In this paper, the author studies the local existence of strong solutions and their possible blow-up in time for a quasilinear system describing the interaction of a short wave induced by an electron field with a long wave representing an extension of the motion of the director field in a nematic liquid crystal’s asymptotic model introduced in [Saxton, R. A., Dynamic instability of the liquid crystal director. In: Current Progress in Hyperbolic Systems (Lindquist, W. B., ed.), Contemp. Math., Vol.100, Amer. Math. Soc., Providence, RI, 1989, pp.325–330] and [Hunter, J. K. and Saxton, R. A., Dynamics of director fields, SIAM J. Appl. Math., 51, 1991, 1498–1521] and studied in [Hunter, J. K. and Zheng, Y., On a nonlinear hyperbolic variational equation I, Arch. Rat. Mech. Anal., 129, 1995, 305–353], [Hunter, J. K. and Zheng, Y., On a nonlinear hyperbolic variational equation II, Arch. Rat. Mech. Anal., 129, 1995, 355–383] and in [Zhang, P. and Zheng, Y., On oscillation of an asymptotic equation of a nonlinear variational wave equation, Asymptotic Anal., 18, 1998, 307–327] and, more recently, in [Bressan, A., Zhang, P. and Zheng, Y., Asymptotic variational wave equations, Arch. Rat. Mech. Anal., 183, 2007, 163–185].

In this paper, the authors give a new proof of Block and Weinberger’s Bochner vanishing theorem built on direct computations in the K-theory of the localization algebra.

The authors prove the gradient convergence of the deep learning-based numerical method for high dimensional parabolic partial differential equations and backward stochastic differential equations, which is based on time discretization of stochastic differential equations (SDEs for short) and the stochastic approximation method for nonconvex stochastic programming problem. They take the stochastic gradient decent method, quadratic loss function, and sigmoid activation function in the setting of the neural network. Combining classical techniques of randomized stochastic gradients, Euler scheme for SDEs, and convergence of neural networks, they obtain the $O(K^{\frac{1}{4}})$ rate of gradient convergence with K being the total number of iterative steps.

The present paper is concerned with the eigenvalue problem for cone degenerate p-Laplacian. First the authors introduce the corresponding weighted Sobolev s-paces with important inequalities and embedding properties. Then by adapting Lusternik-Schnirelman theory, they prove the existence of infinity many eigenvalues and eigenfunctions. Finally, the asymptotic behavior of the eigenvalues is given.

In this paper, the authors study further properties and applications of weighted homology and persistent homology. The Mayer-Vietoris sequence and generalized Bockstein spectral sequence for weighted homology are introduced. For applications, the authors show an algorithm to construct a filtration of weighted simplicial complexes from a weighted network. They also prove a theorem to calculate the mod p ² weighted persistent homology provided with some information on the mod p weighted persistent homology.

The author studies the optimal investment stopping problem in both continuous and discrete cases, where the investor needs to choose the optimal trading strategy and optimal stopping time concurrently to maximize the expected utility of terminal wealth. Based on the work of Hu et al. (2018) with an additional stochastic payoff function, the author characterizes the value function for the continuous problem via the theory of quadratic reflected backward stochastic differential equations (BSDEs for short) with unbounded terminal condition. In regard to the discrete problem, she gets the discretization form composed of piecewise quadratic BSDEs recursively under Markovian framework and the assumption of bounded obstacle, and provides some useful a priori estimates about the solutions with the help of an auxiliary forward-backward SDE system and Malliavin calculus. Finally, she obtains the uniform convergence and relevant rate from discretely to continuously quadratic reflected BSDE, which arise from corresponding optimal investment stopping problem through above characterization.

The author proposes a two-dimensional generalization of Constantin-Lax-Majda model. Some results about singular solutions are given. This model might be the first step toward the singular solutions of the Euler equations. Along the same line (vorticity formulation), the author presents some further model equations. He possibly models various aspects of difficulties related with the singular solutions of the Euler and Navier-Stokes equations. Some discussions on the possible connection between turbulence and the singular solutions of the Navier-Stokes equations are made.

In this paper, the author proves that the spacelike self-shrinker which is closed with respect to the Euclidean topology must be flat under a growth condition on the mean curvature by using the Omori-Yau maximum principle.

In this paper, for the Lorentz manifold M ² × ℝ with M ² a 2-dimensional complete surface with nonnegative Gaussian curvature, the authors investigate its spacelike graphs over compact, strictly convex domains in M ², which are evolving by the non-parametric mean curvature flow with prescribed contact angle boundary condition, and show that solutions converge to ones moving only by translation.

In this paper, the authors investigate the optimal conversion rate at which land use is irreversibly converted from biodiversity conservation to agricultural production. This problem is formulated as a stochastic control model, then transformed into a HJB equation involving free boundary. Since the state equation has singularity, it is difficult to directly derive the boundary value condition for the HJB equation. They provide a new method to overcome the difficulty via constructing another auxiliary stochastic control problem, and impose a proper boundary value condition. Moreover, they establish the existence and uniqueness of the viscosity solution of the HJB equation. Finally, they propose a stable numerical method for the HJB equation involving free boundary, and show some numerical results.