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Paneitz Operator
In the mathematics, mathematical field of differential geometry, the Paneitz operator is a fourth-order differential operator defined on a Riemannian manifold of dimension ''n''. It is named after Stephen Paneitz, who discovered it in 1983, and whose preprint was later published posthumously in . In fact, the same operator was found earlier in the context of conformal supergravity by E. Fradkin and A. Tseytlin in 1982 (Phys Lett B 110 (1982) 117 and Nucl Phys B 1982 (1982) 157 ). It is given by the formula :P = \Delta^2 - \delta \left\d + (n-4)Q where Δ is the Laplace–Beltrami operator, ''d'' is the exterior derivative, δ is its formal adjoint, ''V'' is the Schouten tensor, ''J'' is the trace of the Schouten tensor, and the dot denotes tensor contraction on either index. Here ''Q'' is the scalar invariant :(-4, V, ^2+nJ^2+2\Delta J)/4, where Δ is the positive Laplacian. In four dimensions this yields the Q-curvature. The operator is especially important in conformal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
