In this survey we explain a stratification of a Hilbert modular variety $\mathscr{M}_{E}$ in characteristic $p \gt 0$ attached to a totally real number field $E$. This stratification refines the stratification of $\mathscr{M}_{E}$ by Lie type, and has the property that many strata are central leaves in $\mathscr{M}_{E}$, called distinguished central leaves.

In the case when the totally real field $E$ is unramified above $p$, this stratification reduces to the stratification of $\mathscr{M}_{E}$ by $\alpha$-type first introduced by Goren and Oort and studied by Yu, and coincides with the EO stratification on $\mathscr{M}_{E}$. Moreover it is known that every non-supersingular $\alpha$-stratum of $\mathscr{M}_{E}$ is irreducible. To treat the general case where $E$ may be ramified above $p$, a key ingredient is the notion of congruity, a $p$-adic numerical invariant for abelian varieties with real multiplication by $\mathcal{O}_E$ in characteristic $p$. For every Lie stratum $\mathcal{N}_{\underline{e}}$ on $\mathscr{M}_{E}$, this new invariant defines a finite number of locally closed subsets $\mathcal{Q}_{\underline{c}}(\mathcal{N}_{\underline{e}})$, and $\mathcal{N}_{\underline{e}}$ is the disjoint union of these Lie-congruity strata $\mathcal{Q}_{\underline{c}}(\mathcal{N}_{\underline{e}})$ in $\mathcal{N}_{\underline{e}}$.

The incidence relation between the Lie-congruity strata enables one to show that the prime-to-$p$ Hecke correspondences operate transitively on the set of all irreducible components of any distinguished central leaf in $\mathscr{M}_{E}$, see Theorems 7.1, 8.1 and 9.1. The Hecke transitivity implies, according to the method of prime-to-$p$ monodromy of Hecke invariant subvarieties, that every non-supersingular distinguished central leaf in a Hilbert modular variety $\mathscr{M}_{E}$ is irreducible. The last irreducibility result is a key ingredient of the proof the Hecke orbit conjecture for Siegel modular varieties.

We re-visit the asymptotics of a binomial and a Poisson sum which arose as (average) displacement costs when moving randomly placed sensors to anchor positions. The first-order asymptotics of these sums were derived in several stages in a series of recent papers. In this paper, we give a unified approach based on the classical Laplace method with which one can also derive more terms in the asymptotic expansions. Moreover, in a special case, full asymptotic expansions can be given which even hold as identities. This will be proved by a combinatorial approach and systematic ways of computing all coefficients of these identities will be discussed as well.

For a given set $M$ of positive integers, a well-known problem of Motzkin asked to determine the maximal asymptotic density of $M$-sets, denoted by $\mu(M)$, where an $M$-set is a set of non-negative integers in which no two elements differ by an element in $M$. In 1973, Cantor and Gordon found $\mu(M)$ for $|M| \leq 2$. Partial results are known in the case $|M| \geq 3$ including results in the case when $M$ is an infinite set. This number theory problem is also related to various types of coloring problems of the distance graphs generated by $M$. In particular, it is known that the reciprocal of the fractional chromatic number of the distance graph generated by $M$ is equal to the value $\mu(M)$ when $M$ is finite. Motivated by the families $M = \{ a,b,a+b \}$ and $M = \{ a,b,a+b,b-a \}$ discussed by Liu and Zhu, we study two families of sets $M$, namely, $M = \{ a,b,b-a,n(a+b) \}$ and $M = \{ a,b,a+b,n(b-a) \}$. For both of these families, we find some exact values and some bounds on $\mu(M)$. We also find bounds on the fractional and circular chromatic numbers of the distance graphs generated by these families. Furthermore, we determine the exact values of chromatic number of the distance graphs generated by these two families.

We establish some necessary and sufficient conditions for the solvability to a system of a pair of coupled two-sided Sylvester-type tensor equations over the quaternion algebra. We also give an expression of the general solution to the system when it is solvable. As applications, we derive some solvability conditions and expressions of the $\eta$-Hermitian solutions to some systems of coupled two-sided Sylvester-type quaternion tensor equations. Moreover, we provide an example to illustrate the main results of this paper.

Let $\Lambda$, $\Gamma$ be rings and $R = \left( \begin{smallmatrix} \Lambda & 0 \\ M & \Gamma \end{smallmatrix} \right)$ the triangular matrix ring with $M$ a $(\Gamma,\Lambda)$-bimodule. Let $X$ be a right $\Lambda$-module and $Y$ a right $\Gamma$-module. We prove that $(X,0) \oplus (Y \otimes_{\Gamma} M, Y)$ is a silting right $R$-module if and only if both $X_{\Lambda}$ and $Y_{\Gamma}$ are silting modules and $Y \otimes_{\Gamma} M$ is generated by $X$. Furthermore, we prove that if $\Lambda$ and $\Gamma$ are finite dimensional algebras over an algebraically closed field and $X_{\Lambda}$ and $Y_{\Gamma}$ are finitely generated, then $(X,0) \oplus (Y \otimes_{\Gamma} M, Y)$ is a support $\tau$-tilting $R$-module if and only if both $X_{\Lambda}$ and $Y_{\Gamma}$ are support $\tau$-tilting modules, $\operatorname{Hom}_{\Lambda}(Y \otimes_{\Gamma} M, \tau X) = 0$ and $\operatorname{Hom}_{\Lambda}(e\Lambda, Y \otimes_{\Gamma} M) = 0$ with $e$ the maximal idempotent such that $\operatorname{Hom}_{\Lambda}(e\Lambda,X) = 0$.

In this paper, we study the nonlinear parabolic equation with two exponents on complete noncompact Riemannian maniflods. The special types of such equation include the Fisher-KPP equation, the parabolic Allen-Cahn equation and the Newell-Whitehead equation. We get the Souplet-Zhang's gradient estimates for the positive solutions to such equation. We also obtain the Liouville theorem for positive ancient solutions. Our results extend those of Souplet-Zhang (Bull. London. Math. Soc. 38 (2006), 1045--1053) and Zhu (Acta Math. Sci. Ser. B 36 (2016), no. 2, 514--526).

The first aim in the paper is to prove the local exponential asymptotic stability of some entire solutions to bistable reaction diffusion equations via the super-sub solution method. If the integral of the reaction term $f$ over the interval $[0,1]$ is positive, we not only obtain the similar asymptotic stability result found by Yagisita in 2003, but also simplify the proof. The asymptotic stability result for the case $\int^1_{0} f(u) \, du \lt 0$ is also obtained, which is not considered by Yagisita. After that, the asymptotic behavior of entire solutions as $t \to +\infty$ is investigated, since the other side was completely known. Here, the result is established by use of the asymptotic stability of constant solutions and pairs of diverging traveling front solutions, instead of constructing the super-sub solutions as usual. Finally, for the special bistable case $f(u) = u(1-u)(u-\alpha)$, $\alpha \in (0,1)$, we prove the entire solution continuously depends on $\alpha$.

Let $\mathbb{B}_n$ be the open unit ball in $\mathbb{C}^n$. We characterize the spectra of pointwise multipliers $M_u$ acting on Banach spaces of analytic functions on $\mathbb{B}_n$ satisfying some general conditions. These spaces include Bergman-Sobolev spaces $A^p_{\alpha,\beta}$, Bloch-type spaces $\mathcal{B}_{\alpha}$, weighted Hardy spaces $H^p_w$ with Muckenhoupt weights and Hardy-Sobolev Hilbert spaces $H^2_{\beta}$. Moreover, we describe the essential spectra of multipliers in most of the aforementioned spaces, in particular, in those spaces for which the set of multipliers is a subset of the ball algebra.

In this work, we consider a model formulated by a dynamical system and an elliptic variational inequality. We prove the solvability of initial value and periodic problems. Finally, two illustrative examples are given to show the applicability of our theoretical results.

This investigation studies the nonemptiness problems of Wang cubes with two colors. Wang cubes are unit cubes with colored faces, which are generalized from Wang tiles. For a set $B$ of Wang cubs, $\Sigma(B)$ is the set of all global patterns on $\mathbb{Z}^3$ that can be constructed by the cubes in $B$. The nonemptiness problem is to determine whether $\Sigma(B) \neq \emptyset$ or not. Denote by $\mathcal{P}(B)$ the set of all periodic patterns on $\mathbb{Z}^3$ that can be constructed by the cubes in $B$. For Wang cubes, the corresponding Wang's conjecture is that if $\Sigma(B) \neq \emptyset$, then $\mathcal{P}(B) \neq \emptyset$.

We introduce the transition matrices and trace operators to determine whether $\Sigma(B) \neq \emptyset$ and $\mathcal{P}(B) \neq \emptyset$ or not, respectively. A basic set $B$ is called a minimal cycle generator if $\mathcal{P}(B) \neq \emptyset$ but $\mathcal{P}(B') = \emptyset$ for all $B' \subsetneqq B$. By computer computation, there exist $86$ equivalence classes of minimal cycle generators with two colors. By verifying that the basic sets $B$ that contains no minimal cycle generators satisfy $\Sigma(B) = \emptyset$, we prove that the Wang's conjecture for Wang cubes with two colors is true.

It is known that the boundedness of a convex quadratic function over a convex quadratic constraint (c-QP) can be determined by algorithms. In 1985, Terlaky transformed the said boundedness problem into an $l_p$ programming problem and then apply linear programming, while Caron and Obuchowska in 1995 proposed another iterative procedure that checks, repeatedly, the existence of the implicit equality constraints. Theoretical characterization about the boundedness of (c-QP), however, does not have a complete result so far, except for Eaves' theorem, first by Eaves and later by Dostál, which answered the boundedness question only partially for a polyhedral-type of constraints. In this paper, Eaves' theorem is generalized to answer, necessarily and sufficiently, when the general (c-QP) with a convex quadratic constraint (not just a polyhedron) can be bounded from below, with a new insight that it can only be unbounded within an affine subspace.