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Papers by Mansoor Saburov

Research paper thumbnail of Phase transitions for XY-model on the Cayley tree of order three in quantum Markov chain scheme

Comptes Rendus Mathematique, 2011

ABSTRACT In the present paper we study forward Quantum Markov Chains (QMC) defined on a Cayley tr... more ABSTRACT In 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 a phase transition for the XY -model on a Cayley tree of order three in QMC scheme. By the phase transition we mean the existence of two distinct QMC for the given family of interaction operators {K <x,y> }.

Research paper thumbnail of On Quantum Markov Chains on Cayley Tree II: Phase Transitions for the Associated Chain with XY-Model on the Cayley Tree of Order Three

Annales Henri Poincaré, 2011

Research paper thumbnail of On Quantum Markov Chains on Cayley tree III: Ising model

Journal of Statistical Physics

In this paper, we consider the classical Ising model on the Cayley tree of order k and show the e... more In 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 critical temperature coincides with usual critical temperature.

Research paper thumbnail of On cubic equations over p-adic fields

International Journal of Number Theory, 2014

We provide a solvability criterion for a cubic equation in domains Z * p , Zp, Qp. We show that, ... more We provide a solvability criterion for a cubic equation in domains Z * p , Zp, Qp. We show that, in principal, the Cardano method is not always applicable for such equations. Moreover, the numbers of solutions of the cubic equation in domains Z * p , Zp, Qp are provided. Since Fp is a subgroup of Qp, we generalize Serre's and Sun's results concerning with cubic equations over the finite field Fp. Finally, all cubic equations, for which the Cardano method could be applied, are described and the p-adic Cardano formula is provided for those cubic equations.

Research paper thumbnail of Solvability of cubic equations in p-ADIC integers (p > 3)

Siberian Mathematical Journal, 2013

We give a criterion for the existence of solutions to an equation of the form x 3 + ax = b, where... more We give a criterion for the existence of solutions to an equation of the form x 3 + ax = b, where a, b ∈ Q p , in p-adic integers for p > 3. Moreover, in the case when the equation x 3 + ax = b is solvable, we give necessary and sufficient recurrent conditions on a p-adic number x ∈ Z * p under which x is a solution to the equation.

Research paper thumbnail of Single Polygon Counting for <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> Fixed Nodes in Cayley Tree: Two Extremal Cases

We denote a polygon as a connected component in Cayley tree of order 2 containing certain number ... more We 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 component of a Cayley tree and for the second case the polygon contain two fixed

Research paper thumbnail of Phase transitions for XY-model on the Cayley tree of order three in quantum Markov chain scheme

Comptes Rendus Mathematique, 2011

ABSTRACT In the present paper we study forward Quantum Markov Chains (QMC) defined on a Cayley tr... more ABSTRACT In 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 a phase transition for the XY -model on a Cayley tree of order three in QMC scheme. By the phase transition we mean the existence of two distinct QMC for the given family of interaction operators {K &lt;x,y&gt; }.

Research paper thumbnail of On Quantum Markov Chains on Cayley Tree II: Phase Transitions for the Associated Chain with XY-Model on the Cayley Tree of Order Three

Annales Henri Poincaré, 2011

Research paper thumbnail of On Quantum Markov Chains on Cayley tree III: Ising model

Journal of Statistical Physics

In this paper, we consider the classical Ising model on the Cayley tree of order k and show the e... more In 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 critical temperature coincides with usual critical temperature.

Research paper thumbnail of On cubic equations over p-adic fields

International Journal of Number Theory, 2014

We provide a solvability criterion for a cubic equation in domains Z * p , Zp, Qp. We show that, ... more We provide a solvability criterion for a cubic equation in domains Z * p , Zp, Qp. We show that, in principal, the Cardano method is not always applicable for such equations. Moreover, the numbers of solutions of the cubic equation in domains Z * p , Zp, Qp are provided. Since Fp is a subgroup of Qp, we generalize Serre's and Sun's results concerning with cubic equations over the finite field Fp. Finally, all cubic equations, for which the Cardano method could be applied, are described and the p-adic Cardano formula is provided for those cubic equations.

Research paper thumbnail of Solvability of cubic equations in p-ADIC integers (p > 3)

Siberian Mathematical Journal, 2013

We give a criterion for the existence of solutions to an equation of the form x 3 + ax = b, where... more We give a criterion for the existence of solutions to an equation of the form x 3 + ax = b, where a, b ∈ Q p , in p-adic integers for p > 3. Moreover, in the case when the equation x 3 + ax = b is solvable, we give necessary and sufficient recurrent conditions on a p-adic number x ∈ Z * p under which x is a solution to the equation.

Research paper thumbnail of Single Polygon Counting for <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> Fixed Nodes in Cayley Tree: Two Extremal Cases

We denote a polygon as a connected component in Cayley tree of order 2 containing certain number ... more We 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 component of a Cayley tree and for the second case the polygon contain two fixed