In this paper, we apply the local discontinuous Galerkin (LDG) method to solve the classical Keller-Segel (KS) chemotaxis model. The exact solution of the KS chemotaxis model may exhibit blow-up patterns with certain initial conditions, and is not easy to approximate numerically. Moreover, it has been proved that there exists a definition of free energy of the KS system which dissipates with respect to time. We will construct a consistent numerical energy and prove the energy dissipation with the LDG discretization. Several numerical experiments in one and two space dimensions will be given. Especially, for solutions with blow-up (converge to Dirac delta functions), the densities of KS model are computed to be strictly positive in the numerical experiments and the energies are also numerically observed to be strictly positive and decreasing as are seen in the figures. Therefore, the scheme is stable for the KS model with blow-up solutions.

We establish theoretical recovery guarantees of a family of Riemannian optimization algorithms for low rank matrix recovery, which is about recovering an mimes n rank mimes n matrix from mimes n number of linear measurements. The algorithms are first interpreted as iterative hard thresholding algorithms with subspace projections. Based on this connection, we show that provided the restricted isometry constant mimes n of the sensing operator is less than mimes n, the Riemannian gradient descent algorithm and a restarted variant of the Riemannian conjugate gradient algorithm are guaranteed to converge linearly to the underlying rank mimes n matrix if they are initialized by one step hard thresholding. Empirical evaluation shows that the algorithms are able to recover a low rank matrix from nearly the minimum number of measurements necessary.

We study the differential polynomial rings which are defined using the special geometry of the moduli spaces of Calabi-Yau threefolds. The higher genus topological string amplitudes are expressed as polynomials in the generators of these rings, giving them a global description in the moduli space. At particular loci, the amplitudes yield the generating functions of Gromov-Witten invariants. We show that these rings are isomorphic to the rings of quasi modular forms for threefolds with duality groups for which these are known. For the other cases, they provide generalizations thereof. We furthermore study an involution which acts on the quasi modular forms. We interpret it as a duality which exchanges two distinguished expansion loci of the topological string amplitudes in the moduli space. We construct these special polynomial rings and match them with known quasi modular forms for non-compact Calabi-Yau geometries and their

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