Parabolic Bergman space $\berg[p]$ is a Banach space of all $p$-th integrable solutions of a parabolic equation $(\partial/\partial t + (-\Delta)^{\alpha})u = 0$ on the upper half space, where $0<\alpha\leq1$ and $1\leq p<\infty$. In this note, we consider the Toeplitz operator from $\berg[p]$ to $\berg[q]$ where $p\leq q$, and discuss the condition that it be compact.

It is proved that all bounded John disks are local bilipschitz images of quasidisks. This makes it easy to prove many necessary conditions for John disks

We give two kinds of direct sum decompositions of the class of infinitely differentiabe functions. They are related to the kernel of the higher order partial derivative operator and the polynomial space.

We present a certain family of strongly $1$-trivial Montesinos knots, and show that if a well-known conjecture on Seifert surgery is valid, then the family contains all strongly $1$-trivial Montesinos knots.

We consider Riesz decomposition theorem for superbiharmonic functions in the punctured ball. In fact, we show that under certain growth condition on surface integrals, superbiharmonic functions are represented as a sum of potentials and biharmonic functions.

We study the stability of direct images by Frobenius morphisms. We prove that if the cotangent vector bundle of a nonsingular projective surface $X$ is semistable with respect to a numerically positive polarization divisor satisfying certain conditions, then the direct images of the cotangent vector bundle tensored with line bundles on $X$ by Frobenius morphisms are semistable with respect to the polarization. Hence we see that the de Rham complex of $X$ consists of semistable vector bundles if $X$ has the semistable cotangent vector bundle with respect to the polarization with certain mild conditions.

A singular perturbation problem for a scalar bistable nonlocal reaction-diffusion equation is treated. It is rigorously proved that the layer solutions of this nonlocal reaction-diffusion equation converge to solutions of the averaged mean curvature flow on a finite time interval as the singular perturbation parameter tends to zero.

We study the stationary problem of a reaction-diffusion system with a small parameter $\varepsilon$, which approximates the cross-diffusion competition system proposed to study spatial segregation problem between two competing species. The convergence between two systems as $\varepsilon \downarrow 0$ is discussed from analytical and complementarily numerical point of views.