Journal of the Mathematical Society of Japan Articles (Project Euclid)
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Toy models for D. H. Lehmer's conjecture
http://projecteuclid.org/euclid.jmsj/1280496816
<strong>Eiichi BANNAI</strong>, <strong>Tsuyoshi MIEZAKI</strong><p><strong>Source: </strong>J. Math. Soc. Japan, Volume 62, Number 3, 687--705.</p><p><strong>Abstract:</strong><br/>
In 1947, Lehmer conjectured that the Ramanujan τ-function τ( m ) never vanishes for all positive integers m , where τ( m ) are the Fourier coefficients of the cusp form Δ 24 of weight 12. Lehmer verified the conjecture in 1947 for m < 214928639999. In 1973, Serre verified up to m < 10 15 , and in 1999, Jordan and Kelly for m < 22689242781695999.
The theory of spherical t -design, and in particular those which are the shells of Euclidean lattices, is closely related to the theory of modular forms, as first shown by Venkov in 1984. In particular, Ramanujan's τ-function gives the coefficients of a weighted theta series of the E 8 -lattice. It is shown, by Venkov, de la Harpe, and Pache, that τ( m ) = 0 is equivalent to the fact that the shell of norm 2 m of the E 8 -lattice is an 8-design. So, Lehmer's conjecture is reformulated in terms of spherical t -design.
Lehmer's conjecture is difficult to prove, and still remains open. In this paper, we consider toy models of Lehmer's conjecture. Namely, we show that the m -th Fourier coefficient of the weighted theta series of the Z 2 -lattice and the A 2 -lattice does not vanish, when the shell of norm m of those lattices is not the empty set. In other words, the spherical 5 (resp. 7)-design does not exist among the shells in the Z 2 -lattice (resp. A 2 -lattice).
</p>projecteuclid.org/euclid.jmsj/1280496816_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTEquiaffine Darboux frames for codimension 2 submanifolds contained in hypersurfaceshttps://projecteuclid.org/euclid.jmsj/1508918560<strong>Marcos CRAIZER</strong>, <strong>Marcelo J. SAIA</strong>, <strong>Luis F. SÁNCHEZ</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 4, 1331--1352.</p><p><strong>Abstract:</strong><br/> Consider a codimension 1 submanifold $N^n\subset M^{n+1}$, where $M^{n+1}\subset \mathbb{R}^{n+2}$ is a hypersurface. The envelope of tangent spaces of $M$ along $N$ generalizes the concept of tangent developable surface of a surface along a curve. In this paper, we study the singularities of these envelopes. There are some important examples of submanifolds that admit a vector field tangent to $M$ and transversal to $N$ whose derivative in any direction of $N$ is contained in $N$. When this is the case, one can construct transversal plane bundles and affine metrics on $N$ with the desirable properties of being equiaffine and apolar. Moreover, this transversal bundle coincides with the classical notion of Transon plane. But we also give an explicit example of a submanifold that does not admit a vector field with the above property. </p>projecteuclid.org/euclid.jmsj/1508918560_20171025040304Wed, 25 Oct 2017 04:03 EDTGlobal dynamics below excited solitons for the nonlinear Schrödinger equation with a potentialhttps://projecteuclid.org/euclid.jmsj/1508918561<strong>Kenji NAKANISHI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 4, 1353--1401.</p><p><strong>Abstract:</strong><br/>
Consider the nonlinear Schrödinger equation (NLS) with a potential with a single negative eigenvalue. It has solitons with negative small energy, which are asymptotically stable, and, if the nonlinearity is focusing, then also solitons with positive large energy, which are unstable. In this paper we classify the global dynamics below the second lowest energy of solitons under small mass and radial symmetry constraints.
</p>projecteuclid.org/euclid.jmsj/1508918561_20171025040304Wed, 25 Oct 2017 04:03 EDTMaximal regularity of the time-periodic Stokes operator on unbounded and bounded domainshttps://projecteuclid.org/euclid.jmsj/1508918562<strong>Yasunori MAEKAWA</strong>, <strong>Jonas SAUER</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 4, 1403--1429.</p><p><strong>Abstract:</strong><br/>
We investigate the time-periodic Stokes equations with non-homogeneous divergence data in the whole space, the half space, bent half spaces and bounded domains. The solutions decompose into a well-studied stationary part and a purely periodic part, for which we establish $\mathrm{L}^{p}$ estimates. For the whole space and the half space case we use a reduction of the Stokes equations to $(n-1)$ heat equations. Perturbation and localisation methods yield the result on bent half spaces and bounded domains. A one-to-one correspondence between maximal regularity for the initial value problem and time periodic maximal regularity is proven, providing a short proof for the maximal regularity of the Stokes operator avoiding the notion of $\mathcal{R}$-boundedness. The results are applied to a quasilinear model governing the flow of nematic liquid crystals.
</p>projecteuclid.org/euclid.jmsj/1508918562_20171025040304Wed, 25 Oct 2017 04:03 EDTAnalytic continuation of multiple Hurwitz zeta functionshttps://projecteuclid.org/euclid.jmsj/1508918563<strong>Jay MEHTA</strong>, <strong>G. K. VISWANADHAM</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 4, 1431--1442.</p><p><strong>Abstract:</strong><br/>
We obtain the analytic continuation of multiple Hurwitz zeta functions by using a simple and elementary translation formula. We also locate the polar hyperplanes for these functions and express the residues, along these hyperplanes, as coefficients of certain infinite matrices.
</p>projecteuclid.org/euclid.jmsj/1508918563_20171025040304Wed, 25 Oct 2017 04:03 EDTJacquet–Langlands–Shimizu correspondence for theta lifts to $GSp(2)$ and its inner forms I: An explicit functorial correspondencehttps://projecteuclid.org/euclid.jmsj/1508918564<strong>Hiro-aki NARITA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 4, 1443--1474.</p><p><strong>Abstract:</strong><br/> As was first essentially pointed out by Tomoyoshi Ibukiyama, Hecke eigenforms on the indefinite symplectic group $GSp(1,1)$ or the definite symplectic group $GSp^*(2)$ over $\mathbb{Q}$ right invariant by a (global) maximal open compact subgroup are conjectured to have the same spinor $L$-functions as those of paramodular new forms of some specified level on the symplectic group $GSp(2)$ (or $GSp(4)$). This can be viewed as a generalization of the Jacquet–Langlands–Shimizu correspondence to the case of $GSp(2)$ and its inner forms $GSp(1,1)$ and $GSp^*(2)$. In this paper we provide evidence of the conjecture on this explicit functorial correspondence with theta lifts: a theta lift from $GL(2)\times B^{\times}$ to $GSp(1,1)$ or $GSp^*(2)$ and a theta lift from $GL(2)\times GL(2)$ (or $GO(2,2)$) to $GSp(2)$. Here $B$ denotes a definite quaternion algebra over $\mathbb{Q}$. Our explicit functorial correspondence given by these theta lifts are proved to be compatible with archimedean and non-archimedean local Jacquet–Langlands correspondences. Regarding the non-archimedean local theory we need some explicit functorial correspondence for spherical representations of the inner form and non-supercuspidal representations of $GSp(2)$, which is studied in the appendix by Ralf Schmidt. </p>projecteuclid.org/euclid.jmsj/1508918564_20171025040304Wed, 25 Oct 2017 04:03 EDTSelf-dual Wulff shapes and spherical convex bodies of constant width ${\pi}/{2}$https://projecteuclid.org/euclid.jmsj/1508918565<strong>Huhe HAN</strong>, <strong>Takashi NISHIMURA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 4, 1475--1484.</p><p><strong>Abstract:</strong><br/>
For any Wulff shape, its dual Wulff shape is naturally defined. A self-dual Wulff shape is a Wulff shape equaling its dual Wulff shape exactly. In this paper, it is shown that a Wulff shape is self-dual if and only if the spherical convex body induced by it is of constant width ${\pi}/{2}$.
</p>projecteuclid.org/euclid.jmsj/1508918565_20171025040304Wed, 25 Oct 2017 04:03 EDTPositive energy representations of double extensions of Hilbert loop algebrashttps://projecteuclid.org/euclid.jmsj/1508918566<strong>Timothée MARQUIS</strong>, <strong>Karl-Hermann NEEB</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 4, 1485--1518.</p><p><strong>Abstract:</strong><br/> A real Lie algebra with a compatible Hilbert space structure (in the sense that the scalar product is invariant) is called a Hilbert–Lie algebra. Such Lie algebras are natural infinite-dimensional analogues of the compact Lie algebras; in particular, any infinite-dimensional simple Hilbert–Lie algebra $\mathfrak{k}$ is of one of the four classical types $A_J$, $B_J$, $C_J$ or $D_J$ for some infinite set $J$. Imitating the construction of affine Kac–Moody algebras, one can then consider affinisations of $\mathfrak{k}$, that is, double extensions of (twisted) loop algebras over $\mathfrak{k}$. Such an affinisation $\mathfrak{g}$ of $\mathfrak{k}$ possesses a root space decomposition with respect to some Cartan subalgebra $\mathfrak{h}$, whose corresponding root system yields one of the seven locally affine root systems (LARS) of type $A_J^{(1)}$, $B^{(1)}_J$, $C^{(1)}_J$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ or $BC_J^{(2)}$. Let $D\in\mathrm{der}(\mathfrak{g})$ with $\mathfrak{h}\subseteq\mathrm{ker}D$ (a diagonal derivation of $\mathfrak{g}$). Then every highest weight representation $(\rho_{\lambda},L(\lambda))$ of $\mathfrak{g}$ with highest weight $\lambda$ can be extended to a representation $\widetilde{\rho}_{\lambda}$ of the semi-direct product $\mathfrak{g}\rtimes \mathbb{R} D$. In this paper, we characterise all pairs $(\lambda,D)$ for which the representation $\widetilde{\rho}_{\lambda}$ is of positive energy, namely, for which the spectrum of the operator $-i\widetilde{\rho}_{\lambda}(D)$ is bounded from below. </p>projecteuclid.org/euclid.jmsj/1508918566_20171025040304Wed, 25 Oct 2017 04:03 EDTStability and bifurcation for surfaces with constant mean curvaturehttps://projecteuclid.org/euclid.jmsj/1508918567<strong>Miyuki KOISO</strong>, <strong>Bennett PALMER</strong>, <strong>Paolo PICCIONE</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 4, 1519--1554.</p><p><strong>Abstract:</strong><br/>
We give criteria for the existence of smooth bifurcation branches of fixed boundary CMC surfaces in $\mathbb R^3$, and we discuss stability/instability issues for the surfaces in bifurcating branches. To illustrate the theory, we discuss an explicit example obtained from a bifurcating branch of fixed boundary unduloids in ${\mathbb R}^3$.
</p>projecteuclid.org/euclid.jmsj/1508918567_20171025040304Wed, 25 Oct 2017 04:03 EDTReducing subspaces of multiplication operators on weighted Hardy spaces over bidiskhttps://projecteuclid.org/euclid.jmsj/1508918568<strong>Shuhei KUWAHARA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 4, 1555--1563.</p><p><strong>Abstract:</strong><br/>
We consider weighted Hardy spaces over bidisk ${\mathbb D}^2$ which generalize the weighted Bergman spaces $A_\alpha^2({\mathbb D}^2)$. Let $z,w$ be coordinate functions and $M_{z^Nw^N}$ the multiplication by $z^Nw^N$ for a natural number $N$. In this paper, we study the reducing subspaces of $M_{z^Nw^N}$. In particular, we obtain the minimal reducing subspaces of $M_{zw}$.
</p>projecteuclid.org/euclid.jmsj/1508918568_20171025040304Wed, 25 Oct 2017 04:03 EDTOn subadditivity of the logarithmic Kodaira dimensionhttps://projecteuclid.org/euclid.jmsj/1508918569<strong>Osamu FUJINO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 4, 1565--1581.</p><p><strong>Abstract:</strong><br/>
We reduce Iitaka's subadditivity conjecture for the logarithmic Kodaira dimension to a special case of the generalized abundance conjecture by establishing an Iitaka type inequality for Nakayama's numerical Kodaira dimension. Our proof heavily depends on Nakayama's theory of $\omega$-sheaves and $\widehat{\omega}$-sheaves. As an application, we prove the subadditivity of the logarithmic Kodaira dimension for affine varieties by using the minimal model program for projective klt pairs with big boundary divisor.
</p>projecteuclid.org/euclid.jmsj/1508918569_20171025040304Wed, 25 Oct 2017 04:03 EDTA transcendental function invariant of virtual knotshttps://projecteuclid.org/euclid.jmsj/1508918570<strong>Zhiyun CHENG</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 4, 1583--1599.</p><p><strong>Abstract:</strong><br/>
In this work we introduce a new invariant of virtual knots. We show that this transcendental function invariant generalizes several polynomial invariants of virtual knots, such as the writhe polynomial [ 3 ], the affine index polynomial [ 19 ] and the zero polynomial [ 14 ]. Several applications of this new invariant are discussed.
</p>projecteuclid.org/euclid.jmsj/1508918570_20171025040304Wed, 25 Oct 2017 04:03 EDTThe Toledo invariant, and Seshadri constants of fake projective planeshttps://projecteuclid.org/euclid.jmsj/1508918571<strong>Luca F. DI CERBO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 4, 1601--1610.</p><p><strong>Abstract:</strong><br/>
The purpose of this paper is to explicitly compute the Seshadri constants of all ample line bundles on fake projective planes. The proof relies on the theory of the Toledo invariant, and more precisely on its characterization of $\mathbb{C}$-Fuchsian curves in complex hyperbolic spaces.
</p>projecteuclid.org/euclid.jmsj/1508918571_20171025040304Wed, 25 Oct 2017 04:03 EDTHecke pairs of ergodic discrete measured equivalence relations and the Schlichting completionhttps://projecteuclid.org/euclid.jmsj/1508918572<strong>Hisashi AOI</strong>, <strong>Takehiko YAMANOUCHI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 4, 1611--1665.</p><p><strong>Abstract:</strong><br/>
It is shown that for each Hecke pair of ergodic discrete measured equivalence relations, there exists a Hecke pair of groups determined by an index cocycle associated with the given pair. We clarify that the construction of these groups can be viewed as a generalization of a notion of Schlichting completion for a Hecke pair of groups, and show that the index cocycle cited above arises from “adjusted” choice functions for the equivalence relations. We prove also that there exists a special kind of choice functions, preferable choice functions , having the property that the restriction of the corresponding index cocycle to the ergodic subrelation is minimal in the sense of Zimmer. It is then proved that the Hecke von Neumann algebra associated with the Hecke pair of groups obtained above is $\ast$-isomorphic to the Hecke von Neumann algebra associated with the Hecke pair of equivalence relations with which we start.
</p>projecteuclid.org/euclid.jmsj/1508918572_20171025040304Wed, 25 Oct 2017 04:03 EDTSemiclassical Sobolev constants for the electro-magnetic Robin Laplacianhttps://projecteuclid.org/euclid.jmsj/1508918573<strong>Søren FOURNAIS</strong>, <strong>Loïc LE TREUST</strong>, <strong>Nicolas RAYMOND</strong>, <strong>Jean VAN SCHAFTINGEN</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 4, 1667--1714.</p><p><strong>Abstract:</strong><br/>
This paper is devoted to the asymptotic analysis of the optimal Sobolev constants in the semiclassical limit and in any dimension. We combine semiclassical arguments and concentration-compactness estimates to tackle the case when an electro-magnetic field is added as well as a smooth boundary carrying a Robin condition. As a byproduct of the semiclassical strategy, we also get exponentially weighted localization estimates of the minimizers.
</p>projecteuclid.org/euclid.jmsj/1508918573_20171025040304Wed, 25 Oct 2017 04:03 EDTFoundation of symbol theory for analytic pseudodifferential operators, Ihttps://projecteuclid.org/euclid.jmsj/1508918574<strong>Takashi AOKI</strong>, <strong>Naofumi HONDA</strong>, <strong>Susumu YAMAZAKI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 4, 1715--1801.</p><p><strong>Abstract:</strong><br/>
A new symbol theory for pseudodifferential operators in the complex analytic category is given. Here the pseudodifferential operators mean integral operators with real holomorphic microfunction kernels. The notion of real holomorphic microfunctions had been introduced by Sato, Kawai and Kashiwara by using sheaf cohomology theory. Symbol theory for those operators was partly developed by Kataoka and by the first author and it has been effectively used in the analysis of operators of infinite order. However, there was a missing part that links the symbol theory and the cohomological definition of operators, that is, the consistency of the Leibniz–Hörmander rule and the cohomological definition of composition for operators. This link has not been established completely in the existing symbol theory. This paper supplies the link and provides a cohomological foundation of the symbolic calculus of pseudodifferential operators.
</p>projecteuclid.org/euclid.jmsj/1508918574_20171025040304Wed, 25 Oct 2017 04:03 EDTMagidor cardinalshttps://projecteuclid.org/euclid.jmsj/1516957217<strong>Shimon GARTI</strong>, <strong>Yair HAYUT</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 1, 1--23.</p><p><strong>Abstract:</strong><br/>
We define Magidor cardinals as Jónsson cardinals upon replacing colorings of finite subsets by colorings of $\aleph_0$-bounded subsets. Unlike Jónsson cardinals which appear at some low level of large cardinals, we prove the consistency of having quite large cardinals along with the fact that no Magidor cardinal exists.
</p>projecteuclid.org/euclid.jmsj/1516957217_20180126040034Fri, 26 Jan 2018 04:00 ESTHomogeneous affine surfaces: affine Killing vector fields and gradient Ricci solitonshttps://projecteuclid.org/euclid.jmsj/1516957218<strong>Miguel BROZOS-VÁZQUEZ</strong>, <strong>Eduardo GARCÍA-RÍO</strong>, <strong>Peter B. GILKEY</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 1, 25--70.</p><p><strong>Abstract:</strong><br/>
The homogeneous affine surfaces have been classified by Opozda. They may be grouped into 3 families, which are not disjoint. The connections which arise as the Levi-Civita connection of a surface with a metric of constant Gauss curvature form one family; there are, however, two other families. For a surface in one of these other two families, we examine the Lie algebra of affine Killing vector fields and we give a complete classification of the homogeneous affine gradient Ricci solitons. The rank of the Ricci tensor plays a central role in our analysis.
</p>projecteuclid.org/euclid.jmsj/1516957218_20180126040034Fri, 26 Jan 2018 04:00 ESTOn two-weight norm estimates for multilinear fractional maximal functionhttps://projecteuclid.org/euclid.jmsj/1516957219<strong>Benoît Florent SEHBA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 1, 71--94.</p><p><strong>Abstract:</strong><br/>
We prove some Sawyer-type characterizations for multilinear fractional maximal function for the upper triangle case. We also provide some two-weight norm estimates for this operator. As one of the main tools, we use an extension of the usual Carleson Embedding that is an analogue of the P. L. Duren extension of the Carleson Embedding for measures.
</p>projecteuclid.org/euclid.jmsj/1516957219_20180126040034Fri, 26 Jan 2018 04:00 ESTMontesinos knots, Hopf plumbings, and L-space surgerieshttps://projecteuclid.org/euclid.jmsj/1516957220<strong>Kenneth L. BAKER</strong>, <strong>Allison H. MOORE</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 1, 95--110.</p><p><strong>Abstract:</strong><br/>
Using Hirasawa–Murasugi's classification of fibered Montesinos knots we classify the L-space Montesinos knots, providing further evidence towards a conjecture of Lidman–Moore that L-space knots have no essential Conway spheres. In the process, we classify the fibered Montesinos knots whose open books support the tight contact structure on $S^3$. We also construct L-space knots with arbitrarily large tunnel number and discuss the question of whether L-space knots admit essential tangle decompositions in the context of satellite operations and tunnel number.
</p>projecteuclid.org/euclid.jmsj/1516957220_20180126040034Fri, 26 Jan 2018 04:00 ESTLaplace hyperfunctions in several variableshttps://projecteuclid.org/euclid.jmsj/1516957221<strong>Naofumi HONDA</strong>, <strong>Kohei UMETA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 1, 111--139.</p><p><strong>Abstract:</strong><br/>
We establish an edge of the wedge theorem for the sheaf of holomorphic functions with exponential growth at infinity and construct the sheaf of Laplace hyperfunctions in several variables. We also study the fundamental properties of the sheaf of Laplace hyperfunctions.
</p>projecteuclid.org/euclid.jmsj/1516957221_20180126040034Fri, 26 Jan 2018 04:00 ESTA Hardy inequality and applications to reverse Hölder inequalities for weights on $\mathbb{R}$https://projecteuclid.org/euclid.jmsj/1516957222<strong>Eleftherios N. NIKOLIDAKIS</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 1, 141--152.</p><p><strong>Abstract:</strong><br/>
We prove a sharp integral inequality valid for non-negative functions defined on $[0,1]$, with given $L^1$ norm. This is in fact a generalization of the well known integral Hardy inequality. We prove it as a consequence of the respective weighted discrete analogue inequality whose proof is presented in this paper. As an application we find the exact best possible range of $p>q$ such that any non-increasing $g$ which satisfies a reverse Hölder inequality with exponent $q$ and constant $c$ upon the subintervals of $(0,1]$, should additionally satisfy a reverse Hölder inequality with exponent $p$ and in general a different constant $c'$. The result has been treated in [ 1 ] but here we give an alternative proof based on the above mentioned inequality.
</p>projecteuclid.org/euclid.jmsj/1516957222_20180126040034Fri, 26 Jan 2018 04:00 ESTAnalyticity of the Stokes semigroup in $BMO$-type spaceshttps://projecteuclid.org/euclid.jmsj/1516957223<strong>Martin BOLKART</strong>, <strong>Yoshikazu GIGA</strong>, <strong>Takuya SUZUKI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 1, 153--177.</p><p><strong>Abstract:</strong><br/>
We consider the Stokes semigroup in a large class of domains including bounded domains, the half-space and exterior domains. We will prove that the Stokes semigroup is analytic in a certain type of solenoidal subspaces of $BMO$.
</p>projecteuclid.org/euclid.jmsj/1516957223_20180126040034Fri, 26 Jan 2018 04:00 ESTOn Arakawa–Kaneko zeta-functions associated with $GL_2(\mathbb{C})$ and their functional relationshttps://projecteuclid.org/euclid.jmsj/1516957224<strong>Yasushi KOMORI</strong>, <strong>Hirofumi TSUMURA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 1, 179--213.</p><p><strong>Abstract:</strong><br/>
We construct a certain class of Arakawa–Kaneko zeta-functions associated with $GL_2(\mathbb{C})$, which includes the ordinary Arakawa–Kaneko zeta-function. We also define poly-Bernoulli polynomials associated with $GL_2(\mathbb{C})$ which appear in their special values of these zeta-functions. We prove some functional relations for these zeta-functions, which are regarded as interpolation formulas of various relations among poly-Bernoulli numbers. Considering their special values, we prove difference relations and duality relations for poly-Bernoulli polynomials associated with $GL_2(\mathbb{C})$.
</p>projecteuclid.org/euclid.jmsj/1516957224_20180126040034Fri, 26 Jan 2018 04:00 ESTMultiplicativity of the $\mathcal{I}$-invariant and topology of glued arrangementshttps://projecteuclid.org/euclid.jmsj/1516957225<strong>Benoît GUERVILLE-BALLÉ</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 1, 215--227.</p><p><strong>Abstract:</strong><br/>
The invariant $\mathcal{I}(\mathcal{A},\xi,\gamma)$ was first introduced by E. Artal, V. Florens and the author. Inspired by the idea of G. Rybnikov, we obtain a multiplicativity theorem of this invariant under the gluing of two arrangements along a triangle. An application of this theorem is to prove that the extended Rybnikov arrangements form an ordered Zariski pair ( i.e. two arrangements with the same combinatorial information and different ordered topologies). Finally, we extend this method to a family of arrangements and thus we obtain a method to construct new examples of Zariski pairs.
</p>projecteuclid.org/euclid.jmsj/1516957225_20180126040034Fri, 26 Jan 2018 04:00 ESTA product formula for log Gromov–Witten invariantshttps://projecteuclid.org/euclid.jmsj/1516957226<strong>Yuan-Pin LEE</strong>, <strong>Feng QU</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 1, 229--242.</p><p><strong>Abstract:</strong><br/>
The purpose of this short article is to prove a product formula relating the log Gromov–Witten invariants of $V \times W$ with those of $V$ and $W$ in the case the log structure on $V$ is trivial.
</p>projecteuclid.org/euclid.jmsj/1516957226_20180126040034Fri, 26 Jan 2018 04:00 ESTTime periodic problem for the compressible Navier–Stokes equation on $\mathbb{R}^2$ with antisymmetryhttps://projecteuclid.org/euclid.jmsj/1516957227<strong>Kazuyuki TSUDA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 1, 243--281.</p><p><strong>Abstract:</strong><br/>
The compressible Navier–Stokes equation is considered on the two dimensional whole space when the external force is periodic in the time variable. The existence of a time periodic solution is proved for sufficiently small time periodic external force with antisymmetry condition. The proof is based on using the time-$T$-map associated with the linearized problem around the motionless state with constant density. In some weighted $L^\infty$ and Sobolev spaces the spectral properties of the time-$T$-map are investigated by a potential theoretic method and an energy method. The existence of a stationary solution to the stationary problem is also shown for sufficiently small time-independent external force with antisymmetry condition on $\mathbb{R}^2$.
</p>projecteuclid.org/euclid.jmsj/1516957227_20180126040034Fri, 26 Jan 2018 04:00 ESTFermionic formula for double Kostka polynomialshttps://projecteuclid.org/euclid.jmsj/1516957228<strong>Shiyuan LIU</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 1, 283--324.</p><p><strong>Abstract:</strong><br/>
The $X=M$ conjecture asserts that the $1D$ sum and the fermionic formula coincide up to some constant power. In the case of type $A,$ both the $1D$ sum and the fermionic formula are closely related to Kostka polynomials. Double Kostka polynomials $K_{{\boldsymbol\lambda},{\boldsymbol\mu}}(t),$ indexed by two double partitions ${\boldsymbol\lambda},{\boldsymbol\mu},$ are polynomials in $t$ introduced as a generalization of Kostka polynomials. In the present paper, we consider $K_{{\boldsymbol\lambda},{\boldsymbol\mu}}(t)$ in the special case where ${\boldsymbol\mu}=(-,\mu'')$. We formulate a $1D$ sum and a fermionic formula for $K_{{\boldsymbol\lambda},{\boldsymbol\mu}}(t),$ as a generalization of the case of ordinary Kostka polynomials. Then we prove an analogue of the $X=M$ conjecture.
</p>projecteuclid.org/euclid.jmsj/1516957228_20180126040034Fri, 26 Jan 2018 04:00 ESTBasic relative invariants of homogeneous cones and their Laplace transformshttps://projecteuclid.org/euclid.jmsj/1516957229<strong>Hideto NAKASHIMA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 1, 325--344.</p><p><strong>Abstract:</strong><br/>
The purpose of this paper is to show that it is characteristic of symmetric cones among irreducible homogeneous cones that there exists a non-constant relatively invariant polynomial such that its Laplace transform is the reciprocal of a certain polynomial. To prove our theorem, we need the inductive structure of the basic relative invariants of a homogeneous cone. However, we actually work in a more general setting, and consider the inducing of the basic relative invariants from lower rank cones.
</p>projecteuclid.org/euclid.jmsj/1516957229_20180126040034Fri, 26 Jan 2018 04:00 ESTThe selection problem for discounted Hamilton–Jacobi equations: some non-convex caseshttps://projecteuclid.org/euclid.jmsj/1516957230<strong>Diogo A. GOMES</strong>, <strong>Hiroyoshi MITAKE</strong>, <strong>Hung V. TRAN</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 1, 345--364.</p><p><strong>Abstract:</strong><br/>
Here, we study the selection problem for the vanishing discount approximation of non-convex, first-order Hamilton–Jacobi equations. While the selection problem is well understood for convex Hamiltonians, the selection problem for non-convex Hamiltonians has thus far not been studied. We begin our study by examining a generalized discounted Hamilton–Jacobi equation. Next, using an exponential transformation, we apply our methods to strictly quasi-convex and to some non-convex Hamilton–Jacobi equations. Finally, we examine a non-convex Hamiltonian with flat parts to which our results do not directly apply. In this case, we establish the convergence by a direct approach.
</p>projecteuclid.org/euclid.jmsj/1516957230_20180126040034Fri, 26 Jan 2018 04:00 ESTOn the sequential polynomial type of moduleshttps://projecteuclid.org/euclid.jmsj/1516957231<strong>Shiro GOTO</strong>, <strong>Le Thanh NHAN</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 1, 365--385.</p><p><strong>Abstract:</strong><br/>
Let $M$ be a finitely generated module over a Noetherian local ring $R$. The sequential polynomial type $\mathrm{sp}(M)$ of $M$ was recently introduced by Nhan, Dung and Chau, which measures how far the module $M$ is from the class of sequentially Cohen–Macaulay modules. The present paper purposes to give a parametric characterization for $M$ to have $\mathrm{sp}(M)\le s$, where $s\ge -1$ is an integer. We also study the sequential polynomial type of certain specific rings and modules. As an application, we give an inequality between $\mathrm{sp}(S)$ and $\mathrm{sp}(S^G) $, where $S$ is a Noetherian local ring and $G$ is a finite subgroup of $\mathrm{Aut}S$ such that the order of $G$ is invertible in $S$.
</p>projecteuclid.org/euclid.jmsj/1516957231_20180126040034Fri, 26 Jan 2018 04:00 ESTTopology of mixed hypersurfaces of cyclic typehttps://projecteuclid.org/euclid.jmsj/1516957232<strong>Kazumasa INABA</strong>, <strong>Masayuki KAWASHIMA</strong>, <strong>Mutsuo OKA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 1, 387--402.</p><p><strong>Abstract:</strong><br/>
Let $f_{II}({\boldsymbol{z}}, \bar{{\boldsymbol{z}}}) = z_{1}^{a_{1}+b_{1}}\bar{z}_{1}^{b_{1}}z_{2} + \cdots + z_{n-1}^{a_{n-1}+b_{n-1}}\bar{z}_{n-1}^{b_{n-1}}z_{n} + z_{n}^{a_{n}+b_{n}}\bar{z}_{n}^{b_{n}}z_{1}$ be a mixed weighted homogeneous polynomial of cyclic type and $g_{II}({\boldsymbol{z}}) = z_{1}^{a_{1}}z_{2} + \cdots + z_{n-1}^{a_{n-1}}z_{n} + z_{n}^{a_{n}}z_{1}$ be the associated weighted homogeneous polynomial where $a_{j} \geq 1$ and $b_{j} \geq 0$ for $j = 1, \dots, n$. We show that two links $S^{2n-1}_{\varepsilon} \cap f_{II}^{-1}(0)$ and $S^{2n-1}_{\varepsilon} \cap g_{II}^{-1}(0)$ are diffeomorphic and their Milnor fibrations are isomorphic.
</p>projecteuclid.org/euclid.jmsj/1516957232_20180126040034Fri, 26 Jan 2018 04:00 ESTA characterization of regular points by Ohsawa–Takegoshi extension theoremhttps://projecteuclid.org/euclid.jmsj/1516957233<strong>Qi'an GUAN</strong>, <strong>Zhenqian LI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 1, 403--408.</p><p><strong>Abstract:</strong><br/>
In this article, we present that the germ of a complex analytic set at the origin in $\mathbb{C}^n$ is regular if and only if the related Ohsawa–Takegoshi extension theorem holds. We also obtain a necessary condition of the $L^2$ extension of bounded holomorphic sections from singular analytic sets.
</p>projecteuclid.org/euclid.jmsj/1516957233_20180126040034Fri, 26 Jan 2018 04:00 ESTWeighted Bott–Chern and Dolbeault cohomology for LCK-manifolds with potentialhttps://projecteuclid.org/euclid.jmsj/1516957234<strong>Liviu ORNEA</strong>, <strong>Misha VERBITSKY</strong>, <strong>Victor VULETESCU</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 1, 409--422.</p><p><strong>Abstract:</strong><br/>
A locally conformally Kähler (LCK) manifold is a complex manifold, with a Kähler structure on its universal covering $\tilde M$, with the deck transform group acting on $\tilde M$ by holomorphic homotheties. One could think of an LCK manifold as of a complex manifold with a Kähler form taking values in a local system $L$, called the conformal weight bundle . The $L$-valued cohomology of $M$ is called Morse–Novikov cohomology ; it was conjectured that (just as it happens for Kähler manifolds) the Morse–Novikov complex satisfies the $dd^c$-lemma, which (if true) would have far-reaching consequences for the geometry of LCK manifolds. In particular, this version of $dd^c$-lemma would imply existence of LCK potential on any LCK manifold with vanishing Morse–Novikov class of its $L$-valued Hermitian symplectic form. The $dd^c$-conjecture was disproved for Vaisman manifolds by Goto. We prove that the $dd^c$-lemma is true with coefficients in a sufficiently general power of $L$ on any Vaisman manifold or LCK manifold with potential.
</p>projecteuclid.org/euclid.jmsj/1516957234_20180126040034Fri, 26 Jan 2018 04:00 ESTDynamics and the Godbillon–Vey class of $C^1$ foliationshttps://projecteuclid.org/euclid.jmsj/1524038663<strong>Steven HURDER</strong>, <strong>Rémi LANGEVIN</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 2, 423--462.</p><p><strong>Abstract:</strong><br/>
Let $\mathcal{F}$ be a codimension-one, $C^2$-foliation on a manifold $M$ without boundary. In this work we show that if the Godbillon–Vey class $GV(\mathcal{F}) \in H^3(M)$ is non-zero, then $\mathcal{F}$ has a hyperbolic resilient leaf. Our approach is based on methods of $C^1$-dynamical systems, and does not use the classification theory of $C^2$-foliations. We first prove that for a codimension-one $C^1$-foliation with non-trivial Godbillon measure, the set of infinitesimally expanding points $E(\mathcal{F})$ has positive Lebesgue measure. We then prove that if $E(\mathcal{F})$ has positive measure for a $C^1$-foliation, then $\mathcal{F}$ must have a hyperbolic resilient leaf, and hence its geometric entropy must be positive. The proof of this uses a pseudogroup version of the Pliss Lemma. The first statement then follows, as a $C^2$-foliation with non-zero Godbillon–Vey class has non-trivial Godbillon measure. These results apply for both the case when $M$ is compact, and when $M$ is an open manifold.
</p>projecteuclid.org/euclid.jmsj/1524038663_20180418040438Wed, 18 Apr 2018 04:04 EDTSpacelike Dupin hypersurfaces in Lorentzian space formshttps://projecteuclid.org/euclid.jmsj/1524038664<strong>Tongzhu LI</strong>, <strong>Changxiong NIE</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 2, 463--480.</p><p><strong>Abstract:</strong><br/>
Similar to the definition in Riemannian space forms, we define the spacelike Dupin hypersurface in Lorentzian space forms. As conformal invariant objects, spacelike Dupin hypersurfaces are studied in this paper using the framework of the conformal geometry of spacelike hypersurfaces. Further we classify the spacelike Dupin hypersurfaces with constant Möbius curvatures, which are also called conformal isoparametric hypersurface.
</p>projecteuclid.org/euclid.jmsj/1524038664_20180418040438Wed, 18 Apr 2018 04:04 EDTOn the Galois structure of arithmetic cohomology II: ray class groupshttps://projecteuclid.org/euclid.jmsj/1524038665<strong>David BURNS</strong>, <strong>Asuka KUMON</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 2, 481--517.</p><p><strong>Abstract:</strong><br/>
We investigate the explicit Galois structure of ray class groups. We then derive consequences of our results concerning both the validity of Leopoldt’s Conjecture and the existence of families of explicit congruence relations between the values of Dirichlet $L$-series at $s=1$.
</p>projecteuclid.org/euclid.jmsj/1524038665_20180418040438Wed, 18 Apr 2018 04:04 EDTHomogenisation on homogeneous spaceshttps://projecteuclid.org/euclid.jmsj/1524038666<strong>Xue-Mei LI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 2, 519--572.</p><p><strong>Abstract:</strong><br/>
Motivated by collapsing of Riemannian manifolds and inhomogeneous scaling of left invariant Riemannian metrics on a real Lie group $G$ with a sub-group $H$, we introduce a family of interpolation equations on $G$ with a parameter $\epsilon>0$, interpolating hypo-elliptic diffusions on $H$ and translates of exponential maps on $G$ and examine the dynamics as $\epsilon\to 0$. When $H$ is compact, we use the reductive homogeneous structure of Nomizu to extract a converging family of stochastic processes (converging on the time scale $1/\epsilon$), proving the convergence of the stochastic dynamics on the orbit spaces $G/H$ and their parallel translations, providing also an estimate on the rate of the convergence in the Wasserstein distance. Their limits are not necessarily Brownian motions and are classified algebraically by a Peter–Weyl’s theorem for real Lie groups and geometrically using a weak notion of the naturally reductive property; the classifications allow to conclude the Markov property of the limit process. This can be considered as “taking the adiabatic limit” of the differential operators $\mathcal{L}^\epsilon=(1/\epsilon) \sum_k (A_k)^2+(1/\epsilon) A_0+Y_0$ where $Y_0, A_k$ are left invariant vector fields and $\{A_k\}$ generate the Lie-algebra of $H$.
</p>projecteuclid.org/euclid.jmsj/1524038666_20180418040438Wed, 18 Apr 2018 04:04 EDTBirational maps preserving the contact structure on $\mathbb{P}^3_\mathbb{C}$https://projecteuclid.org/euclid.jmsj/1524038667<strong>Dominique CERVEAU</strong>, <strong>Julie DÉSERTI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 2, 573--615.</p><p><strong>Abstract:</strong><br/>
We study the group of polynomial automorphisms of $\mathbb{C}^3$ (resp. birational self-maps of $\mathbb{P}^3_\mathbb{C}$) that preserve the contact structure.
</p>projecteuclid.org/euclid.jmsj/1524038667_20180418040438Wed, 18 Apr 2018 04:04 EDTCurvilinear coordinates on generic conformally flat hypersurfaces and constant curvature 2-metricshttps://projecteuclid.org/euclid.jmsj/1524038668<strong>Francis E. BURSTALL</strong>, <strong>Udo HERTRICH-JEROMIN</strong>, <strong>Yoshihiko SUYAMA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 2, 617--649.</p><p><strong>Abstract:</strong><br/>
There is a one-to-one correspondence between associated families of generic conformally flat (local-)hypersurfaces in 4-dimensional space forms and conformally flat 3-metrics with the Guichard condition. In this paper, we study the space of conformally flat 3-metrics with the Guichard condition: for a conformally flat 3-metric with the Guichard condition in the interior of the space, an evolution of orthogonal (local-)Riemannian 2-metrics with constant Gauss curvature $-1$ is determined; for a 2-metric belonging to a certain class of orthogonal analytic 2-metrics with constant Gauss curvature $-1$, a one-parameter family of conformally flat 3-metrics with the Guichard condition is determined as evolutions issuing from the 2-metric.
</p>projecteuclid.org/euclid.jmsj/1524038668_20180418040438Wed, 18 Apr 2018 04:04 EDTNeedle decompositions and isoperimetric inequalities in Finsler geometryhttps://projecteuclid.org/euclid.jmsj/1524038669<strong>Shin-ichi OHTA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 2, 651--693.</p><p><strong>Abstract:</strong><br/>
Klartag recently gave a beautiful alternative proof of the isoperimetric inequalities of Lévy–Gromov, Bakry–Ledoux, Bayle and Milman on weighted Riemannian manifolds. Klartag's approach is based on a generalization of the localization method (so-called needle decompositions) in convex geometry, inspired also by optimal transport theory. Cavalletti and Mondino subsequently generalized the localization method, in a different way more directly along optimal transport theory, to essentially non-branching metric measure spaces satisfying the curvature-dimension condition. This class in particular includes reversible (absolutely homogeneous) Finsler manifolds. In this paper, we construct needle decompositions of non-reversible (only positively homogeneous) Finsler manifolds, and show an isoperimetric inequality under bounded reversibility constants. A discussion on the curvature-dimension condition $\mathrm{CD}(K,N)$ for $N=0$ is also included, it would be of independent interest.
</p>projecteuclid.org/euclid.jmsj/1524038669_20180418040438Wed, 18 Apr 2018 04:04 EDTRank two jump loci for solvmanifolds and Lie algebrashttps://projecteuclid.org/euclid.jmsj/1524038670<strong>Ştefan PAPADIMA</strong>, <strong>Laurenţiu PAUNESCU</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 2, 695--709.</p><p><strong>Abstract:</strong><br/>
We consider representation varieties in $SL_2$ for lattices in solvable Lie groups, and representation varieties in $\mathfrak{sl}_2$ for finite-dimensional Lie algebras. Inside them, we examine depth 1 characteristic varieties for solvmanifolds, respectively resonance varieties for cochain Differential Graded Algebras of Lie algebras. We prove a general result that leads, in both cases, to the complete description of the analytic germs at the origin, for the corresponding embedded rank 2 jump loci.
</p>projecteuclid.org/euclid.jmsj/1524038670_20180418040438Wed, 18 Apr 2018 04:04 EDTA functional equation with Borel summable solutions and irregular singular solutionshttps://projecteuclid.org/euclid.jmsj/1524038671<strong>Sunao ŌUCHI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 2, 711--731.</p><p><strong>Abstract:</strong><br/>
A Functional equation $\sum_{i=1}^{m}a_{i}(z)u(\varphi_{i}(z))=f(z)$ is considered. First we show the existence of solutions of formal power series. Second we study the homogeneous equation $(f(z)\equiv 0)$ and construct formal solutions containing exponential factors. Finally it is shown that there exists a genuine solution in a sector whose asymptotic expansion is a formal solution, by using the theory of Borel summability of formal power series. The equation has similar properties to those of irregular singular type in the theory of ordinary differential equations.
</p>projecteuclid.org/euclid.jmsj/1524038671_20180418040438Wed, 18 Apr 2018 04:04 EDTSpaces of nonnegatively curved surfaceshttps://projecteuclid.org/euclid.jmsj/1524038672<strong>Taras BANAKH</strong>, <strong>Igor BELEGRADEK</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 2, 733--756.</p><p><strong>Abstract:</strong><br/>
We determine the homeomorphism type of the space of smooth complete nonnegatively curved metrics on $S^2$, $RP^2$, and $\mathbb{C}$ equipped with the topology of $C^\gamma$ uniform convergence on compact sets, when $\gamma$ is infinite or is not an integer. If $\gamma=\infty$, the space of metrics is homeomorphic to the separable Hilbert space. If $\gamma$ is finite and not an integer, the space of metrics is homeomorphic to the countable power of the linear span of the Hilbert cube. We also prove similar results for some other spaces of metrics including the space of complete smooth Riemannian metrics on an arbitrary manifold.
</p>projecteuclid.org/euclid.jmsj/1524038672_20180418040438Wed, 18 Apr 2018 04:04 EDTExponential mixing for generic volume-preserving Anosov flows in dimension threehttps://projecteuclid.org/euclid.jmsj/1524038673<strong>Masato TSUJII</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 2, 757--821.</p><p><strong>Abstract:</strong><br/>
Let $M$ be a closed 3-dimensional Riemann manifold and let $3\le r\le \infty$. We prove that there exists an open dense subset in the space of $C^r$ volume-preserving Anosov flows on $M$ such that all the flows in it are exponentially mixing.
</p>projecteuclid.org/euclid.jmsj/1524038673_20180418040438Wed, 18 Apr 2018 04:04 EDTArithmetic exceptionality of generalized Lattès mapshttps://projecteuclid.org/euclid.jmsj/1524038674<strong>Ömer KÜÇÜKSAKALLI</strong>, <strong>Hurşit ÖNSİPER</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 2, 823--832.</p><p><strong>Abstract:</strong><br/>
We consider the arithmetic exceptionality problem for the generalized Lattès maps on $\mathbf{P}^2$. We prove an existence result for maps arising from the product $E \times E$ of elliptic curves $E$ with CM.
</p>projecteuclid.org/euclid.jmsj/1524038674_20180418040438Wed, 18 Apr 2018 04:04 EDTReflections at infinity of time changed RBMs on a domain with Liouville brancheshttps://projecteuclid.org/euclid.jmsj/1524038675<strong>Zhen-Qing CHEN</strong>, <strong>Masatoshi FUKUSHIMA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 2, 833--852.</p><p><strong>Abstract:</strong><br/>
Let $Z$ be the transient reflecting Brownian motion on the closure of an unbounded domain $D\subset \mathbb{R}^d$ with $N$ number of Liouville branches. We consider a diffuion $X$ on $\overline{D}$ having finite lifetime obtained from $Z$ by a time change. We show that $X$ admits only a finite number of possible symmetric conservative diffusion extensions $Y$ beyond its lifetime characterized by possible partitions of the collection of $N$ ends and we identify the family of the extended Dirichlet spaces of all $Y$ (which are independent of time change used) as subspaces of the space $\mathrm{BL}(D)$ spanned by the extended Sobolev space $H_e^1(D)$ and the approaching probabilities of $Z$ to the ends of Liouville branches.
</p>projecteuclid.org/euclid.jmsj/1524038675_20180418040438Wed, 18 Apr 2018 04:04 EDTThe graded structure induced by operators on a Hilbert spacehttps://projecteuclid.org/euclid.jmsj/1524038676<strong>Kunyu GUO</strong>, <strong>Xudi WANG</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 2, 853--875.</p><p><strong>Abstract:</strong><br/>
In this paper we define a graded structure induced by operators on a Hilbert space. Then we introduce several concepts which are related to the graded structure and examine some of their basic properties. A theory concerning minimal property and unitary equivalence is then developed. It allows us to obtain a complete description of $\mathcal{V}^\ast(M_{z^k})$ on any $H^2(\omega)$. It also helps us to find that a multiplication operator induced by a quasi-homogeneous polynomial must have a minimal reducing subspace. After a brief review of multiplication operator $M_{z+w}$ on $H^2(\omega,\delta)$, we prove that the Toeplitz operator $T_{z+\overline{w}}$ on $H^2(\mathbb{D}^2)$, the Hardy space over the bidisk, is irreducible.
</p>projecteuclid.org/euclid.jmsj/1524038676_20180418040438Wed, 18 Apr 2018 04:04 EDTErratum to “$L^p$ measure of growth and higher order Hardy–Sobolev–Morrey inequalities on $\mathbb{R}^{N}$”https://projecteuclid.org/euclid.jmsj/1524038677<strong>Patrick J. RABIER</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 2, 877--877.</p>projecteuclid.org/euclid.jmsj/1524038677_20180418040438Wed, 18 Apr 2018 04:04 EDTModuli of regular singular parabolic connections with given spectral type on smooth projective curveshttps://projecteuclid.org/euclid.jmsj/1527795357<strong>Michi-aki INABA</strong>, <strong>Masa-Hiko SAITO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 3, 879--894.</p><p><strong>Abstract:</strong><br/>
We define a moduli space of stable regular singular parabolic connections with given spectral type on smooth projective curves and show the smoothness of the moduli space and give a relative symplectic structure on the moduli space. Moreover, we define the isomonodromic deformation on this moduli space and prove the geometric Painlevé property of the isomonodromic deformation.
</p>projecteuclid.org/euclid.jmsj/1527795357_20180719040046Thu, 19 Jul 2018 04:00 EDTOn the solutions of quadratic Diophantine equations IIhttps://projecteuclid.org/euclid.jmsj/1529309019<strong>Manabu MURATA</strong>, <strong>Takashi YOSHINAGA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 3, 895--919.</p><p><strong>Abstract:</strong><br/>
A quantity concerning the solutions of a quadratic Diophantine equation in $n$ variables coincides with a mass of a special orthogonal group of a quadratic form in dimension $n-1$, via the mass formula due to Shimura. We show an explicit formula for the quantity, assuming the maximality of a lattice in the $(n-1)$-dimensional quadratic space. The quantity is determined by the computation of a group index and of the mass of the genus of maximal lattices in that quadratic space. As applications of the result, we give the number of primitive solutions for the sum of $n$ squares with 6 or 8 and also the quantity in question for the sum of 10 squares.
</p>projecteuclid.org/euclid.jmsj/1529309019_20180719040046Thu, 19 Jul 2018 04:00 EDTFinite-particle approximations for interacting Brownian particles with logarithmic potentialshttps://projecteuclid.org/euclid.jmsj/1529309020<strong>Yosuke KAWAMOTO</strong>, <strong>Hirofumi OSADA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 3, 921--952.</p><p><strong>Abstract:</strong><br/>
We prove the convergence of $N$-particle systems of Brownian particles with logarithmic interaction potentials onto a system described by the infinite-dimensional stochastic differential equation (ISDE). For this proof we present two general theorems on the finite-particle approximations of interacting Brownian motions. In the first general theorem, we present a sufficient condition for a kind of tightness of solutions of stochastic differential equations (SDE) describing finite-particle systems, and prove that the limit points solve the corresponding ISDE. This implies, if in addition the limit ISDE enjoy a uniqueness of solutions, then the full sequence converges. We treat non-reversible case in the first main theorem. In the second general theorem, we restrict to the case of reversible particle systems and simplify the sufficient condition. We deduce the second theorem from the first. We apply the second general theorem to $\mathrm{Airy}_\beta$ interacting Brownian motion with $\beta=1, 2, 4$, and the Ginibre interacting Brownian motion. The former appears in the soft-edge limit of Gaussian (orthogonal/unitary/symplectic) ensembles in one spatial dimension, and the latter in the bulk limit of Ginibre ensemble in two spatial dimensions, corresponding to a quantum statistical system for which the eigen-value spectra belong to non-Hermitian Gaussian random matrices. The passage from the finite-particle stochastic differential equation (SDE) to the limit ISDE is a sensitive problem because the logarithmic potentials are long range and unbounded at infinity. Indeed, the limit ISDEs are not easily detectable from those of finite dimensions. Our general theorems can be applied straightforwardly to the grand canonical Gibbs measures with Ruelle-class potentials such as Lennard-Jones 6-12 potentials and and Riesz potentials.
</p>projecteuclid.org/euclid.jmsj/1529309020_20180719040046Thu, 19 Jul 2018 04:00 EDTOn the fundamental group of a smooth projective surface with a finite group of automorphismshttps://projecteuclid.org/euclid.jmsj/1528790543<strong>Rajendra Vasant GURJAR</strong>, <strong>Bangere P. PURNAPRAJNA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 3, 953--974.</p><p><strong>Abstract:</strong><br/>
In this article we prove new results on fundamental groups for some classes of fibered smooth projective algebraic surfaces with a finite group of automorphisms. The methods actually compute the fundamental groups of the surfaces under study upto finite index. The corollaries include an affirmative answer to Shafarevich conjecture on holomorphic convexity, Nori’s well-known question on fundamental groups and free abelianness of second homotopy groups for these surfaces. We also prove a theorem that bounds the multiplicity of the multiple fibers of a fibration for any algebraic surface with a finite group of automorphisms $G$ in terms of the multiplicities of the induced fibration on $X/G$. If $X/G$ is a $\mathbb{P}^1$-fibration, we show that the multiplicity actually divides $|G|$. This theorem on multiplicity, which is of independent interest, plays an important role in our theorems.
</p>projecteuclid.org/euclid.jmsj/1528790543_20180719040046Thu, 19 Jul 2018 04:00 EDTChern classes of logarithmic derivations for free divisors with Jacobian ideal of linear typehttps://projecteuclid.org/euclid.jmsj/1527795358<strong>Xia LIAO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 3, 975--988.</p><p><strong>Abstract:</strong><br/>
Let $X$ be a nonsingular variety defined over an algebraically closed field of characteristic $0$, and $D$ be a free divisor with Jacobian ideal of linear type. We compute the Chern class of the sheaf of logarithmic derivations along $D$ and compare it with the Chern–Schwartz–MacPherson class of the hypersurface complement. Our result establishes a conjecture by Aluffi raised in [ Alu12b ].
</p>projecteuclid.org/euclid.jmsj/1527795358_20180719040046Thu, 19 Jul 2018 04:00 EDTBlow-analytic equivalence versus contact bi-Lipschitz equivalencehttps://projecteuclid.org/euclid.jmsj/1528790544<strong>Lev BIRBRAIR</strong>, <strong>Alexandre FERNANDES</strong>, <strong>Vincent GRANDJEAN</strong>, <strong>Terence GAFFNEY</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 3, 989--1006.</p><p><strong>Abstract:</strong><br/>
The main result of this note is that two blow-analytically equivalent real analytic plane function germs are sub-analytically bi-Lipschitz contact equivalent.
</p>projecteuclid.org/euclid.jmsj/1528790544_20180719040046Thu, 19 Jul 2018 04:00 EDTDerived equivalence of Ito–Miura–Okawa–Ueda Calabi–Yau 3-foldshttps://projecteuclid.org/euclid.jmsj/1529309021<strong>Alexander KUZNETSOV</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 3, 1007--1013.</p><p><strong>Abstract:</strong><br/>
We prove derived equivalence of Calabi–Yau threefolds constructed by Ito–Miura–Okawa–Ueda as an example of non-birational Calabi–Yau varieties whose difference in the Grothendieck ring of varieties is annihilated by the affine line.
</p>projecteuclid.org/euclid.jmsj/1529309021_20180719040046Thu, 19 Jul 2018 04:00 EDTCompact foliations with finite transverse LS categoryhttps://projecteuclid.org/euclid.jmsj/1528790547<strong>Steven HURDER</strong>, <strong>Paweł WALCZAK</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 3, 1015--1046.</p><p><strong>Abstract:</strong><br/>
We prove that if $F$ is a foliation of a compact manifold $M$ with all leaves compact submanifolds, and the transverse saturated category of $F$ is finite, then the leaf space $M/F$ is compact Hausdorff. The proof is surprisingly delicate, and is based on some new observations about the geometry of compact foliations. The transverse saturated category of a compact Hausdorff foliation is always finite, so we obtain a new characterization of the compact Hausdorff foliations among the compact foliations as those with finite transverse saturated category.
</p>projecteuclid.org/euclid.jmsj/1528790547_20180719040046Thu, 19 Jul 2018 04:00 EDTExamples of four dimensional cusp singularitieshttps://projecteuclid.org/euclid.jmsj/1529309022<strong>Hiroyasu TSUCHIHASHI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 3, 1047--1062.</p><p><strong>Abstract:</strong><br/>
We give some examples of four dimensional cusp singularities which are not of Hilbert modular type. We construct them, using quadratic cones and subgroups of reflection groups.
</p>projecteuclid.org/euclid.jmsj/1529309022_20180719040046Thu, 19 Jul 2018 04:00 EDTAlexander invariants of ribbon tangles and planar algebrashttps://projecteuclid.org/euclid.jmsj/1529309023<strong>Celeste DAMIANI</strong>, <strong>Vincent FLORENS</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 3, 1063--1084.</p><p><strong>Abstract:</strong><br/>
Ribbon tangles are proper embeddings of tori and cylinders in the 4-ball $B^4$, “bounding” 3-manifolds with only ribbon disks as singularities. We construct an Alexander invariant $\mathbf{A}$ of ribbon tangles equipped with a representation of the fundamental group of their exterior in a free abelian group $G$. This invariant induces a functor in a certain category $\mathbf{R}ib_G$ of tangles, which restricts to the exterior powers of Burau–Gassner representation for ribbon braids, that are analogous to usual braids in this context. We define a circuit algebra $\mathbf{C}ob_G$ over the operad of smooth cobordisms, inspired by diagrammatic planar algebras introduced by Jones [ Jon99 ], and prove that the invariant $\mathbf{A}$ commutes with the compositions in this algebra. On the other hand, ribbon tangles admit diagrammatic representations, through welded diagrams. We give a simple combinatorial description of $\mathbf{A}$ and of the algebra $\mathbf{C}ob_G$, and observe that our construction is a topological incarnation of the Alexander invariant of Archibald [ Arc10 ]. When restricted to diagrams without virtual crossings, $\mathbf{A}$ provides a purely local description of the usual Alexander poynomial of links, and extends the construction by Bigelow, Cattabriga and the second author [ BCF15 ].
</p>projecteuclid.org/euclid.jmsj/1529309023_20180719040046Thu, 19 Jul 2018 04:00 EDTA system of conjugate functions on parabolic Bloch spaceshttps://projecteuclid.org/euclid.jmsj/1529309024<strong>Yôsuke HISHIKAWA</strong>, <strong>Masaharu NISHIO</strong>, <strong>Masahiro YAMADA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 3, 1085--1102.</p><p><strong>Abstract:</strong><br/>
The parabolic Bloch space is the set of all solutions $u$ of the parabolic operator $L^{(\alpha)}$ with the finite Bloch norm $\| u \|_{\mathcal{B}_{\alpha} (\sigma)}$. In this paper, we introduce $L^{(\alpha)}$-conjugates of parabolic Bloch functions, and investigate several properties. As an application, we give an isomorphism theorem on parabolic Bloch spaces.
</p>projecteuclid.org/euclid.jmsj/1529309024_20180719040046Thu, 19 Jul 2018 04:00 EDTScalar curvature of self-shrinkerhttps://projecteuclid.org/euclid.jmsj/1529309025<strong>Zhen GUO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 3, 1103--1110.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider the scalar curvature of a self-shrinker and get the gap theorem of the scalar curvature. We get also a relationship between the upper bound of the square of the length of the second fundamental form and the Ricci mean value.
</p>projecteuclid.org/euclid.jmsj/1529309025_20180719040046Thu, 19 Jul 2018 04:00 EDTFree probability for purely discrete eigenvalues of random matriceshttps://projecteuclid.org/euclid.jmsj/1529892023<strong>Benoit COLLINS</strong>, <strong>Takahiro HASEBE</strong>, <strong>Noriyoshi SAKUMA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 3, 1111--1150.</p><p><strong>Abstract:</strong><br/>
In this paper, we study random matrix models which are obtained as a non-commutative polynomial in random matrix variables of two kinds: (a) a first kind which have a discrete spectrum in the limit, (b) a second kind which have a joint limiting distribution in Voiculescu’s sense and are globally rotationally invariant. We assume that each monomial constituting this polynomial contains at least one variable of type (a), and show that this random matrix model has a set of eigenvalues that almost surely converges to a deterministic set of numbers that is either finite or accumulating to only zero in the large dimension limit. For this purpose we define a framework (cyclic monotone independence) for analyzing discrete spectra and develop the moment method for the eigenvalues of compact (and in particular Schatten class) operators. We give several explicit calculations of discrete eigenvalues of our model.
</p>projecteuclid.org/euclid.jmsj/1529892023_20180719040046Thu, 19 Jul 2018 04:00 EDTElliptic fibrations on K3 surfaces and Salem numbers of maximal degreehttps://projecteuclid.org/euclid.jmsj/1529892024<strong>Xun YU</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 3, 1151--1163.</p><p><strong>Abstract:</strong><br/>
We study the maximal Salem degree of automorphisms of K3 surfaces via elliptic fibrations. In particular, we establish a characterization of such maximum in terms of elliptic fibrations with infinite automorphism groups. As an application, we show that any supersingular K3 surface in odd characteristic has an automorphism the entropy of which is the natural logarithm of a Salem number of degree 22.
</p>projecteuclid.org/euclid.jmsj/1529892024_20180719040046Thu, 19 Jul 2018 04:00 EDTComposing generic linearly perturbed mappings and immersions/injectionshttps://projecteuclid.org/euclid.jmsj/1528790548<strong>Shunsuke ICHIKI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 3, 1165--1184.</p><p><strong>Abstract:</strong><br/>
Let $N$ (resp., $U$) be a manifold (resp., an open subset of ${\mathbb{R}}^m$). Let $f:N\to U$ and $F:U\to {\mathbb{R}}^\ell$ be an immersion and a $C^{\infty}$ mapping, respectively. Generally, the composition $F\circ f$ does not necessarily yield a mapping transverse to a given subfiber-bundle of $J^1(N,\mathbb{R}^\ell)$. Nevertheless, in this paper, for any $\mathcal{A}^1$-invariant fiber, we show that composing generic linearly perturbed mappings of $F$ and the given immersion $f$ yields a mapping transverse to the subfiber-bundle of $J^1(N,\mathbb{R}^\ell)$ with the given fiber. Moreover, we show a specialized transversality theorem on crossings of compositions of generic linearly perturbed mappings of a given mapping $F:U\to \mathbb{R}^\ell$ and a given injection $f:N\to U$. Furthermore, applications of the two main theorems are given.
</p>projecteuclid.org/euclid.jmsj/1528790548_20180719040046Thu, 19 Jul 2018 04:00 EDTCommon reducing subspaces of several weighted shifts with operator weightshttps://projecteuclid.org/euclid.jmsj/1529892025<strong>Caixing GU</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 3, 1185--1225.</p><p><strong>Abstract:</strong><br/>
We characterize common reducing subspaces of several weighted shifts with operator weights. As applications, we study the common reducing subspaces of the multiplication operators by powers of coordinate functions on Hilbert spaces of holomorphic functions in several variables. The identification of reducing subspaces also leads to structure theorems for the commutants of von Neumann algebras generated by these multiplication operators. This general approach applies to weighted Hardy spaces, weighted Bergman spaces, Drury–Arveson spaces and Dirichlet spaces of the unit ball or polydisk uniformly.
</p>projecteuclid.org/euclid.jmsj/1529892025_20180719040046Thu, 19 Jul 2018 04:00 EDT