- #1
St41n
- 32
- 0
Can I do this:
sup_(x,y) f(x,y) = sup_y sup_x f(x,y) ?
sup_(x,y) f(x,y) = sup_y sup_x f(x,y) ?
The supremum w.r.t. 2 parameters, also known as the double supremum, is the smallest upper bound of a set of numbers that depends on two variables. It is denoted by supx,y and is defined as the largest value that is less than or equal to all numbers in the set.
The supremum w.r.t. 2 parameters is different from the supremum w.r.t. 1 parameter in that the latter is only dependent on one variable, while the former is dependent on two variables. This means that the supremum w.r.t. 2 parameters is a more specific upper bound for a set of numbers.
Yes, the supremum w.r.t. 2 parameters can be negative. The supremum is the largest value that is less than or equal to all numbers in the set, and this can include negative numbers. However, the supremum w.r.t. 2 parameters cannot be negative if the set of numbers contains only positive numbers.
The double supremum is commonly used in mathematical optimization problems where two variables need to be optimized simultaneously. It is also used in economics, physics, and engineering to determine the maximum value of a system or process that is dependent on two variables.
Yes, there is a relationship between the supremum w.r.t. 2 parameters and the infimum w.r.t. 2 parameters. The supremum w.r.t. 2 parameters is the largest upper bound, while the infimum w.r.t. 2 parameters is the smallest lower bound for a set of numbers. These two values are related as the supremum w.r.t. 2 parameters is equal to the negative of the infimum w.r.t. 2 parameters, and vice versa.