Some vocabulary

[JeffM]

This is probably basic stuff in some not very advanced branch of mathematics, and once I have some vocabulary, I shall probably have further questions, but I do not even know what topic I have stumbled into.

To start, I have a system of n functions

[MATH]y_j = f_j(x_1,\ ... x_n),\ \text { where } n \in \mathbb Z,\ 1 \le j \le n, \text { and } x_j,\ y_j \in \mathbb S_j, \text { a set of numbers.}[/MATH]
In general, the sets are bounded.

Is there a name for this kind of system?

Is there a simple notation for it?

Now, if these functions are differentiable, is there a simple way to denote the set of partial derivatives?

What branch of math works with these systems?

Once we have cleared that up, I shall probably have more questions about how such systems behave.

 

[Cubist]

Hi, have you made any progress on this problem?

I think the general name for this system might be a solution set (click for wikipedia link).

There is a Matlab function solve(click) that performs this kind of calculation. It takes a different action depending on the supplied functions. Initially it tries to solve symbolically. If this does not work then one numeric solution is returned via the function vpasolve(click). This latter function can be called directly which allows you to pass in a "starting guess" and according to this page(click) it may use a Newton–Raphson approach, again depending on the functions. A Newton approach would seem to suit if the functions are differentiable.

Just a thought:- A simpler definition of the problem could be obtained if the "y" values are subtracted from their corresponding function, so that all the functions values then become equivalent to zero. So don't immediatly dismiss any leads/ references that might quote the problem in this way.

Good luck with your research.

 

[Cubist]

[MATH]y_j = f_j(x_1,\ ... x_n),\ \text { where } n \in \mathbb Z,\ 1 \le j \le n, \text { and } x_j,\ y_j \in \mathbb S_j, \text { a set of numbers.}[/MATH]
In general, the sets are bounded.

Actually are the sets formed by ranges, or are they discrete values?

 

[JeffM]

Cubist

Thanks for taking the time to look at my question. Yes, I realize that it may be convenient to write f(x, y) = 0 rather than f(x) = y.

I am not really interested in numerical solutions because my problems are generic.

So my first problem is that I have a system in which the n x's and the n y's are all bounded non-negative integers. I want to show that any numeric relation that is true in such a system is also true of two corresponding systems that are infinitely differentiable, one involving only polynomials and the other involving only sine (and perhaps for convenience cosine) functions. In particular, I want local extrema in the integer system to be the only local extrema in the differentiable systems.

Obviously, this is relatively simple if n = 1, but the notation becomes very cumbersome when I generalize. Furthermore, if someone has already done this, I can stop worrying about it and just use those results.

 

[Cubist]

I thought there might be more to this than I had assumed!

I'm not sure that I can help much further. I found a site that talks about a matrix of partial derivatives (but probably in a different context) and they use a shorthand that presents a vector of functions, see https://mathinsight.org/derivative_matrix. I don't know if this is a standard notation, but the website is in tex format so that it could be adopted fairly easily if it suits your need.

Are you using a polynomial fit (or series expansion) and Fourier series to obtain the corresponding systems? Please don't feel obliged to answer if this question is too basic!