On b-bistochastic quadratic stochastic operators
- Farrukh Mukhamedov^{1}Email author and
- Ahmad Fadillah Embong^{1}
https://doi.org/10.1186/s13660-015-0744-y
© Mukhamedov and Embong 2015
Received: 17 November 2014
Accepted: 27 June 2015
Published: 21 July 2015
Abstract
In the present paper we introduce a new notion of order called b-order. Then we define a bistochasticity quadratic stochastic operator (q.s.o.) with respect to the b-order, and call it a b-bistochastic q.s.o. We include several properties of the b-bistochastic q.s.o. and descriptions of all b-bistochastic q.s.o. defined on a two dimensional simplex.
Keywords
MSC
1 Introduction
The history of quadratic stochastic operators can be traced back to Bernstein’s work [1]. Nowadays, scientists are interested in these operators, since they have a lot of applications, especially in population genetics [2, 3]. Moreover, the quadratic stochastic operators were also used as a crucial source of analysis for the study of dynamical properties and modelings in many different fields such as biology [4–8], physics [9, 10], economics, and mathematics [3, 10–12].
In other words, each q.s.o. describes the sequence of generations in terms of probability distributions if the values of \(P_{ij,k}\) and the distribution of the current generation are given. In [13], it is given by a self-contained exposition of the recent achievements and open problems in the theory of the q.s.o. The main problem in the nonlinear operator theory is to study the behavior of nonlinear operators. Nowadays, there is only a small number of studies on dynamical phenomena on higher dimensional systems that are presently comprehended, even though they are very important. In the case of a q.s.o., the difficulty of the problem depends on the given cubic matrix \((P_{ijk})^{m}_{i,j,k=1}\). An asymptotic behavior of the q.s.o. even on the small dimensional simplex is complicated [12, 14–17].
The concept of majorization was established in 1929 [18] even though the idea was introduced much earlier by Lorenz [19], Dalton [20], and Schur [21]. This kind of theory was very important from an economic point of view, which resulted from the gaps in the income or wealth distribution in society. Later, it led to idea of Lorenz curve and principle of transfers. Moreover, Schur’s work on Hadamard’s determinant inequality also contributed to the development of majorization [21].
The idea of majorization kept occurring in other fields, such as chemistry and physics, but it was attributed by different names such as ‘x is more mixed than y’, ‘x is more chaotic than y’ and ‘x is more disordered than y’. One of the examples is given by [22].
Further, Parker and Ram [23] introduced a new order called majorization and they were referring to the majorization that was popularized by Hardy, Littlewood and Polya, classical majorization. This new order opened a path for the study to generalize the theory of majorization of Hardy et al. [18]. The new majorization has an advantage as compared to the classical one since it can be defined as a partial order on sequences. While classical majorization is not an antisymmetric order (because any sequence is majorized by any of its permutations), it is only defined as a preorder on a sequence [23].
Furthermore, one of the best known methods to solve optimization is the greedy method. This method is preferred because of space- and time-efficiency. It also yields crucial classes of optimization and usually provides a proper estimation to the optimal solution. Many of the studies in greed [24–27] introduced special classes of optimization problems and provided their algorithms. Matroids and greedoids modeling were used in the past to approach greedy-solvable problems. Unfortunately there were not many problems that can be generalized. Hence, in [23], it was proven that the concept of majorization has a direct relation with the greedy method. Moreover, the same scholars also provided good examples in solving greed problems such as continuous knapsack, storage of files on tape, and job sequencing [23]. Note that Stott Parker and Prasad Ram were focussing the descriptions of the order’s classes defined on linear systems only. Hence, we are interested in the investigation of the case of quadratic ones.
In [28] a definition of bistochastic q.s.o. was proposed in terms of classical majorization (see [29]). Namely, a q.s.o. is called bistochastic (also called doubly stochastic) if \(V({\mathbf{x}}) \prec {\mathbf{x}}\) for all x taken from the \(n-1\) dimensional simplex. In [28, 30], the necessary and sufficient conditions were given for the bistochasticity of a q.s.o. In general, the descriptions of such a kind of operators are still an open problem.
Therefore, in the present paper, we are motivated to use majorization introduced in [23] to define a bistochasticity q.s.o. In order to differentiate between the terms majorization and classical majorization, we call majorization a b-order, while classical majorization is called majorization only. The main goal of this paper is to describe all such kind of operators on a two dimensional simplex.
2 b-Order and b-bistochastic operators
Let \({\mathbf{x}},{\mathbf{y}}\in S^{n-1}\). We say that x is b-ordered by y (\({\mathbf{x}}\leq^{b} {\mathbf{y}}\)) if and only if \(\mathcal{U}_{k}({\mathbf{x}}) \leq\mathcal{U}_{k}({\mathbf{y}})\), for all \(k \in\{1,\dots, n-1\}\).
- (i)
\({\mathbf{x}}\leq^{b} {\mathbf{x}}\),
- (ii)
\({\mathbf{x}}\leq^{b} {\mathbf{y}}\), \({\mathbf{y}}\leq^{b} {\mathbf{x}}\Longrightarrow {\mathbf{x}}= {\mathbf{y}}\),
- (iii)
\({\mathbf{x}}\leq^{b} {\mathbf{y}}\), \({\mathbf{y}}\leq^{b} {\mathbf{z}}\Longrightarrow {\mathbf{x}}\leq^{b} {\mathbf{z}}\).
- (i)
One has \({\mathbf{x}}\leq^{b} {\mathbf{y}}\) if and only if \(\lambda {\mathbf{x}}\leq^{b} \lambda {\mathbf{y}}\) for any \(\lambda> 0\).
- (ii)
If \({\mathbf{x}}\leq^{b} {\mathbf{y}}\) and \(\lambda\leq\mu\) then \(\lambda {\mathbf{x}}\leq^{b} \mu {\mathbf{y}}\).
Any operator V with \(V(S^{n-1})\subset S^{n-1}\) is called stochastic and if V is satisfied \(V({\mathbf{x}})\leq^{b}{\mathbf{x}}\) for all \({\mathbf{x}}\in S^{n-1}\), then it is called b-bistochastic. The following include discussions on nonlinear b-bistochastic operators. The simplest nonlinear operators are quadratic ones. Therefore, we restrict ourselves to such a kind of operators.
Remark 2.1
In [28] a q.s.o. was introduced and studied with the property \(V({\mathbf{x}}) \prec {\mathbf{x}}\) for all \({\mathbf{x}}\in S^{n-1}\). Such an operator is called bistochastic. In our definition, we are taking the b-order instead of the majorization. Note that if one takes absolute continuity instead of the b-order, then the b-bistochastic operator reduces to a Volterra q.s.o. [31–34].
Let us recall some preliminaries.
Remark 2.2
Let \(g(x)=mx+c \), then \(g(x) \leq0\) (respectively, \(g(x)\geq0\)) for all \(x \in[0,1]\) if and only if \(c \leq0\) (respectively, \(c \geq0\)) and \(m+c \leq0\) (respectively, \(m+c\geq0\)).
A point \({\mathbf{x}}_{0}\) is called a fixed point of F if one has \(F({\mathbf{x}}_{0})={\mathbf{x}}_{0}\).
Definition 2.3
- 1.
\({\mathbf{x}}_{0}\) is called attractive if every eigenvalue of the Jacobian eigenvalues satisfies \(|\lambda |<1\);
- 2.
\({\mathbf{x}}_{0}\) is called repelling if every eigenvalue of the Jacobian eigenvalues satisfies \(|\lambda |>1\).
- (i)
\(\overline{{\mathbf{x}}}_{k} = (\underbrace{0,0,\dots,0}_{k},\frac{1}{n-k}, \frac{1}{n-k}, \dots, \frac{1}{n-k})\), where \(k=\{0,\dots,n-1\}\),
- (ii)
\({\mathbf{e}}_{m} = (\underbrace{0,0, \dots, 0, 1}_{m}, 0,\dots, 0) \), \(m\in\{1,\dots,n\}\).
3 Properties of b-bistochastic q.s.o.
In this section, we provide some basic properties of a b-bistochastic q.s.o. in a general setting. In the following, we need an auxiliary result.
Lemma 3.1
- (i)
\(C\leq0\) and
- (ii)
\(A_{k}+C\leq0\), \(k=\overline{1,n}\).
The proof is obvious.
Now we are ready to formulate several properties of a b-bistochastic q.s.o.
Theorem 3.2
- (i)
\(\sum_{m=1}^{k}\sum_{i,j=1}^{n}P_{ij,m} \leq kn\), \(k \in\{1,\dots,n \}\);
- (ii)
\(P_{ij,k} =0\) for all \(i,j\in\{k+1,\dots,n\}\) where \(k\in\{1,\dots,n-1\}\);
- (iii)
\(P_{nn,n}=1\);
- (iv)for every \({\mathbf{x}}\in S^{n-1}\) one has$$\begin{aligned}& V({\mathbf{x}})_{k} = \sum_{l=1}^{k}P_{ll,k}x^{2}_{l} + 2 \sum_{l=1}^{k}\sum _{j=l+1}^{n}P_{lj,k}x_{l}x_{j} \quad \textit{where } k = \overline{1,n-1}, \\& V({\mathbf{x}})_{n} = x_{n}^{2} + \sum _{l=1}^{n-1}P_{ll,n}x^{2}_{l} + 2 \sum_{l=1}^{n-1}\sum _{j=l+1}^{n}P_{lj,n}x_{l}x_{j}; \end{aligned}$$
- (v)
\(P_{lj,l} \leq\frac{1}{2}\) for all \(j\geq l+1\), \(l\in\{1,\dots,n-1\}\).
Proof
Hence, \(P_{ij,k} =0\), for all \(i,j\in\{k+1,\dots,n\}\), where \(k=\overline{1,n-1}\).
The proof of (iii) is evident.
Now, by Lemma 3.1 one finds that \(P_{lj,l} \leq \frac{1}{2}\).
This completes the proof. □
By \(\mathcal{V}_{b}\) we denote the set of all b-bistochastic q.s.o.
Proposition 3.3
The set \(\mathcal{V}_{b}\) is convex.
Proof
4 Limiting behavior of b-bistochastic q.s.o.
In this section, we are going to study the limiting behavior of a b-bistochastic q.s.o.
Theorem 4.1
Let V be a b-bistochastic q.s.o. defined on \(S^{n-1}\), then for every \({\mathbf{x}}\in S^{n-1}\) the limit \(\lim_{m \rightarrow\infty} V^{(m)}({\mathbf{x}})\) exists.
Proof
Corollary 4.2
Let V be a b-bistochastic q.s.o. on \(S^{n-1}\), and let \(\lim_{m \rightarrow\infty} V^{m}({\mathbf{x}})= \overline{{\mathbf{x}}}\), then \(\overline{{\mathbf{x}}}\) is a fixed point of V.
Proposition 4.3
Let V be a b-bistochastic q.s.o., then \({\mathbf{x}}=(0,0,\dots,1)\) is its fixed point.
Proof
From the last expression, we immediately get the following lemma.
Lemma 4.4
Theorem 4.5
The fixed point \((0,0,\dots,0)\) is not repelling.
Proof
Corollary 4.6
If \(P_{lj,l} < \frac{1}{2}\) for all \(l\in\{1,\dots,n-1\}\), \(j\geq l+1\), then the fixed point \({\mathbf{x}}= (0,0,\dots,0)\) is attracting.
From the results, it is natural to ask the following question: Is there a trajectory of a b-bistochastic q.s.o. which converges to a fixed point different from \((0,0,\dots,1)\)?
We want to consider this question in a one dimensional setting.
Let us denote by \(\operatorname{Fix}(V)\) the set of all fixed points of V belonging to the simplex \(S^{n-1}\).
The following theorem describes the limiting behavior of a b-bistochastic on a one dimensional setting.
Theorem 4.7
Proof
Furthermore, using Corollary 4.2, we know that the limit of b-bistochasticity converges to a fixed point, thus we need to consider several cases.
Case 3. If \(a=1\) and \(b = \frac{1}{2}\), then one obviously gets \(\lim_{m \rightarrow\infty} V^{(m)}({\mathbf{x}}) ={\mathbf{x}}\).
This completes the proof. □
5 Description of b-bistochastic q.s.o. on 2D simplex
In this section we are going to describe all b-bisochastic q.s.o. defined on a two dimensional simplex. Before doing that, we need the following auxiliary facts.
Lemma 5.1
- (I)
\(a\geq0 \);
- (II)\(a<0 \) and one of the following is satisfied:
- (i)
\(b \leq0\);
- (ii)
\(b \geq-2a\);
- (iii)
\(b^{2}-4ac\leq0\).
- (i)
Lemma 5.2
- (I)Let the critical point \((x_{0},y_{0})\) belongs to \({\mathbb{D}}\), then
- (i)
if \((x_{0},y_{0})\) is a maximum point and \(f(x_{0},y_{0}) \leq 0\), then \(f(x,y)\leq0\) for all \(x,y\in {\mathbb{D}}\);
- (ii)
if \((x_{0},y_{0})\) is a saddle point and \(f(x_{0},y_{0}) \leq0\), then \(f(x,y)\leq0\) for all \(x,y\in {\mathbb{D}}\);
- (iii)
if \((x_{0},y_{0})\) is a minimum point, then \(f(x,y)\leq0\) for all \(x,y\in {\mathbb{D}}\).
- (i)
- (II)
Let \((x_{0},y_{0})\notin {\mathbb{D}}\), then one has \(f(x,y)\leq0\) for all \(x,y\in {\mathbb{D}}\).
The proof immediately follows from the fact that the given function is either paraboloid or saddle roof. Note that in (II) g reaches its maximum on the boundaries.
The main result of this paper is the following theorem.
Theorem 5.3
- (i)
\(F_{1} = E_{1} = D_{1} = F_{2} = 0\);
- (ii)
\(B_{1} \leq \frac{1}{2}\), \(C_{1} \leq\frac{1}{2}\), \(E_{2} \leq \frac{1}{2}\);
- (iii)
\(C_{1}+C_{2} \leq\frac{1}{2}\),
- (I)
\(a\geq0 \);
- (II)\(a<0 \) and one of the following is satisfied:
- (1)
\(b \leq0\);
- (2)
\(b \geq-2a\);
- (3)
\(b^{2}-4ac\leq0\).
- (1)
Proof
The conditions (i) and (ii) immediately follow from Theorem 3.2, which are equivalent to (5.2).
First, using the fact \(g(x_{1},x_{2})\) is not linear, we need to investigate g for its extremums on the boundaries (i.e. Side 1: \(x_{1} = 0\), Side 2: \(x_{2} = 0\) and Side 3: \(x_{2}=1-x_{1}\)) first and then in the internal region. So, we are going to study the function g on each side one by one.
- (I)
\(a\geq0 \);
- (II)\(a<0 \) and one of the following is satisfied to meet (5.4):
- (1)
\(b \leq0\);
- (2)
\(b \geq-2a\);
- (3)
\(b^{2}-4ac\leq0\).
- (1)
Now we consider the internal region i.e. \(\mathbb{D} = \{(x_{1},x_{2})| 0\leq x_{1}+x_{2}\leq1\}\).
- (a)
a maximum point if \(g_{x_{1}x_{1}} < 0 \) and \(g_{x_{1}x_{1}}g_{x_{2}x_{2}} - (g_{x_{1}x_{2}})^{2}>0\);
- (b)
a minimum point if \(g_{x_{1}x_{1}} > 0 \) and \(g_{x_{1}x_{1}}g_{x_{2}x_{2}} - (g_{x_{1}x_{2}})^{2}>0\);
- (c)
a saddle point if \(g_{x_{1}x_{1}}g_{x_{2}x_{2}} - (g_{x_{1}x_{2}})^{2}<0\).
Furthermore, in order to cover all possible values of R, N, and Q that they shall take, we examine several cases:
Case I: \(R > 0\) and \(RN - Q^{2} > 0\);
Case II: \(R < 0\) and \(RN - Q^{2} > 0\);
Case III: \(RN < 0\);
Case IV: \(RN>0\) and \(RN-Q^{2} < 0\);
Case V: \(R = 0\);
Case VI: \(N=0\);
Case VII: \(Q=0\);
Case VIII: \(RN-Q^{2} = 0 \).
We want to highlight that the values of P and M are positive due to (ii) and (iii). The investigation of each case is done separately.
Case I. Assume that \(R>0\) and \(RN-Q^{2} > 0\), then one immediately gets the critical point is minimum. Due to Lemma 5.2 we get \(g \leq0\). Correspondingly, (5.4) is true in this case.
Case IV. Let \(RN>0\) and \(RN-Q^{2} < 0\). By a similar method to Case III, one proves that (5.4) holds.
The last subcase \(N<0\) may proceed in a similar manner.
Case VI. We let \(N=0\) and the proof is analogous to Case V.
Case VIII. In the last case, we let \(MN-Q^{2} = 0 \). It is clear that we have \(2x_{2}(Q^{2} - NR) = MQ - PR \) and \(2x_{1} (RN - Q^{2}) = MN - PQ \). This gives us \(MQ - PR = 0\) and \(MN - PQ = 0 \) or otherwise \(g_{x_{1}} =0 \) and \(g_{x_{2}} =0\) do not have any solution (i.e. the critical point). Therefore, the maximum is reached on the boundaries, which gives \(g(x_{1},x_{2}) \leq 0\) for all \((x_{1},x_{2}) \in {\mathbb{D}}\) (see the investigations on Side 1, Side 2, and Side 3).
On the other hand, if \(MQ - PR = 0\) and \(MN - PQ = 0 \), then one has infinitely many solutions. Using the fact \(M \geq0 \) and \(P \geq0 \), then we have two possible subcases, which are (i) \(N <0\), \(Q<0\), \(R< 0 \) and (ii) \(N>0\), \(Q>0\), \(R>0 \). Taking into account the first subcase (i) and the argument in the Case II, one infers that \(g(x_{1},x_{2}) \leq0\) for all \((x_{1},x_{2}) \in {\mathbb{D}}\).
Briefly, we show that for all cases in the internal region, \(g(x_{1},x_{2})\leq0\) for all \((x_{1},x_{2})\in {\mathbb{D}}\). In addition, the reverse can be proved by the same way. This completes the proof. □
Remark 5.4
Note that such a kind of description of a bistochastic q.s.o. is not known in the literature. Our result fully describes all b-bistochasic q.s.o. on a two dimensional setting. The theorem proved allows one to find all extreme points of the set of a b-bistochastic q.s.o. on 2D simplex, which can be considered as one of the future studies that can be done. Moreover, it gives insight in and preliminary information on the direction of a higher dimensional setting.
Declarations
Acknowledgements
The present study was conducted with the supports from the grants ERGS13-024-0057, FRGS14-135-0376 of Malaysian Ministry of Education. The first author, FM, would like to thank the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, for offering a Junior Associate Scheme fellowship. Finally, the authors would like to thank Dr. Mansoor Saburov for his useful suggestions to improve the content of the paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Bernstein, SN: The solution of a mathematical problem concerning the theory of heredity. Ucheniye-Zapiski N.-I. Kaf. Ukr. Otd. Mat. 1, 83-115 (1924) (in Russian) Google Scholar
- Lotka, AJ: Undamped oscillations derived from the law of mass action. J. Am. Chem. Soc. 42, 1595-1599 (1920) View ArticleGoogle Scholar
- Lyubich, YI: Mathematical structures in population genetics. In: Biomathematics, vol. 22. Springer, Berlin (1992) Google Scholar
- Hofbauer, J, Hutson, V, Jansen, W: Coexistence for systems governed by difference equations of Lotka-Volterra type. J. Math. Biol. 25, 553-570 (1987) MathSciNetView ArticleGoogle Scholar
- Hofbauer, J, Sigmund, K: Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge (1998) View ArticleGoogle Scholar
- May, RM: Simple mathematical models with very complicated dynamics. Nature 261, 459-467 (1976) View ArticleGoogle Scholar
- May, RM, Oster, GF: Bifurcations and dynamics complexity in simple ecological models. Am. Nat. 110, 573-599 (1976) View ArticleGoogle Scholar
- Narendra, SG, Samaresh, CM, Elliott, WM: On the Volterra and other nonlinear models of interacting populations. Rev. Mod. Phys. 43, 231-276 (1971) View ArticleGoogle Scholar
- Plank, M, Losert, V: Hamiltonian structures for the n-dimensional Lotka-Volterra equations. J. Math. Phys. 36, 3520-3543 (1995) MathSciNetView ArticleGoogle Scholar
- Takeuchi, Y: Global Dynamical Properties of Lotka-Volterra Systems. World Scientific, Singapore (1996) View ArticleGoogle Scholar
- Ulam, SM: A Collection of Mathematical Problems. Interscience, New York (1960) Google Scholar
- Vallander, SS: On the limit behavior of iteration sequence of certain quadratic transformations. Sov. Math. Dokl. 13, 123-126 (1972) MathSciNetGoogle Scholar
- Ganikhodzhaev, R, Mukhamedov, F, Rozikov, U: Quadratic stochastic operators and processes: results and open problems. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 14, 270-335 (2011) MathSciNetView ArticleGoogle Scholar
- Ganikhodzhaev, NN, Zanin, DV: On a necessary condition for the ergodicity of quadratic operators defined on a two-dimensional simplex. Russ. Math. Surv. 59, 571-572 (2004) MathSciNetView ArticleGoogle Scholar
- Mukhamedov, F, Saburov, M, Qaralleh, I: On \(\xi^{s}\)-quadratic stochastic operators on two dimensional simplex and their behavior. Abstr. Appl. Anal. 2013, Article ID 942038 (2013) MathSciNetGoogle Scholar
- Mukhamedov, F, Qaralleh, I, Rozali, WNFAW: On \(\xi^{(a)}\)-quadratic stochastic operators on 2D simplex. Sains Malays. 43, 1275-1281 (2014) Google Scholar
- Zakharevich, MI: On a limit behavior and ergodic hypothesis for quadratic mappings of a simplex. Russ. Math. Surv. 33, 207-208 (1978) MathSciNetView ArticleGoogle Scholar
- Hardy, GH, Littlewood, JE, Polya, G: Inequalities. Cambridge University Press, Cambridge (1978) Google Scholar
- Lorenz, MO: Methods of measuring the concentration of wealth. Publ. Am. Stat. Assoc. 9(70), 209-219 (1905) Google Scholar
- Dalton, H: The measurement of the inequality of incomes. Econ. J. 30, 348-361 (1920) View ArticleGoogle Scholar
- Schur, I: Über eine Klasse von Mittelbildungen mit Anwendungen auf die Determinanten Theorie. Sitzungsber. Berl. Math. Ges., vol. 22, pp. 9-20 (1923) Google Scholar
- Zylka, C, Vojta, G: Thermodynamic proofs of algebraic inequalities. Phys. Lett. A 152(3), 163-164 (1991) MathSciNetView ArticleGoogle Scholar
- Parker, DS, Ram, P: Greed and Majorization. Computer Science Department, University of California, Los Angeles (1996) Google Scholar
- Helman, P, Moret, BM, Shapiro, HD: An exact characterization of greedy structures. SIAM J. Discrete Math. 6(2), 274-283 (1993) MathSciNetView ArticleGoogle Scholar
- Korte, B, Lovász, L: Mathematical structures underlying greedy algorithms. In: Fundamentals of Computation Theory, pp. 205-209. Springer, Berlin (1981) View ArticleGoogle Scholar
- Lawler, EL: Submodular functions and polymatroid optimization. In: Combinatorial Optimization: Annotated Bibliographies, pp. 32-38 (1985) Google Scholar
- Lovász, L: Submodular functions and convexity. In: Mathematical Programming the State of the Art, pp. 235-257. Springer, Berlin (1983) View ArticleGoogle Scholar
- Ganikhodzhaev, RN: On the definition of quadratic bistochastic operators. Russ. Math. Surv. 48, 244-246 (1993) MathSciNetView ArticleGoogle Scholar
- Marshall, AW, Olkin, I, Arnold, BC: Inequalities: Theory of Majorization and Its Applications. Springer, Berlin (2011) View ArticleGoogle Scholar
- Ganikhodzhaev, R, Mukhamedov, F, Saburov, M: G-Decompositions of matrices and related problems I. Linear Algebra Appl. 436, 1344-1366 (2012) MathSciNetView ArticleGoogle Scholar
- Mukhamedov, F: On infinite dimensional Volterra operators. Russ. Math. Surv. 55, 1161-1162 (2000) MathSciNetView ArticleGoogle Scholar
- Mukhamedov, F, Taha, MH: On Volterra and orthogonality preserving quadratic stochastic operators. Miskolc Math. Notes (in press) Google Scholar
- Volterra, V: Lois de fluctuation de la population de plusieurs espàces coexistant dans le même milieu. Association Franc. Lyon 1926 (1927), 96-98 (1926) Google Scholar
- Volterra, V: Leçons sur la théorie mathématique de la lutte pour la vie. Gauthier-Villars, Paris (1931) Google Scholar