The combination technique and some generalisations

The combination technique has repeatedly been shown to be an effective tool for the approximation with sparse grid spaces. Little is known about the reasons of this effectiveness and in some cases the combination technique can even break down. It is known, however, that the combination technique produces an exact result in the case of a projection into a sparse grid space if the involved partial projections commute.

The performance of the combination technique is analysed using a projection framework and the C/S decomposition. Error bounds are given in terms of angles between the spanning subspaces or the projections onto these subspaces. Based on this analysis modified combination coefficients are derived which are optimal in a certain sense and which can substantially extend the applicability and performance of the combination technique.     

Level shift operators for open quantum systems

Level shift operators describe the second-order displacement of eigenvalues under perturbation. They play a central role in resonance theory and ergodic theory of open quantum systems at positive temperatures. We exhibit intrinsic properties of level shift operators, properties which stem from the structure of open quantum systems at positive temperatures and which are common to all such systems. They determine the geometry of resonances bifurcating from eigenvalues of positive temperature Hamiltonians and they relate the Gibbs state, the kernel of level shift operators, and zero energy resonances. We show that degeneracy of energy levels of the small part of the open quantum system causes the Fermi Golden Rule Condition to be violated and we analyze ergodic properties of such systems.     

电力市场中合同电量与竞争电量交易比例的研究

在单边开放的区域电力市场中,合理的合同电量与竞争电量交易比例是保证电力市场有效运行的一个重要环节。竞争电量所占的比例将主要取决于当前发电公司的市场行为。首先使用BP神经网络对电力需求弹性系数进行了预测,然后以长期电力市场均衡为目标函数,考虑贵州电网发电机组的可用容量与负荷预测的误差,以及贵州输电线路的可靠性诸因素,推导出合同电量与竞争电量交易比例,经过与南方区域电力市场目前运营规则规定的交易比例比较,该比例是合理的,可以规避电力市场价格波动等带来的风险。     

Solution bundle for a class of impulsive differential inclusions on Banach spaces

In this paper a class of impulsive differential inclusions is investigated. The existence of solution bundle is proved. And we also construct a nonlinear semigroup of operators on cb(E) (closed-bounded subset of E) to describe the set of attainable states.     

Contractive multifunctions, fixed point inclusions and iterated multifunction systems

We study the properties of multifunction operators that are contractive in the Covitz-Nadler sense. In this situation, such operators T possess fixed points satisfying the relation x∈Tx. We introduce an iterative method involving projections that guarantees convergence from any starting point x₀∈X to a point x∈XT, the set of all fixed points of a multifunction operator T. We also prove a continuity result for fixed point sets XT as well as a “generalized collage theorem” for contractive multifunctions. These results can then be used to solve inverse problems involving contractive multifunctions. Two applications of contractive multifunctions are introduced: (i) integral inclusions and (ii) iterated multifunction systems.     

Nonlinear control of systems with multiple equilibria and unknown sinusoidal disturbance

In this paper, nonlinear systems having multiple equilibrium points and low order dynamics are investigated. Roll motions of ships are studied by means of modern nonlinear techniques to exemplify the behavior of such nonlinear systems in the case when they are under the influence of external sinusoidal disturbances with unknown amplitudes. The main objective is to analyze the performance of this system at different operating conditions, including those giving rise to chaos, and to design a controller with an overparameterized structure to stabilize the system at the origin. A nonlinear recursive backstepping controller is proposed and the transient performance is investigated. Lyapunov-based techniques are used to force systematic following of a reference model while introducing a nonlinear parameter estimator to guarantee adaptivity. Robustness problems as well as ways to tune the controller parameters are examined. Simulation results are submitted for the uncontrolled and controlled cases, verifying the effectiveness of the proposed controller. Finally, a discussion and conclusions are given with possible future extensions.     

Periodic and subharmonic solutions of a class of subquadratic second-order Hamiltonian systems

Some solvability conditions of periodic and subharmonic solutions are obtained for a class of subquadratic nonautonomous second-order Hamiltonian systems by the minimax methods in critical point theory.     

On linear and nonlinear Rossby waves in an ocean

This paper deals with recent developments of linear and nonlinear Rossby waves in an ocean. Included are also linear Poincaré, Rossby, and Kelvin waves in an ocean. The dispersion diagrams for Poincaré, Kelvin and Rossby waves are presented. Special attention is given to the nonlinear Rossby waves on a β-plane ocean. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a modified nonlinear Schrödinger equation. The solution of this equation represents solitary waves in a dispersive medium. In other words, the envelope of the amplitude of the waves has a soliton structure and these envelope solitons propagate with the group velocity of the Rossby waves. Finally, a nonlinear analytical model is presented for long Rossby waves in a meridional channel with weak shear. A new nonlinear wave equation for the amplitude of large Rossby waves is derived in a region where fluid flows over the recirculation core. It is shown that the governing amplitude equations for the inner and outer zones are both KdV type, where weak nonlinearity is balanced by weak dispersion. In the inner zone, the nonlinear amplitude equation has a new term proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude, and this term occurs to account for a nonlinearity due to the flow over the vortex core. The solution of the amplitude equations with the linear shear flow represents the solitary waves. The present study deals with the lowest mode (n=1) analysis. An extension of the higher modes (n?2) of this work will be made in a subsequent paper.     

Systems of generalized quasivariational inclusions problems with applications to variational analysis and optimization problems

In this paper, we study an existence theorem of systems of generalized quasivariational inclusions problem. By this result, we establish the existence theorems of solutions of systems of generalized equations, systems of generalized vector quasiequilibrium problem, collective variational fixed point, systems of generalized quasiloose saddle point, systems of minimax theorem, mathematical program with systems of variational inclusions constraints, mathematical program with systems of equilibrium constraints and systems of bilevel problem and semi-infinite problem with systems of equilibrium problem constraints. This research was supported by the National Science Council of the Republic of China.     

Dynamical dark energy models – dynamical system approach

We study the Friedmann-Robertson-Walker model with dynamical dark energy modelled in terms of the equation of state p _X = w _X (a(z)) ρ _X in which the coefficient w _X is parameterized by the scale factor a or redshift z. We use methods of qualitative analysis of differential equations to investigate the space of all admissible solutions for all initial conditions on the two-dimensional phase plane. We show advantages of representing this dynamics as a motion of a particle in the one-dimensional potential V(a). One of the features of this reduction is the possibility of investigating how typical big rip singularities are in the future evolution of the model. The properties of potential function V can serve as a tool for qualitative classification of all evolution paths. Some important features like resolution of the acceleration problem can be simply visualized as domains on the phase plane. Then one is able to see how large is the class of solutions (labelled by the inset of the initial conditions) leading to the desired property.