The purpose of this paper is to demonstrate that the index of a vector field on a smooth manifold is an intrinsic invariant of the manifold. The paper is divided into three sections. The first section introduces the Inverse Function Theorem and provides the proof for the Implicit Function Theorem, laying the foundation for our discussion. In the second section we will introduce the concept of a...

Object of the present paper is to study pseudo Hermitian magnetic curves in (κ,µ) manifold admitting Zamkovoy connection. We give main classification theorem for pseudo-Hermitian curve. Again we find curvature and torsion a (κ,µ)-manifold

We recall the complex structure on generalized loop spaces Wk,2(S,X), where S is a compact real manifold with boundary and X manifold, prove Hartogs-type extension theorem for holomorphic maps from certain domains in spaces.

This paper is devoted to the study of the stability of limit cycles of a nonlinear delay differential equation with a distributed delay. The equation arises from a model of population dynamics describing the evolution of a pluripotent stem cells population. We study the local asymptotic stability of the unique nontrivial equilibrium of the delay equation and we show that its stability can be lo...

In this study, a model of active control system with time delay feedback is investigated, and time delays were set in control loop. An efficient method was used to analyze the stability of the system. By solving the characteristic equation, the local stability and the existence of codimension one bifurcation (Hopf bifurcation and fold bifurcation) were obtained. Regarding the time delay as bifu...

1. Background: the H. Federer-W. Fleming theory of currents 3 2. Codimension 1 regularity theory 6 2.1. Excess and -regularity theorem 6 2.2. Lipschitz approximation and comparison with harmonic functions 7 2.3. Monotonicity and tangent cones 8 2.4. Persistence of singularities and dimension reduction 8 3. Codimension n > 1 regularity theory 10 3.1. The basic examples and the new difficulties 1...

In this paper, a system of three globally coupled limit cycle oscillators with a linear time-delayed coupling are investigated. Considering the delay as a parameter, we also study the effect of time delay on the dynamics. Next, Hopf bifurcations induced by time delays using the normal form theory and center manifold reduction are obtained. Based on the symmetric Hopf bifurcation theorem, we inv...

We study a class of discrete-time bidirectional ring neural network model with delay. We discuss the asymptotic stability of the origin and the existence of Neimark-Sacker bifurcations, by analyzing the corresponding characteristic equation. Employing M-matrix theory and the Lyapunov functional method, global asymptotic stability of the origin is derived. Applying the normal form theory and the...

We show that for generic Riemannian metrics on a simply-connected closed spin manifold of dimension ≥ 5 the dimension of the space of harmonic spinors is no larger than it must be by the index theorem. The same result holds for periodic fundamental groups of odd order. The proof is based on a surgery theorem for the Dirac spectrum which says that if one performs surgery of codimension ≥ 3 on a ...

The purpose of this paper is to study bifurcation of an Internet congestion control algorithm, namely REM (Random Exponential Marking) algorithm, with communication delay. By choosing the delay constant as a bifurcation parameter, we prove that REM algorithm exhibits Hopf bifurcation. The formulas for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic so...