Orthogonal Polynomials in the Spectral Analysis of Markov Processes

In pioneering work in the 1950s, S. Karlin and J. McGregor showed that probabilistic aspects of certain Markov processes can be studied by analyzing orthogonal eigenfunctions of associated operators. In the decades since, many authors have extended and deepened this surprising connection between orthogonal polynomials and stochastic processes. This book gives a comprehensive analysis of the spectral representation of the most important one-dimensional Markov processes, namely discrete-time birth-death chains, birth-death processes and diffusion processes. It brings together the main results from the extensive literature on the topic with detailed examples and applications. Also featuring an introduction to the basic theory of orthogonal polynomials and a selection of exercises at the end of each chapter, it is suitable for graduate students with a solid background in stochastic processes as well as researchers in orthogonal polynomials and special functions who want to learn about applications of their work to probability.

Solutions of Sylvester equation in -modular operators

UDC 517.9 We study the solvability of the Sylvester equation and the operator equation in the general setting of the adjointable operators between Hilbert -modules. Based on the Moore – Penrose inverses of the associated operators, we propose necessary and sufficient conditions for the existence of solutions to these equations, and obtain the general expressions of the solutions in the solvable cases. We also provide an approach to the study of the positive solutions for a special case of Lyapunov equation.

The operator approach to the truncated multidimensional moment problem

We study the truncated multidimensional moment problem with a general type of truncations. The operator approach to the moment problem is presented. The case where the associated operators form a commuting self-adjoint tuple is characterized in terms of the given moments. The case of the dimensional stability is characterized in terms of the prescribed moments as well. Some sufficient conditions for the solvability of the moment problem are presented. A construction of the corresponding solution is described by algorithms. Numerical examples of the construction are provided.

An Invitation to Operator-Based Statistics

This article deals with operator-based statistics and its advantages. It first provides an overview of the historical and pedagogical aspects of operator-based statistics before explaining the underlying practical and theoretical motivations, along with synthetic and conceptual arguments. In particular, it develops the operator-based approach for factor multivariate analysis (and for their asymptotic studies) and offers several examples that show the value of operators in statistics. The discussion focuses on covariance operators, Hankel and Toeplitz operators, regression operators, measure-associated operators, tensor operators, and some other important categories. The article also describes noncommutative or quantum statistics and concludes with some reflections on the key notions and formulations of a "unified statistics" and projectors in (classical) statistics.

Extended Supersymmetric Structures in Gapped and Superconducting Graphene

In view of the many quantum field theoretical descriptions of graphene in 2 + 1 dimensions, we present another field theoretical feature of graphene, in the presence of defects. Particularly, we shall be interested in gapped graphene in the presence of a domain wall and also for superconducting graphene in the presence of a vortex. As we explicitly demonstrate, the gapped graphene electrons that are localized on the domain wall are associated with four N = 2 one-dimensional supersymmetries, with each pair combining to form an extended N = 4 supersymmetry with non-trivial topological charges. The case of superconducting graphene is more involved, with the electrons localized on the vortex being associated with n one-dimensional supersymmetries, which in turn combine to form an N = 2n extended supersymmetry with non-trivial topological charges. As we shall prove, all supersymmetries are unbroken, a feature closely related to the number of the localized fermions and also to the exact form of the associated operators. In addition, the corresponding Witten index is invariant under compact and odd perturbations.