8,291 research outputs found
Discrete complex analysis on planar quad-graphs
We develop a linear theory of discrete complex analysis on general
quad-graphs, continuing and extending previous work of Duffin, Mercat, Kenyon,
Chelkak and Smirnov on discrete complex analysis on rhombic quad-graphs. Our
approach based on the medial graph yields more instructive proofs of discrete
analogs of several classical theorems and even new results. We provide discrete
counterparts of fundamental concepts in complex analysis such as holomorphic
functions, derivatives, the Laplacian, and exterior calculus. Also, we discuss
discrete versions of important basic theorems such as Green's identities and
Cauchy's integral formulae. For the first time, we discretize Green's first
identity and Cauchy's integral formula for the derivative of a holomorphic
function. In this paper, we focus on planar quad-graphs, but we would like to
mention that many notions and theorems can be adapted to discrete Riemann
surfaces in a straightforward way.
In the case of planar parallelogram-graphs with bounded interior angles and
bounded ratio of side lengths, we construct a discrete Green's function and
discrete Cauchy's kernels with asymptotics comparable to the smooth case.
Further restricting to the integer lattice of a two-dimensional skew coordinate
system yields appropriate discrete Cauchy's integral formulae for higher order
derivatives.Comment: 49 pages, 8 figure
On the Positivity of the Discrete Green's Function for Unstructured Finite Element Discretizations in Three Dimensions
The aim of this paper is twofold. First, we prove estimates for a
regularized Green's function in three dimensions. We then establish new
estimates for the discrete Green's function and obtain some positivity results.
In particular, we prove that the discrete Green's functions with singularity in
the interior of the domain cannot be bounded uniformly with respect of the mesh
parameter . Actually, we show that at the singularity the discrete Green's
function is of order , which is consistent with the behavior of the
continuous Green's function. In addition, we also show that the discrete
Green's function is positive and decays exponentially away from the
singularity. We also provide numerically persistent negative values of the
discrete Green's function on Delaunay meshes which then implies a discrete
Harnack inequality cannot be established for unstructured finite element
discretizations
The two-, three- and four-gluon sector of QCD in the Landau gauge
Due to the nonperturbative masslessness of the ghost field, ghost loops that
contribute to gluon Green's functions in the Landau gauge display infrared
divergences, akin to those one would encounter in a conventional perturbative
treatment. This is in sharp contrast with gluon loops, in which the
perturbative divergences are tamed by the dynamical generation of a gluon mass
acting as an effective infrared cutoff. In this paper, after reviewing the full
nonperturbative origin of this divergence in the two-gluon sector, we discuss
its implications for the three- and four-gluon sector, showing in particular
that some of the form factors characterizing the corresponding Green's
functions are bound to diverge in the infrared.Comment: 13 pages, 7 figures. Talk given at Discrete 2014 - Fourth Symposium
on Prospects in the Physics of Discrete Symmetries. 2-6 December, 2014 -
King's College, London, Englan
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