Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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  • 2024-03-01

    Asymptotic behavior of solutions to stochastic 3D globally modified Navier-Stokes equations with unbounded delays

    Cung The Anh, Vu Manh Toi, Phan Thi Tuyet

    Abstract : This paper studies the existence of weak solutions and the stability of stationary solutions to stochastic 3D globally modified Navier-Stokes equations with unbounded delays in the phase space $BCL_{-\infty}(H)$. We first prove the existence and uniqueness of weak solutions by using the classical technique of Galerkin approximations. Then we study stability properties of stationary solutions by using several approach methods. In the case of proportional delays, some sufficient conditions ensuring the polynomial stability in both mean square and almost sure senses will be provided.

  • 2023-01-01

    Cross-intercalates and geometry of short extreme points in the Latin polytope of degree 3

    Bokhee Im, Jonathan D. H. Smith

    Abstract : The polytope of tristochastic tensors of degree three, the Latin polytope, has two kinds of extreme points. Those that are at a maximum distance from the barycenter of the polytope correspond to Latin squares. The remaining extreme points are said to be short. The aim of the paper is to determine the geometry of these short extreme points, as they relate to the Latin squares. The paper adapts the Latin square notion of an intercalate to yield the new concept of a cross-intercalate between two Latin squares. Cross-intercalates of pairs of orthogonal Latin squares of degree three are used to produce the short extreme points of the degree three Latin polytope. The pairs of orthogonal Latin squares fall into two classes, described as parallel and reversed, each forming an orbit under the isotopy group. In the inverse direction, we show that each short extreme point of the Latin polytope determines four pairs of orthogonal Latin squares, two parallel and two reversed.

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  • 2023-09-01

    On non-displaceable Lagrangian submanifolds in two-step flag varieties

    Yoosik Kim

    Abstract : We prove that the two-step flag variety $\mathcal{F}\ell(1,n;n+1)$ carries a non-displaceable and non-monotone Lagrangian Gelfand--Zeitlin fiber diffeomorphic to $S^3 \times T^{2n-4}$ and a continuum family of non-displaceable Lagrangian Gelfand--Zeitlin torus fibers when $n > 2$.

  • 2023-09-01

    Continuum-wise expansiveness for $C^1$ generic vector fields

    Manseob Lee

    Abstract : It is shown that every continuum-wise expansive $C^1$ generic vector field $X$ on a compact connected smooth manifold $M$ satisfies Axiom A and has no cycles, and every continuum-wise expansive homoclinic class of a $C^1$ generic vector field $X$ on a compact connected smooth manifold $M$ is hyperbolic. Moreover, every continuum-wise expansive $C^1$ generic divergence-free vector field $X$ on a compact connected smooth manifold $M$ is Anosov.

  • 2023-05-01

    Two-sided estimates for transition probabilities of symmetric Markov chains on ${ \mathbb{Z} }^d$

    Zhi-He Chen

    Abstract : In this paper, we are mainly concerned with two-sided estimates for transition probabilities of symmetric Markov chains on ${ \mathbb{Z}  }^d$, whose one-step transition probability is comparable to $|x-y|^{-d}\phi_j(|x-y|)^{-1}$ with $\phi_j$ being a positive regularly varying function on $[1,\infty)$ with index $\alpha\in [2,\infty)$. For upper bounds, we directly apply the comparison idea and the Davies method, which considerably improves the existing arguments in the literature; while for lower bounds the relation with the corresponding continuous time symmetric Markov chains are fully used. In particular, our results answer one open question mentioned in the paper by Murugan and Saloff-Coste (2015).

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  • 2024-01-01

    Periodic surface homeomorphisms and contact structures

    Dheeraj Kulkarni, Kashyap Rajeevsarathy, Kuldeep Saha

    Abstract : In this article, we associate a contact structure to the conjugacy class of a periodic surface homeomorphism, encoded by a combinatorial tuple of integers called a marked data set. In particular, we prove that infinite families of these data sets give rise to Stein fillable contact structures with associated monodromies that do not factor into products to positive Dehn twists. In addition to the above, we give explicit constructions of symplectic fillings for rational open books analogous to Mori's construction for honest open books. We also prove a sufficient condition for the Stein fillability of rational open books analogous to the positivity of monodromy for honest open books due to Giroux and Loi-Piergallini.

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  • 2023-09-01

    Li-Yau gradient estimates on closed manifolds under Bakry-\'Emery Ricci curvature conditions

    Xing Yu Song, Ling Wu

    Abstract : In this paper, motivated by the work of Q.~S.~Zhang in [25], we derive optimal Li-Yau gradient bounds for positive solutions of the $f$-heat equation on closed manifolds with Bakry-\'Emery Ricci curvature bounded below.

  • 2023-05-01

    Gradient type estimates for linear elliptic systems from composite materials

    Youchan Kim, Pilsoo Shin

    Abstract : In this paper, we consider linear elliptic systems from composite materials where the coefficients depend on the shape and might have the discontinuity between the subregions. We derive a function which is related to the gradient of the weak solutions and which is not only locally piecewise H\"{o}lder continuous but locally H\"{o}lder continuous. The gradient of the weak solutions can be estimated by this derived function and we also prove the local piecewise gradient H\"{o}lder continuity which was obtained by the previous results.

  • 2023-01-01

    Left invariant Lorentzian metrics and curvatures on non-unimodular Lie groups of dimension three

    Ku Yong Ha, Jong Bum Lee

    Abstract : For each connected and simply connected three-dimensional non-unimodular Lie group, we classify the left invariant Lorentzian metrics up to automorphism, and study the extent to which curvature can be altered by a change of metric. Thereby we obtain the Ricci operator, the scalar curvature, and the sectional curvatures as functions of left invariant Lorentzian metrics on each of these groups. Our study is a continuation and extension of the previous studies done in [3] for Riemannian metrics and in [1] for Lorentzian metrics on unimodular Lie groups.

  • 2024-01-01

    Gorenstein $FP_{n}$-injective modules with respect to a semidualizing bimodule

    Zhiqiang Cheng, Guoqiang Zhao

    Abstract : Let $S$ and $R$ be rings and $_{S}C_{R}$ a semidualizing bimodule. We introduce the notion of $G_C$-$FP_n$-injective modules, which generalizes $G_C$-$FP$-injective modules and $G_C$-weak injective modules. The homological properties and the stability of $G_C$-$FP_n$-injective modules are investigated. When $S$ is a left $n$-coherent ring, several nice properties and new Foxby equivalences relative to $G_C$-$FP_n$-injective modules are given.

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May, 2024
Vol.61 No.3

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