Complex singularities and PDEs

In this paper we give a review on the computational methods used to characterize the complex singularities developed by some relevant PDEs. We begin by reviewing the singularity tracking method based on the analysis of the Fourier spectrum. We then introduce other methods generally used to detect the hidden singularities. In particular we show some applications of the Pad\'e approximation, of the Kida method, and of Borel-Polya method. We apply these techniques to the study of the singularity formation of some nonlinear dispersive and dissipative one dimensional PDE of the 2D Prandtl equation, of the 2D KP equation, and to Navier-Stokes equation for high Reynolds number incompressible flows in the case of interaction with rigid boundaries

Route to chaos in the weakly stratified Kolmogorov flow

We consider a two-dimensional fluid exposed to Kolmogorov’s forcing cos(ny) and heated from above. The stabilizing effects of temperature are taken into account using the Boussinesq approximation. The fluid with no temperature stratification has been widely studied and, although relying on strong simplifications, it is considered an important tool for the theoretical and experimental study of transition to turbulence. In this paper, we are interested in the set of transitions leading the temperature stratified fluid from the laminar solution [U∝cos(ny),0, T ∝ y] to more complex states until the onset of chaotic states. We will consider Reynolds numbers 0 < Re ≤ 30, while the Richardson numbers shall be kept in the regime of weak stratifications (Ri ≤ 5 × 10 −3 ). We shall first review the non-stratified Kolmogorov flow and find a new period-tripling bifurcation as the precursor of chaotic states. Introducing the stabilizing temperature gradient, we shall observe that higher Re are required to trigger instabilities. More importantly, we shall see new states and phenomena: the newly discovered period-tripling bifurcation is supercritical or subcritical according to Ri; more period-tripling and doubling bifurcations may depart from this new state; strong enough stratifications trigger new regions of chaotic solutions and, on the drifting solution branch, non-chaotic bursting solutions

Radiation hardened transistor characteristics for applications at LHC and beyond

The high radiation environment at the LHC will require the use of radiation hardened microelectronics for the readout of inner detectors. Two such technologies are a Harris bulk CMOS process and the DMILL mixed technology process. Transistors have been fabricated in both of these and have been tested before and after irradiation to 10 Mrads, the total dose expected in the innermost silicon microstrip layers. Several processing runs of Harris transistors have been carried out and samples from one have also been irradiated to 100 Mrads. A preamplifier-shaper circuit, to be used for readout of the CMS microstrip tracker, has been tested and the noise performance is compared with individual transistors

Complex singularity analysis for vortex layer flows

We study the evolution of a 2D vortex layer at high Reynolds number. Vortex layer flows are characterized by intense vorticity concentrated around a curve. In addition to their intrinsic interest, vortex layers are relevant configurations because they are regularizations of vortex sheets. In this paper, we consider vortex layers whose thickness is proportional to the square-root of the viscosity. We investigate the typical roll-up process, showing that crucial phases in the initial flow evolution are the formation of stagnation points and recirculation regions. Stretching and folding characterizes the following stage of the dynamics, and we relate these events to the growth of the palinstrophy. The formation of an inner vorticity core, with vorticity intensity growing to infinity for larger Reynolds number, is the final phase of the dynamics. We display the inner core's self-similar structure, with the scale factor depending on the Reynolds number. We reveal the presence of complex singularities in the solutions of Navier-Stokes equations; these singularities approach the real axis with increasing Reynolds number. The comparison between these singularities and the Birkhoff-Rott singularity seems to suggest that vortex layers, in the limit, behave differently from vortex sheets

Cytokine-induced instabilities in a reaction–diffusion-chemotaxis model of Multiple Sclerosis: Bifurcation analysis and well-posedness

In this paper, we develop a model for the evolution of the Multiple Sclerosis pathology that considers the modulatory influence of cytokines on the activation rate of macrophages. Our starting point is the reaction–diffusion-chemotaxis model proposed in 4, and we modify the macrophage activation mechanism. What triggers the immune cells into an active state is still debated in the medical literature. In this paper, we explore the hypothesis, e.g., Lassmann, (2018), that cytokines mediate the activation mechanism. Our primary focus is on the rigorous analysis of instabilities responsible for the formation of demyelinating lesions and on the qualitative properties of the solution. Through a weakly nonlinear analysis, we characterize the chemotaxis-driven Turing instability and construct the stationary patterns that emerge from this instability. Using biologically relevant parameter values, we show that the asymptotic solutions of our model system reproduce the concentric demyelinating rings, confluent plaques, and preactive lesions observed in Balò sclerosis and type III Multiple Sclerosis. Furthermore, we explore the initiation and progression of demyelinated plaques through extensive numerical simulations on two-dimensional domains. Our findings reveal that the alternative scenario proposed here results in a less aggressive pathology characterized by reduced inflammation levels and significantly slower disease progression. Under the appropriate regularity conditions on the initial data, we prove the existence of a unique global solution to our proposed system. This study provides insights into the role of cytokines in the pathogenesis of Multiple Sclerosis, shedding light on the disease's dynamics and offering potential avenues for therapeutic interventions

Analysis of complex singularities in high-Reynolds-number Navier-Stokes solutions

Numerical solutions of the laminar Prandtl boundary-layer and Navier-Stokes equations are considered for the case of the two-dimensional uniform flow past an impulsively-started circular cylinder. We show how Prandtl's solution develops a finite time separation singularity. On the other hand Navier-Stokes solution is characterized by the presence of two kinds of viscous-inviscid interactions that can be detected by the analysis of the enstrophy and of the pressure gradient on the wall. Moreover we apply the complex singularity tracking method to Prandtl and Navier-Stokes solutions and analyze the previous interactions from a different perspective

Usefulness of regional right ventricular and right atrial strain for prediction of early and late right ventricular failure following a left ventricular assist device implant: A machine learning approach

Background: Identifying candidates for left ventricular assist device surgery at risk of right ventricular failure remains difficult. The aim was to identify the most accurate predictors of right ventricular failure among clinical, biological, and imaging markers, assessed by agreement of different supervised machine learning algorithms. Methods: Seventy-four patients, referred to HeartWare left ventricular assist device since 2010 in two Italian centers, were recruited. Biomarkers, right ventricular standard, and strain echocardiography, as well as cath-lab measures, were compared among patients who did not develop right ventricular failure (N = 56), those with acute–right ventricular failure (N = 8, 11%) or chronic–right ventricular failure (N = 10, 14%). Logistic regression, penalized logistic regression, linear support vector machines, and naïve Bayes algorithms with leave-one-out validation were used to evaluate the efficiency of any combination of three collected variables in an “all-subsets” approach. Results: Michigan risk score combined with central venous pressure assessed invasively and apical longitudinal systolic strain of the right ventricular–free wall were the most significant predictors of acute–right ventricular failure (maximum receiver operating characteristic–area under the curve = 0.95, 95% confidence interval = 0.91–1.00, by the naïve Bayes), while the right ventricular–free wall systolic strain of the middle segment, right atrial strain (QRS-synced), and tricuspid annular plane systolic excursion were the most significant predictors of Chronic-RVF (receiver operating characteristic–area under the curve = 0.97, 95% confidence interval = 0.91–1.00, according to naïve Bayes). Conclusion: Apical right ventricular strain as well as right atrial strain provides complementary information, both critical to predict acute–right ventricular failure and chronic–right ventricular failure, respectively

Classical and quantum vortex leapfrogging in two-dimensional channels

The leapfrogging of coaxial vortex rings is a famous effect which has been noticed since the times of Helmholtz. Recent advances in ultra-cold atomic gases show that the effect can now be studied in quantum fluids. The strong confinement which characterises these systems motivates the study of leapfrogging of vortices within narrow channels. Using the two-dimensional point vortex model, we show that in the constrained geometry of a two-dimensional channel the dynamics is richer than in an unbounded domain: alongside the known regimes of standard leapfrogging and the absence of it, we identify new regimes of image-driven leapfrogging and periodic orbits. Moreover, by solving the Gross-Pitaevskii equation for a Bose-Einstein condensate, we show that all four regimes exist for quantum vortices too. Finally, we discuss the differences between classical and quantum vortex leapfrogging which appear when the quantum healing length becomes significant compared to the vortex separation or the channel size, and when, due to high velocity, compressibility effects in the condensate becomes significant