Sentence examples similar to a minimum is attained from inspiring English sources

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.

Proof As mentioned earlier, here we only need to show that the above theorem holds for s = 3. Define now m ( t ) : = min 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 3.1. It follows that m ( t ) = u x ( t, ξ ( t ) ). Clearly, u x x ( t, ξ ( t ) ) = 0 since u ( t, ⋅ ) ∈ H 3 ( S ) ⊂ C 2 ( S ).

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)).

Since (prod _{j=0}^k L^{q}(Delta,mu _j)) is uniformly convex, this minimum is attained at a unique (u_0in U) (see, e.g., [6, p. 22]), and (4.3) gives (u_0=v).

Any path for which this minimum is attained represents a minimum‐bottleneck path (for u and v).

Their limit is a measure, called gradient Young measure (GYM), and describes the effective energy density Wqc, the quasiconvex envelope of the original energy density W. This gives rise to a relaxed minimisation problem (R) which is well-posed in the sense that the minimum is 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.

In particular, we prove that, for every given norm, there are arbitrarily small perturbations of it for which the minimum is attained.

Furthermore, since V is symmetric around (t=frac{1}{2}), it follows that it is nonincreasing on ([0,frac{1}{2}]), its minimum is attained at (t=frac{1}{2}) and its maximum is attained at (t=0) and (t=1).

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.