On the minimization of Dirichlet eigenvalues of the Laplace operator

We study the variational problem \inf \{\lambda_k(\Omega): \Omega\ \textup{open in}\ \R^m,\ |\Omega| < \infty, \ \h(\partial \Omega) \le 1 \}, where $\lambda_k(\Omega)$ is the $k$'th eigenvalue of the Dirichlet Laplacian acting in $L^2(\Omega)$, \h(\partial \Omega) is the $(m-1)$- dimensional Hausdorff measure of the boundary of $\Omega$, and $|\Omega|$ is the Lebesgue measure of $\Omega$. If $m=2$, and $k=2,3, \cdots$, then there exists a convex minimiser $\Omega_{2,k}$. If $m \ge 2$, and if $\Omega_{m,k}$ is a minimiser, then $\Omega_{m,k}^*:= \textup{int}(\overline{\Omega_{m,k}})$ is also a minimiser, and $\R^m\setminus \Omega_{m,k}^*$ is connected. Upper bounds are obtained for the number of components of $\Omega_{m,k}$. It is shown that if $m\ge 3$, and $k\le m+1$ then $\Omega_{m,k}$ has at most $4$ components. Furthermore $\Omega_{m,k}$ is connected in the following cases : (i) $m\ge 2, k=2,$ (ii) $m=3,4,5,$ and $k=3,4,$ (iii) $m=4,5,$ and $k=5,$ (iv) $m=5$ and $k=6$. Finally, upper bounds on the number of components are obtained for minimisers for other constraints such as the Lebesgue measure and the torsional rigidity.Comment: 16 page

Weyl formulas for annular ray-splitting billiards

We consider the distribution of eigenvalues for the wave equation in annular (electromagnetic or acoustic) ray-splitting billiards. These systems are interesting in that the derivation of the associated smoothed spectral counting function can be considered as a canonical problem. This is achieved by extending a formalism developed by Berry and Howls for ordinary (without ray-splitting) billiards. Our results are confirmed by numerical computations and permit us to infer a set of rules useful in order to obtain Weyl formulas for more general ray-splitting billiards

Limitation of entanglement due to spatial qubit separation

We consider spatially separated qubits coupled to a thermal bosonic field that causes pure dephasing. Our focus is on the entanglement of two Bell states which for vanishing separation are known as robust and fragile entangled states. The reduced two-qubit dynamics is solved exactly and explicitly. Our results allow us to gain information about the robustness of two-qubit decoherence-free subspaces with respect to physical parameters such as temperature, qubit-bath coupling strength and spatial separation of the qubits. Moreover, we clarify the relation between single-qubit coherence and two-qubit entanglement and identify parameter regimes in which the terms robust and fragile are no longer appropriate.Comment: 7 pages, 3 figures; revised version, accepted for publication in Europhys. Let

Molecular wires acting as quantum heat ratchets

We explore heat transfer in molecular junctions between two leads in the absence of a finite net thermal bias. The application of an unbiased, time-periodic temperature modulation of the leads entails a dynamical breaking of reflection symmetry, such that a directed heat current may emerge (ratchet effect). In particular, we consider two cases of adiabatically slow driving, namely (i) periodic temperature modulation of only one lead and (ii) temperature modulation of both leads with an ac driving that contains a second harmonic, thus generating harmonic mixing. Both scenarios yield sizeable directed heat currents which should be detectable with present techniques. Adding a static thermal bias, allows one to compute the heat current-thermal load characteristics which includes the ratchet effect of negative thermal bias with positive-valued heat flow against the thermal bias, up to the thermal stop-load. The ratchet heat flow in turn generates also an electric current. An applied electric stop-voltage, yielding effective zero electric current flow, then mimics a solely heat-ratchet-induced thermopower (``ratchet Seebeck effect''), although no net thermal bias is acting. Moreover, we find that the relative phase between the two harmonics in scenario (ii) enables steering the net heat current into a direction of choice.Comment: 9 pages, 8 figure

Anomalous slow fidelity decay for symmetry breaking perturbations

Symmetries as well as other special conditions can cause anomalous slowing down of fidelity decay. These situations will be characterized, and a family of random matrix models to emulate them generically presented. An analytic solution based on exponentiated linear response will be given. For one representative case the exact solution is obtained from a supersymmetric calculation. The results agree well with dynamical calculations for a kicked top.Comment: 4 pages, 2 figure

The k-Point Random Matrix Kernels Obtained from One-Point Supermatrix Models

The k-point correlation functions of the Gaussian Random Matrix Ensembles are certain determinants of functions which depend on only two arguments. They are referred to as kernels, since they are the building blocks of all correlations. We show that the kernels are obtained, for arbitrary level number, directly from supermatrix models for one-point functions. More precisely, the generating functions of the one-point functions are equivalent to the kernels. This is surprising, because it implies that already the one-point generating function holds essential information about the k-point correlations. This also establishes a link to the averaged ratios of spectral determinants, i.e. of characteristic polynomials

A Novel Algorithm for the Determination of Bacterial Cell Volumes That is Unbiased by Cell Morphology

The determination of cell volumes and biomass offers a means of comparing the standing stocks of auto- and heterotrophic microbes of vastly different sizes for applications including the assessment of the flux of organic carbon within aquatic ecosystems. Conclusions about the importance of particular genotypes within microbial communities (e.g., of filamentous bacteria) may strongly depend on whether their contribution to total abundance or to biomass is regarded. Fluorescence microscopy and image analysis are suitable tools for determining bacterial biomass that moreover hold the potential to replace labor-intensive manual measurements by fully automated approaches. However, the current approaches to calculate bacterial cell volumes from digital images are intrinsically biased by the models that are used to approximate the morphology of the cells. Therefore, we developed a generic contour based algorithm to reconstruct the volumes of prokaryotic cells from two-dimensional representations (i.e., microscopic images) irrespective of their shape. Geometric models of commonly encountered bacterial morphotypes were used to verify the algorithm and to compare its performance with previously described approaches. The algorithm is embedded in a freely available computer program that is able to process both raw (8-bit grayscale) and thresholded (binary) images in a fully automated manne

A Novel Algorithm for the Determination of Bacterial Cell Volumes That is Unbiased by Cell Morphology

The determination of cell volumes and biomass offers a means of comparing the standing stocks of auto- and heterotrophic microbes of vastly different sizes for applications including the assessment of the flux of organic carbon within aquatic ecosystems. Conclusions about the importance of particular genotypes within microbial communities (e.g., of filamentous bacteria) may strongly depend on whether their contribution to total abundance or to biomass is regarded. Fluorescence microscopy and image analysis are suitable tools for determining bacterial biomass that moreover hold the potential to replace labor-intensive manual measurements by fully automated approaches. However, the current approaches to calculate bacterial cell volumes from digital images are intrinsically biased by the models that are used to approximate the morphology of the cells. Therefore, we developed a generic contour based algorithm to reconstruct the volumes of prokaryotic cells from two-dimensional representations (i.e., microscopic images) irrespective of their shape. Geometric models of commonly encountered bacterial morphotypes were used to verify the algorithm and to compare its performance with previously described approaches. The algorithm is embedded in a freely available computer program that is able to process both raw (8-bit grayscale) and thresholded (binary) images in a fully automated manner

Anomalous Seismic Amplitudes Measured in the Los Angeles Basin Interpreted as a Basin-Edge Diffraction Catastrophe

The Los Angeles Basin Passive Seismic Experiment (labpse) involved the installation of an array of 18 seismic stations along a line crossing the Los Angeles basin from the foothills of the San Gabriel Mountains through the Puente Hills to the coast. At 3–5 km spacing between stations the array has much higher resolution than the permanent network of stations in southern California. This resolution was found to be important for analyzing the factors that govern the amplitude variation across the basin. We inverted spectra of P- and S-body-wave seismograms from local earthquakes (M_L 2.1–4.8) for site effects, attenuation, and corner frequency factor using a standard model that assumes geometric spreading varying as inverse distance, exponential attenuation, and an ω^2 source model. The S-wave attenuation was separable into basin and bedrock contributions. In addition to the body-wave analysis, S-wave coda were analyzed for coda Q and coda-determined site effects. We find S- wave Q (Q_S) in bedrock is higher than in the basin. High-frequency Q_S is higher than low-frequency Q_S. Coda Q (Q_c) is higher than Q_S. P-wave Q (Q_P) was not separable into basement and bedrock values, so we determined an average value only. The corner frequencies for P and S waves were found to be nearly the same. The standard model fit over 97% of the S-wave data, but data from six clustered events incident along the basin edge within a restricted range of incidence and azimuth angles generated anomalous amplitudes of up to a factor of 5 higher than predicted. We test whether such basin-edge focusing might be modeled by catastrophe theory. After ruling out site, attenuation, and radiation effects, we conclude a caustic modeled as a diffraction catastrophe could explain both the frequency and spatial dependence of the anomalous variation