# On orthogonality preserving quadratic stochastic operators

## Abstract

A quadratic stochastic operator (in short QSO) is usually used to present the time evolution of differing species in biology. Some quadratic stochastic operators have been studied by Lotka and Volterra. In the present paper, we first give a simple characterization of Volterra QSO in terms of absolutely continuity of discrete measures. Further, we introduce a notion of orthogonal preserving QSO, and describe such kind of operators defined on two dimensional simplex. It turns out that orthogonal preserving QSOs are permutations of Volterra QSO. The associativity of genetic algebras generated by orthogonal preserving QSO is studied too.

- Authors:

- Department of Computational and Theoretical Sciences, Faculty of Science International Islamic University Malaysia, P.O. Box 141, 25710 Kuantan, Pahang Malaysia (Malaysia)

- Publication Date:

- OSTI Identifier:
- 22391655

- Resource Type:
- Journal Article

- Journal Name:
- AIP Conference Proceedings

- Additional Journal Information:
- Journal Volume: 1660; Journal Issue: 1; Conference: ICoMEIA 2014: International Conference on Mathematics, Engineering and Industrial Applications 2014, Penang (Malaysia), 28-30 May 2014; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0094-243X

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; BIOLOGICAL EVOLUTION; MATHEMATICAL OPERATORS; MATHEMATICAL SOLUTIONS; STOCHASTIC PROCESSES; TWO-DIMENSIONAL CALCULATIONS; VOLTERRA INTEGRAL EQUATIONS

### Citation Formats

```
Mukhamedov, Farrukh, and Taha, Muhammad Hafizuddin Mohd.
```*On orthogonality preserving quadratic stochastic operators*. United States: N. p., 2015.
Web. doi:10.1063/1.4915690.

```
Mukhamedov, Farrukh, & Taha, Muhammad Hafizuddin Mohd.
```*On orthogonality preserving quadratic stochastic operators*. United States. doi:10.1063/1.4915690.

```
Mukhamedov, Farrukh, and Taha, Muhammad Hafizuddin Mohd. Fri .
"On orthogonality preserving quadratic stochastic operators". United States. doi:10.1063/1.4915690.
```

```
@article{osti_22391655,
```

title = {On orthogonality preserving quadratic stochastic operators},

author = {Mukhamedov, Farrukh and Taha, Muhammad Hafizuddin Mohd},

abstractNote = {A quadratic stochastic operator (in short QSO) is usually used to present the time evolution of differing species in biology. Some quadratic stochastic operators have been studied by Lotka and Volterra. In the present paper, we first give a simple characterization of Volterra QSO in terms of absolutely continuity of discrete measures. Further, we introduce a notion of orthogonal preserving QSO, and describe such kind of operators defined on two dimensional simplex. It turns out that orthogonal preserving QSOs are permutations of Volterra QSO. The associativity of genetic algebras generated by orthogonal preserving QSO is studied too.},

doi = {10.1063/1.4915690},

journal = {AIP Conference Proceedings},

issn = {0094-243X},

number = 1,

volume = 1660,

place = {United States},

year = {2015},

month = {5}

}

DOI: 10.1063/1.4915690

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