Sentence examples for minimal solutions to from inspiring English sources

The conditions for the existence of the maximal and minimal solutions to the system were established.

By means of the monotone iterative technique and combining with suitable conditions, the existence of the maximal and minimal solutions to the fractional differential equation is obtained.

In Section 4, we first examine the existence of maximal and minimal solutions to (operatorname{GVI}(C,Gamma)) without requiring the involved mapping to have topped (bottomed) values.

Finally, through the monotone iterative technique, we not only obtain the maximal and minimal solutions to the fractional differential equation but also establish iterative schemes for approximating the solutions, which start from the known simple linear functions.

However, to obtain all the minimal solutions to (Acircmathbf{x} = mathbf{b}) is in general a computationally difficult task because the number of minimal solutions could be exponentially large with respect to the input size.

Unfortunately, determining all these interval-valued vectors requires an enumeration of all minimal solutions to a system of 0-1 integer linear inequalities, the number of which could be exponentially large.

Let (u) be the minimal solution to (2.1) in (L^2(Omega,e^{psi -varphi })).

In Step 2: Compute the minimal solution to the system of equations.

A Genetic Algorithm is employed to determine the global minimal solution to the multimodal objective function, which can be difficult to achieve by traditional gradient-directed search methods.

Given a set-valued optimization problem ( S P − ⪯ ), an element x ¯ ∈ X is called a minimal solution to ( S P − ⪯ ) if ( F ( x ) ⪯ F ( x ¯ )  for some  x ∈ X ) ⟹ F ( x ¯ ) ⪯ F ( x ).

Therefore (v) is the minimal solution to (bar{partial }v=beta :=e^psi (alpha +u,bar{partial }psi )) in (L^2(Omega,e^{-varphi })) and by Hörmander's estimate begin{aligned} int _Omega |v|^2e^{-varphi }dlambda le int _Omega |beta |^2_{ipartial bar{partial }varphi }e^{-varphi }dlambda.