Sentence examples for implies along with from inspiring English sources

For every x ∈ D ( T ), we have by the monotonicity of T lim inf n → ∞ 〈 T x n, x n 〉 ≥ 〈 − λ 0 c ∗, x 〉 + 〈 T x, x 0 − x 〉, which implies along with (3.6) 〈 − λ 0 c ∗ − T x, x 0 − x 〉 > 0. By the maximal monotonicity of T, we get a contradiction.

First, if x ^ changes sign in I0 then x ^ cannot be an extremal solution of problem (10) because it cannot be compared with the solution x ≡ 0. If, on the other hand, x ^ ≥ 0 in I0 then the differential equation yields x ^ ′ ≤ 0 a.e. on I0, which implies, along with the initial condition x ^ - π 2 = 0, that x ^ ( t ) = 0 for all t ∈ I0.

If s 0 = 0, then we have ε ∥ x n ∥ 2 ≤ ε 〈 J x n, x n 〉 + 〈 T t x n, x n 〉 ≤ 〈 A s n x n, x n 〉 + s n Λ ψ ( S ), which implies along with equation (2.7) ε lim sup n → ∞ ∥ x n ∥ 2 ≤ lim n → ∞ s n Λ ψ ( S ) = 0, where S is an upper bound for the sequence { ∥ x n ∥ }.

This implies that, along with appropriate context and judgment, the DFBETA statistic is a useful tool for diagnosing heterogeneity.

This implies that, along with other preventing measures, maintenance of good air quality in critical areas in hospitals is helpful to reduce the incidence of these infections.

This implies that, along with the provision of care, adult child care-givers have to spend time organising the informal care-giving network, co-ordinating care activities, and coping with disagreements among the informal helpers.

These modeling results imply that, along with better process understanding of base emission factor controls for volatile organic compounds, improvements in global fields of solar surface radiation fluxes in warm climate zones will be needed to reduce major uncertainties in isoprene source fluxes.

(2) Sampling rate determination Through selecting the preferred features, users can observe the variation of the implied information along with the reducing sampling rate.

This along with (j) implies inequality (40).

Moreover, Gronwall's inequality leads to begin{aligned} y(t) leq& C varepsilon_{1}+varepsilon_{2})^{1/2} int _{0}^{t}biglVert psi(cdot,t bigrVert ^{2}_{L^{infty}(0,l },mathrm{d}texp biggl{ C int_{0}^{t}biglVert psi (cdot,t bigrVert ^{2}_{L^{infty}(0,l },mathrm{d} t biggr} leq& C varepsilon_{1}+ varepsilon_{2})^{1/2}, end{aligned} which along with (3.28) implies (3.25).

This along with Theorem 3.2 implies that is bounded.