目标流形为Heisenberg群能量极小映射的逆Poincaré不等式

研究目标流形为Heisenberg群能量极小映射的性质, 得到关于能量极小映射的逆Poincaré不等式.     

Poincaré polynomials of moduli spaces of stable maps into flag manifolds

By using the Bialynicki-Birula decomposition and holomorphic Lefschetz formula, we calculate the Poincaré polynomials of the moduli spaces in low degrees.     

Pseudo-rotations of the open annulus

In this paper, we study pseudo-rotations of the open annulus, i.e. conservative homeomorphisms of the open annulus whose rotation set is reduced to a single irrational number (the angle of the pseudo-rotation). We prove in particular that, for every pseudo-rotation h of angle ρ, the rigid rotation of angle ρ is in the closure of the conjugacy class of h. We also prove that pseudo-rotations are not persistent in C^r topology for any r ≥ 0.     

The Bifurcation of the Equatorial Periodic Orbit of the Planar Polynomial Vector Fields

In this paper, we have obtained the necessary and sufficient condition for the set of points at infinity on the plane R ² to be a periodic orbit which is called an equatorial periodic orbit of a planar vector field X(x), and the formulae about the multiplicity of the equatorial periodic orbit of X(x). We have also proved that the main result of [9] is erroneous with regard to the formulae. This work is supported by the NSF of China     

Combined use of the finite element and finite superelement methods

Results of the theoretical and numerical studies of an algorithm based on the combined use of the finite element and finite superelement methods are presented. Estimates of the errors for one of the variants of the method applied to solving the Laplace equation are obtained. The method can be used to solve a problem concerning the skin layer appearing due to high velocities.     

Indices of Toeplitz tuples on pseudoregular domains

This paper proves an index theorem of Toeplitz tuples on pseudoregular domains in Cn. Geometrically, the index of Toeplitz tuple TΦn is (-1)n time wrapping number of Φn around the origin. As one of the applications of the index theorem, we completely characterize the automorphism groups of Toeplitz algebras on Poincaré domain. As another application, it is shown that C*(Ω)C*(Bn) for every Poincare domain Ω in Cn(n≠2). It is also noticed that C*(Ω)C*(B2) if and only if the Poincaré conjecture is true for Ω.     

The Poincaré–Bendixson theorem for a class of delay equations with state-dependent delay and monotonic feedback

We consider the real-valued differential equation

$x^{\prime }(t)=f(x(t),x(t?d(x_t)))$

with state-dependent delay, where f is strictly monotonic in its second argument. We describe a class of such equations for which a version of the Poincaré–Bendixson theorem holds.     

Analogue of the Cayley–Sylvester formula and the Poincaré series for the algebra of invariants of n-ary form

The number of linearly independent homogeneous invariants of degree k for the n-ary form of order d is calculated. Also, a formula for the Poincaré series for the algebra of invariants of the n-ary form is found.     

Analysis of a Shil’nikov Type Homoclinic Bifurcation

The bifurcation associated with a homoclinic orbit to saddle-focus including a pair of pure imaginary eigenvalues is investigated by using related homoclinic bifurcation theory. It is proved that, in a neighborhood of the homoclinic bifurcation value, there are countably infinite saddle-node bifurcation values, period-doubling bifurcation values and double-pulse homoclinic bifurcation values. Also, accompanied by the Hopf bifurcation, the existence of certain homoclinic connections to the periodic orbit is proved.     

Pro-torus actions on Poincaré duality spaces

In this paper, it is shown that some of the results of torus actions on Poincaré duality spaces, Borel’s dimension formula and topological splitting principle to local weights, hold if ‘torus’ is replaced by ‘pro-torus’.     

A bijection between the adjoint and coadjoint orbits of a semidirect product

We prove that there exists a geometric bijection between the sets of adjoint and coadjoint orbits of a semidirect product, provided a similar bijection holds for particular subgroups. We also show that under certain conditions the homotopy types of any two orbits in bijection with each other are the same. We apply our theory to the examples of the affine group and the Poincaré group, and discuss the limitations and extent of this result to other groups.     

Multivalued synchronization by Poincaré coupling

This work presents multivalued chaotic synchronization via coupling based on the Poincaré plane. The coupling is carried out by an underdamped signal, triggered every crossing event of the trajectory of the master system through a previously defined Poincaré plane. A master–slave system is explored, and the synchronization between the systems is detected via the auxiliary system approach and the maximum conditional Lyapunov exponent. Due to the response to specific conditions two phenomena may be obtained: univalued and multivalued synchronization. Since the Lyapunov exponent is not enough to detect these two phenomena, the distance between the pieces of trajectories of the slave and auxiliary systems with different initial conditions is also used as a tool for the detection of multivalued synchronization. Computer simulations using the benchmark chaotic systems of Lorenz and Rössler are used to exemplify the approach proposed.     

Integral representations and the generalized Poincaré inequality on Carnot groups

We give some integral representations of the form f(x) = P(f)+K(?f) on two-step Carnot groups, where P(f) is a polynomial and K is an integral operator with a specific singularity. We then obtain the weak Poincaré inequality and coercive estimates as well as the generalized Poincaré inequality on the general Carnot groups.     

Absolute continuity of the Sobolev type functions on metric spaces

Under study is the absolute continuity of the functions satisfying the Poincaré inequality on s-regular metric spaces.     

Degenerate Orbit Flip Homoclinic Bifurcations with Higher Dimensions

Bifurcations of a degenerate homoclinic orbit with orbit flip in high dimensional system are studied. By establishing a local coordinate system and a Poincaré map near the homoclinic orbit, the existence and uniqueness of 1–homoclinic orbit and 1–periodic orbit are given. Also considered is the existence of 2–homoclinic orbit and 2–periodic orbit. In additon, the corresponding bifurcation surfaces are given. Project supported by the National Natural Science Foundation of China (No: 10171044), the Natural Science Foundation of Jiangsu Province (No: BK2001024), the Foundation for University Key Teachers of the Ministry of Education of China