Showing papers in "Tohoku Mathematical Journal in 2011"
TL;DR: In this paper, the authors modify the definition of Ricci curvature of Ollivier of Markov chains on graphs to study the properties of general graphs, Cartesian product of graphs, random graphs, and some special class of graphs.
Abstract: We modify the definition of Ricci curvature of Ollivier of Markov chains on graphs to study the properties of the Ricci curvature of general graphs, Cartesian product of graphs, random graphs, and some special class of graphs.
251 citations
TL;DR: In this article, the authors studied the regularity of optimal transport maps on Riemannian manifolds, in relation with the geometric conditions of Ma-Trudinger-Wang and the geometry of the cut locus.
Abstract: In this paper we continue the investigation of the regularity of optimal transport maps on Riemannian manifolds, in relation with the geometric conditions of Ma-Trudinger-Wang and the geometry of the cut locus. We derive some sufficient and some necessary conditions to ensure that the optimal transport map is always continuous. In dimension two, we can sharpen our result into a necessary and sufficient condition. We also provide some sufficient conditions for regularity, and review existing results.
57 citations
TL;DR: In this paper, the authors define an invariant of a pair of a 6-dimensional symplectic manifold with vanishing 1st Chern class and its relatively spin Lagrangian submanifold with vanishing Maslov index.
Abstract: In this paper we define an invariant of a pair of a 6 dimensional symplectic manifold with vanishing 1st Chern class and its relatively spin Lagrangian submanifold with vanishing Maslov index. This invariant is a function on the set of the path connected components of bounding cochains (solutions of the $A_{\infty}$ version of the Maurer-Cartan equation of the filtered $A_{\infty}$ algebra associated to the Lagrangian submanifold). In the case when the Lagrangian submanifold is a rational homology sphere, it becomes a numerical invariant. This invariant depends on the choice of almost complex structures. The way how it depends on the almost complex structures is described by a wall crossing formula which involves a moduli space of pseudo-holomorphic spheres.
57 citations
TL;DR: In this article, it was shown that QUE cannot hold on a compact boundaryless Riemannian manifold (M,g) of dimension two if the eigenfunctions of the square root of minus the Laplacian are saturated.
Abstract: We provide a necessary and sufficient condition that $L^p$-norms, $2
39 citations
TL;DR: In this article, it was shown that the toric variety of root systems is the fine moduli space of stable projective lines investigated by Losev and Manin and applied this result to root systems of type $A.
Abstract: A root system $R$ of rank $n$ defines an $n$-dimensional smooth projective toric variety $X(R)$ associated with its fan of Weyl chambers. We give a simple description of the functor of $X(R)$ in terms of the root system $R$ and apply this result in the case of root systems of type $A$ to give a new proof of the fact that the toric variety $X(A_n)$ is the fine moduli space $\overline{L}_{n+1}$ of stable $(n+1)$-pointed chains of projective lines investigated by Losev and Manin.
39 citations
TL;DR: In this article, the authors apply sheaf-theoretical methods to monodromy zeta functions of Milnor fibrations and show that these functions can be generalized to polynomial functions over any toric variety.
Abstract: We apply sheaf-theoretical methods to monodromy zeta functions of Milnor fibrations. Classical formulas due to Kushnirenko, Varchenko and Oka, etc. on polynomials over the complex affine space will be generalized to polynomial functions over any toric variety. Moreover our results enable us to calculate the monodromy zeta functions of any constructible sheaf.
39 citations
TL;DR: The lattice of Muchnik degrees as mentioned in this paper provides an elegant and useful framework for the classification of certain foundationally interesting problems which are algorithmically unsolvable, such as tiling problems and Z d -subshifts.
Abstract: The study of mass problems and Muchnik degrees was originally motivated by Kolmogorov’s non-rigorous 1932 interpretation of intuitionism as a calculus of problems. The purpose of this paper is to summarize recent investigations into the lattice of Muchnik degrees of nonempty effectively closed sets in Euclidean space. Let Ew be this lattice. We show that Ew provides an elegant and useful framework for the classification of certain foundationally interesting problems which are algorithmically unsolvable. We exhibit some specific degrees in Ew which are associated with such problems. In addition, we present some structural results concerning the lattice Ew. One of these results answers a question which arises naturally from the Kolmogorov interpretation. Finally, we show how Ew can be applied in symbolic dynamics, toward the classification of tiling problems and Z d -subshifts of finite type.
36 citations
TL;DR: In this paper, a reaction-diusion evolution equation perturbed by a Gaussian noise was studied and asymptotics to all orders in a small parameter in front of the noise were given, with uniform estimates on the remainders.
Abstract: We study a reaction-diusion evolution equation perturbed by a Gaussian noise. Here the leading operator is the innitesimal generator of a C0-semigroup of strictly negative type, the nonlinear term has at most polynomial growth and is such that the whole system is dissipative. The corresponding It^o stochastic equation describes a process on a Hilbert space with dissipative nonlinear drift and a Gaussian noise. Under smoothness assumptions on the non-linearity, asymptotics to all orders in a small parameter in front of the noise are given, with uniform estimates on the remainders. Applications to nonlinear SPDEs with a linear term in the drift given by a Laplacian in a bounded domain are included. As a particular example we consider the small noise asymptotic expansions for the stochastic FitzHugh-Nagumo equations of neurobiology around deterministic solutions. 2010 Mathematics Subject Classication. Primary 35K57 , 35R60, 35C20 ; Secondary 92B20
27 citations
TL;DR: In this article, the existence of connecting orbits for the Fujita equation with a critical or supercritical exponent was studied and it was shown that for certain ranges of the exponent, there exists a homoclinic connection from positive steady states to zero.
Abstract: We study the existence of connecting orbits for the Fujita equation with a critical or supercritical exponent. For certain ranges of the exponent we prove the existence of heteroclinic connections from positive steady states to zero and a homoclinic orbit with respect to zero.
26 citations
TL;DR: In this paper, the authors give useful and simple criteria for determining the singularity of wave fronts near cuspidal edges near the singular curvatures of cusidal edges in wave fronts.
Abstract: We give useful and simple criteria for determining $D_4^\pm$ singularities of wave fronts. As an application, we investigate behaviors of singular curvatures of cuspidal edges near $D_4^+$ singularities.
26 citations
TL;DR: Under an infinitesimal version of the Bishop-Gromov relative volume comparison condition for a measure on an Alexandrov space, this article proved a topological splitting theorem for Riemannian manifold with singularities of nonnegative (Bakry-Emery) Ricci curvature.
Abstract: Under an infinitesimal version of the Bishop-Gromov relative volume comparison condition for a measure on an Alexandrov space, we prove a topological splitting theorem of Cheeger-Gromoll type. As a corollary, we prove an isometric splitting theorem for Riemannian manifolds with singularities of nonnegative (Bakry-Emery) Ricci curvature.
TL;DR: In this paper, the cleanliness condition was introduced for rank 1 sheaves and a conjectural Riemann-Roch type formula was proposed for the case of rank 2 sheaves.
Abstract: This article is devoted to studying the ramification of Galois torsors and of $\ell$-adic sheaves in characteristic $p>0$ (with $\ell
e p$). Let $k$ be a perfect field of characteristic $p>0$, $X$ a smooth, separated and quasi-compact $k$-scheme, $D$ a simple normal crossing divisor on $X$, $U=X-D$, $\Lambda$ a finite local ${\mathbb Z}_{\ell} $-algebra and ${\mathscr F}$ a locally constant constructible sheaf of $\Lambda$-modules on $U$. We introduce a {\em boundedness} condition on the ramification of ${\mathscr F}$ along $D$, and study its main properties, in particular, some specialization properties that lead to the fundamental notion of cleanliness and to the definition of the {\em characteristic cycle} of ${\mathscr F}$. The cleanliness condition extends the one introduced by Kato for rank 1 sheaves. Roughly speaking, it means that the ramification of ${\mathscr F}$ along $D$ is controlled by its ramification at the generic points of $D$. Under this condition, we propose a conjectural Riemann-Roch type formula for ${\mathscr F}$. Some cases of this formula have been previously proved by Kato and by the second author (T. S.).
TL;DR: The definition, the scope and the time homogeneity of WMSPs are explored, and a new class of processes is discussed, mirror semi-Markov processes, arising from the information retrieval on the Web.
Abstract: We propose and discuss a new class of processes, web Markov skeleton processes (WMSP), arising from the information retrieval on the Web. The framework of WMSP covers various known classes of processes, and it contains also important new classes of processes. We explore the definition, the scope and the time homogeneity of WMSPs, and discuss in detail a new class of processes, mirror semi-Markov processes. In the last section we briefly review some applications of WMSPs in computing page importance on the Web.
TL;DR: In this paper, it was shown that any faithful quasi-free actions of a finite group on the Cuntz algebra are mutually conjugate, and that they are asymptotically representable.
Abstract: We show that any faithful quasi-free actions of a finite group on the Cuntz algebra $\mathcal{O}_\infty$ are mutually conjugate, and that they are asymptotically representable.
TL;DR: In this article, the stability of a stationary solution for the Lugiato-Lefever equation with the periodic boundary condition in one space dimension was studied, which is a damped and driven nonlinear Schrodinger equation introduced to model the optical cavity.
Abstract: We study the stability of a stationary solution for the Lugiato-Lefever equation with the periodic boundary condition in one space dimension, which is a damped and driven nonlinear Schrodinger equation introduced to model the optical cavity. In this paper, we prove the Strichartz estimates for the linear damped Schrodinger equation with potential and external forcing and investigate the stability of certain stationary solutions under the initial perturbation within the framework of $L^2$.
TL;DR: In this article, the singular values of certain Siegel functions were shown to generate a class polynomial over an imaginary quadratic field of discriminant less than or equal to 7.
Abstract: Let $K$ be an imaginary quadratic field of discriminant less than or equal to $-7$ and $K_{(N)}$ be its ray class field modulo $N$ for an integer $N$ greater than $1$. We prove that the singular values of certain Siegel functions generate $K_{(N)}$ over $K$ by extending the idea of our previous work. These generators are not only the simplest ones conjectured by Schertz, but also quite useful in the matter of computation of class polynomials. We indeed give an algorithm to find all conjugates of such generators by virtue of the works of Gee and Stevenhagen.
TL;DR: In this article, the Gauss-Codazzi equation for conformally flat hypersurfaces in $S^4$ is shown to be a soliton equation, and a dressing action from soliton theory is used to construct geometric Ribaucour transforms of these surfaces.
Abstract: E. Cartan proved that conformally flat hypersurfaces in $S^{n+1}$ for $n>3$ have at most two distinct principal curvatures and locally envelop a one-parameter family of $(n-1)$-spheres. We prove that the Gauss-Codazzi equation for conformally flat hypersurfaces in $S^4$ is a soliton equation, and use a dressing action from soliton theory to construct geometric Ribaucour transforms of these hypersurfaces. We describe the moduli of these hypersurfaces in $S^4$ and their loop group symmetries. We also generalise these results to conformally flat $n$-immersions in $(2n-2)$-spheres with flat and non-degenerate normal bundle.
TL;DR: In this article, the authors extend Calabi ansatz over Kahler-Einstein manifolds to SasakiEinstein manifold and prove the existence of a complete scalar-flat Kahler metric on the toric Kahler cone manifold constructed from a toric diagram with a constant height.
Abstract: We extend Calabi ansatz over Kahler-Einstein manifolds to Sasaki-Einstein manifolds. As an application we prove the existence of a complete scalar-flat Kahler metric on Kahler cone manifolds over Sasaki-Einstein manifolds. n particular there exists a complete scalar-flat Kahler metric on the toric Kahler cone manifold constructed from a toric diagram with a constant height.
TL;DR: The modular towers of Lubin-Tate and Drinfeld provide period rings leading to an equivalence between a category of certain $\mathrm{GL}_h(K)$-equivariant vector bundles on Drinfeld's upper half space of dimension $h-1$ and a class of certain $D^*$-vector bundles on the projective space.
Abstract: Let $K$ be a nonarchimedean local field, let $h$ be a positive integer, and denote by $D$ the central division algebra of invariant $1/h$ over $K$. The modular towers of Lubin-Tate and Drinfeld provide period rings leading to an equivalence between a category of certain $\mathrm{GL}_h(K)$-equivariant vector bundles on Drinfeld's upper half space of dimension $h-1$ and a category of certain $D^*$-equivariant vector bundles on the $(h-1)$-dimensional projective space.
TL;DR: In this article, the authors study the behavior of semistability under tensor product in various settings: vector bundles, euclidean and hermitian lattices (alias Humbert forms or Arakelov bundles), multifiltered vector spaces.
Abstract: We study the behaviour of semistability under tensor product in various settings: vector bundles, euclidean and hermitian lattices (alias Humbert forms or Arakelov bundles), multifiltered vector spaces.
One approach to show that semistable vector bundles in characteristic zero are preserved by tensor product is based on the notion of nef vector bundles. We revisit this approach and show how far it can be transferred to hermitian lattices. J.-B. Bost conjectured that semistable hermitian lattices are preserved by tensor product. Using properties of nef hermitian lattices, we establish an inequality in that direction.
We axiomatize our method in the general context of monoidal categories, and then give an elementary proof of the fact that semistable multifiltered vector spaces (which play a role in diophantine approximation) are preserved by tensor product.
TL;DR: In this paper, the authors generalize the Toponogov comparison theorem to a complete Riemannian manifold with smooth convex boundary, where a geodesic triangle is replaced by an open (geodesic) triangle standing on the boundary of the manifold.
Abstract: The aim of our article is to generalize the Toponogov comparison theorem to a complete Riemannian manifold with smooth convex boundary. A geodesic triangle will be replaced by an open (geodesic) triangle standing on the boundary of the manifold, and a model surface will be replaced by the universal covering surface of a cylinder of revolution with totally geodesic boundary.
TL;DR: The Oda's Strong Factorization Conjecture states that a proper birational map between nonsingular toric varieties can be decomposed as a sequence of smooth toric blowups followed by a series of inverses of such maps as discussed by the authors.
Abstract: The Oda's Strong Factorization Conjecture states that a proper birational map between smooth toric varieties can be decomposed as a sequence of smooth toric blowups followed by a sequence of smooth toric blowdowns. This article describes an algorithm that conjecturally constructs such a decomposition. Several reductions and simplifications of the algorithm are presented and some special cases of the conjecture are proved. 1. Introduction. The general strong factorization problem asks if a proper birational map between nonsingular varieties (in characteristic zero) can be factored into a sequence of blowups with nonsingular centers followed by a sequence of inverses of such maps. Oda (5) posed the same problem for toric varieties and toric birational maps. Since toric varieties are defined by combinatorial data, the conjecture for toric varieties also takes a combinatorial form. A nonsingular toric variety is determined by a nonsingular fan and a smooth toric blowup corresponds to a smooth star subdivision of the fan. The conjecture then is:
TL;DR: In this article, the authors consider the stochastic ranking process with space-time dependent jump rates for the particles and prove that the joint empirical distribution of jump rate and scaled position converges almost surely to a deterministic distribution, and also the tagged particle processes converge almost surely, in the infinite particle limit.
Abstract: We consider the stochastic ranking process with space-time dependent jump rates for the particles. The process is a simplified model of the time evolution of the rankings such as sales ranks at online bookstores. We prove that the joint empirical distribution of jump rate and scaled position converges almost surely to a deterministic distribution, and also the tagged particle processes converge almost surely, in the infinite particle limit. The limit distribution is characterized by a system of inviscid Burgers-like integral-partial differential equations with evaporation terms, and the limit process of a tagged particle is a motion along a characteristic curve of the differential equations except at its Poisson times of jumps to the origin. 2000 Mathematics Subject Classification. Primary 60K35; Secondary 35C05, 82C22.
TL;DR: In this article, the authors give the boundedness of Carleson type maximal operators with rough kernel, which improves some known results, such as the $L √ p$ boundedness.
Abstract: In this paper, the authors give the $L^p$ boundedness of a class of the Carleson type maximal operators with rough kernel, which improves some known results.
TL;DR: In this article, value distribution and normality problems with regard to 2-element shared sets are considered. But they do not consider the case where each function in the family shares a 3-element set with its derivative.
Abstract: It is known that a family of meromorphic functions is normal if each function in the family shares a 3-element set with its derivative. In this paper we consider value distribution and normality problems with regard to 2-element shared sets. First we construct an example, by use of the Weierstrass doubly periodic functions, to show that a 3-element shared set can not be reduced to a 2-element shared set in general. We obtain a new criterion of normal families and new Picard-type theorems. The proofs make use of some results in complex dynamics. More examples are constructed to show that our assumptions are necessary.
TL;DR: In this paper, a complete analysis of the spectrum of the unique self-adjoint extension of the sub-Laplacian on the one-dimensional Heisenberg group is given.
Abstract: We give a complete analysis of the spectrum of the unique self-adjoint extension of the sub-Laplacian on the one-dimensional Heisenberg group.
TL;DR: In this paper, the Stokes' type theorems on complete non-compact manifolds were extended to complete noncompact manifold with complete Laplacian and also the Kelvin-Nevanlinna-Royden criterion for $p$-parabolicity.
Abstract: We present some new Stokes' type theorems on complete non-compact manifolds that extend, in different directions, previous works by Gaffney and Karp and also the so called Kelvin-Nevanlinna-Royden criterion for $p$-parabolicity. Applications to comparison and uniqueness results involving the $p$-Laplacian are deduced.
TL;DR: In this article, the authors classified the surfaces with a holomorphic foliation (with singularities) on which a rational curve exists, and showed that these surfaces are not Kählerian.
Abstract: Motivated by recent results on non-Kählerian compact complex surfaces with small second Betti number, we classify those on which a holomorphic foliation (with singularities) exists. Introduction. According to Kodaira, a compact connected complex surface S belongs to the class VII0 if it is minimal and its first Betti number b1(S) is equal to 1. It is still an open and fundamental problem to get a classification of these surfaces, which are not Kählerian and hence rather elusive. Let us shortly recall some advances about this problem [Nak], [DOT], [Tel], in order to place and to motivate our result. Because of the important rôle of the second Betti number in the following discussion, it is convenient to denote by VIIn0, n ∈ N , the class of VII0 surfaces S with b2(S) = n, and to set VII0 = ⊔ n>0 VII n 0. Surfaces of class VII0 have been completely classified in a series of works by Kodaira, Inoue, Bogomolov, Li-Yau-Zheng and Teleman. Let us henceforth concentrate our attention to surfaces of class VII0 . Around 1977, Kato [Kat] discovered a large collection of VII0 surfaces, nowadays called Kato surfaces (a.k.a. surfaces with a global spherical shell). They are, in some sense, generalizations of the classical Hopf surfaces (which belong to class VII0), and a significant number of papers has been dedicated to them, so that Kato surfaces may be today considered as “well known” surfaces. No other examples of VII0 surfaces have been discovered so far, and indeed some authors courageously conjecture that every VII0 surface should be a Kato surface. An important result in that direction has been proved by Nakamura [Na1], [Na2], in some particular cases, and then Dloussky-Oeljeklaus-Toma [DOT], in the general case: if S is a surface of class VIIn0 (n > 0) and contains n rational curves, then S is a Kato surface (the converse also being true, by construction). That result motivates the search for rational curves on VII0 surfaces. In recent years, Teleman developed a general strategy for finding those rational curves, using methods of gauge theory [Tel]. Up to now, his strategy has been successfull for small values of the second Betti number: it is proved in [Te1] and [Te2] that every surface of class VII0 or VII 2 0 contains at least one rational curve. We shall give in Section 1 more details on this spectacular result. 2010 Mathematics Subject Classification. Primary 32J15; Secondary 37F75.
TL;DR: In this paper, the Sato-Segal-Wilson construction of KP solutions is adapted to conformally immersed tori in the 4-sphere via quaternionic holomorphic geometry.
Abstract: The quaternionic KP hierarchy is the integrable hierarchy of p.d.e obtained by replacing the complex numbers with the quaternions in the standard construction of the KP hierarchy and its solutions; it is equivalent to what is often called the Davey-Stewartson II hierarchy. This article studies its relationship with the theory of conformally immersed tori in the 4-sphere via quaternionic holomorphic geometry. The Sato-Segal-Wilson construction of KP solutions is adapted to this setting and the connection with quaternionic holomorphic curves is made. We then compare three different notions of "spectral curve": the QKP spectral curve; the Floquet multiplier spectral curve for the related Dirac operator; and the curve parameterising Darboux transforms of a conformal 2-torus in the 4-sphere.