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 EDTThe Equivalence Theorem of Kinetic Solutions and Entropy Solutions for Stochastic Scalar Conservation Lawshttp://projecteuclid.org/euclid.tjm/1452806058<strong>Dai NOBORIGUCHI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 38, Number 2, 575--587.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove the equivalence of kinetic solutions and entropy solutions for the initial-boundary value problem with a non-homogeneous boundary condition for a multi-dimensional scalar first-order conservation law with a multiplicative noise. We somewhat generalized the definitions of kinetic solutions and of entropy solutions given in Kobayasi and Noboriguchi [8] and Bauzet, Vallet and Wittobolt [1], respectively.
</p>projecteuclid.org/euclid.tjm/1452806058_20160114161406Thu, 14 Jan 2016 16:14 ESTOn Functional Relations for Witten Multiple Zeta-functionshttp://projecteuclid.org/euclid.tjm/1459367257<strong>Soichi IKEDA</strong>, <strong>Kaneaki MATSUOKA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, , 22 pages.</p><p><strong>Abstract:</strong><br/>
In this paper we discuss functional relations for the multi-variable version of Witten zeta-functions associated with the Lie algebra $\mathfrak{sl}(r)$ ($r=3, 4, 5$).
</p>projecteuclid.org/euclid.tjm/1459367257_20160330154744Wed, 30 Mar 2016 15:47 EDTA Class Number Problem for the Cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}(\sqrt{5})$http://projecteuclid.org/euclid.tjm/1459367258<strong>Takuya AOKI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, , 13 pages.</p><p><strong>Abstract:</strong><br/>
Let $K_n$ be the $n$-th layer of the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}(\sqrt{5})$ and $h_n$ the class number of $K_n$.
We prove that, if $\ell$ is a prime number less than $6\cdot10^4$, then $\ell$ does not divide $h_n$ for any non-negative integer $n$.
</p>projecteuclid.org/euclid.tjm/1459367258_20160330154744Wed, 30 Mar 2016 15:47 EDTInterface Regularity of the Solutions to Maxwell Systems on Riemannian Manifoldshttp://projecteuclid.org/euclid.tjm/1459367259<strong>Makoto KANOU</strong>, <strong>Tomohiko SATO</strong>, <strong>Kazuo WATANABE</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, , 18 pages.</p><p><strong>Abstract:</strong><br/>
In this paper we study the interface regularity of the solutions to the differential systems defined by differential forms (for example, stationary Maxwell systems) on $N(\geq 3)$-dimensional Riemannian manifolds.
Our results are natural extensions of the results of \textit{Interface regularity of the solutions for the rotation free and the divergence free systems} and \textit{Interface vanishing for solutions to Maxwell and Stokes systems}.
</p>projecteuclid.org/euclid.tjm/1459367259_20160330154744Wed, 30 Mar 2016 15:47 EDTIsomorphism Classes of Modules over Iwasawa Algebra with $\lambda =4$http://projecteuclid.org/euclid.tjm/1459367260<strong>Kazuaki MURAKAMI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, , 32 pages.</p><p><strong>Abstract:</strong><br/>
We classify the isomorphism classes of finitely generated torsion $\mathcal{O}_E[[T]]$-modules which are free over $\mathcal{O}_E$ of rank $4$, where $\mathcal{O}_E$ is the ring of the integers of a local field $E$.
We apply this classification to the Iwasawa module associated to the cyclotomic $\mathbb{Z}_p$-extension of an imaginary quadratic field.
</p>projecteuclid.org/euclid.tjm/1459367260_20160330154744Wed, 30 Mar 2016 15:47 EDTTopological Symmetry Groups of Complete Bipartite Graphshttp://projecteuclid.org/euclid.tjm/1459367261<strong>Kathleen HAKE</strong>, <strong>Blake MELLOR</strong>, <strong>Matt PITTLUCK</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, , 24 pages.</p><p><strong>Abstract:</strong><br/>
The symmetries of complex molecular structures can be modeled by the {\em topological symmetry group} of the underlying embedded graph.
It is therefore important to understand which topological symmetry groups can be realized by particular abstract graphs.
This question has been answered for complete graphs \cite{fmn3}; it is natural next to consider complete bipartite graphs.
In previous work we classified the complete bipartite graphs that can realize topological symmetry groups isomorphic to $A_4$, $S_4$ or $A_5$ \cite{me}; in this paper we determine which complete bipartite graphs have an embedding in $S^3$ whose topological symmetry group is isomorphic to $\Z_m$, $D_m$, $\Z_r \x \Z_s$ or $(\Z_r \x \Z_s) \ltimes \Z_2$.
</p>projecteuclid.org/euclid.tjm/1459367261_20160330154744Wed, 30 Mar 2016 15:47 EDTOn Classification of Quandles of Cyclic Typehttp://projecteuclid.org/euclid.tjm/1459367262<strong>Seiichi KAMADA</strong>, <strong>Hiroshi TAMARU</strong>, <strong>Koshiro WADA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, , 15 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we study quandles of cyclic type, which form a particular subclass of finite quandles.
The main result of this paper describes the set of isomorphism classes of quandles of cyclic type in terms of certain cyclic permutations.
By using our description, we give a direct classification of quandles of cyclic type with cardinality up to 12.
</p>projecteuclid.org/euclid.tjm/1459367262_20160330154744Wed, 30 Mar 2016 15:47 EDTA Study of Submanifolds of the Complex Grassmannian Manifold with Parallel Second Fundamental Formhttp://projecteuclid.org/euclid.tjm/1459367263<strong>Isami KOGA</strong>, <strong>Yasuyuki NAGATOMO</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, , 13 pages.</p><p><strong>Abstract:</strong><br/>
We prove an extension of a theorem of A. Ros on \textit{a characterization of seven compact Kaehler submanifolds by holomorphic pinching} [5] to certain submanifolds of the complex Grassmannian manifolds.
</p>projecteuclid.org/euclid.tjm/1459367263_20160330154744Wed, 30 Mar 2016 15:47 EDTA Note on the Galois Brumer-Stark Conjecturehttp://projecteuclid.org/euclid.tjm/1459367264<strong>Jiro NOMURA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, , 11 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove, for primes $l$ satisfying some conditions, the $l$-parts of the Galois Brumer-Stark conjecture, which is formulated by Dejou and Roblot for Galois CM-extensions with dihedral or generalized quaternion Galois group of specified degrees.
</p>projecteuclid.org/euclid.tjm/1459367264_20160330154744Wed, 30 Mar 2016 15:47 EDTUniform Blow-up Rate for Nonlocal Diffusion-like Equations with Nonlocal Nonlinear Sourcehttp://projecteuclid.org/euclid.tjm/1459367265<strong>Jiashan ZHENG</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, , 16 pages.</p><p><strong>Abstract:</strong><br/>
We present new blow-up results for nonlocal reaction-diffusion equations with nonlocal nonlinearities.
The nonlocal source terms we consider are of several types, and are relevant to various models in physics and engineering.
They may involve an integral of an unknown function, either in space, in time, or both in space and time, or they may depend on localized values of the solution.
We first show the existence and uniqueness of the solution to problem relying on contraction mapping fixed point theorem.
Then, the comparison principles for problem are established through a standard method.
Finally, for the radially symmetric and non-increasing initial data, we give a complete classification in terms of global and single point blow-up according to the parameters.
Moreover, the blow-up rates are also determined in each case.
</p>projecteuclid.org/euclid.tjm/1459367265_20160330154744Wed, 30 Mar 2016 15:47 EDTOn the $C^\alpha$-convergence of the Solution of the Chern-Ricci Flow on Elliptic Surfaceshttp://projecteuclid.org/euclid.tjm/1459367266<strong>Masaya KAWAMURA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, , 10 pages.</p><p><strong>Abstract:</strong><br/>
We will study the Chern-Ricci flow on non-K\"ahler properly elliptic surfaces.
These surfaces are compact complex surfaces whose first Betti number is odd, Kodaira dimension is equal to 1 and admit an elliptic fibration to a smooth compact curve.
We will show that a solution of the Chern-Ricci flow converges in $C^\alpha$-topology on these elliptic surfaces by choosing a special initial metric.
</p>projecteuclid.org/euclid.tjm/1459367266_20160330154744Wed, 30 Mar 2016 15:47 EDTHigher-order Convolution Identities for Cauchy Numbershttp://projecteuclid.org/euclid.tjm/1459367267<strong>Takao KOMATSU</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, , 15 pages.</p><p><strong>Abstract:</strong><br/>
Euler's famous formula written in symbolic notation as $(B_0+B_0)^n=-n B_{n-1}-(n-1)B_n$ was extended to $(B_{l_1}+\cdots+B_{l_m})^n$ for $m\ge 2$ and arbitrary fixed integers $l_1,\dots,l_m\ge 0$.
In this paper, we consider the higher-order recurrences for Cauchy numbers $(c_{l_1}+\cdots+c_{l_m})^n$, where the $n$-th Cauchy number $c_n$ ($n\ge 0$) is defined by the generating function $x/\ln(1+x)=\sum_{n=0}^\infty c_n x^n/n!$.
In special, we give an explicit expression in the case $l_1=\cdots=l_m=0$ for any integers $n\ge 1$ and $m\ge 2$.
We also discuss the case for Cauchy numbers of the second kind $\widehat c_n$ in similar ways.
</p>projecteuclid.org/euclid.tjm/1459367267_20160330154744Wed, 30 Mar 2016 15:47 EDTAsymptotic Behavior of Solutions for Semilinear Volterra Diffusion Equations with Spatial Inhomogeneity and Advectionhttp://projecteuclid.org/euclid.tjm/1459367268<strong>Yusuke YOSHIDA</strong>, <strong>Yoshio YAMADA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, , 22 pages.</p><p><strong>Abstract:</strong><br/>
This paper is concerned with semilinear Volterra diffusion equations with spatial inhomogeneity and advection.
We intend to study the effects of interaction among diffusion, advection and Volterra integral under spatially inhomogeneous environments.
Since the existence and uniqueness result of global-in-time solutions can be proved in the standard manner, our main interest is to study their asymptotic behavior as $t\to \infty$.
For this purpose, we study the related stationary problem by the monotone method and establish some sufficient conditions on the existence of a unique positive solution.
Its global attractivity is also studied with use of a suitable Lyapunov functional.
</p>projecteuclid.org/euclid.tjm/1459367268_20160330154744Wed, 30 Mar 2016 15:47 EDTRemarks on $r$-planes in Complete Intersectionshttp://projecteuclid.org/euclid.tjm/1459367269<strong>Chikashi MIYAZAKI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, , 9 pages.</p><p><strong>Abstract:</strong><br/>
This paper investigates the families of smooth complete intersections containing $r$-planes in projective spaces.
We are going in a primitive way to shed some light on a point and an $r$-plane containing the point in a complete intersection from the viewpoint of projective geometry.
</p>projecteuclid.org/euclid.tjm/1459367269_20160330154744Wed, 30 Mar 2016 15:47 EDT$B_w^u$-function Spaces and Their Interpolationhttp://projecteuclid.org/euclid.tjm/1459367270<strong>Eiichi NAKAI</strong>, <strong>Takuya SOBUKAWA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, , 34 pages.</p><p><strong>Abstract:</strong><br/>
We introduce $B_w^u$-function spaces which unify Lebesgue, Morrey-Campanato, Lipschitz, $B^p$, $\CMO$, local Morrey-type spaces, etc., and investigate the interpolation property of $B_w^u$-function spaces.
We also apply it to the boundedness of linear and sublinear operators, for example, the Hardy-Littlewood maximal and fractional maximal operators, singular and fractional integral operators with rough kernel, the Littlewood-Paley operator, Marcinkiewicz operator, and so on.
</p>projecteuclid.org/euclid.tjm/1459367270_20160330154744Wed, 30 Mar 2016 15:47 EDTGreenberg's Conjecture for the Cyclotomic $\mathbb{Z}_{2}$-extension of Certain Number Fields of Degree Fourhttp://projecteuclid.org/euclid.tjm/1471873310<strong>Naoki KUMAKAWA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 1, 1--15.</p><p><strong>Abstract:</strong><br/>
The purpose of this paper is to construct infinite families of real abelian number fields $K$ of degree four with $\lambda_{2}(K)= \mu_{2}(K) =0$ and $\nu_{2}(K) >0$.
</p>projecteuclid.org/euclid.tjm/1471873310_20160822094156Mon, 22 Aug 2016 09:41 EDTMaps Which Preserve a Certain Norm Condition between the Exponential Groups of Uniform Algebrashttp://projecteuclid.org/euclid.tjm/1471873311<strong>Tatsuya NOGAWA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 1, 39--44.</p><p><strong>Abstract:</strong><br/>
Let $\A_j$ be a uniform algebra with a Choquet boundary $Ch\A_j$, $j = 1, 2$.
In this paper we prove that if $\phi : \exp\A_1 \to \exp\A_2$ is a surjection and satisfies the equality
\begin{equation*}
\max \left\{ \left\| \frac{\phi (f)}{\phi (g)} -1 \right\|_\infty, \left\| \frac{\phi (g)}{\phi (f)} -1 \right\|_\infty \right \}
=\max \left\{ \left\| \frac{f}{g} -1 \right\|_\infty, \left\| \frac{g}{f} -1 \right\|_\infty \right \}
\end{equation*}
for any $f, g \in \exp\A_1$, then $\phi$ is of the form
\begin{equation*}
\phi(f)(y) = \left\{ \begin{array}{ll}
\phi(1)(y) f(\varphi(y))^{\kappa(y)} & \text{for}~ y \in K, \\
\phi(1)(y) \overline{f(\varphi(y))}^{\kappa(y)} & \text{for}~ y \in Ch\A_2 \setminus K \\
\end{array} \right.
\end{equation*}
for any $f \in \exp\A_1$, where $\kappa$ is a continuous function from $Ch\A_2$ into $\{ 1, -1 \}$, $\varphi$ is a homeomorphism from $Ch\A_2$ onto $Ch\A_1$ and $K$ is a clopen subset of $Ch\A_2$.
</p>projecteuclid.org/euclid.tjm/1471873311_20160822094156Mon, 22 Aug 2016 09:41 EDTThe Dual Jacobian of a Generalised Hyperbolic Tetrahedron, and Volumes of Prismshttp://projecteuclid.org/euclid.tjm/1471873312<strong>Alexander KOLPAKOV</strong>, <strong>Jun MURAKAMI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 1, 45--67.</p><p><strong>Abstract:</strong><br/>
We derive an analytic formula for the dual Jacobian matrix of a generalised hyperbolic tetrahedron.
Two cases are considered: a mildly truncated and a prism truncated tetrahedron.
The Jacobian for the latter arises as an analytic continuation of the former, that falls in line with a similar behaviour of the corresponding volume formulae.
Also, we obtain a volume formula for a hyperbolic $n$-gonal prism: the proof requires the above mentioned Jacobian, employed in the analysis of the edge lengths behaviour of such a prism, needed later for the Schläfli formula.
</p>projecteuclid.org/euclid.tjm/1471873312_20160822094156Mon, 22 Aug 2016 09:41 EDTNested Square Roots and Poincaré Functionshttp://projecteuclid.org/euclid.tjm/1471873313<strong>Noboru AOKI</strong>, <strong>Shota KOJIMA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 1, 241--269.</p><p><strong>Abstract:</strong><br/>
We are concerned with finitely nested square roots which are roots of iterations of a real quadratic polynomial $x^2-c$ with $c\geq 2$, and the limits of such nested square roots.
We investigate how they are related to a Poincaré function $f(x)$ satisfying the functional equation $f(sx)=f(x)^2-c$, where $s=1+\sqrt{1+4c}$.
Our main theorems can be viewed as a natural generalization of the work of Wiernsberger and Lebesgue for the case $c=2$.
The key ingredients of the proof are some analytic properties of $F(x)$, which have been intensively studied by the second author using infinite compositions.
</p>projecteuclid.org/euclid.tjm/1471873313_20160822094156Mon, 22 Aug 2016 09:41 EDTTrace Operator for 2-microlocal Besov Spaces with Variable Exponentshttp://projecteuclid.org/euclid.tjm/1471873314<strong>Takahiro NOI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 1, 293--327.</p><p><strong>Abstract:</strong><br/>
Moura, Neves and Schneider proved the trace theorem for 2-microlocal Besov spaces with variable integrability and smoothness, where the summability parameter was constant (Math.\,Nachr.,286 (2013) 1240--1254).
In this paper, we extend the trace theorem for the case that the summability parameter is also a variable exponent.
</p>projecteuclid.org/euclid.tjm/1471873314_20160822094156Mon, 22 Aug 2016 09:41 EDTNew Trigonometric Identities and Reciprocity Laws of Generalized Dedekind Sumshttp://projecteuclid.org/euclid.tjm/1484903126<strong>Genki SHIBUKAWA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 2, 329--349.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove new trigonometric identities, which are product-to-sum type formulas for the higher derivatives of the cotangent and cosecant functions.
Furthermore, from specializations of our formulas, we derive various known and new reciprocity laws of generalized Dedekind sums.
</p>projecteuclid.org/euclid.tjm/1484903126_20170120040714Fri, 20 Jan 2017 04:07 ESTThe Capitulation Problem for Certain Cyclic Quartic Number Fieldshttp://projecteuclid.org/euclid.tjm/1484903127<strong>Abdelmalek AZIZI</strong>, <strong>Idriss JERRARI</strong>, <strong>Mohammed TALBI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 2, 351--359.</p><p><strong>Abstract:</strong><br/>
Let $K$ be a cyclic quartic number field such that its 2-class group is of type $(2,4)$, $K_2^{(1)}$ be the Hilbert 2-class field of $K$, $K_2^{(2)}$ be the Hilbert 2-class field of $K_2^{(1)}$ and $G=\text{Gal}(K_2^{(2)}/K)$ be the Galois group of $K_2^{(2)}/K$.
Our goal is to study the capitulation problem of 2-ideal classes of $K$ and to determine the structure of $G$.
</p>projecteuclid.org/euclid.tjm/1484903127_20170120040714Fri, 20 Jan 2017 04:07 ESTThe Uniqueness Theorem for the Heat Equation on the Heisenberg Grouphttp://projecteuclid.org/euclid.tjm/1484903128<strong>Yasuyuki OKA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 2, 361--371.</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to establish the uniqueness theorem for the Cauchy problem for the heat equation with the Tikhonov condition on the Heisenberg group.
To do this, we give Green's formula and show the existence of a Lipschitz cut-off function on the Heisenberg group in accordance with the idea in [7].
</p>projecteuclid.org/euclid.tjm/1484903128_20170120040714Fri, 20 Jan 2017 04:07 ESTNon-Hopf Hypersurfaces in 2-dimensional Complex Space Formshttp://projecteuclid.org/euclid.tjm/1484903129<strong>Mayuko KON</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 2, 373--387.</p><p><strong>Abstract:</strong><br/>
In this paper we give a geometric characterization of non-Hopf hypersurfaces in the complex space form $M^2(c)$ under a condition on the shape operator.
We also classify pseudo-parallel real hypersurfaces of $M^2(c)$.
</p>projecteuclid.org/euclid.tjm/1484903129_20170120040714Fri, 20 Jan 2017 04:07 ESTA Rigidity Theorem for Proper Holomorphic Mappings between Generalized Pseudoellipsoidshttp://projecteuclid.org/euclid.tjm/1484903130<strong>Atsushi HAYASHIMOTO</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 2, 389--421.</p><p><strong>Abstract:</strong><br/>
Let $E(\alpha) \subset \mathbb{C}^{m+1}$ and $E(\beta) \subset \mathbb{C}^{n+1}$ be generalized pseudoellipsoids.
Assume that the inequality $m<n$ holds.
They are parametrized by $N$-tuples of positive integers $\alpha=(\alpha_1, \dots, \alpha_N)$ and $\beta=(\beta_1, \dots, \beta_N)$.
(See introduction for the definition of a generalized pseudoellipsoid)
Assume that there exists a proper holomorphic mapping between them.
In this article, two facts are proved.
Firstly, under the assumptions of the existence of such a mapping, certain nondegeneracy conditions of a submatrix of the Jacobian matrix and additional inequalities on dimensions, the parameters $(\alpha_1, \dots, \alpha_N)$ and $(\beta_1, \dots, \beta_N)$ coincide; $\alpha_1=\beta_1, \dots, \alpha_N=\beta_N$ after re-ordering if necessary.
Secondly, such a proper holomorphic mapping is a linear embedding up to automorphisms of a source and a target domains.
</p>projecteuclid.org/euclid.tjm/1484903130_20170120040714Fri, 20 Jan 2017 04:07 ESTToeplitz Operators and the Roe-Higson Type Index Theorem in Riemannian Surfaceshttp://projecteuclid.org/euclid.tjm/1484903131<strong>Tatsuki SETO</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 2, 423--439.</p><p><strong>Abstract:</strong><br/>
Let $M$ be a non-compact complete Riemannian manifold of dimension two and $N$ a circle in $M$.
We assume that $M$ is partitioned by $N$.
We define a unital $C^{\ast}$-algebra $C_{b}^{\ast}(M)$, which is slightly larger than the Roe algebra of $M$.
We also construct $[u_{\phi}]$ in $K_{1}(C_{b}^{\ast}(M))$, which is a counter part of Roe's odd index class.
We prove that Connes' pairing of Roe's cyclic one-cocycle with $[u_{\phi}]$ is equal to the Fredholm index of a Toeplitz operator on $N$.
It is a part of an extension of the Roe-Higson index theorem to even-dimensional partitioned manifolds.
</p>projecteuclid.org/euclid.tjm/1484903131_20170120040714Fri, 20 Jan 2017 04:07 ESTOn Hausdorff Dimension of Certain Sets Arising from Diophantine Approximations for Complex Numbershttp://projecteuclid.org/euclid.tjm/1484903132<strong>Zhengyu CHEN</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 2, 441--458.</p><p><strong>Abstract:</strong><br/>
We discuss the Hausdorff dimension of certain sets related to Diophantine approximations over an imaginary quadratic field $\mathbb{Q}(\sqrt{d})$.
We show that, for an infinite subset $\mathcal{A}$ of $\mathbb{Z}[\omega] \backslash \{0\}$, the set of $z \in \mathbb{C}$ with $|z-a/r| < 1 / |r|^{1+\rho}$ having infinitely many solutions of $a \in \mathbb{Z}[\omega]$ and $r \in \mathcal{A}$ with some $\rho > 0$ has Hausdorff dimension $2(1+\gamma) / (1+\rho)$, where $\gamma$ is the sup of $h$ such that $\sum_{r \in \mathcal{A}} 1/(|r|^{2})^{h}$ diverges.
This result is a version of a result by G. Harman for complex numbers without the coprime condition.
In particular, this result implies a version of the classical Jarnik-Besicovitch result when we take $\mathcal{A} = \mathbb{Z}[\omega]\backslash\{0\}$.
We also discuss the Hausdorff dimension of the set of complex numbers which have infinitely many solutions to the Diophantine inequality concerning the Duffin-Schaeffer conjecture over $\mathbb{Q}(\sqrt{d})$.
</p>projecteuclid.org/euclid.tjm/1484903132_20170120040714Fri, 20 Jan 2017 04:07 ESTOn the Centralizer Algebras of the Primitive Unitary Reflection Group of Order 96http://projecteuclid.org/euclid.tjm/1484903133<strong>Masashi KOSUDA</strong>, <strong>Manabu OURA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 2, 469--482.</p><p><strong>Abstract:</strong><br/>
Among the unitary reflection groups, the one on the title is singled out by its importance in, for example, coding theory and number theory.
In this paper we examine the semi-simple structure of the centralizer algebra in the tensor representation, and show that the dimensions of the centralizers coincide with the numbers of some combinatorial objects.
</p>projecteuclid.org/euclid.tjm/1484903133_20170120040714Fri, 20 Jan 2017 04:07 ESTReidemeister Torsion and Dehn Surgery on Twist Knotshttp://projecteuclid.org/euclid.tjm/1484903134<strong>Anh T. TRAN</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 2, 517--526.</p><p><strong>Abstract:</strong><br/>
We compute the Reidemeister torsion of the complement of a twist knot in $S^3$ and that of the 3-manifold obtained by a $\frac{1}{q}$-Dehn surgery on a twist knot.
</p>projecteuclid.org/euclid.tjm/1484903134_20170120040714Fri, 20 Jan 2017 04:07 ESTAsymptotically Unweighted Shifts, Hypercyclicity, and Linear Chaoshttp://projecteuclid.org/euclid.tjm/1484903135<strong>Ayuko NATSUME</strong>, <strong>Masahiko TANIGUCHI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 2, 527--536.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce weighted backward shifts, which are asymptotically unweighted, and give several conditions for such operators on the classical $\ell^p$ spaces to be hypercyclic and chaotic.
</p>projecteuclid.org/euclid.tjm/1484903135_20170120040714Fri, 20 Jan 2017 04:07 ESTA Sufficient Condition for Orbits of Hermann Actions to be Weakly Reflectivehttp://projecteuclid.org/euclid.tjm/1484903136<strong>Shinji OHNO</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 2, 537--564.</p><p><strong>Abstract:</strong><br/>
In this paper, we give sufficient conditions for orbits of Hermann actions to be weakly reflective in terms of symmetric triads, that is a generalization of irreducible root systems.
Using these sufficient conditions, we obtain new examples of weakly reflective submanifolds in compact symmetric spaces.
</p>projecteuclid.org/euclid.tjm/1484903136_20170120040714Fri, 20 Jan 2017 04:07 ESTOn the Moduli Space of Pointed Algebraic Curves of Low Genus III ---Positive Characteristic---http://projecteuclid.org/euclid.tjm/1484903137<strong>Tetsuo NAKANO</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 2, 565--582.</p><p><strong>Abstract:</strong><br/>
In his classical work, Pinkham discovered a beautiful theorem on the moduli space of pointed algebraic curves with a fixed Weierstrass gap sequence at the marked point.
Namely, the complement of a Weierstrass gap sequence in the set of non-negative integers is a numerical semigroup, and he described such a moduli space in terms of the negative part of the miniversal deformation space of the monomial curve of this semigroup.
Unfortunately, his theorem holds only in characteristic 0 and does not hold in positive characteristic in general.
In this paper, we will study his theorem in positive characteristic, and give a fairly sharp condition for his theorem to hold in positive characteristic up to genus 4.
As an application, we present a complete analysis of his theorem in positive characteristic in the low genus case.
</p>projecteuclid.org/euclid.tjm/1484903137_20170120040714Fri, 20 Jan 2017 04:07 ESTGauss Sums on the Iwahori-Hecke Algebras of Type $A$http://projecteuclid.org/euclid.tjm/1491465731<strong>Yasushi GOMI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 3, 583--596.</p><p><strong>Abstract:</strong><br/>
In this paper, we determine $\tilde{\tau}_q\bigl(\chi^\lambda_q\bigr)$, the Gauss sums on the Iwahori-Hecke algebras of type $A$ for irreducible characters $\chi^\lambda_q$, which are $q$-analogues of those on the symmetric groups.
We also explicitly determine the values of the corresponding trace function $\psi^{(n)}_q=\sum_{\lambda \vdash n} \tilde{\tau}_q\bigl(\chi^\lambda_q\bigr) \chi^\lambda_q$.
</p>projecteuclid.org/euclid.tjm/1491465731_20170406040226Thu, 06 Apr 2017 04:02 EDTMixed Quantum Double Construction of Subfactorshttp://projecteuclid.org/euclid.tjm/1491465732<strong>Satoshi GOTO</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 3, 597--617.</p><p><strong>Abstract:</strong><br/>
We generalize the quantum double construction of subfactors to that from arbitrary flat connections on 4-partite graphs and call it the \textit{mixed quantum double construction}.
If all the four graphs of the original 4-partite graph are connected, it is easy to see that this construction produces Ocneanu's asymptotic inclusion
of both subfactors generated by the original flat connection horizontally and vertically.
The construction can be applied for example to the non-standard flat connections which appear in the construction of the Goodman-de la Harpe-Jones subfactors or to those obtained by the composition of flat part of any biunitary connections as in N. Sato's paper~[40].
An easy application shows that the asymptotic inclusions of the Goodman-de la Harpe-Jones subfactors are isomorphic to those of the Jones subfactors of type $A_n$ except for the cases of orbifold type.
If two subfactors $A\subset B$ and $A \subset C$ have common $A$-$A$ bimodule systems, we can construct a flat connection in general.
Then by applying our construction to the flat connection, we obtain the asymptotic inclusion of both $A\subset B$ and $A \subset C$.
We also discuss the case when the original 4-partite graph contains disconnected graphs and give some such examples.
General phenomena when disconnected graphs appear are explained by using bimodule systems.
</p>projecteuclid.org/euclid.tjm/1491465732_20170406040226Thu, 06 Apr 2017 04:02 EDTFitting Ideals of Iwasawa Modules and of the Dual of Class Groupshttp://projecteuclid.org/euclid.tjm/1475723094<strong>Cornelius GREITHER</strong>, <strong>Masato KURIHARA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 3, 619--642.</p><p><strong>Abstract:</strong><br/>
In this paper we study some problems related to a refinement of Iwasawa theory, especially questions about the Fitting ideals of several natural Iwasawa modules and of the dual of the class groups, as a sequel to our previous papers [8], [3].
Among other things, we prove that the annihilator of $\mathbb{Z}_{p}(1)$ times the Stickelberger element is not in the Fitting ideal of the dualized Iwasawa module if the $p$-component of the bottom Galois group is elementary $p$-abelian with $p$-rank $\geq 4$.
Our results can be applied to the case that the base field is $\mathbb{Q}$.
</p>projecteuclid.org/euclid.tjm/1475723094_20170406040226Thu, 06 Apr 2017 04:02 EDTExplicit Forms of Cluster Variables on Double Bruhat Cells $G^{u,e}$ of Type Chttp://projecteuclid.org/euclid.tjm/1475723089<strong>Yuki KANAKUBO</strong>, <strong>Toshiki NAKASHIMA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 3, 643--678.</p><p><strong>Abstract:</strong><br/>
Let $G=Sp_{2r}({\mathbb C})$ be a simply connected simple algebraic group over $\mathbb{C}$ of type $C_r$, $B$ and $B_-$ its two opposite Borel subgroups, and $W$ the associated Weyl group.
For $u$, $v\in W$, it is known that the coordinate ring ${\mathbb C}[G^{u,v}]$ of the double Bruhat cell $G^{u,v}=BuB\cup B_-vB_-$ is isomorphic to an upper cluster algebra $\overline{\mathcal{A}}(\textbf{i})_{{\mathbb C}}$ and the generalized minors $\Delta(k;\textbf{i})$ are the cluster variables of ${\mathbb C}[G^{u,v}]${5}.
In the case $v=e$, we shall describe the generalized minor $\Delta(k;\textbf{i})$ explicitly.
</p>projecteuclid.org/euclid.tjm/1475723089_20170406040226Thu, 06 Apr 2017 04:02 EDTOn the Construction of Continued Fraction Normal Series in Positive Characteristichttp://projecteuclid.org/euclid.tjm/1475723093<strong>Dong Han KIM</strong>, <strong>Hitoshi NAKADA</strong>, <strong>Rie NATSUI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 3, 679--694.</p><p><strong>Abstract:</strong><br/>
Motivated by the famous Champernowne construction of a normal number, R.~Adler, M.~Keane, and M.~Smorodinsky constructed a normal number with respect to the simple continued fraction transformation.
In this paper, we follow their idea and construct a normal series for the Artin continued fraction expansion in positive characteristic.
A normal series for L\"uroth expansion is also discussed.
</p>projecteuclid.org/euclid.tjm/1475723093_20170406040226Thu, 06 Apr 2017 04:02 EDTConstruction of Double Grothendieck Polynomials of Classical Types using IdCoxeter Algebrashttp://projecteuclid.org/euclid.tjm/1491465733<strong>Anatol N. KIRILLOV</strong>, <strong>Hiroshi NARUSE</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 3, 695--728.</p><p><strong>Abstract:</strong><br/>
We construct double Grothendieck polynomials of classical types which are essentially equivalent to but simpler than the polynomials defined by A.~N. Kirillov in arXiv:1504.01469 and identify them with the polynomials defined by T.~Ikeda and H.~Naruse in Adv. Math. (2013) for the case of maximal Grassmannian permutations.
We also give geometric interpretation of them in terms of algebraic localization map and give explicit combinatorial formulas.
</p>projecteuclid.org/euclid.tjm/1491465733_20170406040226Thu, 06 Apr 2017 04:02 EDTCoxeter Elements of the Symmetric Groups Whose Powers Afford the Longest Elementshttp://projecteuclid.org/euclid.tjm/1491465734<strong>Masashi KOSUDA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 3, 729--742.</p><p><strong>Abstract:</strong><br/>
The purpose of this paper is to present a condition for the power of a Coxeter element of $\mathfrak{S}_n$ to become the longest element.
To be precise, given a product $C$ of $n-1$ distinct adjacent transpositions of $\mathfrak{S}_n$ in any order, we describe a condition for $C$ such that the $(n/2)$-th power $C^{n/2}$ of $C$ becomes the longest element, in terms of the Amida diagrams.
</p>projecteuclid.org/euclid.tjm/1491465734_20170406040226Thu, 06 Apr 2017 04:02 EDTDouble Kostka polynomials and Hall bimodulehttp://projecteuclid.org/euclid.tjm/1475723088<strong>Shiyuan LIU</strong>, <strong>Toshiaki SHOJI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 3, 743--776.</p><p><strong>Abstract:</strong><br/>
Double Kostka polynomials $K_{\boldsymbol{\lambda},\boldsymbol{\mu}}(t)$ are polynomials in $t$, indexed by double partitions $\boldsymbol{\lambda}, \boldsymbol{\mu}$.
As in the ordinary case, $K_{\boldsymbol{\lambda}, \boldsymbol{\mu}}(t)$ is defined in terms of Schur functions $s_{\boldsymbol{\lambda}}(x)$ and Hall--Littlewood functions $P_{\boldsymbol{\mu}}(x;t)$.
In this paper, we study combinatorial properties of $K_{\boldsymbol{\lambda},\boldsymbol{\mu}}(t)$ and $P_{\boldsymbol{\mu}}(x;t)$.
In particular, we show that the Lascoux--Sch\"utzenberger type formula holds for $K_{\boldsymbol{\lambda},\boldsymbol{\mu}}(t)$ in the case where $\boldsymbol{\mu} = (-,\mu'')$.
Moreover, we show that the Hall bimodule $\mathscr{M}$ introduced by Finkelberg-Ginzburg-Travkin is isomorphic to the ring of symmetric functions (with two types of variables) and the natural basis $\mathfrak{u}_{\boldsymbol{\lambda}}$ of $\mathscr{M}$ is sent to $P_{\boldsymbol{\lambda}}(x;t)$ (up to scalar) under this isomorphism.
This gives an alternate approach for their result.
</p>projecteuclid.org/euclid.tjm/1475723088_20170406040226Thu, 06 Apr 2017 04:02 EDTThe Direct Image Sheaf $f_*(O_X)$http://projecteuclid.org/euclid.tjm/1475723086<strong>Kentaro MITSUI</strong>, <strong>Iku NAKAMURA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 3, 777--782.</p><p><strong>Abstract:</strong><br/>
We prove $f_*(O_X)=O_S$ for a proper flat surjective morphism $f:X\to S$ of noetherian schemes under a mild condition.
</p>projecteuclid.org/euclid.tjm/1475723086_20170406040226Thu, 06 Apr 2017 04:02 EDTOn Mono-nodal Trees and Genus One Dessins of Pakovich-Zapponi Typehttp://projecteuclid.org/euclid.tjm/1475723085<strong>Hiroaki NAKAMURA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 3, 783--795.</p><p><strong>Abstract:</strong><br/>
In this paper, we classify Grothendieck dessins of X-shaped plane trees defined over the rationals.
</p>projecteuclid.org/euclid.tjm/1475723085_20170406040226Thu, 06 Apr 2017 04:02 EDTLogarithmic Solutions of the Fifth Painlevé Equation near the Originhttp://projecteuclid.org/euclid.tjm/1475723087<strong>Shun SHIMOMURA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 3, 797--825.</p><p><strong>Abstract:</strong><br/>
For the fifth Painlevé equation near the origin we present two kinds of logarithmic solutions, which are represented, respectively, by convergent series with multipliers admitting asymptotic expansions in descending logarithmic powers and by those with multipliers polynomial in logarithmic powers.
It is conjectured that the asymptotic multipliers are also polynomials in logarithmic powers.
These solutions are constructed by iteration on certain rings of exponential type series.
</p>projecteuclid.org/euclid.tjm/1475723087_20170406040226Thu, 06 Apr 2017 04:02 EDTKummer Theories for Algebraic Tori and Normal Basis Problemhttp://projecteuclid.org/euclid.tjm/1475723092<strong>Noriyuki SUWA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 3, 827--862.</p><p><strong>Abstract:</strong><br/>
We discuss the inverse Galois problem with normal basis, concerning Kummer theories for algebraic tori, in the framework of group schemes.
The unit group scheme of a group algebra plays an important role in this article, as was pointed out by Serre~[8].
We develop our argument not only over a field but also over a ring, considering integral models of Kummer theories for algebraic tori.
</p>projecteuclid.org/euclid.tjm/1475723092_20170406040226Thu, 06 Apr 2017 04:02 EDTGeneralized Poincaré Condition and Convergence of Formal Solutions of Some Nonlinear Totally Characteristic Equationshttp://projecteuclid.org/euclid.tjm/1475723091<strong>Hidetoshi TAHARA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 3, 863--883.</p><p><strong>Abstract:</strong><br/>
This paper discusses a holomorphic nonlinear singular partial differential equation $(t \partial_t)^mu=F(t,x,\{(t \partial_t)^j \partial_x^{\alpha}u \}_{j+\alpha \leq m, j<m})$ that is of nonlinear totally characteristic type.
The Newton Polygon at $x=0$ of the equation is defined, and by means of this polygon we define a generalized Poincaré condition (GP) and a condition (R) that the equation has a regular singularity at $x=0$.
Under these conditions, (GP) and (R), it is proved that every formal power series solution is convergent in a neighborhood of the origin.
</p>projecteuclid.org/euclid.tjm/1475723091_20170406040226Thu, 06 Apr 2017 04:02 EDTInvariance of the Drinfeld Pairing of a Quantum Grouphttp://projecteuclid.org/euclid.tjm/1475723090<strong>Toshiyuki TANISAKI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 3, 885--900.</p><p><strong>Abstract:</strong><br/>
We give two alternative proofs of the invariance of the Drinfeld pairing under the action of the braid group.
One uses the Shapovalov form, and the other uses a characterization of the universal $R$-matrix.
</p>projecteuclid.org/euclid.tjm/1475723090_20170406040226Thu, 06 Apr 2017 04:02 EDTToward Noether's Problem for the Fields of Cross-ratioshttp://projecteuclid.org/euclid.tjm/1491465735<strong>Hiroshi TSUNOGAI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 3, 901--922.</p><p><strong>Abstract:</strong><br/>
In this article, we consider an analogue of Noether's problem for the fields of cross-ratios, and discuss on a rationality problem which connects this with Noether's problem.
We show that the affirmative answer of the analogue implies the affirmative answer for Noether's Problem for any permutation group with odd degree.
We also obtain some negative results for various permutation groups with even degree.
</p>projecteuclid.org/euclid.tjm/1491465735_20170406040226Thu, 06 Apr 2017 04:02 EDTOrbital Integrals on Unitary Hyperbolic Spaces Over $\frak p$-adic Fieldshttp://projecteuclid.org/euclid.tjm/1491465736<strong>Masao TSUZUKI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 39, Number 3, 923--975.</p><p><strong>Abstract:</strong><br/>
For a given étale quadratic algebra $E$ over a $\mathfrak{p}$-adic field $F$, we establish a transfer of unramified test functions on the symmetric space $\mathrm{GL}(2,F)\backslash\mathrm{GL}(2,E)$ to those on a unitary hyperbolic space so that the orbital integrals match.
This is an important step toward a comparison of relative trace formulas of these symmetric spaces, which would yield an example of a non-tempered analogue of a refined global Gross-Prasad conjecture.
</p>projecteuclid.org/euclid.tjm/1491465736_20170406040226Thu, 06 Apr 2017 04:02 EDTA Geometric Model of Mixing Lyapunov Exponents Inside Homoclinic Classes in Dimension Threehttp://projecteuclid.org/euclid.tjm/1502179213<strong>Xiaolong LI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 40, Number 1, 1--43.</p><p><strong>Abstract:</strong><br/>
For $C^1$ diffeomorphisms of three dimensional closed manifolds, we provide a geometric model of mixing Lyapunov exponents inside a homoclinic class of a periodic saddle $p$ with non-real eigenvalues.
Suppose $p$ has stable index two and the sum of the largest two Lyapunov exponents is greater than $\log(1-\delta)$, then $\delta$-weak contracting eigenvalues are obtained by an arbitrarily small $C^1$ perturbation.
Using this result, we give a sufficient condition for stabilizing a homoclinic tangency within a given $C^1$ perturbation range.
</p>projecteuclid.org/euclid.tjm/1502179213_20170808040030Tue, 08 Aug 2017 04:00 EDTThe Doeblin-Lenstra Conjecture for a Complex Continued Fraction Algorithmhttp://projecteuclid.org/euclid.tjm/1502179214<strong>Nicola M. R. OSWALD</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 40, Number 1, 45--52.</p><p><strong>Abstract:</strong><br/>
We prove the analogue of the Doeblin-Lenstra conjecture for a certain complex continued fraction algorithm introduced by Julius Hurwitz [5] in 1895 and further investigated by Shigeru Tanaka [12] in 1985 using ergodic theoretical means.
</p>projecteuclid.org/euclid.tjm/1502179214_20170808040030Tue, 08 Aug 2017 04:00 EDTEquivariant Bauer-Furuta invariants on Some Connected Sums of 4-manifoldshttp://projecteuclid.org/euclid.tjm/1502179215<strong>Chanyoung SUNG</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 40, Number 1, 53--63.</p><p><strong>Abstract:</strong><br/>
On some connected sums of 4-manifolds with natural actions of finite groups, we use equivariant Bauer-Furuta invariant to deduce the existence of solutions of Seiberg-Witten equations invariant under the group actions.
For example, for any integer $k\geq 2$ we show that the connected sum of $k$ copies of a 4-manifold $M$ with nontrivial Bauer-Furuta invariant has a nontrivial $\mathbb{Z}_k$-equivariant Bauer-Furuta invariant for the obviously glued Spin$^c$ structure, where the $\mathbb{Z}_k$-action cyclically permutes $k$ summands of $M$.
This contrasts with the fact that ordinary Bauer-Furuta invariants of such connected sums are all trivial for any sufficiently large $k$, when $b_1(M)=0$.
</p>projecteuclid.org/euclid.tjm/1502179215_20170808040030Tue, 08 Aug 2017 04:00 EDTOne-parameter Families of Homeomorphisms, Topological Monodromies, and Foliationshttp://projecteuclid.org/euclid.tjm/1502179216<strong>Kenjiro SASAKI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 40, Number 1, 65--81.</p><p><strong>Abstract:</strong><br/>
The homological monodromy of a degeneration whose singular fiber has at most normal crossings was described by C. H. Clemens.
In his work, local monodromies were described in detail.
It is actually a classical result that the local monodromy around a node is a Dehn twist.
For higher-dimensional case, we describe local monodromies alternatively:
On a local smooth fiber of dimension $n \geq 2$, we construct $n+1$ singular foliations and then describe the action of the local monodromy on each leaf.
Here the $i$th singular foliation is used for describing its action on the $i$th face of the boundary of a local smooth fiber.
</p>projecteuclid.org/euclid.tjm/1502179216_20170808040030Tue, 08 Aug 2017 04:00 EDTOn the Finiteness Properties of Local Cohomology Modules for Regular Local Ringshttp://projecteuclid.org/euclid.tjm/1502179217<strong>Monireh SEDGHI</strong>, <strong>Kamal BAHMANPOUR</strong>, <strong>Reza NAGHIPOUR</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 40, Number 1, 83--96.</p><p><strong>Abstract:</strong><br/>
Let $\frak a$ denote an ideal in a regular local (Noetherian) ring $R$ and let $N$ be a finitely generated $R$-module with support in $V(\frak a)$.
The purpose of this paper is to show that all homomorphic images of the $R$-modules $\mathrm{Ext}^j_R(N, H^i_{\frak a}(R))$ have only finitely many associated primes, for all $i, j\geq 0$, whenever $\dim R \leq4$ or $\dim R/ \frak a \leq 3$ and $R$ contains a field.
In addition, we show that if $\dim R=5$ and $R$ contains a field, then the $R$-modules $\mathrm{Ext}^j_R(N, H^i_{\frak a}(R))$ have only finitely many associated primes, for all $i, j\geq 0$.
</p>projecteuclid.org/euclid.tjm/1502179217_20170808040030Tue, 08 Aug 2017 04:00 EDTCommon Fixed Points for Nonlinear $(\psi ,\varphi)_{s}$-weakly $C$-contractive Mappings in Partially Ordered $b$-metric Spaceshttp://projecteuclid.org/euclid.tjm/1502179218<strong>Jamal REZAEI ROSHAN</strong>, <strong>Mina DINARVAND</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 40, Number 1, 97--121.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce the concept of $(\psi ,\varphi)_{s}$-weakly $C$-contractive mappings in the setup of partially ordered $b$-metric spaces and investigate some fixed point and common fixed point results for such mappings.
Our main results generalize several well-known comparable results in the recent literature.
Furthermore, we furnish some suitable examples and an applications of a common solution for a system of integral equations to illustrate the effectiveness and usability of our obtained results.
</p>projecteuclid.org/euclid.tjm/1502179218_20170808040030Tue, 08 Aug 2017 04:00 EDTOn Zeta Elements and Refined Abelian Stark Conjectureshttp://projecteuclid.org/euclid.tjm/1502179219<strong>Alice LIVINGSTONE BOOMLA</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 40, Number 1, 123--151.</p><p><strong>Abstract:</strong><br/>
We apply recent methods of Burns, Kurihara and Sano in [5] to study connections between the values at \(s=0\) of the higher derivatives of abelian \(L\)-functions of number fields and the higher Fitting ideals of the canonical Selmer groups of \(\mathbb{G}_m\).
Whereas Burns, Kurihara and Sano apply these methods to the setting of the `Rubin-Stark conjecture', we study the `evaluators' defined in a more general setting by Emmons and Popescu in [7] and by Vallieres in [14].
This allows us to conjecture that the ideals formed from the images of the evaluators can be described precisely in terms of the higher Fitting ideals of the canonical Selmer groups of \(\mathbb{G}_m\).
Moreover, we are able to prove that this conjecture follows from the equivariant Tamagawa number conjecture.
</p>projecteuclid.org/euclid.tjm/1502179219_20170808040030Tue, 08 Aug 2017 04:00 EDTA Note on the Shuffle Variant of Jeśmanowicz' Conjecturehttp://projecteuclid.org/euclid.tjm/1502179220<strong>Zsolt RÁBAI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 40, Number 1, 153--163.</p><p><strong>Abstract:</strong><br/>
Let $(a,b,c)$ be a primitive Pythagorean triple.
In 1956, Jeśmanowicz conjectured that the equation $a^{x}+b^{y}=c^{z}$ has the unique solution $(x,y,z)=(2,2,2)$ in positive integers.
In 2010 Miyazaki proposed a similar problem.
He conjectured that if $(a,b,c)$ is again a primitive Pythagorean triple with $b$ even, then the equation $c^{x}+b^{y}=a^{z}$ with $x$, $y$ and $z$ positive integers has the unique solution $(x,y,z)=(1,1,2)$ if $c=b+1$ and no solutions if $c>b+1$.
He also proved that his conjecture is true if $c \equiv 1 \pmod{b}$.
We extend Miyazaki's result to the case $c \equiv 1 \pmod{b/2^{\textrm{ord}_{2}(b)}}$.
</p>projecteuclid.org/euclid.tjm/1502179220_20170808040030Tue, 08 Aug 2017 04:00 EDTHistoric Behaviour for Random Expanding Maps on the Circlehttp://projecteuclid.org/euclid.tjm/1502179221<strong>Yushi NAKANO</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 40, Number 1, 165--184.</p><p><strong>Abstract:</strong><br/>
F. Takens constructed a residual subset of the state space consisting of initial points with historic behaviour for expanding maps on the circle.
We prove that this statistical property of expanding maps on the circle is preserved under small random perturbations.
The proof is given by establishing a random Markov partition, which follows from a random version of Shub's Theorem on topological conjugacy with the folding maps.
</p>projecteuclid.org/euclid.tjm/1502179221_20170808040030Tue, 08 Aug 2017 04:00 EDTOn the Stokes Matrices of the $tt^*$-Toda Equationhttp://projecteuclid.org/euclid.tjm/1502179222<strong>Stefan Andrew HOROCHOLYN</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 40, Number 1, 185--202.</p><p><strong>Abstract:</strong><br/>
We derive a formula for the signature of the symmetrized Stokes matrix $\mathcal{S}+\mathcal{S}^\mathrm{T}$ for the $tt^*$-Toda equation, reminiscent of a formula of Beukers and Heckmann for the generalized hypergeometric equation.
The condition $\mathcal{S}+\mathcal{S}^\mathrm{T} > 0$ is prominent in the work of Cecotti and Vafa on the $tt^*$ equation; using our formula, we show that the Stokes matrices $\mathcal{S}$ satisfying this condition are parameterized by the points of an open convex polytope.
</p>projecteuclid.org/euclid.tjm/1502179222_20170808040030Tue, 08 Aug 2017 04:00 EDTThe Defects of Power Series in the Unit Disk with Hadamard Gapshttp://projecteuclid.org/euclid.tjm/1502179223<strong>Narufumi TSUBOI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 40, Number 1, 203--222.</p><p><strong>Abstract:</strong><br/>
We show a sufficient condition for the defect $\delta (0,f)$ of an analytic function $f(z) = 1 + \sum_{k=1}^{\infty} c_k z^{n_k}$ in the unit disk with Hadamard gaps to vanish.
As a consequence, we find that such $f(z)$ whose characteristic function is sufficiently large has no finite defective value.
</p>projecteuclid.org/euclid.tjm/1502179223_20170808040030Tue, 08 Aug 2017 04:00 EDTTrajectory-harps and Horns Applied to the Study of the Ideal Boundary of a Hadamard Kähler Manifoldhttp://projecteuclid.org/euclid.tjm/1502179224<strong>Qingsong SHI</strong>, <strong>Toshiaki ADACHI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 40, Number 1, 223--236.</p><p><strong>Abstract:</strong><br/>
A trajectory-harps is made of a trajectory for a Kähler magnetic field $\mathbb{B}_{\kappa}$ and an associated variation of geodesics, and a trajectory-horn is made of a geodesic and an associated variation of trajectories.
On a Hadamard Kähler manifold $M$ we study thickness and string-angles of trajectory-harps, and study tube-lengths and tube-angles of trajectory-horns.
As an application of these we show that two distinct points on the compactification of $M$ with geometric ideal boundary can be joined by a trajectory for $\mathbb{B}_{\kappa}$ if the strength $|\kappa|$ is less than the upper bound of sectional curvatures of $M$.
</p>projecteuclid.org/euclid.tjm/1502179224_20170808040030Tue, 08 Aug 2017 04:00 EDTA Remark on the Deformation Equivalence Classes of Hopf Surfaceshttp://projecteuclid.org/euclid.tjm/1502179225<strong>Shota MURAKAMI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 40, Number 1, 237--245.</p><p><strong>Abstract:</strong><br/>
Let $S$ be a compact complex surface.
With the exception of some complex surfaces, it is known that there are only finitely many deformation types of complex surfaces with the same homotopy type as $S$.
Let $\mathcal{H}(S)$ be the set of deformation equivalence classes of complex surfaces homotopy equivalent to $S$.
We evaluate $\# \mathcal{H}(S)$ when $S$ is a Hopf surface.
As a corollary, we construct a sequence of Hopf surfaces $(S_n)$ such that although $\# \mathcal{H}(S_n)$ is finite for all $n$, the sequence $(\# \mathcal{H}(S_n))$ is unbounded.
</p>projecteuclid.org/euclid.tjm/1502179225_20170808040030Tue, 08 Aug 2017 04:00 EDTCharacterization of Non-F-split del Pezzo Surfaces of Degree 2http://projecteuclid.org/euclid.tjm/1502179226<strong>Natsuo SAITO</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 40, Number 1, 247--253.</p><p><strong>Abstract:</strong><br/>
We investigate a smooth del Pezzo surface of degree 2 which is not Frobenius split.
We give a characterization of non-F-split del Pezzo surfaces of degree 2 which exist only if the characteristic of the ground field is 2 or 3.
Moreover, we prove that the set of centers of the blow-ups on $\mathbb{P}^{2}$ which gives a non-F-split del Pezzo surface is projectively equivalent to the only complete 7-arc over $\mathbb{F}_{9}$ if the characteristic is 3.
</p>projecteuclid.org/euclid.tjm/1502179226_20170808040030Tue, 08 Aug 2017 04:00 EDTOn Sums of Averages of Generalized Ramanujan Sumshttp://projecteuclid.org/euclid.tjm/1502179227<strong>Isao KIUCHI</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 40, Number 1, 255--275.</p><p><strong>Abstract:</strong><br/>
We shall consider some formulas for weighted averages of the generalized Ramanujan sum
$\sum_{d|\gcd(k,n)}f(d)g\left({k}/{d}\right)h\left({n}/{d}\right)$
for any arithmetical functions $f,\ g$ and $h$, with the weights concerning completely multiplicative functions, completely additive functions and others.
</p>projecteuclid.org/euclid.tjm/1502179227_20170808040030Tue, 08 Aug 2017 04:00 EDTCertain Forms Violate the Hasse Principlehttp://projecteuclid.org/euclid.tjm/1502179228<strong>Dong Quan Ngoc NGUYEN</strong>. <p><strong>Source: </strong>Tokyo Journal of Mathematics, Volume 40, Number 1, 277--299.</p><p><strong>Abstract:</strong><br/>
A family of smooth geometrically irreducible curves violates the Hasse principle if they have local points everywhere, but they possesses no global points.
In this paper, we show how to construct non-constant algebraic families of forms of degree $4k$ that violate the Hasse principle.
Some examples of non-constant algebraic families of forms of degrees 12 and 24 that violate the Hasse principle are given to illustrate the method.
</p>projecteuclid.org/euclid.tjm/1502179228_20170808040030Tue, 08 Aug 2017 04:00 EDT