David Gomez-Ullate

IE University

The work of Adler provides necessary and sufficient conditions for the Wronskian of a given sequence of eigenfunctions of Schr\"odinger's equation to have constant sign in its domain of definition. We extend this result by giving explicit formulas for the number of real zeros of the Wronskian of an arbitrary sequence of eigenfunctions. Our results...

We prove that every rational extension of the quantum harmonic oscillator that is exactly solvable by polynomials is monodromy free, and therefore can be obtained by applying a finite number of state-deleting Darboux transformations on the harmonic oscillator. Equivalently, every exceptional orthogonal polynomial system of Hermite type can be obtai...

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...

It has been recently discovered that exceptional families of Sturm-Liouville orthogonal polynomials exist, that generalize in some sense the classical polynomials of Hermite, Laguerre and Jacobi. In this paper we show how new families of exceptional orthogonal polynomials can be constructed by means of multiple-step algebraic Darboux transformation...

In this paper, we present a novel algorithm called the Hybrid Search algorithm that integrates the Zermelo's Navigation Initial Value Problem with the Ferraro-Mart\'in de Diego-Almagro algorithm to find the optimal route for a vessel to reach its destination. Our algorithm is designed to work in both Euclidean and spherical spaces and utilizes a he...

Based on data gathered by echo-sounder buoys attached to drifting fish-aggregating devices (dFADs) across tropical oceans, we applied a machine learning protocol to examine the temporal trends of tuna-school associations with drifting objects both in comparison to previous studies, and in the context of the ‘ecological trap’ theory. Using a binary...

Condition monitoring industrial machinery by using data mining in industrial projects has been extensively extended in the last few years. Applying data science to industrial processes should be straightforward in theory, but very few instances in the literature deal with the actual practical issues encountered while carrying out industrial data sc...

Background We estimated the association between the level of restriction in nine different fields of activity and SARS-CoV-2 transmissibility in Spain, from 15 September 2020 to 9 May 2021. Methods A stringency index (0–1) was created for each Spanish province (n = 50) daily. A hierarchical multiplicative model was fitted. The median of coefficien...

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 classification (ON/OFF) problem for each device. However, the training datasets gathered by smart meters do not contain these labels, but on...

In this work we introduce NeoCam , an open source hardware-software platform for video-based monitoring of preterms infants in Neonatal Intensive Care Units (NICUs). NeoCam includes an edge computing device that performs video acquisition and processing in real-time. Compared to other proposed solutions, it has the advantage of handling data mo...

Many recent studies have focused on the automatic classification of electrocardiogram (ECG) signals using deep learning (DL) methods. Most rely on existing complex DL methods, such as transfer learning or providing the models with carefully designed extracted features based on domain knowledge. A common assumption is that the deeper and more comple...

Non-Intrusive Load Monitoring (NILM) aims to predict the status or consumption of domestic appliances in a household only by knowing the aggregated power load. NILM can be formulated as regression problem or most often as a classification problem. Most datasets gathered by smart meters allow to define naturally a regression problem, but the corresp...

Based on the data gathered by echo-sounder buoys attached to drifting Fish Aggregating Devices (dFADs) across tropical oceans, the current study applies a Machine Learning protocol to examine the temporal trends of tuna schools' association to drifting objects. Using a binary output, metrics typically used in the literature were adapted to account...

Based on the data gathered by echo-sounder buoys attached to drifting Fish Aggregating Devices (dFADs) across tropical oceans, the current study applies a Machine Learning protocol to examine the temporal trends of tuna schools' association to drifting objects. Using a binary output, metrics typically used in the literature were adapted to account...

In this paper, we show how to construct exceptional orthogonal polynomials (XOP) using isospectral deformations of classical orthogonal polynomials. The construction is based on confluent Darboux transformations, where repeated factorizations at the same eigenvalue are allowed. These factorizations allow us to construct Sturm–Liouville problems wit...

Background We estimated the association between the level of restriction in nine different fields of activity and SARS-CoV-2 transmissibility in Spain, from 15 September 2020 to 9 May 2021. Methods A stringency index (0 to 1) was created for mobility, social distancing, commerce, indoor and outdoor bars and restaurants, culture and leisure, worshi...

The use of dFADs by tuna purse-seine fisheries is widespread across oceans, and the echo-sounder buoys attached to these dFADs provide fishermen with estimates of tuna biomass aggregated to them. This information has potential for gaining insight into tuna behaviour and abundance, but has traditionally been difficult to process and use. The current...

We show a method to construct isospectral deformations of classical orthogonal polynomials. The construction is based on confluent Darboux transformations, and it allows to construct Sturm-Liouville problems with polynomial eigenfunctions that have an arbitrary number of continuous parameters. We propose to call these new orthogonal polynomial syst...

Echo-sounder data registered by buoys attached to drifting FADs provide a very valuablesource of information on populations of tuna and their behaviour. This value increases whenthese data are supplemented with oceanographic data coming from CMEMS. We use thesesources to develop Tuna-AI, a Machine Learning model aimed at predicting tuna biomassunde...

Non-Intrusive Load Monitoring (NILM) is generally framed as a supervised learning problem whose input is the time series for aggregated power load of a household and whose output is the time series for the consumption of an individual appliance. Often the interest lies in predicting whether an appliance is ON or OFF, rather than its power usage. In...

We provide a complete classification and an explicit representation of rational solutions to the fourth Painlevé equation PIV and its higher order generalizations known as the A2n-Painlevé or Noumi-Yamada systems. The construction of solutions makes use of the theory of cyclic dressing chains of Schrödinger operators. Studying the local expansions...

In this paper we revisit exceptional Hermite polynomials from the point of view of spectral theory, following the work initiated by Lance Littlejohn. Adapting a result of Deift, we provide an alternative proof of the completeness of these polynomial families. In addition, using equivalence of Hermite Wronskians we characterize the possible gap sets...

Exceptional orthogonal polynomials are families of orthogonal polynomials that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing for polynomial sequences that miss a finite number of ''exceptional'' degrees. In this paper we introduce a new cons...

Fall is a prominent issue due to its severe consequences both physically and mentally. Fall detection and prevention is a critical area of research because it can help elderly people to depend less on caregivers and allow them to live and move more independently. Using electrocardiograms (ECG) signals independently for fall detection and activity c...

This note presents the classification of ladder operators corresponding to the class of rational extensions of the harmonic oscillator. We show that it is natural to endow the class of rational extensions and the corresponding intertwining operators with the structure of a category ℝ𝔼𝕏𝕋. The combinatorial data for this interpretation is realized as...

In this paper we revisit exceptional Hermite polynomials from the point of view of spectral theory, following the work initiated by Lance Littlejohn. Adapting a result of Deift, we provide an alternative proof of the completeness of these polynomial families. In addition, using equivalence of Hermite Wronskians we characterize the possible gap sets...

Non-Intrusive Load Monitoring (NILM) aims to predict the status or consumption of domestic appliances in a household only by knowing the aggregated power load. NILM can be formulated as regression problem or most often as a classification problem. Most datasets gathered by smart meters allow to define naturally a regression problem, but the corresp...

We provide a complete classification and an explicit representation of rational solutions to the fourth Painlev\'e equation PIV and its higher order generalizations known as the $A_{2n}$-Painlev\'e or Noumi-Yamada systems. The construction of solutions makes use of the theory of cyclic dressing chains of Schr\"odinger operators. Studying the local...

Although the solutions of Painlev\'e equations are transcendental in the sense that they cannot be expressed in terms of known elementary functions, there do exist rational solutions for specialized values of the equation parameters. A very successful approach in the study of rational solutions to Painlev\'e equations involves the reformulation of...

Exceptional orthogonal polynomials are families of orthogonal polynomials that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing for polynomial sequences that are missing a finite number of "exceptional" degrees. In this note we sketch the const...

These are the lecture notes for a course on exceptional polynomials taught at the AIMS-Volkswagen Stiftung Workshop on Introduction to Orthogonal Polynomials and Applications that took place in Douala (Cameroon) from October 5–12, 2018. They summarize the basic results and construction of exceptional poynomials, developed over the past 10 years. In...

This paper focuses on the construction of rational solutions for the ‐Painlevé system, also called the Noumi‐Yamada system, which are considered the higher order generalizations of PIV. In this even case, we introduce a method to construct the rational solutions based on cyclic dressing chains of Schrödinger operators with potentials in the class o...

These are the lecture notes for a course on exceptional polynomials taught at the \textit{AIMS-Volkswagen Stiftung Workshop on Introduction to Orthogonal Polynomials and Applications} that took place in Douala (Cameroon) from October 5-12, 2018. They summarize the basic results and construction of exceptional poynomials, developed over the past ten...

Exceptional orthogonal polynomials are complete families of orthogonal polynomials that arise as eigenfunctions of a Sturm–Liouville problem. Antonio Durán discovered a gap in the original proof of completeness for exceptional Hermite polynomials, that has propagated to analogous results for other exceptional families. In this paper we provide an a...

Exceptional orthogonal polynomials are complete families of orthogonal polynomials that arise as eigenfunctions of a Sturm-Liouville problem. Antonio Dur\'an discovered a gap in the original proof of completeness for exceptional Hermite polynomials, that has propagated to analogous results for other exceptional families. In this paper we provide an...

Non-intrusive load monitoring (NILM) is a technique aimed to detect which appliances are turned on in a certain building, by only knowing the aggregated electric load. Currently, there is a special interest in NILM thanks to its multiple applications and its low cost implementation. Several state-of-the-art techniques are being applied in this disc...

This note presents the classification of ladder operators corresponding to the class of rational extensions of the harmonic oscillator. We show that it is natural to endow the class of rational extensions and the corresponding intertwining operators with the structure of a category REXT. The combinatorial data for this interpretation is realized as...

In several reinforcement learning (RL) scenarios such as security settings, there may be adversaries trying to interfere with the reward generating process for their own benefit. We introduce Threatened Markov Decision Processes (TMDPs) as a framework to support an agent against potential opponents in a RL context. We also propose a level-k thinkin...

Drifting Fish Aggregating Devices (dFADs) are small drifting platforms with an attached solar powered buoy that report their position with daily frequency via GPS. We use data of 9,440 drifting objects provided by a buoys manufacturing company, to test the predictions of surface current velocity provided by two of the main models: the NEMO model us...

We present a new approach to determine the rational solutions of the higher order Painleve equations associated to periodic dressing chain systems. We obtain new sets of solutions, giving determinantal representations indexed by specific Maya diagrams in the odd case or universal characters in the even case.

This is the first paper of a series whose aim is to reach a complete classification and an explicit representation of rational solutions to the higher order generalizations of $\textrm{PIV}$ and $\textrm{PV}$, also known as the $A_N$-Painlev\'e or the Noumi-Yamada system. This paper focuses on the construction of rational solutions for the $A_{2n}$...

It was recently conjectured that every system of exceptional orthogonal polynomials is related to a classical orthogonal polynomial system by a sequence of Darboux transformations. In this paper we prove this conjecture, which paves the road to a complete classification of all exceptional orthogonal polynomials. In some sense, this paper can be reg...

When considering different allocations of the marketing budget of a firm, some predictions, that correspond to scenarios similar to others observed in the past, can be made with more confidence than others, that correspond to more innovative strategies. Selecting a few relevant features of the predicted probability distribution leads to a multi-obj...

In a previous paper we derived equivalence relations for pseudo-Wronskian determinants of Hermite polynomials. In this paper we obtain the analogous result for Laguerre and Jacobi polynomials. The equivalence formulas are richer in this case since rational Darboux transformations can be defined for four families of seed functions, as opposed to onl...

We propose a robust implementation of the Nerlove--Arrow model using a Bayesian structural time series model to explain the relationship between advertising expenditures of a country-wide fast-food franchise network with its weekly sales. Thanks to the flexibility and modularity of the model, it is well suited to generalization to other markets or...

We derive identities between determinants whose entries are Hermite polynomials. These identities have a combinatorial interpretation in terms of Maya diagrams, partitions and Durfee rectangles, and they are the product of an equivalence class of rational Darboux transformations. Since the determinants have different orders, we analyze the problem...

It was recently conjectured that every system of exceptional orthogonal polynomials is related to classical orthogonal polynomials by a sequence of Darboux transformations. In this paper we prove this conjecture, which paves the road to a complete classification of all exceptional orthogonal polynomials. In some sense, this paper can be regarded as...

The bispectral anti-isomorphism is applied to differential operators involving elements of the stabilizer ring to produce explicit formulas for all difference operators having any of the Hermite exceptional orthogonal polynomials as eigenfunctions with eigenvalues that are polynomials in .

The bispectral anti-isomorphism is applied to differential operators involving elements of the stabilizer ring to produce explicit formulas for all difference operators having any of the Hermite exceptional orthogonal polynomials as eigenfunctions with eigenvalues that are polynomials in $x$.

We study the performance of infotaxis search strategy measured by the rate of success and mean search time, under changes in the environment parameters such as diffusivity, rate of emission or wind velocity. We also investigate the drop of performance caused by an innacurate modelling of the environment. Our findings show that infotaxis remains rob...

In olfactory search an immobile target emits chemical molecules at constant rate. The molecules are transported by the medium, which is assumed to be turbulent. Considering a searcher able to detect such chemical signals and whose motion follows the infotaxis strategy, we study the statistics of the first-passage time to the target when the searche...

In olfactory search an immobile target emits chemical molecules at constant rate. The molecules are transported by the medium, which is assumed to be turbulent. Considering a searcher able to detect such chemical signals and whose motion follows the infotaxis strategy, we study the statistics of the first-passage time to the target when the searche...

Considering successive extensions of primary translationally shape invariant potentials, we enlarge the Krein-Adler theorem to mixed chains of state adding and state-deleting Darboux-B\"acklund transformations. It allows us to establish novel bilinear Wronskian and determinantal identities for classical orthogonal polynomials.

In this paper we state and prove some properties of the zeros of exceptional Jacobi and Laguerre polynomials. Generically, 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 inter...

In this paper we perform a statistical analysis of the high-frequency returns of the IBEX35 Madrid stock exchange index. We find that its probability distribution seems to be stable over different time scales, a stylized fact observed in many different financial time series. However, an in-depth analysis of the data using maximum likelihood estimat...

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 int...

We provide an example of how the complex dynamics of a recently introduced model can be understood via a detailed analysis of its associated Riemann surface. Thanks to this geometric description an explicit formula for the period of the orbits can be derived, which is shown to depend on the initial data and the continued fraction expansion of a sim...

We survey some recent developments in the theory of orthogonal polynomials defined by differential equations. The key finding is that there exist orthogonal polynomials defined by 2nd order differential equations that fall outside the classical families of Jacobi, Laguerre, and Hermite polynomials. Unlike the classical families, these new examples,...

Back in 1967, Clifford Gardner, John Greene, Martin Kruskal and Robert Miura published a seminal paper in Physical Review Letters which was to become a cornerstone in the theory of integrable systems. In 2006, the authors of this paper received the AMS Steele Prize. In this award the AMS pointed out that `In applications of mathematics, solitons an...

We adapt the notion of the Darboux transformation to the context of polynomial Sturm–Liouville problems. As an application, we characterize the recently described Xm Laguerre polynomials in terms of an isospectral Darboux transformation. We also show that the shape invariance of these new polynomial families is a direct consequence of the permutabi...

We prove an extension of Bochner’s classical result that characterizes the classical polynomial families as eigenfunctions of a second-order differential operator with polynomial coefficients. The extended result involves considering differential operators with rational coefficients and the requirement is that they have a numerable sequence of poly...

This is a call for contributions to a special issue of Journal of Physics A: Mathematical and Theoretical dedicated to integrability and nonlinear phenomena. The motivation behind this special issue is to summarize in a single comprehensive publication, the main aspects (past and present), latest developments, different viewpoints and the direction...

This is a call for contributions to a special issue of Journal of Physics A: Mathematical and Theoretical dedicated to integrability and nonlinear phenomena. The motivation behind this special issue is to summarize in a single comprehensive publication, the main aspects (past and present), latest developments, different viewpoints and the direction...

This is a call for contributions to a special issue of Journal of Physics A: Mathematical and Theoretical dedicated to integrability and nonlinear phenomena. The motivation behind this special issue is to summarize in a single comprehensive publication, the main aspects (past and present), latest developments, different viewpoints and the direction...

We present two infinite sequences of polynomial eigenfunctions of a Sturm-Liouville problem. As opposed to the classical orthogonal polynomial systems, these sequences start with a polynomial of degree one. We denote these polynomials as $X_1$-Jacobi and $X_1$-Laguerre and we prove that they are orthogonal with respect to a positive definite inner...

We investigate the dynamics defined by the following set of three coupled first-order ODEs: It is shown that the system can be reduced to quadratures which can be expressed in terms of elementary functions. Despite the integrable character of the model, the general solution is a multiple-valued function of time (considered as a complex variable),...

Mechanisms are elucidated underlying the existence of dynamical systems whose generic solutions approach asymptotically (at large time) isochronous evolutions: all their dependent variables tend asymptotically to functions periodic with the same fixed period. We focus on two such mechanisms, emphasizing their generality and illustrating each of the...

Our goal in this paper is to extend the theory of quasi-exactly solvable Schrödinger operators beyond the Lie-algebraic class. Let be the space of nth degree polynomials in one variable. We first analyze exceptional polynomial subspaces , which are those proper subspaces of invariant under second-order differential operators which do not preserve ....

A new class of many-body models is identified and investigated. Just as those we recently discovered, these many-body problems are solvable provided the initial data satisfy certain constraints: for such data the solution of the initial-value problem can be achieved via algebraic operations, such as finding the zeros of known polynomials. Entirely...

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 τ. In this work we consider a family of systems whose solutions can be expressed as the inversion of a single hyperelliptic integra...

We present evidence to suggest that the study of one-dimensional quasi-exactly solvable (QES) models in quantum mechanics should be extended beyond the usual sl(2) approach. The motivation is twofold: We first show that certain quasi-exactly solvable potentials constructed with the sl(2) Liealgebraic method allow for a new larger portion of the spe...

Two novel classes of many-body models with nonlinear interactions "of goldfish type" are introduced. They are solvable provided the initial data satisfy a single constraint (in one case; in the other, two constraints): i. e., for such initial data the solution of their initial-value problem can be achieved via algebraic operations, such as finding...

In this paper we derive structure theorems that characterize the spaces of linear and non-linear differential operators that preserve finite dimensional subspaces generated by polynomials in one or several variables. By means of the useful concept of deficiency, we can write explicit basis for these spaces of differential operators. In the case of...

We introduce and discuss a simple Hamiltonian dynamical system, interpretable as a 3-body problem in the complex plane and providing the prototype of a mechanism explaining the transition from regular to irregular motions as travel on Riemann surfaces. The interest of this phenomenology -- illustrating the onset in a deterministic context of irregu...

Calogero's goldfish N -body problem describes the motion of N point particles subject to mutual interaction with velocity-dependent forces under the action of a constant magnetic field transverse to the plane of motion. When all coupling constants are equal to one, the model has the property that for generic initial data, all motions of the system...

We describe a class of algebraically solvable SUSY models by considering the deformation of invariant polynomial flags by means of the Darboux transformation. The algebraic deformations corresponding to the addition of a bound state to a shape-invariant potential are particularly interesting. The polynomial flags in question are indexed by a deform...

We reply to the comment on our recent paper made by Drs Sinha and Roy. We agree that the backward Darboux transformation method used in our paper is equivalent to the approach based on CES potentials, but we stress that the emphasis and the results of our paper are different.

We describe a class of algebraically solvable SUSY models by considering the deformation of invariant polynomial flags by means of the Darboux transformation. The algebraic deformations corresponding to the addition of a bound state to a shape-invariant potential are particularly interesting. The polynomial flags in question are indexed by a deform...

In this paper we discuss a family of toy models for many-body interactions including velocity-dependent forces. By generalizing a construction due to Calogero, we obtain a class of N-body problems in the plane which have periodic orbits for a large class of initial conditions. The two- and three-body cases (N = 2, 3) are exactly solvable, with all...

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 ap...

We investigate the backward Darboux transformations (addition of the lowest bound state) of shape-invariant potentials on the line, and classify the subclass of algebraic deformations, those for which the potential and the bound states are simple elementary functions. A countable family, m = 0, 1, 2, ..., of deformations exists for each family of s...