The previous partial derivations of the HC function have been derived to the fifth degree of freedom n, so in that case, for r = 1, we obtain the following coefficients along with psi (polygamma) function (3.1).
Hierarchical phrase-based approaches use synchronous grammar rules, which simultaneously expand partial derivations of corresponding sentences in two languages.
By calculating the second partial derivations of TCb, we get h 11 = ∂ 2 TC b ∂ r 2 = h b + π Q e − λ D r, h 12 = h 21 = ∂ 2 TC b ∂ Q ∂ r = D h b + π λ Q 2 e − λ D r − h Q D λ Q + 1 e − λ D Q Open image in new window h 22 = ∂ 2 TC b ∂ Q 2 = 2 A b D Q 3 + 2 D 2 h b + π λ 2 Q 3 e − λ D r + 2 D h b λ Q r − D λ 1 Q 2 + λ D Q + λ 2 2 D 2 e − λ D Q Open image in new window.
The richness of the southern soils is attributable to their partial derivation from the basalts of the Deccan, the physiographic plateau region that constitutes most of peninsular India.
Then F k is a partial Jordan derivation of Jensen type with respect to k-th variable and T k is a partial generalized Jordan derivation of Jensen type with respect to k-th variable (related to F k ).
These partial derivatives in respect to variables of previous layers are calculated according to the composite function derivation rules (9) (10) and formed by products of partial derivatives of outer and inner functions [10].
Let A 1, …, A n be normed algebras over the complex field ℂ and let X be a Banach module over A 1, …, A n - 1 and A n. Then (i) A mapping d k : A 1 × A 2 × ⋯ × A n → X is called a k-th partial Jordan derivation of Jensen type if 2 d k x 1, …, γ a k + γ b k 2, …, x n = γ d k ( x 1, …, a k, …, x n ) + γ d k ( x 1, …, b k, …, x n ) .
However, small bias exists and the application of PD method is limited within the derivation of partial differential form of the equation, which impede the attribution analysis in hydrology.
