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Mansoor Saburov

Publications and source records attributed to Mansoor Saburov.

At least 19 recordsLinked to original sources
arXiv2024

The Price of Cognition and Replicator Equations in Parallel Neural Networks

In this paper, we are aiming to propose a novel mathematical model that studies the dynamics of synaptic damage in terms of concentrations of toxic neuropeptides/neurotransmitters during neurotransmission processes. Our primary objective is to employ Wardrop's first and second principles within a neural network of the brain. In order to comprehensively incorporate Wardrop's first and second principles into the neural network of the brain, we introduce two novel concepts: \textit{neuropeptide's (neurotransmitter's) equilibrium} and \textit{synapses optimum}. The \textit{neuropeptide/neurotransmitter equilibrium} refers to \textit{a distribution of toxic neuropeptides/neurotransmitters that leads to uniform damage across all synaptic links}. Meanwhile, \textit{synapses optimum} is \textit{the most desirable distribution of toxic neuropeptides/neurotransmitters that minimizes the cumulative damage experienced by all synapses}. In the context of a neural network within the brain, an analogue of the price of anarchy is \textit{the price of cognition} which is \textit{the most unfavorable ratio between the overall impairment caused by toxic neuropeptide's (neurotransmitter's) equilibrium in comparison to the optimal state of synapses (synapses optimum)}. To put it differently, \textit{the price of cognition} measures \textit{the loss of cognitive ability resulting from increased concentrations of toxic neuropeptides/neurotransmitters}. Additionally, a replicator equation is proposed within this framework that leads to the establishment of the synapses optimum during the neurotransmission process.

q-bio.NC↗
arXiv2022

Stable and Historic Behavior in Replicator Equations Generated by Similar-Order Preserving Mappings

One could observe drastically different dynamics of zero-sum and non-zero-sum games under replicator equations. In zero-sum games, heteroclinic cycles naturally occur whenever the species of the population supersede each other in a cyclic fashion (like for the Rock-Paper-Scissors game). In this case, the highly erratic oscillations may cause the divergence of the time averages. In contrast, it is a common belief that the most ``reasonable" replicator equations of non-zero-sum games satisfy ``The Folk Theorem of Evolutionary Game Theory" which asserts that (i) a Nash equilibrium is a rest point; (ii) a stable rest point is a Nash equilibrium; (iii) a strictly Nash equilibrium is asymptotically stable; (iv) any interior convergent orbit evolves to a Nash equilibrium. In this paper, we propose two distinct vast classes of replicator equations generated by similar-order preserving mappings which exhibit stable as well as mean historic behavior. In the latter case, the time averages of the orbit will slowly oscillate during the evolution of the system and do not converge to any limit. This will eventually cause the divergence of higher-order repeated time averages.

math.DS↗
arXiv2017

Periodic $p$-adic Gibbs measures of $q$-states Potts model on Cayley tree: The chaos implies the vastness of $p$-adic Gibbs measures

We study the set of $p$-adic Gibbs measures of the $q$-states Potts model on the Cayley tree of order three. We prove the vastness of the periodic $p$-adic Gibbs measures for such model by showing the chaotic behavior of the correspondence Potts--Bethe mapping over $\mathbb{Q}\_p$ for $p\equiv 1 \ (\rm{mod} \ 3)$. In fact, for $0 < |θ-1|\_p < |q|\_p^2 < 1$, there exists a subsystem that isometrically conjugate to the full shift on three symbols. Meanwhile, for $0 < |q|\_p^2 \leq |θ-1|\_p < |q|\_p < 1$, there exists a subsystem that isometrically conjugate to a subshift of finite type on $r$ symbols where $r \geq 4$. However, these subshifts on $r$ symbols are all topologically conjugate to the full shift on three symbols. The $p$-adic Gibbs measures of the same model for the cases $p=2,3$ and the corresponding Potts--Bethe mapping are also discussed.Furthermore, for $0 < |θ-1|\_p < |q|\_p < 1,$ we remark that the Potts--Bethe mapping is not chaotic when $p=2,\ p=3$ and $p\equiv 2 \ (\rm{mod} \ 3)$ and we could not conclude the vastness of the periodic $p$-adic Gibbs measures. In a forthcoming paper with the same title, we will treat the case $0 < |q|\_p \leq |θ-1|\_p < 1$ for all $p$.

math.DS↗
arXiv2017

Elliptic Quadratic Operator Equations

In the present paper is devoted to the study of elliptic quadratic operator equations over the finite dimensional Euclidean space. We provide necessary and sufficient conditions for the existence of solutions of elliptic quadratic operator equations. The iterative Newton-Kantorovich method is also presented for stable solutions.

math.FA↗
arXiv2016

On the Lebesgue nonlinear transformations

In this paper, we introduce a quadratic stochastic operators on the set of all probability measures of a measurable space. We study the dynamics of the Lebesgue quadratic stochastic operator on the set of all Lebesgue measures of the set [0,1]. Namely, we prove the regularity of the Lebesgue quadratic stochastic operators

math.DS↗
arXiv2015

On $P$-adic Ising-Vannimenus model on an arbitrary order Cayley tree

In this paper, we continue an investigation of the $p$-adic Ising-Vannimenus model on the Cayley tree of an arbitrary order $k$ $(k\geq 2$). We prove the existence of $p$-adic quasi Gibbs measures by analyzing fixed points of multi-dimensional $p$-adic system of equations. We are also able to show the uniqueness of translation-invariant $p$-adic Gibbs measure. Finally, it is established the existence of the phase transition for the Ising-Vannimenus model depending on the order $k$ of the Cayley tree and the prime $p$. Note that the methods used in the paper are not valid in the real setting, since all of them are based on $p$-adic analysis and $p$-adic probability measures.

math.DS↗
arXiv2014

Local Descriptions of Roots of Cubic Equations over P-adic Fields

The most frequently asked question in the $p-$adic lattice models of statistical mechanics is that whether a root of a polynomial equation belongs to domains $\mathbb{Z}_p^{*}, \ \mathbb{Z}_p\setminus\mathbb{Z}_p^{*}, \ \mathbb{Z}_p, \ \mathbb{Q}_p\setminus\mathbb{Z}_p^{*}, \ \mathbb{Q}_p\setminus\left(\mathbb{Z}_p\setminus\mathbb{Z}_p^{*}\right), \ \mathbb{Q}_p\setminus\mathbb{Z}_p, \ \mathbb{Q}_p $ or not. However, this question was open even for lower degree polynomial equations. In this paper, we give local descriptions of roots of cubic equations over the $p-$adic fields for $p>3$.

math.NT↗
arXiv2013

On Quantum Markov Chains on Cayley tree III: Ising model

In this paper, we consider the classical Ising model on the Cayley tree of order k and show the existence of the phase transition in the following sense: there exists two quantum Markov states which are not quasi-equivalent. It turns out that the found critical temperature coincides with usual critical temperature.

math-ph↗
arXiv2013

On $ξ^{(s)}$-Quadratic Stochastic Operators on two Dimensional simplex and their behavior

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. The general problem in the nonlinear operator theory is to study the behavior of operators. This problem was not fully finished even for quadratic stochastic operators which are the simplest nonlinear operators. To study this problem, it was investigated several classes of QSO. In this paper, we study $ξ^{(s)}$--QSO defined on 2D simplex. We first classify $ξ^{(s)}$--QSO into 20 non-conjugate classes. Further, we investigate the dynamics of three classes of such operators.

math.DS↗
arXiv2013

Mutation and Chaos in Nonlinear Models of Heredity

In this short communication, we shall explore a nonlinear discrete dynamical system that naturally occurs in population systems to describe a transmission of a trait from parents to their offspring. We consider a Mendelian inheritance for a single gene with three alleles and assume that to form a new generation, each gene has a possibility to mutate, that is, to change into a gene of the other kind. We investigate the derived models. A numerical simulation assists us to get some clear picture about chaotic behaviors of such models.

math.DS↗
arXiv2013

Mendelian and Non-Mendelian Quadratic Operators

In this paper, we attempt to provide mathematical models of Mendelian and Non-Mendelian inheritances of the bisexual population system having Fisher's {\textbf{1:1}} principle. In our model, we always assume that distributions of the same phenotype of female and male populations are equal. We study the evolution of a Mendelian trait. As an application of a non-Mendelian inheritance, we construct a quadratic stochastic operator that describes transmission of {\textbf{ABO}} and Rh blood groups.

math.DS↗
arXiv2012

On unification of the strong convergence theorems for a finite family of total asymptotically nonexpansive mappings in Banach spaces

In this paper, we unify all know iterative methods by introducing a new explicit iterative scheme for approximation of common fixed points of finite families of total asymptotically $I$-nonexpansive mappings. Note that such a scheme contains as a particular case of the method introduced in [C.E. Chidume, E.U. Ofoedu, \textit{Inter. J. Math. & Math. Sci.} \textbf{2009}(2009) Article ID 615107, 17p]. We construct examples of total asymptotically nonexpansive mappings which are not asymptotically nonexpansive. Note that no such kind of examples were known in the literature. We prove the strong convergence theorems for such iterative process to a common fixed point of the finite family of total asymptotically $I-$nonexpansive and total asymptotically nonexpansive mappings, defined on a nonempty closed convex subset of uniformly convex Banach spaces. Moreover, our results extend and unify all known results.

math.FA↗
arXiv2012

On cubic equations over $P-$adic field

We provide a solvability criteria for a depressed cubic equation in domains $\bz_p^{*},\bz_p,\bq_p$. We show that, in principal, the Cardano method is not always applicable for such equations. Moreover, the numbers of solutions of the depressed cubic equation in domains $\bz_p^{*},\bz_p,\bq_p$ are provided. Since $\bbf_p\subset\bq_p,$ we generalize J.-P. Serre's \cite{JPSJ} and Z.H.Sun's \cite{ZHS1,ZHS3} results concerning with depressed cubic equations over the finite field $\bbf_p$. Finally, all depressed cubic equations, for which the Cardano method could be applied, are described and the $p-$adic Cardano formula is provided for those cubic equations.

math.NT↗
arXiv2011

$G-$Decompositions of Matrices and Related Problems I

In the present paper we introduce a notion of $G-$decompositions of matrices. Main result of the paper is that a symmetric matrix $A_m$ has a $G-$decomposition in the class of stochastic (resp. substochastic) matrices if and only if $A_m$ belongs to the set ${\mathbf{U}}^m$ (resp. ${\mathbf{U}}_m$). To prove the main result, we study extremal points and geometrical structures of the sets ${\mathbf{U}}^m$, ${\mathbf{U}}_m$. Note that such kind of investigations enables to study Birkhoff's problem for quadratic $G-$doubly stochastic operators.

math.CO↗
arXiv2010

On Quantum Markov Chains on Cayley tree II: Phase transitions for the associated chain with XY-model on the Cayley tree of order three

In the present paper we study forward Quantum Markov Chains (QMC) defined on a Cayley tree. Using the tree structure of graphs, we give a construction of quantum Markov chains on a Cayley tree. By means of such constructions we prove the existence of a phase transition for the XY-model on a Cayley tree of order three in QMC scheme. By the phase transition we mean the existence of two now quasi equivalent QMC for the given family of interaction operators $\{K_{ }\}$.

math-ph↗