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It says that the propagation of singularities follows the bicharacteristic flow of the principal symbol of . The theorem appeared 1972 in a work on Fourier integral operators by Johannes Jisse Duistermaat and Lars Hörmander and since then there have been many generalizations which are known under the name propagation of singularities.
Isomonodromic deformations were first studied by Richard Fuchs, with early pioneering contributions from Lazarus Fuchs, Paul Painlevé, René Garnier, and Ludwig Schlesinger. Inspired by results in statistical mechanics , a seminal contribution to the theory was made by Michio Jimbo , Tetsuji Miwa , and Kimio Ueno , who studied cases involving ...
Since the physical behavior of singularities is unknown, if singularities can be observed from the rest of spacetime, causality may break down, and physics may lose its predictive power. The issue cannot be avoided, since according to the Penrose–Hawking singularity theorems, singularities are inevitable in physically reasonable situations.
An important reason why singularities cause problems in mathematics is that, with a failure of manifold structure, the invocation of Poincaré duality is also disallowed. A major advance was the introduction of intersection cohomology, which arose initially from attempts to
Null singularities: These singularities occur on light-like or null surfaces. An example might be found in certain types of black hole interiors, such as the Cauchy horizon of a charged (Reissner–Nordström) or rotating black hole. A singularity can be either strong or weak:
The singularities of solutions of these equations are The point , and; The point 0 for types III, V and VI, and; The point 1 for type VI, and; Possibly some movable poles; For type I, the singularities are (movable) double poles of residue 0, and the solutions all have an infinite number of such poles in the complex plane.
In algebraic geometry, a Du Val singularity, also called simple surface singularity, Kleinian singularity, or rational double point, is an isolated singularity of a complex surface which is modeled on a double branched cover of the plane, with minimal resolution obtained by replacing the singular point with a tree of smooth rational curves, with intersection pattern dual to a Dynkin diagram of ...
Consider a smooth real-valued function of two variables, say f (x, y) where x and y are real numbers.So f is a function from the plane to the line. The space of all such smooth functions is acted upon by the group of diffeomorphisms of the plane and the diffeomorphisms of the line, i.e. diffeomorphic changes of coordinate in both the source and the target.