# On Regularity of Positive Quadratic Doubly Stochastic Operators

On Regularity of Positive Quadratic Doubly Stochastic Operators ISSN 0001-4346, Mathematical Notes, 2018, Vol. 103, No. 2, pp. 328–333. © Pleiades Publishing, Ltd., 2018. SHORT COMMUNICATIONS 1, 2 ** M. Saburov International Islamic University Malaysia, Kuantan, Malaysia College of Engineering and Technology, American University of the Middle East, Egaila, Kuwait Received January 25, 2017 DOI: 10.1134/S0001434618010376 Keywords: quadratic doubly stochastic operator, cubic stochastic matrix, regularity. 1. INTRODUCTION A linear Markov chain is a discrete time stochastic process whose transitions depend only on the current state of the process. A nonlinear Markov chain is a discrete time stochastic process whose transitions may depend on both the current state and the current distribution of the process (see [1]). These processes arise naturally in the study of the limit behavior of a large number of weakly interacting Markov processes. The ergodic theory of the linear Markov chain is very well established. Let us ﬁrst recall some necessary deﬁnitions. m m m Let I := {1,··· ,m} be a ﬁnite set and {e } be the standard basis of R . Suppose that R is m k k=1 equipped with the l −norm x := |x |,where x =(x ,··· ,x ) ∈ R . 1 k 1 m k=1 We http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematical Notes Springer Journals

# On Regularity of Positive Quadratic Doubly Stochastic Operators

, Volume 103 (2) – Mar 14, 2018
6 pages

Publisher
Subject
Mathematics; Mathematics, general
ISSN
0001-4346
eISSN
1573-8876
D.O.I.
10.1134/S0001434618010376
Publisher site
See Article on Publisher Site

### Abstract

ISSN 0001-4346, Mathematical Notes, 2018, Vol. 103, No. 2, pp. 328–333. © Pleiades Publishing, Ltd., 2018. SHORT COMMUNICATIONS 1, 2 ** M. Saburov International Islamic University Malaysia, Kuantan, Malaysia College of Engineering and Technology, American University of the Middle East, Egaila, Kuwait Received January 25, 2017 DOI: 10.1134/S0001434618010376 Keywords: quadratic doubly stochastic operator, cubic stochastic matrix, regularity. 1. INTRODUCTION A linear Markov chain is a discrete time stochastic process whose transitions depend only on the current state of the process. A nonlinear Markov chain is a discrete time stochastic process whose transitions may depend on both the current state and the current distribution of the process (see [1]). These processes arise naturally in the study of the limit behavior of a large number of weakly interacting Markov processes. The ergodic theory of the linear Markov chain is very well established. Let us ﬁrst recall some necessary deﬁnitions. m m m Let I := {1,··· ,m} be a ﬁnite set and {e } be the standard basis of R . Suppose that R is m k k=1 equipped with the l −norm x := |x |,where x =(x ,··· ,x ) ∈ R . 1 k 1 m k=1 We

### Journal

Mathematical NotesSpringer Journals

Published: Mar 14, 2018

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