Normal surface theory, a tool to represent surfaces in a triangulated 3-manifold combinatorially, is ubiquitous in computational 3-manifold theory. In this paper, we investigate a relaxed notion of normal surfaces where we remove the quadrilateral conditions. This yields normal surfaces that are no longer embedded. We prove that it is NP-hard to decide whether such a surface is immersed. Our pr...

We give a brief introduction to re-publication of Chapter 2 of the author’s 1964 Dissertation. The result has come to be known as the Neimark-Sacker bifurcation theorem. Some of the applications of the theorem are cited and a brief overview of the reduction of the mapping to normal form is given in 3-dimensions. By introducing weighted monomials the reduction method is carried out without the u...

Following Almgren’s construction of the center manifold in his Big regularity paper, we show the C regularity of area-minimizing currents in the neighborhood of points of density one, without using the nonparametric theory. This study is intended as a first step towards the understanding of Almgren’s construction in its full generality. 0. Introduction In this note we consider area-minimizing i...

An analytic pair of dimension n and center V is a pair (V,M) where M is a complex manifold of (complex) dimension n and V ⊂ M is a closed totally real analytic submanifold of dimension n. To an analytic pair (V,M) we associate the class U (V,M) of the functions u : M → [0,π/4[ which are plurisubharmonic in M and such that u(p)= 0 for each p ∈ V . If U (V,M) admits a maximal function u, the trip...

Abstract This topic presents a study on host–parasitoid model with Holling type III functional response. In population dynamics, when host density rises, the parasitoid response initially accelerates due to parasitoid’s improved searching efficiency. However, above certain threshold, will reach saturation level influence of reducing handling time. Thus, we incorporated into characterize such ph...

The plant-pollinator model is a common widely researched by scholars in population dynamics. In fact, its complex dynamical behaviors are universally and simply expressed as class of delay differential-difference equations. this paper, based on several early models, we consider with two combined delays to further describe the mutual constraints between populations under different time qualitati...

In its original form, the Atiyah-Singer Index Theorem equates two global quantities of a closed manifold, one analytic (the index of an elliptic operator) and one topological (a characteristic number). Because it relates invariants from different branches of mathematics, the Index Theorem has many applications and extensions to differential geometry, K-theory, mathematical physics, and other fi...