By Ciza Thomas

The booklet on complicated platforms, sustainability, and innovation explores a wide set of rules and provides a number of the cutting-edge study during this box concisely in six chapters. In a posh method, it truly is tough to grasp precisely how the person parts give a contribution to an saw habit and the level of every component's contributions. it's the interactions of the person elements that make certain the emergent functionalities. This makes it obscure and are expecting the habit of advanced structures and for this reason the consequences of any thoughts during this box. This necessitates for the emergence of a brand new age of strategies with the main target on person orientation and sustainability. This publication explores the various complicated platforms and their dependence at the atmosphere to supply a long term point of view, helping techniques and assisting a sustainable society. The meant viewers of this booklet will more often than not encompass researchers, learn scholars, and practitioners within the box of complicated platforms and sustainability.

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**Example text**

If �(�) = �(��), then ��(�) = ����(�). If �(�) = ����(��), then ��(�) = If �(�) = ���(��), then ��(�) = for � ∈ (0,1) (�(� + �))! � + � � ��(� + �), � ∈ ℕ . (��)! (�(� − 1))! � − 1 � ��(� − 1). (��)! Proof: From Eq. Ln (k ); 0 0 (10) = r k Ln (k ). (11) thus, we obtain Qn (k ) = [ Dnkq (t )]t = t (nk )! 0 This proves the first part. r m +k Ln (m + k ). (13) 0 Hence, we arrive at the part two, Qn (k ) = (n (m + k ))! m +k r Ln (m + k ). (nk )! (14) Finally, to prove the last part, Dnkq (t ) |t = t 0 = Dnk ( I n c ( r t )) |t = t = Dn (k -1) c ( r t )) |t = t 0 (n (k - 1))!

1 1 n! Ln (3) - Ln (2) Ln (1) = 0, (2n )! 2 2 (2n )! (4n )! 1 1 (2n )! Ln (4) - Ln (3) Ln (2) = 0, (3n )! 2 2 (3n )! (5n )! 1 1 (3n )! Ln (5) - Ln (4) Ln (3) = 0. (4n )! 2 2 (4n )! (26) Solving the above system, we obtain the following outcome for � ∈ (0,1]: Ln (t ) » ù 1é 1 1 1 1 t2 + t3 + t4 + t5 ú. ê1 + t + 2ë G(2n + 1) G(3n + 1) G(4n + 1) G(5n + 1) û (27) In general, we attain the solution Ln (t ) » 1 En ,1 (t ). 3. System 3 Suppose the following cost function: F (t , c ) = D 2n c (t ) + Dn c (t ) - c (t ) + I n c (t ) =0 Initial condition : c (0) = 1, c ¢(0) = -1.

25), we obtain the following system: (3n )! (2n )! 1 Ln (3) + Ln (2) -Ln (1) + Ln (0) = 0, n! n! n! (4n )! (3n )! n! Ln (4) + Ln (3) - Ln (2) + Ln (1) = 0, (2n )! (2n )! (2n )! (5n )! (4n )! (2n )! Ln (5) + Ln (4) - Ln (3) + Ln (2) = 0, (3n )! (3n )! (3n )! (6n )! (5n )! (3n )! Ln (6) + Ln (5) - Ln (4) + Ln (3) = 0. (4n )! (4n )! (4n )! (31) Solving the above system, we get the following outcome for � ∈ (0,1): é ù 1 1 1 1 Ln (t ) » ê1 - t + t2 t3 + t4 t5 ú. G n + G n + G n + G n + (2 1) (3 1) (4 1) (5 1) ë û (32) In general, we attain the solution Ln (t ) » En ,1 (-t ).