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## Vol. 49,2020

### No. 1

- AOMOTO, Kazuhiko; MACHIDA, Yoshinori;
*Hypergeometric integrals associated with hypersphere arrangements and Cayley-Menger determinants.*- Hokkaido Mathematical Journal, 49 (2020) pp.1-85
- BANNAI, Shinzo; TOKUNAGA, Hiro-o; YAMAMOTO, Momoko;
*Rational points of elliptic surfaces and the topology of cubic-line, cubic-conic-line arrangements.*- Hokkaido Mathematical Journal, 49 (2020) pp.87-108
- GIANG, Ha Huong;
*Uniqueness theorem of meromorphic mappings of a complete Kähler manifold into a projective space.*- Hokkaido Mathematical Journal, 49 (2020) pp.109-127
- MŁOTKOWSKI, Wojciech; OBATA, Nobuaki;
*On quadratic embedding constants of star product graphs.*- Hokkaido Mathematical Journal, 49 (2020) pp.129-163
- WANG, Dinghuai; ZHOU, Jiang; TENG, Zhidong;
*Sharp estimates for commutators of bilinear operators on Morrey type spaces.*- Hokkaido Mathematical Journal, 49 (2020) pp.165-199

### Fulltext

PDF### Abstract

The $n$-dimensional hypergeometric integrals associated with a hypersphere arrangement $S$ are formulated by the pairing of $n$-dimensional twisted cohomology $H_\nabla^n (X, \Omega^\cdot (*S))$ and its dual. Under the condition of general position we present an explicit representation of the standard form by a special (NBC) basis of the twisted cohomology (contiguity relation in positive direction), the variational formula of the corresponding integral in terms of special invariant $1$-forms $\theta_J$ written by Calyley-Menger minor determinants, and a connection relation of the unique twisted $n$-cycle identified with the unbounded chamber to a special basis of twisted $n$-cycles identified with bounded chambers. Gauss-Manin connections are formulated and are explicitly presented in two simplest cases. In the appendix contiguity relation in negative direction is presented in terms of Cayley-Menger determinants.

MSC(Primary) | 14F40 |
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MSC(Secondary) | 33C70; 14H70; |

Uncontrolled Keywords | hypergeometric integral; hypersphere arrangement; twisted rational de Rham cohomology; Cayley-Menger determinant; contiguity relation; Gauss-Manin connection; |

### Fulltext

PDF### Abstract

In this paper, we continue the study of the relation between rational points of rational elliptic surfaces and the topology of plane curves. As an application, we give first examples of Zariski pairs of cubic-line arrangements that do not involve inflectional tangent lines.

MSC(Primary) | 14J27 |
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MSC(Secondary) | 14H30; 14H50; |

Uncontrolled Keywords | Elliptic surface; rational points; cubic-line arrangement; Zariski pair; |

### Fulltext

PDF### Abstract

In this article, we prove a uniqueness theorem for meromorphic mappings of a complete Kähler manifold $M$ into $\mathbb{P}^n(\mathbb{C})$ sharing hyperplanes in general position with a general condition on the intersections of the inverse images of these hyperplanes.

MSC(Primary) | 32H30 |
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MSC(Secondary) | 32A22; 30D35; |

Uncontrolled Keywords | uniqueness theorem; meromorphic mapping; complete Kähler manifold; |

### Fulltext

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A connected graph $G$ is of QE class if it admits a quadratic embedding in a Hilbert space, or equivalently if the distance matrix is conditionally negative definite, or equivalently if the quadratic embedding constant $\mathrm{QEC}(G)$ is non-positive. For a finite star product of (finite or infinite) graphs $G=G_1\star\cdots \star G_r$ an estimate of $\mathrm{QEC}(G)$ is obtained after a detailed analysis of the minimal solution of a certain algebraic equation. For the path graph $P_n$ an implicit formula for $\mathrm{QEC}(P_n)$ is derived, and by limit argument $\mathrm{QEC}(\mathbb{Z})=\mathrm{QEC}(\mathbb{Z}_+)=-1/2$ is shown. During the discussion a new integer sequence is found.

MSC(Primary) | 05C50 |
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MSC(Secondary) | 05C12; 05C76; |

Uncontrolled Keywords | conditionally negative definite matrix; distance matrix; quadratic embedding; QE constant; star product graph; |

### Fulltext

PDF### Abstract

Denote by $T$ and $I_{\alpha}$ the bilinear Calderón-Zygmund operators and bilinear fractional integrals, respectively. In this paper, it is proved that if $b_{1},b_{2}\in {\rm CMO}$ (the BMO-closure of $C^{\infty}_{c}(\mathbb{R}^n)$), $[\Pi \vec{b},T]$ and $[\Pi\vec{b},I_{\alpha}]$ $(\vec{b}=(b_{1},b_{2}))$ are all compact operators from $\mathcal{M}^{p_{0}}_{\vec{P}}$ (the norm of $\mathcal{M}^{p_{0}}_{\vec{P}}$ is strictly smaller than $2-$fold product of the Morrey norms) to $M^{q_{0}}_{q}$ for some suitable indices $p_{0},p_{1},p_{2}$ and $q_{0},q$. Specially, we also show that if $b_{1}=b_{2}$, then $b_{1}, b_{2}\in {\rm CMO}$ is necessary for the compactness of $[\Pi\vec{b},I_{\alpha}]$ on Morrey space.

MSC(Primary) | 42B20 |
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MSC(Secondary) | 47B07; 42B99; 47G99; |

Uncontrolled Keywords | Bilinear Calderón-Zygmund operator; Bilinear fractional integral operator; Characterization; Compactness; Commutator; |