Sentence examples for by the unique solution from inspiring English sources

We propose a new fairness criterion, utility proportional fairness, which is characterized by the unique solution to this problem.

We denote by the unique solution, in, to (3.3).

Denote by the unique solution of (3.4); namely, (3.5).

For every, denote by the (unique) solution of the problem (2.27).

The final parameter estimate is given by the unique solution (up to a constant factor) of Equation 9.

Let (lambdain _{c}mathcal{R}_{nu}(mathbb {T})) and (sin mathbb {T}), then the generalized exponential function (widehat{e}_{lambda}(cdot, s)) on time scale (mathbb {T}) is denoted by the unique solution of the initial value problem left { begin{array}l} x^{nabla}(t)=lambda x t),quad tin mathbb {T}_{kappa}; x s)=1.

Denote by x the unique solution of the Cauchy problem left { textstylebegin{array}{l} ^{mathrm{c}}D_{t}^{alpha}x t)=lambda x t)+f t,x t)),quad tin J, x 0)=y(0), end{array}displaystyle right.

Denote by x α the unique solution of (2.2).

By Theorem 2.1, the unique solution (x t)) of the SDE (4) will remain in (mathbb{R}^{4}_).

This and inequalities (3.13) mean that the multi-valued function (mathbf{u}_{0}) defined by (3.3) is the unique solution of inequality (3.7) (see Definition 3.3) and, moreover, it is the weak solution to the homogenized problem (3.4).

Theorem 7. Let Assumption 1 be satisfied and assume that constant K satisfies: 2Kk < 1. Denote by u(t) the unique solution of problem (3) in Theorem 1. Assume that A k and f k in problem (5) are specified, respectively, by Ā k and f ^ k in (39) and denote by vk,j, j = 0, 1,..., n, the unique solution of problem (5) in Theorem 2. Let Assumption 3 be satisfied.