That may be a weird question (indices) (Nevermind)

[The20]

Hello.
This may be a weird question, but i don't know where else to ask. Please assume Latex notation where applicable:
Assume i have a function that takes n arguments x_1 to x_n (so the whole thing looks like f(x_1, x_2, ..., x_n)) and a number of points P_i which are n+1 dimensional. Further assume i number n of the dimensions x_1 to x_n. Now, if i want to show the k-th dimension of point P_i, should i write x_{i,k} or x_{k,i}?
I feel like i should put the index of the point first and the index of the dimension second, but i'm not sure.

[Layra-chan]

Let me see if I understand what you're trying to do. You have a function f on variables x_1 through x_n. You have a bunch of points P_i in an n+1 dimensional space, and of those n+1 dimensions you've labeled n of them 1 through n, corresponding to the variables in the function f.
.
Now for a given point P_i, you want the kth coordinate?

Well, in that case your first problem is writing x_{stuff} for that. Semantic overloading is fine for computers but not for communicating with people, even if that person is yourself. x_k is a variable; it has no further components.

What you're looking for is the kth coordinate of the point P_i, so you write it as (P_i)_k. P_i is a semantic unit, so keep it in one piece unless you hate your readers. In fact, you might even want to use, say, the Greek alphabet for marking the different points, P_alpha, P_beta, so that we can tell "Roman letters indicate coordinate index, Greek letters indicate which point".

[The20]

Layra-chan

Let me see if I understand what you're trying to do. You have a function f on variables x_1 through x_n. You have a bunch of points P_i in an n+1 dimensional space, and of those n+1 dimensions you've labeled n of them 1 through n, corresponding to the variables in the function f.
.
Now for a given point P_i, you want the kth coordinate?

Well, basically i want the n-dimensional version of "y_i is the value that was read when we put x_i in the black box. Here's this function f(x) which gives us hat{y}_i if we put in x_i for x."
In hindsight i probably shouldn't be working with points at all, but even if i say y(x_1,x_2,...) i run into the same problem.
(I need to write a short introduction to regression analysis if you want to know what the application to all this is. I'm also still thinking about dropping the multidimensional approach and simply explaining it with one independent variable.)

Layra-chan

Well, in that case your first problem is writing x_{stuff} for that. Semantic overloading is fine for computers but not for communicating with people, even if that person is yourself. x_k is a variable; it has no further components.

What you're looking for is the kth coordinate of the point P_i, so you write it as (P_i)_k. P_i is a semantic unit, so keep it in one piece unless you hate your readers. In fact, you might even want to use, say, the Greek alphabet for marking the different points, P_alpha, P_beta, so that we can tell "Roman letters indicate coordinate index, Greek letters indicate which point".

I doubt anyone will ever read this. Aside from my (reviewers/referees?), they have to.

[Layra-chan]

They're certainly not going to read it if you don't figure out what you're actually trying to say. I still am not sure what you're trying to say, and I'm not sure you are either.

The problem is that you're not distinguishing between your coordinates and your actual data points. Is x_i a point or a coordinate or a direction? Figure this out before you do anything else, and no, the answer can't be more than one of those.

Here's what you're supposed to do. First, you establish your domain, this n+1 dimensional space. Then on this space you establish your coordinate system, selecting n+1 independent axes and naming these axes1 through n+1. Now you can talk about points in this space in terms of a set of n+1 numbers that say where the point is along each of these axes..
Now you can define a function f such that for a point X given by (x_1,...,x_{n+1}), f(X) = [stuff in terms of x_1,...,x_n]. Here, X is a dummy point whose only purpose is to explain what f does.

Now suppose you have a point P_i. You want to find f(P_i), so you need to find the coordinates of P_i. So you say that P_i is given by (p_{i1}, p_{i2},..., p_{in}, p_{i(n+1)}), and now you can say
f(P_i) = [stuff in terms of p_{i1},...,p_{in}]

[The20]

I changed it to a single variable and went from using P_i to just using (x_i,y_i)

[Suicidesoldier#1]

I have a feeling that trying to determine the thermal index in regards to the dimensional cross-reference principle is relativley irrelevant considering the participle principle to the whatchamacallit.