Sentence examples for the above implication from inspiring English sources

It is clear that the inverse of the above implication does not hold.

The bifunction h : Γ × R n → R ∪ is said to be η-pseudomonotone on Γ if, for any x, y ∈ Γ, h ( x ; η ( y, x ) ) ≥ 0 ⇒ h ( y ; η ( x, y ) ) ≤ 0. The above implication is equivalent to the following implication: h ( y ; η ( x, y ) ) > 0 ⇒ h ( x ; η ( y, x ) ) < 0. Next, we present some properties for pseudoinvexity.

end{cases} Assume that there exists (r in[0,1)) such that psi(r d x,Tx leq d x,y quad Longrightarrowquad H(Tx,Ty leq rM_{d} x,y) for all (x,yin X), where M_{d} x,y)=max biggl{ d x,y),d x,Tx),d y,Ty), frac {d x,Ty)+d y,Tx)}{2} biggr}. Then T has a fixed point. We remark that the right-hand side of the above implication is known as Ćirić type contractive condition, see [10, 11, 18].

None of the above implications is reversible.

While the above implications underlie the learners' interactional feedback exchanges in the learning space, interaction exchanges in the social space should not be discouraged by instructors, especially in the CAPR.

In addition to the above methodological implication, this study needs to mention that it is not easy for us to maintain a social balance (so, sustainability) between economic development and environmental protection.

The converses of above implications in Proposition 3 are not always true as we have seen in the following examples.

The above policy implications are short-run remedial measures.

The detailed investigations on the above two implications are in progress.

Based on the above findings, the implications and suggestions are provided for future research.

The comparative analysis from the above observation and implications for future case-based reasoning systems are presented.