Two different direct methods are proposed to solve Cauchy singular integral equations on the real line. The aforementioned methods differ in order to be able to prove their convergence which depends on the smoothness of the known term function in the integral equation.

It is known that executing a perfect shifted QR step via the implicit QR algorithm may not result in a deflation of the perfect shift. Typically, several steps are required before deflation actually takes place. This deficiency can be remedied by determining the similarity transformation via the associated eigenvector. Similar techniques have been deduced for the QZ algorithm and for the rational QZ algorithm. In this paper we present a similar approach for executing a perfect shifted QZ step on a general rank structured pencil instead of a specific rank structured one, e.g., a Hessenberg--Hessenberg pencil. For this, we rely on the rank structures present in the transformed matrices. A theoretical framework is presented for dealing with general rank structured \rev{pencils} and deflating subspaces. We present the corresponding algorithm allowing} to deflate simultaneously a block of eigenvalues rather than a single one. We define the level-rho poles and show that these poles are maintained executing the deflating algorithm. Numerical experiments illustrate the robustness of the presented approach showing the importance of using the improved scaled residual approach.

Magnetoencephalography (MEG) is a valuable non-invasive neurophysiology technique for investigation of brain function and dysfunction. In this chapter, we will discuss the main characteristics of MEG signals, and the great potential it offers for scientific interrogation in psychology, cognitive neuroscience, neurology, and neuropsychiatry. Starting from the physical properties of MEG recordings, the chapter will highlight the main advantages of utilizing MEG in neuroscience (that is a combination of very high temporal resolution and good spatial resolution) and will summarize the current status of MEG in research and clinical settings. To make this topic more relatable to widely available electroencephalography (EEG), we will present several comparisons of MEG with EEG. The objective of the present chapter is to provide a broad overview of the principle concepts and strengths of MEG, aimed at newcomers to the field.

We prove the nonlinear asymptotic stability of stably stratified solutions to the Incompressible Porous Media equation (IPM) for initial perturbations in ?H1- (R2) ? ?H s(R2) with s > 3 and for any 0 < < 1. Such result improves the existing literature, where the asymptotic stability is proved for initial perturbations belonging at least to H20(R2). More precisely, the aim of the article is threefold. First, we provide a simplified and improved proof of global-in-time well-posedness of the Boussinesq equations with strongly damped vorticity in H1- (R2)? ?Hs(R2) with s > 3 and 0 < < 1. Next, we prove the strong convergence of the Boussinesq system with damped vorticity towards (IPM) under a suitable scaling. Lastly, the asymptotic stability of stratified solutions to (IPM) follows as a byproduct. A symmetrization of the approximating system and a careful study of the anisotropic properties of the equations via anisotropic Littlewood-Paley decomposition play key roles to obtain uniform energy estimates. Finally, one of the main new and crucial points is the integrable time decay of the vertical velocity ku2(t)kL1(R2) for initial data only in ?H 1- (R2) ? ?H s(R2) with s > 3.

We consider here a cell-centered finite difference approximation of the Richards equation in three dimensions, averaging for interface values the hydraulic conductivity, a highly nonlinear function, by arithmetic, upstream and harmonic means. The nonlinearities in the equation can lead to changes in soil conductivity over several orders of magnitude and discretizations with respect to space variables often produce stiff systems of differential equations. A fully implicit time discretization is provided by backward Euler one-step formula; the resulting nonlinear algebraic system is solved by an inexact Newton Armijo-Goldstein algorithm, requiring the solution of a sequence of linear systems involving Jacobian matrices. We prove some new results concerning the distribution of the Jacobians eigenvalues and the explicit expression of their entries. Moreover, we explore some connections between the saturation of the soil and the ill conditioning of the Jacobians. The information on eigenvalues justifies the effectiveness of some preconditioner approaches which are widely used in the solution of Richards equation. We also propose a new software framework to experiment with scalable and robust preconditioners suitable for efficient parallel simulations at very large scales. Performance results on a literature test case show that our framework is very promising in the advance toward realistic simulations at extreme scale.

PRISMA is a hyperspectral pushbroom sensor, launched by the Italian Space Agency in 2019. PRISMA collects the reflected Earth signal from VNIR to the SWIR with 230 spectral bands with a variable FWHM according to the prism dispersion element. This work intends to develop a procedure suitable to monitor the consistency of photon and thermal noise components across a times series of L1 radiance images collected on different Mediterranean scenarios (i.e. rural and coastal). To improve the retrieval of the useful signal and the random noise on PRISMA images the spatial variability of the scenes has been considered in the new version of the HYperspectral Noise Parameters Estimation (HYNPE) algorithm. The procedure, tested on two PRISMA time series, has assessed quite stable and coherent values for the retrieved noise coefficients, not significantly affected by seasonal radiance variations and scene characteristics

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Image scaling methods allows us to obtain a given image at a different, higher (upscaling) or lower (downscaling), resolution with the aim of preserving as much as possible the original content and the quality of the image. In this paper, we focus on interpolation methods for scaling three-dimensional grayscale images. Within a unified framework, we introduce two different scaling methods, respectively based on the Lagrange and filtered de la Vall\'ee Poussin type interpolation at the 1st kind's Chebyshev zeros. In both cases, using a non-standard sampling model, we take (via tensor product) the associated trivariate polynomial interpolating the input image. It represents a continuous approximate 3D image to resample at the desired resolution. Using discrete linf and l2 norms, we theoretically estimate the error achieved in output, showing how it depends on the error in input and on the smoothness of the specific image we are processing. Finally, taking the special case of medical images as a case study, we experimentally compare the performances of the proposed methods among them and with the classical multivariate cubic and Lanczos interpolation methods.

The present paper is dedicated to the study of the discrepancies encountered in electron temperature (Te) measurements carried out with electron cyclotron emission (ECE) and Thomson scattering (TS) diagnostics in the core of the JET tokamak. A large database of discharges has been collected, including high-performance scenarios performed with deuterium only and deuterium-tritium mixtures. Discrepancies have been found between core Te measurements taken with an X-mode ECE interferometer (TECE) and a LIDAR TS system (TLID) for Te > 5 keV. Depending on the plasma scenario, TECE has been found to be systematically higher or lower than TLID. Discrepancies have also been observed between the peaks of the ECE spectrum in the second (X2) and third (X3) harmonic domains, even in high optical thickness conditions. These discrepancies can be interpreted as evidence of the presence of non-Maxwellian features in the electron energy distribution function (EEDF). In order to investigate the relation between the shape of the EEDF and the measured discrepancies, a model for bipolar perturbations of Maxwellian EEDF has been developed. The model allows analytical calculations of ECE absorption and emission coefficients; hence, the comparison of modeled ECE spectra with experimental data. The different experimental results observed for the various JET scenarios have been found to be qualitatively reproducible by adapting the model parameters, suggesting that bipolar distortions of the bulk EEDF could play a role in giving rise to the reported discrepancies between ECE and TS measurements.

The mammalian nervous system is made up of an extraordinary array of diverse cells that form intricate functional connections. The programs underlying cell lineage specification, identity and function of the neuronal subtypes are managed by regulatory proteins and RNAs, which coordinate the succession of steps in a stereotyped temporal order. In the central nervous system (CNS), motor neurons (MNs) are responsible for controlling essential functions such as movement, breathing, and swallowing by integrating signal transmission from the cortex, brainstem, and spinal cord (SC) towards peripheral muscles. A prime role in guiding the progression of progenitor cells towards the MN fate has been largely attributed to protein factors. More recently, the relevance of a class of regulatory RNAs abundantly expressed in the CNS - the long noncoding RNAs (lncRNAs) - has emerged overwhelmingly. LncRNA-driven gene expression control is key to regulating any step of MN differentiation and function, and its derangement profoundly impacts neuronal pathophysiology. Here, we uncover a novel function for the neuronal isoform of HOTAIRM1 (nHOTAIRM1), a lncRNA specifically expressed in the SC. Using a model system that recapitulates spinal MN (spMN) differentiation, we show that nHOTAIRM1 intervenes in the binary cell fate decision between MNs and interneurons, acting as a pro-MN factor. Furthermore, human iPSC-derived spMNs without nHOTAIRM1 display altered neurite outgrowth, with a significant reduction of both branch and junction numbers. Finally, the expression of genes essential for synaptic connectivity and neurotransmission is also profoundly impaired when nHOTAIRM1 is absent in spMNs. Mechanistically, nHOTAIRM1 establishes both direct and indirect interactions with a number of target genes in the cytoplasm, being a novel post-transcriptional regulator of MN biology. Overall, our results indicate that the lncRNA nHOTAIRM1 is essential for the specification of MN identity and the acquisition of proper morphology and synaptic activity of post-mitotic MNs.

During embryonic development, naive pluripotent epiblast cells transit to a formative state. The formative epiblast cells form a polarised epithelium, exhibit distinct transcriptional and epigenetic profiles and acquire competence to differentiate into all somatic and germline lineages. However, we have limited understanding of how the transition to a formative state is molecularly controlled. Here we used murine ESC models to show that ESRRB is both required and sufficient to activate formative genes. Genetic inactivation of Esrrb leads to illegitimate expression of mesendoderm and extraembryonic markers, impaired formative expression and failure to self-organise in 3D. Functionally, this results in impaired ability to generate Formative Stem cells and primordial germ cells in the absence of Esrrb. Computational modelling and genomic analyses revealed that ESRRB occupies key formative genes in naive cells and throughout the formative state. In so doing, ESRRB kickstarts the formative transition, leading to timely and unbiased capacity for multi-lineage differentiation.

Path graphs are intersection graphs of paths in a tree. We start from the characterization of path graphs by Monma and Wei (1986) [14] and we reduce it to some 2-coloring subproblems, obtaining the first characterization that directly leads to a polynomial recognition algorithm. Then we introduce the collection of the attachedness graphs of a graph and we exhibit a list of minimal forbidden 2-edge colored subgraphs in each of the attachedness graph.

Homophily is the principle whereby "similarity breeds connections."We give a quantitative formulation of this principle within networks. Given a network and a labeled partition of its vertices, the vector indexed by each class of the partition, whose entries are the number of edges of the subgraphs induced by the corresponding classes, is viewed as the observed outcome of the random vector described by picking labeled partitions at random among labeled partitions whose classes have the same cardinalities as the given one. This is the recently introduced random coloring model for network homophily. In this perspective, the value of any homophily score ?, namely, a nondecreasing real-valued function in the sizes of subgraphs induced by the classes of the partition, evaluated at the observed outcome, can be thought of as the observed value of a random variable. Consequently, according to the score ?, the input network is homophillic at the significance level ? whenever the one-sided tail probability of observing a value of ? at least as extreme as the observed one is smaller than ?. Since, as we show, even approximating ? is an NP-hard problem, we resort to classical tails inequality to bound ? from above. These upper bounds, obtained by specializing ?, yield a class of quantifiers of network homophily. Computing the upper bounds requires the knowledge of the covariance matrix of the random vector, which was not previously known within the random coloring model. In this paper we close this gap. Interestingly, the matrix depends on the input partition only through the cardinalities of its classes and depends on the network only through its degrees. Furthermore all the covariances have the same sign, and this sign is a graph invariant. Plugging this structure into the bounds yields a meaningful, easy to compute class of indices for measuring network homophily. As demonstrated in real-world network applications, these indices are effective and reliable, and may lead to discoveries that cannot be captured by the current state of the art.

The paper deals with the numerical solution of Cauchy Singular Integral Equations based on some non standard polynomial quasi-projection of de la Vallée Poussin type. Such kind of approximation presents several advantages over classical Lagrange interpolation such as the uniform boundedness of the Lebesgue constants, the near-best order of uniform convergence to any continuous function, and a strong reduction of Gibbs phenomenon. These features will be inherited by the proposed numerical method which is stable and convergent, and provides a near-best polynomial approximation of the sought solution by solving a well conditioned linear system. The numerical tests confirm the theoretical error estimates and, in case of functions subject to Gibbs phenomenon, they show a better local approximation compared with analogous Lagrange projection methods.

As known, polynomial interpolation is not advisable in the case of equidistant nodes, given the exponential growth of the Lebesgue constants and the consequent stability problems. In [1] Floater and Hormann introduce a family of rational interpolants (briefly FH interpolants) depending on a fixed integer parameter d >= 1. They are based on any configuration of the nodes in [a, b], have no real poles and approximation order O(h^{d+1}) for functions in C^{d+2}[a, b], where h denotes the maximum distance between two consecutive nodes. FH interpolants turn out to be very useful for equidistant or quasi-equidistant configurations of nodes when the Lebesgue constants present only a logarithmic growth as the number of nodes increases [2, 3]. In this talk, we introduce a generalization of FH interpolants depending on an additional parameter ? ? N. If ? = 1 we get the classical FH interpolants, but taking ? > 1 we succeed in getting uniformly bounded Lebesgue constants for quasi-equidistant configurations of nodes. Moreover, in comparison with the original FH interpolants, we show that the new interpolants present a much better error prole when the function is less smooth.

We consider the problem of interpolating a given function on arbitrary configurations of nodes in a compact interval, with a special focus on the case of equidistant or quasi-equidistant nodes. In this case, instead of polynomial interpolation, a family of rational interpolants introduced by Floater and Hormann in [2] turns out to be very useful . Such interpolants (briefly FH interpolants) generalize Berrut's rational interpolation [1] introducing a fixed integer parameter d >= 1 to speed up the convergence getting, in theory, arbitrarily high approximation orders. In this talk we will further generalize by presenting a whole new family of rational interpolants that depend on an additional parameter ? ? N. When ? = 1 we get the original FH interpolants. For ? > 1 we will see that the new interpolants share a lot of the interesting properties of the original FH interpolants (no real poles, baryentric-type representation, high rates of approximation). But, in addition, we get uniformly bounded Lebesgue constants and a more localized approximation of less smooth functions, compared to the original FH interpolation.

Taking wedge products of the p distinct principal null directions (PNDs) associated with the eigen-bivectors of the Weyl tensor associated with the Petrov classification, when linearly independent, one is able to express them in terms of the eigenvalues governing this decomposition. We study here algebraic and differential properties of such p-forms by completing previous geometrical results concerning type I spacetimes and extending that analysis to algebraically special spacetimes with at least two distinct PNDs. A number of vacuum and nonvacuum spacetimes are examined to illustrate the general treatment.

The class of Petrov type I curvature tensors is further divided into those for which the span of the set of distinct principal null directions has dimension four (maximally spanning type I) or dimension three (nonmaximally spanning type I). Explicit examples are provided for both vacuum and nonvacuum spacetimes.