Sentence examples for almost surely that from inspiring English sources

Theorem 4.1 shows that when (sigma_{i}>frac {beta_{i}}{sqrt{2(d+alpha_{i}+r_{i})}}), (i=1,2), two infectious diseases of system (4) die out almost surely, that is to say, large white noise stochastic disturbance can lead to the two epidemics to be extinct.

The answer is almost surely that he would reject our demand that he make a choice.

where v i k denotes the kth column of (frac {n_{i}^{p} boldsymbol {Phi }_{i}^{H}}{sqrt {P_{p}}}), where (varepsilon _{{ik}}^{u}) indicates the orthogonality between channels of different users, and it is well known that when the number of M→∞, it is almost surely that (varepsilon _{{ik}}^{u} to 0).

Substituting the inequalities (2.5) into (2.4), and letting t sufficiently large, it is obtained almost surely that e ε t log V ( x ( t ) ) ≤ log V ( ξ ( 0 ) ) + p − 1 δ e ε ( t + 1 ) log ( t + 1 ) + I, where I = ∫ 0 t e ε s [ c T F ( s, x s ) c T x ( s ) + p − 1 2 ( c T G ( s, x s ) c T x ( s ) ) 2 + ε log V ( x ( s ) ) ] d s.

Hence, from (25) and for (mathcal {F}_{t} -measurable claims X in (mathbb {L}^{2}(mathbb {F}_)), it holds for all (Pin mathcal {P}_{[underline {a},overline {a}]}), P-almost surely, that for all t∈[0,T].

In the regime where μ is large enough so that K<∞ almost surely and that the process has a positive probability of survival, we show that ωs<∞ if and only if s∈[0,s0] for some s0>1 and we study the properties of these functions.

The public has an extraordinary memory for absurdities, and a man who does anything that appeals to a city editor's inexplicable sense of humor will almost surely find that everything happening to him after that is an anticlimax.

As a result, the message sent by any school that adopts an intelligent design policy will almost surely be that it intends to promote the religious belief that an intelligent designer created the universe.

The Fed can and almost surely will reverse that rate hike sometime soon, but the favorable effects of that reversal on spending will take time to materialize.

Then, for any δ > 0 sufficiently small, mild solution x ( t, ξ, λ ) of (1) is almost surely exponentially stable, that is, there exists 0 < α < p η such that lim sup t → + ∞ 1 t log ∥ x ( t, ξ, λ ) ∥ H ≤ − α 2 p, a.s., (25).

CBS won that category convincingly in February -- its average of 14 million viewers during prime time beat its next closest competitor, NBC, by 1.5 million -- and will almost surely triumph in that measure for the full season as well.