We give bounds for various homological invariants (including Castelnuovo-Mumford regularity, degrees of local cohomology, and injective dimension) of finitely generated VI-modules in the non-describing characteristic case. It turns out that the formulas of these bounds for VI-modules are the same as the formulas of corresponding bounds for FI-modules.

Using the Nakayama functor, we construct an equivalence from a Serre quotient category of a category of finitely generated modules to a category of finite-dimensional modules. We then apply this result to the categories FI_G and VI_q, and answer positively an open question of Nagpal on representation stability theory.

We introduce new boundary conditions for differential forms on symplectic manifolds with boundary. These boundary conditions, dependent on the symplectic structure, allows us to write down elliptic boundary value problems for both second-order and fourth-order symplectic Laplacians and establish Hodge theories for the cohomologies of primitive forms on manifolds with boundary. We further use these boundary conditions to define a relative version of the primitive cohomologies and to relate primitive cohomologies with Lefschetz maps on manifolds with boundary. As we show, these cohomologies of primitive forms can distinguish certain Kahler structures of Kahler manifolds with boundary.

Processing streaming data as they arrive is often necessary for high dimensional data analysis. In this paper, we analyze the convergence of a subspace online PCA iteration, as a followup of the recent work of Li, Wang, Liu, and Zhang [Math. Program., Ser. B, DOI 10.1007/s10107-017-1182-z] who considered the case for the most significant principal component only, i.e., a single vector. Under the sub-Gaussian assumption, we obtain a finite-sample error bound that closely matches the minimax information lower bound of Vu and Lei [Ann. Statist. 41:6 (2013), 2905-2947].

In this paper, we study intermittent behaviors of coupled piecewise-expanding map lattices with two nodes and a weak coupling. We show that the successive phase transition between ordered and disordered phases occurs for almost every orbit. That is, we prove $\liminf_{n\rightarrow \infty}| x_1(n)-x_2(n)|=0$ and $\limsup_{n\rightarrow \infty}| x_1(n)-x_2(n)|\ge c_0>0$, where $x_1(n), x_2(n)$ correspond to the coordinates of two nodes at the iterative step $n$. We also prove the same conclusion for weakly coupled tent-map lattices with any multi-nodes.