Sentence examples for helpful proof from inspiring English sources

We also thank Dr. Flaminia for her helpful proof reading.

The authors thank David Main (Crops and Livestock Research Centre, Charlottetown, PEI), Leonardo Galindo (University of Alberta, Edmonton, AB) and Elsa Reimer and Andrzej Walichnowski (Cereal Research Centre, Winnipeg, MB) for their technical assistance and helpful proof reading.

First, we give some lemmas that will be helpful in the proof of the main result.

Now, we recollect some known definitions and results from the literature which are helpful in the proof of our main results.

One effort made was integration with our expression array data which did not proof helpful; hence pathway analysis of genes in this region would be arbitrary.

Computers may also be helpful in completing proofs when there are a large number of cases to be considered.

From the definition of q-integral and the properties of series, we can get the following results on q-integral, which are helpful in the proofs of our main results.

Similarly as done for derivatives, an operator I q n can be defined as ( I q 0 f ) ( x ) = f ( x ) and ( I q n f ) ( x ) = I q ( I q n − 1 f ) ( x ), n ∈ N. From the definition of q-integral and the properties of series, we can get the following results concerning q-integral, which are helpful in the proofs of our main results.

Similarly as done for derivatives, an operator I q n can be defined, ( I q 0 f ) ( x ) = f ( x ) and ( I q n f ) ( x ) = I q ( I q n − 1 f ) ( x ), n ∈ N. From the definition of q-integral and the properties of series, we can get the following results concerning q-integral, which are helpful in the proofs of our main results.

We thank Anirban Chowdhury for helpful discussion and proof-reading the manuscript.

We give special thanks to Ashley Dagley, and Neil Motter for expert technical assistance, and Dennis Dolny, Robert Skinner, Linda F. Hayward and Daniel Judd for helpful suggestions and proofing the manuscript.