Sentence examples similar to proof given for from inspiring English sources

The concept of minimal element is crucial in the proofs given for Caristi's fixed point theorem.

In the past decades, Caristi's fixed point theorem has been generalized and extended in several directions, and the proofs given for Caristi's result varied and used different techniques, we refer the readers to [1 15].

This proof is a straightforward extension of the proof given in [3] for the lower bound of 2-input A-operations that form an MCM block.

A similar proof is given for S4.

A convergence proof is given for the random waveform case.

The proof is given for the previous named "approximate formulas" (labeled as Formula II).

A constructive proof is given for the existence of load functions.

A new proof is given for a result due to N. Grønbæk on characterizing of amenability for Beurling algebras.

A new proof is given for the result (due to E. Kirchberg) that K0 (and hence K) fails the CSEP.

We close this section with the following lemma whose elementary proof is given for the sake of completeness.

A theoretical proof is given for the occurrence of Marotto's chaotic behavior, if the magnitudes of the interconnection coefficients are large enough and at least one of the activation functions has two simple real roots.