An Integral Spectral Representation of the Propagator for the Wave Equation in the Kerr Geometry

We consider the scalar wave equation in the Kerr geometry for Cauchy data which is smooth and compactly supported outside the event horizon. We derive an integral representation which expresses the solution as a superposition of solutions of the radial and angular ODEs which arise in the separation of variables. In particular, we prove completeness of the solutions of the separated ODEs. This integral representation is a suitable starting point for a detailed analysis of the long-time dynamics of scalar waves in the Kerr geometry.Comment: 41 pages, 4 figures, minor correction

Optimal sequential fingerprinting: Wald vs. Tardos

We study sequential collusion-resistant fingerprinting, where the fingerprinting code is generated in advance but accusations may be made between rounds, and show that in this setting both the dynamic Tardos scheme and schemes building upon Wald's sequential probability ratio test (SPRT) are asymptotically optimal. We further compare these two approaches to sequential fingerprinting, highlighting differences between the two schemes. Based on these differences, we argue that Wald's scheme should in general be preferred over the dynamic Tardos scheme, even though both schemes have their merits. As a side result, we derive an optimal sequential group testing method for the classical model, which can easily be generalized to different group testing models.Comment: 12 pages, 10 figure

Efficient Triangle Counting in Large Graphs via Degree-based Vertex Partitioning

The number of triangles is a computationally expensive graph statistic which is frequently used in complex network analysis (e.g., transitivity ratio), in various random graph models (e.g., exponential random graph model) and in important real world applications such as spam detection, uncovering of the hidden thematic structure of the Web and link recommendation. Counting triangles in graphs with millions and billions of edges requires algorithms which run fast, use small amount of space, provide accurate estimates of the number of triangles and preferably are parallelizable. In this paper we present an efficient triangle counting algorithm which can be adapted to the semistreaming model. The key idea of our algorithm is to combine the sampling algorithm of Tsourakakis et al. and the partitioning of the set of vertices into a high degree and a low degree subset respectively as in the Alon, Yuster and Zwick work treating each set appropriately. We obtain a running time $O \left(m + \frac{m^{3/2} \Delta \log{n}}{t \epsilon^2} \right)$ and an $\epsilon$ approximation (multiplicative error), where $n$ is the number of vertices, $m$ the number of edges and $\Delta$ the maximum number of triangles an edge is contained. Furthermore, we show how this algorithm can be adapted to the semistreaming model with space usage $O\left(m^{1/2}\log{n} + \frac{m^{3/2} \Delta \log{n}}{t \epsilon^2} \right)$ and a constant number of passes (three) over the graph stream. We apply our methods in various networks with several millions of edges and we obtain excellent results. Finally, we propose a random projection based method for triangle counting and provide a sufficient condition to obtain an estimate with low variance.Comment: 1) 12 pages 2) To appear in the 7th Workshop on Algorithms and Models for the Web Graph (WAW 2010

Musashi and Plasticity of Xenopus and Axolotl Spinal Cord Ependymal Cells

The differentiated state of spinal cord ependymal cells in regeneration-competent amphibians varies between a constitutively active state in what is essentially a developing organism, the tadpole of the frog Xenopus laevis, and a quiescent, activatable state in a slowly growing adult salamander Ambystoma mexicanum, the Axolotl. Ependymal cells are epithelial in intact spinal cord of all vertebrates. After transection, body region ependymal epithelium in both Xenopus and the Axolotl disorganizes for regenerative outgrowth (gap replacement). Injury-reactive ependymal cells serve as a stem/progenitor cell population in regeneration and reconstruct the central canal. Expression patterns of mRNA and protein for the stem/progenitor cell-maintenance Notch signaling pathway mRNA-binding protein Musashi (msi) change with life stage and regeneration competence. Msi-1 is missing (immunohistochemistry), or at very low levels (polymerase chain reaction, PCR), in both intact regeneration-competent adult Axolotl cord and intact non-regeneration-competent Xenopus tadpole (Nieuwkoop and Faber stage 62+, NF 62+). The critical correlation for successful regeneration is msi-1 expression/upregulation after injury in the ependymal outgrowth and stump-region ependymal cells. msi-1 and msi-2 isoforms were cloned for the Axolotl as well as previously unknown isoforms of Xenopus msi-2. Intact Xenopus spinal cord ependymal cells show a loss of msi-1 expression between regeneration-competent (NF 50–53) and non-regenerating stages (NF 62+) and in post-metamorphosis froglets, while msi-2 displays a lower molecular weight isoform in non-regenerating cord. In the Axolotl, embryos and juveniles maintain Msi-1 expression in the intact cord. In the adult Axolotl, Msi-1 is absent, but upregulates after injury. Msi-2 levels are more variable among Axolotl life stages: rising between late tailbud embryos and juveniles and decreasing in adult cord. Cultures of regeneration-competent Xenopus tadpole cord and injury-responsive adult Axolotl cord ependymal cells showed an identical growth factor response. Epidermal growth factor (EGF) maintains mesenchymal outgrowth in vitro, the cells are proliferative and maintain msi-1 expression. Non-regeneration competent Xenopus ependymal cells, NF 62+, failed to attach or grow well in EGF+ medium. Ependymal Msi-1 expression in vivo and in vitro is a strong indicator of regeneration competence in the amphibian spinal cord

Pseudospectral Calculation of the Wavefunction of Helium and the Negative Hydrogen Ion

We study the numerical solution of the non-relativistic Schr\"{o}dinger equation for two-electron atoms in ground and excited S-states using pseudospectral (PS) methods of calculation. The calculation achieves convergence rates for the energy, Cauchy error in the wavefunction, and variance in local energy that are exponentially fast for all practical purposes. The method requires three separate subdomains to handle the wavefunction's cusp-like behavior near the two-particle coalescences. The use of three subdomains is essential to maintaining exponential convergence. A comparison of several different treatments of the cusps and the semi-infinite domain suggest that the simplest prescription is sufficient. For many purposes it proves unnecessary to handle the logarithmic behavior near the three-particle coalescence in a special way. The PS method has many virtues: no explicit assumptions need be made about the asymptotic behavior of the wavefunction near cusps or at large distances, the local energy is exactly equal to the calculated global energy at all collocation points, local errors go down everywhere with increasing resolution, the effective basis using Chebyshev polynomials is complete and simple, and the method is easily extensible to other bound states. This study serves as a proof-of-principle of the method for more general two- and possibly three-electron applications.Comment: 23 pages, 20 figures, 2 tables, Final refereed version - Some references added, some stylistic changes, added paragraph to matrix methods section, added last sentence to abstract

QUANTIZATION OF A CLASS OF PIECEWISE AFFINE TRANSFORMATIONS ON THE TORUS

We present a unified framework for the quantization of a family of discrete dynamical systems of varying degrees of "chaoticity". The systems to be quantized are piecewise affine maps on the two-torus, viewed as phase space, and include the automorphisms, translations and skew translations. We then treat some discontinuous transformations such as the Baker map and the sawtooth-like maps. Our approach extends some ideas from geometric quantization and it is both conceptually and calculationally simple.Comment: no. 28 pages in AMSTE

Equivalence of Local and Separable Realizations of the Discontinuity-Inducing Contact Interaction and Its Perturbative Renormalizability

We prove that the separable and local approximations of the discontinuity-inducing zero-range interaction in one-dimensional quantum mechanics are equivalent. We further show that the interaction allows the perturbative treatment through the coupling renormalization. Keywords: one-dimensional system, generalized contact interaction, renormalization, perturbative expansion. PACS Nos: 3.65.-w, 11.10.Gh, 31.15.MdComment: ReVTeX 7pgs, doubl column, no figure, See also the website http://www.mech.kochi-tech.ac.jp/cheon

Dynamical Masses of Young Star Clusters in NGC 4038/4039

In order to estimate the masses of the compact, young star clusters in the merging galaxy pair, NGC 4038/4039 (``the Antennae''), we have obtained medium and high resolution spectroscopy using ISAAC on VLT-UT1 and UVES on VLT-UT2 of five such clusters. The velocity dispersions were estimated using the stellar absorption features of CO at 2.29 microns and metal absorption lines at around 8500 \AA, including lines of the Calcium Triplet. The size scales and light profiles were measured from HST images. From these data and assuming Virial equilibrium, we estimated the masses of five clusters. The resulting masses range from 6.5 x 10^5 to 4.7 x 10^6 M_sun. These masses are large, factor of a few to more than 10 larger than the typical mass of a globular cluster in the Milky Way. The mass-to-light ratios for these clusters in the V- and K-bands in comparison with stellar synthesis models suggest that to first order the IMF slopes are approximately consistent with Salpeter for a mass range of 0.1 to 100 M_sun. However, the clusters show a significant range of possible IMF slopes or lower mass cut-offs and that these variations may correlate with the interstellar environment of the cluster. Comparison with the results of Fokker-Planck simulations of compact clusters with properties similar to the clusters studied here, suggest that they are likely to be long-lived and may lose a substantial fraction of their total mass. This mass loss would make the star clusters obtain masses which are comparable to the typical mass of a globular cluster.Comment: 16 pages, 12 figures, A&A accepte