Update appendix-a.md

Author: pavly-gerges

discrete-maths/appendix-a.md

@@ -3,11 +3,16 @@
 <div align=center><img src="https://electrostat-lab.github.io/Mathematics-I/discrete-maths/archive/algorithm-analysis-using-machines.jpg" width=550 height=850/></div>
 
 ## The following is the symbolic designation legend to aid in the subsequent mathematical analysis: 
-- $f(N)$: the resultant clock-complexity of the designated closure.
-- $N_c$: another symbolic designation for the resultant clock-complexity of a specified closure.
-- $C(N)$: the clock-complexity of the superclosure of the designated closure.
-- $N'_{c_n}$: the clock-complexity of the $n^{th}$ subclosure of the designated closure.
+
 
 ## The following is the generalized formula: 
 
-$$f(N) = N_c = C(N) * \sum_{n=0}^N N'\_{c_n} = C(N) * (N'\_{c_0} + N'\_{c_1} + N'\_{c_2} + ... + N'\_{c_{(N-2)}} + N'\_{c_{(N-1)}} + N'\_{c_{(N)}})$$
+$$Since, N_c = \prod_{i=1}^I E_{e_i}$$
+
+$$C_c = N_c * \sum_{n=1}^N {\tau}\_n$$
+
+- Such that, ${\tau}\_n = C'_c$, and ${{\tau}^{'}}\_n = {C^{''}}_c$, and so on; as it represents the transition between machinery states, so this is a recursive formula re-evaluating on the most inner closures.
+
+$$Then, C_c = \prod_{i=1}^I E_{e_i} * \sum_{n=0}^N {\tau}\_n = (E_{e_1} * E_{e_2} * ... * E_{e_{I-1}} * E_{e_{I}}) * ({\tau}\_{1} + {\tau}\_{2}  + ... + {\tau}\_{(N-1)} + {\tau}\_{(N)})$$
+
+$$And, t_c = (C_c/F_{CPU}) + {\epsilon}$$