12,131 research outputs found

    Star Integrals, Convolutions and Simplices

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    We explore single and multi-loop conformal integrals, such as the ones appearing in dual conformal theories in flat space. Using Mellin amplitudes, a large class of higher loop integrals can be written as simple integro-differential operators on star integrals: one-loop nn-gon integrals in nn dimensions. These are known to be given by volumes of hyperbolic simplices. We explicitly compute the five-dimensional pentagon integral in full generality using Schl\"afli's formula. Then, as a first step to understanding higher loops, we use spline technology to construct explicitly the 6d6d hexagon and 8d8d octagon integrals in two-dimensional kinematics. The fully massive hexagon and octagon integrals are then related to the double box and triple box integrals respectively. We comment on the classes of functions needed to express these integrals in general kinematics, involving elliptic functions and beyond.Comment: 23 page

    A Framework for Modeling Subgrid Effects for Two-Phase Flows in Porous Media

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    In this paper, we study upscaling for two-phase flows in strongly heterogeneous porous media. Upscaling a hyperbolic convection equation is known to be very difficult due to the presence of nonlocal memory effects. Even for a linear hyperbolic equation with a shear velocity field, the upscaled equation involves a nonlocal history dependent diffusion term, which is not amenable to computation. By performing a systematic multiscale analysis, we derive coupled equations for the average and the fluctuations for the two-phase flow. The homogenized equations for the coupled system are obtained by projecting the fluctuations onto a suitable subspace. This projection corresponds exactly to averaging along streamlines of the flow. Convergence of the multiscale analysis is verified numerically. Moreover, we show how to apply this multiscale analysis to upscale two-phase flows in practical applications

    Integrable lattice spin models from supersymmetric dualities

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    Recently, there has been observed an interesting correspondence between supersymmetric quiver gauge theories with four supercharges and integrable lattice models of statistical mechanics such that the two-dimensional spin lattice is the quiver diagram, the partition function of the lattice model is the partition function of the gauge theory and the Yang-Baxter equation expresses the identity of partition functions for dual pairs. This correspondence is a powerful tool which enables us to generate new integrable models. The aim of the present paper is to give a short account on a progress in integrable lattice models which has been made due to the relationship with supersymmetric gauge theories.Comment: 35 pages, preliminary versio

    Determinant and Weyl anomaly of Dirac operator: a holographic derivation

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    We present a holographic formula relating functional determinants: the fermion determinant in the one-loop effective action of bulk spinors in an asymptotically locally AdS background, and the determinant of the two-point function of the dual operator at the conformal boundary. The formula originates from AdS/CFT heuristics that map a quantum contribution in the bulk partition function to a subleading large-N contribution in the boundary partition function. We use this holographic picture to address questions in spectral theory and conformal geometry. As an instance, we compute the type-A Weyl anomaly and the determinant of the iterated Dirac operator on round spheres, express the latter in terms of Barnes' multiple gamma function and gain insight into a conjecture by B\"ar and Schopka.Comment: 11 pages; new comments and references added, typos correcte

    An optimal penalty method for a hyperbolic system modeling the edge plasma transport in a tokamak

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    The penalization method is used to take account of obstacles, such as the limiter, in a tokamak. Because of the magnetic confinement of the plasma in a tokamak, the transport occurs essentially in the direction parallel to the magnetic field lines. We study a 1D nonlinear hyperbolic system as a simplified model of the plasma transport in the area close to the wall. A penalization which cuts the flux term of the momentum is studied. We show numerically that this penalization creates a Dirac measure at the plasma-limiter interface which prevents us from defining the transport term in the usual distribution sense. Hence, a new penalty method is proposed for this hyperbolic system. For this penalty method, an asymptotic expansion and numerical tests give an optimal rate of convergence without spurious boundary layer. Another two-fields penalization has also been implemented and the numerical convergence analysis when the penalization parameter tends to 00 reveals the presence of a boundary layer

    Gravitational waves from compact binaries in post-Newtonian accurate hyperbolic orbits

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    We derive from first principles third post-Newtonian (3PN) accurate Keplerian-type parametric solution to describe PN-accurate dynamics of non-spinning compact binaries in hyperbolic orbits. Orbital elements and functions of the parametric solution are obtained in terms of the conserved orbital energy and angular momentum in both Arnowitt-Deser-Misner type and modified harmonic coordinates. Elegant checks are provided that include a modified analytic continuation prescription to obtain our independent hyperbolic parametric solution from its eccentric version. A prescription to model gravitational wave polarization states for hyperbolic compact binaries experiencing 3.5PN-accurate orbital motion is presented that employs our 3PN-accurate parametric solution
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