The classification work [5], [8] left unsettled only those anomalous isoparametric hypersurfaces with four principal curvatures and multiplicity pair {4, 5}, {6, 9}, or {7, 8} in the sphere. By systematically exploring the ideal theory in commutative algebra in conjunction with the geometry of isoparametric hypersurfaces, we show that an isoparametric hypersurface with four principal curvatures and multiplicities {4, 5} in S19 is homogeneous, and, moreover, an isoparametric hypersurface with four principal curvatures and multiplicities {6, 9} in S31 is either the inhomogeneous one constructed by Ferus, Karcher, and M¨unzner, or the one that is homogeneous. This classification reveals the striking resemblance between these two rather different types of isoparametric hypersurfaces in the homogeneous category, even though the one with multiplicities {6, 9} is of the type constructed by Ferus, Karcher, and M¨unzner and the one with multiplicities {4, 5} stands alone. The quaternion and the octonion algebras play a fundamental role in their geometric structures. A unifying theme in [5], [8], and the present sequel to them is Serre’s criterion of normal varieties. Its technical side pertinent to our situation that we developed in [5], [8] and extend in this sequel is instrumental. The classification leaves only the case of multiplicity pair {7, 8} open.

We consider an incompressible viscous flow without surface tension in a finite-depth domain of two dimensions, with free top boundary and fixed bottom boundary. This system is governed by the Navier-Stokes equations in this moving domain and the transport equation on the moving boundary. In this paper, we construct a stable numerical scheme to simulate the evolution of this system by discontinuous Galerkin method, and discuss the error analysis of the fluid under certain assumptions. Our formulation is mainly based on the geometric structure introduced earlier in the literature, and the natural energy estimate, which is rarely used in the numerical study of this system before.

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We define a partition of Z into intervals {$I$_{j}} and prove the Littlewood-Paley inequality ‖$f$‖_{$p$}≦$C$_{p}‖$Sf$‖_{$p$}, 2≦$p$<∞. Here$f$is a function on [o, 2π) and $$Sf = (\sum |\Delta _j |^2 )^{1/2} , \hat \Delta j = \hat f\chi _{Ij} $$ . This is a new example of a partition having the Littlewood-Paley property since the {$I$_{j}} are not of the type obtained by iterating lacunary partitions finitely many times.