Wikipedia:Reference desk/Archives/Mathematics/2010 February 7

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February 7

Peano form and divided differences / mean value theorem

Hi all, first time here, I've been recommended by my friend who says collectively you're far too clever for your own good!

I was hoping someone could help me with a couple of problems I'm having with Peano kernel theorems - firstly to fill in the missing step in the end of my proof: I can't quite seem to finish the problem off.

For f[,] denoting the divided difference, I've expressed the divided difference f[0,1,2,4] in the form f[0,1,2,4]=$\int _{0}^{4}K(\theta )f'''(\theta )d\theta$, where K is the Peano kernel and f is reasonably behaved (e.g. in $C^{3}[0,4]$). Now I need to show "by integrating $K(\theta )$ analytically and using the mean value theorem, that f[0,1,2,4]=${\frac {1}{6}}f'''{\epsilon }$, some $\epsilon \in [0,4]$."

I already know a proof of this result using the intermediate value theorem: this is meant to be an alternative method of proof, and unfortunately I can't quite see what I'm meant to be getting at: I don't know quite what it means with 'by integrating K analytically' - the only thing I could think to do that they might be referring to is use integration by parts and even then, it seems very illogical to integrate k and differentiate f, when f is already at the correct derivative and K is order 2 (it's piecewise-quadratic if I've calculated it correctly) so surely we'd want to differentiate K - a typo, do you think? Anyway, I tried integrating $f'''$, and I thought I might have been headed in the right direction but I couldn't quite seem to pull any proof together, could anyone help?

Any suggestions at all as to how you could do this by MVT would be greatly appreciated: I presume I need to get something of the form ${\frac {(f''(4)-f''(0))}{4}}$ on the LHS, but even then I'd get the wrong coefficient, since I want a 1/6 for $f'''(\epsilon )$.

Secondly, I've got a fairly similar problem I'm likewise stuck on unfortunately - let f be a function in $C^{4}[0,1]$ and let $\epsilon$ be any fixed point in [0,1]. Calculate the coefficients $\alpha ,\beta ,\gamma ,\delta$ such that the approximant $f'''(\epsilon )\simeq \alpha f(0)+\beta f(1)+\gamma f'(0)+\delta f'(1)$ is exact for all cubic polynomials. Prove that the inequality

$|f'''(\epsilon )-(\alpha f(0)+\beta f(1)+\gamma f'(0)+\delta f'(1))|\leq ({\frac {1}{2}}-\epsilon +2\epsilon ^{3}-\epsilon ^{4})\|f^{(4)}\|_{\infty }$ is satisfied.

I've managed the first bit I think - $f'''(\epsilon )$ should just be 6 times the leading coefficient in the cubic, so then $\alpha =12,\beta =-12,\gamma =6,\delta =6$ - but then, where do I go from there? I can see how you'd use $|\int _{0}^{1}K(\theta )f'''(\theta )d\theta |\leq \|K\|_{1}\|f^{(3)}\|_{\infty }$, but again, I'm struggling with the details.

Please - suggestions, anyone? If you only have time to lend a hand with one of the problems then that'd still be a great help :) Please help soon if possible, I have a huge workload to hand in and these are the only problems I haven't managed to complete, it'd be nice to get a full house! Thankyou very very much in advance! Estrenostre (talk) 02:06, 7 February 2010 (UTC)

For your first question, use the integral MVT in the form : $\int fg=f(x)\int g$ as given on the Mean value theorem page. You just need to verify that $\int K=1/6$. Tinfoilcat (talk) 13:03, 7 February 2010 (UTC)

Also, that last inequality you have written down looks wrong to me. Imagine f was cubic: then the infinity-norm of $f^{(4)}$ is zero, but the integral on the left hand side could be non-zero. I think you need $f^{(3)}$ on the right. Tinfoilcat (talk) 13:45, 7 February 2010 (UTC)

Ah, but if 'f' is cubic then the approximant is exact (that's how I calculated the coefficients), so then the LHS is always 0 in the cubic :) Thankyou very much for the help on the first problem by the way! Estrenostre (talk) 16:03, 7 February 2010 (UTC)

I meant the *last* inequality you wrote down, that is $|\int _{0}^{1}K(\theta )f'''(\theta )d\theta |\leq \|K\|_{1}\|f^{(4)}\|_{\infty }$ Tinfoilcat (talk) 17:34, 7 February 2010 (UTC)

Ah of course, how silly of me! You're quite right, thanks :) Edited now. Estrenostre (talk) 21:02, 7 February 2010 (UTC)

You've now modified the ineq that you said you wanted to prove, but not the inequality I said was wrong. I think you should check carefully what's going on with this question. Tinfoilcat (talk) 21:55, 7 February 2010 (UTC)

Yes, I did notice immediately after - just identified the wrong bit of LaTeX, hard to read at a glance -but then my internet connection proceeded to break before I could fix it, argh. Sorted now :) Any thoughts now the question is finally set up properly? Estrenostre (talk) 23:13, 7 February 2010 (UTC)

Never mind, I've got it sorted! It was a fairly standard solution, the issue was that my lecturer didn't include 'mod' bars around the K(theta) in his definition of the kernel, but they're necessary for the solution. Thanks very much for the help, Tinfoilcat :) Estrenostre (talk) 16:25, 8 February 2010 (UTC)

Circumcircles, Triangles and Similarity

Hey guys (sry for my non-professional English), here's a link for visualizing the construction.

• http://img710.imageshack.us/img710/2999/neuesbilduy.jpg

The triangles DYZ, EXZ and FXY are similar to each other, K,L,M are the midpoints of the circumcircles of each triangle. Angles: ZDY = ZXE = FXY and YZD = EZX = YFX (can be also seen in the given link). How can I now proof that the new triangle KLM is also similar to other triangles?

--85.178.18.74 (talk) 14:56, 7 February 2010 (UTC)

This is a problem of the current Bundeswettbewerb Mathematik, the German mathematical Olympiad, so please don't discuss it till 1. March. Sorry to all that answered this question. --Schnark (talk) 09:18, 11 February 2010 (UTC)

Alpha function?

Euler defined the beta function and the gamma function. I am curious about what happened to the alpha function ? Bo Jacoby (talk) 18:30, 7 February 2010 (UTC).

The beta function article says Jacques Binet gave the function its name. It doesn't say in the gamma function article but I don't think Euler named it either. So what I think you need to do is find some other function of Euler's without a name and name it Euler's alpha function. ;-) Dmcq (talk) 23:43, 7 February 2010 (UTC)

Tangentially to this, do we have an article for Euler-alpha equations? -- The Anome (talk) 14:32, 8 February 2010 (UTC)

Sets containing themselves

It seems to me that Russel's Paradox requires us to do away with the notion of a universal set, but not necessarily of a set which contains itself. Are there any well-studied axiomatizations of set theory in which a set is allowed to contain itself? --129.116.47.49 (talk) 21:14, 7 February 2010 (UTC)

It doesn't even require you to do away with the universal set. All you have to do away with is anything that allows you to form the (paradoxical) Russell set. If you have the axiom schema of specification, then the existence of the universal set entails the existence of the Russell set, but this can be fixed by weakening specification rather than by doing away with the universe. Quine's New Foundations is an example of a set theory that does this. Our article Non-well-founded set theory mentions some alternative set theories which allow non-well-founded sets (of which sets containing themselves are the simplest example) but do not (I think) allow a universal set. Algebraist 21:28, 7 February 2010 (UTC)

Right; there are many known extension of ZF⁻, that is, ZF without the axiom of foundation. These generally retain the "no universal set" character of ZF, but they lose the "every set is in V" character. One of the most well known is the anti-foundation axiom. — Carl (CBM · talk) 23:04, 7 February 2010 (UTC)

To clarify, they lose "every set is in V" provided by V you mean the wellfounded pure sets. That's a common usage but not universal — Kunen for example has an expository chapter where he temporarily refers to the wellfounded sets as WF, and formulates the axiom of foundation as V=WF.

There are several different anti-foundation axioms. The Aczel one has the property that it has a fairly canonical intended interpretation (headed digraphs with a certain non-repetition property on their subgraphs). A long time ago I did some work on an axiom due to Boffa that in some sense attempts to maximize the ontology (for example it implies that the self-singletons form a proper class, whereas the Aczel axiom implies that there's a unique self-singleton). I should dig up those notes and see if there's anything publishable there. --Trovatore (talk) 23:11, 7 February 2010 (UTC)

Thanks for the answers, guys.

I want to understand the anti-foundation axiom, but, while I know what graphs are (ie. a set of vertices and edges connecting them), but I don't understand what that article means by "accessible pointed". Can someone explain that? --129.116.47.49 (talk) 23:17, 7 February 2010 (UTC)

I expanded Aczel's anti-foundation axiom some. Please feel free to help expand that stub as you learn more about the area... — Carl (CBM · talk) 23:28, 7 February 2010 (UTC)

Thanks, that makes sense. --129.116.47.49 (talk) 23:43, 7 February 2010 (UTC)

129.116.47.49, when you said universal set, do you mean a object in a structure that contains everything in that structure, or the philosophical notion of "absolutely everything"? I'm still struggling to understand what exactly is "absolutely everything", since the first time I saw Russell's paradox when "set of all sets" was mentioned, I thought we picked from the set of absolutely everything those things that are sets. Although contemporary set theory is not at all like this, the meta theory seems to be doing what I just said (by saying the class of all models). Money is tight (talk) 02:24, 8 February 2010 (UTC)

I mean a set which contains every set (and therefore, by virtue of being a set, must contain itself). Standard set theory does not allow for the existence of such a set, although that doesn't prevent one from speaking of the notion of the collection of all sets; you just can't refer to that collection itself as a set. --129.116.47.49 (talk) 00:04, 9 February 2010 (UTC)

Well, that's fairly orthogonal to the usual sort of set-theory-without-foundation. If you want a universal set, you need stringent, and rather difficult-to-motivate, restrictions on the axiom of separation. The best known example of such a theory is New Foundations. There is no general agreement on whether NF is even consistent (but no contradiction has been found). One researcher I talked to, who has done a lot of work on the question, believes that either it is inconsistent, or it's fairly weak, probably about the strength of type theory if I recall correctly. But he says that there has not been any recent progress on the question, and that "none is expected". --Trovatore (talk) 00:27, 12 February 2010 (UTC)