Wikipedia:Reference desk/Archives/Mathematics/2016 August 31

= August 31 =

Extension by definitions
Are the concepts extension by definition and conservativity theorem the same thing? If yes, I think the articles should be connected. 178.43.12.92 (talk) 12:09, 31 August 2016 (UTC)
 * I don't know enough to answer that question, but I would suggest that you discuss on one or both of the article talk pages. Robert McClenon (talk) 14:19, 31 August 2016 (UTC)

Representation of Ellipsoid
I know that we can define an ellipsoid by its center and a set of orthongonal vecotrs. Can we also uniquely define an ellipsoid in the following manner:

Let $$B$$ be some orthogonal basis of $$\R^n$$. Let $$c$$ be the ellipsoid's center. Assume now that the lengths of the vectors in $$B$$ satisfy that $$\forall v\in B : c+v\in \partial E$$, (where $$\partial E$$ is the ellipsoid's boundary).

Does the pair $$(c,B)$$ define the ellipsoid uniquely?

If so, what is the matrix $$A$$ that satisfies $$E=\{x\in\mathbb R^n| (x-c)^T A (x-c)\leq 1\}$$ ?

Thanks in advnance! 213.8.204.74 (talk) 12:23, 31 August 2016 (UTC)


 * This doesn't work: consider $$(c,B)=\bigl((0,0),\{(1,1),(-1,1)\}\bigr)$$. Certainly a circle works, but there are also two infinite families of answers with the semi-minor axis approaching 1: one brushing the given points with two nearly-vertical lines, and another brushing them with a single nearly-horizontal line.  --c (talk) 12:40, 31 August 2016 (UTC)


 * Thank you for the counterexample! Are there some restrictions under which it does uniquely defines an ellipsoid? 213.8.204.74 (talk) 13:09, 31 August 2016 (UTC)


 * You could assert that the vectors in B are normal to your ellipsoid's surface, which would force the vectors to correspond (at least in 2 dimensions) to the semi-major and major axes of the ellipsoid.--Jasper Deng (talk) 17:10, 5 September 2016 (UTC)


 * This restriction corresponds to the OP's "I know that we can define an ellipsoid by its center and a set of orthogonal vectors." That is to say, you need to be able to freely choose the orthogonal vectors B (up to order and sign) and the centre.  Alternatively, if you cannot choose B, you need to be able to specify the matrix A.  —Quondum 19:33, 5 September 2016 (UTC)

The Church-Turing thesis: are there any candidates for real or imagined alternatives?
I've been trying to go a bit deeper in understanding Turing machines, the entscheidungsproblem, etc. The Stanford Encyclopedia of philosophy has a good article on the Church-Turing thesis. That article points out that the thesis is really not proven and perhaps can't be proven. What they are saying is that it is conceivable (and perhaps this can never be disproven but only supported through the lack of evidence to the contrary) that there are other "effective" mechanical or biological processes that could solve problems that Turing machines can't solve. My question is: have there ever been any serious proposals about what such procedures could be like? I've heard people say that quantum computers don't follow the Turing machine model because they introduce randomness but according to the Stanford article on Turing machines (and also the classic book Introduction to Automata theory, languages, and computation) non-deterministic Turing machines are still equivalent to traditional Turing machines. I've heard of alternative computer architectures (non-Von Neumann) but don't know much about them but I assume they are still Turing Machines(?) I've been trying to think of what an alternative might be. The way I see it is the only alternative I could conceive is if there were some way to literally search an infinite space, not a space that could increase without bound but a literal infinite space... but that sounds like total fantasy, I can't conceive of how that would work. Any comments, thoughts, pointers would be welcome. --MadScientistX11 (talk) 13:25, 31 August 2016 (UTC)


 * In principle, continuous systems can compute things that discrete systems can't (see The wave equation with computable initial data such that its unique solution is not computable by Pour-El and Richards). Also, in a general relativistic universe it's not obviously impossible for there to exist a worldline of infinite length that lies entirely in the past of some event. In particular, in a Kerr–Newman spacetime (except in the Schwarzschild or superextremal cases), if you go through the wormhole, you see the entire infinite future of the universe you left in a finite time. Probably both of these situations are unrealistic (continuum mechanics probably breaks down at some scale). -- BenRG (talk) 00:34, 1 September 2016 (UTC)
 * The article Hypercomputation has some more information. -- BenRG (talk) 01:05, 1 September 2016 (UTC)
 * Excellent! Just the kind of thing I was wondering about and had no idea even where to look. thanks much. --MadScientistX11 (talk) 14:18, 1 September 2016 (UTC)