Sentence examples for extensions of solutions from inspiring English sources

Also analytic and continuous extensions of solutions to homogeneous Dirac equations are studied, and it is shown that for each cell of harmonicity considered here there exist solutions to each iterate of the Dirac operator which may not be extended beyond any point of the closure of the cell.

The goal of this study is to identify the unknown boundary condition u ( 1, t ) at x = 1 by using the over measured data u ( x 0, t ) = ψ 1 and u x ( x 0, t ) = ψ 2. The key point here is the unique extensions of solutions on [ 0, x 0 ] to the closed interval [ 0, 1 ] which are implied by the uniqueness of the solutions.

Some results of this paper improve and extend previous theorems given by Gao, and five examples are given to show the extension of solutions of the system of complex difference equations.

Theorems about the existence, uniqueness or extension of solutions of initial value problems, and about intersections of such solutions with barriers γ i can be found for example in [[35], Chapter 5].

Some results obtained in this paper improve and extend the previous theorems given by Gao, and five examples show the extension of solutions of the system of complex difference equation.

This makes our new framework a natural extension of solution concepts in conventional game theory.

The integral equations are investigated with regard to the existence of a minimal and a maximal solution, extension of the solutions, and the generation of the solutions by successive approximations.

Since then, various extensions of these solutions set characterizations to nondifferentiable convex programs, infinite-dimensional convex programs, and multi-objective convex programs have been given in [2 4].

The main point here is the unique extensions of the solutions on [ 0, x 0 ] to the closed interval [ 0, 1 ] which are implied by the uniqueness of the solutions.

Further, from the extension theorem of solutions, we have.

Following step 1, step 2, and the extension principle of solutions, there exists a positive integer n large enough such that bar{delta}_{m_{n}}geqfrac{A_{2}}{A_{2}+epsilon}.