Global minimization of large-scale constrained concave quadratic problems by separable programming

The global minimization of a large-scale linearly constrained concave quadratic problem is considered. The concave quadratic part of the objective function is given in terms of the nonlinear variablesx R ⁿ , while the linear part is in terms ofy R ^k. For large-scale problems we may havek much larger thann. The original problem is reduced to an equivalent separable problem by solving a multiple-cost-row linear program with 2n cost rows. The solution of one additional linear program gives an incumbent vertex which is a candidate for the global minimum, and also gives a bound on the relative error in the function value of this incumbent. Ana priori bound on this relative error is obtained, which is shown to be 0.25, in important cases. If the incumbent is not a satisfactory approximation to the global minimum, a guaranteed-approximate solution is obtained by solving a single liner zero–one mixed integer programming problem. This integer problem is formulated by a simple piecewise-linear underestimation of the separable problem.This research was supported by the Division of Computer Research, National Science Foundation under Research Grant DCR8405489.Dedicated to Professor George Dantzig in honor of his 70th Birthday.     

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.     

Geometry of quadratic differential systems in the neighborhood of infinity

In this article we give a complete global classification of the class QS_ess of planar, essentially quadratic differential systems (i.e. defined by relatively prime polynomials and whose points at infinity are not all singular), according to their topological behavior in the vicinity of infinity. This class depends on 12 parameters but due to the action of the affine group and re-scaling of time, the family actually depends on five parameters. Our classification theorem (Theorem 7.1) gives us a complete dictionary connecting very simple integer-valued invariants which encode the geometry of the systems in the vicinity of infinity, with algebraic invariants and comitants which are a powerful tool for computer algebra computations helpful in the route to obtain the full topological classification of the class QS of all quadratic differential systems.     

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.     

Regularity for Certain Nonlinear Parabolic Systems

Abstract

By means of an inequality of Poincaré type, a weak Harnack inequality for the gradient of a solution and an integral inequality of Campanato type, it is shown that a solution to certain degenerate parabolic system is locally Hölder continuous. The system is a generalization of p-Laplacian system. Using a difference quotient method and Moser type iteration it is then proved that the gradient of a solution is locally bounded. Finally using the iteration and scaling it is shown that the gradient of the solution satisfies a Campanato type integral inequality and is locally Hölder continuous.     

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.     

On two-parameter bifurcation analysis of switched system composed of Duffing and van der Pol oscillators

The behaviors of system which alternate between Duffing oscillator and van der Pol oscillator are investigated to explore the influence of the switches on dynamical evolutions of system. Switches related to the state and time are introduced, upon which a typical switched model is established. Poincaré map of the whole switched system is defined by suitable local sections and local maps, and the formal expression of its Jacobian matrix is obtained. The location of the fixed point and associated Floquet multipliers are calculated, based on which two-parameter bifurcation sets of the switched system are obtained, dividing the parameter space into several regions corresponding to different types of attractors. It is found that cascading of period-doubling bifurcations may lead the system to chaos, while fold bifurcations determine the transition between period-3 solution and chaotic movement.     

A generalization of Poincaré's theorem for recurrence systems

A new proof and a genuine generalization to systems of first order equations is given from Poincaré classical theorem on ratio asymptotics of solutions of higher order recurrence equations. The asymptotic behavior of a fundamental system of solutions is obtained.     

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