The numerical solution of the Helmholtz equation for wave propagation problems in underwater acoustics

The Helmholtz Equation (-delta-K(2)n(2))u=0 with a variable index of refraction, n, and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems. A numerical algorithm was developed and a computer code implemented that can effectively solve this equation in the intermediate frequency range. The equation is discretized using the finite element method, thus allowing for the modeling of complicated geometrices (including interfaces) and complicated boundary conditions. A global radiation boundary condition is imposed at the far field boundary that is exact for an arbitrary number of propagating modes. The resulting large, non-selfadjoint system of linear equations with indefinite symmetric part is solved using the preconditioned conjugate gradient method applied to the normal equations. A new preconditioner is developed based on the multigrid method. This preconditioner is vectorizable and is extremely effective over a wide range of frequencies provided the number of grid levels is reduced for large frequencies. A heuristic argument is given that indicates the superior convergence properties of this preconditioner

On accuracy conditions for the numerical computation of waves

The Helmholtz equation (Delta + K(2)n(2))u = f with a variable index of refraction n, and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems. Such problems can be solved numerically by first truncating the given unbounded domain and imposing a suitable outgoing radiation condition on an artificial boundary and then solving the resulting problem on the bounded domain by direct discretization (for example, using a finite element method). In practical applications, the mesh size h and the wave number K, are not independent but are constrained by the accuracy of the desired computation. It will be shown that the number of points per wavelength, measured by (Kh)(-1), is not sufficient to determine the accuracy of a given discretization. For example, the quantity K(3)h(2) is shown to determine the accuracy in the L(2) norm for a second-order discretization method applied to several propagation models

Segment-specific expression of 2P domain potassium channel genes in human nephron.

BackgroundThe 2P domain potassium (K2P) channels are a recently discovered ion channel superfamily. Structurally, K2P channels are distinguished by the presence of two pore forming loops within one channel subunit. Functionally, they are characterized by their ability to pass potassium across the physiologic voltage range. Thus, K2P channels are also called open rectifier, background, or leak potassium channels. Patch clamp studies of renal tubules have described several open rectifier potassium channels that have as yet eluded molecular identification. We sought to determine the segment-specific expression of transcripts for the 14 known K2P channel genes in human nephron to identify potential correlates of native leak channels.MethodsHuman kidney samples were obtained from surgical cases and specific nephron segments were dissected. RNA was extracted and used as template for the generation of cDNA libraries. Real-time polymerase chain reaction (PCR) (TaqMan) was used to analyze gene expression.ResultsWe found significant (P < 0.05) expression of K2P10 in glomerulus, K2P5 in proximal tubule and K2P1 in cortical thick ascending limb of Henle's loop (cTAL) and in distal nephron segments. In addition, we repeatedly detected message for several other K2P channels with less abundance, including K2P3 and K2P6 in glomerulus, K2P10 in proximal tubule, K2P5 in thick ascending limb of Henle's loop, and K2P3, K2P5, and K2P13 in distal nephron segments.ConclusionK2P channels are expressed in specific segments of human kidney. These results provide a step toward assigning K2P channels to previously described native renal leaks

Yu-Shiba-Rusinov states in phase-biased S-QD-S junctions

We study the effects of a phase difference on Yu-Shiba-Rusinov (YSR) states in a spinful Coulomb-blockaded quantum dot contacted by a superconducting loop. In the limit where charging energy is larger than the superconducting gap, we determine the subgap excitation spectrum, the corresponding supercurrent, and the differential conductance as measured by a normal-metal tunnel probe. In absence of a phase difference only one linear combination of the superconductor lead electrons couples to the spin, which gives a single YSR state. With finite phase difference, however, it is effectively a two-channel scattering problem and therefore an additional state emerges from the gap edge. The energy of the phase-dependent YSR states depend on the gate voltage and one state can cross zero energy twice inside the valley with odd occupancy. These crossings are shifted by the phase difference towards the charge degeneracy points, corresponding to larger exchange couplings. Moreover, the zero-energy crossings give rise to resonant peaks in the differential conductance with magnitude $4e^2/h$. Finally, we demonstrate that the quantum fluctuations of the dot spin do not alter qualitatively any of the results.Comment: 13 pages, 7 figure