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This paper demonstrates how to explicitly construct a bidiagonal factorization of the banded recurrence matrix that appears in mixed multiple orthogonality on the step-line in terms of the coeffcients of the mixed multiple orthogonal polynomials. The construction is based on the \(LU\) factorization of the moment matrix and Christoffel transformations applied to the matrix of measures and the associated mixed multiple orthogonal polynomials., Comment: 14 pages

This work explores classical discrete multiple orthogonal polynomials, including Hahn, Meixner of the first and second kinds, Kravchuk, and Charlier polynomials, with an arbitrary number of weights. Explicit expressions for the recursion coefficients of Hahn multiple orthogonal polynomials are derived. By leveraging the multiple Askey scheme and the recently discovered explicit hypergeometric representation of type I multiple Hahn polynomials, corresponding explicit hypergeometric representations are provided for the type I polynomials and recursion coefficients of all the aforementioned descendants within the Askey scheme. Additionally, integral representations for these families within the Hahn class in the Askey scheme are presented. The multiple Askey scheme is further completed by providing the corresponding limits for the weights, polynomials, and recurrence coefficients., Comment: 37 pages, 5 figures

This paper addresses two primary objectives in the realm of classical multiple orthogonal polynomials with an arbitrary number of weights. Firstly, it establishes new and explicit hypergeometric expressions for type I Hahn multiple orthogonal polynomials. Secondly, applying the residue theorem and the Mellin transform, the paper derives contour integral representations for several families of orthogonal polynomials. Specifically, it presents contour integral formulas for both type I and type II multiple orthogonal polynomials in the Laguerre of the first kind, Jacobi-Pi\~neiro, and Hahn families. The evaluation of these integrals leads to explicit hypergeometric representations., Comment: 21 pages

In this work we apply Riemann-Hilbert problem to study two families of orthogonal polynomials on the unit circle, named of modified Jacobi and Bessel. We derive first and second order differential equations, for these families of orthogonal polynomials and functions of second kind, as well as nonlinear equations that the Verblunsky coefficients satisfy.

This paper serves as an introduction to banded totally positive matrices, exploring various characterizations and associated properties. A significant result within is the demonstration that the collection of such matrices forms a semigroup, notably including a subset permitting positive bidiagonal factorization. Moreover, the paper applies this concept to investigate step line normality concerning the degrees of associated recursion polynomials. It presents a spectral Favard theorem, ensuring the existence of measures, thereby guaranteeing that these recursion polynomials represent mixed multiple orthogonal polynomials that maintain normality on the step line indices.

This paper delves into classical multiple orthogonal polynomials with an arbitrary number of weights, including Jacobi-Pi\~neiro, Laguerre of both first and second kinds, as well as multiple orthogonal Hermite polynomials. Novel explicit expressions for nearest-neighbor recurrence coefficients, as well as the step line case, are provided for all these polynomial families. Furthermore, new explicit expressions for type I multiple orthogonal polynomials are derived for Laguerre of the second kind and also for multiple Hermite polynomials.

Sequences of bivariate orthogonal polynomials written as vector polynomials of increasing size satisfy a couple of three term relations with matrix coefficients. In this work, introducing a time-dependent parameter, we analyse a Lax-type pair system for the coefficients of the three term relations. We also deduce several characterizations relating the Lax-type pair, the shape of the weight, Stieltjes function, moments, a differential equation for the weight, and the bidimensional Toda-type systems.

For a general number $p\geq 2$ of measures, we provide explicit expressions for the Jacobi-Pi\~neiro and Laguerre of the first kind multiple orthogonal polynomials of type I, presented in terms of multiple hypergeometric functions.

This paper explores a factorization using bidiagonal matrices of the recurrence matrix of Hahn multiple orthogonal polynomials. The factorization is expressed in terms of ratios involving the generalized hypergeometric function ${}_3F_2$ and is proven using recently discovered contiguous relations. Moreover, employing the multiple Askey scheme, a bidiagonal factorization is derived for the Hahn descendants, including Jacobi-Pi\~neiro, multiple Meixner (kinds I and II), multiple Laguerre (kinds I and II), multiple Kravchuk, and multiple Charlier, all represented in terms of hypergeometric functions. For the cases of multiple Hahn, Jacobi-Pi\~neiro, Meixner of kind II, and Laguerre of kind I, where there exists a region where the recurrence matrix is nonnegative, subregions are identified where the bidiagonal factorization becomes a positive bidiagonal factorization., Comment: 14 pages, 2 figures

This paper investigates stochastic finite matrices and the corresponding finite Markov chains constructed using recurrence matrices for general families of orthogonal polynomials and multiple orthogonal polynomials. The paper explores the spectral theory of transition matrices, utilizing both orthogonal and multiple orthogonal polynomials. Several properties are derived, including classes, periodicity, recurrence, stationary states, ergodicity, expected recurrence times, time-reversed chains, and reversibility. Furthermore, the paper uncovers factorization in terms of pure birth and pure death processes. The case study focuses on hypergeometric orthogonal polynomials, where all the computations can be carried out effectively. Particularly within the Askey scheme, all descendants under Hahn (excluding Bessel), such as Hahn, Jacobi, Meixner, Kravchuk, Laguerre, Charlier, and Hermite, present interesting examples of recurrent reversible birth and death finite Markov chains. Additionally, the paper considers multiple orthogonal polynomials, including multiple Hahn, Jacobi-Pi\~neiro, Laguerre of the first kind, and Meixner of the second kind, along with their hypergeometric representations and derives the corresponding recurrent finite Markov chains and time-reversed chains., Comment: 40 pages, 3 figures

Banded bounded matrices, which represent non normal operators, of oscillatory type that admit a positive bidiagonal factorization are considered. To motivate the relevance of the oscillatory character the Favard theorem for Jacobi matrices is revisited and it is shown that after an adequate shift of the Jacobi matrix one gets an oscillatory matrix. In this work we present a spectral theorem for this type of operators and show how the theory of multiple orthogonal polynomials apply., Comment: arXiv admin note: substantial text overlap with arXiv:2203.13578

In this work we show how to get advantage from the Riemann--Hilbert analysis in order to obtain information about the matrix orthogonal polynomials and functions of second kind associated with a weight matrix. We deduce properties for the recurrence relation coefficients from differential properties of the weight matrix. We take the matrix polynomials of Hermite, Laguerre and Jacobi type as a case study.

Spiking Neural Networks (SNNs) have attracted recent interest due to their energy efficiency and biological plausibility. However, the performance of SNNs still lags behind traditional Artificial Neural Networks (ANNs), as there is no consensus on the best learning algorithm for SNNs. Best-performing SNNs are based on ANN to SNN conversion or learning with spike-based backpropagation through surrogate gradients. The focus of recent research has been on developing and testing different learning strategies, with hand-tailored architectures and parameter tuning. Neuroevolution (NE), has proven successful as a way to automatically design ANNs and tune parameters, but its applications to SNNs are still at an early stage. DENSER is a NE framework for the automatic design and parametrization of ANNs, based on the principles of Genetic Algorithms (GA) and Structured Grammatical Evolution (SGE). In this paper, we propose SPENSER, a NE framework for SNN generation based on DENSER, for image classification on the MNIST and Fashion-MNIST datasets. SPENSER generates competitive performing networks with a test accuracy of 99.42% and 91.65% respectively.

In this work we survey on connections of Markov chains and the theory of multiple orthogonality. Here we mainly concentrate on give a procedure to generate stochastic tetra diagonal Hessenberg matrices, coming from some specific families of multiple orthogonal, such as the ones of Jacobi--Pi\~neiro and Hypergeometric Lima--Loureiro. We show that associated with a positive tetra diagonal nonnegative bounded Hessenberg matrix we can construct two stochastic tetra diagonal ones. These two stochastic tridiagonal nonnegative Hessenberg matrices are shown to be, enlightened by the Poincar\'e theorem, limit transpose of each other.

Explicit expressions for the Hahn multiple polynomials of type I, in terms of Kamp\'e de F\'eriet hypergeometric series, are given. Orthogonal and biorthogonal relations are proven. Then, part of the Askey scheme for multiple orthogonal polynomials type I is completed. In particular, explicit expressions in terms of generalized hypergeometric series and Kamp\'e de F\'eriet hypergeometric series, are given for the multiple orthogonal polynomials of type I for the Jacobi-Pi\~neiro, Meixner I, Meixner II, Kravchuk, Laguerre I, Laguerre II and Charlier families., Comment: 27 pages. Minor typos corrected

Spectral and factorization properties of oscillatory matrices leads to a spectral Favard theorem for bounded banded matrices, that admit a positive bidiagonal factorization, in terms of sequences of mixed multiple orthogonal polynomials with respect to a set positive Lebesgue-Stieltjes measures. A mixed multiple Gauss quadrature formula with corresponding degrees of precision is given., Comment: 33 pages

Recently a spectral Favard theorem for bounded banded lower Hessenberg matrices that admit a positive bidiagonal factorization was presented. In this paper conditions, in terms of continued fractions, for an oscillatory tetradiagonal Hessenberg matrix to have such positive bidiagonal factorization are found. Oscillatory tetradiagonal Toeplitz matrices are taken as a case study of matrix that admits a positive bidiagonal factorization. Moreover, it is proved that oscillatory banded Hessenberg matrices are organized in rays, with the origin of the ray not having the positive bidiagonal factorization and all the interior points of the ray having such positive bidiagonal factorization., Comment: 20 pages This is the second part of the splitting of the paper arXiv:2203.13578 into three

Recently a spectral Favard theorem for bounded banded lower Hessenberg matrices that admit a positive bidiagonal factorization was presented. These type of matrices are oscillatory. In this paper the Lima-Loureiro hypergeometric multiple orthogonal polynomials and the Jacobi-Pi\~neiro multiple orthogonal polynomials are discussed at the light of this bidiagonal factorization for tetradiagonal matrices. The Darboux transformations of tetradiagonal Hessenberg matrices is studied and Christoffel formulas for the elements of the bidiagonal factorization are given, i.e., the bidiagonal factorization is given in terms of the recursion polynomials evaluated at the origin., Comment: This is the third part of the splitting of the paper arXiv:2203.13578 into three. 15 pages and 1 figure

We consider matrix orthogonal polynomials related to Jacobi type matrices of weights that can be defined in terms of a given matrix Pearson equation. Stating a Riemann-Hilbert problem we can derive first and second order differential relations that these matrix orthogonal polynomials and the second kind functions associated to them verify. For the corresponding matrix recurrence coefficients, non-Abelian extensions of a family of discrete Painlev\'e d-PIV equations are obtained for the three term recurrence relation coefficients.

We study two families of orthogonal polynomials with respect to the weight function $w(t)(t^2-\|x\|^2)^{\mu-\frac12}$, $\mu > -\frac 12$, on the cone $\{(x,t): \|x\| \le t, \, x \in \mathbb{R}^d, t >0\}$ in $\mathbb{R}^{d+1}$. The first family consists of monomial polynomials $\mathsf{V}_{\mathbf{k},n}(x,t) = t^{n-|\mathbf{k}|} x^\mathbf{k} + \cdots$ for $\mathbf{k} \in \mathbb{N}_0^d$ with $|\mathbf{k}| \le n$, which has the least $L^2$ norm among all polynomials of the form $t^{n-|\mathbf{k}|} x^\mathbf{k} + \mathsf{P}$ with $\deg \mathsf{P} \le n-1$, and we will provide an explicit construction for $\mathsf{V}_{\mathbf{k},n}$. The second family consists of orthogonal polynomials defined by the Rodrigues type formulas when $w$ is either the Laguerre weight or the Jacobi weight, which satisfies a generating function in both cases. The two families of polynomials are partially biorthogonal., Comment: 24 pp

On the of the applications in the realm of the Internet-of-Things (IoT) is real-time localization of assets in specific application environments where satellite based global positioning is unviable. Numerous approaches for localization relying on wireless sensor mesh systems have been evaluated, but the recent Bluetooth Low Energy (BLE) 5.1 direction finding features based on Angle-of-Arrival (AoA) promise a low-cost solution for this application. In this paper, we present an implementation of a BLE 5.1 based circular antenna array, and perform two experimental evaluations over the quality of the retrieved data sampled from the array. Specifically, we retrieve samples of the phase value of the Constant Tone Extension which enables the direction finding functionalities through calculation of phase differences between antenna pairs. We evaluate the quality of the sampled phase data in an anechoic chamber, and in a real-world environment using a setup composed of four BLE beacons., Comment: 2022 13th International Symposium on Communication Systems, Networks and Digital Signal Processing (CSNDSP)

A spectral Favard theorem for bounded banded lower Hessenberg matrices that admit a positive bidiagonal factorization is found. The large knowledge on the spectral and factorization properties of oscillatory matrices leads to this spectral Favard theorem in terms of sequences of multiple orthogonal polynomials of types I and II with respect to a set of positive Lebesgue-Stieltjes~measures. Also a multiple Gauss quadrature is proven and corresponding degrees of precision are found. This spectral Favard theorem is applied to Markov chains with $(p+2)$-diagonal transition matrices, i.e. beyond birth and death, that admit a positive stochastic bidiagonal factorization. In the finite case, the Karlin-McGregor spectral representation is given. It is shown that the Markov chains are recurrent and explicit expressions in terms of the orthogonal polynomials for the stationary distributions are given. Similar results are obtained for the countable infinite Markov chain. Now the Markov chain is not necessarily recurrent, and it is characterized in terms of the first measure. Ergodicity of the Markov chain is discussed in terms of the existence of a mass at $1$, which is an eigenvalue corresponding to the right and left eigenvectors., Comment: 38 pages. This a very much improved version of the initial one. We have divided the original paper in three parts. Two more parts to follow: arXiv:2210.10728, Positive bidiagonal factorization of tetradiagonal Hessenberg matrices arXiv:2210.10727, Bidiagonal factorization of tetradiagonal matrices and Darboux transformations

Lichens, organisms resulting from a symbiosis between a fungus and an algae, are frequently used as age estimators, especially in recent geological deposits and archaeological structures, using the correlation between lichen size and age. Current non-automated manual lichen and measurement (with ruler, calipers or using digital image processing tools) is a time-consuming and laborious process, especially when the number of samples is high. This work presents a workflow and set of image acquisition and processing tools developed to efficiently identify lichen thalli in flat rocky surfaces, and to produce relevant lichen size statistics (percentage cover, number of thalli, their area and perimeter). The developed workflow uses a regular digital camera for image capture along with specially designed targets to allow for automatic image correction and scale assignment. After this step, lichen identification is done in a flow comprising assisted image segmentation and classification based on interactive foreground extraction tool (GrabCut) and automatic classification of images using Simple Linear Iterative Clustering (SLIC) for image segmentation and Support Vector Machines (SV) and Random Forest classifiers. Initial evaluation shows promising results. The manual classification of images (for training) using GrabCut show an average speedup of 4 if compared with currently used techniques and presents an average precision of 95\%. The automatic classification using SLIC and SVM with default parameters produces results with average precision higher than 70\%. The developed system is flexible and allows a considerable reduction of processing time, the workflow allows it applicability to data sets of new lichen populations.