Discussion on the Leibniz rule and Laplace transform of fractional derivatives using series representation

ABSTRACT

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. On this basis, the Lebiniz rule and Laplace transform of fractional calculus is investigated. 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. After pointing out such problems, 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.     

Visibility graphs of fractional Wu–Baleanu time series

ABSTRACT

We study time series generated by the parametric family of fractional discrete maps introduced by Wu and Baleanu, presenting an alternative way of introducing these maps. For the values of the parameters that yield chaotic time series, we have studied the Shannon entropy of the degree distribution of the natural and horizontal visibility graphs associated to these series. In these cases, the degree distribution can be fitted with a power law. We have also compared the Shannon entropy and the exponent of the power law fitting for the different values of the fractionary exponent and the scaling factor of the model. Our results illustrate a connection between the fractionary exponent and the scaling factor of the maps, with the respect to the onset of the chaos.     

高中化学教科书中概念定义类型的分析与思考——从高考争议试题谈起

2018年高考理科综合试卷（全国I卷）单项选择题第8题存在2个正确答案，引起了争议，其实质是对教科书中"酶"的概念界定的争议。借此契机，以人教版高中化学教科书为例，分析了教科书中化学概念定义类型及特点，发现教科书概念定义类型丰富，但对同一概念的定义类型较为单一，造成某些概念内涵或外延不清晰，这是引起此次争议的重要原因；其次从概念定义类型与课程主题内容关系的角度考虑，教科书中"酶"的概念实质上是介绍作为"酶"的一类蛋白质的功能和作用，而并非揭示酶的构成，因此也不能片面地判定教科书中"酶"的概念存在错误；最后为教科书概念表达和教师教学提供了建议。     

A new algorithm for computing the multivariate Faà di Bruno’s formula

A new algorithm for computing the multivariate Faà di Bruno’s formula is provided. We use a symbolic approach based on the classical umbral calculus that turns the computation of the multivariate Faà di Bruno’s formula into a suitable multinomial expansion. We propose a MAPLE procedure whose computational times are faster compared with the ones existing in the literature. Some illustrative applications are also provided.     

不同pH条件下方程式的书写规则*

不同pH条件下,化学反应方程式书写时需要考虑体系中的主要物种。当有共轭酸碱对存在时,其平衡移动又提供H⁺或OH^-以配平方程式。详细讨论了这类方程式书写时可以出现的和不可以出现的物种。对近年来高考和化学竞赛中涉及的方程式进行了分析和探讨,对部分方程式答案加以完善,并以此探讨了方程式书写的一般规则。     

Numerical solutions of random mean square Fisher-KPP models with advection

This paper deals with the construction of numerical stable solutions of random mean square Fisher-Kolmogorov-Petrosky-Piskunov (Fisher-KPP) models with advection. The construction of the numerical scheme is performed in two stages. Firstly, a semidiscretization technique transforms the original continuous problem into a nonlinear inhomogeneous system of random differential equations. Then, by extending to the random framework, the ideas of the exponential time differencing method, a full vector discretization of the problem addresses to a random vector difference scheme. A sample approach of the random vector difference scheme, the use of properties of Metzler matrices and the logarithmic norm allow the proof of stability of the numerical solutions in the mean square sense. In spite of the computational complexity, the results are illustrated by comparing the results with a test problem where the exact solution is known.     

Two Unconditional Stable Schemes for Simulation of Heat Equation on Manifold Using DEC

To predict the heat diffusion in a given region over time, it is often necessary to find the numerical solution for heat equation. However, the computational domain of classical numerical methods are limited to flat spacetime. With the techniques of discrete differential calculus, we propose two unconditional stable numerical schemes for simulation heat equation on space manifoldand time. The analysis of their stability and error is accomplished by the use of maximum principle.     

地方高校化工专业学生创新创业能力提升模式探索

针对当前地方高校化工类专业创新创业人才供需现状,结合地方高校实际,提出了化工专业学生虚拟公司设计实训、校内仿真实训、校内外实践基地一体化创新创业能力提升实践教学模式,具有"内外结合、虚实结合、阶梯式推进"等特色。经探索,采用该模式可走出一条具有地方高校特色的实践教学新路径,可形成工科专业实践教学的示范模式,对于提升地方高校的竞争力具有积极意义。