Some Topological Approaches for Generalized Rough Sets via Ideals

The idea of neighborhood systems is induced from the geometric idea of “near,” and it is primitive in the topological structures. Now, the idea of neighborhood systems has been extensively applied in rough set theory. The master contribution of this manuscript is to generate various topologies by means of the concepts of j -adhesion neighborhoods and ideals. Then, we define a new rough set model derived from these topologies and discussed main features. We show that these topologies are finer than those given in the previous ones under arbitrary binary relations. In addition, we elucidate that these topologies are finer than those topologies initiated based on different neighborhoods and ideals under reflexive relations. Several examples are provided to validate that our model is better than the previous ones.

Positive topological entropy of Reeb flows on spherizations

Let M be a closed manifold whose based loop space Ω (M) is “complicated”. Examples are rationally hyperbolic manifolds and manifolds whose fundamental group has exponential growth. Consider a hypersurface Σ in T*M which is fiberwise starshaped with respect to the origin. Choose a function H : T*M → ℝ such that Σ is a regular energy surface of H, and let ϕt be the restriction to Σ of the Hamiltonian flow of H.Theorem 1. The topological entropy of ϕt is positive.This result has been known for fiberwise convex Σ by work of Dinaburg, Gromov, Paternain, and Paternain–Petean on geodesic flows. We use the geometric idea and the Floer homological technique from [19], but in addition apply the sandwiching method. Theorem 1 can be reformulated as follows.Theorem 1'. The topological entropy of any Reeb flow on the spherization SM of T*M is positive.For q ∈ M abbreviate Σq = Σ ∩ Tq*M. The following corollary extends results of Morse and Gromov on the number of geodesics between two points.Corollary 1. Given q ∈ M, for almost every q′ ∈ M the number of orbits of the flow ϕt from Σq to Σq′ grows exponentially in time.In the lowest dimension, Theorem 1 yields the existence of many closed, orbits.Corollary 2. Let M be a closed surface different from S2, ℝP2, the torus and the Klein bottle. Then ϕt carries a horseshoe. In particular, the number of geometrically distinct closed orbits grows exponentially in time.

Inflections on the Bedroom Floor

When introducing mathematical concepts in the classroom, students' interest and appreciation are heightened when examples are drawn from everyday life. Recently, while discussing points of inflection with a calculus class, I was challenged to find an example. Surveying the lecture theater, I was unable to find a single example of this very simple geometric idea.

Invariant manifolds in singular perturbation problems for ordinary differential equations

SynopsisBased on Fenichel's geometric idea, invariant manifold theory is applied to singular perturbation problems. This approach clarifies the nature of outer and inner solutions. A specific condition is given to ensure the existence of heteroclinic connections between normally hyperbolic invariant manifolds. A method to approximate the connections is also presented.

Research Report: Forming Geometric Concepts

Geometric idea are vital: we live in a three-dimensional world. Yet, for many children, in truction in geometry in the elementary school revolves around only two points: recognition of hape and development of vocabulary. Re earch provide some evidence on how these points are taught but also indicate other ideas that can be developed.

Recreation: Binary Grids

A problem that arises in one area of mathematics is often found to be related to another area or to be translatable into the language of another area. If this sort of translation is feasible or can be discovered, then it often has the effect of providing insights and symbolic clarity. Very often the problem arises from the need to translate an essentially geometric idea into some form of nonpictorial record.