Weakly reducible H′-splittings of 3-manifolds

We introduce the [Formula: see text]-splittings for 3-manifolds as follows. For a compact connected surface [Formula: see text] properly embedded in a compact connected orientable 3-manifold [Formula: see text], if [Formula: see text] decomposes [Formula: see text] into two handlebodies [Formula: see text] and [Formula: see text], then [Formula: see text] is called an [Formula: see text]-splitting for [Formula: see text]. Clearly, when [Formula: see text] is closed, this is just the Heegaard splitting for [Formula: see text]; when [Formula: see text] is with boundary, the [Formula: see text]-splitting for [Formula: see text] is different from the Heegaard splitting for [Formula: see text]. In this paper, we first show that any compact connected orientable 3-manifold admits an [Formula: see text]-splitting, then generalize Casson–Gordon theorem on weakly reducible Heegaard splitting to the [Formula: see text]-splitting case in the following version: if [Formula: see text] is a weakly reducible [Formula: see text]-splitting for a compact connected orientable 3-manifold [Formula: see text], then (1) [Formula: see text] contains an incompressible closed surface of positive genus or (2) the [Formula: see text]-splitting [Formula: see text] is reducible or (3) there is an essential 2-sphere [Formula: see text] in [Formula: see text] such that [Formula: see text] is a collection of essential disks in [Formula: see text] and [Formula: see text] is an incompressible and not boundary parallel planar surface in [Formula: see text] with at least two boundary components, where [Formula: see text] or (4) [Formula: see text] is stabilized.

Pre-service physics teachers’ mental models about the electric field

This paper identifies pre-service physics teachers’ mental models of the concept of the electric field. The models were determined by means of five contexts all of which were supported with sets of experiments. The contexts examined were (1) the effect of the electric field on the insulator, (2) the comparison of the conductor and insulator in the electric field, (3) the effect of the electric field on the neutral conductor and insulator, (4) the effect of the electric field on the conductor liquid, and (5) the effect of the conductor and insulators materials forming a closed surface on the electric field. Semi-structured interviews related to the contexts were conducted with the 22 pre-service physics teachers. The data collected throughout the interviews were put to content analysis and thus, the pre-service teachers’ mental models were identified. In total, six mental models were identified. One model was a scientific model (Scientific Model of the Electric Field (SMEF)) and five of which were unscientific models (Magnetic-Based Field Model (MBFM), Mechanical Wave Model (MWM), Material Independent Field Model (MIFM), Force-Free Field Model (FFFM) and Force-Based Field Model (FBFM)) were identified. It became apparent as a result of document analysis that several unscientific mental models were also included in resource books. Approximately one and a half years later, almost all students were interviewed again about the contexts so as to find whether or not their models were permanent or not. Following the interviews, their mental models were found to be quite permanent and to be time-independent.

Implementing Self Organising Map to Organise the Unstructured Data

Surface reconstruction is significant in reverse engineering because it should present the correct surface with minimum error using the data available. It has become a challenging process when the data are in the unstructured type and the existing methods are still suffering from accuracy issues. The unstructured data will produce an incorrect surface because there is no connectivity information among the data. So, the unstructured data should undergo the organising process to obtain the correct shape. The Self Organising Map (SOM) has been extensively applied in previous works to solve surface reconstruction problems. However, the performance of the SOM models has remained uncertain. It can be evaluated and tested using different types of data sets. The objectives of this research are to examine the performance and to determine the weaknesses of SOM models. 2D SOM, 3D SOM, and Cube Kohonen (CK) SOM models are investigated and tested using three data sets in this research. As shown in the experimental results, the CKSOM model has proved to perform better because it can represent the correct closed surface with the lowest minimum error.

The Topology of the Configuration Space of a Mathematical Model for Cycloalkenes

As a mathematical model for cycloalkenes, we consider equilateral polygons whose interior angles are the same except for those of the both ends of the specified edge. We study the configuration space of such polygons. It is known that for some case, the space is homeomorphic to a sphere. The purpose of this chapter is threefold: First, using the h-cobordism theorem, we prove that the above homeomorphism is in fact a diffeomorphism. Second, we study the best possible condition for the space to be a sphere. At present, only a sphere appears as a topological type of the space. Then our third purpose is to show the case when a closed surface of positive genus appears as a topological type.

On geodesically reversible Finsler manifolds

A Finsler manifold is said to be geodesically reversible if the reversed curve of any geodesic remains a geometrical geodesic. Well-known examples of geodesically reversible Finsler metrics are Randers metrics with closed [Formula: see text]-forms. Another family of well-known examples are projectively flat Finsler metrics on the [Formula: see text]-sphere that have constant positive curvature. In this paper, we prove some geometrical and dynamical characterizations of geodesically reversible Finsler metrics, and we prove several rigidity results about a family of the so-called Randers-type Finsler metrics. One of our results is as follows: let [Formula: see text] be a Riemannian–Finsler metric on a closed surface [Formula: see text], and [Formula: see text] be a small antisymmetric potential on [Formula: see text] that is a natural generalization of [Formula: see text]-form (see Sec. 1). If the Randers-type Finsler metric [Formula: see text] is geodesically reversible, and the geodesic flow of [Formula: see text] is topologically transitive, then we prove that [Formula: see text] must be a closed [Formula: see text]-form. We also prove that this rigidity result is not true for the family of projectively flat Finsler metrics on the [Formula: see text]-sphere of constant positive curvature.

Existence of Incompressible Vortex-Class Phenomena and Variational Formulation of Raleigh–Plesset Cavitation Dynamics

The following article extends a decomposition to the Navier–Stokes Equations (NSEs) demonstrated in earlier studies by corresponding author, in order to now demonstrate the existence of a vortex elliptical set inherent to the NSEs. These vortice elliptical sets are used to comment on the existence of solutions relative to the NSEs and to identify a potential manner of investigation into the classical Millennial Problem encompassed in Fefferman’s presentation. The article also presents the utilization of a recently developed versatile variational framework by both authors in order to study a related fluid-mechanics phenomena, namely the Raleigh–Plesset equations, which are ultimately obtained from the NSEs. The article develops, for the first time, a Lagrangian density functional for a closed surface which when minimized produced the Raleigh–Plesset equations. The article then proceeds with the demonstration that the Raleigh–Plesset equations may be obtained from thisenergy functional and identifies the energy dissipation predicted by the proposed Lagrangian density. The importance of the novel Raleigh–Plesset functional in the greater scheme of fluid mechanics is commented upon.

A Li–Yau inequality for the 1-dimensional Willmore energy

By the classical Li–Yau inequality, an immersion of a closed surface in ℝ n {\mathbb{R}^{n}} with Willmore energy below 8 ⁢ π {8\pi} has to be embedded. We discuss analogous results for curves in ℝ 2 {\mathbb{R}^{2}} , involving Euler’s elastic energy and other possible curvature functionals. Additionally, we provide applications to associated gradient flows.

Nematic–Isotropic Phase Transition in Liquid Crystals: A Variational Derivation of Effective Geometric Motions

In this work, we study the nematic–isotropic phase transition based on the dynamics of the Landau–De Gennes theory of liquid crystals. At the critical temperature, the Landau–De Gennes bulk potential favors the isotropic phase and nematic phase equally. When the elastic coefficient is much smaller than that of the bulk potential, a scaling limit can be derived by formal asymptotic expansions: the solution gradient concentrates on a closed surface evolving by mean curvature flow. Moreover, on one side of the surface the solution tends to the nematic phase which is governed by the harmonic map heat flow into the sphere while on the other side, it tends to the isotropic phase. To rigorously justify such a scaling limit, we prove a convergence result by combining weak convergence methods and the modulated energy method. Our proof applies as long as the limiting mean curvature flow remains smooth.

Near-to-Far Field Transformation in FDTD: A Comparative Study of Different Interpolation Approaches

Equivalence theorems in electromagnetic field theory stipulate that farfield radiation pattern/scattering profile of a source/scatterer can be evaluated from fictitious electric and magnetic surface currents on an equivalent imaginary surface enclosing the source/scatterer. These surface currents are in turn calculated from tangential (to the equivalent surface) magnetic and electric fields, respectively. However, due to the staggered-in-space placement of electric and magnetic fields in FDTD Yee cell, selection of a single equivalent surface harboring both tangential electric and magnetic fields is not feasible. The work-around is to select a closed surface with tangential electric (or magnetic) fields and interpolate the neighboring magnetic (or electric) fields to bring approximate magnetic (or electric) fields onto the same surface. Interpolation schemes available in the literature include averaging, geometric mean and the mixed-surface approach. In this work, we compare FDTD farfield scattering profiles of a dielectric cube calculated from surface currents that are obtained using various interpolation techniques. The results are benchmarked with those obtained from integral equation solvers available in the commercial packages FEKO and HFSS.