1,498 research outputs found
Structure-Preserving Discretization of Fractional Vector Calculus using Discrete Exterior Calculus
Fractional vector calculus is the building block of the fractional partial
differential equations that model non-local or long-range phenomena, e.g.,
anomalous diffusion, fractional electromagnetism, and fractional
advection-dispersion. In this work, we reformulate a type of fractional vector
calculus that uses Caputo fractional partial derivatives and discretize this
reformulation using discrete exterior calculus on a cubical complex in the
structure-preserving way, meaning that the continuous-level properties
and
hold exactly on the
discrete level. We discuss important properties of our fractional discrete
exterior derivatives and verify their second-order convergence in the root mean
square error numerically. Our proposed discretization has the potential to
provide accurate and stable numerical solutions to fractional partial
differential equations and exactly preserve fundamental physics laws on the
discrete level regardless of the mesh size.Comment: 25 pages, 4 figure
Subdivision Directional Fields
We present a novel linear subdivision scheme for face-based tangent
directional fields on triangle meshes. Our subdivision scheme is based on a
novel coordinate-free representation of directional fields as halfedge-based
scalar quantities, bridging the finite-element representation with discrete
exterior calculus. By commuting with differential operators, our subdivision is
structure-preserving: it reproduces curl-free fields precisely, and reproduces
divergence-free fields in the weak sense. Moreover, our subdivision scheme
directly extends to directional fields with several vectors per face by working
on the branched covering space. Finally, we demonstrate how our scheme can be
applied to directional-field design, advection, and robust earth mover's
distance computation, for efficient and robust computation
On the stability of linear fractional difference systems
A fractional linear system is defined by differential or difference equations of
non-integer order. A well-known result about the stability of fractional differential systems will be extended to discrete-time systems defined by fractional difference equations. This will be accomplished using time scales, which permit to unify continuous and discrete-time systems
Fractional Loop Group and Twisted K-Theory
We study the structure of abelian extensions of the group of
-differentiable loops (in the Sobolev sense), generalizing from the case of
central extension of the smooth loop group. This is motivated by the aim of
understanding the problems with current algebras in higher dimensions. Highest
weight modules are constructed for the Lie algebra. The construction is
extended to the current algebra of supersymmetric Wess-Zumino-Witten model. An
application to the twisted K-theory on is discussed.Comment: Final version in Commun. Math. Phy
- …