Sentence examples similar to manipulation proof from inspiring English sources

It may be a traditional, bound-and-numbered paper book, or it may be a manipulation-proof online notation system.

Recalling the condition equation (49), we have the following relation along the trajectories of system: mathfrak{J}_{2} (t )< 0. Then we obtain the following equation through some similar algebraic manipulations of the proof in Theorem 3.1: int_{0}^{T}y^{T} (t )y (t ),dt< sigma gamma e^{gamma T} int_{0}^{T}w^{T} (t )w (t ),dt.

This would have the Treasury determine whether a currency is "fundamentally misaligned" against the dollar, implying a lower standard of proof than "manipulation".

After some manipulations of (34), the proof of Theorem 3 is completed.

Based on the two analytic forms of (T_{i}^{ell}(x)) given in (2.11) and (2.12) and if we perform similar manipulations as in the proof of Theorem 1, then we get the desired result.

Substitute (33) into (32), after some mathematical manipulations according to the proof of Theorem 1 in [24], we can obtain: begin{aligned} mathcal{A}(p_{i})&=prod_{j=0}^{N}expleft{-pilambda_{mathrm{s}}delta_{j} left[frac{gamma_{t}, p_{j}}{p_{i}minleft(1,d^{-alpha}right)}right]^{frac{2}{alpha}}right.

But the New York Fed thought the reports amounted to market chatter and did not provide definitive proof of widespread manipulation.

Proof of such manipulation was in short supply until last week, when federal regulators released memorandums written by lawyers at Enron that offered new evidence that power producers used the crisis to gouge consumers.

Obviously, we can then apply Theorem 2.5 here for obtaining the desired result after a simple manipulation, and this completes the proof.

A typical 18th-century derivation used a term-by-term manipulation similar to the algebraic proof given above, and as late as 1811, Bonnycastle's textbook An Introduction to Algebra uses such an argument for geometric series to justify the same maneuver on 0.999.

The set and the skirt I built from her discarded piles of publishers' proofs — no digital manipulation was involved.