Functions that introduce new degrees of freedom?

In summary, the conversation discusses the concept of functions that introduce new degrees of freedom, specifically in the context of a 2-dimensional space being mapped to a 3-dimensional space. The conversation also touches on the applicability of this concept in different fields, such as set theory and programming. The conversation ends with a discussion about the possibility of representing new variables and commodities through functions in a model that takes into account technological changes.
  • #1
Loud Red
4
0
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 [tex]\Bbb{R}^{2}[/tex]. Is there any function that generally exists as: [tex]f: \Bbb{R}^{n} \rightarrow \Bbb{R}^{n+1}[/tex]? So in my scenario, it would be [tex]f: \Bbb{R}^{2} \rightarrow \Bbb{R}^{3}[/tex]...

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?
 
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  • #2
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?
 
  • #3
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!
 
  • #4
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...
 
  • #5
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!
 
  • #7
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 [tex]c_{m}[/tex] or whatever).

To shake things up even further, there is a large class of capital commodities (so I would have a matrix [tex]c_{mn}[/tex] 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.
 
  • #8
Why can't functions be discontinuous? :confused:
 

1. What are "functions that introduce new degrees of freedom"?

"Functions that introduce new degrees of freedom" refer to mathematical functions that add new independent variables or parameters to a system or equation. These added variables allow for more complexity and flexibility in the system, and can lead to a better understanding of the underlying mechanisms at work.

2. How do these functions impact scientific research?

These functions can greatly impact scientific research by providing a way to study and analyze complex systems. By introducing new degrees of freedom, scientists can gain a deeper understanding of the underlying mechanisms and relationships within a system, which can lead to new insights and discoveries.

3. Can you provide an example of a function that introduces new degrees of freedom?

One example of a function that introduces new degrees of freedom is the Fourier transform. This mathematical function converts a function of time into a function of frequency, adding a new degree of freedom to the system. This allows scientists to analyze and understand complex signals in fields such as physics, engineering, and biology.

4. What are the benefits of using functions that introduce new degrees of freedom?

The use of these functions can lead to a deeper understanding of complex systems, allowing for more accurate predictions and better decision making. They also provide a way to simplify and analyze complex data, making it easier to identify patterns and relationships within the system.

5. Are there any limitations to using functions that introduce new degrees of freedom?

While these functions can be powerful tools in scientific research, they also have limitations. Introducing too many degrees of freedom can lead to overfitting and inaccurate results. Additionally, these functions may not always be able to accurately capture the complexity of a system, and may require further refinement or adjustments to be truly effective.

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