The phrase "minimum is attained" is grammatically correct and can be used in written English.
It is typically used to describe a situation where the lowest possible value or amount has been reached or achieved. Here is an example of how it could be used in a sentence: "After much effort and hard work, the company's goal of reducing production costs to a minimum was finally attained."
Moreover, in the case of Schur-convex functions, the minimum is attained when all the independent variables are equal.
The JS is non-negative and its minimum is attained by k-out-of-n systems, which are the least complex systems.
When this global minimum is attained at a point z, then we have a best proximity point for which (d z, Sz) = d(A, B)).
In particular, we prove that, for every given norm, there are arbitrarily small perturbations of it for which the minimum is attained.
Let m ( t ) : = inf x ∈ S [ u x ( t, x ) ], t ∈ [ 0, T ). and let ξ ( t ) ∈ S be a point where this minimum is attained by using Lemma 2.4.
Thus, r j = min u ∈ S n − 1 h ( T p, i K j, u ) = h ( T p, i K j, u j ), where u j ∈ S n − 1 is any point where this minimum is attained.
Money Income of Households Distribution by Income level and Selected Characteristics: 2006, in U.S. Census Bureau, Statistical Abstract of the United States: 2009, p. 443, household median incomes follow an inverted-U pattern relative to the age of the householder, with the minimum being attained for households whose householder was 65 years old and over.
The solid lines show the MSE as a function of the shrinkage intensity (in 0.05 increments), and in every case, the minimum was attained near the value indicated in Figure 2.
It is shown that the two minima are attained under step and linear response functions, respectively.
The performance varies with different ({a_{1}^{2}}), and the minimum CRLB is attained when ({a_{1}^{2}} = {a_{2}^{2}}) in both the RIPS-u and the RIPS-sq.
In the case when the minimum value is attained at the point (1-t), we have by the definition of the modulus of uniform smoothness that frac{rho_{X}(t)}{t}geqfrac{f 1-t -2}{2t}{t}geqfrac{f 1-t -2}{2t-(1-t)^{p}{t}geqfrac{f 1-t -2}{2t
