The mean acoustic intensity, m, in the quiet Sun seems to increase with depth, as shown in Fig. 3. The depth dependence of m may be related to the variation of acoustic absorption with depth and the fraction of the acoustic spatiotemporal spectrum that is detected in viewing different depths. (5) The spatial resolution of constructed images decreases with depth. The time-distance relations flatten with increasing depth, so the phase gradient over the observing annulus decreases. The difference in phases used to image adjacent spatial points becomes small, and a point in space is imaged with a broad point-spread function. Another cause is that shorter-wavelength modes cannot be detected in viewing a deeper region.Finally, we mention some important questions that call for further study. What is the degree of cancellation of signals from points other than the target point in this phased detection? What are the horizontal and vertical spatial resolutions corresponding to this 'computational acoustic lens'? What is the correct interpretation of the mean intensity, m, and its apparent variation with depth? Can this method reconstruct the complex index of refraction of acoustic waves at each target point, using the phase information in addition to the intensity? Finally, could one use this method to detect active regions before they emerge, or active regions at the other side of the Sun?Ⅺ

Solids dispersed in a drying drop will migrate to the edge of the drop and form a solid ring. This phenomenon produces ringlike stains and occurs for a wide range of surfaces, solvents, and solutes. Here we show that the migration is caused by an outward flow within the drop that is driven by the loss of solvent by evaporation and geometrical constraint that the drop maintain an equilibrium droplet shape with a fixed boundary. We describe a theory that predicts the flow velocity, the rate of growth of the ring, and the distribution of solute within the drop. These predictions are compared with our experimental results.

We consider the viscous motion of a thin, axisymmetric column of fluid with a free surface. A one-dimensional equation of motion for the velocity and the radius is derived from the Navier-Stokes equation. We compare with recent experiments on the breakup of a liquid jet and on the bifurcation of a drop suspended from an orifice. The equations form singularities as the fluid neck is pinching off. The nature of the singularities is investigated in detail.

Acute kidney injury (AKI) has been reported in up to 25% of critically-ill patients with SARS-CoV-2 infection, especially in those with underlying comorbidities. AKI is associated with high mortality rates in this setting, especially when renal replacement therapy is required. Several studies have highlighted changes in urinary sediment, including proteinuria and hematuria, and evidence of urinary SARS-CoV-2 excretion, suggesting the presence of a renal reservoir for the virus. The pathophysiology of COVID-19 associated AKI could be related to unspecific mechanisms but also to COVID-specific mechanisms such as direct cellular injury resulting from viral entry through the receptor (ACE2) which is highly expressed in the kidney, an imbalanced renin-angotensin-aldosteron system, pro-inflammatory cytokines elicited by the viral infection and thrombotic events. Non-specific mechanisms include haemodynamic alterations, right heart failure, high levels of PEEP in patients requiring mechanical ventilation, hypovolemia, administration of nephrotoxic drugs and nosocomial sepsis. To date, there is no specific treatment for COVID-19 induced AKI. A number of investigational agents are being explored for antiviral/immunomodulatory treatment of COVID-19 and their impact on AKI is still unknown. Indications, timing and modalities of renal replacement therapy currently rely on non-specific data focusing on patients with sepsis. Further studies focusing on AKI in COVID-19 patients are urgently warranted in order to predict the risk of AKI, to identify the exact mechanisms of renal injury and to suggest targeted interventions.

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Abstract. A ¿-level iterative procedure for solving tbe algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed. The work estimate for this procedure is proportional to the number of unknowns, an optimal order result. General geometry is permitted for the underlying domain, but the shape plays a role in the rate of convergence through elliptic regularity. Finally, a short discussion of the use of this method for parabolic problems is presented.

Summary. We derive and analyze the hierarchical basis-multigrid method for solving discretizations of self-adjoint, elliptic boundary value problems using piecewise linear triangular finite elements. The method is analyzed as a block symmetric GauB-Seidel iteration with inner iterations, but it is strongly related to 2-level methods, to the standard multigrid V-cycle, and to earlier Jacobi-like hierarchical basis methods. The method is very robust, and has a nearly optimal convergence rate and work estimate. It is especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.