Quantum two-state dynamics driven by stationary non-Markovian discrete noise: Exact results

We consider the problem of stochastic averaging of a quantum two-state dynamics driven by non-Markovian, discrete noises of the continuous time random walk type (multistate renewal processes). The emphasis is put on the proper averaging over the stationary noise realizations corresponding, e.g., to a stationary environment. A two-state non-Markovian process with an arbitrary non-exponential distribution of residence times (RTDs) in its states with a finite mean residence time provides a paradigm. For the case of a two-state quantum relaxation caused by such a classical stochastic field we obtain the explicit exact, analytical expression for the averaged Laplace-transformed relaxation dynamics. In the limit of Markovian noise (implying an exponential RTD), all previously known results are recovered. We exemplify new more general results for the case of non-Markovian noise with a biexponential RTD. The averaged, real-time relaxation dynamics is obtained in this case by numerically exact solving of a resulting algebraic polynomial problem. Moreover, the case of manifest non-Markovian noise with an infinite range of temporal autocorrelation (which in principle is not accessible to any kind of perturbative treatment) is studied, both analytically (asymptotic long-time dynamics) and numerically (by a precise numerical inversion of the Laplace-transformed averaged quantum relaxation).     

Reversible pharmacokinetic profiles of canrenoic acid and its biotransformed product. Canrenone in the rat

Pharmacokinetic profiles of canrenoic acid (CRA) and canrenone (CR), the reversible metabolite of CRA, were studied in the rat after intraportal (pv) administration in comparison with those after intravenous (iv) administration using an interconversion model. In the clearances for the irreversible loss. CL20 of CR was larger than CL10 of CRA. Nevertheless, the real plasma clearance of CR was less than that of CRA. The distribution volume V1 of CRA was almost close to the real distribution volume Vss.real.D of CRA at the steady state. The hepatic available fraction of CRA, FH1 and sequential hepatic available fraction of generated metabolite, FH2, were estimated. Simultaneous computer multi-line fitting of plasma concentration-time data was carried out and the adequacy of pharmacokinetic parameters in this model was tested using the iterative nonlinear least-squares regression program, MULTI.     

Minimax approximation for the decomposition of energy denominators in Laplace-transformed Møller-Plesset perturbation theories

We implement the minimax approximation for the decomposition of energy denominators in Laplace-transformed Moller-Plesset perturbation theories. The best approximation is defined by minimizing the Chebyshev norm of the quadrature error. The application to the Laplace-transformed second order perturbation theory clearly shows that the present method is much more accurate than other numerical quadratures. It is also shown that the error in the energy decays almost exponentially with respect to the number of quadrature points.     

Resolution and uniqueness of estimated parameters of a model of thin filament regulation in solution

The estimation of chemical kinetic rate constants for any non-trivial model is complex due to the nonlinear effects of second order chemical reactions. We developed an algorithm to accomplish this goal based on the Damped Least Squares (DLS) inversion method and then tested the effectiveness of this method on the McKillop–Geeves (MG) model of thin filament regulation. The kinetics of MG model is defined by a set of nonlinear ordinary differential equations (ODEs) that predict the evolution of troponin–tropomyosin–actin and actin–myosin states. The values of the rate constants are estimated by integrating these ODEs numerically and fitting them to a series of stopped-flow pyrene fluorescence transients of myosin-S1 fragment binding to regulated actin in solution. The accuracy and robustness of the estimated rate constants are evaluated for DLS and two other methods, namely quasi-Newton (QN) and simulated annealing (SA). The comparison of these methods revealed that SA provides the best estimates of the model parameters because of its global optimization scheme. However it converges slowly and does quantify the uniqueness of the estimated parameters. On the other hand the QN method converges rapidly but only if the initial guess of the parameters is close to the optimum values, otherwise it diverges. Overall, the DLS method proves to be the most convenient method. It converges fast and was able to provide excellent estimates of kinetic parameters. Furthermore, DLS provides the model resolution matrix, which quantifies the interdependence of model parameters thereby evaluating the uniqueness of their estimated values. This property is essential for estimating of the dependence of the model parameters on experimental conditions (e.g. Ca²⁺ concentration) when it is assessed from noisy experimental data such as pyrene fluorescence from stopped-flow transients. The advantages of the DLS method observed in this study should be further examined in other physicochemical systems to firmly establish the observed effectiveness of DSL vs. the other parameter estimation methods.     

Numerical simulations of phase separation dynamics in a water-oil-surfactant system

We have studied numerically the dynamics of the microphase separation of a water-oil-surfactant system. We developed an efficient and accurate numerical method for solving the two-dimensional time-dependent Ginzburg-Landau model with two order parameters. The numerical method is based on a conservative, second-order accurate, and implicit finite-difference scheme. The nonlinear discrete equations were solved by using a nonlinear multigrid method. There is, at most, a first-order time step constraint for stability. We demonstrated numerically the convergence of our scheme and presented simulations of phase separation to show the efficiency and accuracy of the new algorithm.     

QSIMPS: A response to the analytical challenge of the pharmacokinetic revolution

Pharmacokinetics is a collection of equations (and concepts) which can transform blood and/or urine concentration—time data for a drug into numerical parameters that are useful as practical instruments for deciding certain questions concerning the efficacy and safety of the drug. To generate the parameters requires the analysis of hundreds of samples; experiments to compare parameters require the analysis of thousands of samples. Typically, tens of thousands of analyses are needed to generate the pharmacokinetic data required to bring a new drug on the market. This paper describes QSIMPS, a collection of software and hardware for the rapid, automated collection, processing and analysis of gas chromatography—mass spectrometry—selected ion monitoring data acquired for use in pharmacokinetic studies.     

Meshless simulation of equilibrium swelling/deswelling of pH‐sensitive hydrogels

Hydrogels have been widely used in microelectromechanical systems (MEMS) and Bio‐MEMS devices. In this article, the equilibrium swelling/deswelling of the pH‐stimulus cylindrical hydrogel in the microchannel is studied and simulated by the meshless method. The multi‐field coupling model, called multi‐effect‐coupling pH‐stimulus (MECpH) model, is presented and used to describe the chemical field, electric field, and the mechanical field involved in the problem. The partial differential equations (PDEs) describing these three fields are either nonlinear or coupled together. This multi‐field coupling and high nonlinear characteristics produce difficulties for the conventional numerical methods (e.g., the finite element method or the finite difference method), so an alternative—meshless method is developed to discretize the PDEs, and the efficient iteration technique is adopted to solve the nonlinear problem. The computational results for the swelling/deswelling diameter of the hydrogel under the different pH values are firstly compared with experimental results, and they have a good agreement. The influences of other parameters on the mechanical properties of the hydrogel are also investigated in detail. It is shown that the multi‐field coupling model and the developed meshless method are efficient, stable, and accurate for simulation of the properties of the stimuli‐sensitive hydrogel. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 326–337, 2006     

AN INDIRECT METHOD FOR DETERMINATION OF PARAMETERS IN THE FORECAST MODEL

An indirect method for determining some parameters in a forecast model has been developed, by which the program of the original model with little modification can be used as a subroutine of the inversion system and the optimum estimate of the parameters can be obtained by calling the subroutine time after time. It is specially useful for the sophisticated numerical weather prediction model. It may also be utilized to improve the forecast timely in case of obvious differences between the recent observations and the forecasts. The effect of the method is verified by numerical simulation tests with simple models.     

Multidimensional stability analysis of a family of biparametric iterative methods: CMMSE2016

In this paper, we present a multidimensional real dynamical study of the Ostrowsky–Chun family of iterative methods to solve systems of nonlinear equations. This family was defined initially for solving scalar equations but, in general, scalar methods can be transferred to make them suitable to solve nonlinear systems. The complex dynamical behavior of the rational operator associated to a scalar method applied to low-degree polynomials has shown to be an efficient tool for analyzing the stability and reliability of the methods. However, a good scalar dynamical behavior does not guarantee a good one in multidimensional case. We found different real intervals where both parameters can be defined assuring a completely stable performance and also other regions where it is dangerous to select any of the parameters, as undesirable behavior as attracting elements that are not solution of the problem to be solved appear. This performance is checked on a problem of chemical wave propagation, Fisher’s equation, where the difference in numerical results provided by those elements of the class with good stability properties and those showed to be unstable, is clear.     

Energy dependent decay rates of Lennard-Jones clusters for use in nucleation theory

Decay rates of small clusters (containing between 10 and 40 Lennard-Jones atoms) are determined by molecular dynamics simulations. The cluster is defined by the condition that the atoms must lie within a specified distance of their center of mass, and initial isothermal states are generated using a Metropolis Monte Carlo method. Plots of the logarithm of the survival fraction against time are found to be nonlinear, indicating that the decay of constant temperature clusters is non-Markovian and depends on the collision rate with a thermalizing gas. However, when the clusters are banded according to their energies, exponential decay is seen. The energy dependent decay rates from simulations agree to within a factor of 2 with those estimated from equilibrium considerations (using free energies from thermodynamic integration and assuming a Gaussian energy distribution), indicating that clusters defined in this way can be used in Markovian rate equations. During nucleation, the cluster energy distribution is shifted from its equilibrium value, leading to a reduction in the nucleation rate by a temperature dependent factor of 100 or more, in the absence of a thermalizing carrier gas.     

Adaptive nonlinear control of a continuous stirred tank reactor

The paper deals with continuous-time nonlinear adaptive control of a continuous stirred tank reactor (CSTR). Control strategy is based on the application of a controller consisting of a linear and a nonlinear part. The static nonlinear part is derived as an inversion and exponential approximation of measured or simulated input-output data. Design of the dynamic linear part is based on an approximation of nonlinear elements in the control loop by a continuous-time external linear model with directly estimated parameters. In the control design procedure, polynomial approach with the pole assignment method was used. The nonlinear adaptive control was tested by simulations on a nonlinear model of a CSTR with a consecutive exothermic reaction.     

A novel thermodynamic state recursion method for description of nonideal nonlinear chromatographic process of frontal analysis

A novel thermodynamic state recursion (TSR) method, which is based on nonequilibrium thermodynamic path described by the Lagrangian-Eulerian representation, is presented to simulate the whole chromatographic process of frontal analysis using the spatial distribution of solute bands in time series like as a series of images. TSR differs from the current numerical methods using the partial differential equations in Eulerian representation. The novel method is used to simulate the nonideal, nonlinear hydrophobic interaction chromatography (HIC) processes of lysozyme and myoglobin under the discrete complex boundary conditions. The results show that the simulated breakthrough curves agree well with the experimental ones. The apparent diffusion coefficient and the Langmuir isotherm parameters of the two proteins in HIC are obtained by the state recursion inverse method. Due to its the time domain and Markov characteristics, TSR is applicable to the design and online control of the nonlinear multicolumn chromatographic systems.     

Estimation of confidence intervals for radial emissivity and optimization of data treatment techniques in Abel inversion

A novel method is described for finding confidence intervals for radially resolved emissivity derived from Abel inversion. The essence of the method is that the measured lateral emission profile (here a profile is defined as a line-of-sight spatial map) is split into two components, namely (a) an estimated noise-free lateral emission profile that mimics the lateral emission profile in the absence of measurement noise and (b) a noise profile that estimates the magnitude of noise along the lateral profile. Through random-number generation, noise with a magnitude defined by the noise profile is then artificially added to the estimated noise-free lateral profile. If this noise-addition and subsequent Abel inversion is iterated for a sufficient number of cycles, a statistical distribution of the Abel-inverted radial profiles can be obtained and confidence intervals can thus be estimated. It was verified that an effective means for the estimation of the noise-free lateral profile involves approximation of the lateral emission profile through polynomial fitting. A noise profile can then be obtained by assuming that the residual between the experimental lateral profile and the fitted profile is due solely to noise. The validity of this methodology was verified with seven lateral test profiles. In addition, the parameters that are commonly used in data treatment by Abel inversion are optimized. It was found that 100–300 data points are sufficient for accurate Abel inversion using the Nestor-Olsen inversion algorithm; more data points provide only minimal improvement in the inversion. Also, it was found that most commonly observed lateral emission profiles can be satisfactorily approximated by polynomials with a degree of fifteen. Further, polynomial fitting can be used to partially filter out noise in the lateral profile. Depending on the shape of the lateral profile and the magnitude of the noise, the optimal polynomial order, in general, ranges from ten to fifteen. For higher-degree polynomials, the effectiveness of a polynomial as a noise filter decreases and severely degrades the quality of the Abel-inverted radial profile.     

Role of geometrical dimensions in electrophoresis applications with orthogonal fields

The role of geometrical dimensions in electrophoresis applications with axial and orthogonal (secondary) electric fields is investigated using a rectangular capillary channel. In particular, the role of the applied orthogonal electrical field in controlling key parameters involved in the effective diffusivity and effective (axial) velocity of the solute is identified. Such mathematically friendly relationships are obtained by applying the method of spatial averaging to the solute species continuity equation; this is accomplished after the role of the capillary geometrical dimensions on the applied electrical field equations has been studied. Moreover, explicit analytical expressions are derived for the effective parameters, i.e., diffusivity and convective velocity as functions of the applied (orthogonal) electric field. Previous attempts (see Sauer et al., 1995) have only led to equations for these parameters that require numerical solution and, therefore, limited the use of such results to practical applications. These may include, for example, the design of separation processes as well as environmental applications such as soil reclamation and wastewater treatment. An illustration of how a secondary electrical field can aid in reducing the optimal separation time is included.     

Parametric uncertainties and influence of the dead volume representation in modelling simulated moving bed separation processes

In this study, a systematic numerical procedure for identifying the model parameters of simulated moving bed (SMB) separation processes is developed. The parameters are first estimated by minimizing a weighted least-squares criterion using experimental data from batch experiments, e.g. the time evolution of the concentration of elution peaks. Then, a cross-validation is achieved using data from experiments in SMB operation. At this stage, the importance of a careful modelling of the dead volumes within the SMB process is highlighted. In addition, confidence intervals on the estimated parameters and on the predicted concentration profiles are evaluated.     

An efficient algorithm for solving nonlinear equations with a minimal number of trial vectors: applications to atomic-orbital based coupled-cluster theory

The conjugate residual with optimal trial vectors (CROP) algorithm is developed. In this algorithm, the optimal trial vectors of the iterations are used as basis vectors in the iterative subspace. For linear equations and nonlinear equations with a small-to-medium nonlinearity, the iterative subspace may be truncated to a three-dimensional subspace with no or little loss of convergence rate, and the norm of the residual decreases in each iteration. The efficiency of the algorithm is demonstrated by solving the equations of coupled-cluster theory with single and double excitations in the atomic orbital basis. By performing calculations on H(2)O with various bond lengths, the algorithm is tested for varying degrees of nonlinearity. In general, the CROP algorithm with a three-dimensional subspace exhibits fast and stable convergence and outperforms the standard direct inversion in iterative subspace method.     

Modeling of penetrant diffusion in glassy polymers with an integral sorption deborah number

A mathematical model was developed to explain the anomalous penetrant diffusion behavior in glassy polymers. The model equations were derived by using the linear irreversible thermodynamics theory and the kinematic relations in continuum mechanics, showing the coupling between the polymer mechanical behavior and penetrant transport. The Maxwell model was used as the stress–strain constitutive equation, from which the polymer relaxation time was defined. An integral sorption Deborah number was proposed as the ratio of the characteristic relaxation time in the glassy region to the characteristic diffusion time in the swollen region. With this definition, an integral sorption process was characterized by a single Deborah number and the controlling mechanism was identified in terms of the value of the Deborah number. The model equations were two coupled nonlinear differential equations. A finite difference method was developed for solving the model equations. Numerical simulation of integral sorption of penetrants in glassy polymers was performed. The simulation results show that (1) the present model can predict Case II transport behavior as well as the transition from Case II to Fickian diffusion and (2) the integral sorption Deborah number is a major parameter affecting the transition. © 1993 John Wiley & Sons, Inc.     

Numerical methods for light propagation in large LC cells: a new approach

Accurate numerical methods for the propagation of light in large 3D samples with strong lateral variation of the director field require prohibitive amounts of time. We consider and compare a standard spectral method and the Finite Difference in Frequency Domain method, showing that the CPU time can be reduced by one or two orders of magnitude using a perturbation approach or a recently developed Reduced Order Method. The equations obtained are applied to liquid crystal cells with in-plane switching, illuminated by a large incoherent source. The developed formalism, based on numerically exact equations, is particularly suitable for treating magnetic or optically active media and for extending to such media the well known approximations based on the 4 × 4 (Berreman) or 2 × 2 (Jones) matrices.     

Numerical methods for light propagation in large LC cells: a new approach

Accurate numerical methods for the propagation of light in large 3D samples with strong lateral variation of the director field require prohibitive amounts of time. We consider and compare a standard spectral method and the Finite Difference in Frequency Domain method, showing that the CPU time can be reduced by one or two orders of magnitude using a perturbation approach or a recently developed Reduced Order Method. The equations obtained are applied to liquid crystal cells with in-plane switching, illuminated by a large incoherent source. The developed formalism, based on numerically exact equations, is particularly suitable for treating magnetic or optically active media and for extending to such media the well known approximations based on the 4 ×4 (Berreman) or 2 ×2 (Jones) matrices.