- Ankara, Turkey
Hasan Akin
Harran University, Mathematics, Faculty Member
In the present paper, we consider the Ising model with mixed spin- (1, 1/2) on the second order Cayley tree. For this model, a construction of splitting Gibbs measures is given that allows us to establish the existence of the phase... more
In the present paper, we consider the Ising model with mixed spin- (1, 1/2) on the second order Cayley tree. For this model, a construction of splitting Gibbs measures is given that allows us to establish the existence of the phase transition (non-uniqueness of Gibbs measures). We point out that, in the phase transition region, the considered model exhibits three translation-invariant Gibbs measures in the ferromagnetic and anti-ferromagnetic regimes, respectively, while the classical Ising model does not possess such Gibbs measures in the anti-ferromagnetic setting. It turns out, that like the classical Ising model, we can find a disordered Gibbs measure, therefore, its non-extremity and extremity are investigated by means of tree-indexed Markov chains.
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The aim of this paper is to continue the investigation into the set of translation-invariant splitting Gibbs measures (TISGMs) for Ising model having the mixed spin (1,1/2) (shortly, (1,1/2)-MSIM) on a Cayley tree of arbitrary order. In... more
The aim of this paper is to continue the investigation into the set of translation-invariant splitting Gibbs measures (TISGMs) for Ising model having the mixed spin (1,1/2) (shortly, (1,1/2)-MSIM) on a Cayley tree of arbitrary order. In our previous work [Akın and Mukhamedov, J. Stat. Mech. (2022) 053204], we provided a thorough explanation of the TISGMs, and studied the extremality of disordered phases using a Markov chain with a tree index on a semi-finite Cayley tree with order two. In this paper, we construct the TISGMs and tree-indexed Markov chains associated with to the model. Considering a tree-indexed Markov chain on a Cayley tree of any order, we clarify the extremality of the related disordered phases. By using the Kesten-Stigum condition (KSC), we investigate non-extremality of the disordered phases by means of the eigenvalues of the stochastic matrix associated with (1,1/2)-MSIM on a CT of order k≥2.
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We study the phase diagram for Potts model on a Cayley tree with competing nearest-neighbor interactions J 1, prolonged next-nearest-neighbor interactions J p and one-level next-nearest-neighbor interactions J o . Vannimenus proved that... more
We study the phase diagram for Potts model on a Cayley tree with competing nearest-neighbor interactions J 1, prolonged next-nearest-neighbor interactions J p and one-level next-nearest-neighbor interactions J o . Vannimenus proved that the phase diagram of Ising model with J o =0 contains a modulated phase, as found for similar models on periodic lattices, but the multicritical Lifshitz point
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In this study, an interactive Ising model having the nearest and prolonged next-nearest neighbors defined on a Cayley tree is considered. Inspired by the results obtained for the one-dimensional Ising model, we will construct the... more
In this study, an interactive Ising model having the nearest and prolonged next-nearest neighbors defined on a Cayley tree is considered. Inspired by the results obtained for the one-dimensional Ising model, we will construct the partition function and then calculate the free energy of the Ising model having the prolonged next nearest and nearest neighbor interactions and external field on a two-order Cayley tree using the self-similarity of the semi-infinite Cayley tree. The phase transition problem for the Ising system is investigated under the given conditions.
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In the present paper we study nonlinear dynamics of quantum quadratic operators acting on the algebra of $2\times 2$ matrices $\bm_2(\bc)$. First, we describe quadratic operators with Haar state, i.e. characterizations of quantum... more
In the present paper we study nonlinear dynamics of quantum quadratic operators acting on the algebra of $2\times 2$ matrices $\bm_2(\bc)$. First, we describe quadratic operators with Haar state, i.e. characterizations of quantum convolutions, Kadison-Schwartz operators are given. Then we study stability of dynamics of quadratic convolutions.
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Охарактеризованы бистохастические операторы Кадисона-Шварца, действующие на $M_2(\mathbb C)$. Полученная характеризация позволяет найти положительные операторы, не являющиеся операторами Кадисона-Шварца. Кроме того, приведены примеры... more
Охарактеризованы бистохастические операторы Кадисона-Шварца, действующие на $M_2(\mathbb C)$. Полученная характеризация позволяет найти положительные операторы, не являющиеся операторами Кадисона-Шварца. Кроме того, приведены примеры операторов Кадисона-Шварца, не являющихся вполне положительными.
In the present paper, we consider a notion of orthogonal preserving nonlinear operators. We introduce π-Volterra quadratic operators finite and infinite dimensional settings. It is proved that any orthogonal preserving quadratic operator... more
In the present paper, we consider a notion of orthogonal preserving nonlinear operators. We introduce π-Volterra quadratic operators finite and infinite dimensional settings. It is proved that any orthogonal preserving quadratic operator on finite dimensional simplex is π-Volterra quadratic operator. In infinite dimensional setting, we describe all π-Volterra operators in terms orthogonal preserving operators.
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The universal behaviors of a rational dynamical system associated with the Vannimenus–Ising model having two coupling constants on a Cayley tree of order three are studied. Cobweb diagrams and related map iterates for some relevant... more
The universal behaviors of a rational dynamical system associated with the Vannimenus–Ising model having two coupling constants on a Cayley tree of order three are studied. Cobweb diagrams and related map iterates for some relevant parameters are investigated. The local stability of fixed points is discussed and illustrated through cobweb diagrams. We deal with quantitative universality, such as orbit diagrams and Lyapunov exponents for a class of rational maps. We show that our model is periodic using orbit diagrams and relevant Lyapunov exponents.
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In this work we introduce and study a new family of one dimensional nonlinear cellular automaton which we name as quadratic cellular automata over ternary fields (3 Z). This family is defined by using the quadratic forms as local... more
In this work we introduce and study a new family of one dimensional nonlinear cellular automaton which we name as quadratic cellular automata over ternary fields (3 Z). This family is defined by using the quadratic forms as local transition functions. Further, we define hybrid quadratic cellular automata. Under periodic, null, and reflective boundary conditions it is shown that for some special local rules, hybrid quadratic cellular automata provide a good source for generating pseudo random numbers. The pseudo random numbers generated by hybrid quadratic cellular automata has passed the preliminary tests such as serial, runs, frequency, and poker.
In this paper we describe bistochastic Kadison-Schawrz operators acting on M_2(C). Such a description allows us to find positive, but not Kadison-Schwarz operators. Moreover, by means of that characterization we construct Kadison-Schawrz... more
In this paper we describe bistochastic Kadison-Schawrz operators acting on M_2(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, which are not completely positive.
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In this paper, we consider the Potts model with competing interactions on the Cayley tree of order three. Also we assume the state space set of the Potts model as Φ = {1,2,3,4} while it has taken as Φ′ = {1,2,3} in the previous studies... more
In this paper, we consider the Potts model with competing interactions on the Cayley tree of order three. Also we assume the state space set of the Potts model as Φ = {1,2,3,4} while it has taken as Φ′ = {1,2,3} in the previous studies [3, 8, 9]. we construct the Gibbs measures corresponding to the model by using Markov random field method. We calculate the critical curve such that there is a phase transition for the model. We extend the results introduced in the references [3, 8].
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In 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... more
In 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.
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In the paper we prove that a quadratic stochastic process satisfies the ergodic principle if and only if the associated Markov process satisfies one.
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In 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... more
In 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) proofs. One of the main point of this paper is to propose a measure-theoretical approach the considered model. We find certain conditions for the existence of Gibbs measures corresponding to the model. Then we establish the existence of the phase transition. Moreover, the free energies of the found Gibbs measures are calculated.
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In the paper we consider two positive contractions T,S:L^1(A,τ)⟶ L^1(A,τ) such that T≤ S, here (A,) is a semi-finite JBW-algebra. If there is an n_0∈N such that S^n_0-T^n_0<1. Then we prove that S^n-T^n<1 holds for every n≥ n_0.
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In the present paper we study a phase transition problem for the Potts model with three competing interactions, the nearest neighbors, the second neighbors and triples of neighbors and non-zero external field on Cayley tree of order two.... more
In the present paper we study a phase transition problem for the Potts model with three competing interactions, the nearest neighbors, the second neighbors and triples of neighbors and non-zero external field on Cayley tree of order two. We prove that for some parameter values of the model there is phase transition. We reduce the problem of describing by limiting Gibbs measures to the problem of solving a system of nonlinear functional equations. We extend the results obtained by Ganikhodjaev and Rozikov [Math. Phys. Anal. Geom., 2009, 12, No. 2, 141-156] on phase transition for the Ising model to the Potts model setting.
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In this paper we consider invertible one-dimensional linear cellular automata (CA hereafter) defined on a finite alphabet of cardinality p^k, i.e. the maps T_f[l,r]:Z^Z_p^k→Z^Z_p^k which are given by T_f[l,r](x) = (y_n)_n=-∞^∞ , y_n =... more
In this paper we consider invertible one-dimensional linear cellular automata (CA hereafter) defined on a finite alphabet of cardinality p^k, i.e. the maps T_f[l,r]:Z^Z_p^k→Z^Z_p^k which are given by T_f[l,r](x) = (y_n)_n=-∞^∞ , y_n = f(x_n+l, ..., x_n+r) =ri=l∑λ _ix_n+i(mod p^k), x=(x_n)_n=-∞^∞∈Z^Z_p^k and f:Z^r-l+1_p^k→Z_p^k, over the ring Z_p^k (k ≥ 2 and p is a prime number), where gcd(p,λ_r)=1 and p| λ_i for all i ≠ r (or gcd(p, λ_l)=1 and p|λ_i for all i≠ l). Under some assumptions we prove that any right (left) permutative, invertible one-dimensional linear CA T_f[l,r] and its inverse are strong mixing. We also prove that any right(left) permutative, invertible one-dimensional linear CA is Bernoulli automorphism without making use of the natural extension previously used in the literature.
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We consider the Ising model on a 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... more
We consider the Ising model on a 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 and found a new modulated phase, in addition to the paramagnetic, ferromagnetic, antiferromagnetic phases and a (+ + - -) periodic phase. Vannimenus's results based on 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 several phases, partition function.
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In this paper, we study linear cellular automata (CAs) on Cayley tree of order 2 over the field F_p (the set of prime numbers modulo p). We construct the rule matrix corresponding to finite cellular automata on Cayley tree. Further, we... more
In this paper, we study linear cellular automata (CAs) on Cayley tree of order 2 over the field F_p (the set of prime numbers modulo p). We construct the rule matrix corresponding to finite cellular automata on Cayley tree. Further, we analyze the reversibility problem of this cellular automaton for some given values of a,b,c,d∈F_p∖{0} and the levels n of Cayley tree. We compute the measure-theoretical entropy of the cellular automata which we define on Cayley tree. We show that for CAs on Cayley tree the measure entropy with respect to uniform Bernoulli measure is infinity.
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In 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... more
In 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 fixed points of a given function. Simultaneously, the attractors of the dynamical system have been found. We have come to a conclusion that the considered mapping is topologically conjugate to the symbolic shift which implies its chaoticity and as an application, we have established the existence of periodic p-adic Gibbs measures for the p-adic Ising-Vannemenus model.
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In 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,... more
In 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 are considered. We also investigate reverse recurrence equations which have application in the study of p-adic Gibbs measures. Note that our results also provide the existence of unique solutions of nonlinear functional equations as well.
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In this paper we analytically study the recurrence equations of an Ising model with three competing interactions on a Cayley tree of order three. We exactly describe paramagnetic and ferromagnetic phases of the Ising model. We obtain some... more
In this paper we analytically study the recurrence equations of an Ising model with three competing interactions on a Cayley tree of order three. We exactly describe paramagnetic and ferromagnetic phases of the Ising model. We obtain some rigorous results: critical temperatures and curves, number of phases, partition function. Ganikhodjaev et al. [J. Concrete and Applicable Mathematics, 9 (1), 26-34 (2011)] have numerically studied the Ising model on a second-order Cayley tree. We compare the numerical results to exact solutions of mentioned model.
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In this paper we study the topological and metric directional entropy of Z^2-actions by generated additive cellular automata (CA hereafter), defined by a local rule f[l, r], l, r∈Z, l≤ r, i.e. the maps T_f[l, r]: Z^Z_m→Z^Z_m which are... more
In this paper we study the topological and metric directional entropy of Z^2-actions by generated additive cellular automata (CA hereafter), defined by a local rule f[l, r], l, r∈Z, l≤ r, i.e. the maps T_f[l, r]: Z^Z_m→Z^Z_m which are given by T_f[l, r](x) =(y_n)_ -∞^∞, y_n = f(x_n+l, ..., x_n+r) = ∑_i=l^rλ_ix_i+n(mod m), x=(x_n)_ n=-∞^∞∈Z^Z_m, and f: Z_m^r-l+1→Z_m, over the ring Z_m (m ≥ 2), and the shift map acting on compact metric space Z^Z_m, where m (m ≥2) is a positive integer. Our main aim is to give an algorithm for computing the topological directional entropy of the Z^2-actions generated by the additive CA and the shift map. Thus, we ask to give a closed formula for the topological directional entropy of Z^2-action generated by the pair (T_f[l, r], σ) in the direction θ that can be efficiently and rightly computed by means of the coefficients of the local rule f as similar to [Theor. Comput. Sci. 290 (2003) 1629-1646]. We generalize the results obtained by Akı n [The topo...
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In 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 ρ∈_p. Such a measure is... more
In 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 ρ∈_p. Such a measure is called generalized p-adic quasi Gibbs measure. When ρ equals to p-adic exponent, then it coincides with the p-adic Gibbs measure. When ρ=p, then it coincides with p-adic quasi Gibbs measure. Therefore, we investigate two regimes with respect to the value of |ρ|_p. Namely, in the first regime, one takes ρ=_p(J) for some J∈_p, in the second one |ρ|_p<1. In each regime, we first find conditions for the existence of generalized p-adic quasi Gibbs measures. Furthermore, in the first regime, we establish the existence of the phase transition under some conditions. In the second regime, when ||̊_p,|q|_p≤ p^-2 we prove the existence of a quasi phase transition. It turns out that if ||̊_p<|q-1|_p^2<1 and √(-3)∈_p, then one finds the existence of th...
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In this paper, we consider an Ising model with three competing interactions on a triangular chandelier-lattice (TCL). We describe the existence, uniqueness, and non-uniqueness of translation-invariant Gibbs measures associated with the... more
In this paper, we consider an Ising model with three competing interactions on a triangular chandelier-lattice (TCL). We describe the existence, uniqueness, and non-uniqueness of translation-invariant Gibbs measures associated with the Ising model. We obtain an explicit formula for Gibbs measures with a memory of length 2 satisfying consistency conditions. It is proved rigorously that the model exhibits phase transitions only for given values of the coupling constants. As a consequence of our approach, the dichotomy between alternative solutions of Hamiltonian models on TCLs is solved. Finally, two numerical examples are given to illustrate the usefulness and effectiveness of the proposed theoretical results.
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We consider a non-commutative real analogue of Akcoglu and Suche-ston's result about the mixing properties of positive L 1-contractions of the L 1-space associated with a measure space with probability measure. This result generalizes... more
We consider a non-commutative real analogue of Akcoglu and Suche-ston's result about the mixing properties of positive L 1-contractions of the L 1-space associated with a measure space with probability measure. This result generalizes an analogous result obtained for the L 1-space associated with a finite (complex) W *-algebras.
In this present paper, the recurrence equations of an Ising model with three coupling constants on a third-order Cayley tree are obtained. Paramagnetic and ferromagnetic phases associated with the Ising model are characterized. Types of... more
In this present paper, the recurrence equations of an Ising model with three coupling constants on a third-order Cayley tree are obtained. Paramagnetic and ferromagnetic phases associated with the Ising model are characterized. Types of phases and partition functions corresponding to the model are rigorously studied. Exact solutions of the mentioned model are compared with the numerical results given in Ganikhodjaev et al [J. Concrete and Applicable
Mathematics, 9 (1), (2011), 26-34]..
Mathematics, 9 (1), (2011), 26-34]..