Hokkaido Mathematical Journal

No. 3

SHIMIZU, Tatsuro;

On self-intersection of singularity sets of fold maps.

Hokkaido Mathematical Journal, 50 (2021) pp.297-308

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Abstract

In this article, we give equations involving homology classes given by the self-intersection of singularity sets of fold maps. As an application, we give an obstruction to existence of fold maps up to cobordism.

Keywords	singularity; fold map; self intersection; Thom polynomial;
msc2020(primary)	57R45; 57N80; 57R25;

DAYANTSOLMON, Dagva; GALTBAYAR, Artbazar;

Non-relativistic Pauli–Fierz Hamiltonian for less than two photons.

Hokkaido Mathematical Journal, 50 (2021) pp.309-326

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Abstract

We consider the Pauli–Fierz model, which describes a particle (an electron) coupled to the quantized electromagnetic field and limit the number of photons to less than 2. By computing the resolvent explicitly, we located the spectrum of the Hamiltonian mass. Our results do not depend on the coupling constant $e$ nor on the infrared cutoff parameter $R$.

Keywords	Pauli–Fierz Hamiltonian; mass renormalization; dressed electron states;
msc2020(primary)	81V10;
msc2020(secondary)	81T16; 47A10; 47A75;

CHEN, Xue;

Imaginary Verma modules for the twisted affine Nappi-Witten Lie algebra $\widetilde{H}_{4}[\sigma]$.

Hokkaido Mathematical Journal, 50 (2021) pp.327-344

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Abstract

An irreducible criterion of the imaginary Verma modules for the twisted affine Nappi-Witten Lie algebra $\widetilde{H}_{4}[\sigma]$ is determined. For a (reducible) imaginary Verma module, explicit characterizations of the maximal submodule, imaginary vacuum vectors, and the composition structure are also given.

Keywords	Twisted affine Nappi-Witten Lie algebra; Imaginary Verma modules; Imaginary vacuum vectors;
msc2020(primary)	17B10; 17B65;

SUZUKI, Ryuichi; UMEDA, Noriaki;

Blow-up at space infinity for a quasilinear parabolic equation with space-dependent reaction.

Hokkaido Mathematical Journal, 50 (2021) pp.345-408

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Abstract

We consider a nonnegative solution $u$ of the Cauchy problem for a quasilinear parabolic equation $u_t=\Delta u^m+\mu(x)u^p$ with the initial data $u_0(x)\,(\not\equiv 0)$ satisfying $\| \tilde\mu u_0\|_{L^{\infty}(\mathbf R^N)}<\infty$, where nonnegative function $\mu(x)$ satisfies some condition and $\tilde\mu=\mu^{1/(p-1)}$. We give sufficient conditions on $u_0$ for a weighted solution $\tilde\mu u$ to blow up at space infinity and for a direction $\psi \in \mathbf S^{N-1}$ to be a blow-up direction of $\tilde\mu u$. We also show that such a weighted solution $\tilde\mu u$ blows up completely at the blow-up time of $\tilde\mu u$.

Keywords	blow-up; quasilinear parabolic equation; space infinity; minimal blow-up time; space-dependent reaction; Cauchy problem;
msc2020(primary)	35B40; 35B44; 35B60; 35K15; 35K59; 35K65;

HATTORI, Tae; KASUE, Atsushi; OHKUBO, Motoki;

Some function theoretic properties of nonlinear resistive networks.

Hokkaido Mathematical Journal, 50 (2021) pp.409-454

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Abstract

We consider nonlinear resistive networks. The equivalence of the Liouville property, the Khas'minskii condition and the weak maximum principle for operators of Laplacian with potential is proved, and a number of criteria for these properties are given. The parabolicity of networks is also discussed.

Keywords	nonlinear network; Laplace operator; Liouville property; weak maximum principle; Khas'minskii condition; parabolicity;
msc2020(primary)	31C20; 05C63;

KONDO, Hirofumi;

A Nullstellensatz for ideals of $C^\infty$ functions in dimension 2.

Hokkaido Mathematical Journal, 50 (2021) pp.455-462

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Abstract

Suppose that an ideal $J$ of $C^\infty$ functions on an open subset of $\mathbf{R}^2$ is a Łojasiewicz ideal. We describe the set of $C^\infty$ functions vanishing on the zeros of $J$ explicitly using $J$ in an open neighborhood of each point in zeros of $J$, it can be obtained by taking real radical and closure starting from $J$ repeatedly for a finite number of times. This gives an another affirmative answer to Bochnak's conjecture in dimension 2, which is first done by Risler.

Keywords	Nullstellensatz; zero property; real radical; closed ideal; Łojasiewicz ideal;
msc2020(primary)	26E05; 26E10; 46E25;
msc2020(secondary)	11E25; 32C05; 14P15;