时间依赖的模离散拉普拉斯动力学中的长寿命地毯状瞬态
Long-Lived Carpet-Like Transients in Time-Dependent Modular Discrete Laplacian Dynamics
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中文总结 AI 辅助
本研究通过计算实验发现，周期性非二元模数插入可维持长寿命地毯状瞬态，其中三元插入效果最稳健，且密度对二元尾长度呈非单调依赖。
中文摘要 AI 辅助
二元模拉普拉斯动力学通过二进尺度上的重复复制、增长和坍缩来组织。我们研究孤立和周期性重复的非二元更新如何修改这种组织，以及它们是否能维持密集占据的几何结构。跨多个有限种子、邻域掩码、插入模数和周期性调度的计算实验表明，常数素数模数保留了预期的 \\(p\\)-adic 复制层级。单个非二元插入主要起到有效种子替换的作用：随后的演化保持二元类似，但可能在坍缩-恢复循环的整数相位偏移处恢复，仅产生适度的有限时间致密化。对于周期性调度 \\([2,k,2^s]^\infty\\)（其中 \\(2^s\\) 表示 \\(s\\) 次连续的二元更新），出现了定性不同的响应。奇数模数插入可以维持长寿命、空间相干的地毯状状态，具有相对较高且稳定的占据率，而偶数模数插入保持二元类似。三元插入给出最广泛且最稳健的高密度响应。对于 \\(k=3\\)，密度对二元尾长度呈非单调依赖，在 \\(s=7,15,23\\) 附近出现低谷，揭示了对二元纪元结构的相位敏感交互。跨模数同步也会发生：对于四度对角 Neumann 和 von Neumann 掩码，\\(k=5,7,9\\) 在随后的二元相位反复坍缩到相同的完整配置。一个奇偶性论证解释了这种记忆丧失。增加种子范围通常提高占据率，同时降低尾长度敏感性。总体而言，周期性模数插入充当相位选择、几何依赖的机制，将 \\(p\\)-adic 组织的复制转化为长寿命的地毯状动力学。
英文摘要
Binary modular Laplacian dynamics is organized by repeated replication, growth, and collapse on dyadic scales. We study how isolated and periodically repeated non-binary updates modify this organization and whether they can sustain densely occupied geometric structures. Computational experiments across multiple finite seeds, neighborhood masks, inserted moduli, and periodic schedules show that constant prime moduli retain the expected \(p\)-adic replication hierarchy. A single non-binary insertion mainly acts as an effective-seed replacement: subsequent evolution remains binary-like, but may resume at an integer phase shift of the collapse--recovery cycle, with only modest finite-time densification. A qualitatively different response occurs for periodic schedules \([2,k,2^s]^\infty\), where \(2^s\) denotes \(s\) consecutive binary updates. Odd-modulus insertions can sustain long-lived, spatially coherent carpet-like regimes with comparatively high and stable occupation, whereas even-modulus insertions remain binary-like. Ternary insertion gives the broadest and most robust high-density response. For \(k=3\), density depends non-monotonically on the binary-tail length, with troughs near \(s=7,15,23\) that reveal phase-sensitive interaction with the binary epoch structure. Cross-modulus synchronization also occurs: for degree-four diagonal Neumann and von Neumann masks, \(k=5,7,9\) repeatedly collapse onto the same complete configuration at subsequent binary phases. A parity argument explains this loss of memory. Increasing seed extent generally raises occupation while reducing tail-length sensitivity. Overall, periodic modular insertions act as phase-selective, geometry-dependent mechanisms transforming \(p\)-adically organized replication into long-lived carpet-like dynamics.
发表机构
- Institute of Informatics, John Paul II Catholic University of Lublin（卢布林约翰保罗二世天主教大学信息学研究所）
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