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 : 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 : 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.
Abstract : We introduce the notions of symbolic expansivity and symbolic shadowing for homeomorphisms on non-metrizable compact spaces which are generalizations of expansivity and shadowing, respectively, for metric spaces. The main result is to generalize the Smale's spectral decomposition theorem to symbolically expansive homeomorphisms with symbolic shadowing on non-metrizable compact Hausdorff totally disconnected spaces.
Abstract : Let $C$ be a curve and $V \to C$ an orthogonal vector bundle of rank $r$. For $r \le 6$, the structure of $V$ can be described using tensor, symmetric and exterior products of bundles of lower rank, essentially due to the existence of exceptional isomorphisms between $\mathrm{Spin} (r , \mathbb C)$ and other groups for these $r$. We analyze these structures in detail, and in particular use them to describe moduli spaces of orthogonal bundles. Furthermore, the locus of isotropic vectors in $V$ defines a quadric subfibration $Q_V \subset \mathbb P V$. Using familiar results on quadrics of low dimension, we exhibit isomorphisms between isotropic Quot schemes of $V$ and certain ordinary Quot schemes of line subbundles. In particular, for $r \le 6$ this gives a method for enumerating the isotropic subbundles of maximal degree of a general $V$, when there are finitely many.
Abstract : We describe the moduli space of Higgs pairs on an irreducible nodal curve of arithmetic genus one and its geometric structures in terms of the Hitchin map and a flat degeneration of the moduli space of Higgs bundles on an elliptic curve.
Abstract : This work is a continuation of Crisp's work on automorphism groups of CLTTF Artin groups, where the defining graph of a CLTTF Artin group is connected, large-type, and triangle-free. More precisely, we provide an explicit presentation of the automorphism group of an edge-separated CLTTF Artin group whose defining graph has no separating vertices.
Abstract : We study various aspects of the structure and representation theory of singular Artin monoids. This includes a number of generalizations of the desingularization map and explicit presentations for certain finite quotient monoids of diagrammatic nature. The main result is a categorification of the classical desingularization map for singular Artin monoids associated to finite Weyl groups using BGG category $\mathcal{O}$.
Abstract : In this paper, we introduce the notion of the stability of automorphic forms for the general linear group and relate the stability of automorphic forms to the moduli space of real tori and the Jacobian real locus.
Abstract : Let $E$ be an elliptic curve defined over $\mathbb{Q}$ of conductor $N$, $c$ the Manin constant of $E$, and $m$ the product of Tamagawa numbers of $E$ at prime divisors of $N$. Let $K$ be an imaginary quadratic field where all prime divisors of $N$ split in $K$, $P_K$ the Heegner point in $E(K)$, and ${\rm III}(E/K)$ the Shafarevich-Tate group of $E$ over $K$. Let $2u_K$ be the number of roots of unity contained in $K$. Gross and Zagier conjectured that if $P_K$ has infinite order in $E(K)$, then the integer $ c \cdot m \cdot u_K \cdot |{\rm III}(E/K)|^{\frac{1}{2}}$ is divisible by $|E(\mathbb{Q})_{\rm{tor}} |$. In this paper, we prove that this conjecture is true if $E(\mathbb{Q})_{\rm{tor}} \cong \mathbb{Z}/2\mathbb{Z}$ or $\mathbb{Z}/4\mathbb{Z}$ except for two explicit families of curves. Further, we show these exceptions can be removed under Stein--Watkins conjecture.
Aslı Güçlükan İlhan, Sabri Kaan Gürbüzer
J. Korean Math. Soc. 2022; 59(5): 963-986
https://doi.org/10.4134/JKMS.j220104
Sudhakar Kumar Chaubey, Young Jin Suh
J. Korean Math. Soc. 2023; 60(2): 341-358
https://doi.org/10.4134/JKMS.j220057
Gyu Whan Chang, Jun Seok Oh
J. Korean Math. Soc. 2023; 60(2): 407-464
https://doi.org/10.4134/JKMS.j220271
Hojoo Lee
J. Korean Math. Soc. 2023; 60(1): 71-90
https://doi.org/10.4134/JKMS.j220095
Liwen Gao, Yan Lin, Shuhui Yang
J. Korean Math. Soc. 2024; 61(2): 207-226
https://doi.org/10.4134/JKMS.j210768
Zhiqiang Cheng, Guoqiang Zhao
J. Korean Math. Soc. 2024; 61(1): 29-40
https://doi.org/10.4134/JKMS.j220398
Ya Gao , Yanling Gao, Jing Mao , Zhiqi Xie
J. Korean Math. Soc. 2024; 61(1): 183-205
https://doi.org/10.4134/JKMS.j230283
Dejan \'Cebi\'c , Neboj\v sa M. Ralevi\'c
J. Korean Math. Soc. 2022; 59(6): 1067-1082
https://doi.org/10.4134/JKMS.j210607
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