Hokkaido Mathematical Journal

No. 3

NODA, Takahiro;

On a certain invariant of differential equations associated with nilpotent graded Lie algebras.

Hokkaido Mathematical Journal, 47 (2018) pp.445-464

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Abstract

In this paper, we provide a new invariant for partial differential equations (PDEs) under contact transformations by using nilpotent graded Lie algebras. By virtue of this invariant, various geometric behavior of PDEs can be understood. As a typical class, we clarify geometric behavior of second-order PDEs in terms of our invariant.

MSC(Primary)	58A15
MSC(Secondary)	58A17;
Uncontrolled Keywords	Invariant of differential equations; (Linear) differential systems; Nilpotent graded Lie algebras;

GÜNEY DUMAN, Merve; ÖĞÜT, Ümmügülsüm; KESKİN, Refik;

Generalized Lucas Numbers of the form $wx^{2}$ and $wV_{m}x^{2}$.

Hokkaido Mathematical Journal, 47 (2018) pp.465-480

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Abstract

Let $P\geq 3$ be an integer. Let $(V_{n})$ denote generalized Lucas sequence defined by $V_{0}=2$, $V_{1}=P$, and $V_{n+1}=PV_{n}-V_{n-1}$ for $n\geq 1$. In this study, when $P$ is odd, we solve the equation $V_{n}=wx^{2}$ for some values of $w$. Moreover, when $P$ is odd, we solve the equation $V_{n}=wkx^{2}$ with $k \mid P$ and $k \gt 1$ for $w=3,11,13$. Lastly, we solve the equation $V_{n}=wV_{m}x^{2}$ for $w=7,11,13$.

MSC(Primary)	11B37
MSC(Secondary)	11B39;
Uncontrolled Keywords	Generalized Lucas sequence; Generalized Fibonacci sequence; congruence; square terms in Lucas sequences;

SHI, Jiangtao; HOU, Ruchen; ZHANG, Cui;

The influence of nonnormal noncyclic subgroups on the structure of finite groups.

Hokkaido Mathematical Journal, 47 (2018) pp.481-486

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Abstract

We obtain a complete classification of finite groups in which all noncyclic proper subgroups are nonnormal, and we apply this classification to investigate some structures of finite groups.

MSC(Primary)	20D05
MSC(Secondary)	20D10;
Uncontrolled Keywords	noncyclic subgroup; nonnormal; nonabelian simple group;

KAKIZAWA, Ryôhei;

The existence of Leray-Hopf weak solutions with linear strain.

Hokkaido Mathematical Journal, 47 (2018) pp.487-500

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Abstract

This paper deals with the global existence of weak solutions to the initial value problem for the Navier-Stokes equations in $\mathbb{R}^{n}$ ($n \in \mathbb{Z}$, $n\geq 2$). Concerning initial data of the form $Ax+v(0)$, where $A \in M_{n}(\mathbb{R})$ and $v(0) \in L^{2}_{\sigma}(\mathbb{R}^{n})$, the weak solutions are properly-defined with the aid of the alternativity of the trilinear from $(Ax\cdot\nabla)v\cdot\varphi$. Furthermore, we construct the Leray-Hopf weak solution which satisfies not only the Navier-Stokes equations but also the energy inequality via the Galerkin approximation. From the viewpoint of quadratic forms, the Gronwall-Bellman inequality admits the uniform boundedness of the approximate solution.

MSC(Primary)	35Q30
MSC(Secondary)	76D03; 76D05;
Uncontrolled Keywords	Navier-Stokes equations; Leray-Hopf weak solutions; Linear strain;

FARWIG, Reinhard; SCHULZ, Raphael; TANIUCHI, Yasushi;

Spatial Asymptotic Profiles of Solutions to the Navier-Stokes System in a Rotating Frame with Fast Decaying Data.

Hokkaido Mathematical Journal, 47 (2018) pp.501-529

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Abstract

The nonstationary Navier-Stokes system for a viscous, incompressible fluid influenced by a Coriolis force in the whole space ${\mathbb R}^3$ is considered at large distances. The solvability of the corresponding integral equations of these equations in weighted $L^\infty$-spaces is established. Furthermore, the leading terms of the asymptotic profile of the solution at fixed time $t \gt 0$ for $|x| \gt t$ and far from the axis of rotation are investigated.

MSC(Primary)	76U05
MSC(Secondary)	76D05; 35B40; 35Q30; 35Q35;
Uncontrolled Keywords	Rotating Navier-Stokes equations; Coriolis operator; mild solutions; weighted $L^\infty$-spaces; rate of spatial decay;

ITO, Takaaki;

A characterization for tropical polynomials being the minimum finishing time of project networks.

Hokkaido Mathematical Journal, 47 (2018) pp.531-544

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Abstract

A tropical polynomial is called $R$-polynomial if it can be realized as the minimum finishing time of a project network. $R$-polynomials satisfy the term extendability condition, and correspond to simple graphs. We give a characterization of $R$-polynomials in terms of simple graphs.

MSC(Primary)	15A80
MSC(Secondary)	06A07; 05C69;
Uncontrolled Keywords	max-plus algebra; tropical algebra; discrete event system;

BIRBRAIR, Lev; COSTA, João Carlos Ferreira; FILHO, Edvalter Da Silva Sena;

Topological bi-$\mathcal{K}$-equivalence of pairs of map germs.

Hokkaido Mathematical Journal, 47 (2018) pp.545-556

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Abstract

Let $P^{k}(n,p \times q)$ be the set of all pairs of real polynomial map germs $(f, g) : (\mathbb{R}^{n},0) \rightarrow (\mathbb{R}^{p} \times \mathbb{R}^{q} ,0)$ with degree of $ f_1 , \dots, f_p ,$ $g_1 ,\dots, g_q$ less than or equal to $k \in \N$. The main result of this paper shows that the set of equivalence classes of $P^{k}(n,p \times q)$, with respect to bi-$C^{0}$-$\mathcal{K}$-equivalence, is finite.

MSC(Primary)	32S15
MSC(Secondary)	32S05;
Uncontrolled Keywords	Topological contact equivalence; finiteness theorem; topological classification; pairs of map germs;

MASOUDI, Yousef; NADJAFIKHAH, Mehdi;

Moving frames and conservation laws of a Lagrangian invariant under the Hyperbolic Rotation-Translation group.

Hokkaido Mathematical Journal, 47 (2018) pp.557-579

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Abstract

Noether's First Theorem guarantees conservation laws provided that the Lagrangian is invariant under a Lie group action. In this paper, via the concept of Killing vector fields and the Minkowski metric, we first construct an important Lie group, known as Hyperbolic Rotation-Translation group. Then, according to Gonçalves and Mansfield's method, we obtain the invariantized Euler-Lagrange equations and the space of conservation laws in terms of vectors of invariants and the adjoint representation of a moving frame, for Lagrangians, which are invariant under Hyperbolic Rotation-Translation (or HRT) group action, in the case where the independent variables are not invariant.

MSC(Primary)	35L65
MSC(Secondary)	58E30; 70S10; 53B30; 53C50; 58D19; 43A65;
Uncontrolled Keywords	Conservation laws; Moving frames; Differential invariants; Normalized differential invariants; Syzygies; Killing vector fields;

FU, Haiping; PENG, Jianke;

Rigidity theorems for compact Bach-flat manifolds with positive constant scalar curvature.

Hokkaido Mathematical Journal, 47 (2018) pp.581-605

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Abstract

In this paper, we prove some rigidity theorems for compact Bach-flat $n$-manifold with the positive constant scalar curvature. In particular, our conditions in Theorem 2 have the additional properties of being sharp.

MSC(Primary)	53C20
MSC(Secondary)	53C24;
Uncontrolled Keywords	Bach-flat; constant curvature space; Weyl curvature tensor; trace-free Riemannian curvature tensor;

KURATA, Hisayasu; YAMASAKI, Maretsugu;

Discrete Green Potentials with Finite Energy.

Hokkaido Mathematical Journal, 47 (2018) pp.607-624

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Abstract

For a hyperbolic infinite network, it is well-known that Green potentials with finite energy are Dirichlet potentials. Conversely, if a Dirichlet potential has non-positive Laplacian, then it is a Green potential with finite energy. In this paper, we study whether a Dirichlet potential can be expressed as a difference of two Green potentials with finite energy. Comparisons of the Dirichlet sum of a function and that of its Laplacian play important roles in our study. As a by-product, we obtain a Riesz decomposition of a function whose Laplacian is a Dirichlet function.

MSC(Primary)	31C20
MSC(Secondary)	31C25;
Uncontrolled Keywords	discrete potential theory; Dirichlet potential; Green potential; Riesz representation; discrete Laplacian;

MAO, Jing; XIANG, Ni;

Estimates for the first eigenvalue of the drifting Laplacian on embedded hypersurfaces.

Hokkaido Mathematical Journal, 47 (2018) pp.625-636

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Abstract

For an $(n-1)$-dimensional compact orientable smooth metric measure space $\big(M,g,e^{-f}dv_{g}\big)$ embedded in an $n$-dimensional compact orientable Riemannian manifold $N$, we successfully give a lower bound for the first nonzero eigenvalue of the drifting Laplacian on $M$, provided the Ricci curvature of $N$ is bounded from below by a positive constant and the weighted function $f$ on $M$ satisfies two constraints.

MSC(Primary)	35P15
MSC(Secondary)	53C42;
Uncontrolled Keywords	Ricci curvature; eigenvalues; drifting Laplacian; smooth metric measure spaces;

OHNO, Shinji; SAKAI, Takashi; URAKAWA, Hajime;

Rigidity of transversally biharmonic maps between foliated Riemannian manifolds.

Hokkaido Mathematical Journal, 47 (2018) pp.637-654

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Abstract

On a smooth foliated map from a complete, possibly non-compact, foliated Riemannian manifold into another foliated Riemannian manifold of which transversal sectional curvature is non-positive, we will show that, if it is transversally biharmonic and has the finite energy and finite bienergy, then it is transversally harmonic.

MSC(Primary)	58E20
MSC(Secondary)	53C43;
Uncontrolled Keywords	foliation; divergence theorem; transversally harmonic; transversally biharmonic;

MASSEY, David B.;

IPA-deformations of functions on affine space.

Hokkaido Mathematical Journal, 47 (2018) pp.655-676

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Abstract

We investigate deformations of functions on affine space, deformations in which the changes specialize to a distinguished point in the zero-locus of the original function. Such deformations – deformations with isolated polar activity – enable us to obtain nice results on the cohomology of the Milnor fiber of the original function.

MSC(Primary)	32S05
MSC(Secondary)	32S60; 32S30; 32S55; 32S50; 32S15;
Uncontrolled Keywords	hypersurface; deformations; isolated polar activity; relative polar curve; relative conormal; hypercohomology;