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 EDTOn 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 EDTMeasure-valued solutions to the complete Euler systemhttps://projecteuclid.org/euclid.jmsj/1531469370<strong>Jan BŘEZINA</strong>, <strong>Eduard FEIREISL</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 4, 1227--1245.</p><p><strong>Abstract:</strong><br/>
We introduce the concept of dissipative measure-valued solution to the complete Euler system describing the motion of an inviscid compressible fluid. These solutions are characterized by a parameterized (Young) measure and a dissipation defect in the total energy balance. The dissipation defect dominates the concentration errors in the equations satisfied by the Young measure. A dissipative measure-valued solution can be seen as the most general concept of solution to the Euler system retaining its structural stability. In particular, we show that a dissipative measure-valued solution necessarily coincides with a classical one on its life span provided they share the same initial data.
</p>projecteuclid.org/euclid.jmsj/1531469370_20181023220059Tue, 23 Oct 2018 22:00 EDTThe Gordian distance of handlebody-knots and Alexander biquandle coloringshttps://projecteuclid.org/euclid.jmsj/1532678872<strong>Tomo MURAO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 4, 1247--1267.</p><p><strong>Abstract:</strong><br/>
We give lower bounds for the Gordian distance and the unknotting number of handlebody-knots by using Alexander biquandle colorings. We construct handlebody-knots with Gordian distance $n$ and unknotting number $n$ for any positive integer $n$.
</p>projecteuclid.org/euclid.jmsj/1532678872_20181023220059Tue, 23 Oct 2018 22:00 EDTRogers dilogarithms of higher degree and generalized cluster algebrashttps://projecteuclid.org/euclid.jmsj/1532678873<strong>Tomoki NAKANISHI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 4, 1269--1304.</p><p><strong>Abstract:</strong><br/>
In connection with generalized cluster algebras we introduce a certain generalization of the celebrated Rogers dilogarithm, which we call the Rogers dilogarithms of higher degree. We show that there is an identity of these generalized Rogers dilogarithms associated with any period of seeds of a generalized cluster algebra.
</p>projecteuclid.org/euclid.jmsj/1532678873_20181023220059Tue, 23 Oct 2018 22:00 EDTWhitney regularity and Thom condition for families of non-isolated mixed singularitieshttps://projecteuclid.org/euclid.jmsj/1532678874<strong>Christophe EYRAL</strong>, <strong>Mutsuo OKA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 4, 1305--1336.</p><p><strong>Abstract:</strong><br/>
We investigate the equisingularity question for 1-parameter deformation families of mixed polynomial functions $f_t({\boldsymbol{z}},\bar{{\boldsymbol{z}}})$ from the Newton polygon point of view. We show that if the members $f_t$ of the family satisfy a number of elementary conditions, which can be easily described in terms of the Newton polygon, then the corresponding family of mixed hypersurfaces $f_t^{-1}(0)$ is Whitney equisingular (and hence topologically equisingular) and satisfies the Thom condition.
</p>projecteuclid.org/euclid.jmsj/1532678874_20181023220059Tue, 23 Oct 2018 22:00 EDTAccurate trajectory-harps for Kähler magnetic fieldshttps://projecteuclid.org/euclid.jmsj/1531469371<strong>Toshiaki ADACHI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 4, 1337--1346.</p><p><strong>Abstract:</strong><br/>
In preceding papers we gave estimates on string-lengths, string-cosines and zenith angles of trajectory-harps under the condition that sectional curvatures of the underlying manifold are bounded from above. In this paper we study the cases that equalities hold in these estimates. Refining the previous proofs we give conditions that trajectory-harps are congruent to trajectory-harps on a complex space form.
</p>projecteuclid.org/euclid.jmsj/1531469371_20181023220059Tue, 23 Oct 2018 22:00 EDTVertex operator algebras, minimal models, and modular linear differential equations of order 4https://projecteuclid.org/euclid.jmsj/1535616221<strong>Yusuke ARIKE</strong>, <strong>Kiyokazu NAGATOMO</strong>, <strong>Yuichi SAKAI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 4, 1347--1373.</p><p><strong>Abstract:</strong><br/>
In this paper we classify vertex operator algebras with three conditions which arise from Virasoro minimal models: (A) the central charge and conformal weights are rational numbers, (B) the space spanned by characters of all simple modules of a vertex operator algebra coincides with the space of solutions of a modular linear differential equation of order $4$ and (C) the dimensions of first three weight subspaces of a VOA are $1, 0$ and $1$, respectively. It is shown that vertex operator algebras which we concern have central charges $c=-46/3, -3/5, -114/7, 4/5$, and are isomorphic to minimal models for $c=-46/3, -3/5$ and ${\mathbb{Z}}_2$-graded simple current extensions of minimal models for $c=-114/7, 4/5$.
</p>projecteuclid.org/euclid.jmsj/1535616221_20181023220059Tue, 23 Oct 2018 22:00 EDTEnergy decay and diffusion phenomenon for the asymptotically periodic damped wave equationhttps://projecteuclid.org/euclid.jmsj/1536220816<strong>Romain JOLY</strong>, <strong>Julien ROYER</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 4, 1375--1418.</p><p><strong>Abstract:</strong><br/>
We prove local and global energy decay for the asymptotically periodic damped wave equation on the Euclidean space. Since the behavior of high frequencies is already mostly understood, this paper is mainly about the contribution of low frequencies. We show in particular that the damped wave behaves like a solution of a heat equation which depends on the H-limit of the metric and the mean value of the absorption index.
</p>projecteuclid.org/euclid.jmsj/1536220816_20181023220059Tue, 23 Oct 2018 22:00 EDTLocal polar invariants and the Poincaré problem in the dicritical casehttps://projecteuclid.org/euclid.jmsj/1538380983<strong>Yohann GENZMER</strong>, <strong>Rogério MOL</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 4, 1419--1451.</p><p><strong>Abstract:</strong><br/>
We develop a study on local polar invariants of planar complex analytic foliations at $(\mathbb{C}^{2},0)$, which leads to the characterization of second type foliations and of generalized curve foliations, as well as to a description of the $GSV$-index. We apply it to the Poincaré problem for foliations on the complex projective plane $\mathbb{P}^{2}_{\mathbb{C}}$, establishing, in the dicritical case, conditions for the existence of a bound for the degree of an invariant algebraic curve $S$ in terms of the degree of the foliation $\mathcal{F}$. We characterize the existence of a solution for the Poincaré problem in terms of the structure of the set of local separatrices of $\mathcal{F}$ over the curve $S$. Our method, in particular, recovers the known solution for the non-dicritical case, $\deg(S) \leq \deg (\mathcal{F}) + 2$.
</p>projecteuclid.org/euclid.jmsj/1538380983_20181023220059Tue, 23 Oct 2018 22:00 EDTA family of cubic fourfolds with finite-dimensional motivehttps://projecteuclid.org/euclid.jmsj/1532678875<strong>Robert LATERVEER</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 4, 1453--1473.</p><p><strong>Abstract:</strong><br/>
We prove that cubic fourfolds in a certain 10-dimensional family have finite-dimensional motive. The proof is based on the van Geemen–Izadi construction of an algebraic Kuga–Satake correspondence for these cubic fourfolds, combined with Voisin’s method of “spread”. Some consequences are given.
</p>projecteuclid.org/euclid.jmsj/1532678875_20181023220059Tue, 23 Oct 2018 22:00 EDTCritical nonlinear Schrödinger equations in higher space dimensionshttps://projecteuclid.org/euclid.jmsj/1532678876<strong>Nakao HAYASHI</strong>, <strong>Chunhua LI</strong>, <strong>Pavel I. NAUMKIN</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 4, 1475--1492.</p><p><strong>Abstract:</strong><br/>
We study the critical nonlinear Schrödinger equations \[ i\partial _{t}u+\frac{1}{2}\Delta u = \lambda \vert u\vert^{{2}/{n}}u, \quad (t,x) \in \mathbb{R}^{+}\times \mathbb{R}^{n}, \] in space dimensions $n\geq 4$, where $\lambda \in \mathbb{R}$. We prove the global in time existence of solutions to the Cauchy problem under the assumption that the absolute value of Fourier transform of the initial data is bounded below by a positive constant. Also we prove the two side sharp time decay estimates of solutions in the uniform norm.
</p>projecteuclid.org/euclid.jmsj/1532678876_20181023220059Tue, 23 Oct 2018 22:00 EDTVolume minimization and conformally Kähler, Einstein–Maxwell geometryhttps://projecteuclid.org/euclid.jmsj/1536220817<strong>Akito FUTAKI</strong>, <strong>Hajime ONO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 4, 1493--1521.</p><p><strong>Abstract:</strong><br/>
Let $M$ be a compact complex manifold admitting a Kähler structure. A conformally Kähler, Einstein–Maxwell metric (cKEM metric for short) is a Hermitian metric $\tilde g$ on $M$ with constant scalar curvature such that there is a positive smooth function $f$ with $g = f^{2} \tilde g$ being a Kähler metric and $f$ being a Killing Hamiltonian potential with respect to $g$. Fixing a Kähler class, we characterize such Killing vector fields whose Hamiltonian function $f$ with respect to some Kähler metric $g$ in the fixed Kähler class gives a cKEM metric $\tilde g = f^{-2}g$. The characterization is described in terms of critical points of certain volume functional. The conceptual idea is similar to the cases of Kähler–Ricci solitons and Sasaki–Einstein metrics in that the derivative of the volume functional gives rise to a natural obstruction to the existence of cKEM metrics. However, unlike the Kähler–Ricci soliton case and Sasaki–Einstein case, the functional is neither convex nor proper in general, and often has more than one critical points. The last observation matches well with the ambitoric examples studied earlier by LeBrun and Apostolov–Maschler.
</p>projecteuclid.org/euclid.jmsj/1536220817_20181023220059Tue, 23 Oct 2018 22:00 EDTDiffusion with nonlocal Robin boundary conditionshttps://projecteuclid.org/euclid.jmsj/1538553644<strong>Wolfgang ARENDT</strong>, <strong>Stefan KUNKEL</strong>, <strong>Markus KUNZE</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 4, 1523--1556.</p><p><strong>Abstract:</strong><br/>
We investigate a second order elliptic differential operator $A_{\beta, \mu}$ on a bounded, open set $\Omega\subset\mathbb{R}^{d}$ with Lipschitz boundary subject to a nonlocal boundary condition of Robin type. More precisely we have $0\leq \beta\in L^{\infty}(\partial\Omega)$ and $\mu\colon\partial\Omega\to{\mathscr{M}}(\overline{\Omega})$, and boundary conditions of the form $$ \partial_{\nu}^{{\mathscr{A}}}u(z)+\beta(z)u(z)=\int_{\overline{\Omega}}u(x)\mu(z)(\mathrm{d}x), \quad z\in\partial\Omega, $$ where $\partial_{\nu}^{{\mathscr{A}}}$ denotes the weak conormal derivative with respect to our differential operator. Under suitable conditions on the coefficients of the differential operator and the function $\mu$ we show that $A_{\beta, \mu}$ generates a holomorphic semigroup $T_{\beta,\mu}$ on $L^{\infty}(\Omega)$ which enjoys the strong Feller property. In particular, it takes values in $C(\overline{\Omega})$. Its restriction to $C(\overline{\Omega})$ is strongly continuous and holomorphic. We also establish positivity and contractivity of the semigroup under additional assumptions and study the asymptotic behavior of the semigroup.
</p>projecteuclid.org/euclid.jmsj/1538553644_20181023220059Tue, 23 Oct 2018 22:00 EDTAnalysis of elastic symbols with the Cauchy integral and construction of asymptotic solutionshttps://projecteuclid.org/euclid.jmsj/1538553645<strong>Hideo SOGA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 4, 1557--1580.</p><p><strong>Abstract:</strong><br/>
This paper deals with the elastic wave equation $(D_t^2 - L(x, D_{x'}, D_{x_n})) u(t, x', x_n)=0$ in the half-space $x_n>0$. In the constant coefficient case, it is known that the solution is represented by using the Cauchy integral $\int_c e^{ix_n\zeta} (I-L(\xi', \zeta))^{-1} d\zeta$. In this paper this representation is extended to the variable coefficient case, and an asymptotic solution with the similar Cauchy integral is constructed. In this case, the terms $\partial_x^\alpha \int_c e^{ix_n\zeta} (I-L(x,\xi',\zeta))^{-1} d\zeta$ appear in the inductive process. These do not become lower terms necessarily, and therefore the principal part of asymptotic solution is a little different from the form in the constant coefficient case.
</p>projecteuclid.org/euclid.jmsj/1538553645_20181023220059Tue, 23 Oct 2018 22:00 EDTSpectrum for compact operators on Banach spaceshttps://projecteuclid.org/euclid.jmsj/1536890442<strong>Luis BARREIRA</strong>, <strong>Davor DRAGIČEVIĆ</strong>, <strong>Claudia VALLS</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 1, 1--17.</p><p><strong>Abstract:</strong><br/>
For a two-sided sequence of compact linear operators acting on a Banach space, we consider the notion of spectrum defined in terms of the existence of exponential dichotomies under homotheties of the dynamics. This can be seen as a natural generalization of the spectrum of a matrix—the set of its eigenvalues. We give a characterization of all possible spectra and explicit examples of sequences for which the spectrum takes a form not occurring in finite-dimensional spaces. We also consider the case of a one-sided sequence of compact linear operators.
</p>projecteuclid.org/euclid.jmsj/1536890442_20190124220103Thu, 24 Jan 2019 22:01 ESTLinks with trivial $Q$-polynomialhttps://projecteuclid.org/euclid.jmsj/1538640044<strong>Yasuyuki MIYAZAWA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 1, 19--42.</p><p><strong>Abstract:</strong><br/>
The $Q$-polynomial is an invariant of the isotopy type of an unoriented link defined by Brandt, Lickorish, Millett, and Ho around 1985. It is shown that there exist infinitely many prime knots and links with trivial $Q$-polynomial, and so the $Q$-polynomial does not detect trivial links.
</p>projecteuclid.org/euclid.jmsj/1538640044_20190124220103Thu, 24 Jan 2019 22:01 ESTTopological canal foliationshttps://projecteuclid.org/euclid.jmsj/1539590425<strong>Gilbert HECTOR</strong>, <strong>Rémi LANGEVIN</strong>, <strong>Paweł WALCZAK</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 1, 43--63.</p><p><strong>Abstract:</strong><br/>
Regular canal surfaces of $\mathbb{R}^3$ or $\mathbb{S}^3$ admit foliations by circles: the characteristic circles of the envelope. In order to build a foliation of $\mathbb{S}^3$ with leaves being canal surfaces, one has to relax the condition “canal” a little ( “weak canal condition” ) in order to accept isolated umbilics. Here, we define a topological condition which generalizes this “weak canal” condition imposed on leaves, and classify the foliations of compact orientable 3-manifolds we can obtain this way.
</p>projecteuclid.org/euclid.jmsj/1539590425_20190124220103Thu, 24 Jan 2019 22:01 ESTVanishing theorems of $L^2$-cohomology groups on Hessian manifoldshttps://projecteuclid.org/euclid.jmsj/1540541021<strong>Shinya AKAGAWA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 1, 65--89.</p><p><strong>Abstract:</strong><br/>
We show vanishing theorems of $L^2$-cohomology groups of Kodaira–Nakano type on complete Hessian manifolds by introducing a new operator $\partial'_F$. We obtain further vanishing theorems of $L^2$-cohomology groups $L^2H^{p,q}_{\bar{\partial}}(\Omega)$ on a regular convex cone $\Omega$ with the Cheng–Yau metric for $p>q$.
</p>projecteuclid.org/euclid.jmsj/1540541021_20190124220103Thu, 24 Jan 2019 22:01 ESTEquivalence of Littlewood–Paley square function and area function characterizations of weighted product Hardy spaces associated to operatorshttps://projecteuclid.org/euclid.jmsj/1540368034<strong>Xuan Thinh DUONG</strong>, <strong>Guorong HU</strong>, <strong>Ji LI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 1, 91--115.</p><p><strong>Abstract:</strong><br/>
Let $L_1$ and $L_2$ be nonnegative self-adjoint operators acting on $L^2(X_1)$ and $L^2(X_2)$, respectively, where $X_1$ and $X_2$ are spaces of homogeneous type. Assume that $L_1$ and $L_2$ have Gaussian heat kernel bounds. This paper aims to study some equivalent characterizations of the weighted product Hardy spaces $H^{p}_{w,L_{1},L_{2}}(X_{1}\times X_{2})$ associated to $L_{1}$ and $L_{2}$, for $p \in (0, \infty)$ and the weight $w$ belongs to the product Muckenhoupt class $ A_{\infty}(X_{1} \times X_{2})$. Our main result is that the spaces $H^{p}_{w,L_{1},L_{2}}(X_{1}\times X_{2})$ introduced via area functions can be equivalently characterized by the Littlewood–Paley $g$-functions and $g^{\ast}_{\lambda_{1}, \lambda_{2}}$-functions, as well as the Peetre type maximal functions, without any further assumption beyond the Gaussian upper bounds on the heat kernels of $L_1$ and $L_2$. Our results are new even in the unweighted product setting.
</p>projecteuclid.org/euclid.jmsj/1540368034_20190124220103Thu, 24 Jan 2019 22:01 ESTBerkes' limit theoremhttps://projecteuclid.org/euclid.jmsj/1540368039<strong>Satoshi TAKANOBU</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 1, 117--145.</p><p><strong>Abstract:</strong><br/>
In Berkes' striking paper of the early 1990s, he presented another limit theorem different from the central limit theorem for a lacunary trigonometric series not satisfying Erdős' lacunary condition. In this paper, we upgrade his result to the limit theorem having high versatility, which we would call Berkes' limit theorem. By this limit theorem, it is explained in a unified way that Fukuyama–Takahashi's counterexample and Takahashi's counterexample are all convergent to limiting distributions of the same type as Berkes.
</p>projecteuclid.org/euclid.jmsj/1540368039_20190124220103Thu, 24 Jan 2019 22:01 ESTOn unconditional well-posedness for the periodic modified Korteweg–de Vries equationhttps://projecteuclid.org/euclid.jmsj/1540541018<strong>Luc MOLINET</strong>, <strong>Didier PILOD</strong>, <strong>Stéphane VENTO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 1, 147--201.</p><p><strong>Abstract:</strong><br/>
We prove that the modified Korteweg–de Vries equation is unconditionally well-posed in $H^s({\mathbb{T}})$ for $s\ge 1/3$. For this we gather the smoothing effect first discovered by Takaoka and Tsutsumi with an approach developed by the authors that combines the energy method, with Bourgain's type estimates, improved Strichartz estimates and the construction of modified energies.
</p>projecteuclid.org/euclid.jmsj/1540541018_20190124220103Thu, 24 Jan 2019 22:01 ESTLocal time penalizations with various clocks for one-dimensional diffusionshttps://projecteuclid.org/euclid.jmsj/1541408432<strong>Christophe PROFETA</strong>, <strong>Kouji YANO</strong>, <strong>Yuko YANO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 1, 203--233.</p><p><strong>Abstract:</strong><br/>
We study some limit theorems for the law of a generalized one-dimensional diffusion weighted and normalized by a non-negative function of the local time evaluated at a parametrized family of random times (which we will call a clock ). As the clock tends to infinity, we show that the initial process converges towards a new penalized process, which generally depends on the chosen clock. However, unlike with deterministic clocks, no specific assumptions are needed on the resolvent of the diffusion. We then give a path interpretation of these penalized processes via some universal $\sigma$-finite measures.
</p>projecteuclid.org/euclid.jmsj/1541408432_20190124220103Thu, 24 Jan 2019 22:01 ESTPartitioning subsets of generalised scattered ordershttps://projecteuclid.org/euclid.jmsj/1542704621<strong>Chris LAMBIE-HANSON</strong>, <strong>Thilo WEINERT</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 1, 235--257.</p><p><strong>Abstract:</strong><br/>
In 1956, 48 years after Hausdorff provided a comprehensive account on ordered sets and defined the notion of a scattered order, Erdős and Rado founded the partition calculus in a seminal paper. The present paper gives an account of investigations into generalisations of scattered linear orders and their partition relations for both singletons and pairs. We consider analogues for these order-types of known partition theorems for ordinals or scattered orders and prove a partition theorem from assumptions about cardinal characteristics. Together, this continues older research by Erdős, Galvin, Hajnal, Larson and Takahashi and more recent investigations by Abraham, Bonnet, Cummings, Džamonja, Komjáth, Shelah and Thompson.
</p>projecteuclid.org/euclid.jmsj/1542704621_20190124220103Thu, 24 Jan 2019 22:01 ESTPseudo Kobayashi hyperbolicity of subvarieties of general type on abelian varietieshttps://projecteuclid.org/euclid.jmsj/1542704620<strong>Katsutoshi YAMANOI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 1, 259--298.</p><p><strong>Abstract:</strong><br/>
We prove that the Kobayashi pseudo distance of a closed subvariety $X$ of an abelian variety $A$ is a true distance outside the special set $\operatorname{Sp}(X)$ of $X$, where $\operatorname{Sp}(X)$ is the union of all positive dimensional translated abelian subvarieties of $A$ which are contained in $X$. More strongly, we prove that a closed subvariety $X$ of an abelian variety is taut modulo $\operatorname{Sp}(X)$; Every sequence $f_n:{\mathbb{D}}\to X$ of holomorphic mappings from the unit disc ${\mathbb{D}}$ admits a subsequence which converges locally uniformly, unless the image $f_n(K)$ of a fixed compact set $K$ of ${\mathbb{D}}$ eventually gets arbitrarily close to $\operatorname{Sp}(X)$ as $n$ gets larger. These generalize a classical theorem on algebraic degeneracy of entire curves in irregular varieties.
</p>projecteuclid.org/euclid.jmsj/1542704620_20190124220103Thu, 24 Jan 2019 22:01 ESTThe combinatorics of Lehn's conjecturehttps://projecteuclid.org/euclid.jmsj/1538640045<strong>Alina MARIAN</strong>, <strong>Dragos OPREA</strong>, <strong>Rahul PANDHARIPANDE</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 1, 299--308.</p><p><strong>Abstract:</strong><br/>
Let $S$ be a nonsingular projective surface equipped with a line bundle $H$. Lehn's conjecture is a formula for the top Segre class of the tautological bundle associated to $H$ on the Hilbert scheme of points of $S$. Voisin has recently reduced Lehn's conjecture to the vanishing of certain coefficients of special power series. The first result here is a proof of the vanishings required by Voisin by residue calculations (A. Szenes and M. Vergne have independently found the same proof). Our second result is an elementary solution of the parallel question for the top Segre class on the symmetric power of a nonsingular projective curve $C$ associated to a higher rank vector bundle $V$ on $C$. Finally, we propose a complete conjecture for the top Segre class on the Hilbert scheme of points of $S$ associated to a higher rank vector bundle on $S$ in the $K$-trivial case.
</p>projecteuclid.org/euclid.jmsj/1538640045_20190124220103Thu, 24 Jan 2019 22:01 ESTPositive factorizations of symmetric mapping classeshttps://projecteuclid.org/euclid.jmsj/1542704619<strong>Tetsuya ITO</strong>, <strong>Keiko KAWAMURO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 1, 309--327.</p><p><strong>Abstract:</strong><br/>
We study a question of Etnyre and Van Horn-Morris whether a symmetric mapping class admitting a positive factorization is a lift of a quasipositive braid. We answer the question affirmatively for mapping classes satisfying certain cyclic conditions.
</p>projecteuclid.org/euclid.jmsj/1542704619_20190124220103Thu, 24 Jan 2019 22:01 ESTOn $n$-trivialities of classical and virtual knots for some unknotting operationshttps://projecteuclid.org/euclid.jmsj/1541667932<strong>Noboru ITO</strong>, <strong>Migiwa SAKURAI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 1, 329--347.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce a new nontrivial filtration, called F-order, for classical and virtual knot invariants; this filtration produces filtered knot invariants, which are called finite type invariants similar to Vassiliev knot invariants. Finite type invariants introduced by Goussarov, Polyak, and Viro are well-known, and we call them finite type invariants of GPV-order. We show that for any positive integer $n$ and for any classical knot $K$, there exist infinitely many of nontrivial classical knots, all of whose finite type invariants of GPV-order $\le n-1$, coincide with those of $K$ (Theorem 1). Further, we show that for any positive integer $n$, there exists a nontrivial virtual knot whose finite type invariants of our F-order $\le n-1$ coincide with those of the trivial knot (Theorem 2). In order to prove Theorem 1 (Theorem 2, resp.), we define an $n$-triviality via a certain unknotting operation, called virtualization (forbidden moves, resp.), and for any positive integer $n$, find an $n$-trivial classical knot (virtual knot, resp.).
</p>projecteuclid.org/euclid.jmsj/1541667932_20190124220103Thu, 24 Jan 2019 22:01 EST