Markov数的一种推广

Appearance of Balancing and related number sequences in steady state probabilities of some Markov chains

Balancing and Lucas-balancing numbers are solutions of a Diophantine equation and satisfy a second order homogeneous recurrence relation. Interestingly, these numbers can be seen as numerators and denominators in the steady state probabilities of a class of transition probability matrices of Markov chains. An identity relating the balancing numbers and the silver ratio can be obtained as a byproduct.

k-马尔科夫数的序

Orderings of k-Markov Numbers

本文研究k-马尔科夫数的序问题，证明其满足Aigner的猜想，基于簇代数理论。

On the Markov numbers: fixed numerator, denominator, and sum conjectures

The Markov numbers are the positive integer solutions of the Diophantine equation $x^2 + y^2 + z^2 = 3xyz$. Already in 1880, Markov showed that all these solutions could be generated along a binary tree. So it became quite usual (and useful) to index the Markov numbers by the rationals between 0 and 1 which stand at the same place in the Stern-Brocot binary tree. The Frobenius conjecture claims that each Markov number appears at most once in the tree. In particular, if the conjecture is true, the order of Markov numbers would establish a new strict order on the rationals. Aigner suggested thre…

广义k-马尔科夫数的序

Orderings of Generalized k-Markov Numbers

本文研究了广义k-马尔科夫数沿特定线性增长的性质，扩展了普通情况下的研究，发现k增大时，数沿随机线性增长的可能性增加，支持了k版本的弗罗贝尼乌斯唯一性猜想。

On the ordering of the Markov numbers

The Markov numbers are the positive integers that appear in the solutions of the equation $x^2+y^2+z^2=3xyz$. These numbers are a classical subject in number theory and have important ramifications in hyperbolic geometry, algebraic geometry and combinatorics. It is known that the Markov numbers can be labeled by the lattice points $(q,p)$ in the first quadrant and below the diagonal whose coordinates are coprime. In this paper, we consider the following question. Given two lattice points, can we say which of the associated Markov numbers is larger? A complete answer to this question would solv…

Continued fractions and orderings on the Markov numbers

Markov numbers are integers that appear in the solution triples of the Diophantine equation, $x^2+y^2+z^2=3xyz$, called the Markov equation. A classical topic in number theory, these numbers are related to many areas of mathematics such as combinatorics, hyperbolic geometry, approximation theory and cluster algebras. There is a natural map from the rational numbers between zero and one to the Markov numbers. In this paper, we prove two conjectures seen in Martin Aigner's book, Markov's theorem and 100 years of the uniqueness conjecture, that determine an ordering on subsets of the Markov numbe…

广义Markov数的SL(2,Z)-矩阵化

SL(2,Z)-matrixizations of generalized Markov numbers

本文利用$SL(2, \mathbb{Z})$中的$k$-广义Cohn矩阵和$k$-Markov-monodromy矩阵族，揭示了广义Markov数的组合结构，恢复了方程正整数解的树结构，并给出了几何与组合解释及计算算法。

提升算子的马尔可夫链

Markov chains for promotion operators

我们考虑Schuetzenberger推广算子在有限偏序集的线性扩张集L上的推广。这就产生了L上的强连通图。在早期的工作（arXiv：1205.7074）中，我们研究了这些线性扩展上的基于提升的马尔可夫链，推广了Tsetlin库上的结果。利用R-平凡幺半群的理论，我们得到了一般偏序集为根森林时转移阵的特征值。我们首先调查这些结果，然后提出明确的界限上的混合时间和猜想特征值公式更一般的偏序集。我们还提出了推广到任意子集的对称群。

有限图上的局部马尔可夫游走

Locally Markov walks on finite graphs

本文提出局部马尔可夫游走的概念，分析其平稳分布和混合时间，证明其不可约性和遍历性，并揭示其在完全图上的截止现象。

进位过程与欧拉数的推广

A generalization of carries process and Eulerian numbers

本文研究了Holte的奇异矩阵的一个推广，即非标准数制中进位的马尔可夫链的转移概率矩阵。平稳分布的显式描述的数字，可以看作是一个推广的欧拉数和MacMahon数。我们还表明，类似的性质，甚至举行的计数系统与负基地。

Markov数的镜像变形

A mirror deformation of Markov numbers

本文提出了一种Markov数的$q$-变形，通过镜像突变规则生成所有镜像Markov数，并给出了其几何实现和相关变形的公式。

无向图空间上不可约马尔可夫链

Irreducible Markov Chains on spaces of graphs with fixed degree-color sequences

本文研究了一种彩色扩展的简单切换马尔可夫链，用于采样具有固定度色序列的图集。通过证明简单颜色保持切换可连接该空间，为测试块划分网络数据统计模型拟合提供了不可约马尔可夫链的基础。

关于 $q$-变形 Markov 数：Cohn 矩阵与带权边的完美匹配

On $q$-deformed Markov numbers. Cohn matrices and perfect matchings with weighted edges

本文提出经典 Markov 数的自然 $q$-变形，证明其唯一性及多项式性质，并借助 $q$-变形 Cohn 矩阵与带权边蛇图完美匹配给出计算和组合解释。

关于随机有向无环图的 Markov 等价类的数量与大小

On the number and size of Markov equivalence classes of random directed acyclic graphs

本文研究了均匀随机有向无环图的 Markov 等价类，利用塔分解证明了等价类与图的比值趋于正常数，且典型等价类大小有界并具有超多项式尾部。