Physical Sciences and Mathematics Commons^™

Shake Slice And Shake Concordant Links, Anthony Bosman

Faculty Publications

© 2020 World Scientific Publishing Company. We can construct a 4-manifold by attaching 2-handles to a 4-ball with framing r along the components of a link in the boundary of the 4-ball. We define a link as r-shake slice if there exists embedded spheres that represent the generators of the second homology of the 4-manifold. This naturally extends r-shake slice, a generalization of slice that has previously only been studied for knots, to links of more than one component. We also define a relative notion of shaker-concordance for links and versions with stricter conditions on the embedded spheres that we …

Classification Of Rectifying Space-Like Submanifolds In Pseudo-Euclidean Spaces, Bang Yen Chan, Yun Myung Oh

Faculty Publications

The notions of rectifying subspaces and of rectifying submanifolds were introduced in [B.-Y. Chen, Int. Electron. J. Geom 9 (2016), no. 2, 1–8]. More precisely, a submanifold in a Euclidean m-space Em is called a rectifying submanifold if its position vector field always lies in its rectifying subspace. Several fundamental properties and classification of rectifying submanifolds in Euclidean space were obtained in [B.-Y. Chen, op. cit.]. In this present article, we extend the results in [B.-Y. Chen, op. cit.] to rectifying space- like submanifolds in a pseudo-Euclidean space with arbitrary codimension. In particular, we completely classify all rectifying space-like submanifolds …

Characterization Of Rectifying And Sphere Curves In R^3, Yun Myung Oh, Julie Logan

Faculty Publications

Studies of curves in 3D-space have been developed by many geometers and it is known that any regular curve in 3D space is completely determined by its curvature and torsion, up to position. Many results have been found to characterize various types of space curves in terms of conditions on the ratio of torsion to curvature. Under an extracondition on the constant curvature, Y.L. Seo and Y. M. Oh found the series solution when the ratio of torsion to curvature is a linear function. Furthermore, this solution is known to be a rectifying curve by B. Y. Chen’s work. This …

A General Elliptic Nonlinear System Of Multiple Functions With Application, Timothy Robertson, Joon Hyuk Kang

Faculty Publications

The purpose of this paper is to give a sufficient condition for the existence, nonexistence and uniqueness of positive solutions to a rather general type of elliptic system of the Dirichlet problem on a bounded domain Ω in Rn . We also investigate the effects of perturbation on the positive solutions to the system. The techniques used in this paper are upper-lower solutions, eigenvalues of operators, the maximum principles and spectrum estimates. The arguments also rely on some detailed properties for the solution of logistic equations. This result yields an algebraically computable criterion for the positive coexistence of competing species …

A General Elliptic Nonlinear System Of Two Functions With Application, Timothy Robertson, Joon Hyuk Kang

Faculty Publications

The purpose of this paper is to give a sufficient condition for the existence and nonexistence of positive solutions to a rather general type of elliptic system of the Dirichlet problem on the bounded domain Ω in Rn. Also considered are the effects of perturbations on the coexistence state and uniqueness. The techniques used in this paper are upper-lower solutions, eigenvalues of operators, maximum principles and spectrum estimates. The arguments also rely on some detailed properties for the solution of logistic equations. These results yield an algebraically computable criterion for the positive coexistence of competing species of animals in many …

Growth Conditions For Uniqueness Of Smooth Positive Solutions To An Elliptic Model, Joon Hyuk Kang

Faculty Publications

The uniqueness of positive solution to the elliptic model

∆u + u[a + g(u, v)] = 0 in Ω, ∆v + v[a + h(u, v)] = 0 in Ω, u = v = 0 on ∂Ω,

were investigated.

A Curve Satisfying T/K=S With Constant K>0, Yun Myung Oh, Ye Lim Seo

Faculty Publications

In the present paper, we investigate a space curve in which the curvature is constant and the torsion is a linear function. The aim of this paper is to find an explicit formula for this space curve when the ratio of the torsion to the curvature is a linear function when the curvature is constant.

Smooth Positive Solutions To An Elliptic Model With C2 Functions, Joon Hyuk Kang

Faculty Publications

Two species of animals are competing or cooperating in the same environment. Under what conditions do they coexist peacefully? Or under what conditions does either one of the two species become extinct, that is, is either one of the two species excluded by the other? We investigate this phenomena in mathematical point of view.

In this paper we concentrate on coexistence solutions of the competition or cooperation model

This system is the general model for the steady state of a competitive or cooperative interacting system depending on growth conditions for and . The techniques used in this paper are elliptic …

Measuring A Circle: A Math Lesson For Grades 5-10, Robert C. Moore

Faculty Publications

This article is designed to promote teaching methods that engage students in active learning and result in deep conceptual understanding by offering a sample lesson to help students (grades 5-10, ages 10-15) answer questions about and gain a deeper understanding of how to measure the circumference and area of a circle.

Riemannian Submersions And Lagrangian Isometric Immerson 1, Yun Myung Oh

Faculty Publications

In [1], it has shown that if a Riemannian manifold admits a non- trivial Riemannian submersion with total geodesic fibers, then it cannot be isometrically immersed in any Riemannian manifold of non-positive sectional curvature as a minimal submanifold. In this paper, we consider a nontrivial Riemannian submersion and investigate some properties on Lagrangian iso- metric immersions using the submersion invariant.

Non-Negative Steady State Solutions To An Elliptic Biological Model, Brian Ibanez, Joon Hyuk Kang, Jungho Lee

Faculty Publications

The non-existence and existence of positive solutions for the generalized predator-prey biological model for two species of animals Δu + ug(u,v) = 0 in Ω, Δv + vh(u,v) = 0 in Ω, u = v = 0 on ∂Ω, is investigated in this paper. The techniques used in this paper are from elliptic theory, the upper-lower solution method, the maximum principles and spectrum estimates. The arguments also rely on detailed properties of solutions to logistic equations. © 2009 Academic Publications.

A Construction Of Lagrangian Submanifolds In Complex Euclidean Spaces With Legendre Curves, Yun Myung Oh

Faculty Publications

In [1], B. Y. Chen provided a new method to construct Lagrangian surfaces in C2 by using Legendre curves in S3(1)C2. In this paper, we investigate the similar methods to construct some Lagrangian submanifolds in complex Euclidean spaces Cn (n≥b3).

Anatomy Of A Chaotic Attractor: Subtle Model-Predicted Patterns Revealed In Population Data, Aaron A. King, R. F. Costantino, J. M. Cushing, Shandelle M. Henson, Robert A. Desharnais, Brian Dennis

Faculty Publications

Mathematically, chaotic dynamics are not devoid of order but display episodes of near-cyclic temporal patterns. This is illustrated, in interesting ways, in the case of chaotic biological populations. Despite the individual nature of organisms and the noisy nature of biological time series, subtle temporal patterns have been detected. By using data drawn from chaotic insect populations, we show quantitatively that chaos manifests itself as a tapestry of identifiable and predictable patterns woven together by stochasticity. We show too that the mixture of patterns an experimentalist can expect to see depends on the scale of the system under study.

Explaining And Predicting Patterns In Stochastic Population Systems, Shandelle M. Henson, Aaron A. King, R. F. Costantino, J. M. Cushing, Brian Dennis, Robert A. Desharnais

Faculty Publications

Lattice effects in ecological time-series are patterns that arise because of the inherent discreteness of animal numbers. In this paper, we suggest a systematic approach for predicting lattice effects. We also show that an explanation of all the patterns in a population time-series may require more than one deterministic model, especially when the dynamics are complex.

A Sufficient Condition For The Uniqueness Of Positive Steady State To A Reaction Diffusion System, Joon Hyuk Kang, Yun Oh

Faculty Publications

In this paper, we concentrate on the uniqueness of the positive solution for the general elliptic system { Δu + u(g1(u) - g 2(v)) = 0 in R+ × Ω, Δν + v(h 1(u) - h2(v)) = 0 u|∂Ω = v| ∂Ω = 0. This system is the general model for the steady state of a competitive interacting system. The techniques used in this paper are upper-lower solutions, maximum principles and spectrum estimates. The arguments also rely on some detailed properties for the solution of logistic equations.