论文标题
基于数字扩展的集成定理
A theorem on integration based on the digital expansion
论文作者
论文摘要
实际数字的二进制辐射扩展可用于编码任何一系列硬币投掷的结果，这一事实在数字理论，测量理论和统计物理学之间提供了有趣的联系。受这一事实的启发，为单个变量的可区分函数的确定积分建立了一般结果，该功能允许任何此类积分都可以按双系列编写。该定理可以直接应用于各种物理兴趣的积分，并得出实数和实现功能的新系列扩展。我们将定理应用于在古典哈密顿系统的一个维度中的运动方程式的整合，重点是分析非线性摆。
The binary radix expansion of a real number can be used to code the outcome of any series of coin tosses, a fact that provides an intriguing link between number theory, measure theory and statistical physics. Inspired by this fact, a general result is established for the definite integral of a differentiable function of a single variable that allows any such integral to be exactly written in terms of a double series. The theorem can be directly applied to a wide variety of integrals of physical interest and to derive new series expansions of real numbers and real-valued functions. We apply the theorem to the integration of the equation of motion in one dimension of classical Hamiltonian systems, focusing in the analysis of the nonlinear pendulum.
