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Papers by Ganikhodjaev Nasir

Research paper thumbnail of On the ergodic principle for Markov and quadratic Stochastic Processes and its relations

In the paper we prove that a quadratic stochastic process satisfies the ergodic principle if and ... more In the paper we prove that a quadratic stochastic process satisfies the ergodic principle if and only if the associated Markov process satisfies one.

Research paper thumbnail of Periodic Extreme Gibbs Measures with Memory Length 2 of Vannimenus Model

In this paper we investigate extreme Gibbs measures with period p for the Vannimenus model.

Research paper thumbnail of On the ergodic principle for Markov and quadratic stochastic processes and its relations

Linear Algebra and Its Applications, 2006

In the paper we prove that a quadratic stochastic process satisfies the ergodic principle if and ... more In the paper we prove that a quadratic stochastic process satisfies the ergodic principle if and only if the associated Markov process satisfies one.

Research paper thumbnail of Modulated Phase of a Potts Model with Competing Binary Interactions on a Cayley Tree

Journal of Statistical Physics, 2009

We study the phase diagram for Potts model on a Cayley tree with competing nearest-neighbor inter... 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 is at zero temperature. Later Mariz et al. generalized this result for Ising model with J o ≠0 and recently Ganikhodjaev et al. proved similar result for the three-state Potts model with J o =0. We consider Potts model with J o ≠0 and show that for some values of J o the multicritical Lifshitz point be at non-zero temperature. We also prove that as soon as the same-level interactionJ o is nonzero, the paramagnetic phase found at high temperatures for J o =0 disappears, while Ising model does not obtain such property. To perform this study, an iterative scheme similar to that appearing in real space renormalization group frameworks is established; it recovers, as particular case, previous work by Ganikhodjaev et al. for J o =0. At vanishing temperature, the phase diagram is fully determined for all values and signs of J 1,J p and J o . At finite temperatures several interesting features are exhibited for typical values of J o /J 1.

Research paper thumbnail of On Pure Phases of the Vannimenus Model

Abstract One of the main problems of the Statistical Physics is to describe all Gibbs measures co... more Abstract One of the main problems of the Statistical Physics is to describe all Gibbs measures corresponding to the given Hamiltonian. It is well known that such measures form a nonempty convex compact subset in the set of all probabilistic measures. The purpose ...

Research paper thumbnail of On the ergodic principle for Markov and quadratic Stochastic Processes and its relations

In the paper we prove that a quadratic stochastic process satisfies the ergodic principle if and ... more In the paper we prove that a quadratic stochastic process satisfies the ergodic principle if and only if the associated Markov process satisfies one.

Research paper thumbnail of On extreme Gibbs measures of the Vannimenus model

Journal of Statistical Mechanics-theory and Experiment, 2011

One of the main problems of statistical physics is that of describing all Gibbs measures correspo... more One of the main problems of statistical physics is that of describing all Gibbs measures corresponding to a given Hamiltonian. It is well known that such measures form a nonempty convex compact subset in the set of all probability measures. The purpose of this paper is to investigate extreme Gibbs measures of the Vannimenus model.

Research paper thumbnail of Periodic Extreme Gibbs Measures with Memory Length 2 of Vannimenus Model

In this paper we investigate extreme Gibbs measures with period p for the Vannimenus model.

Research paper thumbnail of On extreme Gibbs measures of the Vannimenus model

Journal of Statistical Mechanics-theory and Experiment, 2011

One of the main problems of statistical physics is that of describing all Gibbs measures correspo... more One of the main problems of statistical physics is that of describing all Gibbs measures corresponding to a given Hamiltonian. It is well known that such measures form a nonempty convex compact subset in the set of all probability measures. The purpose of this paper is to investigate extreme Gibbs measures of the Vannimenus model.

Research paper thumbnail of Modulated Phase of a Potts Model with Competing Binary Interactions on a Cayley Tree

Journal of Statistical Physics, 2009

We study the phase diagram for Potts model on a Cayley tree with competing nearest-neighbor inter... 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 is at zero temperature. Later Mariz et al. generalized this result for Ising model with J o ≠0 and recently Ganikhodjaev et al. proved similar result for the three-state Potts model with J o =0. We consider Potts model with J o ≠0 and show that for some values of J o the multicritical Lifshitz point be at non-zero temperature. We also prove that as soon as the same-level interactionJ o is nonzero, the paramagnetic phase found at high temperatures for J o =0 disappears, while Ising model does not obtain such property. To perform this study, an iterative scheme similar to that appearing in real space renormalization group frameworks is established; it recovers, as particular case, previous work by Ganikhodjaev et al. for J o =0. At vanishing temperature, the phase diagram is fully determined for all values and signs of J 1,J p and J o . At finite temperatures several interesting features are exhibited for typical values of J o /J 1.

Research paper thumbnail of On Pure Phases of the Vannimenus Model

Abstract One of the main problems of the Statistical Physics is to describe all Gibbs measures co... more Abstract One of the main problems of the Statistical Physics is to describe all Gibbs measures corresponding to the given Hamiltonian. It is well known that such measures form a nonempty convex compact subset in the set of all probabilistic measures. The purpose ...

Research paper thumbnail of On Entropy Transmission for Quantum Channels

In this paper a notion of entropy transmission of quantum channels is introduced as a natural ext... more In 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)B(H)B(H) into itself, where HHH is an infinite dimensional Hilbert space. Using a representation theorem of ucp mapping we associate to every ucp map a uniquely determined state,

Research paper thumbnail of On the ergodic principle for Markov and quadratic stochastic processes and its relations

Linear Algebra and Its Applications, 2006

In the paper we prove that a quadratic stochastic process satisfies the ergodic principle if and ... more In the paper we prove that a quadratic stochastic process satisfies the ergodic principle if and only if the associated Markov process satisfies one.

Research paper thumbnail of On the ergodic principle for Markov and quadratic Stochastic Processes and its relations

In the paper we prove that a quadratic stochastic process satisfies the ergodic principle if and ... more In the paper we prove that a quadratic stochastic process satisfies the ergodic principle if and only if the associated Markov process satisfies one.

Research paper thumbnail of Periodic Extreme Gibbs Measures with Memory Length 2 of Vannimenus Model

In this paper we investigate extreme Gibbs measures with period p for the Vannimenus model.

Research paper thumbnail of On the ergodic principle for Markov and quadratic stochastic processes and its relations

Linear Algebra and Its Applications, 2006

In the paper we prove that a quadratic stochastic process satisfies the ergodic principle if and ... more In the paper we prove that a quadratic stochastic process satisfies the ergodic principle if and only if the associated Markov process satisfies one.

Research paper thumbnail of Modulated Phase of a Potts Model with Competing Binary Interactions on a Cayley Tree

Journal of Statistical Physics, 2009

We study the phase diagram for Potts model on a Cayley tree with competing nearest-neighbor inter... 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 is at zero temperature. Later Mariz et al. generalized this result for Ising model with J o ≠0 and recently Ganikhodjaev et al. proved similar result for the three-state Potts model with J o =0. We consider Potts model with J o ≠0 and show that for some values of J o the multicritical Lifshitz point be at non-zero temperature. We also prove that as soon as the same-level interactionJ o is nonzero, the paramagnetic phase found at high temperatures for J o =0 disappears, while Ising model does not obtain such property. To perform this study, an iterative scheme similar to that appearing in real space renormalization group frameworks is established; it recovers, as particular case, previous work by Ganikhodjaev et al. for J o =0. At vanishing temperature, the phase diagram is fully determined for all values and signs of J 1,J p and J o . At finite temperatures several interesting features are exhibited for typical values of J o /J 1.

Research paper thumbnail of On Pure Phases of the Vannimenus Model

Abstract One of the main problems of the Statistical Physics is to describe all Gibbs measures co... more Abstract One of the main problems of the Statistical Physics is to describe all Gibbs measures corresponding to the given Hamiltonian. It is well known that such measures form a nonempty convex compact subset in the set of all probabilistic measures. The purpose ...

Research paper thumbnail of On the ergodic principle for Markov and quadratic Stochastic Processes and its relations

In the paper we prove that a quadratic stochastic process satisfies the ergodic principle if and ... more In the paper we prove that a quadratic stochastic process satisfies the ergodic principle if and only if the associated Markov process satisfies one.

Research paper thumbnail of On extreme Gibbs measures of the Vannimenus model

Journal of Statistical Mechanics-theory and Experiment, 2011

One of the main problems of statistical physics is that of describing all Gibbs measures correspo... more One of the main problems of statistical physics is that of describing all Gibbs measures corresponding to a given Hamiltonian. It is well known that such measures form a nonempty convex compact subset in the set of all probability measures. The purpose of this paper is to investigate extreme Gibbs measures of the Vannimenus model.

Research paper thumbnail of Periodic Extreme Gibbs Measures with Memory Length 2 of Vannimenus Model

In this paper we investigate extreme Gibbs measures with period p for the Vannimenus model.

Research paper thumbnail of On extreme Gibbs measures of the Vannimenus model

Journal of Statistical Mechanics-theory and Experiment, 2011

One of the main problems of statistical physics is that of describing all Gibbs measures correspo... more One of the main problems of statistical physics is that of describing all Gibbs measures corresponding to a given Hamiltonian. It is well known that such measures form a nonempty convex compact subset in the set of all probability measures. The purpose of this paper is to investigate extreme Gibbs measures of the Vannimenus model.

Research paper thumbnail of Modulated Phase of a Potts Model with Competing Binary Interactions on a Cayley Tree

Journal of Statistical Physics, 2009

We study the phase diagram for Potts model on a Cayley tree with competing nearest-neighbor inter... 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 is at zero temperature. Later Mariz et al. generalized this result for Ising model with J o ≠0 and recently Ganikhodjaev et al. proved similar result for the three-state Potts model with J o =0. We consider Potts model with J o ≠0 and show that for some values of J o the multicritical Lifshitz point be at non-zero temperature. We also prove that as soon as the same-level interactionJ o is nonzero, the paramagnetic phase found at high temperatures for J o =0 disappears, while Ising model does not obtain such property. To perform this study, an iterative scheme similar to that appearing in real space renormalization group frameworks is established; it recovers, as particular case, previous work by Ganikhodjaev et al. for J o =0. At vanishing temperature, the phase diagram is fully determined for all values and signs of J 1,J p and J o . At finite temperatures several interesting features are exhibited for typical values of J o /J 1.

Research paper thumbnail of On Pure Phases of the Vannimenus Model

Abstract One of the main problems of the Statistical Physics is to describe all Gibbs measures co... more Abstract One of the main problems of the Statistical Physics is to describe all Gibbs measures corresponding to the given Hamiltonian. It is well known that such measures form a nonempty convex compact subset in the set of all probabilistic measures. The purpose ...

Research paper thumbnail of On Entropy Transmission for Quantum Channels

In this paper a notion of entropy transmission of quantum channels is introduced as a natural ext... more In 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)B(H)B(H) into itself, where HHH is an infinite dimensional Hilbert space. Using a representation theorem of ucp mapping we associate to every ucp map a uniquely determined state,

Research paper thumbnail of On the ergodic principle for Markov and quadratic stochastic processes and its relations

Linear Algebra and Its Applications, 2006

In the paper we prove that a quadratic stochastic process satisfies the ergodic principle if and ... more In the paper we prove that a quadratic stochastic process satisfies the ergodic principle if and only if the associated Markov process satisfies one.