Sentence examples for by the weaker condition from inspiring English sources

Furthermore, when replacing (V1) by the weaker condition ( V 1 ′ ), this situation becomes more delicate due to the lack of compactness.

(iv) The condition (lambda_{i}in k,1)), for all i is replaced by the weaker condition (lambda_{0}in k,1)).

Another way of sharpening Theorem 1.6 is given in Proposition 5.11, where the condition of finite presentability for the cover of (mathfrak{G }) is replaced by the weaker Condition FP _2).

We shall see that, under fairly general assumptions, this idea allows one to replace (0.1) by the weaker condition r ( K ) ≤ 2 q T (0.2). and, thus, significantly improve the convergence conditions established, in particular, in [6 9, 12].

As a consequence, the same thesis can be deduced replacing, in case (c), that '(( X,d ) ) is complete and (g(X)) is closed' by the weaker condition '(g(X)) is d-complete', and the proof is obtained by verbatim.

The following theorem shows that if the moment condition of Theorem 3.1 is replaced by a stronger condition E | X | p log | X | < ∞, then condition (3.1) can be replaced by the weaker condition (3.7).

Theorem 2.3 is the analogous of Theorem 1 of Rana et al. [32] on partial metrics, except that the conditions (20) and the fact that a, b ∈ [0, 1], are replaced by the weaker conditions (C2), (C3) and a, b ∈ [ 0, 1 2 ].

Mercer [7] made a significant improvement by replacing (1) with the weaker condition that the variances of the two sequences are equal: (sum_{i=1}^{n} p_{i} (x_{i} - bar{x})^{2} = sum_{i=1}^{n} p_{i} (y_{i} - bar{y})^{2}).

We replaced the condition fbigl(0^bigr)=fbigl(0^bigr), (3.2) where fbigl(0^bigr):=lim_{xto0^ f(x), qquad fbigl(0^bigr):= lim_{xto0^ f(x), in [35], Theorem 4.1, by the weakest condition that f is q-regular at zero because (3.2) is only needed to guarantee that (lim_{ntoinfty } f(q^{n-1/2})=lim_{ntoinfty} f -q^{n-1/2})) and this holds if -q^{n-1/2}ular at zero.

This is, however, not the case, because condition (ii) excludes the special case r n ≡ 0. To overcome this drawback, we shall show the same result by replacing condition (ii) with the weak condition: l i m s u p n → ∞ r n <1 ⇔ l i m i n f n → ∞ δ n >0.

Could we weaken the control condition by the strictly weaker condition : lim n → ∞ sup x ∈ C ̃ ∥ T ( t n ) x - T ( t n - 1 ) x ∥ = 0 ?