Non‐stationary autoregressive processes with infinite variance

Consider an AR(p) process , where {?_t} is a sequence of i.i.d. random variables lying in the domain of attraction of a stable law with index 0<α<2. This time series {Y_t} is said to be a non‐stationary AR(p) process if at least one of its characteristic roots lies on the unit circle. The limit distribution of the least squares estimator (LSE) of for {Y_t} with infinite variance innovation {?_t} is established in this paper. In particular, by virtue of the result of Kurtz and Protter (1991) of stochastic integrals, it is shown that the limit distribution of the LSE is a functional of integrated stable process. Simulations for the estimator of β and its limit distribution are also given.     

Motions of a sphere in a time-dependent stokes flow: A generalization of Faxen’s law

A general solution of the unsteady Stokes equation in spherical coordinates is derived for flow in the exterior of a sphere, and then applied to study the arbitrary unsteady motion of a rigid sphere in an unbounded single fluid domain which is undergoing a time-dependent mean flow. Calculation of the hydrodynamic force and torque on the sphere leads to a generalization of the Faxen’s law to time-dependent flow fields which satisfy the unsteady Stokes equation. For illustrative purposes, we consider the relative motion of gas bubbles which undergo very rapid oscillations so that the generalized Faxen’s law derived for a solid sphere can be applied. We also demonstrate that our results reduce to those of Faxen for the steady flow limit.     

Robustness of the autoregressive spectral estimate for linear processes with infinite variance

Consider a discrete-time linear process { x _t }, a one-sided moving average of independent identically distributed random variables {ε _t }, with the common distribution in the domain of attraction of a symmetric stable law of index δ∈ (0, 2) and the moving-average coefficients b ( j ) such that ε _t is invertible in terms of the present and possibly infinite past values of { x _t }. By treating { x _t } as if it is second-order stationary, a normalized spectral density function f (μ) is defined in terms of the b ( j ) and, having observed x ₁, ..., x _T , an autoregression of order k is fitted by the well-known Yule–Walker and least squares methods and the normalized autoregressive spectral estimators are constructed. On letting k ←∞ as T ←∞, but sufficiently slowly, these estimators are shown to be uniformly consistent for f (μ), the convergence rate being T ^−1/φ, φ > δ. The finite sample behaviour is investigated by a simulation study which also examines possible effects of considering 'non-invertible' models.     

Studies in the control of tubular reactors—II Stabilization by modal control

The lumped parameter model that was developed in Part I is linearized to obtain the linear dynamical model of the system near an unstable steady state. When concentration and temperature measurements are possible along the reactor length and their number is the same as the number of collocation points, modal state-feedback controllers are designed to relocate the largest eigenvalues to negative values and thus locally stabilize an unstable steady state. Transient calculations of the non-linear system equations are preformed and the domain of attraction of the stabilized steady state is examined for different locations of the eigenvalues of the closed loop system. It is seen that for both problems I and II the domain of attraction becomes very large when the unstable and one stable eigenvalue are shifted near the third largest one. This is true in spite of the large differences in their dynamical characteristics.     

Effect of Kinesin-5 Tail Domain on Motor Dynamics for Antiparallel Microtubule Sliding

Kinesin-5 motor consists of two pairs of heads and tail domains, which are situated at the opposite ends of a common stalk. The two pairs of heads can bind to two antiparallel microtubules (MTs) and move on the two MTs independently towards the plus ends, sliding apart the two MTs, which is responsible for chromosome segregation during mitosis. Prior experimental data showed that the tails of kinesin-5 Eg5 can modulate the dynamics of single motors and are critical for multiple motors to generate high steady forces to slide apart two antiparallel MTs. To understand the molecular mechanism of the tails modulating the ability of Eg5 motors, based on our proposed model the dynamics of the single Eg5 with the tails and that without the tails moving on single MTs is studied analytically and compared. Furthermore, the dynamics of antiparallel MT sliding by multiple Eg5 motors with the tails and that without the tails is studied numerically and compared. Both the analytical results for single motors and the numerical results for multiple motors are consistent with the available experimental data.     

Effect of lewis number on the transient behavior of catalytic gas—solid systems

The influence of Lewis number, ρ?_pD_e/?λ_e, on the transient behavior of exothermic irreversible first-order catalytic reaction inside a spherical porous pellet is analyzed taking into account external heat and mass transport as well as initial conditions corresponding to a pellet at uniform temperature and devoid of reactant.The destabilizing effect of low Lewis number and the existence of a limit cycle are confirmed and a qualitative interpretation is put forward.In regions of multiple stationary states, the Lewis number modifies substantially the domains of attraction of the stable solutions. Moreover, for certain values of this group and over the whole range of realistic initial conditions, the system remains entirely within the domain of attraction of just one of the steady states.     

The Variance Ratio Test with Stable Paretian Errors

This paper examines the distribution of the overlapping variance ratio (OVR) statistic when the errors are distributed with thick tails as described by the family of stable Paretian distributions. The asymptotic distribution of the OVR statistic, which depends on the characteristic exponent, can be estimated using simulation. It is found that the convergence of the distribution of the OVR statistic to its asymptotic limit is extremely slow. Thus, the asymptotic results will not be able to provide any useful approximation in finite samples. To facilitate the OVR statistic as a test for the random walk hypothesis, the tail quantiles are estimated for several finite sample sizes.     

Transient analysis of exothermic reactions within catalyst pellets. Effect of initial conditions

The transient behavior of exothermic, irreversible, first-order catalytic reaction inside a spherical, porous pellet is analyzed using mixed type boundary conditions to represent external resistances. Calculations are performed for Lewis numbers, ?C_pD_e/?λ_e, equal to unity and a pellet initially at uniform temperature and devoid of reactant, which is considered realistic. Phenomena occurring during the evolution of transients in the region of steady external transport limitation are described.In ranges where more than one steady state exists, the effect of the initial conditions on the attainment of the different steady states is analyzed. The results include information about transient behavior when the initial condition is at the boundary between stable domains of attraction. For certain ranges of parameters it is shown that the initial conditions studied may fall outside of the domain of attraction of one of the stable steady states, demonstrating that information about domains of attraction as well as steady states should be taken into account when designing catalytic systems.     

Design of Fuzzy Control for Maintaining a CSTR in the Steep Section of Its Nonlinear Characteristic

The operation of a continuous stirred tank reactor (CSTR) in the steep domain of its nonlinear process characteristic is studied, using catalyzed hydrogen peroxide decomposition as a model process because of its transparent nonlinear dynamics. Unstable equilibrium states and corresponding limit cycles exist in the steep domain, the latter representing dynamically stable modes of operation where safer control is expected due to the lower educt concentration in the reactor. For CSTR control in the steep domain, a fuzzy approach is applied based on simulations and experimental understanding of the stable and unstable states of the reactor. Simulations are carried out for the set‐point control of an unstable equilibrium state, and the tracking control of the corresponding stable limit cycle. The simulation results show that operation in the limit cycle requires much less control energy than that of the corresponding unstable equilibrium state in the case of disturbances, e. g., due to unavoidable modelling errors.     

Semiparametric Estimation in Time‐Series Regression with Long‐Range Dependence

Abstract. We consider semiparametric estimation in time‐series regression in the presence of long‐range dependence in both the errors and the stochastic regressors. A central limit theorem is established for a class of semiparametric frequency domain‐weighted least squares estimates, which includes both narrow‐band ordinary least squares and narrow‐band generalized least squares as special cases. The estimates are semiparametric in the sense that focus is on the neighbourhood of the origin, and only periodogram ordinates in a degenerating band around the origin are used. This setting differs from earlier studies on time‐series regression with long‐range dependence, where a fully parametric approach has been employed. The generalized least squares estimate is infeasible when the degree of long‐range dependence is unknown and must be estimated in an initial step. In that case, we show that a feasible estimate which has the same asymptotic properties as the infeasible estimate, exists. By Monte Carlo simulation, we evaluate the finite‐sample performance of the generalized least squares estimate and the feasible estimate.     

Generalized age-adjusted effective modulus method for creep in composite beam structures. Part I — Theory

The age-adjusted effective modulus method for approximate analysis of structures exhibiting linear aging creep law and underlying Ba?ant's theorem is generalized to a form which is more accurate for composite beam structures in flexure. A rigorous derivation of a linear algebraic relationship between the internal force histories and the deformation histories of the beam is given using the integral operator symbolics.     

Synthesis and Solution Properties of a Double‐Tailed Quaternary Ammonium Surfactant with a Protrudent Head Group

Modification of the molecular structure of surfactants is an effective method for exploring their self‐assembly. A double‐tailed quaternary ammonium surfactant with a protrudent head group, namely 2‐octyldecyltrimethylammonium bromide (2‐ODTAB) was synthesized, and the solution properties were investigated by surface tension, dynamic light scattering, and cryogenic TEM. A comparative study was also performed on the traditional double‐tailed homologue surfactants dioctyldimethylammonium bromide (8–8), didecyldimethylammonium bromide (10–10), and didodecyldimethylammonium bromide (12–12). The results showed that 2‐ODTAB was more effective at lowering surface tension and in forming stable vesicles than traditional double‐tailed surfactants with similar alkyl chain length. The reason is attributed to the improved structure of 2‐ODTAB, in which the two alkyl tails are connected to the ionic head group by one carbon atom. This structure imparts more freedom to the head group and thus favors formation of more stable aggregates at low concentration. In addition, the lower limit of the alkyl chain length of the double‐tailed surfactants for forming stable vesicles was illustrated.     

Feasible separation regions for distillation II: A generalized distillation limit

The boundary of the feasible separation region consists of several curves that are related to specific types of operating modes of the column. Some of these curves create a well‐known product composition multitude, whereas other curves form a generalized distillation limit. The generalized distillation limit demarcates the sloppy splits (i.e., separations in which the composition of at least one product lies inside the composition space) from regions not accessible by distillation and depends on the thermodynamic state of the feed (a mixture of vapor and liquid in equilibrium or saturated vapor/liquid) and column equipment (total/partial condenser and total/partial reboiler). The mathematical equations describing the generalized distillation limit are obtained based on the relationships between the curves (which form the generalized distillation limit) and specific types of operating modes of the column as well as the material balances for the enriching and stripping columns. Furthermore, the vapor and liquid pinch‐point curves, which go through the feed composition point, are not dependent on the thermodynamic state of the feed and column equipment. In addition, an algorithm for determining the generalized distillation limit is obtained. © 2017 American Institute of Chemical Engineers AIChE J, 2017     

Analysis of axially dispersed systems with general boundary conditions—II: Solution for dispersion in the appended sections

The axial dispersion problem formulated in Part I[1] of this series is solved here for “case 1” which corresponds to the situation of finite, non-zero dispersion in both semi-infinite appended sections. The solutions are obtained in terms of the spectrum of the operator consisting of a continuous spectrum and frequently a finite set of discrete eigenvalues. Transient solutions are presented for a first order reaction in the axially dispersed reactor for a variety of initial conditions demonstrating where and where not the Danckwerts boundary conditions may be valid. It is shown that “controlling” eigenvalues may be exploited from persisting tails of transient concentration measurements on tracer experiments to obtain dispersion parameters.     

On Uniqueness of Moving Average Representations of Heavy‐tailed Stationary Processes

The paper solves the open problem of identification of two‐sided moving average representations with i.i.d. summands, for stationary processes in non‐Gaussian domains of attraction of α‐stable laws. This shows the possibility to identify nonparametrically both the sequence of two‐sided moving average coefficients and the distribution of the underlying heavy‐tailed i.i.d. process.     

On a Mixture GARCH Time‐Series Model

Abstract. Recently, there has been a lot of interest in modelling real data with a heavy‐tailed distribution. A popular candidate is the so‐called generalized autoregressive conditional heteroscedastic (GARCH) model. Unfortunately, the tails of GARCH models are not thick enough in some applications. In this paper, we propose a mixture generalized autoregressive conditional heteroscedastic (MGARCH) model. The stationarity conditions and the tail behaviour of the MGARCH model are studied. It is shown that MGARCH models have tails thicker than those of the associated GARCH models. Therefore, the MGARCH models are more capable of capturing the heavy‐tailed features in real data. Some real examples illustrate the results.     

Morphological effects on the transverse permeability of arrays of aligned fibers

The effect of micro-structure on the effective transverse permeability (K) of unidirectional arrays of cylindrical fibers is investigated computationally using the Boundary Element Method (BEM). By converting the original problem into an integral equation on the boundaries of the domain of interest, the BEM allows the straightforward discretization of complex, multi-connected domains with a fraction of the effort required by domain methods. The equations of Stokes flow are solved in cells containing up to 36 individual fibers located either in uniform arrays (square and hexagonal) or in random locations and the effective permeability is calculated from the corresponding pressure drop and flow rate. Random placement of particles results in the generation of individual microstructures whose permeability shows a scatter around a mean value. This scatter increases significantly as the porosity (ϕ) of the cell is reduced. Statistical averages are compared to analytical and numerical predictions for the permeability of a perfect square array. It is found that a fully random structure exhibits a permeability slightly higher than a perfect square array for very high porosity values (ϕ > 0.8), with this trend disappearing for ϕ ≤ 0.8 and the averages coinciding with the result of the perfect square array. Additionally, the effect of random perturbations in the location and size of fibers around mean values corresponding to perfect square packing of mono-sized fibers is investigated. It is found that such variations have little effect on K at the high-porosity end (ϕ > 0.8), but they result in a noticeable increase in the effective permeability (as compared to a perfect square array) at the limit of low porosity (ϕ > 0.5).     

Empirical Likelihood for a Long Range Dependent Process Subordinated to a Gaussian Process

This article develops empirical likelihood methodology for a class of long range dependent processes driven by a stationary Gaussian process. We consider population parameters that are defined by estimating equations in the time domain. It is shown that the standard block empirical likelihood (BEL) method, with a suitable scaling, has a non‐standard limit distribution based on a multiple Wiener–Itô integral. Unlike the short memory time series case, the scaling constant involves unknown population quantities that may be difficult to estimate. Alternative versions of the empirical likelihood method, involving the expansive BEL (EBEL) methods are considered. It is shown that the EBEL renditions do not require an explicit scaling and, therefore, remove this undesirable feature of the standard BEL. However, the limit law involves the long memory parameter, which may be estimated from the data. Results from a moderately large simulation study on finite sample properties of tests and confidence intervals based on different empirical likelihood methods are also reported.