Sentence examples for using the analytic methods from inspiring English sources

The main purpose of this paper is, using the analytic methods and the properties of Gauss sums, to study this problem and give a sharp upper bound estimate for (1.1).

The main purpose of this paper is, using the analytic methods and the properties of character sums, to study the computational problem of one kind of hybrid mean value involving the quadratic Gauss sums and a new sum analogous to Kloosterman sums, and to give an interesting hybrid mean value formula for it.

The main purpose of this paper is using the analytic methods and the properties of Gauss sums to study the computational problem of one kind of fourth power mean of two-term exponential sums, and to give an interesting identity and asymptotic formula for it.

The main purpose of this paper is using the analytic methods and the properties of Gauss sums to study the computational problem of a new sum analogous to quadratic Gauss sums, and to give an interesting asymptotic formula for its 2k th power mean.

The first extensive study of muscle as a viscoelastic material (i.e., using the analytic methods of rheology) was carried out on single fibers and small bundles of fibers from frog skeletal striated muscles [1].

Applications of maneb, diazinon, malathion, acephate, and chlorpyrifos account for 78% of the kilograms of pesticides applied whose urinary metabolites were detectable using the analytic methods available in this study (Table 1).

The main purpose of this paper is using the analytic method and the properties of Kloosterman sums to study this problem, and prove the following result.

The main purpose of this paper is using the analytic method and the properties of Gauss sums to study the 2k th power mean of G ( χ, m, n, c ; p ), and to give a sharp asymptotic formula for it.

Hardness and Young's modulus of GaN thin films, calculated from the load-displacement (P h) records by using the analytic method developed by Oliver and Pharr [12], are 19.3 ± 1.1 and 286.1 ± 25.3 GPa, respectively.

The main purpose of this paper is using the analytic method and the orthogonal properties of the character sums to study the computational problem of one kind hybrid power mean of the generalized Gauss sums (operatorname{mod} p), an odd prime with (pequiv 1operatorname{mod} 3), and give some exact computational formulas for this kind hybrid power mean.

The main purpose of this paper is using the analytic method and the properties of Kloosterman sums to study the estimate problem of the mean value ∑ a ∈ A χ ( a ), and give a sharp upper bound estimate for it, where χ denotes any non-principal even character mod p. MSC 11L40.