College of Art and Sciences Universiti Utara Sintok Kedah, 0060, MALAYSIA

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1 Iteratioal Joural of Pure ad Applied Mathematics Volume 2 No , ISSN: (prited versio); ISSN: (o-lie versio) url: doi: /ijpam.v2i3.5 PAijpam.eu GENERALIZED TWO-HYBRID ONE-STEP IMPLICIT THIRD DERIVATIVE BLOCK METHOD FOR THE DIRECT Abstract: SOLUTION OF SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS Mohammad Alkasassbeh, Zuri Omar 2,2 Departmet of Mathematics College of Art ad Scieces Uiversiti Utara Sitok Kedah, 0060, MALAYSIA I this article, a geeral two-hybrid oe-step implicit third derivative block method is developed for the direct solutio of the secod order iitial value problems through iterpolatio ad collocatio approach. To derive this method, the approximate basis fuctio is iterpolated at the values {x,x +r} while its secod ad third derivatives are collocated at all poits {x,x +r, x +s,x +} i the give iterval. The ew developed method produces better accuracy if compared to the existig methods whe solvig the same problems. AMS Subject Classificatio: 35J25, 35Q60, 65L06, 65L05 Key Words: hybrid block method, secod order ordiary differetial equatio, direct solutio, third derivative, iterpolatio ad collocatio. Itroductio I this article, we are iterested i solvig the followig secod order ordiary differetial equatio y = f(x,y,y ), y(a) = y 0, y (a) = y, a x b. () There are may methods available for approximatig (). Oe of the methods Received: September 30, 206 Revised: November 2, 206 Published: February 7, 207 c 207 Academic Publicatios, Ltd. url:

2 498 M. Alkasassbeh, Z. Omar is called a block method which iitially developed by [2] to provide startig values for predictor-corrector schemes. The usage of this method was further exteded by [] ad [5] to get better approximatio for solvig (). However, block methods are sometimes cofied by zero-stability barrier (see [6]). Hece, several mathematicias itroduced hybrid block methods to overcome this drawback of block methods. I hybrid block methods step ad off-step poits are combied to form a sigle block for solvig ODEs, see [5],[9],[0]. I order to ehace the accuracy of the solutio, [3] suggested secod derivative methods which ca solve stiff ODEs. For the same reaso, [2] also proposed a Simpso s type secod derivative method to approximate the solutio of first order stiff ODEs system. Based o the previous work metioed above, we attempt to develop a ew geeralized two-hybrid oe-step third derivative implicit method for solvig secod order ODEs directly usig iterpolatio ad collocatio approach which ca improve the accuracy. 2. Developmet of the Method The followig power series is used as a basis fuctio for a approximate solutio to () y(x) = 2v+u j=0 a j ( x x ) j, (2) h where u ad v are the umber of iterpolatio ad collocatio poits respectively. The, the secod ad third derivatives are 2v+u y (x) = j=2 2v+u y (x) = j=3 a j j! h 2 (j 2)! (x x ) j 2 = f(x,y,y ), (3) h a j j! h 3 (j 3)! (x x ) j 3 = g(x,y,y ). (4) h Iterpolatig (2) at x +û, for û = {0,r} ad collocatig (3) ad (4) at all poits x +ˆv, for ˆv = {0,r,s,} where {0 < r < s < }, ad the combiig the

3 GENERALIZED TWO-HYBRID ONE-STEP IMPLICIT resulted equatios gives a system of equatios i matrix form a 0 y r r 2 r 3 r 4 r 9 2 a y +r h 2 2 6r 2r !r 7 a 2 f h 2 h 2 h 2 7!h 2 2 6s 2s !s 7 a 3 f +r h 2 h 2 h 2 7!h ! 2 a h 2 h 2 h 2 7!h 6 2 a 5 = f +s f +. (5) h 3 a 6 g 6 24r 9!r h 3 h 3 6!h 3 a 7 g +r 6 24s 9!s a 8 g +s h 3 h 3 6!h ! a 9 g + h 3 h 3 6!h 3 Solvig the ukow coefficiets a js i (5) usig matrix maipulatio ad substitutig them back ito Equatio (2) yields y(x) = α 0 y +α r y +r +h 2 [β r f +r +β s f +s + β i f +i ]+h 3 [γ r g +r +γ s g +s + γ i g +i ], (6) where = 0,,2,...,N, h = x x is the costat step size for the partitio π N of the iterval [a,b] which is give by π N = [a = x 0 < x <... < x N < x N = b], α 0,α r,β 0,β r,β s,β,γ 0,γ r,γ s ad γ are udetermied costats listed i Appedix I. Differetiatig (6) with respect to x produces y (x) = x α 0y + x α ry +r +h 2 [ x β rf +r + x β sf +s + +h 3 [ x γ rg +r + x γ sg +s + x β if +i ] x γ ig +i ]. (7) Evaluatig (6) at the o iterpolatig poits {x +s,x + }ad (7) at all poits x +i, i = {0,r,s,} produces the followig geeral equatios i block form A (0) Y m+ = A () Y m + B (i) F m+i + D (i) G m+i, (8)

4 500 M. Alkasassbeh, Z. Omar where rh sh A (0) = , A () = h , B (0) B (0) 26 B (0) = B (0) B (0), B () = B (0) B (0) D (0) D (0) 26 D (0) = D (0) D (0), D () = D (0) D (0) 66 B () B () 2 B () 3 B () 2 B () 22 B () 23 B () 3 B () 32 B () 33 B () 4 B () 42 B () 43 B () 5 B () 52 B () 53 B () 6 B () 62 B () 63 D () D () 2 D () 3 D () 2 D () 22 D () 32 D () 3 D () 32 D () 33 D () 4 D () 42 D () 43 D () 5 D () 52 D () 53 D () 6 D () 62 D () 63,. Y m+ = [ y +r,y +s,y +,y +r,y +s,y +] T, Y m = [ y s,y r,y,y s,y r,y ] T, F m = [ ] T f 5,f 4,f 3,f 2,f,f, F m+ = [ ] T f +r,f +s,f +, G m = [ ] T g 5,g 4,g 2,g,g, G m+ = [ ] T g +r,g +s,g +. II. Here, the o-zero etries of B (0),B (),D (0) ad D () are listed i Appedix

5 GENERALIZED TWO-HYBRID ONE-STEP IMPLICIT Aalysis of the Method 3.. Zero Stability Defiitio. The hybrid block method formula (8) is said to be zero stable if o root (R z ) of the first characteristic equatio ρ(r) has modulus greater tha oe i.e R z ad if R z = the the multiplicity of R z must ot exceed two. To show that the roots of the first characteristic equatio satisfies the prior defiitio we assume that {r,s} (0,) ad hece ρ(r) =det([ra (0) A () ]) = 0 R rh 0 R sh ρ(r) = 0 0 R R h = 0, R R which implies whose solutios are R 4 (R ) 2 = 0, R = R 2 = R 3 = R 4 = 0, R 5 = R 6 =. Therefore, the developed method is zero stable Order of the Method The liear operator ˆL associated with the hybrid block methods formula (8) accordig to [3] ad [4] is said to be of order p if ˆL{y(x);h} = A (0) Y m A () Y m+ B (i) F m+i expadig i Taylor series ad combiig like terms D (i) G m+i, ˆL{y(x); h} = C i h i y (i) = 0, (9)

6 502 M. Alkasassbeh, Z. Omar where C 0 = C =... = C p+ = 0 ad C p+2 0. The term C p+2 is called the error costat ad the local trucatio error is give by : t +k = C p+2 y p+2 h p+2 (x )+O(h p+3 ). Equatio (9) ca be expressed as show below E 0 E 2 0 E 3 E 4 = 0 0, E 5 0 E 6 0 where the values E,E 2,E 3,E 4,E 5 ad E 6 are listed i Appedix III. Comparig the coefficiets of y i ad h i produces C 0 = C =... = C 9 = 0 with vector of error costats C 0 = r 6 s 2 (3r 4 0r 3 s 2 0r 3 +9r 2 s 3 +36r 2 s 2 +9r 2 36rs 3 36rs 2 +42s 2 ) s 7 (9r 2 s 4 36r 2 s 3 +42r 2 s 0rs 5 +36rs 4 36rs 3 +3s 5 0s 4 +9s 3 ) (42r K 2 s 4 36r 2 s 3 +9r 2 s 36rs 4 +36rs 3 0rs+9s 3 0s 2 +3) 2r 5 s 2 (5r 4 5r 3 s 2 5r 3 +2r 2 s 3 +48r 2 s 2 +2r 2 42rs 3 42rs 2 +42s 2 ) 2s 6 (2r 2 s 4 42r 2 s 3 +42r 2 s 5rs 5 +48rs 4 42rs 3 +5s 5 5s 4 +2s 3 ) (84r 2 s 4 84r 2 s 3 +24r 2 s 84rs 4 +96rs 3 30rs+24s 3 30s 2 +0),, where K = , which implies that the order p of this method is Cosistecy Defiitio 2. A block method is said to be cosistet if its order p is greater tha oe. Sice the order p = 8 of the hybrid block method from the above aalysis that is greater tha oe hece the cosistecy property is satisfied Covergece Theorem 3 (Herici, 962). Cosistecy ad zero stability are sufficiet coditios for a liear multi-step method to be coverget.

7 GENERALIZED TWO-HYBRID ONE-STEP IMPLICIT The hybrid block formula (8) is coverget sice it fulfils both the cosistecy ad zero stability coditios. 4. Numerical Examples I this sectio the accuracy of the geeral two hybrid oe-step implicit hybrid block formula (8) with order 8 is examied o three test problems, with step size h = 00. The computed results are the compared with the latest methods. Problem. f(x,y,y ) = 00y, y(0) =, y (0) = 0. Exact Solutio. y = e 0x, with h = 00. Source. [6]. X Exact Computed Error i Error value solutio solutio ew method for [6] (+00) (-) (-4) (-0) (-4) (-0) (-3) (-9) (-3 ).55924(-9) (-3) (-9) (-3) (-9) (-3) (-9) (-3) (-9) (-3) (-9) (-2) (-9) (-2) (-9) Table : Compariso of the proposed method with [6]. Problem 2. f(x,y,z) = x(y ) 2, y(0) =, y (0) = 2. Exact Solutio. y = +l( 2+x 2 x ), with h = 00. Source. [7].

8 504 M. Alkasassbeh, Z. Omar X Exact Computed Error i Error value solutio solutio ew method for [7] (-6).02(-5) (-6).938(-4) (-5).69(-3) (-6) 4.034(-3) (-5).482(-2) (-5) (-2) (-4) 6.566(-2) (-4).2952(-) (-4) 2.643(-) (-3) (-) Table 2: Compariso of the proposed method with [7]. Problem 3. f(x,y,y ) = y, y(0) = 0, y () =. Exact Solutio. y = e x, with h = 00. Source. [8]. X Exact Computed Error i Error value solutio solutio ew method for [8] (-7) (-3) (-6).6425(-2) (-6) (-2) (-6) (-2) (-5 ).338(-) (-5) 2.037(-) (-5) (-) (-5) (-) (-5) 6.78(-) (-4 ) 8.23(-) Table 3: Compariso of the proposed method with [8]. 5. Coclusio A geeral two hybrid oe-step block method of uiform order 8 has bee developed for the direct solutio of geeral secod order ODEs. The developed method is tested o three problems. Numerical aalysis shows that the developed method is cosistet ad zero stable which implies its covergece. Besides havig excellet properties of the umerical method, the umerical results reveals that the ew method has perform better tha the existig

9 GENERALIZED TWO-HYBRID ONE-STEP IMPLICIT methods. Refereces [] P. Herici, Discrete Variable Methods i ODE, Joh Wiley ad Sos, New York (962). [2] W.E. Mile, Numerical Solutio of Differetial Equatios, Joh Wiley ad Sos, New York (953). [3] S.O. Fatula, Numerical Methods for Iitial Value Problems i Ordiary Differetial Equatio, Academic Press, New York (988). [4] J.D. Lambert, Computatioal Methods i ODEs, Joh Wiley ad Sos, New York (973). [5] W. Gragg, H.J. Stetter, Geeralized multistep predictor-corrector methods, J. Assoc. Comput. Mach., (964), [6] F.M. Kolawole, Cotiuous hybrid block stomer cowell methods for solutios of secod order ordiary differetial equatios, Joural of Mathematical ad Computatioal Sciece, 4 (204), [7] A. O. Adesaya, M. K. Fasasi, ad T. A. Aake, Three steps hybrid block method for the solutio of geeral secod order ordiary differetial equatios, J.P. Joural of Applied Mathematics, 8 (203), -. [8] O. Olaega, D. O. Awoyemi, B. G. Oguware, ad F. O. Obarhua, Cotiuous doublehybrid poit method for the solutio of secod order ordiary differetial equatios, Iteratioal Joural of Advaced Scietific ad Techical Research, 5 (205), [9] W.H. Eright, Secod derivative multistep methods for stiff ordiary differetial equatios, SIAM J. Numer. Aal., (974), [0] S. N. Jator, Solvig secod order iitial value problems by a hybrid multistep method without predictors, Applied Mathematics ad Computatio, 27 (200), [] C. W. Gear, Hybrid methods for iitial value problems i ordiary differetial equatios, SIAM Joural of Numerical Aalysis, 2 (964), [2] R.K. Sahi, S.N. Jator, N. A. Kha, a simpso s-type secod derivative method for stiff systems, Iteratioal joural of pure ad applied mathematics, 8 (202), [3] F. Ngwae, S. Jator, Block hybrid-secod derivative method for stiff systems, It. J. Pure Appl. Math., 80 (202), [4] L. F. Shampie, H. A. Watts, Block implicit oe-step methods, Mathematics of Computatio, 23 (969), [5] J.D. Rosser, A Ruge-kutta for all seasos, SIAM, Rev., 9 (967), [6] G.G. Dahlquist, Numerical itegratio of ordiary differetial equatios, Math. Scad., 4 (956), [7] Gupta, G.K. Implemetig secod-derivative multistep methods usig Nordsieck polyomial represetatio, Math. Comp., 32 (978), 3-8.

10 506 M. Alkasassbeh, Z. Omar Appedix I α 0 = (x x+hr) (hr), α r = (x x ), (hr) β 0 = ( (((x x ) 2 /2 ((x x ) 4 (3r 2 s 2 +4r 2 s+3r 2 +4rs 2 +4rs+3s 2 ))/ (2h 2 r 2 s 2 )+((x x ) 9 (r +s+rs))/(36h 7 r 3 s 3 )+((x x ) 5 (r 3 s 3 + 4r 3 s 2 +4r 3 s+r 3 +4r 2 s 3 +8r 2 s 2 +4r 2 s+4rs 3 +4rs 2 +s 3 ))/(0h 3 r 3 s 3 ) +(hr(x x )( 0r 5 s 0r 5 +36r 4 s 2 +53r 4 s+36r 4 36r 3 s 3 08r 3 s 2 08r 3 s 36r 3 +90r 2 s 3 +24r 2 s 2 +90r 2 s+68rs 3 +68rs 2 882s 3 ))/ (2520s 3 ) ((x x ) 8 (4r 2 s+4r 2 +4rs 2 +rs+4r+4s 2 +4s))/(56h 6 r 3 s 3 ) ((x x ) 6 (4r 3 s 2 +7r 3 s+4r 3 +4r 2 s 3 +20r 2 s 2 +20r 2 s+4r 2 +7rs 3 +20rs 2 +7rs+4s 3 +4s 2 ))/(30h 4 r 3 s 3 )+((x x ) 7 (r 3 s+r 3 +4r 2 s 2 +8r 2 s+4r 2 +rs 3 +8rs 2 +8rs+r+s 3 +4s 2 +s))/(2h 5 r 3 s 3 ) ), β r = ( (((x x ) 8 (7r 3 +7r 2 s+7r 2 8rs 2 3rs 8r+4s 2 +4s))/(56h 6 r 3 (r s) 3 (r ) 3 ) (hr(x x )(05r 6 385r 5 s 385r r 4 s r 4 s +468r 4 80r 3 s 3 836r 3 s 2 836r 3 s 80r r 2 s r 2 s r 2 s 966rs 3 966rs s 3 ))/(2520(r s) 3 (r ) 3 ) ((x x ) 6 ( 7r 3 s 2 28r 3 s 7r 3 +5r 2 s 3 +3r 2 s 2 +3r 2 s+5r 2 +5rs 3 +4rs 2 +5rs 4s 3 4s 2 ))/(30h 4 r 3 (r s) 3 (r ) 3 )+((x x ) 9 (2r s+2rs 3r 2 ))/ (36h 7 r 3 (r s) 3 (r ) 3 )+((x x ) 7 ( 7r 3 s 7r 3 +2r 2 s 2 2r 2 s+2r 2 + 2rs 3 +7rs 2 +7rs+2r s 3 4s 2 s))/(2h 5 r 3 (r s) 3 (r ) 3 ) (s(x x ) 5 (7r 3 s+7r 3 5r 2 s 2 7r 2 s 5r 2 +rs 2 +rs+s 2 ))/(0h 3 r 3 (r s) 3 (r ) 3 ) (s 2 (x x ) 4 (5r 3s+5rs 7r 2 ))/(2h 2 r 2 (r s) 3 (r ) 3 )) ), β s = ( ((((x x ) 7 ( 2r 3 s+r 3 2r 2 s 2 7r 2 s+4r 2 +7rs 3 +2rs 2 7rs+r+ 7s 3 2s 2 2s))/(2h 5 s 3 (r s) 3 (s ) 3 )+((x x ) 6 (5r 3 s 2 +5r 3 s 4r 3 7r 2 s 3 +3r 2 s 2 +4r 2 s 4r 2 28rs 3 +3rs 2 +5rs 7s 3 +5s 2 ))/(30h 4 s 3 (r s) 3 (s ) 3 )+((x x ) 9 (r 2s 2rs+3s 2 ))/(36h 7 s 3 (r s) 3 (s ) 3 ) ((x x ) 8 ( 8r 2 s+4r 2 +7rs 2 3rs+4r+7s 3 +7s 2 8s))/(56h 6 s 3 (r s) 3 (s ) 3 ) (hr 5 (x x )( 20r 4 s+0r 4 +75r 3 s 2 +25r 3 s 36r 3 63r 2 s 3 243r 2 s 2 +08r 2 s+36r rs 3 +38rs 2 98rs 294s 3 +20s 2 )) /(2520s 3 (r s) 3 (s ) 3 )+(r(x x ) 5 ( 5r 2 s 2 +r 2 s+r 2 +7rs 3 7rs 2 + rs+7s 3 5s 2 ))/(0h 3 s 3 (r s) 3 (s ) 3 ) (r 2.(x x ) 4 (3r 5s 5rs+

11 GENERALIZED TWO-HYBRID ONE-STEP IMPLICIT s 2 ))/(2h 2 s 2 (r s) 3 (s ) 3 ))) ), β = ( (((x x ) 8 (4r 2 s 8r 2 +4rs 2 3rs+7r 8s 2 +7s+7))/(56h 6 (r ) 3 (s ) 3 ) ((x x ) 7 (r 3 s 2r 3 +4r 2 s 2 7r 2 s 2r 2 +rs 3 7rs 2 +2rs+7r 2s 3 2s 2 +7s))/(2h 5 (r ) 3 (s ) 3 ) ((x x ) 6 ( 4r 3 s 2 +5r 3 s+5r 3 4r 2 s 3 +4r 2 s 2 +3r 2 s 7r 2 +5rs 3 +3 rs 2 28rs+5s 3 7s 2 ))/(30h 4 (r ) 3 (s ) 3 )+((x x ) 9 (2r+2s rs 3))/(36h 7 (r ) 3 (s ) 3 )+(hr 5 (x x )(0r 4 s 20r 4 36r 3 s 2 +25r 3 s+75r 3 +36r 2 s 3 +08r 2 s 2 243r 2 s 63r 2 98rs rs rs+20s 3 294s 2 ))/(2520(r ) 3 (s ) 3 ) (rs(x x ) 5 (r 2 s 2 +r 2 s 5r 2 +rs 2 7rs+7r 5s 2 +7s))/(0h 3 (r ) 3 (s ) 3 ) (r 2 s 2 (x x ) 4 (5r+5s 3rs 7))/(2h 2 (r ) 3 (s ) 3 )) ), γ 0 = ( (((x x )(x x+hr)(8h 7 r 6 s 5h 7 r 7 +8h 7 r 6 8h 7 r 5 s 2 72h 7 r 5 s 8h 7 r 5 +84h 7 r 4 s 2 +84h 7 r 4 s 26h 7 r 3 s 2 5h 6 r 6 x+5h 6 r 6 x + 8h 6 r 5 sx 8h 6 r 5 sx +8h 6 r 5 x 8h 6 r 5 x 8h 6 r 4 s 2 x+8h 6 r 4 s 2 x 72h 6 r 4 sx+72h 6 r 4 sx 8h 6 r 4 x+8h 6 r 4 x +84h 6 r 3 s 2 x 84h 6 r 3 s 2 x +84h 6 r 3 sx 84h 6 r 3 sx 26h 6 r 2 s 2 x+26h 6 r 2 s 2 x 5h 5 r 5 x 2 +0h 5 r 5 xx 5h 5 r 5 x 2 +8h 5 r 4 sx 2 36h 5 r 4 sxx +8h 5 r 4 sx 2 +8h 5 r 4 x 2 36h 5 r 4 xx +8h 5 r 4 x 2 8h 5 r 3 s 2 x 2 +36h 5 r 3 s 2 xx 8h 5 r 3 s 2 x 2 72h5 r 3 sx 2 +44h 5 r 3 sxx 72h 5 r 3 sx 2 8h5 r 3 x 2 +36h 5 r 3 xx 8h 5 r 3 x 2 +84h5 r 2 s 2 x 2 68h 5 r 2 s 2 xx +84h 5 r 2 s 2 x 2 +84h5 r 2 sx 2 68 h 5 r 2 sxx +84h 5 r 2 sx h5 rs 2 x 2 588h 5 rs 2 xx +294h 5 rs 2 x 2 5h4 r 4 x 3 +5h 4 r 4 x 2 x 5h 4 r 4 xx 2 +5h 4 r 4 x 3 +8h 4 r 3 sx 3 54h 4 r 3 sx 2 x +54h 4 r 3 sxx 2 8h 4 r 3 sx 3 +8h 4 r 3 x 3 54h 4 r 3 x 2 x +54h 4 r 3 xx 2 8h 4 r 3 x 3 8h 4 r 2 s 2 x 3 +54h 4 r 2 s 2 x 2 x 54h 4 r 2 s 2 xx 2 +8h 4 r 2 s 2 x 3 72h 4 r 2 sx 3 +26h 4 r 2 sx 2 x 26h 4 r 2 sxx 2 +72h4 r 2 sx 3 8h4 r 2 x 3 +54h 4 r 2 x 2 x 54h 4 r 2 xx 2 +8h4 r 2 x 3 336h4 rs 2 x h 4 rs 2 x 2 x 008h 4 r s 2 xx h4 rs 2 x 3 336h4 rsx h 4 rsx 2 x 008h 4 rsxx h4 rsx 3 26h 4 s 2 x h 4 s 2 x 2 x 378h 4 s 2 xx 2 +26h 4 s 2 x 3 5h 3 r 3 x h 3 r 3 x 3 x 30h 3 r 3 x 2 x 2 +20h 3 r 3 xx 3 5h 3 r 3 x 4 +8h 3 r 2 sx 4 72h 3 r 2 sx 3 x +08h 3 r 2 sx 2 x 2 72h 3 r 2 sxx 3 +8h 3 r 2 sx 4 +8h 3 r 2 x 4 72h 3 r 2 x 3 x + 08h 3 r 2 x 2 x 2 72h3 r 2 xx 3 +8h3 r 2 x 4 +08h3 rs 2 x 4 432h 3 rs 2 x 3 x +648 h 3 rs 2 x 2 x 2 432h3 rs 2 xx 3 +08h3 rs 2 x h3 rsx 4 728h 3 rsx 3 x +

12 508 M. Alkasassbeh, Z. Omar 2592h 3 rsx 2 x 2 728h3 rsxx h3 rsx 4 +08h3 rx 4 432h 3 rx 3 x +648h 3 rx 2 x 2 432h 3 rxx 3 +08h 3 rx 4 +68h 3 s 2 x 4 672h 3 s 2 x 3 x +008 h 3 s 2 x 2 x 2 672h 3 s 2 xx 3 +68h 3 s 2 x 4 +68h 3 sx 4 672h 3 sx 3 x +008h 3 sx 2 x 2 672h 3 sxx 3 +68h 3 sx 4 5h 2 r 2 x 5 +25h 2 r 2 x 4 x 50h 2 r 2 x 3 x 2 +50h 2 r 2 x 2 x 3 25h2 r 2 xx 4 +5h2 r 2 x 5 50h2 rsx h 2 rsx 4 x 500h 2 rsx 3 x h2 rsx 2 x 3 750h2 rsxx 4 +50h2 rsx 5 50h2 rx h 2 rx 4 x 500h 2 rx 3 x h2 rx 2 x 3 750h2 rxx 4 +50h2 rx 5 60h2 s 2 x h 2 s 2 x 4 x 600h 2 s 2 x 3 x h 2 s 2 x 2 x 3 300h 2 s 2 xx 4 +60h 2 s 2 x 5 240h 2 sx h 2 sx 4 x 2400h 2 sx 3 x h 2 sx 2 x 3 200h 2 sxx h 2 sx 5 60 h 2 x h 2 x 4 x 600h 2 x 3 x h2 x 2 x 3 300h2 xx 4 +60h2 x 5 +55hrx6 330hrx 5 x +825hrx 4 x 2 00hrx3 x hrx2 x 4 330hrxx5 +55hrx6 +90hsx 6 540hsx 5 x +350hsx 4 x 2 800hsx3 x hsx2 x 4 540hsx x 5 +90hsx 6 +90hx 6 540hx 5 x +350hx 4 x 2 800hx 3 x hx 2 x 4 540hxx 5 +90hx 6 35x x 6 x 735x 5 x x 4 x 3 225x 3 x x 2 x 5 245xx 6 +35x 7 ))/(2520h 6 r 2 s 2 ) ), γ r = ( ((x x ) 9 /(72h 6 r 2 (r s) 2 (r ) 2 ) ((x x ) 6 (r 2 s 2 +4r 2 s+r 2 rs 3 2rs 2 +rs 2s 3 2s 2 ))/(30h 3 r 2 (r s) 3 (r ) 2 ) (s 2 (x x ) 4 )/(2hr(r s) 2 (r ) 2 )+(h 2 r 2 (x x )(5r 4 5r 3 s 5r 3 +2r 2 s 2 +48r 2 s+2r 2 42rs 2 42rs+42s 2 ))/(260(r s) 2 (r ) 2 ) ((x x ) 8 (r 2 +rs+2r 2s 2 2s))/(56h 5 r 2 (r s) 3 (r ) 2 )+((x x ) 7 (2r 2 s+2r 2 rs 2 +2rs +r s 3 4s 2 s))/(42h 4 r 2 (r s) 3 (r ) 2 ) (s(x x ) 5 ( 2r 2 s 2r 2 +2rs 2 +rs+s 2 ))/(20h 2 r 2 (r s) 3 (r ) 2 )) ), γ s = ( ((r 2 s(x x ) 4 (r ))/(2h(s )(rs s 2 ) 2 (r +s rs ))+((x x ) 6 (hr 9 s 3 +hr 9 s 2 2hr 9 s hr 8 s 4 3hr 8 s 2 +4hr 8 s hr 7 s 4 3hr 7 s 3 +4hr 7 s 2 +8hr 6 s 4 4hr 6 s 2 4hr 6 s 8hr 5 s 4 +3hr 5 s 3 +3hr 5 s 2 +2hr 5 s+hr 4 s 4 h r 4 s 2 +hr 3 s 4 hr 3 s 3 ))/(30h 4 r 3 s 2 (r s) 2 (r ) 2 (s ) 2 (rs s 2 )(r +s rs )) ((x x ) 8 (2hr 8 s 2 2hr 8 s hr 7 s 3 3hr 7 s 2 +4hr 7 s hr 6 s 4 +2hr 6 s 3 hr 6 s 2 +3hr 5 s 4 +hr 5 s 2 4hr 5 s 3hr 4 s 4 2hr 4 s 3 +3hr 4 s 2 +2hr 4 s+ hr 3 s 4 +hr 3 s 3 2hr 3 s 2 ))/(56h 6 r 3 s(r s)(r ) 2 (s )(r +s rs )(r 2 s 2 r 2 s 3 +2rs 4 2rs 3 s 5 +s 4 ))+((x x )(5h 3 r 3 s 2 5h 3 r 3 s 4h 3 r 2 s 3 9h 3 r 2 s 2 +33h 3 r 2 s+9h 3 r s 4 +87h 3 r s 3 9h 3 r s 2 87h 3 r s 63h 3

13 GENERALIZED TWO-HYBRID ONE-STEP IMPLICIT r 0 s 4 20h 3 r 0 s 3 +5h 3 r 0 s 2 +3h 3 r 0 s+77h 3 r 9 s 4 +79h 3 r 9 s h 3 r 9 s 2 72h 3 r 9 s 243h 3 r 8 s 4 +9h 3 r 8 s 3 +26h 3 r 8 s 2 +8h 3 r 8 s+62h 3 r 7 s 4 02h 3 r 7 s 3 60h 3 r 7 s 2 42h 3 r 6 s 4 +42h 3 r 6 s 3 ))/(2520hrs 2 (r s) 2 (r ) 2 (s ) 2 (rs s 2 )(r +s rs ))+((x x ) 5 (2hr 9 s 3 hr 9 s 2 hr 9 s 2hr 8 s 4 3hr 8 s 3 +2hr 8 s 2 +3hr 8 s+4hr 7 s 4 hr 7 s 3 3hr 7 s+hr 6 s 3 2hr 6 s 2 +hr 6 s 4hr 5 s 4 +3hr 5 s 3 +hr 5 s 2 +2hr 4 s 4 2hr 4 s 3 ))/(20h 3 r 3 s 2 (r s) 2 (r )(s )(rs s 2 )(r +s rs ) 2 )+((x x ) 9 (hr 7 s 2 hr 7 s hr 6 s 3 2hr 6 s 2 +3hr 6 s+3hr 5 s 3 3hr 5 s 3hr 4 s 3 +2hr 4 s 2 +h r 4 s+hr 3 s 3 hr 3 s 2 ))/(72h 7 r 3 s 2 (r s) 2 (r )(s )(rs s 2 )(r 2 s 2 2r 2 s+r 2 2rs 2 +4rs 2r+s 2 2s+)) ((x x ) 7 (hr 9 s 2 hr 9 s+hr 8 s 3 hr 8 s 2hr 7 s 4 3hr 7 s 3 3hr 7 s 2 +8hr 7 s+4hr 6 s 4 +4hr 6 s 3 8hr 6 s 4hr 5 s 3 +3hr 5 s 2 +hr 5 s 4hr 4 s 4 +3hr 4 s 3 +hr 4 s+2hr 3 s 4 hr 3 s 3 h r 3 s 2 ))/(42h 5 rs 2 (r s)(r )(s ) 2 (rs s 2 )(r +s rs )(r 2 s r 3 s r 3 +r 4 ))) ), γ = ( (((x x )(5h 3 r 3 s 4 5h 3 r 3 s 3 33h 3 r 2 s 5 +9h 3 r 2 s 4 +4h 3 r 2 s h 3 r s 6 +9h 3 r s 5 87h 3 r s 4 9h 3 r s 3 3h 3 r 0 s 7 5h 3 r 0 s 6 +20h 3 r 0 s 5 +63h 3 r 0 s 4 +72h 3 r 9 s h 3 r 9 s 7 79h 3 r 9 s 6 77h 3 r 9 s 5 8h 3 r 8 s 9 26h 3 r 8 s 8 9h 3 r 8 s h 3 r 8 s 6 +60h 3 r 7 s 9 +02h 3 r 7 s 8 62h 3 r 7 s 7 42h 3 r 6 s 9 +42h 3 r 6 s 8 ))/(2520hrs 2 (r s) 2 (r ) 2 (s ) 2 (rs s 2 )(r +s rs )) ((x x ) 4 (hr 8 s 4 hr 8 s 5 +3hr 7 s 6 2h r 7 s 5 hr 7 s 4 3hr 6 s 7 +3hr 6 s 5 +hr 5 s 8 +2hr 5 s 7 3hr 5 s 6 hr 4 s 8 +hr 4 s 7 ))/(2h 2 r 2 (r s)(r ) 2 (s ) 2 (rs s 2 ) 2 (r +s rs )) ((x x ) 6 (hr 9 s 4 2hr 9 s 5 +hr 9 s 3 +4hr 8 s 6 3hr 8 s 5 hr 8 s 3 +4hr 7 s 6 3hr 7 s 5 h r 7 s 4 4hr 6 s 8 4hr 6 s 7 +8hr 6 s 5 +2hr 5 s 9 +3hr 5 s 8 +3hr 5 s 7 8hr 5 s 6 h r 4 s 9 +hr 4 s 7 hr 3 s 9 +hr 3 s 8 ))/(30h 4 r 3 s 2 (r s) 2 (r ) 2 (s ) 2 (rs s 2 ) (r +s rs )) ((x x ) 5 (2hr 9 s 4 hr 9 s 5 hr 9 s 6 +3hr 8 s 7 +2hr 8 s 6 3hr 8 s 5 2hr 8 s 4 3hr 7 s 8 hr 7 s 6 +4hr 7 s 5 +hr 6 s 9 2hr 6 s 8 +hr 6 s 7 +h r 5 s 9 +3hr 5 s 8 4hr 5 s 7 2hr 4 s 9 +2hr 4 s 8 ))/20h 3 r 3 s 2 (r s) 2 (r )(s ) (rs s 2 )(r +s rs ) 2 ) ((x x ) 9 (hr 7 s 3 hr 7 s 4 +3hr 6 s 5 2hr 6 s 4 hr 6 s 3 3hr 5 s 6 +3hr 5 s 4 +hr 4 s 7 +2hr 4 s 6 3hr 4 s 5 hr 3 s 7 +hr 3 s 6 ))/(72 h 7 r 3 s 2 (r s) 2 (r )(s )(rs s 2 )(r 2 s 2 2r 2 s+r 2 2rs 2 +4rs 2r+ s 2 2s+)) (s 2 (r s) 2 (x x ) 8 (2r+2s+))/(56h 5 (r )(r+s rs

14 50 M. Alkasassbeh, Z. Omar )(r 2 s 2 r 2 s 3 +2rs 4 2rs 3 s 5 +s 4 ))+((x x ) 7 (hr 9 s 3 hr 9 s 4 hr 8 s 5 +hr 8 s 3 +8hr 7 s 6 3hr 7 s 5 3hr 7 s 4 2hr 7 s 3 8hr 6 s 7 +4hr 6 s 5 +4hr 6 s 4 +hr 5 s 8 +3hr 5 s 7 4hr 5 s 6 +hr 4 s 9 +3hr 4 s 7 4hr 4 s 6 hr 3 s 9 hr 3 s 8 +2hr 3 s 7 ))/(42h 5 rs 2 (r s)(r )(s ) 2 (rs s 2 )(r +s rs )(r 2 s r 3 s r 3 +r 4 )) ). Appedex II B (0) 6 = h2 r s 3 ( 0r 5 s 0r 5 +36r 4 s 2 +53r 4 s+36r 4 36r 3 s 3 08r 3 s 2 08r 3 s 36r 3 +90r 2 s 3 +24r 2 s 2 +90r 2 s+68rs 3 +68rs 2 882s 3), B (0) 26 = h2 s r 3 ( 36r 3 s 3 +90r 3 s 2 +68r 3 s 882r 3 +36r 2 s 4 08r 2 s r 2 s 2 +68r 2 s 0rs 5 +53rs 4 08rs 3 +90rs 2 0s 5 +36s 4 36s 3), B (0) 36 = h r 3 s 3 ( 882r 3 s 3 +68r 3 s 2 +90r 3 s 36r 3 +68r 2 s 3 +24r 2 s 2 08r 2 s+36r 2 +90rs 3 08rs 2 +53rs 0r 36s 3 +36s 2 0s ), B (0) 46 = hr 420s 3 ( 5r 5 s 5r 5 +6r 4 s 2 +23r 4 s+6r 4 4r 3 s 3 40r 3 s 2 40 r 3 s 4r 3 +28r 2 s 3 +28r 2 s+56rs 3 +56rs 2 20s 3), B (0) 56 = hs ( 4r 3 420r 3 s 3 28r 3 s 2 56r 3 s+20r 3 6r 2 s 4 +40r 2 s 3 56r 2 s+ 5rs 5 23rs 4 +40rs 3 28rs 2 +5s 5 6s 4 +4s 3), B (0) 66 = h ( 20r 3 420r 3 s 3 s 3 56r 3 s 2 28r 3 s+4r 3 56r 2 s 3 +40r 2 s 6r 2 28rs 3 +40rs 2 23rs+5r +4s 3 6s 2 +5s) ). B () = h 2 r (r s) 3 (r ) 3 ( 05r 6 385r 5 s 385r r 4 s r 4 s+ 468r 4 80r 3 s 3 836r 3 s 2 836r 3 s 80r r 2 s r 2 s r 2 s 966rs 3 966rs s 3), B () 2 = h 2 r s 3 (r s) 3 (s ) 3 ( 20r 4 s+0r 4 +75r 3 s 2 +25r 3 s 36r 3 63r 2 s 3

15 GENERALIZED TWO-HYBRID ONE-STEP IMPLICIT r 2 s 2 +08r 2 s+36r rs 3 +38rs 2 98rs 294s 3 +20s 2), B () 3 = h 2 r (r ) 3 (s ) 3 ( 0r 4 s 20r 4 36r 3 s 2 +25r 3 s+75r 3 +36r 2 s 3 +08r 2 s 2 243r 2 s 63r 2 98rs 3 +38rs rs+20s 3 294s 2), B () 2 = h 2 s 6 ( 63r r 3 (r s) 3 (r ) 3 s r 3 s 294r 3 +75r 2 s 3 243r 2 s 2 ) +38r 2 s+20r 2 20rs 4 +25rs 3 +08rs 2 98rs+0s 4 36s 3 +36s 2), B () 22 = h 2 s (r s) 3 (s ) 3 ( 80r 3 s r 3 s 2 966r 3 s+378r r 2 s 4 836r 2 s 3 +05s r 2 s 2 966r 2 s 385rs rs 4 836rs rs 2 385s s 4 80s 3), B () 23 = h 2 s (r ) 3 (s ) 3 ( 36r 3 s 2 98r 3 s+20r 3 36r 2 s 3 +08r 2 s r 2 s 294r 2 +0rs 4 +25rs 3 243rs rs 20s 4 +75s 3 63s 2), B () 3 = h r 3 (r s) 3 (r ) 3 ( 294r 3 s r 3 s 63r 3 +20r 2 s 3 +38r 2 s 2 243r 2 s+75r 2 98rs 3 +08rs 2 +25rs 20r +36s 3 36s 2 +0s ), B () 32 = h s 3 (r s) 3 (s ) 3 ( 20r 3 s 2 98r 3 s+36r 3 294r 2 s 3 +38r 2 s 2 +08r 2 s 36r rs 3 243rs 2 +25rs+0r 63s 3 +75s 2 20s ), B () 33 = h (r ) 3 (s ) 3 ( 378r 3 s r 3 s 2 720r 3 s+80r r 2 s 3 248r 2 s r 2 s 468r 2 720rs rs 2 457rs+385r +80 s 3 468s s 05 ), B () 4 = hr 420(r s) 3 (r ) 3 ( 05r 6 350r 5 s 350r r 4 s 2 +87r 4 s+388r 4 40r 3 s 3 342r 3 s 2 342r 3 s 40r r 2 s r 2 s r 2 s 574rs 3 574rs 2 +20s 3), B () 42 = hr 5 420s 3 (r s) 3 (s ) 3 ( 0r 4 s+5r 4 +35r 3 s 2 +0r 3 s 6r 3 28r 2 s 3 98r 2 s 2 +46r 2 s+4r 2 +98rs 3 +42rs 2 70rs 98s 3 +70s 2), B () 43 = hr 5 420(r ) 3 (s ) 3 ( 5r 4 s 0r 4 6r 3 s 2 +0r 3 s+35r 3 +4r 2 s 3

16 52 M. Alkasassbeh, Z. Omar +46r 2 s 2 98r 2 s 28r 2 70rs 3 +42rs 2 +98rs+70s 3 98s 2), B () 5 = hs 5 420r 3 (r s) 3 (r ) 3 ( 28r 3 s 2 +98r 3 s 98r 3 +35r 2 s 3 98r 2 s r 2 s+70r 2 0rs 4 +0rs 3 +46rs 2 70rs+5s 4 6s 3 +4s 2), B () 52 = hs ( 40r 3 420(r s) 3 (s ) 3 s r 3 s 2 574r 3 s+20r r 2 s 4 342r 2 s 3 +05s r 2 s 2 574r 2 s 350rs 5 +87rs 4 342rs rs 2 350s s 4 40s 3), B () 53 = hs 5 420(r ) 3 (s ) 3 ( 4r 3 s 2 70r 3 s+70r 3 6r 2 s 3 +46r 2 s 2 +42r 2 s 98r 2 +5rs 4 +0rs 3 98rs 2 +98rs 0s 4 +35s 3 28s 2), B () 6 = h 420r 3 (r s) 3 (r ) 3 ( 98r 3 s 2 +98r 3 s 28r 3 +70r 2 s 3 +42r 2 s 2 98r 2 s+35r 2 70rs 3 +46rs 2 +0rs 0r +4s 3 6s 2 +5s ), B () 62 = h ( 70r 3 420s 3 (r s) 3 (s ) 3 s 2 70r 3 s+4r 3 98r 2 s 3 +42r 2 s r 2 s 6r 2 +98rs 3 98rs 2 +0rs+5r 28s 3 +35s 2 0s ), B () 63 = h 420(r ) 3 (s ) 3 ( 20r 3 s r 3 s 2 490r 3 s+40r r 2 s 3 554r 2 s r 2 s 388r 2 490rs rs 2 87rs+350 r +40s 3 388s s 05 ). D (0) 6 = h3 r s 2 ( 5r 4 8r 3 s 8r 3 +8r 2 s 2 +72r 2 s+8r 2 84rs 2 84rs +26s 2), D (0) 26 = h3 s r 2 ( 8r 2 s 2 84r 2 s+26r 2 8rs 3 +72rs 2 84rs+5s 4 8s 3 +8s 2), D (0) 36 = h3 2520r 2 s 2 ( 26r 2 s 2 84r 2 s+8r 2 84rs 2 +72rs 8r +8s 2 8s +5), D (0) 46 = h2 r 2 840s 2 ( 5r 4 6r 3 s 6r 3 +4r 2 s 2 +56r 2 s+4r 2 56rs 2 56rs +70s 2),

17 GENERALIZED TWO-HYBRID ONE-STEP IMPLICIT D (0) 56 = h2 s 2 840r 2 ( 4r 2 s 2 56r 2 s+70r 2 6rs 3 +56rs 2 56rs+5s 4 6s 3 +4s 2), D (0) 66 = h2 840r 2 s 2 ( 70r 2 s 2 56r 2 s+4r 2 56rs 2 +56rs 6r +4s 2 6s+5 ). D () = h 3 r 3 260(r s) 2 (r ) 2 ( 5r 4 5r 3 s 5r 3 +2r 2 s 2 +48r 2 s+2r 2 42rs 2 42rs+42s 2), D () 2 = h 3 r s 2 (r s) 2 (s ) 2 ( 8r 42s+36rs 9r 2 s 8r 2 +5r 3), D () 3 = h 3 r (r ) 2 (s ) 2 ( 5r 3 8r 2 s 9r 2 +8rs 2 +36rs 42s 2), D () 2 = h 3 s r 2 (r s) 2 (r ) 2 ( 42r 8s 36rs+9rs 2 +8s 2 5s 3), D () 22 = h 3 s 3 260(r s) 2 (s ) 2 ( 2r 2 s 2 42r 2 s+42r 2 5rs 3 +48rs 2 42rs+5s 4 5s 3 +2s 2), D () 23 = h 3 s (r ) 2 (s ) 2 ( 8r 2 s 42r 2 8rs 2 +36rs+5s 3 9s 2), D () 3 = h r 2 (r s) 2 (r ) 2 ( 9r +8s 36rs+42rs 2 8s 2 5 ), D () 32 = h s 2 (r s) 2 (s ) 2 ( 8r +9s 36rs+42r 2 s 8r 2 5 ), D () 33 = h 3 260(r ) 2 (s ) 2 ( 42r 2 s 2 42r 2 s+2r 2 42rs 2 +48rs 5r +2s 2 5s+5 ), D () 4 = h 2 r 2 840(r s) 2 (r ) 2 ( 5r 4 40r 3 s 40r 3 +28r 2 s 2 +2r 2 s+28r 2 84rs 2 84rs+70s 2), D () 42 = h 2 r 5 840s 2 (r s) 2 (s ) 2 ( 4r 28s+28rs 8r 2 s 6r 2 +5r 3), D () 43 = h 2 r 5 840(r ) 2 (s ) 2 ( 5r 3 6r 2 s 8r 2 +4rs 2 +28rs 28s 2),

18 54 M. Alkasassbeh, Z. Omar D () 5 = h 2 s 5 840r 2 (r s) 2 (r ) 2 ( 28r 4s 28rs+8rs 2 +6s 2 5s 3) D () 52 = h 2 s 2 840(r s) 2 (s ) 2 ( 28r 2 s 2 84r 2 s+70r 2 40rs 3 +2rs 2 84rs+5s 4 40s 3 +28s 2), D () 53 = h 2 s 5 840(r ) 2 (s ) 2 ( 4r 2 s 28r 2 6rs 2 +28rs+5s 3 8s 2), D () 6 = h 2 840r 2 (r s) 2 (r ) 2 ( 8r +6s 28rs+28rs 2 4s 2 5 ), D () 62 = h 2 840s 2 (r s) 2 (s ) 2 ( 6r +8s 28rs+28r 2 s 4r 2 5 ), D () 63 = h 2 840(r ) 2 (s ) 2 ( 70r 2 s 2 84r 2 s+28r 2 84rs 2 +2rs 40r +28s 2 40s+5 ). Appidex III E = ( (hi r i )y (i) y hry 2520s 3 (h 2 y (2) r 2 (882s 3 +0r 5 (+s) 68rs 2 (+s)+36r 3 (+s) 3 6r 2 s(5+4s+5s 2 ) r 4 (36+ 53s+36s 2 ))) h i+2 y (i+2) [ 2520( +r) 3 (r s) 3 ) (r)i (r 2 (05r s 3 385r 5 (+s) 966rs 2 (+s)+6r 2 s(20+403s+20s 2 )+r 4 ( s+468s 2 ) 36r 3 (5+5s+5s 2 +5s 3 )) (2520(r s) 3 ( +s) 3 s 3 )) (r 6 (0r 4 ( +2s)+42s 2 ( 5+7s)+r 3 (36 25s 75s 2 ) 6rs( s+42s 2 )+9r 2 ( 4 2s+27s 2 +7s 3 ))(s) i (2520( +r) 3 ( +s) 3 ) (r 6 (0r 4 ( 2+s)+42s 2 ( 7+5s)+r 3 (75+25s 36s 2 ) 6rs( 42 23s+33s 2 )+9r 2 ( 7 27s+2s 2 +4s 3 ))] (2520s 2 ) (h3 y (3) r 3 (5r s 2 8r 3 (+s) 84rs(+s)+8r 2 (+4s+s 2 )))+ h i+3 y (i+3) [ (260( +r) 2 (r s) 2 )) ri ((r 3 (5r 4 +42s 2 5r 3 (+s) 42rs(+s)+2r 2 (+4s+s 2 ))) (2520(r s) 2 ( +s) 2 s 2 ) si ((r 6 (5r 3 42s 9r 2 (2+s)+8r (+2s)))) (2520( +r) 2 ( +s) 2 ) ((r6 (5r 3 42s 2 +8rs(2+s) 9r 2 (+2s))) ),

19 GENERALIZED TWO-HYBRID ONE-STEP IMPLICIT E 2 = E 3 = [ (hi s i )y (i) y hsy (2520r 3 ) (h2 y (2) s 2 (2s 3 (8 8s+5s 2 ) 2r 2 s (4+2s 9s 2 +3s 3 )+6r 3 (47 28s 5s 2 +6s 3 )+rs 2 ( 90+08s 53s 2 +0s 3 ))) h i+2 y (i+2) [ (2520( +r) 3 r 3 (r s) 3 ) (r)i ((s 6 (2r 3 (4 2s +3s 2 ) 2s 2 (8 8s+5s 2 )+rs(98 08s 25s 2 +20s 3 ) 3r 2 (70+46s 8s 2 +25s 3 ))))+ (2520(r s) 3 ( +s) 3 ) (s)i ((s 2 (s 3 (80 468s+385s 2 05s 3 ) +6r 3 ( 63+6s 20s 2 +30s 3 ) 6r 2 s( 6+403s 306s 2 +78s 3 )+rs 2 ( s 457s s 3 )))) (2520( +r) 3 ( +s) 3 )) (((s6 (s 2 ( 63+75s 20s 2 )+6r 3 (35 33s+6s 2 ) 6r 2 (49 23s 8s 2 +6s 3 )+rs( s +25s 2 +0s 3 ))))] [ (2520r 2 ) (h3 y (3) (s 3 (6r 2 (2 4s+3s 2 ) 6rs(4 2s+3s 2 ) +s 2 (8 8s+5s 2 ))))] h i+3 y (i+3) 840 [ (2520( +r) 2 r 2 (r s) 2 ) ri ((s 6 (r( 42+36s 9s 2 )+s(8 8s+5s 2 )))) (260(r s) 2 ( +s) 2 )) si (((s 3 (6r 2 (7 7s+2s 2 ) 3rs ] (4 6s+5s 2 )+s 2 (2 5s+5s 2 )))) (s6 ( 8r( 2+s)s+6r 2 ( 7+3s)+s 2 ( 9+5s))), ( (2520( +r) 2 ( +s) 2 ) ] (hi )y(i) y hy 2520r 3 s (h 2 y (2) 3 (2s(5 8s+8s 2 )+r(0 53s +08s 2 90s 3 ) 2r 2 (3 9s+2s 2 +4s 3 )+6r 3 (6 5s 28s 2 +47s 3 )) h i+2 y (i+2) [ (2520( +r) 3 r 3 (r s) 3 ) ri ((2r 3 (3 2s+4s 2 ) 2s(5 8s +8s 2 ) 3r 2 (25 8s+46s 2 +70s 3 )+r(20 25s 08s 2 +98s 3 )))+ (2520(r s) 3 ( +s) 3 s 3 ) si ((s( 20+75s 63s 2 )+6r 3 (6 33s+35s 2 ) 6r 2 (6 8s 23s 2 +49s 3 )+r(0+25s 243s s 3 )))+ (2520( +r) 3 ( +s) 3 ) ((05 385s+468s 2 80s 3 +6r 3 ( 30+20s 6s 2 +63s 3 ) 6r 2 ( s 403s 2 +6s 3 )+r( s 836s s 3 )))] (2520r 2 s 2 ) (h3 y (3) (5 8s+8s 2 6r(3 2s+4s 2 )+6r 2 (3 4s+2s 2 ))) [(2520( +r) 2 r 2 (r s) 2 ) ri ((5 8s+8s 2 +r( 9+36s 42s 2 )))+ (2520(r s) 2 ( +s) 2 s 2 ) si ((5 9s+8r( +2s) 6r 2 ( 3+7s))) h i+3 y (i+3) (260( +r) 2 ( +s) 2 ) (( 5+5s 2s2 6r 2 (2 7s+7s 2 )+3r(5 6s+4s 2 )))] ), E 4 = ( (hi r i )y (i+) y (420s 3 ) (hy(2) r(20s 3 +5r 5 (+s) 56rs 2 (+s) r 4 (6+23s+6s 2 ) 28r 2 (s+s 3 )+2r 3 (7+20s+20s 2 +7s 3 ))) h i+ y (i+2) [ (420( +r) 3 (r s) 3 ) ri (r(05r 6 +20s 3 350r 5 (+s) 574rs 2

20 56 M. Alkasassbeh, Z. Omar (+s)+4r 2 s(35+s+35s 2 )+r 4 (388+87s+388s 2 ) 2r 3 (70+67s +67s 2 +70s 3 )))+ (420(r s) 3 ( +s) 3 s 3 )) si ( ((r 5 (5r 4 ( +2s)+4s 2 ( 5+7s) +r 3 (6 0s 35s 2 ) 4rs( 5+3s+7s 2 )+2r 2 ( 7 23s+49s 2 +4s 3 )))) + (420( +r) 3 ( +s) 3 )) ( r5 (5r 4 ( 2+s)+4s 2 ( 7+5s)+r 3 (35+0s 6s 2 ) +4rs(7+3s 5s 2 )+2r 2 ( 4 49s+23s 2 +7s 3 )))] (840s 2 ) (h2 y (3) (r 2 (5r 4 +70s 2 6r 3 (+s) 56rs(+s)+4r 2 (+4s+s 2 ))))+ h i+2 y (i+3) [ (840( +r) 2 (r s) 2 )) ri ( ((r 2 (5r 4 +70s 2 40r 3 (+s) 84rs(+s)+28r 2 (+4s+s 2 ))))+ (840(r s) 2 ( +s) 2 s 2 ) si ((r 5 (5r 3 28s 8r 2 (2+s)+4r(+2s)))) ) + (840( +r) 2 ( +s) 2 ) ((r5 (5r 3 28s 2 +4rs(2+s) 8r 2 (+2s))))], ( s E 5 = i (hi )y (i+) y (420r 3 )) (hy(2) s(s 3 (4 6s+5s 2 )+4r 3 (5 4s 2s 2 +s 3 ) 8r 2 s(7 5s 2 +2s 3 )+rs 2 ( 28+40s 23s 2 +5s 3 ))) [ (420( +r) 3 r 3 (r s) 3 ) ri ((s 5 (s 2 ( 4+6s 5s 2 )+4r 3 (7 7s+2s 2 ) 7r 2 (0+6s 4s 2 +5s 3 )+2rs(35 23s 5s 2 +5s 3 ))))+ (420(r s) 3 ( +s) 3 )) s i (s(s 3 (40 388s+350s 2 05s 3 )+4r 3 ( 5+4s 35s 2 +0s 3 ) 2r 2 s( s 67s 2 +94s 3 )+rs 2 ( s 87s s 3 )))+ (420( +r) 3 ( +s) 3 )) ( s5 (s 2 ( 28+35s 0s 2 )+4r 3 (5 5s+s 2 ) +r 2 ( 98+42s+46s 2 6s 3 )+rs(98 98s+0s 2 +5s 3 )))] (840r 2 ) (h 2 y (3) (s 2 (4r 2 (5 4s+s 2 ) 8rs(7 7s+2s 2 )+s 2 (4 6s+5s 2 )))) + h i+2 y (i+3) [ (840( +r) 2 r 2 (r s) 2 ) ri ((s 5 ( 4r(7 7s+2s 2 )+s(4 6s +5s 2 ))))+ (840(r s) 2 ( +s) 2 )) si ( ((s 2 (4r 2 (5 6s+2s 2 ) 4rs(2 28s+ ) 0s 2 )+s 2 (28 40s+5s 2 ))))+ (s5 (4r 2 ( 2+s) 4rs( 7+4s)+s 2 ( 8+5s))) (840( +r) 2 ( +s) 2 ) ], h i+ y (i+2) E 6 = ( (hi )y(i+) y (420r 3 s 3 )) (hy(2) (s(5 6s+4s 2 )+r(5 23s+40s 2 28s 3 ) 8r 2 (2 5s+7s 3 )+4r 3 ( 2s 4s 2 +5s 3 ))) h i+ y (i+2) [ (420( +r) 3 r 3 (r s) 3 ) ri ((s( 5+6s 4s 2 )+4r 3 (2 7s+7s 2 ) 7r 2 (5 4s+6s 2 +0s 3 )+2r(5 5s 23s 2 +35s 3 )))+ (420(r s) 3 ( +s) 3 s 3 ) si ((s( 0 +35s 28s 2 )+4r 3 ( 5s+5s 2 )+r 2 ( 6+46s+42s 2 98s 3 )+r(5+0s 98s 2 +98s 3 )))+ (420( +r) 3 ( +s) 3 )) ((05 350s+388s2 40s 3 +4r 3 ( 0 +35s 4s 2 +5s 3 ) 2r 2 ( 94+67s 777s s 3 )+r( s

21 GENERALIZED TWO-HYBRID ONE-STEP IMPLICIT s s 3 )))] (840r 2 s 2 ) (hi+2 y (3) (5 6s+4s 2 +4r 2 ( 4s+5s 2 ) 8r(2 7s+7s 2 ))) h i+2 y (i+3) [ (840( +r) 2 r 2 (r s) 2 ) ri ((5 6s+4s 2 4r (2 7s+7s 2 )))+ (840(r s) 2 ( +s) 2 s 2 ) si ((5+r 2 (4 28s) 8s+4r( 4+7s))) + (840( +r) 2 ( +s) 2 ) (( 5+40s 28s2 4r 2 (2 6s+5s 2 )+4r(0 28s+ 2s 2 ))) ).

22 58