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Given a matrix-valued function A(x) which uniformly positive definite for every x, having components a ij, the operator = (()) + + is elliptic. This is the most general form of a second-order divergence form linear elliptic differential operator.
In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. Intuitively, in the simplified two and three dimensional case, the joint distribution forms an ellipse and an ellipsoid , respectively, in iso-density plots.
For example, in the simplest kind of Monge–Ampère equation, the determinant of the hessian matrix of a function is prescribed: det D 2 u = f . {\displaystyle \det D^{2}u=f.} As follows from Jacobi's formula for the derivative of a determinant, this equation is elliptic if f is a positive function and solutions satisfy the constraint of being ...
In matrix notation, we can let () be an matrix valued function of and () be a -dimensional column vector-valued function of , and then we may write (the divergence form as) = + + One may assume, without loss of generality, that the matrix is symmetric (that is, for all ,,, () = (). We make that assumption in the rest of this article.
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Finding the Jones matrix, J(α, β, γ), for an arbitrary rotation involves a three-dimensional rotation matrix. In the following notation α , β and γ are the yaw, pitch, and roll angles (rotation about the z-, y-, and x-axes, with x being the direction of propagation), respectively.
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