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交叉提交 (展示 3 之 3 条目 )

[1] arXiv:2508.04706 (交叉列表自 math.GM) [中文pdf, pdf, html, 其他]

标题： 非均匀网格上差分方程的存在性结果通过上下解方法 显示英文标题

标题： Existence Result for Difference Equations on Non-Uniform Grids via Upper and Lower Solution Method

Shalmali Bandyopadhyay, Kimser Lor

主题： 一般数学 (math.GM)

本文利用上下解方法，建立了在非均匀时间网格上离散二阶边值问题的存在性理论。 我们考虑形式为$u^{\Delta\Delta}(t_{i-1}) + f(t_i, u(t_i), u^\Delta(t_{i-1})) = 0$的差分方程，在非均匀时间网格${t_0, t_1, \ldots, t_{n+2}}$上，具有混合边界条件$u^\Delta(t_0) = 0$和$u(t_{n+2}) = g(t_{n+2})$。 这将之前关于齐次边界条件的工作扩展到了非齐次情况，需要一个复杂的泛函分析框架来处理产生的仿射函数空间。 我们的方法采用了一种分解策略，将边界效应与微分结构分离，从而能够应用布劳威尔不动点定理，证明解介于上函数和下函数之间的存在性。 显示英文摘要

This paper establishes an existence theory for discrete second-order boundary value problems on non-uniform time grids using the upper and lower solution method. We consider difference equations of the form $u^{\Delta\Delta}(t_{i-1}) + f(t_i, u(t_i), u^\Delta(t_{i-1})) = 0$ on a non-uniform time grid ${t_0, t_1, \ldots, t_{n+2}}$ with mixed boundary conditions $u^\Delta(t_0) = 0$ and $u(t_{n+2}) = g(t_{n+2})$. This extends previous work on homogeneous boundary conditions to the non-homogeneous case, requiring a sophisticated functional analytic framework to handle the resulting affine function spaces. Our approach employs a decomposition strategy that separates boundary effects from the differential structure, enabling the application of Brouwer's Fixed Point Theorem to establish existence with solutions bounded between upper and lower functions.

[2] arXiv:2508.04708 (交叉列表自 math.GM) [中文pdf, pdf, html, 其他]

标题： 代数框架用于洛朗级数上的离散动力系统 显示英文标题

标题： Algebraic Framework for Discrete Dynamical Systems over Laurent Series

Ramamonjy Aandriamifidisoa, Loukman Ben Saindou

评论： 10页

主题： 一般数学 (math.GM)

我们推广了离散代数动力系统的框架\cite{Andriamifidisoa2014}，使其适用于\(\Z^r\)上的Laurent多项式和级数，从而能够对双向离散系统进行建模。 通过重新定义空间\(\Dprime\)和\(\Aprime\)，引入一个双线性映射（在第3节中定义为标量积），并扩展移位算子，我们保留了\cite{Andriamifidisoa2014}的对偶性和伴随性质。 这些性质通过示例和一个关于双向序列变换的数据处理案例研究进行了严格证明和说明。 与 Oberst\cite{Ob90}相比，我们的代数方法强调Laurent级数的结构，为多维系统提供了一个简化的框架。 这项工作解决了\cite{Andriamifidisoa2014}提出的一个开放问题，并在多维数据处理中有所应用，例如图像滤波和控制理论。 显示英文摘要

We generalize the framework of discrete algebraic dynamical systems \cite{Andriamifidisoa2014} to Laurent polynomials and series over \(\Z^r\), enabling the modeling of bidirectional discrete systems. By redefining the spaces \(\Dprime\) and \(\Aprime\), introducing a bilinear mapping (defined as the scalar product in Section 3), and extending the shift operator, we preserve the duality and adjoint properties of \cite{Andriamifidisoa2014}. These properties are rigorously proved and illustrated through examples and a data processing case study on bidirectional sequence transformations. In contrast to Oberst \cite{Ob90}, our algebraic approach emphasizes the structure of Laurent series, providing a streamlined framework for multidimensional systems. This work addresses an open question from \cite{Andriamifidisoa2014} and has applications in multidimensional data processing, such as image filtering and control theory.

[3] arXiv:2508.04709 (交叉列表自 math.GM) [中文pdf, pdf, html, 其他]

标题： 在高斯PDMF空间中求解模糊线性系统 显示英文标题

标题： Solving fuzzy linear systems in Gaussian PDMF space

Chuang Zheng

评论： 28页，15图

主题： 一般数学 (math.GM)

在论文的第一部分，我们求解半模糊线性系统（SFLS）$A \mathbf{\tilde{x}}= \mathbf{\tilde{b}}$，其中$A\in \mathbb{R}^{m\times n}$是一个实值矩阵，$\mathbf{\tilde{b}}$是一个模糊数向量，而$\mathbf{\tilde{x}}$是未知的模糊数向量。$\mathbf{\tilde{b}}$和$\mathbf{\tilde{x}}$的元素都属于一个模糊数空间$\mathcal{X}$，即高斯概率密度隶属函数（Gaussian-PDMF）空间。 我们提出Cramer法则来计算方阵$A$的解，并发现其解集是一个$5(n-R(A))$维的仿射空间，其中$A\in \mathbb{R}^{m\times n}$和$R(A)$是$A$的秩。 RREF矩阵$A$的解的显式形式被陈述以确保建模的可用性。 在论文的第二部分，我们求解全模糊线性系统（FFLS）$\mathbf{\tilde{A}}\mathbf{\tilde{x}}=\mathbf{\tilde{b}}$，其中$\mathbf{\tilde{A}}$是一个所有元素都在$\mathcal{X}$中的模糊矩阵。我们分析其解集，并在模糊RREF矩阵下给出解的参数形式。然后通过将其限制到环$\mathcal{X}$的单位群，将高斯消去法适应于模糊矩阵和系统，并证明经过初等行变换后的解集等价。我们还通过将$\mathbf{\tilde{A}}$的元素限制到形成域的$\mathcal{X}$的一个子集，建立FFLS与SFLS之间的联系。在第三部分，给出了两个数值例子来说明我们的方法。本文的所有结果都是明确的，因为模糊数的隶属函数所属的高斯-PDMF空间$\mathcal{X}$具有完整的代数结构。所提出的框架为使用具有不确定性和模糊性的模糊线性系统解决数学模型提供了一个可行且系统化的工具。 显示英文摘要

In the first part of the paper, we solve the semi-fuzzy linear system (SFLS) $A \mathbf{\tilde{x}}= \mathbf{\tilde{b}}$, where $A\in \mathbb{R}^{m\times n}$ is a real-valued matrix, $\mathbf{\tilde{b}}$ is a fuzzy number vector, and $\mathbf{\tilde{x}}$ is the unknown fuzzy number vector. The elements of both $\mathbf{\tilde{b}}$ and $\mathbf{\tilde{x}}$ belong to a fuzzy number space $\mathcal{X}$, namely the Gaussian probability density membership function (Gaussian-PDMF) space. We present the Cramer's rule to calculate the solution with square matrix $A$ and find out that its solution set is a $5(n-R(A))$ dimensional affine space with $A\in \mathbb{R}^{m\times n}$ and $R(A)$ being the rank of $A$. The explicit form of the solution for RREF matrix $A$ is stated to ensure usability for modeling. In the second part of the paper, we solve the fully-fuzzy linear system (FFLS) $\mathbf{\tilde{A}}\mathbf{\tilde{x}}=\mathbf{\tilde{b}}$, where $\mathbf{\tilde{A}}$ is a fuzzy matrix with all components in $\mathcal{X}$. We analyze its solution set and present the parametric form of solutions under the fuzzy RREF matrix. We then adapt Gaussian elimination method to fuzzy matrices and systems by restricting it to the unit group of ring $\mathcal{X}$, proving the equivalence of solution sets after elementary row operations. We also establish the connection between FFLS and SFLS by confining elements of $\mathbf{\tilde{A}}$ to a subset of $\mathcal{X}$ that forms a field. In the third part, two numerical examples are given to illustrated our method. All results in this paper are explicit since the Gaussian-PDMF space $\mathcal{X}$, to which the membership function of the fuzzy number belongs, possesses a complete algebraic structure. The proposed framework offers a feasible and systematical tool for solving the mathematical models using fuzzy linear systems with uncertainty and fuzziness.

替换提交 (展示 2 之 2 条目 )

[4] arXiv:2204.07643 (替换) [中文pdf, pdf, html, 其他]

标题： 迈向黎曼假设的证明 显示英文标题

标题： Towards a proof of the Riemann Hypothesis

Guilherme Rocha de Rezende

评论： 18页 13图

主题： 一般数学 (math.GM)

在本文中，我们提出对著名的幅角原理的重新审视，这可能会导致黎曼假设的解决。 我们正在寻找合作者。 显示英文摘要

In this article we propose a revisitation of the well-known argument principle that may lead to the solution of the Riemann hypothesis. We are looking for collaborators.

[5] arXiv:2411.11851 (替换) [中文pdf, pdf, html, 其他]

标题： 树的给定度量维数下的原子键连接性和扎格雷布指数的界 显示英文标题

标题： Bounds on Atom-Bond Connectivity and Zagreb Indices in Trees with a Given Metric Dimension

Waqar Ali, Mohamad Nazri Bin Husin, Muhammad Faisal Nadeem, Muqaddas Jabin

主题： 一般数学 (math.GM)

设 $\mathbb{G} = (\mathcal{V}, \mathcal{E})$是一个简单连通图，其中 $\mathcal{V}$和 $\mathcal{E}$分别表示顶点集和边集。 第一Zagreb指标定义为 $\mathcal{M}_{1}(\mathbb{G}) = \sum_{v \in \mathcal{V}} \zeta_{\mathbb{G}}(v)^2$，而第二Zagreb指标由 $\mathcal{M}_{2}(\mathbb{G}) = \sum_{uv \in \mathcal{E}} \zeta_{\mathbb{G}}(u)\, \zeta_{\mathbb{G}}(v)$给出，其中 $\zeta_{\mathbb{G}}(v)$表示顶点 $v$的度。 另一个显著的基于度数的不变量是原子-键连接（ABC）指数，它在化学图论中被引入，并由 \[ ABC(\mathbb{G}) = \sum_{uv \in \mathcal{E}} \sqrt{\frac{\zeta_{\mathbb{G}}(u) + \zeta_{\mathbb{G}}(v) - 2}{\zeta_{\mathbb{G}}(u)\, \zeta_{\mathbb{G}}(v)}}. \] 一个基本的图参数，度量维数指的是在分辨集中最小的顶点数，该集合可以根据距离唯一地区分所有其他顶点。 在本工作中，我们研究度量维数对树类中Zagreb和ABC指数的影响。 我们推导了针对$\mathcal{M}_1$和 $\mathcal{M}_2$的紧界——上下界，并提供了ABC指数的上界，所有这些都用树的阶数及其度量维数来表示。 此外，我们确定了达到这些界限的极端树结构。 这些发现强调了度量维数在塑造拓扑描述符中的作用，并对理论图分析和分子化学的实际应用做出了贡献。 显示英文摘要

Let $\mathbb{G} = (\mathcal{V}, \mathcal{E})$ be a simple connected graph, where $\mathcal{V}$ and $\mathcal{E}$ denote the vertex and edge sets, respectively. The first Zagreb index is defined as $\mathcal{M}_{1}(\mathbb{G}) = \sum_{v \in \mathcal{V}} \zeta_{\mathbb{G}}(v)^2$, while the second Zagreb index is given by $\mathcal{M}_{2}(\mathbb{G}) = \sum_{uv \in \mathcal{E}} \zeta_{\mathbb{G}}(u)\, \zeta_{\mathbb{G}}(v)$, where $\zeta_{\mathbb{G}}(v)$ represents the degree of vertex $v$. Another notable degree-based invariant is the atom-bond connectivity (ABC) index, introduced in chemical graph theory, and defined by \[ ABC(\mathbb{G}) = \sum_{uv \in \mathcal{E}} \sqrt{\frac{\zeta_{\mathbb{G}}(u) + \zeta_{\mathbb{G}}(v) - 2}{\zeta_{\mathbb{G}}(u)\, \zeta_{\mathbb{G}}(v)}}. \] A fundamental graph parameter, the metric dimension, refers to the minimum number of vertices in a resolving set that uniquely distinguishes all other vertices based on distances. In this work, we investigate the influence of metric dimension on the Zagreb and ABC indices within the class of trees. We derive sharp bounds-both upper and lower for $\mathcal{M}_1$ and $\mathcal{M}_2$, and provide an upper bound for the ABC index, all expressed in terms of the tree's order and its metric dimension. Furthermore, we identify the extremal tree structures that attain these bounds. These findings underscore the role of metric dimension in shaping topological descriptors and contribute both to theoretical graph analysis and practical applications in molecular chemistry.