Scribed by (22)24) as: e(k + 1) = e(k) + e(k) e(k
Scribed by (22)24) as: e(k + 1) = e(k) + e(k) e(k) = W (k) = -e(k) W (k)e(k ) e(k ) W (k ) T, exactly where e(k + 1) may be de(22) (23)W (k )e(k – 1) + ( – 1)w (k) = e(k – 1) + ( – 1)w (k) e(k – 1) + – 1 w (k)(24)The expression of e(k ) is usually obtained from (23) and (24), as shown in (25) as: e(k) = -e(k )e(k ) 2 W (k )(25)YTX-465 MedChemExpress Mathematics 2021, 9,7 ofAt exactly the same time, we can know the equivalent expression of (21), as shown beneath:1 V (k ) = two e(k)[e(k) + 2e(k)](26)Substitute (25) into (26) to have the below: V (k) =1-e(k)= 1 2 e2 ( k ) 2 = 1 e2 (k)e(k ) two W (k ) two e(k ) 4 W (k ) 2 e(k ) two W (k )-e(k) – e2 (k)e(k ) 2 W (k) 2 e(k ) two W (k ) 2+ 2e(k)(27)e(k ) W (k )-According towards the Lyapunov stability theorem, the technique is steady if V (k) 0 at any sampling time. It might be observed from (27) that , e2 (k), and than 0, so a enough situation for V (k) 0 is:e(k) two W (k ) two e(k ) 2 W (k ) two e(k ) two W (k )are all numbers greater- 2 0 = 0 e(k) two W (k)(28)whereis described as follows:e(k ) two W (k )=e(k ) two w (k )+e(k ) 2 w (k )+e(k ) two w (k )(29)Theorem 2. The variable is defined as sup , where sup = supku(k ) two . W (k )The other variable isdescribed as sup , exactly where sup = supky(k ) 2 . u(k )The convergence of your APNF model is assured if2 sup supthe learning rate satisfies 0 Proof. In the chain rule: (30)e(k ) W (k )=e(k ) y(k ) u(k ) y(k ) u(k ) W (k )(31)substitute (31) into (24) and (25) to have: W (k) = -e(k) y(k) u(k) W (k) e(k) = -e(k) Substitute (33) into (26) to acquire:e(k ) y(k ) u(k ) two e -e(k) y(k) y(k ) u(k ) W (k ) 2 (k) 4 y(k ) four u(k ) 4 two e e = 1 2 e2 (k) y(k) – e2 (k) y(k) two (k) two u(k) 2 W (k) two (k) two two y(k ) 2 u(k ) 2 e = e2 (k) y(k) 1 (k) two u(k) 2 W (k) 2 [- two y(k ) 2 u(k ) two e + 1 y(k) 2 (k) 2 u(k) 2 W (k) 2 e(k ) y(k ) u(k ) two y(k) u(k ) W (k ) 2 e(k ) y(k ) u(k )(32) (33)V (k) =1-e(k)y(k ) u(k ) two + 2e(k) u(k ) W (k ) 2 y(k ) two u(k ) 2 u(k ) 2 W (k )(34)Mathematics 2021, 9,eight ofAccording to (ten), we are able to know the following:u(k ) w (k ) w (k) x (k) w (k) u(k ) w (k ) w (k) x (k) w (k) u(k ) w (k ) w (k) x (k) w (k)= i = +=Pu(i ) wi (i ) wi (i ) w (i ) w (k) w (k) x (k) w (k) + x (k) w (k) u(i ) wi (i ) wi (i ) w (i ) w (k) w (k) x (k) w (k) + x (k) w (k)(35)= i = +=P(36)= i = +=Pu(i ) wi (i ) wi (i ) w (i ) w (k) w (k) x (k) w (k) + x (k ) w (k)(37)It could be obtained from (35)37):u(k ) W (k )=u(k ) u(k ) T u(k ) w (k ) w (k ) w (k )(38)According to (9), the maximum impact factor from the symmetric catastrophe function is w (k), which is usually obtained as shown under:wi ( k ) i (k = w (k)+w (w)+()- )w (k) wi (k )++ k wi ( k ) wi = w (k)+w(k) w (k) wi ( k ) (k)+ wi ( k ) wi ( k ) = wi ( k ) w (k )+(- )[w (k )+w (k )] wi ( k ) w (k = w (k)+(- i)w) (k)+w (k) wi (k )–i f w (k ) 0, w (k) 0 i f w (k) 0, w (k ) 0 i f w (k ) 0, w (k) 0 i f w (k) 0, w (k) wi (k)|i,, =(39)The corresponding derivative might be obtained from (39):wi (k ) w j, (k )i, j , , ++, , –=wi (k ) 1 wi (k ) w j (k )+wi (k ) wi (k ) wi (k) 2 w j (k )(40)The meaning of (40) is that the symmetric catastrophe functions w (k), w (k) and w (k ) are used to derive w (k), w (k), and w (k), respectively, in 4 cases (corresponding for the four instances of (39): ++, +-, -+, –). In accordance with (40), when w (k) w (k ) 0, w (k) 0, w (k) may be written as:w ( k ) w,++ (k )=1 w (k )+w (k )+(- )w (k )-w ( k )[w (k)+w (k)+(- )w (k)]=1 w (k)++-w ( k ) w (k )++(41)In the exact same way, we are able to Alizarin complexone Protocol acquire other formulas, as shown below:w ( k ) w,– (k )=1 w (k )—w ( k.