Table of contents

Foreword

The present volume of Journal of Physics: Conference Series contains the Proceedings of The 37-th International Conference on Quantum Probability and Related Topics (QP37), held in the Faculty of Science of the International Islamic University Malaysia, Kuantan, Malaysia, in August 22-26, 2016.

The intrinsic multi-disciplinary character of Quantum Probability is emphasized by its fruitful interaction with different aspects of contemporary mathematics. In addition to the traditional topics of Quantum Probability such as the recently discovered bridge with the classical theory of Orthogonal Polynomials, Quantum Field Theory, Quantum statistical Mechanics, the theory of Quantum Markov semigroups and of Quantum Markov chains and fields, Quantum Stochastic Differential Equations, operator algebras and dynamical systems, the conference has stimulated active communications with the theory of Quadratic operators, both in its classical and quantum aspects, which has a stronghold in the Faculty of Science of International Islamic University Malaysia (Kuantan).

The constructive interaction between pure and applied mathematics is another characteristic feature that has accompanied Quantum Probability in its almost fifty years of developments since the early 1970's. This is reflected, in the present volume, by a substantial presence of applications to several concrete models, not only of the mathematical techniques developed within Quantum Probability, but also in a multiplicity of other techniques which range from soliton theory, Fourier analysis, the theory of Boundary Value problems, ....

The history of the 36 Quantum Probability conferences preceding the present one, illustrated by a collection of 33 proceedings volumes, shows that in several occasions these interactions between apparently far research directions have given rise to fruitful new points of view that have enriched all disciplines involved with new perspectives, new results and new problems.

The program of conference included invited lectures (40 minutes) and poster section of more than 80 participants, including invited speakers from India, Italy, Japan, Jordan, Malaysia, The Republic of Korea, Singapore, Taiwan, Tunis, UK, USA and Uzbekistan. 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. It is the strongest hope of the organizers that similar unexpected developments will be stimulated also by the present conference.

Guest Editors

Luigi Accardi, Farrukh Mukhamedov, Pah Chin Hee

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 main thesis of the present paper is that: Quantum Probability is not a generalization of classical probability, but it is a deeper level of it. Classical random variables have an intrinsic (microscopic) non–commutative structure that generalize usual quantum theory. The study of this generalization is the core of the non–linear quantization program.

The equivalence between the notions of "2-Meixner" and "classic Meixner" random variables is proved first. A derivation of the classic Meixner random variables, using double commutators involving the semi-quantum operators, is presented next.

We discuss some recent results on the structure of quantum Markov semigroups of weak coupling limit type and their stationary states. In particular, we identify the minimal central projection in the fixed point algebra where they act in a trivial way and show, when they admit a single Bohr frequency, that all invariant states are convex combinations of equilibrium states and arbitrary states supported in the above minimal central projection.

We introduce a new notion of recurrence and transience for the Quantum Markov Chains which are weaker than Accardi-Koroliuk's recurrence and transience, respectively. We investigate some its properties.

In the framework of the Hudson - Parthasarathy quantum stochastic calculus, we employ some recent selection results to prove that the function space of the matrix elements of solutions to quantum stochastic differential inclusion (QSDI) is arcwise connected both locally and globally.

In this paper, we consider the Ising-XY model with competing interactions on the Cayley tree of order two. This model can be seen as a non-commutative (i.e. J-XY -interactions on next-neighbor vertices) perturbation of the classical Ising model on the Cayley tree. For the considered model we establish the existence of three translation-invariant quantum Markov chains. We notice that if the XY -interactions vanish, i.e. J = 0, then one gets the Ising model. If the classical Ising model vanishes in the considered model, then we obtain XY -model for which it turns out there exists only one translation invariant QMC.

In this paper, we define (n, k)-Gaussian quadratic stochastic operators and investigate the trajectory behavior of (3, 3)-Gaussian quadratic stochastic operators. It is established the necessary and sufficient conditions for regularity of these operators.

In 2010, N. Ganikhodjaev proposed the models of ABO and Rh blood groups of Malaysian people. Based on some numerical simulations, it was showed that the evolution of ABO blood groups of Malaysian people has a unique stable equilibrium. In this paper, we analytically prove that the Ganikhodjaev model of ABO blood groups has a unique fixed point.

This paper is a continuation of our previous studies on nonlinear consensus which unifies and generalizes all previous results. We consider a nonlinear protocol for a structured time-varying synchronous multi-agent system. We present an opinion sharing dynamics of the multi-agent system as a trajectory of non-autonomous polynomial stochastic operators associated with multidimensional stochastic hyper-matrices. We show that the multi-agent system eventually reaches to a nonlinear consensus if either one of the following two conditions is satisfied: (i) every member of the group people has a positive subjective distribution on the given task after some revision steps or (ii) all entries of some multidimensional stochastic hyper-matrix are positive.

In the present paper, we consider orthogonal preserving quadratic stochastic operators defined on infinite dimensional simplex. We provide a full descriptions of such kind of operators. Moreover, certain examples are given.

The genetic and evolution algebras generally are non-associative algebra. The concept of evolution and genetic algebras were introduced to answer the question what non-Mendelian genetics offers to mathematics. This paper we review some results of evolution and genetic algebras.

A constructive approach to the classification and invariance problems, with respect to basis changes, of the finite dimensional algebras is offered. A construction of an invariant open, dense (in the Zariski topology) subset of the space of structure constants of algebras is given. A classification of all algebras with structure constants from this dense set is given by providing canonical representatives of their orbits. A finite system of generators for the corresponding field of invariant rational functions of structure constants is shown.

An algebraic analog of the Fundamental Basis Theorem of geometry is offered with a pure algebraic proof involving the famous Waring's problem for polynomials. Unlike the geometry case the offered system of invariant differential operators is commuting, which is a new result even in the classical geometry of surfaces. Moreover the algebraic analog works in more general settings then does the Fundamental Basis Theorem of geometry.

Let X_k be the space of analytic, critical circle homeomorphisms with an irrational rotation number ρ_k = [k, k,..., k,...]. It is shown [S. Ostlund, D. Rand, J. Sethna and E. Siggia, Physica D,8 (1983) 303] that, the renormalization operator defined on X_k has a unique fixed point T₀. In this paper, we study the properties of orbits of a critical point and build a potential for T₀.

In this paper, we consider uniformly mean ergodic and uniformly asymptotical stable Markov operators on ordered Banach spaces. In terms of the ergodicity coefficient, we show the equivalence of uniform and weak mean ergodicities of Markov operators. This result allowed us to establish a category theorem for uniformly mean ergodic Markov operators. Furthermore, using properties of the ergodicity coefficient, we develop the perturbation theory for uniformly asymptotical stable Markov chains in the abstract scheme.

In this paper we establish a vector version the "zero-two" law for positive contractions in the Orlicz-Kantorovich spaces.

In this paper, we study the dynamics of the Potts-Bethe mapping associated with the p–adic q–state Potts model over the Cayley tree of order three. Namely, we establish the regularity of the Potts-Bethe mapping for the case p ≡ 2 (mod 3) with p ≥ 5.

An asymptotic formula for the Lesbegue constant of the Riesz means of Fourier-Laplace series on the sphere obtained in this paper.

For the Ising model on the Cayley tree of order k, we construct new sets of non translation-invariant Gibbs measures.

In the paper, we consider the λ-model with spin values {1, 2, 3} on the Cayley tree of order two. We describe ground states of the model.

This paper aims to analyze the stability of nonlinear localized modes in dipolar Bose-Einstein condensates in optical lattice by anti-continuum approach. It is relatively convenient to employ the anti-continuum method for solving discrete nonlinear Schrödinger equation. This method has shown the ability to produce similar result of analysis of the system stability as produced by other methods. The dynamical stability of the system has been investigated within certain permissible range of parameters.

The existence and stability of bright odd strongly localized modes in discrete binary media with cubic and quintic nonlinearities are investigated mainly by applying anti-continuum method. There are three different types of bright localized modes being studied; symmetric, antisymmetric and shifted odd modes. However, certain analytical analysis is carried out first applying the anti-continuum limit assumption to estimate initial excitations in the media. It can be deduced that, the quintic nonlinearity is a significant factor for stability in antisymmetric and shifted odd modes.

The flat top soliton is a localized solution of cubic - quintic Nonlinear Shrödinger equation (C-Q NLSE) that can propagate preserving its shape. In this work we consider the interaction of soliton with weak external localized potentials. It is shown that depending on an initial velocity the soliton may be reflected or transmitted by potential. The approximate analytical results based on varational approach qualitatively confirmed by numerical simulations.

This paper is devoted to the analytical and numerical study of the solitons interaction and scattering on the external delta potential in the weakly nonlocal nonlinear media. Using generalized form of Nonlinear Schrödinger Equation (NLSE) which is Cubic-Quintic NLSE in weakly nonlocal nonlinear media, we applied the variational approximation method to derive the equations for soliton parameters evolution during scattering process. Then a direct numerical simulation of NLSE is used to check the validity of approximations, considering the soliton initially located far from potential. Depending on initial velocity of the soliton, the phenomenon of reflection and transmission of the soliton through the potential have been shown. The critical values of the velocity separating these two scenarios have been identified.

In this paper we prove a precise equiconvergence relation between index of the Bochner-Riesz means of the expansions and power of the singularity of the distributions with compact support.

The objective of this study is to evaluate the empirical performance of interpolation techniques in risk-neutral density (RND) estimation. Firstly, the empirical performance is evaluated by using statistical analysis based on the implied mean and the implied variance of RND. Secondly, the interpolation performance is measured based on pricing error. We propose using the leave-one-out cross-validation (LOOCV) pricing error for interpolation selection purposes. The statistical analyses indicate that there are statistical differences between the interpolation techniques:second-order polynomial, fourth-order polynomial and smoothing spline. The results of LOOCV pricing error shows that interpolation by using fourth-order polynomial provides the best fitting to option prices in which it has the lowest value error.

A research study on boundary layer flow of a dusty viscous fluid past a stretching sheet of another dusty viscous quiescent fluid is conducted. The upper lighter fluid imposes downward on a lower heavier fluid. The existence of solid particles in a dusty fluid either naturally or deliberately in the form of dust, ash or soot suspended in the fluid may influence both upper and lower fluid flow characteristic. Therefore, this study investigates the effects of specific physical parameters such as fluid particle interaction, stretching rate, suction and injection. The governing partial differential equations are transformed into ordinary differential equations by using similarity transformation. The resulting differential equations are solved numerically by using Finite Difference Method with Richardson Extrapolation. The behaviors of physical parameters on velocity profile as well as skin frictions are presented graphically.

In this paper, we introduce a square root function over the p-adic field _p. This enables to explicitly prescribe roots of the quadratic equation over _p. Namely, we calculate the p–adic absolute value and the first digit of roots of the quadratic equation over _p.

Neural framework has for quite a while been known for its ability to handle a complex nonlinear system without a logical model and can learn refined nonlinear associations gives. Theoretically, the most surely understood computation to set up the framework is the backpropagation (BP) count which relies on upon the minimization of the mean square error (MSE). However, this algorithm is not totally efficient in the presence of outliers which usually exist in dynamic data. This paper exhibits the modelling of quadriceps muscle model by utilizing counterfeit smart procedures named consolidated backpropagation neural network nonlinear autoregressive (BPNN-NAR) and backpropagation neural network nonlinear autoregressive moving average (BPNN-NARMA) models in view of utilitarian electrical incitement (FES). We adapted particle swarm optimization (PSO) approach to enhance the performance of backpropagation algorithm. In this research, a progression of tests utilizing FES was led. The information that is gotten is utilized to build up the quadriceps muscle model. 934 preparing information, 200 testing and 200 approval information set are utilized as a part of the improvement of muscle model. It was found that both BPNN-NAR and BPNN-NARMA performed well in modelling this type of data. As a conclusion, the neural network time series models performed reasonably efficient for non-linear modelling such as active properties of the quadriceps muscle with one input, namely output namely muscle force.

This article has been retracted by IOP Publishing following an allegation that this article may contain tortured phrases [1].

IOP Publishing has investigated and agrees the article contains a number of nonsensical phrases that feature throughout the paper and lack of context surrounding the use of nanoparticles [2], to the extent that the article makes very little sense. This casts serious doubt over the legitimacy of the article.

IOP Publishing wishes to credit PubPeer commenters [3] for bringing the issue to our attention.

The authors neither agree nor disagree to this retraction.

[1] Cabanac G, Labbe C, Magazinov A, 2021, Tortured phrases: A dubious writing style emerging in science. Evidence of critical issues affecting established journals, arXiv:2107.06751v1

[2] Saadi Bin Ahmad Kamaruddin et al 2017 J. Phys.: Conf. Ser.819 012029

[3] https://pubpeer.com/publications/22276FF313DD4E624B73B2CAD7EEB0

Retraction published: 03 May 2023