On the Leibniz rule and Laplace transform for fractional derivatives

Taylor series is a useful mathematical tool when describing and constructing a function. With the series representation, some properties of fractional calculus can be revealed clearly. This paper investigates two typical applications: Lebiniz rule and Laplace transform. It is analytically shown that the commonly used Leibniz rule cannot be applied for Caputo derivative. Similarly, the well-known Laplace transform of Riemann-Liouville derivative is doubtful for n-th continuously differentiable function. By the aid of this series representation, the exact formula of Caputo Leibniz rule and the explanation of Riemann-Liouville Laplace transform are presented. Finally, three illustrative examples are revisited to confirm the obtained results

Convergence Theory of Learning Over-parameterized ResNet: A Full Characterization

ResNet structure has achieved great empirical success since its debut. Recent work established the convergence of learning over-parameterized ResNet with a scaling factor $\tau=1/L$ on the residual branch where $L$ is the network depth. However, it is not clear how learning ResNet behaves for other values of $\tau$. In this paper, we fully characterize the convergence theory of gradient descent for learning over-parameterized ResNet with different values of $\tau$. Specifically, with hiding logarithmic factor and constant coefficients, we show that for $\tau\le 1/\sqrt{L}$ gradient descent is guaranteed to converge to the global minma, and especially when $\tau\le 1/L$ the convergence is irrelevant of the network depth. Conversely, we show that for $\tau>L^{-\frac{1}{2}+c}$, the forward output grows at least with rate $L^c$ in expectation and then the learning fails because of gradient explosion for large $L$. This means the bound $\tau\le 1/\sqrt{L}$ is sharp for learning ResNet with arbitrary depth. To the best of our knowledge, this is the first work that studies learning ResNet with full range of $\tau$.Comment: 31 page

On the Timbre of the Piano Music

This paper explains clearly the conception of timbre, and analyses theoretically the reasons why the timbre of piano changes. By closely connected with teaching work, the author corrects some errors in the application of timbre and offers some good advice in mastering this concept in teaching. Key words： Frequency, ground note, harmonic, compound tone, timbre Résumé: Ce texte explique la définitin de timbre dans un langue simple et claire , et ensuite analyse , sur le plan théorique , la raison pour laquelle le timbre du piano varie . Etroitement lié à la pratique de l’enseignement , ce texte redresse quelques erreurs de la conception du timbre dans l’application , tout en donnant des conseils utiles pour bien la maîtriser dans l’enseignement . Mots-clés: fréquence , ton fondamental , harmonique , ton composé , timbre 摘要：文章圍繞音色的定義周密地進行了深入淺出地解讀，進而從理論上分析了鋼琴音色產生變化的原因。文章密切聯繫教學實際，糾正了在音色概念應用上出現的一些偏差，並針對教學中應如何把握這一概念提出了一些有益的建議。 關鍵詞：頻率；基音；泛音；複合音；音

Optimal Borehole Energy Storage Charging Strategy in a Low Carbon Space Heat System

Domestic heating is the major demand of energy systems, which can bring significant uncertainties to system operation and shrink the security margin. From this aspect, the borehole system, as an interseasonal heating storage, can effectively utilize renewable energy to provide heating to ease the adverse impact from domestic heating. This paper proposes an optimal charging strategy for borehole thermal storage by harvesting energy from photovoltaic (PV) generation in a low-carbon space heating system. The system optimizes the heat injection generated by air source heat pump in the charging seasons to charge the borehole, which provides high inlet temperature for ground source heat pump to meet space heating demand in discharging seasons. The borehole is modeled by partial differential equations, solved by the finite-element method at both 2D and 3D for volume simulation. The pattern search optimization is used to resolve the model. The case study illustrates that with the optimal charging strategies, less heat flux injection can help the borehole to reach a higher temperature so that the heating system is more efficient compared with boilers. This paper can benefit communities with seasonable borehole storage to provide clean but low-cost heating and also maximize PV penetration.</p