In this article, we classify all standard invariants that can arise from a composed inclusion of an A 3 with an A 4 subfactor. More precisely, if N P is an A 3 subfactor and P M is an A 4 subfactor, then only four standard invariants can arise from the composed inclusion N M. We answer a question posed by Bisch and Haagerup in 1994. The techniques of this paper also show that there are exactly four standard invariants for the composed inclusion of two A 4 subfactors.

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We give an elementary proof of the existence of an asymptotic expansion in powers of$k$of the Bergman kernel associated to$L$^{$k$}, where$L$is a positive line bundle over a compact complex manifold. We also give an algorithm for computing the coefficients in the expansion.

Convex bodies with identical John and LYZ ellipsoids are characterized. This solves an important problem from convex geometry posed by G. Zhang. As applications, several sharp affine isoperimetric inequalities are established.

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