Abstract—A Kalman filter (KF) **estimator** has been formulated using a sequence of **reduced**-**order** models representing a whole batch behavior for providing the estimates of dynamic composition in a ternary batch distillation process operated in an **optimal**-reflux policy. A set of full-**order** models is firstly obtained by linearizing around different pseudo-steady state operating conditions along batch **optimal** profiles. They are further **reduced** their orders to achieve their observability and controllability individually by using a model reduction method. The performances of the **reduced**-**estimator** have been investigated and compared with those of a conventional nonlinear **estimator**. Simulation results have demonstrated that the performances of the proposed **estimator** are reasonably good and almost identical to the conventional one in all cases.

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for some biasing constant , . To alleviate the problem of bias in Ridge regression, Singh et al (1986) proposed an Almost Unbiased Ridge **estimator** (AUGRR) using the Jackknife technique which was actually introduced by Quenouille(1956).The Ridge regression **estimator** has undergone several modifications over the years. Batah et al (2008) introduced the modified Jackknife Ridge **estimator** (MJRE) by combining the ideas of GRE and Jackknife Ridge estimators. Batah (2011) again introduced another variant of the Jackknife **estimator** known as the Generalized Jackknife Ridge **estimator** (GJR) together with its associated Generalized Jackknife Ordinary Ridge **estimator** (GOJR).The Generalized Ridge **estimator** (GRE) leads to a reduction in the sampling variance whereas the Jackknife Ridge **estimator** (JRE) leads to the reduction of bias. Khurana, Chaubey & Chandra (2012) suggested a new Ridge **estimator** called the second- **order** Jackknife Ridge **estimator** (J2R) to further reduce bias. In this paper, a generalization for the nth-**order** Jackknife Ridge **estimator** is proposed for further reduction of bias. A convergence criterion is designed to select the **optimal** n so that an **estimator** with minimum n is selected for the most **reduced** bias and minimum variance. In canonical form model (1) can be written as

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these difficulties. More specifically, §2 describes the proper orthogonal decomposi- tion (POD) technique, also known as the Karhunen-Lo`eve procedure, that is used to obtain low dimensional dynamic models of distributed parameter systems. The POD method, which is well known in statistical and pattern recognition fields [2], has been shown to be an effective tool for the analysis of complex systems such as turbulence flows, shear flows, and weather prediction (see e.g., [3] and the references therein). Roughly speaking, POD is an **optimal** technique of finding a basis that spans an ensemble of data, collected from an experiment or a numerical simulation of a dynamical system, in the sense that when these basis functions are used in a Galerkin procedure, they will yield a finite dimensional system with the smallest possible degrees of freedom. Thus this method may well be suited to treat **optimal** control and parameter estimation of distributed parameter sys- tems. In §3, we describe our successful use of POD techniques as a **reduced** basis method for computation of feedback controls and compensators in a high pres- sure CVD reactor. More specifically, we present a proof-of-concept computational implementation of this method with a simplified growth example of group III-V compounds that includes multiple species and controls, gas phase reactions (no surface reactions), and time dependent tracking signals that are consistent with pulsed vapor reactant inputs. In §4 state estimation and feedback tracking con- trol methods for nonlinear systems are presented. The methods, which are based on the “state-dependent Riccati equations”, allow the construction of nonlinear estimators and nonlinear feedback tracking controls for a wide class of systems including high pressure CVD systems considered here. The performance of the nonlinear **estimator** and tracking control will be presented on a flight dynamics simulation example. Finally, §5 contains our overall conclusions.

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Conditional Value-at-Risk (CVaR) is one of the commonly used risk meas- ures. The paper shows that the **optimal** **estimator** of CVaR is strong consis- tency if the first-**order** moment of the population exists. We subsequently carry out numerical simulations to test the conclusion. We use the results to make an empirical analysis of Shenzhen A shares.

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result enables one to construct a standard asymptotic confidence region on the true parameters. However, the first-**order** approximation might be inaccurate with samples of the sizes encountered in many applications and hence it might yield a substantial gap between the true and the nominal coverage probabilities in practice. On the other hand, it is well known that bootstrap generally provides asymptotic refinements to the coverage probabilities of confidence regions under regularity conditions, see Beran (1988), Hall (1986, 1992), and Horowitz (1997, 2001). However, the standard theory of the bootstrap can not be directly applied to the confidence regions based on the QR **estimator** because the statistic of interest is not a smooth function of sample moments that has an Edgeworth expansion. 1 In his important recent contribution, Horowitz

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We study a replenishment model that the retailer shares his leftovers on his direct online sales channel and traditional retail channel. When one of the sales channels is out of stock and another has leftovers, the latter shares its leftovers with the former. Commonly, there is competition between two channels. There- fore, we take the consumers’ behavior including retailer inconvenience cost of traditional retail channel, and risks and delivery lead time of direct online sales channel into consideration. Then, we use the function of consumer utility to measure the Demand distribution function. Furthermore, the **optimal** replenish- ment models in independent decision-making and in lateral transshipment are established separately. Then the results of two models are compared by **order** quantity and retailer’s profit. What’s more, it is found that lateral transshipment between dual sales channel inventory can reduce the **order** quantity and increase the retailer’s profit when the prices and demand distribution satisfy certain condi- tions. Finally, the numerical analysis of the model is carried out. The validity of the research results is proved by examples.

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unknown. However, in the context of a Monte Carlo study it is desirable since estimation performance is not impacted by the noise introduced through a data driven bandwidth selection. See Jones and Signorini (1997) for an approach that is similar to ours. Table 1 provides average absolute bias (B) and average mean squared error (MSE) for each **estimator** and each density considered for n = 200, 400 respectively. 3

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Terminal dampers are an essential component to multi-zone VAV HVAC systems, without them the air flow would be entirely controlled at the main air handler. The COBE ROM was simplified to a single zone building so the terminal damper feedback loop was not modeled directly but its main function was captured and incorporated into another aspect of the model. In multi-zone system, the terminal box dampers control the flow supplied to the zone, and in part controlling the amount of heating or cooling introduced to the zone. Each zone will have different heating or cooling loads and the main AHU is controlled such that the largest load will still be met. In zones that call for less heating or cooling, the terminal dampers are modulated to avoid over-conditioning the space. The **optimal** way to control the terminal dampers is by allowing the highest flow rate into the room which reduces the system’s overall pressure losses. If all the zones are being adequately conditioned and the terminal dampers are set at their minimum position, the main AHU’s VFD reduces the overall supply air flow rate. In the single zone COBE model, when the zone is being over-conditioned, a feedback signal is sent to the main air handler to reduce the supply air flow. The modulation does

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two coherent states of the field, each paired with an atomic state in a superposition of the excited and ground states, but with opposite phases. When we let γ 6 = 0, but still small, the resulting spontaneous emission events correspond to the system switching between the two stable states [19]. Van Handel and Mabuchi [11] examined the phase bistable state from the perspective of pro- jection filtering. They defined a three-dimensional, nonlinear manifold: one dimension measures the relative populations of two gaussian field states (paired with the corresponding atomic state appropriate for their phase), and the other two dimensions reflect the positions of the two gaussians. After an excellent, clear derivation of the filtering equation and rules for projecting it, they project the master equation onto this nonlinear manifold and derive a nonlinear set of stochastic differential equations to use as a simple filter. This filter behaves almost identically to the **optimal** filter, despite its simplicity, which implies that the underlying dynamics are fundamentally quite simple. If we take the two gaussian field modes to be fixed, the resulting 1-dimensional system is identical to the filter for a stationary Markovian jump process (the Wonham filter). Our analysis of the geometry of this system in Chapter 4 focuses on the geometry of the switching behavior; we should expect that the dynamics we derive will require the inclusion of the state position variables to characterize the transitions.

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representations. In **order** to facilitate the calculations of the La norms, the con tinuous-tim e part o f the hybrid system is approximated by a discrete-time sys tem with arbitrarily fast sampling. This can be done in a chosen (sensible) frequency range as accurately as desired by hold-input discretization. In this approximation the former hybrid system is replaced by an N-periodic discrete- time system, with the small (fast) sampling time chosen to be a submultiple x/N of the controller sampling time x. By lifting the N-periodic control system a tim e-invariant discrete-tim e transfer-function representation is obtained. A similar approach has been used for a controller discretization problem in Keller and Anderson (1992), for which it becomes easy to evaluate the norms. This technique of fast sampling will allow us to approximate the integrals of (3.4.1), taken over one (slow) sampling period x, by the average o f their N sampled values over the period x for N sufficiently large. To establish the valid ity of this procedure, we shall show that these sums converge, as N — kx 3, to the integrals (3.4.1), using the definition o f Riemann integral. This proof o f convergence, in turn, requires that the impulse responses of the operators de fined in (3.4.2) be continuous and exponentially stable. The following lemma establishes this resu lt (The proof is straightforward and is omitted).

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[7] Alemohammad, N., Rezakhah, S. and Alizadeh, S.H. (2016) Markov Switching Component Garch Model Stability and Forecasting. Communications in Statistics Theory and Statistics , 45, 4332-4348. https://doi.org/10.1080/03610926.2013.841934 [8] Irungu, I., Mwita, P. and Waititu, A. (2018) Consistency of the Model **Order**

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Probably, the ﬁeld in which more eﬀort has been paid to the development of real-time simulation techniques is that of computational surgery [5–10]. This is because surgery training systems are equipped with haptic peripherals, those that provide the user with realistic touch sensations (force feedback). Just like some 25 pictures per second are necessary for a realistic perception of movement in ﬁlms, haptic feedback needs for some 500 Hz to 1 kHz in **order** to achieve the necessary realism. The diﬃculty of the task is thus readily understood: to perform 500–1000 simulations of highly non-linear solids (soft tissues are frequently assumed to be hyperelastic), possibly suﬀering contact, cutting, etc. Among the very few truly non-linear surgery simulators developed so far, one can cite [10–13]. Essentially, the former employ some type of explicit, lumped mass, Lagrangian ﬁnite elements to perform the simulations, possibly including an intensive usage of GPUs.

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rediction is an important aspect of relationship analysis in any scientific study. It not only reflects the adequacy of underlying model but also assist in making a suitable choice among the competing models. Generally, predictions of study variable in linear models are obtained either for actual values or for average-values but not for both simultaneously. Section-2 describes the model specification and the predictors. Section-3 deals with the properties of the estimators. We have derived the expression for the prediction of mean value of the study variable for minimax and Stein-minimax **estimator** separately and their results are presented in the form of theorems in Section-4. In Section-5 comparative study have been made. Proof of the theorems are given in Section-6.

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which is not possible as p l < q. Thus there does not exist a non- **reduced** ring with p l q non-units and |J(R)| = q. Proposition 2.2. Let p, q, and r be distinct primes, p < q < r and n = pqr. Then there does not exist any finite, non-**reduced** ring R with n non-units satisfying the following:

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Generally, the functional mechanism of environmental taxes and charges are similar, which influence the production/consumption behaviour of polluters by altering prices [11]. Taxes are regarded as the best approach to alter prices to match marginal societal costs [12]. In **order** to realize a higher level of efficiency, the state is expected to internalize externalities into the tax system via tools such as "pollution tax" [13]. Taxing based on the harm brought about by an activity represents an actual instrument for the control as it taxes the people who actually partake in these activities [14]. This however does not translate to zero emission. Decreasing emission has an associated cost, and the aim is to reduce it to levels deemed acceptable to society [15]. A proper emissions tax mechanism can be used to decrease emissions by providing car owners with incentives to change their vehicles or better maintain it. This will also mitigate the number of cars on the road [16].

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- For many of these models, point forecasts or point estimator of the series and components obtained with linear model remains approximately optimal.. - The decomposition of NL mod[r]

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ν = 2 case is the only one he considers. When ν > 2, we see that the rule-of-thumb constants are increasing in the dimensionality of the problem. The basic idea behind this is given that higher- **order** kernels reduce bias, larger bandwidths are needed to minimize AMISE. However, note that the increase is not uniform over ν. For example, the **optimal** bandwidth for q = 2 and ν = 6 for the Gaussian kernel is 1.1318 and the rule-of-thumb bandwidth for that same kernel with ν = 8 is 1.1235. There appears to be an interplay between the roughness and variance which goes into calculating the rule-of-thumb bandwidth.

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Seismic probabilistic risk assessment (SPRA) aims to evaluate the risk of a nuclear power plants (NPP) through various failure scenarios initiated by seismic events. One structure that is of particular importance for SPRA at NPPs is the auxiliary building due to its location and function. Auxiliary buildings are typically located near the containment building, which houses the reactor, and contain critical nonstructural components (NCs) such as systems for radioactive waste, chemical and volume control, and emergency cooling (NRC 2016). Due to the high rigidity of typical auxiliary buildings, operational failure of these NCs is significantly more likely than failure of the structure due to seismic events. However, to evaluate the failure probabilities of NCs under epistemic and aleatory uncertainties associated with the behavior of both the structure and NCs, simple yet sufficiently accurate **reduced**-**order** models (ROMs) are needed due to the computational challenges of accomplishing the task with finite element (FE) models. Reduction in model complexity, however, may lead to the loss of certain dynamic characteristics of the 3D structure (Sezen et al. 2015, and Althoff 2017).

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Four generalised steady-state **reduced**-**order** models for separation processes in packed columns are developed and compared in this work. The models are based on the two film theory of mass transfer and the more rigorous of them have as a starting point one of the so called rate based methods. The mass and energy transfer rates across the vapour liquid interface are evaluated by means of different approximate solutions of the Maxwell-Stefan equations for steady-state, unidirectional mass transfer. The differential equations of the models are converted into algebraic equations through the application of the orthogonal collocation procedure on the spatial variable. The resulting system of algebraic equations is subsequently solved using a modification of the Powell hybrid method. Three case studies dealing with distillation columns are presented but the models are easily modified to work with other separation processes (e.g., absorption).

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Optimal policies for inventory control problems usually obey the following rule: If it is optimal to order an amount u when the stock level is x, then Uie optimal order from.. This [r]