Tokyo Journal of Mathematics Articles (Project Euclid)

The Homogeneous Slice Theorem for the Complete Complexification of a Proper Complex Equifocal Submanifold

Thu, 05 Aug 2010 15:41 EDT

Naoyuki KOIKE

Source: Tokyo J. of Math., Volume 33, Number 1, 1--30.

Abstract:
The notion of a complex equifocal submanifold in a Riemannian symmetric space of non-compact type has been recently introduced as a generalization of isoparametric hypersurfaces in the hyperbolic space. As its subclass, the notion of a proper complex equifocal submanifold has been introduced. Some results for a proper complex equifocal submanifold have been recently obtained by investigating the lift of its complete complexification to some path space. In this paper, we give a new construction of the complete complexification of a proper complex equifocal submanifold and, by using the construction, show that leaves of focal distributions of the complete complexification are the images by the normal exponential map of principal orbits of a certain kind of pseudo-orthogonal representation on the normal space of the corresponding focal submanifold.

$H$-Supermagic Strength of Some Graphs

Mon, 31 Jan 2011 09:18 EST

P. JEYANTHI, P. SELVAGOPAL

Source: Tokyo J. of Math., Volume 33, Number 2, 499--507.

Abstract:
A simple graph $G=(V,E)$ admits an $H$-covering if every edge in $E$ belongs to a subgraph of $G$ isomorphic to $H$. We say that $G$ is $H$-magic if there is a total labeling $f:V\cup E\rightarrow\{1,2,3,\ldots,|V|+|E|\}$ such that for each subgraph $H'=(V',E')$ of $G$ isomorphic to $H$, $s(f)=\sum_{v\in V'} f(v)+\sum_{e\in E'} f(e)$ is constant. When $f(V)=\{1,2,\ldots,|V|\}$, then $G$ is said to be $H$-supermagic. In this case, the $H$-supermagic strength of $G$ is defined as the minimum of all $s(f)$ where the minimum is taken over all $H$-supermagic labelings $f$ of $G$, and is denoted by $SM_H(G)$. In this paper we find the $C_k$-supermagic strength of k-polygonal snakes of any length and $H$-supermagic strength of a chain of an arbitrary 2-connected simple graph $H$. Also we make a conjecture regarding the $P_h$-supermagic strength of $P_n$ for $2 \leq h \leq n$.

Braid Presentation of Spatial Graphs

Mon, 31 Jan 2011 09:18 EST

Ken KANNO, Kouki TANIYAMA

Source: Tokyo J. of Math., Volume 33, Number 2, 509--522.

Abstract:
We define braid presentation of edge-oriented spatial graphs as a natural generalization of braid presentation of oriented links. We show that every spatial graph has a braid presentation. For an oriented link, it is known that the braid index is equal to the minimal number of Seifert circles. We show that an analogy does not hold for spatial graphs.

The IH-complex of Spatial Trivalent Graphs

Mon, 31 Jan 2011 09:18 EST

Atsushi ISHII, Kengo KISHIMOTO

Source: Tokyo J. of Math., Volume 33, Number 2, 523--535.

Abstract:
We define the IH-complex on the set of spatial trivalent graphs by using the IH-move, which is a local spatial move appeared in a study of knotted handlebodies. The IH-distance between two spatial trivalent graphs is defined by the minimal number of IH-moves needed to transform one into the other. It gives a distance function on the IH-complex. We give a lower bound for the IH-distance, and evaluate it.

On the Limit of the Colored Jones Polynomial of a Non-simple Link

Mon, 31 Jan 2011 09:18 EST

Mayuko YAMAZAKI, Yoshiyuki YOKOTA

Source: Tokyo J. of Math., Volume 33, Number 2, 537--551.

Abstract:
We compute the limit of the colored Jones invariant of a prime link, which gives the first evidence for Volume Conjecture of a link whose complement decomposes into two hyperbolic pieces.

Heat Kernel Estimates for Random Walks on Some Kinds of One-dimensional Continuum Percolation Clusters

Thu, 11 Aug 2011 10:54 EDT

Jun MISUMI

Source: Tokyo J. of Math., Volume 34, Number 1, 1--17.

Abstract:
We consider random walks on random graphs determined by a some kind of continuum percolation on $\mathbf{R}$. The vertex set of the random graph is given by the Poisson points conditioned that all points of $\mathbf{Z}$ are contained. The edge set of the random graph is determined by the random radii of the spheres centered at each points. We give heat kernel estimates for the random walks under the condition on the moment of the random radii. We will also discuss random walks on continuum percolation clusters in $\mathbf{R}^d$, $d\ge 2$.

On the Global Monodromy of a Fibration of the Fermat Surface of Odd Degree $n$

Thu, 11 Aug 2011 10:54 EDT

Kazushi AHARA, Ikuko AWATA

Source: Tokyo J. of Math., Volume 34, Number 1, 19--52.

Abstract:
The purpose of this paper is to investigate the global topological monodromy of a certain fibration of the Fermat surface without using numerical analysis by computer.

Abundance of Non-uniform Hyperbolicity in Bifurcations of Surface Endomorphisms

Thu, 11 Aug 2011 10:54 EDT

Hiroki TAKAHASI

Source: Tokyo J. of Math., Volume 34, Number 1, 53--113.

Abstract:
We study an interplay between homoclinic behavior and singularities in surface endomorphisms. We show that appropriate rescalings near homoclinic orbits intersecting fold singularities yield families of non-invertible Hénon-like maps. Then we construct positive measure sets of parameters corresponding to maps which exhibit nonuniformly hyperbolic behavior. This implies an extension of the celebrated theorem of Benedicks and Carleson, and that of Mora and Viana to surface endomorphisms.

The Best Constant of $L^p$ Sobolev Inequality Corresponding to Dirichlet Boundary Value Problem II

Thu, 11 Aug 2011 10:54 EDT

Yorimasa OSHIME, Kohtaro WATANABE

Source: Tokyo J. of Math., Volume 34, Number 1, 115--133.

Abstract:
Let $M = 2m$ $(m = 1,2,\ldots)$. In [1] the best constant of $L^p$ Sobolev inequality $$ \sup_{-1\leq x\leq 1}\vert u(x)\vert\leq C\Biggl( \int_{-1}^{1}\vert u^{(M)}(x)\vert^{p}dx\Biggr)^{1/p} $$ was obtained for $u$ satisfying $u, u^{(M)} \in L^{p}(-1,1)$ and $u^{(2i)}(\pm 1) = 0$ $(0\leq i\leq [(M-1)/2])$. On the other hand, for the case $M$ is odd, up to now, only the case $M =1$ was treated for technical difficulty; see [2]. This paper treats the case $M =3$ with different two approach, one is based on the property of the function associated with certain Green function and another is on the property of function space. For the latter approach, symmetrizations of functions play an important role.

Witten Multiple Zeta Values Attached to $\mathfrak{sl}(4)$

Thu, 11 Aug 2011 10:54 EDT

Jianqiang ZHAO, Xia ZHOU

Source: Tokyo J. of Math., Volume 34, Number 1, 135--152.

Abstract:
In this paper we shall prove that every Witten multiple zeta value of weight $w>3$ attached to $\mathfrak{sl}(4)$ at nonnegative integer arguments is a finite $\mathbb{Q}$-linear combination of MZVs of weight $w$ and depth three or less, except for the nine irregular cases where the Riemann zeta value $\zeta(w-2)$ and the double zeta values of weight $w-1$ and depth $<3$ are also needed.

Surgical Distance between Lens Spaces

Thu, 11 Aug 2011 10:54 EDT

Kazuhiro ICHIHARA, Toshio SAITO

Source: Tokyo J. of Math., Volume 34, Number 1, 153--164.

Abstract:
It is well-known that any pair of closed orientable 3-manifolds are related by a finite sequence of Dehn surgeries on knots. Furthermore Kawauchi showed that such knots can be taken to be hyperbolic. In this article, we consider the minimal length of such sequences connecting a pair of 3-manifolds, in particular, a pair of lens spaces.

On Genelarized DS-diagram and Moves

Thu, 11 Aug 2011 10:54 EDT

Masaharu KOUNO

Source: Tokyo J. of Math., Volume 34, Number 1, 165--183.

Abstract:
DS-diagram and flow spine are good tools for studying 3-manifolds ([5], [8]). In this paper, we introduce the concept of generalized DS-diagram and study its properties. We define two types of moves that change generalized DS-diagrams but do not change their associated manifolds. We prove that any two generalized DS-diagrams such that their associated manifolds are homeomorphic to each other can be deformed into each other by a finite sequence of moves of the types.

On (FC)-sequences and Mixed Multiplicities of Multi-graded Algebras

Thu, 11 Aug 2011 10:54 EDT

Duong Quoc VIET, Truong Thi Hong THANH

Source: Tokyo J. of Math., Volume 34, Number 1, 185--202.

Abstract:
Let $S=\bigoplus_{n_1,\ldots, n_s \geq 0}S_{(n_1,\ldots, n_s)}$ be a finitely generated standard multi-graded algebra over a Noetherian local ring $A$. This paper investigates the positivity of mixed multiplicities of $S$ and characterizes them in terms of Hilbert-Samuel multiplicities. As an application, we get some results on the mixed multiplicities of ideals that covers the main results in [Vi] and [TV].

Geometric Limits and Length Bounds on Curves

Thu, 11 Aug 2011 10:54 EDT

Teruhiko SOMA

Source: Tokyo J. of Math., Volume 34, Number 1, 203--219.

Abstract:
In this paper, we present the new proof of the Length Upper Bounds Theorem on curves in surfaces, which is crucial in the proof of Ending Lamination Conjecture by Minsky et al. Our proof is based on arguments in Bowditch [Bow2] but we use geometric limit arguments fully.

A Note on the $k$-Buchsbaum Property of Symbolic Powers of Stanley-Reisner Ideals

Thu, 11 Aug 2011 10:54 EDT

Nguyên Công MINH, Yukio NAKAMURA

Source: Tokyo J. of Math., Volume 34, Number 1, 221--227.

Abstract:
Let $I$ be the Stanley-Reisner ideal of pure simplicial complex $\Delta$ of dimension one. We shall give a formula for $S/I^{(r)}$ to be a $k$-Buchsbaum ring for each $r>0$, where $I^{(r)}$ is the $r$-th symbolic power of $I$. The main result is an improvement of the previous result in [MN] on the $k$-Buchsbaumness of $S/I^{(r)}$.

Diffeomorphism Classes of Real Bott Manifolds

Thu, 11 Aug 2011 10:54 EDT

Admi NAZRA

Source: Tokyo J. of Math., Volume 34, Number 1, 229--260.

Abstract:
A real Bott manifold is obtained as the orbit space of the $n$-torus $T^n$ by a free action of an elementary abelian $2$-group $(\mathbb{Z}_2)^n$. This paper deals with the classification of 5-dimensional real Bott manifolds and studies certain types of $n$-dimensional real Bott manifolds ($n\geq 6$).

Existence of Invariant Planes in a Complex Projective 3-Space under Discrete Projective Transformation Groups

Thu, 11 Aug 2011 10:54 EDT

Masahide KATO

Source: Tokyo J. of Math., Volume 34, Number 1, 261--285.

Abstract:
Let $\Gamma$ be a finitely generated discrete subgroup of $\mathrm{PGL}(4,\mathbf{C})$ acting on $\mathbf{P}^3$. Suppose that $\Gamma$ leaves invariant a surface in $\mathbf{P}^3$. Then, except for a few cases, we can find a plane which is invariant by a finite index subgroup of $\Gamma$. The exceptional cases will be determined explicitly.

Erratum to``New Techniques for Classifying Williams Solenoids'' (Tokyo Journal of Mathematics, Vol. 30, No. 1, pp. 139--157, June 2007)

Thu, 11 Aug 2011 10:54 EDT

Marcy BARGE, Richard SWANSON

Source: Tokyo J. of Math., Volume 34, Number 1, 287--288.

On the Existence of a Darling-Kac Set for the Renormalized Rauzy Map

Mon, 30 Jan 2012 08:50 EST

Kae INOUE, Hitoshi NAKADA

Source: Tokyo J. of Math., Volume 34, Number 2, 289--302.

Abstract:
It is well-known that the renormalized Rauzy map is conservative and ergodic. In this paper, we show that a Darling-Kac set exists for the renormalized Rauzy map. This implies the pointwise dual ergodicity of this map.

Generating the Mapping Class Group of a Punctured Surface by Involutions

Mon, 30 Jan 2012 08:50 EST

Naoyuki MONDEN

Source: Tokyo J. of Math., Volume 34, Number 2, 303--312.

Abstract:
Let $\Sigma_{g,b}$ denote a closed oriented surface of genus $g$ with $b$ punctures and let $\mathrm{Mod}(\Sigma_{g,b})$ denote its mapping class group. Kassabov showed that $\mathrm{Mod}(\Sigma_{g,b})$ is generated by 4 involutions if $g>7$ or $g=7$ and $b$ is even, 5 involutions if $g>5$ or $g=5$ and $b$ is even, and 6 involutions if $g>3$ or $g=3$ and $b$ is even. We proved that $\mathrm{Mod}(\Sigma_{g,b})$ is generated by 4 involutions if $g=7$ and $b$ is odd, and 5 involutions if $g=5$ and $b$ is odd.

On the Deuring-Shafarevich Formula

Mon, 30 Jan 2012 08:50 EST

Daisuke SHIOMI

Source: Tokyo J. of Math., Volume 34, Number 2, 313--318.

Abstract:
In this paper, we will give a new proof of the Deuring-Shafarevich formula, which asserts a relation between the $p$-ranks of Jacobi varieties. We analyze the zeta functions of global function fields to prove the formula, without using tools of the algebraic geometry.

On Expressions of Theta Series by $\eta$-products

Mon, 30 Jan 2012 08:50 EST

Akihiko OKAMOTO

Source: Tokyo J. of Math., Volume 34, Number 2, 319--326.

Abstract:
In this paper, we give a certain identity between an $\eta$-product of weight 1 and theta series associated with a pair of binary quadratic forms. We also have explicit description of Siegel's theorem by an $\eta$-product. For quadratic forms $Q_1$ and $Q_2$ which are in the same genus, we express the difference $\vartheta_{Q_1}(\tau)-\vartheta_{Q_2}(\tau)$ by an $\eta$-product.

Asymptotic Behavior of Solutions to the Semilinear Wave Equation with Time-dependent Damping

Mon, 30 Jan 2012 08:50 EST

Kenji NISHIHARA

Source: Tokyo J. of Math., Volume 34, Number 2, 327--343.

Abstract:
We consider the Cauchy problem for the semilinear wave equation with time-dependent damping $$ \left\{ \begin{array}{@{}ll} u_{tt} - \Delta u + b(t)u_t=f(u)\,, & (t,x) \in {\bf R}^+ \times {\bf R}^N \\ (u,u_t)(0,x) = (u_0,u_1)(x)\,, & x \in {\bf R}^N\,. \end{array}\right. \eqno{(*)} $$ hen $b(t)=(t+1)^{-\beta}$ with $0\le \beta <1$, the damping is effective and the solution $u$ to ($*$) behaves as that to the corresponding parabolic problem. When $f(u)=O(|u|^{\rho})$ as $u \to 0$ with $1<\rho < \frac{N+2}{[N-2]_+}$(the Sobolev exponent), our main aim is to show the time-global existence of solutions for small data in the supercritical exponent $\rho>\rho_F(N):=1+2/N$. We also obtain some blow-up results on the solution within a finite time, so that the smallness of the data is essential to get global existence in the supercritical exponent case.

On Some Functional-differential Inequalities Related to the Exponential Mapping

Mon, 30 Jan 2012 08:50 EST

Włodzimierz FECHNER

Source: Tokyo J. of Math., Volume 34, Number 2, 345--352.

Abstract:
We consider real-valued twice differentiable functions defined on an open interval. Our main result states that if a function $f$ satisfies some inequalities then a map $x\mapsto f(x)\exp(-cx)$ is convex, where $c$ is an arbitrary point of $\mathbf{R}$ or of $\mathbf{R}\setminus (c_1,c_2)$ for some real $c_1, c_2$.

Some Relations among Apostol-Vu Double Zeta Values for Coordinatewise Limits at Non-positive Integers

Mon, 30 Jan 2012 08:50 EST

Takuya OKAMOTO

Source: Tokyo J. of Math., Volume 34, Number 2, 353--366.

Abstract:
We consider Apostol-Vu double zeta values for coordinatewise limits at non-positive integers, and we give some relations among Riemann's zeta values, Euler-Zagier double zeta values and Apostol-Vu double zeta values for all coordinatewise limits at non-positive integers. Using the relations, we also give relations among multiple Bernoulli numbers.

Integer Points and Independent Points on the Elliptic Curve $y^2=x^3-p^kx$

Mon, 30 Jan 2012 08:50 EST

Yasutsugu FUJITA, Nobuhiro TERAI

Source: Tokyo J. of Math., Volume 34, Number 2, 367--381.

Abstract:
Let $E_k$ be the elliptic curve given by $y^2=x^3-p^k x$, where $p$ is a prime number and $k \in \{1,2,3\}$. In this paper, we first give a necessary and sufficient condition for the rank of $E_k(\mathbf{Q})$ to equal one or two, respectively, and in the rank two case, explicitly describe independent points of free part of the Mordell-Weil group $E_k(\mathbf{Q})$. Secondly, we show several subfamilies of $E_k$ whose integer points and ranks can be completely determined.

Nested Subclasses of the Class of $\alpha$-selfdecomposable Distributions

Mon, 30 Jan 2012 08:50 EST

Makoto MAEJIMA, Yohei UEDA

Source: Tokyo J. of Math., Volume 34, Number 2, 383--406.

Abstract:
A probability distribution $\mu$ on $\mathbf{R}^d$ is selfdecomposable if its characteristic function $\widehat\mu(z), z\in\mathbf{R}^d$, satisfies that for any $b>1$, there exists an infinitely divisible distribution $\rho_b$ satisfying $\widehat\mu(z) = \widehat\mu(b^{-1}z)\widehat\rho_b(z)$. This concept has been generalized to the concept of $\alpha$-selfdecomposability by many authors in the following way. Let $\alpha\in\mathbf{R}$. An infinitely divisible distribution $\mu$ on $\mathbf{R}^d$ is $\alpha$-selfdecomposable, if for any $b>1$, there exists an infinitely divisible distribution $\rho_b$ satisfying $\widehat\mu(z) = \widehat\mu(b^{-1}z)^{b^{\alpha}}\widehat\rho_b(z)$. By denoting the class of all $\alpha$-selfdecomposable distributions on $\mathbf{R}^d$ by $L^{(\alpha)}(\mathbf{R}^d)$, we define in this paper a sequence of nested subclasses of $L^{(\alpha)}(\mathbf{R}^d)$, and investigate several properties of them by two ways. One is by using limit theorems and the other is by using mappings of infinitely divisible distributions.

On Stronger Versions of Brumer's Conjecture

Mon, 30 Jan 2012 08:50 EST

Masato KURIHARA

Source: Tokyo J. of Math., Volume 34, Number 2, 407--428.

Abstract:
Let $k$ be a totally real number field and $L$ a CM-field such that $L/k$ is finite and abelian. In this paper, we study a stronger version of Brumer's conjecture that the Stickelberger element times the annihilator of the group of roots of unity in $L$ is in the Fitting ideal of the ideal class group of $L$, and also study the dual version. We mainly study the Teichmüller character component, and determine the Fitting ideal in a certain case. We will see that these stronger versions hold in a certain case. It is known that the stronger version (SB) does not hold in general. We will prove in this paper that the dual version (DSB) does not hold in general, either.

Cobordism of Algebraic Knots Defined by Brieskorn Polynomials

Mon, 30 Jan 2012 08:50 EST

Vincent BLANLŒIL, Osamu SAEKI

Source: Tokyo J. of Math., Volume 34, Number 2, 429--443.

Abstract:
In this paper we study the cobordism of algebraic knots associated with weighted homogeneous polynomials, and in particular Brieskorn polynomials. Under some assumptions we prove that the associated algebraic knots are cobordant if and only if the Brieskorn polynomials have the same exponents.

The $^{*}$-Ricci Tensor for Hypersurfaces in $\mathbb{C}\mathbb{P}^n$ and $\mathbb{C}\mathrm{H}^n$

Mon, 30 Jan 2012 08:50 EST

Thomas A. IVEY, Patrick J. RYAN

Source: Tokyo J. of Math., Volume 34, Number 2, 445--471.

Abstract:
We update and refine the work of T. Hamada concerning *-Einstein hypersurfaces in $\mathbb{C}\mathbb{P}^n$ and $\mathbb{C}\mathrm{H}^n$. We also address existence questions using the methods of moving frames and exterior differential systems.

Birational Maps of Moduli Spaces of Vector Bundles on $K3$ Surfaces

Mon, 30 Jan 2012 08:50 EST

Masanori KIMURA, Kōta YOSHIOKA

Source: Tokyo J. of Math., Volume 34, Number 2, 473--491.

Abstract:
In this note, we construct a birational map of a moduli space of stable sheaves on a $K3$ surface induced by a reflection functor.

Pictures and Littlewood-Richardson Crystals

Mon, 30 Jan 2012 08:50 EST

Toshiki NAKASHIMA, Miki SHIMOJO

Source: Tokyo J. of Math., Volume 34, Number 2, 493--506.

Abstract:
We shall describe the one-to-one correspondence between the set of pictures and the set of Littlewood-Richardson crystals.

On Wronskian Determinant Formulas of the General Hypergeometric Functions

Mon, 30 Jan 2012 08:50 EST

Hironobu KIMURA

Source: Tokyo J. of Math., Volume 34, Number 2, 507--524.

Abstract:
The general hypergeometric functions of confluent type are studied. We establish a link between the general hypergeometric functions defined by 1-dimensional integrals and those defined by multi-dimensional integrals. The key point is to form an intermediate Wronskian determinant for the 1-dimensional ones and to rewrite it into a multi-dimensional integral using the generalized Veronese map.

An Existence Theorem for the Steady Navier-Stokes Problem in Higher Dimensions

Mon, 30 Jan 2012 08:50 EST

Antonio RUSSO, Alfonsina TARTAGLIONE

Source: Tokyo J. of Math., Volume 34, Number 2, 525--533.

Abstract:
We extend a well-known result of H. Fujita and H. Morimoto [1] to exterior domains of $\mathbb{R}^m$, with $m\ge 4$.

The Adams Inequality on Weighted Morrey Spaces

Mon, 30 Jan 2012 08:50 EST

Takeshi IIDA, Yasuo Komori-FURUYA, Enji SATO

Source: Tokyo J. of Math., Volume 34, Number 2, 535--545.

Abstract:
We introduce new weight classes, and extend the Adams inequality to weighted Morrey spaces. We also investigate the boundedness of the modified fractional integral operator from weighted Morrey spaces to Lipschitz or BMO spaces.

Asymptotic Expansions of Solutions to the Heat Equations with Initial Value in the Dual of Gel'fand-Shilov Spaces

Mon, 30 Jan 2012 08:50 EST

Yasuyuki OKA

Source: Tokyo J. of Math., Volume 34, Number 2, 547--567.

Abstract:
We will derive the asymptotic expansions of the solutions $U(x,t)$ to the heat equation with $\left(\mathcal{S}^r_r\right)^{\prime}(\mathbf{R}^d)$, $r\geq 1/2$, initial value, where $\left(\mathcal{S}^r_r\right)^{\prime}(\mathbf{R}^d)$ is the dual space of the Gel'fand-Shilov space $\mathcal{S}^r_r(\mathbf{R}^d$. Moreover, we show that, when $1/2\leq r\leq 1$, these asymptotic expansions satisfy the strong asymptotic condition on some circle $D_R=\{t\in\mathbf{C}\ |\ \mathrm{Re}\ t^{-1}>R^{-1}\}$. Therefore, we find that these asymptotic series for $\left(\mathcal{S}^r_r\right)^{\prime}(\mathbf{R}^d)$ initial value are Borel summable by means of A. D. Sokal's result on the Borel summability. As an application, we show the asymptotic expansions of the Weyl transform with Planck's constant $\hbar$ in some state, which are refinement of a classical limit of the quantum mechanical expectation values expressed by the Weyl transform.

Some Relationships between the Geometry of the Tangent Bundle and the Geometry of the Riemannian Base Manifold

Thu, 19 Jul 2012 08:36 EDT

Guillermo HENRY, Guillermo KEILHAUER

Source: Tokyo J. of Math., Volume 35, Number 1, 1--15.

Abstract:
We compute the curvature tensor of the tangent bundle of a Riemannian manifold endowed with a natural metric and we get some relationships between the geometry of the base manifold and the geometry of the tangent bundle.

Excellent Extensions and Global Cotorsion Dimensions

Thu, 19 Jul 2012 08:36 EDT

Shang WENLIANG

Source: Tokyo J. of Math., Volume 35, Number 1, 17--22.

Abstract:
In this paper, we mainly investigate the global cotorsion dimension under the excellent extension of rings. We show that if $S$ is an excellent extension of $R$, then $\mathrm{cot.D}(S)=\mathrm{cot.D}(R)$. Furthermore, some known results, such as Corollary 3.8 and 3.12, can be also obtained as direct corollaries of our theorem.

Upper Bounds for the Arithmetical Ranks of Monomial Ideals

Thu, 19 Jul 2012 08:36 EDT

Pietro MONGELLI

Source: Tokyo J. of Math., Volume 35, Number 1, 23--34.

Abstract:
We prove some generalization of a lemma by Schmitt and Vogel which yields the arithmetical rank in cases that could not be settled by the existing methods. Our results are based on divisibility conditions and exploit both combinatorial and linear algebraic considerations. They mainly apply to ideals generated by monomials.

Existence and Stability of Almost Periodic Solutions of Nonlinear Damped Equations of a Suspended String

Thu, 19 Jul 2012 08:36 EDT

Hitomi HATTORI, Masaru YAMAGUCHI

Source: Tokyo J. of Math., Volume 35, Number 1, 35--61.

Abstract:
In this paper we shall show the existence and the stability of almost periodic solutions of the boundary value problem to a nonlinear suspended string equation with a linear damping term and an almost periodic weakly nonlinear forcing term. We treat both weak solutions and strong solutions. Also we show the existence of time global solutions of the initial boundary value problem to the equation.

Leray's Inequality for Fluid Flow in Symmetric Multi-connected Two-dimensional Domains

Thu, 19 Jul 2012 08:36 EDT

Reinhard FARWIG, Hiroko MORIMOTO

Source: Tokyo J. of Math., Volume 35, Number 1, 63--70.

Abstract:
We consider the stationary Navier-Stokes equations with nonhomogeneous boundary condition in a domain with several boundary components. If the boundary value satisfies only the necessary flux condition (GOC), Leray's inequality does not holds true in general and we cannot prove the existence of a solution. But for a 2-D domain which is symmetric with respect to a line and where the data is also symmetric, C. Amick showed the existence of solutions by reduction to absurdity; later H. Fujita proved Leray-Fujita's inequality and hence the existence of symmetric solutions. In this paper we give a new short proof of Leray-Fujita's inequality and hence a proof of the existence of weak solutions.

Weighted Composition Operators on $C(X)$ and $\mathrm{Lip}_c(X,\alpha)$

Thu, 19 Jul 2012 08:36 EDT

Maliheh HOSSEINI, Fereshteh SADY

Source: Tokyo J. of Math., Volume 35, Number 1, 71--84.

Abstract:
Let $A$ and $B$ be subalgebras of $C(X)$ and $C(Y)$, respectively, for some topological spaces $X$ and $Y$. An arbitrary map $T: A\rightarrow B$ is said to be multiplicatively range-preserving if for every $f,g\in A$, $(fg)(X)=(TfTg)(Y)$, and $T$ is said to be separating if $TfTg=0$ whenever $fg=0$. For a given metric space $X$ and $\alpha\in (0,1]$, let Lip$_c(X,\alpha)$ be the algebra of all complex-valued functions on $X$ satisfying the Lipschitz condition of order $\alpha$ on each compact subset of $X$. In this note we first investigate the general form of multiplicatively range-preserving maps from $C(X)$ onto $C(Y)$ for realcompact spaces $X$ and $Y$ (not necessarily compact or locally compact) and then we consider such preserving maps from Lip$_c(X, \alpha)$ onto Lip$_c(Y,\beta)$ for metric spaces $X$ and $Y$ and $\alpha, \beta\in (0,1]$. We show that in both cases multiplicatively range-preserving maps are weighted composition operators which induce homeomorphisms between $X$ and $Y$. We also give a description of a linear separating map $T: A\rightarrow C(Y)$, where $A$ is either $C(X)$ for a normal space $X$ or Lip$_c(X,\alpha)$ for a metric space $X$ and $0<\alpha\le1$ and $Y$ is an arbitrary Hausdorff space.

Rational Solutions of Difference Painlevé Equations

Thu, 19 Jul 2012 08:36 EDT

Shun SHIMOMURA

Source: Tokyo J. of Math., Volume 35, Number 1, 85--95.

Abstract:
We capture all the rational solutions of some difference Painlevé equations of P$_{\mathrm{I}}$ and P$_{\mathrm{II}}$ types. For non-autonomous cases, it is shown that all the rational solutions of the difference P$_{\mathrm{II}}$ are ones generated by successive application of auto-Bäcklund transformations to the seed solution vanishing identically, and that the other equations of P$_{\mathrm{I}}$ type admit no rational solutions. For autonomous cases, all the nontrivial rational solutions are obtained, and they exist under a certain condition on a fixed point of the equation. If such a condition is not satisfied, there exist solutions that are rational in an exponential function.

Small-time Existence of a Strong Solution of Primitive Equations for the Ocean

Thu, 19 Jul 2012 08:36 EDT

Hirotada HONDA, Atusi TANI

Source: Tokyo J. of Math., Volume 35, Number 1, 97--138.

Abstract:
Primitive equations derived originally by Richardson in 1920's have been considered as the model equations describing the motion of atmosphere, ocean and coupled atmosphere and ocean. In this paper, we discuss the free boundary problem of the primitive equations for the ocean in three-dimensional strip with surface tension. Using the so-called $p$-coordinates and a coordinate transformation similar to that in [2] in order to fix the time-dependent domain, we prove temporally local existence of the unique strong solution to the transformed problem in Sobolev-Slobodetskiĭ spaces.

Indices Isotypiques des Éléments Cyclotomiques

Thu, 19 Jul 2012 08:36 EDT

Tatiana BELIAEVA, Jean-Robert BELLIARD

Source: Tokyo J. of Math., Volume 35, Number 1, 139--164.

Abstract:
Given $F$ a real abelian field, $p$ an odd prime and $\chi$ any Dirichlet character of $F$, we give a method for computing the $\chi$-index $(H^1(G_S,\mathbf{Z}_p(r))^\chi: C^F(r)^\chi)$ where the Tate twist $r$ is an odd integer $r\geq 3$, the group $C^F(r)$ is the group of higher circular units, $G_S$ is the Galois group over $F$ of the maximal $S$ ramified algebraic extension of $F$, and $S$ is the set of places of $F$ dividing $p$. This $\chi$-index can now be computed in terms only of elementary arithmetic of finite fields $\mathbf{F}_\ell$. Our work generalizes previous results by Kurihara who used the assumption that the order of $\chi$ divides $p-1$.

Gauss Sums on Finite Groups

Thu, 19 Jul 2012 08:36 EDT

Yasushi GOMI, Taiki MAEDA, Ken-ichi SHINODA

Source: Tokyo J. of Math., Volume 35, Number 1, 165--179.

Abstract:
We shall discuss Gauss sums on finite groups and give several examples including the case of the complex reflection groups $G(m,r,n)$, and hence finite symmetric groups, and also finite Weyl groups.

The Effect of External Fields in the Theory of Liquid Crystals

Thu, 19 Jul 2012 08:36 EDT

Junichi ARAMAKI

Source: Tokyo J. of Math., Volume 35, Number 1, 181--211.

Abstract:
We consider the response of external field to the theory of liquid crystals. We treat the Landau-de Gennes functional with the Dirichlet boundary condition for the director field which may be non-constant. We show that there exist two families of critical points such that one carries out the superheating fields of superconductors and the other one carries out strong stability. We also show that under some conditions, strong field does not bring the pure nematic state which is different response from superconductors.

Continued Fractions and Gauss' Class Number Problem for Real Quadratic Fields

Thu, 19 Jul 2012 08:36 EDT

Fuminori KAWAMOTO, Koshi TOMITA

Source: Tokyo J. of Math., Volume 35, Number 1, 213--239.

Abstract:
The main purpose of this article is to present a numerical data which shows relations between real quadratic fields of class number 1 and a mysterious behavior of the period of simple continued fraction expansion of certain quadratic irrationals. For that purpose, we define a class number, a fundamental unit,a discriminant and a Yokoi invariant for a non-square positive integer, and then see that a generalization of theorems of Siegel and of Yokoi holds. These and a theorem of Friesen and Halter-Koch imply several interesting conjectures for solving Gauss' class number problem for real quadratic fields.

Covers in 4-uniform Intersecting Families with Covering Number Three

Thu, 19 Jul 2012 08:36 EDT

Shuya CHIBA, Michitaka FURUYA, Ryota MATSUBARA, Masanori TAKATOU

Source: Tokyo J. of Math., Volume 35, Number 1, 241--251.

Abstract:
Let $k$ be an integer. In [3, 4], Frankl, Ota and Tokushige proved that the maximum number of three-covers of a $k$-uniform intersecting family with covering number three is $k^3 - 3k^2 + 6k -4$ for $k=3$ or $k \ge 9$, but the case $4 \le k \le 8$ remained open. In this paper, we prove that the same holds for $k=4$, and show that a 4-uniform family with covering number three which has 36 three-covers is uniquely determined.

Deformations of a Holomorphic Map and Its Degeneracy Locus

Wed, 23 Jan 2013 09:29 EST

Madoka EBIHARA

Source: Tokyo J. of Math., Volume 35, Number 2, 253--277.

Abstract:
Let $f: X \to Y$ be a surjective holomorphic map of compact complex manifolds and $\Delta$ the degeneracy locus of $f$. In this paper we shall discuss relationship between infinitesimal deformations of $f$ and the corresponding infinitesimal displacements of $\Delta$ in $Y$. We shall prove that two kinds of Kodaira-Spencer maps are compatible under certain assumptions. As an application of our main theorem, deformations of quadric bundles shall be discussed.

Weyl's Theorems and Extensions of Bounded Linear Operators

Wed, 23 Jan 2013 09:29 EST

Pietro AIENA, Muneo CHŌ, Lingling ZHANG

Source: Tokyo J. of Math., Volume 35, Number 2, 279--289.

Abstract:
A bounded operator $T\in L(X)$, $X$ a Banach space, is said to satisfy Weyl's theorem if the set of all spectral points that do not belong to the Weyl spectrum coincides with the set of all isolated points of the spectrum which are eigenvalues and having finite multiplicity. In this article we give sufficient conditions for which Weyl's theorem for an extension $\overline T$ of $T$ (respectively, for $T$) entails that Weyl's theorem holds for $T$ (respectively, for $\overline T$).

A Note on Traces of Singular Moduli

Wed, 23 Jan 2013 09:29 EST

Ja Kyung KOO, Dong Hwa SHIN

Source: Tokyo J. of Math., Volume 35, Number 2, 291--296.

Abstract:
We generalize Osburn's work ([6]) about a congruence for traces defined in terms of Hauptmoduli associated to certain genus zero groups of higher levels.

Generalized Besov Spaces and Their Applications

Wed, 23 Jan 2013 09:29 EST

Takeshi KAWAZOE, Hatem MEJJAOLI

Source: Tokyo J. of Math., Volume 35, Number 2, 297--320.

Abstract:
We define and study the Bessel potential and inhomogeneous Besov spaces associated with the Dunkl operators on $\mathbf{R}^d$. As applications on these spaces we construct the Sobolev type embedding theorem and the paraproduct operators associated with the Dunkl operators, as similar to those defined by Bony. We also establish Strichartz type estimates for the Dunkl-Schrödinger equation and finally we study the problem of well posedness of the generalized heat equation.

Positive Solutions for Non-cooperative Singular $p$-Laplacian Systems

Wed, 23 Jan 2013 09:29 EST

D. D. HAI

Source: Tokyo J. of Math., Volume 35, Number 2, 321--331.

Abstract:
We prove the existence of\ positive solutions for the $p$-Laplacian system \[ \left\{ \begin{array}{@{\,}c@{\enskip}c} -\Delta _{p}u_{1}=\lambda f_{1}(u_{2}) &\text{in\enskip}\Omega \,, \\ -\Delta _{p}u_{2}=\lambda f_{2}(u_{1}) &\text{in\enskip}\Omega \,, \\ \ \ \ \quad u_{1}=u_{2}=0 & \,\enskip \text{on\enskip}\partial \Omega \,, \end{array} \right. \] where $\Delta _{p}u=\mbox{div}(|\nabla u|^{p-2}\nabla u),p>1, \Omega$ is a bounded domain in $\mathbf{R}^{n}$ with smooth boundary $\partial \Omega ,f_{i}:(0,\infty) \rightarrow \mathbf{R}$ are possibly singular at 0 and are not required to be positive or nondecreasing, and $\lambda $ is a large parameter.

A Simple Proof of the Functional Relation for the Lerch Type Tornheim Double Zeta Function

Wed, 23 Jan 2013 09:29 EST

Takashi NAKAMURA

Source: Tokyo J. of Math., Volume 35, Number 2, 333--337.

Abstract:
In this paper, we give a simple proof of the functional relation for the Lerch type Tornheim double zeta function. By using it, we obtain simple proofs of some explicit evaluation formulas for double $L$-values.

Existence and Non-existence of a Finite Invariant Measure

Wed, 23 Jan 2013 09:29 EST

Stanley EIGEN, Arshag HAJIAN, Yuji ITO, Vidhu S. PRASAD

Source: Tokyo J. of Math., Volume 35, Number 2, 339--358.

Abstract:
About fifty years ago, questions on the existence and non-existence of finite invariant measures were studied by various authors and from different directions. In this article, we examine these classical results and prove directly that all the conditions introduced by these authors are equivalent to each other. We begin at the fundamental level of a recurrent transformation whose properties can be strengthened to obtain the aforementioned classical results for the existence of a finite invariant measure. We conclude with the introduction of a new property, Strongly Weakly Wandering (sww) sequences, the existence of which is equivalent to the non-existence of a finite invariant measure. It is shown that every sww sequence is also an Exhaustive Weakly Wandering (eww) sequence for ergodic transformations. Although all ergodic transformations with no finite invariant measure are known to have eww sequences, there are exceedingly few actual examples for which explicit eww sequences can be exhibited. The significance of sww sequences is that it provides a condition which is easier to verify than the condition for eww sequences (Proposition 4.5). In a second paper, we will continue these studies and also connect them to some of the more recent derived conditions for finite invariant measures. The impetus for this work, began with the late Professor Shizuo Kakutani, with whom the authors worked and had many fruitful discussions on these topics.

Surfaces with Constant Chebyshev Angle

Wed, 23 Jan 2013 09:29 EST

Carlos M. C. RIVEROS, Armando M. V. CORRO

Source: Tokyo J. of Math., Volume 35, Number 2, 359--366.

Abstract:
In this paper we consider surfaces with negative Gaussian curvature parametrized by a generalized Chebyshev net with constant Chebyshev angle in the Euclidean 3-space. We characterize these surfaces in terms of a meromorphic function which satisfies a certain differential equation. Moreover, we show that these surfaces have the geometric property that the asymptotic lines have the same sign of geodesic curvature. As an application we obtain for each constant Chebyshev angle a four-parameter family of complete surfaces.

Sasaki-Einstein Metrics on $S^2 \times S^3$

Wed, 23 Jan 2013 09:29 EST

Mitsuhiro IMADA

Source: Tokyo J. of Math., Volume 35, Number 2, 367--373.

Abstract:
In [9], Boyer and Galicki introduced a contact reduction method in the context of Sasakian manifolds, which produces 5-dimentional Sasaki-Einstein manifolds from a 7-sphere. In this paper, we compute very explicitly the metric obtained from the above mentioned reduction via a projection, $S^3 \times S^3 \to S^2 \times S^3$, and show that this metric is the homogeneous Kobayashi-Tanno metric.

A Characterization of a Multiple Weights Class

Wed, 23 Jan 2013 09:29 EST

Takeshi IIDA

Source: Tokyo J. of Math., Volume 35, Number 2, 375--383.

Abstract:
Moen introduced a multiple weights class. Moen proved that the multiple weights condition of vector of weights implies each weight function satisfies a certain $A_{p}$ condition. In 2010, Chen and Xue improved sufficency under an unnatural condition. However we can remove this condition and prove necessity. We also give a typical example of the multiple weights class.

Isometries and Maps Compatible with Inverted Jordan Triple Products on Groups

Wed, 23 Jan 2013 09:29 EST

Osamu HATORI, Go HIRASAWA, Takeshi MIURA, Lajos MOLNÁR

Source: Tokyo J. of Math., Volume 35, Number 2, 385--410.

Abstract:
Motivated by the famous Mazur-Ulam theorem in this paper we study algebraic properties of isometries between metric groups. We present some general results on so-called $d$-preserving maps between subsets of groups and apply them in several directions. We consider $d$-preserving maps on certain groups of continuous functions defined on compact Hausdorff spaces and describe the structure of isometries between groups of functions mapping into the circle group $\mathbb T$. Finally, we show a generalization of the Mazur-Ulam theorem for commutative groups and present a metric characterization of normed real-linear spaces among commutative metric groups.

Ideal Class Groups of CM-fields with Non-cyclic Galois Action

Wed, 23 Jan 2013 09:29 EST

Masato KURIHARA, Takashi MIURA

Source: Tokyo J. of Math., Volume 35, Number 2, 411--439.

Abstract:
Suppose that $L/k$ is a finite and abelian extension such that $k$ is a totally real base field and $L$ is a CM-field. We regard the ideal class group $\mathrm{Cl}_{L}$ of $L$ as a $\mathrm{Gal}(L/k)$-module. As a sequel of the paper [8] by the first author, we study a problem whether the Stickelberger element for $L/k$ times the annihilator ideal of the roots of unity in $L$ is in the Fitting ideal of $\mathrm{Cl}_{L}$, and also a problem whether it is in the Fitting ideal of the Pontrjagin dual $(\mathrm{Cl}_{L})^{\vee}$. We systematically construct extensions $L/k$ for which these properties do not hold, and also give numerical examples.

On a Distribution Property of the Residual Order of $a \pmod{p}$ with a Quadratic Residue Condition

Wed, 23 Jan 2013 09:29 EST

Koji CHINEN, Chikako TAMURA

Source: Tokyo J. of Math., Volume 35, Number 2, 441--459.

Abstract:
Let $a$ be a positive integer with $a\geq 2$ and $Q_a(k,l)$ be the set of odd prime numbers $p$ such that the residual order of $a$ in $\mathbf{Z}/p\mathbf{Z}^\times$ is congruent to $l \bmod k$. The natural density of the set $Q_a(q,0)$ ($q$ is a prime) is already known. In this paper, we consider the set $S_{a,b}(k,l)$, which consists of the primes $p$ that belong to $Q_a(k,l)$ and satisfy $\big(\frac{b}{p}\big)=1$, where $\big(\frac{b}{p}\big)$ is the Legendre symbol and $b$ is a fixed integer. Heuristically, the natural density of $S_{a,b}(k,l)$ is expected to be half of that of $Q_a(k,l)$, but it is not true for some choices of $a$ and $b$. In this paper, we determine the natural density of $S_{a,b}(k,l)$ for $(k,l)=(2,j), (q,0), (4,l)$, where $j=0,1$, $q$ is an odd prime and $l=0,2$.

Isomorphism among Families of Weighted $K3$ Hypersurfaces

Wed, 23 Jan 2013 09:29 EST

Masanori KOBAYASHI, Makiko MASE

Source: Tokyo J. of Math., Volume 35, Number 2, 461--467.

Abstract:
It is known that there are exactly 95 weighted projective spaces having Gorenstein $K3$ surfaces as anticanonical divisors, some of which have isometric Picard lattices for generic members. For each set of such families, an explicit birational correspondence coming from a torus action is constructed in this paper. As a result the number of `essentially different' families of weighted Gorenstein $K3$ surfaces is 75.

Pseudo-Anosov Maps and Pairs of Filling Simple Closed Geodesics on Riemann Surfaces

Wed, 23 Jan 2013 09:29 EST

Chaohui ZHANG

Source: Tokyo J. of Math., Volume 35, Number 2, 469--482.

Abstract:
Let $S$ be a Riemann surface of finite area with at least one puncture $x$. Let $a\subset S$ be a simple closed geodesic. In this paper, we show that for any pseudo-Anosov map $f$ of $S$ that is isotopic to the identity on $S\cup \{x\}$, the pair $(a, f^m(a))$ of geodesics fills $S$ for $m\geq 3$. We also study the cases of $0<m\leq 2$ and show that if $(a,f^2(a))$ does not fill $S$, then there is only one geodesic $b$ such that $b$ is disjoint from both $a$ and $f^2(a)$. In fact, $b=f(a)$ and $\{a,f(a)\}$ forms the boundary of an $x$-punctured cylinder on $S$. As a consequence, we show that if $a$ and $f(a)$ are not disjoint, then $(a,f^m(a))$ fills $S$ for any $m\geq 2$.

On Minimal Number of Singular Fibers in a Genus-2 Lefschetz Fibration

Wed, 23 Jan 2013 09:29 EST

Naoyuki MONDEN

Source: Tokyo J. of Math., Volume 35, Number 2, 483--490.

Abstract:
We show that the minimal number of singular fibers in a genus-2 Lefschetz fibration over a closed surface of genus $h$ is equal to 5 if $h\geq 3$, 5 or 6 if $h=2$ and 6 or 7 if $h=1$.

Curvature Pinching for Complete Kaehler Submanifolds of a Complex Projective Space

Wed, 23 Jan 2013 09:29 EST

Yoshio MATSUYAMA

Source: Tokyo J. of Math., Volume 35, Number 2, 491--497.

Abstract:
A classification of complete Kaehler submanifolds $M_{n}$ in $P_{n+p}(C)$ with scalar curvature $\rho > n^{2}$ is given, resolving a conjecture of K. Ogiue.

On Composite Twisted Torus Knots

Wed, 23 Jan 2013 09:29 EST

Kanji MORIMOTO

Source: Tokyo J. of Math., Volume 35, Number 2, 499--503.

Abstract:
In the present note, we will show that there are infinitely many composite twisted torus knots.

Metric Fibrations from Simply Connected Rank---One Projective Spaces

Wed, 23 Jan 2013 09:29 EST

Richard H. ESCOBALES

Source: Tokyo J. of Math., Volume 35, Number 2, 505--512.

Abstract:
In this paper we classify all non-trivial Riemannian submersions with connected fibers from any of the simply connected, rank-one projective spaces. The result follows from results of Gromoll, Grove, Wilking, Becker, Casson, Gottlieb, Schultz, Ucci, and Wolf, together with results of the author.

On the Kernel of the Reciprocity Map of Simple Normal Crossing Varieties over Finite Fields

Wed, 23 Jan 2013 09:29 EST

Rin SUGIYAMA

Source: Tokyo J. of Math., Volume 35, Number 2, 513--526.

Abstract:
In this paper, we study the kernel of the reciprocity map of certain simple normal crossing varieties over a finite field and give an example of a simple normal crossing surface whose reciprocity map is not injective for any finite scalar extension.