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.     

Homological Perturbation Theory and Mirror Symmetry

We explain how deformation theories of geometric objects such as complex structures,Poisson structures and holomorphic bundle structures lead to differential Gerstenhaber or Poisson al-gebras.We use homological perturbation theory to construct A∞ algebra structures on the cohomology,and their canonically defined deformations.Such constructions are used to formulate a version of A∞ algebraic mirror symmetry.     

Students' understanding of the notion of function in dynamic geometry environments

To make optimal use of computational environments, one must understand how students interact with the environments and how students' mathematical thinking is reflected and affected by their use of the environments. Similarly, to make sense of research on students' thinking and learning, one must understand how the environments and contexts used in the research may affect the conclusions one derives.The research on students' learning of functions has approached the topic in terms of symbols and graphs (see, for example, Leinhardt et al. (1990) for a review of work up to that point; Harel and Dubinsky (1992) for a collection of research; and Dugdale et. al. (1995), for some recent thinking about implications for curriculum reform using technology). Dynamic geometry environments (DGEs) like Cabri Geometry or Geometer's Sketchpad, offer us an opportunity to get a new perspective on these old and important issues. DGEs let students build geometrical constructions and then drag certain objects around the screen in a continuous manner while observing how the entire construction responds dynamically. In this way DGEs model functional relationships that are not specified by symbols or represented by graphs.Based on interviews with undergraduate mathematics majors, this paper presents preliminary observations that confirm some old results and raise some new questions about students' notions of function.     

High school students’ understandings of geometric transformations in the context of a technological environment

This study investigated the nature of students’ understandings of geometric transformations, which included translations, reflections, rotations, and dilations, in the context of the technological tool, The Geometer’s Sketchpad. The researcher implemented a seven-week instructional unit on geometric transformations within an Honors Geometry class. Students’ conceptions of transformations as functions were analyzed using the APOS theory and were informed by an analysis of students’ interpretations and uses of representations of geometrical objects using the constructs of drawing and figure. The analysis suggests students’ understandings of key concepts including domain, variables and parameters, and relationships and properties of transformations were critical for supporting the development of deeper understandings of transformations as functions.     

Constructing strength three covering arrays with augmented annealing

A covering arrayCA(N;t,k,v) is an N×k array such that every N×t sub-array contains all t-tuples from v symbols at least once, where t is the strength of the array. One application of these objects is to generate software test suites to cover all t-sets of component interactions. Methods for construction of covering arrays for software testing have focused on two main areas. The first is finding new algebraic and combinatorial constructions that produce smaller covering arrays. The second is refining computational search algorithms to find smaller covering arrays more quickly. In this paper, we examine some new cut-and-paste techniques for strength three covering arrays that combine recursive combinatorial constructions with computational search; when simulated annealing is the base method, this is augmented annealing. This method leverages the computational efficiency and optimality of size obtained through combinatorial constructions while benefiting from the generality of a heuristic search. We present a few examples of specific constructions and provide new bounds for some strength three covering arrays.     

Diagrams,Tensors and Geometric Reasoning

Geometry and in particular projective geometry (and its corresponding invariant theory) deals a lot with structural properties of geometric objects and their interrelations. This papers describes how concepts of tensor calculus can be used to express geometric invariants and how, in particular, diagrammatic notation can be used to deal with invariants in a highly intuitive way. In particular we explain how geometries like euclidean or spherical geometry can be dealt with in this framework. Dedicated to the memory of Victor Klee, and in particular to his striving for conceptual simplicity     

A construction of classifying spaces for p-adic group actions

One major open problem in geometric topology is the Hilbert-Smith conjecture. A natural approach to this conjecture is to work on classifying spaces of p-adic integers. However, the well-known Milnor's construction of classifying space of p-adic integers is not locally connected, hence will not help to solve the conjecture, and the other known constructions are very complex. The goal of this paper is to give a new construction of classifying spaces for p-adic group actions.     

An O(n log m) algorithm for the Josephus Problem

The Josephus Problem can be described as follows: There are n objects arranged in a circle. Beginning with the first object, we move around the circle and remove every m th object. As each object is removed, the circle closes in. Eventually, all n objects will have been removed from the circle. The order in which the objects are removed induces a permutation on the integers 1 through n. Knuth has described two O(n log n) algorithms for generating this permuation. The problem of determining a more efficient algorithm for generating the permutation is left open. In this paper we give an O(n log m) algorithm.     

Abū al-Wafā’ Latinus? A study of method

This article studies the legacy in the West of Abū al-Wafā’s Book on those geometric constructions which are necessary for craftsmen. Although two-thirds of the geometric constructions in the text also appear in Renaissance works, a joint analysis of original solutions, diagram lettering, and probability leads to a robust finding of independent discovery. The analysis shows that there is little chance that the similarities between the contents of Abū al-Wafā’s Book and the works of Tartaglia, Marolois, and Schwenter owe anything to historical transmission. The commentary written by Kamāl al-Dīn Ibn Yūnus seems to have had no Latin legacy, either.     

A Feasible Algorithm for Locating Concave and Convex Zones of Interval Data and Its Use in Statistics-Based Clustering

Often, we need to divide n objects into clusters based on the value of a certain quantity x. For example, we can classify insects in the cotton field into groups based on their size and other geometric characteristics. Within each cluster, we usually have a unimodal distribution of x, with a probability density ρ(x) that increases until a certain value x ₀ and then decreases. It is therefore natural, based on ρ(x), to determine a cluster as the interval between two local minima, i.e., as a union of adjacent increasing and decreasing segments. In this paper, we describe a feasible algorithm for solving this problem.     

VB-structures and generalizations

Annals of Global Analysis and Geometry - Motivated by properties of higher tangent lifts of geometric structures, we introduce concepts of weighted structures for various geometric objects on a...     

The interval Branch-and-Prune algorithm for the discretizable molecular distance geometry problem with inexact distances

The Distance Geometry Problem in three dimensions consists in finding an embedding in ${\mathbb{R}^3}$ of a given nonnegatively weighted simple undirected graph such that edge weights are equal to the corresponding Euclidean distances in the embedding. This is a continuous search problem that can be discretized under some assumptions on the minimum degree of the vertices. In this paper we discuss the case where we consider the full-atom representation of the protein backbone and some of the edge weights are subject to uncertainty within a given nonnegative interval. We show that a discretization is still possible and propose the iBP algorithm to solve the problem. The approach is validated by some computational experiments on a set of artificially generated instances.     

Finite-type integrable geometric structures

In this paper, we consider finite-type geometric structures of arbitrary order and solve the integrability problem for these structures. This problem is equivalent to the integrability problem for the corresponding G-structures. The latter problem is solved by constructing the structure functions for G-structures of order ≥1. These functions coincide with the well-known ones for the first-order G-structures, although their constructions are different. We prove that a finite-type G-structure is integrable if and only if the structure functions of the corresponding number of its first prolongations are equal to zero. Applications of this result to second-and third-order ordinary differential equations are noted. __________ Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 1, Geometry of Integrable Models, 2004.     

Dragging as a conceptual tool in dynamic geometry environments

Some ‘drag-to-fit’ solutions given by student teachers to three geometric construction problems in a dynamic geometry environment (DGE) are analysed. The responses of a group of experienced mathematics teachers to the question whether or not such solutions can be considered ‘legitimate’ are then discussed. This raises fundamental questions concerning the concept of legitimacy, the relationship between DGEs and Formal Axiomatic Euclidean Geometry, the nature of ‘conceptual tools’ in different geometric environments, and the functions of dragging in DGEs. The authors argue that, if dragging is viewed as a conceptual tool, then certain drag-to-fit solutions, although soft constructions, may still be considered as conceptually legitimate and therefore valid. Finally, some important questions are raised concerning the impact that teachers’ different attitudes towards legitimacy might have on students’ learning through DGEs.     

Constructing Ramsey graphs from Boolean function representations

Explicit construction of Ramsey graphs or graphs with no large clique or independent set, has remained a challenging open problem for a long time. While Erdös’ probabilistic argument shows the existence of graphs on 2n vertices with no clique or independent set of size 2 ⁿ , the best explicit constructions achieve a far weaker bound. There is a connection between Ramsey graph constructions and polynomial representations of Boolean functions due to Grolmusz; a low degree representation for the OR function can be used to explicitly construct Ramsey graphs [17,18]. We generalize the above relation by proposing a new framework. We propose a new definition of OR representations: a pair of polynomials represent the OR function if the union of their zero sets contains all points in {0, 1} ⁿ except the origin. We give a simple construction of a Ramsey graph using such polynomials. Furthermore, we show that all the known algebraic constructions, ones to due to Frankl-Wilson [12], Grolmusz [18] and Alon [2] are captured by this framework; they can all be derived from various OR representations of degree O(√n) based on symmetric polynomials. Thus the barrier to better Ramsey constructions through such algebraic methods appears to be the construction of lower degree representations. Using new algebraic techniques, we show that better bounds cannot be obtained using symmetric polynomials.     

On the use of the term diastēma in ancient Greek constructions

I examine the use of the term diastēma by Greek geometers in both plane and spherical constructions. I show that while diastēma may be translated as radius in plane constructions, this will not work on the sphere. These investigations have some implications for how we think of construction in Greek mathematics in general.     

Derived Completions in Stable Homotopy theory

We introduce a notion of derived completion applicable to arbitrary homomorphisms of commutative S-algebras, and work out some of its properties, including invariance results, a spectral sequence proceeding from purely algebraic information to the geometric results, and analysis of relationships with earlier constructions. We also provide some examples. The construction has applications in algebraic K-theory.     

The construction of a special equilateral triangle

This article deals with the construction of an equilateral triangle that must satisfy the following special constraint conditions. If the equilateral triangle is denoted by ΔABC, then the radii of the inscribed circle, the three escribed circles of ΔABC, and the circumcircle of ΔABC all must have positive integral radii. The inscribed circle radius is required to be 1 unit. The three escribed circles that have equal radii must have 3 units each, and the circumcircle of the triangle must have 2 units. All these requirements may seem outlandish. The aim is to teach crucial Geometric principles that Geometric designs must take into account before the constructions are implemented. This article hopefully may be useful to students of College Geometry as well as teachers.     

Recent advances on the Discretizable Molecular Distance Geometry Problem

The Molecular Distance Geometry Problem (MDGP) consists in finding an embedding in R³${\mathbb{R}}^3$ of a nonnegatively weighted simple undirected graph with the property that the Euclidean distances between embedded adjacent vertices must be the same as the corresponding edge weights. The Discretizable Molecular Distance Geometry Problem (DMDGP) is a particular subset of the MDGP which can be solved using a discrete search occurring in continuous space; its main application is to find three-dimensional arrangements of proteins using Nuclear Magnetic Resonance (NMR) data. The model provided by the DMDGP, however, is too abstract to be directly applicable in proteomics. In the last five years our efforts have been directed towards adapting the DMDGP to be an ever closer model of the actual difficulties posed by the problem of determining protein structures from NMR data. This survey lists recent developments on DMDGP related research.