Tokyo Journal of Mathematics Articles (Project Euclid)
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The Homogeneous Slice Theorem for the Complete Complexification of a Proper Complex Equifocal Submanifold
http://projecteuclid.org/euclid.tjm/1279719575
<strong>Naoyuki KOIKE</strong><p><strong>Source: </strong>Tokyo J. of Math., Volume 33, Number 1, 1--30.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.tjm/1279719575_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTOn the Krull-Schmidt Decomposition of Mordell-Weil Groupshttps://projecteuclid.org/euclid.tjm/1515466831<strong>Daniel MACIAS CASTILLO</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 40, Number 2, 353--378.</p><p><strong>Abstract:</strong><br/>
Let $A$ be an abelian variety defined over a number field $k$ and $p$ a prime number. Under some natural and not-too-stringent conditions on $A$ and $p$ we show that certain invariants associated to Iwasawa-theoretic $p$-adic Selmer groups control the Krull-Schmidt decompositions of the $p$-adic completions of the groups of points of $A$ over finite extensions of $k$.
</p>projecteuclid.org/euclid.tjm/1515466831_20180108220047Mon, 08 Jan 2018 22:00 ESTThe Growth Rates of Ideal Coxeter Polyhedra in Hyperbolic 3-Spacehttps://projecteuclid.org/euclid.tjm/1515466832<strong>Jun NONAKA</strong>, <strong>Ruth KELLERHALS</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 40, Number 2, 379--391.</p><p><strong>Abstract:</strong><br/>
In~[7], Kellerhals and Perren conjectured that the growth rates of the reflection groups given by compact hyperbolic Coxeter polyhedra are always Perron numbers. We prove that this conjecture holds in the context of ideal Coxeter polyhedra in $\mathbb{H}^3$. Our methods allow us to bound from below the growth rates of composite ideal Coxeter polyhedra by the growth rates of its ideal Coxeter polyhedral constituents.
</p>projecteuclid.org/euclid.tjm/1515466832_20180108220047Mon, 08 Jan 2018 22:00 ESTRavels Arising from Montesinos Tangleshttps://projecteuclid.org/euclid.tjm/1515466833<strong>Erica FLAPAN</strong>, <strong>Allison N. MILLER</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 40, Number 2, 393--420.</p><p><strong>Abstract:</strong><br/>
A ravel is a spatial graph which is non-planar but contains no non-trivial knots or links. We characterize when a Montesinos tangle can become a ravel as the result of vertex closure with and without replacing some number of crossings by vertices.
</p>projecteuclid.org/euclid.tjm/1515466833_20180108220047Mon, 08 Jan 2018 22:00 ESTNorm Conditions on Maps between Certain Subspaces of Continuous Functionshttps://projecteuclid.org/euclid.tjm/1515466834<strong>Razieh Sadat GHODRAT</strong>, <strong>Fereshteh SADY</strong>, <strong>Arya JAMSHIDI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 40, Number 2, 421--437.</p><p><strong>Abstract:</strong><br/>
For a locally compact Hausdorff space $X$, let $C_0(X)$ be the Banach space of continuous complex-valued functions on $X$ vanishing at infinity endowed with the supremum norm $\|\cdot\|_X$. We show that for locally compact Hausdorff spaces $X$ and $Y$ and certain (not necessarily closed) subspaces $A$ and $B$ of $C_0(X)$ and $C_0(Y)$, respectively, if $T:A \longrightarrow B$ is a surjective map satisfying one of the norm conditions i) $\|(Tf)^s (Tg)^t\|_Y=\|f^s g^t\|_X$, or ii) $\|\, |Tf|^s+|Tg|^t\,\|_Y=\|\, |f|^s+|g|^t\, \|_X$, \noindent for some $s,t\in \mathbb{N}$ and all $f,g\in A$, then there exists a homeomorphism $\varphi: \mathrm{ch}(B) \longrightarrow \mathrm{ch}(A)$ between the Choquet boundaries of $A$ and $B$ such that $|Tf(y)|=|f(\varphi(y))|$ for all $f\in A$ and $y\in \mathrm{ch}(B)$. We also give a result for the case where $A$ is closed (or, in general, satisfies a special property called Bishop's property) and $T:A \longrightarrow B$ is a surjective map satisfying the inclusion $R_\pi((Tf)^s (Tg)^t) \subseteq R_\pi(f^s g^t)$ of peripheral ranges. As an application, we characterize such maps between subspaces of the form $A_1f_1+A_2f_2+\cdots+A_nf_n$, where for each $1\le i \le n$, $A_i$ is a uniform algebra on a compact Hausdorff space $X$ and $f_i$ is a strictly positive continuous function on $X$. Our results in case (ii) improve similar results in~[30], for subspaces rather than uniform algebras, without the additional assumption that $T$ is $\mathbb{R}^+$-homogeneous.
</p>projecteuclid.org/euclid.tjm/1515466834_20180108220047Mon, 08 Jan 2018 22:00 ESTA Lower Bound of the Dimension of the Vector Space Spanned by the Special Values of Certain Functionshttps://projecteuclid.org/euclid.tjm/1515466835<strong>Minoru HIROSE</strong>, <strong>Makoto KAWASHIMA</strong>, <strong>Nobuo SATO</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 40, Number 2, 439--479.</p><p><strong>Abstract:</strong><br/>
Let $K$ be a number field. Fix a finite set of analytic functions $\bold{f}_{\infty}:=\{f_{1,\infty}(x),\ldots,f_{s,\infty}(x) \}$ defined on $\{x\in \mathbb{C} \mid |x|>1\}$ (resp. $\mathbb{C}_p$-valued functions $\bold{f}_{p}:=\{f_{1,p}(x),\ldots,f_{s,p}(x) \}$ defined on $\{x\in \mathbb{C}_p \mid |x|_p>1\}$). For $\beta\in K$, we denote the $K$-vector space spanned by $f_{1,\infty}(\beta),\ldots,f_{s,\infty}(\beta)$ by $V_K(\bold{f}_{\infty},\beta)$ (resp. $f_{1,p}(\beta),\ldots,f_{s,p}(\beta)$ by $V_K(\bold{f}_{p},\beta)$). In this article, under some assumptions for $\bold{f}_{\infty}$ (resp. $\bold{f}_{p}$), we give an estimation of a lower bound of the dimension of $V_K(\bold{f}_{\infty},\beta)$ (resp. $V_K(\bold{f}_{p},\beta)$) (see Theorem~2.4 for Archimedean case and Theorem~8.6 for $p$-adic case). Applying our estimation, we give a lower bound of the dimension of the $K$-vector space spanned by the special values of the Lerch functions over a number field in $\mathbb{C}$ (see Theorem~1.1 and Remark~1.2) and the $p$-adic analog of the above result (see Theorem~1.3 and Remark~1.4). Furthermore, we also give a lower bound of the $K$-vector space spanned by the special values of certain $p$-adic functions related with $p$-adic Hurwitz zeta function (see Theorem~1.5).
</p>projecteuclid.org/euclid.tjm/1515466835_20180108220047Mon, 08 Jan 2018 22:00 ESTSecond Sectional Classes of Polarized Three-foldshttps://projecteuclid.org/euclid.tjm/1515466836<strong>Yoshiaki FUKUMA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 40, Number 2, 481--494.</p><p><strong>Abstract:</strong><br/>
Let $X$ be a smooth complex projective variety of dimension $3$ and $L$ an ample line bundle on $X$. In this paper we study the second sectional class $\mathrm{cl}_{2}(X,L)$ of $(X,L)$. First we show the inequality $\mathrm{cl}_{2}(X,L)\geq L^{3}-1$, and we characterize $(X,L)$ with $-1\leq \mathrm{cl}_{2}(X,L)-L^{3}\leq 3$. Furthermore the classification of pairs $(X,L)$ with small second sectional classes is obtained. We also classify $(X,L)$ with $2L^{3}\geq \mathrm{cl}_{2}(X,L)$.
</p>projecteuclid.org/euclid.tjm/1515466836_20180108220047Mon, 08 Jan 2018 22:00 ESTA Function Determined by a Hypersurface of Positive Characteristichttps://projecteuclid.org/euclid.tjm/1515466837<strong>Kosuke OHTA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 40, Number 2, 495--515.</p><p><strong>Abstract:</strong><br/>
Let $R=k\jump{X_1, \dots ,X_{n+1}}$ be a formal power series ring over a perfect field $k$ of characteristic $p>0$, and let $\mathfrak{m} = (X_1 , \dots , X_{n+1})$ be the maximal ideal of $R$. Suppose $0\neq f \in\mathfrak{m}$. In this paper, we introduce a function $\xi_{f}(x)$ associated with a hypersurface $R/(f)$ defined on the closed interval $[0,1]$ in $\mathbb{R}$. The Hilbert-Kunz multiplicity and the F-signature of $R/(f)$ appear as the values of our function $\xi_{f}(x)$ on the interval's endpoints. The F-signature of the pair, denoted by $s(R,f^{t})$, was defined by Blickle, Schwede and Tucker. Our function $\xi_{f}(x)$ is integrable, and the integral $\int_{t}^{1}\xi_{f}(x)dx$ is just $s(R,f^{t})$ for any $t\in[0,1]$.
</p>projecteuclid.org/euclid.tjm/1515466837_20180108220047Mon, 08 Jan 2018 22:00 ESTHarmonic Analysis on the Space of $p$-adic Unitary Hermitian Matrices, Mainly for Dyadic Casehttps://projecteuclid.org/euclid.tjm/1515466838<strong>Yumiko HIRONAKA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 40, Number 2, 517--564.</p><p><strong>Abstract:</strong><br/>
We are interested in harmonic analysis on $p$-adic homogeneous spaces based on spherical functions. In the present paper, we investigate the space $X$ of unitary hermitian matrices of size $m$ over a ${\mathfrak p}$-adic field $k$ mainly for dyadic case, and give the unified description with our previous papers for non-dyadic case. The space becomes complicated for dyadic case, and the set of integral elements in $X$ has plural Cartan orbits. We introduce a typical spherical function $\omega(x;z)$ on $X$, study its functional equations, which depend on $m$ and the ramification index $e$ of $2$ in $k$, and give its explicit formula, where Hall-Littlewood polynomials of type $C_n$ appear as a main term with different specialization according as the parity $m = 2n$ or $2n+1$, but independent of $e$. By spherical transform, we show the Schwartz space ${\mathcal S}(K\backslash X)$ is a free Hecke algebra ${\mathcal H}(G,K)$-module of rank $2^n$, and give parametrization of all the spherical functions on $X$ and the explicit Plancherel formula on ${\mathcal S}(K\backslash X)$. The Plancherel measure does not depend on $e$, but the normalization of $G$-invariant measure on $X$ depends.
</p>projecteuclid.org/euclid.tjm/1515466838_20180108220047Mon, 08 Jan 2018 22:00 ESTOn Concentration Phenomena of Least Energy Solutions to Nonlinear Schrödinger Equations with Totally Degenerate Potentialshttps://projecteuclid.org/euclid.tjm/1515466839<strong>Shun KODAMA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 40, Number 2, 565--603.</p><p><strong>Abstract:</strong><br/>
We study concentration phenomena of the least energy solutions of the following nonlinear Schrödinger equation: \[ h^2 \Delta u - V(x) u + f( u ) = 0 \quad \text{in} \ \mathbb{R}^N, \ u>0, \ u \in H^1(\mathbb{R}^N)\,, \] for a totally degenerate potential $V$. Here $h>0$ is a small parameter, and $f$ is an appropriate, superlinear and Sobolev subcritical nonlinearity. In~[16], Lu and Wei proved that when the parameter $h$ approaches zero, the least energy solutions concentrate at the most centered point of the totally degenerate set $\Omega = \{ x \in \mathbb{R}^N \mid V(x) = \min_{ y \in \mathbb{R}^N } V(y) \}$ when $f(u) = u^p$. In this paper, we show that this kind of result holds for more general $f$. In particular, our proof does not need a so-called uniqueness-nondegeneracy assumption (see, the next-to-last paragraph in Section~1) on the limiting equation (2.6) in Section~2. Furthermore, in~[16] Lu and Wei made a technical assumption for $V$, that is, \[ V(x) - \min_{y \in \mathbb{R}^N} V(y) \geq C d(x, \partial \Omega)^2 \quad \text{for} \ x \in \Omega^{\rm c}\,, \] where $C$ is a positive constant, but our proof does not need this assumption. In our proof, we employ a modification of the argument which has been developed by del Pino and Felmer in~[9] using Schwarz's symmetrization.
</p>projecteuclid.org/euclid.tjm/1515466839_20180108220047Mon, 08 Jan 2018 22:00 ESTFundamental Solutions of the Knizhnik-Zamolodchikov Equation of One Variable and the Riemann-Hilbert Problemhttps://projecteuclid.org/euclid.tjm/1516935614<strong>Shu OI</strong>, <strong>Kimio UENO</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 20 pages.</p><p><strong>Abstract:</strong><br/>
In this article, we show that the generalized inversion formulas of the multiple polylogarithms of one variable, which are generalizations of the inversion formula of the dilogarithm, characterize uniquely the multiple polylogarithms under certain conditions. This means that the multiple polylogarithms are constructed from the multiple zeta values. We call such a problem of determining certain functions a recursive Riemann-Hilbert problem of additive type. Furthermore we show that the fundamental solutions of the KZ equation of one variable are uniquely characterized by the connection relation between the fundamental solutions of the KZ equation normalized at $z=0$ and $z=1$ under some assumptions. Namely the fundamental solutions of the KZ equation are constructed from the Drinfel'd associator. We call this problem a Riemann-Hilbert problem of multiplicative type.
</p>projecteuclid.org/euclid.tjm/1516935614_20180125220106Thu, 25 Jan 2018 22:01 ESTA Diffusion Process with a Random Potential Consisting of Two Contracted Self-Similar Processeshttps://projecteuclid.org/euclid.tjm/1511221565<strong>Yuki SUZUKI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 30 pages.</p><p><strong>Abstract:</strong><br/>
We study a limiting behavior of a one-dimensional diffusion process with a random potential. The potential consists of two independent contracted self-similar processes with different indices for the right and the left hand sides of the origin. Brox (1986) and Schumacher (1985) studied a diffusion process with a Brownian potential, and showed, roughly speaking, after a long time with high probability the process is at the bottom of a valley. Their result was extended to a diffusion process in an asymptotically self-similar random environment by Kawazu, Tamura and Tanaka (1989). Our model is a variant of their models. But we show, roughly speaking, after a long time it is possible that our process is not at the bottom of a valley. We also study asymptotic behaviors of the minimum process and the maximum process of our process.
</p>projecteuclid.org/euclid.tjm/1511221565_20180125220106Thu, 25 Jan 2018 22:01 ESTThe Reductivity of Spherical Curves Part II: 4-gonshttps://projecteuclid.org/euclid.tjm/1516935618<strong>Yui ONODA</strong>, <strong>Ayaka SHIMIZU</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 13 pages.</p><p><strong>Abstract:</strong><br/>
The reductivity of a spherical curve represents how reduced it is. It is unknown if there exists a spherical curve whose reductivity is four. In this paper we give an unavoidable set for spherical curves with reductivity four, that is, we give a set of parts of spherical curves such that every spherical curve with reductivity four has at least one of the parts, by considering 4-gons.
</p>projecteuclid.org/euclid.tjm/1516935618_20180125220106Thu, 25 Jan 2018 22:01 ESTOn the Non-existence of Static Pluriclosed Metrics on Non-Kähler Minimal Complex Surfaceshttps://projecteuclid.org/euclid.tjm/1513566015<strong>Masaya KAWAMURA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 12 pages.</p><p><strong>Abstract:</strong><br/>
The pluriclosed flow is an example of Hermitian flows generalizing the Kähler-Ricci flow. We classify static pluriclosed solutions of the pluriclosed flow on non-Kähler minimal compact complex surfaces. We show that there are no static pluriclosed metrics on Kodaira surfaces, non-Kähler minimal properly elliptic surfaces and Inoue surfaces.
</p>projecteuclid.org/euclid.tjm/1513566015_20180125220106Thu, 25 Jan 2018 22:01 ESTApplications of an Inverse Abel Transform for Jacobi Analysis: Weak-$L^1$ Estimates and the Kunze-Stein Phenomenonhttps://projecteuclid.org/euclid.tjm/1511221566<strong>Takeshi KAWAZOE</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 36 pages.</p><p><strong>Abstract:</strong><br/>
For the Jacobi hypergroup $({\bf R}_+,\Delta,*)$, the weak-$L^1$ estimate of the Hardy-Littlewood maximal operator was obtained by W. Bloom and Z. Xu, later by J. Liu, and the endpoint estimate for the Kunze-Stein phenomenon was obtained by J. Liu. In this paper we shall give alternative proofs based on the inverse Abel transform for the Jacobi hypergroup. The point is that the Abel transform reduces the convolution $*$ to the Euclidean convolution. More generally, let $T$ be the Hardy-Littlewood maximal operator, the Poisson maximal operator or the Littlewood-Paley $g$-function for the Jacobi hypergroup, which are defined by using $*$. Then we shall give a standard shape of $Tf$ for $f\in L^1(\Delta)$, from which its weak-$L^1$ estimate follows. Concerning the endpoint estimate of the Kunze-Stein phenomenon, though Liu used the explicit form of the kernel of the convolution, we shall give a proof without using the kernel form.
</p>projecteuclid.org/euclid.tjm/1511221566_20180125220106Thu, 25 Jan 2018 22:01 ESTMorse--Bott Inequalities for Manifolds with Boundaryhttps://projecteuclid.org/euclid.tjm/1513566016<strong>Ryuma ORITA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 18 pages.</p><p><strong>Abstract:</strong><br/>
In the present paper, we define Morse--Bott functions on manifolds with boundary which are generalizations of Morse functions and show Morse--Bott inequalities for these manifolds.
</p>projecteuclid.org/euclid.tjm/1513566016_20180125220106Thu, 25 Jan 2018 22:01 ESTOn a Variational Problem Arising from the Three-component FitzHugh-Nagumo Type Reaction-Diffusion Systemshttps://projecteuclid.org/euclid.tjm/1513566017<strong>Takashi KAJIWARA</strong>, <strong>Kazuhiro KURATA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 44 pages.</p><p><strong>Abstract:</strong><br/>
We study a variational problem arising from the three-component Fitzhugh-Nagumo type reaction diffusion systems and its shadow systems. In [15], Oshita studied the two-component systems. He revealed that a minimizer of energy corresponding to the problem oscillates under an appropriate condition and also studied its stability. Moreover, he mentioned its energy estimate without a proof. We investigate the behavior of a minimizer corresponding to the three-component problem, its stability and its energy estimate and extend some results of Oshita to the three-component systems and its shadow systems. In particular, we give a necessary and sufficient condition that the minimizer highly oscillates as $ \epsilon \to 0 $. Also, we establish a precise order of the energy estimate of the minimizer as $ \epsilon \to 0 $. In the proof of the energy estimate, we propose a new interpolation inequality.
</p>projecteuclid.org/euclid.tjm/1513566017_20180125220106Thu, 25 Jan 2018 22:01 ESTAsymptotic Behavior of Solutions to the One-dimensional Keller-Segel System with Small Chemotaxishttps://projecteuclid.org/euclid.tjm/1516935619<strong>Yumi YAHAGI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 17 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, the one-dimensional Keller-Segel system defined on a bounded interval with the Neumann boundary conditions is considered. The system describes the phenomenon such that the cellular slime molds form an aggregation by the chemotaxis movement. In the case of small chemotaxis, the asymptotic behavior of solutions to the system are analyzed, as the time development, by using the Fourier series. Some of numerical examples are also given.
</p>projecteuclid.org/euclid.tjm/1516935619_20180125220106Thu, 25 Jan 2018 22:01 ESTWeak-type Estimates in Morrey Spaces for Maximal Commutator and Commutator of Maximal Functionhttps://projecteuclid.org/euclid.tjm/1513566018<strong>Amiran GOGATISHVILI</strong>, <strong>Rza MUSTAFAYEV</strong>, <strong>Müjdat AǦCAYAZI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 26 pages.</p><p><strong>Abstract:</strong><br/>
In this paper it is shown that the Hardy-Littlewood maximal operator $M$ is not bounded on Zygmund-Morrey space $\mathcal{M}_{L(\log L),\lambda}$, $0 < \lambda < n$, but $M$ is still bounded on $\mathcal{M}_{L(\log L),\lambda}$ for radially decreasing functions. The boundedness of the iterated maximal operator $M^2$ from $\mathcal{M}_{L(\log L),\lambda}$ to weak Zygmund-Morrey space $\mathcal{W \! M}_{L(\log L),\lambda}$ is proved. The class of functions for which the maximal commutator $C_b$ is bounded from $\mathcal{M}_{L(\log L),\lambda}$ to $\mathcal{W \! M}_{L(\log L),\lambda}$ are characterized. It is proved that the commutator of the Hardy-Littlewood maximal operator $M$ with function $b \in \text{BMO}(\mathbb{R}^n)$ such that $b^- \in L_{\infty}(\mathbb{R}^n)$ is bounded from $\mathcal{M}_{L(\log L),\lambda}$ to $\mathcal{W \! M}_{L(\log L),\lambda}$. New pointwise characterizations of $M_{\alpha} M$ by means of norm of Hardy-Littlewood maximal function in classical Morrey spaces are given.
</p>projecteuclid.org/euclid.tjm/1513566018_20180125220106Thu, 25 Jan 2018 22:01 ESTA Sufficient Condition That $J(X^*)=J(X)$ Holds for a Banach Space $X$https://projecteuclid.org/euclid.tjm/1513566019<strong>Naoto KOMURO</strong>, <strong>Kichi-Suke SAITO</strong>, <strong>Ryotaro TANAKA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 5 pages.</p><p><strong>Abstract:</strong><br/>
It is shown that the James constant of the space $\mathbb{R}^2$ endowed with a $\pi/2$-rotation invariant norm coincides with that of its dual space. As a corollary, we have the same statement on symmetric absolute norms on $\mathbb{R}^2$.
</p>projecteuclid.org/euclid.tjm/1513566019_20180125220106Thu, 25 Jan 2018 22:01 ESTOn the Unique Solvability of Nonlinear Fuchsian Partial Differential Equationshttps://projecteuclid.org/euclid.tjm/1516935620<strong>Dennis B. BACANI</strong>, <strong>Jose Ernie C. LOPE</strong>, <strong>Hidetoshi TAHARA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 15 pages.</p><p><strong>Abstract:</strong><br/>
We consider a singular nonlinear partial differential equation of the form $$ (t\partial_t)^mu= F \Bigl( t,x,\bigl\{(t\partial_t)^j \partial_x^{\alpha}u \bigr\}_{(j,\alpha) \in I_m} \Bigr) $$ with arbitrary order $m$ and $I_m=\{(j,\alpha) \in \mathbb{N} \times \mathbb{N}^n \,;\, j+|\alpha| \leq m, j<m \}$ under the condition that $F(t,x,\{z_{j,\alpha} \}_{(j,\alpha) \in I_m})$ is continuous in $t$ and holomorphic in the other variables, and it satisfies $F(0,x,0) \equiv 0$ and $(\partial F/\partial z_{j,\alpha})(0,x,0) \equiv 0$ for any $(j,\alpha) \in I_m \cap \{|\alpha|>0 \}$. In this case, the equation is said to be a nonlinear Fuchsian partial differential equation. We show that if $F(t,x,0)$ vanishes at a certain order as $t$ tends to $0$ then the equation has a unique solution with the same decay order.
</p>projecteuclid.org/euclid.tjm/1516935620_20180125220106Thu, 25 Jan 2018 22:01 ESTOn the Semi-simple Case of the Galois Brumer-Stark Conjecture for Monomial Groupshttps://projecteuclid.org/euclid.tjm/1511221567<strong>Xavier-François ROBLOT</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 11 pages.</p><p><strong>Abstract:</strong><br/>
In a previous work, we stated a conjecture, called the Galois Brumer-Stark conjecture, that generalizes the (abelian) Brumer-Stark conjecture to Galois extensions. Other generalizations of the Brumer-Stark conjecture to non-abelian Galois extensions are due to Nickel. Nomura proved that the Brumer-Stark conjecture implies the weak non-abelian Brumer-Stark conjecture of Nickel when the group is monomial. In this paper, we use the methods of Nomura to prove that the Brumer-Stark conjecture implies the Galois Brumer-Stark conjecture for monomial groups in the semi-simple case.
</p>projecteuclid.org/euclid.tjm/1511221567_20180125220106Thu, 25 Jan 2018 22:01 ESTGeometric Aspects of $p$-angular and Skew $p$-angular Distanceshttps://projecteuclid.org/euclid.tjm/1516935621<strong>Jamal ROOIN</strong>, <strong>Somayeh HABIBZADEH</strong>, <strong>Mohammad Sal MOSLEHIAN</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 20 pages.</p><p><strong>Abstract:</strong><br/>
Corresponding to the concept of $p$-angular distance $\alpha_p[x,y]:=\left\lVert\lVert x\rVert^{p-1}x-\lVert y\rVert^{p-1}y\right\rVert$, we first introduce the notion of skew $p$-angular distance $\beta_p[x,y]:=\left\lVert \lVert y\rVert^{p-1}x-\lVert x\rVert^{p-1}y\right\rVert$ for non-zero elements of $x, y$ in a real normed linear space and study some of significant geometric properties of the $p$-angular and the skew $p$-angular distances. We then give some results comparing two different $p$-angular distances with each other. Finally, we present some characterizations of inner product spaces related to the $p$-angular and the skew $p$-angular distances. In particular, we show that if $p>1$ is a real number, then a real normed space $\mathcal{X}$ is an inner product space, if and only if for any $x,y\in \mathcal{X}\smallsetminus{\lbrace 0\rbrace}$, it holds that $\alpha_p[x,y]\geq\beta_p[x,y]$.
</p>projecteuclid.org/euclid.tjm/1516935621_20180125220106Thu, 25 Jan 2018 22:01 ESTOn the Plus and the Minus Selmer Groups for Elliptic Curves at Supersingular Primeshttps://projecteuclid.org/euclid.tjm/1516935622<strong>Takahiro KITAJIMA</strong>, <strong>Rei OTSUKI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 31 pages.</p><p><strong>Abstract:</strong><br/>
Let $p$ be an odd prime number, and $E$ an elliptic curve defined over a number field. Suppose that $E$ has good reduction at any prime lying above $p$, and has supersingular reduction at some prime lying above $p$. In this paper, we construct the plus and the minus Selmer groups of $E$ over the cyclotomic $\mathbb Z_p$-extension in a more general setting than that of B.D. Kim, and give a generalization of a result of B.D. Kim on the triviality of finite $\Lambda$-submodules of the Pontryagin duals of the plus and the minus Selmer groups, where $\Lambda$ is the Iwasawa algebra of the Galois group of the $\mathbb Z_p$-extension.
</p>projecteuclid.org/euclid.tjm/1516935622_20180125220106Thu, 25 Jan 2018 22:01 ESTMaximal Operator and its Commutators on Generalized Weighted Orlicz-Morrey Spaceshttps://projecteuclid.org/euclid.tjm/1513566020<strong>Fatih DERINGOZ</strong>, <strong>Vagif S. GULIYEV</strong>, <strong>Sabir G. HASANOV</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 23 pages.</p><p><strong>Abstract:</strong><br/>
In the present paper, we shall give necessary and sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator and its commutators on generalized weighted Orlicz-Morrey spaces $M^{\Phi,\varphi}_{w}({\mathbb R}^n)$. The main advance in comparison with the existing results is that we manage to obtain conditions for the boundedness not in integral terms but in less restrictive terms of supremal operators and we do not need $\Delta_2$-condition for the boundedness of the maximal operator. We also consider the vector-valued boundedness of the Hardy-Littlewood maximal operator.
</p>projecteuclid.org/euclid.tjm/1513566020_20180125220106Thu, 25 Jan 2018 22:01 ESTOn the Integral Representation of Binary Quadratic Forms and the Artin Conditionhttps://projecteuclid.org/euclid.tjm/1511221568<strong>Chang LV</strong>, <strong>Junchao SHENTU</strong>, <strong>Yingpu DENG</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 14 pages.</p><p><strong>Abstract:</strong><br/>
For diophantine equations of the form $ax^2+bxy+cy^2+g=0$ over $\mathbb{Z}$ whose coefficients satisfy some assumptions, we show that a condition with respect to the Artin reciprocity map, which we call the Artin condition,is the only obstruction to the local-global principle for integral solutions of the equation. Some concrete examples are presented.
</p>projecteuclid.org/euclid.tjm/1511221568_20180125220106Thu, 25 Jan 2018 22:01 ESTA Perturbed CR Yamabe Equation on the Heisenberg Grouphttps://projecteuclid.org/euclid.tjm/1513566021<strong>Takanari SAOTOME</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 26 pages.</p><p><strong>Abstract:</strong><br/>
We will study the CR Yamabe equation for a partially integrable CR structure on the Heisenberg group which is deformed from the standard CR structure. By using the Lyapunov-Schmidt reduction, it is shown that a perturbed standard solution of the CR Yamabe equation is a solution of the deformed CR Yamabe equation, under certain conditions of the deformation. Especially, the deformed CR structure is only partially integrable, in general.
</p>projecteuclid.org/euclid.tjm/1513566021_20180125220106Thu, 25 Jan 2018 22:01 ESTFrom Colored Jones Invariants to Logarithmic Invariantshttps://projecteuclid.org/euclid.tjm/1511221569<strong>Jun MURAKAMI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 23 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we express the logarithmic invariant of knots in terms of derivatives of the colored Jones invariants. Logarithmic invariant is defined by using the Jacobson radicals of the restricted quantum group $\overline{\mathcal U}_\xi(sl_2)$ where $\xi$ is a root of unity. We also propose a version of the volume conjecture stating a relation between the logarithmic invariants and the hyperbolic volumes of the cone manifolds along a knot, which is proved for the figure-eight knot.
</p>projecteuclid.org/euclid.tjm/1511221569_20180125220106Thu, 25 Jan 2018 22:01 ESTComplex Interpolation of Certain Closed Subspaces of Generalized Morrey Spaceshttps://projecteuclid.org/euclid.tjm/1516935625<strong>Denny Ivanal HAKIM</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 28 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we give a description about the first and second complex interpolation between $L^\infty$ and the generalized Morrey spaces. Our result can be viewed as a supplement of the complex interpolation of generalized Morrey spaces, discussed in [9]. We also give an explicit description of some closed subspaces of generalized Morrey spaces and their complex interpolation spaces.
</p>projecteuclid.org/euclid.tjm/1516935625_20180125220106Thu, 25 Jan 2018 22:01 ESTQuasi Contact Metric Manifolds with Constant Sectional Curvaturehttps://projecteuclid.org/euclid.tjm/1516935626<strong>Fereshteh MALEK</strong>, <strong>Rezvan HOJATI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 11 pages.</p><p><strong>Abstract:</strong><br/>
A quasi contact metric manifold is a natural generalization of a contact metric manifold based on the geometry of the corresponding quasi Kahler cones. In this paper we prove that if a quasi contact metric manifold has constant sectional curvature $c$, then $c=1$; additionally, if the characteristic vector field is Killing, then the manifold is Sasakian. These facts are some generalizations of Olszak's theorem to quasi contact metric manifolds.
</p>projecteuclid.org/euclid.tjm/1516935626_20180125220106Thu, 25 Jan 2018 22:01 ESTOn the Polynomial Quadruples with the Property $D(-1;1)$https://projecteuclid.org/euclid.tjm/1511221570<strong>Marija BLIZNAC TREBJEŠANIN</strong>, <strong>Alan FILIPIN</strong>, <strong>Ana JURASIĆ</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 14 pages.</p><p><strong>Abstract:</strong><br/>
In this paper we prove, under some assumptions, that every polynomial $D(-1)$-triple in $\mathbb{Z}[X]$ can only be extended to a polynomial $D(-1;1)$-quadruple in $\mathbb{Z}[X]$ by polynomials $d^{\pm}$. More precisely, if $\{a,b,c;d\}$ is a polynomial $D(-1;1)$-quadruple, then $$d=d^{\pm}=-(a+b+c)+2(abc\pm rst)\,,$$ where $r$, $s$ and $t$ are polynomials from $\mathbb{Z}[X]$ with positive leading coefficients that satisfy $ab-1=r^2$, $ac-1=s^2$ and $bc-1=t^2$.
</p>projecteuclid.org/euclid.tjm/1511221570_20180125220106Thu, 25 Jan 2018 22:01 ESTTruncated Bernoulli-Carlitz and Truncated Cauchy-Carlitz Numbershttps://projecteuclid.org/euclid.tjm/1511221571<strong>Takao KOMATSU</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 16 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we define the truncated Bernoulli-Carlitz numbers and the truncated Cauchy-Carlitz numbers as analogues of hypergeometric Bernoulli numbers and hypergeometric Cauchy numbers, and as extensions of Bernoulli-Carlitz numbers and the Cauchy-Carlitz numbers. These numbers can be expressed explicitly in terms of incomplete Stirling-Carlitz numbers.
</p>projecteuclid.org/euclid.tjm/1511221571_20180125220106Thu, 25 Jan 2018 22:01 ESTMean Values of the Barnes Double Zeta-functionhttps://projecteuclid.org/euclid.tjm/1513566022<strong>Takashi MIYAGAWA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 16 pages.</p><p><strong>Abstract:</strong><br/>
In the study of order estimation of the Riemann zeta-function $ \zeta(s) = \sum_{n=1}^\infty n^{-s} $, solving Lindel\"{o}f hypothesis is an important theme. As one of the relationships, asymptotic behavior of mean values has been studied. Furthermore, the theory of the mean values is also noted in the double zeta-functions, and the mean values of the Euler-Zagier type of double zeta-function and Mordell-Tornheim type of double zeta-function were studied. In this paper, we prove asymptotic formulas for mean square values of the Barnes double zeta-function $ \zeta_2 (s, \alpha ; v, w ) = \sum_{m=0}^\infty \sum_{n=0}^\infty (\alpha+vm+wn)^{-s} $ with respect to $ \mathrm{Im}(s) $ as $ \mathrm{Im}(s) \rightarrow + \infty $.
</p>projecteuclid.org/euclid.tjm/1513566022_20180125220106Thu, 25 Jan 2018 22:01 ESTParabolic Flows on Almost Complex Manifoldshttps://projecteuclid.org/euclid.tjm/1516935627<strong>Masaya KAWAMURA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 14 pages.</p><p><strong>Abstract:</strong><br/>
We define two parabolic flows on almost complex manifolds, which coincide with the pluriclosed flow and the Hermitian curvature flow respectively on complex manifolds. We study the relationship between these parabolic evolution equations on a compact almost Hermitian manifold.
</p>projecteuclid.org/euclid.tjm/1516935627_20180125220106Thu, 25 Jan 2018 22:01 ESTReal Hypersurfaces of Complex Quadric in Terms of Star-Ricci Tensorhttps://projecteuclid.org/euclid.tjm/1513566023<strong>Xiaomin CHEN</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 15 pages.</p><p><strong>Abstract:</strong><br/>
In this article, we introduce the notion of star-Ricci tensors in the real hypersurfaces of complex quadric $Q^m$. It is proved that there exist no Hopf hypersurfaces in $Q^m,m\geq3$, with commuting star-Ricci tensor or parallel star-Ricci tensor. As a generalization of star-Einstein metric, star-Ricci solitons on $M$ are considered. In this case we show that $M$ is an open part of a tube around a totally geodesic $\mathbb{C}P^\frac{m}{2}\subset Q^{m},m\geq4$.
</p>projecteuclid.org/euclid.tjm/1513566023_20180125220106Thu, 25 Jan 2018 22:01 ESTPrecise Asymptotic Formulae for the First Hitting Times of Bessel Processeshttps://projecteuclid.org/euclid.tjm/1511221572<strong>Yuji HAMANA</strong>, <strong>Hiroyuki MATSUMOTO</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 13 pages.</p><p><strong>Abstract:</strong><br/>
We study the first hitting time to $b$ of a Bessel process with index $\nu$ starting from $a$, which is denoted by $\tau_{a,b}^{(\nu)}$, in the case when $0<b<a$. When $\nu>1$ and $\nu-1/2$ is not an integer, we obtain that $\mathbb{P}(t<\tau_{a,b}^{(\nu)}<\infty)$ is asymptotically equal to $\kappa_1^{(\nu)}t^{-\nu}+\kappa_2^{(\nu)} t^{-\nu-1}$ as $t\to\infty$ for some explicit constants $\kappa_1^{(\nu)}$ and $\kappa_2^{(\nu)}$. The constant $\kappa_1^{(\nu)}$ is known and the aim is to get $\kappa_2^{(\nu)}$. Combining our result with the known facts, we obtain the precise asymptotic formula for every index $\nu$.
</p>projecteuclid.org/euclid.tjm/1511221572_20180125220106Thu, 25 Jan 2018 22:01 ESTLittlewood-Paley $g^{\ast}_{\lambda,\mu}$-function and Its Commutator on Non-homogeneous Generalized Morrey Spaceshttps://projecteuclid.org/euclid.tjm/1511221573<strong>Miaomiao WANG</strong>, <strong>Shaoxian MA</strong>, <strong>Guanghui LU</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 10 pages.</p><p><strong>Abstract:</strong><br/>
Let $\mu$ be a non-negative Radon measure on $\mathbb{R}^{d}$ which may be a non-doubling measure. In this paper, the authors prove that the Littlewood-Paley $g^{\ast}_{\lambda,\mu}$-function is bounded on the generalized Morrey space $\mathcal{L}^{p,\phi}(\mu)$, and also obtain that the commutator $g^{\ast}_{\lambda,\mu,b}$ generated by the Littlewood-Paley function $g^{\ast}_{\lambda,\mu}$ and the regular bounded mean oscillation space $(=$RBMO$)$, which is due to X. Tolsa, is bounded on $\mathcal{L}^{p,\phi}(\mu)$. As a corollary, the authors prove that the commutator $g^{\ast}_{\lambda,\mu,b}$ is bounded on the Morrey space $\mathcal{M}^{p}_{q}(\mu)$ defined by Sawano and Tanaka when we take $\phi(t)=t^{1-\frac{p}{q}}$ with $1<p<q<\infty$.
</p>projecteuclid.org/euclid.tjm/1511221573_20180125220106Thu, 25 Jan 2018 22:01 ESTNotes on a $p$-adic Exponential Map for the Picard Grouphttps://projecteuclid.org/euclid.tjm/1513566024<strong>Wataru KAI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 15 pages.</p><p><strong>Abstract:</strong><br/>
For proper flat schemes over complete discrete valuation rings of mixed characteristic, we construct an isomorphism of certain subgroups of the Picard group and the first cohomology group of the structure sheaf. When the Picard functor is representable and smooth, our construction recovers and gives finer information to the isomorphism coming from its formal completion. An alternative proof of an old theorem of Mattuck is given.
</p>projecteuclid.org/euclid.tjm/1513566024_20180125220106Thu, 25 Jan 2018 22:01 ESTA Note on Tropical Curves and the Newton Diagrams of Plane Curve Singularitieshttps://projecteuclid.org/euclid.tjm/1511221574<strong>Takuhiro TAKAHASHI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 11 pages.</p><p><strong>Abstract:</strong><br/>
For an isolated singularity which is Newton non-degenerate and also convenient, the Milnor number can be computed from the complement of its Newton diagram in the first quadrant by using Kouchnirenko's formula. In this paper, we consider tropical curves dual to subdivisions of this complement for a plane curve singularity, and show that there exists a tropical curve by which we can count the Milnor number. Our formula may be regarded as a tropical version of the well-known formula by the real morsification due to A'Campo and Gusein-Zade.
</p>projecteuclid.org/euclid.tjm/1511221574_20180125220106Thu, 25 Jan 2018 22:01 ESTOn Connected Component Decompositions of Quandleshttps://projecteuclid.org/euclid.tjm/1511221575<strong>Yusuke IIJIMA</strong>, <strong>Tomo MURAO</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 20 pages.</p><p><strong>Abstract:</strong><br/>
We give a formula of the connected component decomposition of the Alexander quandle: $\mathbb{Z}[t^{\pm1}]/(f_1(t),\ldots, f_k(t))=\bigsqcup^{a-1}_{i=0}\mathrm{Orb}(i)$, where $a=\gcd (f_1(1),\ldots, f_k(1))$. We show that the connected component $\mathrm{Orb}(i)$ is isomorphic to $\mathbb{Z}[t^{\pm1}]/J$ with an explicit ideal $J$. By using this, we see how a quandle is decomposed into connected components for some Alexander quandles. We introduce a decomposition of a quandle into the disjoint union of maximal connected subquandles. In some cases, this decomposition is obtained by iterating a connected component decomposition. We also discuss the maximal connected sub-multiple conjugation quandle decomposition.
</p>projecteuclid.org/euclid.tjm/1511221575_20180125220106Thu, 25 Jan 2018 22:01 ESTOn Relative Hyperbolicity for a Group and Relative Quasiconvexity for a Subgrouphttps://projecteuclid.org/euclid.tjm/1513566025<strong>Yoshifumi MATSUDA</strong>, <strong>Shin-ichi OGUNI</strong>, <strong>Saeko YAMAGATA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 30 pages.</p><p><strong>Abstract:</strong><br/>
We consider two families of subgroups of a group. Assume that each subgroup which belongs to one family is contained in some subgroup which belongs to the other family. We then discuss relations of relative hyperbolicity for the group with respect to the two families, respectively. If the group is hyperbolic relative to the two families, respectively, then we consider relations of relative quasiconvexity for a subgroup of the group with respect to the two families, respectively.
</p>projecteuclid.org/euclid.tjm/1513566025_20180125220106Thu, 25 Jan 2018 22:01 ESTHomology of the Complex of All Non-trivial Nilpotent Subgroups of a Finite Non-solvable Grouphttps://projecteuclid.org/euclid.tjm/1513566026<strong>Nobuo IIYORI</strong>, <strong>Masato SAWABE</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 8 pages.</p><p><strong>Abstract:</strong><br/>
Let $G$ be a finite non-solvable group. We study homology of the complex $\mathcal{N}(G)$ of all non-trivial nilpotent subgroups of $G$. The determination of $H_{n}(\mathcal{N}(G))$ is reduced to that of homology of its subcomplex $\mathcal{N}_{\pi_{1}}(G)$ consisting of all nilpotent $\pi_{1}$-subgroups, where $\pi_{1}$ is the connected component of the prime graph of $G$ containing 2. Furthermore, $\mathcal{N}_{\pi_{1}}(G)$ is connected if $G$ possesses no strongly embedded subgroups.
</p>projecteuclid.org/euclid.tjm/1513566026_20180125220106Thu, 25 Jan 2018 22:01 ESTA Calculation of the Hyperbolic Torsion Polynomial of a Pretzel Knothttps://projecteuclid.org/euclid.tjm/1513566027<strong>Takayuki MORIFUJI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 6 pages.</p><p><strong>Abstract:</strong><br/>
In this short note, we calculate the highest degree term of the hyperbolic torsion polynomial of a pretzel knot with three tangles. It gives a supporting evidence for a conjecture of Dunfield, Friedl and Jackson that the hyperbolic torsion polynomial determines the genus and fiberedness of a hyperbolic knot.
</p>projecteuclid.org/euclid.tjm/1513566027_20180125220106Thu, 25 Jan 2018 22:01 ESTConformal Slant Riemannian Maps to Kähler Manifoldshttps://projecteuclid.org/euclid.tjm/1516935630<strong>Mehmet Akif AKYOL</strong>, <strong>Bayram ŞAHIN</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 13 pages.</p><p><strong>Abstract:</strong><br/>
As a generalization of slant submanifolds and slant Riemannian maps, we introduce conformal slant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds. We give non-trivial examples, investigate the geometry of foliations and obtain decomposition theorems by using the existence of conformal Riemannian maps. Moreover, we also investigate the harmonicity of such maps and find necessary and sufficient conditions for conformal slant Riemannian maps to be totally geodesic.
</p>projecteuclid.org/euclid.tjm/1516935630_20180125220106Thu, 25 Jan 2018 22:01 ESTReal Hypersurfaces with $^{*}$-Ricci Solitons of Non-flat Complex Space Formshttps://projecteuclid.org/euclid.tjm/1519009215<strong>Xiaomin CHEN</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 19 pages.</p><p><strong>Abstract:</strong><br/>
Kaimakamis and Panagiotidou in [11] introduced the notion of $^*$-Ricci soliton and studied the real hypersurfaces of a non-flat complex space form admitting a $^*$-Ricci soliton whose potential vector field is the structure vector field. In this article, we consider a real hypersurface of a non-flat complex space form which admits a $^*$-Ricci soliton whose potential vector field belongs to the principal curvature space and the holomorphic distribution.
</p>projecteuclid.org/euclid.tjm/1519009215_20180218220038Sun, 18 Feb 2018 22:00 ESTModuli of Surface Diffeomorphisms with Cubic Tangencieshttps://projecteuclid.org/euclid.tjm/1519009219<strong>Shinobu HASHIMOTO</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 27 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we study conjugacy invariants for 2-dimensional diffeomorphisms with homoclinic cubic tangencies (two-sided tangencies of the lowest order) under certain open conditions. Ordinary arguments used in past studies of conjugacy invariants associated with one-sided tangencies do not work in the two-sided case. We present a new method which is applicable to the two-sided case.
</p>projecteuclid.org/euclid.tjm/1519009219_20180218220038Sun, 18 Feb 2018 22:00 EST$L^1$ and $L^\infty$-boundedness of Wave Operators for Three Dimensional Schrödinger Operators with Threshold Singularitieshttps://projecteuclid.org/euclid.tjm/1520305215<strong>Kenji YAJIMA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 22 pages.</p><p><strong>Abstract:</strong><br/>
It is known that wave operators for three dimensional Schrödinger operators $-\Delta + V$ with threshold singularities are bounded in $L^p(\mathbb{R}^3)$ for $1<p<3$ in general and, for $1<p<\infty$ if and only if zero energy resonances are absent and all zero energy eigenfunctions $\phi$ of $-\Delta + V$ satisfy $\int V(x)x^\alpha \phi(x) dx=0$ for $|\alpha|\leq 1$. We prove here that they are bounded in $L^1(\mathbb{R}^3)$ if and only if zero energy resonances are absent. We also show that they are bounded in $L^\infty(\mathbb{R}^3)$ if no resonances are present and all zero energy eigenfunctions $\phi(x)$ satisfy $\int_{\mathbb{R}^3} x^\alpha V(x)\phi(x)dx=0$ for $0\leq |\alpha|\leq 2$. This fills the unknown parts of the $L^p$-boundedness problem for wave operators of three dimensional Schrödinger operators.
</p>projecteuclid.org/euclid.tjm/1520305215_20180305220040Mon, 05 Mar 2018 22:00 ESTAlgebraic Structure of the Lorentz and of the Poincaré Lie Algebrashttps://projecteuclid.org/euclid.tjm/1540800031<strong>Pablo ALBERCA BJERREGAARD</strong>, <strong>Dolores MARTÍN BARQUERO</strong>, <strong>Cándido MARTÍN GONZÁLEZ</strong>, <strong>Daouda NDOYE</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 41, Number 2, 305--346.</p><p><strong>Abstract:</strong><br/>
We start with the Lorentz algebra $\text{\large$\mathfrak{L}$}=\text{\large$\mathfrak{o}$}_{\mathbb{R}}(1,3)$ over the reals and find a suitable basis $B$ such that the structure constants relative to it are integers. Thus we consider the $\mathbb{Z}$-algebra $\text{\large$\mathfrak{L}$}_{\mathbb{Z}}$ which is free as a $\mathbb{Z}$-module of which $B$ is $\mathbb{Z}$-basis. This allows us to define the Lorentz type algebra $\text{\large$\mathfrak{L}$}_K:=\text{\large$\mathfrak{L}$}_{\mathbb{Z}}\otimes_{\mathbb{Z}} K$ over any field $K$. In a similar way, we consider Poincaré type algebras over any field $K$. In this paper we study the ideal structure of Lorentz and of Poincaré type algebras over different fields. It turns out that Lorentz type algebras are simple if and only if the ground field has no square root of $-1$. Thus, they are simple over the reals but not over the complex. Also, if the ground field is of characteristic $2$ then Lorentz and Poincaré type algebras are neither simple nor semisimple. We extend the study of simplicity of the Lorentz algebra to the case of a ring of scalars where we have to use the notion of $\text{\large$\mathfrak{m}$}$-simplicity (relative to a maximal ideal $\text{\large$\mathfrak{m}$}$ of the ground ring of scalars). The Lorentz type algebras over a finite field $\mathbb{F}_q$ where $q=p^n$ and $p$ is odd are simple if and only if $n$ is odd and $p$ of the form $p=4k+3$. In case $p=2$ then the Lorentz type algebras are not simple. Once we know the ideal structure of the algebras, we get some information of their automorphism groups. For the Lorentz type algebras (except in the case of characteristic 2) we describe the affine group scheme of automorphisms and the derivation algebras. For the Poincaré algebras we restrict this program to the case of an algebraically closed field of characteristic other than 2.
</p>projecteuclid.org/euclid.tjm/1540800031_20190205220059Tue, 05 Feb 2019 22:00 ESTOn the Functional Relations for the Euler-Zagier Multiple Zeta-functionshttps://projecteuclid.org/euclid.tjm/1533520816<strong>Soichi IKEDA</strong>, <strong>Kaneaki MATSUOKA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 41, Number 2, 477--485.</p><p><strong>Abstract:</strong><br/>
The purpose of this paper is to formulate problem about existence or non-existence of functional relations for the Euler-Zagier multiple zeta-functions and solve this problem. This problem is a functional analogue of the problem about existence or non-existence of relations among the multiple zeta values. By our results, we can solve a functional analogue of the problem about the dimension of the $\mathbb{Q}$-vector space spanned by the multiple zeta values.
</p>projecteuclid.org/euclid.tjm/1533520816_20190205220059Tue, 05 Feb 2019 22:00 ESTGenus 3 Curves Whose Jacobians Have Endomorphisms by $\mathbb{Q}(\zeta _7 +\bar{\zeta}_7 )$, IIhttps://projecteuclid.org/euclid.tjm/1540800039<strong>Jerome William HOFFMAN</strong>, <strong>Dun LIANG</strong>, <strong>Zhibin LIANG</strong>, <strong>Ryotaro OKAZAKI</strong>, <strong>Yukiko SAKAI</strong>, <strong>Haohao WANG</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 34 pages.</p><p><strong>Abstract:</strong><br/>
In this work we consider constructions of genus three curves $Y$ such that $\text{End}(\text{Jac} (Y))\otimes \mathbb{Q}$ contains the totally real cubic number field $\mathbb{Q}(\zeta _7 +\bar{\zeta}_7 )$. We construct explicit three-dimensional families whose general member is a nonhyperelliptic genus 3 curve with this property. The case when $Y$ is hyperelliptic was studied in \textsc{J. W. Hoffman, H. Wang}, $7$-gons and genus $3$ hyperelliptic curves, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales., Serie A. Matemàticas \textbf{107} (2013), 35--52, and some nonhyperelliptic curves were constructed in \textsc{J. W. Hoffman, Z. Liang, Y. Sakai, H. Wang}, Genus $3$ curves whose Jacobians have endomorphisms by $\mathbb{Q}(\zeta _7 +\bar{\zeta}_7 )$, J. Symb. Comp. \textbf{74} (2016), 561--577.
</p>projecteuclid.org/euclid.tjm/1540800039_20190307215627Thu, 07 Mar 2019 21:56 ESTBounds for Multiple Recurrence Rate and Dimensionhttps://projecteuclid.org/euclid.tjm/1533520820<strong>Michihiro HIRAYAMA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 15 pages.</p><p><strong>Abstract:</strong><br/>
We give an upper bound for the rate of simultaneous recurrence for some commuting family of measure preserving transformations. This provides a generalization of the Boshernitzan quantitative recurrence theorem for single transformation. We also give a lower bound for the Hausdorff dimension of the invariant measures for such transformations and for countable group actions.
</p>projecteuclid.org/euclid.tjm/1533520820_20190307215627Thu, 07 Mar 2019 21:56 ESTSpecial Lagrangian Submanifolds and Cohomogeneity One Actions on the Complex Projective Spacehttps://projecteuclid.org/euclid.tjm/1552013761<strong>Masato ARAI</strong>, <strong>Kurando BABA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 30 pages.</p><p><strong>Abstract:</strong><br/>
We construct examples of cohomogeneity one special Lagrangian submanifolds in the cotangent bundle over the complex projective space, whose Calabi-Yau structure was given by Stenzel. For each example, we describe the condition of special Lagrangian as an ordinary differential equation. Our method is based on a moment map technique and the classification of cohomogeneity one actions on the complex projective space classified by Takagi.
</p>projecteuclid.org/euclid.tjm/1552013761_20190307215627Thu, 07 Mar 2019 21:56 ESTA Generating Function to Generalize the Sum Formula for Quadruple Zeta Valueshttps://projecteuclid.org/euclid.tjm/1533520821<strong>Tomoya MACHIDE</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 27 pages.</p><p><strong>Abstract:</strong><br/>
In the present paper, we prove an identity for the generating function of the quadruple zeta values with the action of the matrix ring $\mathrm{M}_{4}(\mathbb{Z})$. Taking homogeneous parts on both sides of the identity and substituting appropriate values for the variables, we obtain the sum formula for quadruple zeta values. We also obtain its weighted analogues, which include the formulas for this case proved by Guo and Xie (2009, J. Number Theory 129, 2747--2765) and by Ong, Eie, and Liaw (2013, Int. J. Number Theory 9, 1185--1198).
</p>projecteuclid.org/euclid.tjm/1533520821_20190307215627Thu, 07 Mar 2019 21:56 ESTAn Unknotting Index for Virtual Knotshttps://projecteuclid.org/euclid.tjm/1533520823<strong>Kirandeep KAUR</strong>, <strong>Seiichi KAMADA</strong>, <strong>Akio KAWAUCHI</strong>, <strong>Prabhakar MADETI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 14 pages.</p><p><strong>Abstract:</strong><br/>
In this paper we introduce the notion of an unknotting index for virtual knots. We give some examples of computation by using writhe invariants, and discuss a relationship between the unknotting index and the virtual knot module. In particular, we show that for any non-negative integer $n$ there exists a virtual knot whose unknotting index is $(1,n)$.
</p>projecteuclid.org/euclid.tjm/1533520823_20190307215627Thu, 07 Mar 2019 21:56 ESTPsyquandles, Singular Knots and Pseudoknotshttps://projecteuclid.org/euclid.tjm/1540800040<strong>Sam NELSON</strong>, <strong>Natsumi OYAMAGUCHI</strong>, <strong>Radmila SAZDANOVIC</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 25 pages.</p><p><strong>Abstract:</strong><br/>
We generalize the notion of biquandles to \textit{psyquandles} and use these to define invariants of oriented singular links and pseudolinks. In addition to psyquandle counting invariants, we introduce Alexander psyquandles and corresponding invariants such as Alexander psyquandle polynomials and Alexander-Gröbner psyquandle invariants of oriented singular knots and links. We consider the relationship between Alexander psyquandle colorings of pseudolinks and $p$-colorings of pseudolinks. As a special case we define a generalization of the Alexander polynomial for oriented singular links and pseudolinks we call the \textit{Jablan polynomial} and compute the invariant for all pseudoknots with up to five crossings and all 2-bouquet graphs with up to 6 classical crossings.
</p>projecteuclid.org/euclid.tjm/1540800040_20190307215627Thu, 07 Mar 2019 21:56 ESTA Cyclic Cocycle and Relative Index Theorems on Partitioned Manifoldshttps://projecteuclid.org/euclid.tjm/1540800041<strong>Tatsuki SETO</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 18 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we extend Roe's cyclic $1$-cocycle to relative settings. We also prove two relative index theorems for partitioned manifolds by using its cyclic cocycle, which are generalizations of index theorems on partitioned manifolds. One of these theorems is a variant of Theorem 3.3 in ``\textit{Relative-partitioned index theorem}'' by M. Karami, A.H.S. Sadegh and M.E. Zadeh.
</p>projecteuclid.org/euclid.tjm/1540800041_20190307215627Thu, 07 Mar 2019 21:56 ESTHopf-homoclinic Bifurcations and Heterodimensional Cycleshttps://projecteuclid.org/euclid.tjm/1533520824<strong>Shuntaro TOMIZAWA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 21 pages.</p><p><strong>Abstract:</strong><br/>
We consider a $C^r$ diffeomorphism having a Hopf point with $r\ge 5$. If there exists a homoclinic orbit associated with the Hopf point, we say that the diffeomorphism has a \emph{Hopf-homoclinic cycle}. In this paper we prove that every $C^r$ diffeomorphism having a Hopf-homoclinic cycle can be $C^r$ approximated by diffeomorphisms with heterodimensional cycles. Moreover, we study stabilizations of such heterodimensional cycles.
</p>projecteuclid.org/euclid.tjm/1533520824_20190307215627Thu, 07 Mar 2019 21:56 ESTCompact Commutators of Calderón-Zygmund and Generalized Fractional Integral Operators with a Function in Generalized Campanato Spaces on Generalized Morrey Spaceshttps://projecteuclid.org/euclid.tjm/1533520825<strong>Ryutaro ARAI</strong>, <strong>Eiichi NAKAI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 26 pages.</p><p><strong>Abstract:</strong><br/>
We consider the commutators $[b,T]$ and $[b,I_{\rho}]$, where $T$ is a Calderón-Zygmund operator, $I_{\rho}$ is a generalized fractional integral operator and $b$ is a function in Campanato spaces with variable growth condition. It is known that these commutators are bounded on generalized Morrey spaces with variable growth condition. In this paper we discuss the compactness of these commutators.
</p>projecteuclid.org/euclid.tjm/1533520825_20190307215627Thu, 07 Mar 2019 21:56 ESTSeveral Properties of Multiple Hypergeometric Euler Numbershttps://projecteuclid.org/euclid.tjm/1552013762<strong>Takao KOMATSU</strong>, <strong>Wenpeng ZHANG</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 20 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce the higher order hypergeometric Euler numbers and show several interesting expressions. In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli, Cauchy and Euler numbers. One advantage of hypergeometric numbers, including Bernoulli, Cauchy and Euler hypergeometric numbers, is the natural extension of determinant expressions of the numbers. As applications, we can get the inversion relations such that Euler numbers are elements in the determinant.
</p>projecteuclid.org/euclid.tjm/1552013762_20190307215627Thu, 07 Mar 2019 21:56 ESTPrincipal Curvatures of Homogeneous Hypersurfaces in a Grassmann Manifold $\widetilde{\text{Gr}}_{ 3}(\text{Im}\mathbb{O})$ by the $G_2$-actionhttps://projecteuclid.org/euclid.tjm/1552013763<strong>Kanako ENOYOSHI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 14 pages.</p><p><strong>Abstract:</strong><br/>
We compute the principal curvatures of homogeneous hypersurfaces in a Grassmann manifold $\widetilde{\text{Gr}}_{ 3}(\text{Im}\mathbb{O})$ by the $G_2$-action. As applications, we show that there is a unique orbit which is an austere submanifold, and that there are just two orbits which are proper biharmonic homogeneous hypersurfaces. We also show that the austere orbit is a weakly reflective submanifold.
</p>projecteuclid.org/euclid.tjm/1552013763_20190307215627Thu, 07 Mar 2019 21:56 ESTCoincidence Between Two Binary Recurrent Sequences of Polynomials Arising from Diophantine Tripleshttps://projecteuclid.org/euclid.tjm/1552013764<strong>Takafumi MIYAZAKI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 9 pages.</p><p><strong>Abstract:</strong><br/>
A set of positive integers is called a Diophantine tuple if the product of any two elements in the set increased by 1 is a perfect square. A conjecture in this field asserts that any Diophantine triple can be uniquely extended to a Diophantine quadruple in some sense. This problem is reduced to study the coincidence between certain two binary recurrent sequences of integers. As an analogy of this, we consider a similar coincidence on the polynomial ring in one variable over the integers, and determine it completely. Our result is regarded as a generalization of a result in the paper ``Complete solution of the polynomial version of a problem of Diophantus'' by A. Dujella, C. Fuchs in Journal of Number Theory 106 (2004) on the polynomial variant of Diophantine tuples.
</p>projecteuclid.org/euclid.tjm/1552013764_20190307215627Thu, 07 Mar 2019 21:56 ESTRelation between Combinatorial Ricci Curvature and Lin-Lu-Yau's Ricci Curvature on Cell Complexeshttps://projecteuclid.org/euclid.tjm/1552013765<strong>Kazuyoshi WATANABE</strong>, <strong>Taiki YAMADA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 21 pages.</p><p><strong>Abstract:</strong><br/>
In this paper we compare the combinatorial Ricci curvature on cell complexes and Lin-Lu-Yau's Ricci curvature defined on graphs. On a cell complex, the combinatorial Ricci curvature is introduced by the Bochner-Weitzenb\"{o}ck formula. A cell complex corresponds to a graph such that the vertices are cells and the edges are vectors on the cell complex. We compare these two kinds of Ricci curvatures by the coupling and the Kantorovich duality.
</p>projecteuclid.org/euclid.tjm/1552013765_20190307215627Thu, 07 Mar 2019 21:56 ESTGeometric Aspects of Lucas Sequences, Ihttps://projecteuclid.org/euclid.tjm/1552013766<strong>Noriyuki SUWA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Advance publication, 62 pages.</p><p><strong>Abstract:</strong><br/>
We present a way of viewing Lucas sequences in the framework of group scheme theory. This enables us to treat the Lucas sequences from a geometric and functorial viewpoint, which was suggested by Laxton $\langle$On groups of linear recurrences, I$\rangle$ and by Aoki-Sakai $\langle$Mod $p$ equivalence classes of linear recurrence sequences of degree 2$\rangle$.
</p>projecteuclid.org/euclid.tjm/1552013766_20190307215627Thu, 07 Mar 2019 21:56 EST