Journal of the Mathematical Society of Japan Articles (Project Euclid)
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The latest articles from Journal of the Mathematical Society of Japan 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 EDTMon, 25 Apr 2011 09:24 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Toy models for D. H. Lehmer's conjecture
http://projecteuclid.org/euclid.jmsj/1280496816
<strong>Eiichi BANNAI</strong>, <strong>Tsuyoshi MIEZAKI</strong><p><strong>Source: </strong>J. Math. Soc. Japan, Volume 62, Number 3, 687--705.</p><p><strong>Abstract:</strong><br/>
In 1947, Lehmer conjectured that the Ramanujan τ-function τ( m ) never vanishes for all positive integers m , where τ( m ) are the Fourier coefficients of the cusp form Δ 24 of weight 12. Lehmer verified the conjecture in 1947 for m < 214928639999. In 1973, Serre verified up to m < 10 15 , and in 1999, Jordan and Kelly for m < 22689242781695999.
The theory of spherical t -design, and in particular those which are the shells of Euclidean lattices, is closely related to the theory of modular forms, as first shown by Venkov in 1984. In particular, Ramanujan's τ-function gives the coefficients of a weighted theta series of the E 8 -lattice. It is shown, by Venkov, de la Harpe, and Pache, that τ( m ) = 0 is equivalent to the fact that the shell of norm 2 m of the E 8 -lattice is an 8-design. So, Lehmer's conjecture is reformulated in terms of spherical t -design.
Lehmer's conjecture is difficult to prove, and still remains open. In this paper, we consider toy models of Lehmer's conjecture. Namely, we show that the m -th Fourier coefficient of the weighted theta series of the Z 2 -lattice and the A 2 -lattice does not vanish, when the shell of norm m of those lattices is not the empty set. In other words, the spherical 5 (resp. 7)-design does not exist among the shells in the Z 2 -lattice (resp. A 2 -lattice).
</p>projecteuclid.org/euclid.jmsj/1280496816_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTExamples of four dimensional cusp singularitieshttps://projecteuclid.org/euclid.jmsj/1529309022<strong>Hiroyasu TSUCHIHASHI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 3, 1047--1062.</p><p><strong>Abstract:</strong><br/>
We give some examples of four dimensional cusp singularities which are not of Hilbert modular type. We construct them, using quadratic cones and subgroups of reflection groups.
</p>projecteuclid.org/euclid.jmsj/1529309022_20180719040046Thu, 19 Jul 2018 04:00 EDTAlexander invariants of ribbon tangles and planar algebrashttps://projecteuclid.org/euclid.jmsj/1529309023<strong>Celeste DAMIANI</strong>, <strong>Vincent FLORENS</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 3, 1063--1084.</p><p><strong>Abstract:</strong><br/>
Ribbon tangles are proper embeddings of tori and cylinders in the 4-ball $B^4$, “bounding” 3-manifolds with only ribbon disks as singularities. We construct an Alexander invariant $\mathbf{A}$ of ribbon tangles equipped with a representation of the fundamental group of their exterior in a free abelian group $G$. This invariant induces a functor in a certain category $\mathbf{R}ib_G$ of tangles, which restricts to the exterior powers of Burau–Gassner representation for ribbon braids, that are analogous to usual braids in this context. We define a circuit algebra $\mathbf{C}ob_G$ over the operad of smooth cobordisms, inspired by diagrammatic planar algebras introduced by Jones [ Jon99 ], and prove that the invariant $\mathbf{A}$ commutes with the compositions in this algebra. On the other hand, ribbon tangles admit diagrammatic representations, through welded diagrams. We give a simple combinatorial description of $\mathbf{A}$ and of the algebra $\mathbf{C}ob_G$, and observe that our construction is a topological incarnation of the Alexander invariant of Archibald [ Arc10 ]. When restricted to diagrams without virtual crossings, $\mathbf{A}$ provides a purely local description of the usual Alexander poynomial of links, and extends the construction by Bigelow, Cattabriga and the second author [ BCF15 ].
</p>projecteuclid.org/euclid.jmsj/1529309023_20180719040046Thu, 19 Jul 2018 04:00 EDTA system of conjugate functions on parabolic Bloch spaceshttps://projecteuclid.org/euclid.jmsj/1529309024<strong>Yôsuke HISHIKAWA</strong>, <strong>Masaharu NISHIO</strong>, <strong>Masahiro YAMADA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 3, 1085--1102.</p><p><strong>Abstract:</strong><br/>
The parabolic Bloch space is the set of all solutions $u$ of the parabolic operator $L^{(\alpha)}$ with the finite Bloch norm $\| u \|_{\mathcal{B}_{\alpha} (\sigma)}$. In this paper, we introduce $L^{(\alpha)}$-conjugates of parabolic Bloch functions, and investigate several properties. As an application, we give an isomorphism theorem on parabolic Bloch spaces.
</p>projecteuclid.org/euclid.jmsj/1529309024_20180719040046Thu, 19 Jul 2018 04:00 EDTScalar curvature of self-shrinkerhttps://projecteuclid.org/euclid.jmsj/1529309025<strong>Zhen GUO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 3, 1103--1110.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider the scalar curvature of a self-shrinker and get the gap theorem of the scalar curvature. We get also a relationship between the upper bound of the square of the length of the second fundamental form and the Ricci mean value.
</p>projecteuclid.org/euclid.jmsj/1529309025_20180719040046Thu, 19 Jul 2018 04:00 EDTFree probability for purely discrete eigenvalues of random matriceshttps://projecteuclid.org/euclid.jmsj/1529892023<strong>Benoit COLLINS</strong>, <strong>Takahiro HASEBE</strong>, <strong>Noriyoshi SAKUMA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 3, 1111--1150.</p><p><strong>Abstract:</strong><br/>
In this paper, we study random matrix models which are obtained as a non-commutative polynomial in random matrix variables of two kinds: (a) a first kind which have a discrete spectrum in the limit, (b) a second kind which have a joint limiting distribution in Voiculescu’s sense and are globally rotationally invariant. We assume that each monomial constituting this polynomial contains at least one variable of type (a), and show that this random matrix model has a set of eigenvalues that almost surely converges to a deterministic set of numbers that is either finite or accumulating to only zero in the large dimension limit. For this purpose we define a framework (cyclic monotone independence) for analyzing discrete spectra and develop the moment method for the eigenvalues of compact (and in particular Schatten class) operators. We give several explicit calculations of discrete eigenvalues of our model.
</p>projecteuclid.org/euclid.jmsj/1529892023_20180719040046Thu, 19 Jul 2018 04:00 EDTElliptic fibrations on K3 surfaces and Salem numbers of maximal degreehttps://projecteuclid.org/euclid.jmsj/1529892024<strong>Xun YU</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 3, 1151--1163.</p><p><strong>Abstract:</strong><br/>
We study the maximal Salem degree of automorphisms of K3 surfaces via elliptic fibrations. In particular, we establish a characterization of such maximum in terms of elliptic fibrations with infinite automorphism groups. As an application, we show that any supersingular K3 surface in odd characteristic has an automorphism the entropy of which is the natural logarithm of a Salem number of degree 22.
</p>projecteuclid.org/euclid.jmsj/1529892024_20180719040046Thu, 19 Jul 2018 04:00 EDTComposing generic linearly perturbed mappings and immersions/injectionshttps://projecteuclid.org/euclid.jmsj/1528790548<strong>Shunsuke ICHIKI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 3, 1165--1184.</p><p><strong>Abstract:</strong><br/>
Let $N$ (resp., $U$) be a manifold (resp., an open subset of ${\mathbb{R}}^m$). Let $f:N\to U$ and $F:U\to {\mathbb{R}}^\ell$ be an immersion and a $C^{\infty}$ mapping, respectively. Generally, the composition $F\circ f$ does not necessarily yield a mapping transverse to a given subfiber-bundle of $J^1(N,\mathbb{R}^\ell)$. Nevertheless, in this paper, for any $\mathcal{A}^1$-invariant fiber, we show that composing generic linearly perturbed mappings of $F$ and the given immersion $f$ yields a mapping transverse to the subfiber-bundle of $J^1(N,\mathbb{R}^\ell)$ with the given fiber. Moreover, we show a specialized transversality theorem on crossings of compositions of generic linearly perturbed mappings of a given mapping $F:U\to \mathbb{R}^\ell$ and a given injection $f:N\to U$. Furthermore, applications of the two main theorems are given.
</p>projecteuclid.org/euclid.jmsj/1528790548_20180719040046Thu, 19 Jul 2018 04:00 EDTCommon reducing subspaces of several weighted shifts with operator weightshttps://projecteuclid.org/euclid.jmsj/1529892025<strong>Caixing GU</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 3, 1185--1225.</p><p><strong>Abstract:</strong><br/>
We characterize common reducing subspaces of several weighted shifts with operator weights. As applications, we study the common reducing subspaces of the multiplication operators by powers of coordinate functions on Hilbert spaces of holomorphic functions in several variables. The identification of reducing subspaces also leads to structure theorems for the commutants of von Neumann algebras generated by these multiplication operators. This general approach applies to weighted Hardy spaces, weighted Bergman spaces, Drury–Arveson spaces and Dirichlet spaces of the unit ball or polydisk uniformly.
</p>projecteuclid.org/euclid.jmsj/1529892025_20180719040046Thu, 19 Jul 2018 04:00 EDTMeasure-valued solutions to the complete Euler systemhttps://projecteuclid.org/euclid.jmsj/1531469370<strong>Jan BŘEZINA</strong>, <strong>Eduard FEIREISL</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 4, 1227--1245.</p><p><strong>Abstract:</strong><br/>
We introduce the concept of dissipative measure-valued solution to the complete Euler system describing the motion of an inviscid compressible fluid. These solutions are characterized by a parameterized (Young) measure and a dissipation defect in the total energy balance. The dissipation defect dominates the concentration errors in the equations satisfied by the Young measure. A dissipative measure-valued solution can be seen as the most general concept of solution to the Euler system retaining its structural stability. In particular, we show that a dissipative measure-valued solution necessarily coincides with a classical one on its life span provided they share the same initial data.
</p>projecteuclid.org/euclid.jmsj/1531469370_20181023220059Tue, 23 Oct 2018 22:00 EDTThe Gordian distance of handlebody-knots and Alexander biquandle coloringshttps://projecteuclid.org/euclid.jmsj/1532678872<strong>Tomo MURAO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 4, 1247--1267.</p><p><strong>Abstract:</strong><br/>
We give lower bounds for the Gordian distance and the unknotting number of handlebody-knots by using Alexander biquandle colorings. We construct handlebody-knots with Gordian distance $n$ and unknotting number $n$ for any positive integer $n$.
</p>projecteuclid.org/euclid.jmsj/1532678872_20181023220059Tue, 23 Oct 2018 22:00 EDTRogers dilogarithms of higher degree and generalized cluster algebrashttps://projecteuclid.org/euclid.jmsj/1532678873<strong>Tomoki NAKANISHI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 4, 1269--1304.</p><p><strong>Abstract:</strong><br/>
In connection with generalized cluster algebras we introduce a certain generalization of the celebrated Rogers dilogarithm, which we call the Rogers dilogarithms of higher degree. We show that there is an identity of these generalized Rogers dilogarithms associated with any period of seeds of a generalized cluster algebra.
</p>projecteuclid.org/euclid.jmsj/1532678873_20181023220059Tue, 23 Oct 2018 22:00 EDTWhitney regularity and Thom condition for families of non-isolated mixed singularitieshttps://projecteuclid.org/euclid.jmsj/1532678874<strong>Christophe EYRAL</strong>, <strong>Mutsuo OKA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 4, 1305--1336.</p><p><strong>Abstract:</strong><br/>
We investigate the equisingularity question for 1-parameter deformation families of mixed polynomial functions $f_t({\boldsymbol{z}},\bar{{\boldsymbol{z}}})$ from the Newton polygon point of view. We show that if the members $f_t$ of the family satisfy a number of elementary conditions, which can be easily described in terms of the Newton polygon, then the corresponding family of mixed hypersurfaces $f_t^{-1}(0)$ is Whitney equisingular (and hence topologically equisingular) and satisfies the Thom condition.
</p>projecteuclid.org/euclid.jmsj/1532678874_20181023220059Tue, 23 Oct 2018 22:00 EDTAccurate trajectory-harps for Kähler magnetic fieldshttps://projecteuclid.org/euclid.jmsj/1531469371<strong>Toshiaki ADACHI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 4, 1337--1346.</p><p><strong>Abstract:</strong><br/>
In preceding papers we gave estimates on string-lengths, string-cosines and zenith angles of trajectory-harps under the condition that sectional curvatures of the underlying manifold are bounded from above. In this paper we study the cases that equalities hold in these estimates. Refining the previous proofs we give conditions that trajectory-harps are congruent to trajectory-harps on a complex space form.
</p>projecteuclid.org/euclid.jmsj/1531469371_20181023220059Tue, 23 Oct 2018 22:00 EDTVertex operator algebras, minimal models, and modular linear differential equations of order 4https://projecteuclid.org/euclid.jmsj/1535616221<strong>Yusuke ARIKE</strong>, <strong>Kiyokazu NAGATOMO</strong>, <strong>Yuichi SAKAI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 4, 1347--1373.</p><p><strong>Abstract:</strong><br/>
In this paper we classify vertex operator algebras with three conditions which arise from Virasoro minimal models: (A) the central charge and conformal weights are rational numbers, (B) the space spanned by characters of all simple modules of a vertex operator algebra coincides with the space of solutions of a modular linear differential equation of order $4$ and (C) the dimensions of first three weight subspaces of a VOA are $1, 0$ and $1$, respectively. It is shown that vertex operator algebras which we concern have central charges $c=-46/3, -3/5, -114/7, 4/5$, and are isomorphic to minimal models for $c=-46/3, -3/5$ and ${\mathbb{Z}}_2$-graded simple current extensions of minimal models for $c=-114/7, 4/5$.
</p>projecteuclid.org/euclid.jmsj/1535616221_20181023220059Tue, 23 Oct 2018 22:00 EDTEnergy decay and diffusion phenomenon for the asymptotically periodic damped wave equationhttps://projecteuclid.org/euclid.jmsj/1536220816<strong>Romain JOLY</strong>, <strong>Julien ROYER</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 4, 1375--1418.</p><p><strong>Abstract:</strong><br/>
We prove local and global energy decay for the asymptotically periodic damped wave equation on the Euclidean space. Since the behavior of high frequencies is already mostly understood, this paper is mainly about the contribution of low frequencies. We show in particular that the damped wave behaves like a solution of a heat equation which depends on the H-limit of the metric and the mean value of the absorption index.
</p>projecteuclid.org/euclid.jmsj/1536220816_20181023220059Tue, 23 Oct 2018 22:00 EDTLocal polar invariants and the Poincaré problem in the dicritical casehttps://projecteuclid.org/euclid.jmsj/1538380983<strong>Yohann GENZMER</strong>, <strong>Rogério MOL</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 4, 1419--1451.</p><p><strong>Abstract:</strong><br/>
We develop a study on local polar invariants of planar complex analytic foliations at $(\mathbb{C}^{2},0)$, which leads to the characterization of second type foliations and of generalized curve foliations, as well as to a description of the $GSV$-index. We apply it to the Poincaré problem for foliations on the complex projective plane $\mathbb{P}^{2}_{\mathbb{C}}$, establishing, in the dicritical case, conditions for the existence of a bound for the degree of an invariant algebraic curve $S$ in terms of the degree of the foliation $\mathcal{F}$. We characterize the existence of a solution for the Poincaré problem in terms of the structure of the set of local separatrices of $\mathcal{F}$ over the curve $S$. Our method, in particular, recovers the known solution for the non-dicritical case, $\deg(S) \leq \deg (\mathcal{F}) + 2$.
</p>projecteuclid.org/euclid.jmsj/1538380983_20181023220059Tue, 23 Oct 2018 22:00 EDTA family of cubic fourfolds with finite-dimensional motivehttps://projecteuclid.org/euclid.jmsj/1532678875<strong>Robert LATERVEER</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 4, 1453--1473.</p><p><strong>Abstract:</strong><br/>
We prove that cubic fourfolds in a certain 10-dimensional family have finite-dimensional motive. The proof is based on the van Geemen–Izadi construction of an algebraic Kuga–Satake correspondence for these cubic fourfolds, combined with Voisin’s method of “spread”. Some consequences are given.
</p>projecteuclid.org/euclid.jmsj/1532678875_20181023220059Tue, 23 Oct 2018 22:00 EDTCritical nonlinear Schrödinger equations in higher space dimensionshttps://projecteuclid.org/euclid.jmsj/1532678876<strong>Nakao HAYASHI</strong>, <strong>Chunhua LI</strong>, <strong>Pavel I. NAUMKIN</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 4, 1475--1492.</p><p><strong>Abstract:</strong><br/>
We study the critical nonlinear Schrödinger equations \[ i\partial _{t}u+\frac{1}{2}\Delta u = \lambda \vert u\vert^{{2}/{n}}u, \quad (t,x) \in \mathbb{R}^{+}\times \mathbb{R}^{n}, \] in space dimensions $n\geq 4$, where $\lambda \in \mathbb{R}$. We prove the global in time existence of solutions to the Cauchy problem under the assumption that the absolute value of Fourier transform of the initial data is bounded below by a positive constant. Also we prove the two side sharp time decay estimates of solutions in the uniform norm.
</p>projecteuclid.org/euclid.jmsj/1532678876_20181023220059Tue, 23 Oct 2018 22:00 EDTVolume minimization and conformally Kähler, Einstein–Maxwell geometryhttps://projecteuclid.org/euclid.jmsj/1536220817<strong>Akito FUTAKI</strong>, <strong>Hajime ONO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 4, 1493--1521.</p><p><strong>Abstract:</strong><br/>
Let $M$ be a compact complex manifold admitting a Kähler structure. A conformally Kähler, Einstein–Maxwell metric (cKEM metric for short) is a Hermitian metric $\tilde g$ on $M$ with constant scalar curvature such that there is a positive smooth function $f$ with $g = f^{2} \tilde g$ being a Kähler metric and $f$ being a Killing Hamiltonian potential with respect to $g$. Fixing a Kähler class, we characterize such Killing vector fields whose Hamiltonian function $f$ with respect to some Kähler metric $g$ in the fixed Kähler class gives a cKEM metric $\tilde g = f^{-2}g$. The characterization is described in terms of critical points of certain volume functional. The conceptual idea is similar to the cases of Kähler–Ricci solitons and Sasaki–Einstein metrics in that the derivative of the volume functional gives rise to a natural obstruction to the existence of cKEM metrics. However, unlike the Kähler–Ricci soliton case and Sasaki–Einstein case, the functional is neither convex nor proper in general, and often has more than one critical points. The last observation matches well with the ambitoric examples studied earlier by LeBrun and Apostolov–Maschler.
</p>projecteuclid.org/euclid.jmsj/1536220817_20181023220059Tue, 23 Oct 2018 22:00 EDTDiffusion with nonlocal Robin boundary conditionshttps://projecteuclid.org/euclid.jmsj/1538553644<strong>Wolfgang ARENDT</strong>, <strong>Stefan KUNKEL</strong>, <strong>Markus KUNZE</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 4, 1523--1556.</p><p><strong>Abstract:</strong><br/>
We investigate a second order elliptic differential operator $A_{\beta, \mu}$ on a bounded, open set $\Omega\subset\mathbb{R}^{d}$ with Lipschitz boundary subject to a nonlocal boundary condition of Robin type. More precisely we have $0\leq \beta\in L^{\infty}(\partial\Omega)$ and $\mu\colon\partial\Omega\to{\mathscr{M}}(\overline{\Omega})$, and boundary conditions of the form $$ \partial_{\nu}^{{\mathscr{A}}}u(z)+\beta(z)u(z)=\int_{\overline{\Omega}}u(x)\mu(z)(\mathrm{d}x), \quad z\in\partial\Omega, $$ where $\partial_{\nu}^{{\mathscr{A}}}$ denotes the weak conormal derivative with respect to our differential operator. Under suitable conditions on the coefficients of the differential operator and the function $\mu$ we show that $A_{\beta, \mu}$ generates a holomorphic semigroup $T_{\beta,\mu}$ on $L^{\infty}(\Omega)$ which enjoys the strong Feller property. In particular, it takes values in $C(\overline{\Omega})$. Its restriction to $C(\overline{\Omega})$ is strongly continuous and holomorphic. We also establish positivity and contractivity of the semigroup under additional assumptions and study the asymptotic behavior of the semigroup.
</p>projecteuclid.org/euclid.jmsj/1538553644_20181023220059Tue, 23 Oct 2018 22:00 EDTAnalysis of elastic symbols with the Cauchy integral and construction of asymptotic solutionshttps://projecteuclid.org/euclid.jmsj/1538553645<strong>Hideo SOGA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 70, Number 4, 1557--1580.</p><p><strong>Abstract:</strong><br/>
This paper deals with the elastic wave equation $(D_t^2 - L(x, D_{x'}, D_{x_n})) u(t, x', x_n)=0$ in the half-space $x_n>0$. In the constant coefficient case, it is known that the solution is represented by using the Cauchy integral $\int_c e^{ix_n\zeta} (I-L(\xi', \zeta))^{-1} d\zeta$. In this paper this representation is extended to the variable coefficient case, and an asymptotic solution with the similar Cauchy integral is constructed. In this case, the terms $\partial_x^\alpha \int_c e^{ix_n\zeta} (I-L(x,\xi',\zeta))^{-1} d\zeta$ appear in the inductive process. These do not become lower terms necessarily, and therefore the principal part of asymptotic solution is a little different from the form in the constant coefficient case.
</p>projecteuclid.org/euclid.jmsj/1538553645_20181023220059Tue, 23 Oct 2018 22:00 EDTSpectrum for compact operators on Banach spaceshttps://projecteuclid.org/euclid.jmsj/1536890442<strong>Luis BARREIRA</strong>, <strong>Davor DRAGIČEVIĆ</strong>, <strong>Claudia VALLS</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 1, 1--17.</p><p><strong>Abstract:</strong><br/>
For a two-sided sequence of compact linear operators acting on a Banach space, we consider the notion of spectrum defined in terms of the existence of exponential dichotomies under homotheties of the dynamics. This can be seen as a natural generalization of the spectrum of a matrix—the set of its eigenvalues. We give a characterization of all possible spectra and explicit examples of sequences for which the spectrum takes a form not occurring in finite-dimensional spaces. We also consider the case of a one-sided sequence of compact linear operators.
</p>projecteuclid.org/euclid.jmsj/1536890442_20190124220103Thu, 24 Jan 2019 22:01 ESTLinks with trivial $Q$-polynomialhttps://projecteuclid.org/euclid.jmsj/1538640044<strong>Yasuyuki MIYAZAWA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 1, 19--42.</p><p><strong>Abstract:</strong><br/>
The $Q$-polynomial is an invariant of the isotopy type of an unoriented link defined by Brandt, Lickorish, Millett, and Ho around 1985. It is shown that there exist infinitely many prime knots and links with trivial $Q$-polynomial, and so the $Q$-polynomial does not detect trivial links.
</p>projecteuclid.org/euclid.jmsj/1538640044_20190124220103Thu, 24 Jan 2019 22:01 ESTTopological canal foliationshttps://projecteuclid.org/euclid.jmsj/1539590425<strong>Gilbert HECTOR</strong>, <strong>Rémi LANGEVIN</strong>, <strong>Paweł WALCZAK</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 1, 43--63.</p><p><strong>Abstract:</strong><br/>
Regular canal surfaces of $\mathbb{R}^3$ or $\mathbb{S}^3$ admit foliations by circles: the characteristic circles of the envelope. In order to build a foliation of $\mathbb{S}^3$ with leaves being canal surfaces, one has to relax the condition “canal” a little ( “weak canal condition” ) in order to accept isolated umbilics. Here, we define a topological condition which generalizes this “weak canal” condition imposed on leaves, and classify the foliations of compact orientable 3-manifolds we can obtain this way.
</p>projecteuclid.org/euclid.jmsj/1539590425_20190124220103Thu, 24 Jan 2019 22:01 ESTVanishing theorems of $L^2$-cohomology groups on Hessian manifoldshttps://projecteuclid.org/euclid.jmsj/1540541021<strong>Shinya AKAGAWA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 1, 65--89.</p><p><strong>Abstract:</strong><br/>
We show vanishing theorems of $L^2$-cohomology groups of Kodaira–Nakano type on complete Hessian manifolds by introducing a new operator $\partial'_F$. We obtain further vanishing theorems of $L^2$-cohomology groups $L^2H^{p,q}_{\bar{\partial}}(\Omega)$ on a regular convex cone $\Omega$ with the Cheng–Yau metric for $p>q$.
</p>projecteuclid.org/euclid.jmsj/1540541021_20190124220103Thu, 24 Jan 2019 22:01 ESTEquivalence of Littlewood–Paley square function and area function characterizations of weighted product Hardy spaces associated to operatorshttps://projecteuclid.org/euclid.jmsj/1540368034<strong>Xuan Thinh DUONG</strong>, <strong>Guorong HU</strong>, <strong>Ji LI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 1, 91--115.</p><p><strong>Abstract:</strong><br/>
Let $L_1$ and $L_2$ be nonnegative self-adjoint operators acting on $L^2(X_1)$ and $L^2(X_2)$, respectively, where $X_1$ and $X_2$ are spaces of homogeneous type. Assume that $L_1$ and $L_2$ have Gaussian heat kernel bounds. This paper aims to study some equivalent characterizations of the weighted product Hardy spaces $H^{p}_{w,L_{1},L_{2}}(X_{1}\times X_{2})$ associated to $L_{1}$ and $L_{2}$, for $p \in (0, \infty)$ and the weight $w$ belongs to the product Muckenhoupt class $ A_{\infty}(X_{1} \times X_{2})$. Our main result is that the spaces $H^{p}_{w,L_{1},L_{2}}(X_{1}\times X_{2})$ introduced via area functions can be equivalently characterized by the Littlewood–Paley $g$-functions and $g^{\ast}_{\lambda_{1}, \lambda_{2}}$-functions, as well as the Peetre type maximal functions, without any further assumption beyond the Gaussian upper bounds on the heat kernels of $L_1$ and $L_2$. Our results are new even in the unweighted product setting.
</p>projecteuclid.org/euclid.jmsj/1540368034_20190124220103Thu, 24 Jan 2019 22:01 ESTBerkes' limit theoremhttps://projecteuclid.org/euclid.jmsj/1540368039<strong>Satoshi TAKANOBU</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 1, 117--145.</p><p><strong>Abstract:</strong><br/>
In Berkes' striking paper of the early 1990s, he presented another limit theorem different from the central limit theorem for a lacunary trigonometric series not satisfying Erdős' lacunary condition. In this paper, we upgrade his result to the limit theorem having high versatility, which we would call Berkes' limit theorem. By this limit theorem, it is explained in a unified way that Fukuyama–Takahashi's counterexample and Takahashi's counterexample are all convergent to limiting distributions of the same type as Berkes.
</p>projecteuclid.org/euclid.jmsj/1540368039_20190124220103Thu, 24 Jan 2019 22:01 ESTOn unconditional well-posedness for the periodic modified Korteweg–de Vries equationhttps://projecteuclid.org/euclid.jmsj/1540541018<strong>Luc MOLINET</strong>, <strong>Didier PILOD</strong>, <strong>Stéphane VENTO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 1, 147--201.</p><p><strong>Abstract:</strong><br/>
We prove that the modified Korteweg–de Vries equation is unconditionally well-posed in $H^s({\mathbb{T}})$ for $s\ge 1/3$. For this we gather the smoothing effect first discovered by Takaoka and Tsutsumi with an approach developed by the authors that combines the energy method, with Bourgain's type estimates, improved Strichartz estimates and the construction of modified energies.
</p>projecteuclid.org/euclid.jmsj/1540541018_20190124220103Thu, 24 Jan 2019 22:01 ESTLocal time penalizations with various clocks for one-dimensional diffusionshttps://projecteuclid.org/euclid.jmsj/1541408432<strong>Christophe PROFETA</strong>, <strong>Kouji YANO</strong>, <strong>Yuko YANO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 1, 203--233.</p><p><strong>Abstract:</strong><br/>
We study some limit theorems for the law of a generalized one-dimensional diffusion weighted and normalized by a non-negative function of the local time evaluated at a parametrized family of random times (which we will call a clock ). As the clock tends to infinity, we show that the initial process converges towards a new penalized process, which generally depends on the chosen clock. However, unlike with deterministic clocks, no specific assumptions are needed on the resolvent of the diffusion. We then give a path interpretation of these penalized processes via some universal $\sigma$-finite measures.
</p>projecteuclid.org/euclid.jmsj/1541408432_20190124220103Thu, 24 Jan 2019 22:01 ESTPartitioning subsets of generalised scattered ordershttps://projecteuclid.org/euclid.jmsj/1542704621<strong>Chris LAMBIE-HANSON</strong>, <strong>Thilo WEINERT</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 1, 235--257.</p><p><strong>Abstract:</strong><br/>
In 1956, 48 years after Hausdorff provided a comprehensive account on ordered sets and defined the notion of a scattered order, Erdős and Rado founded the partition calculus in a seminal paper. The present paper gives an account of investigations into generalisations of scattered linear orders and their partition relations for both singletons and pairs. We consider analogues for these order-types of known partition theorems for ordinals or scattered orders and prove a partition theorem from assumptions about cardinal characteristics. Together, this continues older research by Erdős, Galvin, Hajnal, Larson and Takahashi and more recent investigations by Abraham, Bonnet, Cummings, Džamonja, Komjáth, Shelah and Thompson.
</p>projecteuclid.org/euclid.jmsj/1542704621_20190124220103Thu, 24 Jan 2019 22:01 ESTPseudo Kobayashi hyperbolicity of subvarieties of general type on abelian varietieshttps://projecteuclid.org/euclid.jmsj/1542704620<strong>Katsutoshi YAMANOI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 1, 259--298.</p><p><strong>Abstract:</strong><br/>
We prove that the Kobayashi pseudo distance of a closed subvariety $X$ of an abelian variety $A$ is a true distance outside the special set $\operatorname{Sp}(X)$ of $X$, where $\operatorname{Sp}(X)$ is the union of all positive dimensional translated abelian subvarieties of $A$ which are contained in $X$. More strongly, we prove that a closed subvariety $X$ of an abelian variety is taut modulo $\operatorname{Sp}(X)$; Every sequence $f_n:{\mathbb{D}}\to X$ of holomorphic mappings from the unit disc ${\mathbb{D}}$ admits a subsequence which converges locally uniformly, unless the image $f_n(K)$ of a fixed compact set $K$ of ${\mathbb{D}}$ eventually gets arbitrarily close to $\operatorname{Sp}(X)$ as $n$ gets larger. These generalize a classical theorem on algebraic degeneracy of entire curves in irregular varieties.
</p>projecteuclid.org/euclid.jmsj/1542704620_20190124220103Thu, 24 Jan 2019 22:01 ESTThe combinatorics of Lehn's conjecturehttps://projecteuclid.org/euclid.jmsj/1538640045<strong>Alina MARIAN</strong>, <strong>Dragos OPREA</strong>, <strong>Rahul PANDHARIPANDE</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 1, 299--308.</p><p><strong>Abstract:</strong><br/>
Let $S$ be a nonsingular projective surface equipped with a line bundle $H$. Lehn's conjecture is a formula for the top Segre class of the tautological bundle associated to $H$ on the Hilbert scheme of points of $S$. Voisin has recently reduced Lehn's conjecture to the vanishing of certain coefficients of special power series. The first result here is a proof of the vanishings required by Voisin by residue calculations (A. Szenes and M. Vergne have independently found the same proof). Our second result is an elementary solution of the parallel question for the top Segre class on the symmetric power of a nonsingular projective curve $C$ associated to a higher rank vector bundle $V$ on $C$. Finally, we propose a complete conjecture for the top Segre class on the Hilbert scheme of points of $S$ associated to a higher rank vector bundle on $S$ in the $K$-trivial case.
</p>projecteuclid.org/euclid.jmsj/1538640045_20190124220103Thu, 24 Jan 2019 22:01 ESTPositive factorizations of symmetric mapping classeshttps://projecteuclid.org/euclid.jmsj/1542704619<strong>Tetsuya ITO</strong>, <strong>Keiko KAWAMURO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 1, 309--327.</p><p><strong>Abstract:</strong><br/>
We study a question of Etnyre and Van Horn-Morris whether a symmetric mapping class admitting a positive factorization is a lift of a quasipositive braid. We answer the question affirmatively for mapping classes satisfying certain cyclic conditions.
</p>projecteuclid.org/euclid.jmsj/1542704619_20190124220103Thu, 24 Jan 2019 22:01 ESTOn $n$-trivialities of classical and virtual knots for some unknotting operationshttps://projecteuclid.org/euclid.jmsj/1541667932<strong>Noboru ITO</strong>, <strong>Migiwa SAKURAI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 1, 329--347.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce a new nontrivial filtration, called F-order, for classical and virtual knot invariants; this filtration produces filtered knot invariants, which are called finite type invariants similar to Vassiliev knot invariants. Finite type invariants introduced by Goussarov, Polyak, and Viro are well-known, and we call them finite type invariants of GPV-order. We show that for any positive integer $n$ and for any classical knot $K$, there exist infinitely many of nontrivial classical knots, all of whose finite type invariants of GPV-order $\le n-1$, coincide with those of $K$ (Theorem 1). Further, we show that for any positive integer $n$, there exists a nontrivial virtual knot whose finite type invariants of our F-order $\le n-1$ coincide with those of the trivial knot (Theorem 2). In order to prove Theorem 1 (Theorem 2, resp.), we define an $n$-triviality via a certain unknotting operation, called virtualization (forbidden moves, resp.), and for any positive integer $n$, find an $n$-trivial classical knot (virtual knot, resp.).
</p>projecteuclid.org/euclid.jmsj/1541667932_20190124220103Thu, 24 Jan 2019 22:01 ESTGluing construction of compact $\operatorname{Spin}(7)$-manifoldshttps://projecteuclid.org/euclid.jmsj/1551322823<strong>Mamoru DOI</strong>, <strong>Naoto YOTSUTANI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 2, 349--382.</p><p><strong>Abstract:</strong><br/>
We give a differential-geometric construction of compact manifolds with holonomy $\operatorname{Spin}(7)$ which is based on Joyce's second construction of compact $\operatorname{Spin}(7)$-manifolds and Kovalev's gluing construction of compact $G_2$-manifolds. We provide several examples of compact $\operatorname{Spin}(7)$-manifolds, at least one of which is new. Here in this paper we need orbifold admissible pairs $(\overline{X}, D)$ consisting of a compact Kähler orbifold $\overline{X}$ with isolated singular points modelled on $\mathbb{C}^4/\mathbb{Z}_4$, and a smooth anticanonical divisor $D$ on $\overline{X}$. Also, we need a compatible antiholomorphic involution $\sigma$ on $\overline{X}$ which fixes the singular points on $\overline{X}$ and acts freely on the anticanoncial divisor $D$. If two orbifold admissible pairs $(\overline{X}_1, D_1)$, $(\overline{X}_2, D_2)$ and compatible antiholomorphic involutions $\sigma_i$ on $\overline{X}_i$ for $i=1,2$ satisfy the gluing condition, we can glue $(\overline{X}_1 \setminus D_1)/\langle\sigma_1\rangle$ and $(\overline{X}_2 \setminus D_2)/\langle\sigma_2\rangle$ together to obtain a compact Riemannian 8-manifold $(M, g)$ whose holonomy group $\operatorname{Hol}(g)$ is contained in $\operatorname{Spin}(7)$. Furthermore, if the $\widehat{A}$-genus of $M$ equals 1, then $M$ is a compact $\operatorname{Spin}(7)$-manifold, i.e. a compact Riemannian manifold with holonomy $\operatorname{Spin}(7)$.
</p>projecteuclid.org/euclid.jmsj/1551322823_20190423040044Tue, 23 Apr 2019 04:00 EDTBoundary Harnack principle and elliptic Harnack inequalityhttps://projecteuclid.org/euclid.jmsj/1551150260<strong>Martin T. BARLOW</strong>, <strong>Mathav MURUGAN</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 2, 383--412.</p><p><strong>Abstract:</strong><br/>
We prove a scale-invariant boundary Harnack principle for inner uniform domains over a large family of Dirichlet spaces. A novel feature of our work is that we do not assume volume doubling property for the symmetric measure.
</p>projecteuclid.org/euclid.jmsj/1551150260_20190423040044Tue, 23 Apr 2019 04:00 EDTA new stability notion of closed hypersurfaces in the hyperbolic spacehttps://projecteuclid.org/euclid.jmsj/1551085233<strong>Marco Antonio Lázaro VELÁSQUEZ</strong>, <strong>Henrique Fernandes DE LIMA</strong>, <strong>Jonatan Floriano DA SILVA</strong>, <strong>Arlandson Matheus Silva OLIVEIRA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 2, 413--428.</p><p><strong>Abstract:</strong><br/>
In this article, we establish the notion of strong $(r,k,a,b)$-stability related to closed hypersurfaces immersed in the hyperbolic space $\mathbb{H}^{n+1}$, where $r$ and $k$ are nonnegative integers satisfying the inequality $0 \leq k<r \leq n-2$ and $a$ and $b$ are real numbers (at least one nonzero). In this setting, considering some appropriate restrictions on the constants $a$ and $b$, we show that geodesic spheres are strongly $(r,k,a,b)$-stable. Afterwards, under a suitable restriction on the higher order mean curvatures $H_{r+1}$ and $H_{k+1}$, we prove that if a closed hypersurface into the hyperbolic space $\mathbb{H}^{n+1}$ is strongly $(r,k,a,b)$-stable, then it must be a geodesic sphere, provided that the image of its Gauss mapping is contained in the chronological future (or past) of an equator of the de Sitter space.
</p>projecteuclid.org/euclid.jmsj/1551085233_20190423040044Tue, 23 Apr 2019 04:00 EDTCompletely positive isometries between matrix algebrashttps://projecteuclid.org/euclid.jmsj/1551085234<strong>Masamichi HAMANA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 2, 429--449.</p><p><strong>Abstract:</strong><br/>
Let $\varphi$ be a linear map between operator spaces. To measure the intensity of $\varphi$ being isometric we associate with it a number, called the isometric degree of $\varphi$ and written $\mathrm{id}(\varphi)$, as follows. Call $\varphi$ a strict $m$-isometry with $m$ a positive integer if it is an $m$-isometry, but is not an $(m+1)$-isometry. Define $\mathrm{id}(\varphi)$ to be 0, $m$, and $\infty$, respectively if $\varphi$ is not an isometry, a strict $m$-isometry, and a complete isometry, respectively. We show that if $\varphi:M_n\to M_p$ is a unital completely positive map between matrix algebras, then $\mathrm{id}(\varphi) \in \{0,\,1,\,2,\,\dots,\,[({n-1})/{2}],\,\infty\}$ and that when $n\ge 3$ is fixed and $p$ is sufficiently large, the values $1,\,2,\,\dots,\,[({n-1})/{2}]$ are attained as $\mathrm{id}(\varphi)$ for some $\varphi$. The ranges of such maps $\varphi$ with $1 \le \mathrm{id}(\varphi)<\infty$ provide natural examples of operator systems that are isometric, but not completely isometric, to $M_n$. We introduce and classify, up to unital complete isometry, a certain family of such operator systems.
</p>projecteuclid.org/euclid.jmsj/1551085234_20190423040044Tue, 23 Apr 2019 04:00 EDTThe logarithmic derivative for point processes with equivalent Palm measureshttps://projecteuclid.org/euclid.jmsj/1551690078<strong>Alexander I. BUFETOV</strong>, <strong>Andrey V. DYMOV</strong>, <strong>Hirofumi OSADA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 2, 451--469.</p><p><strong>Abstract:</strong><br/>
The logarithmic derivative of a point process plays a key rôle in the general approach, due to the third author, to constructing diffusions preserving a given point process. In this paper we explicitly compute the logarithmic derivative for determinantal processes on $\mathbb{R}$ with integrable kernels, a large class that includes all the classical processes of random matrix theory as well as processes associated with de Branges spaces. The argument uses the quasi-invariance of our processes established by the first author.
</p>projecteuclid.org/euclid.jmsj/1551690078_20190423040044Tue, 23 Apr 2019 04:00 EDTNotes on the bicategory of W*-bimoduleshttps://projecteuclid.org/euclid.jmsj/1551085235<strong>Yusuke SAWADA</strong>, <strong>Shigeru YAMAGAMI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 2, 471--481.</p><p><strong>Abstract:</strong><br/>
Categories of W*-bimodules are shown in an explicit and algebraic way to constitute an involutive W*-bicategory.
</p>projecteuclid.org/euclid.jmsj/1551085235_20190423040044Tue, 23 Apr 2019 04:00 EDTUpper bounds for the dimension of tori acting on GKM manifoldshttps://projecteuclid.org/euclid.jmsj/1551430829<strong>Shintarô KUROKI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 2, 483--513.</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to give an upper bound for the dimension of a torus $T$ which acts on a GKM manifold $M$ effectively. In order to do that, we introduce a free abelian group of finite rank, denoted by $\mathcal{A}(\Gamma,\alpha,\nabla)$, from an (abstract) $(m,n)$-type GKM graph $(\Gamma,\alpha,\nabla)$. Here, an $(m,n)$-type GKM graph is the GKM graph induced from a $2m$-dimensional GKM manifold $M^{2m}$ with an effective $n$-dimensional torus $T^{n}$-action which preserves the almost complex structure, say $(M^{2m},T^{n})$. Then it is shown that $\mathcal{A}(\Gamma,\alpha,\nabla)$ has rank $\ell(> n)$ if and only if there exists an $(m,\ell)$-type GKM graph $(\Gamma,\widetilde{\alpha},\nabla)$ which is an extension of $(\Gamma,\alpha,\nabla)$. Using this combinatorial necessary and sufficient condition, we prove that the rank of $\mathcal{A}(\Gamma_{M},\alpha_{M},\nabla_{M})$ for the GKM graph $(\Gamma_{M},\alpha_{M},\nabla_{M})$ induced from $(M^{2m},T^{n})$ gives an upper bound for the dimension of a torus which can act on $M^{2m}$ effectively. As one of the applications of this result, we compute the rank associated to $\mathcal{A}(\Gamma,\alpha,\nabla)$ of the complex Grassmannian of 2-planes $G_{2}(\mathbb{C}^{n+2})$ with the natural effective $T^{n+1}$-action, and prove that this action on $G_{2}(\mathbb{C}^{n+2})$ is the maximal effective torus action which preserves the standard complex structure.
</p>projecteuclid.org/euclid.jmsj/1551430829_20190423040044Tue, 23 Apr 2019 04:00 EDTGood tilting modules and recollements of derived module categories, IIhttps://projecteuclid.org/euclid.jmsj/1552035634<strong>Hongxing CHEN</strong>, <strong>Changchang XI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 2, 515--554.</p><p><strong>Abstract:</strong><br/>
Homological tilting modules of finite projective dimension are investigated. They generalize both classical and good tilting modules of projective dimension at most one, and produce recollements of derived module categories of rings in which generalized localizations of rings are involved. To decide whether a good tilting module is homological, a sufficient and necessary condition is presented in terms of the internal properties of the given tilting module. Consequently, a class of homological, non-trivial, infinitely generated tilting modules of higher projective dimension is constructed, and the first example of an infinitely generated $n$-tilting module which is not homological for each $n \ge 2$ is exhibited. To deal with both tilting and cotilting modules consistently, the notion of weak tilting modules is introduced. Thus similar results for infinitely generated cotilting modules of finite injective dimension are obtained, though dual technique does not work for infinite-dimensional modules.
</p>projecteuclid.org/euclid.jmsj/1552035634_20190423040044Tue, 23 Apr 2019 04:00 EDTOn bifurcations of cuspshttps://projecteuclid.org/euclid.jmsj/1551085236<strong>Zbigniew SZAFRANIEC</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 2, 555--567.</p><p><strong>Abstract:</strong><br/>
Let $F_t$, where $t \in \mathbb{R}$, be an analytic family of plane-to-plane mappings with $F_0$ having a critical point at the origin. The paper presents effective algebraic methods of computing the number of those cusp points of $F_t$, where $0 < |t|\ll 1$, emanating from the origin at which $F_t$ has a positive/negative local topological degree.
</p>projecteuclid.org/euclid.jmsj/1551085236_20190423040044Tue, 23 Apr 2019 04:00 EDTStrongly singular bilinear Calderón–Zygmund operators and a class of bilinear pseudodifferential operatorshttps://projecteuclid.org/euclid.jmsj/1551690077<strong>Árpád BÉNYI</strong>, <strong>Lucas CHAFFEE</strong>, <strong>Virginia NAIBO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 2, 569--587.</p><p><strong>Abstract:</strong><br/>
Motivated by the study of kernels of bilinear pseudodifferential operators with symbols in a Hörmander class of critical order, we investigate boundedness properties of strongly singular Calderón–Zygmund operators in the bilinear setting. For such operators, whose kernels satisfy integral-type conditions, we establish boundedness properties in the setting of Lebesgue spaces as well as endpoint mappings involving the space of functions of bounded mean oscillations and the Hardy space. Assuming pointwise-type conditions on the kernels, we show that strongly singular bilinear Calderón–Zygmund operators satisfy pointwise estimates in terms of maximal operators, which imply their boundedness in weighted Lebesgue spaces.
</p>projecteuclid.org/euclid.jmsj/1551690077_20190423040044Tue, 23 Apr 2019 04:00 EDTOn delta invariants and indices of idealshttps://projecteuclid.org/euclid.jmsj/1551085237<strong>Toshinori KOBAYASHI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 2, 589--597.</p><p><strong>Abstract:</strong><br/>
Let $R$ be a Cohen–Macaulay local ring with a canonical module. We consider Auslander's (higher) delta invariants of powers of certain ideals of $R$. Firstly, we shall provide some conditions for an ideal to be a parameter ideal in terms of delta invariants. As an application of this result, we give upper bounds for orders of Ulrich ideals of $R$ when $R$ has Gorenstein punctured spectrum. Secondly, we extend the definition of indices to the ideal case, and generalize the result of Avramov–Buchweitz–Iyengar–Miller on the relationship between the index and regularity.
</p>projecteuclid.org/euclid.jmsj/1551085237_20190423040044Tue, 23 Apr 2019 04:00 EDTAmple canonical heights for endomorphisms on projective varietieshttps://projecteuclid.org/euclid.jmsj/1551690076<strong>Takahiro SHIBATA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 2, 599--634.</p><p><strong>Abstract:</strong><br/>
We define an “ample canonical height” for an endomorphism on a projective variety, which is essentially a generalization of the canonical heights for polarized endomorphisms introduced by Call–Silverman. We formulate a dynamical analogue of the Northcott finiteness theorem for ample canonical heights as a conjecture, and prove it for endomorphisms on varieties of small Picard numbers, abelian varieties, and surfaces. As applications, for the endomorphisms which satisfy the conjecture, we show the non-density of the set of preperiodic points over a fixed number field, and obtain a dynamical Mordell–Lang type result on the intersection of two Zariski dense orbits of two endomorphisms on a common variety.
</p>projecteuclid.org/euclid.jmsj/1551690076_20190423040044Tue, 23 Apr 2019 04:00 EDTThe maximum of the 1-measurement of a metric measure spacehttps://projecteuclid.org/euclid.jmsj/1552982427<strong>Hiroki NAKAJIMA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 2, 635--650.</p><p><strong>Abstract:</strong><br/>
For a metric measure space, we consider the set of distributions of 1-Lipschitz functions, which is called the 1-measurement. On the 1-measurement, we have the Lipschitz order relation introduced by M. Gromov. The aim of this paper is to study the maximum and maximal elements of the 1-measurement of a metric measure space with respect to the Lipschitz order. We present a necessary condition of a metric measure space for the existence of the maximum of the 1-measurement. We also consider a metric measure space that has the maximum of its 1-measurement.
</p>projecteuclid.org/euclid.jmsj/1552982427_20190423040044Tue, 23 Apr 2019 04:00 EDTLong-time existence of the edge Yamabe flowhttps://projecteuclid.org/euclid.jmsj/1552035633<strong>Eric BAHUAUD</strong>, <strong>Boris VERTMAN</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 2, 651--688.</p><p><strong>Abstract:</strong><br/>
This article presents an analysis of the normalized Yamabe flow starting at and preserving a class of compact Riemannian manifolds with incomplete edge singularities and negative Yamabe invariant. Our main results include uniqueness, long-time existence and convergence of the edge Yamabe flow starting at a metric with everywhere negative scalar curvature. Our methods include novel maximum principle results on the singular edge space without using barrier functions. Moreover, our uniform bounds on solutions are established by a new ansatz without in any way using or redeveloping Krylov–Safonov estimates in the singular setting. As an application we obtain a solution to the Yamabe problem for incomplete edge metrics with negative Yamabe invariant using flow techniques. Our methods lay groundwork for studying other flows like the mean curvature flow as well as the porous medium equation in the singular setting.
</p>projecteuclid.org/euclid.jmsj/1552035633_20190423040044Tue, 23 Apr 2019 04:00 EDTSuperharmonic functions of Schrödinger operators and Hardy inequalitieshttps://projecteuclid.org/euclid.jmsj/1554364814<strong>Yusuke MIURA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 3, 689--708.</p><p><strong>Abstract:</strong><br/>
Given a Dirichlet form with generator $\mathcal{L}$ and a measure $\mu$, we consider superharmonic functions of the Schrödinger operator $\mathcal{L} + \mu$. We probabilistically prove that the existence of superharmonic functions gives rise to the Hardy inequality. More precisely, the $L^2$-Hardy inequality is derived from Itô's formula applied to the superharmonic function.
</p>projecteuclid.org/euclid.jmsj/1554364814_20190724220337Wed, 24 Jul 2019 22:03 EDTLeibniz complexity of Nash functions on differentiationshttps://projecteuclid.org/euclid.jmsj/1552896021<strong>Goo ISHIKAWA</strong>, <strong>Tatsuya YAMASHITA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 3, 709--726.</p><p><strong>Abstract:</strong><br/>
The derivatives of Nash functions are Nash functions which are derived algebraically from their minimal polynomial equations. In this paper we show that, for any non-Nash analytic function, it is impossible to derive its derivatives algebraically, i.e., by using linearity and Leibniz rule finite times. In fact we prove the impossibility of such kind of algebraic computations, algebraically by using Kähler differentials. Then the notion of Leibniz complexity of a Nash function is introduced in this paper, as a computational complexity on its derivative, by the minimal number of usages of Leibniz rules to compute the total differential algebraically. We provide general observations and upper estimates on Leibniz complexity of Nash functions, related to the binary expansions, the addition chain complexity, the non-scalar complexity and the complexity of Nash functions in the sense of Ramanakoraisina.
</p>projecteuclid.org/euclid.jmsj/1552896021_20190724220337Wed, 24 Jul 2019 22:03 EDTWeak subsolutions to complex Monge–Ampère equationshttps://projecteuclid.org/euclid.jmsj/1551776436<strong>Vincent GUEDJ</strong>, <strong>Chinh H. LU</strong>, <strong>Ahmed ZERIAHI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 3, 727--738.</p><p><strong>Abstract:</strong><br/>
We compare various notions of weak subsolutions to degenerate complex Monge–Ampère equations, showing that they all coincide. This allows us to give an alternative proof of mixed Monge–Ampère inequalities due to Kołodziej and Dinew.
</p>projecteuclid.org/euclid.jmsj/1551776436_20190724220337Wed, 24 Jul 2019 22:03 EDTSmall covers over wedges of polygonshttps://projecteuclid.org/euclid.jmsj/1552982426<strong>Suyoung CHOI</strong>, <strong>Hanchul PARK</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 3, 739--764.</p><p><strong>Abstract:</strong><br/>
A small cover is a closed smooth manifold of dimension $n$ having a locally standard $\mathbb{Z}_2^n$-action whose orbit space is isomorphic to a simple polytope. In the paper, we classify small covers and real toric manifolds whose orbit space is isomorphic to the dual of the simplicial complex obtainable by a sequence of wedgings from a polygon, using a systematic combinatorial method of puzzles finding toric spaces.
</p>projecteuclid.org/euclid.jmsj/1552982426_20190724220337Wed, 24 Jul 2019 22:03 EDTAsymptotic behavior of lifetime sums for random simplicial complex processeshttps://projecteuclid.org/euclid.jmsj/1556092819<strong>Masanori HINO</strong>, <strong>Shu KANAZAWA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 3, 765--804.</p><p><strong>Abstract:</strong><br/>
We study the homological properties of random simplicial complexes. In particular, we obtain the asymptotic behavior of lifetime sums for a class of increasing random simplicial complexes; this result is a higher-dimensional counterpart of Frieze's $\zeta(3)$-limit theorem for the Erdős–Rényi graph process. The main results include solutions to questions posed in an earlier study by Hiraoka and Shirai about the Linial–Meshulam complex process and the random clique complex process. One of the key elements of the arguments is a new upper bound on the Betti numbers of general simplicial complexes in terms of the number of small eigenvalues of Laplacians on links. This bound can be regarded as a quantitative version of the cohomology vanishing theorem.
</p>projecteuclid.org/euclid.jmsj/1556092819_20190724220337Wed, 24 Jul 2019 22:03 EDTFinite formal model of toric singularitieshttps://projecteuclid.org/euclid.jmsj/1557475349<strong>David BOURQUI</strong>, <strong>Julien SEBAG</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 3, 805--829.</p><p><strong>Abstract:</strong><br/>
We study the formal neighborhoods at rational non-degenerate arcs of the arc scheme associated with a toric variety. The first main result of this article shows that these formal neighborhoods are generically constant on each Nash component of the variety. Furthermore, using our previous work, we attach to every such formal neighborhood, and in fact to every toric valuation, a minimal formal model (in the class of stable isomorphisms) which can be interpreted as a measure of the singularities of the base-variety. As a second main statement, for a large class of toric valuations, we compute the dimension and the embedding dimension of such minimal formal models, and we relate the latter to the Mather discrepancy. The class includes the strongly essential valuations, that is to say those the center of which is a divisor in the exceptional locus of every resolution of singularities of the variety. We also obtain a similar result for monomial curves.
</p>projecteuclid.org/euclid.jmsj/1557475349_20190724220337Wed, 24 Jul 2019 22:03 EDTDiscriminants of classical quasi-orthogonal polynomials with application to Diophantine equationshttps://projecteuclid.org/euclid.jmsj/1558080017<strong>Masanori SAWA</strong>, <strong>Yukihiro UCHIDA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 3, 831--860.</p><p><strong>Abstract:</strong><br/>
We derive explicit formulas for the discriminants of classical quasi-orthogonal polynomials, as a full generalization of the result of Dilcher and Stolarsky (2005). We consider a certain system of Diophantine equations, originally designed by Hausdorff (1909) as a simplification of Hilbert's solution (1909) of Waring's problem, and then create the relationship to quadrature formulas and quasi-Hermite polynomials. We reduce these equations to the existence problem of rational points on a hyperelliptic curve associated with discriminants of quasi-Hermite polynomials, and show a nonexistence theorem for solutions of Hausdorff-type equations by applying our discriminant formula.
</p>projecteuclid.org/euclid.jmsj/1558080017_20190724220337Wed, 24 Jul 2019 22:03 EDTRelative stability associated to quantised extremal Kähler metricshttps://projecteuclid.org/euclid.jmsj/1556179398<strong>Yoshinori HASHIMOTO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 3, 861--880.</p><p><strong>Abstract:</strong><br/>
We study algebro-geometric consequences of the quantised extremal Kähler metrics, introduced in the previous work of the author. We prove that the existence of quantised extremal metrics implies weak relative Chow polystability. As a consequence, we obtain asymptotic weak relative Chow polystability and relative $K$-semistability of extremal manifolds by using quantised extremal metrics; this gives an alternative proof of the results of Mabuchi and Stoppa–Székelyhidi. In proving them, we further provide an explicit local density formula for the equivariant Riemann–Roch theorem.
</p>projecteuclid.org/euclid.jmsj/1556179398_20190724220337Wed, 24 Jul 2019 22:03 EDT$L_p$ regularity theorem for elliptic equations in less smooth domainshttps://projecteuclid.org/euclid.jmsj/1556092821<strong>Yoichi MIYAZAKI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 3, 881--907.</p><p><strong>Abstract:</strong><br/>
We consider a $2m$th-order strongly elliptic operator $A$ subject to Dirichlet boundary conditions in a domain $\Omega$ of $\mathbb{R}^{n}$, and show the $L_{p}$ regularity theorem, assuming that the domain has less smooth boundary. We derive the regularity theorem from the following isomorphism theorems in Sobolev spaces. Let $k$ be a nonnegative integer. When $A$ is a divergence form elliptic operator, $A-\lambda$ has a bounded inverse from the Sobolev space $W^{k-m}_{p}(\Omega)$ into $W^{k+m}_{p}(\Omega)$ for $\lambda$ belonging to a suitable sectorial region of the complex plane, if $\Omega$ is a uniformly $C^{k,1}$ domain. When $A$ is a non-divergence form elliptic operator, $A-\lambda$ has a bounded inverse from $W^{k}_{p}(\Omega)$ into $W^{k+2m}_{p}(\Omega)$, if $\Omega$ is a uniformly $C^{k+m,1}$ domain. Compared with the known results, we weaken the smoothness assumption on the boundary of $\Omega$ by $m-1$.
</p>projecteuclid.org/euclid.jmsj/1556092821_20190724220337Wed, 24 Jul 2019 22:03 EDTOn sharper estimates of Ohsawa–Takegoshi $L^2$-extension theoremhttps://projecteuclid.org/euclid.jmsj/1552377784<strong>Genki HOSONO</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 3, 909--914.</p><p><strong>Abstract:</strong><br/>
We present an $L^2$-extension theorem with an estimate depending on the weight functions for domains in $\mathbb{C}$. When the Hartogs domain defined by the weight function is strictly pseudoconvex, this estimate is strictly sharper than known optimal estimates. When the weight function is radial, we prove that our estimate provides the $L^2$-minimum extension.
</p>projecteuclid.org/euclid.jmsj/1552377784_20190724220337Wed, 24 Jul 2019 22:03 EDTRough flowshttps://projecteuclid.org/euclid.jmsj/1559030414<strong>Ismaël BAILLEUL</strong>, <strong>Sebastian RIEDEL</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 3, 915--978.</p><p><strong>Abstract:</strong><br/>
We introduce in this work a concept of rough driver that somehow provides a rough path-like analogue of an enriched object associated with time-dependent vector fields. We use the machinery of approximate flows to build the integration theory of rough drivers and prove well-posedness results for rough differential equations on flows and continuity of the solution flow as a function of the generating rough driver. We show that the theory of semimartingale stochastic flows developed in the 80's and early 90's fits nicely in this framework, and obtain as a consequence some strong approximation results for general semimartingale flows and provide a fresh look at large deviation theorems for ‘Gaussian’ stochastic flows.
</p>projecteuclid.org/euclid.jmsj/1559030414_20190724220337Wed, 24 Jul 2019 22:03 EDTGeneral formal solutions for a unified family of $P_{\mathrm{J}}$-hierarchies (J=I, II, IV, 34)https://projecteuclid.org/euclid.jmsj/1556092820<strong>Yoko UMETA</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 3, 979--1003.</p><p><strong>Abstract:</strong><br/>
A unified family of $P_{\mathrm{J}}$-hierarchies (J=I, II, IV, 34) with a large parameter is introduced and we construct general formal solutions which are called instanton-type solutions for the system.
</p>projecteuclid.org/euclid.jmsj/1556092820_20190724220337Wed, 24 Jul 2019 22:03 EDTOn an upper bound of $\lambda$-invariants of $\mathbb{Z}_p$-extensions over an imaginary quadratic fieldhttps://projecteuclid.org/euclid.jmsj/1556179397<strong>Kazuaki MURAKAMI</strong>. <p><strong>Source: </strong>Journal of the Mathematical Society of Japan, Volume 71, Number 3, 1005--1026.</p><p><strong>Abstract:</strong><br/>
For an odd prime number $p$, we give an explicit upper bound of $\lambda$-invariants for all $\mathbb{Z}_p$-extensions of an imaginary quadratic field $k$ under several assumptions. We also give an explicit upper bound of $\lambda$-invariants for all $\mathbb{Z}_p$-extensions of $k$ in the case where the $\lambda$-invariant of the cyclotomic $\mathbb{Z}_p$-extension of $k$ is equal to 3.
</p>projecteuclid.org/euclid.jmsj/1556179397_20190724220337Wed, 24 Jul 2019 22:03 EDT