Hokkaido Mathematical Journal

No. 1

BENKHALIFA, Mahmoud;

Whitehead realization problem for 1-connected and 5-dimensional CW-complexes.

Hokkaido Mathematical Journal, 55 (2026) pp.1-20

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Abstract

This paper aims to solve the so-called realization problem of J. H. C. Whitehead in the category $\bf{CW^5_1}$ of 1-connected and 5-dimensional CW-complexes $X$ such that $\text{Tor}(H_{2}(X),\mathbb{Z}_2)=0$. For this purpose we define the notion of $\Gamma$-homomorphisms which are graded homomorphisms $f_*:H_{*}(X)\to H_*(Y)$ satisfying a certain algebraic condition. We prove that if $f_{*}$ is $\Gamma$-homomorphism, then there exists a map $\alpha:X \to Y$ such that $H_{*}(\alpha)=f_*$.

Keywords	Whitehead's exact sequence; characteristic extension; $\Gamma$-automorphisms; homotopy types;
msc2020(primary)	55P15;

CHRISTENSEN, Ole; KIM, Hong Oh; KIM, Rae Young;

Two results on polynomially-generated wavelet frames and nonorthogonal polynomial frames.

Hokkaido Mathematical Journal, 55 (2026) pp.21-36

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Abstract

Redundancy of frames plays an important role in the context of erasures, as one might be able to reconstruct signals even if information is lost. The purpose of the paper is to present two results on overcomplete polynomially-generated frames. In the first part we show that bandlimited wavelet frames, for which the Fourier transform of the window $\psi$ has polynomial behavior in a neighborhood of zero, are remarkably stable towards erasures: given such a frame and any $N\in \mathbb{N}$, there exists an ordering of the frame elements having the property that for any $N\in \mathbb{N}$ the subfamily obtained by selecting each $N$th element itself a frame. The result is surprising, because it is known that band-limited wavelets for which $\widehat{\psi}$ vanishes on a neighborhood of zero never has this property. We illustrate the results with a number of concrete constructions, e.g., showing that it is even possible to construct a Parseval frame with the subsampling property. No such example has been identified in the literature so far. In the second part of the paper we introduce a method that allows to construct an overcomplete frame for Hilbert spaces of the form $L^2(-r,r)$ or $L^2(0,r)$, starting with a Riesz basis for the same space. When applied to standard orthogonal polynomials the construction yields nonorthogonal polynomial frames with attractive features: the frames are linearly independent, have infinite excess, the frame decomposition is simple, and the functions in the frame are ``very close'' to the functions in the given orthogonal system.

Keywords	Frames; redundancy; wavelet frames; orthogonal polynomials; polynomial frames;
msc2020(primary)	42C15; 42C40;

SASAKI, Toru; KAJIWARA, Tsuyoshi; OTANI, Yoji;

Stability of the equilibria of a generalized cell-pathogen-immunity system.

Hokkaido Mathematical Journal, 55 (2026) pp.47-56

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Abstract

In this paper, we prove a stability result for a simple mathematical model that describes virus dynamics as well as predator-prey-substrate interaction in a chemostat. Our model consists of ordinary differential equations with general functional forms. We assume that the functions satisfy some conditions, such as monotonicity and signatures at zero and infinity. Employing the general forms makes the relation among the basic reproduction number, the existence of equilibria and the stability switch clear.

Keywords	differential equations; virus dynamics; stability of equilibrium; predator-prey system;
msc2020(primary)	34D20; 92B99; 92D25;

DENG, Yanlin; DU, Feng; MAO, Jing; ZHAO, Yan;

Sharp eigenvalue estimates and related rigidity theorems.

Hokkaido Mathematical Journal, 55 (2026) pp.57-85

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Abstract

In this paper, sharp bounds for the first nonzero eigenvalues of different type have been obtained. Moreover, when those bounds are achieved, related rigidities can be characterized. More precisely, first, by applying the Bishop-type volume comparison proven in [10], [13] and the Escobar-type eigenvalue comparisons for the first nonzero Steklov eigenvalue of the Laplacian proven in [26], for manifolds with radial sectional curvature upper bound, under suitable preconditions, we can show that the first nonzero Wentzell eigenvalue of the geodesic ball on these manifolds can be bounded from above by that of the geodesic ball with the same radius in the model space (i.e., spherically symmetric manifolds) determined by the curvature bound. Besides, this upper bound for the first nonzero Wentzell eigenvalue can be achieved if and only if these two geodesic balls are isometric with each other. This conclusion can be seen as an extension of eigenvalue comparisons in [9], [26]. Second, we prove a general Reilly formula for the drifting Laplacian, and then use the formula to give a sharp lower bound for the first nonzero Steklov eigenvalue of the drifting Laplacian on compact smooth metric measure spaces with boundary and convex potential function. Besides, this lower bound can be achieved only for the Euclidean ball of the prescribed radius.

Keywords	Laplacian; Drifting Laplacian; Wentzell eigenvalues; Eigenvalue comparisons; Steklov eigenvalues; Reilly formula;
msc2020(primary)	35P15; 53C20; 53C42;

NUÑO-BALLESTEROS, Juan J.; PEÑAFORT SANCHIS, Guillermo; VILLA, Cinzia;

On the analytic structure of double and triple points in the target of finite holomorphic multi-germs.

Hokkaido Mathematical Journal, 55 (2026) pp.87-111

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Abstract

We study the analytic structure of the double and triple point spaces $M_2(f)$ and $M_3(f)$ of finite multi-germs $f: (X,S)\to(\mathbb{C}^{n+1},0)$, based on results of Mond and Pellikaan for the mono-germ case. We show that these spaces are Cohen-Macaulay, provided that certain dimensional conditions are satisfied, and give explicit expressions for their defining ideals in terms of those of their mono-germ branches.

Keywords	Fitting ideals; Cohen-Macaulay; Multiple point; Singular mappings;
msc2020(primary)	14M05; 58K99; 32S25;

TOYOKAWA, Hisayoshi;

Conservative and ergodic $\sigma$-finite invariant measures for Markov operators and Darling-Kac law.

Hokkaido Mathematical Journal, 55 (2026) pp.113-136

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Abstract

We give necessary and sufficient conditions for the existence and finiteness of absolutely continuous conservative and ergodic $\sigma$-finite invariant measures for a Markov operator, via the induced operator and the (Thaler) jump operator with respect to an appropriate sweep-out set. The key of our characterisation is mean constrictivity, introduced in [3], for the induced/jump operator. As an example, a one-parameter family of random maps on the unit interval with a uniformly contraction property is concerned. For those random maps, further statistical laws, such as the Darling-Kac law, are established through the asymptotics of their $\sigma$-finite and infinite invariant measures.

Keywords	$\sigma$-finite invariant measures; Markov operators; ergodicity; mean constrictivity; random dynamical systems; Darling-Kac law; Dynkin-Lamperti arcsine law; infinite ergodic theory;
msc2020(primary)	37A30;
msc2020(secondary)	37H05;

MAEDA, Sadahiro; ADACHI, Toshiaki;

Notes on the redefinition of Berger-spheres.

Hokkaido Mathematical Journal, 55 (2026) pp.137-148

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Abstract

Studying geometric properties of geodesic spheres of sufficiently small radii in a complex hyperbolic space $\mathbb{C}H^n(c)$ of constant holomorphic sectional curvature $c$ $(< 0)$, we present examples of Riemannian manifolds which are diffeomorphic to Euclidean spheres and are not so called Berger spheres. These manifolds are closely related to the redefinition of Berger spheres given in [9]. We next characterize these manifolds considered as real hypersurfaces in $\mathbb{C}H^n(c)$ from the viewpoint of submanifold geometry.

Keywords	Berger-spheres; complex hyperbolic spaces; geodesic spheres; Riemannian homogeneous manifolds; length spectrum; circles; principal curvature vectors;
msc2020(primary)	53B25;
msc2020(secondary)	53C40;

MORIFUJI, Takayuki; TRAN, Anh T.;

Twisted Alexander polynomials of punctured torus bundles with tunnel number one.

Hokkaido Mathematical Journal, 55 (2026) pp.149-165

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Abstract

In this paper, we provide an explicit formula for the twisted Alexander polynomial of a once-punctured torus bundle over the circle with tunnel number one associated with a curve of irreducible representations into $\mathrm{GL}_4(\mathbb{C})$. Consequently, we can derive the Reidemeister torsion on the curve by using this formula.

Keywords	Twisted Alexander polynomial; Reidemeister torsion; Tong-Yang-Ma representation; once-punctured torus bundle; tunnel number one;
msc2020(primary)	57K31;
msc2020(secondary)	57K14; 57M05;