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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]

In this paper we consider function , (where , , ) on three fields: the set of real, -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 . 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 ; for the parameters from second class any trajectory (under ) 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 . For the -adic variable we give a review of known results about dynamical systems of function . 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. [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]

A chain of evolution algebras (CEA) is an uncountable family (depending on time) of evolution algebras on the field of real numbers. The matrix of structural constants of a CEA satisfies the Chapman-Kolmogorov equation. In this paper, we consider three CEAs of three-dimensional real evolution algebras. These CEAs depend on several (non-zero) functions defined on the set of time. For each chain we give a full classification (up to isomorphism) of the algebras depending on the time-parameter. We find concrete functions ensuring that the corresponding CEA contains all possible three-dimensional evolution algebras. [ABSTRACT FROM AUTHOR]

In this paper we investigate generalized Gibbs measure (GGM) for -adic Hard-Core (HC) model with a countable set of spin values on a Cayley tree of order . This model is defined by -adic parameters , . We analyze -adic functional equation which provides the consistency condition for the finite-dimensional generalized Gibbs distributions. Each solutions of the functional equation defines a GGM by -adic version of Kolmogorov's theorem. We define -adic Gibbs distributions as limit of the consistent family of finite-dimensional generalized Gibbs distributions and show that, for our -adic HC model on a Cayley tree, such a Gibbs distribution does not exist. Under some conditions on parameters , and we find the number of translation-invariant and two-periodic GGMs for the -adic HC model on the Cayley tree of order two. [ABSTRACT FROM AUTHOR]