15 research outputs found
Characteristic Evolution and Matching
I review the development of numerical evolution codes for general relativity
based upon the characteristic initial value problem. Progress in characteristic
evolution is traced from the early stage of 1D feasibility studies to 2D
axisymmetric codes that accurately simulate the oscillations and gravitational
collapse of relativistic stars and to current 3D codes that provide pieces of a
binary black hole spacetime. Cauchy codes have now been successful at
simulating all aspects of the binary black hole problem inside an artificially
constructed outer boundary. A prime application of characteristic evolution is
to extend such simulations to null infinity where the waveform from the binary
inspiral and merger can be unambiguously computed. This has now been
accomplished by Cauchy-characteristic extraction, where data for the
characteristic evolution is supplied by Cauchy data on an extraction worldtube
inside the artificial outer boundary. The ultimate application of
characteristic evolution is to eliminate the role of this outer boundary by
constructing a global solution via Cauchy-characteristic matching. Progress in
this direction is discussed.Comment: New version to appear in Living Reviews 2012. arXiv admin note:
updated version of arXiv:gr-qc/050809
Characteristic Evolution and Matching
I review the development of numerical evolution codes for general relativity
based upon the characteristic initial value problem. Progress is traced from
the early stage of 1D feasibility studies to 2D axisymmetric codes that
accurately simulate the oscillations and gravitational collapse of relativistic
stars and to current 3D codes that provide pieces of a binary black spacetime.
A prime application of characteristic evolution is to compute waveforms via
Cauchy-characteristic matching, which is also reviewed.Comment: Published version http://www.livingreviews.org/lrr-2005-1
Condylar response to forward mandibular positioning
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Geometric methods for anisotopic inverse boundary value problems.
Electromagnetic fields have a natural representation as differential forms. Typically the measurement of a field involves an integral over a submanifold of the domain.
Differential forms arise as the natural objects to integrate over submanifolds of each dimension. We will see that the (possibly anisotropic) material response to a field can be naturally associated with a Hodge star operator. This geometric point of view is now well established in computational electromagnetism,
particularly by Kotiuga, and by Bossavit and and others. The essential point is that Maxwell鈥檚 equations can be formulated in a context independent of the ambient Euclidean metric. This approach has theoretical
elegance and leads to simplicity of computation.
In this paper we will review the geometric formulation of the (scalar) anisotropic inverse conductivity problem, amplifying some of the geometric points made in Uhlmann鈥檚
paper in this volume. We will go on to consider generalizations of this anisotropic inverse boundary value problem to systems of Partial Differential Equation, including the result of Joshi and the author on the inverse boundary value problem for harmonic k-forms