In this paper, we develop and analyze
the Runge-Kutta discontinuous Galerkin
(RKDG) method to solve weakly coupled hyperbolic
multi-domain problems. Such problems involve transfer type
boundary conditions with discontinuous fluxes
between different domains, calling for
special techniques to prove stability of the RKDG methods.
We prove both stability and error estimates for our
RKDG methods on simple models, and then apply them to a
biological cell proliferation model \cite{EMSC}. Numerical
results are provided to illustrate the good behavior of
our RKDG methods.
A classification of $\SLn$ contravariant, continuous function-valued valuations on convex bodies is established.
Such valuations are natural extensions of $\SLn$ contravariant $L_p$ Minkowski valuations, the classification of which characterized $L_p$ projection bodies, which are fundamental in the $L_p$ Brunn-Minkowski theory, for $p \geq 1$.
Hence our result will help to better understand extensions of the $L_p$ Brunn-Minkowski theory.
In fact, our results characterize general projection functions which extend $L_p$ projection functions ($p$-th powers of the support functions of $L_p$ projection bodies) to projection functions in the $L_p$ Brunn-Minkowski theory for $0< p < 1$ and in the Orlicz Brunn-Minkowski theory.
Let X, Y be realcompact spaces or completely regular spaces consisting of X, Y -points. Let X, Y be a linear bijective map from X, Y (resp. X, Y ) onto X, Y (resp. X, Y ). We show that if X, Y preserves nonvanishing functions, that is,
Three-fold quasi-homogeneous isolated rational singularity is argued to define a four dimensional N = 2 SCFT. The SeibergWitten geometry is built on the mini-versal deformation of the singularity. We argue in this paper that the corresponding SeibergWitten differential is given by the GelfandLeray form of K. Saitos primitive form. Our result also extends the SeibergWitten solution to include irrelevant deformations.