Generalized Wigner Functions for Damped Systems in Deformation Quantization

Quantization of damped systems usually gives rise to complex spectra and corresponding resonant states, which do not belong to the Hilbert space. Therefore, the standard form of calculating Wigner function (WF) does not work for these systems. In this paper we show that in order to let WF satisfy a *-genvalue equation for the damped systems, one must modify its standard form slightly, and this modification exactly coincides with the resu lts derived from a *-Exponential expansion in deformation quantization.     

Modified Form of Wigner Functions for Non-Hamiltonian Systems

Quantization of non-Hamiltonian systems （such as damped systems） often gives rise to complex spectra and corresponding resonant states, therefore a standard form calculating Wigner functions cannot lead to static quasiprobability distribution functions. We show that a modified form of the Wigner functions satisfies a ＊-genvalue equation and can be derived from deformation quantization for such systems.     

Stationary response of combined linear dynamic systems to stationary excitation

The stationary response of a broad class of combined linear systems to stationary random excitation is determined by the normal mode method. The systems are characterized by a viscously damped simple beam (or string, membrane, thin plate or shell, etc.) connected at discrete points to a multiplicity of viscously damped linear oscillators and/or masses. The solution of the free vibration problem by way of Green functions and the deterministic forced vibration problem by modal analysis for both proportional and non-proportional damping is reviewed. The orthogonality relation for the natural modes of vibration is used to derive a unique relationship between the cross-spectral density functions of the applied forces and the cross-spectral density functions of the generalized forces. Finally, the response spectral density functions and the mean square responses of the beam and oscillators are derived in closed form, exact for the proportionally damped system and approximate for the non-proportionally damped system.     

Shape Analysis and Solution to a Class of Nonlinear Wave Equation with Cubic Term

In this paper, we analyze the relation between the shape of the bounded traveling wave solutions and dissipation coefficient of nonlinear wave equation with cubic term by the theory and method of planar dynamical systems. Two critical values which can characterize the scale of dissipation effect are obtained. If dissipation effect is not less than a certain critical value, the traveling wave solutions appear as kink profile; while if it is less than this critical value, they appear as damped oscillatory. All expressions of bounded traveling wave solutions are presented, including exact expressions of bell and kink profile solitary wave solutions, as well as approximate expressions of damped oscillatory solutions. For approximate damped oscillatory solution, using homogenization principle, we give its error estimate by establishing the integral equation which reflects the relations between the exact and approximate solutions. It can be seen that the error is an infinitesimal decreasing in the exponential form.     

QCD sum rules for pion wave function revisited

We analyze new QCD sum rules for the pion wave function (WF) ?_π (x), obtained recently in [1] in the non-local vacuum condensates method for non-diagonal correlators, and suggest a new approach for extracting WF of the π-meson and the mass and WF of the first resonance. As a result, we obtain approximately the same form of the pion WF by two different methods and for two different ansatzes of non-local quark condensates. We predict the mass of the π′-resonance and obtain the form of its WF.     

表面杂质对NO+CO/Pt(100)反应体系振荡动力学的影响

提出了一个格气模型， 探讨催化表面的惰性杂质对NO+CO/Pt(100)反应体系振荡动力学行为的影响．研究发现表面杂质较少时，反应进程中表面重构能够形成相连通的1*1相和持续的振荡行为．当表面杂质不断增加时，反应进程中1*1相只能形成许多孤立的畴，从而在1*1相形成的随机的局域振荡的空间关联被削弱，因而体系全局的持续振荡行为演变为衰减振荡行为．当表面吸附的CO和NO的扩散速率增加时，局域振荡的空间协同又得到加强，体系又呈现出持续振荡行为．     

A real decoupled method and free interface component mode synthesis methods for generally damped systems

This paper reports on the development of a new transformation method. In contrast to most existing mode transformation methods in which the first-order state-space equation of the damped vibration system is transformed into a decoupled form with complex coefficient matrices, using the decoupled method presented in this paper, the equation of the damped system can be decomposed into a decoupled equation with real coefficient matrices. Two new free interface component mode synthesis methods are also presented. The equivalent full-mode matrix of the damped structure is used to capture the effects of the higher-order modes. Additionally, this work modifies the compatibility conditions at the junctions that are employed in most of the previous component mode synthesis methods for generally damped systems. The first component mode synthesis method is performed in complex space, whereas the second method can be applied in real space. Because the coefficient matrices of the coupled equation constructed by the second component mode synthesis method are all real-valued, the solution of the eigenproblem for this coupled equation can be performed in real space as well. Additionally, numerical examples demonstrate the accuracy and validity of these two component mode synthesis methods.     

Approach to the quantum evolution for underdamped, critically damped, and overdamped driven harmonic oscillators using unitary transformation

Exact solution of the Schrödinger equation is derived for underdamped, critically damped, and overdamped harmonic oscillators with a driving force. A unitary operator transforming Hamiltonian into a simple form is introduced. The transformed Hamiltonian, represented in terms of a modified frequency ω, is identical with the Hamiltonian of the standard harmonic oscillator for the underdamped oscillator, with the Hamiltonian of a free particle for the critically damped oscillator, and with the Hamiltonian of a system with a harmonic parabolic potential for the overdamped oscillator. The eigenvalues of underdamped oscillator are discrete while those of the critically damped and the overdamped oscillators are continuous.     

Absolute work function measurement by using photoelectron spectroscopy

Work function (WF) of a material is not only an intrinsic characteristic of bulk but also a surface property. The measurement and control of WF have been of great concern in many electronic and optical devices as the WF governs charge transfer and charge injection/collection efficiency at interfaces and emission characteristics of conventional charged particle emitters. Photoelectron spectroscopy (PES) has been mainly used to determine surface electronic structure and chemical composition. Despite the common use of this technique to measure WF, there has been a lack of discussion on how to use the PES and what to be considered to determine the absolute WF. The main contribution of this review lies in the discussion of the causes of errors when measuring WF, and provides a guide for reliable WF measurement. Along with the limitations of current measurement technology, we propose future directions for absolute WF measurement.     

Determining the Wiedemann-Franz ratio from the thermal hall conductivity: application to Cu and YBa2Cu3O6.95

The Wiedemann-Franz (WF) ratio compares the thermal and electrical conductivities in a metal. We describe a new way to determine its value, based on the thermal Hall conductivity. The technique is applied to copper and to untwinned YBaCuO. In the latter, we uncover a T-linear dependence and suppression of the Hall-channel WF ratio. We discuss the implications of this suppression. The general suppression of the WF ratio in systems with predominant electron-electron scattering is discussed.     

Spectral Theory of Time Dispersive and Dissipative Systems

We study linear time dispersive and dissipative systems. Very often such systems are not conservative and the standard spectral theory can not be applied. We develop a mathematically consistent framework allowing (i) to constructively determine if a given time dispersive system can be extended to a conservative one; (ii) to construct that very conservative system—which we show is essentially unique. We illustrate the method by applying it to the spectral analysis of time dispersive dielectrics and the damped oscillator with retarded friction. In particular, we obtain a conservative extension of the Maxwell equations which is equivalent to the original Maxwell equations for a dispersive and lossy dielectric medium.     

Substructuring and modal synthesis for damped systems

A new approach to complex mode synthesis for damped systems is presented. The technique involves using two successive transformations of the equations written in first order, state space form. The transformations are applied to each subsystem, with consecutive use being made of free interface undamped modes in the existing physical co-ordinates followed by fixed interface damped modes in the generalized co-ordinates. The system eigenvalues are obtained by the QR method in the EISPACK matrix programs. An advantage of the technique is that it can be applied to systems or subsystems possessing rigid body motion. The new method compares very well in accuracy and applicability with other existing methods.     

Prediction of averaged energy for moderately damped systems with strong coupling

In this paper, the prediction of averaged energy of damped structural-acoustic systems is investigated. A simplified energy finite element method, referred to as EFEM⁰, is developed based on the energy flow analysis (EFA) equations and implemented using the finite volume method. The resulting formulations can be incorporated into statistical energy analysis (SEA) software and extends SEA application to moderately damped systems with strong coupling. The formulations are verified against analytical solutions for a single beam and coupled beams with both strong and weak coupling. A hybrid technique consisting of the EFEM⁰ approximation superimposed on a direct field is used to model moderately damped plates. For lightly damped systems, both methods produce acceptable results. For moderately damped two-dimensional systems, the EFEM⁰ method augmented with the direct field component produces significantly improved results.     

A new analytical technique for vibration analysis of non-proportionally damped beams

Vibrating linear mechanical systems, in particular continuous systems, are often modelled considering proportional damping distributions only, although in many real situations this simplified approach does not describe the dynamics of the system with sufficient accuracy. In this paper an analytical method is given to take into account the effects of a more general viscous damping model, referred to as non-proportional damping, on a class of vibrating continuous systems. A state-form expansion applied in conjunction with a transfer matrix technique is adopted to extract the eigenvalues and to express the eigenfunctions in analytical form, i.e., complex modes corresponding to non-synchronous motions. Numerical examples are included in order to show the efficiency of the proposed method; non-proportional damping distributions of different type, such as internal and external lumped or distributed viscous damping, are tested on non-homogeneous Euler-Bernoulli beams in bending vibration with different boundary conditions. Finally, a discussion on root locus diagrams behaviour and on modal damping ratio significance for non-proportionally damped systems is presented.     

Study of Certain Aspects of Anharmonic,Time-Dependent and Damped Harmonic Oscillator Systems

The present review is devoted to the study of certain aspects of anharmonic, time-dependent and damped oscillator(s) system using different theoretical techniques. A theoretical understanding of these systems is important for application in many problems in physics, mechanics and other fields. We discuss in detail the difficulties in the theoretical analysis of the problem. In particular we discuss here the regular, well-behaved perturbative solution, the large quantum number behaviour of anharmonic oscillator(s) using the technique of coherent states, exact solution of quantum anharmonic oscillators, the electromagnetic radiation emitted by a charged particle executing damped anharmonic oscillator motion using Krylov-Bogoliubov approximation method, use of invariants to obtain solution and coherent states of time-dependent oscillator(s), the derivation of perturbative frequencies of a damped coupled anharmonic oscillators system using suitable canonical transformation in the framework of Hamilton-Jacobi formalism and the quantisation and construction of coherent states of a damped oscillator using time-dependent operators.     

An experimental study of a new multi-bunch damping mechanism in PETRA

A new damping mechanism of multi-bunch oscillations in a storage ring with help of a narrow-band feedback system has been experimentally tested in PETRA. The oscillations of a multi-bunch filling were sucessfully damped. The bandwidth of the active loop was 1.5 kHz instead of 5 MHz needed for standard damper systems.     

A physically realizable whitening filter for cryogenic resonant gravitational-wave antennas

In this paper the problem of a physically realizable whitening-filter (WF) synthesis for cryogenic resonant gravitational-wave antennas is considered in the framework of the optimal linear Kalman-Bucy filtering. A system of equations determining the structure of the Kalman filter for gravitational-wave antennas with a displacement converter is presented. The transfer function of a stationary physically realizable WF is derived. The Kalman approach ensures the possibility of high frequency parameter measurements of Gaussian and non-Gausssian noises in low-dissipation oscillatory systems for observation intervals, which are relatively short in comparison to the relaxation time.     

On the integral representation of states on a C*-algebra

The purpose of this paper is to give some complements to the various extremal decompositions of states on aC*-dynamical system i.e. a pair (A, G) whereA is aC*-algebra andG is a group acting on aA by *-automorphisms. We shall see for instance that the method of decomposition associated with a maximal abelianW*-algebra does not give all the extremal measures in the general case. We also give the explicit form of the greatest lower bound of all the extremal measures and a certain form of continuity of the decomposition. Finally we characterize various systems in the literature (G-abelian algebras, large systems and quasi-large systems) in terms of the equivalence of different notions of ergodicity.     

Multi-symplectic Birkhoffian structure for PDEs with dissipation terms

A generalization of the multi-symplectic form for Hamiltonian systems to self-adjoint systems with dissipation terms is studied. These systems can be expressed as multi-symplectic Birkhoffian equations, which leads to a natural definition of Birkhoffian multi-symplectic structure. The concept of Birkhoffian multi-symplectic integrators for Birkhoffian PDEs is investigated. The Birkhoffian multi-symplectic structure is constructed by the continuous variational principle, and the Birkhoffian multi-symplectic integrator by the discrete variational principle. As an example, two Birkhoffian multi-symplectic integrators for the equation describing a linear damped string are given.