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.     

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.     

Duplicating the cube and other notes on constructions in the hyperbolic plane

The Euclidean classical problem ofDuplicating the Cube is investigated for the first time inhyperbolic geometry, and its partial solution presented. The other two associated problems ofTrisecting an Angle andSquaring the Circle in the hyperbolic plane were solved classically, but a few additions to these solutions are presented, along with some miscellaneous notes about straightedge and compass constructions in the hyperbolic plane.A preliminary version of this paper was presented at the MAA Undergraduate Student Paper Session, August 1987, Salt Lake City AMS/MAA Joint Meetings. Financial support was provided by a J.W.T. Youngs Undergraduate Award in Mathematics (UC Santa Cruz), and UCSD.     

Geometry of Solutions of the N=2 SYM Theory in Harmonic Superspace

The classical equations of motion of the D=4, N=2 supersymmetric Yang–Mills (SYM) theory for Minkowski and Euclidean spaces are analyzed in harmonic superspace. We study dual superfield representations of equations and subsidiary conditions corresponding to classical SYM solutions with different symmetries. In particular, alternative superfield constructions of self-dual and static solutions are described in the framework of the harmonic approach.     

Horospheres and Convex Bodies in n-Dimensional Hyperbolic Space

In n-dimensional Euclidean space, the measure of hyperplanes intersecting a convex domain is proportional to the (n–2)-mean curvature integral of its boundary. This question was considered by Santaló in hyperbolic space. In non-Euclidean geometry the totally geodesic hypersurfaces are not always the best analogue to linear hyperplanes. In some situations horospheres play the role of Euclidean hyperplanes.In dimensions n=2 and 3, Santaló proved that the measure of horospheres intersecting a convex domain is also proportional to the (n–2)-mean curvature integral of its boundary.In this paper we show that this analogy does not generalize to higher dimensions. We express the measure of horospheres intersecting a convex body as a linear combination of the mean curvature integrals of its boundary.     

Robot navigation functions on manifolds with boundary

This paper concerns the construction of a class of scalar valued analytic maps on analytic manifolds with boundary. These maps, which we term navigation functions, are constructed on an arbitrary sphere world—a compact connected subset of Euclidean n-space whose boundary is formed from the disjoint union of a finite number of (n − l)-spheres. We show that this class is invariant under composition with analytic diffeomorphisms: our sphere world construction immediately generates a navigation function on all manifolds into which a sphere world is deformable. On the other hand, certain well known results of S. Smale guarantee the existence of smooth navigation functions on any smooth manifold. This suggests that analytic navigation functions exist, as well, on more general analytic manifolds than the deformed sphere worlds we presently consider.     

An approach to symmetrization via polarization

We prove that the Steiner symmetrization of a function can be approximated in by a sequence of very simple rearrangements which are called polarizations. This result is exploited to develop elementary proofs of many inequalities, including the isoperimetric inequality in Euclidean space. In this way we also obtain new symmetry results for solutions of some variational problems. Furthermore we compare the solutions of two boundary value problems, one of them having a "polarized" geometry and we show some pointwise inequalities between the solutions. This leads to new proofs of well-known functional inequalities which compare the solutions of two elliptic or parabolic problems, one of them having a "Steiner-symmetrized" geometry. The method also allows us to investigate the case of equality in the inequalities. Roughly speaking we prove that the equality sign is valid only if the original problem has the symmetrized geometry.

     

A unified construction for c*· c- and L · L*-geometries in projective spaces

We present two new constructions for c^* · c-geometries. The first provides, for each even prime powerq, a flag-transitive c^* · c-geometry of orderq–1 that is embedded in the projective space PG(3,q) and which is related with the Cameron-Fisher extended grids of odd type. The second construction is valid independently of the parity ofq. Forq even, it produces the same geometry as the first construction, and forq odd, two geometries related with some extended grids constructed by Meixner and Pasini.Next, by using some complementary models for c^* and L in a projective plane, we derive from our construction a new family of L · L^*-geometries embedded in PG(3,q). Forq even, these geometries are flag-transitive.     

About Tracing Problems in Dynamic Geometry

Dynamic Geometry is the field of interactively performing geometric construction on a computer. In addition to simulating ruler-and-compass constructions we allow a drag mode. This drag mode allows to move geometric objects that have at least one degree of freedom. The remaining part of the construction should adjust automatically. Thus, during the motion, we have to trace the resulting paths of all geometric objects. This path tracking problem is known as the Tracing Problem from Dynamic Geometry. It combines the step-by-step procedure of doing geometric constructions with the continuous concept of motions. This study is based on the model for Dynamic Geometry used in the interactive geometry software Cinderella. We give a numerical solution to the Tracing Problem based on continuation methods and a reliable algorithm based on real and complex interval arithmetic. Degenerate situations like the intersection of two identical lines lead to critical points in the configuration space and are treated separately.     

Solution analysis for the pseudomonotone second-order cone linear complementarity problem

In this paper, we are concerned with the set of the solutions and the geometric property of the pseudomonotone second-order cone linear complementarity problems (SOCLCP). Based on Tao’s recent work [Tao, J. Optim. Theory Appl., 159, 41–56 (2013)] on pseudomonotone LCP on Euclidean Jordan algebras, we characterize the set of solutions and also derive intrinsic properties that reveal the underlying geometry of the pseudomonotone SOCLCP.     

Another Constructive Axiomatization of Euclidean Planes

H. Tietze has proved algebraically that the geometry of uniquely determined ruler and compass constructions coincides with the geometry of ruler and set square constructions. We provide a new proof of this result via new universal axiom systems for Euclidean planes of characteristic ≠ 2 in languages containing only operation symbols.     

Two-Sided Approximation of Solutions of Boundary-Value Problems

We propose a general scheme for the two-sided approximation of solutions of boundary-value problems for ordinary differential equations. This scheme involves a number of known and new two-sided methods. In our investigation, we use constructions of the Samoilenko numerical-analytic method together with the procedure of the construction of two-sided methods proposed by Kurpel’ and Shuvar.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 2, pp. 284–288, February, 2005.     

The Nonlinear Schrödinger Equation on Hyperbolic Space

In this article we study some aspects of dispersive and concentration phenomena for the Schrödinger equation posed on hyperbolic space  ⁿ , in order to see if the negative curvature of the manifold gets the dynamics more stable than in the Euclidean case. It is indeed the case for the dispersive properties: we prove that the dispersion inequality is valid, in a stronger form than the one on ? ⁿ . However, the geometry does not have enough of an effect to avoid the concentration phenomena and the picture is actually worse than expected. The critical nonlinearity power for blow-up turns out to be the same as in the euclidean case, and we prove that there are more explosive solutions for critical and supercritical nonlinearities.     

Products of Menger spaces: A combinatorial approach

We construct Menger subsets of the real line whose product is not Menger in the plane. In contrast to earlier constructions, our approach is purely combinatorial. The set theoretic hypothesis used in our construction is far milder than earlier ones, and holds in almost all canonical models of set theory of the real line. On the other hand, we establish productive properties for versions of Menger's property parameterized by filters and semifilters. In particular, the Continuum Hypothesis implies that every productively Menger set of real numbers is productively Hurewicz, and each ultrafilter version of Menger's property is strictly between Menger's and Hurewicz's classic properties. We include a number of open problems emerging from this study.     

Analysis on the minimal representation of O(p,q) I. Realization via conformal geometry

This is the first in a series of papers devoted to an analogue of the metaplectic representation, namely the minimal unitary representation of an indefinite orthogonal group; this representation corresponds to the minimal nilpotent coadjoint orbit in the philosophy of Kirillov–Kostant. We begin by applying methods from conformal geometry of pseudo-Riemannian manifolds to a general construction of an infinite-dimensional representation of the conformal group on the solution space of the Yamabe equation. By functoriality of the constructions, we obtain different models of the unitary representation, as well as giving new proofs of unitarity and irreducibility. The results in this paper play a basic role in the subsequent papers, where we give explicit branching formulae, and prove unitarization in the various models.     

A proportional view: The mathematics of James Glenie (1750–1817)

The mathematical work of James Glenie (1750–1817) was published at irregular intervals during a turbulent life. His ideas, mostly deriving from his time as an Assistant in Mathematics at St Andrews University in Scotland, were developed intermittently over a period of thirty-seven years. His mathematical achievements, underestimated by previous historians, were deeply rooted in Euclidean geometry and his own generalized theory of proportion. Among them are many new geometrical constructions and proofs, a novel demonstration of the binomial theorem, and an alternative approach to the differential calculus.     

On Mascheroni's La geometria del compasso at the beginning of the 19th century

Lorenzo Mascheroni's 1797 work La geometria del compasso, which develops a geometry based solely on compass constructions, is considered by the author as stepping back behind the “demarcation line” of Euclidean geometry. In this work Mascheroni emphasizes the practical aspects of this geometry over a theoretical approach. A century later, in 1899, David Hilbert and his student Michael Feldblum proposed a totally different approach – algebraic and axiomatic – concerning geometric constructions based on various instruments. Taking into account that, at the end of the 18th century, straightedge geometry was also developed, one may ask what happened to the image of instrument-based geometry during the 19th century? By focusing on Mascheroni's book and its reception, this article aims to examine the various views and conceptions of mathematicians with respect to this geometry.     

The role of Steiner hulls in the solution to Steiner tree problems

ASteiner tree problem on the plane is that of finding a minimum lengthSteiner tree connecting a given setK ofterminals and lying within a given regionR of the Euclidean plane; it includes as special cases the Euclidean Steiner minimal tree problem (ESMT), the rectilinear Steiner tree problem (RST), and the Steiner tree problem on graphs (STG). ASteiner hull forK inR generically refers to any subregion ofR known to contain a Steiner tree. This paper gives a survey of the role of Steiner hulls in the Steiner tree problem. The significance of Steiner hulls in the efficient solution of Steiner tree problems is outlined, and then a compendium is given of the known Steiner hull constructions for ESMT, RST, and STG problems.     

On the Topology of Vacuum Spacetimes

We prove that there are no restrictions on the spatial topology of asymptotically flat solutions of the vacuum Einstein equations in (n + 1)-dimensions. We do this by gluing a solution of the vacuum constraint equations on an arbitrary compact manifold Sⁿ \Sigma^n to an asymptotically Euclidean solution of the constraints on \mathbbRⁿ \mathbb{R}^n . For any Sⁿ \Sigma^n which does not admit a metric of positive scalar curvature, this provides for the existence of asymptotically flat vacuum spacetimes with no maximal slices. Our main theorem is a special case of a more general gluing construction for nondegenerate solutions of the vacuum constraint equations which have some restrictions on the mean curvature, but for which the mean curvature is not necessarily constant. This generalizes the construction [16], which is restricted to constant mean curvature data.     

On the “equivalence” rule for flows of perfect gas : PMM vol. 38, n 6, 1974, pp. 1004–1014

An extension of the equivalence of “area” rule [1, 2] is presented. The rule was initially derived for stationary flows of perfect (inviscid and non-heat-conducting) gas past slender fine pointed bodies (or blunted bodies in the hypersonic flow case) whose transverse dimensions are small in comparison with their length. According to that rule the wave drag of a three-dimensional body is equal to the wave drag of an axisymmetric body with the same distribution of cross-sectional areas along the axis. The rule is extended here to stationary and nonstationary flows past nonslender bodies and to internal flows, using the procedure of averaging with respect to the angular variable of a cylindrical system of coordinates. That procedure is, strictly speaking, valid for nearly axisymmetric bodies. However the numerical solutions obtained by the authors for a fairly wide range of external and internal problems show that the generalized equivalence rule is applicable to substantially nonaxisymmetric configurations (^*) (see next page).