Sentence examples for the pair of problems from inspiring English sources

We now prove the following duality results for the pair of problems (HNWP) and (HNWD).

Thus (MP) and (MD) become multiobjective extension of the pair of problems (P1) and (D1) in [7].

(iii) If F x, y; ·) is sublinear and l = 1, then (MP) and (MD) are an extension of the pair of problems studied in Yang et al. [9].

If F x, y; ·) is sublinear, then (MP) and (MD) reduce to the pair of problems (MP2) and (MD2) studied in Mishra et al. [10].

(ii) If F x, y; ·) is sublinear, |J 2| = 0, |K 2| = 0 and l = 1, then (MP) and (MD) reduce to the pair of problems (P1) and (D1) of Mond and Schechter [7].

In this section, we consider some special cases of problems (MP) and (MD) by choosing particular forms of compact convex sets, and the number of objective and constraint functions: (i) If F x, y; ·) is sublinear, then (MP) and (MD) reduce to the pair of problems (MP2) and (MD2) studied in Mishra et al. [10].

Example 5.4 Let α ∈ ( 0, 1 ] Q, a ∈ T L. Consider the pair of initial value problems with Riemann-Liouville and Caputo fractional operators ∇ α a y ( t ) = 4 5, t > a, a C ∇ α y ( t ) = 4 5, t > a, ∇ α − 1 a y ( t ) | t = a = y α − 1, y ( a ) = y 0, where y α − 1, y 0 ∈ R. Note that the initial condition ∇ α − 1 a y ( t ) | t = a = y α − 1 can be interpreted via Corollary 4.8 (see also [14]).

The most common pair of problems is that for reactive processes most systems are: underpumped and the pumping is not uniform.

Given a primal problem (P), there are many ways to formulate the dual problem (DP) such that the weak and strong duality theorems hold true between the primal and dual pair of problems (P) and (DP).

(61) Then the primal and dual pair of problems (CLFP) and (DCLFP) have no duality gap.

Consider the primal and dual pair of problems (CLFP) and (DCLFP).