从三角阵列序列到近对数凹状态转移
From Triangular Array Progression to Near-Log-Concave State Transitions
AI总结：
本文提出一种生成三角阵列的无限整数序列，该阵列具有近对数凹性等特性，可用于物理建模电子跃迁系统，还能生成满足特定状态转移特征的右随机矩阵族。
AI中文摘要：
本文介绍了一种无限整数序列序列，该序列生成一个三角阵列，其中对于所有n≥0，第n行的和为2ⁿ。该三角阵列的每一行都可以实现为无环多重图的度序列。尽管前四行与帕斯卡三角的行重合，但所提出的阵列序列随后变得不对称并偏离二项式系数。该三角阵列包含无限多个无零点的单峰近对数凹序列，其对数凹度偏差在对数尺度下收敛于log(4/3)。该序列在每一代提供一个概率测度，定义了所谓的近对数凹随机变量，生成无限马尔可夫链的状态转移，并包含一个二阶全正托普利茨矩阵。我们研究该构造的香农熵动力学，并提出其在物理学中的一项应用，用于建模表现出一组状态转移特征的系统，这些特征控制电子在不同能态间的跃迁。我们提供了三角阵列在吉洪诺夫立方体上的一种变换，以生成无限族右随机矩阵，所有这些矩阵都满足给定的一组状态转移特征。
英文摘要：
The paper introduces near-log-concave random variables and the state-transitions of a family of Markov chains derived from an algorithmic integer sequence progression. We propose a combinatorial rule that generates an asymmetric triangular array whose $n$-th row sums to $2^{n}$ for all $n \geq 0$, where every row is the degree sequence of a multigraph without loops. The structure hosts infinitely many zero-free unimodal near-log-concave sequences whose log-concavity deviation converges to $\log(4/3)$ under logarithmic scaling. From a probabilistic perspective, the construct defines a sequence of near-log-concave probability mass functions, from which, a structural decoupling of moments emerges: the mean diverges while the variance asymptotically converges to a steady-state limit. We study the Shannon entropy dynamics of the triangular array, and generate infinitely many right-stochastic matrices via coordinate transformations over the Tychonoff cube. Our specific framework indicates a broader methodology for modelling asymmetric stochastic systems, where distributions propagate as non-dispersive discrete wave packets, advancing indefinitely without flattening.
