一类根式不等式的有理化算法与机器证明

文中讨论了一类根式不等式的有理等价问题.证明了这类根式不等式可等价转化为一组有理不等式.建立了一个算法RFD,并用Maple编程实现.对一个给定的这类根式不等式,RFD可自动快速地产生一组有理等价不等式.将RFD算法和差分代换方法相结合,给出了一大类具有相当难度的几何不等式的机器证明.此前该课题仅有的工作是杨路关于二次根式的结果.     

一类构造性几何不等式的机器证明

阐述了一个基于胞腔分解的不等式证明算法．据此算法编制的Maple通用程序能有效地处理含有根式的不等式型定理,对于Bottema等所著《几何不等式》一书中的大部分不等式定理的验证尤其高效．     

常用基本不等式的机器证明

不等式机器证明问题是智能系统领域的难点和热点问题.借助不等式证明软件BOTTEMA,对若干常用的基本不等式成功地实现了机器证明,包括算术、几何与调和平均不等式、排序不等式、Chebyshev不等式、Bernoulli不等式、三角形不等式及Jensen不等式等.所论不等式含有的变元个数是一个不确定的变量,属于Tarski模型外的不等式类型.机器证明得出的结论有时可能是已知结果的推广,其方法本身对同类不等式有示范性,更多的例子表明了该算法和软件的有效性.     

一阶谓词演算定理机器证明的余式方法

本文将一阶谓词泻算的证明转化为代数族的计算，从而获得了代数化的Ｈｅｒｂｒａｎｄ过程，双通过一阶多项式间的求余运算，给出了余式方法并评理服它的完备性。同时，我们证明了归结原理是余式方法的一种特例。     

利用变上限积分构造辅助函数证明一些积分不等式

根据积分不等式自身的特点选择适当的积分区间，利用变上限积分在相应区间上构造出合适的辅助函数，由单调与可导的关系以及积分中值定理或拉格朗日中值定理的巧妙应用确定出该辅助函数的单调性，再由单调性定义得到一些积分不等式并加以证明。     

一类三角形几何不等式的自动证明

讨论了一类只含三角函数的三角形几何不等式的自动证明问题。运用代数方法将其有理化,在不新增加根式的条件下将问题转换为一个二元多项式不等式的证明,设计的基于胞腔分解和实根分离的算法实现了二元多项式不等式的自动证明,输出的证明过程可以手工验证或借助一些数学软件进行理解。实验表明上述算法对一大批具有相当难度,特别是关于三角函数的几何不等式十分高效,并且能够解决三角形内角的任意有理倍数函数的不等式机器证明问题。     

基于三角不等式判定和局部策略的高效邻域覆盖模型

邻域覆盖模型由于其原理简单以及对复杂数据具有较好的处理能力,在分类任务中得到了广泛应用.然而,邻域覆盖模型普遍存在运行效率较低的问题,且缺乏相关研究工作.为解决此问题,在传统邻域覆盖模型中引入距离间的三角不等式关系以提升构建邻域的效率,同时引入局部策略,定义了局部邻域覆盖以提升构建邻域覆盖的效率.为提升运行效率,从两个...     

基于Tableau的定理机器证明系统TableauTAP

使用SWI-PROLOG语言在微机上设计实现了基于tableau的定理证明系统TabIeauTAP。该系统可以证明不含等词的经典逻辑公式童耋譬逻辑公式，通过预处理自动生成tableau规则，因此容易对其功能进行扩展。应用该系统对TPTP的400个逻辑问题进行证明，实验结果表明，TableauTAP在时间和空间效率上都是比较高的。     

基于拼凑替换的定理机器证明的研究与实现

机器定理证明可以避免人工证明容易出现的低级错误,是人工智能的重要方面,有广泛的应用前景;函数式程序设计的设计思想更加接近于数学,在定理证明方面有天然优势.人们证明逻辑推理的过程通过函数式程序实现,并将其证明的步骤显示出来,采用了逻辑推理机器证明.通过思考人脑在证明定理时的思考方式,给出了一个简单易懂的机器证明的方式.首先将证明的已知和结论形式化,将已知设为start,结果为end,已经证明的公理就是road,那么证明的过程就是从start沿着road到达end的过程.实验表明,逻辑证明通过函数式程序实现,达到了预期目的.     

Two general integral inequalities and their applications to stability analysis for systems with time‐varying delay

Integral inequalities have been widely used in stability analysis for systems with time‐varying delay because they directly produce bounds for integral terms with respect to quadratic functions. This paper presents two general integral inequalities from which almost all of the existing integral inequalities can be obtained, such as Jensen inequality, the Wirtinger‐based inequality, the Bessel–Legendre inequality, the Wirtinger‐based double integral inequality, and the auxiliary function‐based integral inequalities. Based on orthogonal polynomials defined in different inner spaces, various concrete single/multiple integral inequalities are obtained. They can produce more accurate bounds with more orthogonal polynomials considered. To show the effectiveness of the new inequalities, their applications to stability analysis for systems with time‐varying delay are demonstrated with two numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

General Barnes-Godunova-Levin type inequalities for Sugeno integral

Integral inequalities play important roles in classical probability and measure theory. Non-additive measure is a generalization of additive probability measure. Sugeno’s integral is a useful tool in several theoretical and applied statistics which has been built on non-additive measure. For instance, in decision theory, the Sugeno integral is a median, which is indeed a qualitative counterpart to the averaging operation underlying expected utility. In this paper, Barnes-Godunova-Levin type inequalities for the Sugeno integral on abstract spaces are studied in a rather general form and, for this, we introduce some new technics for the treatment of concave functions in the Sugeno integration context. Also, several examples are given to illustrate the validity of this inequality. Moreover, a strengthened version of Barnes-Godunova-Levin type inequality for Sugeno integrals on a real interval based on a binary operation ★ is presented.     

Stability analysis of systems with time-varying delay via relaxed integral inequalities

This paper investigates the stability of linear systems with a time-varying delay. The key problem concerned is how to effectively estimate single integral term with time-varying delay information appearing in the derivative of Lyapunov–Krasovskii functional. Two novel integral inequalities are developed in this paper for this estimation task. Compared with the frequently used inequalities based on the combination of Wirtinger-based inequality (or Auxiliary function-based inequality) and reciprocally convex lemma, the proposed ones can provide smaller bounding gap without requiring any extra slack matrix. Four stability criteria are established by applying those inequalities. Based on three numerical examples, the advantages of the proposed inequalities are illustrated through the comparison of maximal admissible delay bounds provided by different criteria.     

Event‐triggered H∞ control for a class of nonlinear networked control systems using novel integral inequalities

This paper is concerned with event‐triggered H_∞ control for a class of nonlinear networked control systems. An event‐triggered transmission scheme is introduced to select ‘necessary’ sampled data packets to be transmitted so that precious communication resources can be saved significantly. Under the event‐triggered transmission scheme, the closed‐loop system is modeled as a system with an interval time‐varying delay. Two novel integral inequalities are established to provide a tight estimation on the derivative of the Lyapunov–Krasovskii functional. As a result, a novel sufficient condition on the existence of desired event‐triggered H_∞ controllers is derived in terms of solutions to a set of linear matrix inequalities. No parameters need to be tuned when controllers are designed. The proposed method is then applied to the robust stabilization of a class of nonlinear networked control systems, and some linear matrix inequality‐based conditions are formulated to design both event‐triggered and time‐triggered H_∞ controllers. Finally, two numerical examples are given to demonstrate the effectiveness of the proposed method. Copyright © 2016 John Wiley & Sons, Ltd.     

Intuitionistic fuzzy Choquet integral operator for multi-criteria decision making

For the real decision making problems, most criteria have inter-dependent or interactive characteristics so that it is not suitable for us to aggregate them by traditional aggregation operators based on additive measures. Thus, to approximate the human subjective decision making process, it would be more suitable to apply fuzzy measures, where it is not necessary to assume additivity and independence among decision making criteria. In this paper, an intuitionistic fuzzy Choquet integral is proposed for multiple criteria decision making, where interactions phenomena among the decision making criteria are considered. First, we introduced two operational laws on intuitionistic fuzzy values. Then, based on these operational laws, intuitionistic fuzzy Choquet integral operator is proposed. Moreover, some of its properties are investigated. It is shown that the intuitionistic fuzzy Choquet integral operator can be represented by some special t-norms and t-conorms, and it is also a generalization of the intuitionistic fuzzy OWA operator and intuitionistic fuzzy weighted averaging operator. Further, the procedure and algorithm of multi-criteria decision making based on intuitionistic fuzzy Choquet integral operator is given under uncertain environment. Finally, a practical example is provided to illustrate the developed approaches.     

Application of Sinc-collocation method for solving a class of nonlinear Fredholm integral equations

In this paper, the numerical solution of nonlinear Fredholm integral equations of the second kind is considered by two methods. The methods are developed by means of the Sinc approximation with the single exponential (SE) and double exponential (DE) transformations. These numerical methods combine a Sinc collocation method with the Newton iterative process that involves solving a nonlinear system of equations. We provide an error analysis for the methods. So far approximate solutions with polynomial convergence have been reported for this equation. These methods improve conventional results and achieve exponential convergence. Some numerical examples are given to confirm the accuracy and ease of implementation of the methods.     

多决策树的模糊积分融合在银行信贷管理系统中的应用

为了提高基于数据挖掘的商业银行信贷管理系统的信贷风险评估水平,将多决策树的Choquet模糊积分融合(MTCFF)模型应用到银行信贷管理系统中。基本思想是采用决策树在已知类型的客户数据上进行挖掘,按照决策树剪枝程度不同形成不同的决策树并产生规则,利用所生成的不同决策树的规则,对未知类型的客户数据进行分类,然后让Choquet模糊积分对多棵决策树的分类结果进行融合,形成最优判断。采用UCI数据库中German客户信用卡数据集进行验证,实验证明Choquet模糊积分的非线性融合效果优于单棵决策树的分类效果,也优于其他线性融合方法,并且Choquet模糊积分要优于Sugeno模糊积分。     

B-polynomial multiwavelets approach for the solution of Abel's integral equation

A numerical method for solving Abel's integral equation as singular Volterra integral equations is presented. The method is based upon Bernstein polynomial (B-polynomial) multiwavelet basis approximations. The properties of B-polynomial multiwavelets are first presented. These properties are then utilized to reduce the singular Volterra integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.