Sentence examples for based on proposition from inspiring English sources
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, it is known that if one can construct an effective algorithm for problem ((mathrm{P}_{rho})) and adjust parameter ρ to force the iterate to asymptotically satisfy (g_{ell }(x)=0), then the solution to (P) can be yielded.
Hence the proof of the Grünbaum conjecture given in [H. König, N. Tomczak-Jaegermann, Norms of minimal projections, J. Funct. Anal. 119 (1994) 253 280] which is based on Proposition 3.1 is incomplete.
Based on Proposition 1, we have (27).
Based on Proposition 4.2, we obtain more direct estimates to prove Theorem 4.1.
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).
