Proceedings of the London Mathematical Society最新文献

We consider the nonlinear Schrödinger equation on the half-line $x⩾0$ with a Robin boundary condition at $x=0$ and with initial data in the weighted Sobolev space $H^{1,1}\left(R_{+}\right)$ . We prove that there exists a global weak solution of this initial-boundary value problem and provide a representation for the solution in terms of the solution of a Riemann-Hilbert problem. Using this representation, we obtain asymptotic formulas for the long-time behavior of the solution. In particular, by restricting our asymptotic result to solutions whose initial data are close to the initial profile of the stationary one-soliton, we obtain results on the asymptotic stability of the stationary one-soliton under any small perturbation in $H^{1,1}\left(R_{+}\right)$ . In the focusing case, such a result was already established by Deift and Park using different methods, and our work provides an alternative approach to obtain such results. We treat both the focusing and the defocusing versions of the equation.

Introduction: The role of the human eye in severe acute respiratory syndrome coronavirus 2 (SARS-COV-2) is still under investigation. The pathophysiology of the ocular findings is arduous when dealing with critically ill Covid-19 patients with comorbidities. Multiorgan involvement and the effects of inflammation, infection and systemic treatment on the retina are complex, and comparison of studies is difficult. Most studies in human patients have investigated the anterior segment, whereas few reports deal with the posterior segment of the eye. The present review aims to evaluate the retinal manifestations and imaging features in COVID-19 patients.

Methods: Studies on the retinal manifestations and retinal imaging in COVID-19 patients published through June 2021 were reviewed. We included cross-sectional and case-control studies, case series, case reports and correspondence in the analysis.

Results: Flame-shaped hemorrhages, cotton wool spots, augmented diameter and tortuosity of retinal vessels were found on funduscopic examination. Peripapillary, macular retinal nerve fiber layer and ganglion cell layer thickness alterations were reported on spectral domain optical coherence tomography. Reduced vessel density of the superficial and deep retinal capillary plexus on optical coherence tomography angiography was reported.

Conclusions: Retinal complications may arise in COVID-19 patients. Although no consensus on presentation is currently available, retinal funduscopy and imaging has shown neuronal and vascular alterations. Systemic neurological complications and microangiopathy are associated with SARS-COV-2; thus, as the retina has a neuronal and vascular component, funduscopy and retinal imaging on COVID-19 patients can provide further insight to SARS-COV-2 disease and the follow-up of patients.

This is a companion paper to earlier work of the authors (Preprint, arXiv:1604.03466, 2016), which interprets the Heegaard Floer homology for a manifold with torus boundary in terms of immersed curves in a punctured torus. We establish a variety of properties of this invariant, paying particular attention to its relation to knot Floer homology, the Thurston norm, and the Turaev torsion. We also give a geometric description of the gradings package from bordered Heegaard Floer homology and establish a symmetry under ${Spin}^c$ conjugation; this symmetry gives rise to genus one mutation invariance in Heegaard Floer homology for closed three-manifolds. Finally, we include more speculative discussions on relationships with Seiberg-Witten theory, Khovanov homology, and ${\mathrm{HF}}^{\pm }$ . Many examples are included.