Wehavepi = 1,p2 = ,pl+p2 =1,whereasp = 2. Had the equations been written in the opposite order, we should have found pi = 2, P2 = 0, Pl + P2 = 2. This illustrates the fact that PI + + Pr, unlike the rank, is not an invariant. 1 Species is defined in the author's paper "Pfaffian Systems of Species One," Trans. Amer. Math. Soc., 35, 356-71 (1933). 2 Cf. Goursat, E., Probleme de Pfaff, Paris, 291 ...

We introduce the terms strong suband super-Gaussianity to refer to the previously introduced class of densities log-concave is x and log-convex in x respectively. We derive relationships among the various definitions of suband super-Gaussianity, and show that strong suband super-Gaussianity are related to the score function being star-shaped upward or downward with respect to the origin. We ill...