A concise quaternion geometry of rotations

This communication compiles propositions concerning the spherical geometry of rotations when represented by unit quaternions. The propositions are thought to establish a two‐way correspondence between geometrical objects in the space of real unit quaternions representing rotations and geometrical objects constituted by directions in the three‐dimensional space subjected to these rotations. In this way a purely geometrical proof of the spherical Ásgeirsson's mean value theorem and a geometrical interpretation of integrals related to the spherical Radon transform of a probability density functions of unit quaternions are accomplished. Copyright © 2004 John Wiley & Sons, Ltd.     

A Note on Geometric Structures of Linear Ordered Orthogonal Arrays and (T,M,S)-Nets of Low Strength

The equivalent geometrical configurations of linear ordered orthogonal arrays are determined when their strengths are 3 and 4. Existence of such geometrical configurations is investigated. They are also useful in the study of (T, M, S)-nets.     

Complex interpolation and geometry of Banach spaces

Summary The coincidence of the real and complex methods of interpolation is investigated. Positive results are established under the presence of geometrical properties which are expressed in terms of vector valued Fourier transforms. The results are applied to complex interpolation of H^p spaces and to the study of geometrical properties of Banach spaces.     

扁球壳在均布压力与均匀温度场联合作用下的屈曲

根据扁壳几何非线性理论,推导了均布压力与均匀温度场联合作用下的扁球壳的位移型几何非线性控制方程.考虑夹紧边界条件,采用打靶法得到了扁球壳轴对称弯曲与屈曲的数值结果.讨论了壳体几何参数对平衡路径、临界荷载的影响.给出了壳体临界几何参数.当几何参数大于临界几何参数时,上、下临界荷载都随几何参数增加而增加.给定几何参数时,考察了不同均匀温度场对壳体上、下临界荷载、临界几何参数以及平衡构型的影响.均匀升温会使上临界荷载显著增加,会使下临界荷载略有减小.均匀变温会使临界几何参数改变.     

Bordism classes represented by multiple point manifolds of immersed manifolds

We present a geometrical version of Herbert’s theorem determining the homology classes represented by the multiple point manifolds of a self-transverse immersion. Herbert’s theorem and generalizations can readily be read off from this result. The simple geometrical proof is based on ideas in Herbert’s paper. We also describe the relationship between this theorem and the homotopy theory of Thom spaces.     

Inpolygons of minimal diameter

The present paper deals with an application of a distributional version of Pontryagin's maximum principle to the following geometrical problem: Find to a given convex polygon an inpolygon of minimal diameter. This geometrical problem is formulated as a problem of optimal control. For some special cases optimal inpolygons are determined.     

Universal geometrical equivalence of the algebraic structures of common signature

This article is a part of our effort to explain the foundations of algebraic geometry over arbitrary algebraic structures [1–8]. We introduce the concept of universal geometrical equivalence of two algebraic structures A and B of a common language L which strengthens the available concept of geometrical equivalence and expresses the maximal affinity between A and B from the viewpoint of their algebraic geometries. We establish a connection between universal geometrical equivalence and universal equivalence in the sense of equality of universal theories.     

Classification of Material <Emphasis Type="Italic">G</Emphasis>-Structures

A geometrical interpretation of the G-structures associated to elastic material bodies is given. In addition, characterizations of their integrability are obtained. Since the lack of integrability is a geometrical measure of the lack of homogeneity, the corresponding inhomogeneity conditions are obtained.     

Normalising geometrical figures: dynamic manipulation and construction of meanings for ratio and proportion

Enlarging-shrinking geometrical figures by 13 year-olds is studied during the implementation of proportional geometric tasks in the classroom. Students worked in groups of two using ‘Turtleworlds’, a piece of geometrical construction software which combines symbolic notation, through a programming language, with dynamic manipulation of geometrical objects by dragging on sliders representing variable values. In this paper we study the students’ normalising activity, as they use this kind of dynamic manipulation to modify ‘buggy’ geometrical figures while developing meanings for ratio and proportion. We describe students’ normative actions in terms of four distinct Dynamic Manipulation Schemes (Reconnaissance, Correlation, Testing, Verification). We discuss the potential of dragging for mathematical insight in this particular computational environment, as well as the purposeful nature of the task which sets up possibilities for students to appreciate the utility of proportional relationships.     

A Bayesian approach to vectorization of object boundaries from digital images and to geometrical uncertainty assessment

A new Bayesian approach is presented for extracting 2D object boundaries with measures of uncertainty. The boundaries are described by minimal closed sequences of segments and arcs, called mixed polygons. The sequence is minimal in the sense that it is able to describe all the geometrical properties of the boundary without being redundant. Based on geometrical measures evaluated on the object boundary model, a prior distribution is introduced in order to favor a mixed polygon with good geometrical properties, avoiding short sides, collinearity between segments, and so on. The estimation process is based on a two‐stage procedure that combines reversible‐jump MCMC (RJMCMC) and classic MCMC methods. The RJMCMC method is viewed as a model selection technique, and it is used to estimate the correct number of sides of the mixed polygon. The MCMC algorithm provides a sample of mixed polygons through which to evaluate the mixed polygon that best approximates the object boundary and its geometrical uncertainty. A convergence criterion for the RJMCMC method is provided. Copyright © 2011 John Wiley & Sons, Ltd.     

Graph isomorphism and equality of simplices

We show that the graph isomorphism problem is equivalent to the problem of recognizing equal simplices in ? ⁿ . This result can lead to new methods in the graph isomorphism problem based on geometrical properties of simplices. In particular, relations between several well-known classes of invariants of graphs and geometrical invariants of simplices are established.     

On the geometry of the Frobenius problem

The geometrical method to find the Frobenius number indicated by Arnold (Funct. Anal. Other. Math. 2, 2007) in the case of 3 generators can be extended to any number of generators. The method provides not only the Frobenius number but also a set of numbers, from which all the nonrepresentable numbers can be generated. In the case of three generators, we show some geometrical implications of the conditions for a semigroup to be symmetric or nonsymmetric.     

The geometry of the secant caustic of a planar curve

The secant caustic of a planar curve M is the image of the singular set of the secant map of M. We analyze the geometrical properties of the secant caustic of a planar curve, i.e. the number of branches of the secant caustic, the parity of the number of cusps and the number of inflexion points in each branch of this set. In particular, we investigate in detail some of the geometrical properties of the secant caustic of a rosette, i.e. a smooth regular oriented closed curve with non-vanishing curvature.     

Blaschke-Type Theorem and Separation of Disjoint Closed Geodesic Convex Sets

In this paper, we deal with analytic and geometrical properties of geodesic convex sets and geodesic paths. We show that Blaschke’s Theorem for convex sets is also true for geodesic convex sets and geodesic paths in a simple polygon. Some geometrical properties of geodesic triangles are presented. Furthermore, separation of geodesic convex sets is shown.     

Dynamic investigation of triangles inscribed in a circle,which tend to an equilateral triangle

We present a geometrical investigation of the process of creating an infinite sequence of triangles inscribed in a circle, whose areas, perimeters and lengths of radii of the inscribed circles tend to a limit in a monotonous manner.

First, using geometrical software, we investigate four theorems that represent interesting geometrical properties, after which we present formal proofs that rest on a combination between different fields of mathematics: trigonometry, algebra and geometry, and the use of the concept of standard deviation that is taken from statistics.     

A hyperspatial representation of a particular set of fuzzy preference relations

In this paper we study a geometrical representation of a special class of fuzzy preference relations by establishing a one-one correspondence between the set of all these relations and a hypercube of an n(n?1)/2 dimensional space.Furthermore, we study some geometrical figures (called H-spheres) and properties of the points which represent the preference relations in the above mentioned euclidean space.     

Characteristics of Pre-Service Primary School Teachers’ Configural Reasoning

The goal of this study is to identify the characteristics of pre-service primary teachers’ configural reasoning, understood as the relationships between concepts and figures set to solve geometrical proof problems. Ninety-seven primary teachers were asked to solve two geometrical proof problems in which a geometrical figure was provided. The results suggest the existence of two levels of the pre-service primary teachers’ discursive organization. The first, when pre-service primary teachers link the geometrical facts to the figure by discursive apprehensions, and the second, when several geometrical facts are related by logical chains to infer new information. The identification of a relevant configuration and the way in which geometrical facts are logically organized from discursive apprehensions are key factors in the shift between these moments. We contend that these factors help to explain how the image component stimulates thought and how conceptual constraints control the formal rigor of the process.     

Extension of the scaled boundary finite element method to geometrical and physical nonlinearity

The scaled boundary finite element method (SBFEM) has been used in many fields of engineering to solve the governing equations in bounded and unbounded 2D as well as 3D domains. In solid mechanics, the semi-analytical solution strategy of the SBFE formulation (numerical in circumferential direction, analytical in radial direction) is based on the assumption of linear elastic material behavior and only small geometrical changes. However, a large group of materials (e.g. rubber) shows geometrical and physical nonlinearity at mechanical loading. In this contribution, the extension of the SBFEM to geometrical and physical nonlinearity is examined. A plane finite element is developed which uses the concept of shape functions constructed by the SBFEM in the framework of a nonlinear finite element analysis. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Theorem Justification and Acquisition in Dynamic Geometry: A Case of Proof by Contradiction

Theorem acquisition and deductive proof have always been core elements in the study and teaching of Euclidean geometry. The introduction of dynamic geometry environments,DGE (e.g., Cabri-Géomètre, Geometer's Sketchpad), into classrooms in the past decade has posed a challenge to this praxis. Student scan experiment through different dragging modalities on geometrical objects that they construct, and consequently infer properties(generalities, theorems) about the geometrical artefacts. Because of the inductive nature of the DGE, the experimental-theoretical gap that exists in the acquisition and justification of geometrical knowledge becomes an important pedagogical and epistemological concern. In this paper, we will describe and study a ‘Cabri proof by contradiction’ of a theorem on cyclic quadrilaterals given by a pair of 16 year-old students in a Hong Kong secondary school. We will discuss how their construction motivates a visual-cognitive scheme on `seeing' proof in DGE, and how this scheme could fit into the theoretical construct of cognitive unity of theorems proposed by Boero, Garuti and Mariotti(1996). The issue of a cognitive duality and its relation to visualization will be raised and discussed. Finally, we propose a possible perspective to bridge the experimental-theoretical gap in DGE by introducing the idea of a dynamic template as a visualizer to geometrical theorem justification and acquisition. This revised version was published online in July 2006 with corrections to the Cover Date.     

Ordering Cellular Spaces with Application to Curves and Knots

This paper concerns the problem of ordering geometrical objects, which have the structure of finite cellular spaces. We introduce cellular codes, cellular indices, and (k,l)-sizes and apply them to ordering 2-cells, plane curves, and projections of knots. Results of this paper can be applied (1) to ordering other geometrical objects with cellular space structures, (2) in psychological and computer tests for comparison of complexity of geometrical objects, and (3) to ordering objects with fractal and chaotic structure, which admit approximation by cellular spaces.