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The Susceptible-Exposed-Infectious-Recovered (SEIR) model is applied in several countries to ascertain the spread of the coronavirus disease 2019 (COVID-19). We consider discrete-time SEIR epidemic model in a closed system which does not account for births or deaths, total population size under consideration is constant. This dynamical system generated by a non-linear evolution operator depending on four parameters. Under some conditions on parameters we reduce the evolution operator to a quadratic stochastic operator (QSO) which maps 3-dimensional simplex to itself. We show that the QSO has uncountable set of fixed points (all laying on the boundary of the simplex). It is shown that all trajectories of the dynamical system (generated by the QSO) of the SEIR model are convergent (i.e. the QSO is regular). Moreover, we discuss the efficiency of the model for Uzbekistan., Comment: 13 pages, 4 figures

In this paper, we focus on studying non-probability Gibbs measures for a Hard Core (HC) model on a Cayley tree of order $k\geq 2$, where the set of integers $\mathbb Z$ is the set of spin values. It is well-known that each Gibbs measure, whether it be a gradient or non-probability measure, of this model corresponds to a boundary law. A boundary law can be thought of as an infinite-dimensional vector function defined at the vertices of the Cayley tree, which satisfies a nonlinear functional equation. Furthermore, every normalisable boundary law corresponds to a Gibbs measure. However, a non-normalisable boundary law can define gradient or non-probability Gibbs measures. In this paper, we investigate the conditions for uniqueness and non-uniqueness of translation-invariant and periodic non-probability Gibbs measures for the HC-model on a Cayley tree of any order $k\geq 2$., 19 pages, 2 figures

In this paper we consider a population consisting of two species, dynamics of which is defined by a quadratic stochastic operator with variable coefficients, making it discontinuous operator at two points. This operator depends on three parameters. It is shown that under suitable conditions on the parameters this operator may have fixed points, convergence of trajectories and there may exist periodic points., 10 pages, 1 figure

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., 32 pages

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., 10 pages, 5 figures

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., 13 pages

For the SOS (solid-on-solid) 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 non-linear functional equation. Recently some translation-invariant and height-periodic (non-normalisable) solutions to the equation are found. Here our aim is to find non-height-periodic and non-normalisable boundary laws for the SOS model. By such a solution one can construct a non-probability Gibbs measure. We find explicitly several non-normalisable boundary laws. Moreover, we reduce the problem to solving of a non-linear, second-order difference equation. We give analytic and numerical analysis of the difference equation., 17 pages, 13 figures

In the present paper we consider a flow, which is a time depending family of two-dimensional algebras. For any two algebras of this flow, formed at different times, we find conditions under which they are isomorphic. We show that the flow contains uncountably pairwise non-isomorphic algebras and there is one commutative algebra among them. Moreover, we compare our algebras with a known classification of 2-dimensional real algebras., Comment: 15 pages and 1 figure

In this short note we study a dynamical system generated by a two-parametric quadratic operator mapping 3-dimensional simplex to itself. This is an evolution operator of the frequencies of gametes in a two-locus system. We find the set of all (a continuum set) fixed points and show that each fixed point is non-hyperbolic. We completely describe the set of all limit points of the dynamical system. Namely, for any initial point (taken from the 3-dimensional simplex) we find an invariant set containing the initial point and a unique fixed point of the operator, such that the trajectory of the initial point converges to this fixed point., Comment: 5 pages

In this paper we study $p$-adic dynamical systems generated by the function $f(x)={a\over x^2}$ in the set of complex $p$-adic numbers. We find an explicit formula for the $n$-fold composition of $f$ for any $n\geq 1$. Using this formula we give fixed points, periodic points, basin of attraction and Siegel disk of each fixed (periodic) point depending on parameters $p$ and $a$., Comment: 11 pages

We study a discrete-time dynamical system of wild mosquito population with parameters: $\beta$ - the birth rate of adults; $\alpha$ - maximum emergence rate; $\mu>0$ - the death rate of adults; $\gamma$ - Allee effects. We prove that if $\gamma\geq\frac{\alpha(\beta-\mu)}{\mu^2}$ then the mosquito population dies and if $\gamma, Comment: 13 pages, 2 figures

We study the discrete-time dynamical systems associated to a stage-structured wild and sterile mosquito population. We describe all fixed points of the evolution operator (which depends on five parameters) of mosquito population and show that depending on the parameters this operator may have unique, two and infinitely many fixed points. Under some general conditions on parameters we determine type of each fixed point and give the limit points of the dynamical system. Moreover, for a special case of parameters we give full analysis of corresponding dynamical system. We give some biological interpretations of our results., Comment: 13 pages, 8 figures

In this paper we study dynamical systems generated by an evolution operator of a dioecious population. This evolution operator is a six-parametric, non-linear operator mapping $[0,1]^2$ to itself. We find all fixed points and under some conditions on parameters we give limit points of trajectories constructed by iterations of the evolution operator., Comment: 7 pages, 2 figures

We consider the Leslie's prey-predator model with discrete-time. This model is given by a non-linear 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., Comment: 18 pages, 34 figures

We construct quadratic stochastic processes (QSP) (also known as Markov processes of cubic matrices) in continuous and discrete times. These are dynamical systems given by (a fixed type, called $��$) stochastic cubic matrices satisfying an analogue of Kolmogorov-Chapman equation (KCE) with respect to a fixed multiplications (called $��$) between cubic matrices. The existence of a stochastic (at each time) solution to the KCE provides the existence of a QSP called a QSP of type $(��| ��)$. In this paper, our aim is to construct and study trajectories of QSPs for specially chosen notions of stochastic cubic matrices and a wide class of multiplications of such matrices (known as Maksimov's multiplications)., 14 pages. arXiv admin note: text overlap with arXiv:1706.07616

It is known that any multiplication of a finite dimensional algebra is determined by a matrix of structural constants. In general, this is a cubic matrix. Difficulty of investigation of an algebra depends on the cubic matrix. Such a cubic matrix defines a quadratic mapping called an evolution operator. In the case of evolution algebras, the cubic matrix consists many zeros allowing to reduce it to a square matrix. In this paper for any finite dimensional algebra we construct a family of evolution algebras corresponding to Jacobian of the evolution operator at a point of the algebra. We obtain some results answering how properties of an algebra depends on the properties of the corresponding family of evolution algebras. Moreover, we consider evolution algebras corresponding to 2 and 3-dimensional nilpotent Leibniz algebras. We prove that such evolution algebras are nilpotent too. Also we classify such evolution algebras., 14 pages

Motivating by the China's five element philosophy (CFEP) we construct a permuted Volterra quadratic stochastic operator acting on the four dimensional simplex. This operator (depending on 10 parameters) is considered as an evolution operator for CFEP. We study the discrete time dynamical system generated by this operator. Mainly our results related to a symmetric operator (depending on one parameter). We show that this operator has a unique fixed point, which is repeller. Moreover, in the case of non-zero parameter, it has two 5-periodic orbits. We divide the simplex to four subsets: the first set consists a single point (the fixed point); the second (resp. third) set is the set of initial points trajectories of which converge to the first (resp. second) 5-periodic orbit; the fourth subset is the set of initial points trajectories of which do not converge and their sets of limit points are infinite and lie on the boundary of the simplex. We give interpretations of our results to CFEP., 16 pages

In the present paper, we give a certain condition on subgroups of the group representation of the Cayley tree such that an invariance property holds. Except for the given condition, we show that the invariance property does not hold.

We consider a four-parametric $(a, b, \alpha, \beta)$ family of Volterra quadratic stochastic operators for a bisexual population (i.e., each organism of the population must belong either to the female sex or the male sex). We show that independently on parameters each such operator has at least two fixed points. Moreover, under some conditions on parameters the operator has infinitely many (continuum) fixed points. Choosing parameters, numerically we show that a fixed point may be any type: attracting, repelling, saddle and non-hyperbolic. We separate five subfamilies of quadratic operators and show that each operator of these subfamilies is regular, i.e. any trajectory constructed by the operator converges to a fixed point.

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 $f(x)={ax\over x^2+a}$. 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 $\mathbb C_p$. 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 $Q_p$. Moreover some periodic orbits of the system are investigated., 11 pages. arXiv admin note: text overlap with arXiv:1703.09001

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 $\ell$-Volterra quadratic stochastic operator mapping two-dimensional simplex to itself. We show that if $cd(c+d)=0$ then (under some conditions on $a,b$) this $\ell$-Volterra operator may have up to three or a countable set of fixed points; if $cd(c+d)\ne 0$ 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., 21 pages, 5 figures

Recently, continuous-time dynamical systems, based on systems of ordinary differential equations, for mosquito populations are studied. In this paper we consider discrete-time dynamical system generated by an evolution quadratic operator of mosquito population and show that this system has two fixed points, which are 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., Comment: 19 pages

Recently, by A. Elduque and A. Labra a new technique and a type of an evolution algebra are introduced. Several nilpotent evolution algebras defined in terms of bilinear forms and symmetric endomorphisms are constructed. The technique then used for the classification of the nilpotent evolution algebras up to dimension five. In this paper we develop this technique for high dimensional evolution algebras. We construct nilpotent evolution algebras of any type. Moreover, we show that, except the cases considered by Elduque and Labra, this construction of nilpotent evolution algebras does not give all possible nilpotent evolution algebras., 13 pages, 1 figure

We consider Markov processes of cubic stochastic (in a fixed sense) matrices which are also called quadratic stochastic process (QSPs). A QSP is a particular case of a continuous-time dynamical system whose states are stochastic cubic matrices satisfying an analogue of the Kolmogorov-Chapman equation (KCE). Since there are several kinds of multiplications between cubic matrices we have to fix first a multiplication and then consider the KCE with respect to the fixed multiplication. Moreover, the notion of stochastic cubic matrix also varies depending on the real models of application. The existence of a stochastic (at each time) solution to the KCE provides the existence of a QSP. In this paper, our aim is to construct QSPs for two specially chosen notions of stochastic cubic matrices and two multiplications of such matrices (known as Maksimov's multiplications). We construct a wide class of QSPs and give some time-dependent behavior of such processes. We give an example with applications to the Biology, constructing a QSP which describes the time behavior (dynamics) of a population with the possibility of twin births., 18 pages

We consider algebras of $m\times m\times m$-cubic matrices (with $m=1,2,\dots$). Since there are several kinds of multiplications of cubic matrices, one has to specify a multiplication first and then define an algebra of cubic matrices (ACM) with respect to this multiplication. We mainly use the associative multiplications introduced by Maksimov. Such a multiplication depends on an associative binary operation on the set of size $m$. We introduce a notion of equivalent operations and show that such operations generate isomorphic ACMs. It is shown that an ACM is not baric. An ACM is commutative iff $m=1$. We introduce a notion of accompanying algebra (which is $m^2$-dimensional) and show that there is a homomorphism from any ACM to the accompanying algebra. We describe (left and right) symmetric operations and give left and right zero divisors of the corresponding ACMs. Moreover several subalgebras and ideals of an ACM are constructed., 11 pages

Consider a bisexual population such that the set of females can be partitioned into finitely many different types indexed by $\{1,2,\dots,n\}$ and, similarly, that the male types are indexed by $\{1,2,\dots,\nu \}$. Recently an evolution algebra of bisexual population was introduced by identifying the coefficients of inheritance of a bisexual population as the structure constants of the algebra. In this paper we study constrained evolution algebra of bisexual population in which type "1" of females and males have preference. For such algebras sets of idempotent and absolute nilpotent elements are known. We consider two particular cases of this algebra, giving more constrains on the structural constants of the algebra. By the first our constrain we obtain an $n+\nu$- dimensional algebra with a matrix of structural constants containing only 0 and 1. In the second case we consider $n=\nu=2$ but with general constrains. In both cases we study dynamical systems generated by the quadratic evolution operators of corresponding constrained algebras. We find all fixed points, limit points and some 2-periodic points of the dynamical systems. Moreover we study several properties of the constrained algebras connecting them to the dynamical systems. We give some biological interpretation of our results., Comment: 27 pages

We investigate the dynamical system generated by the function $\lfloor��x\rfloor$ defined on $\mathbb R$ and with a parameter $��\in \mathbb R$. For each given $m\in \mathbb N$ we show that there exists a region of values of $��$, where the function has exactly $m$ fixed points (which are non-negative integers), also there is another region for $��$, where there are exactly $m+1$ fixed points (which are non-positive integers). Moreover the full set $\mathbb Z$ of integer numbers is the set of fixed points iff $��=1$. We show that depending on $��$ and on the initial point $x$ the limit of the forward orbit of the dynamical system may be one of the following possibilities: (i) a fixed point, (ii) a two-periodic orbit or (iii) $\pm\infty$., 8 pages

In this paper we consider discrete-time dynamical systems generated by gonosomal evolution operators of sex linked inheritance. Mainly we study dynamical systems of a hemophilia, which biologically is a group of hereditary genetic disorders that impair the body's ability to control blood clotting or coagulation, which is used to stop bleeding when a blood vessel is broken. We give an algebraic model of the biological system corresponding to the hemophilia. The evolution of such system is studied by a nonlinear (quadratic) gonosomal operator. In a general setting, this operator is considered as a mapping from $\mathbb R^n$, $n\geq 2$ to itself. In particular, for a gonosomal operator at $n=4$ we explicitly give all (two) fixed points. Then limit points of the trajectories of the corresponding dynamical system are studied. Moreover we consider a normalized version of the gonosomal operator. In the case $n=4$, for the normalized gonosomal operator we show uniqueness of fixed point and study limit points of the dynamical system., Comment: 15 pages

We consider an evolution algebra which corresponds to a bisexual population with a set of females partitioned into finitely many different types and the males having only one type. For such algebras in terms of its structure constants we calculate right and plenary periods of generator elements. Some results on subalgebras of EACP and ideals on low-dimensional EACP are obtained., Comment: 10 pages

We consider an evolution algebra which corresponds to a bisexual population with a set of females partitioned into finitely many different types and the males having only one type. We study basic properties of the algebra. This algebra is commutative (and hence flexible), not associative and not necessarily power associative, in general. Moreover it is not unital. A condition is found on the structural constants of the algebra under which the algebra is associative, alternative, power associative, nilpotent, satisfies Jacobi and Jordan identities. In a general case, we describe the full set of idempotent elements and the full set of absolute nilpotent elements. The set of all operators of left (right) multiplications is described. Under some conditions on the structural constants it is proved that the corresponding algebra is centroidal. Moreover the classification of 2-dimensional and some 3-dimensional algebras are obtained., 15 pages

We consider models with nearest-neighbor interactions and with the set $[0,1]$ of spin values, on a Cayley tree of order $k\geq 1$. We study periodic Gibbs measures of the model with period two. For $k=1$ we show that there is no any periodic Gibbs measure. In case $k\geq 2$ we get a sufficient condition on Hamiltonian of the model with uncountable set of spin values under which the model have not any periodic Gibbs measure with period two. We construct several models which have at least two periodic Gibbs measures., Comment: 24 pages. arXiv admin note: substantial text overlap with arXiv:1202.2542, arXiv:1202.1722, arXiv:1210.7311

We investigate the behavior of trajectories of a $(3,2)$-rational $p$-adic dynamical system in the complex $p$-adic filed ${\mathbb C}_p$, when there exists a unique fixed point $x_0$. We study this $p$-adic dynamical system by dynamics of real radiuses of balls (with the center at the fixed point $x_0$). We show that there exists a radius $r$ depending on parameters of the rational function such that: when $x_0$ is an attracting point then the trajectory of an inner point from the ball $U_r(x_0)$ goes to $x_0$ and each sphere with a radius $>r$ (with the center at $x_0$) is invariant; When $x_0$ is a repeller point then the trajectory of an inner point from a ball $U_r(x_0)$ goes forward to the sphere $S_r(x_0)$. Once the trajectory reaches the sphere, in the next step it either goes back to the interior of $U_r(x_0)$ or stays in $S_r(x_0)$ for some time and then goes back to the interior of the ball. As soon as the trajectory goes outside of $U_r(x_0)$ it will stay (for all the rest of time) in the sphere (outside of $U_r(x_0)$) that it reached first., 12 pages

In this paper we define a chain of $n$-dimensional evolution algebras corresponding to a permutation of $n$ numbers. We show that a chain of evolution algebras (CEA) corresponding to a permutation is trivial (consisting only algebras with zero-multiplication) iff the permutation has not a fixed point. We show that a CEA is a chain of nilpotent algebras (independently on time) iff it is trivial. We construct a wide class of chains of 3-dimensional EAs and a class of symmetric $n$-dimensional CEAs. A construction of arbitrary dimensional CEAs is given. Moreover, for a chain of 3-dimensional EAs 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., Comment: 15 pages

We consider Bernoulli (bond) percolation with parameter $p$ on the Cayley tree of order $k$. We introduce the notion of zebra-percolation that is percolation by paths of alternating open and closed edges. In contrast with standard percolation with critical threshold at $p_c= 1/k$, we show that zebra-percolation occurs between two critical values $p_{{\rm c},1}$ and $p_{{\rm c},2}$ (explicitly given). We provide the specific formula of zebra-percolation function., Comment: 8 pages, 2 figures

An evolution algebra corresponds to a quadratic matrix $A$ of structural constants. It is known the equivalence between nil, right nilpotent evolution algebras and evolution algebras which are defined by upper triangular matrices $A$. We establish a criterion for an $n$-dimensional nilpotent evolution algebra to be with maximal nilpotent index $2^{n-1}+1$. We give the classification of finite-dimensional complex evolution algebras with maximal nilpotent index. Moreover, for any $s=1,...,n-1$ we construct a wide class of $n$-dimensional evolution algebras with nilpotent index $2^{n-s}+1$. We show that nilpotent evolution algebras are not dibaric and establish a criterion for two-dimensional real evolution algebras to be dibaric., 18 pages, this version correct a mistake in Proposition 3.11

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 $n\times n$-matrices with non-negative entries which has at least one zero-row; the second class contains all $n\times n$-matrices with non-negative entries which in each row and in each column has exactly one non-zero entry. These matrices 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., 11 pages

Recently by Casas, Ladra and Rozikov a notion of a 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., Comment: 19 pages. arXiv admin note: text overlap with arXiv:1004.0155

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., 17 pages

We establish the solvability criteria for the equation $x^q=a$ in the field of $p$-adic numbers, for any $q$ in two cases: (i) $q$ is not divisible by $p$; (ii) $q=p$. Using these criteria we show that any $p$-adic number can be represented in finitely many different forms and we describe the algorithms to obtain the corresponding representations. Moreover it is showed that solvability problem of $x^q=a$ for any $q$ can be reduced to the cases (i) and (ii)., 14 pages

In this paper we give a concept of multi-dimensional-time dynamical system (MDTDS). Such dynamical system is generated by a finite family of functions $\{f_i\}$. The multi-dimensional-time space is taken as a free group. Using the subgroups of the free group we define periodic orbits of MDTDS and construct such orbits. We study fixed points of MDTDS and show that the set of the fixed points is the intersection of the set of fixed points of each $f_i$. The $\omega$-limit set of the MDTDS is defined and some properties of the set is studied. Moreover we construct a MDTDS for income of a deposit from several banks and construct a MDTDS on circle. For these MDTDSs we describe the set of fixed and periodic points. We discuss several open problems., Comment: 14 pages, Some misprints are corrected

We investigate the trajectory of an arbitrary $(2,1)$-rational $p$-adic dynamical system in a complex $p$-adic field $\C_p$. (i) In the case where there is no fixed point we show that the $p$-adic dynamical system has a 2-periodic cycle $x_1, x_2$. If it is attracting then it attracts each trajectory which starts from an element of a ball of radius $r=|x_1-x_2|_p$ with the center at $x_1$ or at $x_2$. If the 2-periodic cycle is an indifferent, then in each step the balls transfer to each other. All the other spheres with radius $>r$ and the center at $x_1$ and $x_2$ are invariant independently of the attractiveness of the cycle. (ii) In the case where the fixed point $x_0$ is unique we prove that if the point is attracting then there exists $\delta>0$, such that the basin of attraction for $x_0$ is the ball of radius $\delta$ and the center at $x_0$ and any sphere with radius $\geq \delta$ is invariant. If $x_0$ is an indifferent point then all spheres with the center at $x_0$ are invariant. If $x_0$ is a repelling point then there exits $\delta>0$, such that the trajectory which starts at an element of the ball of radius $\delta$ with the center in $x_0$ leaves this ball, whereas any sphere with radius $\geq \delta$ is invariant. (iii) In case of existence of two fixed points, we show that Siegel disks may either coincide or be disjoint for different fixed points of the dynamical system. Besides, we find the basin of the attractor of the system. Varying the parameters it is proven that there exists an integer $k\geq 2$, and spheres $S_{r_1}(x_i), ..., S_{r_k}(x_i)$ such that the limiting trajectory will be periodically traveling the spheres $S_{r_j}$. For some values of the parameters there are trajectories which go arbitrary far from the fixed points., Comment: 21 pages

We consider algebras with basis numerated by elements of a group $G.$ We fix a function $f$ from $G\times G$ 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 $\hat G$ of $G$ we define a $\hat G$-periodic algebra, which corresponds to a $\hat G$-periodic function $f,$ we establish a criterion for the right nilpotency of a $\hat G$-periodic algebra. In addition, for $G=\mathbb Z$ we describe all $2\mathbb Z$- and $3\mathbb Z$-periodic algebras. Some properties of $n\mathbb Z$-periodic algebras are obtained., Comment: 15 pages

In this paper we study the $p$-adic equation $x^q=a$ over the field of $p$-adic numbers. We construct an algorithm of calculation of criteria of solvability in the case of $q=p^m$ and present a computer program to compute the criteria for fixed value of $m \leq p-1$. Moreover, using this solvability criteria for $q=2,3,4,5,6$, we classify $p$-adic 6-dimensional filiform Leibniz algebras., Comment: 15 pages

The structural constants of an evolution algebra is given by a quadratic matrix $A$. In this work we establish equivalence between nil, right nilpotent evolution algebras and evolution algebras, which are defined by upper triangular matrix $A$. The classification of 2-dimensional complex evolution algebras is obtained. For an evolution algebra with a special form of the matrix $A$ we describe all its isomorphisms and their compositions. We construct an algorithm running under Mathematica which decides if two finite dimensional evolution algebras are isomorphic., 17 pages

In this note we consider several kind of partition functions of one-dimensional models with nearest - neighbor interactions $I_n, n\in \mathbf{Z}$ and spin values $\pm 1$. We derive systems of recursive equations for each kind of such functions. These systems depend on parameters $I_n, n\in \mathbf{Z}$. Under some conditions on the parameters we describe solutions of the systems of recursive equations., 9 pages

The history of the quadratic stochastic operators can be traced back to work of S.Bernshtein (1924). During more than 80 years this theory developed and many papers were published. In recent years it has again become of interest in connection with numerous applications to many branches of mathematics, biology and physics. But most results of the theory were published in non English journals, full text of which are not accessible. In this paper we give a brief description of the results and discuss several open problems., 14 pages

We consider quadratic stochastic operators, which are separable as a product of two linear operators. Depending on properties of these linear operators we classify the set of the separable quadratic stochastic operators: first class of constant operators, second class of linear and third class of nonlinear (separable) quadratic stochastic operators. Since the properties of operators from the first and second classes are well-known, we mainly study properties of the operators of the third class. We describe some Lyapunov functions of the operators and apply them to study $\omega$-limit sets of the trajectories generated by the operators. Also we compare our results with known results of the theory of quadratic operators and give some open problems., Comment: 9 pages

In this article we give a concept of ground subgroup for finite and countable groups. By our definition such a subgroup of a group depends on a given subset of the group and on a given partition of the subset. For finite and free groups we describe some sets of ground subgroups. We apply the ground subgroups to describe ground states of a model of statistical mechanics., Comment: 14 pages

We show that any periodic with respect to normal subgroups (of the group representation of the Cayley tree) of finite index $p$-harmonic function is a constant. For some normal subgroups of infinite index we describe a class of (non-constant) periodic $p$-harmonic functions. If $p\neq2$, the $p$-harmonicity is non-linear, i.e., the linear combination of $p$-harmonic functions need not be $p$-harmonic. In spite of this, we show that linear combinations of the $p$-harmonic functions described for normal subgroups of infinite index are also $p$-harmonic., Comment: 8 pages

We consider $\ell$-Volterra quadratic stochastic operators defined on $(m-1)$-dimensional simplex, where $\ell\in\{0,1,...,m\}$. Under some conditions on coefficients of such operators we describe Lyapunov functions and apply them to obtain upper estimates for the set of $\omega$- limit points of trajectories. We describe a set of fixed points of $\ell$-Volterra operators., Comment: 11 pages