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formerly Central European Journal of Mathematics

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Volume 14, Issue 1


Volume 13 (2015)

Dynamics of doubly stochastic quadratic operators on a finite-dimensional simplex

Rawad Abdulghafor
  • Corresponding author
  • Kulliyyah of Information and Communication Technology, International Islamic University Malaysia, 53100 Kuala Lumpur, Malaysia
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/ Farruh Shahidi / Akram Zeki
  • Kulliyyah of Information and Communication Technology, International Islamic University Malaysia, 53100 Kuala Lumpur, Malaysia
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/ Sherzod Turaev
  • Kulliyyah of Information and Communication Technology, International Islamic University Malaysia, 53100 Kuala Lumpur, Malaysia
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Published Online: 2016-07-26 | DOI: https://doi.org/10.1515/math-2016-0045


The present paper focuses on the dynamics of doubly stochastic quadratic operators (d.s.q.o) on a finite-dimensional simplex. We prove that if a d.s.q.o. has no periodic points then the trajectory of any initial point inside the simplex is convergent. We show that if d.s.q.o. is not a permutation then it has no periodic points on the interior of the two dimensional (2D) simplex. We also show that this property fails in higher dimensions. In addition, the paper also discusses the dynamics classifications of extreme points of d.s.q.o. on two dimensional simplex. As such, we provide some examples of d.s.q.o. which has a property that the trajectory of any initial point tends to the center of the simplex. We also provide and example of d.s.q.o. that has infinitely many fixed points and has infinitely many invariant curves. We therefore came-up with a number of evidences. Finally, we classify the dynamics of extreme points of d.s.q.o. on 2D simplex.

Keywords: Doubly stochastic quadratic operators; Fixed point; Trajectory; Extreme point; Simplex

MSC 2010: 15A51; 15A63; 46T99; 46A55

1 Introduction

Without a doubt, many biological processes can be considered as some nonlinear dynamical systems. From this point of view, the main problem associated with the state of the process in certain time frame is the same as when studying the limit behavior of the trajectories of corresponding dynamical systems. One of the important dynamical systems is the one generated by quadratic stochastic operators (q.s.o. henceforth) on finite-dimensional simplex, which appear in many problems of population genetics [1]. The central problem in this theory is to study the limit behavior of the trajectory(dynamics) of a given initial point. There are many papers that study such operators (see [2] for review). One of the important classes of q.s.o. are Volterra operators [3]. It is proven for this operators that the ω-limit set of any non-fixed initial point from the interior of the simplex belong to the boundary of the simplex. Moreover, ω– limits set (we will definite it later) of any initial point is either a countable set or is a singleton. There are other classes of q.s.o. that are well studied [46]. Another interesting class of q.s.o. is the class of doubly stochastic (d.s.q.o. henceforth) ones. It was firstly defined in [3] in terms of majorization. In was later developed in papers [7, 8] and infinite-dimensional case in [9]. These papers basically studied the structure of the class of d.s.q.o., that is, necessary and sufficient conditions for doubly stochasticity and extreme points of the set of doubly stochastic operators. In particular, it is shown that, up to permutation, there are 37 extreme d.s.q.o. on 2D simplex. It is also worth mentioning that d.s.q.o. are very different than Volterra operators since from the classification theorem for d.s.q.o. [8] it follows that the class of d.s.q.o. and Volterra intersect at the identity operator. Nonetheless, the study of the limit behavior (dynamics) of d.s.q.o is still wide open, except for ergodic theorem for these operators [10]. In fact, the paper [2] asks to investigate the above problems as the class of d.s.q.o. Namely, it asks if a given d.s.q.o. in general is regular, i.e. has convergent trajectory for any initial point. We will answer positively to this question under small assumption. This question is interesting because d.s.q.o. have rich applications. One application is that these kinds of maps appear in population genetics problems as being a sub-class of the general q.s.o. In addition, doubly stochastic maps are widely used in economics and statistics. One can see it in [11]. The present paper, therefore, aims to provide a couple of convergence theorems for general d.s.q.o. Importantly, it does not give any clue where the trajectory converges. Therefore, in addition, we classify the dynamics of extreme d.s.q.o. on 2D simplex. A motivation to study extreme d.s.q.o. comes from the fact that any d.s.q.o. can be written as a convex combination of the extremes [7, 8]. Thus we deem we should start with extreme d.s.q.o. Another motivation comes from the paper [12] where general quadratic stochastic operators having coefficients 1 or 0 were classified on 2D simplex. Extreme d.s.q.o.’s have coefficients 1; 1 2 or 0 [7, 8]. So our result will extend the results in [12] in the class of d.s.q.o. on 2D simplex. The paper is organized as follows. In the next section we give some preliminaries concerning majorization and d.s.q.o. In Section 3, we provide a class of Lyapunov functions for d.s.q.o. and provide a general convergence theorem for d.s.q.o. on finite-dimensional simplex. Section 4 focuses on dynamics classifications of extreme d.s.q.o. on 2D simplex. We note that a couple of examples of extreme d.s.q.o. on 2D simplex have been considered in [13].

2 Preliminaries

In this section we provide some definition based on majorization theory and define doubly stochastic operator.

We define the (m – 1)– dimensional simplex as follows.


The set intSm – 1 = {xSm – 1 : xi > 0} is called the (relative) interior of the simplex. The points ek=(0,0,,1k,,0) are the vertices of the simplex and the scalar vector (1m,1m,,1m) is the center of the simplex.

A quadratic stochastic operator V : Sm–1Sm – 1 is defined as:


Where the coefficients pij,k satisfy the following conditions


If we let Ak = (pij,k )i, j, k = 1, m, then the operator can be given in terms of m matrices and we write


where matrices Ai are non-negative and symmetric. For any x = (x1, x2, …, xm) ∈ Sm–1, we define x↓ = (x[i], x[2], …, x[m]) where x[1]x[2] ≥ ··· ≥ x[m] - nonincreasing rearrangement x. Recall [11, 14] that for two elements x, y of the simplex Sm–1 the element x is majorized by y and written xy ( or yx) if the following condition holds


for any k = 1, m – 1. In fact, this definition is referred to as weak majorization [11], the definition of majorization requires i=1mx[i]=i=1my[i]. However, since we consider vectors only on the simplex, we may drop this condition.

A matrix P = (pij)i, j = 1, m called doubly stochastic (sometimes bistochastic), if


For a doubly stochastic matrix P = (pij), if its entries consist of only 0’s and 1’s, then the matrix is a permutation matrix.

A linear map T : Sm – 1Sm – 1 is said to be T- transform, if T = λI + (1λ)P where I is an identity matrix, P is a permutation matrix which is obtained by swapping only two rows of I and 0 ⩽ λ ⩽ 1.

([11], chapter 2). For the concept of majorization, the following assertions are equivalent for any x, ySm–1.

  1. xy that is i=1kx[i]i=1ky[i],k=1,m1¯.

  2. x = Py for some doubly stochastic matrix P.

  3. The vector x belongs to the convex hull of all m! permutation vectors of y.

  4. The vector x can be obtained by a finite compositions of T- transforms of the vector y, that is, there exist T – transforms T1, T2, …, Tk such that x = T1T2Tky.

  5. The inequality φ(x) ≤ φ(y) holds for any Schur-convex function.

So from the above lemma, it follows that doubly stochasticity of a matrix P is equivalent to Pxx for all xSm − 1. Motivated by this, in [11], the definition of doubly stochastic operator is given as follows.

([15]). A continuous stochastic operator V : Sm – 1Sm–1 is called doubly stochastic, if


Hence if a q.s.o. satisfies the above property, then we call it d.s.q.o.

For example, the identity operator, permutation operators (that is the linear operators with permutation matrix), T- transforms are all doubly stochastic. The following is an example of the d.s.q.o.

Consider V : S2S2 given by


It is straightforward to check that this operator satisfies the condition of doubly stochasticity in the Definition 2.2.

Let V be doubly stochastic operator and x0Sm–1. The set


is called the trajectory starting at x0. Here, V0(x0) = x0 and Vn(x0) = V(Vn – 1(x0)). We denote by ω(x0) the set of limit points of the trajectory starting at x0 and it is said to be the ω- limit set of the trajectory starting at x0. Notice that the ω- limit set of any point is invariant for V by definition.

The point x0 is called p-periodic, if there is a positive integer p such that Vp(x0) = x0 and Vi(x0) ≠ x0i = 1, p – 1. If p = 1, we say that the point is fixed. We just say periodic if the period is irrelevant. By periodic point we always mean a periodic point of a period strictly grater than one.

Notice that the center C=(1m,1m,,1m) is always fixed for a doubly stochastic V. Indeed, by the definition of the double stochasticity, we have V(C) ≺ C. On the other hand, Cx for any vector x from the simplex.

Therefore, CV(C) ≺ C implies .V(C)↓ = C↓ = C. Hence, V(C) = C and C is a fixed point. We will use this fact in subsequent sections.

3 The limit behavior of d.s.q.o

In this section we first provide a class of functionals on the simplex that are Lyapunov functions for a d.s.q.o. Then we prove the convergence theorem for d.s.q.o.

A continuous functional given φ : Sm–1R is said to be a Lyapunov function for the operator V if the limit limn φ(Vn(x0)) exists along the trajectory {x0, V(x0), V2(x0), …, Vn(x0), …}.

The Lyapunov function is considered to be very useful in the study of limit behavior of (discrete) dynamical systems.

A continuous functional given by


is a Lyapunov function for doubly stochastic operator V. Moreover, if xSm–1 and Vx is not the permutation of x, then φ(Vx) < φ(x).

Proof. Use the fact (see [11], prop. F.1, page 78.) that the function ψ(x)=i<jxixj, defined on the simplex, is Schur-concave, i.e. satisfies ψ(x) ≥ ψ(y) whenever xy. Moreover, if x, ySm – 1, and x is not the permutation of y, then ψ(x) > ψ(y). Taking into consideration φ(x) = 1 – 2ψ(x), we obtain that x ≺ y implies φ(x) ≤ φ(y). Since V is doubly stochastic, then Vxx hence φ(Vx) ≺ φ(x) is bounded. Note that the sequence an = φ(Vn(x0)), where V is doubly stochastic and x0Sm–1, is monotone. Hence the limn→∞ φ(Vn(x0)) exists.

Now we show the second part. If xintSm–1, then according to Lemma it follows that V(x) ∈ intSm–1. Since all components of V(x) and x are positive and Vx is not the permutation of x, then by applying the above mentioned fact, we get φ(x) < φ(y). □

Based on the above method one can provide larger class of Lyapunov functions.

Any continuous symmetric convex (or concave) functional is a Lyapunov function for a doubly stochastic operator V.

These theorems are very useful to study the dynamics of individual doubly stochastic operators. We will provide an application of these theorems in the next section.

Assume that a d.s.q.o. V does not have any periodic points on intSm–1. Then the trajectory of any initial point on the interior of simplex is convergent under any d.s.q.o.

Proof. Let V : Sm–1Sm–1 be d.s.q.o. Then we have


It means that


The sequences {i=1k(Vn(x))[i],n=1,2,k=1,m¯} are increasing and bounded, consequently convergent.

The last suggests that the following sequences are also convergent {(Vn(x))[k],n=1,2,k=1,m¯}

Let us denote yk = limn → ∞ .(V nx)[k] and y = (y1, y2, . . ., ym).

If z = (z1, z2, . . ., zm) ∈ ω(x0) then there exists {x.(nj )}, such that (V nj x) → z.

Therefore we have (Vnj x)z. On the other hand (Vnj x)↓ → y, since y is a limit of the sequence (Vnx), n =1,2,... That’s why z = (y1, y2, …, ym) = y. This means that the cardinality of ω(x0) cannot be greater than m!.

Note that since the simplex is compact, then ω(x0) ≠ ∅. Now, assume that the cardinality of it is greater than 1. Then there is a point z in ω(x0) that is not fixed by V. Because the set ω(x0) is V – invariant then any iterate of z under must belong to ω(x0). But this set is finite and therefore there exist natural numbers p, q (and let us assume p > q) such that Vp(z) = Vq(z). But then the point Vq(z) is periodic with period pq. However, by the assumption V does not have any periodic points. Hence m must be 1 and |ω(x0)| = 1. Therefore, ω(x0) may consist of a single point only and this implies that the trajectory starting at x0 tends to that point. □

In the proof of this theorem we did not use the fact that V is d.s.q.o. In fact, it does not have to be quadratic. This theorem holds for any (continuous) doubly stochastic map on a finite-dimensional simplex.

An operator V : Sm-1Sm–1 is a permutation if there exists a permutation matrix such that Vx = PxxSm–1. It is evident that there are exactly m! permutation operators on Sm−1.

Any non-permutation d.s.q.o. V : S2S2 on 2D simplex does not have periodic points on the interior of the simplex.

Proof. Assume V has a periodic point x0intS2 and Vp(x0) = x0 for p > 1. Then VV(x0)Vp(x0)=x0

Hence V(x0) is some permutation of x0.

Let x0 = (x, y, z), x, y, z > 0, x + y + z = 1 and V(x0)=((A1x0,x0),(A2x0,x0),(A3x0,x0))

where (·,·) denotes the usual inner product. Without loss of generality we may assume xyz and (A1x0, x0) ≥ (A2x0, x0) ≥ (A3x0, x0). Then V(x0) = x0 (not some permutation of x0). Because x + y + z = 1 then one can assume xyz and one can also assume the second assumption by changing matrices A1, A2, A3 if necessary. By applying Theorems 2.6 and 2.4 of [8] we can represent each Ak, k = 1, 2, 3 as Ak=Sk+SkT2

where Sk is a row-stochastic matrix and SkT is its transpose. Not that (Akx0,x0)=(Sk+SkT2x0,x0)=12(Skx0,x0)+12(SkTx0,x0)=12(Skx0,x0)+12(x0,Skx0)=(Skx0,x0)

Hence V(x0) = ((S1x0, x0), (S2x0, x0), (S3x0, x0)) = (x, y, z).

Let Sk = {sij, k)i, j = 1, 3. Since V(x0) = x0 then (S1x0, x0) = x. On the other hand, since S1 is row-stochastic we must have si1,1x+si2,1y+si3,1zx(7)

Therefore x=(s1x0,x0)=(s11,1x+s12,1y+s13,1z)x+(s21,1x+s22,1y+s23,1z)y+(s31,1x+s32,1y+s33,1z)zxx+xy+xz=x(x+y+z)=x,

and we must have the equality in (7). Recall that x0=(x,y,z)(13,13,13) as (13,13,13) is a fixed point for V. Hence the equality in (7) is possible only in the following cases: S1=(100100100)(8)

or x=y,andS1=(s11,1s12,10s21,1s22,10s31,1s32,10)(9)

Similar arguments for S3 = (sij,3)i, j = 1, 3 show that si1,3x+si2,3y+si3,3zz(10)

with equality is only possible in the following cases: S3=(001001001)(11)

or y=z,andS3=(0s12,3s13,30s22,3s23,30s32,3s33,3)(12)

Not that (9) and (12) can not hold simultaneously, as it would imply x = y = z hence x0=(13,13,13).

If (8) and (12) hold, then V would fix the first and the third components of x0 = (x, y, z) which implies that it would also fix the second component of x0 = (x, y, z). By the same reasons (9) and (11) can not hold simultaneously. Finally (8) and (11) together would imply that S2=(010010010) and hence V = Id which is not impossible as V is not a permutation operator. □

In the assumption of the theorem we assumed V to be non-permutation but only used the fact that V is not an identity operator. This is because we assumed (A1x0, x0) ≥ (A2x0, x0) ≥ (A3x0, x0).

It is essential that x0intS2, that is, a d.s.q.o. can have periodic points on the boundary of the simplex.

Consider V : S2S2 given by V(x)=y2+2xzV(y)=z2+2xyV(z)=x2+2yz

This operator is d.s.q.o. by the classification theorem 2.6 of [8] or one can easily check that Vxx. One can see that V3(1, 0, 0) = (1, 0, 0), that is, (1, 0, 0) is a 3-periodic point.

Combining the previous two theorems we obtain a very interesting corollary.

Any non-permutation d.s.q.o. has convergent trajectory on the interior of the simplex.

Theorem 3.5 fails in higher dimension. Consider the following example in 3D simplex.

Consider the operator V : S3S3 given by Vx=Dx=(x2+x42,x1+x32,x2+x42,x1+x32)

where D=(012012121120012012120120)

For this operator, one has V2k+1=V1=(x2+x42,x1+x32,x2+x42,x1+x32)

and V2k=V2=(x1+x32,x2+x42,x1+x32,x2+x42)

Therefore, any point is either periodic or becomes periodic after one iteration. Notice that this linear operator is given by a doubly stochastic matrix. Hence Vxx and so V is doubly stochastic. One can make this operator a d.s.q.o. by multiplying each component to x1 + x2 + x3 + x4 = 1.

4 The classification of extreme points of d.s.q.o. on S2

In this section we give classification of dynamics of extreme d.s.q.o.

A d.s.q.o. is extreme if it cannot be written as a convex hull of two different d.s.q.o.

In [7, 8] it was shown that the set of all d.s.q.o. form a convex polytope and the extreme points in 2D simplex were fully described in the following way considering a d.s.q.o. V : S2S2 given by V = (A1|A2|A3) where (A1|A2|A3) are non-negative symmetric matrices such that (this follows from the definition of q.s.o.) their sum is a matrix whose all entries equal to 1. LetU3=A=aiji,j=1,3¯aij0,i,jαaijαforα1,2,3,i,j=13aij=3.

The set of extreme points of a set A we denote by ExtrA.

We recall a couple of facts from [7] and [8].

It is known [7, 8] that d.s.q.o. is extreme A = (aij) ∈ Ext r𝓤3 if and only if aii = 1 or 0, aij = 1 or 12 or 0 and A(012121201212120) and there are 25 extreme matrices.

In [7, 8] A d.s.q.o. V = (A1 |A2|A3) is extreme if and only if at least two of matrices A1, A2, A3 are extreme in 𝓤3.

Using these facts one can find triples (A1|A2|A3) such that V is extreme. There are 37 such triples. For a given extreme d.s.q.o. (A1|A2|A3) if we rearrange matrices A1, A2, A3 we still get extreme d.s.q.o.

Thus, counting permutations, there are 37 * 3! = 222 extreme d.s.q.o.

Now our objective is to classify them in terms of their dynamics. As we mentioned all 222 extreme d.s.q.o. are obtained by permuting component of 37 extreme operators. Thus, to study the dynamics we will be dealing with the same collection of matrices even if we change the components. This implies that it is enough to consider only 37 extreme d.s.q.o. It is worth noting that in general a permuted operator does not have the same dynamics, but it can only be studied as the original operator. For instance, we if consider the identity operator, then all points are fixed. However, if we permute the identity operator (see operator below) then all points are going to be periodic except the center of the simplex. Let’s consider the simplest case - permutation operator given by V(x,y,z)=(y,z,x)

The matrix correspondence for this operator is A1=(0121121120120),A2=(0012001212121),A3=(1121212001200)(13)

Note that V3(x, y, z) = (x, y, z) and therefore every initial point is 3-periodic. Thus, it is not difficult to study the dynamics of this operator. We split the remaining 36 operators into two classes: those who have exactly one matrix from (13) in their matrix representation are in class 2 and the rest in class 1.

It is important that operators from the class 2 must have exactly one matrix from (13). Referring to Ganikhodzhaev and Shahidi (2010)’s Theorem 3.4 if an operator has more than 1 matrix from (13) then it becomes a permutation operator.

Because extreme d.s.q.o. are classified in [8] (Theorem 3.4), then it is not difficult to see that in fact the class 2 has 3 (or 18 up to permutation) operators while the class 1 has 33.

We are going to choose one representative from each class and study the limit behavior of the trajectories. We will see that their dynamics is different.

(Class 1). Consider V : S2S2 given by V1(x)=z2+xy+yzV1(y)=x2+xz+yzV1(z)=y2+xy+xz

Figure 1 shows that the trajectory of initial values of x, y and z for V1 converges to the center (13).

The trajectory of operator V1.
Fig. 1

The trajectory of operator V1.

For V1, the trajectory of any initial point from the interior of the simplex converges to the center of the simplex. Moreover, V1 has no periodic points on intS2.

We note that the convergence part of this theorem automatically follows from Theorems 3.3 and 3.5 of previous section. However, we will show how one apply Theorem 3.1 of previous section.

Proof. We first show that V1 has no periodic point on intS2. Suppose on the contrary, that intS2 is a p-periodic (p > 1) point of V1. Then by definition x0V1x0V12x0....V1px0=x

If follows that each Vkx0,k = 1, p interchange the components of x0. Therefore, if we define φ(x0) = x2 + y2 + z2 for x0 = (x, y, z) we must have φ(Vx0) = φ(x0) by Theorem 3.1.

One can easily compute that φ(Vx0)φ(x0)=zy(xz)2(x+z)

hence –zy(x – z)2(x + z) = 0.

Since x, y, z > 0 the above holds only if x = z. Assume x = z = c, 0 < c < 1 then x0 = (c,1 – 2c, c). One can see that V1(x) = 2c – 3c2 so in order for V1 to permute the components of x0 we must have 2c – 3c2 = c or 2c – 3c2 = 1 – 2c which only happens if c=13 that is when x0 = (x, y, z) = (13,13,13). But (13,13,13) is fixed point and we assumed x0 to be periodic (of period greater than one). Thus, as one can recall Theorem 3.3 from the previous section, that the trajectory of any x0intS2 is convergent. Notice that a convergent point is always fixed by V1.

By solving the system of equations x=z2+xy+yzy=x2+xz+yzz=y2+xy+xz

we find that V1 has a unique fixed point (13,13,13) on intS2. Therefore (13,13,13) must be the convergent point for any x0intS2. □

(Class 2). Consider V : S2S2 given by V2(x)=xy+2xzV2(y)=y2+xy+yz=yV2(z)=x2+z2+yz.

Accordingly, we see that under any iteration y does not change (see Figure 2). For a fixed, simple calculations we show that the trajectory of any point from the line y = c tends to (1c2,c,1c2) (see Figure 2).

The trajectory of operator V2.
Fig. 2

The trajectory of operator V2.

The operator V2 has infinitely many invariant curves (segments). Moreover, V2 has a unique fixed point in each invariant curve and the trajectory of any initial point on a given invariant curve tends to the “center” of the curve.

Proof. Let γc = {(x, y, z) ∈ intS2 | y = c} 0 < c < 1. It is obvious that γc is V2-invariant. By solving the system of equations x=xc+2xzy=y=cz=x2+z2+cz.

where c is fixed we find a unique solution (1c2,c,1c2) which shows that V2 has infinitely many fixed points.

Notice that y = c defines a plane in R3 and γc = intS2 {y = c} is in fact a segment and the point (1c2,c,1c2) is the middle of this segment, so that is why we called it the center of γc.

Now, take an initial point on γc. We know from the previous section that the trajectory is convergent. At the same time it stays in γc (as c is invariant) which has a fixed point. Thus, the trajectory must tend to its unique fixed point, that is to the center (1c2,c,1c2) of γc. □

5 Conclusions and future work

In this paper, we discussed an important class of quadratic stochastic operators, which was defined in terms of majorization of vectors, that is, doubly stochastic quadratic operators (d.s.q.o) on finite-dimensional simplex. The theorem of the convergence of d.s.q.o. is proved. We also showed that it has no periodic point is on the interior of the simplex but it may have (in fact infinitely many) periodic points in higher dimensions of the simplex. Moreover, we classified the extreme points of d.s.q.o. on two dimensional 2D simplex. From the analysis, we found three different classes of 222 extreme points of d.s.q.o. on 2D simplex. We found out that 198 extreme d.s.q.o.’s converge to the center of the simplex. In the second class, we found 18 extreme d.s.q.o., which had property as in Theorem 4.5. In the third classification, we had 6 permutation operators which are extreme d.s.q.o. It is very interesting to study dynamics on infinite-dimensional simplex. The problem would be very different if the infinite-dimensional simplex was not compact, the operator would not necessarily have a fixed point and one should choose a proper metric to study the dynamics.


We would like to thank Kulliyyah of Information and Communication Technology and the Research Management Center in International Islamic University Malaysia for their support. The authors would like to express their gratitude to the anonymous referee for the comprehensive review of the previous draft. Not only did it allow to fill up some gaps it also allowed us to prove deeper results.


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About the article

Received: 2015-04-27

Accepted: 2016-05-23

Published Online: 2016-07-26

Published in Print: 2016-01-01

Citation Information: Open Mathematics, Volume 14, Issue 1, Pages 509–519, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0045.

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© 2016 Abdulghafor etal., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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