Online first

Latest issue

• Select all

• A Note on ξ-Bergman Kernels

  Shijie Bao, Qi’an Guan, Zheng Yuan

  2024, 20(3): 481-506. https://doi.org/10.1007/s11464-023-0021-1

  In the present note, we introduce the ξ-complex singularity exponents, which come from the asymptotic property of ξ-Bergman kernels on sub-level sets of plurisubharmonic functions; give some relations (including a closedness property) among ξ-complex singularity exponents, complex singularity exponents, and jumping numbers; generalize some properties of complex singularity exponents (such as the restriction formula and subadditivity property) to ξ-complex singularity exponents.

• Complex Symmetric Composition Operators on the Hardy Space of the Unit Ball

  Xingtang Dong, Yanqiong Guo, Zehua Zhou

  2024, 20(3): 507-521. https://doi.org/10.1007/s11464-023-0040-y

  In this paper, we introduce a new conjugation induced by a sequence of symmetric unitary n × n matrices on ℂⁿ. And we completely characterize the complex symmetric composition operator C_ϕ on the Hardy space of the unit ball with respect to the conjugation mentioned above. Furthermore, we propose an interesting open question under the background of this problem: whether every square complex matrix is symmetrically unitarily equivalent to its transpose (SUET).

• Combinatorial p-th R-curvature Ricci Flows and Calabi Flows on Surfaces

  Chunlei Liu

  2024, 20(3): 523-545. https://doi.org/10.1007/s11464-023-0094-x

  In this paper, we study the convergence of the solutions to combinatorial p-th R-curvature Ricci flows and combinatorial p-th R-curvature Calabi flows on surfaces. R-curvature was introduced by Ge [Int. Math. Res. Not. IMRN, 2017, 2017(11): 3510–3527] which is a modification of the well-known discrete Gaussian curvature on triangulated manifolds. We show that the long time convergence of the solutions to combinatorial p-th R-curvature Ricci flows on surfaces is equivalent to the existence of constant R-curvature metrics. Furthermore, we show that the solutions to combinatorial p-th R-curvature Calabi flows on surfaces in the Euclidean background geometry and hyperbolic background geometry have the long time convergence if and only if there exist constant R-curvature metrics.

• Global Well-posedness for the Fourth-order Defocusing Cubic Equation with Initial Data Lying in a Critical Sobolev Space

  Miao Chen, Hua Wang, Xiaohua Yao

  2024, 20(3): 547-580. https://doi.org/10.1007/s11464-023-0135-5

  This paper is devoted to studying the Cauchy problem of the fourth-order defocusing, cubic equation iu_t + Δ²u = − |u|²u in critical Sobolev space. We first prove that the problem is locally well-posed in the critical Sobolev space

  {\dot{H}}^{s_c}({\mathbb{R}}^N)

  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{H}^{s_{c}}(\mathbb{R}^{N})$$\end{document}

  , 3 ≤ N ≤ 7. Using the argument of Dodson in [Rev. Mat. Iberoam., 2022, 38(4): 1087–1100], we further prove that the problem is globally well-posed in some critical Sobolev space

  H_p^s({\mathbb{R}}^N)

  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{p}^{s}(\mathbb{R}^{N})$$\end{document}

  for N = 6 and N = 7 with radial initial data.

• Sums of Higher Divisor Functions with Almost Equal Variables

  Miao Lou

  2024, 20(3): 603-615. https://doi.org/10.1007/s11464-023-0038-5

  Let k, ℓ ≥ 3 be integers, and let τ_k(n) denote the k-th divisor function. In this paper, we apply the circle method to obtain an asymptotic formula for the sum

  \sum _{|m_i-X|Y}{\tau }_k(m_1^2+m_2^2+\cdots +m_{\ell }^2)(i=1,2,\dots ,\ell )

  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum_{|m_{i}-X|\leq Y}\ \ \tau_{k}(m_{1}^{2}+m_{2}^{2}+\cdots+m_{\ell}^{2})\ \ (i=1,2,\ldots,\ell)$$\end{document}

  with

  Y=X^{1-\frac{8}{5(k+1)}+20\epsilon }

  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y = {X^{1 - {8 \over {5(k + 1)}} + 20\varepsilon }}$$\end{document}

  tending to infinity. Previously, only the case k = 2 was considered.

• From Right (n + 2)-angulated Categories to n-exangulated Categories

  Jian He, Jing He, Panyue Zhou

  2024, 20(3): 669-688. https://doi.org/10.1007/s11464-023-0121-y

  In this paper, we introduce the notion of a right semi-equivalence for right (n + 2)-angulated categories. Let

  \mathcal{C}

  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr{C}$$\end{document}

  be an n-exangulated category and

  \mathcal{X}

  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr{X}$$\end{document}

  be a strongly covariantly finite subcategory of

  \mathcal{C}

  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr{C}$$\end{document}

  . We prove that the right (n + 2)-angulated category

  \mathcal{C}/\mathcal{X}

  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr{C}/\mathscr{X}$$\end{document}

  has an n-suspension functor that is a right semi-equivalence under a natural assumption. As an application, we show that a right (n + 2)-angulated category has an n-exangulated structure if and only if the n-suspension functor is a right semi-equivalence. Furthermore, we also prove that an n-exangulated category

  \mathcal{C}

  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr{C}$$\end{document}

  has the structure of a right (n + 2)-angulated category with a right semi-equivalence if and only if for any object

  X\in \mathcal{C}

  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X \in \mathscr{C}$$\end{document}

  , the morphism X → 0 is a trivial inflation.

• Jones Polynomial Versus Determinant of Quasi-alternating Links

  Khaled Qazaqzeh

  2024, 20(3): 689-698. https://doi.org/10.1007/s11464-023-0141-7

  We prove that there are only finitely many values of the Jones polynomial of quasi-alternating links of a given determinant. Consequently, we prove that there are only finitely many quasi-alternating links of a given Jones polynomial iff there are only finitely many quasi-alternating links of a given determinant.

• Large Deviations Principle for Stochastic Delay Differential Equations with Super-linearly Growing Coefficients

  Diancong Jin, Ziheng Chen, Tau Zhou

  2024, 20(3): 699-720. https://doi.org/10.1007/s11464-023-0072-3

  We utilize the weak convergence method to establish the Freidlin–Wentzell large deviations principle (LDP) for stochastic delay differential equations (SDDEs) with super-linearly growing coefficients, which covers a large class of cases with non-globally Lipschitz coefficients. The key ingredient in our proof is the uniform moment estimate of the controlled equation, where we handle the super-linear growth of the coefficients by an iterative argument. Our results allow both the drift and diffusion coefficients of the considered equations to super-linearly grow not only with respect to the delay variable but also to the state variable. This work extends the existing results which develop the LDPs for SDDEs with super-linearly growing coefficients only with respect to the delay variable.