Hokkaido Mathematical Journal

No. 3

AUDEH, Wasim; MORADI, Hamid Reza; SABABHEH, Mohammad;

Davidson-Power type and singular value inequalities.

Hokkaido Mathematical Journal, 54 (2025) pp.345-372

Fulltext

PDF

Abstract

This paper has two independent goals. First, we prove a generalized Davidson-Power type inequality that generalizes many recent results in the literature. Then, mean inequalities for the singular values of the product of matrices are discussed.

Abstract

Applications that involve the numerical radius, the norm of accretive-dissipative matrices, and the norm of product of matrices will be shown.

Keywords	Davidson-Power inequality; singular values; unitarily invariant norm; numerical radius; Aluthge transform;
msc2020(primary)	47A30; 15A18;
msc2020(secondary)	47A63; 47A12; 15A60;

ALI, Asma; AHMAD, Bakhtiyar; CHELVAM, T. Tamizh;

Extended zerodivisor graph with genus and crosscap at most two.

Hokkaido Mathematical Journal, 54 (2025) pp.373-393

Fulltext

PDF

Abstract

Let $A$ be a commutative ring with identity and $Z(A)$ the set of all zerodivisors of $A$. The extended zerodivisor graph $\overline{\Gamma}(A)$ of $A$ is defined as the graph with vertex set $Z^\ast(A)=Z(A)\setminus \{0\}$ and two distinct vertices $a$ and $b$ are adjacent if and only if $a^mb^n=0$ with $a^m\neq 0$ and $b^n\neq 0$ for two integers $m\geq 1$ and $n\geq 1.$ In this paper, we classify all finite non-local rings $A$ such that $\overline{\Gamma}(A)$ is of genus and crosscap at most two.

Keywords	Commutative ring; extended zerodivisor graph; zerodivisor graph; genus; crosscap; reduced ring;
msc2020(primary)	05C99; 13A15; 13B99;

GIANNOTTI, Cristina; SPIRO, Andrea; ZOPPELLO, Marta;

Proving the Chow-Rashevskiĭ Theorem à la Rashevskiĭ.

Hokkaido Mathematical Journal, 54 (2025) pp.395-424

Fulltext

PDF

Abstract

We give a new independent proof of a generalised version of the theorem by Rashevskiĭ, which appeared in [Uch. Zapiski Ped. Inst. K. 2 (1938), 83-94] and from which the classical Chow-Rashevskiĭ Theorem follows as a corollary. The proof is structured to allow generalisations to the case of orbits of compositions of flows in absence of group structures, thus appropriate for applications in Control Theory. In fact, the same structure of the proof has been successfully exploited in [C. Giannotti, A. Spiro and M. Zoppello, arXiv 2401.07555 & 2401.07560 (2024)] to determine new controllability criteria for real analytic non-linear control systems. It also yields a corollary, which can be used to derive results under lower regularity assumptions, as it is illustrated by a simple explicit example.

Keywords	Non-integrable distributions; Chow-Rashevskiĭ-Sussmann Theorem; Iterated Lie brackets;
msc2020(primary)	58A30; 34H05; 58J60;

HASEGAWA, Masaru; SAJI, Kentaro;

On cylindrical directions of cross caps and cuspidal edges.

Hokkaido Mathematical Journal, 54 (2025) pp.425-446

Fulltext

PDF

Abstract

We investigate contact of cross caps and cuspidal edges with (right circular) cylinders. We introduce cylindrical directions of cross caps and cuspidal edges which are defined by the kernel field of $A_3$-contact of the surfaces with cylinders.

Keywords	cylindrical direction; cross cap; cuspidal edge; contact mapping;
msc2020(primary)	57R45;
msc2020(secondary)	53A05; 58Kxx;

REZAEI, Shahram;

Attached primes of top generalized local cohomology modules.

Hokkaido Mathematical Journal, 54 (2025) pp.447-457

Fulltext

PDF

Abstract

Let $\frak{a}$ be an ideal of a commutative Noetherian local ring $(R,\frak{m})$ and $M$ and $N$ two finitely generated $R$-modules. Assume that $\operatorname{dim} R = d$ and $\operatorname{pd} M < \infty$. In this paper, we investigate the attached primes of top generalized local cohomology module $\operatorname{Att}_R \operatorname{H}_{\frak{a}}^{d}(M,N)$. In fact, we prove that if $\operatorname{H}_{\frak{a}}^{d}(M,N) \neq 1$ then

Abstract

i) $\operatorname{Att}_R \operatorname{H}_{\frak{a}}^{d}(M,N) \subseteq \lbrace \frak{p} \in \operatorname{Supp}_R N \vert \operatorname{cd}(\frak{a},M,R/\frak{p}) = d\rbrace.$

Abstract

ii) $\lbrace \frak{p} \in \operatorname{Supp}_R N \vert \operatorname{cd}(\frak{a},M,R/\frak{p}) = d ,\, \operatorname{dim} R/\frak{p} \leq d - \operatorname{pd} M \rbrace \subseteq \operatorname{Att}_R \operatorname{H}_{\frak{a}}^{d}(M,N).$

Abstract

iii) $\operatorname{Att}_R(\operatorname{H}_{\frak{a}}^{d}(M,N)) = \lbrace \frak{p} \in \operatorname{Supp}_R N \vert \operatorname{cd}(\frak{a},M, R/\frak{p}) = d , \, \frak{p} = \Ann_R(\operatorname{H}_{\frak{a}}^{d}(M,R/\frak{p}))\rbrace.$

Abstract

iv) $\operatorname{Max} \lbrace \frak{p} \in \operatorname{Supp}_R N \vert \operatorname{cd}(\frak{a},M,R/\frak{p}) = d \rbrace \subseteq \operatorname{Att}_R(\operatorname{H}_{\frak{a}}^{d}(M,N)).$

Abstract

Also, we prove that the containment in (i) is an equality whenever $\operatorname{pd} M < d$ and $\operatorname{dim} N = 1$.

Keywords	annihilator; attached prime; generalized local cohomology;
msc2020(primary)	13D45; 14B15;

INOGUCHI, Jun-ichi; NAITOH, Hiroo;

Grassmann geometry on $H^2\times\mathbb{R}$.

Hokkaido Mathematical Journal, 54 (2025) pp.459-514

Fulltext

PDF

Abstract

We study the Grassmann geometry of surfaces in the reducible Riemannian symmetric space $H^2\times\mathbb{R}$.

Keywords	Grassmann geometry; reducible symmetric space;
msc2020(primary)	53B25; 53C40; 53C30;