Hokkaido Mathematical Journal

No. 1

SAWANO, Yoshihiro; SHIMOMURA, Tetsu; TANAKA}, Hitoshi;
A remark on modified Morrey spaces on metric measure spaces.
Hokkaido Mathematical Journal, 47 (2018) pp.1-15

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Abstract

Morrey norms, which are originally endowed with two parameters, are considered on general metric measure spaces. Volberg, Nazarov and Treil showed that the modified Hardy-Littlewood maximal operator is bounded on Legesgue spaces. The modified Hardy-Littlewood maximal operator is known to be bounded on Morrey spaces on Euclidean spaces, if we introduce the third parameter instead of adopting a natural extension of Morrey spaces. When it comes to geometrically doubling, as long as an auxiliary parameter is introduced suitably, the Morrey norm does not depend on the third parameter and this norm extends naturally the original Morrey norm. If the underlying space has a rich geometric structure, there is still no need to introduce auxiliary parameters. However, an example shows that this is not the case in general metric measure spaces. In this paper, we present an example showing that Morrey spaces depend on the auxiliary parameters.

MSC(Primary) 42B35
MSC(Secondary) 42B25;
Uncontrolled Keywords Morrey space; non-doubling; metric measure spaces;
MIZOTA, Yusuke; NISHIMURA, Takashi;
Lowerable vector fields for a finitely ${\cal L}$-determined multigerm.
Hokkaido Mathematical Journal, 47 (2018) pp.17-23

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We show that the module of lowerable vector fields for a finitely ${\cal L}$-determined multigerm is finitely generated in a constructive way.

MSC(Primary) 58K40
MSC(Secondary) 57R45; 58K20;
Uncontrolled Keywords Lowerable vector field; finitely ${\cal L}$-determined multigerm; preparation theorem; finitely generated module;
ASBOEI, Alireza Khalili; DARAFSHEH, Mohammad Reza; MOHAMMADYARI, Reza;
The influence of order and conjugacy class length on the structure of finite groups.
Hokkaido Mathematical Journal, 47 (2018) pp.25-32

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Let $2^{n}+1 \gt 5$ be a prime number. In this article, we will show $G\cong C_{n}(2)$ if and only if $|G|=|C_{n}(2)|$ and $G$ has a conjugacy class length ${|C_{n}(2)|}/({2^{n}+1})$. Furthermore, we will show Thompson's conjecture is valid under a weak condition for the symplectic groups $C_{n}(2)$.

MSC(Primary) 20D08
MSC(Secondary) 20D60;
Uncontrolled Keywords Finite simple group; conjugacy class length; Thompson's conjecture;

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Abstract

We treat 2D and 3D tumor invasion models with quasi-variational structures, which are composed of two PDEs, one ODE and certain constraint conditions. Although the original model was proposed by M. R. A. Chaplain and A. R. A. Anderson in 2003, the difference between their original model and ours is that the constraint conditions for the distributions of tumor cells and the extracellular matrix are imposed in our model, which give a quasi-variational structure. For 2D and 3D tumor invasion models with quasi-variational structures, we show the existence of global-in-time solutions and consider their large-time behaviors. Especially, for the large-time behaviors, we show that there exists at least one global-in-time solution such that it converges to a constant steady state in an appropriate function space as time goes to $\infty$.

MSC(Primary) 35B40
MSC(Secondary) 35Q92; 35A01;
Uncontrolled Keywords large-time behavior; tumor invasion; quasi-variational structure;
KOGUCHI, Yuto; MATSUMOTO, Keiji; SETO, Fuko;
Schwarz maps associated with the triangle groups $(2,4,4)$ and $(2,3,6)$.
Hokkaido Mathematical Journal, 47 (2018) pp.69-108

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Abstract

We consider the Schwarz maps with monodromy groups isomorphic to the triangle groups $(2,4,4)$ and $(2,3,6)$ and their inverses. We apply our formulas to studies of mean iterations.

MSC(Primary) 14K25
MSC(Secondary) 33C05;
Uncontrolled Keywords Schwarz map; theta function; mean iteration;
KOIKE, Kenji;
The Fermat septic and the Klein quartic as moduli spaces of hypergeometric Jacobians.
Hokkaido Mathematical Journal, 47 (2018) pp.109-141

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We study the Schwarz triangle function with the monodromy group $\Delta(7,7,7)$, and we construct its inverse by theta constants. As consequences, we give uniformizations of the Klein quartic curve and the Fermat septic curve as Shimura curves parametrizing Abelian $6$-folds with endomorphisms $\mathbb{Z}[\zeta_7]$.

MSC(Primary) 14G35
MSC(Secondary) 30F10; 33C05;
Uncontrolled Keywords Shimura curves; Hypergeometric functions; Theta functions;
FAN, Dashan; ZHAO, Fayou;
Certain bilinear operators on Morrey spaces.
Hokkaido Mathematical Journal, 47 (2018) pp.143-159

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Abstract

In this paper, we consider that $T(f,g)$ is a bilinear operator satisfying \begin{equation*} |T(f,g)(x)|\preceq \int_{\mathbb{R}^{n}}\frac{|f(x-ty)g(x-y)|}{|y|^{n}}dy \end{equation*} for $x$ such that $0\notin {\rm supp}~(f(x-t\cdot )) \cap {\rm supp}~(g(x+\cdot ))$. We obtain the boundedness of $T(f,g)$ on the Morrey spaces with the assumption of the boundedness of the operator $T(f,g)$ on the Lebesgues spaces. As applications, we yield that many well known bilinear operators, as well as the first Calderón commutator, are bounded from the Morrey spaces $L^{q,\lambda_{1}}\times L^{r,\lambda_{2}}$ to $L^{p,\lambda}$, where $\lambda /p={\lambda_{1}}/{q}+{\lambda_{2}}/{r}$.

MSC(Primary) 42B20
MSC(Secondary) 42B25; 42B35;
Uncontrolled Keywords Multilinear operators; bilinear Hilbert transform; the first Calderón commutator; Morrey spaces;
BISWAS, Indranil; MJ, Mahan;
An almost complex Castelnuovo de Franchis theorem.
Hokkaido Mathematical Journal, 47 (2018) pp.161-169

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Given a compact almost complex manifold, we prove a Castelnuovo–de Franchis type theorem for it.

MSC(Primary) 32Q60
MSC(Secondary) 14F45;
Uncontrolled Keywords Almost complex structure; Castelnuovo–de Franchis theorem; Riemann surface; fundamental group;
BARBERA, Mariacristina; IMBESI, Maurizio; LA BARBIERA, Monica;
On the symmetric algebras associated to graphs with loops.
Hokkaido Mathematical Journal, 47 (2018) pp.171-190

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We study the symmetric algebra of monomial ideals that arise from graphs with loops. The notion of $s$-sequence is investigated for such ideals in order to compute standard algebraic invariants of their symmetric algebra in terms of the corresponding invariants of special quotients of the polynomial ring related to the graphs.

MSC(Primary) 05C99
MSC(Secondary) 15A78; 13P10;
Uncontrolled Keywords Graphs with loops; graph ideals; symmetric algebra; s-sequences;
EUH, Yunhee; PARK, JeongHyeong; SEKIGAWA, Kouei;
Characteristic function of Cayley projective plane as a harmonic manifold.
Hokkaido Mathematical Journal, 47 (2018) pp.191-203

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Any locally rank one Riemannian symmetric space is a harmonic manifold. We give the characteristic function of a Cayley projective plane as a harmonic manifold. The aim of this work is to show the explicit form of the characteristic function of the Cayley projective plane.

MSC(Primary) 53C25
MSC(Secondary) 53C35;
Uncontrolled Keywords Cayley projective plane; harmonic manifold; characteristic function;
MIZUKAWA, Hiroshi; YAMADA, Hiro-Fumi;
Arithmetic identities for class regular partitions.
Hokkaido Mathematical Journal, 47 (2018) pp.205-221

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Extending the notion of $r$-(class) regular partitions, we define $(r_{1},\dots,r_{m})$-class regular partitions. Partition identities are presented and described by making use of the Glaisher correspondence.

MSC(Primary) 05E10
MSC(Secondary) 05E05;
Uncontrolled Keywords $r$-regular partition; $r$-class regular partition; Glaisher correspondence; Hall-Littlewood symmetric function; character table of symmetric group;
TUMENBAYAR, Khulan; TOKUNAGA, Hiro-o;
Elliptic surfaces and contact conics for a 3-nodal quartic.
Hokkaido Mathematical Journal, 47 (2018) pp.223-244

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Let ${\mathcal Q}$ be an irreducible $3$-nodal quartic and let ${\mathcal C}$ be a smooth conic such that ${\mathcal C} \cap {\mathcal Q}$ does not contain any node of ${\mathcal Q}$ and the intersection multiplicity at $z \in {\mathcal C} \cap {\mathcal Q}$ is even for each $z$. In this paper, we study geometry of ${\mathcal C} + {\mathcal Q}$ through that of integral sections of a rational elliptic surface which canonically arises from ${\mathcal Q}$ and $z \in {\mathcal C} \cap {\mathcal Q}$. As an application, we construct Zariski pairs $({\mathcal C}_1 + {\mathcal Q}, {\mathcal C}_2 + {\mathcal Q})$, where ${\mathcal C}_i$ $(i = 1, 2)$ are smooth conics tangent to ${\mathcal Q}$ at four distinct points.

MSC(Primary) 14J27
MSC(Secondary) 14H30; 14H50;
Uncontrolled Keywords Elliptic surface; section; contact conic; Zariski pair;