In this paper, we investigate simultaneous properties of a convex integrand $γ$ and its dual $δ$. The main results are the following three.

(1) For a $C^\infty$ convex integrand $\gamma: S^n \to \mathbf{R}_+$, its dual convex integrand $\delta: S^n \to \mathbf{R}_+$ is of class $C^\infty$ if and only if $γ$ is a strictly convex integrand.

(2) Let $\gamma: S^n \to \mathbf{R}_+$ be a $C^\infty$ strictly convex integrand. Then, $γ$ is stable if and only if its dual convex integrand $\delta: S^n \to \mathbf{R}_+$ is stable.

(3) Let $\gamma: S^n \to \mathbf{R}_+$ be a $C^\infty$ strictly convex integrand. Suppose that $γ$ is stable. Then, for any $i$ ($0 \le i \le n$), a point $\theta_0 \in S^n$ is a non-degenerate critical point of $γ$ with Morse index $i$ if and only if its antipodal point $-\theta_0 \in S^n$ is a non-degenerate critical point of the dual convex integrand $δ$ with Morse index ($n-i$).

Combinatorial properties of some ideals related to bipartite graphs are examined. A description of the integral closure expressed through the log set of edge ideals of complete bipartite graphs is illustrated together with the fact that nontrivial generalized graph ideals of a strong complete quasi-bipartite graph are integrally closed.

Let $(M,\phi,\xi,\eta,g)$ be a three-dimensional Kenmotsu manifold. In this paper, we prove that the triple $(g,V,\lambda)$ on $M$ is a $*$-Ricci soliton if and only if $M$ is locally isometric to the hyperbolic 3-space $\mathbf{H}^3(-1)$ and $\lambda=0$. Moreover, if $g$ is a gradient $*$-Ricci soliton, then the potential vector field coincides with the Reeb vector field. We also show that the metric of a coKähler 3-manifold is a $*$-Ricci soliton if and only if it is a Ricci soliton.

We show that the normalized Ricci flow $g(t)$ on a smooth closed manifold $M$ existing for all $t \geq 0$ with scalar curvature converging to constant in $L^2$ norm should satisfy $$\liminf_{t\rightarrow \infty} \int_M|\stackrel{\circ}{r}|_{g(t)}^2d\mu_{g(t)} =0,$$ where $\stackrel{\circ}{r}$ is the trace-free part of Ricci tensor. Using this, we give topological obstructions to the existence of such a Ricci flow (even with positive scalar curvature tending to $\infty$ in sup norm) on 4-manifolds.

We consider the initial-value problem for the equivariant Schrödinger maps near a family of harmonic maps. We provide some supplemental arguments for the proof of local well-posedness result by Gustafson, Kang and Tsai in [Duke Math. J. 145(3) 537-583, 2008]. We also prove that the solution near harmonic maps is unique in $C(I;\dot{H}^1(\mathbf{R}^2)\cap \dot{H}^2(\mathbf{R}^2))$ for time interval $I$. In the proof, we give a justification of the derivation of the modified Schrödinger map equation in low regularity settings without smallness of energy.

The notion of block divisibility naturally leads one to introduce unitary cyclotomic polynomials $\Phi_n^*(x)$. They can be written as certain products of cyclotomic poynomials. We study the case where $n$ has two or three distinct prime factors using numerical semigroups, respectively Bachman's inclusion-exclusion polynomials. Given $m \ge 1$ we show that every integer occurs as a coefficient of $\Phi^*_{mn}(x)$ for some $n\ge 1$ following Ji, Li and Moree [9]. Here $n$ will typically have many different prime factors. We also consider similar questions for the polynomials $(x^n-1)/\Phi_n^*(x)$, the inverse unitary cyclotomic polynomials.

Z. Zhou et al. proved that in a Teichmüller equivalence class, there exists an extremal quasiconformal mapping with a weakly non-decreasable dilatation. In this paper, we prove that in an infinitesimal equivalence class, there exists a weakly non-decreasable extremal Beltrami differential.

We show that for an arbitrarily given closed Riemannian manifold $M$ admitting a point $p \in M$ with a single cut point, every closed Riemannian manifold $N$ admitting a point $q \in N$ with a single cut point is diffeomorphic to $M$ if the radial curvatures of $N$ at $q$ are sufficiently close in the sense of $L^1$-norm to those of $M$ at $p$.

In 2004 Oort studied the foliation on the space of $p$-divisible groups. In his theory, special leaves called central streams play an important role. It is still meaningful to investigate central streams, for example, there remain a lot of unknown things on the boundaries of central streams. In this paper, we classify the boundary components of the central stream for a Newton polygon consisting of two segments, where one slope is less than 1/2 and the other slope is greater than 1/2. Moreover we determine the generic Newton polygon of each boundary component using this classification.