Wikipedia:Reference desk/Archives/Mathematics/2008 March 8

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March 8

Index of matrix

So my professor posted a question: Suppose A and B are n × n non-degenerate symmetric matrices and there is an n×n matrix C such that A = C(transpose)BC. Prove that the index of A equals that of B. my question is what in the world is the index of a matrix?199.74.71.147 (talk) 01:38, 8 March 2008 (UTC)

You might see if Sylvester's law of inertia sounds right. Otherwise, the only index I know of is the one described in Atiyah–Singer index theorem, which is probably not what you want since the index of a symmetric matrix should always be 0. JackSchmidt (talk) 02:45, 8 March 2008 (UTC)

I think it had something to do with the number of negative numbers in some matrix which described extrema or something? —Preceding unsigned comment added by 199.74.71.147 (talk) 04:30, 8 March 2008 (UTC)

I've never heard of the index of a matrix - I suggest you ask your professor. --Tango (talk) 12:54, 8 March 2008 (UTC)

If A is a real symmetric matrix then the number of positive eigenvalues of A is called the index of A.--Shahab (talk) 08:39, 9 March 2008 (UTC)

Pseudoinverse of a Matrix

Okay, the first question is that in the Pseudoinverse article here at Wikipedia, there are a couple of identity transformations which, if multiplied by some expression, can be used to expand or reduce expressions. My question is does it matter how or where is the expression multiplied? Is it right multiplication or left multiplication? Do they work for ALL expressions? Does it matter if I multiply by a vector or a matrix.?

The second question is that I already know that $P=AA^{+}$ is a projector (easy to show by definition) but to which space does it project onto? A is any nxm (with $n\geq m$) matrix with real or complex entries and $A^{*}$ is its complex conjugate transpose. P would then be a nxn square matrix. I think that P maps a vector (x1,x2,...,xn) to (x1,x2,...,x_m-1,x_m,0,...0) where the last n-m entries are zero. Is this true? If it is then how to prove it because I can't seem to analytically prove my conjecture.

The third question is, in the same article, we have that $(A^{+})^{+}=A$. A is the same as above (an nxm matrix with $n\geq m$) and this is what I have so far.

$(A^{+})^{+}=(A^{+*}A^{+})^{-1}A^{+*}=(((A^{*}A)^{-1}A^{*})^{*}((A^{*}A)^{-1}A^{*}))^{-1}((A^{*}A)^{-1}A^{*})^{*}$
$=(A(A^{*}A)^{-1}(A^{*}A)^{-1}A^{*})^{-1}A(A^{*}A)^{-1}=(A(A^{*}A)^{-2}A^{*})^{-1}A(A^{*}A)^{-1}$
and from here I don't know what to do. I cannot distribute the inverse sign because I don't know if A will be invertible or not (A is rectangular). So any hint as to how to proceed will be appreciated. ThanksA Real Kaiser (talk) 03:44, 8 March 2008 (UTC)

In response to question one, the identities are true identities, so you can replace the left-hand side with the right-hand side (or vice versa) wherever you want. The clause "the right hand side equals the left hand side multiplied with some expression" is somewhat obscure. To take one example, the identity

A⁺ = A^* A^+* A⁺

says that you can replace A⁺ with A^* A^+* A⁺ wherever the former occurs in some expression. It might be useful to picture this as multiplying on the left by A^* A^+*. These identities come in left-right pairs, so you can expand expressions in either direction.

For question two, the pseudoinverse article indicates (in the applications section) that AA⁺ projects onto the image of A. Is this good enough for your purposes?

As for question three, the fact that (A⁺)⁺ = A follows from the symmetry in the definition of pseudoinverse. I'll retranscribe the four defining properties here, using B instead of A⁺ to make the symmetry easier to see:

ABA = A;

BAB = B;

(AB)^* = AB; and

(BA)^* = BA.

Notice that the properties as a whole remain the same if you swap A and B (and, to be pedantic, also swap m and n). This symmetry immediately implies that B is a pseudoinverse of A if and only if A is a pseudoinverse for B. Michael Slone (talk) 06:54, 8 March 2008 (UTC)

For, the second part, how can I show that A projects onto range(A). And, for the third part, I only have that the definition of $A^{+}=(A^{*}A)^{-1}A^{*}$ so I am trying to prove that $(A^{+})^{+}=A$ using only this definition. That is why I posted those steps. Is there anyway that it can be done?A Real Kaiser (talk) 08:40, 8 March 2008 (UTC)

For the second question, you can use the identity A = AA⁺A to show that the image of A is the same as the image of AA⁺. However, the way in which AA⁺ projects onto the image of A is not quite as straightforward as either padding or replacing entries with zeros. For example, if

$A={\begin{bmatrix}1&0\\0&1\\1&1\end{bmatrix}}$, then $A^{+}={\frac {1}{3}}{\begin{bmatrix}2&-1&1\\-1&2&1\end{bmatrix}}$,

which means that

$AA^{+}={\frac {1}{3}}{\begin{bmatrix}2&-1&1\\-1&2&1\\1&1&2\end{bmatrix}}$.

For the third question, your current approach will not give a proof that (A⁺)⁺ = A for an arbitrary matrix A, since you are treating a special case. The identity you are using, A⁺ = (A^*A)^–1A^*, only holds if A^*A is invertible, which need not be the case. The actual definition of pseudoinverse consists of four identities that a matrix and its pseudoinverse must satisfy, and one can verify that the identities remain unchanged if the names of the matrix and its pseudoinverse are swapped. Assuming that you already know that pseudoinverses exist, that is the proof that (A⁺)⁺ = A. Michael Slone (talk) 03:45, 9 March 2008 (UTC)

Uniqueness of best fit curves

Hi, I was wondering if somebody here could give me some pointers on how to prove the uniqueness of best fit curves basically i have all the formulae derived for polynomial curves up to degree n but I'm not sure how to go about having data points of different x values implies uniqueness Thanks,199.74.71.147 (talk) 04:37, 8 March 2008 (UTC)

Hi again, i've made some progress on the degree 2 polynomial (which is the main thing i need to prove) such that i have a system of equations

sorry i don't know how to do the wikipedia syntax for math, so here the sums are just from i=1 to i=3 for the three (x,y) coordiates so like the first sum is just x1^4+x2^4+x3^4 where x1 x2 and x3 are unique

sum x_i⁴ =lambda1*sum x_i³+lambda2*sum x_i²

sum x_i³ =lambda1*sum x_i²+lambda2*sum x_i¹

sum x_i² =lambda1*sum x+3*lambda2

so what i need to do is just prove that no lambda1 and lambda 2 exist that holds for all three equations and i've been looking at it for a while but can't figure it out199.74.71.147 (talk) 07:00, 8 March 2008 (UTC)

I think the solution would involve showing that the matrix formation of the equation that gives you your best fit curves involves an invertible matrix, and from that has a unique solution. Confusing Manifestation(Say hi!) 21:57, 10 March 2008 (UTC)

natural logarithm

what is a natural lgarithm? —Preceding unsigned comment added by 68.198.96.12 (talk) 06:02, 8 March 2008 (UTC)

See Natural logarithm. --~~MusicalConnoisseur~~ Got Classical? 06:59, 8 March 2008 (UTC)

Euler-Mascheroni constant: betting on its irrationality

Famously, the Euler-Mascheroni constant (γ) is not known to be irrational, let alone transcendental. But I think most mathematicians would be very surprised if it turned out not to be transcendental. Whether or not I am right to think this, what grounds might they have for strongly suspecting that γ is irrational? And transcendental? What sorts of odds would you give, on those questions?

A bit of mathematical epistemology – or mathematical doxastics, in fact.

–_⊥^{¡ɐɔıʇǝo}Noetica!^T– 09:08, 8 March 2008 (UTC)

The measure of the algebraic numbers in the reals is 0, so in comparison preciously few numbers are algebraic. For some numbers that are defined in such a way that you can easily determine they are very close to 0 or 1, there is a difficult proof that they are not just very close, but in fact equal. Apart from such cases, I don't know of any case of a number arising naturally that is known to be algebraic but for which the proof of algebraicity is difficult. There is nothing in the definition of γ that makes you suspect it might be rational. Combining these things makes the conjecture plausible.

There is no lack of numbers that have not been proved transcendental but that everyone would believe to be so: cos 1 + ln 3, sin cos 1, e/π, ad nauseam. I don't know if anything has been written about this.  --Lambiam 10:53, 8 March 2008 (UTC)

Thanks Lambiam. Yes, I vaguely understood about the sort of measure you speak of; I know more about the terminology after consulting your useful link. Please answer this, if it is a well-formed question: What is the Lebesgue measure of the rationals in the algebraics?

With my theme of mathematical epistemology in mind, I am intrigued by your third sentence. Among the "naturally arising" numbers known to be algebraic, those for which the proof is easy must outnumber those for which the proof is hard, right? By a considerable ratio, wouldn't you say?

A lot of this is highly technical in mathematics, and therefore beyond me; but I can still pose questions in "operational" form. Indulge me, please:

Should a prudent and rationally self-interested mathematician-gambler stake her entire fortune $F for a return of 1.000001*$F if γ is transcendental, and of $0 if γ is algebraic? If not, what is the lowest number you would intuitively put in place of 1.000001 to make the bet acceptable?

–_⊥^{¡ɐɔıʇǝo}Noetica!^T– 23:44, 8 March 2008 (UTC)

I don't think that's a well-formed question. As I understand it, the Lebesgue measure is only defined for subsets of $\mathbb {R} ^{n}$. The rational and algebraic numbers have the same cardinality, but that doesn't really tell us much. As for your bet - I don't know for sure about this particular constant, but for a randomly chosen real number (with uniform distribution), there is no return that would make it worthwhile. A randomly chosen real number is almost surely transcendental. --Tango (talk) 23:53, 8 March 2008 (UTC)

After thinking about it, the question we want to ask is: What is the index of the rationals as a subgroup of the algebraics? I would imagine it's (countably) infinite, but can't immediately prove it. --Tango (talk) 23:58, 8 March 2008 (UTC)

Thanks, Tango. I'll explore those technical matters further, beyond what I already know about cardinalities.

As for your point about the bet, of course I understand your general point about "randomly chosen" real numbers, but haven't you applied it wrongly to the betting scenario? If we are utterly certain that a real is "randomly" chosen, wouldn't a bet with a factor of 1.000001 on its being transcendental clearly be good value? A further concern: isn't it rather difficult to formalise, or operationalise, the random selection a real number? A concern beyond that: no matter how we do that formalising or operationalising, why should we think that this has any bearing on the nature of γ? To answer that, we must surely know something rather privileged about γ! Why should we think we have such privileged knowledge?

Interesting? You can perhaps see why I am moved to ask my original questions.

–_⊥^{¡ɐɔıʇǝo}Noetica!^T– 00:21, 9 March 2008 (UTC)

I, of course, had the bet thing backwards - *any* bet would be worth taking for the reasons given. For some reason I was thinking we were betting on it being algebraic, when you clearly wrote it the other way around. I'm not sure what the problem is with choosing a random real number, but it might be easier to choose one at random from the interval [0,1], then the probability density isn't infinitesimal. If we don't have any privileged knowledge of this constant, then the probability is exactly that for a random number. However, we do have quite a lot of knowledge about the number, so the probability probably is different. I would expect our knowledge of the number (for example, the fact that it can be expressed as a pretty simple formula) increases the chance of it being algebraic quite significantly. --Tango (talk) 01:03, 9 March 2008 (UTC)

[Corrected your "better" to "betting". :) ] Ah, I did not make clear what I meant by "privileged knowledge". It is hard for me to formulate without begging one or two questions. Anyway, you quite reasonably took me to mean the ensemble of all knowledge that we have about γ (like that presented in the article, except much more comprehensive).

Now suppose that we had asked exactly the same questions, but about π or e, before it was known that either of these was transcendental. Would the simple relations that either of these are involved in have increased our confidence that they were algebraic? Wouldn't we have been mistaken, to reason like that? (Perhaps not!) And surely those two constants are for present purposes comparable to γ – that is, in the history of mathematical discovery and in their derivations and roles in current mathematics.

I suppose the earliest mathematicians, reasoning in something like the same way, were justifiably amazed to find that √2 was irrational!

–_⊥^{¡ɐɔıʇǝo}Noetica!^T– 01:47, 9 March 2008 (UTC)

I would say that the chance of pi being transcendental given only the information we had before it was proven was less than the chance of a randomly chosen real number being transcendental. The chance of a random real number being transcendental is, effectively, 1. There wouldn't have been so many mathematicians interested in the problem if it was just a matter of proving something we already knew, so I'd say the chance was less than 1. Just because you lose a bet doesn't mean you were wrong to place the bet - you can only base your decision on information you have at the time, so future information doesn't change what was the correct course of action. --Tango (talk) 13:44, 9 March 2008 (UTC)

Tango: Q has infinite index in Q(√2), let alone in the algebraics. A better measure of the 'size' of the rationals in the algebraics might be the degree of the field extension (aka the dimension of the algebraics as a rational vector space). This is also countably infinite. Algebraist 01:57, 9 March 2008 (UTC)

Thanks Algebraist. I'll look that up, too. (But I may not understand it.)

I should say why the notion of randomly selecting a real number seems problematic. Try it! Any finitely specifiable procedure for uniquely identifying a real number, or naming it by anything standard means, must, it seems, require some biased limiting operation. For a start, any positive number you offer me that is less than 10^10^10^10^10^1010^10^10^10^42 is clearly biased: it is so suspiciously low, in the range of the real numbers!

–_⊥^{¡ɐɔıʇǝo}Noetica!^T– 02:23, 9 March 2008 (UTC)

Yes, there is no uniform probability distribution on the reals. That's why Tango changed his/her mind to a random number between 0 and 1. Btw, when you're talking about arbitrary reals, it's not very useful to talk about 'uniquely specifying' or 'naming' them: almost all reals are undefinable! (pinning down what 'definable' means in a non-paradoxical way is a bit tricky, but this'll be true however you do it) Algebraist 04:53, 9 March 2008 (UTC)

That's fine, Algebraist. That's the sort of difficulty I had in mind, but could not formulate rigorously. Two asides: First, there would be no uniform probability distribution on the natural numbers either, would there? Second, why is talk of randomly choosing a real number in the interval 0–1 any more straightforward? Are all those reals "definable" (any way you pin that down)? (Ignore these asides if answering them must assume mathematical apparatus that I clearly do not have at my command!)

Anyway, my question in bold, above, still stands. Perhaps there is no reasonable way to answer it  – or it is not for mathematicians qua mathematicians to answer. Meta-mathematicians? More likely. But still, a very slippery question for anyone, I suspect. It is a matter of epistemology, as I have said; and it is out of philosophical interest that I posed it. I wonder how one would pursue it further?

Thanks, all!

–_⊥^{¡ɐɔıʇǝo}Noetica!^T– 07:41, 9 March 2008 (UTC)

So first, no, a real number that you pick at random between 0 and 1 will almost surely be undefinable, for a notion of definability that you fix in advance. Once you've picked it, it will take infinite time for you to tell us which one it is. But that doesn't particularly bother mathematicians. Take your time. Let's assume an interest/inflation rate of zero so that the value of the bet doesn't evaporate before it can be exchanged.

It's an interesting question, more a question for foundations of probability than anything else, I'd say. One thing I'd throw into the mix is, who is it that's offering to make the bet? Maybe that person knows something about γ that you don't, and that can affect the Bayesian probability by a lot. (This is one of the slippery points in analyzing the Monty Hall problem.) --Trovatore (talk) 08:02, 9 March 2008 (UTC)

Or to put it another way, son, do not bet him, for as sure as you do you are going to get an ear full of cider. --Trovatore (talk) 19:21, 9 March 2008 (UTC)

modular question

how can i solve this problem by step to step? i will make computer program, due to this reason i ve got to learn it's solving method...

ax = c ( mod m )

( example: 5x = 7 ( mod 37) )

could someone explain it with ax = c (mod m).. thank you and best regards... Altan B. —Preceding unsigned comment added by 81.215.233.51 (talk) 12:26, 8 March 2008 (UTC)

Assuming a is coprime to m (as in your example), the obvious thing to do is to find the multiplicative inverse of a mod m and multiply it by c to get the answer. The obvious way to do this is the extended Euclidean algorithm. Algebraist 12:58, 8 March 2008 (UTC)

In the case that a is not coprime to m, there are two possibilities for each factor they share: either c is divisible by it as well, or it isn't. If it is, divide a, c, and m, all through by that common factor and keep going. If it isn't, there's no solution. Black Carrot (talk) —Preceding comment was added at 06:39, 9 March 2008 (UTC)

Free numerical ODE libraries

If you were writing a computer program and wanted to solve an ordinary differential equation from inside it, how would you do it? Would you implement a solution method (e.g. Runge–Kutta) yourself, or use a library? What free, open source libraries are available?

Accuracy guarantees aren't a big concern; I'm not going to use this to design jet engines. =) —Keenan Pepper 17:18, 8 March 2008 (UTC)

I'd be inclined just to write it myself as its more fun that way and you'll probably learn more. Otherwise there are a good few open source libraries out there, if your using java try Apache Commons Maths.--Salix alba (talk) 18:36, 8 March 2008 (UTC)

While you could write it yourself, libraries are likely to have better methods than you would implement. For example, it might have a method to optimally determine the step length in your Runge-Kutta integration. I tend to use Matlab for these kinds of things, but it's not free (though it appears to be to me, since my institution has a license). GNU Octave is free and probably has an ODE library. If you're tied to a specific language, what language? I'm sure others can help you with various libraries in FORTRAN, C, C++, etc. moink (talk) 17:08, 9 March 2008 (UTC)

That's what I was thinking: why re-implement what a library can already do better? I already use Octave, and I had no idea it had ODE solvers, but there they are, so thanks for suggesting that. I also found the ODE solvers in the GNU Scientific Library, so I think that's everything I could need. —Keenan Pepper 19:21, 9 March 2008 (UTC)

Speaking of Octave, does anyone know of an IDE for it that doesn't suck? I googled around a bit and found a couple of mostly abandoned SourceForge projects that no one had touched much for a couple years. That doesn't prove they suck, of course, but it makes me reluctant to invest the time to find out. I know there's an Eclipse plug-in for it, but frankly I wasn't very impressed with Eclipse the last time I seriously tried it (which was more than a year ago). --Trovatore (talk) 07:15, 10 March 2008 (UTC)

You could try TeXmacs. Morana (talk) 11:26, 10 March 2008 (UTC)

Follow up on question of "straight lines"

I had posted the question about "straight lines" above. See "Lines" on March 7 posting. I wanted to follow-up with the following question. As I was driving the other day, I noticed those white (and yellow) lines that are painted on the road ... the markings that serve as a guide to delineate the lane change divisions, the center of the road, the edge of the road, etc. As roads are not perfectly straight (rather they twist and turn), those center yellow lines (for example) are not "straight lines". Rather, they curve along with the twists and turns of the road that they are marking. What is the mathematical / geometry term for that? Is it simply a curved line? Is it really a "line" at all (mathematically speaking)? Or is there a better term? Thanks. (Joseph A. Spadaro (talk) 20:47, 8 March 2008 (UTC))

It is called a median line. An important non-trivial application is to determine the median line of two counties coastlines where the territorial waters or exclusive economic zones etc overlap. There should be a mathematical name for this process of finding the median line between arbitrary curves but I don't know what it is. SpinningSpark 21:28, 8 March 2008 (UTC)

I'd call it a median curve. Considering the edges of the road are equidistant (not parallel, strictly speaking, since they aren't straight), it's a much simpler problem than defining borders between countries. --Tango (talk) 22:36, 8 March 2008 (UTC)

It's still called a median line by those that produce them, whatever you call it. SpinningSpark 19:56, 9 March 2008 (UTC)

See medial axis. --Salix alba (talk) 23:01, 8 March 2008 (UTC)

You may be interested in Voronoi diagrams and related topics. -- 128.104.112.85 (talk) 16:35, 13 March 2008 (UTC)

Solution to a Second Degree DiffEQ

Hello. I was wondering what the general solution to this differential equation is, if it exists:

$y''={\frac {K}{y^{2}}}$

...where K is a constant. It seems that the solution to this equation would give me a general free-fall equation, when you let K be the gravitational constant times the mass of the planet or whatever.

Thanks in advance, Phillip (talk) 22:06, 8 March 2008 (UTC)

Just multiply both sides by y² and integrate (twice). --Tango (talk) 22:37, 8 March 2008 (UTC)

Huh? I don't understand this suggestion. How can you evaluate the first integral?

The method I recently learned for solving this kind of ODE (in which the independent variable, say $t$, does not appear explicitly) involves introducing a new variable $v(t)=y'(t)$ and then expressing $y''(t)$ as

${\frac {d^{2}y}{dt^{2}}}={\frac {dv}{dt}}={\frac {dy}{dt}}{\frac {dv}{dy}}=v{\frac {dv}{dy}}$

This gives you a first-order, separable ODE for $v(y)$, and after you solve that you get another first-order, separable ODE for $y(t)$. —Keenan Pepper 01:13, 9 March 2008 (UTC)

Let me elucidate Tango's response. We have ${\dfrac {d^{2}y}{dx^{2}}}={\frac {K}{y^{2}}}$. We rearrange this to be $y^{2}d^{2}y=Kd^{2}x$. We then integrate both sides twice. -mattbuck (Talk) 01:40, 9 March 2008 (UTC)

That much I understood. What I don't understand is how to evaluate the integral $\int y^{2}{\frac {d^{2}y}{dt^{2}}}dt$ and get something useful. Using the substituion I said, you can put it in the form $\int y^{2}dv$ where $v={\frac {dy}{dt}}$, but that doesn't help. —Keenan Pepper 02:27, 9 March 2008 (UTC)

Oh, I got it. "by parts" was all you needed to say. Turns out to be perfectly equivalent to the method I described, so think about it whichever way you want. —Keenan Pepper 02:38, 9 March 2008 (UTC)

I don't think you actually need to do it by parts, I think you can just say:

$\int \left(\int y^{2}\,dy\right)\,dy=\int \left(\int K\,dt\right)\,dt$

I'll find some pen and paper in a minute and check that... --Tango (talk) 13:49, 9 March 2008 (UTC)

Ok, when I try and do it by parts I get a horrible mess... I think I'll just blindly trust that the "multiply by dt and add an integral sign and ignore the fact that it's complete nonsense" method works when used repeatedly. --Tango (talk) 13:57, 9 March 2008 (UTC)

No no no... There's a reason it's written as ${\frac {d^{2}y}{dt^{2}}}$ instead of ${\frac {d^{2}y}{d^{2}t}}$ (which I don't know how to interpret) or ${\frac {dy^{2}}{dt^{2}}}$ (which would be $\left({\frac {dy}{dt}}\right)^{2}$). The reason is that treating it that way leads to wrong answers.

Consider the equation you wrote.

$\int \left(\int y^{2}dy\right)dy=\int \left(\int Kdt\right)dt$

If you actually do the integrations, you get

${\frac {1}{12}}y^{4}={\frac {1}{2}}Kt^{2}+C_{1}t+C_{2}$

$y(t)=\left(6Kt^{2}+C_{3}t+C_{4}\right)^{1/4}$

But this is not a solution to the original differential equation (try it!), so you must have done something wrong. What you did wrong was misinterpret ${\frac {d^{2}y}{dt^{2}}}$ as something other than the second derivative of $y(t)$.

Now, I thought at one point I tried rearranging the original equation as

$\int y^{2}{\frac {d^{2}y}{dt^{2}}}dt=\int Kdt$

integrating symbolically by parts, and getting the same answer I got with the method I proposed, but now I can't duplicate that, so I must have made an error before. I crossed out what I said.

In summary, just do it the way I proposed (which my textbook also recommends, and I've written up at Autonomous system (mathematics)#Solution techniques), because it eventually gives you the correct answer. —Keenan Pepper 16:56, 9 March 2008 (UTC)

Your point is probably right, but I think your integration is wrong - there should be a term linear in y from the 2nd integral acting on the constant from the first. That won't fix the problem, though. (I'll claim I was trusting Mattbuck, who'll probably claim he was trusting me, and then no-one has to take responsibility for anything!) --Tango (talk) 17:48, 9 March 2008 (UTC)

That's true. —Keenan Pepper 18:24, 9 March 2008 (UTC)

It's ok Tango, I don't blame you, I blame the Flying Spaghetti Monster - his noodly appendages messed it up. -mattbuck (Talk) 18:34, 9 March 2008 (UTC)

First step is to get rid of the constant. Use

$K^{-1/2}x$

as a new independent variable. Now the equation changed from

$y''={\frac {K}{y^{2}}}$

into

$y''\cdot y^{2}=1.$

Second step is to substitute a power series

$y=\sum _{i=0}^{\infty }a_{i}\cdot x^{i}\,$

in the differential equation and get

$\left(\sum _{i=0}^{\infty }(i+2)\cdot (i+1)\cdot a_{i+2}\cdot x^{i}\right)\cdot \left(\sum _{i=0}^{\infty }a_{i}\cdot x^{i}\right)^{2}=1.$

Third step is to compute the terms of this series. Choose $a_{0}$ and $a_{1}$ . Compute $a_{i+2}$ for i=0,1,2,... . Bo Jacoby (talk) 19:26, 9 March 2008 (UTC).

So anyway, I continued with the correct method and got an integral that wasn't particularly nice, but not unreasonable, and I got a solution of the form $f(y)=t$. The problem is that the function $f(y)$ is gnarly, and as far as I know, you can't invert it in general. (Mathematica can't solve it.) For one special case, though, you can invert it, and you get an "escape velocity" solution

$y(t)={\sqrt[{3}]{\frac {9K}{2}}}(t-t_{0})^{2/3}$

The time-reversal of that is also a solution, but other than those two I can't get any in closed form. —Keenan Pepper 18:51, 9 March 2008 (UTC)

Using the method Keenan Pepper just described, I got just as far. My equation is:

$\pm \left(t-t_{0}\right)={\frac {1}{C}}{\sqrt {y(Cy-2K)}}+{\frac {2K}{C^{\tfrac {3}{2}}}}\ln \left|{\sqrt {C(Cy-2K)}}+C{\sqrt {y}}\right|.$

$C$ and $t_{0}$ are arbitrary constants. Perhaps it's somehow solvable by the Lambert W function or something analogical? — Pt _(T) 23:53, 9 March 2008 (UTC)

Note that if $y''-y^{-2}=0$, then the 'energy', $E=2^{-1}{y'}^{2}+y^{-1}$, is a constant of motion because $E'=(2^{-1}{y'}^{2}+y^{-1})'=y'y''-y'y^{-2}=y'(y''-y^{-2})=0.$ This simplifies matters, because substituting $y''=(y^{-1})^{2}=(E-2^{-1}y'^{2})^{2}$ gives a differential equation of the first order in the velocity $v=y'$, namely $v'=(E-2^{-1}v^{2})^{2}$ or, separating variables, $dt=dv(E-2^{-1}v^{2})^{-2}$, so you are left with the integral

$t-t_{0}=\int (E-2^{-1}v^{2})^{-2}dv.$

Bo Jacoby (talk) 12:38, 11 March 2008 (UTC).

Is there a specific name for a town celebrating its 190th anniversary?

TXKay (talk) 22:43, 8 March 2008 (UTC)

Not a real one. Someone that speaks better Latin than me could make one up. See Anniversary#Latin-derived_numerical_names. --Tango (talk) 22:56, 8 March 2008 (UTC)

The best I can come up with is this made-up word: centumnonagintennial. The word nonagintennial is in actual but rare use for a 90th anniversary,[1][2] and I've turned that into 190th by prefixing it with centum meaning 100. Nonangintennial comes from Latin nonaginta meaning 90, and 190 in Latin is centum nonaginta. Squashing a numerical prefix into a word is not entirely un-Latin: decemviri, centumviri.  --Lambiam 00:07, 9 March 2008 (UTC)

Is there a reason they're celebrating that anniversary, instead of waiting a few years for the bicentennial? Black Carrot (talk) 06:34, 9 March 2008 (UTC)

Since when has anyone needed a reason to have a party? --Tango (talk) 13:49, 9 March 2008 (UTC)

Since when has anyone sought a reason to postpone a party?Zain Ebrahim (talk) 14:19, 10 March 2008 (UTC)