Journal of the
Korean Mathematical Society
JKMS
ISSN(Print) 0304-9914
ISSN(Online) 2234-3008
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2023-01-01
Cross-intercalates and geometry of short extreme points in the Latin polytope of degree 3
Bokhee Im, Jonathan D. H. SmithAbstract : 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
Continuum-wise expansiveness for $C^1$ generic vector fields
Manseob LeeAbstract : 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.
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2023-09-01
On non-displaceable Lagrangian submanifolds in two-step flag varieties
Yoosik KimAbstract : 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 $n > 2$.
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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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2023-09-01
Li-Yau gradient estimates on closed manifolds under Bakry-\'Emery Ricci curvature conditions
Xing Yu Song, Ling WuAbstract : 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.
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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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2022-07-01
A note on the zeros of Jensen polynomials
Young-One Kim, Jungseob LeeAbstract : Sufficient conditions for the Jensen polynomials of the derivatives of a real entire function to be hyperbolic are obtained. The conditions are given in terms of the growth rate and zero distribution of the function. As a consequence some recent results on Jensen polynomials, relevant to the Riemann hypothesis, are extended and improved.
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2022-09-01
A generalization of Maynard's results on the Brun-Titchmarsh theorem to number fields
Jeoung-Hwan Ahn, Soun-Hi KwonAbstract : Maynard proved that there exists an effectively computable constant $q_1$ such that if $q \geq q_1$, then $\frac{\log q}{\sqrt{q} \phi(q)} {\rm Li}(x) \ll \pi(x;q,m) \!<\! \frac{2}{\phi(q)} {\mathrm{Li}}(x)$ for $x \geq q^8$. In this paper, we will show the following. Let $\delta_1$ and $\delta_2$ be positive constants with $0< \delta_1, \delta_2 < 1$ and $\delta_1+\delta_2 > 1$. Assume that $L \neq {\mathbb Q}$ is a number field. Then there exist effectively computable constants $c_0$ and $d_1$ such that for $d_L \geq d_1$ and $x \geq \exp \left( 326 n_L^{\delta_1} \left(\log d_L\right)^{1+\delta_2}\right)$, we have $$\left| \pi_C(x) - \frac{|C|}{|G|} {\mathrm{Li}}(x) \right| \leq \left(1- c_0 \frac{\log d_L}{d_L^{7.072}} \right) \frac{|C|}{|G|} {\mathrm{Li}}(x).$$
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2023-09-01
Forbidden theta graph, bounded spectral radius and size of non-bipartite graphs
Shuchao Li, Wanting Sun, Wei WeiAbstract : Zhai and Lin recently proved that if $G$ is an $n$-vertex connected $\theta(1, 2, r+1)$-free graph, then for odd $r$ and $n \geqslant 10r$, or for even $r$ and $n \geqslant 7r$, one has $\rho(G) \le \sqrt{\lfloor\frac{n^2}{4}\rfloor}$, and equality holds if and only if $G$ is $K_{\lceil\frac{n}{2}\rceil, \lfloor\frac{n}{2}\rfloor}$. In this paper, for large enough $n$, we prove a sharp upper bound for the spectral radius in an $n$-vertex $H$-free non-bipartite graph, where $H$ is $\theta(1, 2, 3)$ or $\theta(1, 2, 4)$, and we characterize all the extremal graphs. Furthermore, for $n \geqslant 137$, we determine the maximum number of edges in an $n$-vertex $\theta(1, 2, 4)$-free non-bipartite graph and characterize the unique extremal graph.
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2023-05-01
Gradient type estimates for linear elliptic systems from composite materials
Youchan Kim, Pilsoo ShinAbstract : 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.
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Symmetry of the twisted Gromov-Witten classes of projective line
Hyenho Lho
J. Korean Math. Soc. 2023; 60(3): 479-501
https://doi.org/10.4134/JKMS.j210448 -
Ramanujan continued fractions of order eighteen
Yoon Kyung Park
J. Korean Math. Soc. 2023; 60(2): 395-406
https://doi.org/10.4134/JKMS.j220180 -
Robust portfolio optimization under hybrid CEV and stochastic volatility
Jiling Cao, Beidi Peng, Wenjun Zhang
J. Korean Math. Soc. 2022; 59(6): 1153-1170
https://doi.org/10.4134/JKMS.j210728 -
Stabilized-penalized collocated finite volume scheme for incompressible biofluid flows
Nasserdine Kechkar
, Mohammed LouaarJ. Korean Math. Soc. 2022; 59(3): 519-548
https://doi.org/10.4134/JKMS.j210211
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Infinite families of congruences modulo $2$ for $2$-core and $13$-core partitions
Ankita Jindal, Nabin Kumar Meher
J. Korean Math. Soc. 2023; 60(5): 1073-1085
https://doi.org/10.4134/JKMS.j230031 -
A conjecture of Gross and Zagier: case $E(\mathbb{Q})_{\rm{tor}} \cong \mathbb{Z}/2\mathbb{Z}$ or $\mathbb{Z}/4\mathbb{Z}$
Dongho Byeon, Taekyung Kim, Donggeon Yhee
J. Korean Math. Soc. 2023; 60(5): 1087-1107
https://doi.org/10.4134/JKMS.j230085 -
A note on unicity of meromorphic functions in several variables
yezhou Li, heqing sun
J. Korean Math. Soc. 2023; 60(4): 859-876
https://doi.org/10.4134/JKMS.j220473 -
On solvability of a class of degenerate Kirchhoff equations with logarithmic nonlinearity
U\u{g}ur Sert
J. Korean Math. Soc. 2023; 60(3): 565-586
https://doi.org/10.4134/JKMS.j220129
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