Hokkaido Mathematical Journal

No. 1

HONDA, Atsufumi; NAOKAWA, Kosuke; UMEHARA, Masaaki; YAMADA, Kotaro;

Isometric realization of cross caps as formal power series and its applications.

Hokkaido Mathematical Journal, 48 (2019) pp.1-44

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Abstract

Two cross caps in Euclidean 3-space are said to be infinitesimally isometric if their Taylor expansions of the first fundamental forms coincide by taking a local coordinate system. For a given $C^\infty$ cross cap $f$, we give a method to find all cross caps which are infinitesimally isomeric to $f$. More generally, we show that for a given $C^{\infty}$ metric with singularity having certain properties like as induced metrics of cross caps (called a Whitney metric), there exists locally a $C^\infty$ cross cap infinitesimally isometric to the given one. Moreover, the Taylor expansion of such a realization is uniquely determined by a given $C^{\infty}$ function with a certain property (called characteristic function). As an application, we give a countable family of intrinsic invariants of cross caps which recognizes infinitesimal isometry classes completely.

MSC(Primary)	57R45
MSC(Secondary)	53A05;
Uncontrolled Keywords	cross cap; Whitney umbrella; positive semi-definite metric; isometric deformation; intrinsic invariant;

NUÑO-BALLESTEROS, J. J.; PALLARÉS-TORRES, I.;

A Lê-Greuel type formula for the image Milnor number.

Hokkaido Mathematical Journal, 48 (2019) pp.45-59

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Abstract

Let $f:(\mathbb{C}^n,0)\rightarrow (\mathbb{C}^{n+1},0)$ be a corank 1 finitely determined map germ. For a generic linear form $p:(\mathbb{C}^{n+1},0)\to(\mathbb{C},0)$ we denote by $g:(\mathbb{C}^{n-1},0)\rightarrow (\mathbb{C}^{n},0)$ the transverse slice of $f$ with respect to $p$. We prove that the sum of the image Milnor numbers $\mu_I(f)+\mu_I(g)$ is equal to the number of critical points of $p|_{X_s}:X_s\to\mathbb{C}$ on all the strata of $X_s$, where $X_s$ is the disentanglement of $f$ (i.e., the image of a stabilisation $f_s$ of $f$).

MSC(Primary)	32S30
MSC(Secondary)	32S05; 58K40;
Uncontrolled Keywords	Image Milnor number; Lê-Greuel formula; finite determinacy;

SATO, Shuichi;

Vector valued inequalities and Littlewood-Paley operators on Hardy spaces.

Hokkaido Mathematical Journal, 48 (2019) pp.61-84

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Abstract

We prove certain vector valued inequalities on $\Bbb R^n$ related to Littlewood-Paley theory. They can be used in proving characterization of the Hardy spaces in terms of Littlewood-Paley operators by methods of real analysis.

MSC(Primary)	42B25
MSC(Secondary)	42B30;
Uncontrolled Keywords	Vector valued inequalities; Littlewood-Paley functions; Hardy spaces;

LI, Peijin; WANG, Xiantao;

Lipschitz continuity of $\alpha$-harmonic functions.

Hokkaido Mathematical Journal, 48 (2019) pp.85-97

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Abstract

The aim of this paper is to discuss the Lipschitz continuity of $\alpha$-harmonic functions.

MSC(Primary)	31A05
MSC(Secondary)	35J25;
Uncontrolled Keywords	$\alpha$-harmonic function; $\alpha$-harmonic equation; Lipschitz continuity; majorant;

NAKAI, Eiichi; YONEDA, Tsuyoshi;

Applications of Campanato spaces with variable growth condition to the Navier-Stokes equation.

Hokkaido Mathematical Journal, 48 (2019) pp.99-140

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Abstract

We give new viewpoints of Campanato spaces with variable growth condition for applications to the Navier-Stokes equation. Namely, we formulate a blowup criteria along maximum points of the 3D-Navier-Stokes flow in terms of stationary Euler flows and show that the properties of Campanato spaces with variable growth condition are very useful for this formulation, since variable growth condition can control the continuity and integrability of functions on the neighborhood at each point. Our criterion is different from the Beale-Kato-Majda type and Constantin-Fefferman type criterion. If geometric behavior of the velocity vector field near the maximum point has a kind of stationary Euler flow configuration up to a possible blowup time, then the solution can be extended to be the strong solution beyond the possible blowup time. As another application we also mention the Cauchy problem for the Navier-Stokes equation.

MSC(Primary)	46E35
MSC(Secondary)	35Q30; 76D03; 76D05;
Uncontrolled Keywords	Campanato spaces with variable growth condition; blowup criterion; 3D Navier-Stokes equation; stationary 3D Euler flow; Cauchy problem;

HIROTSU, Takashi;

Brauer groups of Châtelet surfaces over local fields.

Hokkaido Mathematical Journal, 48 (2019) pp.141-154

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Abstract

A Châtelet surface over a field is a typical geometrically rational surface. Its Chow group of zero-cycles has been studied as an important birational invariant by many researchers since the 1970s. Recently, S. Saito and K. Sato obtained a duality between the Chow and Brauer groups from the Brauer-Manin pairing. For a Châtelet surface over a local field, we combine their result with the known calculation of the Chow group to determine the structure and generators of the Brauer group of a regular proper flat model of the surface over the integer ring of the base field.

MSC(Primary)	14G20
MSC(Secondary)	14F22; 14C15; 14J26;
Uncontrolled Keywords	Brauer groups; Chow groups; Châtelet surfaces; Local fields;

BENSSAAD, Meryem; BELHIRECHE, Hanane; SELVADURAY, Steave C.;

Equation system describing the radiation intensity and the air motion with the water phase transition.

Hokkaido Mathematical Journal, 48 (2019) pp.155-193

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Abstract

In this paper we consider the equation system describing the motion of the air and the variation of the radiation intensity and the quantity of water droplets in the air, including also the process of water phase transition. Under a suitable condition we prove the existence and uniqueness of the local solution. By eliminating the approximation by regularization of vapor density and by including the equation of radiation, this result improves previous ones.

MSC(Primary)	35Q35
MSC(Secondary)	76N10; 35Q79;
Uncontrolled Keywords	Model of the atmosphere; phase transitions; radiation intensity; integral semilinear terms;

REZAEI, Shahram;

On the annihilators of formal local cohomology modules.

Hokkaido Mathematical Journal, 48 (2019) pp.195-206

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Abstract

Let $\frak{a}$ denote an ideal in a commutative Noetherian local ring $(R,\frak{m})$ and $M$ a non-zero finitely generated $R$-module of dimension $d$. Let $d:=\dim(M/\frak{a} M)$. In this paper we calculate the annihilator of the top formal local cohomology module $\mathfrak{F}_{\frak{a}}^d (M)$. In fact, we prove that ${\rm Ann}_R(\mathfrak{F}_{\frak{a}}^d (M))={\rm Ann}_R(M/U_R(\frak{a}, M))$, where $$ U_R(\frak{a}, M):=\cup\lbrace N: N\leqslant M \text{ and } \dim(N/\frak{a}N) \lt \dim(M/\frak{a}M) \rbrace. $$ We give a description of $U_R(\frak{a}, M)$ and we will show that $$ {\rm Ann}_R (\mathfrak{F}_{\frak{a}}^d(M)) = {\rm Ann}_R (M/\cap_{\frak{p}_j \in {\rm Assh}_R M \cap {\rm V}(\frak{a})} N_j), $$ where $0=\bigcap_{j=1}^{n} N_{j}$ denotes a reduced primary decomposition of the zero submodule $0$ in $M$ and $N_j$ is a $\frak{p}_j$-primary submodule of $M$, for all $j=1,\dots, n$. Also, we determine the radical of the annihilator of $\mathfrak{F}_{\frak{a}}^d (M)$. We will prove that $$ \sqrt{{\rm Ann}_R(\mathfrak{F}_{\frak{a}}^d (M))} = {\rm Ann}_R(M/G_R(\frak{a}, M)), $$ where $G_R(\frak{a}, M)$ denotes the largest submodule of $M$ such that ${\rm Assh}_R(M)\cap {\rm V}(\frak{a}) \subseteq {\rm Ass}_R(M/G_R(\frak{a}, M))$ and ${\rm Assh}_R(M)$ denotes the set $\{\frak{p} \in {\rm Ass} M:\dim R/\frak{p} = \dim M\}.$

MSC(Primary)	13D45
MSC(Secondary)	13E05;
Uncontrolled Keywords	attached primes; local cohomology; annihilator;

CLOOS, Cai Constantin;

Local well-posedness for the derivative nonlinear Schrödinger Equation in Besov Spaces.

Hokkaido Mathematical Journal, 48 (2019) pp.207-244

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Abstract

It is shown that the cubic derivative nonlinear Schrödinger equation is locally well-posed in Besov spaces $B^{s}_{2,\infty}(\mathbb X)$, $s\ge 1/2$, where we treat the non-periodic setting $\mathbb X=\mathbb R$ and the periodic setting $\mathbb X=\mathbb T$ simultaneously. The proof is based on the strategy of Herr for initial data in $H^{s}(\mathbb T)$, $s\ge 1/2$.

MSC(Primary)	35Q55
MSC(Secondary)
Uncontrolled Keywords	local well-posedness; derivative nonlinear Schrödinger equation; Besov space; multilinear estimates;