248,009 research outputs found
Generalized roof duality and bisubmodular functions
Consider a convex relaxation of a pseudo-boolean function . We
say that the relaxation is {\em totally half-integral} if is a
polyhedral function with half-integral extreme points , and this property is
preserved after adding an arbitrary combination of constraints of the form
, , and where \gamma\in\{0, 1, 1/2} is a
constant. A well-known example is the {\em roof duality} relaxation for
quadratic pseudo-boolean functions . We argue that total half-integrality is
a natural requirement for generalizations of roof duality to arbitrary
pseudo-boolean functions. Our contributions are as follows. First, we provide a
complete characterization of totally half-integral relaxations by
establishing a one-to-one correspondence with {\em bisubmodular functions}.
Second, we give a new characterization of bisubmodular functions. Finally, we
show some relationships between general totally half-integral relaxations and
relaxations based on the roof duality.Comment: 14 pages. Shorter version to appear in NIPS 201
Numerical evaluation of the general massive 2-loop 4-denominator self-mass master integral from differential equations
The differential equation in the external invariant p^2 satisfied by the
master integral of the general massive 2-loop 4-denominator self-mass diagram
is exploited and the expansion of the master integral at p^2=0 is obtained
analytically. The system composed by this differential equation with those of
the master integrals related to the general massive 2-loop sunrise diagram is
numerically solved by the Runge-Kutta method in the complex p^2 plane. A
numerical method to obtain results for values of p^2 at and close to thresholds
and pseudo-thresholds is discussed in details.Comment: Latex, 20 pages, 7 figure
Localized boundary-domain singular integral equations based on harmonic parametrix for divergence-form elliptic PDEs with variable matrix coefficients
This is the post-print version of the Article. The official publised version can be accessed from the links below. Copyright @ 2013 Springer BaselEmploying the localized integral potentials associated with the Laplace operator, the Dirichlet, Neumann and Robin boundary value problems for general variable-coefficient divergence-form second-order elliptic partial differential equations are reduced to some systems of localized boundary-domain singular integral equations. Equivalence of the integral equations systems to the original boundary value problems is proved. It is established that the corresponding localized boundary-domain integral operators belong to the Boutet de Monvel algebra of pseudo-differential operators. Applying the Vishik-Eskin theory based on the factorization method, the Fredholm properties and invertibility of the operators are proved in appropriate Sobolev spaces.This research was supported by the grant EP/H020497/1: "Mathematical Analysis of Localized Boundary-Domain Integral Equations for Variable-Coefficient Boundary Value Problems" from the EPSRC, UK
The Szeg\H{o} kernel in analytic regularity and analytic Fourier Integral Operators
We build a general theory of microlocal (homogeneous) Fourier Integral
Operators in real-analytic regularity, following the general construction in
the smooth case by H\"ormander and Duistermaat. In particular, we prove that
the Boutet-Sj\"ostrand parametrix for the Szeg\H{o} projector at the boundary
of a strongly pseudo-convex real-analytic domain can be realised by an analytic
Fourier Integral Operator. We then study some applications, such as FBI-type
transforms on compact, real-analytic Riemannian manifolds and propagators of
one-homogeneous (pseudo)differential operators
Hadamard Regularization
Motivated by the problem of the dynamics of point-particles in high
post-Newtonian (e.g. 3PN) approximations of general relativity, we consider a
certain class of functions which are smooth except at some isolated points
around which they admit a power-like singular expansion. We review the concepts
of (i) Hadamard ``partie finie'' of such functions at the location of singular
points, (ii) the partie finie of their divergent integral. We present and
investigate different expressions, useful in applications, for the latter
partie finie. To each singular function, we associate a partie-finie (Pf)
pseudo-function. The multiplication of pseudo-functions is defined by the
ordinary (pointwise) product. We construct a delta-pseudo-function on the class
of singular functions, which reduces to the usual notion of Dirac distribution
when applied on smooth functions with compact support. We introduce and analyse
a new derivative operator acting on pseudo-functions, and generalizing, in this
context, the Schwartz distributional derivative. This operator is uniquely
defined up to an arbitrary numerical constant. Time derivatives and partial
derivatives with respect to the singular points are also investigated. In the
course of the paper, all the formulas needed in the application to the physical
problem are derived.Comment: 50 pages, to appear in Journal of Mathematical Physic
Closed-Form Expressions for Irradiance from Non-Uniform Lambertian Luminaires Part I: Linearly-Varying Radiant Exitance
We present a closed-form expression for the irradiance at a point on a surface due to an arbitrary polygonal Lambertian lurninaire with linearly-varying radiant exitance. The solution consists of elementary functions and a single well-behaved special function that can be either approximated directly or computed exactly in terms of classical special functions such as Clausen's integral or the closely related dilogarithm. We first provide a general boundary integral that applies to all planar luminaires and then derive the closed-form expression that applies to arbitrary polygons, which is the result most relevant for global illumination. Our approach is to express the problem as an integral of a simple class of rational functions over regions of the sphere, and to convert the surface integral to a boundary integral using a generalization of irradiance tensors. The result extends the class of available closed-form expressions for computing direct radiative transfer from finite areas to differential areas. We provide an outline of the derivation, a detailed proof of the resulting formula, and complete pseudo-code of the resulting algorithm. Finally, we demonstrate the validity of our algorithm by comparison with Monte Carlo. While there are direct applications of this work, it is primarily of theoretical interest as it introduces much of the machinery needed to derive closed-form solutions for the general case of luminaires with radiance distributions that vary polynomially in both position and direction
Imaging extended sources with coded mask telescopes: Application to the INTEGRAL IBIS/ISGRI instrument
Context. In coded mask techniques, reconstructed sky images are
pseudo-images: they are maps of the correlation between the image recorded on a
detector and an array derived from the coded mask pattern. Aims. The
INTEGRAL/IBIS telescope provides images where the flux of each detected source
is given by the height of the local peak in the correlation map. As such, it
cannot provide an estimate of the flux of an extended source. What is needed is
intensity sky images giving the flux per solide angle as typically done at
other wavelengths. Methods. In this paper, we present the response of the
INTEGRAL IBIS/ISGRI coded mask instrument to extended sources. We develop a
general method based on analytical calculations in order to measure the
intensity and the associated error of any celestial source and validated with
Monte-Carlo simulations. Results. We find that the sensitivity degrades almost
linearly with the source extent. Analytical formulae are given as well as an
easy-to-use recipe for the INTEGRAL user. We check this method on IBIS/ISGRI
data but these results are general and applicable to any coded mask telescope.Comment: 9 pages, 6 figures, Accepted for publication in A&
Character values and Hochschild homology
We present a conjecture (and a proof for G=SL(2)) generalizing a result of J.
Arthur which expresses a character value of a cuspidal representation of a
-adic group as a weighted orbital integral of its matrix coefficient. It
also generalizes a conjecture by the second author proved by Schneider-Stuhler
and (independently) the first author. The latter statement expresses an
elliptic character value as an orbital integral of a pseudo-matrix coefficient
defined via the Chern character map taking value in zeroth Hochschild homology
of the Hecke algebra. The present conjecture generalizes the construction of
pseudo-matrix coefficient using compactly supported Hochschild homology, as
well as a modification of the category of smooth representations, the so called
compactified category of smooth -modules. This newly defined "compactified
pseudo-matrix coefficient" lies in a certain space on which the weighted
orbital integral is a conjugation invariant linear functional, our conjecture
states that evaluating a weighted orbital integral on the compactified
pseudo-matrix coefficient one recovers the corresponding character value of the
representation. We also discuss general properties of that space, building on
works of Waldspurger and Beuzart-Plessis.Comment: A funding acknowledgment is the only change from the previous
version. 26p
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