Asymptotic behaviour of zeros of exceptional Jacobi and Laguerre polynomials

The location and asymptotic behaviour for large n of the zeros of exceptional Jacobi and Laguerre polynomials are discussed. The zeros of exceptional polynomials fall into two classes: the regular zeros, which lie in the interval of orthogonality and the exceptional zeros, which lie outside that interval. We show that the regular zeros have two interlacing properties: one is the natural interlacing between consecutive polynomials as a consequence of their Sturm-Liouville character, while the other one shows interlacing between the zeros of exceptional and classical polynomials. A generalization of the classical Heine-Mehler formula is provided for the exceptional polynomials, which allows to derive the asymptotic behaviour of their regular zeros. We also describe the location and the asymptotic behaviour of the exceptional zeros, which converge for large n to fixed values.Comment: 19 pages, 3 figures, typed in AMS-LaTe

A conjecture on Exceptional Orthogonal Polynomials

Exceptional orthogonal polynomial systems (X-OPS) arise as eigenfunctions of Sturm-Liouville problems and generalize in this sense the classical families of Hermite, Laguerre and Jacobi. They also generalize the family of CPRS orthogonal polynomials. We formulate the following conjecture: every exceptional orthogonal polynomial system is related to a classical system by a Darboux-Crum transformation. We give a proof of this conjecture for codimension 2 exceptional orthogonal polynomials (X2-OPs). As a by-product of this analysis, we prove a Bochner-type theorem classifying all possible X2-OPS. The classification includes all cases known to date plus some new examples of X2-Laguerre and X2-Jacobi polynomials

Thresholding methods in non-intrusive load monitoring

Non-intrusive load monitoring (NILM) is the problem of predicting the status or consumption of individual domestic appliances only from the knowledge of the aggregated power load. NILM is often formulated as a classifcation (ON/OFF) problem for each device. However, the training datasets gathered by smart meters do not contain these labels, but only the electric consumption at every time interval. This paper addresses a fundamental methodological problem in how a NILM problem is posed, namely how the diferent possible thresholding methods lead to diferent classifcation problems. Standard datasets and NILM deep learning models are used to illustrate how the choice of thresholding method afects the output results. Some criteria that should be considered for the choice of such methods are also proposed. Finally, we propose a slight modifcation to current deep learning models for multi-tasking, i.e. tackling the classifcation and regression problems simultaneously. Transfer learning between both problems might improve performance on each of them.Funding for open access publishing: Universidad de Cádiz/CBUA. This research has been financed in part by the Spanish Agencia Estatal de Investigación under grants PID2021-122154NB-I00 and TED2021-129455B-I00, and by a 2021 BBVA Foundation project for research in Mathematics. He also acknowledges support from the EU under the 2014-2020 ERDF Operational Programme and the Department of Economy, Knowledge, Business and University of the Regional Government of Andalusia (FEDER-UCA18-108393)

Dynamical systems on infinitely sheeted Riemann surfaces

This paper is part of a program that aims to understand the connection between the emergence of chaotic behaviour in dynamical systems in relation with the multi-valuedness of the solutions as functions of complex time tau. In this work we consider a family of systems whose solutions can be expressed as the inversion of a single hyperelliptic integral. The associated Riemann surface R -> C = {tau} is known to be an infinitely sheeted covering of the complex time plane, ramified at an infinite set of points whose projection in the tau-plane is dense. The main novelty of this paper is that the geometrical structure of these infinitely sheeted Riemann surfaces is described in great detail, which allows us to study global properties of the flow such as asymptotic behaviour of the solutions, periodic orbits and their stability or sensitive dependence on initial conditions. The results are then compared with a numerical integration of the equations of motion. Following the recent approach of Calogero, the real time trajectories of the system are given by paths on R that are projected to a circle on the complex plane tau. Due to the branching of R, the solutions may have different periods or may be aperiodic

Quasi-exact solvability and the direct approach to invariant subspaces

We propose a more direct approach to constructing differential operators that preserve polynomial subspaces than the one based on considering elements of the enveloping algebra of sl(2). This approach is used here to construct new exactly solvable and quasi-exactly solvable quantum Hamiltonians on the line which are not Lie-algebraic. It is also applied to generate potentials with multiple algebraic sectors. We discuss two illustrative examples of these two applications: we show that the generalized Lame potential possesses four algebraic sectors, and describe a quasi-exactly solvable deformation of the Morse potential which is not Lie-algebraic