If a minimum (maximum) exists, then it verifies( p - gamma = C^{prime}(mu ) ).
A rate maximum exists at commercially attractive low water conditions, and optimisation of the process parameters gives acetic acid with a selectivity in excess of 99% based upon methanol.
The loglikelihood function is globally concave, and hence a global maximum exists.
For every (A, B inoperatorname{CL}(X) ), let H A,B) = begin{cases} max {sup_{xin A}d x,B),sup_{yin B}d y,A) }&mbox{if the maximum exists}; infty&mbox{otherwise.} end{cases} (1) Such a map H is called the generalized Hausdorff metric induced by d.
For every A, B ∈ C L X X ), let H ( A, B ) = { max { sup x ∈ A d ( x, B ), sup y ∈ B d ( y, A ) }, if the maximum exists ; ∞, otherwise.
For every A, B ∈ CL ( X ), let H ( A, B ) = { max { sup x ∈ A d ( x, B ), sup y ∈ B d ( y, A ) } if the maximum exists ; ∞ otherwise.
With ( Z = 0 ), the maximum exists where the curve turns flat, forming a company size of ( S_{Z = 0} ).
This maximum exists as the coefficients of q.s.o. are not greater than 1.
Therefore, we see from (2) that another maximum exists, which is very close, in terms of the log likelihood value, to this (global) maximum.
Since the noise variance is strict positive, standard arguments can be used to show that with our choice of the utility functions, the maximum exists.
