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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