Intra-tablet coating variability for several pharmaceutical tablet shapes

A combined discrete element method (DEM)–Monte Carlo simulation algorithm is used to investigate intra-tablet coating thickness variability for six pharmaceutical tablet shapes. The DEM simulations are used to collect tablet orientations when the tablets enter a specified spray zone. The Monte Carlo simulations are used to “coat” the tablets assuming a uniform spray and orientations chosen randomly from the distribution generated by the DEM simulations. The simulations demonstrate that for the non-spherical tablet shapes investigated here, the coating thickness variability, defined using a coefficient of variation, decreases with the square root of the number of coating trials, then approaches an asymptotic value. The greater the degree of a tablet's preferred orientation, the larger the asymptotic coefficient of variation. In addition, the number of trials required to reach this asymptote decreases for larger asymptotic values. These findings are consistent with theoretical analyses. For the range of parameters investigated, increasing the pan speed decreases the asymptotic coefficient of variation, but increasing the fill level had no consistent effect. The asymptotic coefficients of variation did not correlate with tablet sphericity, aspect ratio, or band-to-tablet area ratio, but the results suggest that a measure of a tablet's symmetry may provide better results. Modifying the angle of the coating spray so that a larger portion of the band is coated during each coating trial results in smaller asymptotic coefficients of variation than when the spray is oriented normal to the tablet bed surface. In addition, the relative order of the tablets with the smallest asymptotic coefficients of variation changes when the spray orientation is changed.     

An Asymptotic F Test for Uncorrelatedness in the Presence of Time Series Dependence

We propose a simple asymptotically F-distributed Portmanteau test for zero autocorrelations in an otherwise dependent time series. By employing the orthonormal series variance estimator of the variance matrix of sample autocovariances, our test statistic follows an F distribution asymptotically under fixed-smoothing asymptotics. The asymptotic F theory accounts for the estimation error in the underlying variance estimator, which the asymptotic chi-squared theory ignores. Monte Carlo simulations reveal that the F approximation is much more accurate than the corresponding chi-squared approximation in finite samples. The asymptotic F test is as easy to use as the chi-squared test: there is no need to obtain critical values by simulations. Furthermore, it has more accurate empirical sizes and substantial power advantages, comparing to other competitors.     

Monte Carlo simulation of terpolymerization: Optimizing the simulation and post-processing times

Monte Carlo simulations are a useful and easy way to understand a polymerization reaction process properly. However, achieving reliable results with Monte Carlo simulations can also lead to prohibitive computational times and a considerable amount of data to be processed afterward. The present study analyses the Monte Carlo simulation of a steady-state terpolymerization process to reduce the overall computational time of the simulation and the post-processing of its results. Different sorting algorithms (Bubble, Insertion, Selection, and Tim) and Python libraries (Joblib and Numba) were used. The chain composition distribution and the micro-structures resultant of different scenarios were assessed by processing the simulated mechanism results. The simulation time results indicate the Tim sorting algorithm as the best to use in the post-processing step and the Numba library as the best suited for both the simulation and the post-processing step.     

A Self‐Normalized Semi‐Parametric Test to Detect Changes in the Long Memory Parameter

We consider the problem of testing for change points in the long memory parameter. The test relies on semi‐parametric estimation of the long memory parameter, which does not require the complete parametric specification of the whole spectrum. A self‐normalizer utilizing a sequence of recursive semi‐parametric estimators is used to make the asymptotic distribution of the test statistic free of the nuisance scale parameter. We study the asymptotic behavior of the proposed test for situations when there is at most one change point and also when there are an unknown number of change points. Monte Carlo simulations are carried out to examine the finite‐sample performance of the proposed test.     

The effects of quasi Gaussian size distributions on dynamic behavior of a one-dimensional granular gas

A dynamic model of a one-dimensional granular gas with quasi Gaussian size distribution is presented, in which the rods are subject to inelastic mutual collisions and thermalized by a viscosity heat bath. The dispersive degree of the rod size distribution can be measured by the standard deviation σ at the same mean value μ. By Monte Carlo simulations, the effect of the dispersion of the quasi Gaussian size distribution on dynamic behavior of the system is investigated in the same inelasticity case for the first time. When the typical relaxation time τ of the driving Brownian process is longer than the mean collision time τ_c, the average energy of the system decays exponentially with time towards a stable asymptotic value, and the energy relaxation time τ_B to a nonequilibrium steady state becomes shorter with increasing values of σ. In the steady state, as σ increases, the velocity distribution deviates more obviously from the Gaussian one, such as the higher kurtosis and the fatter tails of the velocity distribution functions. Furthermore, with the increase of σ, the rods cluster more significantly, and the spatial correlations of density and velocities become stronger.     

DISTRIBUTIONS OF LEAST SQUARES ESTIMATORS OF AUTOREGRESSIVE PARAMETERS FOR A PROCESS WITH COMPLEX ROOTS ON THE UNIT CIRCLE

Abstract. Asymptotic distributions of the autoregressive parameters in the AR(2) model are derived, when the characteristic polynomial has a pair of complex roots on the unit circle. Percentage points are tabulated based on simulations from the asymptotic formulae. The usefulness of the asymptotic results in finite sample situations is investigated by a Monte Carlo study, and an illustrative example is given.     

A sequential procedure to determine the number of breaks in trend with an integrated or stationary noise component

Perron and Yabu (2009a) consider the problem of testing for a break occurring at an unknown date in the trend function of a univariate time series when the noise component can be either stationary or integrated. This article extends their work by proposing a sequential test that allows one to test the null hypothesis of, say, l breaks versus the alternative hypothesis of (l + 1) breaks. The test enables consistent estimation of the number of breaks. In both stationary and integrated cases, it is shown that asymptotic critical values can be obtained from the relevant quantiles of the limit distribution of the test for a single break. Monte Carlo simulations suggest that the procedure works well in finite samples.     

Poisson QMLE of Count Time Series Models

Regularity conditions are given for the consistency of the Poisson quasi‐maximum likelihood estimator of the conditional mean parameter of a count time series model. The asymptotic distribution of the estimator is studied when the parameter belongs to the interior of the parameter space and when it lies at the boundary. Tests for the significance of the parameters and for constant conditional mean are deduced. Applications to specific integer‐valued autoregressive (INAR) and integer‐valued generalized autoregressive conditional heteroscedasticity (INGARCH) models are considered. Numerical illustrations, Monte Carlo simulations and real data series are provided.     

Monte Carlo simulations of phase behavior and microscopic structure for supercritical CO2 and thiophene mixtures

The phase equilibria of thiophene in supercritical carbon dioxide are calculated by Monte Carlo simulations in Gibbs ensemble using a united atom force field. To validate the simulations, binary vapor–liquid coexistence curves were computed for two different temperatures using Monte Carlo simulations. An excellent agreement between simulations and experimental data is obtained. The effects of pressure on structural properties were studied for thiophene–CO₂ binary mixtures. The radial distribution functions and local composition of thiophene in CO₂ were investigated over a range of pressures. A weak dependence of thiophene structural properties with pressure was observed in supercritical phase. Local solution structure of thiophene in supercritical CO₂ was studied by computing angular–radial distribution functions and spatial distribution functions with three-dimensional probability distributions. The characteristic angular–radial distributions show a mutually parallel arrangement between thiophene plane and CO₂ molecules within the first solvation shell. Spatial distribution functions (SDFs) results show that CO₂ molecules have two higher probability distributions around thiophene molecules located above and below the thiophene ring.     

Dynamic Monte Carlo Simulations of Oscillatory Reactions

The dynamic Monte Carlo method has been used to simulate the 2 A + B₂ → 2 AB reaction catalyzed by a reconstructing substrate. Oscillatory behavior and spatio-temporal is studied as a function of grid size. Spatio-temporal pattern formation has been simulated in various forms: cellular patterns, target patterns, rotating spirals, and turbulent patterns. Cellular patterns are a manifestation of a local synchronization mechanism in which all reaction fronts periodically extinguish each other. This illustrates that dynamic Monte Carlo simulations form a promising technique and can be used to predict macroscopic kinetic phenomena on a molecular basis.     

Combination of a Monte Carlo approach with the contact time distribution concept for the steady-state modeling of an isothermal heterogeneous coordinated anionic ring opening polymerization reactor

In this paper, the possibility of combining a molecular level Monte Carlo simulation with a chemical engineering model of an isothermal fixed-bed reactor is demonstrated. This approach is applied to the isothermal heterogeneous coordinated anionic ring opening polymerization of ε-caprolactone. The contact time distribution (CTD) concept as defined by Orcutt et al. (Chem. Eng. Prog. Symp. Ser. 38 (58) (1962) 1), Nauman and Collinge (Chem. Eng. Sci. 23 (1968a) 1309) and Shinnar et al. (Chem. Eng. Sci. 27 (1972) 1627) is used. The contact time is the time spent by a reactant within the pores of the pellets constituting the fixed-bed. The classical monodisperse model represents the hydrodynamic and mass transfer in the reactor. The CTD is calculated from this model according to Shinnar et al. theory. From that point, the reactor outlet monomer conversion can be calculated analytically, the ε-caprolactone consumption being a first-order process. Furthermore, the molecular size distribution (MSD) of the oligomers at the reactor outlet is computed on the basis of Monte Carlo simulations. Comparisons with experimental data are given.     

Comparison of Monte Carlo and quasi-Monte Carlo technique in structure and relaxing dynamics of polymer in dilute solution

Structure and dynamics of polymer in solvent solution is an important area of research since the functional properties of polymer are largely dependent on the morphology of the polymers in solution. This structure related properties are especially important in case of surface science where the phase-separated morphology in the micro/nano scale dictates the properties of the product. Modeling polymers in solution is an efficient way to determine the morphology and thus the properties of the products. It saves time as well as helps to design novel materials with desired properties. Polymers in solution systems are generally modeled with bead spring model and Monte Carlo or importance sampling Monte Carlo simulations is used to find the optimal configuration where the energy of the system is minimized. Often in these simulations, random numbers are used in the Monte Carlo steps. Normally random numbers try to form clusters and do not cover the entire dimension of the system. Thus the minimum energy structures obtained from simulations with random numbers are not optimal configuration of the system. In the present work a lattice-based model is used for polymer solution system and importance sampling Monte Carlo is used for simulation. Quasi-random numbers generated from Hammersley sequence sampling (HSS) are used in the simulation steps for stochastic selection polymers and its movements. Quasi-random numbers obtained from HSS are random in nature and they have n-dimensional uniformity. They do not form clusters and the structural configuration obtained using quasi-random numbers are optimal in nature. The optimal configurations of the polymers as obtained from random number and quasi-random number are compared. The result shows that simulation with HSS attains a lower energy state after initial quench. At the late stage of spinodal decomposition, the structure factor decrease-showing Ostwald ripening which is not observed from simulation with random numbers.     

A hybrid stochastic-deterministic algorithm for lattice-gas models of catalytic reactions and the computation of TPD spectra

We present a hybrid numerical approach for modeling surface reactions in the framework of a lattice-gas model with lateral interactions between adsorbed particles. A hybrid multiscale algorithm, which we refer to as Quasi-Equilibrium Kinetic Monte Carlo (QE-KMC), comprises traditional Metropolis Monte Carlo (MMC) simulations of equilibrium systems and standard numerical methods for deterministic ordinary differential equations (ODEs). The functional dependence of these ODEs on the macroscopic state variables (adsorbate coverages) is not explicitly known, but their right-hand sides can be evaluated “on the fly” with prescribed accuracy by means of the MMC simulations. At the time scale of these ODEs it is assumed that an equilibrium statistical distribution of adsorbed particles on an infinite lattice is attained at every moment in time due to infinitely fast surface diffusion. QE-KMC and conventional KMC simulations are used to study the temperature-programmed desorption (TPD) spectra of adsorbed particles. We critically discuss results of previous studies that applied Monte Carlo simulations to describe the TPD spectra in the case of fast adsorbate diffusion and strong lateral interactions. We show that the quasi-equilibrium TPD spectra can be quickly and accurately estimated by the QE-KMC algorithm, while the KMC simulations require much more extensive computational resources to obtain the same results.     

Tests for Linearity in Star Models: Supwald and Lm‐Type Tests

This article investigates approximation and supremum approaches for testing linearity in smooth transition autoregressive (STAR) models. We show that since the approximation of STAR models by Taylor series expansions may not accurately describe the specific transition dynamic when the process is away from the null, LM‐type tests may fail to detect the form of nonlinearity for which they are designed for. Investigating a supremum approach, the article provides the asymptotic distribution of a SupWald test that is obtained by taking the supremum of a Wald statistic over the Cartesian product of the spaces for the transition and threshold parameters. Simulated asymptotic critical values for the resulting tests are provided for a wide range of autoregressive orders and shown to differ across exponential and logistic STAR (ESTAR and LSTAR) models. Monte Carlo experiments show that SupWald tests for ESTAR and LSTAR models outperform LM‐type tests, compares well relative to the recently developed score‐based tests and each SupWald statistic performs the best against the true alternative for which it is formed. SupWald tests also provide results that are consistent with the findings from (independently) estimating and diagnostic testing of STAR models in real exchange rate data.     

Phenol adsorption on porous and non-porous carbons

Phenol physisorption on a series of porous and non-porous amorphous carbons was studied at 298 K. Grand Canonical Monte Carlo computer simulations were performed to simulate phenol adsorption from the gas phase. Phenol is adsorbed in a solid-like state within the pores and there is no well-developed multilayer regime. The ‘t’ plot method was adapted to phenol adsorption and the results obtained are in agreement with the model solids employed. The simulated adsorption isotherms were compared with experimental results obtained for adsorption from aqueous solutions of phenol. BET surface areas were calculated. Other characteristics of the adsorption system analyzed were: adsorption energy distribution functions, density profiles, distribution of molecules according to gas-solid energy, and local isotherms.     

Excess free energy of nanoparticles in a polymer brush

We present an efficient method for direct determination of the excess free energy ΔF of a nanoparticle inserted into a polymer brush. In contrast to Widom's insertion method, the present approach can be efficiently implemented by Monte Carlo or Molecular Dynamics methods also in a dense environment. In the present investigation the method is used to determine the free energy penalty ΔF(R, D) for placing a spherical particle with an arbitrary radius R at different positions D between the grafting plane and the brush surface. Deep inside the brush, or for dense brushes, one finds ΔF ∝ R³ whereas for shallow nanoclusions ΔF ∝ R², regardless of the particle interaction (attractive/repulsive) with the polymer.The pressure and density fields around spherical nanoinclusions in a polymer brush are also investigated. Extensive Monte Carlo simulations show that the force, exerted on the particle by the surrounding brush, depends essentially on the proximity of the nanocolloid particle to the brush surface not only in strength but also with respect to its angular distribution. For shallow nanoinclusions close to the brush surface this angular distribution is shown to result in a growing buoyant force while deep inside the brush this effect is negligible.     

Differential mobility analyser transfer functions in scanning mode

A novel method is described for the calculation of the differential mobility analyser (DMA) transfer function in scanning mode, which is based on the derivation of an analytical solution for the non-diffusive particle trajectories inside the DMA, under exponentially varying electrical field and fully developed laminar flow. The scanning mode transfer functions can substantially improve the measurement accuracy of fast scanning mobility particle sizers (SMPS) as shown by Collins, Cocker, Flagan, and Seinfeld [(2004). The scanning DMA transfer function. Aerosol Science and Technology, 38, 833–850]. Compared to the Monte Carlo simulations described by Collins et al., the method developed here is much more accurate and sufficiently fast to be employed in advanced DMA inversion algorithms.     

Fault detection and diagnosis with parametric uncertainty using generalized polynomial chaos

This paper presents a new methodology to identify and diagnose intermittent stochastic faults occurring in a process. A generalized polynomial chaos (gPC) expansion representing the stochastic inputs is employed in combination with the nonlinear mechanistic model of the process to calculate the resulting statistical distribution of measured variables that are used for fault detection and classification. A Galerkin projection based stochastic finite difference analysis is utilized to transform the stochastic mechanistic equation into a coupled deterministic system of equations which is solved numerically to obtain the gPC expansion coefficients. To detect and recognize faults, the probability density functions (PDFs) and joint confidence regions (JCRs) of the measured variables to be used for fault detection are obtained by substituting samples from a random space into the gPC expansions. The method is applied to a two dimensional heat transfer problem with faults consisting of stochastic changes combined with step change variations in the thermal diffusivity and in a boundary condition. The proposed methodology is compared with a Monte Carlo (MC) simulations based approach to illustrate its advantages in terms of computational efficiency as well as accuracy.     

A symptotic robustness of sequential confidence intervals and the connection with influence functions

The robustness of sequential confidence intervals is studied by considering contamination with probability ε of the basic underlying distribution in a so-called gross errors model. Asymptotic theory is considered when d → 0, where the prescribed length of the interval is 2d, and simultaneously ε ? ε(d) → 0. A general theorem, in a distribution free setting, is given which provides expressions for the asymptotic coverage probability and the asymptotic distribution of the stopping variable. The results depend on the rate of ε(d)/d as d → 0 and on the contaminating distribution. If the latter distribution is degenerate, it turns out that the influence functions of the above mentioned two estimators used in the construction of the procedure, appear in the expressions for the asymptotic coverage probability and the asymptotic distribution of the stopping variable respectively. This shows how the sequential procedure inherits the robustness properties of the estimators concerned and how this is quantified. The general theorem is specialized to two procedures for the estimation of the mean of a symmetric distribution. Results of Monte Carlo studies indicate agreement between the asymptotic theory and the actual behavior of the procedures.