We consider an abstract pair-interaction model in quantum field theory with a coupling constant $\lambda\in\mathbb{R}$ and analyze the Hamiltonian $H(\lambda)$ of the model. In the massive case, there exist constants $\lambda_{\rm c}\lt 0$ and $\lambda_{{\rm c},0}\lt\lambda_{\rm c}$ such that, for each $\lambda \in (\lambda_{{\rm c},0},\lambda_{\rm c})\cup (\lambda_{\rm c},\infty)$, $H(\lambda)$ is diagonalized by a proper Bogoliubov transformation, so that the spectrum of $H(\lambda)$ is explicitly identified, where the spectrum of $H(\lambda)$ for $\lambda>\lambda_{\rm c}$ is different from that for $\lambda\in (\lambda_{{\rm c},0},\lambda_{\rm c})$. As for the case $\lambda\lt\lambda_{{\rm c},0}$, we show that $H(\lambda)$ is unbounded from above and below. In the massless case, $\lambda_{\rm c}$ coincides with $\lambda_{{\rm c},0}$.

For a tropical curve $\Gamma$ and a finite subgroup $K$ of the isometry group of $\Gamma$, we prove, extending the work by Haase, Musiker and Yu ([6]), that the $K$-invariant part of the complete linear system associated to a $K$-invariant effective divisor on $\Gamma$ is finitely generated

Let $(R,\frak{m})$ be a Noetherian local ring, $\frak{a}$ an ideal of $R$ and $M$ a finitely generated $R$-module. In this paper, we study Colocalization of formal local cohomology modules. Here, similar to the local global Principle in local cohomology theory, we investigate artinianness and minimaxness of formal local cohomology modules in terms of their colocalizations. Among other things, we will prove that, for any integer $n$, $\mathfrak{F}_{\frak{a}}^i(M)$ is artinian $R$-module for all $i \lt n$, if and only if $_{\frak{p}}(\mathfrak{F}_{\frak{a}}^i(M)) $ is representable $R_{\frak{p}}$-module for all $i \lt n$ and all $\frak{p} \in \operatorname{Spec}(R)$. Also, $ \mathfrak{F}_{\frak{a}}^i(M) $ is minimax $R$-module for all $i \lt n$, if and only if $ _{\frak{p}}(\mathfrak{F}_{\frak{a}}^i(M)) $ is representable $R_{\frak{p}}$-module for all $i \lt n$ and all $\frak{p} \in \operatorname{Spec}(R)\setminus\lbrace \frak{m}\rbrace$.

We study the fourth order nonlinear Schrödinger equation \begin{equation*} i{\partial }_{t}u-\frac{1}{4}\partial _{x}^{4}u=f(u) ,\quad (t,x)\in \mathbb{R}\times \mathbb{R}, \end{equation*} where $f(u) $ is the power nonlinearity of order $p>5.$ The scattering operator is constructed in a neighborhood of the origin in a sutable weighted Sobolev space.

We present a precise discussion of Golubitsky and Schaeffer's treatment on bifurcation of Euler buckling problem. We discuss smoothness of the problem and derive equations describing the bifurcation set $B$ and the hysteresis set $H$ up to order 3, which enable us to draw their figures approximately under suitable assumptions.