  We establish improved mean value estimates associated with the number of
integer solutions of certain systems of diagonal equations, in some instances
attaining the sharpest conjectured conclusions. This is the first occasion on
which bounds of this quality have been attained for Diophantine systems not of
Vinogradov type. As a consequence of this progress, whenever $u \ge 3v$ we
obtain the Hasse principle for systems consisting of $v$ cubic and $u$
quadratic diagonal equations in $6v+4u+1$ variables, thus attaining the
convexity barrier for this problem.
