Journal of the
Korean Mathematical Society
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ISSN(Online) 2234-3008
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2023-05-01
Two-sided estimates for transition probabilities of symmetric Markov chains on ${ \mathbb{Z} }^d$
Zhi-He ChenAbstract : 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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2023-03-01
Time periodic solutions to a heat equation with linear forcing and boundary conditions
In-Jee Jeong, Sun-Chul KimAbstract : In this study, we consider a heat equation with a variable-coefficient linear forcing term and a time-periodic boundary condition. Under some decay and smoothness assumptions on the coefficient, we establish the existence and uniqueness of a time-periodic solution satisfying the boundary condition. Furthermore, possible connections to the closed boundary layer equations were discussed. The difficulty with a perturbed leading order coefficient is demonstrated by a simple example.
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2023-01-01
Geometry of bilinear forms on a normed space $\mathbb{R}^n$
Sung Guen KimAbstract : For every $n\geq 2$, let $\mathbb{R}^n_{\|\cdot\|}$ be $\mathbb{R}^n$ with a norm $\|\cdot\|$ such that its unit ball has finitely many extreme points more than $2n$. We devote to the description of the sets of extreme and exposed points of the closed unit balls of ${\mathcal L}(^2\mathbb{R}^n_{\|\cdot\|})$ and ${\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})$, where ${\mathcal L}(^2\mathbb{R}^n_{\|\cdot\|})$ is the space of bilinear forms on $\mathbb{R}^n_{\|\cdot\|}$, and ${\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})$ is the subspace of ${\mathcal L}(^2\mathbb{R}^n_{\|\cdot\|})$ consisting of symmetric bilinear forms. Let ${\mathcal F}={\mathcal L}(^2\mathbb{R}^n_{\|\cdot\|})$ or ${\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})$. First we classify the extreme and exposed points of the closed unit ball of ${\mathcal F}$. We also show that every extreme point of the closed unit ball of ${\mathcal F}$ is exposed. It is shown that ${ext}B_{{\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})}={ext}B_{{\mathcal L}(^2\mathbb{R}^n_{\|\cdot\|})}\cap {\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})$ and ${exp}B_{{\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})}={exp}B_{{\mathcal L}(^2\mathbb{R}^n_{\|\cdot\|})}\cap {\mathcal L}_s(^2\mathbb{R}^n_{\|\cdot\|})$, which expand some results of \cite{18, 23, 28, 29, 35, 38, 40, 41, 43}.
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2022-09-01
Dynamic behavior of cracked beams and shallow arches
Semion Gutman, Junhong Ha, Sudeok ShonAbstract : We develop a rigorous mathematical framework for studying dynamic behavior of cracked beams and shallow arches. The governing equations are derived from the first principles, and stated in terms of the subdifferentials of the bending and the axial potential energies. The existence and the uniqueness of the solutions is established under various conditions. The corresponding mathematical tools dealing with vector-valued functions are comprehensively developed. The motion of beams and arches is studied under the assumptions of the weak and strong damping. The presence of cracks forces weaker regularity results for the arch motion, as compared to the beam case.
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2023-07-01
Some algebras having relations like those for the 4-dimensional Sklyanin algebras
Alexandru Chirvasitu, S. Paul SmithAbstract : The 4-dimensional Sklyanin algebras are a well-studied 2-parameter family of non-commutative graded algebras, often denoted $A(E,\tau)$, that depend on a quartic elliptic curve $E \subseteq \mathbb{P}^3$ and a translation automorphism $\tau$ of $E$. They are graded algebras generated by four degree-one elements subject to six quadratic relations and in many important ways they behave like the polynomial ring on four indeterminates except that they are not commutative. They can be seen as ``elliptic analogues'' of the enveloping algebra of $\mathfrak{gl}(2,\mathbb{C})$ and the quantized enveloping algebras $U_q(\mathfrak{gl}_2)$. Recently, Cho, Hong, and Lau conjectured that a certain 2-parameter family of algebras arising in their work on homological mirror symmetry consists of 4-dimensional Sklyanin algebras. This paper shows their conjecture is false in the generality they make it. On the positive side, we show their algebras exhibit features that are similar to, and differ from, analogous features of the 4-dimensional Sklyanin algebras in interesting ways. We show that most of the Cho-Hong-Lau algebras determine, and are determined by, the graph of a bijection between two 20-point subsets of the projective space $\mathbb{P}^3$. The paper also examines a 3-parameter family of 4-generator 6-relator algebras admitting presentations analogous to those of the 4-dimensional Sklyanin algebras. This class includes the 4-dimensional Sklyanin algebras and most of the Cho-Hong-Lau algebras.
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2023-05-01
Temporal decay of solutions for a chemotaxis model of angiogenesis type
Jaewook Ahn, Myeonghyeon KimAbstract : This paper considers a parabolic-hyperbolic-hyperbolic type chemotaxis system in $\mathbb{R}^{d}$, $d\ge3$, describing tumor-induced angiogenesis. The global existence result and temporal decay estimate for a unique mild solution are established under the assumption that some Sobolev norms of initial data are sufficiently small.
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2023-01-01
Left invariant Lorentzian metrics and curvatures on non-unimodular Lie groups of dimension three
Ku Yong Ha, Jong Bum LeeAbstract : 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.
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2024-03-01
Conics in quintic del Pezzo varieties
Kiryong Chung, Sanghyeon LeeAbstract : The smooth quintic del Pezzo variety $Y$ is well-known to be obtained as a linear sections of the Grassmannian variety $\mathrm{Gr}(2,5)$ under the Pl\"ucker embedding into $\mathbb{P}^{9}$. Through a local computation, we show the Hilbert scheme of conics in $Y$ for $\text{dim} Y \ge 3$ can be obtained from a certain Grassmannian bundle by a single blowing-up/down transformation.
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2023-05-01
Time periodic solution for the compressible magneto-micropolar fluids with external forces in $\mathbb{R}^3$
Qingfang Shi, Xinli ZhangAbstract : In this paper, we consider the existence of time periodic solutions for the compressible magneto-micropolar fluids in the whole space $\mathbb{R}^3$. In particular, we first solve the problem in a sequence of bounded domains by the topological degree theory. Then we obtain the existence of time periodic solutions in $\mathbb{R}^3$ by a limiting process.
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2022-11-01
$\mathbb{D}$-solutions of BSDEs with Poisson jumps
Imen HassairiAbstract : In this paper, we study backward stochastic differential equations (BSDEs shortly) with jumps that have Lipschitz generator in a general filtration supporting a Brownian motion and an independent Poisson random measure. Under just integrability on the data we show that such equations admit a unique solution which belongs to class $\mathbb{D}$.
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Construction of a Mattila--Sj\"{o}lin type function over a finite field
Daewoong Cheong, Jinbeom Kim
J. Korean Math. Soc. 2023; 60(4): 799-822
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Boundedness of Calder\'{o}n-Zygmund operators on inhomogeneous product Lipschitz spaces
Shaoyong He, Taotao Zheng
J. Korean Math. Soc. 2022; 59(3): 469-494
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Pseudo-Riemannian Sasaki solvmanifolds
Diego Conti, Federico A. Rossi, Romeo Segnan Dalmasso
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Cup product on relative bounded cohomology
HeeSook Park
J. Korean Math. Soc. 2023; 60(4): 823-833
https://doi.org/10.4134/JKMS.j220345
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Summability in Musielak--Orlicz Hardy spaces
Jun Liu, Haonan Xia
J. Korean Math. Soc. 2023; 60(5): 1057-1072
https://doi.org/10.4134/JKMS.j220646 -
Finite quotients of singular Artin monoids and categorification of the desingularization map
Helena Jonsson, Volodymyr Mazorchuk, Elin Persson Westin, Shraddha Srivastava, Mateusz Stroinski, Xiaoyu Zhu
J. Korean Math. Soc. 2023; 60(6): 1255-1302
https://doi.org/10.4134/JKMS.j230020 -
Maximal domains of solutions for analytic quasilinear differential equations of first order
Chong-Kyu Han, Taejung Kim
J. Korean Math. Soc. 2022; 59(6): 1171-1184
https://doi.org/10.4134/JKMS.j220043 -
Average entropy and asymptotics
Tatyana Barron, Manimugdha Saikia
J. Korean Math. Soc. 2024; 61(1): 91-107
https://doi.org/10.4134/JKMS.j230152
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