + 2(Midpoint Area) / 3 What is the name of this method?

I heard that this method only works for polynomial functions below a certain power (6, I think). Why is that? --JianLi 20:53, 15 February 2006 (UTC)

For a first degree curve (a line) any method will give exact results, even with only a single interval used. For any general curve, you will need an infinite number of intervals, or in other words, you will need to use calculus. StuRat 21:57, 15 February 2006 (UTC)

The method is called Simpson's rule, and it is exact for polynomials of degree three (or less). -- Jitse Niesen (talk) 22:03, 15 February 2006 (UTC)

Thanks, that's the one. The formula given in the article

$\int _{a}^{b}f(x)\,dx\approx \int _{a}^{b}P(x)\,dx={\frac {b-a}{6}}\left[f(a)+4f\left({\frac {a+b}{2}}\right)+f(b)\right].$

is equivalent to the "trapezoid and midpoint" formula above, but it's unusual that neither the wikipedia article nor the wolfram article make mention of the "trapezoid and midpoint" method. --JianLi 00:28, 16 February 2006 (UTC)

This question is best left for MathWorld. Look up Simpsons rule as a specific form of Newton -Cotes approximation formula. The formula '(Trapezoidal Area) + 2(Midpoint Area)/ 3' is an expmale of using a averging way of minimizing the error due to Simpsons rule and the second order Newton-Cotes formula. The error analisys might be a bit harder, but it can be done. It will work for almost all continious polynomials of any degree, but the error migth be large if the degree exceeds the approximation interval, and has many changes in curvature in the interval. The approximation becomes usless with Weinerstrauss monsters. ( continous everwhere, diffrentiable nowhere). --67.160.212.217 02:35, 17 February 2006 (UTC)

Perhaps this is as good a time as any to point out that numerical analysts have developed a broad range of integration methods. Which to choose is part theory, part experiment. Some suprisingly efficient methods, such as Gaussian quadrature, use evaluation at carefully chosen points. Romberg's method is powerful for more general situations. And specifics of the target function can be vital in choosing an appropriate method, especially when the function has troublesome features. The numerical integration article has a brief discussion, but for such a heavily-studied domain the expert literature (or a live expert, if available) is best. --KSmrq^T 02:59, 17 February 2006 (UTC)

February 16

Miniature Golf- Obstacle at an Angle

I have to design a hole-in-one using a bank shot at several angles. I know that angle of incidence = angle of reflection, but is that true when the obstacle is at an angle besides 90 degrees to the horizontal, such as 60, 45,30, 50,etc.

Here's a simple diagram:

                                                 \   ( Angle of Reflection) 
                                                   \
                                                     \                                   /
                                                        \                             /                       (Obstacle- 45 degrees to parallel wall)
                                                           \                      /
                                                             \                /
                                                               \          /
                                                                 \    /

---

--------------------------/----------------------------------------------------(Imaginary Horizontal Parallel to Wall)

                                                               /  
                                                            /
(Angle of Incidence)                           /
                                                      /    
                                                   /
                                                / 
                                               (Start)

Thank you for your help! Please answer if possible as soon as possible?! Go Wikipedia!!!!!!! You may have to use physics, geometry,etc. For the purposes of a middle/ high school project for now I'm assuming under my teacher's directions that there isn't friction and no energy is lost. Signed, Sarepr91

The article specular reflection has a nice diagram of what you're going for. The angle the surface is at relative to something else is irrelevant, since what's being measured is the angle between the normal from the surface and the trajectory of the ball. BTW, I think you're also assuming the ball has no spin. Black Carrot 02:06, 16 February 2006 (UTC)

That's right, only the angle between the ball and the wall (where it hits) matters. StuRat 05:08, 16 February 2006 (UTC)

-sarepr91: my teacher says that the angle between the obstacle and the wall matters and i have to state the theory behind that! please help tell me some resources!! Thanx for your answers by the way.

Perhaps there's something about the situation I missed. As I understand it, you're asking about a ball rolling along a flat horizontal surface, bouncing off something (which is at a known angle to the ball's trajectory), and rolling away along the flat surface, in which case the simple laws of reflection should be enough. Does the ball bounce up into the air? And what's the difference between the 'wall' and the 'obstacle'? Black Carrot 03:36, 17 February 2006 (UTC)

The terms "obstacle" and "wall" are confusing, because in common use these can be synonyms, different names for the same thing. Let us assume that a ball is rolling along a flat, level surface; a billiard table would be an ideal example. It then hits a vertical obstacle, colliding with a velocity that can be separated into two perpendicular components: the velocity heading parallel to the obstacle and that heading straight into it. The bounce reverses the direction of the latter, leaving the parallel component the same. With no friction or other losses this produces the "angle of incidence equals angle of reflection" rule. A popular graphical approach treats the line of travel as going straight through the obstacle with no deflection, as drawn on a piece of paper; but then the paper is folded along the line through the obstacle to simulate the mirror effect. (In sophisticated technical terms this demonstrates that the path satisfies a minimization principle.) The crude ASCII diagram above is also little help; a PNG image would be clearer.

As specific examples: A ball heading straight at the obstacle reverses direction and comes staight back out. A ball rolling along the obstacle continues unchanged. A ball hitting at a small, grazing, angle is barely affected and comes out at a small angle. --KSmrq^T 04:24, 17 February 2006 (UTC)

I don't see why you can't draw a better diagram even with ASCII text, like this:

+---------------------------------------------------+
|                                                   |
+                        /\                         |
 o                      <  >                    O   |
+                        \/                         |
|                                                   |
+---------------------------------------------------+

StuRat 06:28, 17 February 2006 (UTC)

multiplication of negative integers

why multiplication of two negative integers is positive?

for example
   -2*-2 = 4.

mani

Because multiplying anything by a negative number flips it's sign. So, if you start with a negative and multiply by a negative, that flips it's sign to positive. StuRat 06:29, 16 February 2006 (UTC)

We attempt to explain this in the negative number article, in the section on multiplication. Does it answer your question adequately? If not, tell us why not so we can improve the article. —Blotwell 09:36, 16 February 2006 (UTC)

Halloween comes. You decide to give two pence to each child ringing at your door.

A bunch rings, but two of them won't take the money.

Count -2 p per retributed child in your balance. Count -2 children, multiply  : you won 4 p. For US people, the same goes with dimes. --DLL 22:35, 19 February 2006 (UTC)

Same would go with pennies. zappa 16:37, 22 February 2006 (UTC)

The multiplication section of the negative number article has just been rewritten to help people understand this question, so you might want to re-read that section to see if it helps now. -R. S. Shaw 22:00, 20 February 2006 (UTC)

A partial differential equation

Hello Ref desk! I am going to do a calculus test tomorrow, and I am having trouble understanding one of the exercise questions that have been handed out to help us prepare for the test. I guess it's a bit late, but could you help me how to solve it? The problem is:

Decide the general solution to the partial differential equation

$y{\frac {\partial f}{\partial x}}-x{\frac {\partial f}{\partial y}}=0,x>0,y>0$

by introducing the variables s = x² + y², t = x² - y².

I guess I shouldn't have skipped so many lectures...

/130.238.41.167 09:08, 16 February 2006 (UTC)

• To get started, figure out x and y in terms of s and t. Figure out the derivatives. Now re-express the equation using s and t, and using the chain rule. That should get you started. --BluePlatypus 17:37, 16 February 2006 (UTC)

• By visual inspection, the solution is $f(x,y)=C\left(x^{2}+y^{2}\right)+D$, where C and D are constant. I've long forgotten the standard procedure. So is "beslis" the word for "solve" in Dutch ? (Unsigned comment by User:deeptrivia)

"beslis" is "to decide" (or determine) in Dutch. (Compare "besluit" ("decision"), German "beschluss") Then there's "bewijs" ("proof"). --BluePlatypus 16:09, 17 February 2006 (UTC)

Thanks for the answer! I was a bit puzzled at first (what derivatives? I thought) but then I realised how the chain rule could be useful. Unfortunately, it didn't help me on the test... /217.208.26.85 22:44, 17 February 2006 (UTC)

Big Bang

I already asked this on the humanities section on a question about Hume, but I wanted to ask a mathematician if our math does indeed 'break down' seconds after the BB, during it and 'before' it. ...does it fail? break down, or just don't make sense?.and is it known what kind of math is there then? or is there no math? or is there some 'irrational' math?.--Cosmic girl 16:48, 16 February 2006 (UTC)

It's the physics that break down, not the math. --BluePlatypus 17:01, 16 February 2006 (UTC)

thanks :) ..Kainaw told me they do break down though.--Cosmic girl 17:49, 16 February 2006 (UTC)

I agree with BluePlatypus. I suspect that Kainaw meant to say that "the mathematical results of the current physical laws no longer give a satisfactory explanation of the behavior of the universe at times very close to the Big Bang". StuRat 22:20, 16 February 2006 (UTC)

cool, thanks :) ...hey, sorry to be so annoying...but is there any possibility that our MATH (meaning our inbedded human logic...since I believe we can't know if math has any meaning beyond our minds) can be different or not exist or be 'irrational' (for us) beyond or 'outside' the Big Bang? there IS a possibility , right? (a really wacky, useless and absurd possibility but a possibility still)

Oh and also! (I'm annoying I know...but I've nowhere else to ask)WHY do physics 'fail' when 'explaining infinity' or when aproaching infinity?( I read it somewhere but I don't get it)--Cosmic girl 17:19, 17 February 2006 (UTC)

Is the moon still there when nobody is looking?

Does a falling tree make a sound if nobody is listening?

Does god vanish in a puff of logic when nobody is believing?

No one can ever know for sure, but if your answer to any one of the above questions is no (or yes, in case of the third), so why should math or logic vanish if nobody is thinking?

Your second question is less profound yet far more difficult to answer. There are at least as many reasons why the modeling of the world by mathematical formulas (that's physics) fails when any term of theses formulas aproaches infinity as there are types of infinity.

In the context of big bang, there are huge concentrations of energy involved, and we simply cannot, by any experimentation, know what might happen there.

The Infidel 20:02, 17 February 2006 (UTC)

I didn't mean that math 'vanishes' when noone is thinking... I asked IF it's possible that math can be different from the math we have in the reality beyond the BB(if there is such a thing) but I know the difficulty of this question so I'll just stop asking it. =P. --Cosmic girl 21:39, 17 February 2006 (UTC)

The question "why is mathematics useful in describing nature?" is indeed one of the main questions asked by philosophers of mathematics, see the article for some different answers or attempted answers. An especially interesting discussion is given by Eugene Wigner in his paper The Unreasonable Effectiveness of Mathematics in the Natural Sciences. /217.208.26.85 22:58, 17 February 2006 (UTC)

Math, as such, is independent of any reality, it's a kind of transcendent truth. Of corse, what is known of math to any thinking entities afar from our culture can and will differ widely. Our math was largely guided by the needs of machanics. It probably would appear very different if electronics would have been the driving force. There would be different notations, theorems would be stated in a different way, some known results would be unknown due to lack of interest, others, unknown to us, would be well known. But they are valid and true, known or unknown, stated or unstated.

If that's not what your question was about, ask again. The Infidel 23:12, 17 February 2006 (UTC)

Ok, I see your point, but what I meant to ask was more like is 1=1, 2=2 ... (meaning 'rational') beyond our existence...It may seem obvious that it is so, even beyond the big bang, but I'm asking if I'm the only one that sees the possibility of math being irrational (meaning 3=10 or something)beyond the BB...I know it's crazy,and there's no way of knowing it but what does the philosophy of mathematics say about this?does it admit the probability? I don't understand the article very well, it's too technical.--Cosmic girl 23:21, 17 February 2006 (UTC)

Math is characterized as being rational. What you have in mind (3=10 "or something") would not be math, would not be rationally linked to other statements.

Another aspect is: is it possible to have a kind of logic that is so strange that we never can think of it? That question is undecidable.

The Infidel 13:30, 18 February 2006 (UTC)

Exaclty! it wouldnt be 'math' anymore since math is rational...agree, and you rephrased me better, I meant to ask if there can be some sort of 'logic' that would be ilogic for us, or inconcievalbe. thanx...I wasn't aware that that's how I should've phrased the question...maybe at the humanities reference desk.--Cosmic girl 19:11, 18 February 2006 (UTC)

It's probably covered in that philosophy of mathematics article linked above, but that's the kind of question that does divide the mathematical community to some extent, with some of the different sides believing that mathematics is a fundamental thing that is the same approached from any angle (so, for example, alien mathematicians are guaranteed to have similar mathematics to us, just with different names and maybe different foci), some that believe that we only see the maths that happens to explain things best - so there may be plenty of other mathematical systems out there but we'll only figure out the ones that work into our own section of the universe, and that it's just some crazy stuff that happens to work kind of well in approximating the universe. Confusing Manifestation 14:55, 18 February 2006 (UTC)

Actually I agree with the latter...but maybe the former are right...that's the nice thing, whatever. =P--Cosmic girl 19:11, 18 February 2006 (UTC)

I haven't seen the answer here, but I have seen it in other questions on the reference desk. Math works because it starts with a set of axioms and then proves results based on those. The axioms determine the results so given the same set of axioms, the results will be the same. If you choose a different set of axioms things will not be the same. Another simpler way of saying it is math works because we pick the starting rules of the game. The philophical questions come in when you try to decide if the chosen starting rules end up with a system that describes reality accurately. If they don't the system is still consistent, just not "useful" in the common sense. - Taxman ^Talk 17:42, 20 February 2006 (UTC)

How can it be consistent if it doesn't explain reality?--Cosmic girl 18:26, 20 February 2006 (UTC)

Looks like you still need a little clarification... Mathematics isn't about reality, mathematics doesn't need to explain reality, mathematics doesn't try to explain reality, any connection between mathematics and reality is completely coincidental, no mathematical dogs were hurt during the production of reality. Mathematics is about abstract objects, statements regarding such objects and connections between such statements. It is about starting from an arbitary set of axioms and logically deducing conclusions from these axioms. Any set of axioms is valid, although some have special properties which make them more worthwhile to explore. According to one approach, our physical reality is actually a specific set of axioms. So of all the sets of axioms which could be dicussed, only one (or several equivalent ones) describes the physical universe. Any specific details of this specific universe, such as the big bang or whatever, have no implications whatsoever on the discussion of other structures. So there is essentially one axiom system that describes reality, but that doesn't make other systems inconsistent, "bad", uninteresting or whatever. -- Meni Rosenfeld (talk) 18:51, 20 February 2006 (UTC)

The numbers in a balance sheet of a company can be consistent with each other, but still not with reality. The Infidel 18:51, 20 February 2006 (UTC)

[After edit conflict] Hmm, I thought that was self evident, but that just shows different people see things differently and I can't explain myself as well as I thought! Here's an example, see Banach-Tarski paradox. The math used to get that result is consistent, because given the same set of assumptions the same result will always occur. But it doesn't reflect physical reality as we know it. The rest of math can work the same way. It can be internally consistent in that the conclusions follow the precedents, but those conclusions don't have to match observation. If that doesn't work, I'll wait until someone else can explain better. - Taxman ^Talk 18:57, 20 February 2006 (UTC)

February 17

"amount in millions"

Stupid question: when something says 2,553 (amount in millions), what does that mean? 2,553,000,000 or 2,553,000?

In the USA, it means 2,553,000,000. In other countries, including the UK, it may mean 2,553,000. Please tell us what country you are asking this question from. zappa 16:41, 22 February 2006 (UTC)

length of months of indian solar calendars

Please kindly tell me the precise interval of time between two samkrantis of various months in indian solar calendar (exactly the Vikram Sambat).What is the phylosophy behind the mathematics of Vikram sambat.

I don't have an answer, but this question would be more appropriate for the humanities reference desk. —Keenan Pepper 21:22, 18 February 2006 (UTC)

triple integral

someone is going to probably ask a question about a triple integral later today--205.188.116.136 20:21, 17 February 2006 (UTC)

Yeah, probably someone did, somewhere. Not here though. --Trovatore 06:52, 18 February 2006 (UTC)

You are both correct. I asked my Gran a question about Triple integrals just now but she didn't reply - Adrian Pingstone 10:28, 18 February 2006 (UTC)

You are in the same school with the same teacher and you think that your friend dares to ask. Why don't you ? Don' be scared by a "homework to do" answer, just ask very naturally "I learned a lot about triple integrals but there is something that I would like to discuss ..." --DLL 22:25, 19 February 2006 (UTC)

semiprimes

The probability that a number is prime is 1/ln(P), is there a similar result for semiprimes, what if we only allow square free numbers. Can this be extended to 3-almost, 4-almost..... Ozone 05:49, 18 February 2006 (UTC)

See http://mathworld.wolfram.com/Semiprime.html. --DLL 22:19, 19 February 2006 (UTC)

Thanks but that doesn't show a simple result like 1/ln(P). That page sadly does not tell you the asymtotic density. (-Ozone-)

Maybe it is not that simple. We cannot just tell M. Weisstein to make it simpler. try OEIS ?

There ougtha be more comprehensive as well as didactic maths sites. Didactic ones are not comprehensive. Let's try here. --DLL 22:28, 21 February 2006 (UTC)

February 19

The length of a curve when the metric is not positive semidefinite

What is the length of a curve when the metric is not positive semidefinite? I suppose it's

$L=\int _{a}^{b}{\sqrt {\left|\sum _{i,j}a_{ij}{dx^{i} \over dt}{dx^{j} \over dt}\right|\,}}dt\$

but I can't find any source for this definition. The Infidel 11:45, 19 February 2006 (UTC)

It may depend on your application, but I don't see that the length is well-defined unless the context allows you to make a meaningful choice of branch of the square root function in a consistent way. For example, if the rootand is positive or negative consistently over the whole curve ("spacelike" or "timelike" throughout, in Minkowski space terminology) you're sorted (though the length comes out imaginary in one case). —Blotwell 06:44, 20 February 2006 (UTC)

Is this a mathematical definition or is it just what physic students usually do? I know that in engineering, if a length turns out negative they often flip sign without comment (yet many of the houses do not collapse ;-). The Infidel 19:27, 20 February 2006 (UTC)

find the inverse function

Inverse function <=== found it! Dmharvey 15:57, 19 February 2006 (UTC)

LMAO nice one -zappa 16:42, 22 February 2006 (UTC)

Pythagorean-like problem

Find the sum of all $0\leq a\leq 100$ such that

${\frac {1}{a^{2}}}+{\frac {1}{b^{2}}}={\frac {1}{c^{2}}}$

where

$a,b,c\in \mathbb {Z}$

---

I am wondering if there is a way of solving the problem above without enumerating all possible Pythagorean triples in the range, since this problem was taken out of a math competition.

Thanks.

206.172.66.43 17:34, 19 February 2006 (UTC)

Are you thinking this is the same as the Pythagorean problem?

$a^{2}+b^{2}=c^{2}\,\!$

On the contrary, the left side is

${\frac {a^{2}+b^{2}}{a^{2}b^{2}}},$

and we can reciprocate both sides and clear the denominator to get

$a^{2}b^{2}=c^{2}(a^{2}+b^{2})\,\!.$

Clearly it is true that a²+b² must be a perfect square, but the problem is more subtle than that. --KSmrq^T 20:51, 19 February 2006 (UTC)

It's precisely Pythagorean triples $(a,b,d)$ satisfying $d\mid ab$. Let $(a,b,d)=t(a',b',d')$ with (pairwise) coprime $(a',b',d')$. It follows that $d'\mid t$, hence it is essentially sufficient to consider relatively coprime Pythagorean triples $(a',b',d')$ satisfying $a'd'\leq 100$, which are $(3,4,5)$ and $(5,12,13)$.--gwaihir 21:30, 19 February 2006 (UTC)

gwaihir's clue was the final piece of the puzzle I needed to nail this problem. In fact I had hit upon the idea of $d\mid ab$ as a criteria for generating the triples for the original equation. But I was unable to prove that multiples of basic Pythagorean triples where $a'd'\leq 100$ were all that's needed. Essentially, my mental block was on the fact that either the numbers in a Pythagorean triple are all coprime with each other or all three share the same common factor. I kept on trying to find the case where two number out of the three had a common factor that was not divisible by the third, turned out this is impossible, because for a Pythagorean triple $(a,b,c)$:

Assume that $c$ has no common factor with $a$ or $b$, but $a$ and $b$ have a greatest common factor, $d$, that is greater than 1. Then

$c^{2}=a^{2}+b^{2}=(a'd)^{2}+(b'd)^{2}=a'^{2}d^{2}+b'^{2}d^{2}=d^{2}(a'^{2}+b'^{2})$

Clearly, $c$ is also divisible by $d'$, which is a contradiction with the original assumption that $c$ has no common factor with $a$ or $b$.

Assume that $a$ has no common factor with $b$ or $c$, but $b$ and $c$ have a greatest common factor, $d$, that is greater than 1. Then

$c^{2}=a^{2}+b^{2}\Rightarrow a^{2}=c^{2}-b^{2}=(c'd)^{2}+(b'd)^{2}=c'^{2}d^{2}+b'^{2}d^{2}=d^{2}(c'^{2}+b'^{2})$

Clearly, $a$ is also divisible by $d'$, which is a contradiction with the original assumption that $a$ has no common factor with $b$ or $c$.

Symmetrically, it could be shown that it is impossible for $a$ and $c$ to share a greater than 1 common factor that is not divisible by $b$.

Hence, for Pythagorean triples, either all three numbers have a greater-than-one common factor or they are pairwise coprime, that means $gcd(a,b)=1$ and $gcd(b,c)=1$ and $gcd(c,a)=1$ all at the same time! There are no other situations possible in terms of common factors.

With this proof, along with gwaihir's clue, it is not hard to figure out the following possible values for $a$:

15, 30, 45, 60, 75, 90	for the (3, 4, 5) case
65	for the (5, 12, 13) case

Thus, the answer to the original question (the sum of all such $a$) is: 350.

Hurrrrrrrrrrrah, thank you so much for your help, gwaihir and everyone else!

129.97.252.63 05:03, 21 February 2006 (UTC)

My addition of the above numbers gives 380, but more importantly, you forgot the (4,3,5) case. Minor details about counting: For a = 60, there are two solutions: should 60 be counted twice? If yes, what about negative b's and c's?--gwaihir 10:41, 21 February 2006 (UTC)

Waaaaaaaaaaaaah, I'm stupid, I'm stupid. 129.97.252.63 21:04, 21 February 2006 (UTC)

---

OK, using gwaihirs ideas, if a,b have a common factor p we can reduce it as $p^{4}a'^{2}b'^{2}=c^{2}(p^{2}a'^{2}+p^{2}b'^{2})$ with c = pc', a'p=a, b'p=b to $a'^{2}b'^{2}=c'^{2}(a'^{2}+b'^{2})$, so we need only to consider coprime a, b.

If p is a prime factor with p^n dividing a (but not b, as they are coprime), the $c^{2}b^{2}=0(mod\,p^{2n})$ so p^n|c as $b^{2}\neq 0(mod\,p)$. This holds for every p^n| a, so a divides c, there exists u such that au =c. We can rewrite $a^{2}b^{2}=a^{2}u^{2}(a^{2}+b^{2})$ so $b^{2}=u^{2}(a^{2}+b^{2})$ and as all numbers are integers with $u\geq 1$, we have b^2 > b^2 as a contradiction, no solutions exist, the sum is zero.

The Infidel 22:36, 19 February 2006 (UTC)

(15, 20, 12) is a solution.--gwaihir 23:00, 19 February 2006 (UTC)

Yep. JackofOz 23:25, 19 February 2006 (UTC)

Ouch, you'r right. I've lost solutions in the first step when reducing by a common factor. c =pc' is only on possibility, the other is p| a'^2+b'^2. The Infidel 18:59, 20 February 2006 (UTC)

Statistics: probability

I knew I shouldn't have decided to major in a degree that requires a statistics class. I acutally have two questions.

1. If given a variable X with a given, discrete sample space S={0, 1, 2, 3, 4, 5}, with accompanying probabilities of each value occurring, how can you verify that that is a legitimate discrete distribution?
2. If given P(A), P(B), and P(A and B), how do you figure P(B not A)?

Thanks, Hermione1980 18:32, 19 February 2006 (UTC)

First: don't look at our article about probability (no joke)

Second: think of the samples as boxes labled 0 to 5 with different weights that add up to 1.

Third: play around with subsets of these, looking at the total weight

Forth: look at the article measure theory. A probability measure is just a measure where the total set of samples has measure 1.

The Infidel 19:59, 19 February 2006 (UTC)

For question 2, the inclusion-exclusion principle article might interest you. —Blotwell 06:19, 20 February 2006 (UTC)

For the first: all probablilities must be nonnegative and their sum must be exactly 1. For the second: P(B and not A) = P(B) - P(A and B) or something like that. – b_jonas 17:19, 23 February 2006 (UTC)

February 20

Powerball

Powerball says that the odds of matching just the last number, the Powerball, is 1 in 68.96. The article also says that the Powerball goes from the number 1 to 42. So shouldn't the chance of matching the powerball be around 1 in 42? What explains the discrepancy? zafiroblue05 | Talk 00:57, 20 February 2006 (UTC)

• That's the odds for matching the powerball only. So it's 1 in 42 minus the chance of matching one or more of the white numbers. Perhaps this should be pointed out on Powerball, since it's asked often enough that it shows up on the Powerball FAQ. --jpgordon∇∆∇∆ 03:04, 20 February 2006 (UTC)

Trigonometry question

I want to write a program for a robot. One thing I need is something that allows my robot to move to any given location from its current location. Now, I know how to calculate the distance (difference between x and y coordinates each squared and added then the square root of that number) but I don't know how I would calculate the angle. Taking the arctan of the change in y over x won't always work because arctan (-1/1) /= arctan (1/-1). Is there any way around this?

Of course there's a way around it: just add 180 degrees if x is negative. Many programming languages have a special function set aside for this, for example atan2() in C, but that's just a convenience. —Keenan Pepper 02:50, 20 February 2006 (UTC)

FORTRAN, and I suspect many other languages, also have an ATAN2 function. StuRat 05:52, 20 February 2006 (UTC)

In fact, I needed that atan2 function in BASIC once so I implemented it. Here's the code.

   FUNCTION Atan2 (Y, X)
   
   hpi = 1.570796326794897#
   pi = 2 * hpi
   
   IF ABS(Y) < ABS(X) THEN
    IF X > 0 THEN
     Atan2 = ATN(Y / X)
    ELSE
     Atan2 = ATN(Y / X) + pi
    END IF
   ELSE
    IF Y > 0 THEN
     Atan2 = ATN(-X / Y) + hpi
    ELSE
     Atan2 = ATN(-X / Y) - hpi
    END IF
   END IF
   
   END FUNCTION

– b_jonas

Non-integer numeral-system bases

In some Wikipedia articles I've seen the statement that the most "efficient" (in some unstated way) number base is e (2.718...). This raises some questions:

1. Is it reasonable to discuss a non-integer base?
2. Have such bases been studied?
3. Presuming this means position-implies-power system of notation, doesn't that mean that under an irrational base (like e), no rational number would have a finite expression?
4. For instance, in base e an integer would have to be approximated (to as many e-places as desired). Wouldn't counting in base e be rather problematic?
5. In what way would the base e be "efficient"?

Thanks. -R. S. Shaw 04:30, 20 February 2006 (UTC)

I have a slight feeling that the "base" they're talking about is not the base as in the base you write your numbers in (base 2, base 10), but as the base of an exponential (the e in $e^{x}$, the 10 in $10^{100}$). If you use e as the base of an exponential, it's neater to differentiate for example: ${\frac {d}{dx}}e^{x}=e^{x}$. enochlau (talk) 05:40, 20 February 2006 (UTC)

No, they're talking about e as a positional number system base, as in decimal, binary, and e-ary. One example is Computer numbering formats#Why binary?. -R. S. Shaw 10:02, 20 February 2006 (UTC)

To answer #3, no they wouldn't, because e is irrational transcendental. Now, positional notation numeral systems work on exponentials. The number 352 in base 10 is $3\times 10^{2}+5\times 10^{1}+2\times 10^{0}$. 110 in binary (base 2) is $1\times 2^{2}+1\times 2^{1}+0\times 2^{0}=6_{base\ 10}$. In order to write numbers on base e, you'd use the digits 0, 1, and 2, and you'd have to work out the exponents to match the value you're trying to represent. Now, I'm not aware of how well this would work, but considering all the nice properties e has as an exponential base, and since numerical bases work based on the exponent principle, seems reasonable to think that this base would have some very interesting properties. I can't tell which, though, so I'll leave that to someone else :P. Also, we should create base e eventually. ☢ Ҡi∊ff⌇↯ 06:06, 20 February 2006 (UTC)

Right, the integers 0, 1, and 2 can be expressed finitely. But >2 would not. For instance, 3 would equal 1 × e^1 + 0 × e^0 + 0 × e^-1 + 2 × e^-2 + ... to several e-places, 3 (base 10) = 10.0200112000... (base e). -R. S. Shaw 10:02, 20 February 2006 (UTC)

I think non-terminating representations for integers is due to e being transcendental, not irrational. Integers can have terminating representations in irrational bases - see golden ratio base for example. Gandalf61 15:31, 20 February 2006 (UTC)

Good call. Fixed my statement. ☢ Ҡi∊ff⌇↯ 18:13, 20 February 2006 (UTC)

... or indeed an even better example is base sqrt(2), in which every integer has a finite representation that is created by inserting 0s between the digits of its binary representation so 2 is 100, 3 is 101, 4 is 10000 etc. Gandalf61 17:28, 20 February 2006 (UTC)

Any algebraic number has an associated base in which all integers terminate. There is also a "standard form" which allows only certain sequences of digits, for example the golden ratio base in which more than one 1 in a row is forbidden. —Keenan Pepper 17:58, 20 February 2006 (UTC)

(The forbidden sequences are determined by the polynomial of which the algebraic number is a root. The golden mean is a root of the polynomial $x^{2}-x-1$, so "011" can always be replaced by "100". —Keenan Pepper 18:00, 20 February 2006 (UTC))

Donald Knuth discusses such topics in TAOCP Vol. 2:Seminumerical Algorithms. (See the reference at the end of the positional notation article.) A helpful article can be found online in American Scientist. See also The ternary calculating machine of Thomas Fowler. --KSmrq^T 19:31, 20 February 2006 (UTC)

Thanks for the link to the Hayes essay in American Scientist; it is very helpful wrt question 5. (The "efficiency" is an optimum of a measure of little or no practical significance.) The golden ratio base article answers 1 & 2. Q 3 has been clarified (most integers have only infinite expressions in transcendental bases). And yes, counting in base e would be horrible; the "carry" in the addition 2 + 1 would not only affect the position to the left of the units position, but also an infinite number of places to the right. -R. S. Shaw 22:22, 21 February 2006 (UTC)

Yes such systems are studied, usually in the form of $\sum _{n}q_{n}t^{n}$ where $q_{n}$ is a rational number and t is not. For example, Diophantine analysis and Galois theory when t is an algebraic number. I've seen even crazier things; I wrote rational zeta series on one such crazy system. The point is not so much to "count" in base e, but to study the filligree of how rationals "fit totether" with real numbers. I say "filligree" because the "fit together" part is very fractal-ish with things like crazy near-identities, and etc. Borwein, for example, found a very simple approximation for pi, which is accurate to the first 50 billion digits before it breaks down (!!!!) So take that 22/7! Another, simpler example: over on talk:real number theres a never-ending discussion of 1.00000....=0.99999... These kinds of "accidental" degeneracies between number systems are a lot more interesting in non-integer bases, and you get a much better idea of what the "true meaning" of 1.00000....=0.99999... (and its not what most people say it is). linas 07:06, 22 February 2006 (UTC)

Where can I read more about this Borwein approximation? By the way, there's also $(1+9^{-4^{7\times 6}})^{3^{2^{85}}}$ which gives e to 18457734525360901453873570 decimal places. [2] Fredrik Johansson 07:46, 22 February 2006 (UTC)

That approximation for e is just substituing a very large number in $e=\lim _{n\to \infty }{(1+{\frac {1}{n}})^{n}}$. Nothing special about it, you could just as well have written $(1+999^{-999^{999}})^{999^{999^{999}}}$ for an even better approximation. When you finish doing the actual calculation, I'm not sure the universe will be still there... -- Meni Rosenfeld (talk) 09:00, 22 February 2006 (UTC)

Well, there is one thing special about it: it is almost pandigital (has all decimal algarisms once - except for zero) ☢ Ҡi∊ff⌇↯ 07:09, 23 February 2006 (UTC)

Where would it have gone to? JackofOz 06:20, 23 February 2006 (UTC)

Kieff: Hey, you're right, that is interesting. Alas, This doesn't have anything to do with e really.

JackOfOz: Well, being that I'm not a phsycist, I don't know where the universe will be in $10^{10^{25}}$ years. I'm not sure physicists know that either :-) -- Meni Rosenfeld (talk) 12:09, 23 February 2006 (UTC)

Like, are there any other places? JackofOz 10:35, 24 February 2006 (UTC)

Trajectory of a projectile with air drag

Despite the heading, this isn't a question in physics since I'm actually interested in the solution of a specific (type of) set of differential equations. Suppose a projectile is moving in the x-y plane, experiencing a constant gravitational acceleration of magnitude g in the negative direction of the y axis, and air drag proportional to the square of its velocity. This gives rise to the equations:

${\frac {d^{2}x}{dt^{2}}}=-\alpha {\frac {dx}{dt}}{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}$

${\frac {d^{2}y}{dt^{2}}}=-\alpha {\frac {dy}{dt}}{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}-g$

As far as I know, these equations have no (non-trivial) elementary solution. However, does anyone know a way to represent its solution in some other way, perhaps using non-elementary functions or power series? Since this motion has a singularity point, I think there should be some sort of power series involving t, the time since the point of singularity, and $\theta$, the angle to which the direction of velocity converges in the singularity. Perhaps an expansion in $\left({\frac {t-1}{t+1}}\right)$? How about a way to express a solution in terms of the boundary conditions of location and velocity at a specific time? Any help in the matter would be appreciated. -- Meni Rosenfeld (talk) 16:12, 20 February 2006 (UTC)

OK, what do we know:

1. |x'| is decreasing and eventually reaches 0, where it stays.
2. Once (and while) If x'=0, the solution for y can be represented as one of the following:

$y(t)={\begin{cases}y_{0}+{\frac {\log \cos {\sqrt {\alpha g}}(t_{0}-t)}{\alpha }}&t_{0}-{\frac {\pi }{2{\sqrt {\alpha g}}}}<t\leq t_{0}\\y_{0}-{\frac {\log \operatorname {cosh} \,{\sqrt {\alpha g}}(t-t_{0})}{\alpha }}&t\geq t_{0}\end{cases}}$

$y(t)=y_{0}-{\sqrt {\frac {g}{\alpha }}}t$

$y(t)=y_{0}-{\frac {\log \operatorname {sinh} \,{\sqrt {\alpha g}}(t-t_{0})}{\alpha }}$ where $t>t_{0}+{\frac {\operatorname {arcsinh} 1}{\sqrt {\alpha g}}}$

I'm not sure what you're looking for in the interesting region. Arthur Rubin | (talk) 02:21, 24 February 2006 (UTC)

I don't agree that |x'| reaches 0. It does converge to 0, and the solution in the case x'=0 is indeed simple, but alas, I am looking for an exact solution, so this does not solve the problem. As I said, what I am looking for is a way to represent the exact solution in the entire region, in terms of either non-elementary functions, or an infinite series (preferrably a rapidly converging one). -- Meni Rosenfeld (talk) 06:21, 24 February 2006 (UTC)

Sorry, my mistake. That's still the exact solution for |x'| = 0, and I think I can produce the first few terms of series as t approaches +infinity and the singularity from above, but I don't know about exact solutions. We'll see how the first few terms work out. Arthur Rubin | (talk) 08:13, 24 February 2006 (UTC)

If you know an algorithm that can produce arbitarily many terms of the exact power series, that would be great. I doubt there is a closed form for the nth term, but I could be mistaken. What series do you have in mind? Laurent? -- Meni Rosenfeld (talk) 08:21, 24 February 2006 (UTC)

Approach 1: Expand around the singularity

$x(t)=a_{-1}\log t+a_{0}+a_{1}t+a_{2}t^{2}+\ldots$

$y(t)=b_{-1}\log t+b_{0}+b_{1}t+b_{2}t^{2}+\ldots$

which can be solved to find

$x(t)={\frac {\cos \theta }{\alpha }}\log t+x_{0}+g{\frac {\sin 2\theta }{24}}t^{2}-\alpha g^{2}{\frac {5\cos \theta +4\cos 3\theta }{1440}}t^{4}+\ldots$

$y(t)={\frac {\sin \theta }{\alpha }}\log t+y_{0}-g{\frac {5+\cos 2\theta }{24}}t^{2}-\alpha g^{2}{\frac {3\sin \theta +\sin 3\theta }{360}}t^{4}+\ldots$

Approach 2: Convert to polar coordinates:

$(x',y')=\,(r\cos \phi ,r\sin \phi )$

leaving the equations as

$r'=\,-\alpha r^{2}-g\sin \phi$

$\phi '=-{\frac {g\cos \phi }{r}}$

Approach 3: Expand around t=Infinity.

I can't find the proper form of the expression for the series.

Found it! Write:

$\gamma ={\sqrt {\alpha g}}$

$x(t)=a_{0}(t)+a_{1}(t)e^{-\gamma t}+a_{2}(t)e^{-2\gamma t}+\ldots$

$y(t)=b_{0}(t)+b_{1}(t)e^{-\gamma t}+b_{2}(t)e^{-2\gamma t}+\ldots$

where the a_i are polynomials in t. The first few terms, writing

$\tau =\,\gamma t$

are

$x(t)=x_{0}+{\frac {A}{\alpha }}e^{-\tau }+{\frac {A^{3}}{36\alpha }}(2+12B-\tau )e^{-3\tau }+\ldots$

$y(t)=(y_{0}-{\frac {\tau }{\alpha }})+{\frac {A^{2}}{4\alpha }}(4B-\tau )e^{-2\tau }+{\frac {A^{4}}{256\alpha }}(1+16B+256B^{2}-(4+64B)\tau +8\tau ^{2})e^{-4\tau }+\ldots$

How the parameters x₀, y₀, A, and B propagate under shifting time, and how this may relate to other characteristics of the solution, are left as an exercize for the reader. Note also that if A=0, it makes sense for B to become infinite such that A² B is constant, producing the solution above with x'=0.

Arthur Rubin | (talk) 18:43, 24 February 2006 (UTC) and 20:13, 24 February 2006 (UTC)

February 21

ovaloid : definition (finite projective geometry)

There seems to be little talk about ovals , ovoids and ovaloids on this page. Now before I can help make some pages, I would like to clear something up :

an ovaloid is a set of points, no three collinear, in PG(3,q) that is maximal with this property. Now, when one says maximal, do they mean :no set with this property and more points can be found, or do they mean : it cannpot be extended to a bigger set such that no three are collinear.

There is a subtle difference in definition there am I not right?

Thanks,

evilbu

You are certainly right that there is a difference in definition, although I am too ignorant of finite geometry to say whether the two definitions do not happen to be equivalent in this particular case. However, the word maximal (which makes sense with respect to any partial order) would generally be understood to be with respect to the inclusion order on subsets, so from the sentence you gave I would expect the second of your definitions to be correct. If I were writing the sentence and meant the first definition I would have written something like has maximal cardinality. But again, I'm not a finite geometer so no absolute guarantees. —Blotwell 02:25, 23 February 2006 (UTC)

one billion seconds

How many minutes is one billion seconds?

Google: http://www.google.com/search?q=one+billion+seconds+in+minutes (divide by 60) - Fredrik Johansson 18:03, 21 February 2006 (UTC)

Hey, maybe you could write a musical about it. You could call it Mortgage. --Trovatore 18:13, 21 February 2006 (UTC)

There are 60 secs in a min. Try arithmetics, which records elementary properties of certain operations on numerals : 100/6 = 16.6. 1000/6 = 166.66 ... Or try Google for the same operation. --DLL 22:08, 21 February 2006 (UTC)

In other words, divide one billion by 60 to get your answer. StuRat 03:13, 22 February 2006 (UTC)

Very crudley: about 31 years. gelo 07:09, 22 February 2006 (UTC)

Or, considering 365 days = 1 year, 31 years 259 days 1 hour 46 minutes and 40 seconds. Disregard the months because those are too innacurate. ☢ Ҡi∊ff⌇↯ 07:28, 22 February 2006 (UTC)

Gee, so many replies and not even one straight answer :-) It's 16,666,666 minutes and 2 thirds of a minute (40 seconds). -- Meni Rosenfeld (talk) 09:16, 22 February 2006 (UTC)

If you care about years, it may help to know that π seconds is a nanocentury.--gwaihir 11:20, 22 February 2006 (UTC)

You meant "is roughly a nanocentury". There's a 0.5% difference. -- Meni Rosenfeld (talk) 13:21, 22 February 2006 (UTC)

There's roughly a 0.5% difference. SCNR--gwaihir 10:41, 23 February 2006 (UTC)

You're right, there's roughly 12% difference between 0.5% and the difference. -- Meni Rosenfeld (talk) 12:33, 23 February 2006 (UTC)

:-) --gwaihir 13:01, 23 February 2006 (UTC)

How to calculate confidence intervals?

I have normally distributed observations x_i, with i from 1 to N, with a known mean m and standard deviation s, and I would like to compute the upper and lower values of the confidence interval at a parameter p (typically 0.05 for 95% confidence intervals.) What are the formulas for the upper and lower bounds? --James S. 20:30, 21 February 2006 (UTC)

I found http://mathworld.wolfram.com/ConfidenceInterval.html but I don't like the integrals, and what is equation diamond? Apparently the inverse error function is required. Gnuplot has:

     inverf(x)         inverse error function of x

--James S. 22:00, 21 February 2006 (UTC)

The simplest answer is to get a good book on statistics, because they almost always have values for the common confidence intervals (you may have to look at the tables for Student's t-distribution in the row labelled n = infinity). Confusing Manifestation 03:57, 22 February 2006 (UTC)

The question was how to calculate, not how to look up in a table. --171.65.82.76 07:36, 22 February 2006 (UTC)

In which case the unfortunate answer is that you have to use the inverse error function, which is not calculable in terms of elementary functions. Thus, your choices are to look up a table, use an inbuilt function, or use the behaviour of erf to derive an approximation such as a Taylor series. (Incidentally, I think equation "diamond" is meant to be the last part of equation 5.) Confusing Manifestation 16:12, 22 February 2006 (UTC)

Any general statistical package worth its salt (including free ones such as R) will let you calculate the inverse error function mentioned above for any p, as will most spreadsheets (e.g. see the NORMSINV function in MS Excel). -- Avenue 12:07, 23 February 2006 (UTC)

Perl5's Statistics::Distributions module has source code for

 $u=Statistics::Distributions::udistr (.05);
 print "u-crit (95th percentile = 0.05 sig_level) = $u\n";

...from which I paraphase this...

Wrong solution:

 x = -ln(4 * significance_level * (1 - significance_level));
 
 critval = sqrt( x * ( 1.57079628800
               + x * ( 0.03706987906
               + x * (-0.8364353589E-03
               + x * (-0.2250947176E-03
               + x * ( 0.6841218299E-05
               + x * ( 0.5824238515E-05
               + x * (-0.1045274970E-05
               + x * ( 0.8360937017E-07
               + x * (-0.3231081277E-08
               + x * ( 0.3657763036E-10
               + x *   0.6936233982E-12)))))))))));
 
 if (significance_level > 0.5) then {critval = -critval};
 
 ci_top = mean + (standard_deviation/sqrt(N_obs)) * critval/2;
 ci_bot = mean - (standard_deviation/sqrt(N_obs)) * critval/2;

Where significance_level is, e.g., 0.05 for 95% confidence.

--James S. 07:30, 24 February 2006 (UTC)

Correct solution

Please see Talk:Confidence interval. --James S. 08:07, 27 February 2006 (UTC)

February 22

pseudo-BCH formulas

Suppose we have a smooth function f(.) - is it possible to write this operator A = f(X) B f(-X) for operators B and X in terms of a bunch of commutators, like the Campbell Baker Hausdoff formulas? --HappyCamper 05:05, 22 February 2006 (UTC)

I'm not sure what to do with such a question. BCH refers to Lie algebras, tangent spaces at the identity of a Lie group along with the operator sometimes called Lie bracket. How is this supposed to relate to your situation? --KSmrq^T 01:52, 23 February 2006 (UTC)

It's sort of awkward for me to describe, because I'm trying to venture into a new area unknown to me previously...let me try explaining this way:

$e^{-X}Ae^{X}=A+\left[A,X\right]+{\frac {1}{2}}\left[\left[A,X\right],X\right]+\cdots$ - this I know holds. Now, the other day, I read that it can be rewritten as

$e^{-X}Ae^{X}=Ae^{\left.X\right]}$

I am not sure whether the ] refers to "left commutation" or "right commutation", but regardless, it seems to make sense structurally. Now, is it true that

$e^{-X}Ae^{X}=Ae^{\left.X\right]}=e^{\left[-X\right.}A$?

Since there really isn't anything special about the exponential, if I replace the exponential with some sort of arbitrary function, does

$f(-X)Af(X)=Af(\left.X\right])=f(\left[-X\right.)A$ hold? If so, can I write the resulting infinite commutator series with the same structural form as the first line above, but with just different coefficients? --HappyCamper 09:51, 23 February 2006 (UTC)

Actually, the exponential is special. It maps a vector from the Lie algebra to an element of the Lie group, and has important properties in doing so. --KSmrq^T 03:59, 25 February 2006 (UTC)

Oh, I see. So, it would not be possible to replace the exponential with say, a q-exponential, and see what happens? --HappyCamper 18:20, 25 February 2006 (UTC)

Gödel's theorem of incompleteness.

What are the philosophical implications of this theorem?...If there are any...I'm not sure, I'm just wondering.--Cosmic girl 19:29, 22 February 2006 (UTC)

Did you read Gödel's incompleteness theorem? (Completely?) Dmharvey 19:33, 22 February 2006 (UTC)

Yes, I've read it completely! :S..but from what I understood, even if my understanding of it is consistent, I will nevcer find a proof that it is...haha (lame joke).--Cosmic girl 21:16, 22 February 2006 (UTC)

How did I know you would come up with this?

It implies that "There are more things in heaven and earth, Mylady, Than are dreamt of in your philosophy". ;-)

Skip that Gödel stuff for now and go directly to the Turing machine and the Halting problem The Infidel 19:58, 22 February 2006 (UTC)

I thought so, Hamlet...did you really know I was gonna ask this? hahaha...creepy.--Cosmic girl 21:18, 22 February 2006 (UTC)

Yes, I suspected it. The question was already a seed in your mind when you asked about other kinds of maths. Next logical step would be Chomsky. The other branch of your thoughts would take you to the questions about existance. Indian philosophy of rebirth (what of our beeing is (thought to be) reborn) could be of interest here. The Infidel 17:50, 23 February 2006 (UTC)

Why chomksy?...isn't he just a lingüist who talks about politics?...what are his thoughts on 'truth' for example, or on 'god'...I thought Chomsky was only a communist linguüist who has strong opinions when it comes to politics...but I've never encountered any original or interesting philosophical propositions of his...and also...what does karma and hindi philosophy have to do with the theorem? I don't see the conection...I thougt this theorem was about undertinty or something like that.--Cosmic girl 19:18, 23 February 2006 (UTC)

Chomsky because of Chomsky hierarchy in formal languages. (I don't care what he said about politics. Read the newspapers to know the world is crazy. Read history books to know it's always been.)

The second was not so much about karma but about reincarnation, and not about Gödel but about your other question: does a mathematical entity exist for itself and if so, in what sense does it exist (Your wording was different.)

Closely connected to the question of existance is the question of what it is that exists. Hindu philosophers have spent a lot of thought about what it is that (they think is) reborn. Into a different species. In a different time. Deprived of memories. But still in a weird way the same. You don't have to believe all this, but the huge variety of sophisticated distinctions of self is very interesting. (See also Poincaré's reccurrence theorem.)

The Infidel 21:16, 25 February 2006 (UTC)

I asked something about this on my early Wikipedia days. I was wondering if I could use the theorem against Intelligent Design\Theism... I think it makes sense. :P ☢ Ҡi∊ff⌇↯ 21:11, 22 February 2006 (UTC)

How can it be used against ID and Theism?--Cosmic girl 21:18, 22 February 2006 (UTC)

Agree, Kieff, but this is mathematics. There was once a big farce about philosophers using science concepts to create a smoke screen. Analogy helps but it is not falsifiable. --DLL 21:20, 22 February 2006 (UTC)

Well, it's hard to refute an argument you haven't actually given, but I would be surprised if there were any very convincing argument against theism from the Gödel theorems. By the way Gödel himself is generally thought to have believed in God, though the record is less than clear. --Trovatore 21:28, 22 February 2006 (UTC)

By the way, no one has really said much about Cosmic Girl's original question. The theorems are of great philosophical importance in the philosophy of mathematics; applications to philosophy outside phil of math are much dicier.

Even within philosophy of mathematics, while almost everyone agrees the theorems are important, precisely in what way they're important remains contested. My take on it would be something like this: They made it more difficult to give a "formalist" or "logicist" account of mathematics and why it works. Since they didn't make it any more difficult to give a Platonist/realist account, they somewhat enhanced the position of the Platonist viewpoint with respect to its competitors (though without in any way helping to explain "where" or "how" mathematical entities exist independently of our reasoning about them). --Trovatore 22:17, 22 February 2006 (UTC)

I'm kind of lost now...what does the theorem have to say about whether mathematical entities exist independently of our reasoning about them?..does it have to say anything in the 1st place? I don't want to sound patronizing because I'm really stupid when it comes to math... but I believe that there's no possible way of knowing without a doubt, EVER, that mathematical entities exist appart from our reasoning about them.--Cosmic girl 02:01, 23 February 2006 (UTC)

The theorem has nothing at all to say about whether mathematical objects exist; that's not the subject matter of the theorem. What I said was, the theorem makes alternative accounts more difficult. In particular, it screws up Hilbert's program, which was the big formalist thrust pre-Gödel, pretty badly.

My personal view is that wanting to know things about mathematics "without a doubt" is the wrong goal. I'll grant that it's a time-honored one, going back to the Greeks, but I still think it's wrong. We don't know mathematical truths without a doubt, even if we've proved them; the assumptions that we used to prove them could be wrong, or there could be a mistake in the proof. We have more certainty than exists in, say, biology, but it's a difference of degree rather than kind. --Trovatore 03:44, 23 February 2006 (UTC)

What did Gödel find so unconfortable about this that he went crazy? I'm not disturbed by this theorem, maybe I'm not getting something...haha... ( I hope I'm not punished by some power from beyond for this joke).oh and also...what does the mystery of the Aleph have to do with this?.--Cosmic girl 20:45, 23 February 2006 (UTC)

At the time, the mathematicians were getting really worked up about the nature of maths, and they were asking themselves "just how much can we prove with a simple set of axioms?". Unfortunately, GIT says the answer is "no", and mathematicians have already found a whole bunch of things that they would like to prove true (or false, as it may be), but know they never will, but which they sometimes don't feel comfortable stating as axioms because it's impossible to know whether the theorem itself or its contradiction is true (because neither is provable). The article itself mentions a few examples, but the ones that come to mind are the continuum hypothesis (which is probably the "mystery of the aleph" you're talking about) and the axiom of choice - at a rough guess there are probably comparable numbers of papers written that assume the axiom of choice as those that try to avoid it entirely. Confusing Manifestation 02:08, 24 February 2006 (UTC)

Are you sure mathematicians have 'accepted' there are things they can never prove true or false? like what things? and can they 'prove' anything true in the first place? if so, what can they prove?.--Cosmic girl 17:39, 24 February 2006 (UTC)

Several points I have to quarrel with in the above. First, we don't have any theorems of the form "such and such can never be proved or disproved", only of the form "from this specified set of axioms, such and such can never be proved or disproved". So for example we know that the continuum hypothesis, CH, can never be proved or disproved from ZFC, granting that ZFC is consistent. It does not follow that we'll never know whether it's true, only that we'll never know a proof or disproof from ZFC. A current line of work by W. Hugh Woodin provides an argument, convincing to many, that CH is false (which was also what Gödel believed).

As for the axiom of choice, most mathematicians consider it unproblematic these days, though there are still reasons to prefer choice-free proofs when you can find them.

None of this addresses Cosmic girl's question as to why Gödel lost his marbles. I would not assume that had anything to do with his work in mathematical logic. Maybe he'd have gone crazy faster if he hadn't done logic. --Trovatore 03:41, 24 February 2006 (UTC)

wow...I'm so lost now, so you are saying that Gödel meant that say axiom A and axiom B can't prove such and such? what I meant by the question was that IF the theorem meant that we can't know if axiom A and axiom B are 'true' in the first place...well of course they are true, because they are axioms...but I thought that the theorem meant that while axioms can appear to be automatically true to us, they may not be so. and the rest of the stuff I don't get. :| --Cosmic girl 17:37, 24 February 2006 (UTC)

To be honest I don't really understand what you're saying above. No, the theorems don't mean that we can't know whether the axioms are true. No, it's not an "of course" that axioms are true; axioms are automatically provable, not automatically true. You can have a deductive system in which some of the things you can prove are in fact false. One of the repercussions of the incompleteness theorems (please note there are two of them! you keep using the singular) is that truth cannot be identified with provability in the sense that, say, Hilbert would have wanted. There are still those who identify truth with provability, but they have to give up properties of truth that most of us are not so ready to part with.

But you keep asking what the theorems mean. What they mean, literally, is what they actually say, which is a completely concrete claim about what happens in formal deductive systems. I think you should first try to understand the theorems at that level, before trying to understand their importance for philosophy. That will give you a much better nonsense filter, which you badly need in this arena, because there's so much nonsense written on this topic. --Trovatore 21:15, 24 February 2006 (UTC)

You are right, sorry for all the nonsense...=P.--Cosmic girl 03:11, 25 February 2006 (UTC)

A formal system is build with axioms ; no question of truth in an axiom, only practical results.

A consistent system contains no contradictions.

• GIT1 : every consistent system allows to construct - using its axioms and statements built from them – at least a true statement which is not provable (using the same). Thus it is incomplete.

The above unprovable statement may be provable in a richer system, and so on.

A formal theory including basic arithmetical truths and also certain truths about formal provability is “rich enough” to enter the second contest : Zermelo-Fraenkel set theory with the axiom of choice (ZFC) is a good example, as it allows the construction of the ordinal numbers.

• GIT2 : a “rich enough” formal system including a statement of its own consistency is inconsistent.

ZFC does not try and admits its own incompleteness.

Consequences for philosophy : few. For the philo of maths : some problems are undecidable. However, few people play with them ... but they make one of the paths that help maths to evolve. See the articles above and, Cosmic, another question, plz. --DLL 22:04, 24 February 2006 (UTC)

Thank you, your answer was the simplest one,and I will read the articles above.--Cosmic girl 03:11, 25 February 2006 (UTC)

February 23

Econometrics question: Autocorrelation

Hi everyone... am having trouble with a specific aspect of first-order autocorrelation.

The setup is y_t=B*y_(t-1)+u_t ; u_t=p*u_(t-1)+e_t ... y_t and u_t are covariance-stationary processes.

I'm supposed to eventually find cov(y_(t-1), u_t) and I've started off by breaking the covariance into cov(y_t/B , u_t) and cov (-u_t/B , u_t). I'm not sure this is the right approach, am becoming convinced it probably isn't. Anyways, continuing from there I've got the second covariance (with the u's) solved, but the first one, no matter how I break it down, still comes back to something in the form of cov(y_t , u_t).

Any ideas of how to get around this? Thanks very much for any thoughts. rabbit84.92.181.246 00:44, 23 February 2006 (UTC)

repeating decimals

is .999... repeating to infinity equal to 1?

• Yep! See Recurring decimal#The case of 0.99999.... --jpgordon∇∆∇∆ 07:44, 23 February 2006 (UTC)

And also Proof that 0.999... equals 1, and for some additional background real number. -- Meni Rosenfeld (talk) 12:16, 23 February 2006 (UTC)

And if you want your head to explode, see Talk:Proof that 0.999... equals 1, where people identified only by IP addresses try to bring that fact into dispute, and manage to use "phd" as an insult. Confusing Manifestation 12:24, 23 February 2006 (UTC)

You're right, that argument is hilarious. Sad to see that the anon posting there, clearly has some mathematical knowledge, and yet can mix up facts so badly. All the more reason to read real number - The more one knows about where real numbers come from, the less he is likely to make ridiculous claims. -- Meni Rosenfeld (talk) 12:46, 23 February 2006 (UTC)

It's one of the funniest I've ever seen -- all six or so archives as well. Is there a special place for mathematical WP:BJAODN? --jpgordon∇∆∇∆ 03:33, 24 February 2006 (UTC)

What's sadder still is the endless stream of misguided "helpers" who can't see that they're being trolled. --KSmrq^T 18:34, 25 February 2006 (UTC)

Quadratic formula

In the quadratic equation article, it says an alternate form of the quadratic formula is:

$x_{1,2}={\frac {2c}{-b\pm {\sqrt {b^{2}-4ac\ }}}}$

Am I correct in saying that this would not work where c is equal to zero, e.g. in the case "x² - 5x = 0"? That formula would return zero, but the actual answer would be 0 or 5. --210.246.30.87 07:35, 23 February 2006 (UTC)

Congratulations! You have uncovered a longstanding instance of error, delusion, or vandalism. The formula is totally bogus. It went in via this edit (later merged en bloc into quadratic equation). The amazing thing is that this has survived unchallenged for almost two years. -R. S. Shaw 09:49, 23 February 2006 (UTC)

It's not vandalism at all - the formula is perfectly valid. It is used whenever you need to worry about numerical errors from accumulating if you use the so-called "normal" or "canonical" expression for the quadratic formula. --HappyCamper 09:59, 23 February 2006 (UTC)

The expression arises from resolving the radical (is this the correct term for the procedure, as it seems to be an awful mouthful). Here, for example, we have

$x_{+,-}={\frac {-b\pm {\sqrt {b^{2}-4ac\ }}}{2a}}={\frac {\left(-b\pm {\sqrt {b^{2}-4ac\ }}\right)\left(-b\mp {\sqrt {b^{2}-4ac\ }}\right)}{2a\left(-b\mp {\sqrt {b^{2}-4ac\ }}\right)}}={\frac {4ac}{2a\left(-b\mp {\sqrt {b^{2}-4ac}}\right)}}$

Finally, note how some terms magically cancel out on the top and bottom...we get the celebrated (at least in numerical methods and analysis circles)

$x_{+,-}={\frac {2c}{-b\pm {\sqrt {b^{2}-4ac\ }}}}$

of course, valid when a is not zero, and when c is not zero. So, to answer the original question, yes, you are correct to say that the formula fails when c is zero. When c is zero, the conjugate term vanishes:

$\left(-b\mp {\sqrt {b^{2}-4ac\ }}\right)$

Sort of subtle, but do you see how? This should give you a hint to figure out when the alterative formula would be advantageous over the other one...Let me update the article :-) --HappyCamper 10:12, 23 February 2006 (UTC)

But you're not entirely correct to say that the formula incorrectly gives zero when $c=0$. When you take the minus sign it correctly gives 0, because 0 is one of the roots as it always is when $c=0$. When you take the plus sign to find the other root, the denominator becomes zero so the formula gives 0/0, which you recognize as meaning "Now would be a great time to use an alternative formula". So yes, the formula fails, but it doesn't lie to you and tell you an incorrect 0. —Blotwell 12:09, 23 February 2006 (UTC)

In numerical analysis the question of stability of quadratic root-finding is an early cautionary example, illustrating the dangers of naively pulling a formula out of a mathematics text. Assuming a is nonzero and d =b²−4ac is positive, typical code might be

if (b > 0)

 then q := -0.5 * (b + sqrt(d))

 else q := -0.5 * (b - sqrt(d))

r1 := q / a

r2 := c / q

A copysign function could be used instead of the test, for languages that adequately support IEEE floating point. --KSmrq^T 18:52, 25 February 2006 (UTC)

3-torus

This is not a factual question, it is a request. Can anyone generate an animation of the 3-torus? I was thinking something similar to the pentatope one you can see at the right, where each frame is a 3D slice of it. ☢ Ҡi∊ff⌇↯ 10:20, 23 February 2006 (UTC)

Grr...edit conflict....um...this request sort of begs the question: What does a 5d torus look like? Does it even exist? Can 5-D objects have holes in them? Are holes only found in 3-D objects? I wonder if any topologists are here on Wikipedia? Hm... :-) :-) :-) --HappyCamper 10:27, 23 February 2006 (UTC)

Can't say that I know much topology, but...

• I see no reason why 5-D objects shouldn't have holes.
• The torus article, I believe, supplies enough information to decide how does this 3-torus look like, for those able to understand it (myself not included).
• The pentatope is 4-D, making it reasonable to represent it with a 3D animation. But how would you represent a 5-D object? Looks tricky to me.
• Um... whom did you have an edit conflict with? Yourself? I understand, it happens to me all the time.

-- Meni Rosenfeld (talk) 12:29, 23 February 2006 (UTC)

Uh... The 3-Torus is a 4-D object, so an animation would be adequate. I can imagine a 4-D sphere, but a 4-D torus clogs my imagination. ☢ Ҡi∊ff⌇↯ 13:06, 23 February 2006 (UTC)

Ah, I see. So, an animation of the 3-torus would start as a sphere, that would split into two and then join again? ☢ Ҡi∊ff⌇↯ 14:22, 23 February 2006 (UTC)

In general the n-Torus is just S^n-1 X Sⁿ, where Sⁿ is the n-dimensional sphere. (Basically, you take a (n-1)-sphere and rotate is around an axis... think of 3D case, where you rotate a circle around).

All the more reason for me to believe it is a 5-D object... But like I said, I don't really know what I am talking about. -- Meni Rosenfeld (talk) 13:43, 23 February 2006 (UTC)

Ummm..... I think the statement "n-Torus is just S^n-1 X Sⁿ, where Sⁿ is the n-dimensional sphere" is incorrect. In the case n = 3, this would mean that the 3-torus is $S^{2}\times S^{3}$, which is incorrect because the latter space is simply connected whereas the 3-torus is not simply connected (in fact its fundamental group is isomorphic to the sum of three copies of Z). Also it's not entirely clear to me that "the 3-torus is a 4-D object". I can see easily how to embed it in five dimensions: in fact since it's just $S^{1}\times S^{1}\times S^{1}$, we already know that the first $S^{1}\times S^{1}$ can be embedded in three dimensions (that's just an ordinary 2-torus), and then the last $S^{1}$ can be embedded in two dimensions (since it's just an ordinary circle), giving five dimensions altogether. How do you squeeze it into four dimensions? I'm not saying it can't be done; I'm just not a geometer and can't see immediately how it would work. Dmharvey 18:11, 23 February 2006 (UTC)

OK, I think I see now. The statement should be replaced by "the n-torus is just $S^{1}\times T^{n-1}$", where $T^{n}$ is the n-torus. Then you can embed the n-torus in n+1 dimensions as follows. Draw the circle $S^{1}$ in your first two dimensions. We're going to "drag an n-1 torus around the circle". So we have one spare dimension (in the direction of the radius of the circle) plus n-1 dimensions perpendicular to our first two. By induction the n-1 torus that we are dragging requires n dimensions, which is (n-1) + the spare one, so we have enough room. So: a 3-torus needs 4 dimensions, a 4-torus needs five dimensions, etc. That was clear as mud. Sorry. I guess the way to think about it is that in the ordinary 2-torus case, we're dragging a 1-torus (which also happens to be a 1-sphere) around a circle. The correct generalisation is to drag a 2-torus around, not a 2-sphere. Perhaps this helps to visualise the thing, which was the original question :-) Dmharvey 18:28, 23 February 2006 (UTC)

OK, I messed up there... what I meant to say was that the n-torus was $S^{1}\times S^{n-1}$. Mind you, that was still not correct. Tompw 18:49, 23 February 2006 (UTC)

So, what a 3-torus would look like as an animation then? ☢ Ҡi∊ff⌇↯ 08:20, 24 February 2006 (UTC)

I think I can work out what it looks like from one particular "direction", and maybe someone skilled in animation will be able to make an animation for you (I have no idea how to do that). Let's call the dimensions A, B, C, and D. Our main circle (call it M) will be in dimensions A and B. Let R be the radial direction of that circle, so we're going to rotate a 2-torus T which lives in dimensions R+C+D around our circle. Now, I haven't yet specified exactly how to embed T in those three dimensions; there are a few choices, and I think the versions of the 3-torus you get will look kind of different. I'm going to choose the embedding where the axis passing through the "hole" in the middle of the 2-torus T runs along dimension C. Now let's slice the 4-dimensional object we have created. I'm going to take a slice in the A+B+D plane. What this corresponds to is taking a slice of T perpendicular to the C axis and rotating it around the circle M. Such a slice of T looks like two circles with the same centre but different radii.

So here's what the animation looks like: let t (time) run from say 0 to 1. (t is moving along the C axis.) When 0 < t < 1, it looks like an annulus rotated around the circle; or in other words it looks like a "thin 2-torus" contained "inside" a "fat 2-torus". At time t = 1/2 the fat torus has maximal radius and the thin torus has minimal radius. At times t = 0 and t = 1 the two tori coalesce into a single 2-torus, before vanishing completely.

I'm sure other embeddings give interesting results too, I think that's the easiest one to visualise. In the animation you might want to make the outer one transparent and the inner one a different colour so you can see what's going on. Dmharvey 13:27, 24 February 2006 (UTC)

You can represent any n-D object in 3-D or 2-D projections, knowing that you miss a lot ; quasicrystals are a 3-D projection of a 6-D grid. An animation here would be gorgeous. --DLL 21:30, 24 February 2006 (UTC)

A topological 3-torus is trivial to illustrate. Show a point moving inside a solid unit cube. When the point hits a face while moving away from the origin, it teleports to the opposite face, moving toward the origin. For a geometric 3-torus, a specific geometry must be chosen. This can be rather entertaining. Consider the familiar 2-torus, commonly seen as the surface of a donut, depicted in 3D space. It has a geometric form in 4D with the fascinating property that it is "flat", as measured by Riemannian curvature of the surface. (A web search for 'flat torus' finds some nice images.) A comparable option for the 3-torus might map (u,v,w) to the 6D point (cos u,sin u,cos v,sin v,cos w,sin w)/√3; but then a series of projections to 3D or 2D must be chosen. --KSmrq^T 06:07, 26 February 2006 (UTC)

Sub-groups of general linear group

Does there exsist any non-trival subgroup of the (real) General linear group that is not a sub-group of the Orthogonal group? Further, does there exsist such a sub-group that doesn't have the Orthogonal group as a sub-group?

(Alternatively: is there a geometry which is an Affine geometry (which deals with invarients under GL_n), but not a Euclidean geometry (which deals with invarients under O_n)? Further, is there such a geometry which does not "contain" Euclidean geometry?) Tompw 13:40, 23 February 2006 (UTC)

There sure is. For example, take the scalar multiples of the identity matrix by integer powers of 2. This is a group because the product of any two scalar multiples of the identity matrix is another scalar multiple of the identity matrix, and the product of any powers of 2 is another power of 2. None of its elements are orthogonal except the identity matrix itself. —Keenan Pepper 14:16, 23 February 2006 (UTC)

Thanks for that one... any others? (Ignoring those in a similar vein) Tompw 18:43, 23 February 2006 (UTC)

There is plenty to be sure. The symplectic group, the diagonal subgroup, the complex general linear group embedded in the real general linear group, and the group of upper triangular matrices are a few that immediately come to mind. -- Fropuff 20:11, 23 February 2006 (UTC)

Estimation methods

In doing a mock maths paper, one question asked me to estimate the sum of six four-digit numbers by a)the front-end method and b)the "cluster" method. None of us have any idea of what these terms meant, and after some google search(wikipedia search on both terms failed, returning results on topic of statistics), the results shows that the front-end method is to truncate the number to have only two most significant figures, then add them up as usual.(notice in case some number has three digits only as in my example, like 729, it takes 700, not 720). But unfortunately, I still can't find out what the "cluster" method is. Any helps would be appreciated. Thanks. --Lemontea 15:18, 23 February 2006 (UTC)

See if this helps. It's the first Google hit from arithmetic estimation cluster. hydnjo talk 21:17, 23 February 2006 (UTC)

Thanks! It helped a lots. Some follow-up questions: a)From the webpage you provided, does that means both methods are not strictly algorithmic--e.g. it's up to the user to determine which number the group cluster to, how many sig. fig. the front-end method should use? b)If we do front-end method all the way to completion(added all digits), then it is no longer an estimation method, is that right? Finally, it seems wikipedia didn't give any mention of these things, maybe we can add them to the corresponding entries. --Lemontea 02:59, 24 February 2006 (UTC)

It seems that "clustering" is a special case say, "estimate the total weight of 24 containers that nominally weigh 10 pounds each (9.8, 10.1, 10.3, 9.9 ....)", so a reasonable estimate would then be 240 pounds. And yes, if you "front-end" to completion then it is no longer an estimate (neglecting mistakes). The idea behind estimating is to quickly get a ballpark answer to the degree of precision that you want but can also be utilized to get the correct answer via mental arithmetic and amaze your friends.  ;-) This could well be the substance of an article on say, Estimating arithmetic. hydnjo talk 05:29, 24 February 2006 (UTC)

Statical Mechanics

(Transferred from the French Reference Desk)

I know that planck's:

${\mathcal {\,}}E=h\nu (e^{h\nu /kT}-1)^{-1}\,$

mulitiplied by 1/n then do another parts of statics. Then do N*n backing to the original formulas

Same or not between these two cases?

note:for 3D

${\mathcal {\,}}E=3h\nu (e^{h\nu /kT}-1)^{-1}\,$

.

.

.

.

.

.

${\mathcal {\,}}E=(1/3)*3h\nu (e^{h\nu /kT}-1)^{-1}\,$

........then others

Between

${\mathcal {\,}}E=h\nu (e^{h\nu /kT}-1)^{-1}\,$

.......then others in Math.....

.

.

.

.

.

.

The same or Not?

• I see that they are not the same. Because of hv/ (e{hv/ kT}-1) do not need to set somewhat coeffiecients, even in this case.

And setting somewhat numbers does not respect Planck.

The official method, is to be the 2nd I wrote before.--HydrogenSu 23 février 2006 à 16:34 (CET)

February 24

Lagrangian dual of a logarithmic barrier problem

Yes, this is a homework question. Yes, I've put some thought into it myself. I just need some help getting over the last little bit.

Here's the primal problem:

minimize $c^{T}x-43\sum _{i=1}^{m}\ln s_{i}$

subject to $Ax+s=b$, $s>0$

Ok, so first I make the Lagrangian:

$L(x,s,u)=c^{T}x-43\sum _{i=1}^{m}\ln s_{i}+u^{T}(Ax+s-b)$

and I can rearrange this so that x & s are separate:

$L(x,s,u)=-u^{T}b+(c^{T}+u^{T}A)x+u^{T}s-43\sum _{i=1}^{m}\ln s_{i}$

and now I find the dual function

$L^{*}(u)=-u^{T}b+\min _{x}[(c^{T}+u^{T}A)x]+\min _{s>0}(u^{T}s-43\sum _{i=1}^{m}\ln s_{i})$

Now, since x is unrestricted, and the dual function can't go off unbounded below, I know that the dual problem must have the constraint $c^{T}+u^{T}A=0$ which means I can reduce my dual function to

$L^{*}(u)=-u^{T}b+\min _{s>0}(u^{T}s-43\sum _{i=1}^{m}\ln s_{i})$

And... now I'm stuck. What, if anything, do I do with the part that's minimized over s? Is there another constraint in the dual problem, or am I stuck with this, or what? moink 02:08, 24 February 2006 (UTC)

The answer, by the way, is to show that $(u^{T}s-43\sum _{i=1}^{m}\ln s_{i})$ is convex and large as s nears zero and as s increases without bound. Then differentiate that expression w.r.t. s, find the critical point (which must therefore be a global minimum) and substitute that into the Lagrangian. moink 16:49, 28 February 2006 (UTC)

Thanks for coming back with that...I was wondering about it myself. --HappyCamper 15:44, 1 March 2006 (UTC)

Mathematics for the masses

Does anybody know of any puzzles or concepts in mathematics that are:

• Entertaining and educational
• Short
• Easy (People tune out at the sign of a pronumeral)

I work at a Science Centre, and am looking for ideas,

Thanks, --Alexs letterbox 07:56, 24 February 2006 (UTC)

Sure, there are lots. Can't think of any right now, but... Perhaps list of paradoxes is a good place to start looking for all sorts of curiosities. And I think maybe platonic solids match your description. I'll let you know if I think of anything else... -- Meni Rosenfeld (talk) 08:06, 24 February 2006 (UTC)

Take a look at recreational mathematics. :D ☢ Ҡi∊ff⌇↯ 08:18, 24 February 2006 (UTC)

Try cut-the-knot? —Blotwell 10:25, 24 February 2006 (UTC)

I like "guess the number of jelly beans in the jar". It touches on geometry, sampling, and probability (if asked to also estimate the number of each color). And after the contest you all get to eat the jelly beans ! StuRat 11:18, 24 February 2006 (UTC)

Sudoku? Proto||type 12:51, 24 February 2006 (UTC)

We might need an article on pronumeral — does it differ from variable? Gdr 16:28, 24 February 2006 (UTC)

I saw some beautiful graphical proofs of - 1. sum of cubes, 2.pythogorean theorem, etc...which they didnt teach me at school. I found it very nice and inspiring for thinking problems in a very graphical PoV. I could search it up again if u want. -- Rohit 18:07, 24 February 2006 (UTC)

How about fractals? The pictures always look stunning and impressive. --HappyCamper 20:11, 24 February 2006 (UTC)

Sample size

Without having to relearn statistics, how large of a sample would be required to determine, with a normal degree of certainty, valid results about the distribution of topics in Wikipedia (so from a total population of 1 million)? Rmhermen 19:41, 24 February 2006 (UTC)

You'd first have to do a preliminary examination to determine how to determine that, since you don't know what kind of distribution to expect. You'd also have to define the problem in greater detail. --BluePlatypus 20:12, 24 February 2006 (UTC)

How many topics are you thinking of splitting it up into? Is there any overlap? If not, it sounds like the chi-squared test is what you'd use. If so, the only way I know of to find out what sample size you need is to input the data you expect, and try it with progressively larger sample numbers until you get what you want, then take a survey somewhat larger than that in case you guessed wrong on the expected values. Black Carrot 21:30, 24 February 2006 (UTC)

1100 is common number used as a minimum, as it gives about a 3% margin of error over a 90% confidence interval, meaning the results will be within 3% of the actual number 90% of the time. Hopefully a statistician here can show the calculations for this and add details. StuRat 22:45, 25 February 2006 (UTC)

February 25

Pythagorian Triples

Is there a way to prove that there are an infinite number of pythagorian triple families (like 3,4,5 is the family in which 6,8,10 and 9,12,15 belong)? I assume this would involve the proof that there are an infinite number of prime numbers, but you also have to prove that they are in the right proportions. — Ilyanep (Talk) 04:18, 25 February 2006 (UTC)

It's also probably an indirect proof — Ilyanep (Talk) 04:19, 25 February 2006 (UTC)

See Pythagorean triple#Generating Pythagorean triples. It shouldn't be too hard to prove that the triple is primitive if m and n are coprime and one of them is even, and then you're done. —Keenan Pepper 04:31, 25 February 2006 (UTC)

I added a sentence to that article which makes the proof trivial. Since all pythagorean primes are represented by complex numbers of the form $(m+in)^{2}$, one needs only to prove that there are infinite number of complex arguments of such numbers (which is also trivial because their arguments are twice the argument of $m+in$.  Grue  16:36, 25 February 2006 (UTC)

Take any odd integer greater than 1 - call it a. Square it, so you get a². Now express a², which is also odd, as the sum of two integers whose difference is 1 - call these b and c (note that b and c will be coprime). Then

$c^{2}-b^{2}=(c+b)(c-b)=c+b=a^{2}$

So (a,b,c) is a Pythagorean triple - for example, starting with a=13 you get (13,84,85). So each odd integer > 1 gives you a distinct Pythagorean triple. This does not generate all Pythagorean triples, but it does generate an infinite number. Gandalf61 16:44, 25 February 2006 (UTC)

You can also go this more "complicated" method. Here's a sketch. Suppose $x^{2}+y^{2}=z^{2}$. Now divide both sides by $z^{2}$, and set a = x/z, b = y/z. The equation then becomes $a^{2}+b^{2}=1$, which is precisely a unit square. Suppose you have a line that goes from the origin to the circle. If the line has a rational slope, the point of intersection must also be a pair of rational points. So, now, you try and parameterize the original variables in terms of the slope, and this will give you the family of all the possible solutions, and you can show that there are an infinite number of them. --HappyCamper 18:26, 25 February 2006 (UTC)

In my universe, a²+b²=1 is precisely a unit circle. Furthermore, the line with slope 1, which I'm sure is a rational slope, intersects the circle at (¹⁄_√2,¹⁄_√2), which I'm sure is not a rational point. --KSmrq^T 19:04, 25 February 2006 (UTC)

Right. HappyCamper might have been misremembering the following fact. First pick an initial point P on the circle (say (0, 1)), and then consider any line with rational slope passing through P. Except for the tangent line, every such line intersects the circle in exactly one other point whose coordinates are rational; and conversely, if you take a point Q on the circle with rational coordinates, then the line PQ has rational slope. Effectively this is an example of a rational map between the projective line and the circle. Dmharvey 19:22, 25 February 2006 (UTC)

Oh yes, thanks for correcting that critical mistake of mine. A while ago, I used this method to find the all the rational solutions to x^2 + xy + y^2 = 1, which of course, is related to triangles which have angles of 120 degrees instead of 90 degrees. (The 3-5-7 triangle is an example) --HappyCamper 20:50, 25 February 2006 (UTC)

Which nicely brings us to the generation of Pythagorean triples via a rational parameterization of a unit circle. Let the fixed point be (−1,0). Then points on the circle are given by x = (1−t²)/(1+t²), y = 2t/(1+t²). Here t is, indeed, the slope; and clearly a rational slope gives a rational point. More directly, for any rational t we have a Pythagorean triple obtained from 1−t², 2t, 1+t² by clearing denominators. Conversely, a rational point, (x,y), guarantees a rational slope, t = y/(1+x). It suffices to restrict attention to 0 ≤ t ≤ 1, the first quadrant. --KSmrq^T 23:45, 25 February 2006 (UTC)

Golden Times on a clock face

Does anyone have a solution to the following question that would be accessible to an average high school student? "If a rectangle is drawn inside the circle touching where the hands of a clock are pointing, at how many times will the rectangle be 'golden'?" Visit http://en.wikipedia.org/wiki/10:08 for clarification. There is an image at http://en.wikipedia.org/wiki/Talk:10:08 [Don]

Well, the answer is either 22 or 44 times in 12 hours, depending on whether you also allow the hands to point to the short side of the embedded rectangle. Suppose x is the fraction of the circle taken up by the long side of the embedded golden rectangle. Then the times are:

${\frac {12}{11}}(n+x),{\frac {12}{11}}(n+1-x)$ and optionally ${\frac {12}{11}}(n+{\frac {1}{2}}+x),{\frac {12}{11}}(n+{\frac {1}{2}}-x)$ hours since midnight or noon, where n is an integer in the range from 0 to 10. Arthur Rubin | (talk) 17:27, 25 February 2006 (UTC)

Using "SohCahToa" trigonometry, I found that the central angle that intercepts the long side of the embedded golden rectangle is approximately 116.565 degrees. That makes your "x" = 116.565/360 = 0.32379. If we let n=9, the (12/11)(9+x) formula gives 10:10:17.07, one instance of "golden time" that I found earlier by a different means. However, I confess that I do not understand why your formulas work. Please explain how you derived them. Thanks! [Don]

OK, the position of the hour hand at time y hours (measured as a fraction of the circle) is

${\frac {y}{12}}\mod 1$

The position of the minute hand is

$y\mod 1$

so the angle between them is

$\theta (y)={\frac {11}{12}}y\mod 1$

Solving θ(y) = x gives the desired result with the minute hand to the "right" (clockwise) of the hour hand with the two hands forming the long side of a golden rectangle. The other formulas cover the other orientations. Arthur Rubin | (talk) 18:19, 26 February 2006 (UTC)

Arthur, I appreciate your interest in my "Golden Time" inquiry. Unfortunately, I don't understand your latest explanation. If I look at 10:10:17.07 = 10.171409246, as an example, what would y/12 mod 1 and y mod 1 produce? [Don]

I would have thought the puzzle of determining the time from the angle between the hour and minute hands would have been somewhere on Wikipedia, but I can't find it. For the meaning of "mod", see the article on modular arithmetic, ignoring anything relating to multiplication and division. In the instant case (10:10:17.07 above):

y mod 1 = 0.171409246, i.e. the minute hand is 17.1% of a revolution clockwise from noon.

y/12 mod 1 = 0.8476174372, i.e. the hour hand is 84.8% of a revolution clockwise from noon.

(y - y/12) mod 1 = 0.323791809, i.e. the angle between the hands is 116.565 degrees.

Arthur Rubin | (talk) 15:43, 27 February 2006 (UTC)

It was good for me to review "clock" or modular arithmetic. I now understand that mod 1 returns the decimal part of a number. In our problem, it is the fraction of a revolution from noon. Please respond to a couple more questions.

1. How does one "Solve θ(y) = x" or "x = 11/12y mod 1"? Is it appropriate to add an integer "n" to x when dropping the "mod 1" since mod 1 only returns the remainder when dividing a number by 1?

The answer to your second question is "yes". — Arthur Rubin | (talk) 10:20, 3 March 2006 (UTC)

2. How did you know that n should be an integer from 0 to 10 rather than from 0 to 11? [Don}

because n=11 corresponds to y>12,and the clock returns to its original position in 12 hours. — Arthur Rubin | (talk) 10:20, 3 March 2006 (UTC)

Thanks for your help. (I have learned how to log in!) Don don 15:49, 4 March 2006 (UTC)

What sort of a function is X^X?

A variable to a constant is a parabola, a constant to a variable is exponential, but what is a variable to a variable?

It is neither of those. X^X is equivalent to e^{X ln(X)}, but I don't know a name for a class of functions that contains that. —Keenan Pepper 17:22, 25 February 2006 (UTC)

I've never really thought of it before, but perhaps Transcendental function? Beautiful beautiful topic: Differential Galois theory. --HappyCamper 18:36, 25 February 2006 (UTC)

Also, note that the function y = x^c is only a parabola if c=2. In general, if c is a constant integer, the function is called a polynomial. ×Meegs 03:11, 26 February 2006 (UTC)

Since we're on this topic, how do we find the lowest point of $f(x)=x^{x}$? Derivating it we get $f'(x)=x^{x}\ln x+x^{x}=e^{x\ln x}\ln x+e^{x\ln x}=e^{x\ln x}(\ln x+1)$, but I can't go any further. ☢ Ҡi∊ff⌇↯ 05:12, 26 February 2006 (UTC)

Psh, I guess I was too tired when I tried this yesterday.

$e^{x\ln x}(\ln x+1)=0$

This will happen when either $e^{x\ln x}$ or $\ln x+1$ are zero, but this only happens for $\ln x+1$. So we have $\ln x=-1$, which gives us $x=e^{-1}$. Bah! ☢ Ҡi∊ff⌇↯ 05:22, 26 February 2006 (UTC)

Define g(x) = ln (f(x)), where f(x) = x^x. Then, since the logarithm is a monotonically increasing, you can just find the minimum of g(x), which in turn will let you find the minimum of f(x). The answer should be "reasonably nice number". Sort of a nifty trick which is very useful in statistics and detection theory. Not to mention statistical thermodynamics. --HappyCamper 05:23, 26 February 2006 (UTC)

Nice trick. Thanks for that! ☢ Ҡi∊ff⌇↯ 06:28, 26 February 2006 (UTC)

Yes, that trick comes in very handy. Let's say you have a nasty exponentiated multivariable beast. What happens, is that very often, when you take the derivative with respect to a certain variable, a lot of the other irrelevant variables just disappear! --HappyCamper 15:02, 26 February 2006 (UTC)

$e^{x\ln x}(\ln x+1)=x^{x}(\ln x+1)$. Which is strictly increasing and isn't real for x < 0 and undefined at x = 0. —Ruud 05:24, 26 February 2006 (UTC)

I just got to $e^{x\ln x}$ because I thought that'd be important for the solution, but in the end leaving it as x^x would be the same. Bleh. ☢ Ҡi∊ff⌇↯ 06:27, 26 February 2006 (UTC)

On second thought, it seems to be quite an interesting function when x < 0. —Ruud 05:45, 26 February 2006 (UTC)

Agreed. I've been trying to make a 3D plot of it for a while but I don't have any program to do so. ☢ Ҡi∊ff⌇↯ 06:27, 26 February 2006 (UTC)

Looks like an exponentially damped corkscrew for x<0.

-lethe ^{talk +} 07:08, 26 February 2006 (UTC)

Higher Maths Formula

Hi there, I searched for one of the numerous parabola formulas: $y=k(x+a)(x+b)$ formula, but got no hits; I also searched the parabola article. Can anyone give me the specific article for this formula? Thanks, ^KILO-_LIMA 22:47, 25 February 2006 (UTC)

Assuming k,a,b are constants and x is a variable, then it's a form of Quadratic equation. --BluePlatypus 22:59, 25 February 2006 (UTC)

It's called a merkwaardig product in Dutch but I wouldn't know how to translate that into English. Maybe irreducible polynomial is of any help? Cheers, —Ruud 23:01, 25 February 2006 (UTC)

I've heard it being called "Factored form", but I don't think there is a Wikipedia article on it, and no section on it in the quadratic equation article. --Borbrav 01:51, 26 February 2006 (UTC)

The article on polynomials calls it 'factored form'. It's not specific to quadratics. --BluePlatypus 03:17, 26 February 2006 (UTC)

Yes, I have heard it referred-to as the "factored form" as well. Among other things, this form makes it easy to find the roots (the "x-intercepts") of a function. The first section of the article Factoring, entitled "Factoring a Quadratic Equation" should help you in dealing with parabolas. ×Meegs 03:23, 26 February 2006 (UTC)

February 26

Mechanical aproximations of functions

I've been interested on mechanisms to aproximate certain functions in the physical world. For example, you can draw a circle by fixing a point somewhere and rotating a fixed length thing around. An ellipse can be drawn with a string with ends fixed on the foci. Sure, easy, because they can be defined like that anyway. But, I'm looking for methods for a few particular functions, not sure if they are even possible, but it's worth asking...

1. sin x
2. x² (Keenan Pepper's idea will do)
3. 1/x
4. e^x
5. ln x
6. (e^θ,θ) (logarithmic spiral)
7. the catenary

Note that I'm not looking for straight geometric algorithms, like using a ruler\straightedge and compass, or careful measures (except on the case of certain, necessary proportions), but mechanical devices that actually "plot" these curves in their natural movement. Also, they must be table-top gadget things, so saying I could throw a sphere on a plane to plot x² isn't really what I want. These devices can include strings, pulleys, rods, wheels, gears, trails, etc.

Additionally, I'd like to know if there's any interesting, physical method to approximate e, something akin to Buffon's needle?

Well, that'll be all. :P ☢ Ҡi∊ff⌇↯ 13:25, 26 February 2006 (UTC)

Okay, at least for sinx, if you agree to some level of sophistication, you can create a mechanical device which when you rotate it, it moves a long sheet of paper and simultaneously moves a pen in an essentially harmonic motion, thus plotting a sine wave on the paper (I hope you understood what I meant, couldn't find the best words). As for some other curves, and without using such gizmos, I have an idea which I haven't really thought out through, but it just might be crazy enough to work: Suppose you wrap 2 strings around a wheel in opposite directions - so that when the wheel rolls, it releases one string and pulls the other. If the strings are kept in tension, this should keep the difference of lengths of the strings constant. So, by wrapping them around properly placed pulleys\nails, and attaching them to a pen, you can create a curve where the difference of distances between 2 points is constant - Which I believe is a hyperbola, essentially plotting 1/x. With the proper setting, I think this idea can allow plotting a curve where any given linear combination of distances between any number of arbitary points is constant - Opening many possiblities, and with some tweaking perhaps including a parabola. What do you think? -- Meni Rosenfeld (talk) 14:23, 26 February 2006 (UTC)

For the sine, the problem with the rotating wheel is to extract the movement in one axis without distortion. I've been thinking of a few devices, but they end up with either a cycloid or some weird shape I can't name. ☢ Ҡi∊ff⌇↯ 14:53, 26 February 2006 (UTC)

You shouldn't have to reinvent the cam. It already exists. moink 01:38, 3 March 2006 (UTC)

The catenary's an easy one: put a pen on the rim of a circle and roll the circle I'll have to think about the others. Confusing Manifestation 14:31, 26 February 2006 (UTC)

You're mixing it up with the cycloid. Funnily I, too, mistook it for the cycloid at first, but followed the link to make sure. Besides, I'm not sure your idea is easy to implement in practice. -- Meni Rosenfeld (talk) 14:34, 26 February 2006 (UTC)

Yes, sorry, I just checked the article to confirm what I thought. I suppose to make a catenary you could try to find some way to make a "string" that can be frozen in place somehow ... Confusing Manifestation 14:41, 26 February 2006 (UTC)

Nah, not what I'm looking for :P ☢ Ҡi∊ff⌇↯ 14:53, 26 February 2006 (UTC)

I'm not sure about the transcendental functions, but the conic sections should be possible. For the parabola, fix one end of a piece of string to the focus, and let the other end move along a line parallel to the directrix but on the opposite side of the focus. Draw the parabola by moving the pen so that the string remains taut.

The hyperbola is much trickier. I need to think about it longer. —Keenan Pepper 16:16, 26 February 2006 (UTC)

This one will do. Thanks! ☢ Ҡi∊ff⌇↯ 18:16, 26 February 2006 (UTC)

For x², would it work to choose the focus and the directrix and attach a string that is fixed to the focus and can move along the directrix, attach your pencil to the midpoint of the string, pull it taut, and trace? I guess this would work for any conic section, so it would do 1/x as well (simply adjust the eccentricity). -lethe ^{talk +} 16:18, 26 February 2006 (UTC)

Not the directrix itself, but a line parallel to the directrix. The same thing does not work for a hyperbola. —Keenan Pepper 16:51, 26 February 2006 (UTC)

I don't understand. Which parallel line will you choose? A conic section is defined as the locus of points with fixed ratio of distance between the focus and directrix. -lethe ^{talk +} 16:55, 26 February 2006 (UTC)

Any parallel line; it doesn't matter. If the string can slide easily along the line, and the string is taut, then it's perpendicular to the line. How far away the line is only affects how much of the parabola you can draw. —Keenan Pepper 17:01, 26 February 2006 (UTC)

For the parabola, you're right, it doesn't matter how far the parallel line is. But for the other conics it does, you have to use the directrix. As far as not being able to draw very much, I guess we'll have to use an elastic string but make sure that the pencil stays at the right part (hmm... that would be difficult). -lethe ^{talk +} 17:06, 26 February 2006 (UTC)

Wait, elastic? What are you talking about? In my construction, the string must have a fixed length, and the pen can move along the string as long as the string remains taut... —Keenan Pepper 17:13, 26 February 2006 (UTC)

In my construction, the pen remains fixed exactly 1/(1+e) percent of the way from the directrix. This insures that the ratio between the distances is always e. The ratio remains fixed, but the total length will grow to infinity in a hyperbola or parabola, so we need an elastic string. -lethe ^{talk +} 17:19, 26 February 2006 (UTC)

I think my method is more practical. How are you supposed to keep the elastic straight? —Keenan Pepper 17:22, 26 February 2006 (UTC)

Keep the elastic straight by pulling taut. The more serious problem I guess is how you keep the ratio fixed in an elastic string. -lethe ^{talk +} 17:27, 26 February 2006 (UTC)

Maybe some sensitivity to tension: if the two parts of the elastic have equal tension, then the ratio can be kept fixed. -lethe ^{talk +} 17:45, 26 February 2006 (UTC)

Agreed. Also, the fact there'll be a limit to the drawing is irrelevant. I don't intend to use this to infinity, you know. ☢ Ҡi∊ff⌇↯ 17:25, 26 February 2006 (UTC)

I don't need elastic because it goes to infinity, only because the length changes. I didn't mean to suggest that an elastic string will actually allow you to draw an infinite graph in finite time. -lethe ^{talk +} 17:27, 26 February 2006 (UTC)

Sure, but still, the elastic wouldn't be a good idea. I think there must be some more ingenius way without using strings. ☢ Ҡi∊ff⌇↯ 17:36, 26 February 2006 (UTC)

Well if you don't like my elastic, see Keenan's idea. -lethe ^{talk +} 17:45, 26 February 2006 (UTC)

Yeah, his works great. ☢ Ҡi∊ff⌇↯ 18:14, 26 February 2006 (UTC)

For the hyperbola, I think you can do it by fixing one end of the string to one focus and letting the other end move around a circle centered at the other focus. A practical problem is getting the string to move smoothly around the circle without sticking and slipping. —Keenan Pepper 16:56, 26 February 2006 (UTC)

I don't understand why focus/directrix won't work for the hyperbola. It's basically an exact mechanical implementation of the definition. -lethe ^{talk +} 17:11, 26 February 2006 (UTC)

Well, I managed to make a approximation of sine. The method needs some polishing but it works. The others seem to be a lot harder to derive mechanically, so I'm dropping the idea. Still, I wonder if there's any method to approximate e, like Buffon's needle for pi. Any thoughts on that? ☢ Ҡi∊ff⌇↯ 15:29, 2 March 2006 (UTC)

If you have n independent events, each with probability 1/n, then the probability of exactly one of them happening approaches 1/e as n increases. —Keenan Pepper 21:54, 3 March 2006 (UTC)

For one, you may be interested in the tractrix. Dysprosia 10:25, 5 March 2006 (UTC)

x^2 = 2^x

There are three solutions for $x^{2}=2^{x}$, x = 2, x = 4 and the other one is negative. I have absolutely no idea how to find this, and I've tried everything I know. How do we solve things like these? ☢ Ҡi∊ff⌇↯ 14:47, 26 February 2006 (UTC)

Did you mean "how to find that these are the only roots" or "how to find the negative root"? If the former, that's easy by analysing the functions. If the latter, this is an essentially non-algebraic equation so the root can't be given explicitly in terms of elementary functions, but it should be possible to represent it with non-elementary functions (I can try to work that out if you'd like), and you can approximate it with any numerical root-finding algorithm (it is roughly -0.76666469596212309311). Hope that's what you meant... -- Meni Rosenfeld (talk) 14:58, 26 February 2006 (UTC)

Oh, so I was trying in vain... Nice to know. ☢ Ҡi∊ff⌇↯ 15:39, 26 February 2006 (UTC)

You may be looking for Lambert's W function. —Ruud 15:21, 26 February 2006 (UTC)

Yeah, that does it. Thanks! ☢ Ҡi∊ff⌇↯ 15:39, 26 February 2006 (UTC)

This is (sequence A073084 in the OEIS) - Fredrik Johansson 15:24, 26 February 2006 (UTC)

I wonder if anything can be said by letting x = a + bi ... when you do this, you get two sets of equations by equating real and imaginary parts

${\frac {a^{2}-b^{2}}{2^{a}}}=\cos(b\ln 2)$

${\frac {2ab}{2^{a}}}=\sin(b\ln 2)$

Dividing the first by the second, we get

${\frac {2ab}{a^{2}-b^{2}}}=\tan(b\ln 2)$ - and ah! This can be written as a quadratic equation

$a^{2}\tan(b\ln 2)-2ab-b^{2}\tan(b\ln 2)=0\;\,$ and I'm sure something can be said about the admissibility of these roots, the conditions on a and b, et cetera... --HappyCamper 15:45, 26 February 2006 (UTC)

Since b = 0, this doesn't help. Fredrik Johansson 16:18, 26 February 2006 (UTC)

Why is b = 0 ? --HappyCamper 18:54, 26 February 2006 (UTC)

We're considering the function only for real values of x. ☢ Ҡi∊ff⌇↯ 19:00, 26 February 2006 (UTC)

Oh. Hm...a pity. :-) --HappyCamper 19:04, 26 February 2006 (UTC)

A new form of logic?

Goedel's incompleteness theorem only applies to first order logic, right? If so, would that mean that it might be possible to create a different form of logic that is sound, complete, etc. AND is capable of doing what Goedel's incompleteness theorem says that first order logic is incapable of (i.e. is also capable of making a complete and consistent system of math)? —The preceding unsigned comment was added by 86.138.233.25 (talk • contribs) 15:49, 26 February 2006 .

In a sense, yes. Let's specialize to the case of the natural numbers (too hard to talk about all of "math" because it's not clear where math ends). You could have a generalized "logic" that would, by fiat, "conclude" that a statement about the natural numbers is true, just in case it is true. But it wouldn't be very helfpful to you when trying to make deductions, because you wouldn't know whether something was a correct deduction, until you knew whether the conclusion was true.

Bottom line is, if your "logic" is one such that the question of what constitutes a correct deduction is checkable by a fixed computer program, it can't get you out from under the Gödel theorems. (Rant: will everyone please notice there are two Gödel incompleteness theorems?) --Trovatore 16:03, 26 February 2006 (UTC)

I think no. Gödel's incompleteness theorem is very general, and it would probably apply to any extended logic as well, becuase that extended logic would be able to do anything first-order loic can do. – b_jonas 12:22, 27 February 2006 (UTC)

No, it's simply not true that the first incompleteness theorem applies to "any extended logic"; see the example I give above. For a less contrived example, it doesn't apply to second-order logic. A few simple axioms completely determine the whole theory of the natural numbers, in second-order logic.

But while that fact about second-order logic is important for theoretical purposes, it doesn't make much practical difference in terms of trying to figure out what's true about the natural numbers. That's because no one knows how to make deductions, in general, in second-order logic (and checking whether such deductions are correct provably cannot be reduced to a computer program. --Trovatore 14:51, 27 February 2006 (UTC)

Yes. I think it was Kolmogorov who developed non-Godelian logic that is consistent and complete.  Grue  21:23, 3 March 2006 (UTC)

What is this beautiful abstract nonsense trying to say to me?

Could someone please explain what the Nine lemma is, and why it is important? Where is this lemma used, and what is it really trying to say in the article? --HappyCamper 15:58, 26 February 2006 (UTC)

Well, I guess there is some theorem that uses the nine lemma, that's why it's called a lemma. I'm not sure which, but I can tell you that the closely related snake lemma is the key for proving the existence of the derived functors, which are studied in homological algebra. I've always viewed the 3x3 lemma (its name as I learned it) as an application of the snake lemma, but I suppose it must have been used in the proof of a theorem as well. -lethe ^{talk +} 16:35, 26 February 2006 (UTC)

Converting trigonometric expressions into algebraic form

How do I go from $\csc(\arctan \left({\frac {\left(x\right)}{\sqrt {2}}}\right))$ to ${\frac {\sqrt {x^{2}+2}}{x}}$? I've been staring at problems of this sort for a couple days, and I'm at a complete loss. --Theshibboleth 20:19, 26 February 2006 (UTC)

In other words, you're trying to find $csc(\theta )$ where $tan(\theta )={x \over {\sqrt {2}}}$. Picture a right triangle in which one of the angles is $\theta$, the opposite leg is x, and the adjacent leg is ${\sqrt {2}}$. Now it should be easy. —Keenan Pepper 21:39, 26 February 2006 (UTC)

February 27

Parabolas and straight lines

Hi, I am aware of the no-homweork questions, but this question is really beginning to annoy me; and I hope you won't be bothered by helping me complete it! It concerns parabolas and straight lines:

File:Parabolas34.JPG

Question 9(a) and (b) I have done correctly. However I am having problems with (c). It begins:

$-2x^{2}+4x+6=2x+2$ - The $2x+2$ is from the line. Then, becuase this is a quadratic question, it must equal zero, so: $-2x^{2}+2x+4=0$. Then change to a positive $x^{2}$ term: $2x^{2}-2x-4=0$. This then factorises to $(2x-4)(x+1)=0$. So $2x-4=0$ so $x=2$ and $x+1=0$ so $x=-1$. Now this is where the problem occurs. Becuase we have the x-coordinate, put it back into the formula to get the y-coordinate. So using $x=-1$: $y=2x^{2}-2x-4$, $y=2+2-4$, and $y=0$. So the coordinates are $(-1,0)$—but looking at the graph this is already there.

So, using $x=2$: $y=2x^{2}-2x-4$, $y=8-4-4$, and therefore $y=0$. So the coordinates for this one are $x=(2,0)$—but this is clearly not correct by looking at the graph. Does anybody know where I am going wrong? I appreciate to the highest on this question. If you cannot answer, then no problems caused. Thank you very much again. ^KILO-_LIMA 20:33, 27 February 2006 (UTC)

You need to go back to the original equation $-2x^{2}+4x+6=2x+2$ and work from that. In other words, the parabola shown on your graph is $y=-2x^{2}+4x+6$, and the straight line is $y=2x+2$. Chuck 21:00, 27 February 2006 (UTC)

(Chuck Carrol was faster than me in an edit conflict, but I'll post this anyway.)

You're susbtituting the polynomials back to the wrong formula. $x=2$ is indeed the solution, but then $-2x^{2}+4x+6=-2\cdot 2^{2}+4\cdot 2+6={}$$6=2\cdot 2+2=2x+2$.

Anyway, I think this doesn't count a homework question, because you show that you have done efforts to solve it, not just ask us to solve it, and because you ask a concrete question. So, I think there's no problem posting this. – b_jonas 21:06, 27 February 2006 (UTC)

And you were nice about it, which never hurts! — QuantumEleven | (talk) 08:52, 1 March 2006 (UTC)

February 28

G. Cantor

From what I've read, I know that the general consensus among mathematicians is that 'Cantor was right'...and they agree even more because of Quantum Mechanics (I don't know why)... I'm also aware that Cantor was religious and sort of 'prooved' the existence of God with his theory about transfinite numbers and said that God was the 'actual infinite' or 'actus purisimus' or something like that... My question is...If mathematicians accept as truth Cantor's theory, and supposedly it's 'proved' (I don't know how)...then doesn't that mean that all mathematicians are theists?...that would be the logical consecuence...but I'm sure this is not the case...so why...I mean, how do non-theist mathematicians 'escape' Cantor's reasoning?.--Cosmic girl 02:34, 28 February 2006 (UTC)

Cantor's Absolute Infinite doesn't follow logically from his formal mathematical works; in fact, it's inconsistent with them. It's just a beautiful way of integrating his faith with his mathematics. —Keenan Pepper 03:39, 28 February 2006 (UTC)

Hi Pepper!...why is his Absolute Infinite inconsistent with his mathematical works? I guess I know why, but maybe I'm wrong, can you explain me?.--Cosmic girl 04:27, 28 February 2006 (UTC)

Cantor was certainly a believer, but I'm not aware that he ever published a proof of God's existence. Are you sure you aren't confusing him with Gödel? He did have such a purported proof; see Gödel's ontological proof. --Trovatore 03:42, 28 February 2006 (UTC)

Nope...I'm not :) lol.thanks for your answers to the Gödel question by the way! XD --Cosmic girl 04:34, 28 February 2006 (UTC)

The premise of the question is nonsense. Mathematics can only prove mathematical theorems; it can prove nothing about physical reality, much less about metaphysical reality. Furthermore, the reasoning in this question is horrible: It rambles from a handwave "consensus among mathematicians" (who?) that "Cantor was right" (about what?) and speculation that Cantor "sort of proved" (is that vague enough?) the existence of "God" (defined how?) to a question of "escaping" (eh?) Cantor's reasoning. Atheists are a diverse group, but I would expect most to treat such arguments with even less interest and credibility than an alleged sighting of the Loch Ness Monster drawing crop circles. --KSmrq^T 03:50, 28 February 2006 (UTC)

Hahaha...okok! you don't have to yell at me! I'll explain myself... first of all, by consensus among mathematicians I meant that Cantor's theory is regarded as 'true' and 'proved' now, and in his time he was criticized...and I made it clear that I didn't know how is it 'proved', since I don't know the math of it in the first place and I admit it. second...by God I mean 'actual infinity' like Cantor did,and by 'escapinc Cantor's reasoning' I mean that since he said he 'prooved the existence of an actual infinite which he equated with God'...I believe there must be detractors...what I asked was if the detractors had consistent arguments and theorems like Cantor's to 'disprove' 'actual infinity' or 'god'.

ps. take it easy KSmrq...I'm dumb.--Cosmic girl 04:32, 28 February 2006 (UTC)

Cantor did not equate what's ordinarily called the "actual infinite" with God. In fact he went out of his way not to. That's where the word "transfinite" came from; as I understand it: He wanted a word for something that wasn't finite, but wasn't the Infinite either. --Trovatore 04:36, 28 February 2006 (UTC)

He did equate the actual infinite with God...and I believe transfinite numbers stand for numbers that are composed of an infinite ammount of numbers that are infinite as well...not something that isn't finite nor infinite...but correct me if I'm wrong.--Cosmic girl 04:42, 28 February 2006 (UTC)

OK, you're wrong :-). No, he simply did not equate what most people call "the actual infinite" with God. He didn't call it "actual infinite", but rather "transfinite". Your facts are simply wrong here. --Trovatore 04:46, 28 February 2006 (UTC)

Are you sure? ok, I bet you are right...but my understanding was that the actual infinite was a 'real infinity' and transfinite numbers where infinite numbers but only potentially... :S. --Cosmic girl 05:02, 28 February 2006 (UTC)

You're mixing up two (quite different) distinctions: potential-vs-actual, and transfinite-vs-absolute-infinite. The set of all natural numbers, for example, is an "actual" infinity, but not an "absolute" infinity, because there are bigger things (such as the set of all real numbers). The article Absolute Infinite claims that he identified "absolute infinity" with God, but if you look at his writings on it, I think that's an oversimplification. I seriously don't think he thought God was any sort of mathematical object. He was much too traditionally Christian for that. --Trovatore 05:18, 28 February 2006 (UTC)

I see...thanx :).--Cosmic girl 16:57, 28 February 2006 (UTC)

Integration

Hello mathematicians, I've been having fun with Euler's equation and have managed to juggle a homework equation up to the point where I'm left with

$\int {\frac {dy}{\sqrt {1-A(y+1)^{2}}}}$, where A is a constant. This looks to me like I should be getting y equal to something with arcsin or arccos, and need a push in the right direction. Sorry this is a homework question guys, but pretend it isn't

(Can someone put this into maths syntax if possible, ;) ): Thanks --131.251.0.8 10:02, 28 February 2006 (UTC)

Check out Trigonometric substitution. You need to work a little on the integral to bring it to the form discussed there, but it isn't difficult. -- Meni Rosenfeld (talk) 11:36, 28 February 2006 (UTC)

Incidentally, Euler's equation already redirects to Euler equations, and when it comes to Euler it helps to specify things a bit more (see List of topics named after Leonhard Euler to see why) Confusing Manifestation 11:45, 28 February 2006 (UTC)

Actually, scratch that, we have both Euler equations which are for fluid dynamics and Euler's equations for rigid rotation. Could we perhaps have some clarity here? Confusing Manifestation 11:47, 28 February 2006 (UTC)

The user probably searched for the article with a space between Euler and 's, just like in the above link, so he\she didn't find the article. ☢ Ҡi∊ff⌇↯ 12:13, 28 February 2006 (UTC)

Cool, I've done that, and have ended with a dodgy equation in the form:

2cos(ci+i) = cos(ci), with c constant and i the sqrt(-1). Surely there's no solution to cos(i) is there? cosine only applies to real numbers surely. --131.251.0.7 12:23, 28 February 2006 (UTC)

Cosine applies to complex numbers (see also Cosine#Relationship to exponential function), but you shouldn't have gotten there - I guess you used a wrong substitution. Follow the example in Trigonometric substitution which is identical to your case after some simplification. -- Meni Rosenfeld (talk) 13:02, 28 February 2006 (UTC)

Drat, once again my impatience causes my edit to be lost. (Closed window before seeing edit conflict.) Anyway, trigonometric substitution is truly the correct choice. The core trigonometric identity is cos² θ+sin² θ = 1, which we can solve for either trig function. So, letting B²=A, set sin θ = B(y+1); then the denominator neatly simplifies to cos θ. Next, dy must be converted; but since y = −1+(sin θ)/B, clearly dy = (cos θ) dθ/B. Thus the form in the integrand becomes trivial to integrate as a function of θ, and all that remains is to convert θ back to a function of y in the result. We are playing fast and loose here, and ignoring issues that would arise with a definite integral. (What if A is zero? Does the sign of B matter? Do we lose a needed branch by taking the positive square root, cos θ = +√(1−sin² θ)? Should there be limits on the range of y?) Depending on your instructor's tastes, this is probably acceptable for such a homework question.

"square root" of exponential function

Richard Feynman once asked if there was a "natural" "square root" of the exponential function - that is, a function f defined on the reals such that

$f\left(f(x)\right)=e^{x}$

It's "obvious" that there are continuous solutions, and less obviously $C^{\infty }$ solutions, and I've been told there are real analytic solutions, but is there a "natural" solution?

One can find all continuous solutions by letting a be an arbitrary real strictly between 0 and 1, and f an arbitrary monotone increasing continuous function from [0,a] onto [a,1]. One can then extend f uniquely to a solution of the specified equation.

It's not too difficult to do the same with $C^{\infty }$ solutions -- all you need to do is for the initial f to be $C^{\infty }$ and the formal derivatives of f(f(x)) must match the derivatives of e^x at x=0.

Also "obviously", if there is a "natural" solution g of

$g\left(e^{x}\right)=g(x)+1,$

then we could take

$f(x)=g^{-1}\left(g(x)+{\frac {1}{2}}\right)$

Any ideas? Arthur Rubin | (talk) 16:00, 28 February 2006 (UTC)

How many real analytic functions are there? If there's only one, then you could consider that a "natural" solution.

However, I think it's quite possible that there is no single "natural" solution. Iirc, Concrete Mathematics says that there's no single best way to extend Stirling numbers to arbitary real arguments. – b_jonas 21:02, 28 February 2006 (UTC)

I've looked this up in the Art of Computer Programming chapter 4.7 (which is in volume 2), and if I understand it correctly, it says that for an analytic function $f$, there are exaclty two analytic functions $g$ for which $g(g(x))=f(x)$, with some conditions of course. – b_jonas 21:10, 28 February 2006 (UTC)

Interesting question. Confusing. I got the idea to work with polynomial approximations first (so as to give hint of what an analytic solution might look like). So take f(x) to be a polynomial of degree n, and try to match up terms with the polynomial $e_{N}(x)=\sum _{k=0}^{N}x^{k}/k!$ where $N=n^{2}$. Works fine for n=1, breaks for anything larger, since the problem is over-determined: there are $n^{2}+1$ equations for $n+1$ unknowns. One might hope for some "happy accident" that would make most of these equations be degenerate, but I don't see how. I tried this by pencil and paper, and, unless I made a mistake, there is no solution for n=2. So I don't see how there can be an analytic solution, unless I'm overlooking something. linas 01:17, 1 March 2006 (UTC)

That doesn't in itself show the non-existence of an analytic solution. Consider $f(f(x))=x^{2}$; the solution $f(x)=x^{\sqrt {2}}$ is analytic but $a(ax^{2}+bx+c)^{2}+b(ax^{2}+bx+c)+bx+c=x^{2}$ does not have a solution. Here is a list of references to similar problems: [3]. Kusma (討論) 02:07, 1 March 2006 (UTC)

Err, well, I phrased that badly, in several ways. The very first sentance posing this problem shows that there are zillions of analytic functions that "solve" this (pick a real a, s.t. etc.) However, most of these choices will not be differentiable at x=1. And if I force $C^{\infty }$ at x=1, then they won't be differentiable at x=0. Also, all I've managed to demonstrate is that working with monomials is a trap (as it often is). So I'm trying to dream up other sequences of functions that approximate/converge to an answer ... linas 02:01, 1 March 2006 (UTC)

Further follow up. The proof that there exist $C^{\infty }$ solutions is as follows, with some additional effort. Suppose f is $C^{\infty }$ on [0,a].

Write, formally,

$f(x)=\sum _{k=0}^{\infty }{a_{k}x^{k}}$ for x near, but greater than 0, (with a₀ = the a specified above in the continuous solution).

$f(x)=\sum _{k=0}^{\infty }{b_{k}(x-a)^{k}}$ for x near, but less than a.

(These 'equalities' mean that all derivatives are equal, not necessarily that the series converge.)

If, working as formal power series, $f(f(x))=\sum _{k=0}^{\infty }{\frac {x^{k}}{k!}}$, then the resulting extension will be $C^{\infty }$.

Now, given an arbitrary sequence a, (satisfying the constraints that $0<a_{0}<1$ and $0<a_{1}$) we can solve for the corresponding sequence b, and construct a $C^{\infty }$ bridge f, such that $f'(x)\,>0$ for $0\leq x\leq a$, which will then satisfy all the conditions, and can be extended to a full $C^{\infty }$ solution.

Arthur Rubin | (talk) 02:49, 1 March 2006 (UTC)

Furthermore, $f(x)=x^{\sqrt {2}}$ is not analytic in any sense near x=0, as $f''\,$ does not exist there.

Yes, well, silly me. Where I was trying to go with this was to look at sequences of functions, since "naturalness" often manifests itself in terms of some related sequence which has some particularly nice properties. But I'm taking a naive aproach here, clearly there's a raft of literature, which Kusma references. linas 03:07, 1 March 2006 (UTC)