Background: Information regarding the cardiac manifestations of COVID-19 is scarce. We performed a systematic and comprehensive echocardiographic evaluation of consecutive patients hospitalized with COVID-19 infection. Methods: 100 consecutive patients diagnosed with COVID-19 infection underwent complete echocardiographic evaluation within 24 hours of admission and were compared to reference values. Echocardiographic studies included left ventricular (LV) systolic and diastolic function, valve hemodynamics and right ventricular (RV) assessment, as well as lung ultrasound. A second exam was performed in case of clinical deterioration. Results: Thirty two patients (32%) had a normal echocardiogram at baseline. The most common cardiac pathology was RV dilatation and dysfunction (observed in 39% of patients), followed by LV diastolic dysfunction (16%) and LV systolic dysfunction (10%). Patients with elevated troponin (20%) or worse clinical condition did not demonstrate any significant difference in LV systolic function compared to patients with normal troponin or better clinical condition, but had worse RV function. Clinical deterioration occurred in 20% of patients. In these patients, the most common echocardiographic abnormality at follow-up was RV function deterioration (12 patients), followed by LV systolic and diastolic deterioration (in 5 patients). Femoral vein thrombosis (DVT) was diagnosed in 5 of 12 patients with RV failure. Conclusions: In COVID-19 infection, LV systolic function is preserved in the majority of patients, but LV diastolic and RV function are impaired. Elevated troponin and poorer clinical grade are associated with worse RV function. In patients presenting with clinical deterioration at follow-up, acute RV dysfunction, with or without DVT, is more common, but acute LV systolic dysfunction was noted in ≈20%.

Abstract. We consider singular solutions of the L 2 -critical biharmonic nonlinear Schrödinger equation. We prove that the blowup rate is bounded by a quartic-root, the solution approaches a quasi-self-similar profile, and a finite amount of L 2 -norm, which is no less than the critical power, concentrates into the singularity. We also prove the existence of a ground-state solution. We use asymptotic analysis to show that the blowup rate of peak-type singular solutions is slightly faster than that of a quartic-root, and the self-similar profile is given by the ground-state standing wave. These findings are verified numerically (up to focusing levels of 10 8 ) using an adaptive grid method. We also use the spectral renormalization method to compute the ground state of the standing-wave equation, and the critical power for collapse, in one, two, and three dimensions.

We present a general framework for constructing singular solutions of nonlinear evolution equations that become singular on a d-dimensional sphere, where d > 1. The asymptotic profile and blowup rate of these solutions are the same as those of solutions of the corresponding one-dimensional equation that become singular at a point. We provide a detailed numerical investigation of these new singular solutions for the following equations: The nonlinear Schrödinger equation iψ t (t, x) + ∆ψ + |ψ| 2σ ψ = 0 with σ > 2, the biharmonic nonlinear Schrödinger equation iψ t (t, x) − ∆ 2 ψ + |ψ| 2σ ψ = 0 with σ > 4, the nonlinear heat equation ψ t (t, x) − ∆ψ − |ψ| 2σ ψ = 0 with σ > 0, and the nonlinear biharmonic heat equation ψ t (t, x) + ∆ 2 ψ − |ψ| 2σ ψ = 0 with σ > 0.

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Aims We aim to assess changes in routine echocardiographic and longitudinal strain parameters in patients recovering from Coronavirus disease 2019 during hospitalization and at 3-month follow-up. Methods and results Routine comprehensive echocardiography and STE of both ventricles were performed during hospitalization for acute coronavirus disease 2019 (COVID-19) infection as part of a prospective pre-designed protocol and compared with echocardiography performed ∼3 months after recovery in 80 patients, using a similar protocol. Significantly improved right ventricle (RV) fractional area change, longer pulmonary acceleration time, lower right atrial pressure, and smaller RV end-diastolic and end-systolic area were observed at the recovery assessment (P < 0.05 for all). RV global longitudinal strain improved at the follow-up evaluation (23.2 ± 5 vs. 21.7 ± 4, P = 0.03), mostly due to improvement in septal segments. Only eight (10%) patients recovering from COVID-19 infection had abnormal ejection fraction (EF) at follow-up. However, LV related routine (E, E/e′, stroke volume, LV size), or STE parameters did not change significantly from the assessment during hospitalization. A significant proportion [36 (45%)] of patients had some deterioration of longitudinal strain at follow-up, and 20 patients (25%) still had abnormal LV STE ∼3 months after COVID-19 acute infection. Conclusion In patients previously discharged from hospitalization due to COVID-19 infection, RV routine echocardiographic and RV STE parameters improve significantly concurrently with improved RV haemodynamics. In contrast, a quarter of patients still have LV systolic dysfunction based on STE cut-offs. Moreover, LV STE does not improve significantly, implying subclinical LV dysfunction may be part and parcel of recovering from COVID-19 infection.

We use asymptotic analysis and numerical simulations to study peak-type singular solutions of the supercritical biharmonic NLS. These solutions have a quartic-root blowup rate, and collapse with a quasi self-similar universal profile, which is a zero-Hamiltonian solution of a fourth-order nonlinear eigenvalue problem.

We present new singular solutions of the biharmonic nonlinear Schrödinger equationThese solutions collapse with the quasi self-similar ring profile ψQ B , whereis the ring width that vanishes at singularity, rmax(t) ∼ r0L α (t) is the ring radius, and α = 4−σThe blowup rate of these solutions is 1 3+α for 4/d ≤ σ < 4, and slightly faster than 1/4 for σ = 4. These solutions are analogous to the ring-type solutions of the nonlinear Schrödinger equation.