In this paper, we give a survey of various sphere theorems in geometry. These include the topological sphere theorem of Berger and Klingenberg as well as the differentiable version obtained by the authors. These theorems employ a variety of methods, including geodesic and minimal surface techniques as well as Hamilton’s Ricci flow. We also obtain here new results concerning complete manifolds w...

The strong coupling, multivariate, and nonlinear characteristics of the mathematical model a doubly-fed induction generator (DFIG) make it challenging to analyze bifurcation category DFIG. Therefore, in this study, we developed method for analyzing DFIG via topologically equivalent dimensionality reduction neighborhood values based on central manifold theorem realized discrimination Hopf type W...

In this paper we present a new topological tool which allows to prove the existence of Shilnikov homoclinic or heteroclinic solutions. We present an application of this method to the Michelson system y′′′ + y′ + 0.5y = c [16]. We prove that there exists a countable set of parameter values c for which a pair of the Shilnikov homoclinic orbits to the equilibrium points (±c√2, 0, 0) appear. This r...

We introduce the notion of the projective linking number LinkP(Γ, Z) of a compact oriented real submanifold Γ of dimension 2p − 1 in complex projective n-space P with an algebraic subvariety Z ⊂ P − Γ of codimension p. This notion is related to projective winding numbers and quasi-plurisubharmonic functions, and it generalizes directly from P to any projective manifold. Part 1 of this paper est...

We present a method for the solution of the Dynamic Programming Equation (DPE) that arises in an infinite horizon optimal control problem. The method extends to the discrete time version of Al’brecht’s procedure for locally approximating the solution of the Hamilton Jacobi Bellman PDE. Assuming that the dynamics and cost are C(R) and C(R) smooth, respectively, we explicitly find expansions of t...

Theorem 1. Let X be a Poincare pair of dimension n ≥ 5, ζ a stable PL bundle on X, and f : M → X a degree one normal map, where M is a PL manifold. Let σ f ∈ Ω∞Lvq(X, ζX) be the relative signature of f , and suppose we are given a path p from σ f to the base point of Ω ∞Lvq(X, ζX). (We can identify such a path with a Lagrangian in the Poincare object representing σ f , which is well-defined up ...

In this paper, a delayed plankton-nutrient interaction model consisting of phytoplankton, zooplankton and dissolved nutrient is considered. It is assumed that some species of phytoplankton releases toxin (known as toxin producing phytoplankton (TPP)) which is harmful for zooplankton growth and this toxin releasing process follows a discrete time variation. Using delay as bifurcation parameter, ...

This paper proves the following conjecture of Boyer and Zhang: If a small hyperbolic knot in a homotopy sphere has a non-trivial cyclic surgery slope r, then it has an incompressible surface with non-integer boundary slope strictly between r − 1 and r + 1. I state this result below as Theorem 4.1, after giving the background needed to understand it. Corollary 1.1 of Theorem 4.1 is that any smal...

Harmonic maps are natural generalizations of harmonic functions and are critical points of the energy functional defined on the space of maps between two Riemannian manifolds. The Liouville type properties for harmonic maps have been studied extensively in the past years (Cf. [Ch], [C], [EL1], [EL2], [ES], [H], [HJW], [J], [SY], [S], [Y1], etc.). In 1975, Yau [Y1] proved that any harmonic funct...