Mathematics desk
< March 31	<< Mar \| April \| May >>	April 2 >

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

April 1

Cube - Ellipsoid Bijection

Given a cube and an ellipsoid, I am looking for a bijection $f:\mathbb {R} ^{n}\to \mathbb {R} ^{n}$ such that the image of f on the cube is an ellipsoid (unnecessarily the ellipsoid), and the image of f on the ellipsoid is a cube. I think that maybe some special coordinate system or some homeomorphism whould do the job, but I can't figure out which exactly to use... 31.154.81.0 (talk) 06:30, 1 April 2017 (UTC)[reply]

You can trivially map a cube centered on the origin to the unit ball centered on the origin: just scale the vectors by the inverse of the radius of the cube in their direction. Then you can scale, rotate, and translate the ball to be the desired ellipsoid. Is that clear enough? --Tardis (talk) 08:24, 1 April 2017 (UTC)[reply]

What's "the radius of the cube"?

Besides, your transformation seems to transform the cube into an ellipsoid, but not vice versa. (I edited my question a bit, to make it clearer) 31.154.81.0 (talk) 13:30, 1 April 2017 (UTC)[reply]

Tardis apparently missed that you want a (partial) involution. If you want an analytical expression for the transform, it is going to be hard, but their answer still covers most of it.

WLOG assume that cube and ellipsoid are centered and aligned, i.e. the main axes of the ellipsoid are parallel to the cube's sides. (There should be a symmetry by reflexion that can send either of the two in alignment with the other, and you just need to add it to the transform we will define later). Use generalized spherical coordinates

r,\theta _{1},...\theta _{n}

(with the reference axes aligned with the cube and ellipsoid). The ellipsoid coordinates will be described by

F(\theta _{1},...\theta _{n})(r/r_{0})^{2}=1

with

F

some horrible combination (sums and products) of trigonometric functions of the angle coordinate, the cube coordinates will be

r=G(\theta _{1},...\theta _{n})r_{0}

(where

G

is some even more horrible combination that is continuous but not differentiable near the edges of the cube). Then simply define the transform as

T:(r,\theta _{1},...\theta _{n})\rightarrow (K_{F(\theta _{1},...\theta _{n}),G(\theta _{1},...\theta _{n})}(r),\theta _{1},...\theta _{n})

where

K_{a,b}

is any involution that switches

a

and

b

, for instance

K_{a,b}(r)=ab/r

. Tigraan^{Click here to contact me} 13:29, 1 April 2017 (UTC)[reply]

First, could you cite any reference that writes these horrible functions explicitly?

Second, isn't a cube in spherical coordinates defined using a set of inequalities? 31.154.81.0 (talk) 14:10, 1 April 2017 (UTC)[reply]

For the second: I assumed by "cube" you mean the external surfaces (i.e. six squares for the 3-D case), since an ellipsoid is a (hyper-)surface rather than a volume. In that case, cube coordinates are defined by equalities, not inequalities. If you want a bijection between two sets of different dimensions, that is possible but it becomes significantly harder.

For the first: I don't have a reference, but the idea is simply to take the equation in cartesian coordinates and transform it in the generalized coordinates, see N-sphere#Spherical_coordinates. The ellipsoid case shouldn't be too hard and there is probably a textbook that gives it; for the cube case, the hard part is not the equation of a plane, but how to separate the 2N faces of the N-dimension hypercube. Tigraan^{Click here to contact me} 09:31, 3 April 2017 (UTC)[reply]

The "radius of the cube in [a] direction" is the length of the line segment in that direction whose ends are at the origin and on the surface of the cube. Explicitly: given a point ${\mathbf {x} }=(x_{1},x_{2},\ldots ,x_{n})$ , the radius of the cube $[-1,1]^{n}$ in that direction is just ${\frac {|{\mathbf {x} }|}{\max _{i}|x_{i}|}}$ .
I did miss the bit about it being a single function f in each direction, but with the (added) relaxation that the image of f need not be the same as the relevant ellipsoid in its domain, we can trivially satisfy it by mapping the cube to a disjoint ellipsoid, and then combining that with the inverse of that mapping with an appropriate translation to again make the result disjoint.

Is that clearer (and more complete)? --Tardis (talk) 16:07, 1 April 2017 (UTC)[reply]

Unless I misunderstand, your transformation deals with the case that the ellipsoid and the cube are disjoint. Otherwise, it is not clear what to do, since your function is defined on two different domains (the cube and the ellipsoid), and in their intersection there's an undefined behavior. 31.154.81.0 (talk) 17:22, 1 April 2017 (UTC)[reply]

April 1st surprise, game theory

Someone expects to get a surprise from me today. But If I don't surprise him, he would be surprised, unless, he's using the inverse logic and the surprise would be getting a surprise. What should I do to surprise him, do something or not do anything? --Dikipewia (talk) 13:21, 1 April 2017 (UTC)[reply]

The human thing to do is to give them a surprise. In game theory one would just toss a coin and they wouldn't be surprised whatever happened - which is a bummer. Otherwise see Unexpected hanging paradox. Dmcq (talk) 13:30, 1 April 2017 (UTC)[reply]

Ah, but if your friend truly expects a gift then it really wouldn't be a surprise to receive one, would it? So the logical thing to do would be to bring no gift at all. Now of course, if your friend is Sicilian, it gets a little more complicated. Earl of Arundel (talk) 17:12, 1 April 2017 (UTC)[reply]

Tickle him. Then tell him this was "the surprise". This tickling thing as a surprise will surprise him.

PS: dont listen to user:Earl of Arundel

—usernamekiran (talk) 19:51, 1 April 2017 (UTC)[reply]

Raymond Smullyan describes exactly this situation on the first page of What is the Name of This Book?. His brother Emile told him that he was going to fool Raymond on April 1st. At the end of the day he had done nothing, and when Raymond objected, this conversation ensued:

Emile: So, you expected me to fool you, didn't you?
Raymond: Yes.
Emile: But I didn't, did I?
Raymond: No.
Emile: But you expected me to, didn't you?
Raymond: Yes.
Emile: So I fooled you, didn't I?

Smullyan says he lay in bed for a long time afterwards trying to figure out if he had been fooled. CodeTalker (talk) 02:23, 2 April 2017 (UTC)[reply]

This reminds me of a classic: "The prof said he is giving a surprise test next week, with the surprise being the day". However:

1) He can't give the test on Friday, because if we haven't had the test by Thursday, then we would know it was going to be Friday.

2) Knowing this, he can't give it on Thursday, because, if we haven't had the test by Wednesday, and we know we can't have it Friday, then it must be Thursday.

3) Extend the logic to show that it can't on Monday, Tuesday, or Wednesday either. StuRat (talk) 06:09, 3 April 2017 (UTC)[reply]

That is the same as the Unexpected Hanging paradox, as mentioned by Dmcq in the first reply to the question. AndrewWTaylor (talk) 12:25, 3 April 2017 (UTC)[reply]

Thanks. Good to know. StuRat (talk) 20:04, 7 April 2017 (UTC)[reply]

A surprising theorem, whose formulation - has no free variables - and begins with an existential quantifier necessary for the surprise.

Do you have more examples of such theorems, besides the following examples?

There exists a set S (in ZF) for which there exists no function from the set of natural numbers on S (This is a weaker version of Cantor's theorem, however this weak version is not less surprising than Cantor's theorem is).
There exists a proposition, formulated in PA, but unprovable in PA if PA is consistent (This is a weaker version of Goedel's second incompleteness theorem. In my view, this weak version is still surprising, although Goedel's second incompleteness theorem is more surprising of course - because it shows also that Arithmetic cannot be both consistent and complete [and effective] no matter how many new axioms are added to it).

Btw, I'm not looking for theorems whose formulation begins with: "There exists a model...". HOTmag (talk) 19:03, 1 April 2017 (UTC)[reply]

Not exactly sure what you mean by 'surprising', but you don't have to go any further than the axioms to get something with the criterion of starting with an existential quantifier. For example

There exists a set with no elements.
There exists an infinite set.

It seems to me that you must have at least one axiom of this form, otherwise it would be trivial to prove consistency. --RDBury (talk) 14:36, 2 April 2017 (UTC)[reply]

Actually the usual form of first-order logic can prove that there exists at least one thing, for example

(\exists x)x=x

without assuming any axioms at all, or rather only the logical axioms, which we don't usually count as axioms.

There are alternative formulations (so-called free logic) that do not allow this inference without adding non-logical axioms. Most workers reckon that free logic adds technical complications for no real benefit. If you use free logic, then you can allow the empty structure as a model of a theory, whereas if you don't you won't, and I guess allowing the empty model suits some people's idea of tidiness. But there's no obvious benefit to having the empty model. --Trovatore (talk) 20:22, 2 April 2017 (UTC)[reply]

@RDBury, of course you need at least one axiom - starting with an existential quantifier - for proving any theorem starting with an existential quantifier, but I don't see how all of this has anything to do with my question about "surprising" theorems. I don't think the axiom of empty set and the axiom of infinity involve any surprise (because every physical object can intuitively be regarded as an empty set, and everyone knows that every natural number is followed by a greater natural number, so the natural numbers are endless - hence the concept of "endlessness" is not surprising at all, so the next intuitive step should naturally be the concept of an infinite set, which is not supposed to surprise us at all). However, Cantor's theorem (as well as Goedel's second incompleteness theorem), astonished the mathematicians once it was introduced, didn't it? That's why I think the two examples I gave in my first post are really examples of surprising theorems (containing no free variables and beginning with an existential quantifier), aren't they?

Probably, what made you misinterpret me, was the expression: "an existential quantifier necessary for the surprise ", but this expression was only intended to exclude vacuous quantifiers and the like (e.g. if A is a surprising theorem whose formulation contains no free variables and starts with a universal quantifier, then A can be rephrased and start with a vacuous existential quantifier - by adding "there exists c such that" - before that universal quantifier; That's why I made it clear that the existential quantifier must be necessary for the surprise in A, whereas vacuous quantifiers are redundant). HOTmag (talk) 22:39, 2 April 2017 (UTC)[reply]

I guess what I was trying to get at was that the criterion of starting with an existential quantifier doesn't really narrow down the possibilities that much. Applying the word 'surprising' to something is a matter of opinion and many facts that seemed counterintuitive in the past have become more pedestrian over time. For example

There exists a ratio which is not equal to the ratio of two integers.

was extremely controversial 2500 years ago but seems pretty basic now. If you want to go a bit more recent

There exists a function from the reals to the reals which is continuous but nowhere differentiable.

would have seemed impossible until Weierstrass came along, but now it's basic analysis. You could consider the axioms/theorems rejected by the intuitionist or constructivist schools to be surprising, to say the least, to them. Most of these could be stated in a way that takes the form you're talking about. --RDBury (talk) 21:35, 3 April 2017 (UTC)[reply]

Thanx. HOTmag (talk) 18:20, 6 April 2017 (UTC)[reply]

Not sure I understand the question correctly, but we have an article on non-constructive proof. Tigraan^{Click here to contact me} 09:34, 3 April 2017 (UTC)[reply]

Not sure how that article is related to my question, which is about (surprising) theorems rather than about (non-constructive) proofs.

What don't you understand? Do you agree that there are surprising theorems (e.g. Cantor's theorem and Goedel's incompleteness theorems)? Do you agree that some theorems start with an existential quantifier? What's not clear? HOTmag (talk) 10:15, 3 April 2017 (UTC)[reply]

Define "surprising". Tigraan^{Click here to contact me} 10:43, 3 April 2017 (UTC)[reply]

Naturally, a surprise (or astonishment) - being a very unique state of mind, can't have a clear-cut definition (I have some ideas how to vaguely define it though), so when I requested "more examples of surprising theorems", I had to count on the reader's intuition. Anyways, I believe we all agree that some theorems are surprising (e.g. Cantor's theorem and Goedel's incompleteness theorems) while some others are not (e.g. axioms which are also theorems of course). HOTmag (talk) 14:21, 3 April 2017 (UTC)[reply]

Clearly then, I do not have the state of mind required to answer. (BTW an axiom is certainly not a theorem, you must have used the wrong term here.) Tigraan^{Click here to contact me} 16:17, 3 April 2017 (UTC)[reply]

Every axiom is also a theorem, and this is a very simple result of the very definition of "theorem" (which is a proposition having a proof, while every axiom is provable by a proof containing only one proposition which is the axiom itself). HOTmag (talk) 19:56, 3 April 2017 (UTC)[reply]