Vol. 1, No. 1 | Jan – June 2017 SJCMS | P-ISSN: 2520-0755 | Vol. 1 | No. 1 | © 2017 Sukkur IBA 22 Necessary and Sufficient Conditions for Complementary Stochastic Quadratic Operators of Finite-Dimensional Simplex Rawad Abdulghafor, Sherzod Turaev, Akram Zeki, Collage of Information and Communication Technology, International Islamic University Malaysia, 53100,Kuala Lumpur, Malaysia raaac2004@yahoo.com sherzod@iium.edu.my, akramzeki@iium.edu.my Abstract We define a complementary stochastic quadratic operator on finite-dimensional space as a new sub-class of quadratic stochastic operator. We give necessary and sufficient conditions for complementary stochastic quadratic operator. Keywords: Quadratic stochastic operators, complementary stochastic quadratic operator, finite-dimensional space, Sub-stochastic matrix 1. Introduction A quadratic stochastic operator is a general nonlinear model traced back to [1]. A lot of research have been devoted to investigations of various sub-classes of quadratic stochastic operator such as doubly stochastic quadratic operators, dissipative of quadratic stochastic operators, volterra quadratic stochastic operators and extreme doubly stochastic quadratic operators [2]–[7]. In recent years, this theory has become of a great interest in its multiple applications to the problems of population genetics [8]–[10] and control systems [11], [12]. During the past 80 years, nonlinear models have been focused in many researches due to their efficiency as well as their advantages over linear models [13], [14]. In effect, this motivates the study of a new nonlinear model in this paper. The central and main problem in nonlinear models of the family classes of quadratic stochastic operator is to study the limit behaviour their related trajectories. This is true as such nonlinear have complicated structure. This paper focuses on defining a nonlinear model with less complex structure. 2. Preliminaries A quadratic stochastic operator is formed as follows: (Vx)k = ∑ pij,k m i,j=1 xi𝑥𝑗, (1) where 𝑥 ∈ 𝑆𝑚−1 = {𝑥 = (𝑥1,𝑥2,…,𝑥𝑚) ∈ 𝑅 𝑚 ∶ 𝑥𝑖 ≥ 0,∀𝑖 = 1,𝑚,̅̅ ̅̅ ̅̅ ∑𝑥𝑖 𝑚 𝑖=1 = 1} (2) and the coefficients 𝑝𝑖𝑗,𝑘 satisfy the conditions mailto:raaac2004@yahoo.com R. Abdulghafor et al. Necessary and Sufficient Conditions for Complementary Stochastic Quadratic Operators of Finite-Dimensional Simplex (pp. 22 - 27) SJCMS | P-ISSN: 2520-0755 | Vol. 1 | No. 1 | © 2017 Sukkur IBA 23 𝑝𝑖𝑗,𝑘 = 𝑝𝑗𝑖,𝑘 ≥ 0,∑𝑝𝑖𝑗,𝑘 𝑚 𝑘=1 = 1. (3) A quadratic stochastic operator is defined on a free population space. In meaning, suppose that the free space of population involves a set of 𝑚 elements. This set is defined on a simplex as in Equation 2, and it is termed as an (𝑚 − 1)-dimensional simplex. A quadratic stochastic operator assigned the same simplex, 𝑉:𝑆𝑚−1 → 𝑆𝑚−1 , is formed as in Equation 1. The maps among the elements 𝑥𝑖 are considered as a distributed stochastic matrix given by 𝑝𝑖𝑗,𝑘 = (𝑝𝑖𝑗,1,𝑝𝑖𝑗,2,… ,𝑝𝑖𝑗,𝑘) (4) where 𝑝𝑖𝑗,𝑘 is considered under the conditions of Equation 3. In this paper, we define a new model of complementary stochasticity quadratic operators from the general model of quadratic stochastic operator under some derived some conditions included in 𝑝𝑖𝑗,𝑘. The concept of the complementary stochasticity quadratic operators is explored in the next section. 3. Complementary Stochasticity Quadratic Operators As known that 𝑝𝑖𝑗,𝑘 is a stochastic matrix of the distribution matrices (𝑝𝑖𝑗,1,𝑝𝑖𝑗,2,…,𝑝𝑖𝑗,𝑘) in the operator 𝑉(𝑥):𝑆𝑚−1 → 𝑆𝑚−1 as given in Equation 1. Definition: An operator 𝑉(𝑥) is called complementary stochasticity quadratic if has a complementary stochastic matrix 𝑃, then 𝑉(𝑥) = 𝑃𝑥, where 𝑃 is a matrix (m × 𝑚). 𝑃 = [𝑝𝑖𝑗,𝑘] is said to be a complementary stochastic matrix if i) for all 1 ≤ 𝑖 ≠≤ 𝑚, 𝑝 𝑖𝑗,𝑘 = 𝑝 𝑗𝑖,𝑘 = 0 or 𝑝 𝑗𝑖,𝑘 ≠ 0 and 𝑝 𝑗𝑖,𝑘 = 1 − 𝑝 𝑖𝑗,𝑘 . ii) For all1 ≤ 𝑖 ≤ 𝑚, 𝑝 𝑖𝑖,𝑘 = 0 or𝑝 𝑖𝑖,𝑘 = 1 2 . iii) For all distributed matrices𝑃𝑖𝑗,𝑘 = (𝑝 𝑖𝑗,1 ,𝑝 𝑖𝑗,2 ,… ,𝑝 𝑖𝑗,𝑘 ),∑ 𝑝 𝑖𝑗,𝑘 =𝑚𝑖𝑗=1 𝑚,∑ 𝑝 𝑖𝑗,𝑘 = 1𝑚𝑘=1 . Therefore, the new notations for complementary stochasticity quadratic operators for matrices (𝑝𝑖𝑗,1,𝑝𝑖𝑗,2,… ,𝑝𝑖𝑗,𝑘) are 𝑈𝑙𝑜𝑤 = { 𝑝𝑖𝑖,𝑘 = 1 2 ∨ 0,𝑝𝑗𝑖,𝑘 = 1 − 𝑝𝑖𝑗,𝑘 𝑜𝑟 0, ∑ 𝑝𝑖𝑗,𝑘 = 𝑚 𝑚 𝑖𝑗=1 ,∑𝑝𝑖𝑗,𝑘 = 1 𝑚 𝑘=1 } (5) where 𝑝 𝑖𝑖,𝑘 are the diagonal elements limited to either 0 or 1 2 , each symmetric element is stochastic 𝑝 𝑖𝑗,𝑘 + 𝑝 𝑗𝑖,𝑘 = 1 with respect that the elements’ sum of each distributed matrix 𝑝𝑖𝑗,𝑘 begin equal to 𝑚 and the summation of all distributed matrices (𝑝 𝑖𝑗,1 , 𝑝 𝑖𝑗,2 ,… ,𝑝 𝑖𝑗,𝑘 ) is a matrix that has all of its elements equal to 1. The key idea in complementary stochasticity quadratic operators is to make the coefficient of the elements equal to 1 or0. The distributed matrices 𝑝𝑖𝑗,𝑘 = (𝑝𝑖𝑗,1,𝑝𝑖𝑗,2,…,𝑝𝑖𝑗,𝑘) are structured as ( 𝑎11,1 𝑎12,1 … 𝑎1𝑚,1 1 − 𝑎12,1 𝑎22,1 … 𝑎2𝑚,1 ⋮ ⋮ ⋱ ⋮ 1 − 𝑎1𝑚,1 1 − 𝑎2𝑚,1 … 𝑎𝑚𝑚,1 ) + R. Abdulghafor et al. Necessary and Sufficient Conditions for Complementary Stochastic Quadratic Operators of Finite-Dimensional Simplex (pp. 22 - 27) SJCMS | P-ISSN: 2520-0755 | Vol. 1 | No. 1 | © 2017 Sukkur IBA 24 ( 𝑎11,2 𝑎12,2 … 𝑎1𝑚,2 1 − 𝑎12,2 𝑎22,2 … 𝑎2𝑚,2 ⋮ ⋮ ⋱ ⋮ 1 − 𝑎1𝑚,2 1 − 𝑎2𝑚,2 … 𝑎𝑚𝑚,2 ) + ( ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋮ ⋮ ⋱ ⋮ ⋯ ⋯ ⋯ ⋯ ) + ( 𝑎11,𝑚 𝑎12,𝑚 … 𝑎1𝑚,𝑚 1 − 𝑎12,𝑚 𝑎22,𝑚 … 𝑎2𝑚,𝑚 ⋮ ⋮ ⋱ ⋮ 1 − 𝑎1𝑚,𝑚 1 − 𝑎2𝑚,𝑚 … 𝑎𝑚𝑚,𝑚 ) = ( 1 1 ⋯ 1 1 1 ⋯ 1 ⋮ ⋮ ⋱ ⋮ 1 1 ⋯ 1 ) (6) referring to the evaluation operator in Equation 1, the analytical procedure of this method is as follows: ∑ 𝑥𝑖 𝑝𝑖𝑗,𝑘 𝑚 𝑖,𝑗=1 𝑥𝑗 = ( (𝑥1 𝑥2 … 𝑥𝑚) ( 𝑎11,1 𝑎12,1 … 𝑎1𝑚,1 1 − 𝑎12,1 𝑎22,1 … 𝑎2𝑚,1 ⋮ ⋮ ⋱ ⋮ 1 − 𝑎1𝑚,1 1 − 𝑎2𝑚,1 … 𝑎𝑚𝑚,1 ) ( 𝑥1 𝑥2 ⋮ 𝑥𝑚 ) (𝑥1 𝑥2 … 𝑥𝑚) ( 𝑎11,2 𝑎12,2 … 𝑎1𝑚,2 1 − 𝑎12,2 𝑎22,2 … 𝑎2𝑚,2 ⋮ ⋮ ⋱ ⋮ 1 − 𝑎1𝑚,2 1 − 𝑎2𝑚,2 … 𝑎𝑚𝑚,2 ) ( 𝑥1 𝑥2 ⋮ 𝑥𝑚 ) ⋮ ⋮ ⋮ (𝑥1 𝑥2 … 𝑥𝑚) ( 𝑎11,𝑚 𝑎12,𝑚 … 𝑎1𝑚,𝑚 1 − 𝑎12,𝑚 𝑎22,𝑚 … 𝑎2𝑚,𝑚 ⋮ ⋮ ⋱ ⋮ 1 − 𝑎1𝑚,𝑚 1 − 𝑎2𝑚,𝑚 … 𝑎𝑚𝑚,𝑚 ) ( 𝑥1 𝑥2 ⋮ 𝑥𝑚 ) ) where 𝑎𝑖𝑗 is the coefficient among the two points 𝑥𝑖 and 𝑥𝑗 and (𝑥1,𝑥2,… ,𝑥𝑚) are vector of points. Then, the evaluation of the nonlinear operator 𝑉(𝑥) is calculated as follows: ∑ 𝑥𝑖 𝑝𝑖𝑗,𝑘 𝑚 𝑖,𝑗=1 𝑥𝑗 = ( (𝑥1 𝑥2 … 𝑥𝑚) ( 𝑎11,1 𝑎12,1 … 𝑎1𝑚,1 1 − 𝑎12,1 𝑎22,1 … 𝑎2𝑚,1 ⋮ ⋮ ⋱ ⋮ 1 − 𝑎1𝑚,1 1 − 𝑎2𝑚,1 … 𝑎𝑚𝑚,1 ) ( 𝑥1 𝑥2 ⋮ 𝑥𝑚 ) (𝑥1 𝑥2 … 𝑥𝑚) ( 𝑎11,2 𝑎12,2 … 𝑎1𝑚,2 1 − 𝑎12,2 𝑎22,2 … 𝑎2𝑚,2 ⋮ ⋮ ⋱ ⋮ 1 − 𝑎1𝑚,2 1 − 𝑎2𝑚,2 … 𝑎𝑚𝑚,2 ) ( 𝑥1 𝑥2 ⋮ 𝑥𝑚 ) ⋮ ⋮ ⋮ (𝑥1 𝑥2 … 𝑥𝑚) ( 𝑎11,𝑚 𝑎12,𝑚 … 𝑎1𝑚,𝑚 1 − 𝑎12,𝑚 𝑎22,𝑚 … 𝑎2𝑚,𝑚 ⋮ ⋮ ⋱ ⋮ 1 − 𝑎1𝑚,𝑚 1 − 𝑎2𝑚,𝑚 … 𝑎𝑚𝑚,𝑚 ) ( 𝑥1 𝑥2 ⋮ 𝑥𝑚 ) ) (𝑥𝑖) { 𝑉(𝑥1) = 𝑎11,1𝑥1𝑥1 + 𝑎12,1𝑥1𝑥2 + ⋯+ 𝑎1𝑚,1𝑥1𝑥𝑚 + 𝑎21,1𝑥2𝑥1 + 𝑎22,1𝑥2𝑥2 + …+ 𝑎2𝑚,1𝑥2𝑥𝑚 + ⋯…+ 𝑎𝑚1,1𝑥𝑚𝑥1 + 𝑎𝑚2,1𝑥𝑚𝑥2 + ⋯ + 𝑎𝑚𝑚,1𝑥𝑚𝑥𝑚 , 𝑉(𝑥2) = 𝑎11,2𝑥1𝑥1 + 𝑎12,2𝑥1𝑥2 + ⋯+ 𝑎1𝑚,2𝑥1𝑥𝑚 + 𝑎21,2𝑥2𝑥1 + 𝑎22,2𝑥2𝑥2 + …+ 𝑎2𝑚,2𝑥2𝑥𝑚 + ⋯…+ 𝑎𝑚1,2𝑥𝑚𝑥1 + 𝑎𝑚2,2𝑥𝑚𝑥2 + ⋯ + 𝑎𝑚𝑚,2𝑥𝑚𝑥𝑚 , ⋮ = ⋮ + ⋮ + ⋮ + ⋮ + ⋮ + ⋮ + ⋮ + ⋮ 𝑉(𝑥𝑚) = 𝑎11,𝑚𝑥1𝑥1 + 𝑎12,𝑚𝑥1𝑥2 + ⋯+ 𝑎1𝑚,𝑚𝑥1𝑥𝑚 + 𝑎21,𝑚𝑥2𝑥1 + 𝑎22,𝑚𝑥2𝑥2 + …+ 𝑎2𝑚,𝑚𝑥2𝑥𝑚 + ⋯…+ 𝑎𝑚1,𝑚𝑥𝑚𝑥1 + 𝑎𝑚2,𝑚𝑥𝑚𝑥2 + ⋯ + 𝑎𝑚𝑚,𝑚𝑥𝑚𝑥𝑚 , (8) 4. Some Examples Of Complementary Stochasticity Quadratic Operators 1. Example 1, operator 𝑉1(𝑥) with 𝑚 = 3: In the case of distributed matrices 𝑝𝑖𝑗,𝑘, there are three as follows 𝑝𝑖𝑗,𝑘 = { 𝑝𝑖𝑗,1 = ( 0 0 0.9 0 0.5 0.4 0.1 0.6 0.5 ),𝑝𝑖𝑗,2 = ( 0.5 0 0.1 1 0.5 0 0.9 0 0 ),𝑝𝑖𝑗,3 = ( 0.5 1 0 0 0 0.6 0 0.4 0.5 ). Using Equation (7) then we get 𝑉1(𝑥) { 𝑋1 = 𝑥1𝑥3 + 𝑥2𝑥3 + 𝑥2 2 2 + 𝑥3 2 2 𝑋2 = 𝑥1𝑥2 + 𝑥1𝑥3 + 𝑥1 2 2 + 𝑥2 2 2 𝑋3 = 𝑥1𝑥2 + 𝑥2𝑥3 + 𝑥1 2 2 + 𝑥3 2 2 2. Example 2, operator 𝑉2(𝑥) with 𝑚 = 3: In the case of distributed matrices 𝑝𝑖𝑗,𝑘, then there are three and given by 𝑝𝑖𝑗,𝑘 = { 𝑝𝑖𝑗,1 = ( 0 0.5 0.5 0.5 0.5 0 0.5 0 0.5 ),𝑝𝑖𝑗,2 = ( 0.5 0.5 0 0.5 0 0.5 0 0.5 0.5 ), 𝑝𝑖𝑗,3 = ( 0.5 0 0.5 0 0.5 0.5 0.5 0.5 0 ). Similarly, sing Equation (7) then we get R. Abdulghafor et al. Necessary and Sufficient Conditions for Complementary Stochastic Quadratic Operators of Finite-Dimensional Simplex (pp. 22 - 27) SJCMS | P-ISSN: 2520-0755 | Vol. 1 | No. 1 | © 2017 Sukkur IBA 25 𝑉2(𝑥) { 𝑋1 = 𝑥1𝑥2 + 𝑥1𝑥3 + 𝑥2 2 2 + 𝑥3 2 2 𝑋2 = 𝑥1𝑥2 + 𝑥2𝑥3 + 𝑥1 2 2 + 𝑥3 2 2 𝑋3 = 𝑥1𝑥3 + 𝑥2𝑥3 + 𝑥1 2 2 + 𝑥2 2 2 3. Example 3, operator 𝑉3(𝑥) with 𝑚 = 4: In the case of distributed matrices 𝑝𝑖𝑗,𝑘 , there are four = { 𝑝𝑖𝑗,1 = ( 0.5 0 0 0.2 0 0.5 0 0 0 0 0.5 0.6 0.8 0 0.4 0.5 ),𝑝𝑖𝑗,2 = ( 0.5 1 0.7 0 0 0 0 0.6 0.3 0 0.5 0 0 0.4 0 0 ), 𝑝𝑖𝑗,3 = ( 0. 0 0.3 0 0 0 0 0.4 0.7 1 0 0.4 0 0.6 0.6 0 ),𝑝𝑖𝑗,4 = ( 0 0 0 0.8 1 0.5 1 0 0 0 0 0 0.2 0 0 0.5 ) . Using Equation (7) again, we get 𝑉3(𝑥) { 𝑋1 = 𝑥1𝑥4 + 𝑥3𝑥4 + 𝑥1 2 2 + 𝑥2 2 2 + 𝑥3 2 2 + 𝑥4 2 2 𝑋2 = 𝑥1𝑥2 + 𝑥1𝑥3 + 𝑥2𝑥4 + 𝑥1 2 2 + 𝑥3 2 2 𝑋3 = 𝑥1𝑥3 + 𝑥2𝑥3 + 𝑥2𝑥4 + 𝑥3𝑥4 𝑋4 = 𝑥1𝑥2 + 𝑥1𝑥4 + 𝑥2𝑥3 + 𝑥2 2 2 + 𝑥4 2 2 4. Example 4, operator 𝑉4(𝑥) with 𝑚 = 4: In the case of distributed matrices 𝑝𝑖𝑗,𝑘 , there are four 𝑝𝑖𝑗,𝑘 = { 𝑝𝑖𝑗,1 = ( 0 0.6 0.4 0 0.4 0 0 0.6 0.6 0 0 0.4 0 0.4 0.6 0 ),𝑝𝑖𝑗,2 = ( 0 0.4 0.6 0 0.6 0 0 0.4 0.4 0 0 0.6 0 0.6 0.4 0 ), 𝑝𝑖𝑗,3 = ( 0.5 0 0 0.5 0 0.5 0.5 0 0 0.5 0.5 0 0.5 0 0 0.5 ),𝑝𝑖𝑗,4 = ( 0.5 0 0 0.5 0 0.5 0.5 0 0 0.5 0.5 0 0.5 0 0 0.5 ). Using Equation (7) we get 𝑉4(𝑥) { 𝑋1 = 𝑥1𝑥2 + 𝑥1𝑥3 + 𝑥2𝑥4 + 𝑥3𝑥4 𝑋2 = 𝑥1𝑥2 + 𝑥1𝑥3 + 𝑥2𝑥4 + 𝑥3𝑥4 𝑋3 = 𝑥1𝑥4 + 𝑥2𝑥3 + 𝑥1 2 2 + 𝑥2 2 2 + 𝑥3 2 2 + 𝑥4 2 2 𝑋4 = 𝑥1𝑥4 + 𝑥2𝑥3 + 𝑥1 2 2 + 𝑥2 2 2 + 𝑥3 2 2 + 𝑥4 2 2 5. Example 5, operator 𝑉5(𝑥) with 𝑚 = 5: In the case of distributed matrices 𝑝𝑖𝑗,𝑘, then there are five as 𝑝𝑖𝑗,𝑘 = { 𝑝𝑖𝑗,1 = ( 0.5 0 0.9 0 0 0 0 0 0.75 0.6 0.1 0 0 0 0 0 0.25 0 0 0 1 0.4 0 0 0.5) ,𝑝𝑖𝑗,2 = ( 0 0 0.1 0 0 0 0.5 0 0.25 0 0.9 0 0.5 1 0 0 0.75 0 0 0.35 0 0 0 0.65 0 ) , 𝑝𝑖𝑗,3 = ( 0.5 0 0 0.7 0 0 0 0.9 0 0.4 0 0.1 0 0 0.5 0.3 0 0 0.5 0 0 0.6 0.5 0 0 ) ,𝑝𝑖𝑗,4 = ( 0 0.45 0 0.3 1 0.55 0.5 0 0 0 0 0 0.5 0 0.5 0.7 0 0 0 0 0 0 0.5 0 0 ) , 𝑝𝑖𝑗,5 = ( 0 0.55 0 0 0 0.45 0 0.1 0 0 0 0.9 0 0 0 0 0 1 0.5 0.65 0 0 0 0.35 0.5 ) . Using Equation (7) we get (𝑥) { 𝑋1 = 𝑥1𝑥3 + 𝑥1𝑥5 + 𝑥2𝑥4 + 𝑥2𝑥5 + 𝑥1 2 2 + 𝑥5 2 2 𝑋2 = 𝑥1𝑥3 + 𝑥2𝑥4 + 𝑥3𝑥4 + 𝑥4𝑥5 + 𝑥2 2 2 + 𝑥3 2 2 𝑋3 = 𝑥1𝑥4 + 𝑥2𝑥3 + 𝑥2𝑥5 + 𝑥3𝑥5 + 𝑥1 2 2 + 𝑥4 2 2 𝑋4 = 𝑥1𝑥2 + 𝑥1𝑥4 + 𝑥1𝑥5 + 𝑥3𝑥5 + 𝑥2 2 2 + 𝑥3 2 2 𝑋5 = 𝑥1𝑥2 + 𝑥2𝑥3 + 𝑥3𝑥4 + 𝑥4𝑥5 + 𝑥4 2 2 + 𝑥5 2 2 6. Example 6, operator 𝑉6(𝑥) with 𝑚 = 5: R. Abdulghafor et al. Necessary and Sufficient Conditions for Complementary Stochastic Quadratic Operators of Finite-Dimensional Simplex (pp. 22 - 27) SJCMS | P-ISSN: 2520-0755 | Vol. 1 | No. 1 | © 2017 Sukkur IBA 26 In the case of distributed matrices 𝑝𝑖𝑗,𝑘, then there are four as follows 𝑝𝑖𝑗,𝑘 = { 𝑝𝑖𝑗,1 = ( 0.5 0 0 0.5 0 0 0.5 0.5 0 0 0 0.5 0 0 0.5 0.5 0 0 0.5 0 0 0 0.5 0 0.5) ,𝑝𝑖𝑗,2 = ( 0 0 0.65 0 0.35 0 0 0 0.35 0.65 0.35 0 0 0.65 0 0 0.65 0.35 0 0 0.65 0.35 0 0 0 ) , 𝑝𝑖𝑗,3 = ( 0 0 0.35 0 0.65 0 0 0 0.65 0.35 0.65 0 0 0.35 0 0 0.35 0.65 0 0 0.35 0.65 0 0 0 ) ,𝑝𝑖𝑗,4 = ( 0.5 0.5 0 0.5 0 0.5 0 0.5 0 0 0 0.5 0.5 0 0 0.5 0 0 0 0.5 0 0 0 0.5 0 ) , 𝑝𝑖𝑗,5 = ( 0 0.5 0 0 0 0.5 0.5 0 0 0 0 0 0.5 0 0.5 0 0 0 0.5 0.5 0 0 0.5 0.5 0.5) . Using Equation (7) we get (𝑥) { 𝑋1 = 𝑥1𝑥4 + 𝑥2𝑥3 + 𝑥3𝑥5 + 𝑥1 2 2 + 𝑥2 2 2 + 𝑥4 2 2 + 𝑥5 2 2 𝑋2 = 𝑥1𝑥3 + 𝑥1𝑥5 + 𝑥2𝑥4 + 𝑥2𝑥5 + 𝑥3𝑥4 𝑋3 = 𝑥1𝑥3 + 𝑥1𝑥5 + 𝑥2𝑥4 + 𝑥2𝑥5 + 𝑥3𝑥4 𝑋4 = 𝑥1𝑥2 + 𝑥1𝑥4 + 𝑥2𝑥3 + 𝑥4𝑥5 + 𝑥1 2 2 + 𝑥3 2 2 𝑋5 = 𝑥1𝑥2 + 𝑥3𝑥5 + 𝑥4𝑥5 + 𝑥2 2 2 + 𝑥3 2 2 + 𝑥4 2 2 + 𝑥5 2 2 5. 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