Truncation error in grid generation: A case study

Theoretical proofs state that the planar Winslow or homogenous Thompson–Thames–Mastin (hTTM) map is a diffeomorphism, yet numerical solutions to the hTTM equations produce folded grids on the so-called “horseshoe” domain. A quasi-analytic solution to the horseshoe problem is constructed to demonstrate that folding is due to truncation error effects. Higher-order difference methods are also explored. © 1995 John Wiley & Sons, Inc.     

A case study of dynamic visualization and problem solving

This paper reports an example of a situation in which university students had to solve geometrical problems presented to them dynamically using the interactive computerized environment of the ‘MicroWorlds Project Builder’. In the process of the problem solving, the students used ten different solution strategies. The unsuccessful strategies were then classified into three main categories: distracting, reducing and confusing. One student group had to solve the same problem in its non-dynamic version. The results received from both groups were compared and analysed. Analysis of the solution strategies and the process of the categorization revealed that the percentage of success in both groups was similar and in the case of the given problem, the dynamic visual mode of the problem distracted the students’ attention away from proper handling of the solution of the problem.     

Spatial visualization and measurement of area: A case study in spatialized mathematics instruction

The purpose of this study was to explore the influence of spatial visualization skills when students solve area tasks. Spatial visualization is closely related to mathematics achievement, but little is known about how these skills link to task success. We examined middle school students’ representations and solutions to area problems (both non-metric and metric) through qualitative and quantitative task analysis. Task solutions were analyzed as a function of spatial visualization skills and links were made between student solutions on tasks with different goals (i.e., non-metric and metric). Findings suggest that strong spatial visualizers solved the tasks with relative ease, with evidence for conceptual and procedural understanding. By contrast, Low and Average Spatial students more frequently produced errors due to failure to correctly determine linear measurements or apply appropriate formula, despite adequate procedural knowledge. A novel finding was the facilitating role of spatial skills in the link between metric task representation and success in determining a solution. From a teaching and learning perspective, these results highlight the need to connect emergent spatial skills with mathematical content and support students to develop conceptual understanding in parallel with procedural competence.     

Z-cyclic triplewhist tournaments—The noncompatible case,part II

Two odd primes p₁ = 2 u₁ + 1, p₂ = 2 u₂ + 1, u₁, u₂ odd, are said to be noncompatible if b₁ ≠ b₂. Let b_i ≥ 2, i = 1, 2 and denote the set {(p₁, p₂): {p₁, p₂} are noncompatible, p_i < 200} by NC. In Part 1 of this study we established the existence of Z-cyclic triplewhist tournaments on 3p₁p₂ + 1 players for all (p₁, p₂) ϵ NC. Here we extend these results and establish Z-cyclic triplewhist tournaments on 3p₁p₂ + 1 players for all (p₁, p₂) ϵ NC and for all α₁ ≥ 1, α₂ ≥ 1. It is believed that these are the first infinite classes of such triplewhist tournaments. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 189–201, 1997     

Optimal scheduling of reservoir releases during flood: Deterministic optimization problem,part 2, case study

This paper applies the optimization procedure developed in Part 1 to the problem of the optimal scheduling of reservoir releases during flood in the case study concerning the river system of Upper Vistula in Poland. Technical details related to the implementation of the proposed algorithm are discussed.The research reported here has been supported by the Central Basic Research Program CPBP-03.09, Metody Analizy i Uytkowania Aasobow Wodnych, Polish Academy of Sciences, Warsaw, Poland. This support is kindly acknowledged.     

Quadratic convergence in interval arithmetic,part II

The size of the error incurred by one operation in an interval arithmetic procedure depends on the extent to which the operands are dependent, i.e., depend on the same initial variables. In this part we will investigate the effect of such dependence. Our results are applied to prove the quadratic convergence of the centered form and of a method of Hansen and Smith for solving linear algebraic systems.Supported in part by NSF grant GJ-797.     

Dupin hypersurfaces in R 5, part II

Dupin hypersurfaces in R ⁵ with four distinct principal curvatures are studied in the context of Lie sphere geometry.     

Foliated plateau problem,part II: Harmonic maps of foliations

Our basic results concerning harmonic maps are parallel to those in Part I [Gro11] about minimal subvarieties. First we produce compact harmonic foliations by solving in some cases the asymptotic Dirichlet problem. Then we construct transversal measures by adopting the parabolic equation method of Eells and Sampson. Finally we indicate some applications to the rigidity and the pinching problems.     

Variable metric relaxation methods,part II: The ellipsoid method

The deepest, or least shallow, cut ellipsoid method is a polynomial (time and space) method which finds an ellipsoid, representable by polynomial space integers, such that the maximal ellipsoidal distance relaxation method using this fixed ellipsoid is polynomial: this is equivalent to finding a linear transforming such that the maximal distance relaxation method of Agmon, Motzkin and Schoenberg in this transformed space is polynomial. If perfect arithmetic is used, then the sequence of ellipsoids generated by the method converges to a set of ellipsoids, which share some of the properties of the classical Hessian at an optimum point of a function; and thus the ellipsoid method is quite analogous to a variable metric quasi-Newton method. This research was supported in part by the F.C.A.C. of Quebec, and the N.S.E.R.C. of Canada under Grant A 4152.     

Hofer'sL ∞-geometry: energy and stability of Hamiltonian flows,part II

In this paper we first show that the necessary condition introduced in our previous paper is also a sufficient condition for a path to be a geodesic in the group Ham ^c (M) of compactly supported Hamiltonian symplectomorphisms. This applies with no restriction onM. We then discuss conditions which guarantee that such a path minimizes the Hofer length. Our argument relies on a general geometric construction (the gluing of monodromies) and on an extension of Gromov's non-squeezing theorem both to more general manifolds and to more general capacities. The manifolds we consider are quasi-cylinders, that is spaces homeomorphic toM×D ² which are symplectically ruled overD ². When we work with the usual capacity (derived from embedded balls), we can prove the existence of paths which minimize the length among all homotopic paths, provided thatM is semi-monotone. (This restriction occurs because of the well-known difficulty with the theory ofJ-holomorphic curves in arbitraryM.) However, we can only prove the existence of length-minimizing paths (i.e. paths which minimize length amongstall paths, not only the homotopic ones) under even more restrictive conditions onM, for example whenM is exact and convex or of dimension 2. The new difficulty is caused by the possibility that there are non-trivial and very short loops in Ham ^c (M). When such lengthminimizing paths do exist, we can extend the Bialy-Polterovich calculation of the Hofer norm on a neighbourhood of the identity (C ^l-flatness).Although it applies to a more restricted class of manifolds, the Hofer-Zehnder capacity seems to be better adapted to the problem at hand, giving sharper estimates in many situations. Also the capacity-area inequality for split cylinders extends more easily to quasi-cylinders in this case. As applications, we generalise Hofer's estimate of the time for which an autonomous flow is length-minimizing to some manifolds other thanR ²ⁿ , and derive new results such as the unboundedness of Hofer's metric on some closed manifolds, and a linear rigidity result.Oblatum 13-X-1994 & 8-V-1995Partially supported by NSERC grant OGP 0092913 and FCAR grant ER-1199Partially supported by NSF grant DMS 9103033 and NSF Visiting Professorship for Women GER 9350075     

Relative Chebyshev centers in normed linear spaces, part II

Let E be a normed linear space, A a bounded set in E, and G an arbitrary set in E. The relative Chebyshev center of A in G is the set of points in G best approximating A. We have obtained elsewhere general results characterizing the spaces in which the center reduces to a singleton in terms of structural properties related to uniform and strict convexity. In this paper, an analysis of the Chebyshev norm case, which falls outside the scope of the previous analysis, is presented.