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We prove several rigidity theorems for elliptic genera associated to Toeplitz operators on odd dimensional spin manifolds. We also prove an equivariant odd index theorem for Dirac operators with involution parity and the Atiyah-Hirzebruch vanishing theorems for odd dimensional spin manifolds.

In this paper we consider the well-posedness of the compressible Prandtl boundary layer for the Navier–Stokes–Poisson equations with nonslip boundary condition and obtain the linear stability of the approximate solution constructed by boundary layer expansion.

In this paper, we prove a sum set estimate and the exact sum set theorem for unimodular Kac algebras. Combining the characterization of minimizers of the Donoso–Stark uncertainty principle and the Hirschman–Beckner uncertainty principle, we characterize the extremal pairs of Young’s inequality and extremal operators of the Hausdorff–Young inequality for unimodular Kac algebras.