Osaka Journal of Mathematics Articles (Project Euclid)
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The latest articles from Osaka Journal of Mathematics on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTTue, 22 Mar 2011 10:05 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Quotients of bounded homogeneous domains by cyclic groups
http://projecteuclid.org/euclid.ojm/1277298908
<strong>Christian Miebach</strong><p><strong>Source: </strong>Osaka J. Math., Volume 47, Number 2, 331--352.</p><p><strong>Abstract:</strong><br/>
Let $D$ be a bounded homogeneous domain in $\mathbb{C}^{n}$
and let $\varphi$ be an automorphism of $D$ which generates
a discrete subgroup $\Gamma$ of $\Aut_{\mathcal{O}}(D)$. It
is shown that the complex space $D/\Gamma$ is Stein.
</p>projecteuclid.org/euclid.ojm/1277298908_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTFrobenius structures and characters of affine Lie algebrashttps://projecteuclid.org/euclid.ojm/1547607634<strong>Ikuo Satake</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 1, 183--212.</p><p><strong>Abstract:</strong><br/>
The explicit description of the Frobenius structure for the elliptic root system of type $D_4^{(1,1)}$ in terms of the characters of an affine Lie algebra of type $D_4^{(1)}$ is given.
</p>projecteuclid.org/euclid.ojm/1547607634_20190115220101Tue, 15 Jan 2019 22:01 ESTA classification problem on mapping classes on fiber spaces over Teichmüller spaceshttps://projecteuclid.org/euclid.ojm/1554278420<strong>Yingqing Xiao</strong>, <strong>Chaohui Zhang</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 2, 213--227.</p><p><strong>Abstract:</strong><br/>
Let $\tilde{S}$ be an analytically finite Riemann surface which is equipped with a hyperbolic metric. Let $S=\tilde{S}\backslash \{\mbox{one point}\ x\}$. There exists a natural projection $\Pi$ of the $x$-pointed mapping class group Mod$_S^x$ onto the mapping class group Mod$(\tilde{S})$. In this paper, we classify elements in the fiber $\Pi^{-1}(\chi)$ for an elliptic element $\chi\in \mbox{Mod}(\tilde{S})$, and give a geometric interpretation for each element in $\Pi^{-1}(\chi)$. We also prove that $\Pi^{-1}(t_a^n\circ \chi)$ or $\Pi^{-1}(t_a^n\circ \chi^{-1})$ consists of hyperbolic mapping classes provided that $t_a^n\circ \chi$ and $t_a^n\circ \chi^{-1}$ are hyperbolic, where $a$ is a simple closed geodesic on $\tilde{S}$ and $t_a$ is the positive Dehn twist along $a$.
</p>projecteuclid.org/euclid.ojm/1554278420_20190403040038Wed, 03 Apr 2019 04:00 EDTA Block Refinement of the Green-Puig Parameterization of the Isomorphism Types of Indecomposable Moduleshttps://projecteuclid.org/euclid.ojm/1554278421<strong>Morton E. Harris</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 2, 229--236.</p><p><strong>Abstract:</strong><br/>
Let $p$ be a prime integer, let $\mathcal{O}$ be a commutative complete local Noetherian ring with an algebraically closed residue field $k$ of charateristic $p$ and let$G$ be a finite group. Let $P$ be a $p$-subgroup of $G$ and let $X$ be an indecomposable $\mathcal{O} P$-module with vertex $P$. Let $\Lambda (G,P,X)$ denote a set of representatives for the isomorphism classes of indecomposable $\mathcal{O} G$-modules with vertex-source pair $(P,X)$ (so that $\Lambda(G,P,X)$ is a finite set by the Green correspondence). As mentioned in [5, Notes on Section~26], L. Puig asserted that a defect multiplicity module determined by $(P,X)$ can be used to obtain an extended parameterization of $\Lambda(G,P,X)$. In [5, Proposition 26.3], J. Thévenaz completed this program under the hypotheses that $X$ is $\mathcal{O}$-free. Here we use the methods of proof of [5, Theorem 26.3] to show that the $\mathcal{O}$-free hypothesis on $X$ is superfluous. (M. Linckelmann has also proved this, cf. [3]). Let $B$ be a block of $\mathcal{O} G$. Then we obtain a corresponding paramaterization of the $(\mathcal{O} G)B$-modules in $\Lambda(G,P,X)$.
</p>projecteuclid.org/euclid.ojm/1554278421_20190403040038Wed, 03 Apr 2019 04:00 EDTToric manifolds over cyclohedrahttps://projecteuclid.org/euclid.ojm/1554278422<strong>Seonjeong Park</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 2, 237--254.</p><p><strong>Abstract:</strong><br/>
We study the action of the dihedral group on the (equivariant) cohomology of the toric manifolds associated with cycle graphs.
</p>projecteuclid.org/euclid.ojm/1554278422_20190403040038Wed, 03 Apr 2019 04:00 EDTA Markov's theorem for extended welded braids and linkshttps://projecteuclid.org/euclid.ojm/1554278423<strong>Celeste Damiani</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 2, 255--268.</p><p><strong>Abstract:</strong><br/>
Extended welded links are a generalization of Fenn, Rimányi, and Rourke's welded links. Their braided counterpart are extended welded braids, which are closely related to ribbon braids and loop braids. In this paper we prove versions of Alexander and Markov's theorems for extended welded braids and links, following Kamada's approach to the case of welded objects.
</p>projecteuclid.org/euclid.ojm/1554278423_20190403040038Wed, 03 Apr 2019 04:00 EDTUniform well-posedness for a time-dependent Ginzburg-Landau model in superconductivityhttps://projecteuclid.org/euclid.ojm/1554278424<strong>Jishan Fan</strong>, <strong>Bessem Samet</strong>, <strong>Yong Zhou</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 2, 269--276.</p><p><strong>Abstract:</strong><br/>
We study the initial boundary value problem for a time-dependent Ginzburg-Landau model in superconductivity. First, we prove the uniform boundedness of strong solutions with respect to diffusion coefficient 0 < $\epsilon$ < 1 in the case of Coulomb gauge. Our second result is the global existence and uniqueness of the weak solutions to the limit problem when $\epsilon=0$.
</p>projecteuclid.org/euclid.ojm/1554278424_20190403040038Wed, 03 Apr 2019 04:00 EDTCurves with maximally computed Clifford indexhttps://projecteuclid.org/euclid.ojm/1554278425<strong>Takao Kato</strong>, <strong>Gerriet Martens</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 2, 277--288.</p><p><strong>Abstract:</strong><br/>
We say that a curve $X$ of genus $g$ has maximally computed Clifford index if the Clifford index $c$ of $X$ is, for $c>2$, computed by a linear series of the maximum possible degree $d$ < $g$; then $d = 2c+3$ resp. $d = 2c+4$ for odd resp. even $c$. For odd $c$ such curves have been studied in [6]. In this paper we analyze if/how far analoguous results hold for such curves of even Clifford index $c$.
</p>projecteuclid.org/euclid.ojm/1554278425_20190403040038Wed, 03 Apr 2019 04:00 EDTReduced contragredient Lie algebras and PC Lie algebrashttps://projecteuclid.org/euclid.ojm/1554278426<strong>Nagatoshi Sasano</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 2, 289--299.</p><p><strong>Abstract:</strong><br/>
Using the theory of standard pentads, we can embed an arbitrary finite-dimensional reductive Lie algebra and its finite-dimensional completely reducible representation into some larger graded Lie algebra. However, it is not easy to find the structure of the ``larger graded Lie algebra'' from the definition in general cases. Under these, the first aim of this paper is to show that the ``larger graded Lie algebra'' is isomorphic to some PC Lie algebra, which are Lie algebras corresponding to special standard pentads called pentads of Cartan type. The second aim is to find the structure of a PC Lie algebra.
</p>projecteuclid.org/euclid.ojm/1554278426_20190403040038Wed, 03 Apr 2019 04:00 EDTCleft Coextension for symmetric twisted partial coactions on coalgebrashttps://projecteuclid.org/euclid.ojm/1554278427<strong>Q.-G. Chen</strong>, <strong>B. Yang</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 2, 301--322.</p><p><strong>Abstract:</strong><br/>
In this paper, we will introduce the concepts of symmetric twisted partial Hopf coactions, and discuss under which conditions a given symmetric twisted partial Hopf coaction is globalizable. Then we will introduce the notion of partial cleft coextensions which are dual to partial cleft extensions introduced by M. M. S. Alves et.al., and discuss its relation with partial crossed coproducts introduced by the first author of this paper, which covers the classical results in classical Hopf algebra theory.
</p>projecteuclid.org/euclid.ojm/1554278427_20190403040038Wed, 03 Apr 2019 04:00 EDTSTRONG-VISCOSITY SOLUTIONS: CLASSICAL AND PATH-DEPENDENT PDEshttps://projecteuclid.org/euclid.ojm/1554278428<strong>Andrea Cosso</strong>, <strong>Francesco Russo</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 2, 323--373.</p><p><strong>Abstract:</strong><br/>
The aim of the present work is the introduction of a viscosity type solution, called $strong$-$viscosity$ $solution$ emphasizing also a similarity with the existing notion of $strong$ $solution$ in the literature. It has the following peculiarities: it is a purely analytic object; it can be easily adapted to more general equations than classical partial differential equations. First, we introduce the notion of strong-viscosity solution for semilinear parabolic partial differential equations, defining it, in a few words, as the pointwise limit of classical solutions to perturbed semilinear parabolic partial differential equations; we compare it with the standard definition of viscosity solution. Afterwards, we extend the concept of strong-viscosity solution to the case of semilinear parabolic path-dependent partial differential equations, providing an existence and uniqueness result.
</p>projecteuclid.org/euclid.ojm/1554278428_20190403040038Wed, 03 Apr 2019 04:00 EDTHopf bands in arborescent Hopf plumbingshttps://projecteuclid.org/euclid.ojm/1554278429<strong>Filip Misev</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 2, 375--389.</p><p><strong>Abstract:</strong><br/>
For a positive Hopf plumbed arborescent Seifert surface $S$, we study the set of Hopf bands $H\subset S$, up to homology and up to the action of the monodromy. The classification of Seifert surfaces for which this set is finite is closely related to the classification of finite Coxeter groups.
</p>projecteuclid.org/euclid.ojm/1554278429_20190403040038Wed, 03 Apr 2019 04:00 EDTInvariants of the trace map and uniform spectral properties for discrete Sturmian Dirac operatorshttps://projecteuclid.org/euclid.ojm/1554278430<strong>Roberto A. Prado</strong>, <strong>Ruy C. Charão</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 2, 391--416.</p><p><strong>Abstract:</strong><br/>
We establish invariants for the trace map associated to a family of 1D discrete Dirac operators with Sturmian potentials. Using these invariants we prove that the operators have purely singular continuous spectrum of zero Lebesgue measure, uniformly on the mass and parameters that define the potentials. For rotation numbers of bounded density we prove that these Dirac operators have purely $\alpha$-continuous spectrum, as to the Schrödinger case, for some $\alpha \in (0,1)$. To the Sturmian Schrödinger and Dirac models we establish a comparison between invariants of the trace maps, which allows to compare the numbers $\alpha$'s and lower bounds on transport exponents.
</p>projecteuclid.org/euclid.ojm/1554278430_20190403040038Wed, 03 Apr 2019 04:00 EDTHomotopy groups of certain highly connected manifolds via loop space homologyhttps://projecteuclid.org/euclid.ojm/1554278431<strong>Samik Basu</strong>, <strong>Somnath Basu</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 2, 417--430.</p><p><strong>Abstract:</strong><br/>
For $n\geq 2$ we consider $(n-1)$-connected closed manifolds of dimension at most $(3n-2)$. We prove that away from a finite set of primes, the $p$-local homotopy groups of $M$ are determined by the dimension of the space of indecomposable elements in the cohomology ring $H^\ast(M; \mathbb{Q})$. Moreover, we show that these $p$-local homotopy groups can be expressed as a direct sum of $p$-local homotopy groups of spheres. This generalizes some of the results of our earlier work [1].
</p>projecteuclid.org/euclid.ojm/1554278431_20190403040038Wed, 03 Apr 2019 04:00 EDTRemarks on Artin Approximation with constraintshttps://projecteuclid.org/euclid.ojm/1563242417<strong>Dorin Popescu</strong>, <strong>Guillaume Rond</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 3, 431--440.</p><p><strong>Abstract:</strong><br/>
We study various approximation results of solutions of equations $f(x,Y)=0$ where $f(x,Y)\in\mathbb{K}[\![x]\!][Y]^r$ and $x$ and $Y$ are two sets of variables, and where some components of the solutions $y(x)\in\mathbb{K}[\![x]\!]^m$ do not depend on all the variables $x_j$. These problems were highlighted by M. Artin.
</p>projecteuclid.org/euclid.ojm/1563242417_20190715220050Mon, 15 Jul 2019 22:00 EDTResults on the topology of generalized real Bott manifoldshttps://projecteuclid.org/euclid.ojm/1563242418<strong>Raisa Dsouza</strong>, <strong>V. Uma</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 3, 441--458.</p><p><strong>Abstract:</strong><br/>
Generalized Bott manifolds (over $\mathbb C$ and $\mathbb R$) have been defined by Choi, Masuda and Suh in [4]. In this article we extend the results of [7] on the topology of real Bott manifolds to generalized real Bott manifolds. We give a presentation of the fundamental group, prove that it is solvable and give a characterization for it to be abelian. We further prove that these manifolds are aspherical only in the case of real Bott manifolds and compute the higher homotopy groups. Furthermore, using the presentation of the cohomology ring with $\mathbb Z_2$-coefficients, we derive a combinatorial characterization for orientablity and spin structure.
</p>projecteuclid.org/euclid.ojm/1563242418_20190715220050Mon, 15 Jul 2019 22:00 EDTOn the non-periodic stable Auslander-Reiten Heller component for the Kronecker algebra over a complete discrete valuation ringhttps://projecteuclid.org/euclid.ojm/1563242420<strong>Kengo Miyamoto</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 3, 459--496.</p><p><strong>Abstract:</strong><br/>
We consider the Kronecker algebra $A=\mathcal{O}[X,Y]/(X^2,Y^2)$, where $\mathcal{O}$ is a complete discrete valuation ring. Since $A\otimes\kappa$ is a special biserial algebra, where $\kappa$ is the residue field of $\mathcal{O}$, one can compute a complete list of indecomposable $A\otimes \kappa$-modules. For each indecomposable $A\otimes \kappa$-module, we obtain a special kind of $A$-lattices called ``Heller lattices''. In this paper, we determine the non-periodic component of a variant of the stable Auslander--Reiten quiver for the category of $A$-lattices that contains ``Heller lattices''.
</p>projecteuclid.org/euclid.ojm/1563242420_20190715220050Mon, 15 Jul 2019 22:00 EDTUntwisting number and Blanchfield pairingshttps://projecteuclid.org/euclid.ojm/1563242421<strong>Maciej Borodzik</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 3, 497--505.</p><p><strong>Abstract:</strong><br/>
In this note we use Blanchfield forms to study knots that can be turned into an unknot using a single $\overline{t}_{2k}$ move.
</p>projecteuclid.org/euclid.ojm/1563242421_20190715220050Mon, 15 Jul 2019 22:00 EDTDirac operators on the Fefferman spin spaces in almost CR-geometryhttps://projecteuclid.org/euclid.ojm/1563242422<strong>Masayoshi Nagase</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 3, 507--524.</p><p><strong>Abstract:</strong><br/>
A spin structure on a contact Riemannian manifold carries a spin structure on a circle bundle over the manifold. We have interest in the Dirac operators associated with those structures. In terms of a modified Tanno connection, relations between them are studied and some kinds of their explicit expressions are offered.
</p>projecteuclid.org/euclid.ojm/1563242422_20190715220050Mon, 15 Jul 2019 22:00 EDTVirtual link and knot invariants from non-abelian Yang-Baxter 2-cocycle pairshttps://projecteuclid.org/euclid.ojm/1563242423<strong>Marco A. Farinati</strong>, <strong>Juliana García Galofre</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 3, 525--547.</p><p><strong>Abstract:</strong><br/>
Given a set $X$, we provide the algebraic counterpart of the (mixed) Reidemeister moves for virtual knots and links, with semi-arcs labeled by $X$: we define (commutative and noncommutative) invariants with values in groups, using ``2-cocycles", and we also introduce a universal group $U_{nc}^{fg}(X)$ and functions $\pi_f, \pi_g\colon X\times X\to U_{nc}^{fg}(X)$ governing all 2-cocycles in $X$. We exhibit examples of computations -of the group and their invariants- achieved using GAP [7].
</p>projecteuclid.org/euclid.ojm/1563242423_20190715220050Mon, 15 Jul 2019 22:00 EDTToroidal surgeries and the genus of a knothttps://projecteuclid.org/euclid.ojm/1563242424<strong>Mario Eudave-muñoz</strong>, <strong>Araceli Guzmán-tristán</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 3, 549--575.</p><p><strong>Abstract:</strong><br/>
In this paper we give an upper bound for the slopes yielding an incompressible torus by surgery on a hyperbolic knot in the 3-sphere in terms of its genus.
</p>projecteuclid.org/euclid.ojm/1563242424_20190715220050Mon, 15 Jul 2019 22:00 EDTOn a class of Rauzy fractals without the finiteness propertyhttps://projecteuclid.org/euclid.ojm/1563242425<strong>Gustavo A. Pavani</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 3, 577--599.</p><p><strong>Abstract:</strong><br/>
We present some topological and arithmetical aspects of a class of Rauzy fractals $\mathcal{R}_{a,b}$ related to the polynomials of the form $P_{a,b}(x)=x^{3}-ax^{2}-bx-1$, where $a$ and $b$ are integers satisfying $-a+1 \leq b \leq -2$. This class has the property that $0$ lies on the boundary of $\mathcal{R}_{a,b}$. We construct explicit finite automata that recognize the boundaries of these fractals. This allows to establish the number of neighbors of $\mathcal{R}_{a,b}$ in the tiling it generates. Furthermore, we prove that if $2a+3b+4 \leq 0$ then $\mathcal{R}_{a,b}$ is not homeomorphic to a topological disk. We also show that the boundary of the set $\mathcal{R}_{3,-2}$ is generated by two infinite iterated function systems.
</p>projecteuclid.org/euclid.ojm/1563242425_20190715220050Mon, 15 Jul 2019 22:00 EDTLagrangian submanifolds in strict nearly Kähler 6-manifoldshttps://projecteuclid.org/euclid.ojm/1563242426<strong>Hông Vân Lê</strong>, <strong>Lorenz Schwachhöfer</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 3, 601--629.</p><p><strong>Abstract:</strong><br/>
Lagrangian submanifolds in strict nearly Kähler 6-manifolds are related to special Lagrangian submanifolds in Calabi-Yau 6-manifolds and coassociative cones in $G_2$-manifolds. We prove that the mean curvature of a Lagrangian submanifold $L$ in a nearly Kähler manifold $(M, J, g)$ is symplectically dual to the Maslov 1-form on $L$. Using relative calibrations, we derive a formula for the second variation of the volume of a Lagrangian submanifold $L^3$ in a strict nearly Kähler manifold $(M^6, J, g)$ and compare it with McLean's formula for special Lagrangian submanifolds. We describe a finite dimensional local model of the moduli space of compact Lagrangian submanifolds in a strict nearly Kähler 6-manifold. We show that there is a real analytic atlas on $(M^6, J, g)$ in which the strict nearly Kähler structure $(J, g)$ is real analytic. Furthermore, w.r.t. an analytic strict nearly Kähler structure the moduli space of Lagrangian submanifolds of $M^6$ is a real analytic variety, whence infinitesimal Lagrangian deformations are smoothly obstructed if and only if they are formally obstructed. As an application, we relate our results to the description of Lagrangian submanifolds in the sphere $S^6$ with the standard nearly Kähler structure described in [34].
</p>projecteuclid.org/euclid.ojm/1563242426_20190715220050Mon, 15 Jul 2019 22:00 EDTRemarks on Föllmer's pathwise Itô calculushttps://projecteuclid.org/euclid.ojm/1563242427<strong>Yuki Hirai</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 3, 631--660.</p><p><strong>Abstract:</strong><br/>
We extend some results about Föllmer's pathwise Itô calculus that have only been derived for continuous paths to càdlàg paths with quadratic variation. We study some fundamental properties of pathwise Itô integrals with respect to càdlàg integrators, especially associativity and the integration by parts formula. Moreover, we study integral equations with respect to pathwise Itô integrals. We prove that some classes of integral equations, which can be explicitly solved in the usual stochastic calculus, can also be solved within the framework of Föllmer's calculus.
</p>projecteuclid.org/euclid.ojm/1563242427_20190715220050Mon, 15 Jul 2019 22:00 EDTRooted trees with the same plucking polynomialhttps://projecteuclid.org/euclid.ojm/1563242428<strong>Zhiyun Cheng</strong>, <strong>Sujoy Mukherjee</strong>, <strong>Józef H. Przytycki</strong>, <strong>Xiao Wang</strong>, <strong>Seung Yeop Yang</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 3, 661--674.</p><p><strong>Abstract:</strong><br/>
In this paper we address the following question: When do two rooted trees have the same plucking polynomial? The solution provided in the present paper has an algebraic version (Theorem 2.5) and a geometric version (Theorem 1.2). Furthermore, we give a criterion for a sequence of non-negative integers to be realized as a rooted tree.
</p>projecteuclid.org/euclid.ojm/1563242428_20190715220050Mon, 15 Jul 2019 22:00 EDTVector bundles, isoparametric functions and Radon transforms on symmetric spaceshttps://projecteuclid.org/euclid.ojm/1571623217<strong>Yasuyuki Nagatomo</strong>, <strong>Masaro Takahashi</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 4, 675--711.</p><p><strong>Abstract:</strong><br/>
We systematically construct isoparametric functions on compact symmetric spaces using vector bundles and sections of the bundles. We establish a relation between invariants of vector bundles and invariants of hypersurfaces which are the level sets of the isoparametric functions induced by sections of the bundles. We hope that this approach provides a new method for computing invariants of hypersurfaces. The Radon transform is performed to derive isoparametric functions on spheres from our functions.
</p>projecteuclid.org/euclid.ojm/1571623217_20191020220052Sun, 20 Oct 2019 22:00 EDTStrong instability of standing waves for nonlinear Schrödinger equations with attractive inverse power potentialhttps://projecteuclid.org/euclid.ojm/1571623218<strong>Noriyoshi Fukaya</strong>, <strong>Masahito Ohta</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 4, 713--726.</p><p><strong>Abstract:</strong><br/>
We study the strong instability of standing waves $e^{i\omega t}\phi_\omega(x)$ for nonlinear Schr\"{o}dinger equations with an $L^2$-supercritical nonlinearity and an attractive inverse power potential, where $\omega\in\mathbb{R}$ is a frequency, and $\phi_\omega\in H^1(\mathbb{R}^N)$ is a ground state of the corresponding stationary equation. Recently, for nonlinear Schrödinger equations with a harmonic potential, Ohta~(2018) proved that if $\partial_\lambda^2S_\omega(\phi_\omega^\lambda)|_{\lambda=1}\le0$, then the standing wave is strongly unstable, where $S_\omega$ is the action, and $\phi_\omega^\lambda(x)\mathrel{\mathop:}=\lambda^{N/2}\phi_\omega(\lambda x)$ is the scaling, which does not change the $L^2$-norm. In this paper, we prove the strong instability under the same assumption as the above-mentioned in inverse power potential case. Our proof is applicable to nonlinear Schrödinger equations with other potentials such as an attractive Dirac delta potential.
</p>projecteuclid.org/euclid.ojm/1571623218_20191020220052Sun, 20 Oct 2019 22:00 EDTLocal connectedness of the space of punctured torus grouphttps://projecteuclid.org/euclid.ojm/1571623219<strong>Sungbok Hong</strong>, <strong>Jihoon Park</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 4, 727--737.</p><p><strong>Abstract:</strong><br/>
We will give a necessary condition for local connectedness of the space of Kleinian punctured torus group using Bromgerg's local coordinate system and provide a sufficient condition for local connectedness on a dense subset of the necessary condition. That is, the collection of the points where the boundary of the space of punctured torus group is not locally connected is a dense subset of the points satisfying the necessary condition.
</p>projecteuclid.org/euclid.ojm/1571623219_20191020220052Sun, 20 Oct 2019 22:00 EDTTwo theorems on the Fock-Bargmann-Hartogs domainshttps://projecteuclid.org/euclid.ojm/1571623220<strong>Akio Kodama</strong>, <strong>Satoru Shimizu</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 4, 739--757.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove two mutually independent theorems on the family of Fock-Bargmann-Hartogs domains. Let $D_1$ and $D_2$ be two Fock-Bargmann-Hartogs domains in $\mathbb{C}^{N_1}$ and $\mathbb{C}^{N_2}$, respectively. In Theorem 1, we obtain a complete description of an arbitrarily given proper holomorphic mapping between $D_1$ and $D_2$ in the case where $N_1 = N_2$. Also, we shall give a geometric interpretation of Theorem 1. And, in Theorem 2, we determine the structure of $\text{Aut}(D_1\times D_2)$ using the data of $\text{Aut}(D_1)$ and $\text{Aut}(D_2)$ for arbitrary $N_1$ and $N_2$.
</p>projecteuclid.org/euclid.ojm/1571623220_20191020220052Sun, 20 Oct 2019 22:00 EDTThe finite group action and the equivariant determinant of elliptic operators IIIhttps://projecteuclid.org/euclid.ojm/1571623221<strong>Kenji Tsuboi</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 4, 759--785.</p><p><strong>Abstract:</strong><br/>
Let $M$ be an even-dimensional closed oriented manifold and $g$ a periodic automorphism of $M$ of order $p$. In this paper, under the assumption that the fixed points of $g^k\;(1\leq k\leq p-1)$ are isolated, a calculation formula is provided for the homomorphism $I_D:{\Bbb Z}_p\to{\Bbb R}/{\Bbb Z}$ defined in [6] for equivariant twisted signature operators $D$ over $M$. The formula gives a new method to study the periodic automorphisms of oriented manifolds. As examples of the application of the formula, results about the existence of the cyclic group action for 2,4,6-dimensional closed oriented manifolds are obtained.
</p>projecteuclid.org/euclid.ojm/1571623221_20191020220052Sun, 20 Oct 2019 22:00 EDTTeichmüller polynomials of fibered alternating linkshttps://projecteuclid.org/euclid.ojm/1571623222<strong>Robert Billet</strong>, <strong>Livio Liechti</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 4, 787--806.</p><p><strong>Abstract:</strong><br/>
We give an algorithm for computing the Teichmüller polynomial for a certain class of fibered alternating links associated to trees. Furthermore, we exhibit a mutant pair of such links distinguished by the Teichmüller polynomial.
</p>projecteuclid.org/euclid.ojm/1571623222_20191020220052Sun, 20 Oct 2019 22:00 EDTLarge time behavior of global solutions to nonlinear wave equations with frictional and viscoelastic damping termshttps://projecteuclid.org/euclid.ojm/1571623223<strong>Ryo Ikehata</strong>, <strong>Hiroshi Takeda</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 4, 807--830.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the Cauchy problem for a nonlinear wave equation with frictional and viscoelastic damping terms in ${\mathbb R}^{n}$. As is pointed out by [10], in this combination, the frictional damping term is dominant for the viscoelastic one for the global dynamics of the linear equation. In this note we observe that if the initial data is small, the frictional damping term is again dominant even in the nonlinear equation case. In other words, our main result is diffusion phenomena: the solution is approximated by the heat kernel with a suitable constant. Especially, the result obtained for the $n = 3$ case is essentially new. Our proof is based on several estimates for the corresponding linear equations.
</p>projecteuclid.org/euclid.ojm/1571623223_20191020220052Sun, 20 Oct 2019 22:00 EDTRegularity of Schrödinger's functional equation and mean field PDEs for h-path processeshttps://projecteuclid.org/euclid.ojm/1571623224<strong>Toshio Mikami</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 4, 831--842.</p><p><strong>Abstract:</strong><br/>
We show that the solution of Schrödinger's functional equation is measurable in space, kernel and marginals. As an application, we show that the drift vector of the h-path process with given two end point marginals is a measurable function of space, time and marginal at each time. In particular, we show that the coefficients of mean field PDE systems which the marginals satisfy are measurable functions of space, time and marginal.
</p>projecteuclid.org/euclid.ojm/1571623224_20191020220052Sun, 20 Oct 2019 22:00 EDTA generalization of functional limit theorems on the Riemann zeta processhttps://projecteuclid.org/euclid.ojm/1571623225<strong>Satoshi Takanobu</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 4, 843--882.</p><p><strong>Abstract:</strong><br/>
$\zeta(\cdot)$ being the Riemann zeta function, $\zeta_{\sigma}(t) := \frac{\zeta(\sigma + i t)}{\zeta(\sigma)}$ is, for $\sigma > 1$, a characteristic function of some infinitely divisible distribution $\mu_{\sigma}$. A process with time parameter $\sigma$ having $\mu_{\sigma}$ as its marginal at time $\sigma$ is called a Riemann zeta process. Ehm [2] has found a functional limit theorem on this process being a backwards Lévy process. In this paper, we replace $\zeta(\cdot)$ with a Dirichlet series $\eta(\cdot;a)$ generated by a nonnegative, completely multiplicative arithmetical function $a(\cdot)$ satisfying (3), (4) and (5) below, and derive the same type of functional limit theorem as Ehm on the process corresponding to $\eta(\cdot;a)$ and being a backwards Lévy process.
</p>projecteuclid.org/euclid.ojm/1571623225_20191020220052Sun, 20 Oct 2019 22:00 EDTAsymptotic behavior of solutions to the generalized KdV-Burgers equationhttps://projecteuclid.org/euclid.ojm/1571623226<strong>Ikki Fukuda</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 56, Number 4, 883--906.</p><p><strong>Abstract:</strong><br/>
We study the asymptotic behavior of global solutions to the initial value problem for the generalized KdV-Burgers equation. One can expect that the solution to this equation converges to a self-similar solution to the Burgers equation, due to earlier works related to this problem. Actually, we obtain the optimal asymptotic rate similar to those results and the second asymptotic profile for the generalized KdV-Burgers equation.
</p>projecteuclid.org/euclid.ojm/1571623226_20191020220052Sun, 20 Oct 2019 22:00 EDTSpheres not admitting smooth odd-fixed-point actions of $S_5$ and $SL(2, 5)$https://projecteuclid.org/euclid.ojm/1579079107<strong>Masaharu Morimoto</strong>, <strong>Shunsuke Tamura</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 57, Number 1, 1--8.</p><p><strong>Abstract:</strong><br/>
Let $G$ be a finite group and $\Sigma$ a homology sphere with smooth $G$-action. If the $G$-fixed-point set of $\Sigma$ consists of odd-number points then the dimension of $\Sigma$ could be restrictive. In this article we confirm the claim in the cases where $G = S_5$ or $S\!L(2, 5)$.
</p>projecteuclid.org/euclid.ojm/1579079107_20200115040532Wed, 15 Jan 2020 04:05 ESTSemistable fibrations over an elliptic curve with only one singular fibrehttps://projecteuclid.org/euclid.ojm/1579079108<strong>Abel Castorena</strong>, <strong>Margarida Mendes Lopes</strong>, <strong>Gian Pietro Pirola</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 57, Number 1, 9--15.</p><p><strong>Abstract:</strong><br/>
In this work we describe a construction of semistable fibrations over an elliptic curve with one unique singular fibre and we give effective examples using monodromy of curves.
</p>projecteuclid.org/euclid.ojm/1579079108_20200115040532Wed, 15 Jan 2020 04:05 ESTOn diagrams of simplified trisections and mapping class groupshttps://projecteuclid.org/euclid.ojm/1579079109<strong>Kenta Hayano</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 57, Number 1, 17--37.</p><p><strong>Abstract:</strong><br/>
A simplified trisection is a trisection map on a 4--manifold such that, in its critical value set, there is no double point and cusps only appear in triples on innermost fold circles. We give a necessary and sufficient condition for a 3--tuple of systems of simple closed curves in a surface to be a diagram of a simplified trisection in terms of mapping class groups. As an application of this criterion, we show that trisections of spun 4--manifolds due to Meier are diffeomorphic (as trisections) to simplified ones. Baykur and Saeki recently gave an algorithmic construction of a simplified trisection from a directed broken Lefschetz fibration. We also give an algorithm to obtain a diagram of a simplified trisection derived from their construction.
</p>projecteuclid.org/euclid.ojm/1579079109_20200115040532Wed, 15 Jan 2020 04:05 ESTSome remarks on PL collapsible covers of 2-dimensional polyhedrahttps://projecteuclid.org/euclid.ojm/1579079110<strong>Eugenio Borghini</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 57, Number 1, 39--49.</p><p><strong>Abstract:</strong><br/>
We analyze the topology and geometry of a polyhedron of dimension 2 according to the minimum size of a cover by PL collapsible polyhedra. We provide partial characterizations of the polyhedra of dimension 2 that can be decomposed as the union of two PL collapsible subpolyhedra in terms of their simple homotopy type and certain local properties.
</p>projecteuclid.org/euclid.ojm/1579079110_20200115040532Wed, 15 Jan 2020 04:05 ESTExamples of singular toric varieties with certain numerical conditionshttps://projecteuclid.org/euclid.ojm/1579079111<strong>Hiroshi Sato</strong>, <strong>Yusuke Suyama</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 57, Number 1, 51--59.</p><p><strong>Abstract:</strong><br/>
We give various examples of $\mathbb{Q}$-factorial projective toric varieties such that the sum of the squared torus invariant prime divisors is positive. We also determine the generators for the cone of effective $2$-cycles on a toric variety of Picard number two. This result is convenient to explain our examples.
</p>projecteuclid.org/euclid.ojm/1579079111_20200115040532Wed, 15 Jan 2020 04:05 ESTInitial boundary value problem for 3D Boussinesq system with the thermal dampinghttps://projecteuclid.org/euclid.ojm/1579079112<strong>Yanghai Yu</strong>, <strong>Yanbin Tang</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 57, Number 1, 61--83.</p><p><strong>Abstract:</strong><br/>
In this paper we consider the initial boundary value problem for the 3D Boussinesq system with the velocity dissipation and weak damping effect to instead of the dissipation effect for the thermal conductivity and establish the global existence of weak solutions. Furthermore, we prove that the global weak solution is strong and unique under some small initial data condition.
</p>projecteuclid.org/euclid.ojm/1579079112_20200115040532Wed, 15 Jan 2020 04:05 ESTOn Deviations and Spreads of Meromorphic Minimal Surfaceshttps://projecteuclid.org/euclid.ojm/1579079113<strong>Arnold Kowalski</strong>, <strong>Ivan Marchenko</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 57, Number 1, 85--101.</p><p><strong>Abstract:</strong><br/>
In this paper we consider the influence that the number of separated maximum points of the norm of a meromorphic minimal surface (m.m.s) has on the magnitudes of growth and value distribution. We present sharp estimations of spread of m.m.s in terms of Nevanlinna's defect, magnitude of deviation and the number of separated points of the norm of m.m.s. We also give examples showing that the estimates are sharp.
</p>projecteuclid.org/euclid.ojm/1579079113_20200115040532Wed, 15 Jan 2020 04:05 ESTGeneralized Heegaard Splittings and the Disk Complexhttps://projecteuclid.org/euclid.ojm/1579079114<strong>Jungsoo Kim</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 57, Number 1, 103--140.</p><p><strong>Abstract:</strong><br/>
Let $M$ be an orientable, irreducible $3$-manifold and $(\mathcal{V},\mathcal{W};F)$ a weakly reducible, unstabilized Heegaard splitting of $M$ of genus at least three. In this article, we define an equivalence relation $\sim$ on the set of the generalized Heegaard splittings obtained by weak reductions and find special subsets of the disk complex $\mathcal{D}(F)$ named by the ``$equivalent$ $clusters$'', where we can find a canonical function $\Phi$ from the set of equivalent clusters to the set of the equivalent classes for the relation $\sim$. These equivalent classes are more detailed than the isotopy classes of the generalized Heegaard splittings obtained by weak reductions from $F$. In the last section, we prove $\Phi$ is a bijection if the genus of $F$ is three.
</p>projecteuclid.org/euclid.ojm/1579079114_20200115040532Wed, 15 Jan 2020 04:05 ESTIntersection number and some metrics on Teichmüller spacehttps://projecteuclid.org/euclid.ojm/1579079115<strong>Zongliang Sun</strong>, <strong>Hui Guo</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 57, Number 1, 141--149.</p><p><strong>Abstract:</strong><br/>
Let $T(X)$ be the Teichmüller space of a closed surface $X$ of genus $g \geq 2,$ $C(X)$ be the space of geodesic currents on $X,$ and $L: T(X) \to C(X)$ be the embedding introduced by Bonahon which maps a hyperbolic metric to its corresponding Liouville current. In this paper, we compare some quantitative relations and topological behaviors between the intersection number and the Teichmüller metric, the length spectrum metric and Thurston's asymmetric metrics on $T(X),$ respectively.
</p>projecteuclid.org/euclid.ojm/1579079115_20200115040532Wed, 15 Jan 2020 04:05 ESTAn estimate of the first non-zero eigenvalue of the Laplacian by the Ricci curvature on edges of graphshttps://projecteuclid.org/euclid.ojm/1579079116<strong>Taiki Yamada</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 57, Number 1, 151--163.</p><p><strong>Abstract:</strong><br/>
We define the distance between edges of graphs and study the coarse Ricci curvature on edges. We consider the Laplacian on edges based on the Jost-Horak's definition of the Laplacian on simplicial complexes. As one of our main results, we obtain an estimate of the first non-zero eigenvalue of the Laplacian by the Ricci curvature for a regular graph.
</p>projecteuclid.org/euclid.ojm/1579079116_20200115040532Wed, 15 Jan 2020 04:05 ESTIterated circle bundles and infranilmanifoldshttps://projecteuclid.org/euclid.ojm/1579079117<strong>Igor Belegradek</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 57, Number 1, 165--168.</p><p><strong>Abstract:</strong><br/>
We give short proofs of the following two facts: Iterated principal circle bundles are precisely the nilmanifolds. Every iterated circle bundle is almost flat, and hence diffeomorphic to an infranilmanifold.
</p>projecteuclid.org/euclid.ojm/1579079117_20200115040532Wed, 15 Jan 2020 04:05 ESTSemi-discrete linear Weingarten surfaces with Weierstrass-type representations and their singularitieshttps://projecteuclid.org/euclid.ojm/1579079118<strong>Masashi Yasumoto</strong>, <strong>Wayne Rossman</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 57, Number 1, 169--185.</p><p><strong>Abstract:</strong><br/>
We establish what semi-discrete linear Weingarten surfaces with Weierstrass-type representations in $3$-dimensional Riemannian and Lorentzian spaceforms are, confirming their required properties regarding curvatures and parallel surfaces, and then classify them. We then define and analyze their singularities. In particular, we discuss singularities of (1) semi-discrete surfaces with non-zero constant Gaussian curvature, (2) parallel surfaces of semi-discrete minimal and maximal surfaces, and (3) semi-discrete constant mean curvature $1$ surfaces in de Sitter $3$-space. We include comparisons with different previously known definitions of such singularities.
</p>projecteuclid.org/euclid.ojm/1579079118_20200115040532Wed, 15 Jan 2020 04:05 ESTGlobal asymptotics toward the rarefaction waves for solutions to the Cauchy problem of the scalar conservation law with nonlinear viscosityhttps://projecteuclid.org/euclid.ojm/1579079119<strong>Akitaka Matsumura</strong>, <strong>Natsumi Yoshida</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 57, Number 1, 187--205.</p><p><strong>Abstract:</strong><br/>
In this paper, we investigate the asymptotic behavior of solutions to the Cauchy problem for the scalar viscous conservation law where the far field states are prescribed. Especially, we deal with the case when the viscosity is of non-Newtonian type, including a pseudo-plastic case. When the corresponding Riemann problem for the hyperbolic part admits a Riemann solution which consists of single rarefaction wave, under a condition on nonlinearity of the viscosity, it is proved that the solution of the Cauchy problem tends toward the rarefaction wave as time goes to infinity, without any smallness conditions.
</p>projecteuclid.org/euclid.ojm/1579079119_20200115040532Wed, 15 Jan 2020 04:05 ESTClassification of isoparametric submanifolds admitting a reflective focal submanifold in symmetric spaces of non-compact typehttps://projecteuclid.org/euclid.ojm/1579079120<strong>Naoyuki Koike</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 57, Number 1, 207--246.</p><p><strong>Abstract:</strong><br/>
In this paper, we assume that all isoparametric submanifolds have flat section. The main purpose of this paper is to prove that, if a full irreducible complete isoparametric submanifold of codimension greater than one in a symmetric space of non-compact type admits a reflective focal submanifold and if it is of real analytic, then it is a principal orbit of a Hermann type action on the symmetric space. A hyperpolar action on a symmetric space of non-compact type admits a reflective singular orbit if and only if it is a Hermann type action. Hence is not extra the assumption that the isoparametric submanifold admits a reflective focal submanifold. Also, we prove that, if a full irreducible complete isoparametric submanifold of codimension greater than one in a symmetric space of non-compact type satisfies some additional conditions, then it is a principal orbit of the isotropy action of the symmetric space, where we need not impose that the submanifold is of real analytic. We use the building theory in the proof.
</p>projecteuclid.org/euclid.ojm/1579079120_20200115040532Wed, 15 Jan 2020 04:05 ESTSolvability of some integro-differential equations with drifthttps://projecteuclid.org/euclid.ojm/1586160077<strong>Messoud Efendiev</strong>, <strong>Vitali Vougalter</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 57, Number 2, 247--265.</p><p><strong>Abstract:</strong><br/>
We prove the existence in the sense of sequences of solutions for some integro-differential type equations involving the drift term in the appropriate $H^{2}$ spaces using the fixed point technique when the elliptic problems contain second order differential operators with and without Fredholm property. It is shown that, under the reasonable technical conditions, the convergence in $L^{1}$ of the integral kernels yields the existence and convergence in $H^{2}$ of solutions.
</p>projecteuclid.org/euclid.ojm/1586160077_20200406040125Mon, 06 Apr 2020 04:01 EDT2-stratifold spines of closed 3-manifoldshttps://projecteuclid.org/euclid.ojm/1586160078<strong>J.C. Gómez-larrañaga</strong>, <strong>F. González-acuña</strong>, <strong>Wolfgang Heil</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 57, Number 2, 267--277.</p><p><strong>Abstract:</strong><br/>
$2$-stratifolds are a generalization of $2$-manifolds in that there are disjoint simple closed branch curves. We obtain a list of all closed $3$-manifolds that have a $2$-stratifold as a spine.
</p>projecteuclid.org/euclid.ojm/1586160078_20200406040125Mon, 06 Apr 2020 04:01 EDTKnots with Hopf crossing number at most onehttps://projecteuclid.org/euclid.ojm/1586160079<strong>Maciej Mroczkowski</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 57, Number 2, 279--304.</p><p><strong>Abstract:</strong><br/>
We consider diagrams of links in $S^2$ obtained by projection from $S^3$ with the Hopf map and the minimal crossing number for such diagrams. Knots admitting diagrams with at most one crossing are classified. Some properties of these knots are exhibited. In particular, we establish which of these knots are algebraic and, for such knots, give an answer to a problem posed by Fiedler in [3].
</p>projecteuclid.org/euclid.ojm/1586160079_20200406040125Mon, 06 Apr 2020 04:01 EDTRemarks on the derivation of several second order partial differential equations from a generalization of the Einstein equationshttps://projecteuclid.org/euclid.ojm/1586160080<strong>Makoto Nakamura</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 57, Number 2, 305--331.</p><p><strong>Abstract:</strong><br/>
A generalization of the Einstein equations with the cosmological constant is considered for complex line elements. Several second order semilinear partial differential equations are derived from them as semilinear field equations in homogeneous and isotropic spaces. The nonrelativistic limits of the field equations are also considered. The properties of spatial expansion and contraction are studied based on energy estimates of the field equations. Several dissipative and anti-dissipative properties are remarked.
</p>projecteuclid.org/euclid.ojm/1586160080_20200406040125Mon, 06 Apr 2020 04:01 EDTGraph invariants and Betti numbers of real toric manifoldshttps://projecteuclid.org/euclid.ojm/1586160081<strong>Boram Park</strong>, <strong>Hanchul Park</strong>, <strong>Seonjeong Park</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 57, Number 2, 333--356.</p><p><strong>Abstract:</strong><br/>
For a graph $G$, the graph cubeahedron $\square_G$ and the graph associahedron $\triangle_G$ are simple convex polytopes which admit (real) toric manifolds. In this paper, we introduce a graph invariant, called the $b$-number, and show that the $b$-numbers compute the Betti numbers of the real toric manifold $X^\mathbb{R}(\square_G)$ corresponding to $\square_G$. The $b$-number is a counterpart of the notion of $a$-number, introduced by S. Choi and the second named author, which computes the Betti numbers of the real toric manifold $X^\mathbb{R}(\triangle_G)$ corresponding to $\triangle_G$. We also study various relationships between $a$-numbers and $b$-numbers from the viewpoint of toric topology. Interestingly, for a forest $G$ and its line graph $L(G)$, the real toric manifolds $X^\mathbb{R}(\triangle_G)$ and $X^\mathbb{R}(\square_{L(G)})$ have the same Betti numbers.
</p>projecteuclid.org/euclid.ojm/1586160081_20200406040125Mon, 06 Apr 2020 04:01 EDTSpectrum of generalized Hodge-Laplace operators on flat tori and round sphereshttps://projecteuclid.org/euclid.ojm/1586160082<strong>Stine Franziska Beitz</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 57, Number 2, 357--379.</p><p><strong>Abstract:</strong><br/>
We consider generalized Hodge-Laplace operators $\alpha d \delta + \beta \delta d$ for $\alpha, \beta > 0$ on $p$-forms on compact Riemannian manifolds. In the case of flat tori and round spheres of different radii, we explicitly calculate the spectrum of these operators. Furthermore, we investigate under which circumstances they are isospectral.
</p>projecteuclid.org/euclid.ojm/1586160082_20200406040125Mon, 06 Apr 2020 04:01 EDTError analysis for approximations to one-dimensional SDEs via the perturbation methodhttps://projecteuclid.org/euclid.ojm/1586160083<strong>Shigeki Aida</strong>, <strong>Nobuaki Naganuma</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 57, Number 2, 381--424.</p><p><strong>Abstract:</strong><br/>
We study asymptotic error distributions associated with standard approximation scheme for one-dimensional stochastic differential equations driven by fractional Brownian motions. This problem was studied by, for instance, Gradinaru-Nourdin [6], Neuenkirch and Nourdin [14] and the second named author [13]. The aim of this paper is to extend their results to the case where the equations contain drift terms and simplify the proof of estimates of the remainder terms in [13]. To this end, we represent the approximation solution as the solution of the equation which is obtained by replacing the fractional Brownian path with a perturbed path. We obtain the asymptotic error distribution as a directional derivative of the solution by using this expression.
</p>projecteuclid.org/euclid.ojm/1586160083_20200406040125Mon, 06 Apr 2020 04:01 EDTAn estimate for surface measure of small balls in Carnot groupshttps://projecteuclid.org/euclid.ojm/1586160084<strong>Alexey Rudenko</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 57, Number 2, 425--450.</p><p><strong>Abstract:</strong><br/>
We introduce a family of quasidistances in ${\mathbb R}^d$, such that some of them are equivalent to natural distances on Carnot groups. We find the sufficient conditions for the balls w.r.t. a quasidistance from our family to be comparable to ellipsoids. Using comparability to ellipsoids we find asymptotics of surface measure of intersections of small balls with linear submanifolds and the conditions for finiteness of the integral w.r.t. the surface measure of negative power of the distance. We provide several examples of Carnot groups, where comparability to ellipsoids can be shown for natural distances, and therefore we can study the asymptotics and finitness of the integrals explicitly. We also show an example of a Carnot group, where the comparability to ellipsoids does not hold.
</p>projecteuclid.org/euclid.ojm/1586160084_20200406040125Mon, 06 Apr 2020 04:01 EDTTropicalization of 1-tacnodal curves on toric surfaceshttps://projecteuclid.org/euclid.ojm/1586160085<strong>Takuhiro Takahashi</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 57, Number 2, 451--491.</p><p><strong>Abstract:</strong><br/>
A degeneration of a singular curve on a toric surface, called a tropicalization, was constructed by E. Shustin. He classified the degeneration of 1-cuspidal curves using polyhedral complexes called tropical curves. In this paper, we define a tropical version of a 1-tacnodal curve, that is, a curve having exactly one singular point whose topological type is $A_3$, and by applying the tropicalization method, we classify tropical curves which correspond to 1-tacnodal curves.
</p>projecteuclid.org/euclid.ojm/1586160085_20200406040125Mon, 06 Apr 2020 04:01 EDTOn the slope of rational fibered surfaceshttps://projecteuclid.org/euclid.ojm/1586160086<strong>Margarita Castañeda-salazar</strong>, <strong>Alexis G. Zamora</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 57, Number 2, 493--504.</p><p><strong>Abstract:</strong><br/>
Given a rational fibered surface $f:X \to \mathbb{P}^1$ of genus $g$ we prove the inequality $\frac{6n+5}{n+1}-\frac{9n+12}{2g}\le \lambda_f,$ provided that the genus $g$ is sufficiently high with respect to the gonality $2n+3$ of the general fibre.
</p>projecteuclid.org/euclid.ojm/1586160086_20200406040125Mon, 06 Apr 2020 04:01 EDTERRATA: ``On generalized Dold manifolds'' Osaka J. Math. 56 (2019), 75--90https://projecteuclid.org/euclid.ojm/1586160087<strong>Avijit Nath</strong>, <strong>Parameswaran Sankaran</strong>. <p><strong>Source: </strong>Osaka Journal of Mathematics, Volume 57, Number 2, 505--506.</p>projecteuclid.org/euclid.ojm/1586160087_20200406040125Mon, 06 Apr 2020 04:01 EDT