Sentence examples for using the elementary properties from inspiring English sources

The proof is straight forward using the elementary properties of log-convexity in one variable.

By using the elementary properties of the Gaussian c.f., as in the proof of Lemma 10.IV in [3], one gets ( ∫ | t | ≥ n 1 / 4 2 M | d l d t l e − t 2 / 2 | 2 d t ) 1 / 2 = O ( 1 / n ). for l = 0, 1.

The main purpose of this paper is using the elementary methods and the properties of Gauss sums to give a sharp estimate for some character sums.

The main purpose of this paper is, using the elementary method and the properties of the third-order linear recurrence sequence, to unify the above results by proving the following theorem.

The main purpose of this paper is using the elementary method and the properties of the second-order linear recurrence sequence to study these problems and to prove a generalized conclusion.

The main purpose of this paper is, using the elementary and combination methods, to study the arithmetical properties of the Lucas polynomials and to obtain some new and interesting identities for them.

There is another proof using the elementary amalgamation theorem and the elementary chain theorem.

By using the elementary inequality for, we have (2.24).

Using the elementary inequality, ( and ) in (2.24), we get (2.25).

Finally, using the elementary inequality for real numbers (3.19).

Learning rules are derived from an information-theoretic objective function, and simulations show that a local processor trained with these rules and using an appropriate activation function has the elementary properties required.