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Replacement submissions (showing 6 of 6 entries )

[1] arXiv:2510.20832 (replaced) [cn-pdf, pdf, html, other]

Title: The Thomae Function: Fractal Insights Show Chinese title

Title: 托马斯函数：分形见解

Thomas Lamby, Samuel Nicolay

Comments: 4 figures

Subjects: General Mathematics (math.GM)

This article examines the Thomae function, a paradigmatic example of a function that is continuous on the irrationals and discontinuous elsewhere. Defined for a parameter $\theta>0$, it exhibits a rich self-similar structure and intriguing regularity properties. After revisiting its fundamental characteristics, we analyze its H\"older continuity, emphasizing the interplay between its discrete spikes and its behavior on dense subsets of the real line. This study provides a refined perspective on the irregularity of the Thomae function, using classical analytical tools to elucidate its fractal nature. Show Chinese abstract

本文研究了Thomae函数，这是一个典型的函数，在无理数上连续而在其他地方不连续。 对于参数$\theta>0$定义，它表现出丰富的自相似结构和引人入胜的规律性特征。 在回顾其基本特性之后，我们分析了它的Hölder连续性，强调其离散尖峰与其在实数线稠密子集上的行为之间的相互作用。 这项研究使用经典的分析工具，对Thomae函数的不规则性提供了更精细的视角，阐明了其分形性质。

[2] arXiv:2510.20834 (replaced) [cn-pdf, pdf, html, other]

Title: An ε-free rank-6 decoupling estimate for the paraboloid surface Show Chinese title

Title: 一个无ε的六阶解耦估计对于抛物面表面

Pylyp Cherevan

Comments: 38 pages, 0 figures

Subjects: General Mathematics (math.GM)

For the paraboloid decomposition $F=\sum_{\Theta} F_{\Theta}$ with $\Theta\subset{|\xi|\sim\lambda}$ and radius $r=\lambda^{-2/3}$, we prove a log-free estimate $|F|{L^{6}(Q{\lambda})}\lesssim \lambda^{\Sigma_{\lambda}} D^{\Sigma_{D}} \big(\sum_{\Theta}|F_{\Theta}|{L^{6}}^{2}\big)^{1/2}$ as $\lambda\to\infty$, where $D=\lambda^{1/12}$. Key components: (i) broad geometry of rank 3: bilipschitz behavior of normals gives $\max{i<j<k}|n_i\wedge n_j\wedge n_k|\gtrsim \lambda^{-5/4}$, which via a trilinear Kakeya-BCT insertion contributes $+5/36$ in $\lambda$; (ii) kernel estimate: twelve integrations (6 in $t$, 6 in $x^{\prime}$) and measure analysis (Schur and $TT^{}$) yield $|K|{L^2\to L^2}\lesssim \lambda^{-9/2} D^{-3}$; (iii) robust Kakeya: a density threshold $> c{} D$ brings a factor $D$ ($+1/12$ in $\lambda$, $+1$ in $D$); (iv) algebraic shell: excluding a neighborhood $N_{\beta}(P)$ contributes $-1/12$ in $\lambda$ and $-1$ in $D$; (v) tube packing: explanatory only; (vi) narrow cascade: a double $7/8$ rescaling exits the narrow regime and contributes $-5/64$ in $\lambda$ (zero in $D$). Summing exponents: $\Sigma_{\lambda}=5/36-9/2-5/64=-2557/576\approx -4.44<0$ and $\Sigma_{D}=-3+1-1=-3<0$, hence both $\lambda^{\varepsilon}$- and $D^{\varepsilon}$-losses are removed. Show Chinese abstract

对于抛物面分解 $F=\sum_{\Theta} F_{\Theta}$ ，带有 $\Theta\subset{|\xi|\sim\lambda}$ 和半径 $r=\lambda^{-2/3}$，我们证明了一个无对数估计 $|F|{L^{6}(Q{\lambda})}\lesssim \lambda^{\Sigma_{\lambda}} D^{\Sigma_{D}} \big(\sum_{\Theta}|F_{\Theta}|{L^{6}}^{2}\big)^{1/2}$ ，如 $\lambda\to\infty$所示，其中 $D=\lambda^{1/12}$。 关键组成部分：(i) 3阶广义几何：法线的双利普希茨行为给出$\max{i<j<k}|n_i\wedge n_j\wedge n_k|\gtrsim \lambda^{-5/4}$，通过三线性Kakeya-BCT插入在$\lambda$中贡献$+5/36$；(ii) 核估计：十二次积分（6次在$t$，6次在$x^{\prime}$）和测度分析（Schur和$TT^{}$）产生$|K|{L^2\to L^2}\lesssim \lambda^{-9/2} D^{-3}$；(iii) 鲁棒Kakeya：密度阈值$> c{} D$带来因子$D$（在$\lambda$中$+1/12$，在$D$中$+1$）；(iv) 代数壳：排除一个邻域$N_{\beta}(P)$在$\lambda$中贡献$-1/12$并在$D$中贡献$-1$；(v) 管道包装：仅作解释；(vi) 窄级联：一个双重$7/8$缩放退出窄区域并在$\lambda$中贡献$-5/64$（在$D$中为零）。 指数相加：$\Sigma_{\lambda}=5/36-9/2-5/64=-2557/576\approx -4.44<0$和$\Sigma_{D}=-3+1-1=-3<0$，因此两者$\lambda^{\varepsilon}$- 和$D^{\varepsilon}$-损失都被移除。

[3] arXiv:2510.20837 (replaced) [cn-pdf, pdf, html, other]

Title: Information retrieval in big data using cognitive approaches Show Chinese title

Title: 基于认知方法的大数据信息检索

Santanu Acharjee, Ripunjoy Choudhury

Subjects: General Mathematics (math.GM)

Due to the exponential growth of big data in this digital era, an advanced method for effective information retrieval becomes essential. The basic objective of this paper is to propose a topology-based method for cognitive information retrieval (CIR) in big data environments. By using concepts such as cognitive similarity distances, metric spaces, retrieval topologies, etc., this paper aims to propose the semantic alignment between user queries and document repositories. The paper also extends this approach to incorporate logical connectives in cognitive information retrieval. Show Chinese abstract

由于大数据在这一数字时代的指数级增长，一种先进的有效信息检索方法变得至关重要。 本文的基本目标是提出一种基于拓扑结构的方法，用于大数据环境中的认知信息检索（CIR）。 通过使用认知相似性距离、度量空间、检索拓扑等概念，本文旨在提出用户查询与文档存储库之间的语义对齐。 本文还将这种方法扩展，以在认知信息检索中包含逻辑连接词。

[4] arXiv:2510.20841 (replaced) [cn-pdf, pdf, html, other]

Title: Remarks on Redheffer's inequality Show Chinese title

Title: 关于Redheffer不等式的备注

Nagi Suzuki, Shingo Takeuchi

Comments: 10 pages

Subjects: General Mathematics (math.GM)

Redheffer's inequality and its extensions are applied to study the behavior and estimates of the first eigenvalue of $p$-Laplacian with respect to $p$. Furthermore, a Redheffer-type inequality for the generalized trigonometric function is extended to a broader class. Show Chinese abstract

Redheffer不等式及其推广被用于研究$p$-Laplacian在$p$下的第一个特征值的行为和估计。 此外，一种类似于Redheffer的不等式被扩展到更广泛的广义三角函数类中。

[5] arXiv:2510.20842 (replaced) [cn-pdf, pdf, html, other]

Title: Fractional harmonic transform on point cloud manifolds Show Chinese title

Title: 点云流形上的分数调和变换

Jiamian Li, Bing-Zhao Li

Comments: Submitted to ICASSP 2026

Subjects: General Mathematics (math.GM)

Three-dimensional point clouds can be viewed as discrete samples of smooth manifolds, allowing spectral analysis using the Laplace-Beltrami operator (LBO). However, the traditional point cloud manifold harmonic transform (PMHT) is limited by its fixed basis functions and single spectral representation, which restricts its ability to capture complex geometric features. This paper proposes a point cloud manifold fractional harmonic transform (PMFHT), which generalizes PMHT by introducing fractional-order parameters and constructs a continuously adjustable intermediate fractional-order spectral domain between the spatial domain and the frequency domain. This fractional-order framework supports more flexible transformation and filtering operations. Experiments show that choosing different transformation orders can enrich the spectral representation of point clouds and achieve excellent results in tasks such as filtering and feature enhancement. Therefore, PMFHT not only expands the theoretical framework of point cloud spectral analysis, but also provides a powerful new tool for manifold geometry processing. Show Chinese abstract

三维点云可以看作是平滑流形的离散采样，允许使用拉普拉斯-贝尔特拉米算子（LBO）进行谱分析。然而，传统的点云流形谐波变换（PMHT）由于其固定的基函数和单一的谱表示而受到限制，这限制了其捕捉复杂几何特征的能力。本文提出了一种点云流形分数阶谐波变换（PMFHT），通过引入分数阶参数对PMHT进行了推广，并在空间域和频率域之间构建了一个可连续调整的中间分数阶谱域。这种分数阶框架支持更灵活的变换和滤波操作。实验表明，选择不同的变换阶数可以丰富点云的谱表示，并在滤波和特征增强等任务中取得优异的结果。因此，PMFHT不仅扩展了点云谱分析的理论框架，还为流形几何处理提供了一个强大的新工具。

[6] arXiv:2510.20845 (replaced) [cn-pdf, pdf, html, other]

Title: Schur-Convex Curvature on Dihedral Exponential Families and the Golden-Ratio Stationary Point Show Chinese title

Title: 二面体指数族上的Schur-凸曲率和黄金比例平稳点

Michael Arnold Bruna

Comments: 18 pages, 1 figure

Subjects: General Mathematics (math.GM)

We investigate the Schur-complement curvature of D_N-equivariant folded exponential families on the simplex. Our main structural results are: (i) the curvature kappa_Schur(theta) is convex in the log-parameter theta = ln(q); (ii) it admits a unique stationary point at the golden ratio value q* = phi^-2 (in particular for N = 12); and (iii) it obeys a quadratic folded law kappa_Schur = A(N, m_rho^2) I_1^2 + B(N, m_rho^2) (I_2 - I_1^2), with coefficients A, B determined explicitly by the projector metric of radius m_rho^2. Taken together, these results show that convexity and symmetry alone enforce both the location and the functional form of the "golden lock-in." Beyond their intrinsic interest, these findings identify D_12 as the minimal dihedral lattice where parity (mod 2) and three-cycle (mod 3) constraints coexist, producing a structurally stable equilibrium at the golden ratio. This places the golden ratio not as an accident of parameterization but as a necessary consequence of convex geometry under dihedral symmetry. Possible applications include harmonic analysis on group orbits, invariant convex optimization, and the structure of tilings or quasicrystal-like systems. Show Chinese abstract

我们研究了单纯形上D_N-等变折叠指数族的Schur补曲率。 我们的主要结构结果是：(i) 曲率kappa_Schur(theta)在对数参数theta = ln(q)下是凸的；(ii) 在黄金分割值q* = phi^-2处有一个唯一的驻点（特别是当N = 12时）；(iii) 它遵循二次折叠定律kappa_Schur = A(N, m_rho^2) I_1^2 + B(N, m_rho^2) (I_2 - I_1^2)，其中系数A, B由半径m_rho^2的投影仪度量显式确定。 综上所述，这些结果表明，仅凸性和对称性就强制了“黄金锁定”位置和函数形式。 除了它们本身的兴趣外，这些发现确定了D_12作为最小二面体格子，在其中奇偶性（模2）和三循环（模3）约束共存，从而在黄金比例处产生结构稳定的平衡。 这将黄金比例不是作为参数化的偶然现象，而是作为二面体对称下凸几何的必然结果。 可能的应用包括群轨道上的调和分析、不变凸优化以及铺砌或准晶体类似系统的结构。