The purpose of this paper is to construct, following Morse homology theory, integral chain complexes of the canonical Morse-Smale functions on the Grassmannian $Gr(m,n)$ of $m$-dimensional subspaces and the Grassmannian $\widetilde{Gr}(m,n)$ of oriented $m$-dimensional subspaces in the real space ${\mathbb R}^{m+n}$. As an application, the homology groups of those chain complexes are computed straightforwardly, so as to recover the singular homology groups $H_*(Gr(m,n);{\mathbb Z})$ and $H_*(\widetilde{Gr}(2,n);{\mathbb Z})$ for any positive integers $m$ and $n$.

We study the large time asymptotic behavior of solutions to the Cauchy problem for the fractional nonlinear Schrödinger equation \begin{equation*}\left\{ \begin{array}{c}i\partial _{t}u-\frac{1}{\alpha }\left\vert \partial _{x}\right\vert^{\alpha }u=\lambda \left\vert u\right\vert ^{\alpha }u,\text{ }t\gt 0,\text{ } x\in \mathbb{R}\mathbf{,} \ u\left( 0,x\right) =u_{0}\left( x\right) ,\text{ }x\in \mathbb{R}\mathbf{,}\end{array}\right.\end{equation*}where $\lambda \gt 0,$ the fractional derivative $\left\vert \partial_{x}\right\vert ^{\alpha }=\mathcal{F}^{-1}\left\vert \xi \right\vert^{\alpha }\mathcal{F},$ $\alpha \gt \frac{5}{2}.$ This paper is a sequel to our previous papers [17] for $2\lt \alpha \lt \frac{5}{2}$ and [36] for $\alpha =\frac{5}{2}$. We show that solutions decay in time at the rate $t^{-\frac{1}{\alpha }}\left( \log t\right) ^{-\frac{1}{\alpha }}$, namely that the nonlinearity acts as a dissipative term, when $\lambda \gt 0$. This phenomena does not occur for the cubic problem \begin{equation*}\left\{ \begin{array}{c}i\partial _{t}u-\frac{1}{\alpha }\left\vert \partial _{x}\right\vert^{\alpha }u=\lambda \left\vert u\right\vert ^{2}u,\text{ }t\gt 0,\text{ }x\in \mathbb{R}\mathbf{,} \ u\left( 0,x\right) =u_{0}\left( x\right) ,\text{ }x\in \mathbb{R}\mathbf{,}\end{array}\right.\end{equation*}with $0 \lt \alpha \leq 2.$

We introduce $\mathfrak{f}$-stability, a modification of fibration stability of Dervan-Sektnan [11], and show that $\mathfrak{f}$-semistable fibrations have only semi log canonical singularities. Moreover, $\mathfrak{f}$-stability puts restrictions on semi log canonical centers on Fano fibrations.

We introduced three kinds of invariants of a virtual knot called the first, second, and third intersection polynomials in the first paper [5]. We also gave the connected sum formulae of the intersection polynomials in the second paper [6]. In this paper, we give characterizations of intersection polynomials.

For a simple graph $G$, assume that $J(G)$ is the vertex cover ideal of $G$ and $J(G)^{(s)}$ is the $s$-th symbolic power of $J(G)$. We prove that $\operatorname{reg}(J(C)^{(s)})=\operatorname{reg}(J(C)^s)$ for all $s\geq 1$ and for all odd cycle $C$. For a simplicial complex $\Delta$, we show that if $I_{\Delta}^{\vee}$ is weakly polymatroidal (not necessarily generated in one degree) then $\Delta$ is vertex decomposable. Some evidences are provided that the converse conclusion of the above result also holds true if $\Delta$ is pure. Let $W=G^{\pi}$ be a fully clique-whiskering graph. We prove that $J(W)^s$ is weakly polymatroidal for all $s\geq 1$.

Let $H$ be a division ring of finite dimension over its center, let $H[T]$ be the ring of polynomials in a central variable over $H$, and let $H(T)$ be its quotient skew field. We show that every intermediate division ring between $H$ and $H(T)$ is itself of the form $H(f)$ for some $f$ in the center of $H(T)$. This generalizes the classical Lüroth's theorem. More generally, we extend Igusa's theorem characterizing the transcendence degree 1 subfields of rational function fields, from fields to division rings.

We can run the MMP for any divisor on any $\mathbb{Q}$-factorial projective toric variety. We show that two Mori fiber spaces, which are outputs of the above MMP, are connected by finitely many elementary transforms.

In this paper, we extend Zakeri's result in [27] on boundaries of bounded type Siegel disks of some entire functions to some transcendental meromorphic functions as follows: We consider a one parameter family of some transcendental meromorphic functions with one pole, two critical points, one finite asymptotic value zero, and bounded type fixed Siegel disks centered at the origin. We show that if two critical values coincide, then the boundary of the Siegel disk is a quasicircle containing exactly one critical point, and the set $\Omega_1$ of all parameters for which two critical values coincide is countably infinite. We also show that there exist uncountable sets $\Omega_2$ and $\Omega_3$ such that the boundary of the Siegel disk is a quasicircle containing exactly one critical point for any parameter in $\Omega_2$ and the boundary of the Siegel disk is a quasicircle containing exactly two critical points for any parameter in $\Omega_3$. Furthermore, we can construct $\Omega_2$ so that for uncountably many parameters in $\Omega_2$, the critical values which are the images of the critical points outside the boundaries of the Siegel disks are in the Siegel disks, on the boundaries of the Siegel disks, and outside the closures of the Siegel disks.