Sentence examples for main result then from inspiring English sources

First we give preliminaries and the main result, then we present some lemmas, including proving the tightness and identification of the limit law, to prove the main result.

If there is reason to rule out this simple explanation of the main result, then that should be discussed.

In the latter work Gregory collected the main results then known about transforming a very general class of curves into sections of known curves (hence the designation "universal"), finding the areas bounded by such curves, and calculating the volumes of their solids of revolution.

We give combinatorial proofs of two important special cases of our main result, and then generalize these techniques.

In this section, we first state some assumptions and definitions needed in the proof of our main result and then prove the existence of solutions.

The Schur convexity and Schur-geometric convexity of generalized Heronian means involving two parameters are studied, the main result is then used to obtain several interesting and significantly inequalities for generalized Heronian means.

The main results are then applied to optimal design, primarily the stress-constrained minimum weight problem, to illustrate how they can be used to establish existence of solutions and validity of "ε-perturbations" that are common in computational topology optimization.

Titles and abstracts were reviewed to identify eligible studies and their main results were then extracted.

The main results were then compared with the old study where the interview-based version of the MRS scale was used.

The main result proved in [73] then reads as follows.

The main consequence of Gödel's result, then, as far as the problem of proving the independence of the CH is concerned, was that it pointed mathematicians in the direction of adding non-constructible sets to a model of set theory in order to establish the consistency of the negation of the CH.