几何定理机器证明三十年

由于传统的兴趣和多种原因,几何定理的机器证明在自动推理的研究中占有重要的地位.自吴法发表至今30年,几何定理机器证明的研究和实践有了很大的进展.对无序几何命题而言,代数方法、数值方法均能有效地判定其真假,面积法(消点法)、搜索法更能生成其可读的证明.几何不等式机器证明的研究,由于多项式完全判别系统的建立,也有了突破.研究领域已由机器证明扩展为包括几何作图在内的一般几何问题的机器求解,并有了实际的应用.     

基于指标形式张量的微分几何定理机器证明

提出了一个基于指标形式张量的微分几何定理的机器证明算法.该算法将微分几何定理转化成带指标的张量多项式的计算问题,然后通过利用重写规则,挖掘等价条件和分次选取条件等方法大大减少了这个多项式系统的方程个数.再利用这个多项式系统本身和关于哑元的方程三角化这个多项式系统,将所得到的首项代入结论, 从而得到了该定理的机器证明.该算法不仅能够证明基于指标形式张量的微分几何定理,也可以用于张量方程的求解.     

基于多项式组主项解耦消元法的几何定理机器证明

基于多项式组主项解耦消元法 ,将几何定理的假设条件 (多项式组 PS)化为主项只含主变元的三角型多项式组 DTS,可得到定理命题成立的不含变元的非退化条件 ,即充分必要或更接近充分必要的非退化条件 .由于多项式主系数不含变元 ,已不存在 DTS多项式之间的约化问题 ,故方法有普遍意义 .文中例为西姆松定理的机器证明 .     

质点法在SGARP中的新实现

可持续发展的几何自动推理平台(SGARP)支持用户发展多种多样基于规则的机器自动推理或人机交互推理方法,但缺乏处理符号计算的模块,其解题能力仍有待加强.质点法是最近发展的继面积法之后又一个能对可构造型几何命题生成可读机器证明的具有完全性的算法.基于一种在SGARP中快捷实现符号计算功能的方法,对质点法机器证明算法进行了新的实现.新添加的质点法模块使得用户能更便捷地验证更多的几何定理,从而使SGARP能更好地满足用户学习与发展几何机器推理的需求.     

一类q-级数恒等式的新证法

本文利用一个已知的变换公式及其它基本超几何函数的求和公式，给出了一类ｑ级数恒等式的新的更简单的证明，并建立了该类恒等式的一般形式．     

关于如何利用Maple软件进行平面几何命题的验证的补充说明

文 [1 ]中通过定理 :设G是理想I的Gr bner基 ,f是k[x1 ,…xn]中的多项式 ,则f∈I的充分必要条件是f模G的范式为 0 .即通过对理想成员的判定 ,得出了验证几何命题是否成立的下述方法 :1、首先建立坐标系 ,将命题的条件和结论实行代数化 ,即转化成多项式的形式 ;2、求命题的所有条件 (用多项式表示 )所形成的理想I的Gr bner基Gs;3、求命题的结论f模Gs 的范式conc ;4、对范式进行判断 ,若范式为 0 ,说明命题结论成立 (或对 ) ;若范式不为 0 ,则说明命题结论不成立(或错 ) .但笔者认为该方法欠妥 ,还不够完善 ,有时会使我们得到错误的结论 ,…     

欧拉图与矩阵环的多项式恒等式

本文运用Swan证明Amitsur-levitzki定理所用有向路图论方法，获得了交换环上矩阵环所满足的一类新型多项式恒等式，标准多项式恒等式和Chang-Giambruno-Sehgal多项式恒等式是我们所得恒等式的特例。     

圆锥曲线的一类切线的几何画法

下面是一个关于圆的切线判定的平面几何命题 :如图1所示 ,AB是⊙O的直径 ,EB是⊙O的切线 ,直线EA交⊙O于点D ,A ,点C是线段BE的中点 ,那么 :DC是⊙O的切线 .这个命题不仅给出了圆切线的一个几何画法 .而且可引伸出圆锥曲线的一类切线的几何画法 .本文以命题的形式介绍这种方法 .图 21　椭圆切线的一个几何画法命题 1　如图 2所示 ,AB是椭圆的长轴 ,过B的直线l⊥AB ,点D是椭圆上除长轴两端点外任意一点 ,直线AD交直线l于点E ,点C是线段BE的中点 .则DC是椭圆的切线 .证明　如图 2 ,建立直角坐标系 ,设椭圆图方程是x2a2 + y2b2 =1…     

第二类Volterra型积分方程的Chebyshev-Legendre谱配置方法

提出了一种新的求解第二类线性Volterra型积分方程的Chebyshev谱配置方法.该方法分别对方程中积分部分的核函数和未知函数在Chebyshev-Gauss-Lobatto点上进行插值,通过Chebyshev-Legendre变换,把插值多项式表示成Legendre级数形式,从而将积分转换为内积的形式,再利用Legendre多项式的正交性进行计算.利用Chebyshev插值算子在不带权范数意义下的逼近结果,对该方法在理论上给出了L∞范数意义下的误差估计,并通过数值算例验证了算法的有效性和理论分析的正确性.     

三角形旁切圆的几个几何恒等式

关于三角形的内切圆有这样一个几何恒等式:引理1[1] 设I是△ABC的内切圆的圆心,则下列等式恒成立:IA2/AB·AC+IB2/BA·BC+IC2/CA·CB(1)该命题的证明见文[1].在文[1]中作者巧妙的运用了面积证法从而得到引理1.试想,将引理1中的“内切圆”推广到“旁切圆”,是否仍有类似相关的几何恒等式成立?于是得到下述命题:     

CONJECTURING IN OPEN GEOMETRIC SITUATIONS USING DYNAMIC GEOMETRY: AN EXPLORATORY CLASSROOM EXPERIMENT

In this paper I describe a classroom teaching experiment carried out with a class of Year 10 students. This experiment was twofold. On the one hand it was aimed at developing and trying out a new mode of working in the classroom, taking into account the possibilities offered by dynamic geometry software as support in the conjecturing and proving process in geometry. On the other hand, from the research point of view, it provided the possibility of testing out in the classroom a theoretical model describing and interpreting students’ use of Cabri-Géomètre in open geometric problems, with a particular focus on dragging. The main findings relate to the evolution in the use of dragging in Cabri and the production of rich conjectures, which can provide the basis for development and evolution towards the proving process.     

The method of fundamental solutions with eigenfunction expansion method for nonhomogeneous diffusion equation

In this article we describe a numerical method to solve a nonhomogeneous diffusion equation with arbitrary geometry by combining the method of fundamental solutions (MFS), the method of particular solutions (MPS), and the eigenfunction expansion method (EEM). This forms a meshless numerical scheme of the MFS‐MPS‐EEM model to solve nonhomogeneous diffusion equations with time‐independent source terms and boundary conditions for any time and any shape. Nonhomogeneous diffusion equation with complex domain can be separated into a Poisson equation and a homogeneous diffusion equation using this model. The Poisson equation is solved by the MFS‐MPS model, in which the compactly supported radial basis functions are adopted for the MPS. On the other hand, utilizing the EEM the diffusion equation is first translated to a Helmholtz equation, which is then solved by the MFS together with the technique of the singular value decomposition (SVD). Since the present meshless method does not need mesh generation, nodal connectivity, or numerical integration, the computational effort and memory storage required are minimal as compared with other numerical schemes. Test results for two 2D diffusion problems show good comparability with the analytical solutions. The proposed algorithm is then extended to solve a problem with irregular domain and the results compare very well with solutions of a finite element scheme. Therefore, the present scheme has been proved to be very promising as a meshfree numerical method to solve nonhomogeneous diffusion equations with time‐independent source terms of any time frame, and for any arbitrary geometry. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Challenging Theorem Provers with Mathematical Olympiad Problems in Solid Geometry

Mathematical Olympiad problems can be used to evaluate the performance of theorem provers. In the encyclopedia (Wu et al. in The Dictionary of International Mathematical Olympiads, Volume of Geometry. Hebei Children Press, Shijiazhuang, 2012) there are 207 Mathematical Olympiad contest problems in solid geometry collected around the world and ranging over one century, among which 97 problems can be used to test algebraic provers of equality type. Three general-purpose theorem proving methods are used in the test: the characteristic set method, the Gröbner basis method, and the vector algebra method. 91 out of the 97 problems are proved by the provers, and some contest problems are found to need additional specification to be correct. The proving efficiency and geometric interpretability of the additional non-degeneracy conditions for the 91 problems by the three provers are compared.     

Proving Theorems in Elementary Geometry with Clifford Algebraic Method

本文结合吴方法及平面几何的Ｃｌｉｆｏｒｄ代数表示，提出了几何定理机器证明的一种完备的方法．用这种方法证明定理时，三角化的过程及证明的过程通常较以前的方法更简短而且它们是可以几何解释的．     

On Boundary Stability of Wave Equations with Variable Coefficients

In this paper, we consider the boundary stabilization of the wave equation with variable coefficients by Riemmannian geometry method subject to a different geometric condition which is motivated by the geometric multiplier identities. Several (multiplier) identities (inequalities) which have been built for constant wave equation by Kormornik and Zuazua are generalized to the variable coefficient case by some computational techniques in Riemmannian geometry, so that the precise estimates on the exponential decay rate are derived from those inequalitities. Also, the exponential decay for the solutions of semilinear wave equation with variable coefficients is obtained under natural growth and sign assumptions on the nonlinearity. Our method is rather general and can be adapted to other evolution systems with variable coefficients (e.g. elasticity plates) as well.     

High school students’ perceptions of geometrical proofs proving and refuting geometrical claims of the ‘for every … ’ and ‘there exists’ type

Proving and refuting mathematical claims constitute a significant element in the development of deductive thinking. These issues are mainly studied during geometry lessons and very little (if at all) in lessons of other mathematical disciplines. This study deals with high school students’ perceptions of proofs in the geometry. The study explores whether students know when to use a deductive proof and when an example is sufficient for proving or refuting geometrical claims. The findings indicate that in cases of simple claims, the students corroborate them by using a deductive proof. However, when the claim is more complex, the students tend to present both a proof and an example. Moreover, they are unsure whether using an example can constitute a method for proving a mathematical claim, believing that in mathematics everything must be proven. They believe that examples are used merely for illustration purposes rather than as a means of convincing. The research conclusions support the need for deepening and developing the students’ distinction between cases where examples are insufficient and cases where an example is sufficient for proving a claim.     

On the Unavoidable Uncertainty of Truth in Dynamic Geometry Proving

The aim of this note is to discuss some issues posed by the emergency of universal interfaces able to decide on the truth of geometric statements. More specifically, we consider a recent GeoGebra module allowing general users to verify standard geometric theorems. Working with this module in the context of Varignon’s theorem, we were driven—by the characteristics of the GeoGebra interface—to perform a quite detailed study of the very diverse fate of attempting to automatically prove this statement, when using two different construction procedures. We highlight the relevance—for the theorem proving output—of expression power of the dynamic geometry interface, and we show that the algorithm deciding about the truth of some—even quite simple—statements can fall into a not true and not false situation, providing a source of confusion for a standard user and an interesting benchmark for geometers interested in discovering new geometric facts.