Linear deterministic

In conclusion, the solutions E Qt (f)] for the expected values for such stochastic models are the same as the solutions qT (t) for the corresponding deterministic models, and the transfer-intensity matrix H is analogous to the fractional flow rates matrix K of the deterministic model. If the hazard rates are constant in time, we have the stochastic analogues of linear deterministic systems with constant coefficients. If the hazard rates depend on time, we have the stochastic analogues of linear deterministic systems with time-dependent coefficients. 

As we have already noted, the eigenvalues for the linearized deterministic system at the inhomogeneous fixed point, plotted as black dots in Figs. 5.25b,c are computed numerically for the system (5.30) in the deterministic case by using the spatially discretized set of ordinary differential equations. 

One of the first papers to incorporate production capacity constraints in a pricing and inventory model is Lai [88]. Under linear deterministic demand and quadratic production costs, Lai shows that price is more responsive to demand shocks when a stockout is realized or when capacity is soon to be exhausted, and that considering production capacity constraints reduces the asymmetry in price behavior. 

At high Reynolds numbers, the fluctuating pressure field term dominates the fluctuating shear stress term in (12.4.2-6). It results from interactions between the turbulence and the mean velocity gradients, the so-called rapid-pressure contribution, and from the turbulence proper, the so-called slow-pressure contribution. To simulate the rapid-pressure related process. Pope [1985] proposed a linear deterministic model ... 

In the present work, the probability of the containment failure is determined for the critical sections on the basis of the non-linear deterministic analysis of the containment for the various level of the overpressure Ap = 250, 300, 350, 400, 450 kPa . No catastrophic containment rupture at this pressure are evidenced, hence only potential failure mode considered here is the leakage. The function of the failure was considered as in Eqs. (21) and (22) and 10 Monte Carlo simulations are carried out in program FReET (Novak et al., 2003). The probability of containment failure is calculated from the probability of the reliability function RF in the form ... 

Of course, with calculus analytical solutions became tractable only for linear, deterministic problems. For non-linear or probabilistic phenomena, the invention of computational mathematics has introduced an equivalently distinctive set of scientific methods. Paul Humphreys has best presented convincing arguments that computational science... 

It is very important to make classification of dynamic models and choose an appropriate one to provide similarity between model behavior and real characteristics of the material. The following general classification of the models is proposed for consideration deterministic, stochastic or their combination, linear, nonlinear, stationary or non-stationary, ergodic or non-ergodic. 

It is possible to limit our choice for stochastic modeling by stationary, linear, nonlinear, and ergodic models in combination with deterministic function. In this case the following well studied models can be proposed for the accepted concept [1] ... 

Figure 2.15(a) shows the relationship between and Cp for the component characteristics analysed. Note, there are six points at q = 9, Cp = 0. The correlation coefficient, r, between two sets of variables is a measure of the degree of (linear) association. A correlation coefficient of 1 indicates that the association is deterministic. A negative value indicates an inverse relationship. The data points have a correlation coefficient, r = —0.984. It is evident that the component manufacturing variability risks analysis is satisfactorily modelling the occurrence of manufacturing variability for the components tested. 

Despite bearing no direct relation to any physical dynamical system, the onedimensional discrete-time piecewise linear Bernoulli Shift map nonetheless displays many of the key mechanisms leading to deterministic chaos. The map is defined by (see figure 4.2) ... 

Deterministic Randomness. On the one hand, equation 4.8 is a trivial linear difference equation possessing an equally as trivial solution for each initial point Xq Xn = 2"a o (mod 1). Once an initial point is chosen, the future iterates are determined uniquely. As such, this simple system is an intrinsically deterministic one. On the other hand, look again at the binary decimal expansion of a randomly selected a o- This expansion can also be thought of as a particular semi-infinite sequence of coin tosses. 

The Warner function has all the desired asymptotical characteristics, i.e. a linear dependence of f(r) on r at small deformation and a finite length Nlp in the limit of infinite force (Fig. 3). In a non-deterministic flow such as a turbulent flow, it was found useful to model f(r) with an anharmonic oscillator law which permits us to account for the deviation of f(r) from linearity in the intermediate range of chain deformation [34] ... 

The linearization of the deterministic dynamics allows one to solve the equations of motion explicitly. Equation (15) can be rewritten as a first-order equation of motion in the 2/V-dimensional phase space with the coordinates... 

Let us begin by studying the relative dynamics in the Hamiltonian case, that is, for deterministic driving. The shift (42) to the moving origin is a time-dependent canonical transformation [98]. It transforms the linearized Hamiltonian (33) into... 

According to Stuart A. Kauffman (1991) there is no generally accepted definition for the term complexity . However, there is consensus on certain properties of complex systems. One of these is deterministic chaos, which we have already mentioned. An ordered, non-linear dynamic system can undergo conversion to a chaotic state when slight, hardly noticeable perturbations act on it. Even very small differences in the initial conditions of complex systems can lead to great differences in the development of the system. Thus, the theory of complex systems no longer uses the well-known cause and effect principle. 

COROLLARY 8.5 The reversal of the interpreted value language of a linear monadic recursion scheme is a detemdnistic linear context-free language (the language accepted by a deterministic single-turn pda). 

Deterministic models that are non-linear in c will be limited to specific applications. For example, in the generalized IEM (GIEM) model (Tsai and Fox 1995a Tsai and Fox 1998), which is restricted to binary mixing, the mixture fraction appears non-linearly 61... 

We have seen that Lagrangian PDF methods allow us to express our closures in terms of SDEs for notional particles. Nevertheless, as discussed in detail in Chapter 7, these SDEs must be simulated numerically and are non-linear and coupled to the mean fields through the model coefficients. The numerical methods used to simulate the SDEs are statistical in nature (i.e., Monte-Carlo simulations). The results will thus be subject to statistical error, the magnitude of which depends on the sample size, and deterministic error or bias (Xu and Pope 1999). The purpose of this section is to present a brief introduction to the problem of particle-field estimation. A more detailed description of the statistical error and bias associated with particular simulation codes is presented in Chapter 7. 

In addition, very few observations are pristine and basic measurements such as angular deviation of a needle on a display, linear expansion of a fluid, voltages on an electronic device, only represent analogs of the observation to be made. These observations are themselves dependent on a model of the measurement process attached to the particular device. For instance, we may assume that the deviation of a needle on a display connected to a resistance is proportional to the number of charged particles received by the resistance. The model of the measurement is usually well constrained and the analyst should be in control of the deterministic part through calibration, working curves, assessment of non-linearity, etc. If the physics of the measurement is correctly understood, the residual deviations from the experimental calibration may be considered as random deviates. Their assessment is an integral part of the measurement protocol and the moments of these random deviations should be known to the analyst and incorporated in the model. 

The established tools of nonlinear dynamics provide an elaborate and versatile mathematical framework to examine the dynamic properties of metabolic systems. In this context, the metabolic balance equation (Eq. 5) constitutes a deterministic nonlinear dynamic system, amenable to systematic formal analysis. We are interested in the asymptotic, the linear stability of metabolic states, and transitions between different dynamic regimes (bifurcations). For a more detailed account, see also the monographs of Strogatz [290], Kaplan and Glass [18], as well as several related works on the topic [291 293],... 

Equation 5.8 is the matrix form of all deterministic linear models. 

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