Journal of the
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ISSN(Online) 2234-3008
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2023-01-01
Simple zeros of $L$-functions and the Weyl-type subconvexity
Peter Jaehyun Cho
, Gyeongwon OhAbstract : Let $f$ be a self-dual primitive Maass or modular forms for level $4$. For such a form $f$, we define \begin{align*} N_f^s(T)\!:=\!|\{\rho \in \mathbb{C} : |\Im(\rho)| \leq T, \text{ $\rho$ is a non-trivial simple zero of $L_f(s)$} \}|. \end{align*} We establish an omega result for $N_f^s(T)$, which is $N_f^s(T)=\Omega \big( T^{\frac{1}{6}-\epsilon} \big)$ for any $\epsilon>0$. For this purpose, we need to establish the Weyl-type subconvexity for $L$-functions attached to primitive Maass forms by following a recent work of Aggarwal, Holowinsky, Lin, and Qi.
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2023-11-01
The automorphism groups of Artin groups of edge-separated CLTTF graphs
Byung Hee An, Youngjin ChoAbstract : 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.
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2023-11-01
On the $\eta$-parallelism in almost Kenmotsu $3$-manifolds
Jun-ichi Inoguchi, Ji-Eun LeeAbstract : In this paper, we study the $\eta$-parallelism of the Ricci operator of almost Kenmotsu $3$-manifolds. First, we prove that an almost Kenmotsu $3$-manifold $M$ satisfying $\nabla_{\xi}h=-2\alpha h \varphi$ for some constant $\alpha$ has dominantly $\eta$-parallel Ricci operator if and only if it is locally symmetric. Next, we show that if $M$ is an $H$-almost Kenmotsu $3$-manifold satisfying $\nabla_{\xi}h=-2\alpha h \varphi$ for a constant $\alpha$, then $M$ is a Kenmotsu $3$-manifold or it is locally isomorphic to certain non-unimodular Lie group equipped with a left invariant almost Kenmotsu structure. The dominantly $\eta$-parallelism of the Ricci operator is equivalent to the local symmetry on homogeneous almost Kenmotsu $3$-manifolds.
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2023-09-01
Erratum to ``Pseudo-Riemannian Sasaki solvmanifolds'' [J. Korean Math. Soc. 60 (2023), No. 1, pp. 115--141]
Diego Conti, Federico A. Rossi, Romeo Segnan DalmassoAbstract : In this erratum, we offer a correction to [J. Korean Math. Soc. 60 (2023), No. 1, pp. 115--141]. We rectify Theorem 5.7 and Table 1 of the original paper.
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2023-09-01
Infinite families of congruences modulo $2$ for $2$-core and $13$-core partitions
Ankita Jindal, Nabin Kumar MeherAbstract : A partition of $n$ is called a $t$-core partition if none of its hook number is divisible by $t$. In 2019, Hirschhorn and Sellers [5] obtained a parity result for $3$-core partition function $a_3(n)$. Motivated by this result, both the authors [8] recently proved that for a non-negative integer $\alpha$, $a_{3^{\alpha} m}(n)$ is almost always divisible by an arbitrary power of $2$ and $3$ and $a_{t}(n)$ is almost always divisible by an arbitrary power of $p_i^j$, where $j$ is a fixed positive integer and $t= p_1^{a_1}p_2^{a_2}\cdots p_m^{a_m}$ with primes $p_i \geq 5.$ In this article, by using Hecke eigenform theory, we obtain infinite families of congruences and multiplicative identities for $a_2(n)$ and $a_{13}(n)$ modulo $2$ which generalizes some results of Das [2].
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2023-09-01
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 YheeAbstract : 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.
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2023-07-01
A note on unicity of meromorphic functions in several variables
yezhou Li, heqing sunAbstract : Let $f(z)$ be a meromorphic function in several variables satisfying $$\limsup\limits_{r\rightarrow\infty}\frac{\log T(r,f)}{r}=0.$$ We mainly investigate the uniqueness problem on $f$ in $\mathbb{C}^{m}$ sharing polynomial or periodic small function with its difference polynomials from a new perspective. Our main theorems can be seen as the improvement and extension of previous results.
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2023-05-01
On solvability of a class of degenerate Kirchhoff equations with logarithmic nonlinearity
U\u{g}ur SertAbstract : We study the Dirichlet problem for the degenerate nonlocal parabolic equation \[ u_{t}-a\left(\left\Vert \nabla u\right\Vert _{L^2(\Omega)}^{2}\right)\Delta u=C_b\left\Vert u\right\Vert _{L^2(\Omega)}^{\beta}\left\vert u \right\vert^{q\left(x,t\right)-2}u\log|u|+f \quad \text{in $Q_T$}, \] where $Q_{T}:=\Omega \times (0,T)$, $T>0$, $\Omega \subset \mathbb{R}^{N}$, $N\geq 2$, is a bounded domain with a sufficiently smooth boundary, $q(x,t)$ is a measurable function in $Q_{T}$ with values in an interval $[q^{-},q^{+}]\subset(1,\infty)$ and the diffusion coefficient $a(\cdot)$ is a continuous function defined on $\mathbb{R}_+$. It is assumed that $a(s)\to 0$ or $a(s)\to \infty$ as $s\to 0^+$, therefore the equation degenerates or becomes singular as $\|\nabla u(t)\|_{2}\to 0$. For both cases, we show that under appropriate conditions on $a$, $\beta$, $q$, $f$ the problem has a global in time strong solution which possesses the following global regularity property: $\Delta u\in L^2(Q_T)$ and $a(\left\Vert \nabla u\right\Vert _{L^2(\Omega)}^{2})\Delta u\in L^2(Q_T)$.
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2023-07-01
Computations and conservativeness of traces of one-dimensional diffusions
Ali BENAMOR, Rafed MOUSSAAbstract : We compute explicitly traces of one-dimensional diffusion processes. The obtained trace forms can be regarded as Dirichlet forms on graphs. Then we discuss conditions ensuring the trace forms to be conservative. Finally, the obtained results are applied to the Bessel process of order $\nu$.
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2023-11-01
On nonnil-exact sequences and nonnil-commutative diagrams
Wei Zhao, De chuan ZhouAbstract : In this paper, we investigate the nonnil-exact sequences and nonnil-commutative diagrams and show that they behave in a way similar to the classical ones in Abelian categories.
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Temporal decay of solutions for a chemotaxis model of angiogenesis type
Jaewook Ahn, Myeonghyeon Kim
J. Korean Math. Soc. 2023; 60(3): 619-634
https://doi.org/10.4134/JKMS.j220424 -
Foliations from left orders
Hyungryul Baik, Sebastian Hensel, Chenxi Wu
J. Korean Math. Soc. 2022; 59(4): 699-715
https://doi.org/10.4134/JKMS.j210535 -
Zeros of new Bergman kernels
Noureddine Ghiloufi
, Safa SnounJ. Korean Math. Soc. 2022; 59(3): 449-468
https://doi.org/10.4134/JKMS.j200373 -
On relative Cohen-Macaulay modules
Zhongkui Liu, Pengju Ma, Xiaoyan Yang
J. Korean Math. Soc. 2023; 60(3): 683-694
https://doi.org/10.4134/JKMS.j220479
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Forbidden theta graph, bounded spectral radius and size of non-bipartite graphs
Shuchao Li, Wanting Sun, Wei Wei
J. Korean Math. Soc. 2023; 60(5): 959-986
https://doi.org/10.4134/JKMS.j220341 -
Generalized hexagons embedded in metasymplectic spaces
Sebastian Petit, Hendrik Van Maldeghem
J. Korean Math. Soc. 2023; 60(4): 907-929
https://doi.org/10.4134/JKMS.j220528 -
Erratum to ``Static and related critical spaces with harmonic curvature and three Ricci eigenvalues'' [J. Korean Math. Soc. 57 (2020), No. 6, pp. 1435--1449]
Jongsu Kim
J. Korean Math. Soc. 2022; 59(3): 649-650
https://doi.org/10.4134/JKMS.j210761 -
The first positive and negative Dirac eigenvalues on Sasakian manifolds
Eui Chul Kim
J. Korean Math. Soc. 2023; 60(5): 999-1021
https://doi.org/10.4134/JKMS.j220480
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