A Tate-linear structure on a smooth noetherian local formal scheme $T$ over a field $\kappa$ of characteristic $p$ is an isomorphism $T \xrightarrow{\sim} \mathbf{N}_{\mathbb{Q}}/\mathbf{N}$ of sheaves on the fpqc site of $\operatorname{Spec}(\kappa)$, where $\mathbf{N}$ is an fpqc sheaf of torsion free nilpotent on $\operatorname{Spec}(\kappa)$ which admits a central series $\mathbf{N} = \mathbf{N}_{1} \supsetneqq \mathbf{N}_{2} \supsetneqq \cdots \supsetneqq \mathbf{N}_{c+1} = (1)$ such that each subquotient $\mathbf{N}_{i}/\mathbf{N}_{i+1}$ is the Tate $\mathbb{Z}_{p}$-module attached to a $p$-divisible group over $\kappa$, and $\mathbf{N}_{\mathbb{Q}}$ is the Mal'cev completion of $\mathbf{N}$. A smooth formal scheme over $\kappa$ with a Tate-linear structure is called a Tate-linear formal variety over $\kappa$. Examples of Tate-linear formal varieties include $p$-divisible formal groups, biextensions of $p$-divisible formal groups, and formal completions at closed points of central leaves in Siegel modular varieties in characteristic $p$. Tate-linear structures have a remarkable rigidity property: if a reduced irreducible closed formal subscheme $W$ of a Tate linear formal variety $T$ is stable under the action of a group of Tate-linear automorphisms of $T$ which operates strongly nontrivially on $T$, then $W$ is a Tate-linear formal subvariety. Proofs of statements in this survey article can be found in Chapters 5–6 and 10–11 of [11].

We define successive minimal bases (SMBs) for the space of $u^{n}$-division points of a Drinfeld $\mathbb{F}_{q}[t]$-module over a local field, where $u$ is a finite prime of $\mathbb{F}_{q}[t]$ and $n$ is a positive integer. These SMBs share similar properties to those of SMBs of the lattices associated to Drinfeld modules. We study the relations between these SMBs and those of the lattices. Finally, we apply the relations to study the explicit wild ramification subgroup action on an SMB of the space of $u^{n}$-division points and show a function field analogue of Szpiro's conjecture for rank $2$ Drinfeld modules under a certain limited situation.

We study the structure of potent elements in matrix rings with same diagonals and polynomial rings, motivated by Jacobson's theorem of commutativity. A ring shall be said to be PC if every potent element is central. We investigate the structure of PC rings in relation to the commutativity of rings. It is proved that if $R$ is a PC ring of prime characteristic then the polynomial ring over $R$ is also a PC ring. Every periodic PC ring is shown to be commutative.

In this paper, we introduce several classes of non-$\sigma$-porous subsets of a general Lebesgue space. Also, we study some linear dynamics of operators and show that the set of all non-hypercyclic vectors of a sequences of weighted translation operators on $L^{p}$-spaces is not $\sigma$-porous.

In this paper we study a class of asymptotically periodic fractional $p$-Laplacian equations. Under the suitable conditions, the existence of ground state solutions are obtained via the variational method.

We prove the existence of ancient solutions of the exterior problem for parabolic Hessian quotient equations $-u_{t} S_{k,l}(D^{2}u) = 1$ with prescribed asymptotic behavior at infinity. We construct a subsolution to it and use Perron method to finish the proof.

In this paper, we study the geometry of the free-boundary arising from local minimizers of a degenerate version of the Alt–Caffarelli functional. Specifically, we consider local minimizers of the functional $J_{Q}(u,\Omega) := \int_{\Omega} |\nabla u|^{2} + Q(x)^{2} \chi_{\{u>0\}} \, dx$ where $Q(x) = \operatorname{dist}(x,\Gamma)^{\gamma}$ for $\gamma \gt 0$ and $\Gamma$ a submanifold of dimension $0 \leq k \leq n-1$. Previously, it was shown that on $\Gamma$, the free boundary $\partial \{u \gt 0\}$ may be decomposed into a rectifiable set $\mathcal{S}$, which satisfies effective estimates, and a cusp set $\Sigma$ [11]. In this note, we prove that under mild assumptions, in the case $n = 2$ and $\Gamma$ a line, the cusp set $\Sigma$ does not exist. Building upon the work of Arama and Leoni [3], our results apply to the physical case of a variational formulation of the Stokes' wave.

In this paper, a modified alternately linear implicit (MALI) iteration method is derived for solving the non-symmetric coupled algebraic Riccati equation (NCARE). In the MALI iteration algorithm, the coefficient matrices of the linear matrix equations are fixed at each iteration step. In addition, the MALI iteration method utilizes a weighted average of the estimates in both the last step and current step to update the estimates in the next iteration step. Further, we give the convergence theory of the modified algorithm. Last, numerical examples demonstrate the effectiveness and feasibility of the derived algorithm.

In this paper, we will construct an inertial algorithm without using the embedded projection method to find a solution of variational inequality problems in which the cost mapping is not required to be satisfied any pseudomonotonicity. The iterative sequences generated by algorithms under the main assumption $S_{M} \neq \emptyset$ are proved that they converge to a solution of the corresponding problems. In addition, numerical experiments are provided to show the effectiveness of the algorithm.