Wikipedia:Reference desk/Archives/Mathematics/2008 March 29

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March 29

This function as an approximation to tetration?

Hi, what do you think of this function as an approximation to tetration?

${\ ^{b}a={a^{a^{b} \over {a}}}}$

(I hope the formula shows up correctly - let me know if it doesn't.) The two sides of this equation seem to equal each other with certain values. Please try it and let me know what you think. Have you seen this equation (or a similar one) anywhere else? If so, where? I really want to look at websites with further information on this and similar equations. Thank you. 4.242.147.228 (talk) 02:20, 29 March 2008 (UTC)

Well, $a^{a^{b} \over {a}}=a^{a^{b-1}}$. So (assuming b>=2 - the equation is trivially satisfied for b=1, and if b=0, it's only satisfied for a=1 from what I can see), if we take base a logs on both sides, we get $\ ^{b-1}a=a^{b-1}$, and again gives $\ ^{b-2}a=b-1$, so that would seem to be a necessary and sufficient condition for your equality to hold. I can't see any particular reason for that to be the case other than simple coincidence. --Tango (talk) 02:41, 29 March 2008 (UTC)

OK, thanks. I think I didn't read the tetration article thoroughly enough. I will go back and read it some more. You can consider this question answered for now. 4.242.147.174 (talk) 02:54, 29 March 2008 (UTC)

Function or Not?

I was having fun with the program Grapher, which comes with Mac OSX.5 (or Leopard), and I typed in y=x^-1, and got two curved lines that were reflected across the line y=-x. I was wondering 1) What is this called when a function comes up with two lines and 2) Is it a function at all? In theory, the two lines will never have the same y, but different x values because neither can ever tough the y-axis. This thought may be wrong too, but I figured that zero can't work because 0^-1 = 1/0 and you can't divide anything by zero. Could that alone make this not a function? I don't know. --Xtothe3rd (talk) 02:49, 29 March 2008 (UTC)

Great question! It is indeed a function. You correctly note that the domain of the function (all the numbers which could be put into the function) does not include x=0, however functions are allowed to have restricted domains. The only real criteria is that for any given input (within its domain), a function must not have more than one output (y value).

The function $y=x^{-1}$ does have asymptotes along both the horizontal and vertical axes, and is generally described as a hyperbola(although note that most for most hyperbolas, y is not a function of x. I don't know of a general term for a function made up of multiple disjoint parts, as in this case. But many interesting functions have that property. Hope this helps! --TeaDrinker (talk) 03:29, 29 March 2008 (UTC)

A topologist would call it "disconnected", if that helps. --Tango (talk) 15:19, 29 March 2008 (UTC)

This is a Piecewise continuous function. It is composed of two pieces, each of which is continuous. -- Meni Rosenfeld (talk) 21:02, 29 March 2008 (UTC)

Yes, it is, but that term also applies to a continuous function that is in once piece, so it doesn't really narrow it down. --Tango (talk) 21:55, 29 March 2008 (UTC)

Actually, 1/x is continuous everywhere, since the only point it isn't continuous at is a point it isn't defined at. I would use "piecewise continuous" to refer to something like a step function. --Tango (talk) 21:59, 29 March 2008 (UTC)

For the sake of discussion, a specific function being referred to as "piecewise continuous" can be taken to be discontinuous, since otherwise we would have called it continuous.

1/x is continuous in its domain, but not in $\mathbb {R}$. It is piecewise continuous in $\mathbb {R}$ (if you allow being undefined at transition points). -- Meni Rosenfeld (talk) 00:15, 30 March 2008 (UTC)

Of course, it's worth noting that 1/z is continuous and meromorphic on $\mathbb {C}$, with a simple pole at 0. 1/z is most certainly a function, although its continuity is partly because in complex analysis, all infinities are identical. 1/x just seems odd at first if you don't know about functions which have poles. Remember though, the polynomials you are used to dealing with also have poles, just they occur at x = infinity. -mattbuck (Talk) 02:56, 30 March 2008 (UTC)

Indeed - which leads to the natural and intuitive view of 1/z as a half-turn rotation of the Riemann sphere about an axis through +1 and -1. Gandalf61 (talk) 09:12, 30 March 2008 (UTC)

Glm link function choice

Howdy, does anyone know of a general test to choose a link function (among some restricted class--even if it is just two) for generalized linear models? For instance, in a binomial problem, is there a general way to decide if a logit link or a probit link is more appropriate, without dramatically influencing the inference (that is, testing both and taking the one which fits better)? I only know one special case (Box-Cox power transforms for Gaussian models can be realized as choices between link functions) where there is a general way to select a link (and even that could have some influence on inference). Thanks, --TeaDrinker (talk) 03:35, 29 March 2008 (UTC)

In a case like yours, changing the link will most likely have very little effect on any inference, unless you are playing the accept/reject a p-value game, and one links is just over the cutoff, and the other is just under. But things like confidence intervals, etc for parameters are not likely to change a lot. The most salient difference between the two links you mention is that the probit is in some sense a stronger one, in that the asymptotes to ±∞ occur faster and "harder". This property makes it easier to deem data as being outlying due to unusual contributions to the deviance, although using the logit will usually flag the same data, just not as obviously.

For binomial data, there is a theoretically attractive reason to chose the logit, See the generalized linear models article for details, or a good textbook (e.g., McCullagh & Nelder). But that does not imply that link works better for a particular application, it only means some math works out nicer.

It is in general dangerous to use one's data to direct the choosing of a model, then using such model for inference. Essentially, the process of making the modeling choice is not accounted for in the calculations of Types I and II error; the nominal rates can be way off. Baccyak4H (Yak!) 03:42, 30 March 2008 (UTC)

Thank! --TeaDrinker (talk) 20:54, 1 April 2008 (UTC)

Nicomachus, Introduction to Arithmetic

I cannot seem to find a fee, online copy of Nicomachus' "Introduction to Arithmetic". I find it hard to believe that there is no copy.

Do you know where I can find one?

Thanks so much,

R.B. 74.127.254.155 (talk) 05:45, 29 March 2008 (UTC)

It appears that archive.org has a copy, and is in fact linked to in the external links of Nicomachus, Though I don’t know what language it is in. GromXXVII (talk) 11:09, 29 March 2008 (UTC)

It's Greek to me...  --Lambiam 13:50, 29 March 2008 (UTC)

name of generalized cross product construct

In the Cross product#Higher dimensions section they talk about a form of generalizing the cross product to n dimensions, like

$\bigwedge (\mathbf {v} _{1},\cdots ,\mathbf {v} _{n})={\begin{vmatrix}v_{1}{}^{1}&\cdots &v_{1}{}^{n+1}\\\vdots &\ddots &\vdots \\v_{n}{}^{1}&\cdots &v_{n}{}^{n+1}\\\mathbf {e} _{1}&\cdots &\mathbf {e} _{n+1}\end{vmatrix}}$

Is there a specific name for this construct and are there any references about it? The other parts of that section talk about the wedge product, but I went to the article on wedge product doesn't really seem to talk about the above construct. Thanks, --131.215.166.126 (talk) 09:50, 29 March 2008 (UTC)

I wouldn't swear by it, but I think that is the wedge product, just expressed in a different way. It certainly satisfies the properties mentioned in the introduction to wedge product to do with linear independence. --Tango (talk) 15:22, 29 March 2008 (UTC)

It's not exactly the same thing as the wedge product. The wedge product of n vectors in a vector space E of dimension n+1 can be identified with a linear form on E once you choose an orientation on E. To recover a vector of E, you need in turn an Euclidean structure on E. If you make the "right choices", the vector you get after all these indentifications is indeed the same, but as you can see, it's not very canonical. I'd simply call this construct the cross product of n vectors. Bikasuishin (talk) 23:14, 29 March 2008 (UTC)