On a conjecture of A. Magnus concerning the asymptotic behavior of the recurrence coefficients of the generalized Jacobi polynomials

In 1995 Magnus posed a conjecture about the asymptotics of the recurrence coefficients of orthogonal polynomials with respect to the weights on [-1,1] of the form $$(1-x)^\alpha (1+x)^\beta |x_0 - x|^\gamma \times a jump at x_0,$$ with $\alpha, \beta, \gamma>-1$ and $x_0 \in (-1,1)$. We show rigorously that Magnus' conjecture is correct even in a more general situation, when the weight above has an extra factor, which is analytic in a neighborhood of [-1,1] and positive on the interval. The proof is based on the steepest descendent method of Deift and Zhou applied to the non-commutative Riemann-Hilbert problem characterizing the orthogonal polynomials. A feature of this situation is that the local analysis at $x_0$ has to be carried out in terms of confluent hypergeometric functions.Comment: 29 pages, 4 figure

Stability mapping of bipartite tight-binding graphs with losses and gain: PT−{\cal PT}-symmetry and beyond

We consider bipartite tight-binding graphs composed by $N$ nodes split into two sets of equal size: one set containing nodes with on-site loss, the other set having nodes with on-site gain. The nodes are connected randomly with probability $p$. We give a rationale for the relevance of such "throttle/brake" coupled systems (physically open systems) to grasp the stability issues of complex networks in areas such as biochemistry, neurons or economy, for which their modelling in terms of non-hermitian Hamiltonians is still in infancy. Specifically, we measure the connectivity between the two sets with the parameter $\alpha$, which is the ratio of current adjacent pairs over the total number of possible adjacent pairs between the sets. For general undirected-graph setups, the non-hermitian Hamiltonian $H(\gamma,\alpha,N)$ of this model presents pseudo-Hermiticity, where $\gamma$ is the loss/gain strength. However, we show that for a given graph setup $H(\gamma,\alpha,N)$ becomes ${\cal PT}-$symmetric. In both scenarios (pseudo-Hermiticity and ${\cal PT}-$symmetric), depending on the parameter combination, the spectra of $H(\gamma,\alpha,N)$ can be real even when it is non-hermitian. Thus, we numerically characterize the average fractions of real and imaginary eigenvalues of $H(\gamma,\alpha,N)$ as a function of the parameter set $\{\gamma,\alpha,N\}$. We demonstrate, for both setups, that there is a well defined sector of the $\gamma\alpha-$plane (which grows with $N$) where the spectrum of $H(\gamma,\alpha,N)$ is predominantly real.Comment: 10 pages, 9 figure

Position guidelines and evidence base concerning determinants of childhood obesity with a European perspective

Childhood obesity is one of the most pressing global public health issues, with rates increasing fastest in countries at low levels of income. Obesity occurring during childhood is likely to persist throughout the life course, and it is a cause of increased disease risk from the early years of life. This supplement is the result of collaborations involving a large and multidisciplinary group of researchers that were established in the context of the ongoing European Horizon 2020 project Science and Technology in childhood Obesity Policy (STOP). The aim, as in the entire STOP project, is to generate evidence that can support better policies to tackle the problem of childhood obesity in Europe and elsewhere. Quality of life and health well-being concerning children needs to consider personalized, population, and planetary facets to tackle childhood obesity at early stages of life, for in-deep phenotyping, integrating personalized medicine and precision public health interventions at global levels. This supplement contributes to this aim. © 2021 The Authors. Obesity Reviews published by John Wiley & Sons Ltd on behalf of World Obesity Federation

Spectral and localization properties of random bipartite graphs

Bipartite graphs are often found to represent the connectivity between the components of many systems such as ecosystems. A bipartite graph is a set of $n$ nodes that is decomposed into two disjoint subsets, having $m$ and $n-m$ vertices each, such that there are no adjacent vertices within the same set. The connectivity between both sets, which is the relevant quantity in terms of connections, can be quantified by a parameter $\alpha\in[0,1]$ that equals the ratio of existent adjacent pairs over the total number of possible adjacent pairs. Here, we study the spectral and localization properties of such random bipartite graphs. Specifically, within a Random Matrix Theory (RMT) approach, we identify a scaling parameter $\xi\equiv\xi(n,m,\alpha)$ that fixes the localization properties of the eigenvectors of the adjacency matrices of random bipartite graphs. We also show that, when $\xi10$) the eigenvectors are localized (extended), whereas the localization--to--delocalization transition occurs in the interval $1/10<\xi<10$. Finally, given the potential applications of our findings, we round off the study by demonstrating that for fixed $\xi$, the spectral properties of our graph model are also universal.Comment: 17 pages, 10 figure

Lower Cretaceous (Hauterivian-Albian) ammonite biostratigraphy in the Maestrat Basin (E Spain)

Peer reviewedPublisher PD

Turbulent Erosion of Magnetic Flux Tubes

Results from a numerical and analytical investigation of the solution of a nonlinear axially symmetric diffusion equation for the magnetic field are presented for the case when the nonlinear dependence of the diffusivity nu(B) on the magnetic field satisfies basic physical requirements. We find that for sufficiently strong nonlinearity (i.e. for sufficiently strong reduction of nu inside the tube) a current sheet is spontaneously formed around the tube within one diffusion timescale. This sheet propagates inwards with a velocity inversely proportional to the ratio of the field strength just inside the current sheet to the equipartition field strength B0/Be, so the lifetime of a tube with constant internal flux density is increased approximately by a factor not exceeding B0/Be, even for infinitely effective inhibition of turbulence inside the tube. Among the applications of these results we point out that toroidal flux tubes in the solar convective zone are subject to significant flux loss owing to turbulent erosion on a timescale of about 1 month, and that turbulent erosion may be responsible for the formation of a current sheet around a sunspot. It is further proposed that, despite the simplifying assumptions involved, our solutions correctly reflect the essential features of the sunspot decay process.Comment: 17 pages, 11 figure

Towards an easy-to-use D.S.P

The use of digital signal processors is not yet commonly widespread despite their obvious advantages . In this paper we present a certain number of ideas helping case their use. These ideas have been put to work by conceiving the architecture of a processor which is both optimal and easy to use .L'utilisation des processeurs de signaux n'est pas encore très courante, en dépit des avantages qu'ils procurent . Nous présentons ici un certain nombre de réflexions menées dans le sens d'une plus grande facilité d'utilisation de ces processeurs . Ces réflexions ont été concrétisées dans l'élaboration d'une architecture de processeur, à la fois optimisée et d'utilisation aisée