Table of contents

The present volume of the Journal of Physics: Conference Series represents contributions from participants of the International Conference ''Algebra, Analysis and Quantum Probability" (Tashkent, 10-12 September 2015) organized by the Institute of Mathematics and the Faculty of Mechanics and Mathematics of the National University of Uzbekistan (NUUz) in collaboration with University Putra Malaysia (UPM) and International Islamic University Malaysia (IIUM).

The Conference is dedicated to the 100th anniversary of one of the outstanding scientists of Uzbekistan, the founder of the Tashkent scientific school of functional analysis, who has initiated the investigations on operator algebras and quantum probability theory in Uzbekistan - Professor Tashmukhamed Alievich Sarymsakov (10 Sept. 1915 - 19 Dec. 1995). Among the mathematical community Professor T. A. Sarymsakov is widely known for his research in the fields of probability theory, functional analysis, general topology and their applications. A gifted teacher and skilful organizer he had a beneficial effect on the development of many new mathematicians in Uzbekistan. Professor T.A. Sarymsakov, an outstanding organizer of science in Uzbekistan, was one of the founders of the Uzbekistan Academy of Sciences, where from 1943 he was a member and Vice President, and from 1946 to 1952 president of the Academy of Sciences. Professor Sarymsakov successfully combined his fruitful scientific research with teaching and social work. During 1943-1944, 1952-1958 and 1971-1983 he was the rector of Tashkent State University (now the National University of Uzbekistan). He has made a significant contribution to the development of higher education in Uzbekistan, serving from 1959 to 1960 as the Chairman of the State Committee, and from 1960 to 1971 as the Minister of Higher and Secondary Special Education of Uzbekistan.

The main objective of the scientific conference was to facilitate communication and collaboration between mathematicians from different countries and it was focused on contemporary issues in the theory of operator algebras and non-commutative integration, structure theory of non-associative algebras and their applications in the theory of dynamical systems and quantum probability.

The Conference venue was the Mechanics and Mathematics Faculty of the National University of Uzbekistan and Institute of Mathematics.

The program of conference included plenary and invited lectures (40 minutes), short communications (15 minutes) given by more than 100 participants, including invited speakers from Australia, Brazil, France, Germany, Italy, Malaysia, Russia, Spain, USA and Uzbekistan in the following main sections of the conference:

Operator algebras and non-commutative integration;

Structure theory of non-associative algebras;

Quadratic operators and quantum probability;

Theory of dynamical systems and statistical physics.

We would like to thank the Invited Speakers for their significant contributions to the conference. We would also like to thank the members of the International Scientific Committee and the members of the Organizing Committee. We cannot end without expressing our many thanks to our sponsors for their financial support.

Guest Editors:

Shavkat Ayupov, Vladimir Chilin, Nasir Ganikhodjaev, Farrukh Mukhamedov, Isamiddin Rakhimov

All papers published in this volume of Journal of Physics: Conference Series have been peer reviewed through processes administered by the proceedings Editors. Reviews were conducted by expert referees to the professional and scientific standards expected of a proceedings journal published by IOP Publishing.

The paper is devoted to 2-local derivations on matrix algebras over unital semi-prime Banach algebras. For a unital semi-prime Banach algebra A with the inner derivation property we prove that any 2-local derivation on the algebra M_2ⁿ (A), n ≥ 2, is a derivation. We apply this result to AW*-algebras and show that any 2-local derivation on an arbitrary AW*-algebra is a derivation.

Let A be a laterally complete commutative regular algebra and X be a laterally complete A-module. In this paper we introduce a notion of homogeneous and strictly homogeneous A-modules. It is proved that any homogeneous A-module is strictly homogeneous A-module, if the Boolean algebra of all idempotents in A is multi-σ-finite.

In this paper we prove a non-commutative version of the uniform zero-two law for positive contractions of L₁-spaces associated with von Neumann algebras.

The present paper devoted to study of the extreme boundary of the convex compact set of all semi-additive functionals on the four-point compactum. We shall find some classes of extreme points of the space semi-additive functionals OS(4).

In this paper, we consider a classification, with respect to the orthogonal (unitary) change of basis, of finite dimensional algebras. A finite system of invariants, which separates nonequivalent algebras, whose systems of structural constants are from an invariant, open, dense set, is given.

In this paper we classify Leibniz algebras whose associated Lie algebra is four-dimensional Diamond Lie algebra 𝕯 and the ideal generated by squares of elements is represented by one of the finite-dimensional indecomposable D-modules U_n¹, U_n² or W_n¹ or W_n².

The universal enveloping algebra functor UL: Lb → Alg, defined by Loday and Pirashvili [1], is extended to crossed modules. Then we construct an isomorphism between the category of representations of a Leibniz crossed module and the category of left modules over its universal enveloping crossed module of algebras. Note that the procedure followed in the proof for the Lie case cannot be adapted, since the actor in the category of Leibniz crossed modules does not always exist.

In this paper we present a decomposition of HLⁿ(L, L) into a direct sum of some subspaces for a finite dimensional complex semisimple Leibniz algebra L. Furthermore, we provide a more specific decomposition in case n = 2 into two subspaces. We verify that one of those subspaces annihilates for specific Leibniz algebras with liezation ₂ and some others.

The concept of central extensions plays an important in constructing extensions of algebras. This technique has been successfully used in the classification problem of certain classes of algebras. In 1978 Skjelbred and Sund reduced the classification of nilpotent Lie algebras in a given dimension to the study of orbits under the action of automorphism group on the space of second degree cohomology of a smaller Lie algebra with coefficients in a trivial module. Then W. de Graaf applied the Skjelbred and Sund method to the classification problem of low-dimensional nilpotent Lie and associative algebras over some fields. The main purpose of this note is to establish elementary properties of central extensions of associative dialgebras and apply the above mentioned method to the classification of low dimensional nilpotent associative dialgebras.

In the present paper, we consider a class of quadratic stochastic operators (q.s.o.) called 6-bistochastic q.s.o. We study several descriptive properties of b- bistochastic q.s.o. It turns out that, upper triangular stochastic matrix defines a linear b-stochastic operator. This allowed us to find some sufficient conditions on cubic stochastic matrix to be a b—bistochastic q.s.o.

In this paper we analyze a general form of homeomorphisms of the two-dimensional simplex and study the asymptotical behavior of their trajectories.

We knew that a trajectory of a linear stochastic operator associated with a positive square stochastic matrix starting from any initial point from the simplex converges to a unique fixed point. However, in general, the similar result for a quadratic stochastic operator associated with a positive cubic stochastic matrix does not hold true. In this paper, we provide an example for the quadratic stochastic operator with positive coefficients in which its trajectory may converge to different fixed points depending on initial points.

In this paper we consider a class of strictly non-Volterra quadratic stochastic operators defined on the two-dimensional simplex. We show that such operators have a unique fixed point and the set of limit points is either a single point or an infinite set.

A double stochastic operator is a generalization of a double stochastic matrix. In this paper, we study the dynamics of double stochastic operators. We give a criterion for a regularity of a double stochastic operator in terms of absences of its periodic points. We provide some examples to insure that, in general, a trajectory of a double stochastic operator may converge to any interior point of the simplex.

In this paper, we introduce a new class of quadratic stochastic operators so-called Sarymsakov operators. We study the dynamics of such kind of operators. We establish regularity of each operator from this class. Note that there are some quadratic stochastic Sarymsakov operators which are not contraction mappings.

In this paper, we provide the classes of Poisson and Geometric quadratic stochastic operators with countable state space, study the dynamics of these operators and discuss their application to economics.

In this paper analogically as quadratic stochastic operators and processes we define cubic stochastic operator (CSO) and cubic stochastic processes (CSP). These are defined on the set of all probability measures of a measurable space. The measurable space can be given on a finite or continual set. The finite case has been investigated before. So here we mainly work on the continual set. We give a construction of a CSO and show that dynamical systems generated by such a CSO can be studied by studying of the behavior of trajectories of a CSO given on a finite dimensional simplex. We define a CSP and drive differential equations for such CSPs with continuous time.

The main aim of the present paper is to provide a new construction of quantum Markov chain (QMC) on arbitrary order Cayley tree. In that construction, a QMC is defined as a weak limit of finite volume states with boundary conditions, i.e. QMC depends on the boundary conditions. Note that this construction reminds statistical mechanics models with competing interactions on trees. If one considers one dimensional tree, then the provided construction reduces to well-known one, which was studied by the first author. Our construction will allow to investigate phase transition problem in a quantum setting.

We give a condition of extemelity for translation-invariant Gibbs measures of q—state Potts model on a Cayley tree. We'll improve the regions of extremality for some measures considered in [14]. Moreover, some results in [14] are generalized.

For the Ising model on the Cayley tree, we find new weakly periodic Gibbs measures corresponding to normal subgroups of index two in the group representation of the Cayley tree of order k ≤ 5.

In this paper we study the solvability of the Fredholm partial integral equations of second type with degenerate kernels.

In this paper, the Hamiltonian of the system of two fermions on the three-dimensional lattice ³ is considered. Under some conditions to the potential, it is proven that H(k) has four invariant subspaces: in one of them H(k) coincides with H₀(k) and the restrictions of H(k) to the rests three subspaces are unitarily equivalent to each other. Moreover, for the eigenvalues and eigenfunctions of these restricted operators explicit expressions are found.

Two anti-phase bright solitons in a dipolar Bose-Einstein condensate can form stable bound states, known as soliton molecules. In this paper we study the scattering of a two- soliton molecule by external potential, using the simplest and analytically tractable Gaussian potential barriers and wells, in one spatial dimension. Theoretical model is based on the variational approximation for the nonlocal Gross-Pitaevskii equation (GPE). At sufficiently low velocity of the incident molecule we observe quantum reflection from the potential well. Predictions of the mathematical model are compared with numerical simulations of the GPE, and good qualitative agreement between them is demonstrated.

A system of the Burgers equations of the two-velocity hydrodynamics is constructed. We consider the Cauchy problem in the case of a one-dimensional system. We have obtained a formula for solving the Cauchy problem and the estimate of the stability of this solution. It is shown that with disappearance of the kinetic friction coefficient, which is responsible for the energy dissipation, this formula turns to the famous Cauchy problem for the one-dimensional Burgers equation. The existence and uniqueness of solutions to the Cauchy problem for the one-dimensional systems of the Burgers type are proved using the method of weak approximation.

We investigate spectral properties of a four-electron system in the Hubbard Model framework in the v— dimensional lattice Z^v. We prove that the essential spectrum of the system in a quintet state consists of a single segment and the four-electron bound state or four-electron anti-bound state is absent. We show that the essential spectrum of the system in a triplete states is the union of at most three segments. We also prove that four-electron bound states or a four-electron anti-bound state exists in triplete states. We prove that in the system exists three triplete states and their spectrum are different.