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Otabek Khakimov

Publications and source records attributed to Otabek Khakimov.

At least 19 recordsLinked to original sources
arXiv2026

Local Rota--Baxter operators of weight zero on nilpotent evolution algebras with maximal nilindex

Let $E$ be an $n$-dimensional nilpotent evolution algebra of maximal nilindex over a field of characteristic zero. The Rota--Baxter operators of weights zero and one on such algebras were recently classified. In this paper we investigate the local analogue of the weight-zero case. We prove that every local Rota--Baxter operator of weight zero has a rigid diagonal part: the nonzero diagonal coefficients, when they occur, begin after the obstruction index determined by the structural matrix of $E$ and form a final geometric string governed by one scalar. In contrast, the last-column coefficients are arbitrary. This gives an explicit description of the class $\LRB_0(E)$ and shows that it is generally strictly larger than $\RB_0(\E)$. We also prove that the corresponding $s$-local notion produces no new operators. The classification is further interpreted as a finite union of quasi-affine strata, yielding a dimension comparison between $\LRB_0(E)$ and $\RB_0(E)$. Finally, we study ordinary weight-zero Rota--Baxter operators commuting with derivations and automorphisms and describe the resulting conditions in terms of the directed graph associated with $E$.

math.RA↗
arXiv2026

Rota-Baxter operators of nilpotent evolution algebras with maximal nilindex

Nilpotent evolution algebras of maximal nilindex admit a natural basis in which the structure matrix is strictly upper triangular. In this paper we classify Rota{Baxter operators of weights zero and one on such algebras. We prove that every Rota{Baxter operator is upper triangular with respect to a natural basis. For weight zero, a strong rigidity phenomenon occurs: the operators are diagonal up to possible perturbations supported in the last basis vector. For weight one, a richer structure appears, including both triangular and non-triangular families, with the diagonal entries governed by a rational recurrence relation. Our results provide a complete description of Rota{Baxter operators on nilpotent evolution algebras of maximal nilindex.

math.RA↗
arXiv2025

p-adic dynamical systems for $p$-adic $λ$-models on the Cayley trees

We review and propose to use of associated dynamical system to explore the phase transition phenomena in $p$-adic statistical mechanics setting, by means of the renormalization techniques. Main focus of the paper is the $p$-adic $λ$-model on the Cayley tree. We study generalized $p$-adic quasi Gibbs measures for the $λ$-model. These measures associate recurrence equations that determine dynamical systems for the model. A main point is to verify and confirm that the indicated predictions from a dynamical system point of view are indeed true.

math-ph↗
arXiv2022

On $\exp(Der(E))$ of nilpotent evolution algebras

In the present paper, every evolution algebra is endowed with Banach algebra norm. This together with the description of derivations and automorphisms of nilpotent evolution algebras, allows to investigated the set $\exp(Der(E))$. Moreover, it is proved that $\exp(Der(E))$ is a normal subgroup of $Aut(E)$, and its corresponding index is calculated.

math.FA↗
arXiv2021

Projective surjectivity of quadratic stochastic operators $L_1$ and its application

A nonlinear Markov chain is a discrete time stochastic process whose transitions depend on both the current state and the current distribution of the process. The nonlinear Markov chain over a infinite state space can be identified by a continuous mapping (the so-called nonlinear Markov operator) defined on a set of all probability distributions (which is a simplex). In the present paper, we consider a continuous analogue of the mentioned mapping acting on $L^1$-spaces. Main aim of the current paper is to investigate projective surjectivity of quadratic stochastic operators (QSO) acting on the set of all probability measures. To prove the main result, we study the surjectivity of infinite dimensional nonlinear Markov operators and apply them to the projective surjectivity of a QSO. Furthermore, the obtained result has been applied for the existence of positive solution of some Hammerstein integral equations.

math.FA↗
arXiv2019

Chaotic behavior of the $p$-adic Potts-Bethe mapping II

In our previous investigations, we have developed the renormalization group method to $p$-adic $q$-state Potts model on the Cayley tree of order $k$. This method is closely related to the examination of dynamical behavior of the $p$-adic Potts-Bethe mapping which depends on parameters $q,k$. In \cite{MFKh18} we have considered the case when $q$ is not divisible by $p$, and under some conditions it was established that the mapping is conjugate to the full shift. The present paper is a continuation of the mentioned paper, but here we investigate the case when $q$ is divisible by $p$ and $k$ is arbitrary. We are able to fully describe the dynamical behavior of the $p$-adic Potts-Bethe mapping by means of Markov partition. Moreover, the existence of Julia set is established, over which the mapping enables a chaotic behavior. We point out that a similar result is not known in the case of real numbers (with rigorous proofs).

math.DS↗
arXiv2019

$P$-adic monomial equations and their perturbation

In this paper, we describe the set of all solutions of monomial equation $x^k=a$ over $\mathbb Q_p$. Moreover, as an application of the result, we study several perturbations of the considered equation over $p$-adic field.

math.NT↗
arXiv2019

On omega limiting sets of infinite dimensional Volterra operators

In the present paper, we are aiming to study limiting behavior of infinite dimensional Volterra operators. We introduce two classes $\tilde {\mathcal{V}}^+$ and $\tilde{\mathcal{V}}^-$of infinite dimensional Volterra operators. For operators taken from the introduced classes we study their omega limiting sets $ω_V$ and $ω_V^{(w)}$ with respect to $\ell^1$-norm and pointwise convergence, respectively. To investigate the relations between these limiting sets, we study linear Lyapunov functions for such kind of Volterra operators. It is proven that if Volterra operator belongs to $\tilde {\mathcal{V}}^+$, then the sets and $ω_V^{(w)}(\xb)$ coincide for every $\xb\in S$, and moreover, they are non empty. If Volterra operator belongs to $\tilde {\mathcal{V}}^-$, then $ω_V(\xb)$ could be empty, and it implies the non-ergodicity (w.r.t $\ell^1$-norm) of $V$, while it is weak ergodic.

math.DS↗
arXiv2019

On a $p$-Adic Generalized Gibbs Measure for Ising Model on a Cayley Tree

In this paper we consider a $p$-adic Ising model on the Cayley tree of order $k\geq 2$. We give full description of all $p$-adic translation-invariant generalized Gibbs measures for $k=3$. Moreover, we show the existence of phase transition for $p$-adic Ising model for any $k\geq3$ when $p\equiv1(\operatorname{mod }4)$.

math-ph↗
arXiv2018

Derivations and automorphisms of nilpotent evolution algebras with maximal nilindex

In this paper is devoted to nilpotent finite-dimensional evolution algebras E with $dimE^2 = dimE-1$. We described Lie algebras associated with evolution algebras whose nilindex is maximal. Moreover, in terms of this Lie algebra we fully construct nilpotent evolution algebra with maximal index of nilpotency. Furthermore, this result allowed us fully characterize all local and 2-local derivations of the considered evolution algebras. All automorphisms and local automorphisms of the nilpotent evolution algebras with maximal nilindex are found.

math.RA↗
arXiv2017

Gibbs measures and free energies of Ising-Vannimenus Model on the Cayley tree

In this paper, we consider the Ising-Vannimenus model on a Cayley tree for order two with competing nearest-neighbor and prolonged next-nearest neighbor interactions. We stress that the mentioned model was investigated only numerically, without rigorous (mathematical) proofs. One of the main points of this paper is to propose a measure-theoretical approach for the considered model. We find certain conditions for the existence of Gibbs measures corresponding to the model, which allowed to establish the existence of the phase transition. Moreover, the free energies and entropies, associated with translation invariant Gibbs measures, are calculated.

math-ph↗
arXiv2017

Hypercyclic and supercyclic linear operators on non-Archimedean vector spaces

A main objective of the present paper is to develop the theory of hypercyclicity and supercyclicity of linear operators on topological vector space over non-Archimedean valued fields. We show that there does not exist any hypercyclic operator on finite dimensional spaces. Moreover, we give sufficient and necessary conditions of hypercyclicity (resp. supercyclicity) of linear operators on separable $F$-spaces. It is proven that a linear operator $T$ on topological vector space $X$ is hypercyclic (supercyclic) if it satisfies Hypercyclic (resp. Supercyclic) Criterion. We consider backward shifts on $c_0$, and characterize hypercyclicity and supercyclicity of such kinds of shifts. Finally, we study hypercyclicity, supercyclicity of operators $λI+μB$, where $I$ is identity and $B$ is backward shift. We note that there are essential differences between the non-Archimedean and real cases.

math.FA↗
arXiv2016

Chaotic behavior of the $P$-adic Potts-Bethe mapping

In our previous investigations, we have developed the renormalization group method to $p$-adic models on Cayley trees, this method is closely related to the investigation of $p$-adic dynamical systems associated with a given model. In this paper, we study chaotic behavior of the Potts-Bethe mapping. We point out that a similar kind of result is not known in the case of real numbers (with rigorous proofs).

math.DS↗
arXiv2015

On unconventional limit sets of contractive functions on $\mathbb Z_p$

In the present paper, we are going to study metric properties of unconventional limit set of a semigroup $G$ generated by contractive functions $\{f_{i}\}_{i=1}^N$ on the unit ball $\mathbb Z_p$ of $p$-adic numbers. Namely, we prove that the unconventional limit set is compact, perfect and uniformly disconnected. Moreover, we provide an example of two contractions for which the corresponding unconventional limiting set is quasi-symmetrically equivalent to the symbolic Cantor set.

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↗