Sentence examples for follows proposition from inspiring English sources

Proposition 3.8 may be rewritten in a more clear equivalent form as follows: Proposition 3.9.

A simple G-convex space version of the Fan KKM lemma or Theorem 2.7 is as follows: Proposition 4.3.

The contribution of this paper is summarized as follows:  Proposition of a hybrid positioning algorithm based on the combination of ToA, AoA, and RSS fingerprinting.

Thus, the design of the proposed research propositions is as follows: Proposition 1: The new idea of technological innovation has an impact on building a sustainable city.

The above discussion can be summarized as follows: Proposition 2.2 The ternary differentiated authority results in one-way dependency between the multiple principals.

The above conclusions can be summarized as follows: Proposition 1.2 The flat structure results in competition between the multiple principals endogenously.

Following Proposition 9 (see [10]) and Theorem 3.3, we conclude the following.

(i) Following Proposition 5.1, we have (A_{lambda}^{sigma _{k}}=H_{1-lambda}).

Following Proposition 6.1, such an operation satisfies properties similar to those of an abelian group.

The following in 2003 simply follows from Proposition 6.1: Proposition 6.2.

It follows from Proposition 4.1 iii), Proposition 4.4, and Proposition 4.5 that the conclusion is given.