We establish an estimate on weakly $q$-convex domains in $\mathbb{C}^{n}$ which provides a unified approach to various existence results for the $\bar{\partial}$-problem. We also prove a Diederich--Fornaess type result for weakly $q$-convex domains.

Let $\mathcal{G}_{k}$ be the gauge group of the principal $\mathit{SU}(5)$-bundle over $S^{4}$ with second Chern class $k$. We show that there is a $p$-local homotopy equivalence $\mathcal{G}_{k} \simeq \mathcal{G}_{k'}$ for any prime $p$ if and only if $(120,k) = (120,k')$.

We consider the class of manifolds with compact Lie group actions which restrict to GKM-actions on the maximal torus. First, we see their GKM-graphs admit symmetry of the Weyl groups. And then, we study its combinatorial abstraction; starting with abstract GKM-graphs with symmetry, we derive certain properties which reflect topology in a purely combinatorial way.

We introduce generalized almost contact structures which admit the $B$-field transformations on odd dimensional manifolds. We provide the notion of generalized Sasakian structures from the view point of the generalized almost contact structures. We obtain a generalized Sasakian structure on a non-compact manifold which does not arise as a pair of ordinary Sasakian structures. However we show that a generalized Sasakian structure on a compact $3$-dimensional manifold is equivalent to a pair of Sasakian structures with the same metric.

The Hurwitz action of the $n$-braid group $B_{n}$ on the $n$-fold product $(B_{m})^{n}$ of the $m$-braid group $B_{m}$ is studied. Using a natural action of $B_{n}$ on trees with $n$ labeled edges and $n+1$ labeled vertices, we determine all elements of the orbit of every $n$-tuple of the $n$ distinct standard generators of $B_{n+1}$ under the Hurwitz action of $B_{n}$.

We give a simple criterion for slope stability of Fano manifolds $X$ along divisors or smooth subvarieties. As an application, we show that $X$ is slope stable along an ample effective divisor $D\subset X$ unless $X$ is isomorphic to a projective space and $D$ is a hyperplane section. We also give counterexamples to Aubin's conjecture on the relation between the anticanonical volume and the existence of a Kähler--Einstein metric. Finally, we consider the case that $\dim X = 3$; we give a complete answer for slope (semi)stability along divisors of Fano threefolds.

(Co)associative submanifolds in a $G_{2}$-manifold with a free $S^{1}$ or $T^{2}$ action are characterized by submanifolds in the quotient space. Using our method, we construct various examples of (co)associative submanifolds and fibrations on $G_{2}$-manifolds with the $T^{2}$-symmetry such as the cone of the Iwasawa manifold.

In this paper, we investigate a class of multi-group epidemic models with general exposed distribution and relapse. Nonlinear incidence rate is used between compartments. It is showed that global dynamics are completely determined by the threshold parameter $R_{0}$ under suitable conditions. More specifically, the disease will die out if $R_{0} \leq 1$ and that if $R_{0} > 1$, the disease persists in all groups. The approaches used here, are the theory of non-negative matrices, persistence theory in dynamical systems and graph-theoretical approach to the method of Lyapunov functionals. Furthermore, our results demonstrate that heterogeneity and nonlinear incidence rate do not alter the dynamical behavior of the SIR model with general exposed distribution and relapse. On the other hand, our global dynamical results exclude the existence of Hopf bifurcation leading to sustained oscillatory solutions.

Let $\mathbf{k}$ be an algebraically closed field. A polynomial $F \in \mathbf{k}[X, Y]$ is said to be generally rational if, for almost all $\lambda \in \mathbf{k}$, the curve ``$F=\lambda$'' is rational. It is well known that, if $\mathop{\mathrm{char}}\mathbf{k}=0$, $F$ is generally rational iff there exists $G \in \mathbf{k}(X, Y)$ such that $\mathbf{k}(F, G)=\mathbf{k}(X, Y)$. We give analogous results valid in arbitrary characteristic.

In this paper, we will introduce a simplicial complex $\mathrm{T}_{Q}(\mathcal{H})$ defined by a quiver $Q$ and a family $\mathcal{H}$ of paths in $Q$. We call $\mathrm{T}_{Q}(\mathcal{H})$ a path complex of $\mathcal{H}$ in $Q$. Let $G$ be a finite group, and denote by $\mathrm{Sgp}(G)$ and $\mathrm{Coset}(G)$ respectively the totality of subgroups of $G$, and that of left cosets $gL \in G/L$ of subgroups $L$ of $G$. We will particularly focus on quivers $Q_{G}$ and $Q_{\mathit{CG}}$ obtained naturally from posets $\mathrm{Sgp}(G)$ and $\mathrm{Coset}(G)$ ordered by the inclusion-relation. Then various properties of path complexes associated to $Q_{G}$ and $Q_{\mathit{CG}}$ will be studied.

Let $p$ be a prime number. We define the notion of $F$-finiteness of homomorphisms of $\mathbb{F}_{p}$-algebras, and discuss some basic properties. In particular, we prove a sort of descent theorem on $F$-finiteness of homomorphisms of $\mathbb{F}_{p}$-algebras. As a corollary, we prove the following. Let $g\colon B \to C$ be a homomorphism of Noetherian $\mathbb{F}_{p}$-algebras. If $g$ is faithfully flat reduced and $C$ is $F$-finite, then $B$ is $F$-finite. This is a generalization of Seydi's result on excellent local rings of characteristic $p$.

We study Hilbert geometries admitting similar singularities on their boundary to those of a simplex. We show that in an adapted neighborhood of those singularities, two such geometries are bi-Lipschitz. As a corollary we prove that the Hilbert geometry of a convex set is bi-Lipschitz equivalent to a normed vector space if and only if the convex is a polytope.

This work characterizes global quotient stacks---smooth stacks associated to a finite group acting on a manifold---among smooth quotient stacks $[M/G]$, where $M$ is a smooth manifold equipped with a smooth proper action by a Lie group $G$. The characterization is described in terms of the action of the connected component $G_{0}$ on $M$ and is related to (stacky) fundamental group and covering theory. This characterization is then applied to smooth toric Deligne--Mumford stacks, and global quotients among toric DM stacks are then characterized in terms of their associated combinatorial data of stacky fans.

We study an initial boundary value problem on a ball for the isentropic system of compressible Navier--Stokes equations, in particular, a criterion of breakdown of the classical solution. For smooth initial data away from vacuum, it is proved that the classical solution which is spherically symmetric loses its regularity in a finite time if and only if the concentration of mass forms around the center in Lagrangian coordinate system. In other words, in Euler coordinate system, either the density concentrates or vanishes around the center. For the latter case, one possible situation is that a vacuum ball appears around the center and the density may concentrate on the boundary of the vacuum ball simultaneously.

It is well-known that space-like maximal surfaces and time-like minimal surfaces in Lorentz--Minkowski $3$-space $\boldsymbol{R}^{3}_{1}$ have singularities in general. They are both characterized as zero mean curvature surfaces. We are interested in the case where the singular set consists of a light-like line, since this case has not been analyzed before. As a continuation of a previous work by the authors, we give the first example of a family of such surfaces which change type across a light-like line. As a corollary, we also obtain a family of zero mean curvature hypersurfaces in $\boldsymbol{R}^{n+1}_{1}$ that change type across an ($n-1$)-dimensional light-like plane.