Let $M$ be a cyclic $p$-gonal curve with a positive prime number $p$, and let $V$ be the automorphism of order $p$ satisfying $M/ \lt V) \simeq \bf{P}^{1}$. It is well-known that finite subgroups $H$ of $\operatorname{Aut}(\bf{P}^{1})$ are classified into five types. In this paper, we determine the defining equation of $M$ with $H \subset \operatorname{Aut}(M/ \lt V \gt)$ for each type of $H$, and we make a list of hyperelliptic curves of genus 2 and cyclic trigonal curves of genus 5, 7, 9 with $H = Aut(M/ \lt V \gt)$.

We study the spectral gaps of the generalized Kronig-Penney Hamiltonians which possesses two point interactions in the basic period cell. We suppose that each interaction is given by a rotation. We determine whether or not the jth spectral gap of the Hamiltonian is absent for a given $j \in \mathbb{N}$.

Let $\sigma(n)$ denote the sum of divisors function. We prove that if $(2, m) = 1$ and $2m \gt 3^{9}$ then $1^{0} \sigma(2m) \lt \frac{39}{40}e^{\gamma}2m \log \log 2m$, and for all odd integers $m \gt \frac{3^{9}}{2}$, we have $2^{0} \sigma(m) \lt e^{\gamma}m \log \log m$. Moreover, we show that if $\sigma(2m) \lt \frac{3}{4}e^{\gamma}2m \log \log 2m$, for $m \gt m_{0}$ and $(2, m) = 1$, then the inequality $\sigma(2^{\alpha}m) \lt e^{\gamma}2^{\alpha}m \log \log 2^{\alpha}m$ is true for all integers $\alpha \geq 2$ and $m \gt m_{0}$. Robin criterion implies that the Riemann hypothesis is true for these cases.

For any compact strictly pseudoconvex CR manifold $M$ endowed with a contact form $\theta$ we obtain the Bochner type formula $\frac{1}{2}\Delta_{b}(|\nabla^{H}f|^{2}) = |\pi_{H}\nabla^{2}f|^{2} + (\nabla ^{H}f)(\Delta_{b}f) + \rho(\nabla^{H}f, \nabla^{H}f) + 2Lf$ (involving the sublaplacian $\Delta_{b}$ and the pseudohermitian Ricci curvature $\rho$). When $M$ is compact of CR dimension $n$ and $\rho(X,X) + 2A(X,JX) \geq kG_{\theta}(X,X), X \in H(M)$, we derive the estimate $-\lambda \geq 2nk/(2n - 1)$ on each nonzero eigenvalue $\lambda$ of $\Delta_{b}$ satisfying $\mathrm{Eigen}(\Delta_{b}; \lambda) \cap \mathrm{Ker}(T) \neq (0)$ where $T$ is the characteristic direction of $d\theta$.

We will determine the ring structure of the Hochschild cohomology $HH^{\ast}(\mathbb{Z}D_{2n})$ of the integral group ring of the dihedral group $D_{2n}$ of order $2n$.

We investigate the finite support product of forcing notions related to destructible gaps, and prove the existence of a large set of independent destructible gaps under $\diamond$.

In this paper we prove a mixed weighted Strichartz inequality for the solution of $$\partial^{2}_{t} - \Delta_{x} + V(x) + 1)u(t,x) = F(t,x),$$ where $x \in \mathbb{R}^{3}$ and $V$ is a Hölder continuous non-negative potential satisfying the inequality $$V(x) \leq C(1 + |x|)^{-3-\delta}$$ with some constants $C, \delta \gt 0$.

Let $M$ be an $n$-dimensional compact Willmore submanifold in an $(n + p)$-dimensional unit sphere $S^{n+p}$. Denote by $S$ and $H$ the square of the length of the second fundamental form and the mean curvature of $M$. Let $\rho$ be the non-negative function on $M$ defined by $\rho^{2} = S - nH^{2}$ and $K$ be the function which assigns to each point of $M$ the infimum of the sectional curvature at the point. In this paper, first of all, we prove that, if $K$, $H$ and $p$ satisfy $K \geq \frac{p-1}{2p-1} + (n - 2)\frac{H\rho}{\sqrt{n(n-1)}}+H^{2}$, then either $M$ is totally umbilic; or a Willmore torus $W_{1,n- l}$; or the Veronese surface in $S^4$; if the Ricci curvature $R_{ii}$, $H$ and $\rho$ satisfy $R_{ii} \geq (n - 2)+(n - 2)H{\rho} + H^{2}$, for $n \geq 5$, then either $M$ is totally umbilic or a Willmore torus $W_{m,m}$. Secondly, we consider the Willmore submanifold with flat normal connection, we obtain that, if $0 \leq \rho^{2} \leq n$ then eigher $M$ is totally umbilic or a Willmore torus $W_{m,n-m}$; if $K \geq (n-2)+ \frac{H\rho}{\sqrt{n(n-1)}}+ H^{2}$, then $M$ is totally umbilic or $n \leq \rho^{2} \leq np$.

The Serre duality for a holomorphic vector bundle over a compact, complex manifold still holds over a compact, strongly pseudo-convex manifold $M$. This duality theorem is a vector bundle version of the Serre duality obtained by N. Tanaka in [3] for ordinary $(p, q)$-forms on $M$.

Let $H$ be a Weierstrass semigroup, i.e., the set $H(P)$ of integers which are pole orders at $P$ of regular functions on $C \setminus \{P\}$ for some pointed non-singular curve $(C, P)$. In this paper for any Weierstrass semi group $H$ we construct a double covering $\pi:\tilde{C} \to C$ with a ramification point $\tilde{P}$ such that $H(\pi(\tilde{P})) = H$. We also determine the semigroup $H(\tidle{P})$. Moreover, in the case where $H$ starts with 3 we investigate the relation between the semigroup $H(\tilde{P})$ and the Weierstrass semigroup of a total ramification point on a cyclic covering of the projective line with degree 6.