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.
Abstract : We give a classification of real solvable Lie algebras whose non-trivial coadjoint orbits of corresponding to simply connected Lie groups are all of codimension 2. These Lie algebras belong to a well-known class, called the class of MD-algebras.
Abstract : In this paper, we define cup product on relative bounded cohomology, and study its basic properties. Then, by extending it to a more generalized formula, we prove that all cup products of bounded cohomology classes of an amalgamated free product \( G_{1}\ast_{A}G_{2} \) are zero for every positive degree, assuming that free factors \( G_i \) are amenable and amalgamated subgroup \( A \) is normal in both of them. As its consequences, we show that all cup products of bounded cohomology classes of the groups \( \mathbb{Z} \ast \mathbb{Z} \) and \( \mathbb{Z}_{n} \ast_{\mathbb{Z}_{d}}\mathbb{Z}_m \), where \( d \) is the greatest common divisor of \( n \) and \( m \), are zero for every positive degree.
Abstract : Let $\mathbb F_q$ be a finite field with $q$ elements. A function $f: \mathbb F_q^d\times \mathbb F_q^d \to \mathbb F_q$ is called a Mattila--Sj\"{o}lin type function of index $\gamma \in \mathbb R$ if $\gamma$ is the smallest real number such that whenever $|E|\geq Cq^{\gamma}$ for a sufficiently large constant $C$, the set $f(E,E):=\{f(x,y): x, y\in E\}$ is equal to $\mathbb F_q$. In this article, we construct an example of a Mattila--Sj\"{o}lin type function $f$ and provide its index, generalizing the result of Cheong, Koh, Pham and Shen [1].
Abstract : 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.
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.
Abstract : Using the results in the paper [12], we give an estimate for the first positive and negative Dirac eigenvalue on a 7-dimensional Sasakian spin manifold. The limiting case of this estimate can be attained if the manifold under consideration admits a Sasakian Killing spinor. By imposing the eta-Einstein condition on Sasakian manifolds of higher dimensions $2m+1 \geq 9$, we derive some new Dirac eigenvalue inequalities that improve the recent results in [12, 13].
Abstract : 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 $$\left| \pi_C(x) - \frac{|C|}{|G|} {\mathrm{Li}}(x) \right| \leq \left(1- c_0 \frac{\log d_L}{d_L^{7.072}} \right) \frac{|C|}{|G|} {\mathrm{Li}}(x).$$
Abstract : In this paper, we deal with random attractors for dynamical systems forced by a deterministic noise. These kind of systems are modeled as skew products where the dynamics of the forcing process are described by the base transformation. Here, we consider skew products over the Bernoulli shift with the unit interval fiber. We study the geometric structure of maximal attractors, the orbit stability and stability of mixing of these skew products under random perturbations of the fiber maps. We show that there exists an open set $\mathcal{U}$ in the space of such skew products so that any skew product belonging to this set admits an attractor which is either a continuous invariant graph or a bony graph attractor. These skew products have negative fiber Lyapunov exponents and their fiber maps are non-uniformly contracting, hence the non-uniform contraction rates are measured by Lyapnnov exponents. Furthermore, each skew product of $\mathcal{U}$ admits an invariant ergodic measure whose support is contained in that attractor. Additionally, we show that the invariant measure for the perturbed system is continuous in the Hutchinson metric.
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.
Byoung Jin Choi, Jae Hun Kim
J. Korean Math. Soc. 2022; 59(3): 549-570
https://doi.org/10.4134/JKMS.j210239
Chun-Ru Fu , Huan-Nan Shi, Dong-Sheng Wang
J. Korean Math. Soc. 2023; 60(3): 503-520
https://doi.org/10.4134/JKMS.j220039
Eun-Kyung Cho, Su-Ah Kwon, Suil O
J. Korean Math. Soc. 2022; 59(4): 757-774
https://doi.org/10.4134/JKMS.j210605
J. Korean Math. Soc. 2022; 59(2): 337-352
https://doi.org/10.4134/JKMS.j210245
Soyoon Bak, Philsu Kim, Sangbeom Park
J. Korean Math. Soc. 2022; 59(5): 891-909
https://doi.org/10.4134/JKMS.j210701
Hojoo Lee
J. Korean Math. Soc. 2023; 60(1): 71-90
https://doi.org/10.4134/JKMS.j220095
Jiling Cao, Beidi Peng, Wenjun Zhang
J. Korean Math. Soc. 2022; 59(6): 1153-1170
https://doi.org/10.4134/JKMS.j210728
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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