Curve Shortening Flow in Arbitrary Dimensional Euclidian Space

In this paper, curve shortening flow in Euclidian space R^n（n≥3） is studied, and S. Altschuler＇s results about flow for space curves are generalized. We prove that the curve shortening flow converges to a straight line in infinite time if the initial curve is a ramp. We also prove the planar phenomenon when the curve shortening flow blows up.     

Rational holomorphic maps from B~2 into B~N with degree 2 Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

Rational proper holomorphic maps from the unit ball in C2 into the unit ball CN with degree 2 are studied. Any such map must be equivalent to one of the four types of maps.     

A Generalization of Eells-Sampson's Theorem

We generalize the well-known Eells-Sampson's theorem on the global existence and convergence for the heat flow of harmonic maps. The assumption that the curvature of the target manifold N be nonpoeitive is replaced by the weaker one requiring that the universal cover \tilde{N} admit a strictly convex function with quadratic growth.     

A New Proof of Hamilton's Theorem on Harmonic Maps from Manifolds with Boundary

We give a simple proof of the well-known Hamilton's result [1] on the heat flows and harmonic maps from manifolds with boundary using the approach of Ding-Lin [2].     

Effective cones of quotients of moduli spaces of stable -pointed curves of genus zero

Let , the moduli space of -pointed stable genus zero curves, and let be the quotient of by the action of on the last marked points. The cones of effective divisors , , are calculated. Using this, upper bounds for the cones generated by divisors with moving linear systems are calculated, , along with the induced bounds on the cones of ample divisors of and . As an application, the cone is analyzed in detail.

     

On the Heat Flow for Harmonic Maps with Potential

Let (M, g) and (N, h) be twoconnected Riemannian manifolds without boundary (M compact,N complete). Let G C (N): ifu: M N is a smooth map, we consider the functional E _G (u) = (1/2) _M [|du|²– 2G(u)]dV _M and we study its associated heat equation. Inthe compact case, we recover a version of the Eells–Sampson theorem,while for noncompact target manifold N, we establishsuitable hypotheses and ensure global existence and convergence atinfinity. In the second part of the paper, we study phenomena of blowingup solutions.     

On homology of real algebraic varieties

Let be a commutative ring with unity and an -oriented compact nonsingular real algebraic variety of dimension . If is any nonsingular complexification of , then the kernel, which we will denote by , of the induced homomorphism is independent of the complexification. In this work, we study and give some of its applications.

     

On the existence of Skyrme gauge field monopoles

In this paper, we study Skyrme-like monopoles which are topological solitons in three space dimensions. We prove the existence of symmetric monopole solutions in a gauged Skyrme model by a variational method. We also obtain some properties of the energy-minimizing solutions. For when the Skyrme coupling constant κ=0, we arrive at the BPS monopole equations. Furthermore, we obtain the relation between the BPS and non-BPS monopole solutions, and properties of the BPS monopole solution.     

Removability of the singular set of the heat flow of harmonic maps

We show that the singular set of a weak stationary solution of the heat flow of harmonic maps between Riemannian manifolds and , with compact, is removable if it has ``parabolic codimension' greater than two and the initial energy is sufficiently small.

     

On the geometry of a four-parameter rational planar system of difference equations

In this paper, we consider geometric aspects of a rational, planar system of difference equations defined on the open first quadrant and whose behaviour is governed by four independent, non-negative parameters. This system, indexed as (23, 23) in the notation of Ladas (Open problems and conjectures, J. Differential Equ. Appl. 15(3) 2009, pp. 303–323), is one of the 200 systems from Ladas about which little is known. Using geometric techniques, we answer several questions concerning the behaviour of this system.     

Heat Flow for Extrinsic Biharmonic Maps with Small Initial Energy

Let M ^m and N ^{n k} be two compact Riemannian manifolds without boundary. We consider the L ² gradient flow for the energy F(u):= _M |u|². If m 3 or if m= 4 and F(u ₀) is small, we show that the heat flow for extrinsic biharmonic maps exists for all time, and that the solution subconverges to a smooth extrinsic biharmonic map as time goes to infinity.     

Chaotic dynamics of a third-order Newton-type method

The dynamics of a classical third-order Newton-type iterative method is studied when it is applied to degrees two and three polynomials. The method is free of second derivatives which is the main limitation of the classical third-order iterative schemes for systems. Moreover, each iteration consists only in two steps of Newton's method having the same derivative. With these two properties the scheme becomes a real alternative to the classical Newton method. Affine conjugacy class of the method when is applied to a differentiable function is given. Chaotic dynamics have been investigated in several examples. Applying the root-finding method to a family of degree three polynomials, we have find a bifurcation diagram as those that appear in the bifurcation of the logistic map in the interval.     

含四个参数的四次Thue方程

本文研究了含四个参数的四次Thue方程.利用简单的代数数有理逼近方法给出了该方程解的有效上界,从而将参数个数由两个推广到四个.     

Application of rational expansion method for stochastic differential equations

By means of the Hermite transformation, a new general ansätz and the symbolic computation system Maple, we apply the Riccati equation rational expansion method [24] to uniformly construct a series of stochastic non-traveling wave solutions for stochastic differential equations. To illustrate the effectiveness of our method, we take the stochastic mKdV equation as an example, and successfully construct some new and more general solutions. The method can also be applied to solve other nonlinear stochastic differential equation or equations.     

Asymptotic behaviour of solutions of rational difference systems

In this article, we investigate the asymptotic behaviour of solutions of systems of rational difference equations in arbitrary dimensions. We give conditions for the parameters ensuring that the positive solutions of the considered system are bounded, unbounded, increasing, decreasing, and convergent, respectively.     

Construction of circle bifurcations of a two-dimensional spatially periodic flow

The study by Yudovich [V.I. Yudovich, Example of the generation of a secondary stationary or periodic flow when there is loss of stability of the laminar flow of a viscous incompressible fluid, J. Math. Mech. 29 (1965) 587-603] on spatially periodic flows forced by a single Fourier mode proved the existence of two-dimensional spectral spaces and each space gives rise to a bifurcating steady-state solution. The investigation discussed herein provides a structure of secondary steady-state flows. It is constructed explicitly by an expansion that when the Reynolds number increases across each of its critical values, a unique steady-state solution bifurcates from the basic flow along each normal vector of the two-dimensional spectral space. Thus, at a single Reynolds number supercritical value, the bifurcating steady-state solutions arising from the basic solution form a circle.