We present a sketch of the proof of the following theorems: (1) Every 3-manifold has only finitely many homotopy classes of 2-plane fields which carry tight contact structures. (2) Every closed atoroidal 3-manifold carries finitely many isotopy classes of tight contact structures. In this article we explain how to normalize tight contact structures with respect to a fixed triangulation. Using t...

Abstract A diffusive phytoplankton–zooplankton model with nonlinear harvesting is considered in this paper. Firstly, using the as parameter, we get existence and stability of positive steady state, also investigate spatially homogeneous inhomogeneous periodic solutions. Then, by applying normal form theory center manifold theorem, give direction Hopf bifurcation from state. In addition, prove B...

a famous theorem of schur states that for a group $g$ finiteness of $g/z(g)$‎ ‎implies the finiteness of $g'.$ the converse of schur's theorem is an interesting problem which has been considered by some‎ ‎authors‎. ‎recently‎, ‎podoski and szegedy proved the truth of the converse of schur's theorem for capable groups‎. ‎they also established an explicit bound for the index of the center of such...

In smooth manifold theory, the notion of a tangent space makes it possible for differentiation to take place on an abstract manifold. In this paper, the notion of a distribution will be presented which makes it possible for integration to take place on an abstract manifold. The first section introduces terminology and builds intuition via an analogy to the concept of integral curves. The second...

Using a nonlocal version of the center-manifold theorem and normal form reduction, we prove existence small-amplitude generalized solitary-wave solutions modulated to steady gravity-capillary Whitham equation with weak surface tension. Through application theorem, for solitary wave profiles is reduced four-dimensional system ODEs inheriting reversibility. Along particular parameter curves, rela...

<abstract><p>In this paper, we use a semi-discretization method to explore predator-prey model with Michaelis-Menten functional response. Firstly, investigate the local stability of fixed points. Then, by using center manifold theorem and bifurcation theory, demonstrate that system experiences flip Neimark-Sacker at point when one parameters goes through its critical value. To illus...