In this sequel to an earlier article, employing more commutative algebra than previously, we show that an isoparametric hypersurface with four principal curvatures and multiplicities $(3,4)$ in $S^{15}$ is one constructed by Ozeki and Takeuchi and Ferus, Karcher, and Münzner, referred to collectively as of OT-FKM type. In fact, this new approach also gives a considerably simpler proof, both structurally and technically, that an isoparametric hypersurface with four principal curvatures in spheres with the multiplicity constraint $m_2\ge 2m_1-1$ is of OT-FKM type, which left unsettled exactly the four anomalous multiplicity pairs $(4,5)$, $(3,4)$, $(7,8)$, and $(6,9)$, where the last three are closely tied, respectively, with the quaternion algebra, the octonion algebra, and the complexified octonion algebra, whereas the first stands alone in that it cannot be of OT-FKM type. A by-product of this new approach is that we see that Condition B, introduced by Ozeki and Takeuchi in their construction of inhomogeneous isoparametric hypersurfaces, naturally arises. The cases for the multiplicity pairs $(4,5)$, $(6,9)$, and $(7,8)$ remain open now.

A domain $D\subset {\mathbb{C}}_z$ admits the circular slit mapping $P\left(z\right)$ for $a,b\in D$ such that $P\left(z\right)-1/(z-a)$ is regular at $a$ and $P\left(b\right)=0$. We call $p\left(z\right)=log\left|P\right(z\left)\right|$ the $L_1$-principal function and $\alpha =log\left|P\text{'}\right(b\left)\right|$ the $L_1$-constant, and similarly, the radial slit mapping $Q\left(z\right)$ implies the $L_0$-principal function $q\left(z\right)$ and the $L_0$-constant $\beta$. We call $s=\alpha -\beta$ the harmonic span for $(D,a,b)$. We show the geometric meaning of $s$. Hamano showed the variation formula for the $L_1$-constant $\alpha \left(t\right)$ for the moving domain $D\left(t\right)$ in ${\mathbb{C}}_z$ with $t\in B:=\{t\in \mathbb{C}:|t|<\rho \}$. We show the corresponding formula for the $L_0$-constant $\beta \left(t\right)$ for $D\left(t\right)$ and combine these to prove that, if the total space $\mathcal{D}={\bigcup }_{t\in B}(t,D(t\left)\right)$ is pseudoconvex in $B\times {\mathbb{C}}_z$, then $s\left(t\right)$ is subharmonic on $B$. As a direct application, we have the subharmonicity of $logcoshd\left(t\right)$ on $B$, where $d\left(t\right)$ is the Poincaré distance between $a$ and $b$ on $D\left(t\right)$.

We study the depth of classes of binomial edge ideals and classify all closed graphs whose binomial edge ideal is Cohen-Macaulay.

In this paper, we give a generalization of Khovanov-Rozansky homology. We define a homology associated to the quantum $({\mathfrak{sl}}_n,\wedge V_n)$ link invariant, where $\wedge V_n$ is the set of fundamental representations of $U_q\left({\mathfrak{sl}}_n\right)$. In the case of an oriented link diagram composed of $[k,1]$-crossings, we define a homology and prove that the homology is invariant under Reidemeister II and III moves. In the case of an oriented link diagram composed of general $[i,j]$-crossings, we define a normalized Poincaré polynomial of homology and prove that the normalized Poincaré polynomial is a link invariant.

We consider the following conjecture: if $X$ is a smooth and irreducible $n$-dimensional projective variety over a field $k$ of characteristic zero, then there is a dense set of reductions $X_s$ to positive characteristic such that the action of the Frobenius morphism on $H^n(X_s,{\mathcal{O}}_{X_s})$ is bijective. There is another conjecture relating certain invariants of singularities in characteristic zero (the multiplier ideals) with invariants in positive characteristic (the test ideals). We prove that the former conjecture implies the latter one in the case of ambient nonsingular varieties.