Measure contraction properties $MCP(K,N)$ are synthetic Ricci curvature lower bounds for metric measure spaces which do not necessarily have smooth structures. It is known that if a Riemannian manifold has dimension $N$, then $MCP(K,N)$ is equivalent
to Ricci curvature bounded below by $K$. On the other hand, it was observed in cite{Ri} that there is a family of left invariant metrics on the three dimensional Heisenberg group for which the Ricci curvature is not bounded below. Though this family of metric spaces equipped with the Harr measure satisfy $MCP(0,5)$. In this paper, we give sufficient conditions for a $2n+1$ dimensional weakly Sasakian manifold to satisfy $MCP(0,2n+3)$. This extends the above mentioned result on the Heisenberg group in cite{Ri}.
In this paper, we prove that the Jordan-Kinderlehrer-Otto scheme for a family of linear parabolic equations on the flat torus converges uniformly in space.
We give proofs of QR factorization, Choleskys factorization, and LDU factorization using the inverse function theorem. As a consequence, we obtain analytic dependence of these matrix factorizations which does not follow immediately using Gaussian elimination.
We prove a version of differential Harnack inequality for a family of sub-elliptic diffusions on Sasakian manifolds under certain curvature conditions.
Assume that a Hamiltonian system is monotone. In this paper, we give several characterizations on when such a system is Anosov. Assuming that a monotone Hamiltonian system has no conjugate point, we show that there are two distributions which are inv
ariant under the Hamiltonian flow. We show that a monotone Hamiltonian flow without conjugate point is Anosov if and only if these distributions are transversal. We also show that if the reduced curvature of the Hamiltonian system is non-positive, then the flow is Anosov if and only if the reduced curvature is negative somewhere along each trajectory.
