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The purpose of this paper is to develop new types of direct expansion method of moments (DEMM) by using the n/3th moments for simulating nanoparticle Brownian coagulation in the free molecule regime. The feasibilities of new proposed DEMMs with n/3th moments are investigated to describe the evolution of aerosol size distribution, and some of the models will be applied to further simulation of physical processes.

The accuracy and efficiency of some kinds of methods of moments are mainly compared including the quadrature method of moments (QMOM), Taylor-expansion method of moments (TEMOM), the log-normal preserving method of moments proposed by Lee (LMM) and the derived DEMM in this paper. QMOM with 12 quadrature approximation points is taken as a reference to evaluate other methods.

The newly derived models, namely DEMM(4/3,4) and DEMM(2,6), as well as the previous DEMM(2,4), are considered to be qualified models due to their high accuracy and efficiency. They are confirmed to be valid and alternative models to describe the evolution of aerosol size distribution for particle dynamical process involving the n/3th moments.

The n/3th moments, which have clear physical interpretations when n stands for first several integers, are first introduced in the DEMM method for simulating nanoparticle Brownian coagulation in the free molecule regime.

The behaviours of four indeterminate frame-support combinations namely Type I (with fixed supports), Type II (with pinned supports), Type III (with fixed-pinned supports) and Type IV (with fixed-roller supports) frames under the exposure conditions and loads as existing on site were simulated. Two categories of these combinations (I and II) were studied namely single storey-single bay and multiple storey-single bay frames, as illustrated in the case studies treated. A procedure for determining the probability of failure at different sections along the frame types, the range between the probability of failure bounds and the reliability ratings of the frame types were developed based on the kinetic method of plastic moment analysis, minimum weight design method, piecewise method of moment analysis and first order-second moment (FOSM) methods. The analysis results of the Category I frames showed that the Type I frame was most reliable (with the lowest probability of failure range of 0.3269), while the Type II frame was least reliable (with the highest probability of failure range of 0.4918). These results were consistent with those of the Category II frames. The paper aims to discuss these issues.

Collapse mechanisms were generated for four frame-support types and the corresponding plastic moments were determined using both the kinematic plastic analyses and minimum weight design methods. The members were designed and the plastic moments were distributed at sections of constant interval along the frame length to generate corresponding envelopes. A similar process was carried out to determine the elastic moment variables due to the loads. The reliability index and the corresponding probability of failure at each frame section were determined. Then, the probabilities of failure bounds for the frames were then compared to determine the most reliable.

It was observed that there existed a wide margin between the elastic and plastic moments indicating that design of steel structures at the elastic limit does not take full advantage of its strength. Hence, the design can be carried out beyond the elastic limit and within the safety margin given in equation (3). However, the safety of the entire frame is assessed on the basis of range of values between the highest and the lowest probability of failure bounds. The lower this range is (not exceeding 0.5 or 50 per cent), the more reliable the frame is.

The equations developed in this study can only be directly applied to multi storey-single bay frames. However, the reliability-based analysis and design procedure developed can be extended to other types of frames.

A practical approach for analysing steel frames with different supports with the overall goal of producing safe and economical designs has been developed and presented in this paper.

The procedure adopted is very original and can be backed up by existing literature. The piecewise method for analysing moments at various sections along a frame is also innovative. The whole concept can be adopted to determine the reliability of other types of frames such as multiple bay-multistorey frames with different support types.

The purpose of this paper is to find an accurate, efficient and easy-to-implement point estimate method (PEM) for the statistical moments of random systems.

First, by the theoretical and numerical analysis, the approximate reference variables for the frequently used nine types of random variables are obtained; then by combining with the dimension-reduction method (DRM), a new method which consists of four sub-methods is proposed; and finally, several examples are investigated to verify the characteristics of the proposed method.

Two types of reference variables for the frequently used nine types of variables are proposed, and four sub-methods for estimating the moments of responses are presented by combining with the univariate and bivariate DRM.

In this paper, the number of nodes of one-dimensional integrals is determined subjectively and empirically; therefore, determining the number of nodes rationally is still a challenge.

Through the linear transformation, the optimal reference variables of random variables are presented, and a PEM based on the linear transformation is proposed which is efficient and easy to implement. By the numerical method, the quasi-optimal reference variables are given, which is the basis of the proposed PEM based on the quasi-optimal reference variables, together with high efficiency and ease of implementation.

This chapter analyzes the empirical relationship between the pricesetting/consumption behavior and the sources of persistence in inflation and output. First, a small-scale New-Keynesian model (NKM) is examined using the method of moment and maximum likelihood estimators with US data from 1960 to 2007. Then a formal test is used to compare the fit of two competing specifications in the New-Keynesian Phillips Curve (NKPC) and the IS equation, that is, backward- and forward-looking behavior. Accordingly, the inclusion of a lagged term in the NKPC and the IS equation improves the fit of the model while offsetting the influence of inherited and extrinsic persistence; it is shown that intrinsic persistence plays a major role in approximating inflation and output dynamics for the Great Inflation period. However, the null hypothesis cannot be rejected at the 5% level for the Great Moderation period, that is, the NKM with purely forward-looking behavior and its hybrid variant are equivalent. Monte Carlo experiments investigate the validity of chosen moment conditions and the finite sample properties of the chosen estimation methods. Finally, the empirical performance of the formal test is discussed along the lines of the Akaike's and the Bayesian information criterion.

Bearing friction moments are important factors that affect the vibrations of rotor systems. The bearing friction moments are related to the dimension parameters, lubrication conditions and manufacturing errors of support bearings. This work studies the effects of the bearing friction moments on the vibrations of rotor systems.

The rotor is separated into several shaft elements for formulating a flexible rotor. The time-varying friction moment (TFM) is affected by the time-varying contact loads. The vibrations of FRS from the TFM and Palmgren's friction moment (PFM) calculation methods are compared. Moreover, the effects of the rotor offset and radial clearance on the frequency-amplitude characteristics of FRS are studied.

The TFM method is more consistent with the actual operation mechanisms. The rotor offset and radial clearance can significantly affect the nonlinear vibrations of FRS. This work provides a new reference and research method for the vibration analysis of rotor systems considering the friction effects.

The elastohydrodynamic lubrication (EHL), elastic hysteresis and differential sliding are considered. A flexible rotor system (FRS) dynamic model considering the TFM is proposed. The vibrations of FRS from the TFM calculation method and empirical calculation formula are compared. The effects of the rotor offset and radial clearance on the frequency–amplitude characteristics of FRS are studied.

The purpose of this paper is to analyze and compare four numerical methods to estimate the most suitable one which describes wind speed distribution of M’Sila, a province of northern Algeria.

The site chosen in this investigation is characterized by calm winds; in this case, the appropriate wind speed distribution is that of hybrid Weibull.

The four numerical methods used in the present paper are the maximum likelihood method, the graphical method, the moment method and the energy pattern factor method. The hybrid Weibull distributions using the abovementioned approaches are compared with the measured data via three statistical parameters, namely, the correlation coefficient, the root mean square error and the Chi-square error.

The obtained results showed that the moment method is the suitable one in describing month and annual wind speed hybrid Weibull parameters of this region.

The purpose of this study is to present a newly proposed and developed sorting algorithm-based merging weighted fraction Monte Carlo (SAMWFMC) method for solving the population balance equation for the weighted fraction coagulation process in aerosol dynamics with high computational accuracy and efficiency.

In the new SAMWFMC method, the jump Markov process is constructed as the weighted fraction Monte Carlo (WFMC) method (Jiang and Chan, 2021) with a fraction function. Both adjustable and constant fraction functions are used to validate the computational accuracy and efficiency. A new merging scheme is also proposed to ensure a constant-number and constant-volume scheme.

The new SAMWFMC method is fully validated by comparing with existing analytical solutions for six benchmark test cases. The numerical results obtained from the SAMWFMC method with both adjustable and constant fraction functions show excellent agreement with the analytical solutions and low stochastic errors. Compared with the WFMC method (Jiang and Chan, 2021), the SAMWFMC method can significantly reduce the stochastic error in the total particle number concentration without increasing the stochastic errors in high-order moments of the particle size distribution at only slightly higher computational cost.

The WFMC method (Jiang and Chan, 2021) has a stringent restriction on the fraction functions, making few fraction functions applicable to the WFMC method except for several specifically selected adjustable fraction functions, while the stochastic error in the total particle number concentration is considerably large. The newly developed SAMWFMC method shows significant improvement and advantage in dealing with weighted fraction coagulation process in aerosol dynamics and provides an excellent potential to deal with various fraction functions with higher computational accuracy and efficiency.

In this work, an efficient and easy statistical method to find an equivalent discrete distribution for a continuous random variable (r.v.) is proposed. The proposed method is illustrated by applying it to the treatment of the anthropometrical noise factors in the context of Robust Ergonomic Design (RED; Lanzotti 2006; Barone S. and Lanzotti A., 2007).

In this paper, the authors applied the empirical likelihood method, which was originally proposed by Owen, to the copula moment based estimation methods to take advantage of its properties, effectiveness, flexibility and reliability of the nonparametric methods, which have limiting chi-square distributions and may be used to obtain tests or confidence intervals. The authors derive an asymptotically normal estimator of the empirical likelihood based on copula moment estimation methods (ELCM). Finally numerical performance with a simulation experiment of ELCM estimator is studied and compared to the CM estimator, with a good result.

In this paper we applied the empirical likelihood method which originally proposed by Owen, to the copula moment based estimation methods.

We derive an asymptotically normal estimator of the empirical likelihood based on copula moment estimation methods (ELCM). Finally numerical performance with a simulation experiment of ELCM estimator is studied and compared to the CM estimator, with a good result.

In this paper we applied the empirical likelihood method which originally proposed by Owen 1988, to the copula moment based estimation methods given by Brahimi and Necir 2012. We derive an new estimator of copula parameters and the asymptotic normality of the empirical likelihood based on copula moment estimation methods.

Describes a simple algorithm which has been developed for determining exact moments of top event failure probability from the moments of the basic events in a fault tree. This method requires neither Taylor series expansion of top event failure probability function nor its partial derivates to find these moments. This method has subsequently extended to systems with multistate components. Describes a general algorithm for the purpose. Provides several illustrations to highlight its usefulness.