Some Identities of the Higher-Order Type 2 Bernoulli Numbers and Polynomials of the Second Kind

We introduce the higher-order type 2 Bernoulli numbers and polynomials of the second kind.In this paper,we investigate some identities and properties for them in connection with central factorial numbers of the second kind and the higher-order type 2 Bernoulli polynomials.We give some relations between the higher-order type 2 Bernoulli numbers of the second kind and their conjugates.

Quality Analysis and High-Performance Liquid Chromatographic Fingerprint Analysis of New Cultivated Kind of <i>Lonicerae japonicae</i>Flos “Hua Jin 6” from Different Harvest Times

Lonicerae japonicaeFlos (LJF) is widely used in traditional Chinese medicines for the treatment of various diseases, which is now in great demand every year and has a broad development prospect. However, the flowering phase of common LJF varieties is so short, which seriously restricts the development of LJF industry. As a new cultivated kind of Lonicerae japonicae Flos, “Hua Jin 6” has characteristics in long flowering phase and conveniently picking, which makes it have a broad development prospect. The aim of this study is to provide scientific guidance for its suitable harvest period by measuring yield and quality of “Hua Jin 6” from different harvest time. Studies show that flower size had a slowly rising trend from the first day to the seventh day, and then slowly declined or kept stable. There were no significant differences of total phenolic acid contents in different samples from different days, but contents of total flavonoids were on the rise and up to maximum in the ninth day. The contents of total iridoids had an increasing tendency from the first day to the fifth day and then kept relatively stable in other days. We demonstrated that the quality of “Hua Jin 6” is relatively stable and suitable for harvesting in all flower buds white stage in term of HPLC fingerprints. Our findings can make it possible to select the suitable time for different harvest purpose.

Topics of Truth,Kindness and Aesthetics in Chinese Education1

The ideal goal to cultivate creative talents who "seek truth","strive for kindness" and "adore aesthetics " and are equipped with "perfect personality" should be achieved,which is expected to reach brand-new circumstances of "humans development and construction of completeness" and must and should explore "Truth","Kindness" and "Aesthetics" in Chinese education.This article is meant to explore "truth" of Chinese(language,article,literature) in the intellectual education from cognitive science and to promote students to get into quality of exploration;they should explore "kindness" in the moral education in Chinese(language,article,literature) from ethics,which is to urge wakefulness of students' moral consciousness;in addition,"aesthetics" in Chinese(language,article,literature) from aesthetic education is to spur students to construct free soul.

The Research of the Relationship Between Yangming’s Yixue and His Xinxue

Yangming’s Yixue had been developed several times in his life since originated in Longchang.Yangming’s Yixue and his Xinxue are both sides of the same corn.The Yangming’s Yixue can be roughly divided into two periods.There were the early and middle period of Yangming’s Yixue,which is intended to explain the principles of Yi Zhuan(《易传》)with the theory of Xinxue.The finish of Yangming’s Yixue was a second stage in Yangming’s later years.From the perspective of form and content,there are differences that cannot be ignored between Yangming’s early and middle period and his later years.In the first stage of Yangming’s study,the“Virtuousness”(贞)in Book of Changes is the principle and nature in university.In the later period,Yangming had not focused on interpreting the“Yi Zhuan”,but on the interpretation of ZhouYi’s(《周易》)principles to supplement the connotation of the propositions of Xinxue.This article intends to understand the relationship between Yangming’s Yixue and his Xinxue by dividing the different stages of Yangming’s Yixue firstly.Then by focusing on the influence of Yangming’s later years’Yixue on his Xinxue,we can observe the relation between his Yixue and Xinxue.

A “Hard to Die” Series Expansion and Lucas Polynomials of the Second Kind

We show how to use the Lucas polynomials of the second kind in the solution of a homogeneous linear differential system with constant coefficients, avoiding the Jordan canonical form for the relevant matrix.

The Chapman-Richards Distribution and its Relationship to the Generalized Beta

Background: The Chapman-Richards distribution is developed as a special case of the equilibrium solution to the McKendrick-Von Foerster equation. The Chapman-Richards distribution incorporates the vital rate assumptions of the Chapman-Richards growth function, constant mortality and recruitment into the mathematical form of the distribution. Therefore, unlike 'assumed' distribution models, it is intrinsically linked with the underlying vital rates for the forest area under consideration. Methods: It is shown that the Chapman-Richards distribution can be recast as a subset of the generalized beta distribution of the first kind, a rich family of assumed probability distribution models with known properties. These known properties for the generalized beta are then immediately available for the Chapman-Richards distribution, such as the form of the compatible basal area-size distribution. A simple two-stage procedure is proposed for the estimation of the model parameters and simulation experiments are conducted to validate the procedure for four different possible distribution shapes. Results: The simulations explore the efficacy of the two-stage estimation procedure;these cover the estimation of the growth equation and mortality-recruitment derives from the equilibrium assumption. The parameter estimates are shown to depend on both the sample size and the amount of noise imparted to the synthetic measurements. The results vary somewhat by distribution shape, with the smaller, noisier samples providing less reliable estimates of the vital rates and final distribution forms. Conclusions: The Chapman-Richards distribution in its original form, or recast as a generalized beta form, presents a potentially useful model integrating vital rates and stand diameters into a flexible family of resultant distributions shapes. The data requirements are modest, and parameter estimation is straightforward provided the minimal recommended sample sizes are obtained.

Convergence Phenomenon with Fourier Series of tg(x2)and Alike

The Fourier series of the 2π-periodic functions tg(x2)and 1sin(x)and some of their relatives (first of their integrals) are investigated and illustrated with respect to their convergence. These functions are Generalized functions and the convergence is weak convergence in the sense of the convergence of continuous linear functionals defining them. The figures show that the approximations of the Fourier series possess oscillations around the function which they represent in a broad band embedding them. This is some analogue to the Gibbs phenomenon. A modification of Fourier series by expansion in powers cosn(x)for the symmetric part of functions and sin(x)cosn−1(x)for the antisymmetric part (analogous to Taylor series) is discussed and illustrated by examples. The Fourier series and their convergence behavior are illustrated also for some 2π-periodic delta-function-like sequences connected with the Poisson theorem showing non-vanishing oscillations around the singularities similar to the Gibbs phenomenon in the neighborhood of discontinuities of functions. .

Partial Bell Polynomials, Falling and Rising Factorials, Stirling Numbers, and Combinatorial Identities

In the paper,the authors collect,discuss,and find out several connections,equivalences,closed-form formulas,and combinatorial identities concerning partial Bell polynomials,falling factorials,rising factorials,extended binomial coefficients,and the Stirling numbers of the first and second kinds.These results are new,interesting,important,useful,and applicable in combinatorial number theory.

A KIND OF APPROXIMATION OPERATORS AND ITS APPLICATION

In this paper we introduce a kind of kernels called δ-kernels.Based on them linear opera- tors from the continuous mapping space to R^m are constructed.We give an important appli- cation in Optimization after discussing the properties of these operators.

Some Properties of Degenerate r-Dowling Polynomials and Numbers of the Second Kind

The generating functions of special numbers and polynomials have various applications in many fields as well as mathematics and physics.In recent years,some mathematicians have studied degenerate version of them and obtained many interesting results.With this in mind,in this paper,we introduce the degenerate r-Dowling polynomials and numbers associated with the degenerate r-Whitney numbers of the second kind.We derive many interesting properties and identities for them including generating functions,Dobinski-like formula,integral representations,recurrence relations,differential equation and various explicit expressions.In addition,we explore some expressions for them that can be derived from repeated applications of certain operators to the exponential functions,the derivatives of them and some identities involving them.

Determinantal Expressions and Recursive Relations for the Bessel Zeta Function and for a Sequence Originating from a Series Expansion of the Power of Modified Bessel Function of the First Kind

In the paper,by virtue of a general formula for any derivative of the ratio of two differentiable functions,with the aid of a recursive property of the Hessenberg determinants,the authors establish determinantal expressions and recursive relations for the Bessel zeta function and for a sequence originating from a series expansion of the power of modified Bessel function of the first kind.

The description of a new kind and three new types of medusae nematocysts in the Xiamen Harbour, China

A new kind of nematocyst, heteromorphore, and three new types of nematocysts, microbasic spino-mastigophore, microbasic undulata-mastigophore, microbasic laevis-mastigophore, are described from medusae of the Xiamen Harbour and discussed by comparison with the results of other research workers.

To be kind

My dear friends,have you ever read the story'Snow White and the Seven Dwarfs'?I first read it when I was seven.And now I remember it just as well as if I had read it yesterday.I think everyone is so familiar with this story.This story is the one that left the

‘Disciplining Oneself’and‘Disciplining Others’in Social Life:On the Contemporary Significance of Confucius’s Principle of Loyalty and Reciprocity

In a moral sense,there is an essential di erence between“disciplining oneself”and“disciplining others”in that the former means one setting rules for and giving orders to oneself in regard to one’s own conduct,which are unconditionally compulsory,while the latter is exible in the sense that one’s rules and orders for oneself are not compulsory to others and cannot prohibit others from doing something.In principle,Confucius’s sayings,“do not impose upon others that which you would not wish for yourself”and“establish others in seeking to establish yourself and promote others in s eeking to promote yourself”have epitomized the line of demarcation between them.His doctrine of“not imposing upon others that which you would not wish for yourself,”though based on self-preservation and self-interest,represents a standard of conduct of highly universal value.If the overwhelming majority of a society follows it as their standard of conduct,refraining from harming others in order to protect oneself from harm,then the whole society will endure little,if not no,strife and struggle.

Integral Operator Solving Process of the Boundary Value Problem of Abstract Kinetic Equation with the First Kind of Critical Parameter and Generalized Periodic Boundary Conditions

在这份报纸，边界的概念与第一种批评参数珍视抽象运动方程的问题 0 并且概括周期的边界条件在由函数组成，向量在一个一般 Banach 空格珍视的一个 Lebesgue 空格被介绍，然后描述这些的解决方案由 Volterra 的抽象线性不可分的操作符的抽象边界值问题类型。我们把这个过程称为解决过程的不可分的操作员。

高阶Apostol-Bernoulli函数的一些恒等式

Apostol-Bernoulli函数及其推广形式是研究某些特殊函数的基础,在特殊函数理论研究中占有极其重要的地位。研究了高阶Apostol-Bernoulli函数的性质,利用高阶Apostol-Bernoulli函数的指数生成函数与高阶Apostol-Euler函数的指数生成函数的关系,给出了Apostol-Bernoulli函数的两个表述式及一个推论。

ON NEWMAN-TYPE RATIONAL INTERPOLATION TO |x| AT THE CHEBYSHEV NODES OF THE SECOND KIND

Recently Brutman and Passow considered Newman-type rational interpolation to ｜x｜ induced by arbitrary set of symmetric nodes in [-1,1] and gave the general estimation of the approximation error.By their methods one could establish the exact order of approximation for some special nodes. In the present paper we consider the special case where the interpolation nodes are the zeros of the Chebyshev polynomial of the second kind and prove that in this case the exact order of approximation is O（1/n｜nn）

Landweber迭代正则解的最优渐近阶估计

应用紧算子的奇异系统得到了 Landweber迭代正则解的级数表达式 .通过先验选取正则参数 。

ON THE COEFFICIENTS OF DIFFERENTIATED EXPANSIONS AND DERIVATIVES OF CHEBYSHEV POLYNOMIALS OF THE THIRD AND FOURTH KINDS

Two new analytical formulae expressing explicitly the derivatives of Chebyshev polynomials of the third and fourth kinds of any degree and of any order in terms of Chebyshev polynomials of the third and fourth kinds themselves are proved. Two other explicit formulae which express the third and fourth kinds Chebyshev expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of their original expansion coefficients are also given. Two new reduction formulae for summing some terminating hypergeometric functions of unit argument are deduced. As an application of how to use Chebyshev polynomials of the third and fourth kinds for solving high-order boundary value problems, two spectral Galerkin numerical solutions of a special linear twelfth-order boundary value problem are given.