Sentence examples for by using the existence from inspiring English sources

But, as was discussed in Section 3.2.1, varying domains can be brought in indirectly by using the existence predicate, \(E\), and this allows us to introduce actualist quantification definitionally.

Then, by using the existence theorem we construct an iterative algorithm for problem (1.1).

Saejung's original proof makes substantial reference to the axiom of choice by using the existence of Banach limits.

Furthermore, by using the existence results of cone saddle points for vector-valued mappings, he obtained the following result: ∃ z 1 ∈ Min ⋃ y ∈ Y 0 Ma x w f ( X 0, y ) a n d ∃ z 2 ∈ Max ⋃ x ∈ X 0 Mi n w f ( x, Y 0 ).

Unfortunately, a direct interpretation of important features is not possible since the decision boundry for the classification is determined by the non-linear classifier by using the existence and non-existence of features from both positive and negative examples.

By using the established existence and uniqueness theorem, we then introduce an implicit method for seeking a solution of MVIP (1.1).

In case i≠j, the result follows just as above by using the already established existence of 〈S k,S k 〉(t) for all k and t and by noting that (sum _{k,ell in {i,j}}{langle }S_{k},S_{ell }{rangle }={left langle S_{i}+S_{j},S_{i}+S_{j} right rangle }).

First, by using the contraction principle, the existence and uniqueness of mild solutions are given.

and by using the degree theory, the existence of a positive solution of (1.3) is established.

By using the variational method, the existence and multiplicity of positive solutions are obtained.

By using the equivalent formulation, the existence and uniqueness theorems for solutions of the system are established.