740 research outputs found
A Class of Second Order Difference Approximation for Solving Space Fractional Diffusion Equations
A class of second order approximations, called the weighted and shifted
Gr\"{u}nwald difference operators, are proposed for Riemann-Liouville
fractional derivatives, with their effective applications to numerically
solving space fractional diffusion equations in one and two dimensions. The
stability and convergence of our difference schemes for space fractional
diffusion equations with constant coefficients in one and two dimensions are
theoretically established. Several numerical examples are implemented to
testify the efficiency of the numerical schemes and confirm the convergence
order, and the numerical results for variable coefficients problem are also
presented.Comment: 24 Page
Dispersion of particles in an infinite-horizon Lorentz gas
We consider a two-dimensional Lorentz gas with infinite horizon. This
paradigmatic model consists of pointlike particles undergoing elastic
collisions with fixed scatterers arranged on a periodic lattice. It was
rigorously shown that when , the distribution of particles is
Gaussian. However, the convergence to this limit is ultraslow, hence it is
practically unattainable. Here we obtain an analytical solution for the Lorentz
gas' kinetics on physically relevant timescales, and find that the density in
its far tails decays as a universal power law of exponent . We also show
that the arrangement of scatterers is imprinted in the shape of the
distribution.Comment: Article with supplemental material: 10 pages, 4 figure
Regge's Einstein-Hilbert Functional on the Double Tetrahedron
The double tetrahedron is the triangulation of the three-sphere gotten by
gluing together two congruent tetrahedra along their boundaries. As a piecewise
flat manifold, its geometry is determined by its six edge lengths, giving a
notion of a metric on the double tetrahedron. We study notions of Einstein
metrics, constant scalar curvature metrics, and the Yamabe problem on the
double tetrahedron, with some reference to the possibilities on a general
piecewise flat manifold. The main tool is analysis of Regge's Einstein-Hilbert
functional, a piecewise flat analogue of the Einstein-Hilbert (or total scalar
curvature) functional on Riemannian manifolds. We study the
Einstein-Hilbert-Regge functional on the space of metrics and on discrete
conformal classes of metrics
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