关于Lucas多项式平方和的恒等式

利用广义Lucas多项式L n(x,y)的性质,通过构造组合和式T n(x,y;tx2),结合Bernoulli多项式的生成函数和Euler多项式的生成函数,采用分析学中的方法,得到两个有关L2n(x,y)的恒等式.并从这一结果出发,得到了两个推论,推广了相关文献的一些结果.     

生产函数形式与技术替代率

本文用数学方法推证出两个重要结论:1.生产函数之间存在单增变换式的充要条件是它们的技术替代率相同([1]中定理1):2.生产函数的技术替代率相同的充要条件是它们的替代弹性和生产扩张线均相同([2]中定理2)。在此基础上,文中阐述了生产函数形式完全由技术替代率决定。最后,以常用的C—D、CES和近几年提出的新CES和VES等生产函数为例进行了说明。     

阈红利策略下Erlang(2)风险模型罚金函数的相关结果

研究了两步保费率下Erlang(2)风险过程,给出了Gerber-Shiu折现罚函数的相关结果:即给出了罚金函数的两个微积分方程及其解或更新方程.在索赔额为指数分布条件下得到了两个与破产相关的量并计算出了相应的数值结果.     

Smarandache函数的几类相关方程的解

设φ(n),S(n)分别表示正整数n的Euler函数和Smarandache函数,利用初等的方法和技巧,依据Smarandache函数计算公式,给出k的方程φ(p~αm)=S(p~(ακ))的所有解,其中p为素数,α,m为正整数且gcd(m,p)=1,由此得到方程φ(n)=S(n~k)的所有解(n,k)进而确定了满足条件S(n)|σ(n)的全部正整数n.最后,根据莫比乌斯变换反演定理证明了方程φ(n)=∑_(d|n)S(d)仅有两个解,分别为n=2~5和n=3×2~5.     

复合欧拉函数方程φ(φ(n-φ(φ(n))))=4,6的正整数解

研究了复合欧拉函数方程φ(φ(n-φ(φ(n))))=4,6的可解性问题,其中φ(n)为欧拉函数.利用初等数论内容及计算方法分别得到了两个方程的所有正整数解.求解方法简洁有效,避免繁琐的求解过程,方法可用以求解其他类似复合欧拉函数方程.     

一个类似广义Dedekind和S_2(h,n,k)的二次均值

本文的主要目的是利用 Dirichlet L -函数的均值研究 n为奇数时 ,类似广义 Dedekind和 S2 ( h,n,k)的二次均值 ,得到了两个有趣的渐近公式 .     

一些新的数论函数及其均值公式

对于给定的自然数m,我们利用最大公约数和最小公倍数定义数论函数(m,n)和[m,n]/m.本文主要目的是研究这两个新的函数的渐近性质,利用解析方法得到这两个函数的几个渐近公式.     

与Smarandache函数有关的一个方程的解的注记

设φ(n)、S(n)分别表示正整数n的Euler函数和Smarandache函数,白海荣和廖群英在[Smarandache函数的几类相关方程的解,数学学报中文版,2019,62(2):247-254]中称方程φ(n)=∑d|nS(d)只有两个解,分别为n=2^5和n=3×2^5.本文指出,这两个数均不是此方程的解,并指出其出错原因是因为他们对M?bius反转公式的错误理解所造成的.     

利用微分从属定义的积分算子函数族的性质研究

介绍了利用微分从属关系定义的一类函数类v_k[p,A,B]和一类算子函数I_p~λ(μ,η)(z),在上述算子函数的基础上定义了两类积分算子函数F_(p,μ,η)~(n,λ)(z),G_(p,μ,η)~(n,λ)(z),利用微分从属和凸函数理论,得到了积分算子函数F_(p,μ,η)~(n,λ)(z),G_(p,μ,η)~(n,λ)(z)包含于函数类v_k[p,A,B]的条件,结论推广了部分已有的研究成果.     

广义Euler函数φ_2(n)与Euler函数φ(n)混合的一方程

令函数φ(n)为Euler函数,函数φ_e(n)为广义Euler函数,讨论了方程φ_2(φ(n))=φ(φ_2(n)的可解性,利用初等的方法给出了其成立时正整数n的形式或者n需满足的条件.     

On constant elasticity of substitution – Constant elasticity of transformation Directional Distance Functions

This paper aims to present non-linear CES (Constant Elasticity of Substitution)–CET (Constant Elasticity of Transformation) Directional Distance Functions. These measures inherit the structure of the standard Directional Distance Functions and that of the CES–CET technology. These functions allow non-parametric estimation of efficiency scores through linear programming method. Besides, the CES –CET technology gives the opportunity to explore α-returns to scale assumption for the new distance functions. The duality theory is investigated through pseudo profit, cost and revenue functions. The dual standpoint provides non-linear adjusted prices that can occur into non-linear pricing practices. An application is proposed to give an illustrative example of the primal CES–CET Directional Distance Functions.     

生产函数的判断标准

本文针对现行生产函数理论中只有几个生产函数占主导地位从而相对于变化的经济现实显得威力不足的现状 ,结合经济意义给出了一套生产函数的必要性判断标准 .在这方面目前并没有一套成熟的理论 .本文对生产函数的判断标准做了积极的尝试 ,并结合农业的生产提出一个适用于中小企业的生产函数 .     

On a class of quadratic programs

This paper considers a class of quadratic programs where the constraints ae linear and the objective is a product of two linear functions. Assuming the two linear factors to be non-negative, maximization and minimization cases are considered. Each case is analyzed with the help of a bicriteria linear program obtained by replacing the quadratic objective with the two linear functions. Global minimum (maximum) is attained at an efficient extreme point (efficient point) of the feasible set in the solution space and corresponds to an efficient extreme point (efficient point) of the feasible set in the bicriteria space. Utilizing this fact and certain other properties, two finite algorithms, including validations are given for solving the respective problems. Each of these, essentially, consists of solving a sequence of linear programs. Finally, a method is provided for relaxing the non-negativity assumption on the two linear factors of the objective function.     

中国能源资本替代的不确定性

在研究能源经济问题的可计算一般均衡模型中，都考虑能源、资本和劳动的相互替代。通过对中国1978～2000年历史数据回归分析[1]发现在常替代弹性(CES)生产函数和Cobb—Douglas(C-D)生产函数中，劳动投入对生产函数几乎没有影响。剔除劳动投入后使用能源和资本作为投入[1]重新估计了中国的生产函数，在新的生产函数中发现能源和资本间的替代具有很高的不确定性。本在[1]的基础上，设定了五个情景，包括常规发展情景(BAU)，节能情景(EC)及减少温室气体排放5％(ER5)、10％(ER10)和15％(ER15)三个情景探索节能和控制温室气体排放对中国经济的影响。研究发现，尽管存在很大的不确定性，中国需要增加大量的资本去替代能源，所以在确保可持续发展所需的能源和控制温室气体排放上中国经济面临着巨大的挑战。     

Heuristic Methods for Linear Multiplicative Programming

The linear multiplicative programming is the minimization of the product of affine functions over a polyhedral set. The problem with two affine functions reduces to a parametric linear program and can be solved efficiently. For the objective function with more than two affine functions multiplied, no efficient algorithms that solve the problem to optimality have been proposed, however Benson and Boger have proposed a heuristic algorithm that exploits links of the problem with concave minimization and multicriteria optimization. We will propose a heuristic method for the problem as well as its modification to enhance the accuracy of approximation. Computational experiments demonstrate that the method and its modification solve randomly generated problems within a few percent of relative error.     

Combining Functional Data Registration and Factor Analysis

We extend the definition of functional data registration to encompass a larger class of registration models. In contrast to traditional registration models, we allow for registered functions that have more than one primary direction of variation. The proposed Bayesian hierarchical model simultaneously registers the observed functions and estimates the two primary factors that characterize variation in the registered functions. Each registered function is assumed to be predominantly composed of a linear combination of these two primary factors, and the function-specific weights for each observation are estimated within the registration model. We show how these estimated weights can easily be used to classify functions after registration using both simulated data and a juggling dataset. Supplementary materials for this article are available online.     

Multivariate spline functions. I. Construction, properties and computation

A construction is given which allows the Hilbert space treatment of spline functions to be applied to the case of more than one variable, when the basic operator is a linear partial differential one. The particular case of the tensor product polynomial spline in two variables is then studied using a reproducing kernel, and its main properties, including the minimization ones, are deduced. A stable computational method is then given for this spline function, with certain point evaluation functionals. Finally, extensions are discussed, for more general linear functionals, for more general differential operators, and for more than two variables.     

Branch Points, M Fractions and Rational Approximants Generated by Linear Equations

Rational approximants with error terms are obtained from continuedfractions generated by linear equations. A simple method forobtaining the branch points of a function with a quasi-periodiccontinued fraction is illustrated. Continued are derived whichmatch the derivatives of a function at two or more points inthe complex plane. Rational functions that match derivativesand also take on particular values are presented.     

Profit maximization problem for Cobb–Douglas and CES production functions

Production functions are used to model the production activity of enterprises. In this article, we formulate the necessary and sufficient conditions of strict concavity for Cobb–Douglas and constant elasticity of substitution (CES) production functions. These conditions constitute the theoretical foundation for analyzing the profit maximization problem. An optimal solution is constructed in analytical form and some of its properties are described. Three approaches to solving the profit maximization problem are considered and their equivalence is established. For a Cobb–Douglas production function we investigate the dependence of the maximum profit on elasticity coefficients. A similar analysis is carried out also for the CES production function. The article presents a systematic and detailed discussion of the relevant topics. The topic is related to the investigation of innovation activity of enterprises. The theoretical results and the explicit analytical relationships provide a theoretical and algorithmic base for the “Planer” optimization software—a useful product for the analysis of the production activity of enterprises modeled by production function tools.     

An Algorithm for Solving Multicriterion Linear Programming Problems with Examples

The purpose of this paper is to develop a useful technique for solving linear programmes involving more than one objective function. Motivation for solving multicriterion linear programmes is given along with the inherent difficulty associated with obtaining a satisfactory solution set. By applying a linear programming approach for the solution of two person–zero sum games with mixed strategies, it is shown that a linear optimization problem with multiple objective functions can be formulated in this fashion in order to obtain a solution set satisfying all the requirements for an efficient solution of the problem. The solution method is then refined to take into account disparities between the magnitude of the values generated by each of the objective functions and solution preferences as determined by a decision-maker. A summary of the technique is then given along with several examples in order to demonstrate its applicability.