Functions that introduce new degrees of freedom?

[Loud Red]

Functions that "introduce" new degrees of freedom?

OK, I realize this is a wacky question, so forgive me!

BUT I was thinking about it the other day, and suppose I had a 2 dimensional space $$\Bbb{R}^{2}$$. Is there any function that generally exists as: $$f: \Bbb{R}^{n} \rightarrow \Bbb{R}^{n+1}$$? So in my scenario, it would be $$f: \Bbb{R}^{2} \rightarrow \Bbb{R}^{3}$$...

Would this still be part of set theory or has this entered some other field?

Now, an additional question, I am a programmer and I know I can code "IF p=x THEN add a new dimension" or something of the sort. Is there anything in math that has a counterpart to this? Boolean functions?

 

[arildno]

Well, Cantor proved that R has the same cardinality as R^n for finite n.
That is to say, there exists a bijection (a function) between the line and the plane, for example.

Was that what you meant?

 

[Hurkyl]

There are many. For example, f(x, y) = (x, y, 0).

I'm not really sure what you're trying to ask with your second question!

 

[Loud Red]

I guess I am looking at this from a naive qualitative point of view; I was thinking about applying it to human societies specifically (so I would quantify things like "Energy use from natural resources" then at time T when certain conditions are met, this function would introduce a new dimension for the "Energy use from natural resources" then...I suppose I would need a function to create functions...).

As you can see, if you are a programmer, this is very easily programmable. However, from the more mathematical perspective, I have no clue how to approach this.

For me this is uncharted mathematical waters...

 

[Hurkyl]

What's wrong with simply fixing "energy use from natural resources" to be zero until it's "unlocked"?

Are you trying to get the math to act similar to what your program is doing? Then why not try and do a direct translation? E.G. you could define a set whose contents are the possible assignments to all of your variables!

 

[Dragonfall]

A very natural example of a function f:R^2->R^3 is the parameterization of a surface in 3D. i.e.

http://mathworld.wolfram.com/SurfaceParameterization.html

 

[Loud Red]

What's wrong with simply fixing "energy use from natural resources" to be zero until it's "unlocked"?

Are you trying to get the math to act similar to what your program is doing? Then why not try and do a direct translation? E.G. you could define a set whose contents are the possible assignments to all of your variables!

Well, the problem is that when certain conditions are met, the model is supposed to create new things; so for example, there is a generic variable "capital" which is used to make commodities.

But capital changes with respect to technology. But technological change is discrete change, not continuous change.

For every discrete change in technology, which allows a new capital commodity to be formed, I want to represent this new capital commodity by a new variable (say $$c_{m}$$ or whatever).

To shake things up even further, there is a large class of capital commodities (so I would have a matrix $$c_{mn}$$ for each type of commodity and its technological state).

New classes of capital commodities arrive when certain conditions are met (e.g. oil refineries could be considered "one" capital commodity but they didn't exist until oil was being used). And I don't really have too much control over when something is discovered...but I would like to test what conditions make it so.

I suppose this would be "beyond" the scope of mathematics, just a little maybe.

 

[arildno]

Why can't functions be discontinuous?