Wavelength-dependent mathematical modelling of the differential energy change of a photon has been performed inside a proposed hypothetical optical medium.The existence of this medium demands certain mathematical cons...Wavelength-dependent mathematical modelling of the differential energy change of a photon has been performed inside a proposed hypothetical optical medium.The existence of this medium demands certain mathematical constraints,which have been derived in detail.Using reverse modelling,a medium satisfying the derived conditions is proven to store energy as the photon propagates from the entry to exit point.A single photon with a given intensity is considered in the analysis and hypothesized to possess a definite non-zero probability of maintaining its energy and velocity functions analytic inside the proposed optical medium,despite scattering,absorption,fluorescence,heat generation,and other nonlinear mechanisms.The energy and velocity functions are thus singly and doubly differentiable with respect to wavelength.The solution of the resulting second-order differential equation in two variables proves that energy storage or energy flotation occurs inside a medium with a refractive index satisfying the described mathematical constraints.The minimum-value-normalized refractive index profiles of the modelled optical medium for transformed wavelengths both inside the medium and for vacuum have been derived.Mathematical proofs,design equations,and detailed numerical analyses are presented in the paper.展开更多
微分方程的计算求解在计算机工程上有重要的理论意义和应用价值。针对传统数值解法计算复杂度高、解的形式离散等问题,本文基于微分方程的回归方程观点与解法,应用统计回归方法求解二阶常微分方程,并给出基于中心支持向量机(proximal su...微分方程的计算求解在计算机工程上有重要的理论意义和应用价值。针对传统数值解法计算复杂度高、解的形式离散等问题,本文基于微分方程的回归方程观点与解法,应用统计回归方法求解二阶常微分方程,并给出基于中心支持向量机(proximal support vector machine,P-SVM)在常微分方程的初值和边值问题上的近似解求法。通过在目标优化函数中添加偏置项,构建P-SVM回归模型,从而避免大规模求解线性方程组,得到结构简洁的最优解表达式。模型通过最小化训练样本点的均方误差和,在保证精度的同时,有效提高了近似解的计算速度。此外,形式简洁固定的解析解表达式也便于在实际应用中进行定性分析和性质研究。数值试验结果验证了P-SVM方法是一种高效可行的常微分方程求解方法。展开更多
文摘Wavelength-dependent mathematical modelling of the differential energy change of a photon has been performed inside a proposed hypothetical optical medium.The existence of this medium demands certain mathematical constraints,which have been derived in detail.Using reverse modelling,a medium satisfying the derived conditions is proven to store energy as the photon propagates from the entry to exit point.A single photon with a given intensity is considered in the analysis and hypothesized to possess a definite non-zero probability of maintaining its energy and velocity functions analytic inside the proposed optical medium,despite scattering,absorption,fluorescence,heat generation,and other nonlinear mechanisms.The energy and velocity functions are thus singly and doubly differentiable with respect to wavelength.The solution of the resulting second-order differential equation in two variables proves that energy storage or energy flotation occurs inside a medium with a refractive index satisfying the described mathematical constraints.The minimum-value-normalized refractive index profiles of the modelled optical medium for transformed wavelengths both inside the medium and for vacuum have been derived.Mathematical proofs,design equations,and detailed numerical analyses are presented in the paper.
文摘微分方程的计算求解在计算机工程上有重要的理论意义和应用价值。针对传统数值解法计算复杂度高、解的形式离散等问题,本文基于微分方程的回归方程观点与解法,应用统计回归方法求解二阶常微分方程,并给出基于中心支持向量机(proximal support vector machine,P-SVM)在常微分方程的初值和边值问题上的近似解求法。通过在目标优化函数中添加偏置项,构建P-SVM回归模型,从而避免大规模求解线性方程组,得到结构简洁的最优解表达式。模型通过最小化训练样本点的均方误差和,在保证精度的同时,有效提高了近似解的计算速度。此外,形式简洁固定的解析解表达式也便于在实际应用中进行定性分析和性质研究。数值试验结果验证了P-SVM方法是一种高效可行的常微分方程求解方法。
