Constructive struggle in geometry classrooms

The purpose of these notes is to provide examples of large-scale geometry problems that have the potential to promote constructive struggling in high-school geometry classrooms. These problems are suitable for being explored in a dynamic geometry environment. The notes also contain brief sketches of solutions to the two problems.     

Computerunterstütztes Lösen offener raumgeometrischer Aufgaben

Considering the fact that solid geometry has been a neglected subject in mathematics teaching at lower and middle secondary level, there has been almost no chance to develop a “culture” of open, solid geometry problem solving. Suitable software tools for spatial representations, construction and calculation tasks can support the students in developing and solving open problems in solid geometry designed in line with the content of the conventional geometry curriculum. The article presents problems of this kind and explains the computer-aided problem solving processes. Furthermore, initial results of evaluations of practical lessons including computerized treatment of selected open problems are reported. Finally, the general significance is discussed of introducing the computer as a tool in spatial geometry teaching as well as the basic problems involved in an evaluation of computer-assisted teaching in this context, and further development of computer-aided, open problem solving in spatial geometry.     

抛物线的一些性质

近年,在研究射影几何在二次曲线上的运用中,发现有些平面几何问题用射影几何研究更自然、条理更清楚,而用平面几何方法处理则有难度.将二次曲线中的抛物线放在拓广平面上,借助射影几何中的Pascal定理、Steiner定理,给出了抛物线一些有趣的性质.     

Geometrical formulation of Hamiltonian boundary value problems

A geometrical formulation of nonlinear Hamiltonian boundary value problems is presented. It involves a distance geometry which is a generalization of hypercircle geometry for linear boundary value problems. The connection with dual extremum principles is also exhibited.     

A Stochastic/Perturbation Global Optimization Algorithm for Distance Geometry Problems

We present a new global optimization approach for solving exactly or inexactly constrained distance geometry problems. Distance geometry problems are concerned with determining spatial structures from measurements of internal distances. They arise in the structural interpretation of nuclear magnetic resonance data and in the prediction of protein structure. These problems can be naturally formulated as global optimization problems which generally are large and difficult. The global optimization method that we present is related to our previous stochastic/perturbation global optimization methods for finding minimum energy configurations, but has several key differences that are important to its success. Our computational results show that the method readily solves a set of artificial problems introduced by Moré and Wu that have up to 343 atoms. On a set of considerably more difficult protein fragment problems introduced by Hendrickson, the method solves all the problems with up to 377 atoms exactly, and finds nearly exact solution for all the remaining problems which have up to 777 atoms. These preliminary results indicate that this approach has very good promise for helping to solve distance geometry problems.     

共线点与共点线问题的研究

利用向量法、坐标法、仿射变换以及射影几何中的德萨格定理、帕斯卡定理和布利安桑定理,解决初等几何中的共线点和共点线问题.     

Validation of Solutions of Construction Problems in Dynamic Geometry Environments

This paper discusses issues concerning the validation of solutions of construction problems in Dynamic Geometry Environments (DGEs) as compared to classic paper-and-pencil Euclidean geometry settings. We begin by comparing the validation criteria usually associated with solutions of construction problems in the two geometry worlds – the ‘drag test’ in DGEs and the use of only straightedge and compass in classic Euclidean geometry. We then demonstrate that the drag test criterion may permit constructions created using measurement tools to be considered valid; however, these constructions prove inconsistent with classical geometry. This inconsistency raises the question of whether dragging is an adequate test of validity, and the issue of measurement versus straightedge-and-compass. Without claiming that the inconsistency between what counts as valid solution of a construction problem in the two geometry worlds is necessarily problematic, we examine what would constitute the analogue of the straightedge-and-compass criterion in the domain of DGEs. Discovery of this analogue would enrich our understanding of DGEs with a mathematical idea that has been the distinguishing feature of Euclidean geometry since its genesis. To advance our goal, we introduce the compatibility criterion, a new but not necessarily superior criterion to the drag test criterion of validation of solutions of construction problems in DGEs. The discussion of the two criteria anatomizes the complexity characteristic of the relationship between DGEs and the paper-and-pencil Euclidean geometry environment, advances our understanding of the notion of geometrical constructions in DGEs, and raises the issue of validation practice maintaining the pace of ever-changing software.     

Locus properties of the Neuberg cubic

In this paper we explore various locus problems whose solutions involve the Neuberg cubic of the scalene triangle in the plane. We use analytical geometry to show that the Neuberg equation describes the essential part of the locus in each of these problems. In this way we discover new characteristics of the Neuberg cubic that has been at the focus of attention in the recent renaissance of triangle geometry.     

数学机械化进展综述（迎接ICM2002特约文章）

本文介绍数学机械化理论,构造性代数几何,构造性微分代数几何,构造性实代数几何,方程求解,与几何自动推理的主要进展及其若干领域的应用,我们还提出了一些待解决的问题。     

Wavelet modeling of priors on triangles

Parameters in statistical problems often live in a geometry of certain shape. For example, count probabilities in a multinomial distribution belong to a simplex. For these problems, Bayesian analysis needs to model priors satisfying certain constraints imposed by the geometry. This paper investigates modeling of priors on triangles by use of wavelets constructed specifically for triangles. Theoretical analysis and numerical simulations show that our modeling is flexible and is superior to the commonly used Dirichlet prior.     

对话李克正教授：为什么学习代数几何

以与首都师范大学的李克正教授对话的方式,通过其口简要介绍代数几何这门重要学科在我国的早期发展情况,详细回忆李克正教授在国外学习代数几何的过程,以及他回国后的教学与研究成果,并通俗地阐述代数几何方法对于解决重大数论问题所起的关键作用．     

Deformation of Asymptotically Isospectral Metrics

In this note, we introduce the notion of asymptotically isospectral deformation of order . Then, we study the asymptotically isospectral problems with model spaces. Some techniques in the conformal geometry and Kähler geometry are used to investigate the asymptotically isospectral problems with suitable medol spaces.     

The Grand Astrologerʼs platform and ramp: Four problems in solid geometry from Wang Xiaotongʼs ‘Continuation of ancient mathematics’ (7th century AD)

Wang Xiaotong?s Jigu suanjing is primarily concerned with problems in solid and plane geometry leading to cubic equations which are to be solved numerically by the Chinese variant of Horner?s method. The problems in solid geometry give the volume of a solid and certain constraints on its dimensions, and the dimensions are required; we translate and analyze four of these. Three are solved using dissections, while one is solved using reasoning about calculations with very little recourse to geometrical considerations. The problems in Wang Xiaotong?s text cannot be seen as practical problems in themselves, but they introduce mathematical methods which would have been useful to administrators in organizing labor forces for public works.     

Status determination by interior-point methods for convex optimization problems in domain-driven form

Mathematical Programming - We study the geometry of convex optimization problems given in a Domain-Driven form and categorize possible statuses of these problems using duality theory. Our duality...     

On dually flat square metrics

Square metrics arise from several classification problems in Finsler geometry. They are the rare Finsler metrics to be of excellent geometry properties. It is proved that every non-Riemannian dually flat square metric must be Minkowskian if the dimension ≥3. We also obtain a rigidity result in dually flat Matsumoto metrics.     

Computational geometric aspects of rhythm,melody, and voice-leading

Many problems concerning the theory and technology of rhythm, melody, and voice-leading are fundamentally geometric in nature. It is therefore not surprising that the field of computational geometry can contribute greatly to these problems. The interaction between computational geometry and music yields new insights into the theories of rhythm, melody, and voice-leading, as well as new problems for research in several areas, ranging from mathematics and computer science to music theory, music perception, and musicology. Recent results on the geometric and computational aspects of rhythm, melody, and voice-leading are reviewed, connections to established areas of computer science, mathematics, statistics, computational biology, and crystallography are pointed out, and new open problems are proposed.     

Problem posing via “what if not?” strategy in solid geometry — a case study

The study describes the kinds of problems posed by pre-service teachers on the basis of complex solid geometry tasks using the “what if not?” strategy and the educational value of such an activity. Twenty-eight pre-service teachers participated in two workshops in which they had to pose problems on the basis of given problems. Analysis of the posed problems revealed a wide range of problems including those containing a change of one of the numerical data to another specific one, to a proof problem. Different kinds of posed problems enlightened some phenomena such as a bigger frequency of posed problems with another numerical value and a lack of posed problems including formal generalization. We also discuss the educational strengths of problem posing in solid geometry using the “what if not?” strategy, which could make the learner rethink the geometrical concepts he uses while creating new problems, make connections between the given and the new concepts and as a result deepen his understanding of them.     

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.     

Géométrie de la chaînette ou hypergéométrie

Summary Differential calculus is proving inefficient to solve problems of mechanics dealing with ropes, unless one satisfies oneself with rough approximations.Ordinary or Euclidean geometry is also proving useless: The author shows that in order to solve such problems as described before, one has to use a special branch of geometry. Distances are no longer measured along rigid straight rules, but along the catenary curves in question.The fundamental characteristics of this hypergeometry, with some of their more striking results, are described in the following study.     

An integer programming approach for the search of discretization orders in distance geometry problems

Optimization Letters - Discretizable distance geometry problems (DDGPs) constitute a class of graph realization problems where the vertices can be ordered in such a way that the search space of...