24 research outputs found
Existence of families of spacetimes with a Newtonian limit
J\"urgen Ehlers developed \emph{frame theory} to better understand the
relationship between general relativity and Newtonian gravity. Frame theory
contains a parameter , which can be thought of as , where
is the speed of light. By construction, frame theory is equivalent to general
relativity for , and reduces to Newtonian gravity for .
Moreover, by setting \ep=\sqrt{\lambda}, frame theory provides a framework to
study the Newtonian limit \ep \searrow 0 (i.e. ). A number of
ideas relating to frame theory that were introduced by J\"urgen have
subsequently found important applications to the rigorous study of both the
Newtonian limit and post-Newtonian expansions. In this article, we review frame
theory and discuss, in a non-technical fashion, some of the rigorous results on
the Newtonian limit and post-Newtonian expansions that have followed from
J\"urgen's work
Knowledge-based energy functions for computational studies of proteins
This chapter discusses theoretical framework and methods for developing
knowledge-based potential functions essential for protein structure prediction,
protein-protein interaction, and protein sequence design. We discuss in some
details about the Miyazawa-Jernigan contact statistical potential,
distance-dependent statistical potentials, as well as geometric statistical
potentials. We also describe a geometric model for developing both linear and
non-linear potential functions by optimization. Applications of knowledge-based
potential functions in protein-decoy discrimination, in protein-protein
interactions, and in protein design are then described. Several issues of
knowledge-based potential functions are finally discussed.Comment: 57 pages, 6 figures. To be published in a book by Springe
Multidimensional Conservation Laws: Overview, Problems, and Perspective
Some of recent important developments are overviewed, several longstanding
open problems are discussed, and a perspective is presented for the
mathematical theory of multidimensional conservation laws. Some basic features
and phenomena of multidimensional hyperbolic conservation laws are revealed,
and some samples of multidimensional systems/models and related important
problems are presented and analyzed with emphasis on the prototypes that have
been solved or may be expected to be solved rigorously at least for some cases.
In particular, multidimensional steady supersonic problems and transonic
problems, shock reflection-diffraction problems, and related effective
nonlinear approaches are analyzed. A theory of divergence-measure vector fields
and related analytical frameworks for the analysis of entropy solutions are
discussed.Comment: 43 pages, 3 figure
Non-existence and uniqueness results for supercritical semilinear elliptic equations
Non-existence and uniqueness results are proved for several local and
non-local supercritical bifurcation problems involving a semilinear elliptic
equation depending on a parameter. The domain is star-shaped but no other
symmetry assumption is required. Uniqueness holds when the bifurcation
parameter is in a certain range. Our approach can be seen, in some cases, as an
extension of non-existence results for non-trivial solutions. It is based on
Rellich-Pohozaev type estimates. Semilinear elliptic equations naturally arise
in many applications, for instance in astrophysics, hydrodynamics or
thermodynamics. We simplify the proof of earlier results by K. Schmitt and R.
Schaaf in the so-called local multiplicative case, extend them to the case of a
non-local dependence on the bifurcation parameter and to the additive case,
both in local and non-local settings.Comment: Annales Henri Poincar\'e (2009) to appea
Entropy Stable Finite Volume Approximations for Ideal Magnetohydrodynamics
This article serves as a summary outlining the mathematical entropy analysis of the ideal magnetohydrodynamic (MHD) equations. We select the ideal MHD equations as they are particularly useful for mathematically modeling a wide variety of magnetized fluids. In order to be self-contained we first motivate the physical properties of a magnetic fluid and how it should behave under the laws of thermodynamics. Next, we introduce a mathematical model built from hyperbolic partial differential equations (PDEs) that translate physical laws into mathematical equations. After an overview of the continuous analysis, we thoroughly describe the derivation of a numerical approximation of the ideal MHD system that remains consistent to the continuous thermodynamic principles. The derivation of the method and the theorems contained within serve as the bulk of the review article. We demonstrate that the derived numerical approximation retains the correct entropic properties of the continuous model and show its applicability to a variety of standard numerical test cases for MHD schemes. We close with our conclusions and a brief discussion on future work in the area of entropy consistent numerical methods and the modeling of plasmas