Searcharxiv⌕ Search

arXiv subjects

U. A. Rozikov

Publications and source records attributed to U. A. Rozikov.

At least 19 recordsLinked to original sources
arXiv2026

Non-Linear Generalization of the DLR Equations: $q$-Specifications and $q$-Equilibrium Measures

We introduce a {\it non-linear} generalization of the classical Dobrushin-Lanford-Ruelle (DLR) framework by developing the concept of a $q$-specification and the associated $q$-equilibrium measures. These objects arise naturally from a family of non-linear $q$-stochastic operators acting on the space of probability measures. A $q$-equilibrium measure is characterized as a fixed point of such operators, providing a non-linear analogue of the Gibbs equilibrium in the sense of DLR. We establish general conditions ensuring the existence and uniqueness of $q$-equilibrium measures and demonstrate how quasilocality plays a decisive role in their construction. Moreover, we exhibit examples of $q$-specifications with an empty set of $q$-equilibrium measures. We characterize the set of $q$-equilibrium measures by studying the dynamical systems generated by a class of $q$-stochastic operators. As a concrete application, we show that for the one-dimensional Ising model at sufficiently low temperatures, multiple $q$-equilibrium measures may exist, even though the classical Gibbs measure remains unique. Our results reveal that the $q$-specification formalism extends the DLR theory from linear to non-linear settings and opens a new direction in the study of Gibbs measures and equilibrium states of physical systems.

math-ph↗
arXiv2026

Data-Driven Energy-Based Learning via Gibbs Measures on Hierarchical Structures

We introduce a data-driven probabilistic framework for learning systems based on Gibbs measures on hierarchical structures. Unlike standard empirical risk minimization, where a dataset is used to identify a single optimal parameter, our approach transforms the empirical loss function into an interaction potential defining an energy-based model. The resulting Gibbs distribution describes a family of equilibrium learning states generated by the data. We formulate the consistency conditions of the associated finite-volume distributions and derive nonlinear integral fixed-point equations whose solutions characterize the admissible learning states. These equations provide a rigorous connection between empirical loss landscapes and probabilistic inference on trees. For translation-invariant solutions, the problem reduces to the analysis of positive compact operators induced by data-dependent kernels, allowing us to establish existence and uniqueness conditions in the one-dimensional setting. Furthermore, we show that hierarchical learning systems may exhibit phase-transition phenomena: for certain empirical kernels on Cayley trees, multiple Gibbs measures emerge beyond a critical inverse temperature, corresponding to distinct equilibrium prediction regimes. Numerical experiments with non-separable kernels illustrate the appearance of multiple solution branches and demonstrate the coexistence of several data-induced learning states. Our results provide a new perspective on energy-based learning, where data do not merely determine an optimal model through minimization but define an entire probabilistic landscape of possible inference states.

cs.LG↗
arXiv2026

Ideals in Arbitrary Three-Dimensional Algebras

In this paper, we study arbitrary (not necessarily associative) 3-dimensional algebras. Such an algebra A is determined by a basis and the corresponding multiplication table, which is specified by 27 structure constants. We describe all ideals of A, providing an explicit characterization of both 1-dimensional and 2-dimensional ideals. Moreover, we classify 2-dimensional ideals into 4 distinct types. We prove that A either has infinitely many ideals or at most 4. We also show that, in any case, the maximum number of 1-dimensional ideals is 3, while the maximum number of 2-dimensional ideals is 2. Finally, we present a class of algebras with a finite number of ideals that attain this theoretical maximum.

math.RA↗
arXiv2025

Ising Models with Hidden Markov Structure: Applications to Probabilistic Inference in Machine Learning

In this paper, we investigate tree-indexed Markov chains (Gibbs measures) defined by a Hamiltonian that couples two Ising layers: hidden spins \(s(x) \in \{\pm 1\}\) and observed spins \(σ(x) \in \{\pm 1\}\) on a Cayley tree. The Hamiltonian incorporates Ising interactions within each layer and site-wise emission couplings between layers, extending hidden Markov models to a bilayer Markov random field. Specifically, we explore translation-invariant Gibbs measures (TIGM) of this Hamiltonian on Cayley trees. Under certain explicit conditions on the model's parameters, we demonstrate that there can be up to three distinct TIGMs. Each of these measures represents an equilibrium state of the spin system. These measures provide a structured approach to inference on hierarchical data in machine learning. They have practical applications in tasks such as denoising, weakly supervised learning, and anomaly detection. The Cayley tree structure is particularly advantageous for exact inference due to its tractability.

cs.LG↗
arXiv2025

Coupled Ising-Potts Model: Rich Sets of Critical Temperatures and Translation-Invariant Gibbs Measures

We consider a coupled Ising-Potts model on Cayley trees of order $ k \geq 2 $. This model involves spin vectors $ (s, σ) $, and generalizes both the Ising and Potts models by incorporating interactions between two types of spins: $s = \pm 1$ and $σ= 1, \dots, q$. It is applicable to a wide range of systems, including multicomponent alloys, spin glasses, biological systems, networks, and social models. In this paper, we find some translation-invariant splitting Gibbs measures (TISGMs) and show, for $k\geq 2$, that at sufficiently low temperatures, the number of such measures is at least $2^{q}+1$. This is not an exact upper bound; for $k=2$ and $q=5$, we demonstrate that the number of TISGMs reaches the exact bound of 335, which is much larger than $2^5+1=33$. We prove, for $q=5$ that there are 12 critical temperatures at which the number of TISGMs changes, and we provide the exact number of TISGMs for each intermediate temperature. Additionally, we identify temperature regions where three TISGMs, close to the free measure, are either extreme or non-extreme among all Gibbs measures. We also show that the coupled Ising-Potts model exhibits properties absent in the individual Ising and Potts models. In particular, we observe the following new phenomena: 1. In both the Ising and Potts models, if a Gibbs measure exists at some temperature $T_0$, then it exists for all $T<T_0$. However, in the coupled Ising-Potts model, some TISGMs may only exist at intermediate temperatures (neither very low nor very high). 2. The 5-state Potts model has three critical temperatures and up to 31 TISGMs. We show that for $q=5$, the coupled Ising-Potts model has four times as many critical temperatures and approximately 11 times as many TISGMs. Thus, our model modifies the phase structure more rapidly and exhibits a significantly richer class of splitting Gibbs measures.

math.FA↗
arXiv2025

Non-Gibbsian Multivariate Ewens Probability Distributions on Regular Trees

Ewens' sampling formula (ESF) provides the probability distribution governing the number of distinct genetic types and their respective frequencies at a selectively neutral locus under the infinitely-many-alleles model of mutation. A natural and significant question arises: ``Is the Ewens probability distribution on regular trees Gibbsian?" In this paper, we demonstrate that Ewens probability distributions can be regarded as non-Gibbsian distributions on regular trees and derive a sufficient condition for the consistency condition. This study lays the groundwork for a new direction in the theory of non-Gibbsian probability distributions on trees.

math.PR↗
arXiv2024

Extreme Gibbs measures for a Hard-Core-SOS model on Cayley trees

We investigate splitting Gibbs measures (SGMs) of a three-state (wand-graph) hardcore SOS model on Cayley trees of order $ k \geq 2 $. Recently, this model was studied for the hinge-graph with $ k = 2, 3 $, while the case $ k \geq 4 $ remains unresolved. It was shown that as the coupling strength $θ$ increases, the number of translation-invariant SGMs (TISGMs) evolves through the sequence $ 1 \to 3 \to 5 \to 6 \to 7 $. In this paper, for wand-graph we demonstrate that for arbitrary $ k \geq 2 $, the number of TISGMs is at most three, denoted by $ μ_i $, $ i = 0, 1, 2 $. We derive the exact critical value $θ_{\text{cr}}(k)$ at which the non-uniqueness of TISGMs begins. The measure $ μ_0 $ exists for any $θ> 0$. Next, we investigate whether $ μ_i $, $i=0,1,2$ is extreme or non-extreme in the set of all Gibbs measures. The results are quite intriguing: 1) For $μ_0$: - For $ k = 2 $ and $ k = 3 $, there exist critical values $θ_1(k)$ and $θ_2(k)$ such that $ μ_0 $ is extreme if and only if $θ\in (θ_1, θ_2)$, excluding the boundary values $θ_1$ and $θ_2$, where the extremality remains undetermined. - Moreover, for $ k \geq 4 $, $ μ_0 $ is never extreme. 2) For $μ_1$ and $μ_2$ at $k=2$ there is $θ_5<θ_{\text{cr}}(2)=1$ such that these measures are extreme if $θ\in (θ_5, 1)$.

math-ph↗
arXiv2024

A many-loci system with evolution characterized by uncountable linear operators

This paper investigates the evolution of a multi-locus biological system. The evolution of such a system is described by a quadratic stochastic operator (QSO) defined on a simplex. We demonstrate that this QSO can be decomposed into an infinite series of linear operators, each of which maps certain invariant subsets of the simplex to themselves. Furthermore, the entire simplex is the union of these invariant subsets, enabling analytical examination of the dynamical systems produced by the QSO. Finding all limit points of the dynamical system generated by the QSO in terms of the limit points of the linear operators, we provide a comprehensive characterization of the many-loci population dynamics.

math.DS↗
arXiv2024

Discrete-Time Dynamical Systems Generated by a Quadratic Operator

In this paper, we examine a specific class of quadratic operators. For these operators, we identified all fixed points and categorized their types in the general case. Our analysis revealed that there are no attractive fixed points except the origin. Additionally, we investigated the global dynamics in the two-dimensional case and generalized several results obtained for lower-dimensional scenarios

math.DS↗
arXiv2024

Stability analysis of a three-dimensional system of Topp model with diabetes

Mathematical models of glucose, insulin, and pancreatic $β$-cell mass dynamics are essential for understanding the physiological basis of type 2 diabetes. This paper investigates the Topp model's discrete-time dynamics to represent these interactions. We perform a comprehensive analysis of the system's trajectory, examining both local and global behavior. First, we establish the invariance of the positive trajectory and analyze the existence of fixed points. Then, we conduct a complete stability analysis, determining the local and global asymptotic stability of these fixed points. Finally, numerical examples validate the effectiveness and applicability of our theoretical findings. Additionally, we provide biological interpretations of our results.

math.DS↗
arXiv2024

Dynamical systems of an infinite-dimensional non-linear operator

We investigate discrete-time dynamical systems generated by an infinite-dimensional non-linear operator that maps the Banach space $l_1$ to itself. It is demonstrated that this operator possesses up to seven fixed points. By leveraging the specific form of our operator, we illustrate that analyzing the operator can be simplified to a two-dimensional approach. Subsequently, we provide a detailed description of all fixed points, invariant sets for the two-dimensional operator and determine the set of limit points for its trajectories. These results are then applied to find the set of limit points for trajectories generated by the infinite-dimensional operator.

math.DS↗
arXiv2024

Classification in a rotational flow of two-dimensional algebras

In this paper, we examine a time-dependent family of two-dimensional algebras. We investigate the conditions under which any two algebras from this family, formed at different times, are isomorphic. Our findings reveal that the flow comprises of uncountable pairwise non-isomorphic algebras, including one commutative algebra. Additionally, we compare our results with a previously established classification of 2-dimensional real algebras.

math.AC↗
arXiv2023

Gradient Gibbs measures of an SOS model with alternating magnetism on Cayley trees

The work is devoted to gradient Gibbs measures (GGMs) of a SOS model with countable set $\mathbb Z$ of spin values and having alternating magnetism on Cayley trees. This model is defined by a nearest-neighbor gradient interaction potential. Using Külske-Schriever argument, based on boundary law equations, we give several $q$-height-periodic translations invariant GGMs for $q=2,3,4$.

math-ph↗
arXiv2023

Dynamical systems of Möbius transformation: real, $p$-adic and complex variables

In this paper we consider function $f(x)={x+a\over bx+c}$, (where $b\ne 0$, $c\ne ab$, $x\ne -{c\over b}$) on three fields: the set of real, $p$-adic and complex numbers. We study dynamical systems generated by this function on each field separately and give some comparison remarks. For real variable case we show that the real dynamical system of the function depends on the parameters $(a,b,c)\in \mathbb R^3$. Namely, we classify the parameters to three sets and prove that: for the parameters from first class each point, for which the trajectory is well defined, is a periodic point of $f$; for the parameters from second class any trajectory (under $f$) converges to one of fixed points (there may be up to two fixed points); for the parameters from third class any trajectory is dense in $\mathbb R$. For the $p$-adic variable we give a review of known results about dynamical systems of function $f$. Then using a recently developed method we give simple new proofs of these results and prove some new ones related to trajectories which do not converge. For the complex variables we give a review of known results.

math.DS↗
arXiv2023

Gonosomal algebras and associated discrete-time dynamical systems

In this paper we study the discrete-time dynamical systems associated with gonosomal algebras used as algebraic model in the sex-linked genes inheritance. We show that the class of gonosomal algebras is disjoint from the other non-associative algebras usually studied (Lie, alternative, Jordan, associative power). To each gonosomal algebra, with the mapping $x\mapsto\frac{1}{2}x^{2}$, an evolution operator $W$ is associated that gives the state of the offspring population at the birth stage, then from $W$ we define the operator $V$ which gives the frequency distribution of genetic types. We study discrete-time dynamical systems generated by these two operators, in particular we show that the various stability notions of the equilibrium points are preserved by passing from $W$ to $V$. Moreover, for the evolution operators associated with genetic disorders in the case of a diallelic gonosomal lethal gene we give complete analysis of fixed and limit points of the dynamical systems.

math.DS↗
arXiv2023

Kittel's molecular zipper model on Cayley trees

Kittel's 1D model represents a natural DNA with two strands as a (molecular) zipper, which may separated as the temperature is varied. We define multidimensional version of this model on a Cayley tree and study the set of Gibbs measures. We reduce description of Gibbs measures to solving of a non-linear functional equation, with unknown functions (called boundary laws) defined on vertices of the Cayley tree. Each boundary law defines a Gibbs measure. We give general formula of free energy depending on the boundary law. Moreover, we find some concrete boundary laws and corresponding Gibbs measures. Explicit critical temperature for occurrence a phase transition (non-uniqueness of Gibbs measures) is obtained.

math-ph↗
arXiv2022

Bubble coalescence in interacting system of DNA molecules

We consider two models of interacting DNA molecules: \textit{First} is (four parametric) bubble coalescence model in interacting DNAs (shortly: BCI-DNA). \textit{Second} is (three parametric) bubble coalescence model in a condensed DNA molecules (shortly BCC-DNA). To study bubble coalescence thermodynamics of BCI-DNA and BCC-DNA models we use methods of statistical physics. Namely, we define Hamiltonian of each model and give their translation-invariant Gibbs measures (TIGMs). For the first model we find parameters such that corresponding Hamiltonian has up to three TIGMs (three phases of system) biologically meaning existence of three states: ``No bubble coalescence'', ``Dominated soft zone'', ``Bubble coalescence''. For the second model we show that for any (admissible) parameters this model has unique TIGM. This is a state where ``No bubble coalescence'' phase dominates.

cond-mat.stat-mech↗