Towards an Efficient Finite Element Method for the Integral Fractional Laplacian on Polygonal Domains

We explore the connection between fractional order partial differential equations in two or more spatial dimensions with boundary integral operators to develop techniques that enable one to efficiently tackle the integral fractional Laplacian. In particular, we develop techniques for the treatment of the dense stiffness matrix including the computation of the entries, the efficient assembly and storage of a sparse approximation and the efficient solution of the resulting equations. The main idea consists of generalising proven techniques for the treatment of boundary integral equations to general fractional orders. Importantly, the approximation does not make any strong assumptions on the shape of the underlying domain and does not rely on any special structure of the matrix that could be exploited by fast transforms. We demonstrate the flexibility and performance of this approach in a couple of two-dimensional numerical examples

Continuation for thin film hydrodynamics and related scalar problems

This chapter illustrates how to apply continuation techniques in the analysis of a particular class of nonlinear kinetic equations that describe the time evolution through transport equations for a single scalar field like a densities or interface profiles of various types. We first systematically introduce these equations as gradient dynamics combining mass-conserving and nonmass-conserving fluxes followed by a discussion of nonvariational amendmends and a brief introduction to their analysis by numerical continuation. The approach is first applied to a number of common examples of variational equations, namely, Allen-Cahn- and Cahn-Hilliard-type equations including certain thin-film equations for partially wetting liquids on homogeneous and heterogeneous substrates as well as Swift-Hohenberg and Phase-Field-Crystal equations. Second we consider nonvariational examples as the Kuramoto-Sivashinsky equation, convective Allen-Cahn and Cahn-Hilliard equations and thin-film equations describing stationary sliding drops and a transversal front instability in a dip-coating. Through the different examples we illustrate how to employ the numerical tools provided by the packages auto07p and pde2path to determine steady, stationary and time-periodic solutions in one and two dimensions and the resulting bifurcation diagrams. The incorporation of boundary conditions and integral side conditions is also discussed as well as problem-specific implementation issues

Measurement of ZZ production in leptonic final states at {\surd}s of 1.96 TeV at CDF

In this paper we present a precise measurement of the total ZZ production cross section in pp collisions at {\surd}s= 1.96 TeV, using data collected with the CDF II detector corresponding to an integrated luminosity of approximately 6 fb-1. The result is obtained by combining separate measurements in the four-charged (lll'l'), and two-charged-lepton and two-neutral-lepton (llvv) decay modes of the Z. The combined measured cross section for pp {\to} ZZ is 1.64^(+0.44)_(-0.38) pb. This is the most precise measurement of the ZZ production cross section in 1.96 TeV pp collisions to date.Comment: submitted to Phys. Rev. Let

Insights into the High-energy γ-ray Emission of Markarian 501 from Extensive Multifrequency Observations in the Fermi Era

We report on the γ-ray activity of the blazar Mrk501 during the first 480 days of Fermi operation. We find that the average Large Area Telescope (LAT) -ray spectrum of Mrk501 can be well described by a single power-law function with a photon index of 1.78 0.03. While we observe relatively mild flux variations with the Fermi-LAT (within less than a factor of two), we detect remarkable spectral variability where the hardest observed spectral index within the LAT energy range is 1.52 0.14, and the softest one is 2.51 0.20. These unexpected spectral changes do not correlate with the measured flux variations above 0.3GeV. In this paper, we also present the first results from the 4.5 month long multifrequency campaign (2009 March 15August 1) on Mrk501, which included the Very Long Baseline Array (VLBA), Swift, RXTE, MAGIC, and VERITAS, the F-GAMMA, GASP-WEBT, and other collaborations and instruments which provided excellent temporal and energy coverage of the source throughout the entire campaign. The extensive radio to TeV data set from this campaign provides us with the most detailed spectral energy distribution yet collected for this source during its relatively low activity. The average spectral energy distribution of Mrk501 is well described by the standard one-zone synchrotron self-Compton (SSC) model. In the framework of this model, we find that the dominant emission region is characterized by a size r0.1pc (comparable within a factor of few to the size of the partially resolved VLBA core at 15-43 GHz), and that the total jet power (C1044ergs1) constitutes only a small fraction (<103) of the Eddington luminosity. The energy distribution of the freshly accelerated radiating electrons required to fit the time-averaged data has a broken power-law form in the energy range 0.3 GeV-10 TeV, with spectral indices 2.2 and 2.7 below and above the break energy of 20GeV. We argue that such a form is consistent with a scenario in which the bulk of the energy dissipation within the dominant emission zone of Mrk501 is due to relativistic, proton-mediated shocks. We find that the ultrarelativistic electrons and mildly relativistic protons within the blazar zone, if comparable in number, are in approximate energy equipartition, with their energy dominating the jet magnetic field energy by about two orders of magnitude

Explosive crystallization in the presence of melting

An experimental investigation of explosive crystallization (EC) of thin amorphous Ge films deposited on a solid substrate is performed, and a theory of EC front propagation accompanied by melting in a class of films that includes Ge films is developed. The experiments show that the propagation of a planar EC front is possible for a certain range of substrate temperatures and film thicknesses. It is found that for substrate temperatures larger than a certain threshold, the macroscopically planar front leaves behind a columnar microstructure in the crystal. The theory of EC front propagation is based on the experimental observation that the propagating front exhibits a thin layer of Ge melt between the amorphous and crystalline phases. A uniformly propagating planar front solution is determined, whose propagation speed is found as a function of the substrate temperature and the heat loss parameter that, in turn, depends on the film thickness. A linear stability analysis of the uniformly propagating EC front with a melting layer is performed. It is found that in a certain interval of substrate temperatures the EC front undergoes a monotonic morphological instability with a preferred wave number that explains the formation of the columnar structures observed in experiments. We also perform a nonlinear analysis describing the evolution of the morphological instability. The interval of substrate temperatures for which the instability is observed, as well as the wavelength of the columnar structure, are found to be in good agreement with experimental observations.N