Nonstationary process

In the study of nonstationary processes described by partial differential equations of parabolic and hyperbolic types... 

Homogeneous difference schemes with weights. In a common setting it seems natural to expect that a difference scheme capable of describing this or that nonstationary process would be suitable for the relevant stationary process, that is, for du/dt = 0 we should have at our disposal a difference scheme from a family of homogeneous conservative schemes, whose use permits us to solve the equation Lu + / = 0. 

Deterministic trend models are based on the assumption that the trend of a time series can be approximated closely by simple mathematical functions of time over the entire span of the series. The most common representation of a deterministic trend is by means of polynomials or of transcendental functions. The time series from which the trend is to be identified is assumed to be generated by a nonstationary process where the nonstationarity results from a deterministic trend. A classical model is the regression or error model (Anderson, 1971) where the observed series is treated as the sum of a systematic part or trend and a random part or irregular. This model can be written as... 

Time Factor Prior to Occurrence of a Thermal Explosion (Induction Periods). In the study of spontaneous explosions occurring in closed vessels, a well defined induction period frequently elapses prior to the development of an actual explosion. The length of this time interval has been observed to be anywhere from a few seconds to several minutes, depending upon the experimental conditions employed. Such observations are not surprising, in view of the fact that in order for an explosion to occur a build-up either of the internal energy or of chain carriers is first required. The rate of such a nonstationary process would then be expected to determine the duration of these pre-explosion times. For the case of a purely thermal explosion, the over-all rate of heat release, dq/di, prior to explosion is given by... 

V. L. Tataurov, V. P. Ivanov, in Nonstationary Processes in Catalysis, Izd. Inst. Katal. Sib. Otd. Akad. Nauk SSSR, Novosibirsk, 1979, Part 2, p. 170. 

For the description of nonstationary processes in a periodic reactor, the CBR is used. The equations for gas-phase and surface component concentrations are described by the ODE system. 

Balyshev, O. A. and Tairov, E. A., "Analysis of Transient and Nonstationary Processes in Pipeline Systems (Theoretical and Experimental Aspects)", 164 p. Nauka. Sib. pre-dpriyatie RAN, Novosibirsk (1998). (in Russian). 

A distinction is drawn in equation (21.1) between stochastic errors that are randomly distributed about a mean value of zero, errors caused by the lack of fit of a model, and experimental bias errors that are propagated through the model. The problem of interpretation of impedance data is therefore defined to consist of two parts one of identification of experimental errors, which includes assessment of consistency with the Kramers-Kronig relations (see Chapter 22), and one of fitting (see Chapter 19), which entails model identification, selection of weighting strategies, and examination of residual errors. The error analysis provides information that can be incorporated into regression of process models. The experimental bias errors, as referred to here, may be caused by nonstationary processes or by instrumental artifacts. 

The complexity index can certainly be calculated for graphs with pendant vertices. In this case one takes into account the fact that the pendant vertices do not change the mechanism complexity because the basic graph topology (number of cycles, number of vertices in the graph s cyclic part, the cycle connectedness, etc.) is left unchanged. The presence of pendant vertices becomes of importance in the case of nonstationary processes. For stationary (or... 

In the case of Fourier analysis, the coherence critical value is independent of the processes to be compared, if they sufficiently well follow a linear description [1, 15]. This independency, however, holds exactly only in the limit of long time series. As wavelet analysis is a localized measure, this condition is not fullfilled. Hence, for different AR[1] processes (from white noise to almost nonstationary processes), we found a marginal dependency on the process parameters. 

Generally, this accords with the data available on the nonstationary process [14]. Hence, the electrochenucal reduction of silicon (IV) species in halide— hexafluorosilicate melts is accompanied by formation of rather stable... 

Practical applications of GC methods for assessing catalytic activity and studying heterogeneous reaction are outrunning the development of the theory of nonstationary processes under pulse conditions. Although numerous models describing different types of both reversible and irreversible reactions have been elaborated, many theoretical problems are still far from solution. 

Lampard, D. G. (1954). Generalization of the Wiener-Khinchine theorem to nonstationary processes. J. Appl. Phys., 25, 802-3. 

Several techniques have been developed for the investigation of the rate of nucleation. Earlier methods were based on galvanostatic techniques. The first to use this technique were Samartzev and Evstropiev. Later it was developed and refined in experiments of Schottky and Gutzow but, because of the high sensitivity of the nucleation process on overpotential, the results are difficult to interpret. Potentiostatic techniques gave more comprehensible results. The simplest potentiostatic technique, used by Kaischew, Scheludko, and Bliznakov and by Scheludko and Bliznakov, is to apply an overpotential pulse to the electrode and to measure the time at which current is observed to flow. The technique allows an estimate to be made of the time lag n needed for the formation of the first nucleus. The identification of Ti with meets some difficulties connected with nonstationary processes discussed in detail by Toschew. ... 

Eqs, 67 and 68 show that all quantities required for reliability applications can be computed from the following spectral characteristics of complexvalued nonstationary processes ii, m = 1,2,...,... 

In postulating the stationarity of the stochastic process, very strong assumptions regarding the structure of the process are made. Once these assumptions are dropped, the process can become nonstationary in many different ways. In the framework of the spectral analysis of nonstationary processes, Priestley (see, e.g., Priestley 1999) introduced the evolutionary power spectral density (EPDS) function. The EPSD function has essentially the same type of physical interpretation of the PSD function of stationary processes. The main difference is that whereas the PSD function describes the power-frequency distribution for the whole stationary process, the EPSD function is time dependent and describes the local power-frequency distribution at each instant time. The theory of EPSD function is the only one which preserves this physical interpretation for the nonstationary processes. Moreover, since the spectrum may be estimated by fairly simple numerical techniques, which do not require any specific assumption of the structure of the process, this model, based on the EPSD function, is nowadays the most adopted model for the analysis of structures subjected to nonstationary processes as the seismic motion due to earthquakes. 

In the Priestley spectral representation of nonstationary processes, a sample of the nonstationary stochastic process is defined by the Fourier-Stieltjes integral as follows ... 

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