Tohoku Mathematical Journal Articles (Project Euclid)
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The latest articles from Tohoku Mathematical Journal 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, 19 Apr 2011 09:33 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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$\boldsymbol{Q}$-factorial Gorenstein toric Fano varieties with large Picard number
http://projecteuclid.org/euclid.tmj/1270041023
<strong>Benjamin Nill</strong>, <strong>Mikkel Øbro</strong><p><strong>Source: </strong>Tohoku Math. J. (2), Volume 62, Number 1, 1--15.</p><p><strong>Abstract:</strong><br/>
In dimension $d$, ${\boldsymbol Q}$-factorial Gorenstein toric Fano varieties with
Picard number $\rho_X$ correspond to simplicial reflexive polytopes with $\rho_X +
d$ vertices. Casagrande showed that any $d$-dimensional simplicial reflexive
polytope has at most $3 d$ and $3d-1$ vertices if $d$ is even and odd, respectively.
Moreover, for $d$ even there is up to unimodular equivalence only one such polytope
with $3 d$ vertices, corresponding to the product of $d/2$ copies of a del Pezzo
surface of degree six. In this paper we completely classify all $d$-dimensional
simplicial reflexive polytopes having $3d-1$ vertices, corresponding to
$d$-dimensional ${\boldsymbol Q}$-factorial Gorenstein toric Fano varieties with
Picard number $2d-1$. For $d$ even, there exist three such varieties, with two being
singular, while for $d > 1$ odd there exist precisely two, both being nonsingular
toric fiber bundles over the projective line. This generalizes recent work of the
second author.
</p>projecteuclid.org/euclid.tmj/1270041023_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTWeighted Hamiltonian stationary Lagrangian submanifolds and generalized
Lagrangian mean curvature flows in toric almost Calabi-Yau manifoldshttp://projecteuclid.org/euclid.tmj/1474652263<strong>Hikaru Yamamoto</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 68, Number 3, 329--347.</p><p><strong>Abstract:</strong><br/>
In this paper, we generalize examples of Lagrangian mean curvature flows
constructed by Lee and Wang in $\mathbb{C}^m$ to toric almost Calabi-Yau
manifolds. To be more precise, we construct examples of weighted Hamiltonian
stationary Lagrangian submanifolds in toric almost Calabi-Yau manifolds and
solutions of generalized Lagrangian mean curvature flows starting from these
examples. We allow these flows to have some singularities and topological
changes.
</p>projecteuclid.org/euclid.tmj/1474652263_20160923133752Fri, 23 Sep 2016 13:37 EDTHomotopy theory of mixed Hodge complexeshttp://projecteuclid.org/euclid.tmj/1474652264<strong>Joana Cirici</strong>, <strong>Francisco Guillén</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 68, Number 3, 349--375.</p><p><strong>Abstract:</strong><br/>
We show that the category of mixed Hodge complexes admits a Cartan-Eilenberg
structure, a notion introduced by Guillén-Navarro-Pascual-Roig leading to
a good calculation of the homotopy category in terms of (co)fibrant objects.
Using Deligne's décalage, we show that the homotopy categories associated
with the two notions of mixed Hodge complex introduced by Deligne and Beilinson
respectively, are equivalent. The results provide a conceptual framework from
which Beilinson's and Carlson's results on mixed Hodge complexes and extensions
of mixed Hodge structures follow easily.
</p>projecteuclid.org/euclid.tmj/1474652264_20160923133752Fri, 23 Sep 2016 13:37 EDTCrossed actions of matched pairs of groups on tensor categorieshttp://projecteuclid.org/euclid.tmj/1474652265<strong>Sonia Natale</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 68, Number 3, 377--405.</p><p><strong>Abstract:</strong><br/>
We introduce the notion of $(G, \Gamma)$-crossed action on a tensor category,
where $(G, \Gamma)$ is a matched pair of finite groups. A tensor category is
called a $(G, \Gamma)$-crossed tensor category if it is endowed with a $(G,
\Gamma)$-crossed action. We show that every $(G,\Gamma)$-crossed tensor category
$\mathcal{C}$ gives rise to a tensor category $\mathcal{C}^{(G, \Gamma)}$ that
fits into an exact sequence of tensor categories $\operatorname{Rep} G \longrightarrow
\mathcal{C}^{(G, \Gamma)} \longrightarrow \mathcal{C}$. We also define the notion of a
$(G, \Gamma)$-braiding in a $(G, \Gamma)$-crossed tensor category, which is
connected with certain set-theoretical solutions of the QYBE. This extends the
notion of $G$-crossed braided tensor category due to Turaev. We show that if
$\mathcal{C}$ is a $(G, \Gamma)$-crossed tensor category equipped with a $(G,
\Gamma)$-braiding, then the tensor category $\mathcal{C}^{(G, \Gamma)}$ is a
braided tensor category in a canonical way.
</p>projecteuclid.org/euclid.tmj/1474652265_20160923133752Fri, 23 Sep 2016 13:37 EDTMatrix valued orthogonal polynomials for Gelfand pairs of rank onehttp://projecteuclid.org/euclid.tmj/1474652266<strong>Gert Heckman</strong>, <strong>Maarten van Pruijssen</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 68, Number 3, 407--437.</p><p><strong>Abstract:</strong><br/>
In this paper we study matrix valued orthogonal polynomials of one variable
associated with a compact connected Gelfand pair $(G,K)$ of rank one, as a
generalization of earlier work by Koornwinder [30] and subsequently by Koelink,
van Pruijssen and Roman [28], [29] for the pair (SU(2)$\times$SU(2), SU(2)), and
by Grünbaum, Pacharoni and Tirao [13] for the pair (SU(3), U(2)). Our
method is based on representation theory using an explicit determination of the
relevant branching rules. Our matrix valued orthogonal polynomials have the
Sturm-Liouville property of being eigenfunctions of a second order matrix
valued linear differential operator coming from the Casimir operator, and in
fact are eigenfunctions of a commutative algebra of matrix valued linear
differential operators coming from the $K$-invariant elements in the universal
enveloping algebra of the Lie algebra of $G$.
</p>projecteuclid.org/euclid.tmj/1474652266_20160923133752Fri, 23 Sep 2016 13:37 EDTThe Lê-Greuel formula for functions on analytic spaceshttp://projecteuclid.org/euclid.tmj/1474652267<strong>Roberto Callejas-Bedregal</strong>, <strong>Michelle F. Z. Morgado</strong>, <strong>Marcelo J. Saia</strong>, <strong>José Seade</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 68, Number 3, 439--456.</p><p><strong>Abstract:</strong><br/>
In this article we give an extension of the Lê-Greuel formula to the
general setting of function germs $(f,g)$ defined on a complex analytic variety
$X$ with arbitrary singular set, where $f = (f_1,\ldots,f_k): (X,\underline{0})
\to (\mathbb{C}^k,\underline{0})$ is generically a submersion with respect to
some Whitney stratification on $X$. We assume further that the dimension of the
zero set $V(f)$ is larger than 0, that $f$ has the Thom $a_f$-property with
respect to this stratification, and $g: (X,\underline{0}) \to (\mathbb{C},0)$
has an isolated critical point in the stratified sense, both on $X$ and on
$V(f)$.
</p>projecteuclid.org/euclid.tmj/1474652267_20160923133752Fri, 23 Sep 2016 13:37 EDTRichness of Smith equivalent modules for finite gap Oliver groupshttp://projecteuclid.org/euclid.tmj/1474652268<strong>Toshio Sumi</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 68, Number 3, 457--469.</p><p><strong>Abstract:</strong><br/>
Let $G$ be a finite group not of prime power order. Two real $G$-modules $U$ and
$V$ are $\mathcal{P}(G)$-connectively Smith equivalent if there exists a
homotopy sphere with smooth $G$-action such that the fixed point set by $P$ is
connected for all Sylow subgroups $P$ of $G$, it has just two fixed points, and
$U$ and $V$ are isomorphic to the tangential representations as real $G$-modules
respectively. We study the $\mathcal{P}(G)$-connective Smith set for a finite
Oliver group $G$ of the real representation ring consisting of all differences
of $\mathcal{P}(G)$-connectively Smith equivalent $G$-modules, and determine
this set for certain nonsolvable groups $G$.
</p>projecteuclid.org/euclid.tmj/1474652268_20160923133752Fri, 23 Sep 2016 13:37 EDTUmbilical surfaces of products of space formshttp://projecteuclid.org/euclid.tmj/1474652269<strong>Jaime Orjuela</strong>, <strong>Ruy Tojeiro</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 68, Number 3, 471--486.</p><p><strong>Abstract:</strong><br/>
We give a complete classification of umbilical surfaces of arbitrary codimension
of a product $\mathbb{Q}_{k_1}^{n_1}\times \mathbb{Q}_{k_2}^{n_2}$ of space
forms whose curvatures satisfy $k_1+k_2\neq 0$.
</p>projecteuclid.org/euclid.tmj/1474652269_20160923133752Fri, 23 Sep 2016 13:37 EDTThe equivariant $K$-theory and cobordism rings of divisive weighted projective
spaceshttp://projecteuclid.org/euclid.tmj/1486177213<strong>Megumi Harada</strong>, <strong>Tara S. Holm</strong>, <strong>Nigel Ray</strong>, <strong>Gareth Williams</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 68, Number 4, 487--513.</p><p><strong>Abstract:</strong><br/>
We apply results of Harada, Holm and Henriques to prove that the Atiyah-Segal
equivariant complex $K$-theory ring of a divisive weighted projective space
(which is singular for non-trivial weights) is isomorphic to the ring of
integral piecewise Laurent polynomials on the associated fan. Analogues of this
description hold for other complex-oriented equivariant cohomology theories, as
we confirm in the case of homotopical complex cobordism, which is the universal
example. We also prove that the Borel versions of the equivariant $K$-theory and
complex cobordism rings of more general singular toric varieties, namely those
whose integral cohomology is concentrated in even dimensions, are isomorphic to
rings of appropriate piecewise formal power series. Finally, we confirm the
corresponding descriptions for any smooth , compact, projective toric
variety, and rewrite them in a face ring context. In many cases our results
agree with those of Vezzosi and Vistoli for algebraic $K$-theory, Anderson and
Payne for operational $K$-theory, Krishna and Uma for algebraic cobordism, and
Gonzalez and Karu for operational cobordism; as we proceed, we summarize the
details of these coincidences.
</p>projecteuclid.org/euclid.tmj/1486177213_20170203220030Fri, 03 Feb 2017 22:00 ESTCalabi–Yau 3-folds of Borcea–Voisin type and elliptic fibrationshttp://projecteuclid.org/euclid.tmj/1486177214<strong>Andrea Cattaneo</strong>, <strong>Alice Garbagnati</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 68, Number 4, 515--558.</p><p><strong>Abstract:</strong><br/>
We consider Calabi–Yau 3-folds of Borcea–Voisin type, i.e. Calabi–Yau 3-folds
obtained as crepant resolutions of a quotient $(S\times E)/(\alpha_S\times
\alpha_E)$, where $S$ is a K3 surface, $E$ is an elliptic curve, $\alpha_S\in
\operatorname{Aut}(S)$ and $\alpha_E\in \operatorname{Aut}(E)$ act on the period
of $S$ and $E$ respectively with order $n=2,3,4,6$. The case $n=2$ is very
classical, the case $n=3$ was recently studied by Rohde, the other cases are
less known. First, we construct explicitly a crepant resolution, $X$, of
$(S\times E)/(\alpha_S\times \alpha_E)$ and we compute its Hodge numbers; some
pairs of Hodge numbers we found are new. Then, we discuss the presence of
maximal automorphisms and of a point with maximal unipotent monodromy for the
family of $X$. Finally, we describe the map $\mathcal{E}_n: X \rightarrow
S/\alpha_S$ whose generic fiber is isomorphic to $E$.
</p>projecteuclid.org/euclid.tmj/1486177214_20170203220030Fri, 03 Feb 2017 22:00 ESTHomogeneous Ricci soliton hypersurfaces in the complex hyperbolic spaceshttp://projecteuclid.org/euclid.tmj/1486177215<strong>Takahiro Hashinaga</strong>, <strong>Akira Kubo</strong>, <strong>Hiroshi Tamaru</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 68, Number 4, 559--568.</p><p><strong>Abstract:</strong><br/>
A Lie hypersurface in the complex hyperbolic space is an orbit of a cohomogeneity
one action without singular orbit. In this paper, we classify Ricci soliton Lie
hypersurfaces in the complex hyperbolic spaces.
</p>projecteuclid.org/euclid.tmj/1486177215_20170203220030Fri, 03 Feb 2017 22:00 ESTConvergence of measures penalized by generalized Feynman-Kac transformshttp://projecteuclid.org/euclid.tmj/1486177216<strong>Daehong Kim</strong>, <strong>Masakuni Matsuura</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 68, Number 4, 569--589.</p><p><strong>Abstract:</strong><br/>
We prove the existence of limiting laws for symmetric stable-like processes
penalized by generalized Feynman-Kac functionals and characterize them by the
gauge functions and the ground states of Schrödinger type operators.
</p>projecteuclid.org/euclid.tmj/1486177216_20170203220030Fri, 03 Feb 2017 22:00 ESTConstruction of sign-changing solutions for a subcritical problem on the four
dimensional half spherehttp://projecteuclid.org/euclid.tmj/1486177217<strong>Rabeh Ghoudi</strong>, <strong>Kamal Ould Bouh</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 68, Number 4, 591--605.</p><p><strong>Abstract:</strong><br/>
This paper is devoted to studying the nonlinear problem with subcritical exponent
$(S_\varepsilon) : -\Delta_g u+2u = K|u|^{2-\varepsilon}u$, in $ S^4_+ $,
${\partial u}/{\partial\nu} =0$, on $\partial S^4_+,$ where $g$ is the standard
metric of $S^4_+$ and $K$ is a $C^3$ positive Morse function on
$\overline{S_+^4}$. We construct some sign-changing solutions which blow up at
two different critical points of $K$ in interior. Furthermore, we construct
sign-changing solutions of $(S_\varepsilon)$ having two bubbles and blowing up
at the same critical point of $K$.
</p>projecteuclid.org/euclid.tmj/1486177217_20170203220030Fri, 03 Feb 2017 22:00 ESTHolomorphic submersions onto Kähler or balanced manifoldshttp://projecteuclid.org/euclid.tmj/1486177218<strong>Lucia Alessandrini</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 68, Number 4, 607--619.</p><p><strong>Abstract:</strong><br/>
We study many properties concerning weak Kählerianity on compact complex
manifolds which admits a holomorphic submersion onto a Kähler or a
balanced manifold. We get generalizations of some results of Harvey and Lawson
(the Kähler case), Michelsohn (the balanced case), Popovici (the sG case)
and others.
</p>projecteuclid.org/euclid.tmj/1486177218_20170203220030Fri, 03 Feb 2017 22:00 ESTWintgen ideal submanifolds of codimension two, complex curves, and Möbius
geometryhttp://projecteuclid.org/euclid.tmj/1486177219<strong>Tongzhu Li</strong>, <strong>Xiang Ma</strong>, <strong>Changping Wang</strong>, <strong>Zhenxiao Xie</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 68, Number 4, 621--638.</p><p><strong>Abstract:</strong><br/>
Wintgen ideal submanifolds in space forms are those ones attaining the equality
pointwise in the so-called DDVV inequality which relates the scalar curvature,
the mean curvature and the scalar normal curvature. Using the framework of
Möbius geometry, we show that in the codimension two case, the mean
curvature spheres of the Wintgen ideal submanifold correspond to a 1-isotropic
holomorphic curve in a complex quadric. Conversely, any 1-isotropic complex
curve in this complex quadric describes a 2-parameter family of spheres whose
envelope is always a Wintgen ideal submanifold of codimension two at the regular
points. Via a complex stereographic projection, we show that our
characterization is equivalent to Dajczer and Tojeiro's previous description of
these submanifolds in terms of minimal surfaces in the Euclidean space.
</p>projecteuclid.org/euclid.tmj/1486177219_20170203220030Fri, 03 Feb 2017 22:00 ESTA note on the Kakeya maximal operator and radial weights on the planehttp://projecteuclid.org/euclid.tmj/1486177220<strong>Hiroki Saito</strong>, <strong>Yoshihiro Sawano</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 68, Number 4, 639--649.</p><p><strong>Abstract:</strong><br/>
We obtain an estimate of the operator norm of the weighted Kakeya
(Nikodým) maximal operator without dilation on $L^2(w)$. Here we assume
that a radial weight $w$ satisfies the doubling and supremum condition. Recall
that, in the definition of the Kakeya maximal operator, the rectangle in the
supremum ranges over all rectangles in the plane pointed in all possible
directions and having side lengths $a$ and $aN$ with $N$ fixed. We are
interested in its eccentricity $N$ with $a$ fixed. We give an example of a
non-constant weight showing that $\sqrt{\log N}$ cannot be removed.
</p>projecteuclid.org/euclid.tmj/1486177220_20170203220030Fri, 03 Feb 2017 22:00 ESTReal analytic complete non-compact surfaces in Euclidean space with finite total
curvature arising as solutions to ODEshttp://projecteuclid.org/euclid.tmj/1493172124<strong>Peter Gilkey</strong>, <strong>Chan Yong Kim</strong>, <strong>JeongHyeong Park</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 1, 1--23.</p><p><strong>Abstract:</strong><br/>
We use the solution space of a pair of ODEs of at least second order to construct
a smooth surface in Euclidean space. We describe when this surface is a proper
embedding which is geodesically complete with finite total Gauss curvature. If
the associated roots of the ODEs are real and distinct, we give a universal
upper bound for the total Gauss curvature of the surface which depends only on
the orders of the ODEs and we show that the total Gauss curvature of the surface
vanishes if the ODEs are second order. We examine when the surfaces are
asymptotically minimal.
</p>projecteuclid.org/euclid.tmj/1493172124_20170425220211Tue, 25 Apr 2017 22:02 EDTA remark on Jacquet-Langlands correspondence and invariant $s$http://projecteuclid.org/euclid.tmj/1493172125<strong>Kazutoshi Kariyama</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 1, 25--33.</p><p><strong>Abstract:</strong><br/>
Let $F$ be a non-Archimedean local field, and let $G$ be an inner form of
$\mathrm{GL}_N(F)$ with $N \ge 1$. Let $\boldsymbol{\mathrm{JL}}$ be the
Jacquet--Langlands correspondence between $\mathrm{GL}_N(F)$ and $G$. In this
paper, we compute the invariant $s$ associated with the essentially
square-integrable representation $\boldsymbol{\mathrm{JL}}^{-1}(\rho)$ for a
cuspidal representation $\rho$ of $G$ by using the recent results of Bushnell
and Henniart, and we restate the second part of a theorem given by Deligne,
Kazhdan, and Vignéras in terms of the invariant $s$. Moreover, by using
the parametric degree, we present a proof of the first part of the theorem.
</p>projecteuclid.org/euclid.tmj/1493172125_20170425220211Tue, 25 Apr 2017 22:02 EDTSurfaces of globally $F$-regular type are of Fano typehttp://projecteuclid.org/euclid.tmj/1493172126<strong>Shinnosuke Okawa</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 1, 35--42.</p><p><strong>Abstract:</strong><br/>
We prove that a projective surface of globally $F$-regular type defined over a
field of characteristic zero is of Fano type.
</p>projecteuclid.org/euclid.tmj/1493172126_20170425220211Tue, 25 Apr 2017 22:02 EDTOn the generalized Wintgen inequality for Legendrian submanifolds in Sasakian
space formshttp://projecteuclid.org/euclid.tmj/1493172127<strong>Ion Mihai</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 1, 43--53.</p><p><strong>Abstract:</strong><br/>
The generalized Wintgen inequality was conjectured by De Smet, Dillen,
Verstraelen and Vrancken in 1999 for submanifolds in real space forms. It is
also known as the DDVV conjecture. It was proven recently by Lu (2011) and by Ge
and Tang (2008), independently. The present author established a generalized
Wintgen inequality for Lagrangian submanifolds in complex space forms in 2014.
In the present paper we obtain the DDVV inequality, also known as generalized
Wintgen inequality, for Legendrian submanifolds in Sasakian space forms. Some
geometric applications are derived. Also we state such an inequality for contact
slant submanifolds in Sasakian space forms.
</p>projecteuclid.org/euclid.tmj/1493172127_20170425220211Tue, 25 Apr 2017 22:02 EDTOn the reduction modulo $p$ of Mahler equationshttp://projecteuclid.org/euclid.tmj/1493172128<strong>Julien Roques</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 1, 55--65.</p><p><strong>Abstract:</strong><br/>
The guiding thread of the present work is the following result, in the vain of
Grothendieck's conjecture for differential equations : if the reduction modulo
almost all prime $p$ of a given linear Mahler equation with coefficients in
$\mathbb{Q}(z)$ has a full set of algebraic solutions, then this equation has a
full set of rational solutions. The proof of this result, given at the very end
of the paper, relies on intermediate results of independent interest about
Mahler equations in characteristic zero as well as in positive
characteristic.
</p>projecteuclid.org/euclid.tmj/1493172128_20170425220211Tue, 25 Apr 2017 22:02 EDTOn the universal deformations for ${\rm SL}_2$-representations of knot
groupshttp://projecteuclid.org/euclid.tmj/1493172129<strong>Masanori Morishita</strong>, <strong>Yu Takakura</strong>, <strong>Yuji Terashima</strong>, <strong>Jun Ueki</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 1, 67--84.</p><p><strong>Abstract:</strong><br/>
Based on the analogies between knot theory and number theory, we study a
deformation theory for ${\rm SL}_2$-representations of knot groups, following
after Mazur's deformation theory of Galois representations. Firstly, by
employing the pseudo-${\rm SL}_2$-representations, we prove the existence of the
universal deformation of a given ${\rm SL}_2$-representation of a finitely
generated group $\Pi$ over a perfect field $k$ whose characteristic is not 2. We
then show its connection with the character scheme for ${\rm
SL}_2$-representations of $\Pi$ when $k$ is an algebraically closed field. We
investigate examples concerning Riley representations of 2-bridge knot groups
and give explicit forms of the universal deformations. Finally we discuss the
universal deformation of the holonomy representation of a hyperbolic knot group
in connection with Thurston's theory on deformations of hyperbolic
structures.
</p>projecteuclid.org/euclid.tmj/1493172129_20170425220211Tue, 25 Apr 2017 22:02 EDTOn linear deformations of Brieskorn singularities of two variables into generic
mapshttp://projecteuclid.org/euclid.tmj/1493172130<strong>Kazumasa Inaba</strong>, <strong>Masaharu Ishikawa</strong>, <strong>Masayuki Kawashima</strong>, <strong>Tat Thang Nguyen</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 1, 85--111.</p><p><strong>Abstract:</strong><br/>
In this paper, we study deformations of Brieskorn polynomials of two variables
obtained by adding linear terms consisting of the conjugates of complex
variables and prove that the deformed polynomial maps have only indefinite fold
and cusp singularities in general. We then estimate the number of cusps
appearing in such a deformation. As a corollary, we show that a deformation of a
complex Morse singularity with real linear terms has only indefinite folds and
cusps in general and the number of cusps is 3.
</p>projecteuclid.org/euclid.tmj/1493172130_20170425220211Tue, 25 Apr 2017 22:02 EDTPeriodic magnetic curves in Berger sphereshttp://projecteuclid.org/euclid.tmj/1493172131<strong>Jun-ichi Inoguchi</strong>, <strong>Marian Ioan Munteanu</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 1, 113--128.</p><p><strong>Abstract:</strong><br/>
It is an interesting question whether a given equation of motion has a periodic
solution or not, and in the positive case to describe it. We investigate
periodic magnetic curves in elliptic Sasakian space forms and we obtain a
quantization principle for periodic magnetic flowlines on Berger spheres. We
give a criterion for periodicity of magnetic curves on the unit sphere
${\mathbb{S}}^3$.
</p>projecteuclid.org/euclid.tmj/1493172131_20170425220211Tue, 25 Apr 2017 22:02 EDTNotes on 'Infinitesimal derivative of the Bott class and the Schwarzian
derivatives'http://projecteuclid.org/euclid.tmj/1493172132<strong>Taro Asuke</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 1, 129--139.</p><p><strong>Abstract:</strong><br/>
The derivatives of the Bott class and those of the Godbillon-Vey class with
respect to infinitesimal deformations of foliations, called infinitesimal
derivatives, are known to be represented by a formula in the projective
Schwarzian derivatives of holonomies [3], [1]. It is recently shown that these
infinitesimal derivatives are represented by means of coefficients of transverse
Thomas-Whitehead projective connections [2]. We will show that the formula can
be also deduced from the latter representation.
</p>projecteuclid.org/euclid.tmj/1493172132_20170425220211Tue, 25 Apr 2017 22:02 EDTWillmore surfaces in spheres via loop groups III: on minimal surfaces in space
formshttp://projecteuclid.org/euclid.tmj/1493172133<strong>Peng Wang</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 1, 141--160.</p><p><strong>Abstract:</strong><br/>
The family of Willmore immersions from a Riemann surface into $S^{n+2}$ can be
divided naturally into the subfamily of Willmore surfaces conformally equivalent
to a minimal surface in $\mathbb{R}^{n+2}$ and those which are not conformally
equivalent to a minimal surface in $\mathbb{R}^{n+2}$. On the level of their
conformal Gauss maps into
$Gr_{1,3}(\mathbb{R}^{1,n+3})=SO^+(1,n+3)/SO^+(1,3)\times SO(n)$ these two
classes of Willmore immersions into $S^{n+2}$ correspond to conformally harmonic
maps for which every image point, considered as a 4-dimensional Lorentzian
subspace of $\mathbb{R}^{1,n+3}$, contains a fixed lightlike vector or where it
does not contain such a ``constant lightlike vector''. Using the loop group
formalism for the construction of Willmore immersions we characterize in this
paper precisely those normalized potentials which correspond to conformally
harmonic maps containing a lightlike vector. Since the special form of these
potentials can easily be avoided, we also precisely characterize those
potentials which produce Willmore immersions into $S^{n+2}$ which are not
conformal to a minimal surface in $\mathbb{R}^{n+2}$. It turns out that our
proof also works analogously for minimal immersions into the other space
forms.
</p>projecteuclid.org/euclid.tmj/1493172133_20170425220211Tue, 25 Apr 2017 22:02 EDTOn a characterization of unbounded homogeneous domains with boundaries of light
cone typehttp://projecteuclid.org/euclid.tmj/1498269621<strong>Jun-ichi Mukuno</strong>, <strong>Yoshikazu Nagata</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 2, 161--181.</p><p><strong>Abstract:</strong><br/>
We determine the automorphism groups of unbounded homogeneous domains with
boundaries of light cone type. Furthermore we present a group-theoretic
characterization of one of the domains. As a corollary we prove the
non-existence of compact quotients of the homogeneous domain. We also give a
counterexample of the characterization.
</p>projecteuclid.org/euclid.tmj/1498269621_20170623220041Fri, 23 Jun 2017 22:00 EDTSome new properties concerning BLO martingaleshttp://projecteuclid.org/euclid.tmj/1498269622<strong>Eiichi Nakai</strong>, <strong>Gaku Sadasue</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 2, 183--194.</p><p><strong>Abstract:</strong><br/>
Some new properties concerning BLO martingales are given. The BMO-BLO boundedness
of martingale maximal functions and Bennett type characterization of BLO
martingales are shown. Also, a non-negative BMO martingale that is not in BLO is
constructed.
</p>projecteuclid.org/euclid.tmj/1498269622_20170623220041Fri, 23 Jun 2017 22:00 EDTRemarks on motives of abelian typehttp://projecteuclid.org/euclid.tmj/1498269623<strong>Charles Vial</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 2, 195--220.</p><p><strong>Abstract:</strong><br/>
A motive over a field $k$ is of abelian type if it belongs to the thick and rigid
subcategory of Chow motives spanned by the motives of abelian varieties over
$k$. This paper contains three sections of independent interest. First, we show
that a motive which becomes of abelian type after a base field extension of
algebraically closed fields is of abelian type. Given a field extension $K/k$
and a motive $M$ over $k$, we also show that $M$ is finite-dimensional if and
only if $M_K$ is finite-dimensional. As a corollary, we obtain
Chow–Künneth decompositions for varieties that become isomorphic to an
abelian variety after some field extension. Second, let $\varOmega$ be a universal
domain containing $k$. We show that Murre's conjectures for motives of abelian
type over $k$ reduce to Murre's conjecture (D) for products of curves over
$\varOmega$. In particular, we show that Murre's conjecture (D) for products of
curves over $\varOmega$ implies Beauville's vanishing conjecture on abelian
varieties over $k$. Finally, we give criteria on Chow groups for a motive to be
of abelian type. For instance, we show that $M$ is of abelian type if and only
if the total Chow group of algebraically trivial cycles
$\mathrm{CH}_*(M_\varOmega)_\mathrm{alg}$ is spanned, via the action of
correspondences, by the Chow groups of products of curves. We also show that a
morphism of motives $f: N \rightarrow M$, with $N$ finite-dimensional, which
induces a surjection $f_* : \mathrm{CH}_*(N_\varOmega)_\mathrm{alg} \rightarrow
\mathrm{CH}_*(M_\varOmega)_\mathrm{alg}$ also induces a surjection $f_* :
\mathrm{CH}_*(N_\varOmega)_\mathrm{hom} \rightarrow
\mathrm{CH}_*(M_\varOmega)_\mathrm{hom}$ on homologically trivial cycles.
</p>projecteuclid.org/euclid.tmj/1498269623_20170623220041Fri, 23 Jun 2017 22:00 EDTA fake projective plane via 2-adic uniformization with torsionhttp://projecteuclid.org/euclid.tmj/1498269624<strong>Daniel Allcock</strong>, <strong>Fumiharu Kato</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 2, 221--237.</p><p><strong>Abstract:</strong><br/>
We adapt the theory of non-Archimedean uniformization to construct a smooth
surface from a lattice in ${\rm PSL}_3(\mathbb{Q}_2)$ that has nontrivial
torsion. It turns out to be a fake projective plane, commensurable with
Mumford's fake plane yet distinct from it and the other fake planes that arise
from 2-adic uniformization by torsion-free groups. As part of the proof, and of
independent interest, we compute the homotopy type of the Berkovich space of our
plane.
</p>projecteuclid.org/euclid.tmj/1498269624_20170623220041Fri, 23 Jun 2017 22:00 EDTNon-hyperbolic unbounded Reinhardt domains: non-compact automorphism group,
Cartan's linearity theorem and explicit Bergman kernelhttp://projecteuclid.org/euclid.tmj/1498269625<strong>Atsushi Yamamori</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 2, 239--260.</p><p><strong>Abstract:</strong><br/>
In the study of the holomorphic automorphism groups, many researches have been
carried out inside the category of bounded or hyperbolic domains. On the
contrary to these cases, for unbounded non-hyperbolic cases, only a few results
are known about the structure of the holomorphic automorphism groups. Main
result of the present paper gives a class of unbounded non-hyperbolic Reinhardt
domains with non-compact automorphism groups, Cartan's linearity theorem and
explicit Bergman kernels. Moreover, a reformulation of Cartan's linearity
theorem for finite volume Reinhardt domains is also given.
</p>projecteuclid.org/euclid.tmj/1498269625_20170623220041Fri, 23 Jun 2017 22:00 EDTRobin problems with indefinite and unbounded potential, resonant at $-\infty$,
superlinear at $+\infty$http://projecteuclid.org/euclid.tmj/1498269626<strong>Nikolaos S. Papageorgiou</strong>, <strong>Vicenţiu D. Rădulescu</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 2, 261--286.</p><p><strong>Abstract:</strong><br/>
We consider a semilinear Robin problem with an indefinite and unbounded potential
and a reaction which exhibits asymmetric behavior as $x\rightarrow\pm\infty$.
More precisely it is sublinear near $-\infty$ with possible resonance with
respect to the principal eigenvalue of the negative Robin Laplacian and it is
superlinear at $+\infty$. Resonance is also allowed at zero with respect to any
nonprincipal eigenvalue. We prove two multiplicity results. In the first one, we
obtain two nontrivial solutions and in the second, under stronger regularity
conditions on the reaction, we produce three nontrivial solutions. Our work
generalizes the recent one by Recova-Rumbos (Nonlin. Anal. 112 (2015),
181--198).
</p>projecteuclid.org/euclid.tmj/1498269626_20170623220041Fri, 23 Jun 2017 22:00 EDTContiguity relations of Lauricella's $F_D$ revisitedhttp://projecteuclid.org/euclid.tmj/1498269627<strong>Yoshiaki Goto</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 2, 287--304.</p><p><strong>Abstract:</strong><br/>
We study contiguity relations of Lauricella's hypergeometric function $F_D$, by
using the twisted cohomology group and the intersection form. We derive
contiguity relations from those in the twisted cohomology group and give the
coefficients in these relations by the intersection numbers. Furthermore, we
construct twisted cycles corresponding to a fundamental set of solutions to the
system of differential equations satisfied by $F_D$, which are expressed as
Laurent series. We also give the contiguity relations of these solutions.
</p>projecteuclid.org/euclid.tmj/1498269627_20170623220041Fri, 23 Jun 2017 22:00 EDTReplacing the lower curvature bound in Toponogov's comparison theorem by a weaker
hypothesishttp://projecteuclid.org/euclid.tmj/1498269628<strong>James J. Hebda</strong>, <strong>Yutaka Ikeda</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 2, 305--325.</p><p><strong>Abstract:</strong><br/>
Toponogov's triangle comparison theorem and its generalizations are important
tools for studying the topology of Riemannian manifolds. In these theorems, one
assumes that the curvature of a given manifold is bounded from below by the
curvature of a model surface. The models are either of constant curvature, or,
in the generalizations, rotationally symmetric about some point. One concludes
that geodesic triangles in the manifold correspond to geodesic triangles in the
model surface which have the same corresponding side lengths, but smaller
corresponding angles. In addition, a certain rigidity holds: Whenever there is
equality in one of the corresponding angles, the geodesic triangle in the
surface embeds totally geodesically and isometrically in the manifold.
In this paper, we discuss a condition relating the geometry of a Riemannian
manifold to that of a model surface which is weaker than the usual curvature
hypothesis in the generalized Toponogov theorems, but yet is strong enough to
ensure that a geodesic triangle in the manifold has a corresponding triangle in
the model with the same corresponding side lengths, but smaller corresponding
angles. In contrast, it is interesting that rigidity fails in this setting.
</p>projecteuclid.org/euclid.tmj/1498269628_20170623220041Fri, 23 Jun 2017 22:00 EDTSeidel elements and potential functions of holomorphic disc countinghttps://projecteuclid.org/euclid.tmj/1505181621<strong>Eduardo González</strong>, <strong>Hiroshi Iritani</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 3, 327--368.</p><p><strong>Abstract:</strong><br/>
Let $M$ be a symplectic manifold equipped with a Hamiltonian circle action and
let $L$ be an invariant Lagrangian submanifold of $M$. We study the problem of
counting holomorphic disc sections of the trivial $M$-bundle over a disc
with boundary in $L$ through degeneration. We obtain a conjectural relationship
between the potential function of $L$ and the Seidel element associated to the
circle action. When applied to a Lagrangian torus fibre of a semi-positive toric
manifold, this degeneration argument reproduces a conjecture (now a theorem) of
Chan-Lau-Leung-Tseng [8, 9] relating certain correction terms appearing in the
Seidel elements with the potential function.
</p>projecteuclid.org/euclid.tmj/1505181621_20170911220035Mon, 11 Sep 2017 22:00 EDTA note on stable sheaves on Enriques surfaceshttps://projecteuclid.org/euclid.tmj/1505181622<strong>Kōta Yoshioka</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 3, 369--382.</p><p><strong>Abstract:</strong><br/>
We shall give a necessary and sufficient condition for the existence of stable
sheaves on Enriques surfaces based on results of Kim, Yoshioka, Hauzer and Nuer.
For unnodal Enriques surfaces, we also study the relation of virtual Hodge
“polynomial” of the moduli stacks.
</p>projecteuclid.org/euclid.tmj/1505181622_20170911220035Mon, 11 Sep 2017 22:00 EDTAtomic decompositions of weighted Hardy spaces with variable exponentshttps://projecteuclid.org/euclid.tmj/1505181623<strong>Kwok-Pun Ho</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 3, 383--413.</p><p><strong>Abstract:</strong><br/>
We establish the atomic decompositions for the weighted Hardy spaces with
variable exponents. These atomic decompositions also reveal some intrinsic
structures of atomic decomposition for Hardy type spaces.
</p>projecteuclid.org/euclid.tmj/1505181623_20170911220035Mon, 11 Sep 2017 22:00 EDTThe maximal ideal cycles over normal surface singularities with ${\Bbb
C}^*$-actionhttps://projecteuclid.org/euclid.tmj/1505181624<strong>Masataka Tomari</strong>, <strong>Tadashi Tomaru</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 3, 415--430.</p><p><strong>Abstract:</strong><br/>
The maximal ideal cycles and the fundamental cycles are defined on the
exceptional sets of resolution spaces of normal complex surface singularities.
The former (resp. later) is determined by the analytic (resp. topological)
structure of the singularities. We study such cycles for normal surface
singularities with ${\Bbb C}^*$-action. Assuming the existence of a reduced
homogeneous function of the minimal degree, we prove that these two cycles
coincide if the coefficients on the central curve of the exceptional set of the
minimal good resolution coincide.
</p>projecteuclid.org/euclid.tmj/1505181624_20170911220035Mon, 11 Sep 2017 22:00 EDTGauss maps of toric varietieshttps://projecteuclid.org/euclid.tmj/1505181625<strong>Katsuhisa Furukawa</strong>, <strong>Atsushi Ito</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 3, 431--454.</p><p><strong>Abstract:</strong><br/>
We investigate Gauss maps of (not necessarily normal) projective toric varieties
over an algebraically closed field of arbitrary characteristic. The main results
are as follows: (1) The structure of the Gauss map of a toric variety is
described in terms of combinatorics in any characteristic. (2) We give a
developability criterion in the toric case. In particular, we show that any
toric variety whose Gauss map is degenerate must be the join of some toric
varieties in characteristic zero. (3) As applications, we provide two
constructions of toric varieties whose Gauss maps have some given data (e.g.,
fibers, images) in positive characteristic.
</p>projecteuclid.org/euclid.tmj/1505181625_20170911220035Mon, 11 Sep 2017 22:00 EDTA remark on almost sure global well-posedness of the energy-critical defocusing
nonlinear wave equations in the periodic settinghttps://projecteuclid.org/euclid.tmj/1505181626<strong>Tadahiro Oh</strong>, <strong>Oana Pocovnicu</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 3, 455--481.</p><p><strong>Abstract:</strong><br/>
In this note, we prove almost sure global well-posedness of the energy-critical
defocusing nonlinear wave equation on $\mathbb{T}^d$, $d = 3, 4,$ and $5$, with
random initial data below the energy space.
</p>projecteuclid.org/euclid.tmj/1505181626_20170911220035Mon, 11 Sep 2017 22:00 EDTBoundedness of the maximal operator on Musielak-Orlicz-Morrey spaceshttps://projecteuclid.org/euclid.tmj/1512183626<strong>Fumi-Yuki Maeda</strong>, <strong>Takao Ohno</strong>, <strong>Tetsu Shimomura</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 4, 483--495.</p><p><strong>Abstract:</strong><br/>
We give the boundedness of the maximal operator on Musielak-Orlicz-Morrey spaces, which is an improvement of [7, Theorem 4.1]. We also discuss the sharpness of our conditions.
</p>projecteuclid.org/euclid.tmj/1512183626_20171201220054Fri, 01 Dec 2017 22:00 ESTBounds on the Tamagawa numbers of a crystalline representation over towers of cyclotomic extensionshttps://projecteuclid.org/euclid.tmj/1512183627<strong>Antonio Lei</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 4, 497--524.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the Tamagawa numbers of a crystalline representation over a tower of cyclotomic extensions under certain technical conditions on the representation. In particular, we show that we may improve the asymptotic bounds given in the thesis of Arthur Laurent in certain cases.
</p>projecteuclid.org/euclid.tmj/1512183627_20171201220054Fri, 01 Dec 2017 22:00 ESTHolomorphic isometric embeddings of the projective line into quadricshttps://projecteuclid.org/euclid.tmj/1512183628<strong>Oscar Macia</strong>, <strong>Yasuyuki Nagatomo</strong>, <strong>Masaro Takahashi</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 4, 525--545.</p><p><strong>Abstract:</strong><br/>
We discuss holomorphic isometric embeddings of the projective line into quadrics using the generalisation of the theorem of do Carmo-Wallach in [14] to provide a description of their moduli spaces up to image and gauge equivalence. Moreover, we show rigidity of the real standard map from the projective line into quadrics.
</p>projecteuclid.org/euclid.tmj/1512183628_20171201220054Fri, 01 Dec 2017 22:00 ESTMonodromy representations of hypergeometric systems with respect to fundamental series solutionshttps://projecteuclid.org/euclid.tmj/1512183629<strong>Keiji Matsumoto</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 4, 547--570.</p><p><strong>Abstract:</strong><br/>
We study the monodromy representation of the generalized hypergeometric differential equation and that of Lauricella's $F_C$ system of hypergeometric differential equations. We use fundamental systems of solutions expressed by the hypergeometric series. We express non-diagonal circuit matrices as reflections with respect to root vectors with all entries 1. We present a simple way to obtain circuit matrices.
</p>projecteuclid.org/euclid.tmj/1512183629_20171201220054Fri, 01 Dec 2017 22:00 ESTA polynomial defined by the $SL(2;\mathbb{C})$-Reidemeister torsion for a homology 3-sphere obtained by a Dehn surgery along a $(2p,q)$-torus knothttps://projecteuclid.org/euclid.tmj/1512183630<strong>Teruaki Kitano</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 4, 571--583.</p><p><strong>Abstract:</strong><br/>
Let $K$ be a $(2p,q)$-torus knot. Here $p$ and $q$ are coprime odd positive integers. Let $M_n$ be a 3-manifold obtained by a $1/n$-Dehn surgery along $K$. We consider a polynomial $\sigma_{(2p,q,n)}(t)$ whose zeros are the inverses of the Reidemeister torsion of $M_n$ for $\mathit{SL}(2;\mathbb{C})$-irreducible representations under some normalization. Johnson gave a formula for the case of the $(2,3)$-torus knot under another normalization. We generalize this formula for the case of $(2p,q)$-torus knots by using Tchebychev polynomials.
</p>projecteuclid.org/euclid.tmj/1512183630_20171201220054Fri, 01 Dec 2017 22:00 ESTSpectral zeta functions of graphs and the Riemann zeta function in the critical striphttps://projecteuclid.org/euclid.tmj/1512183631<strong>Fabien Friedli</strong>, <strong>Anders Karlsson</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 4, 585--610.</p><p><strong>Abstract:</strong><br/>
We initiate the study of spectral zeta functions $\zeta_X$ for finite and infinite graphs $X$, instead of the Ihara zeta function, with a perspective towards zeta functions from number theory and connections to hypergeometric functions. The Riemann hypothesis is shown to be equivalent to an approximate functional equation of graph zeta functions. The latter holds at all points where Riemann's zeta function $\zeta(s)$ is non-zero. This connection arises via a detailed study of the asymptotics of the spectral zeta functions of finite torus graphs in the critcal strip and estimates on the real part of the logarithmic derivative of $\zeta(s)$. We relate $\zeta_{\mathbb{Z}}$ to Euler's beta integral and show how to complete it giving the functional equation $\xi_{\mathbb{Z}}(1-s)=\xi_{\mathbb{Z}}(s)$. This function appears in the theory of Eisenstein series although presumably with this spectral intepretation unrecognized. In higher dimensions $d$ we provide a meromorphic continuation of $\zeta_{\mathbb{Z}^d}(s)$ to the whole plane and identify the poles. From our aymptotics several known special values of $\zeta(s)$ are derived as well as its non-vanishing on the line $Re(s)=1$. We determine the spectral zeta functions of regular trees and show it to be equal to a specialization of Appell's hypergeometric function $F_1$ via an Euler-type integral formula due to Picard.
</p>projecteuclid.org/euclid.tmj/1512183631_20171201220054Fri, 01 Dec 2017 22:00 ESTSchottky via the punctual Hilbert schemehttps://projecteuclid.org/euclid.tmj/1512183632<strong>Martin G. Gulbrandsen</strong>, <strong>Martí Lahoz</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 4, 611--619.</p><p><strong>Abstract:</strong><br/>
We show that a smooth projective curve of genus $g$ can be reconstructed from its polarized Jacobian $(X, \Theta)$ as a certain locus in the Hilbert scheme $\mathrm{Hilb}^d(X)$, for $d=3$ and for $d=g+2$, defined by geometric conditions in terms of the polarization $\Theta$. The result is an application of the Gunning-Welters trisecant criterion and the Castelnuovo-Schottky theorem by Pareschi-Popa and Grushevsky, and its scheme theoretic extension by the authors.
</p>projecteuclid.org/euclid.tmj/1512183632_20171201220054Fri, 01 Dec 2017 22:00 ESTMinimal timelike surfaces in a certain homogeneous Lorentzian 3-manifoldhttps://projecteuclid.org/euclid.tmj/1512183633<strong>Sungwook Lee</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 4, 621--635.</p><p><strong>Abstract:</strong><br/>
The 2-parameter family of certain homogeneous Lorentzian 3-manifolds which includes Minkowski 3-space, de Sitter 3-space, and Minkowski motion group is considered. Each homogeneous Lorentzian 3-manifold in the 2-parameter family has a solvable Lie group structure with left invariant metric. A generalized integral representation formula which is the unification of representation formulas for minimal timelike surfaces in those homogeneous Lorentzian 3-manifolds is obtained. The normal Gauß map of minimal timelike surfaces in those homogeneous Lorentzian 3-manifolds and its harmonicity are discussed.
</p>projecteuclid.org/euclid.tmj/1512183633_20171201220054Fri, 01 Dec 2017 22:00 ESTOn the most expected number of components for random linkshttps://projecteuclid.org/euclid.tmj/1512183634<strong>Kazuhiro Ichihara</strong>, <strong>Ken-ichi Yoshida</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 69, Number 4, 637--641.</p><p><strong>Abstract:</strong><br/>
We consider a random link, which is defined as the closure of a braid obtained from a random walk on the braid group. For such a random link, the expected value for the number of components was calculated by Jiming Ma. In this paper, we determine the most expected number of components for a random link, and further, consider the most expected partition of the number of strings for a random braid.
</p>projecteuclid.org/euclid.tmj/1512183634_20171201220054Fri, 01 Dec 2017 22:00 ESTExponentially weighted Polynomial approximation for absolutely continuous functionshttps://projecteuclid.org/euclid.tmj/1520564416<strong>Kentaro Itoh</strong>, <strong>Ryozi Sakai</strong>, <strong>Noriaki Suzuki</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 70, Number 1, 1--15.</p><p><strong>Abstract:</strong><br/>
We discuss a polynomial approximation on $\mathbb{R}$ with a weight $w$ in $\mathcal{F}(C^{2} +)$ (see Section 2). The de la Vallée Poussin mean $v_n(f)$ of an absolutely continuous function $f$ is not only a good approximation polynomial of $f$, but also its derivatives give an approximation for the derivative $f'$. More precisely, for $1 \leq p \leq \infty$, we have $\lim_{n \rightarrow \infty}\|(f - v_{n}(f))w\|_{L^{p}(\mathbb{R})} =0$ and $\lim_{n \rightarrow \infty}\|(f' - v_{n}(f)')w\|_{L^{p}(\mathbb{R})} =0$ whenever $f''w \in L^{p}(\mathbb{R})$.
</p>projecteuclid.org/euclid.tmj/1520564416_20180308220031Thu, 08 Mar 2018 22:00 ESTOn Frobenius manifolds from Gromov–Witten theory of orbifold projective lines with $r$ orbifold pointshttps://projecteuclid.org/euclid.tmj/1520564417<strong>Yuuki Shiraishi</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 70, Number 1, 17--37.</p><p><strong>Abstract:</strong><br/>
We prove that the Frobenius structure constructed from the Gromov–Witten theory for an orbifold projective line with at most $r$ orbifold points is uniquely determined by the WDVV equations with certain natural initial conditions.
</p>projecteuclid.org/euclid.tmj/1520564417_20180308220031Thu, 08 Mar 2018 22:00 ESTWorpitzky partitions for root systems and characteristic quasi-polynomialshttps://projecteuclid.org/euclid.tmj/1520564418<strong>Masahiko Yoshinaga</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 70, Number 1, 39--63.</p><p><strong>Abstract:</strong><br/>
For a given irreducible root system, we introduce a partition of (coweight) lattice points inside the dilated fundamental parallelepiped into those of partially closed simplices. This partition can be considered as a generalization and a lattice points interpretation of the classical formula of Worpitzky.
This partition, and the generalized Eulerian polynomial, recently introduced by Lam and Postnikov, can be used to describe the characteristic (quasi)polynomials of Shi and Linial arrangements. As an application, we prove that the characteristic quasi-polynomial of the Shi arrangement turns out to be a polynomial. We also present several results on the location of zeros of characteristic polynomials, related to a conjecture of Postnikov and Stanley. In particular, we verify the “functional equation” of the characteristic polynomial of the Linial arrangement for any root system, and give partial affirmative results on “Riemann hypothesis” for the root systems of type $E_6, E_7, E_8$, and $F_4$.
</p>projecteuclid.org/euclid.tmj/1520564418_20180308220031Thu, 08 Mar 2018 22:00 ESTThe rates of the $L^p$-convergence of the Euler-Maruyama and Wong-Zakai approximations of path-dependent stochastic differential equations under the Lipschitz conditionhttps://projecteuclid.org/euclid.tmj/1520564419<strong>Shigeki Aida</strong>, <strong>Takanori Kikuchi</strong>, <strong>Seiichiro Kusuoka</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 70, Number 1, 65--95.</p><p><strong>Abstract:</strong><br/>
We consider the rates of the $L^p$-convergence of the Euler-Maruyama and Wong-Zakai approximations of path-dependent stochastic differential equations under the Lipschitz condition on the coefficients. By a transformation, the stochastic differential equations of Markovian type with reflecting boundary condition on sufficiently good domains are to be associated with the equations concerned in the present paper. The obtained rates of the $L^p$-convergence are the same as those in the case of the stochastic differential equations of Markovian type without boundaries.
</p>projecteuclid.org/euclid.tmj/1520564419_20180308220031Thu, 08 Mar 2018 22:00 ESTStochastic calculus for Markov processes associated with semi-Dirichlet formshttps://projecteuclid.org/euclid.tmj/1520564420<strong>Chuan-Zhong Chen</strong>, <strong>Li Ma</strong>, <strong>Wei Sun</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 70, Number 1, 97--119.</p><p><strong>Abstract:</strong><br/>
We present a new Fukushima type decomposition in the framework of semi-Dirichlet forms. This generalizes the result of Ma, Sun and Wang [17, Theorem 1.4] by removing the condition (S). We also extend Nakao's integral to semi-Dirichlet forms and derive Itô's formula related to it.
</p>projecteuclid.org/euclid.tmj/1520564420_20180308220031Thu, 08 Mar 2018 22:00 ESTSharp $L^p$-bounds for the martingale maximal functionhttps://projecteuclid.org/euclid.tmj/1520564421<strong>Adam Osȩkowski</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 70, Number 1, 121--138.</p><p><strong>Abstract:</strong><br/>
The paper studies sharp weighted $L^p$ inequalities for the martingale maximal function. Proofs exploit properties of certain special functions of four variables and self-improving properties of $A_p$ weights.
</p>projecteuclid.org/euclid.tmj/1520564421_20180308220031Thu, 08 Mar 2018 22:00 ESTA coupling of Brownian motions in the $\mathcal{L}_0$-geometryhttps://projecteuclid.org/euclid.tmj/1520564422<strong>Takafumi Amaba</strong>, <strong>Kazumasa Kuwada</strong>. <p><strong>Source: </strong>Tohoku Mathematical Journal, Volume 70, Number 1, 139--174.</p><p><strong>Abstract:</strong><br/>
Under a complete Ricci flow, we construct a coupling of two Brownian motions such that their $\mathcal{L}_0$-distance is a supermartingale. This recovers a result of Lott [J. Lott, Optimal transport and Perelman's reduced volume, Calc. Var. Partial Differential Equations 36 (2009), no. 1, 49–84.] on the monotonicity of $\mathcal{L}_0$-distance between heat distributions.
</p>projecteuclid.org/euclid.tmj/1520564422_20180308220031Thu, 08 Mar 2018 22:00 EST