We consider complex projective space with its Fubini–Study metric and the X-ray transform defined by integration over its geodesics. We identify the kernel of this transform acting on symmetric tensor fields.

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The subject of this paper is a Jacobian, introduced by F. Lazzeri (unpublished), associated with every compact oriented Riemannian manifold whose dimension is twice an odd number. We study the Torelli and Schottky problem for Lazzeri's Jacobian of flat tori and we compare Lazzeri's Jacobian of Kähler manifolds with other Jacobians.

A class of upwind flux splitting methods in the Euler equations of compressible flow is considered in this paper. Using the property that Euler flux F(U) is a homogeneous function of degree one in U, we reformulate the splitting fluxes with F^{+}=A^{+} U, F^{-}=A^{-} U, and the corresponding matrices are either symmetric or symmetrizable and keep only non-negative and non-positive eigenvalues. That leads to the conclusion that the first order schemes are positive in the sense of Lax-Liu [18], which implies that it is L^2-stable in some suitable sense. Moreover, the second order scheme is a stable perturbation of the first order scheme, so that the positivity of the second order schemes is also established, under a CFL-like condition. In addition, these splitting methods preserve the positivity of density and energy.