Hokkaido Mathematical Journal

Vol. 53,2024

    No. 3

    KITTANEH, Fuad; MORADI, Hamid Reza; SABABHEH, Mohammad;
    On the numerical radius of the product of Hilbert space operators.
    Hokkaido Mathematical Journal, 53 (2024) pp.395-420

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    Abstract

    In this paper, we discuss inequalities for the numerical radius of the product of two Hilbert space operators. The significance of the obtained results is in the way they interpolate and improve several known inequalities in the literature. The equality cases are also discussed.

    Keywords Numerical radius; operator norm; triangle inequality; positive operator;
    msc2020(primary) 47A30;
    msc2020(secondary) 47A12; 15A60;
    KLIMEK, Slawomir; McBRIDE, Matt; SAKAI, Kaoru;
    Implementations of Derivations on the Quantum Annulus.
    Hokkaido Mathematical Journal, 53 (2024) pp.421-441

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    Abstract

    We construct compact parametrix implementations of covariant derivations on the quantum annulus.

    Keywords Derivations; unbounded operators; Fredholm operators; $C^*$-algebras;
    msc2020(primary) 46L57;
    msc2020(secondary) 46L89;

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    Abstract

    Source algebras of block ideals are one of the main subjects in the modular representation theory of finite groups. Especially their bimodule structures attract much interest. In this paper, we shall treat block ideals with defect groups isomorphic to extraspecial $p$-groups of order $p^3$ and exponent $p$. We shall first analyze bimodule structures of these block ideals; we shall give a direct sum decomposition. We shall then prove that the images of the transfer maps on the cohomology rings of defect groups defined by the source algebras coincide with the cohomology rings of the block ideals in concern.

    Keywords block ideals of finite group algebras; source algebras; cohomology of block ideals;
    msc2020(primary) 20C20;
    IKEHATA, Ryo;
    A role of potential on $L^{2}$-estimates for some evolution equations.
    Hokkaido Mathematical Journal, 53 (2024) pp.463-484

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    Abstract

    In this paper, we discuss a role of the potential term on the $L^2$ estimate of the solution itself for wave equations with/without a damping term. In the case of free waves, it is known ([10]) that the $L^2$-norm of the solution itself generally increases to infinity (as $t \to \infty$) in $1$ and $2$ dimensions. However, in this paper, we report that such a grow-up property can be controlled by adding a potential term with a generous condition. This idea can also be applied to damped wave equations with potential terms, especially in one dimension, where faster energy decay rates are observed than in the case of ordinary damped wave equaions. Applications to heat and plate equations with a potential can also be treated. In this paper, the low dimensional case is a main target.

    Keywords Wave; heat and plate equations; potential; Cauchy problem; $L^{2}$-bound; damping; fast decay;
    msc2020(primary) 35L05;
    msc2020(secondary) 35B40; 35B45;
    OGAWA, Satoshi;
    Linearization of transition functions along a certain class of Levi-flat hypersurfaces.
    Hokkaido Mathematical Journal, 53 (2024) pp.485-510

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    Abstract

    We pose a normal form of transition functions along some Levi-flat hypersurfaces obtained by suspension. By focusing on methods in circle dynamics and linearization theorems, we give a sufficient condition to obtain a normal form as a geometrical analogue of Arnol'd's linearization theorem.

    Keywords Levi-flat; KAM-theory; Linearization;
    msc2020(primary) 32V30;
    NGOC, Tran Duc; CHI, Kieu Phuong; QUANG, Si Duc;
    Meromorphic mappings from a Kähler manifold into a projective space sharing different families of hyperplanes.
    Hokkaido Mathematical Journal, 53 (2024) pp.511-530

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    Abstract

    Let $M$ be a complete Kähler Manifold, whose universal covering is biholomorphic to a ball in $\mathbb{C}^m$. We prove that if two linearly nondegenerate meromorphic mappings $f$ and $g$ from $M$ into $\mathbb{P}^n(\mathbb{C})$ share two different families of hyperplanes $\{H_j\}_{j=1}^q$ and $\{L_j\}_{j=1}^q$ without multiplicity then there is a linear projective transformation $\mathcal L$ of $\mathbb{P}^n(\mathbb{C})$ into itself such that $\mathcal L(g)\equiv f$ and $\mathcal L(L_j)=H_j$ $(1\le j\le q)$ for $q$ large enough.

    Keywords Kähler manifold; uniqueness theorem; meromorphic mapping; hyperplane;
    msc2020(primary) 32H30; 32A22;
    msc2020(secondary) 30D35;

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    Abstract

    Under the C-admissible condition, Hoshi and Mochizuki proved a geometric version of the Grothendieck conjecture for the geometric outer monodromy representation associated to the configuration space of a hyperbolic curve. In the present paper, under conditions concerning the number of cusps of the hyperbolic curve and the dimension of the configuration space, we remove the C-admissible condition from the result of Hoshi and Mochizuki.

    Keywords configuration space of a hyperbolic curve; geometric version of the Grothendieck conjecture; combinatorial anabelian geometry;
    msc2020(primary) 14H30;
    msc2020(secondary) 14H10;