Which branch(es) of mathematics predominate in these equations?

[Cinitiator]

Homework Statement

Which branch(es) of mathematics predominate in these equations (see pic below)?

Homework Equations

The Attempt at a Solution

 

[conquest]

It might help to define big S small s and possibly the weird bigger then sign with subscript j (ordering of some kind?).

It looks now like some set theory with some (possibly partial) ordering. The use of 'arg' suggests some complex function theory or just complex algebra.

 

[Dickfore]

maybe Graph Theory.

 

[Cinitiator]

It might help to define big S small s and possibly the weird bigger then sign with subscript j (ordering of some kind?).

It looks now like some set theory with some (possibly partial) ordering. The use of 'arg' suggests some complex function theory or just complex algebra.

As far as I know, the big S is a set. I'm also wondering about the strange bigger than sign too. Which branch of mathematics does it come from? I can recognize lots of set theory notation here, but some notation (such as the strange greater than sign) doesn't make sense to me.

 

[Office_Shredder]

If you want us to help you with the notation it would be really useful if you post the full context under which this is occurring

 

[Cinitiator]

If you want us to help you with the notation it would be really useful if you post the full context under which this is occurring

I've lost the original source - all I know is that it's from an economics paper. What I'm particularly interested in are these strange greater than signs, as well as max and min operators. Which branch do they come from? Well, I'm familiar with max and min parameters, but not when they have some parameters under them.

 

[Office_Shredder]

The greater than sign is probably some previously defined ordering on a set

min means exactly what you think it means. The fact that there is an i=1,2 underneath it means take the minimum of the expression where the value of i can be either 1 or 2.

For example
$$\min_{i=1,2,3} i = 1$$
$$\min_{i=3,4,5} 1/(i-6) = -1$$

 

[Cinitiator]

The greater than sign is probably some previously defined ordering on a set

min means exactly what you think it means. The fact that there is an i=1,2 underneath it means take the minimum of the expression where the value of i can be either 1 or 2.

For example
$$\min_{i=1,2,3} i = 1$$
$$\min_{i=3,4,5} 1/(i-6) = -1$$

Thanks a lot, this helped. However, what exactly do you mean by 'some previously defined ordering on a set'? I know that you can use Cartesian product to create n-tuple relations out of 2 sets and give them order, and that you can simply use n-tuples as well as sequences to express an ordered set of data. Is it related to these concepts, or is it something completely different?

Also, which branch of mathematics do the min and max operators come from?