Some remarks on the isoperimetric problem for the higher eigenvalues of the Robin and Wentzell Laplacians

We consider the problem of minimising the $k$th eigenvalue, $k \geq 2$, of the ($p$-)Laplacian with Robin boundary conditions with respect to all domains in $\mathbb{R}^N$ of given volume $M$. When $k=2$, we prove that the second eigenvalue of the $p$-Laplacian is minimised by the domain consisting of the disjoint union of two balls of equal volume, and that this is the unique domain with this property. For $p=2$ and $k \geq 3$, we prove that in many cases a minimiser cannot be independent of the value of the constant $\alpha$ in the boundary condition, or equivalently of the volume $M$. We obtain similar results for the Laplacian with generalised Wentzell boundary conditions $\Delta u + \beta \frac{\partial u}{\partial \nu} + \gamma u = 0$.Comment: 16 page

An Optimal Dimensionality Multi-shell Sampling Scheme with Accurate and Efficient Transforms for Diffusion MRI

This paper proposes a multi-shell sampling scheme and corresponding transforms for the accurate reconstruction of the diffusion signal in diffusion MRI by expansion in the spherical polar Fourier (SPF) basis. The sampling scheme uses an optimal number of samples, equal to the degrees of freedom of the band-limited diffusion signal in the SPF domain, and allows for computationally efficient reconstruction. We use synthetic data sets to demonstrate that the proposed scheme allows for greater reconstruction accuracy of the diffusion signal than the multi-shell sampling schemes obtained using the generalised electrostatic energy minimisation (gEEM) method used in the Human Connectome Project. We also demonstrate that the proposed sampling scheme allows for increased angular discrimination and improved rotational invariance of reconstruction accuracy than the gEEM schemes.Comment: 4 pages, 4 figures presented at ISBI 201

Scattering of Woods-Saxon Potential in Schrodinger Equation

The scattering solutions of the one-dimensional Schrodinger equation for the Woods-Saxon potential are obtained within the position-dependent mass formalism. The wave functions, transmission and reflection coefficients are calculated in terms of Heun's function. These results are also studied for the constant mass case in detail.Comment: 14 page

Barbaetis: A New Genus of Eastern Nearctic Mayflies (Ephemeroptera: Baetidae)

The new genus Barbaetis Waltz and McCafferty, and new species Barbaetis benfieldi Kennedy are described from larvae collected from the New River, Virginia. Barbaetis is easily told from Baetis by the presence of procoxal osmobranchia. Cladistics of B. benfieldi, related Pseudocloeon species, and the lutheri and pavidus complexes of Baetis are presented and indicate the need for further taxonomic revision. The habitat of B. benfieldi is described in terms of several ecological parameters. The new species demonstrates a univoltine life history with postembryonic development restricted to a short springtime period

Low Momentum Scattering in the Dirac Equation

It is shown that the amplitude for reflection of a Dirac particle with arbitrarily low momentum incident on a potential of finite range is -1 and hence the transmission coefficient T=0 in general. If however the potential supports a half-bound state at k=0 this result does not hold. In the case of an asymmetric potential the transmission coefficient T will be non-zero whilst for a symmetric potential T=1.Comment: 12 pages; revised to include additional references; to be published in J Phys

Photoionization of the Ne-like Si4+ ion in ground and metastable states in the 110–184-eV photon energy range

We present measurements of the absolute photoionization cross section of the neonlike Si4+ ion over the 110–184 eV photon energy range. The measurements were performed using two independent merged-beam setups at the super-ACO and ASTRID synchrotron-radiation facilities, respectively. Signals produced in the photoionization of the 2p subshell of the Si4+ ion both from the 2p6 1S0 ground state and the 2p53s 3P0,2 metastable levels were observed. Calculations of the 2p photoionization cross sections were carried out using a multi-configuration Dirac-Fock code. They give results in good agreement with the measured spectra. Comparison with other available theoretical results is also presented