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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 EDTA uniqueness of periodic maps on surfaceshttp://projecteuclid.org/euclid.jmsj/1477327233<strong>Susumu HIROSE</strong>, <strong>Yasushi KASAHARA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 68, Number 4, 1777--1787.</p><p><strong>Abstract:</strong><br/>
Kulkarni showed that, if $g$ is greater than $3$, a periodic map on an oriented surface $\Sigma_g$ of genus $g$ with order not smaller than $4g$ is uniquely determined by its order, up to conjugation and power. In this paper, we show that, if $g$ is greater than $30$, the same phenomenon happens for periodic maps on the surfaces with orders more than $8g/3$, and, for any integer $N$, there is $g > N$ such that there are periodic maps of $\Sigma_g$ of order $8g/3$ which are not conjugate up to power each other. Moreover, as a byproduct of our argument, we provide a short proof of Wiman's classical theorem: the maximal order of periodic maps of $\Sigma_g$ is $4g+2$.
</p>projecteuclid.org/euclid.jmsj/1477327233_20161024124030Mon, 24 Oct 2016 12:40 EDTEquivariant weight filtration for real algebraic varieties with actionhttp://projecteuclid.org/euclid.jmsj/1477327234<strong>Fabien PRIZIAC</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 68, Number 4, 1789--1818.</p><p><strong>Abstract:</strong><br/>
We show the existence of a weight filtration on the equivariant homology of real algebraic varieties equipped with a finite group action, by applying group homology to the weight complex of McCrory and Parusiński. If the group is of even order, we can not extract additive invariants directly from the induced spectral sequence.
Nevertheless, we construct finite additive invariants in terms of bounded long exact sequences, recovering Fichou's equivariant virtual Betti numbers in some cases. In the case of the two-elements group, we recover these additive invariants by using globally invariant chains and the equivariant version of Guillén and Navarro Aznar's extension criterion.
</p>projecteuclid.org/euclid.jmsj/1477327234_20161024124030Mon, 24 Oct 2016 12:40 EDTDarboux curves on surfaces Ihttp://projecteuclid.org/euclid.jmsj/1484730016<strong>Ronaldo GARCIA</strong>, <strong>Rémi LANGEVIN</strong>, <strong>Paweł WALCZAK</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 1, 1--24.</p><p><strong>Abstract:</strong><br/>
In 1872, G. Darboux defined a family of curves on surfaces of $\mathbb{R}^3$ which are preserved by the action of the Möbius group and share many properties with geodesics. Here, we characterize these curves under the view point of Lorentz geometry and prove that they are geodesics in a 3-dimensional sub-variety of a quadric $\Lambda^4$ contained in the 5-dimensional Lorentz space $\mathbb{R}^5_1$ naturally associated to the surface. We construct a new conformal object: the Darboux plane-field $\mathcal{D}$ and give a condition depending on the conformal principal curvatures of the surface which guarantees its integrability. We show that $\mathcal{D}$ is integrable when the surface is a special canal.
</p>projecteuclid.org/euclid.jmsj/1484730016_20170118040048Wed, 18 Jan 2017 04:00 ESTAutomorphicity and mean-periodicityhttp://projecteuclid.org/euclid.jmsj/1484730017<strong>Thomas OLIVER</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 1, 25--51.</p><p><strong>Abstract:</strong><br/>
If $C$ is a smooth projective curve over a number field $k$, then, under fair hypotheses, its $L$-function admits meromorphic continuation and satisfies the anticipated functional equation if and only if a related function is $\mathfrak{X}$-mean-periodic for some appropriate functional space $\mathfrak{X}$. Building on the work of Masatoshi Suzuki for modular elliptic curves, we will explore the dual relationship of this result to the widely believed conjecture that such $L$-functions should be automorphic. More precisely, we will directly show the orthogonality of the matrix coefficients of $GL_{2g}$-automorphic representations to the vector spaces $\mathcal{T}(h(\mathcal{S},\{k_i\},s))$, which are constructed from the Mellin transforms $f(\mathcal{S},\{k_i\},s)$ of certain products of arithmetic zeta functions $\zeta(\mathcal{S},2s)\prod_{i}\zeta(k_i,s)$, where $\mathcal{S}\rightarrow {\rm Spec}(\mathcal{O}_k)$ is any proper regular model of $C$ and $\{k_i\}$ is a finite set of finite extensions of $k$. To compare automorphicity and mean-periodicity, we use a technique emulating the Rankin–Selberg method, in which the function $h(\mathcal{S},\{k_i\},s))$ plays the role of an Eisenstein series, exploiting the spectral interpretation of the zeros of automorphic $L$-functions.
</p>projecteuclid.org/euclid.jmsj/1484730017_20170118040048Wed, 18 Jan 2017 04:00 ESTDeformations of Killing spinors on Sasakian and 3-Sasakian manifoldshttp://projecteuclid.org/euclid.jmsj/1484730018<strong>Craig van COEVERING</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 1, 53--91.</p><p><strong>Abstract:</strong><br/>
We consider some natural infinitesimal Einstein deformations on Sasakian and 3-Sasakian manifolds. Some of these are infinitesimal deformations of Killing spinors and further some integrate to actual Killing spinor deformations. In particular, on 3-Sasakian 7 manifolds these yield infinitesimal Einstein deformations preserving 2, 1, or none of the 3 independent Killing spinors. Toric 3-Sasakian manifolds provide non-trivial examples with integrable deformation preserving precisely 2 Killing spinors. Thus in contrast to the case of parallel spinors the dimension of Killing spinors is not preserved under Einstein deformations but is only upper semi-continuous.
</p>projecteuclid.org/euclid.jmsj/1484730018_20170118040048Wed, 18 Jan 2017 04:00 ESTMore on 2-chains with 1-shell boundaries in rosy theorieshttp://projecteuclid.org/euclid.jmsj/1484730019<strong>SunYoung KIM</strong>, <strong>Junguk LEE</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 1, 93--109.</p><p><strong>Abstract:</strong><br/>
In [ 4 ], B. Kim, and the authors classified 2-chains with 1-shell boundaries into either RN (renamable)-type or NR (non renamable)-type 2-chains up to renamability of support of subsummands of a 2-chain and introduced the notion of chain-walk, which was motivated from graph theory: a directed walk in a directed graph is a sequence of edges with compatible condition on initial and terminal vertices between sequential edges. We consider a directed graph whose vertices are 1-simplices whose supports contain $0$ and edges are plus/minus of $2$-simplices whose supports contain $0$. A chain-walk is a 2-chain induced from a directed walk in this graph. We reduced any 2-chains with 1-shell boundaries into chain-walks having the same boundaries.
In this paper, we reduce any 2-chains of 1-shell boundaries into chain-walks of the same boundary with support of size $3$. Using this reduction, we give a combinatorial criterion determining whether a minimal 2-chain is of RN- or NR-type. For a minimal RN-type 2-chains, we show that it is equivalent to a 2-chain of Lascar type (coming from model theory) if and only if it is equivalent to a planar type 2-chain.
</p>projecteuclid.org/euclid.jmsj/1484730019_20170118040048Wed, 18 Jan 2017 04:00 ESTA construction of diffusion processes associated with sub-Laplacian on CR manifolds and its applicationshttp://projecteuclid.org/euclid.jmsj/1484730020<strong>Hiroki KONDO</strong>, <strong>Setsuo TANIGUCHI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 1, 111--125.</p><p><strong>Abstract:</strong><br/>
A diffusion process associated with the real sub-Laplacian $\Delta_b$, the real part of the complex Kohn–Spencer Laplacian $\square_b$, on a strictly pseudoconvex CR manifold is constructed via the Eells–Elworthy–Malliavin method by taking advantage of the metric connection due to Tanaka and Webster. Using the diffusion process and the Malliavin calculus, the heat kernel and the Dirichlet problem for $\Delta_b$ are studied in a probabilistic manner. Moreover, distributions of stochastic line integrals along the diffusion process will be investigated.
</p>projecteuclid.org/euclid.jmsj/1484730020_20170118040048Wed, 18 Jan 2017 04:00 EST$L^{p}$ measure of growth and higher order Hardy–Sobolev–Morrey inequalities on $\Bbb{R}^{N}$http://projecteuclid.org/euclid.jmsj/1484730021<strong>Patrick J. RABIER</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 1, 127--151.</p><p><strong>Abstract:</strong><br/>
When the growth at infinity of a function $u$ on $\Bbb{R}^{N}$ is compared with the growth of $|x|^{s}$ for some $s\in \Bbb{R},$ this comparison is invariably made pointwise. This paper argues that the comparison can also be made in a suitably defined $L^{p}$ sense for every $1\leq p$ < $\infty$ and that, in this perspective, inequalities of Hardy, Sobolev or Morrey type account for the fact that sub $|x|^{-N/p}$ growth of $\nabla u$ in the $L^{p}$ sense implies sub $|x|^{1-N/p}$ growth of $u$ in the $L^{q}$ sense for well chosen values of $q.$
By investigating how sub $|x|^{s}$ growth of $\nabla ^{k}u$ in the $L^{p}$ sense implies sub $|x|^{s+j}$ growth of $\nabla ^{k-j}u$ in the $L^{q}$ sense for (almost) arbitrary $s\in \Bbb{R}$ and for $q$ in a $p$-dependent range of values, a family of higher order Hardy/Sobolev/Morrey type inequalities is obtained, under optimal integrability assumptions.
These optimal inequalities take the form of estimates for $\nabla^{k-j}(u-\pi _{u}),$ $1\leq j\leq k,$ where $\pi _{u}$ is a suitable polynomial of degree at most $k-1,$ which is unique if and only if $s$ < $-k.$ More generally, it can be chosen independent of $(s,p)$ when $s$ remains in the same connected component of $\Bbb{R}\backslash \{-k,\ldots,-1\}.$
</p>projecteuclid.org/euclid.jmsj/1484730021_20170118040048Wed, 18 Jan 2017 04:00 ESTJoint universality for Lerch zeta-functionshttp://projecteuclid.org/euclid.jmsj/1484730022<strong>Yoonbok LEE</strong>, <strong>Takashi NAKAMURA</strong>, <strong>Łukasz PAŃKOWSKI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 1, 153--161.</p><p><strong>Abstract:</strong><br/>
For $0$ < $\alpha,$ $\lambda \leq 1$, the Lerch zeta-function is defined by $L(s;\alpha, \lambda) := \sum_{n=0}^\infty e^{2\pi i\lambda n} (n+\alpha)^{-s}$, where $\sigma$ > $1$. In this paper, we prove joint universality for Lerch zeta-functions with distinct $\lambda_1,\ldots,\lambda_m$ and transcendental $\alpha$.
</p>projecteuclid.org/euclid.jmsj/1484730022_20170118040048Wed, 18 Jan 2017 04:00 ESTClassification of log del Pezzo surfaces of index threehttp://projecteuclid.org/euclid.jmsj/1484730023<strong>Kento FUJITA</strong>, <strong>Kazunori YASUTAKE</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 1, 163--225.</p><p><strong>Abstract:</strong><br/>
A normal projective non-Gorenstein log-terminal surface $S$ is called a log del Pezzo surface of index three if the three-times of the anti-canonical divisor $-3K_S$ is an ample Cartier divisor. We classify all log del Pezzo surfaces of index three. The technique for the classification is based on the argument of Nakayama.
</p>projecteuclid.org/euclid.jmsj/1484730023_20170118040048Wed, 18 Jan 2017 04:00 ESTSurface diffeomorphisms with connected but not path-connected minimal sets containing arcshttp://projecteuclid.org/euclid.jmsj/1484730024<strong>Hiromichi NAKAYAMA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 1, 227--239.</p><p><strong>Abstract:</strong><br/>
In 1955, Gottschalk and Hedlund introduced in their book that Jones constructed a minimal homeomorphism whose minimal set is connectd but not path-connected and contains infinitely many arcs. However the homeomorphism is defined only on this set. In 1991, Walker first constructed a homeomorphism of $S^1\times \mathbf{R}$ with such a minimal set. In this paper, we will show that Walker's homeomorphism cannot be a diffeomorphism (Theorem 2). Furthermore, we will construct a $C^\infty$ diffeomorphism of $S^1\times \mathbf{R}$ with a compact connected but not path-connected minimal set containing arcs (Theorem 1) by using the approximation by conjugation method.
</p>projecteuclid.org/euclid.jmsj/1484730024_20170118040048Wed, 18 Jan 2017 04:00 ESTOn the fundamental groups of non-generic $\mathbb{R}$-join-type curves, IIhttp://projecteuclid.org/euclid.jmsj/1484730025<strong>Christophe EYRAL</strong>, <strong>Mutsuo OKA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 1, 241--262.</p><p><strong>Abstract:</strong><br/>
We study the fundamental groups of (the complements of) plane complex curves defined by equations of the form $f(y)=g(x)$, where $f$ and $g$ are polynomials with real coefficients and real roots (so-called $\mathbb{R}$-join-type curves). For generic (respectively, semi-generic) such polynomials, the groups in question are already considered in [ 6 ] (respectively, in [ 3 ]). In the present paper, we compute the fundamental groups of $\mathbb{R}$-join-type curves under a simple arithmetic condition on the multiplicities of the roots of $f$ and $g$ without assuming any (semi-)genericity condition.
</p>projecteuclid.org/euclid.jmsj/1484730025_20170118040048Wed, 18 Jan 2017 04:00 ESTThe Chabauty and the Thurston topologies on the hyperspace of closed subsetshttp://projecteuclid.org/euclid.jmsj/1484730026<strong>Katsuhiko MATSUZAKI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 1, 263--292.</p><p><strong>Abstract:</strong><br/>
For a regularly locally compact topological space $X$ of $\rm T_0$ separation axiom but not necessarily Hausdorff, we consider a map $\sigma$ from $X$ to the hyperspace $C(X)$ of all closed subsets of $X$ by taking the closure of each point of $X$. By providing the Thurston topology for $C(X)$, we see that $\sigma$ is a topological embedding, and by taking the closure of $\sigma(X)$ with respect to the Chabauty topology, we have the Hausdorff compactification $\widehat X$ of $X$. In this paper, we investigate properties of $\widehat X$ and $C(\widehat X)$ equipped with different topologies. In particular, we consider a condition under which a self-homeomorphism of a closed subspace of $C(X)$ with respect to the Chabauty topology is a self-homeomorphism in the Thurston topology.
</p>projecteuclid.org/euclid.jmsj/1484730026_20170118040048Wed, 18 Jan 2017 04:00 ESTSequentially Cohen–Macaulay Rees algebrashttp://projecteuclid.org/euclid.jmsj/1484730027<strong>Naoki TANIGUCHI</strong>, <strong>Tran Thi PHUONG</strong>, <strong>Nguyen Thi DUNG</strong>, <strong>Tran Nguyen AN</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 1, 293--309.</p><p><strong>Abstract:</strong><br/>
This paper studies the question of when the Rees algebras associated to arbitrary filtration of ideals are sequentially Cohen–Macaulay. Although this problem has been already investigated by [ CGT ], their situation is quite a bit of restricted, so we are eager to try the generalization of their results.
</p>projecteuclid.org/euclid.jmsj/1484730027_20170118040048Wed, 18 Jan 2017 04:00 ESTThe analytic torsion of the finite metric cone over a compact manifoldhttp://projecteuclid.org/euclid.jmsj/1484730028<strong>Luiz HARTMANN</strong>, <strong>Mauro SPREAFICO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 1, 311--371.</p><p><strong>Abstract:</strong><br/>
We give an explicit formula for the $L^2$ analytic torsion of the finite metric cone over an oriented compact connected Riemannian manifold. We provide an interpretation of the different factors appearing in this formula. We prove that the analytic torsion of the cone is the finite part of the limit obtained collapsing one of the boundaries, of the ratio of the analytic torsion of the frustum to a regularising factor. We show that the regularising factor comes from the set of the non square integrable eigenfunctions of the Laplace Beltrami operator on the cone.
</p>projecteuclid.org/euclid.jmsj/1484730028_20170118040048Wed, 18 Jan 2017 04:00 ESTOn stability of Leray's stationary solutions of the Navier–Stokes system in exterior domainshttp://projecteuclid.org/euclid.jmsj/1484730029<strong>Hajime KOBA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 1, 373--396.</p><p><strong>Abstract:</strong><br/>
This paper studies the stability of a stationary solution of the Navier–Stokes system in $3$-D exterior domains. The stationary solution is called a Leray's stationary solution if the Dirichlet integral is finite. We apply an energy inequality and maximal $L^p$-in-time regularity for Hilbert space-valued functions to derive the decay rate with respect to time of energy solutions to a perturbed Navier–Stokes system governing a Leray's stationary solution.
</p>projecteuclid.org/euclid.jmsj/1484730029_20170118040048Wed, 18 Jan 2017 04:00 ESTRotational beta expansion: ergodicity and soficnesshttp://projecteuclid.org/euclid.jmsj/1484730030<strong>Shigeki AKIYAMA</strong>, <strong>Jonathan CAALIM</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 1, 397--415.</p><p><strong>Abstract:</strong><br/>
We study a family of piecewise expanding maps on the plane, generated by composition of a rotation and an expansive similitude of expansion constant $\beta$. We give two constants $B_1$ and $B_2$ depending only on the fundamental domain that if $\beta$ > $B_1$ then the expanding map has a unique absolutely continuous invariant probability measure, and if $\beta$ > $B_2$ then it is equivalent to $2$-dimensional Lebesgue measure. Restricting to a rotation generated by $q$-th root of unity $\zeta$ with all parameters in $\mathbb{Q}(\zeta,\beta)$, the map gives rise to a sofic system when $\cos(2\pi/q) \in \mathbb{Q}(\beta)$ and $\beta$ is a Pisot number. It is also shown that the condition $\cos(2\pi/q) \in \mathbb{Q}(\beta)$ is necessary by giving a family of non-sofic systems for $q=5$.
</p>projecteuclid.org/euclid.jmsj/1484730030_20170118040048Wed, 18 Jan 2017 04:00 ESTAn index formula for a bundle homomorphism of the tangent bundle into a vector bundle of the same rank, and its applicationshttp://projecteuclid.org/euclid.jmsj/1484730031<strong>Kentaro SAJI</strong>, <strong>Masaaki UMEHARA</strong>, <strong>Kotaro YAMADA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 1, 417--457.</p><p><strong>Abstract:</strong><br/>
In a previous work, the authors introduced the notion of ‘coherent tangent bundle’, which is useful for giving a treatment of singularities of smooth maps without ambient spaces. Two different types of Gauss–Bonnet formulas on coherent tangent bundles on $2$-dimensional manifolds were proven, and several applications to surface theory were given.
Let $M^n$ ($n\ge 2$) be an oriented compact $n$-manifold without boundary and $TM^n$ its tangent bundle. Let $\mathcal{E}$ be a vector bundle of rank $n$ over $M^n$, and $\phi:TM^n\to \mathcal{E}$ an oriented vector bundle homomorphism. In this paper, we show that one of these two Gauss–Bonnet formulas can be generalized to an index formula for the bundle homomorphism $\phi$ under the assumption that $\phi$ admits only certain kinds of generic singularities.
We shall give several applications to hypersurface theory. Moreover, as an application for intrinsic geometry, we also give a characterization of the class of positive semi-definite metrics (called Kossowski metrics) which can be realized as the induced metrics of the coherent tangent bundles.
</p>projecteuclid.org/euclid.jmsj/1484730031_20170118040048Wed, 18 Jan 2017 04:00 ESTOn sharp bilinear Strichartz estimates of Ozawa–Tsutsumi typehttp://projecteuclid.org/euclid.jmsj/1492653636<strong>Jonathan BENNETT</strong>, <strong>Neal BEZ</strong>, <strong>Chris JEAVONS</strong>, <strong>Nikolaos PATTAKOS</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 2, 459--476.</p><p><strong>Abstract:</strong><br/>
We provide a comprehensive analysis of sharp bilinear estimates of Ozawa–Tsutsumi type for solutions $u$ of the free Schrödinger equation, which give sharp control on $|u|^2$ in classical Sobolev spaces. In particular, we generalise their estimates in such a way that provides a unification with some sharp bilinear estimates proved by Carneiro and Planchon–Vega, via entirely different methods, by seeing them all as special cases of a one-parameter family of sharp estimates. The extremal functions are solutions of the Maxwell–Boltzmann functional equation and hence Gaussian. For $u^2$ we argue that the natural analogous results involve certain dispersive Sobolev norms; in particular, despite the validity of the classical Ozawa–Tsutsumi estimates for both $|u|^2$ and $u^2$ in the classical Sobolev spaces, we show that Gaussians are not extremisers in the latter case for spatial dimensions strictly greater than two.
</p>projecteuclid.org/euclid.jmsj/1492653636_20170419220101Wed, 19 Apr 2017 22:01 EDTValue distribution of leafwise holomorphic maps on complex laminations by hyperbolic Riemann surfaceshttp://projecteuclid.org/euclid.jmsj/1492653637<strong>Atsushi ATSUJI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 2, 477--501.</p><p><strong>Abstract:</strong><br/>
We discuss the value distribution of Borel measurable maps which are holomorphic along leaves of complex laminations. In the case of complex lamination by hyperbolic Riemann surfaces with an ergodic harmonic measure, we have a defect relation appearing in Nevanlinna theory. It gives a bound of the number of omitted hyperplanes in general position by those maps.
</p>projecteuclid.org/euclid.jmsj/1492653637_20170419220101Wed, 19 Apr 2017 22:01 EDTOn products in a real moment-angle manifoldhttp://projecteuclid.org/euclid.jmsj/1492653638<strong>Li CAI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 2, 503--528.</p><p><strong>Abstract:</strong><br/>
In this paper we give a necessary and sufficient condition for a (real) moment-angle complex to be a topological manifold. The cup and cap products in a real moment-angle manifold are studied. Consequently, the cohomology ring (with coefficients integers) of a polyhedral product by pairs of disks and their bounding spheres is isomorphic to that of a differential graded algebra associated to $K$ and the dimensions of the disks.
</p>projecteuclid.org/euclid.jmsj/1492653638_20170419220101Wed, 19 Apr 2017 22:01 EDTMultilinear Fourier multipliers with minimal Sobolev regularity, IIhttp://projecteuclid.org/euclid.jmsj/1492653639<strong>Loukas GRAFAKOS</strong>, <strong>Akihiko MIYACHI</strong>, <strong>Hanh VAN NGUYEN</strong>, <strong>Naohito TOMITA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 2, 529--562.</p><p><strong>Abstract:</strong><br/>
We provide characterizations for boundedness of multilinear Fourier multiplier operators on Hardy or Lebesgue spaces with symbols locally in Sobolev spaces. Let $H^q(\mathbb R^n)$ denote the Hardy space when $0 \lt q \le 1$ and the Lebesgue space $L^q(\mathbb R^n)$ when $1 \lt q \le \infty$. We find optimal conditions on $m$-linear Fourier multiplier operators to be bounded from $H^{p_1}\times \cdots \times H^{p_m}$ to $L^p$ when $1/p=1/p_1+\cdots +1/p_m$ in terms of local $L^2$-Sobolev space estimates for the symbol of the operator. Our conditions provide multilinear analogues of the linear results of Calderón and Torchinsky [ 1 ] and of the bilinear results of Miyachi and Tomita [ 17 ]. The extension to general $m$ is significantly more complicated both technically and combinatorially; the optimal Sobolev space smoothness required of the symbol depends on the Hardy–Lebesgue exponents and is constant on various convex simplices formed by configurations of $m2^{m-1}+1$ points in $[0,\infty)^m$.
</p>projecteuclid.org/euclid.jmsj/1492653639_20170419220101Wed, 19 Apr 2017 22:01 EDTAn example of Schwarz map of reducible Appell's hypergeometric equation $E_2$ in two variableshttp://projecteuclid.org/euclid.jmsj/1492653640<strong>Keiji MATSUMOTO</strong>, <strong>Takeshi SASAKI</strong>, <strong>Tomohide TERASOMA</strong>, <strong>Masaaki YOSHIDA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 2, 563--595.</p><p><strong>Abstract:</strong><br/>
We study an Appell hypergeometric system $E_2$ of rank four which is reducible, and show that its Schwarz map admits geometric interpretations: the map can be considered as the universal Abel–Jacobi map of a 1-parameter family of curves of genus 2.
</p>projecteuclid.org/euclid.jmsj/1492653640_20170419220101Wed, 19 Apr 2017 22:01 EDTDimension formulas of paramodular forms of squarefree level and comparison with inner twisthttp://projecteuclid.org/euclid.jmsj/1492653641<strong>Tomoyoshi IBUKIYAMA</strong>, <strong>Hidetaka KITAYAMA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 2, 597--671.</p><p><strong>Abstract:</strong><br/>
In this paper, we give an explicit dimension formula for the spaces of Siegel paramodular cusp forms of degree two of squarefree level. As an application, we propose a conjecture on symplectic group version of Eichler–Jacquet–Langlands type correspondence. It is a generalization of the previous conjecture of the first named author for prime levels published in 1985, where inner twists corresponding to binary quaternion hermitian forms over definite quaternion algebras were treated. Our present study contains also the case of indefinite quaternion algebras. Additionally, we give numerical examples of $L$ functions which support the conjecture. These comparisons of dimensions and examples give also evidence for conjecture on a certain precise lifting theory. This is related to the lifting theory from pairs of elliptic cusp forms initiated by Y. Ihara in 1964 in the case of compact twist, but no such construction is known in the case of non-split symplectic groups corresponding to quaternion hermitian groups over indefinite quaternion algebras and this is new in that sense.
</p>projecteuclid.org/euclid.jmsj/1492653641_20170419220101Wed, 19 Apr 2017 22:01 EDTAsymptotic expansion of resolvent kernels and behavior of spectral functions for symmetric stable processeshttp://projecteuclid.org/euclid.jmsj/1492653642<strong>Masaki WADA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 2, 673--692.</p><p><strong>Abstract:</strong><br/>
We give a precise behavior of spectral functions for symmetric stable processes applying the asymptotic expansion of resolvent kernels.
</p>projecteuclid.org/euclid.jmsj/1492653642_20170419220101Wed, 19 Apr 2017 22:01 EDTPseudograph and its associated real toric manifoldhttp://projecteuclid.org/euclid.jmsj/1492653643<strong>Suyoung CHOI</strong>, <strong>Boram PARK</strong>, <strong>Seonjeong PARK</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 2, 693--714.</p><p><strong>Abstract:</strong><br/>
Given a simple graph $G$, the graph associahedron $P_G$ is a convex polytope whose facets correspond to the connected induced subgraphs of $G$. Graph associahedra have been studied widely and are found in a broad range of subjects. Recently, S. Choi and H. Park computed the rational Betti numbers of the real toric variety corresponding to a graph associahedron under the canonical Delzant realization. In this paper, we focus on a pseudograph associahedron which was introduced by Carr, Devadoss and Forcey, and then discuss how to compute the Poincaré polynomial of the real toric variety corresponding to a pseudograph associahedron under the canonical Delzant realization.
</p>projecteuclid.org/euclid.jmsj/1492653643_20170419220101Wed, 19 Apr 2017 22:01 EDTA note on bounded-cohomological dimension of discrete groupshttp://projecteuclid.org/euclid.jmsj/1492653644<strong>Clara LÖOH</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 2, 715--734.</p><p><strong>Abstract:</strong><br/>
Bounded-cohomological dimension of groups is a relative of classical cohomological dimension, defined in terms of bounded cohomology with trivial coefficients instead of ordinary group cohomology. We will discuss constructions that lead to groups with infinite bounded-cohomological dimension, and we will provide new examples of groups with bounded-cohomological dimension equal to 0. In particular, we will prove that every group functorially embeds into an acyclic group with trivial bounded cohomology.
</p>projecteuclid.org/euclid.jmsj/1492653644_20170419220101Wed, 19 Apr 2017 22:01 EDTCartan matrices and Brauer's $k(B)$-conjecture IVhttp://projecteuclid.org/euclid.jmsj/1492653645<strong>Benjamin SAMBALE</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 2, 735--754.</p><p><strong>Abstract:</strong><br/>
In this note we give applications of recent results coming mostly from the third paper of this series. It is shown that the number of irreducible characters in a $p$-block of a finite group with abelian defect group $D$ is bounded by $|D|$ (Brauer's $k(B)$-conjecture) provided $D$ has no large elementary abelian direct summands. Moreover, we verify Brauer's $k(B)$-conjecture for all blocks with minimal non-abelian defect groups. This extends previous results by various authors.
</p>projecteuclid.org/euclid.jmsj/1492653645_20170419220101Wed, 19 Apr 2017 22:01 EDTThe category of reduced orbifolds in local chartshttp://projecteuclid.org/euclid.jmsj/1492653646<strong>Anke D. POHL</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 2, 755--800.</p><p><strong>Abstract:</strong><br/>
It is well-known that reduced smooth orbifolds and proper effective foliation Lie groupoids form equivalent categories. However, for certain recent lines of research, equivalence of categories is not sufficient. We propose a notion of maps between reduced smooth orbifolds and a definition of a category in terms of marked proper effective étale Lie groupoids such that the arising category of orbifolds is isomorphic (not only equivalent) to this groupoid category.
</p>projecteuclid.org/euclid.jmsj/1492653646_20170419220101Wed, 19 Apr 2017 22:01 EDTLifting puzzles and congruences of Ikeda and Ikeda–Miyawaki liftshttp://projecteuclid.org/euclid.jmsj/1492653647<strong>Neil DUMMIGAN</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 2, 801--818.</p><p><strong>Abstract:</strong><br/>
We show how many of the congruences between Ikeda lifts and non-Ikeda lifts, proved by Katsurada, can be reduced to congruences involving only forms of genus 1 and 2, using various liftings predicted by Arthur's multiplicity conjecture. Similarly, we show that conjectured congruences between Ikeda–Miyawaki lifts and non-lifts can often be reduced to congruences involving only forms of genus 1, 2 and 3.
</p>projecteuclid.org/euclid.jmsj/1492653647_20170419220101Wed, 19 Apr 2017 22:01 EDTContact of a regular surface in Euclidean 3-space with cylinders and cubic binary differential equationshttp://projecteuclid.org/euclid.jmsj/1492653648<strong>Toshizumi FUKUI</strong>, <strong>Masaru HASEGAWA</strong>, <strong>Kouichi NAKAGAWA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 2, 819--847.</p><p><strong>Abstract:</strong><br/>
We investigate the contact types of a regular surface in the Euclidean 3-space $\mathbb{R}^3$ with right circular cylinders. We present the conditions for existence of cylinders with $A_1$, $A_2$, $A_3$, $A_4$, $A_5$, $D_4$, and $D_5$ contacts with a given surface. We also investigate the kernel field of $A_{\ge 3}$-contact cylinders on the surface. This is defined by a cubic binary differential equation and we classify singularity types of its flow in the generic context.
</p>projecteuclid.org/euclid.jmsj/1492653648_20170419220101Wed, 19 Apr 2017 22:01 EDTIntegral transformation of Heun's equation and some applicationshttp://projecteuclid.org/euclid.jmsj/1492653649<strong>Kouichi TAKEMURA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 2, 849--891.</p><p><strong>Abstract:</strong><br/>
It is known that the Fuchsian differential equation which produces the sixth Painlevé equation corresponds to the Fuchsian differential equation with different parameters via Euler's integral transformation, and Heun's equation also corresponds to Heun's equation with different parameters, again via Euler's integral transformation. In this paper we study the correspondences in detail. After investigating correspondences with respect to monodromy, it is demonstrated that the existence of polynomial-type solutions corresponds to apparency of a singularity. For the elliptical representation of Heun's equation, correspondence with respect to monodromy implies isospectral symmetry. We apply the symmetry to finite-gap potentials and express the monodromy of Heun's equation with parameters which have not yet been studied.
</p>projecteuclid.org/euclid.jmsj/1492653649_20170419220101Wed, 19 Apr 2017 22:01 EDTExtension theorem for rough paths via fractional calculushttp://projecteuclid.org/euclid.jmsj/1499846512<strong>Yu ITO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 3, 893--912.</p><p><strong>Abstract:</strong><br/>
On the basis of fractional calculus, we introduce an integral of weakly controlled paths, which is a generalization of integrals in the context of rough path analysis. As an application, we provide an alternative proof of Lyons' extension theorem for geometric Hölder rough paths together with an explicit expression of the extension map.
</p>projecteuclid.org/euclid.jmsj/1499846512_20170712040232Wed, 12 Jul 2017 04:02 EDTSome consequences from Proper Forcing Axiom together with large continuum and the negation of Martin's Axiomhttp://projecteuclid.org/euclid.jmsj/1499846513<strong>Teruyuki YORIOKA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 3, 913--943.</p><p><strong>Abstract:</strong><br/>
Recently, David Asperó and Miguel Angel Mota discovered a new method of iterated forcing using models as side conditions. The side condition method with models was introduced by Stevo Todorčević in the 1980s. The Asperó–Mota iteration enables us to force some $\Pi_2$-statements over $H(\aleph_2)$ with the continuum greater than $\aleph_2$. In this article, by using the Asperó–Mota iteration, we prove that it is consistent that $\mho$ fails, there are no weak club guessing ladder systems, $\mathfrak{p}= \textup{add}(\mathcal{N}) = 2^{\aleph_0}>\aleph_2$ and $\textup{MA}_{\aleph_1}$ fails.
</p>projecteuclid.org/euclid.jmsj/1499846513_20170712040232Wed, 12 Jul 2017 04:02 EDTPoset pinball, GKM-compatible subspaces, and Hessenberg varietieshttp://projecteuclid.org/euclid.jmsj/1499846514<strong>Megumi HARADA</strong>, <strong>Julianna TYMOCZKO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 3, 945--994.</p><p><strong>Abstract:</strong><br/>
This paper has three main goals. First, we set up a general framework to address the problem of constructing module bases for the equivariant cohomology of certain subspaces of GKM spaces. To this end we introduce the notion of a GKM-compatible subspace of an ambient GKM space. We also discuss poset-upper-triangularity, a key combinatorial notion in both GKM theory and more generally in localization theory in equivariant cohomology. With a view toward other applications, we present parts of our setup in a general algebraic and combinatorial framework. Second, motivated by our central problem of building module bases, we introduce a combinatorial game which we dub poset pinball and illustrate with several examples. Finally, as first applications, we apply the perspective of GKM-compatible subspaces and poset pinball to construct explicit and computationally convenient module bases for the $S^1$-equivariant cohomology of all Peterson varieties of classical Lie type, and subregular Springer varieties of Lie type $A$. In addition, in the Springer case we use our module basis to lift the classical Springer representation on the ordinary cohomology of subregular Springer varieties to $S^1$-equivariant cohomology in Lie type $A$.
</p>projecteuclid.org/euclid.jmsj/1499846514_20170712040232Wed, 12 Jul 2017 04:02 EDTGeometry of the Gromov product: Geometry at infinity of Teichmüller spacehttp://projecteuclid.org/euclid.jmsj/1499846515<strong>Hideki MIYACHI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 3, 995--1049.</p><p><strong>Abstract:</strong><br/>
This paper is devoted to studying transformations on metric spaces. It is done in an effort to produce qualitative version of quasi-isometries which takes into account the asymptotic behavior of the Gromov product in hyperbolic spaces. We characterize a quotient semigroup of such transformations on Teichmüller space by use of simplicial automorphisms of the complex of curves, and we will see that such transformation is recognized as a “coarsification” of isometries on Teichmüller space which is rigid at infinity. We also show a hyperbolic characteristic that any finite dimensional Teichmüller space does not admit (quasi)-invertible rough-homothety.
</p>projecteuclid.org/euclid.jmsj/1499846515_20170712040232Wed, 12 Jul 2017 04:02 EDTDoubly transitive groups and cyclic quandleshttp://projecteuclid.org/euclid.jmsj/1499846516<strong>Leandro VENDRAMIN</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 3, 1051--1057.</p><p><strong>Abstract:</strong><br/>
We prove that for $n \gt 2$ there exists a quandle of cyclic type of size $n$ if and only if $n$ is a power of a prime number. This establishes a conjecture of S. Kamada, H. Tamaru and K. Wada. As a corollary, every finite quandle of cyclic type is an Alexander quandle. We also prove that finite doubly transitive quandles are of cyclic type. This establishes a conjecture of H. Tamaru.
</p>projecteuclid.org/euclid.jmsj/1499846516_20170712040232Wed, 12 Jul 2017 04:02 EDTFractional integral operators with homogeneous kernels on Morrey spaces with variable exponentshttp://projecteuclid.org/euclid.jmsj/1499846517<strong>Kwok-Pun HO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 3, 1059--1077.</p><p><strong>Abstract:</strong><br/>
We establish the mapping properties of the fractional integral operators with homogeneous kernels on Morrey spaces with variable exponents.
</p>projecteuclid.org/euclid.jmsj/1499846517_20170712040232Wed, 12 Jul 2017 04:02 EDTOn usual, virtual and welded knotted objects up to homotopyhttp://projecteuclid.org/euclid.jmsj/1499846518<strong>Benjamin AUDOUX</strong>, <strong>Paolo BELLINGERI</strong>, <strong>Jean-Baptiste MEILHAN</strong>, <strong>Emmanuel WAGNER</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 3, 1079--1097.</p><p><strong>Abstract:</strong><br/>
We consider several classes of knotted objects, namely usual, virtual and welded pure braids and string links, and two equivalence relations on those objects, induced by either self-crossing changes or self-virtualizations. We provide a number of results which point out the differences between these various notions. The proofs are mainly based on the techniques of Gauss diagram formulae.
</p>projecteuclid.org/euclid.jmsj/1499846518_20170712040232Wed, 12 Jul 2017 04:02 EDTShifted products of Fourier coefficients of Siegel cusp forms of degree twohttp://projecteuclid.org/euclid.jmsj/1499846519<strong>Winfried KOHNEN</strong>, <strong>Jyoti SENGUPTA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 3, 1099--1103.</p><p><strong>Abstract:</strong><br/>
We prove a non-negativity result for shifted products of two Fourier coefficients of a Siegel Hecke eigenform of degree two not in the Maass space.
</p>projecteuclid.org/euclid.jmsj/1499846519_20170712040232Wed, 12 Jul 2017 04:02 EDTModules over quantized coordinate algebras and PBW-baseshttp://projecteuclid.org/euclid.jmsj/1499846520<strong>Toshiyuki TANISAKI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 3, 1105--1156.</p><p><strong>Abstract:</strong><br/>
Around 1990 Soibelman constructed certain irreducible modules over the quantized coordinate algebra. A. Kuniba, M. Okado, Y. Yamada [ 8 ] recently found that the relation among natural bases of Soibelman's irreducible module can be described using the relation among the PBW-type bases of the positive part of the quantized enveloping algebra, and proved this fact using case-by-case analysis in rank two cases. In this paper we will give a realization of Soibelman's module as an induced module, and give a unified proof of the above result of [ 8 ]. We also verify Conjecture 1 of [ 8 ] about certain operators on Soibelman's module.
</p>projecteuclid.org/euclid.jmsj/1499846520_20170712040232Wed, 12 Jul 2017 04:02 EDTStructure and equivalence of a class of tube domains with solvable groups of automorphismshttp://projecteuclid.org/euclid.jmsj/1499846521<strong>Satoru SHIMIZU</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 3, 1157--1177.</p><p><strong>Abstract:</strong><br/>
In the study of the holomorphic equivalence problem for tube domains, it is fundamental to investigate tube domains with polynomial infinitesimal automorphisms. To apply Lie group theory to the holomorphic equivalence problem for such tube domains $T_\Omega$, investigating certain solvable subalgebras of $\frak g(T_{\Omega})$ plays an important role, where $\frak g(T_{\Omega})$ is the Lie algebra of all complete polynomial vector fields on $T_\Omega$. Related to this theme, we discuss in this paper the structure and equivalence of a class of tube domains with solvable groups of automorphisms. Besides, we give a concrete example of a tube domain whose automorphism group is solvable and contains nonaffine automorphisms.
</p>projecteuclid.org/euclid.jmsj/1499846521_20170712040232Wed, 12 Jul 2017 04:02 EDTHypergroup structures arising from certain dual objects of a hypergrouphttp://projecteuclid.org/euclid.jmsj/1499846522<strong>Herbert HEYER</strong>, <strong>Satoshi KAWAKAMI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 3, 1179--1195.</p><p><strong>Abstract:</strong><br/>
In the present paper hypergroup structures are investigated on distinguished dual objects related to a given hypergroup $K$, especially to a semi-direct product hypergroup $K = H \rtimes_\alpha G$ defined by an action $\alpha$ of a locally compact group $G$ on a commutative hypergroup $H$. Typical dual objects are the sets of equivalence classes of irreducible representations of $K$, of infinite-dimensional irreducible representations of type I hypergroups $K$, and of quasi-equivalence classes of type $\text{II}_1$ factor representations of non-type I hypergroups $K$. The method of proof relies on the notion of a character of a representation of $K = H \rtimes_\alpha G$.
</p>projecteuclid.org/euclid.jmsj/1499846522_20170712040232Wed, 12 Jul 2017 04:02 EDTA class of minimal submanifolds in sphereshttp://projecteuclid.org/euclid.jmsj/1499846523<strong>Marcos DAJCZER</strong>, <strong>Theodoros VLACHOS</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 3, 1197--1212.</p><p><strong>Abstract:</strong><br/>
We introduce a class of minimal submanifolds $M^n$, $n\geq 3$, in spheres $\mathbb{S}^{n+2}$ that are ruled by totally geodesic spheres of dimension $n-2$. If simply-connected, such a submanifold admits a one-parameter associated family of equally ruled minimal isometric deformations that are genuine. As for compact examples, there are plenty of them but only for dimensions $n=3$ and $n=4$. In the first case, we have that $M^3$ must be a $\mathbb{S}^1$-bundle over a minimal torus $T^2$ in $\mathbb{S}^5$ and in the second case $M^4$ has to be a $\mathbb{S}^2$-bundle over a minimal sphere $\mathbb{S}^2$ in $\mathbb{S}^6$. In addition, we provide new examples in relation to the well-known Chern-do Carmo–Kobayashi problem since taking the torus $T^2$ to be flat yields minimal submanifolds $M^3$ in $\mathbb{S}^5$ with constant scalar curvature.
</p>projecteuclid.org/euclid.jmsj/1499846523_20170712040232Wed, 12 Jul 2017 04:02 EDTDegenerations and fibrations of Riemann surfaces associated with regular polyhedra and soccer ballhttp://projecteuclid.org/euclid.jmsj/1499846524<strong>Ryota HIRAKAWA</strong>, <strong>Shigeru TAKAMURA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 3, 1213--1233.</p><p><strong>Abstract:</strong><br/>
To each of regular polyhedra and a soccer ball, we associate degenerating families ( degenerations ) of Riemann surfaces. More specifically: To each orientation-preserving automorphism of a regular polyhedron (and also of a soccer ball), we associate a degenerating family of Riemann surfaces whose topological monodromy is the automorphism. The complete classification of such degenerating families is given. Besides, we determine the Euler numbers of their total spaces. Furthermore, we affirmatively solve the compactification problem raised by Mutsuo Oka — we explicitly construct compact fibrations of Riemann surfaces that compactify the above degenerating families. Their singular fibers and Euler numbers are also determined.
</p>projecteuclid.org/euclid.jmsj/1499846524_20170712040232Wed, 12 Jul 2017 04:02 EDTAbsolutely $k$-convex domains and holomorphic foliations on homogeneous manifoldshttp://projecteuclid.org/euclid.jmsj/1499846525<strong>Maurício CORRÊA Jr.</strong>, <strong>Arturo FERNÁNDEZ-PÉREZ</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 3, 1235--1246.</p><p><strong>Abstract:</strong><br/>
We consider a holomorphic foliation $\mathcal{F}$ of codimension $k\geq 1$ on a homogeneous compact Kähler manifold $X$ of dimension $n \gt k$. Assuming that the singular set ${\rm Sing}(\mathcal{F})$ of $\mathcal{F}$ is contained in an absolutely $k$-convex domain $U\subset X$, we prove that the determinant of normal bundle $\det(N_{\mathcal{F}})$ of $\mathcal{F}$ cannot be an ample line bundle, provided $[n/k]\geq 2k+3$. Here $[n/k]$ denotes the largest integer $\leq n/k.$
</p>projecteuclid.org/euclid.jmsj/1499846525_20170712040232Wed, 12 Jul 2017 04:02 EDTForcing-theoretic aspects of Hindman's Theoremhttp://projecteuclid.org/euclid.jmsj/1499846526<strong>Jörg BRENDLE</strong>, <strong>Luz María GARCÍA ÁVILA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 3, 1247--1280.</p><p><strong>Abstract:</strong><br/>
We investigate the partial order $(\mathrm{FIN})^\omega$ of infinite block sequences, ordered by almost condensation, from the forcing-theoretic point of view. This order bears the same relationship to Hindman's Theorem as $\mathcal{P}(\omega) /\mathrm{fin}$ does to Ramsey's Theorem. While $(\mathcal{P}(\omega) / \mathrm{fin})^2$ completely embeds into $(\mathrm{FIN})^\omega$, we show this is consistently false for higher powers of $\mathcal{P} (\omega) / \mathrm{fin}$, by proving that the distributivity number $\mathfrak{h}_3$ of $(\mathcal{P} (\omega) /\mathrm{fin})^3$ may be strictly smaller than the distributivity number $\mathfrak{h}_{\mathrm{FIN}}$ of $(\mathrm{FIN})^\omega$. We also investigate infinite maximal antichains in $(\mathrm{FIN})^\omega$ and show that the least cardinality $\mathfrak{a}_{\mathrm{FIN}}$ of such a maximal antichain is at least the smallest size of a nonmeager set of reals. As a consequence, we obtain that $\mathfrak{a}_{\mathrm{FIN}}$ is consistently larger than $\mathfrak{a}$, the least cardinality of an infinite maximal antichain in $\mathcal{P} (\omega) / \mathrm{fin}$.
</p>projecteuclid.org/euclid.jmsj/1499846526_20170712040232Wed, 12 Jul 2017 04:02 EDTAnalytic semigroups for the subelliptic oblique derivative problemhttp://projecteuclid.org/euclid.jmsj/1499846527<strong>Kazuaki TAIRA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 69, Number 3, 1281--1330.</p><p><strong>Abstract:</strong><br/>
This paper is devoted to a functional analytic approach to the {\em subelliptic} oblique derivative problem for second-order, elliptic differential operators with a complex parameter $\lambda$. We prove an existence and uniqueness theorem of the homogeneous oblique derivative problem in the framework of $L^{p}$ Sobolev spaces when $\vert\lambda\vert$ tends to $\infty$. As an application of the main theorem, we prove generation theorems of analytic semigroups for this subelliptic oblique derivative problem in the $L^{p}$ topology and in the topology of uniform convergence. Moreover, we solve the long-standing open problem of the asymptotic eigenvalue distribution for the subelliptic oblique derivative problem. In this paper we make use of Agmon's technique of treating a spectral parameter $\lambda$ as a second-order elliptic differential operator of an extra variable on the unit circle and relating the old problem to a new one with the additional variable.
</p>projecteuclid.org/euclid.jmsj/1499846527_20170712040232Wed, 12 Jul 2017 04:02 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 EDT