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Hello mathematical analysts, I have a question for your idle brains

Given two unit vectors vectors and a function , how much can you tell about the nature of this function if you are told only two pieces of information:

1. is piece-wise smooth.
2. is invariant under a basis transformation.

minkwe wrote:Hello mathematical analysts, I have a question for your idle brains :)

Given two unit vectors vectors and a function , how much can you tell about the nature of this function if you are told only two pieces of information:

1. is piece-wise smooth.
2. is invariant under a basis transformation.

Hi Michel

Can you be a bit more precise?

Are x and y unit vectors in R^2 or R^3?

By basis transformation, do you mean a rotation about the origin?

Hi,
I was hoping to be as general as possible. But for the sake of being precise, let us say R^3. By basis transformation, I include rotation or translation of just the basis vectors, ie, they could be pointing in different directions, or the origin could be translated to an arbitrary position in R^3.

It seems to me all one can say about such a function is that it is piecewise smooth and invariant under basis transformations in .

You have specified the domain of the function, which is (because and are unit vectors), but left the co-domain wide open:



Until you specify what is, there is not much restriction on the nature of . In other words, it could be almost anything you like.

minkwe wrote:Hi,
I was hoping to be as general as possible. But for the sake of being precise, let us say R^3. By basis transformation, I include rotation or translation of just the basis vectors, ie, they could be pointing in different directions, or the origin could be translated to an arbitrary position in R^3.

So the vectors x, y are not restricted to unit length? The function f maps R^3 x R^3 to R (I suppose) and is invariant under affine transformations of R^3.

Not difficult to see that the function must be constant.

Joy,

Hint 1: if the is invariant under basis transformation, then f must also be a function of a joint property of the two unit vectors that is invariant under a basis transformation and what properties of two unit vectors do we know about that have that property. Geometric product?

You can probably guess where I'm going with this now.

minkwe wrote:Joy,

Hint 1: if the is invariant under basis transformation, then f must also be a function of a joint property of the two unit vectors that is invariant under a basis transformation and what properties of two unit vectors do we know about that have that property. Geometric product?

You can probably guess where I'm going with this now.

Yes.

What if I added that instead of just being piecewise-smooth, the function must be "differentiable at every point". Because, I want a small perturbation of the directions of the unit vectors, to cause only a small perturbation of the result of the function?

Can we say anything about periodicity, if we know that must be a function of the geometric product of the vectors?