Process stationary

The adaptive estimation of the pseudo-inverse parameters a n) consists of the blocks C and E (Fig. 1) if the transformed noise ( ) has unknown properties. Bloek C performes the restoration of the posterior PDD function w a,n) from the data a (n) + (n). It includes methods and algorithms for the PDD function restoration from empirical data [8] which are based on empirical averaging. Beeause the noise is assumed to be a stationary process with zero mean value and the image parameters are constant, the PDD function w(a,n) converges, at least, to the real distribution. The posterior PDD funetion is used to built a back loop to block B and as a direct input for the estimator E. For the given estimation criteria f(a,d) an optimal estimation a (n) can be found from the expression... 

The condition that the process a(t) is a stationary process is equivalent to the requirement tiiat all the distribution fimctions for a t) are invariant under time translations. This has as a consequence that W a, t) is independent of t and that 1 2(0, t 2, 2) depeirds on t = 2 -1. An even stationary process [4] has the additional requirement that its distribution fimctions are invariant under time reflection. For 1 2, this implies fV2(a 02> t) = 2 2 1 caWcd microscopic reversibility. It means that the quantities are even... 

Using W2 = 17jP2, (A3.2.81 and (A3.2.9) may be used to satisfy the Smoluchowski equation, (A3.2.2). another necessary property for a stationary process. Thus u(t) is an example of a stationary Gaussian-Markov... 

Y. A. Mitropolsky, Nan-stationary processes in nonlinear oscillatory systems, English translation by Air Technical Intelligence Center, Ohio. 

That is, without some measure of control, the process is unstable and will diverge from the set point. Unlike a stationary process, which will achieve a steady state at some point, a non-stationary process requires some sort of integral control. 

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. 

Following Ref. 2 the correlation function of a stationary process K(t) may be presented in the following form ... 

The correlation time, given by l/ eff> may be calculated exactly in the following way. Let us define the normalized correlation function of a stationary process by... 

At high temperatures or in the presence of catalysts, hydroperoxide decomposes at a high rate, so that, after t t, inhibited oxidation becomes a quasi-stationary process with balanced rates of ROOH formation and decomposition. In this case, kdr 1, where kd is the overall specific rate of ROOH decomposition with allowance made for its decomposition... 

Pipeless plants are an alternative to the traditional recipe-driven multipurpose batch plants with fixed piping between the units. In this production concept, the batches of material are moved around between stationary processing stations in mobile vessels. The processing steps are performed at different single purpose or multipurpose stationary units but the material remains in the same vessel throughout the production process. The transportation of the mobile vessels can be realized by a transportation system that is fixed to the vessels or by automated guided vehicles (AGV) that pick up the vessels only to perform a transfer order [1]. 

On the other hand, in comparison to traditional recipe-driven multipurpose batch plants, new technical requirements arise from the use of mobile units. An important pre-requisite for a safe and automatic production are reliable docking systems that provide failure-free connections between the mobile vessels and the stationary processing stations. In the docking process of the mobile vessels the connection of pipes, of electric power and of signal processing equipment is necessary. The vessels therefore must be placed accurately. If vessels of different size are used, the connections must be flexible enough to cope with these. 

We consider now the situation that component A and B are not volatile and that the volume of the stagnant film is small compared with the bulk of the liquid. This means that the bulk concentration of component A and B can be assumed to be constant. In fact we consider a quasi stationary process. Then the boundary conditions of equations (23), (24) and (25) are ... 

Poly(acrylic acid) is not soluble in its monomer and in the course of the bulk polymerization of acrylic acid the polymer separates as a fine powder. The conversion curves exhibit an initial auto-acceleration followed by a long pseudo-stationary process ( 3). This behaviour is very similar to that observed earlier in the bulk polymerization of acrylonitrile. The non-ideal kinetic relationships determined experimentally in the polymerization of these two monomers are summarized in Table I. It clearly appears that the kinetic features observed in both systems are strikingly similar. In addition, the poly(acrylic acid) formed in bulk over a fairly broad range of temperatures (20 to 76°C) exhibits a high degree of syndiotacticity and can be crystallized readily (3). 

In emphasizing the need for satisfying the equipartition theorem, the linear response theory provides a connection for stationary processes through the fluctuation-dissipation theorem. 

It is assumed that the noise voltage n(t) is the result of a real stationary process (Davenport and Root, 1958) with zero mean. Because it can be shown that the spectral density function S(f) is the Fourier transform of the autocorrelation function of the noise, it follows that the rms noise is given by... 

Because the fluid is in equilibrium, any ensemble average property should not change with time. Hence, the ensemble average of (u(tf)u(t")> depends only on the relative difference of time, t — t". That is, it is a stationary process. On transforming the time variables to f and r = tr — f" (rather like the centre of diffusion coefficient transformation of Chap. 9, Sect. 2), the Green—Kubo expression for the diffusion coefficient is obtained [453, 490],... 

These results have also been obtained by Berne, Boon, and Rice3 as previously mentioned. For a stationary process, we have the relation... 

As remarked in II.3, strictly stationary processes do not exist in nature, let alone in the laboratory, but they may be approximately realized when a process lasts much longer than the phenomena one is interested in. One condition is that it lasts much longer than the autocorrelation time. Processes without a finite tc never forget that they have been switched on in the past and can therefore not be treated as approximately stationary. 

The diagonal elements represent autocorrelations, the off-diagonal elements are cross-correlations In case of a zero-average stationary process this equation reduces to... 

Exercise. The above model can be extended into an infinite sequence of zeroes and ones by stringing such triplets together. Is this sequence a stationary process ... 

Even when a system is in a steady state other than equilibrium certain physical quantities may be stationary Markov processes. An example are the current fluctuations in the circuit of fig. 7 when a battery is added, which maintains a constant potential difference and therefore a non-zero average current. Another example is a Brownian particle in a homogeneous gravitational field its vertical velocity is a stationary process, but not its position. 

These processes are non-stationary because the condition singled out a certain time t0. Yet their transition probability depends on the time interval alone as it is the same as the transition probability of the underlying stationary process. Non-stationary Markov processes whose transition probability depends on the time difference alone are called homogeneous processes. 10 They usually occur as subensembles of stationary Markov processes in the way described here. However, the Wiener process defined in 2 is an example of a homogeneous process that cannot be embedded in a stationary Markov process. 

Such matrices are called stochastic matrices ) and have been studied by Perron and Frobenius. It is clear that T has a left eigenvector (1,1,..., 1) with eigenvalue 1 and therefore a right eigenvector ps such that Tps = ps, which is the Pi(y) of the stationary process. It is not necessarily a physical equilibrium state, but may, e.g., represent a steady state in which a constant flow is maintained. The principal task of the theory is to show that for any initial p(0)... 

Remark. A great deal of attention has been paid in recent years to non-equilibrium stationary processes that are unstable and also extended in space. They can give rise to different phases that exist side by side, so that translation symmetry is broken. The name dissipative structures has been coined for them, and the prime examples are the Benard cells and the Zhabotinski reactions, but they also occur in biology and meteorology. However, these are features of the macroscopic equations. They are only relevant for fluctuation theory inasmuch as the fluctuation becomes very large at the point where the instability sets in. The critical fluctuations in XIII.5 are an example. There are many more varieties, in particular in the case of more variables. 

A procedure similar to the condensate separation in the imperfect Bose gas was employed by Lifshitz and Pitaevski [78]. The diagrammatic technique allows us to calculate the reaction rate and steady-state joint correlation functions. A separation of a condensate from terms with k = 0 cannot be done without particle production (p = 0), in which case nA, tiq —> 0 as t —> oo. In this respect the formalism presented by Lushnikov [111] for the non-stationary processes is of certain interest. 

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