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We consider two models of interacting DNA molecules: First is (four parametric) bubble coalescence model in interacting DNAs (shortly: BCI-DNA). 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. [ABSTRACT FROM AUTHOR]

For the solid-on-solid (SOS) model with an external field and with spin values from the set of all integers on a Cayley tree, each (gradient) Gibbs measure corresponds to a boundary law (an infinite-dimensional vector function defined on vertices of the Cayley tree) satisfying a nonlinear functional equation. Recently some translation-invariant and height-periodic (non-normalizable) solutions to the equation are found. Here, our aim is to find non-height-periodic and non-normalizable boundary laws for the SOS model. By such a solution one can construct a non-probability Gibbs measure. We find explicitly several non-normalizable boundary laws. Moreover, we reduce the problem to solving of a nonlinear, second-order difference equation. We give analytic and numerical analyses of the difference equation. [ABSTRACT FROM AUTHOR]

Kittel's 1D model represents a natural DNA with two strands as a (molecular) zipper, which may be 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 nonlinear functional equation, with unknown functions (called boundary laws) defined on vertices of the Cayley tree. Each boundary law defines a Gibbs measure. We give a 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 of a phase transition (non-uniqueness of Gibbs measures) is obtained. [ABSTRACT FROM AUTHOR]

We consider a hard core (HC) model with a countable set ℤ of spin values on the Cayley tree. This model is defined by a countable set of parameters λ i > 0 , i ∈ ℤ ∖ { 0 }. For all possible values of parameters, we give limit points of the dynamical system generated by a function which describes the consistency condition for finite-dimensional measures. Also, we prove that every periodic Gibbs measure for the given model is either translation-invariant or periodic with period two. Moreover, we construct uncountable set of Gibbs measures for this HC model. [ABSTRACT FROM AUTHOR]

In this paper, we give a systematic review of the theory of Gibbs measures of Potts model on Cayley trees (developed since 2013) and discuss many applications of the Potts model to real world situations: mainly biology, physics, and some examples of alloy behavior, cell sorting, financial engineering, flocking birds, flowing foams, image segmentation, medicine, sociology, etc. [ABSTRACT FROM AUTHOR]

We consider a new model which consists of a DNA together with a RNA. Here, we assume that DNA is from a mammal or bird but RNA comes from a virus. To study thermodynamic properties of this model, we use methods of statistical mechanics, namely, the theory of Gibbs measures. We use these measures to describe phases (states) of the DNA–RNA system. Using a Markov chain (corresponding to Gibbs measure) we give conditions (on temperature) of DNA–RNA renaturation. [ABSTRACT FROM AUTHOR]

We consider the Leslie's prey–predator model with discrete time. This model is given by a nonlinear evolution operator depending on five parameters. We show that this operator has two fixed points and define type of each fixed point depending on the parameters. Finding two invariant sets of the evolution operator, we study the dynamical systems generated by the operator on each invariant set. Depending on the parameters, we classify the dynamics between a predator and a prey of the Leslie's model. [ABSTRACT FROM AUTHOR]

We consider a discrete-time dynamical system generated by a nonlinear operator (with four real parameters a , b , c , d) of ocean ecosystem. We find conditions on the parameters under which the operator is reduced to a ℓ -Volterra quadratic stochastic operator mapping two-dimensional simplex to itself. We show that if c d (c + d) = 0 , then (under some conditions on a , b) this ℓ -Volterra operator may have up to three or a countable set of fixed points; if c d (c + d) ≠ 0 , then the operator has up to three fixed points. Depending on the parameters, the fixed points may be attracting, repelling or saddle points. The limit behaviors of trajectories of the dynamical system are studied. It is shown that independently on values of parameters and on initial (starting) point, all trajectories converge. Thus, the operator (dynamical system) is regular. We give some biological interpretations of our results. [ABSTRACT FROM AUTHOR]

We consider algebras with basis numerated by elements of a group G. We fix a function f from G × G to a ground field and give a multiplication of the algebra which depends on f. We study the basic properties of such algebras. In particular, we find a condition on f under which the corresponding algebra is a Leibniz algebra. Moreover, for a given subgroup of G we define a -periodic algebra, which corresponds to a -periodic function f, we establish a criterion for the right nilpotency of a -periodic algebra. In addition, for G=ℤ we describe all 2ℤ- and 3ℤ-periodic algebras. Some properties of nℤ-periodic algebras are obtained. [ABSTRACT FROM AUTHOR]

We consider models with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order k ≥ 1. We show that periodic Gibbs measures are either translation-invariant or periodic with period two. We describe two-periodic Gibbs measures of the model. For k = 1 we show that there is no any periodic Gibbs measure. In case k ≥ 2 we get a sufficient condition on Hamiltonian of the model with uncountable set of spin values under which the model has no periodic Gibbs measure. We construct several models which have at least two periodic Gibbs measures. [ABSTRACT FROM AUTHOR]

This paper is devoted to the description of structure of evolution algebras of "chicken" population (EACP). Such an algebra is determined by a rectangular matrix of structural constants. Using the Jordan form of the matrix of structural constants we obtain a simple description of complex EACP. We give the classification of three-dimensional complex EACP. Moreover, some (n + 1)-dimensional EACP are described. [ABSTRACT FROM AUTHOR]

The contour argument was introduced by Peierls for two dimensional Ising model. Peierls benefited from the particular symmetries of the Ising model. For non-symmetric models the argument was developed by Pirogov and Sinai. It is very general and rather difficult. Intuitively clear that the Peierls argument does work for any symmetric model. But contours defined in Pirogov–Sinai theory do not work if one wants to use Peierls argument for more general symmetric models. We give a new definition of contour which allows relatively easier proof to the main result of the Pirogov–Sinai theory for symmetric models. Namely, our contours allow us to apply the classical Peierls argument (with contour removal operation). [ABSTRACT FROM AUTHOR]

We consider a nearest-neighbor solid-on-solid (SOS) model, with several spin values 0, 1,..., m, m ≥ 2, and zero external field, on a Cayley tree of order k (with k + 1 neighbors). The SOS model can be treated as a natural generalization of the Ising model (obtained for m = 1). We mainly assume that m = 2 (three spin values) and study translation-invariant (TI) and "splitting" (S) Gibbs measures (GMs). (Splitting GMs have a particular Markov-type property specific for a tree.) Furthermore, we focus on symmetric TISGMs, with respect to a "mirror" reflection of the spins. [For the Ising model (where m = 1), such measures are reduced to the "disordered" phase obtained for free boundary conditions, see Refs. 9, 10.] For m = 2, in the antiferromagnetic (AFM) case, a symmetric TISGM (and even a general TISGM) is unique for all temperatures. In the ferromagnetic (FM) case, for m = 2, the number of symmetric TISGMs and (and the number of general TISGMs) varies with the temperature: this gives an interesting example of phase transition. Here we identify a critical inverse temperature, $\beta^1_{\rm c} (=T_{\rm c}^{\rm STISG})\in (0, \infty)$ such that $\forall\ 0\le \beta\le \beta^1_{\rm c}$, there exists a unique symmetric TISGM μ* and $\forall\ \beta > \beta^1_{\rm c}$ there are exactly three symmetric TISGMs: $\mu^*_+$ (a "bottom" symmetric TISGM), $\mu^*_{\rm m}$ (a "middle" symmetric TISGM) and $\mu^*_-$ (a "top" symmetric TISGM). For $\beta > \beta^1_{\rm c}$ we also construct a continuum of distinct, symmertric SGMs which are non-TI. Our second result gives complete description of the set of periodic Gibbs measures for the SOS model on a Cayley tree. A complete description of periodic GMs means a characterisation of such measures with respect to any given normal subgroup of finite index in the representation group of the tree. We show that (i) for an FM SOS model, for any normal subgroup of finite index, each periodic SGM is in fact TI. Further, (ii) for an AFM SOS model, for any normal subgroup of finite index, each periodic SGM is either TI or has period two (i.e. is a chess-board SGM). [ABSTRACT FROM AUTHOR]