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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]

In this paper, we study a two-state Hard-Core (HC) model with activity λ > 0 on a Cayley tree of order k ≥ 2. It is known that there are λcr, λ cr 0 , and λ cr ′ such that • for λ ≤ λcr this model has a unique Gibbs measure μ*, which is translation invariant. The measure μ* is extreme for λ < λ cr 0 and not extreme for λ > λ cr ′ ; • for λ > λcr there exist exactly three 2-periodic Gibbs measures, one of which is μ*, the other two are not translation invariant and are always extreme. The extremity of these periodic measures was proved using the maximality and minimality of the corresponding solutions of some equation, which ensures the consistency of these measures. In this paper, we give a brief overview of the known Gibbs measures for the HC-model and an alternative proof of the extremity of 2-periodic measures for k = 2, 3. Our proof is based on the tree reconstruction method. [ABSTRACT FROM AUTHOR]

Methyl 2-O-benzoyl-3,5-di-O-benzyl-α-L-talofuranoside was synthesized from methyl 3,5-di-O-benzyl-1,2-O-isopropylidene-α-L-talofuranoside which was successively treated with 20% hydrochloric acid in aqueous methanol and benzoyl chloride in anhydrous pyridine. Prospects of using methyl 2-O-benzoyl-3,5-di-O-benzyl-α-L-talofuranoside for the synthesis of new modified nucleosides were demonstrated. [ABSTRACT FROM AUTHOR]

We consider a two-parametric quadratic stochastic operator (QSO) mapping 3-dimensional simplex to itself. To date, for very few QSOs complete description of trajectories (limit points) is known. For our two-parametric QSO we completely describe the set of all limit points of trajectories. Namely, we find unique fixed point and the set of all (a continuum set) 2-periodic points and show that each such a point is non-hyperbolic. Moreover, for any initial point (taken from the 3-dimensional simplex) we find an invariant set containing the initial point and a unique 2-periodic orbit of the operator, such that the trajectory of initial point converges to this 2-periodic orbit. [ABSTRACT FROM AUTHOR]

A series of S-methylated derivatives of 2-thiouracil were prepared and then phosphonylated with diethyl chlorophosphate. The reaction conditions were optimized, the structure of the obtained compounds was determined by 1Н, 13С, 31Р NMR spectroscopy methods. [ABSTRACT FROM AUTHOR]

In this paper, we study the HC-model with a countable set Z of spin values on a Cayley tree of order k ≥ 2 . This model is defined by a countable set of parameters (that is, the activity function λ i > 0 , i ∈ Z ). A functional equation is obtained that provides the consistency condition for finite-dimensional Gibbs distributions. Analyzing this equation, the following results are obtained: Let Λ = ∑ i λ i . For Λ = + ∞ there is no translation-invariant Gibbs measure (TIGM) and no two-periodic Gibbs measure (TPGM); For Λ < + ∞ , the uniqueness of TIGM is proved; Let Λ cr (k) = k k (k - 1) k + 1 . If 0 < Λ ≤ Λ cr , then there is exactly one TPGM that is TIGM; For Λ > Λ cr , there are exactly three TPGMs, one of which is TIGM. [ABSTRACT FROM AUTHOR]

We describe fixed points of an infinite-dimensional nonlinear operator related to a hard-core (HC) model with a countable set of spin values on a Cayley tree. This operator is defined by a countable set of parameters , , . We find a sufficient condition on these parameters under which the operator has a unique fixed point. When this condition is not satisfied, we show that the operator may have up to five fixed points. We also prove that every fixed point generates a normalizable boundary law and therefore defines a Gibbs measure for the given HC model. [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]

For the solid-on-solid model with spin values from the set of all integers on a Cayley tree we give gradient Gibbs measures (GGMs). Such a measure corresponds to a boundary law (which is an infinite-dimensional vector-valued function defined on vertices of the Cayley tree) satisfying an infinite system of functional equations. We give several concrete GGMs of boundary laws which are independent from vertices of the Cayley tree and (as an infinite-dimensional vector) have periodic, (non-)mirror-symmetric coordinates. Namely, the particular class of height-periodic boundary laws of period q ≤ 5 is studied, where solutions are classified by their period and (two-)mirror-symmetry. [ABSTRACT FROM AUTHOR]

We study the Blume–Capel model with a countable set of spin values and a force of interaction between the nearest neighbors on a Cayley tree of order . The following results are obtained. Let , , be the temperature. For , there exist no translation invariant Gibbs measures or -periodic Gibbs measures. For , we prove the uniqueness of a translation-invariant Gibbs measure. Let and . If , then there exists exactly one -periodic Gibbs measure that is translation invariant. For , there exist exactly three -periodic Gibbs measures, one of which is a translation-invariant Gibbs measure. [ABSTRACT FROM AUTHOR]

We consider a population consisting of two species whose dynamics is determined by a quadratic stochastic operator with variable coefficients, which makes it a discontinuous operator at two points. This operator depends on three parameters. The set of these parameters is divided into seven subsets. For each subset of parameters, we find fixed points, periodic points, and the set of limit points of trajectories generated by the respective quadratic stochastic operators. [ABSTRACT FROM AUTHOR]

We study the Potts model in translation-invariant and periodic external fields on the Cayley tree of order . For the Potts model in a translation-invariant external field for , the nonuniqueness of the translation-invariant and periodic Gibbs measure is shown. It is proved that for the Potts model in an external field that is not translation-invariant, translation-invariant Gibbs measures do not exist on the Cayley tree of order . Periodic Gibbs measures are also studied for the Potts model in a periodic external field. We prove that under certain conditions, the number of such measures can be at least three. [ABSTRACT FROM AUTHOR]

There is one-to-one correspondence between quadratic operators (mapping R m to itself) and cubic matrices. It is known that any quadratic operator corresponding to a stochastic (in a fixed sense) cubic matrix preserves the standard simplex. In this paper we find conditions on the (non-stochastic) cubic matrix ensuring that corresponding quadratic operator preserves simplex. Moreover, we construct several quadratic non-stochastic operators which generate chaotic dynamical systems on the simplex. These chaotic behaviors are splitted meaning that the simplex is partitioned into uncountably many invariant (with respect to quadratic operator) subsets and the restriction of the dynamical system on each invariant set is chaos in the sense of Devaney. [ABSTRACT FROM AUTHOR]

In this paper we study -adic dynamical systems generated by the function in the set of complex -adic numbers. We find an explicit formula for the -fold composition of for any . Using this formula we give fixed points, periodic points, basin of attraction and Siegel disk of each fixed (periodic) point depending on parameters and . [ABSTRACT FROM AUTHOR]

We define a DNA molecule as a sequence of the numbers 1 and 2 and embed it on a path of a Cayley tree such that each vertex of the Cayley tree belongs to only one DNA and each DNA has its own countable set of neighboring DNA. The Hamiltonian of this set of DNA is a model with two spin values regarded as DNA base pairs. We describe translation-invariant Gibbs measures (TIGMs) of the model on the Cayley tree of order two and use them to study the thermodynamic properties of the model of DNA. We show that there is a critical temperature such that if the temperature , then there is a unique TIGM, and if , then there are three TIGMs. Each TIGM gives a phase of the set of DNA. In the cases of very high and very low temperatures, we find stationary distributions and typical configurations of the model. [ABSTRACT FROM AUTHOR]

In this paper, we consider a potential on general infinite trees with q spin values and nearest-neighbor p-adic interactions given by a stochastic matrix. We show the uniqueness of the associated Markov chain (splitting Gibbs measures) under some sufficient conditions on the stochastic matrix. Moreover, we find a family of stochastic matrices for which there are at least two p-adic Markov chains on an infinite tree (in particular, on a Cayley tree). When the p-adic norm of q is greater (resp. less) than the norm of any element of the stochastic matrix then it is proved that the p-adic Markov chain is bounded (resp. is not bounded). Our method uses a classical boundary law argument carefully adapted from the real case to the p-adic case, by a systematic use of some nice peculiarities of the ultrametric (p-adic) norms. [ABSTRACT FROM AUTHOR]

We consider a family of (2, 2)-rational functions given on the set of complex p-adic field C p . Each such function f has two distinct fixed points x 1 = x 1 (f) , x 2 = x 2 (f) . We study p-adic dynamical systems generated by the (2, 2)-rational functions. We prove that x 1 is always an indifferent fixed point for f, i.e., x 1 is a center of some Siegel disk S I (x 1) . Depending on the parameters of the function f, the type of the fixed point x 2 may be any possibility: indifferent, attractive, repelling. We find the Siegel disk or the basin of attraction of the fixed point x 2 , when x 2 is indifferent or attractive, respectively. When x 2 is repelling we find an open ball any point of which is repelled from x 2 . Moreover, we study relations between the sets S I (x 1) and S I (x 2) when x 2 is indifferent. For each (2, 2)-rational function on C p there are two points x ^ 1 = x ^ 1 (f) , x ^ 2 = x ^ 2 (f) ∈ C p which are zeros of its denominator. We give explicit formulas of radii of spheres (with the center at the fixed point x 1 ) containing some points such that the trajectories (under actions of f) of the points after a finite step come to x ^ 1 or x ^ 2 . We study periodic orbits of the dynamical system and find an invariant set, which contains all periodic orbits. Moreover, we study ergodicity properties of the dynamical system on each invariant sphere. Under some conditions we show that the system is ergodic if and only if p = 2 . [ABSTRACT FROM AUTHOR]

We consider a population with two equal dominated species, dynamics of which is defined by an one-dimensional piecewise-continuous, two parametric function. It is shown that for any non-zero parameters this function has two fixed points and several periodic points. We prove that all periodic (in particular fixed) points are repelling, and find an invariant set which asymptotically involves the trajectories of any initial point except fixed and periodic ones. We showed that the orbits are unstable and chaotic because Lyapunov exponent is non-negative. The limit sets analyzed by bifurcation diagrams. We give biological interpretations of our results. [ABSTRACT FROM AUTHOR]

We describe the set of all -rational functions given on the set of complex -adic field and having a unique fixed point. We study -adic dynamical systems generated by such -rational functions and show that the fixed point is indifferent and therefore the convergence of the trajectories is not the typical case for the dynamical systems. We obtain Siegel disks of these dynamical systems. Moreover an upper bound for the set of limit points of each trajectory is given. For each -rational function on there is a point which is zero in its denominator. We give explicit formulas of radii of spheres (with the center at the fixed point) containing some points that the trajectories (under actions of ) of the points after a finite step come to . For a class of -rational functions defined on the set of -adic numbers we study ergodicity properties of the corresponding dynamical systems. We show that if then the -adic dynamical system reduced on each invariant sphere is not ergodic with respect to Haar measure. For , under some conditions we prove non ergodicity and show that there exists a sphere on which the dynamical system is ergodic. Finally, we give a characterization of periodic orbits and some uniformly local properties of the rational functions. [ABSTRACT FROM AUTHOR]

We consider a four-parameter family of non-Volterra operators defined on the two-dimensional simplex and show that, with one exception, each such operator has a unique fixed point. Depending on the parameters, we establish the type of this fixed point. We study the set of ω -limiting points for each trajectory and show that this set can be a single point or can contain a 2-periodic trajectory. [ABSTRACT FROM AUTHOR]

Recently, continuous-time dynamical systems of mosquito populations have been studied. In this paper, we consider a discrete-time dynamical system, generated by an evolution quadratic operator of a mosquito population, and show that this system has two fixed points, which become saddle points under some conditions on the parameters of the system. We construct an evolution algebra, taking its matrix of structural constants equal to the Jacobian of the quadratic operator at a fixed point. Idempotent and absolute nilpotent elements, simplicity properties, and some limit points of the evolution operator corresponding to the evolution algebra are studied. We give some biological interpretations of our results. [ABSTRACT FROM AUTHOR]

We show that any (1, 2)-rational function with a unique fixed point is topologically conjugate to a (2, 2)-rational function or to the function f(x) = ax/x2 + a. The case (2, 2) was studied in our previous paper, here we study the dynamical systems generated by the function f on the set of complex p-adic field ℂp. We show that the unique fixed point is indifferent and therefore the convergence of the trajectories is not the typical case for the dynamical systems. We construct the corresponding Siegel disk of these dynamical systems. We determine a sufficiently small set containing the set of limit points. It is given all possible invariant spheres.We show that the p-adic dynamical system reduced on each invariant sphere is not ergodic with respect to Haar measure on the set of p-adic numbers ℚp.Moreover some periodic orbits of the system are investigated. [ABSTRACT FROM AUTHOR]

We study translation-invariant Gibbs measures for the ferromagnetic Potts model with q states on the Cayley tree of order k and generalize some earlier results. We consider the question of the extremality of the known translation-invariant Gibbs measures for the Potts model with three states on the Cayley tree of order k = 3. [ABSTRACT FROM AUTHOR]

We define a DNA as a sequence of $$\pm 1$$ 's and embed it on a path of Cayley tree. Using group representation of the Cayley tree, we give a hierarchy of a countable set of DNAs each of which 'lives' on the same Cayley tree. This hierarchy has property that each vertex of the Cayley tree belongs only to one of DNA. Then we give a model (energy, Hamiltonian) of this set of DNAs by an analogue of Ising model with three spin values (considered as DNA base pairs) on a set of admissible configurations. To study thermodynamic properties of the model of DNAs we describe corresponding translation invariant Gibbs measures (TIGM) of the model on the Cayley tree of order two. We show that there is a critical temperature $$T_{\mathrm{c}}$$ such that (i) if temperature $$T>T_{\mathrm{c}}$$ then there exists unique TIGM; (ii) if $$T=T_{\mathrm{c}}$$ then there are two TIGMs; (iii) if $$T

Adapting some methods for real-valued Gibbs measures on Cayley trees to the p-adic case, we construct several p-adic distributions on the set ℤ of p-adic integers. In addition, we give conditions under which these p-adic distributions become p-adic measures (i.e., bounded distributions). [ABSTRACT FROM AUTHOR]

We consider models with four competing interactions ( external field, nearest neighbor, second neighbor, and three neighbors) and an uncountable set [0, 1] of spin values on the Cayley tree of order two. We reduce the problem of describing the splitting Gibbs measures of the model to the problem of analyzing solutions of a nonlinear integral equation and study some particular cases for Ising and Potts models. We also show that periodic Gibbs measures for the given models either are translation invariant or have the period two. We present examples where periodic Gibbs measures with the period two are not unique. [ABSTRACT FROM AUTHOR]

We consider translation-invariant splitting Gibbs measures (TISGMs) for the q-state Potts model on a Cayley tree of order two. Recently a full description of the TISGMs was obtained, and it was shown in particular that at sufficiently low temperatures their number is $$2^{q}-1$$ . In this paper for each TISGM $$\mu $$ we explicitly give the set of boundary conditions such that limiting Gibbs measures with respect to these boundary conditions coincide with $$\mu $$ . [ABSTRACT FROM AUTHOR]

For the Potts model on the Cayley tree, we obtain some explicit formulas for the free energies and entropies in the case of vector-valued boundary conditions. These formulas include translation-invariant, periodic, and Dobrushin-like boundary conditions and also those corresponding to weakly periodic Gibbs measures. [ABSTRACT FROM AUTHOR]

We study translation-invariant splitting Gibbs measures (TISGMs, tree-indexed Markov chains) for the fertile three-state hard-core models with activity $$\lambda >0$$ on the Cayley tree of order $$k\ge 1$$ . There are four such models: wrench, wand, hinge, and pipe. These models arise as simple examples of loss networks with nearest-neighbor exclusion. It is known that (i) for the wrench and pipe cases $$\forall \lambda >0$$ and $$k\ge 1$$ , there exists a unique TISGM; (ii) for hinge (resp. wand) case at $$k=2$$ if $$\lambda <\lambda _\mathrm{cr}=9/4$$ (resp. $$\lambda <\lambda _\mathrm{cr}=1$$ ), there exists a unique TISGM, and for $$\lambda > 9/4$$ (resp. $$\lambda >1$$ ), there exist three TISGMs. In this paper we generalize the result (ii) for any $$k\ge 2$$ , i.e., for hinge and wand cases we find the exact critical value $$\lambda _\mathrm{cr}(k)$$ with properties mentioned in (ii). Moreover, we find some regions for the $$\lambda $$ parameter ensuring that a given TISGM is extreme or non-extreme in the set of all Gibbs measures. For the Cayley tree of order two, we give explicit formulae and some numerical values. [ABSTRACT FROM AUTHOR]

We consider the solid-on-solid model, with spin values $$0,1,2$$ , on the Cayley tree of order two (binary tree). We treat both ferromagnetic and antiferromagnetic coupling, with interactions which are proportional to the absolute value of the spin differences. We present a classification of all translation-invariant phases (splitting Gibbs measures) of the model. These measures are labeled by solutions to a nonlinear vector-valued functional equation. We show uniqueness in the case of antiferromagnetic interactions, and existence of up to seven phases in the case of ferromagnetic interactions, where the number of phases depends on the interaction strength. Next we investigate whether these states are extremal or non-extremal in the set of all Gibbs measures, when the coupling strength is varied, whenever they exist. We show that two states are always extremal, two states are always non-extremal, while three of the seven states make transitions between extremality and non-extremality. We provide explicit bounds on those transition values, making use of algebraic properties of the models, and an adaptation of the method of Martinelli, Sinclair, Weitz. [ABSTRACT FROM AUTHOR]

For the $$q$$ -state Potts model on a Cayley tree of order $$k\ge 2$$ it is well-known that at sufficiently low temperatures there are at least $$q+1$$ translation-invariant Gibbs measures which are also tree-indexed Markov chains. Such measures are called translation-invariant splitting Gibbs measures (TISGMs). In this paper we find all TISGMs, and show in particular that at sufficiently low temperatures their number is $$2^{q}-1$$ . We prove that there are $$[q/2]$$ (where $$[a]$$ is the integer part of $$a$$ ) critical temperatures at which the number of TISGMs changes and give the exact number of TISGMs for each intermediate temperature. For the binary tree we give explicit formulae for the critical temperatures and the possible TISGMs. While we show that these measures are never convex combinations of each other, the question which of these measures are extremals in the set of all Gibbs measures will be treated in future work. [ABSTRACT FROM AUTHOR]

We find conditions on ideals of an algebra under which the algebra is dibaric. Dibaric algebras have not non-zero homomorphisms to the set of the real numbers. We introduce a concept of bq-homomorphism (which is given by two linear maps f, g of the algebra to the set of the real numbers) and show that an algebra is dibaric if and only if it admits a non-zero bq-homomorphism. Using the pair ( f, g) we define conservative algebras and establish criteria for a dibaric algebra to be conservative. Moreover, the notions of a Bernstein algebra and an algebra induced by a linear operator are introduced and relations between these algebras are studied. For dibaric algebras we describe a dibaric algebra homomorphism and study their properties by bq-homomorphisms of the dibaric algebras. We apply the results to the (dibaric) evolution algebra of a bisexual population. For this dibaric algebra we describe all possible bq-homomorphisms and find conditions under which the algebra of a bisexual population is induced by a linear operator. Moreover, some properties of dibaric algebra homomorphisms of such algebras are studied. [ABSTRACT FROM AUTHOR]

We consider the Ising model on a rooted Cayley tree of order two with nearest neighbor interactions and competing next nearest neighbor interactions restricted to spins belonging to the same branch of the tree. This model was studied by Vannimenus who found a new modulated phase, in addition to the paramagnetic, ferromagnetic, antiferromagnetic phases and a (+ + - -) periodic phase. Vannimenus's results are based on an analysis of the recurrence equations (relating the partition function of an n − generation tree to the partition function of its subsystems containing ( n −1) generations) and most results are obtained numerically. In this paper we analytically study the recurrence equations and obtain some exact results: critical temperatures and curves, number of phases, partition function. [ABSTRACT FROM AUTHOR]

We investigate the behavior of trajectories of a (3, 2)-rational p-adic dynamical system in the complex p-adic field ℂ, when there exists a unique fixed point x. We study this p-adic dynamical system by dynamics of real radiuses of balls (with the center at the fixed point x). We show that there exists a radius r depending on parameters of the rational function such that: when x is an attracting point then the trajectory of an inner point from the ball U( x) goes to x and each sphere with a radius > r (with the center at x) is invariant; When x is a repeller point then the trajectory of an inner point from a ball U( x) goes forward to the sphere S( x). Once the trajectory reaches the sphere, in the next step it either goes back to the interior of U( x) or stays in S( x) for some time and then goes back to the interior of the ball. As soon as the trajectory goes outside of U( x) it will stay (for all the rest of time) in the sphere (outside of U( x)) that it reached first. [ABSTRACT FROM AUTHOR]

We show that the nearest neighbor Ising model on the Cayley tree exhibits new temperature-driven phase transitions. These transitions occur at various inverse temperatures different from the critical one. They are characterised by a change in the number of Gibbs states as well as by a drastic change of the behavior of free energies at these new transition points. We also consider the model in the presence of an external field and compute the free energies of translation invariant and some alternating boundary conditions. [ABSTRACT FROM AUTHOR]

Recently in [2] a notion of chain of evolution algebras is introduced. This chain is a dynamical system the state of which at each given time is an evolution algebra. The sequence of matrices of the structural constants for this chain of evolution algebras satisfies the Chapman-Kolmogorov equation. In this paper we construct 25 distinct examples of chains of two-dimensional evolution algebras. For all of these 25 chains we study the behavior of the baric property, the behavior of the set of absolute nilpotent elements and dynamics of the set of idempotent elements depending on the time. [ABSTRACT FROM AUTHOR]

In this paper we consider a model with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order k ≥ 2. To study translation-invariant Gibbs measures of the model we drive an nonlinear functional equation. For k = 2 and 3 under some conditions on parameters of the model we prove non-uniqueness of translation-invariant Gibbs measures (i.e., there are phase transitions). [ABSTRACT FROM AUTHOR]

We study the Potts model on the Cayley tree. We demonstrate that for this model with a zero external field, periodic Gibbs measures on some invariant sets are translation invariant. Furthermore, we find the conditions under which the Potts model with a nonzero external field admits periodic Gibbs measures. [ABSTRACT FROM AUTHOR]

The notions of the Gibbs measure and of the Markov random field are known to coincide in the real case. But in the p-adic case, the class of p-adic Markov random fields is broader than that of p-adic Gibbs measures. We construct p-adic Markov random fields (on finite graphs) that are not p-adic Gibbs measures. We define a p-adic Markov random field on countable graphs and show that the set of such fields is a nonempty closed subspace in the set of all p-adic probability measures [ABSTRACT FROM AUTHOR]

We present, for the Ising model on the Cayley tree, some explicit formulae of the free energies (and entropies) according to boundary conditions (b.c.). They include translation-invariant, periodic, Dobrushin-like b.c., as well as those corresponding to (recently discovered) weakly periodic Gibbs states. The weakly periodic measures are defined through a partition of the Cayley tree that induces a 4-edge-coloring on that tree. We compute the density of each color. We use these densities for computations of free energies corresponding to a weakly periodic b.c. [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 $k\geqslant 1$. It is known that the 'splitting Gibbs measures' of the model can be described by solutions of a nonlinear integral equation. For arbitrary $k\geqslant 2$ we find a sufficient condition under which the integral equation has unique solution, hence under the condition the corresponding model has unique splitting Gibbs measure. [ABSTRACT FROM AUTHOR]

We describe all linear operators which maps n - 1-dimensional simplex of idempotent measures to itself. Such operators divided to two classes: the first class contains all n × n-matrices with non-negative entries which has at least one zero-row; the second class contains all n × n-matrices with non-negative entries which in each row and in each column has exactly one non-zero entry. Thesematrices play a role of the stochastic matrices in case of idempotent measures. For both classes of linear maps we find fixed points. We also study the dynamical systems generated by the linear maps of the set of idempotent measures. [ABSTRACT FROM AUTHOR]

We study the hard-core model on the Cayley tree. We show that this model admits only periodic Gibbs measures with the period two. We find sufficient conditions for all weakly periodic Gibbs measures to be translation invariant. [ABSTRACT FROM AUTHOR]

In this paper we construct several models with nearest-neighbor interactions and with the set [0,1] of spin values, on a Cayley tree of order k≥2. We prove that each of the constructed model has at least two translational-invariant Gibbs measures. [ABSTRACT FROM AUTHOR]

We completely describe the Siegel discs and attractors for the p-adic dynamical system f(x) = x + ax on the space of complex p-adic numbers. [ABSTRACT FROM AUTHOR]

We introduce the notion of Volterra quadratic stochastic operators of a two-sex population. The description of the fixed points of Volterra quadratic stochastic operators of a two-sex population is reduced to the description of the fixed points of Volterra-type operators. Several Lyapunov functions are constructed for the Volterra quadratic stochastic operators of a two-sex population. By using these functions, we obtain an upper bound for the ω-limit set of trajectories. It is shown that the set of all Volterra quadratic stochastic operators of a two-sex population is a convex compact set, and the extreme points of this set are found. Volterra quadratic stochastic operators of a two-sex population that have a 2-periodic orbit (trajectory) are constructed. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate a class of separable quadratic stochastic operators. Each separable quadratic stochastic operator (SQSO) depends on two quadratic matrices A and B, which have some relations. In this paper we proved that for each skew symmetric matrix A the corresponding SQSO is a linear operator. We also proved that non linear Volterra QSOs are not SQSOs. For a fixed matrix A we also discussed some properties of the set of all the corresponding matrices B of SQSOs. [ABSTRACT FROM AUTHOR]

In this article we study algebraic structures of function spaces defined by graphs and state spaces equipped with Gibbs measures by associating evolution algebras. We give a constructive description of associating evolution algebras to the function spaces (cell spaces) defined by graphs and state spaces and Gibbs measure µ. For finite graphs we find some evolution subalgebras and other useful properties of the algebras. We obtain a structure theorem for evolution algebras when graphs are finite and connected. We prove that for a fixed finite graph, the function spaces has a unique algebraic structure since all evolution algebras are isomorphic to each other for whichever Gibbs measures assigned. When graphs are infinite graphs then our construction allows a natural introduction of thermodynamics in studying of several systems of biology, physics and mathematics by theory of evolution algebras. [ABSTRACT FROM AUTHOR]

We propose a model on the Cayley tree and prove that a uncountable set of Ĝ-periodic Gibbs measures exists for this model, in contrast to models studied previously. [ABSTRACT FROM AUTHOR]

In this note we consider several kind of partition functions of one-dimensional models with nearest - neighbor interactions I, n ∈ Z and spin values ±1. We derive systems of recursive equations for each kind of such functions. These systems depend on parameters I, n ∈ Z. Under some conditions on the parameters we describe solutions of the systems of recursive equations. [ABSTRACT FROM AUTHOR]