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Buffer operators can be utilized to improve the smooth degree of the raw data sequence, and to increase the simulation accuracy of the model. The purpose of this paper is to analyze the cause of increase in the simulation accuracy of the buffer operator.

This paper probed into the modeling mechanism of several typical buffer operators such as the arithmetic buffer operators, the buffer operators with monotonic function and weighted buffer operators. The paper also gives an example of the buffer operator sequence.

The results indicate that after applying an infinite buffer operator, whether the authors adopt a weakening buffer operator or a strengthen buffer operator, the raw sequence can be changed into a constant sequence. Because the discrete GM(1,1) model can completely simulate constant sequence, the simulation accuracy is 100 percent. Because the discrete GM(1,1) model is the accurate form of the GM(1,1) model, after applying an infinite buffer operator, the GM(1,1) model can have a very high simulation accuracy. The buffer operator model can increase the simulation accuracy of both the GM(1,1) model and the discrete GM(1,1) model.

The paper analyses the cause of increasing simulation accuracy of the buffer operator model. The paper may indicate that possible results can be studied in the future. All the buffer operator models have similar properties. After applying an infinite buffer operator, the raw sequence can be changed into a constant sequence. A fixed-point axiom may be the basic cause.

The purpose of this paper is to perfect the axiom systems of buffer operator via adding the axiom of invariable trend.

Based on the three axioms of buffer operator, for any given data sequence of system behavior and any set of data satisfying the axiom of fixed point, it is proved that there always exists a buffer operator satisfying that the set of data is the buffer sequence of the given data sequence, and a specific constructor method of buffer operator is provided. Finally, the axiom of invariable trend is proposed to add in the axiom systems of buffer operator.

The results are convincing that although the raw sequence suffered from certain disturbance may be enlarged or reduced, the trend is in line with the original law. All predictions must be on the premise of this trend to forecast, or prediction will be considered invalid.

The method exposed in the paper can be used to construct a specific buffer operator between two sequences satisfying the axiom of fixed point.

The paper succeeds in providing a kind of universal constructor method for buffer operator, and adding the axiom of invariable trend to perfect the axiom systems of buffer operator and ensure the consistency of variation trend between the predicted values and the actual values.

The purpose of this paper is to analyze the spectral characteristics of moving average operator and to propose a novel time-frequency hybrid sequence operator.

Firstly, the complex data is converted into frequency domain data by Fourier transform. An appropriate frequency domain operator is constructed to eliminate the impact of disturbance. Then, the inverse Fourier transform transforms the frequency domain data in which the disturbance is removed, into time domain data. Finally, an appropriate moving average operator of N items is selected based on spectral characteristics to eliminate the influence of periodic factors and noise.

Through the spectrum analysis of the real-time data sensed and recorded by microwave sensors, the spectral characteristics and the ranges of information, noise and shock disturbance factors in the data can be clarified.

The real-time data analysis results for a drug component monitoring show that the hybrid sequence operator has a good effect on suppressing disturbances, periodic factors and noise implied in the data.

Firstly, the spectral analysis of moving average operator and the novel time-frequency hybrid sequence operator were presented in this paper. For complex data, the ideal effect is difficult to achieve by applying the frequency domain operator or time domain operator alone. The more satisfactory results can be obtained by time-frequency hybrid sequence operator.

According to the basic principle of grey system and third axiom buffer operator, aiming at the problem of disturbance, some new weakening buffer operators are established by analytical skills in the process of prediction. The problem of some contradictions between quantitative analysis and qualitative analysis existing in pretreatment for vibration data sequences is resolved effectively. An example shows that the kind of new weakening buffer operators increase the forecast precision of data forecast model remarkably. The aim of this paper is to attempt to resolve the problem of some contradictions between quantitative analysis and qualitative analysis existing in pretreatment for vibration data sequences.

In view of the problem of some contradictions between quantitative analysis and qualitative analysis existing in pretreatment for vibration data sequences, according to the basic principle of grey system and third axiom buffer operator, some new weakening buffer operators are established by analytical skills. As an example, the kind of new weakening buffer operators can increase the forecast precision of data forecast model remarkably.

The results show that the new weakening buffer operators can increase the forecast precision of data forecast model remarkably.

The new weakening buffer operators exposed in the paper can be used to resolve the problem of some contradictions between quantitative analysis and qualitative analysis existing in pretreatment for vibration data sequences and increase the forecast precision of data forecast model remarkably.

The paper succeeds in increasing the forecast precision of data forecast model remarkably.

In order to handle high and rapid production demands, printed circuit board (PCB) manufacturers have employed high‐speed surface mount machines into their assembly lines. These machines have abilities of fast component placements, but provide challenges for process engineers to optimise the component placement sequences and feeder arrangements via effective planning. A computer program was developed based on operating concepts using genetic algorithm, to solve for various component placement sequencing planning of the high‐speed chipshooter. Genetic algorithms are a class of general purpose search methods based on the concepts of genetic evolution and survival of the fittest. The program provides information on component placement sequences and feeder arrangements for optimal assembly times. Initial tests have shown that the size of the parent space affects the convergence of the solutions during iterations. Finally, comparisons of results have shown improvement over those previously obtained by other researchers.

The mobile communication industry in China is vulnerable to competition, industry regulation, macroeconomy and so on, which leads to service income's volatility and non-stationarity. Traditional income prediction models fail to take account of these factors, thus resulting in a low precision. The purpose of this paper is to to set up a new mobile communication service income prediction model based on grey system theory to overcome the inconformity between traditional models and qualitative analysis.

At first, mobile telecommunication service income is divided into number of users (NU) and average revenue per user (ARPU) prediction, respectively. Then, grey buffer operators are introduced to preprocess the time series according to their features and tendencies to eliminate the effect of shock disturbance. As a result, two grey models based on GM(1, 1) are constructed to forecast NU and ARPU, and thus the service income is obtained. At last, a case on Zhujiang mobile communication company is studied. The result proves that the proposed method is not only more accurate, but also could discover the turning point of income.

The results are convincing: it is more effective and accurate to employ grey buffer operator theory to predict the mobile communication service income compared with other methods. Besides, this method is applicable to cases with less data samples and faster development.

It's common to come across a system with less data and poor information. At this case, the grey prediction method exposed in the paper can be used to forecast the future trend which will give the predictors advice to achieve fine outcomes. Buffer operators can reduce the effect of shock disturbance and the GM(1, 1) model has the advantages of exploiting information using only a couple of data.

Considering the fast development of China's mobile communication in recent years, only limited data can be acquired to predict the future, which will definitely reduce the prediction precision using traditional models. The paper succeeds in introducing GM(1, 1) model based on grey buffer operators into the income prediction and the outcome proves that it has higher prediction precision and extensive application.

The purpose of this paper is to analyze the mechanism and filter efficacy of accumulation generation operator (AGO)/inverse accumulation generation operator (IAGO) in the frequency domain.

The AGO/IAGO in time domain will be transferred to the frequency domain by the Fourier transform. Based on the consistency of the mathematical expressions of the AGO/IAGO in the gray system and the digital filter in digital signal processing, the equivalent filter model of the AGO/IAGO is established. The unique methods in digital signal processing systems “spectrum analysis” of AGO/IAGO are carried out in the frequency domain.

Through the theoretical study and practical example, benefit of spectrum analysis is explained, and the mechanism and filter efficacy of AGO/IAGO are quantitatively analyzed. The study indicated that the AGO is particularly suitable to act on the system's behavior time series in which the long period parts is the main factor. The acted sequence has good effect of noise immunity.

The AGO/IAGO has a wonderful effect on the processing of some statistical data, e.g. most of the statistical data related to economic growth, crop production, climate and atmospheric changes are mainly affected by long period factors (i.e. low-frequency data), and most of the disturbances are short-period factors (high-frequency data). After processing by the 1-AGO, its high frequency content is suppressed, and its low frequency content is amplified. In terms of information theory, this two-way effect improves the signal-to-noise ratio greatly and reduces the proportion of noise/interference in the new sequence. Based on 1-AGO acting, the information mining and extrapolation prediction will have a good effect.

The authors find that 1-AGO has a wonderful effect on the processing of data sequence. When the 1-AGO acts on a data sequence X, its low-pass filtering effect will benefit the information fluctuations removing and high-frequency noise/interference reduction, so the data shows a clear exponential change trends. However, it is not suitable for excessive use because its equivalent filter has poles at the non-periodic content. But, because of pol effect at zero frequency, the 1-AGO will greatly amplify the low-frequency information parts and suppress the high-frequency parts in the information at the same time.

This paper attempts to extend classic absolute degree of grey incidence so that the extended model can be used for grey number sequences.

The classic absolute degree of grey incidence was extended according to basic principles of grey incidence analysis. First, modelling methods and theories of the classic grey incidences were summarized. Then, the zeroing starting operator in grey incidence was extended for grey sequence. Third, the parameters in classic incidence degree were redefined, and an absolute degree of grey incidence for grey number sequences was proposed. The degree can not only be applied to grey number sequence, but also contains the uncertain information of analysis result. Fourth, two non‐linear programming models were constructed to estimate the grey coverage interval of absolute degree of incidences. Finally, the comparison method of grey numbers was introduced for sorting the different absolute degrees of incidences.

A theoretically feasible absolute degree of grey incidence was constructed for grey sequence. A case study showed that the proposed incidence degree was an effective method for grey sequence incidence analysis.

The method exposed in the paper can be used for grey sequences clustering, grey decision making, multi‐attribute decision making theory, uncertain target recognition and other related fields.

The paper succeeded in establishing an incidence analysis model for grey sequences which was still a research gap in grey system theory.

The purpose of this paper is to develop new simple logics and translations for hierarchical model checking. Hierarchical model checking is a model-checking paradigm that can appropriately verify systems with hierarchical information and structures.

In this study, logics and translations for hierarchical model checking are developed based on linear-time temporal logic (LTL), computation-tree logic (CTL) and full computation-tree logic (CTL*). A sequential linear-time temporal logic (sLTL), a sequential computation-tree logic (sCTL), and a sequential full computation-tree logic (sCTL*), which can suitably represent hierarchical information and structures, are developed by extending LTL, CTL and CTL*, respectively. Translations from sLTL, sCTL and sCTL* into LTL, CTL and CTL*, respectively, are defined, and theorems for embedding sLTL, sCTL and sCTL* into LTL, CTL and CTL*, respectively, are proved using these translations.

These embedding theorems allow us to reuse the standard LTL-, CTL-, and CTL*-based model-checking algorithms to verify hierarchical systems that are modeled and specified by sLTL, sCTL and sCTL*.

The new logics sLTL, sCTL and sCTL* and their translations are developed, and some illustrative examples of hierarchical model checking are presented based on these logics and translations.

The purpose of this paper is to summarize the progress in grey system research during 2000-2015, so as to present some important new concepts, models, methods and a new framework of grey system theory.

The new thinking, new models and new methods of grey system theory and their applications are presented in this paper. It includes algorithm rules of grey numbers based on the “kernel” and the degree of greyness of grey numbers, the concept of general grey numbers, the synthesis axiom of degree of greyness of grey numbers and their operations; the general form of buffer operators of grey sequence operators; the four basic models of grey model GM(1,1), such as even GM, original difference GM, even difference GM, discrete GM and the suitable sequence type of each basic model, and suitable range of most used grey forecasting models; the similarity degree of grey incidences, the closeness degree of grey incidences and the three-dimensional absolute degree of grey incidence of grey incidence analysis models; the grey cluster model based on center-point and end-point mixed triangular whitenization functions; the multi-attribute intelligent grey target decision model, the two stages decision model with grey synthetic measure of grey decision models; grey game models, grey input-output models of grey combined models; and the problems of robust stability for grey stochastic time-delay systems of neutral type, distributed-delay type and neutral distributed-delay type of grey control, etc. And the new framework of grey system theory is given as well.

The problems which remain for further studying are discussed at the end of each section. The reader could know the general picture of research and developing trend of grey system theory from this paper.

A lot of successful practical applications of the new models to solve various problems have been found in many different areas of natural science, social science and engineering, including spaceflight, civil aviation, information, metallurgy, machinery, petroleum, chemical industry, electrical power, electronics, light industries, energy resources, transportation, medicine, health, agriculture, forestry, geography, hydrology, seismology, meteorology, environment protection, architecture, behavioral science, management science, law, education, military science, etc. These practical applications have brought forward definite and noticeable social and economic benefits. It demonstrates a wide range of applicability of grey system theory, especially in the situation where the available information is incomplete and the collected data are inaccurate.

The reader is given a general picture of grey systems theory as a new model system and a new framework for studying problems where partial information is known; especially for uncertain systems with few data points and poor information. The problems remaining for further studying are identified at the end of each section.