Sentence examples for based on propositions from inspiring English sources

Cardano's presentation followed the Islamic tradition of solving one instance of every possible case and then giving geometric justifications for his procedures, based on propositions from Euclid's Elements.

The proof of Theorem 1.1 is based on Propositions 4.1 and 4.2.

Based on Propositions 5 and 6, we have the following result.

For completeness of this article, we provide the proof of Theorem 1.1 in detail, based on Propositions 2.1-2.2.

The dominance theory is evolved based on propositions, including dominance, being dominated, transitivity, and equability (Brauers and Zavadskas 2012).

For each transmission mode m with dedicated Ma, Ka and Na,i, calculate the optimal transmit power P t * and the corresponding BPJ-EE according to the bandwidth W* = Wmax and ergodic capacity estimation based on Propositions 3 and 4. Step 2. Choose the optimal m* with optimal M a *, K a * and N a, i * with the maximum BPJ-EE.

Based on Proposition 11, one can develop essentially the same techniques that have been indicated above for the periodic problem (1), (2).

Based on Proposition 1, one understands that only a fraction of SBSs are qualified to enhance the secrecy capacity of a certain MUE.

Based on Proposition 1, we have (27).

Based on Proposition 3.1, the conditions to guarantee (tau^ > 0) for several growth models are derived.

Based on Proposition 4.2, we obtain more direct estimates to prove Theorem 4.1.