A DIFFERENTIABLE SPHERE THEOREM WITH PINCHING INTEGRAL RICCI CURVATURE

In this article, we introduce the Hausdorff convergence to derive a differentiable sphere theorem which shows an interesting rigidity phenomenon on some kind of manifolds.

ON LEVIN TYPE COMPARISON THEOREMS FOR CERTAIN SECOND ORDER DIFFERENTIAL EQUATIONS

In this paper we establish Levin type comparison theorems for certain second order differential equations. The results obtained here generalize and extend some of the earlier ones related to the Levin's comparison theorems.

OSCILLATION THEOREMS FOR SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS WITH DAMPING

Some new oscillation theorems are established for the second order nonlinear differential equations with damping of the form where p（t） and q（t） are allowed to change sign on [t0,∞).

A differentiable sphere theorem with positive Ricci curvature and reverse volume pinching

Let Mn be a compact, simply connected n (≥3)-dimensional Riemannian manifold without bound-ary and Sn be the unit sphere Euclidean space Rn+1. We derive a differentiable sphere theorem whenever themanifold concerned satisfies that the sectional curvature KM is not larger than 1, while Ric(M)≥n+2 4 and the volume V (M) is not larger than (1 + η)V (Sn) for some positive number η depending only on n.

An improved impulsive control approach to nonlinear systems with time-varying delays

A scheme for the impulsive control of nonlinear systems with time-varying delays is investigated in this paper. Based on the Lyapunov-like stability theorem for impulsive functional differential equations （FDEs）, some sufficient conditions are presented to guarantee the uniform asymptotic stability of impulsively controlled nonlinear systems with time-varying delays. These conditions are more effective and less conservative than those obtained. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed method.

On Skew Triangular Matrix Rings

In this paper, we prove the existence and uniqueness of positive solutions for a system of multi-order fractional differential equations. The system is used to represent constitutive relation for viscoelastic model of fractional differential equations. Our results are based on the fixed point theorems of increasing operator and the cone theory, some illustrative examples are also presented.

Extended tanh-function Method for Solving Traveling Wave Solutions of Nonlinear Kundu Equation

In this paper, we study the existence of multiple positive periodic solutions for the second order differential equation x′′（t） ＋ p（t）x′（t） ＋ q（t）x（t） = f（t, x（t））.By using Krasnoselskii fixed point theorem, we establish some criteria for the existence and multiple positive periodic solutions for this differential equation.

On the Spectrum of a Class of Differential Operators and Embedding Theorems

The author considers the embedding problem of weighted Sobolev spaces Hnp in weighted Ls spaces Ls,r,and some sufficient conditions and necessary conditions are given, when weight functions satisfy certain conditions.The author uses the results obtained to the qualitative analysis of the spectrum of 2n-order weighted differential operator,and gives some sufficient conditions and necessary conditions to ensure that the spectrum is discrete.

Observer design for induction motor： an approach based on the mean value theorem

In this paper, observer design for an induction motor has been investigated. The peculiarity of this paper is the synthesis of a mono-Luenberger observer for highly coupled system. To transform the nonlinear error dynamics for the induction motor into the linear parametric varying （LPV） system, the differential mean value theorem combined with the sector nonlinearity transformation has been used. Stability conditions based on the Lyapunov function lead to solvability of a set of linear matrix inequalities. The proposed observer guarantees the global exponential convergence to zero of the estimation error. Finally, the simulation results are given to show the performance of the observer design.

Classical state feedback controller for nonlinear systems using mean value theorem： closed Ioop-FOC of PMSM motor application

The problem of state feedback controllers for a class of Takagi-Sugeno （T-S） Lipschitz nonlinear systems is investigated. A simple systematic and useful synthesis method is proposed based on the use of the differential mean value theorem （DMVT） and convex theory. The proposed design approach is based on the mean value theorem （MVT） to express the nonlinear error dynamics as a convex combination of known matrices with time varying coefficients as linear parameter varying （LPV） systems. Using the Lyapunov theory, stability conditions are obtained and expressed in terms of linear matrix inequalities （LMIs）. The controller gains are then obtained by solving linear matrix inequalities. The effectiveness of the proposed approach for closed loop-field oriented control （CL-FOC） of permanent magnet synchronous machine （PMSM） drives is demonstrated through an illustrative simulation for the proof of these approaches. Furthermore, an extension for controller design with parameter uncertainties and perturbation performance is discussed.

Some Theorems in Affine Differential Geometry

In this paper we prove that an affine hypersphere with scalar curvature zero in a unimodular affine space of dimensionn+1 must be contained either in an elliptic paraboloid or in an affine image of the hypersurfacex 1 x 2...x n+1=const. We prove also that an affine complete, affine maximal surface is an elliptic paraboloid if its affine normals omit 4 or more directions in general position.

Mean Curvature Flow of Arbitrary Codimension in Complex Projective Spaces

Recently,Pipoli and Sinestrari[Pipoli,G.and Sinestrari,C.,Mean curvature flow of pinched submanifolds of CPn,Comm.Anal.Geom.,25,2017,799-846]initiated the study of convergence problem for the mean curvature flow of small codimension in the complex projective space CPm.The purpose of this paper is to develop the work due to Pipoli and Sinestrari,and verify a new convergence theorem for the mean curvature flow of arbitrary codimension in the complex projective space.Namely,the authors prove that if the initial submanifold in CPm satisfies a suitable pinching condition,then the mean curvature flow converges to a round point in finite time,or converges to a totally geodesic submanifold as t→∞.Consequently,they obtain a differentiable sphere theorem for submanifolds in the complex projective space.

A Rigidity Phenomenon on Riemannian Manifolds with Reverse Excess Pinching

The authors introduce the Hausdorff convergence to discuss the differentiable sphere theorem with excess pinching. Finally, a type of rigidity phenomenon on Riemannian manifolds is derived.

ON ALGEBRICO-DIFFERENTIAL EQUATIONS-SOLVING

The char-set method of polynomial equations-solving is naturally extended to the differential case which gives rise to an algorithmic method of solving arbitrary systems of algebrico-differential equations.As an illustration of the method,the Devil's Problem of Pommaret is solved in details.

ON THE TOPOLOGY,VOLUME,DIAMETER AND GAUSS MAP IMAGE OF SUBMANIFOLDS IN A SPHERE

In this paper, the author uses Gauss map to study the topology, volume and diameter of submanifolds in a sphere. It is proved that if there exist ε, 1≥ε > 0 and a fixed unit simple p-vector a such that the Gauss map g of an n-dimensional complete and connected submanifold M in Sn+p satisfies (g, a) ≥ε, then M is diffeomorphic to Sn, and the volume and diameter of M satisfy εnvol(Sn) ≤vol(M) ≤ vol(Sn)/ε and επ ≤diam(M) ≤ π/ε, respectively. The author also characterizes the case where these inequalities become equalities. As an application, a differential sphere theorem for compact submanifolds in a sphere is obtained.

Existence of p-mean Almost Periodic Mild Solution for Fractional Stochastic Neutral Functional Differential Equation

A class of fractional stochastic neutral functional differential equation is analyzed in this paper.With the utilization of the fractional calculations,semigroup theory,fixed point technique and stochastic analysis theory,a sufficient condition of the existence for p-mean almost periodic solution is obtained,which are supported by two examples.