The ((E,q -summability method is regular.
This method is regular [21] under condition (2.2).
s n → s ⇒ t n H → s, as n → ∞, Δ H method is regular, ⇒ E n q ( t n H ) = t n E H → s, as n → ∞, ( E, q ) method is regular, ⇒ ( E q ⋅ Δ H ) method is regular.
Therefore, we have s n → s ⇒ E n 1 ( s n ) = τ n = 1 2 n ∑ r = 0 n ( n r ) s r → s as n → ∞, E n 1 method is regular, ⇒ C n 1 ( E n 1 ( s n ) ) = C n 1 E n 1 → s as n → ∞, C n 1 method is regular, ⇒ C n 1 E n 1 method is regular.
If t n N E → s as n → ∞, then the infinite series ∑ n = 0 ∞ u n is said to be summable ( N, p n ) ( E, q ) to the sum s. s n → s ⇒ ( E, q ) ( s n ) = E n q = ( 1 + q ) − n ∑ k = 0 n ( k n ) q n − k s k → s, as n → ∞, ( E, q ) method is regular, ⇒ ( ( N, p n ) ( E, q ) ( s n ) ) = t n N E → s, as n → ∞, ( N, p n ) method is regular, ⇒ ( N, p n ) ( E, q ) method is regular.
If t n C N → s as n → ∞, then the infinite series ∑ n = 0 ∞ a n or the sequence { s n } is said to be summable C 1 ⋅ N p to the sum s if lim n → ∞ t n C N exists and is equal to s. s n → s ⇒ N p ( s n ) = t n N = P n − 1 ∑ ν = 0 n p n − ν s ν → s, as n → ∞, N p method is regular, ⇒ C 1 ( N p ( s n ) ) = t n C N → s, as n → ∞, C 1 method is regular, ⇒ C 1 ⋅ N p method is regular.
Microscopy revealed that nanoparticles generated by physical methods were regular in shape and efficiently dispersed, while the chemical reduction produced highly aggregated nanoparticles.
Theorem 1 The sequential method I is regular and subsequential.
Sequential method I is regular, i.e., if lim x n = ℓ, then I - lim x n = ℓ.
The method ((mathcal{B}_{i})) is regular if and only if the following conditions hold true (see, for details, [26] and [27]): (i) (Vert mathcal{B} Vert = sup_{n,irightarrowinfty}sum_{k=0}^{infty} vert b_{n,k}(i) vert <infty); (ii) (lim_{nrightarrowinfty}b_{n,k}(i =0) uniformly in i for each (kinmathbb{N}); (iii) (lim_{nrightarrowinfty}sum_{k=0}^{infty}b_{n,k}(i =1) uniformly in i. .
The first method is the regular decoupling field which generates from the unified approach for FBSDEs.
