arXiv subjects
Publications and source records attributed to Nasir Ganikhodjaev.
For Ising model on the Cayley tree of order five and six we present new weakly periodic (non-periodic) Gibbs measures corresponding to normal subgroups of indices two in the group representation of the Cayley tree.
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
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.
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.
In the paper a Volterra quadratic stochastic operators of three dimensional simplex into itself is considered.The full description of ergodic properties such operators is given.
Applying non-ergodic quadratic stochastic operator the continual family of weak ergodic non-homogeneous Markov chains is constructed.
We have found an exact phase diagram of the Potts model with next nearest neighbor interactions on the Bethe lattice of order two. The diagram consists of five phases: ferromagnetic, paramagnetic, modulated, antiphase and paramodulated, all meeting at the Lifshitz point i.e. $p=1/3$. We report on a new phase which we denote as paramodulated, found at low temperatures and characterized by 2-periodic points of an one dimensional dynamical system lying inside the modulated phase. Such a phase, inherent in the Potts model has no analogues in the Ising setting.
In this paper, correlation inequalities which have been considered on Ising model are extended to q-Potts model. It is considered on generalized Potts model with interaction of any number of spins. We replace the set of spin values $F=\{1,2,..., q\}$ by the centered set $F=\{-(q-1)/2,-(q-3)/2,... ,(q-3)/2,(q-1)/2\}$. Let $N$ be the subset of one-dimensional lattice with $n$ vertices, $\g=(\s_1,\s_2,...,\s_n):N \to F^c$ be a configuration where ${(\s_i)}_\g$ is the number which appears as the ith spin (component) in $\g$ and $\s_i$ be a random variable whose value at $\g$ is ${(\s_i)}_\g$. Define $\s^R=\prod_{i \in R}\s_i$ for any list $R$ where any $i \in R$ implies that $i \in N$. We first prove that $<\s^R > \ge 0$ then we prove that for any two lists $R$ and $S$, we have $<\s^R \s^S >- < \s^R > < \s^S > \ge 0$.
In this paper a notion of entropy transmission of quantum channels is introduced as a natural extension of Ohya's entropy. Here by quantum channel is meant unital completely positive mappings (ucp) of $B(H)$ into itself, where $H$ is an infinite dimensional Hilbert space. Using a representation theorem of ucp mapping we associate to every ucp map a uniquely determined state, and prove that entropy of ucp map is less then Ohya's entropy of the associated state.
In the present paper the three state Potts model with competing binary interactions (with couplings $J$ and $J_p$) on the second order Bethe lattice is considered. The recurrent equations for the partition functions are derived. When $J_p=0$, by means of a construction of a special class of limiting Gibbs measures, it is shown how these equations are related with the surface energy of the Hamiltonian. This relation reduces the problem of describing the limit Gibbs measures to find of solutions of a nonlinear functional equation. Moreover, the set of ground states of the one-level model is completely described. Using this fact, one finds Gibbs measures (pure phases) associated with the translation-invariant ground states. The critical temperature is exactly found and the phase diagram is presented. The free energies corresponding to translations-invariant Gibbs measures are found. Certain physical quantities are calculated as well.
In the paper we prove that a quadratic stochastic process satisfies the ergodic principle if and only if the associated Markov process satisfies one.