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Boundary value problems of conjugate and generalized k-holomorphic functions in ℂ2 ℂ2中共轭函数和广义k-荷态函数的边界值问题
IF 1 4区 数学 Q1 MATHEMATICS Pub Date : 2024-08-27 DOI: 10.1007/s10473-024-0511-6
Yanyan Cui, Chaojun Wang, Yonghong Xie, Yuying Qiao

In this paper, conjugate k-holomorphic functions and generalized k-holomorphic functions are defined in the two-dimensional complex space, and the corresponding Riemann boundary value problems and the inverse problems are discussed on generalized bicylinders. By the characteristics of the corresponding functions and boundary properties of the Cauchy type singular integral operators with conjugate k-holomorphic kernels, the general solutions and special solutions of the corresponding boundary value problems are studied in a detailed fashion, and the integral expressions of the solutions are obtained.

本文定义了二维复空间中的共轭 k-holomorphic 函数和广义 k-holomorphic 函数,并讨论了广义双轴上相应的黎曼边界值问题和逆问题。通过相应函数的特征和具有共轭 k-holomorphic 核的 Cauchy 型奇异积分算子的边界性质,详细研究了相应边界值问题的通解和特解,并得到了解的积分表达式。
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引用次数: 0
A singular Dirichlet problem for the Monge-Ampère type equation Monge-Ampère 型方程的奇异 Dirichlet 问题
IF 1 4区 数学 Q1 MATHEMATICS Pub Date : 2024-08-27 DOI: 10.1007/s10473-024-0520-5
Zhijun Zhang, Bo Zhang

We consider the singular Dirichlet problem for the Monge-Ampère type equation ({rm det} D^2 u=b(x)g(-u)(1+|nabla u|^2)^{q/2}, u<0, x in Omega, u|_{partial Omega}=0), where Ω is a strictly convex and bounded smooth domain in ℝn, q ∈ [0, n +1), gC (0, ∞) is positive and strictly decreasing in (0, ∞) with (limlimits_{srightarrow 0^+}g(s)=infty), and bC (Ω) is positive in Ω. We obtain the existence, nonexistence and global asymptotic behavior of the convex solution to such a problem for more general b and g. Our approach is based on the Karamata regular variation theory and the construction of suitable sub-and super-solutions.

我们考虑 Monge-Ampère 型方程的奇异 Dirichlet 问题 ({rm det} D^2 u=b(x)g(-u)(1+|nabla u|^2)^{q/2}, u<;0, x in Omega, u|_{partial Omega}=0), 其中 Ω 是 ℝn 中一个严格凸且有界的光滑域, q∈ [0, n +1), g∈ C∞ (0、∞)为正且在(0,∞)中严格递减,且(limlimits_{sarrow 0^+}g(s)=infty),且 b∈ C∞ (Ω) 在Ω中为正。我们的方法基于卡拉马塔正则变异理论和合适的子解与超解的构造,得到了更一般的 b 和 g 的凸解的存在性、不存在性和全局渐近行为。
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引用次数: 0
The Schur test of compact operators 紧凑算子的舒尔检验
IF 1 4区 数学 Q1 MATHEMATICS Pub Date : 2024-08-27 DOI: 10.1007/s10473-024-0524-1
Qijian Kang, Maofa Wang

Infinite matrix theory is an important branch of function analysis. Every linear operator on a complex separable infinite dimensional Hilbert space corresponds to an infinite matrix with respect a orthonormal base of the space, but not every infinite matrix corresponds to an operator. The classical Schur test provides an elegant and useful criterion for the boundedness of linear operators, which is considered a respectable mathematical accomplishment. In this paper, we prove the compact version of the Schur test. Moreover, we provide the Schur test for the Schatten class S2. It is worth noting that our main results can be applicable to the general matrix without limitation on non-negative numbers. We finally provide the Schur test for compact operators from lp into lq.

无穷矩阵理论是函数分析的一个重要分支。就空间的正交基而言,复可分离无限维希尔伯特空间上的每个线性算子都对应于一个无限矩阵,但并非每个无限矩阵都对应于一个算子。经典的舒尔检验为线性算子的有界性提供了一个优雅而有用的标准,被认为是一项令人尊敬的数学成就。在本文中,我们证明了舒尔检验的紧凑版本。此外,我们还提供了 Schatten 类 S2 的舒尔检验。值得注意的是,我们的主要结果可以适用于一般矩阵,而不受非负数的限制。最后,我们提供了从 lp 到 lq 的紧凑算子的舒尔检验。
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引用次数: 0
Global convergence of a cautious projection BFGS algorithm for nonconvex problems without gradient Lipschitz continuity 针对无梯度 Lipschitz 连续性非凸问题的谨慎投影 BFGS 算法的全局收敛性
IF 1 4区 数学 Q1 MATHEMATICS Pub Date : 2024-08-27 DOI: 10.1007/s10473-024-0506-3
Gonglin Yuan, Xiong Zhao, Jiajia Yu

A cautious projection BFGS method is proposed for solving nonconvex unconstrained optimization problems. The global convergence of this method as well as a stronger general convergence result can be proven without a gradient Lipschitz continuity assumption, which is more in line with the actual problems than the existing modified BFGS methods and the traditional BFGS method. Under some additional conditions, the method presented has a superlinear convergence rate, which can be regarded as an extension and supplement of BFGS-type methods with the projection technique. Finally, the effectiveness and application prospects of the proposed method are verified by numerical experiments.

本文提出了一种用于求解非凸无约束优化问题的谨慎投影 BFGS 方法。该方法无需梯度 Lipschitz 连续性假设即可证明其全局收敛性以及更强的一般收敛结果,比现有的修正 BFGS 方法和传统 BFGS 方法更符合实际问题。在一些附加条件下,所提出的方法具有超线性收敛率,可以看作是对采用投影技术的 BFGS 类方法的扩展和补充。最后,通过数值实验验证了所提方法的有效性和应用前景。
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引用次数: 0
Lévy area analysis and parameter estimation for fOU processes via non-geometric rough path theory 通过非几何粗糙路径理论对 fOU 过程进行莱维区分析和参数估计
IF 1 4区 数学 Q1 MATHEMATICS Pub Date : 2024-08-27 DOI: 10.1007/s10473-024-0501-8
Zhongmin Qian, Xingcheng Xu

This paper addresses the estimation problem of an unknown drift parameter matrix for a fractional Ornstein-Uhlenbeck process in a multi-dimensional setting. To tackle this problem, we propose a novel approach based on rough path theory that allows us to construct pathwise rough path estimators from both continuous and discrete observations of a single path. Our approach is particularly suitable for high-frequency data. To formulate the parameter estimators, we introduce a theory of pathwise Itô integrals with respect to fractional Brownian motion. By establishing the regularity of fractional Ornstein-Uhlenbeck processes and analyzing the long-term behavior of the associated Lévy area processes, we demonstrate that our estimators are strongly consistent and pathwise stable. Our findings offer a new perspective on estimating the drift parameter matrix for fractional Ornstein-Uhlenbeck processes in multi-dimensional settings, and may have practical implications for fields including finance, economics, and engineering.

本文探讨了在多维环境下分数奥恩斯坦-乌伦贝克过程的未知漂移参数矩阵的估计问题。为了解决这个问题,我们提出了一种基于粗糙路径理论的新方法,这种方法允许我们从单一路径的连续和离散观测中构建路径粗糙路径估计器。我们的方法尤其适用于高频数据。为了提出参数估计值,我们引入了关于分数布朗运动的路径伊托积分理论。通过建立分数 Ornstein-Uhlenbeck 过程的规则性并分析相关莱维区过程的长期行为,我们证明了我们的估计值具有很强的一致性和路径稳定性。我们的发现为在多维环境中估计分数奥恩斯坦-乌伦贝克过程的漂移参数矩阵提供了一个新视角,并可能对金融、经济和工程等领域产生实际影响。
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引用次数: 0
Toeplitz operators between weighted Bergman spaces over the half-plane 半平面上加权伯格曼空间之间的托普利兹算子
IF 1 4区 数学 Q1 MATHEMATICS Pub Date : 2024-08-27 DOI: 10.1007/s10473-024-0504-5
Lixia Feng, Yan Li, Zhiyu Wang, Liankuo Zhao

In this paper, by characterizing Carleson measures, we investigate a class of bounded Toeplitz operator between weighted Bergman spaces with Békollé weights over the half-plane for all index choices.

在本文中,我们通过描述 Carleson 度量,研究了在所有索引选择下,半平面上具有贝可莱权重的加权伯格曼空间之间的一类有界托普利茨算子。
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引用次数: 0
On the Cauchy problem for the generalized Boussinesq equation with a damped term 关于带阻尼项的广义布森斯克方程的考希问题
IF 1 4区 数学 Q1 MATHEMATICS Pub Date : 2024-08-27 DOI: 10.1007/s10473-024-0508-1
Xiao Su, Shubin Wang

This paper is devoted to the Cauchy problem for the generalized damped Boussinesq equation with a nonlinear source term in the natural energy space. With the help of linear time-space estimates, we establish the local existence and uniqueness of solutions by means of the contraction mapping principle. The global existence and blow-up of the solutions at both subcritical and critical initial energy levels are obtained. Moreover, we construct the sufficient conditions of finite time blow-up of the solutions with arbitrary positive initial energy.

本文主要研究自然能量空间中带有非线性源项的广义阻尼布森斯克方程的考希问题。借助线性时空估计,我们利用收缩映射原理建立了解的局部存在性和唯一性。在亚临界和临界初始能量水平上,我们得到了解的全局存在性和炸毁性。此外,我们还构建了任意正初始能量下解的有限时间炸毁的充分条件。
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引用次数: 0
A compact embedding result for nonlocal Sobolev spaces and multiplicity of sign-changing solutions for nonlocal Schrödinger equations 非局部索波列夫空间的紧凑嵌入结果和非局部薛定谔方程符号变化解的多重性
IF 1 4区 数学 Q1 MATHEMATICS Pub Date : 2024-08-27 DOI: 10.1007/s10473-024-0512-5
Xu Zhang, Hao Zhai, Fukun Zhao

For any s ∈ (0, 1), let the nonlocal Sobolev space Xs(ℝN) be the linear space of Lebesgue measure functions from ℝN to ℝ such that any function u in Xs(ℝN) belongs to L2(ℝN) and the function

$$(x,y)longmapstobig(u(x)-u(y)big)sqrt{K(x-y)}$$

is in L2(ℝN, ℝN). First, we show, for a coercive function V(x), the subspace

$$E:=bigg{uin X^s(mathbb{R}^N):int_{mathbb{R}^N}V(x)u^2{rm d}x<+inftybigg}$$

of Xs(ℝN) is embedded compactly into Lp(ℝN) for (pin[2,2_s^*)), where (2_s^*) is the fractional Sobolev critical exponent. In terms of applications, the existence of a least energy sign-changing solution and infinitely many sign-changing solutions of the nonlocal Schrödinger equation

$$-{cal{L}_K}u+V(x)u=f(x,u), xin mathbb{R}^N$$

are obtained, where (-{cal{L}_K}) is an integro-differential operator and V is coercive at infinity.

对于任意 s∈ (0,1),让非局部 Sobolev 空间 Xs(ℝN) 是从 ℝN 到 ℝ 的 Lebesgue 度量函数的线性空间,使得 Xs(ℝN) 中的任何函数 u 都属于 L2(ℝN),并且函数$$(x、y)longmapstobig(u(x)-u(y)big)sqrt{K(x-y)}$$ 在 L2(ℝN, ℝN) 中。首先,我们证明,对于胁迫函数 V(x),子空间$E:=bigg{uin X^s(mathbb{R}^N):int_{/mathbb{R}^N}V(x)u^2{rm d}x<+inftybigg}$$的Xs(ℝN)子空间紧凑地嵌入到Lp(ℝN)中,为(pin[2,2_s^*)),其中(2_s^*)是分数索博列夫临界指数。在应用方面,得到了非局部薛定谔方程$$-{cal{L}_K}u+V(x)u=f(x,u),xinmathbb{R}^N$$的最小能量符号变化解和无穷多个符号变化解的存在,其中(-{cal{L}_K})是一个整微分算子,V在无穷处是强制的。
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引用次数: 0
Toeplitz determinants in one and higher dimensions 一维及更高维度的托普利兹行列式
IF 1 4区 数学 Q1 MATHEMATICS Pub Date : 2024-08-27 DOI: 10.1007/s10473-024-0517-0
Surya Giri, S. Sivaprasad Kumar

In this study, we derive the sharp bounds of certain Toeplitz determinants whose entries are the coefficients of holomorphic functions belonging to a class defined on the unit disk (mathbb{U}). Furthermore, these results are extended to a class of holomorphic functions on the unit ball in a complex Banach space and on the unit polydisc in ℂn. The obtained results provide the bounds of Toeplitz determinants in higher dimensions for various subclasses of normalized univalent functions.

在本研究中,我们推导了某些托普利兹行列式的尖锐边界,这些行列式的项是属于定义在单位圆盘 (mathbb{U})上的一类全纯函数的系数。此外,这些结果还扩展到了复巴纳赫空间中单位球和ℂn 中单位多圆盘上的一类全纯函数。所得到的结果为各种归一化等价函数子类提供了更高维度的托普利兹行列式的边界。
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引用次数: 0
The stable reconstruction of strongly-decaying block sparse signals 强衰减块状稀疏信号的稳定重建
IF 1 4区 数学 Q1 MATHEMATICS Pub Date : 2024-08-27 DOI: 10.1007/s10473-024-0509-0
Yifang Yang, Jinping Wang

In this paper, we reconstruct strongly-decaying block sparse signals by the block generalized orthogonal matching pursuit (BgOMP) algorithm in the l2-bounded noise case. Under some restraints on the minimum magnitude of the nonzero elements of the strongly-decaying block sparse signal, if the sensing matrix satisfies the the block restricted isometry property (block-RIP), then arbitrary strongly-decaying block sparse signals can be accurately and steadily reconstructed by the BgOMP algorithm in iterations. Furthermore, we conjecture that this condition is sharp.

本文利用块广义正交匹配追寻算法(BgOMP)在 l2 有界噪声情况下重建强衰减块稀疏信号。在强衰落块稀疏信号非零元素最小幅度的一些限制条件下,如果传感矩阵满足块受限等距特性(block-RIP),那么任意强衰落块稀疏信号都可以通过 BgOMP 算法在迭代中精确稳定地重建。此外,我们还猜想这一条件是尖锐的。
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引用次数: 0
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Acta Mathematica Scientia
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