Geometric Objects In Terms Research Articles

Discrete formulation for the dynamics of rods deforming in space

The movement of rods in an Euclidean space can be described as a field theory on a principal bundle. The dynamics of a rod is governed by partial differential equations that may have a variational origin. If the corresponding smooth Lagrangian density is invariant by some group of transformations, there exist the corresponding conserved Noether currents. Generally, numerical schemes dealing with PDEs fail to reflect these conservation properties. We describe the main ingredients needed to create, from the smooth Lagrangian density, a variational principle for discrete motions of a discrete rod, with the corresponding conserved Noether currents. We describe all geometrical objects in terms of elements on the linear Atiyah bundle using a reduced forward difference operator. We show how this introduces a discrete Lagrangian density that models the discrete dynamics of a discrete rod. The presented tools are general enough to represent a discretization of any variational theory in principal bundles, and its simplicity allows us to perform an iterative integration algorithm to compute the discrete rod evolution in time, starting from any predefined configurations of all discrete rod elements at initial times.

An Investigation of 8th Grade Students’ Knowledge on Geometrical Objects in Terms of Van Heile Levels of Understanding Geometry

In this study, it was aimed to determine 8th grade students’ Van Hiele levels of understanding geometry and to examine their knowledge on geometric objects in terms of Van Hiele levels of understanding geometry. Survey model, a quantitative research design was used in the research. The study group consists of 161 students from three different schools of National Ministry of Education located in the districts of Melikgazi and Kocasinan in Kayseri province. The Van Hiele Geometry Test and the Geometric objects Achievement Test developed by the researcher was used to collect the data of the study. Some of the results obtained from the research are as follows: The Van Hiele levels of understanding geometry of the students who participated in the research were found to be lower compared to what they are expected to have at the present. There was a high correlation between the scores of the students on both tests. Significant mean differences were found among the schools in terms of both the Van Hiele Geometry Test and the Geometric Objects Achievement Test. When the data were analyzed by gender variable, there were significant differences in favor of girl students for both tests.

Review Article

The isolated horizon formalism recently introduced by Ashtekar et al. aims at providing a quasi-local concept of a black hole in equilibrium in an otherwise possibly dynamical spacetime. In this formalism, a hierarchy of geometrical structures is constructed on a null hypersurface. On the other side, the 3+1 formulation of general relativity provides a powerful setting for studying the spacetime dynamics, in particular gravitational radiation from black hole systems. Here we revisit the kinematics and dynamics of null hypersurfaces by making use of some 3+1 slicing of spacetime. In particular, the additional structures induced on null hypersurfaces by the 3+1 slicing permit a natural extension to the full spacetime of geometrical quantities defined on the null hypersurface. This four-dimensional point of view facilitates the link between the null and spatial geometries. We proceed by reformulating the isolated horizon structure in this framework. We also reformulate previous works, such as Damour's black hole mechanics, and make the link with a previous 3+1 approach of black hole horizon, namely the membrane paradigm. We explicit all geometrical objects in terms of 3+1 quantities, putting a special emphasis on the conformal 3+1 formulation. This is in particular relevant for the initial data problem of black hole spacetimes for numerical relativity. Illustrative examples are provided by considering various slicings of Schwarzschild and Kerr spacetimes.

Projective differential geometry of higher reductions of the two-dimensional Dirac equation

We investigate reductions of the two-dimensional Dirac equation imposed by the requirement of the existence of a differential operator D n of order n mapping its eigenfunctions to adjoint eigenfunctions. For first order operators these reductions (and multicomponent analogs thereof) lead to the Lame equations descriptive of orthogonal coordinate systems. Our main observation is that nth order reductions coincide with the projective-geometric ‘Gauss–Codazzi’ equations governing special classes of line congruences in the projective space P 2 n−1 , which is the projectivised kernel of D n . In the second order case this leads to the theory of W-congruences in P 3 which belong to a linear complex, while the third order case corresponds to isotropic congruences in P 5. Higher reductions are compatible with odd order flows of the Davey–Stewartson hierarchy. All these flows preserve the kernel D n , thus defining nontrivial geometric evolutions of line congruences. Multicomponent generalizations are also discussed. The correspondence between geometric picture and the theory of integrable systems is established; the definition of the class of reductions and all geometric objects in terms of the multicomponent KP hierarchy is presented. Generating forms for reductions of arbitrary order are constructed.

Executive control of cognitive processes in task switching.

In 4 experiments, participants alternated between different tasks or performed the same task repeatedly. The tasks for 2 of the experiments required responding to geometric objects in terms of alternative classification rules, and the tasks for the other 2 experiments required solving arithmetic problems in terms of alternative numerical operations. Performance was measured as a function of whether the tasks were familiar or unfamiliar, the rules were simple or complex, and visual cues were present or absent about which tasks should be performed. Task alternation yielded switching-time costs that increased with rule complexity but decreased with task cuing. These factor effects were additive, supporting a model of executive control that has goal-shifting and rule-activation stages for task switching. It appears that rule activation takes more time for switching from familiar to unfamiliar tasks than for switching in the opposite direction.

Geometric Constructions in the Digital Plane

We adapt several important properties from affine geometry so that they become applicable in the digital plane. Each affine property is first reformulated as a property about line transversals. Known results about transversals are then used to derive Helly type theorems for the digital plane. The main characteristic of a Helly type theorem is that it expresses a relation holding for a collection of geometric objects in terms of simpler relations holding for some of the subcollections. For example, we show that in the digital plane a collection of digital lines is parallel if and only if each of its 2-membered subcollections consists of two parallel digital lines. The derived Helly type theorems lead to many applications in digital image processing. For example, they provide an appropriate setting for verifying whether lines detected in a digital image satisfy the constraints imposed by a perspective projection. The results can be extended to higher dimensions or to other geometric systems, such as projective geometry.

Learning of Geometric Concepts in a Logo Environment

To investigate the effects of computer programming in Logo on specific geometric conceptualizations of primary grade children, 48 third graders were randomly assigned to either a Logo or a control group. The Logo group was given 26 weeks of instruction in a Logo environment. The children were then interviewed to ascertain their conceptualizations of angles, shapes, and motions. In both groups children's notions of angle and angle measure were multifaceted and included a number of misconceptions, although performance was uniformly higher for the Logo group. The Logo children were more aware than the control children of the components of geometric shapes and were more likely to conceptualize geometric objects in terms of the actions or procedures used to construct them.

Supersymmetric models for quarks and leptons with nonlinearly realized E

We propose three supersymmetric nonlinear sigma models with global symmetry E/sub 8/. The models can accommodate three left-handed families of quarks and leptons without incurring the Adler-Bell-Jackiw anomaly with respect to either the standard SU(3) x SU(2) x U(1) gauge group, or the SU(5), or SO(10) grand unifying gauge group. They also predict unambiguously a right-handed, fourth family of quarks and leptons. In order to explore the structure of the models, we develop a differential-form formulation of the Kahler manifolds, resulting in general expressions for the curvature tensors and other geometrical objects in terms of the structure constants of the algebra, and the squashing parameters. These results, in turn, facilitate a general method for determining the Lagrangian to quartic order, and so the structure of the inherent four-fermion interactions of the models. We observe that the Kahlerian condition d..omega.. = 0 on the fundamental two-form ..omega.. greatly reduces the number of the independent squashing parameters. We also point out two plausible mechanisms for symmetry breaking, involving gravity.