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Existence of $P$-adic quasi Gibbs measure for countable state Potts model on the Cayley treeApr 12 2012In the present paper we provide a new construction of measure, called $p$-adic quasi Gibbs measure, for countable state of $p$-adic Potts model on the Cayley tree. Such a construction depends on a parameter $\frak{p}$ and wights. In particular case, i.e. ... More

On tensor products of weak mixing vector sequences and their applications to uniquely $E$-weak mixing $C^*$- dynamical systemsSep 03 2010May 02 2011We prove that, under certain conditions, uniform weak mixing (to zero) of the bounded sequences in Banach space implies uniform weak mixing of its tensor product. Moreover, we prove that ergodicity of tensor product of the sequences in Banach space implies ... More

Renormalization method in $p$-adic $λ$-model on the Cayley treeSep 22 2014In this present paper, it is proposed the renormalization techniques in the investigation of phase transition phenomena in $p$-adic statistical mechanics. We mainly study $p$-adic $\l$-model on the Cayley tree of order two. We consider generalized $p$-adic ... More

On Entropy Transmission for Quantum ChannelsMar 26 2007May 31 2007In this paper a notion of entropy transmission of quantum channels is introduced as a natural extension of Ohya's entropy. Here by quantum channel is meant unital completely positive mappings (ucp) of $B(H)$ into itself, where $H$ is an infinite dimensional ... More

Weighted ergodic theorem for contractions of Orlicz-Kantorovich lattice $L_{M}(\hat{\nabla},\hatμ$May 16 2013In the present paper we prove Besocovich weighted ergodic theorem for positive contractions acting on Orlich-Kantorovich space. Our main tool is the use of methods of measurable bundles of Banach-Kantorovich lattices.

On characterizations of bistochastic Kadison-Schwarz operators on $M_2(\mathbb{C})$Jan 05 2016In this paper we describe bistochastic Kadison-Schawrz operators acting on $M_2(\mathbb{C})$. Such a description allows us to find positive, but not Kadison-Schwarz operators. Moreover, by means of that characterization we construct Kadison-Schawrz operators, ... More

On one polynomial $p$-adic dynamical systemDec 25 2007In the paper we describe basin of attraction and the Siegel discs of the $p$-adic dynamical system $f(x)=x^{2n+1}+ax^{n+1}$ over complex $p$-adic field.

On noncommutative weighted local ergodic theoremsJan 15 2007Jul 27 2007In the present paper we consider a von Neumann algebra M with a faithful normal semi-finite trace $\t$, and $\{\alpha_ t\} $ a strongly continuous extension to $L^p(M,\t)$ of a semigroup of absolute contractions on $L^1 (M,\tau)$. By means of a non-commutative ... More

On Kadison-Schwarz type quantum quadratic operators on $\bm_2(\mathbb{C})$Nov 15 2012Apr 20 2013In the present paper we study description of Kadison-Schwarz type quantum quadratic operators acting from $\bm_2(\mathbb{C})$ into $\bm_2(\mathbb{C})\o\bm_2(\mathbb{C})$. Note that such kind of operator is a generalization of quantum convolution. By means ... More

A note on noncommutative unique ergodicity and weighted meansMar 03 2008Sep 22 2008In this paper we study unique ergodicity of $C^*$-dynamical system $(\ga,T)$, consisting of a unital $C^*$-algebra $\ga$ and a Markov operator $T:\ga\mapsto\ga$, relative to its fixed point subspace, in terms of Riesz summation which is weaker than Cesaro ... More

Gibbs measures and free energies of the Ising-Vannimenus Model on the Cayley treeApr 03 2015In this paper, we consider Ising-Vannimenus model on a Cayley tree for order two with competing nearest-neighbor, prolonged next-nearest neighbor interactions. We stress that the mentioned model was investigated only numerically, without rigorous (mathematical) ... More

On Gibbs Measures of Models with Competing Ternary and Binary Interactions and Corresponding Von Neumann Algebras IIOct 06 2005In the present paper the Ising model with competing binary ($J$) and binary ($J_1$) interactions with spin values $\pm 1$, on a Cayley tree of order 2 is considered. The structure of Gibbs measures for the model considered is studied. We completely describe ... More

On homotopy of volterrian quadratic stochastic operatorsDec 18 2007In the present paper we introduce a notion of homotopy of two Volterra operators which is related to fixed points of such operators. It is establish a criterion when two Volterra operators are homotopic, as a consequence we obtain that the corresponding ... More

Ergodic properties of Bogoliubov automorphisms in free probabilityMay 19 2009Oct 06 2009We show that some $C^*$--dynamical systems obtained by "quantizing" classical ones on the free Fock space, enjoy very strong ergodic properties. Namely, if the classical dynamical system $(X, T, \m)$ is ergodic but not weakly mixing, then the resulting ... More

On Limit theorems in $JW$- algebrasApr 27 2009Oct 06 2009In the present paper, we study bundle convergence in $JW$- algebra and prove certain ergodic theorems with respect to such convergence. Moreover, conditional expectations of reversible $JW$-algebras are considered. Using such expectations, the convergence ... More

Open Quantum Random Walks, Quantum Markov Chains and RecurrenceAug 03 2016Sep 12 2017In the present paper, we construct QMCs associated with Open Quantum Random Walks such that the transition operator of the chain is defined by OQRW and the restriction of QMC to the commutative subalgebra coincides with the distribution $P_\rho$ of OQRW. ... More

On pure quasi quantum quadratic operators of M_2(C)Jun 11 2013In the present paper we study quasi quantum quadratic operators (q.q.o) acting on the algebra of $2\times 2$ matrices $M_2(C)$. It is known that a channel is called pure if it sends pure states to pure ones. In this papers, we introduce a weaker condition, ... More

A Solution of variational inequality problem for a finite family of nonexpansive mappings in Hilbert spacesMay 24 2010In this paper we prove the strong convergence of the explicit iterative process to a common fixed point of the finite family of nonexpansive mappings defined on Hilbert space, which solves the the variational inequality on the fixed points set.

A few remarks on mixing properties of $C^*$-dynamical systemsOct 16 2005We consider strictly ergodic and strictly weak mixing $C^*$-dynamical systems. We prove that the system is strictly weak mixing if and only if its tensor product is strictly ergodic, moreover strictly weak mixing too. We also investigate some other mixing ... More

On unconventional limit sets of contractive functions on $\mathbb Z_p$Oct 13 2015In the present paper, we are going to study metric properties of unconventional limit set of a semigroup $G$ generated by contractive functions $\{f_{i}\}_{i=1}^N$ on the unit ball $\mathbb Z_p$ of $p$-adic numbers. Namely, we prove that the unconventional ... More

On Rational $P$-Adic Dyanamical SystemsNov 08 2005Feb 25 2006In the paper we investigate the behavior of trajectory of rational $p$-adic dynamical system in complex $p$-adic filed $\C_p$. It is studied Siegel disks and attractors of such dynamical systems. We show that Siegel disks may either coincide or disjoin ... More

On the "Zero-Two" Law for Positive Contractions on Banach-Kantorovich Lattice $L^p(\nabla,μ)$Oct 16 2005In the present paper we prove the "zero-two" law for positive contractions of lattices $L^p(\nabla,\mu)$ of Banach-Kantorovich, constructed by the measure $\mu$ with values in the ring of all measurable functions.

Relative ergodic properties of C*-dynamical systemsSep 21 2012Mar 27 2015We study various ergodic properties of C*-dynamical systems inspired by unique ergodicity. In particular we work in a framework allowing for ergodic properties defined relative to various subspaces, and in terms of weighted means. Our main results are ... More

On unification of the strong convergence theorems for a finite family of total asymptotically nonexpansive mappings in Banach spacesMay 27 2010Apr 08 2012In this paper, we unify all know iterative methods by introducing a new explicit iterative scheme for approximation of common fixed points of finite families of total asymptotically $I$-nonexpansive mappings. Note that such a scheme contains as a particular ... More

Weighted ergodic theorems for Banach-Kantorovich lattice $L_{p}(\hat{\nabla},\hatμ)$Oct 06 2011In the present paper we prove weighted ergodic theorems and multiparameter weighted ergodic theorems for positive contractions acting on $L_p(\hat{\nabla},\hat{\mu})$. Our main tool is the use of methods of measurable bundles of Banach-Kantorovich lattices. ... More

On some remarks on the Ising model with competing interactions on a Cayley treeOct 06 2005In the present paper the Ising model with competing binary $J$ and $J_1$ interactions with spin values $\pm 1$, on a Cayley tree is considered. We study translation-invatiant Gibbs measures and corresponding free energies ones.

Dissipative generators, divisible dynamical maps and Kadison-Schwarz inequalityAug 15 2019We introduce a concept of Kadison-Schwarz divisible dynamical maps. It turns out that it is a natural generalization of the well known CP-divisibility which characterizes quantum Markovian evolution. It is proved that Kadison-Schwarz divisible maps are ... More

Strict weak mixing of some C*-dynamical systems based on free shiftsDec 01 2006We define a stronger property than unique ergodicity with respect to the fixed-point subalgebra previously investigated by Abadie and Dykema. Such a property is denoted as F-strict weak mixing (F stands for the Markov projection onto the fixed-point operator ... More

On infinite dimensional Volterra type operatorsFeb 26 2009In this paper we study Volterra type operators on infinite dimensional simplex. It is provided a sufficient condition for Volterra type operators to be bijective. Furthermore it is shoved that the condition is not necessary.

Stability and monotonicity of Lotka-Volterra type operatorsDec 17 2009In the present paper, we study Lotka-Volterra (LV) type operators defined in finite dimensional simplex. We prove that any LV type operator is a surjection of the simplex. After, we introduce a new class of LV-type operators, called $M$LV type. We prove ... More

On multiparameter Weighted ergodic theorem for Noncommutative L_{p}-spacesNov 13 2006Oct 08 2007In the paper we consider $T_{1},..., T_{d}$ absolute contractions of von Neumann algebra $\M$ with normal, semi-finite, faithful trace, and prove that for every bounded Besicovitch weight $\{a(\kb)\}_{\kb\in\bn^d}$ and every $x\in L_{p}(\M)$, ($p>1$) ... More

On Non-Linear Markov Operators: surjectivity vs orthogonal preserving propertyJan 06 2017In the present paper, we consider nonlinear Markov operators, namely polynomial stochastic operators. We introduce a notion of orthogonal preserving polynomial stochastic operators. The purpose of this study is to show that surjectivity of nonlinear Markov ... More

Infinite Dimensional Orthogonal Preserving Quadratic Stochastic OperatorsApr 16 2017In the present paper, we study infinite dimensional orthogonal preserving quadratic stochastic operators (OP QSO). A full description of OP QSOs in terms of their canonical form and heredity coefficient's values is provided. Furthermore, some properties ... More

Uniqueness of fixed points of $ b$-bistochastic quadratic stochastic operators and associated nonhomogenous Markov chainsJan 17 2016In the present paper, we consider a class of quadratic stochastic operators (q.s.o.) called $ b- $bistochastic q.s.o. We include several properties of $ b- $bistochastic q.s.o. and their dynamical behavior. One of the main findings in this paper is the ... More

Uniform ergodicities and perturbation bounds of Markov chains on ordered Banach spacesMay 12 2016May 13 2016It is known that Dobrushin's ergodicity coefficient is one of the effective tools in the investigations of limiting behavior of Markov processes. Several interesting properties of the ergodicity coefficient of a positive mapping defined on ordered Banach ... More

Single Polygon Counting for $m$ Fixed Nodes in Cayley Tree: Two Extremal CasesApr 14 2010We denote a polygon as a connected component in Cayley tree of order 2 containing certain number of fix vertices. We found an exact formula for a polygon counting problem for two cases, in which, for the first case the polygon contain a full connected ... More

On $P$-adic Ising-Vannimenus model on an arbitrary order Cayley treeOct 12 2015In this paper, we continue an investigation of the $p$-adic Ising-Vannimenus model on the Cayley tree of an arbitrary order $k$ $(k\geq 2$). We prove the existence of $p$-adic quasi Gibbs measures by analyzing fixed points of multi-dimensional $p$-adic ... More

On the chaotic behavior of a generalized logistic $p$-adic dynamical systemFeb 23 2007In the paper we describe basin of attraction $p$-adic dynamical system $G(x)=(ax)^2(x+1)$. Moreover, we also describe the Siegel discs of the system, since the structure of the orbits of the system is related to the geometry of the $p$-adic Siegel discs. ... More

On Quantum Markov Chains on Cayley tree II: Phase transitions for the associated chain with XY-model on the Cayley tree of order threeNov 10 2010In the present paper we study forward Quantum Markov Chains (QMC) defined on a Cayley tree. Using the tree structure of graphs, we give a construction of quantum Markov chains on a Cayley tree. By means of such constructions we prove the existence of ... More

Quantum Markov fields on graphsNov 09 2009Jan 18 2010We introduce generalized quantum Markov states and generalized d-Markov chains which extend the notion quantum Markov chains on spin systems to that on $C^*$-algebras defined by general graphs. As examples of generalized d-Markov chains, we construct ... More

On Quantum Markov Chains on Cayley tree I: uniqueness of the associated chain with XY-model on the Cayley tree of order twoApr 21 2010Oct 18 2010In the present paper we study forward Quantum Markov Chains (QMC) defined on Cayley tree. A construction of such QMC is provided, namely we construct states on finite volumes with boundary conditions, and define QMC as a weak limit of those states which ... More

Construction of a new class of quantum Markov fieldsDec 16 2016In the present paper, we propose a new construction of quantum Markov fields on arbitrary connected, infinite, locally finite graphs. The construction is based on a specific tessellation on the considered graph, that allows us to express the Markov property ... More

The strong "zero-two" law for positive contractions of Banach-Kantorovich L_p-latticesFeb 25 2015In the present paper we study majorizable operators acting on Banach-Kantorovich $L_p$-lattices, constructed by a measure $m$ with values in the ring of all measurable functions. Then using methods of measurable bundles of Banach-Kantorovich lattices, ... More

On Ground States and Phase Transitions of $λ$-Model on the Cayley TreeOct 22 2016In the paper, we consider the $\lambda$-model with spin values $\{1, 2, 3\}$ on the Cayley tree of order two. We first describe ground states of the model. Moreover, we also proved the existence of translation-invariant Gibb measures for the $\lambda$-model ... More

$G-$Decompositions of Matrices and Related Problems IOct 27 2010May 20 2011In the present paper we introduce a notion of $G-$decompositions of matrices. Main result of the paper is that a symmetric matrix $A_m$ has a $G-$decomposition in the class of stochastic (resp. substochastic) matrices if and only if $A_m$ belongs to the ... More

On the ergodic principle for Markov and quadratic Stochastic Processes and its relationsNov 10 2005In the paper we prove that a quadratic stochastic process satisfies the ergodic principle if and only if the associated Markov process satisfies one.

On chaotic behavior of the $P$-adic generalized Ising mapping and its applicationJun 05 2017In the present paper, by conducting research on the dynamics of the $p$-adic generalized Ising mapping corresponding to renormalization group associated with the $p$-adic Ising-Vannemenus model on a Cayley tree, we have determined the existence of the ... More

On chaos of a cubic $p$-adic dynamical systemAug 23 2006In the paper we describe basin of attraction of the $p$-adic dynamical system $f(x)=x^3+ax^2$. Moreover, we also describe the Siegel discs of the system, since the structure of the orbits of the system is related to the geometry of the $p$-adic Siegel ... More

Positive solutions of nonlinear integral equations and surjectivity of Non-Linear Markov Operators On Infinite Dimensional SimplexJun 11 2018In the present paper, we consider the solvability of positive solutions of nonlinear integral equations by means of investigating non-linear Markov operators. To solve the problem we find necessary and sufficient condition for the surjectivity of nonlinear ... More

N Infinite Dimensional Quadratic Volterra OperatorsOct 16 2005In this paper we study a class of quadratic operators named by Volterra operators on infinite dimensional space. We prove that such operators have infinitely many fixed points and the set of Volterra operators forms a convex compact set. In addition, ... More

On Quantum Markov Chains on Cayley tree III: Ising modelNov 26 2013Nov 27 2013In this paper, we consider the classical Ising model on the Cayley tree of order k and show the existence of the phase transition in the following sense: there exists two quantum Markov states which are not quasi-equivalent. It turns out that the found ... More

Phase transitions for Quantum Markov Chains associated with Ising type models on a Cayley treeMay 15 2016The main aim of the present paper is to prove the existence of a phase transition in quantum Markov chain (QMC) scheme for the Ising type models on a Cayley tree. Note that this kind of models do not have one-dimensional analogous, i.e. the considered ... More

A Formulation of Rényi Entropy on $C^*$-AlgebrasMay 09 2019Jun 05 2019The entropy of probability distribution defined by Shannon has several extensions. R\'enyi entropy is one of the general extensions of Shannon entropy and is widely used in engineering, physics, and so on. On the other hand, the quantum analogue of Shannon ... More

On Volterra and orthoganality preserving quadratic stochastic operatorsJan 14 2014A quadratic stochastic operator (in short QSO) is usually used to present the time evolution of differing species in biology. Some quadratic stochastic operators have been studied by Lotka and Volterra. In the present paper, we first give a simple characterization ... More

On $ξ^{(s)}$-Quadratic Stochastic Operators on two Dimensional simplex and their behaviorMay 28 2013Sep 10 2013A quadratic stochastic operator (in short QSO) is usually used to present the time evolution of differing species in biology. Some quadratic stochastic operators have been studied by Lotka and Volterra. The general problem in the nonlinear operator theory ... More

On Marginal Markov Processes of Quantum Quadratic Stochastic ProcessesApr 24 2009Apr 28 2009In the paper it is defined two marginal Markov processes on von Neumann algebras $\cm$ and $\cm\o\cm$, respectively, corresponding to given quantum quadratic stochastic process (q.q.s.p.). It is proved that such marginal processes uniquely determines ... More

On ergodic type theorems for strictly weak mixing C^*-dynamical systemsOct 17 2005Dec 18 2007We prove that unique ergodicity of tensor product of $C^*$-dynamical system implies its strictly weak mixing. By means of this result a uniform weighted ergodic theorem with respect to $S$-Besicovitch sequences for strictly weak mixing dynamical systems ... More

On a recursive equation over $p$-adic fieldMay 09 2006In the paper we completely describe the set of all solutions of a recursive equation, arising from the Bethe lattice models over $p$-adic numbers.

On dominant contractions and a generalization of the zero-two lawApr 13 2010Zaharopol proved the following result: let $T,S:L^1(X,{\cf},\m)\to L^1(X,{\cf},\m)$ be two positive contractions such that $T\leq S$. If $\|S-T\|<1$ then $\|S^n-T^n\|<1$ for all $n\in\bn$. In the present paper we generalize this result to multi-parameter ... More

On the three state Potts model with competing interactions on the Bethe latticeJul 06 2006In the present paper the three state Potts model with competing binary interactions (with couplings $J$ and $J_p$) on the second order Bethe lattice is considered. The recurrent equations for the partition functions are derived. When $J_p=0$, by means ... More

Genetic Volterra algebras and their derivationsJun 06 2017The present paper is devoted to genetic Volterra algebras. We first study characters of such algebras. We fully describe associative genetic Volterra algebras, in this case all derivations are trivial. In general setting, i.e. when the algebra is not ... More

Ergodic properties of nonhomogeneous Markov chains defined on ordered Banach spaces with a baseNov 04 2013It is known that the Dobrushin's ergodicity coefficient is one of the effective tools to study a behavior of non-homogeneous Markov chains. In the present paper, we define such an ergodicity coefficient of a positive mapping defined on ordered Banach ... More

On phase transition for one dimensional countable state $P$-adic Potts modelJun 28 2011In the present paper we shall consider countable state $p$-adic Potts model on $Z_+$. A main aim is to establish the existence of the phase transition for the model. In our study, we essentially use one dimensionality of the model. To show it we reduce ... More

On dynamical systems and phase transitions for $Q+1$-state $P$-adic Potts model on the Cayley treeNov 05 2010In the present paper, we introduce a new kind of $p$-adic measures for $q+1$-state Potts model, called {\it $p$-adic quasi Gibbs measure}. For such a model, we derive a recursive relations with respect to boundary conditions. Note that we consider two ... More

On quantum quadratic operators of $\bm_2(\mathbb{C})$ and their dynamicsFeb 26 2009Nov 10 2010In the present paper we study nonlinear dynamics of quantum quadratic operators (q.q.o) acting on the algebra of $2\times 2$ matrices $\bm_2(\bc)$. First, we describe q.q.o. with Haar state as well as quadratic operators with the Kadison-Schwarz property. ... More

Derivations and automorphisms of nilpotent evolution algebras with maximal nilindexMay 25 2018Jun 11 2018In this paper is devoted to nilpotent finite-dimensional evolution algebras E with $dimE^2 = dimE-1$. We described Lie algebras associated with evolution algebras whose nilindex is maximal. Moreover, in terms of this Lie algebra we fully construct nilpotent ... More

On $L_1$-Weak Ergodicity of nonhomogeneous discrete Markov processes and its applicationsMay 03 2011Apr 08 2012In the present paper we investigate the $L_1$-weak ergodicity of nonhomogeneous discrete Markov processes with general state spaces. Note that the $L_1$-weak ergodicity is weaker than well-known weak ergodicity. We provide a necessary and sufficient condition ... More

Chaotic behavior of the $P$-adic Potts-Bethe mappingDec 31 2016In our previous investigations, we have developed the renormalization group method to $p$-adic models on Cayley trees, this method is closely related to the investigation of $p$-adic dynamical systems associated with a given model. In this paper, we study ... More

Weak and strong convergence of an implicit iterative process with errors for a finite family of asymptotically quasi $I-$nonexpansive mappings in Banach spaceOct 15 2010In this paper we prove the weak and strong convergence of the implicit iterative process with errors to a common fixed point of a finite family $\{T_j\}_{i=1}^N$ of asymptotically quasi $I_j-$nonexpansive mappings as well as a family of $\{I_j\}_{j=1}^N$ ... More

Hypercyclic and supercyclic linear operators on non-Archimedean vector spacesFeb 16 2017A main objective of the present paper is to develop the theory of hypercyclicity and supercyclicity of linear operators on topological vector space over non-Archimedean valued fields. We show that there does not exist any hypercyclic operator on finite ... More

On a Class of Rational $P$-Adic Dynamical SystemsNov 08 2005Feb 25 2006In this paper we investigate the behavior of trajectories of one class of rational $p$-adic dynamical systems in complex $p$-adic field $\C_p$. We studied Siegel disks and attractors of such dynamical systems. We found the basin of the attractor of the ... More

Self-adjoint cyclically compact operators and their applicationsFeb 09 2015This paper is devoted to self-adjoint cyclically compact operators on Hilbert--Kaplansky module over a ring of bounded measurable functions. The spectral theorem for such a class of operators are given. We apply this result to partial integral equations ... More

Open Quantum Random Walks and associated Quantum Markov ChainsAug 03 2016In the present paper, we establish a connection between Open Quantum Random Walks and the Quantum Markov Chains. In particular, we construct two kinds of Quantum Markov Chains associated with Open Quantum Random Walks. Moreover, we study the recurrence ... More

On Gibbs Measures of $P$-Adic Potts Model on the Cayley TreeOct 06 2005Feb 25 2006We consider a nearest-neighbor $p$-adic Potts (with $q\geq 2$ spin values and coupling constant $J\in \Q_p$) model on the Cayley tree of order $k\geq 1$. It is proved that a phase transition occurs at $k=2$, $q\in p\mathbb{N}$ and $p\geq 3$ (resp. $q\in ... More

On Inhomogeneous $p$-Adic Potts Model on a Cayley TreeOct 06 2005Feb 25 2006We consider a nearest-neighbor inhomogeneous $p$-adic Potts (with $q\geq 2$ spin values) model on the Cayley tree of order $k\geq 1$. The inhomogeneity means that the interaction $J_{xy}$ couplings depend on nearest-neighbors points $x, y $ of the Cayley ... More

Conditionally expectations and martingales in noncommutative $L_p$-spaces associated with center-valued tracesMar 29 2016In this paper we prove the existence of conditional expectations in the noncommutative $L_p(M,\Phi)$ spaces associated with center-valued traces. Moreover, their description is also provided. As an application of the obtained results, we establish the ... More

Measurable bundles of $C^*$-dynamical systems and its applicationsJan 20 2012In the present paper we investigate $L_0$-valued states and Markov operators on $ C^*$-algebras over $L_0$. In particular, we give representations for $L_0$-valued state and Markov operators on $ C^*$ algebras over $L_0$, respectively, as measurable bundles ... More

Volterra Evolution Algebras and Their GraphsApr 17 2019In this paper, we introduce Volterra evolution algebras which are evolution algebras whose structural matrices are described by skew symmetric matrices. A main result of the present paper gives a connection between such kind of algebras with ergodicities ... More

Phase transitions for $P$-adic Potts model on the Cayley tree of order threeAug 16 2012In the present paper, we study a phase transition problem for the $q$-state $p$-adic Potts model over the Cayley tree of order three. We consider a more general notion of $p$-adic Gibbs measure which depends on parameter $\rho\in\bq_p$. Such a measure ... More

Quantum Markov States on Cayley treesFeb 08 2019It is known that any locally faithful quantum Markov state (QMS) on one dimensional setting can be considered as a Gibbs state associated with Hamiltonian with commuting nearest-neighbor interactions. In our previous results, we have investigated quantum ... More

On equation $x^q=a$ over $\bq_p$Jun 29 2011Apr 20 2013In this paper we provide a solvability criterion for the monomial equation $x^q=a$ over $Q_p$ for any natural number $q$.

On description of bistochastic Kadison-Schwarz operators on M_2(C)May 30 2010In this paper we describe bistochastic Kadison-Schawrz operators on $M_2(\mathbb{C})$. Such a description allows us to find positive, but not Kadison-Schwarz operators. Moreover, by means of that characterization we construct Kadison-Schawrz operators, ... More

On non-Archimedean recurrence equations and their applicationsFeb 18 2014In the present paper we study stability of recurrence equations (which in particular case contain a dynamics of rational functions) generated by contractive functions defined on an arbitrary non-Archimedean algebra. Moreover, multirecurrence equations ... More

Local derivations on subalgebras of $τ$-measurable operators with respect to semi-finite von Neumann algebrasOct 07 2014This paper is devoted to local derivations on subalgebras on the algebra $S(M, \tau)$ of all $\tau$-measurable operators affiliated with a von Neumann algebra $M$ without abelian summands and with a faithful normal semi-finite trace $\tau.$ We prove that ... More

Measurable bundles of Banach algebrasApr 13 2014In the present paper we investigate Banach--Kantorovich algebras over faithful solid subalgebras of algebras measurable functions. We prove that any Banach--Kantorovich algebra over faithful solid subalgebras of algebra measurable functions represented ... More

Elliptic Quadratic Operator EquationsJan 08 2017In the present paper is devoted to the study of elliptic quadratic operator equations over the finite dimensional Euclidean space. We provide necessary and sufficient conditions for the existence of solutions of elliptic quadratic operator equations. ... More

Phase diagram of the three states Potts model with next nearest neighbor interactions on the Bethe latticeMar 18 2008We have found an exact phase diagram of the Potts model with next nearest neighbor interactions on the Bethe lattice of order two. The diagram consists of five phases: ferromagnetic, paramagnetic, modulated, antiphase and paramodulated, all meeting at ... More

A Note on Dominant Contractions of Jordan AlgebrasJun 18 2008Oct 07 2009In the paper we consider two positive contractions $T,S:L^{1}(A,\tau)\longrightarrow L^{1}(A,\tau)$ such that $T\leq S$, here $(A,\t)$ is a semi-finite $JBW$-algebra. If there is an $n_{0}\in\mathbb{N}$ such that $\|S^{n_{0}}-T^{n_{0}}\|<1$. Then we prove ... More

A Formulation of Rényi Entropy on $C^*$-AlgebrasMay 09 2019The entropy of probability distribution defined by Shannon has several extensions. R\'enyi entropy is one of the general extensions of Shannon entropy and is widely used in engineering, physics, and so on. On the other hand, the quantum analogue of Shannon ... More

On Uniqueness of Gibbs Measures for $P$-Adic Nonhomogeneous $ł$-Model on the Cayley TreeOct 06 2005Feb 25 2006We consider a nearest-neighbor $p$-adic $\l$-model with spin values $\pm 1$ on a Cayley tree of order $k\geq 1$. We prove for the model there is no phase transition and as well as the unique $p$-adic Gibbs measure is bounded if and only if $p\geq 3$. ... More

Gibbs measures and free energies of Ising-Vannimenus Model on the Cayley treeApr 06 2017In this paper, we consider the Ising-Vannimenus model on a Cayley tree for order two with competing nearest-neighbor and prolonged next-nearest neighbor interactions. We stress that the mentioned model was investigated only numerically, without rigorous ... More

On a generalized uniform zero-two law for positive contractions of non-commutative $L_1$-spaces and its vector-valued extensionMar 25 2016First, Ornstein and Sucheston proved that for a given positive contraction $T:L_1\to L_1$ there exists $m\in N$ such that $\big\|T^{m+1}-T^m\|<2$ then $$ \lim_{n\to\infty}\|T^{n+1}-T^n\|=0. $$ Such a result was labeled as "zero-two" law. In the present ... More

On cubic equations over $P-$adic fieldApr 08 2012We provide a solvability criteria for a depressed cubic equation in domains $\bz_p^{*},\bz_p,\bq_p$. We show that, in principal, the Cardano method is not always applicable for such equations. Moreover, the numbers of solutions of the depressed cubic ... More

On stability properties of positive contractions of $L^1$-spaces accosiated with finite von Neumann algebrasNov 12 2005In the paper we extent the notion of Dobrushin coefficient of ergodicity for positive contractions defined on $L^1$-space associated with finite von Neumann algebra, and in terms of this coefficient we prove stability results for $L^1$-contractions.

On Mixing and Completely Mixing Properties of Positive $L^1$-Contractions of Finite Von Neumann AlgebrasOct 16 2005Akcoglu and Suchaston proved the following result: Let $T:L^1(X,{\cf},\m)\to L^1(X,{\cf},\m)$ be a positive contraction. Assume that for $z\in L^1(X,{\cf},\m)$ the sequence $(T^nz)$ converges weakly in $L^1(X,{\cf},\m)$, then either $\lim\limits_{n\to\infty}\|T^nz\|=0$ ... More

Periodic and Weakly Periodic Ground States for the $λ$-Model on Cayley TreeDec 18 2016In this paper we consider the $\lambda$-model on the Cayley tree of order two. We describe periodic and weakly periodic ground states for the considered model.

On Ground States and Phase Transition for $λ$-Model with the Competing Potts Interactions on Cayley TreesApr 11 2019In this paper, we consider the $\lambda$-model with nearest neighbor interactions and with competing Potts interactions on the Cayley tree of order-two. We notice that if $\lambda$-function is taken as a Potts interaction function, then this model contains ... More

On Phase Transitions for $P$-Adic Potts Model with Competing Interactions on a Cayley TreeDec 07 2005Feb 25 2006In the paper we considere three state $p$-adic Potts model with competing interactions on a Cayley tree of order two. We reduce a problem of describing of the $p$-adic Gibbs measures to the solution of certain recursive equation, and using it we will ... More

Phase Transitions for quantum Ising model with competing XY -interactions on a Cayley treeFeb 08 2019The main aim of the present paper is to establish the existence of a phase transition for the quantum Ising model with competing XY interactions within the quantum Markov chain (QMC) scheme. In this scheme, we employ the $C^*$-algebraic approach to the ... More

Classification Tree Diagrams in Health Informatics ApplicationsFeb 09 2014Health informatics deal with the methods used to optimize the acquisition, storage and retrieval of medical data, and classify information in healthcare applications. Healthcare analysts are particularly interested in various computer informatics areas ... More