发表机构
School of Mathematics and Statistics, Huaiyin Normal University; Great Bay University; School of Management, Zhejiang University（淮阴师范学院数学与统计学院; 大湾区大学; 浙江大学管理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文发现经典黄金分割与在线租赁问题的联系，提出高维黄金分割定义，并设计多重在线租赁的最优控制与平衡策略，证明其等价于调和型黄金分割方法，并引入尾数加权法求解高维方程。
AI 中文摘要
我们发现经典黄金分割与在线租赁问题之间存在联系：当滑雪板数量从一双增加到两双时，最优竞争算法是在[0,s]区间内租两双直到黄金分割点(√5-1)s/2，然后购买一双，最后在时间s购买第二双，达到最优竞争比(5+√5)/4。在此基础上，我们探索高维黄金分割和多重在线租赁。对于黄金分割，我们基于将单位线段分割成m段的两个等价命题，定义了几何型和调和型高维黄金分割τ^(m)和T^(m)。我们建立了几何型比例λ和调和型基ω的方程，并研究了它们的代数特性。将k-广义斐波那契数列扩展为双无限m-广义斐波那契数列，我们建立了它们之间的关系。我们引入了生成序列和斐波那契矩阵等新概念，并研究了其子矩阵。诸如涉及斐波那契矩阵的行列式公式等结果证明了其在研究广义斐波那契数列中的有效性。对于在线租赁，我们考虑m双滑雪板的多重在线租赁。我们设计了一种控制和平衡策略，并通过竞争分析证明了其最优性。我们表明最优策略正是m维调和型黄金分割方法。由于对于高于四维的情况可能不存在显式解，我们提出了一个连续松弛问题。我们探索使用其松弛解来近似最优解或最大实根。我们提出了尾数加权法来求解方程，通过误差分析实现了高精度。
英文摘要
We discover a connection between the classical golden section and the online leasing problem: when the number of skis increases from one to two, the optimal competitive algorithm is to rent two pairs until the golden section point$(\sqrt{5}-1)s/2$ within [0,s], then buy one pair, and finally buy the second at time $s$, achieving the optimal competitive ratio of $(5+\sqrt{5})/4$. Building on this, we explore higher-dimensional golden sections and multiple online leasing. For the golden section, we define geometric-type and harmonic-type higher-dimensional golden sections, $τ^{(m)}$ and $T^{(m)}$, based on two equivalent propositions dividing the unit segment into $m$ segments. We establish equations for the geometric-type ratio $λ$ and harmonic-type basis $ω$, and investigate their algebraic characteristics. Extending the $k$-generalized Fibonacci sequence to a bi-infinite $m$-generalized Fibonacci sequence, we establish their relationship. We introduce novel concepts including generating sequences and the Fibonacci matrix, investigating its submatrices. Results such as a determinant formula involving the Fibonacci matrix demonstrate its effectiveness in studying generalized Fibonacci sequences. For online leasing, we consider multiple online leasing of $m$ pairs of skis. We design a control and balancing strategy and prove it is optimal via competitive analysis. We show the optimal strategy is precisely the $m$-dimensional harmonic-type golden section method. Since explicit solutions may not exist for dimensions higher than four, we propose a continuous relaxation problem. We explore using its relaxed solution to approximate the optimal solution or the largest real root. We propose the mantissa weighting method to solve the equation, achieving high accuracy through error analysis.
Comments83 pages, 5 figures
