Raster (Arithmetic) Overlay
Raster overlay superimposes at least two input raster layers to produce an output layer. Each cell in the output layer is calculated from the corresponding pixels in the input layers. To do this, the layers must line up perfectly; they must have the same pixel resolution and spatial extent. If they do not align, they can be adjusted to fit by the preprocessing functions discussed in Chapter 3. Once preprocessed, raster overlay is flexible, efficient, quick, and offers more overlay possibilities than vector overlay.
Homeopathic preparations are referred to as "homeopathics"  or "remedies". Practitioners rely on two types of reference when prescribing: materia medica and repertories. A homeopathic materia medica is a collection of "drug pictures", organized alphabetically. These entries describe the symptom patterns associated with individual preparations. A homeopathic repertory is an index of disease symptoms that lists preparations associated with specific symptoms. In both cases different compilers may dispute particular inclusions.  The first symptomatic homeopathic materia medica was arranged by Hahnemann. The first homeopathic repertory was Georg Jahr's Symptomenkodex , published in German in 1835, and translated into English as the Repertory to the more Characteristic Symptoms of Materia Medica by Constantine Hering in 1838. This version was less focused on disease categories and would be the forerunner to later works by James Tyler Kent.   Repertories, in particular, may be very large.
The problem of detecting gravitational radiation is receiving considerable attention with the construction of new detectors in the United States, Europe, and Japan. The theoretical modeling of the wave forms that would be produced in particular systems will expedite the search for and analysis of detected signals. The characteristic formulation of GR is implemented to obtain an algorithm capable of evolving black holes in 3D asymptotically flat spacetimes. Using compactification techniques, future null infinity is included in the evolved region, which enables the unambiguous calculation of the radiation produced by some compact source. A module to calculate the waveforms is constructed and included in the evolution algorithm. This code is shown to be second-order convergent and to handle highly non-linear spacetimes. In particular, we have shown that the code can handle spacetimes whose radiation is equivalent to a galaxy converting its whole mass into gravitational radiation in one second. We further use the characteristic formulation to treat the region close to the singularity in black hole spacetimes. The code carefully excises a region surrounding the singularity and accurately evolves generic black hole spacetimes with apparently unlimited stability.