The authors will use a method in metric geometry to show an L ^p-estimate for gradient of the weak solutions to elliptic equations with discontinuous coefficients, even the BMO semi-norms of the coefficients are not small. They also extend them to the weak solutions to parabolic equations.

A new class of backward particle systems with sequential interaction is proposed to approximate the mean-field backward stochastic differential equations. It is proven that the weighted empirical measure of this particle system converges to the law of the McKean-Vlasov system as the number of particles grows. Based on the Wasserstein metric, quantitative propagation of chaos results are obtained for both linear and quadratic growth conditions. Finally, numerical experiments are conducted to validate our theoretical results.

For quasilinear hyperbolic systems on general networks with time-periodic boundary-interface conditions with a dissipative structure, the existence and stability of the time-periodic classical solutions are discussed.

Menasco showed that a closed incompressible surface in the complement of a non-split prime alternating link in S ³ contains a circle isotopic in the link complement to a meridian of the links. Based on this result, he was able to argue the hyperbolicity of non-split prime alternating links in S ³. Adams et al. showed that if F ⊂ S × I L is an essential torus, then F contains a circle which is isotopic in S × I \ L to a meridian of L. The author generalizes his result as follows: Let S be a closed orientable surface, L be a fully alternating link in S × I. If F ⊂ S × I \ L is a closed essential surface, then F contains a circle which is isotopic in S × I \ L to a meridian of L.

The authors analyze continuity equations with Stratonovich stochasticity, $\partial \rho + {\rm{di}}{{\rm{v}}_h}\left[ {\rho \circ \left( {u(t,x) + \sum\limits_{i = 1}^N {{a_i}(x){{\dot W}_i}(t)} } \right)} \right] = 0$ defined on a smooth closed Riemannian manifold M with metric h. The velocity field u is perturbed by Gaussian noise terms Ẇ₁(t), …, Ẇ _N(t) driven by smooth spatially dependent vector fields a ₁(x), …, a _N(x) on M. The velocity u belongs to L _t ¹ W _x ^1,2 with div _h u bounded in L _t,x ^p for p > d + 2, where d is the dimension of M (they do not assume div _h u ∈ L _t,x ^∞). For carefully chosen noise vector fields a _i (and the number N of them), they show that the initial-value problem is well-posed in the class of weak L ² solutions, although the problem can be ill-posed in the deterministic case because of concentration effects. The proof of this “regularization by noise” result is based on a L ² estimate, which is obtained by a duality method, and a weak compactness argument.

This paper concerns the linearization problem on rational maps of degree d ≥ 2 and polynomials of degree d > 2 from the perspective of non-linearizability. The authors introduce a set ${{\cal C}_\infty }$ of irrational numbers and show that if $\alpha \in {{\cal C}_\infty }$, then any rational map is not linearizable and has infinitely many cycles in every neighborhood of the fixed point with multiplier $\lambda = {{\rm{e}}^{2\pi {\rm{i}}\alpha }}$. Adding more constraints to cubic polynomials, they discuss the above problems by polynomial-like maps. For the family of polynomials, with the help of Yoccoz’s method, they obtain its maximum dimension of the set in which the polynomials are non-linearizable.

Using Hodge theory and Banach fixed point theorem, Liu and Zhu developed a global method to deal with various problems in deformation theory. In this note, the authors generalize Liu-Zhu’s method to treat two deformation problems for non-Kähler manifolds. They apply the $\partial \overline \partial $-Hodge theory to construct a deformation formula for (p, q)-forms of compact complex manifold under deformations, which can be used to study the Hodge number of complex manifold under deformations. In the second part of this note, by using the $\partial \overline \partial $-Hodge theory, they provide a simple proof of the unobstructed deformation theorem for the non-Kähler Calabi-Yau $\partial \overline \partial $-manifolds.

In this paper, the authors introduce a new definition of ∞-tilting (resp. cotilting) subcategories with infinite projective dimensions (resp. injective dimensions) in an extriangulated category. They give a Bazzoni characterization of ∞-tilting (resp. cotilting) subcategories. Also, they obtain a partial Auslander-Reiten correspondence between ∞-tilting (resp. cotilting) subcategories and coresolving (resp. resolving) subcategories with an $\mathbb{E}$-projective generator (resp. $\mathbb{E}$-injective cogenerator) in an extriangulated category.