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In this paper, we strengthen the splitting theorem proved in [14, 15] and provide a different approach using ideas from the weak KAM theory., 21 pages

In this paper, we prove that principal circle bundles over the complex projective space equipped with the standard Sasakian structures are volume rigid among all $K$-contact manifolds satisfying positivity conditions of tensors involing the Tanaka-Webster curvature., 27 pages

The liver performs many essential metabolic functions, which can be studied using computational models of hepatocytes. Here we present HepatoDyn, a highly detailed dynamic model of hepatocyte metabolism. HepatoDyn includes a large metabolic network, highly detailed kinetic laws, and is capable of dynamically simulating the redox and energy metabolism of hepatocytes. Furthermore, the model was coupled to the module for isotopic label propagation of the software package IsoDyn, allowing HepatoDyn to integrate data derived from 13C based experiments. As an example of dynamical simulations applied to hepatocytes, we studied the effects of high fructose concentrations on hepatocyte metabolism by integrating data from experiments in which rat hepatocytes were incubated with 20 mM glucose supplemented with either 3 mM or 20 mM fructose. These experiments showed that glycogen accumulation was significantly lower in hepatocytes incubated with medium supplemented with 20 mM fructose than in hepatocytes incubated with medium supplemented with 3 mM fructose. Through the integration of extracellular fluxes and 13C enrichment measurements, HepatoDyn predicted that this phenomenon can be attributed to a depletion of cytosolic ATP and phosphate induced by high fructose concentrations in the medium., Author Summary Despite the key role of hepatocytes in carbohydrate and lipid homeostasis, available dynamic models of hepatocyte metabolism tend to be limited to a single pathway and/or are based on assumptions of constant concentrations of key metabolites involved in redox and energy metabolism (ATP, NAD, NADPH etc.). Furthermore, most dynamic models are unable to integrate information from 13C based experiments. 13C based experiments allow us to infer the relative activity of alternative pathways and hence are highly useful for indicating flux distributions. To overcome these limitations, we developed HepatoDyn, a dynamic model of hepatic metabolism. HepatoDyn uses a large metabolic network including key pathways such as glycolysis, the Krebs cycle, the pentose phosphate pathway and fatty acid metabolism, and dynamically models the concentrations of metabolites involved in the redox and energy metabolism of hepatocytes. In addition, the model was coupled to the label propagation module of the package IsoDyn, allowing it to integrate data from 13C based experiments to assist in the parametrization process. These features make HepatoDyn a powerful tool for studying the dynamics of hepatocyte metabolism.

We prove matrix and scalar differential Harnack inequalities for linear parabolic equations on Riemannian and K\"ahler manifolds., Comment: 15 pages

Measure contraction property is a synthetic Ricci curvature lower bound for metric measure spaces. We consider Sasakian manifolds with non-negative Tanaka-Webster Ricci curvature equipped with the metric measure space structure defined by the sub-Riemannian metric and the Popp measure. We show that these spaces satisfy the measure contraction property $MCP(0,N)$ for some positive integer $N$. We also show that the same result holds when the Sasakian manifold is equipped with a family of Riemannian metrics extending the sub-Riemannian one., Comment: 30 pages

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., 15 pages

We give proofs of QR factorization, Cholesky's 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., 6 pages

We prove a version of differential Harnack inequality for a family of sub-elliptic diffusions on Sasakian manifolds under certain curvature conditions., Comment: 27 pages

We prove a generalization of the Li-Yau estimate for a board class of second order linear parabolic equations. As a consequence, we obtain a new Cheeger-Yau inequality and a new Harnack inequality for these equations. We also prove a Hamilton-Li-Yau estimate, which is a matrix version of the Li-Yau estimate, for these equations. This results in a generalization of Huisken's monotonicity formula for a family of evolving hypersurfaces. Finally, we also show that all these generalizations are sharp in the sense that the inequalities become equalities for a family of fundamental solutions, which however different from the Gaussian heat kernels on which the equality was achieved in the classical case., 31 pages

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 invariant 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., 34 pages, some typos are fixed in the new version

In this paper, we study the Ma-Trudinger-Wang (MTW) conditions for cost functions $c$ which are of the form $c=l\circ d$, where $d$ is a Riemannian distance function with constant sectional curvature. In this case, the MTW conditions are equivalent to some computable conditions on the function $l$. As a corollary, we give some new costs on Riemannian manifolds of constant negative curvature for which the MTW conditions are satisfied., 17 pages

The Ma-Trudinger-Wang curvature --- or cross-curvature --- is an object arising in the regularity theory of optimal transportation. If the transportation cost is derived from a Hamiltonian action, we show its cross-curvature can be expressed in terms of the associated Jacobi fields. Using this expression, we show the least action corresponding to a harmonic oscillator has zero cross-curvature, and in particular satisfies the necessary and sufficient condition \Athreew\ for the continuity of optimal maps. We go on to study gentle perturbations of the free action by a potential, and deduce conditions on the potential which guarantee either that the corresponding cost satisfies the more restrictive condition \Athrees\ of Ma, Trudinger and Wang, or in some cases has positive cross-curvature. In particular, the quartic potential of the anharmonic oscillator satisfies \Athrees\ in the perturbative regime., 18 pages