Time-dependent generalized polynomial chaos

Generalized polynomial chaos (gPC) has non-uniform convergence and tends to break down for long-time integration. The reason is that the probability density distribution (PDF) of the solution evolves as a function of time. The set of orthogonal polynomials associated with the initial distribution will therefore not be optimal at later times, thus causing the reduced efficiency of the method for long-time integration. Adaptation of the set of orthogonal polynomials with respect to the changing PDF removes the error with respect to long-time integration. In this method new stochastic variables and orthogonal polynomials are constructed as time progresses. In the new stochastic variable the solution can be represented exactly by linear functions. This allows the method to use only low order polynomial approximations with high accuracy. The method is illustrated with a simple decay model for which an analytic solution is available and subsequently applied to the three mode Kraichnan–Orszag problem with favorable results.     

Four Wave Mixing spectroscopy at the interface between the time and frequency domains

We demonstrate a new approach to Four Wave Mixing spectroscopy involving simultaneous measurements at time and frequency domains, where spectral selectivity is achieved by phase matching filtering, and the time resolution is obtained within a single ultra-short pulse. We analyze the Four Wave Mixing signal, and show that our method is capable for discrimination between different spectroscopic pathways of vibrational coherences modulating the scattered signal.     

The solutions of the generalized classical and quantum harmonic oscillators with time dependent mass, frequency, two-photon parameter and external force: The squeezing effects

The exact analytical solutions of a generalized classical harmonic oscillator with time dependent mass, frequency, two-photon parameter and external forces are obtained. By using the invariance property of the scaled Wronskian, these solutions are used to obtain the solutions of the quantum mechanical counterpart of the oscillator under Heisenberg picture. In order to discuss the applications of these solutions of the quantum mechanical oscillator, we calculate the exact analytical expressions for the second order variances of both the canonically conjugate quadratures in terms of the time dependent mass, frequency and two-photon parameter. However, these variances calculated in terms of the initial coherent state do not depend on the time dependent driven terms. We argued that the time dependent frequency is on the way of the exact analytical solutions and hence it is kept constant throughout the investigation. We, however, discuss few situations of physical interest where the mass is varying in time. The special circumstance where all the parameters are time independent is used to discuss the squeezing effect in both the quadratures. It is seen that the parameter g involving the two-photon interaction term produces squeezing effects. With the increase of interaction time, the squeezing in both the quadrature components exhibit collapse and revival phenomena for g < ω (frequency of the oscillator). The squeezing of X-quadrature is completely ruled out for g > ω. However, the squeezing for P-quadrature is possible for small interaction time. The squeezing patterns of the X and P-quadratures are also discussed for pulsating mass, and for mass increasing with time. The squeezing is also discussed when the mass is increasing exponentially in time. It is envisage that the solutions could be used in the investigation of quantum statistical properties of the radiation field coupled to the said oscillator.     

带时间窗的某物流配送车辆调度问题的方案优化分析

采用带时间窗的车辆调度优化物流配送系统,建立了带时间窗的多目标约束条件下的派车优化问题的数学模型,并针对某大型连锁超市物流配送中心的车辆调度问题进行了求解,经过对比分析,新方案实现了一定程度多目标优化.     

时间标度上的Taylor公式及链式法则

致力于研究时间标度上二元函数的链式法则,以期其在最优控制上有广泛的应用.同时,对一元函数的泰勒公式给出一种新的较为简单的证明方法.     

An application of vector GARCH model in semiconductor demand planning

Demand planning has been the key to supply chain management in semiconductor industry. With an appropriate weight assignment scheme, the planning accuracy resulting from combinational forecasts can be improved by merging several individual candidate methods. In this paper we discuss the applicability of vector generalized autoregressive conditional heteroskedasticity (GARCH) model to determine the optimal combinational weights of component forecasts, where the conditional variances and correlations of forecast errors from candidate methods are represented and estimated by a maximum-likelihood procedure. The asymptotical properties of parameter estimators for GARCH model are investigated by simulation experiments. An example of the proposed method to real time series of electronic products demonstrates that this weight-varying combinational method produces less prediction errors, compared to other commonly used forecasting approaches that are based on single model selection criteria or fixed weights.     

Basic existence, uniqueness and approximation results for positive solutions to nonlinear dynamic equations on time scales

This paper focuses on the qualitative and quantitative properties of solutions to certain nonlinear dynamic equations on time scales. We present some new sufficient conditions under which these general equations admit a unique, positive solution. These positive (and hence non-oscillatory) solutions: extend across unbounded intervals; and tend to a finite limit as the independent variable becomes large and positive. Our methods include: Banach’s fixed-point theorem, including the method of Picard iterations; and weighted norms and metrics in the time scale setting. Due to the wide-ranging nature of dynamic equations on time scales our results are novel: for ordinary differential equations; for difference equations; for combinations of the two areas; and for general time scales — this is demonstrated via some examples. Furthermore, we state an open problem of interest.     

Periodic solutions of functional dynamic equations with infinite delay

In this paper, sufficient criteria are established for the existence of periodic solutions of some functional dynamic equations with infinite delays on time scales, which generalize and incorporate as special cases many known results for differential equations and for difference equations when the time scale is the set of the real numbers or the integers, respectively. The approach is mainly based on the Krasnosel’ski? fixed point theorem, which has been extensively applied in studying existence problems in differential equations and difference equations but rarely applied in studying dynamic equations on time scales. This study shows that one can unify such existence studies in the sense of dynamic equations on general time scales.     

Horizontal fast excitation in delayed van der Pol oscillator

This paper investigates the interaction effect of horizontal fast harmonic parametric excitation and time delay on self-excited vibration in van der Pol oscillator. We apply the method of direct partition of motion to derive the main autonomous equation governing the slow dynamic of the oscillator. The method of averaging is then performed on the slow dynamic to obtain a slow flow which is analyzed for equilibria and periodic motion. This analysis provides analytical approximations of regions in parameter space where periodic self-excited vibrations can be eliminated. A numerical study is performed on the original oscillator and compared to analytical approximations. It was shown that in the delayed case, horizontal fast harmonic excitation can eliminate undesirable self-excited vibrations for moderate values of the excitation frequency. In contrast, the case without delay requires large excitation frequency to eliminate such motions. This work has application to regenerative behavior in high-speed milling.     

Group delay induced instabilities and Hopf bifurcations, of a controlled double pendulum

Digital filters, frequently used in active control of mechanical systems, enable one to improve the signal-to-noise ratio and the control performance, but introduce group delays into the control loops simultaneously. In order to gain an insight into the effects of a digital filter on a controlled mechanical system, this paper presents the stability switches and the corresponding Hopf bifurcations of a double pendulum system with the linear quadratic control having a digital filter via theoretical analysis, numerical simulations and experiments. In this study, the digital filters are used to remove the undesired noise of high frequency, which is embedded in the control signal, and are modeled as the components of pure time delay during the theoretical analysis and numerical simulations. The study shows that a digital filter with moderate specifications can not only improve the vibration reduction effectively, but also save the energy consumption of the servo-motor remarkably. However, over demanding specifications will make the group delay of the filter exceed a critical value and cause either a divergent motion or a self-excited vibration through a Hopf bifurcation, the occurrence of which depends on both the stability and the size of the basin of attraction of the bifurcating periodic motion. The experimental results well coincide with the theoretical and numerical ones, and strongly support the simplification of the digital filters as the components of pure time delay. Finally, some suggestions are made to avoid the group delay induced instability.