Sentence examples for a pair of solutions from inspiring English sources

Let and be a pair of solutions to the nonhomogeneous -harmonic equation (1.1) in a domain.

Let and be a pair of solutions to (1.3) in a domain.

Let ( u, v ) be a pair of solutions of (1.2) up to a constant, then ( u, v ) satisfies (1.3).

Let and be a pair of solutions to the nonhomogeneous -harmonic equation (1.1) in a bounded domain.

A series of norm comparison theorems for a pair of solutions to (1.6) were established in [5].

It is well known that the Bäcklund transformation is a transformation that is related to a pair of solutions of a nonlinear equation.

Let and be a pair of solution to the conjugate -harmonic tensor in.

It is easy to see that ((u, v in C[0, 1]times C[0, 1]) is a pair of solution to the system (1) if and only if ((u, v)) is a pair of solution of the following nonlinear integral system: left { textstylebegin{array}l} u(t)=lambdaint_{0}^{1}G_{alpha}(t, s)f s, v s)),ds, v(t)=muint_{0}^{1}G_{beta}(t, s g s, u(s)),ds.

Let a pair of solution circles be denoted as CA and CB (the pink circles in Figure 6), and let their tangent points with the three given circles be denoted as A1, A2, A3, and B1, B2, B3, respectively.

Each addition problem was presented centrally above a pair of solution probes.

Step 4. Crossover Selecting a pairs of solutions in step 3 randomly to use them as the parents for crossover operations.