For an invariant subspace $M_1$ of the Hardy space $H^2$ over the bidisk $\mathbb{D}^2$, write $N_1=H^2 \ominus M_1$. Let $\Omega(M_1)=M_1\ominus(z M_1+w M_1)$ and $\widetilde\Omega(N_1)=\{f\in N_1: z f, w f\in M_1\}$. Then $\Omega(M_1)\not=\{0\}$, and $\Omega(M_1), \widetilde\Omega(N_1)$ are key spaces to study the structure of $M_1$. It is known that there is a nonzero $f_0\in M_1$ such that $M_2=M_1\ominus \mathbb{C} \cdot f_0$ is an invariant subspace. It is described the structures of $\Omega(M_2), \widetilde\Omega(N_2)$ using the words of $\Omega(M_1), \widetilde\Omega(N_1)$ and $f_0$. To do so, it occur many cases. We shall give examples for each cases.

This paper is a continuation of the previous paper [9]. Let $M_1$ be an invariant subspace of $H^2$ over the bidisk. Then there exists a nonzero $f_0$ in $M_1$ such that $M_2:=M_1\ominus \mathbb{C} \cdot f_0$ is also an invariant subspace. A relationship is given the ranks of the cross commutators $[R^*_z,R_w]$ on $M_1$ and $M_2$. We also give a relationship of the ranks of the cross commutators $[S_w,S^*_z]$ on $H^2\ominus M_1$ and $H^2\ominus M_2$.

We study the Cauchy problem for a multi-dimensional scalar conservation law with a multiplicative noise. Our aim is to give the well-posedness of an $L^1$-solution characterized by a kinetic formulation under appropriate assumptions. In particular, we focus on the existence of such a solution.

Real hypersurfaces satisfying the condition $\phi l = l \phi$, $(l = R( . , \xi)\xi)$, have been studied by many authors under at least one more condition, since the class of these hypersurfaces is quite tough to be classified. The aim of the present paper is the classification of real hypersurfaces in complex hyperbolic plane $\mathbb{C}H^{2}$ satisfying a generalization of $\phi l = l \phi$ under an additional restriction on a specific function.

Due to Rentschler, Miyanishi and Kojima, the invariant ring for a $\mathrm{G}_a$-action on the affine plane over an arbitrary field is generated by one coordinate. In this note, we give a new short proof for this result using the automorphism theorem of Jung and van der Kulk.

It is known that gyrocommutative gyrogroups and K-loops are equivalent. This is a self-contained paper that presents the equivalence.