Identical initial conditions

In this section, the numerical solutions of the MINLP-model and of the MILP-model as presented in Sections 7.4 and 7.5 are compared with respect to their solution quality (measured by the objective values) and the required solution effort (measured by the computing time). In order to compare the MILP-solution with the MINLP-solution, the optimized values for the start times of polymerizations tn, the recipe assignments W, and the total holdups Mnr are inserted into the MINLP-model and the objective is calculated. To guarantee comparability of the results, the models were stated with identical initial conditions, namely t° = 0, = 2 Vk, pf = 0 Vs, and ra = 0.4 Vs (i.e., the variables defined at the beginning of the corresponding time axes are fixed to the indicated values). For the algorithmic solution procedure, all variables were initialized by 1 (i.e., the search for optimal values starts at values of 1 ), and none of the solvers was specifically customized. 

On another occasion, O, a similar consideration would apply but of course G(t) would have to be replaced by a different function, G t), since even if we started with microscopically identical initial conditions for the system within V, we assume in the interests of physical realism that such knowledge of the boundary system (which could be the rest of the universe) is impossible. 

For identical initial conditions, the difference between a measured profile and the error-function solution is related to the last (nonlinear) term in Eq. 4.43. When diffusivity is a function of local concentration, the concentration profile tends to be relatively flat at a concentration where D(c) is large and relatively steep where D(c) is small (this is demonstrated in Exercise 4.2). Asymmetry of the diffusion profile in a diffusion couple is an indicator of a concentration-dependent diffusivity. 

Recent work by Pritchard has concentrated on a state-to-state description of unimolecular reactions229 and an examination by classical trajectory methods of the effects of overall molecular rotation on the unimolecular rate. The latter calculations have revealed a most interesting aspect of computing in chaotic systems, namely, that the same algorithm gives different results on different machines for a trajectory with identical initial conditions, or even on the same machine with different releases of the same compiler. However, the ensemble average behavior, with an ensemble comprising 100 or more trajectories, is acceptably the same each time.230... 

Figs. 11.2-11.4 show fields of axial velocity in the central plane of the channel at three instants after the initialization of the run. Two simulations performed on respectively 4 (TCI) and 8 processors (TC2) with identical initial conditions and meshes are compared. The characteristics of all pre-... 

Fig. 30. The hyperradius p of the cluster, defined in Eq. (10), in A vs. time, in ps, for the same cluster impacting a cold surface at two slightly different low supersonic velocities of 450 and 500 ms respectively. This figure illustrates how a small increment in the energy of impact changes the outcome dramatically. The cluster (which has identical initial conditions in both runs) evaporates just one monomer at the lower energy but fully shatters at the higher energy.

Differential equations allow biochemical networks to be described on the level of individual reaction steps like enzymatic catalysis, or transcription site binding. Differential equations, however, rely on the assumption that molecular concentrations vary continuously and deterministically. These assumptions may not always hold, especially in cases where some types of molecules participate with low number [Arkin, Ross, and McAdams 1998], As a consequence, the same system with almost identical initial conditions and environmental inputs can exhibit different behavior. 

Finally we note that in the special ensemble of systems considered (i.e., with identical initial conditions e (0), a(0)) the electric and magnetic field strengths e and b may be written... 

The sensitivity of the axisymmetric combustor flow dynamics to the actual choice of inlet velocity conditions was also examined. Figure 11.3 compares the results of initializing the simulations with the turbulent-pipe or LM-6000 swirling conditions and otherwise identical initial conditions S = 0.56, Uo — 100 m/s, STP). The flow visualizations depict the significant effects on the combustor vortex dynamics of changing the specifics of the velocity profiles used to initialize the LES, with noticeably more-axisymmetric features observed in the flow features for the LM-6000 case. The LM-6000 initial velocity conditions (Fig. 11.2a) involve a peak tangential velocity component located farther away from the axis and a more moderate radial gradient of the axial velocity. A clear consequence of these initial condition specifics, apparent in Fig. 11.3a, is that the LM-6000... 

If we carry out such a calculation for a large number of trajectories at the same fixed energy , we obtain a unique map of the global dynamics at that energy. The map is unique because each point on it uniquely specifies a single trajectory in phase space, and trajectories in phase space do not intersea at any given instant in time. If they did intersect, they would have to arise from identical initial conditions, and then claissical mechanics would no longer be a causal theory ... 

However, in various coating and granulation experiments a change in the shape of the distribution is observed, for example a dispersion of the distribution. In Fig. 7.45 the results of five different coating experiments are shown, which were conducted in Wurster equipment, a conventional fluidized bed apparatus (FB) in top and bottom spray configuration, and a spouted bed apparatus (SB) in top and bottom spray configuration. Although identical initial conditions were used and the process conditions are comparable (see Tab. 7.6), different final distributions are achieved. This effect cannot be explained by the common model of Eq. 7.34, but it underlines the influence that different types of equipment with different flow patterns may have on the process result. 

Under these assumptions, the formulation of cell division as an FPT problem requires that the following be specified (i) a stochastic model for how cell size, s, increases with time, t, between divisions (ii) the function of the cell size, s, that attains critical or threshold value, 6, at division and (iii) appropriate initial conditions, including the initial distribution of cell sizes. While in all previous cases considered, every member of the ensemble (i.e., each cell in the population) was assumed to be subjected to identical initial conditions, in this section we relax that condition to allow different cells to experience different initial conditions (i.e., different freshly divided cells having different sizes). This is an added source of stochasticity, namely, extrinsic noise in addition to the intrinsic fluctuations encoded in stochastic growth for a given initial condition. 

Creep modeling A stress-strain diagram is a significant source of data for a material. In metals, for example, most of the needed data for mechanical property considerations are obtained from a stress-strain diagram. In plastic, however, the viscoelasticity causes an initial deformation at a specific load and temperature and is followed by a continuous increase in strain under identical test conditions until the product is either dimensionally out of tolerance or fails in rupture as a result of excessive deformation. This type of an occurrence can be explained with the aid of the Maxwell model shown in Fig. 2-24. 

In ESL, the initial conditions for concentrations A, B, C, and D are followed by the table of measured data, time (min.) versus titrated volume (mL). In the DYNAMIC section, the program statements are identical to those of the model equations. Here, a function generator, FGENl, is used to describe the tabular data. The objective function to be minimised is defined as follows... 

Assuming that the target interface can be modeled as a quiescent, sharp boundary, with Eq. (30) initially at equilibrium there is zero net flux of species Red across the interface and each phase has a uniform composition of Red, CRed, (where the integer i = 1 or 2). The initial condition is identical to Eqs. (11) and (12). 

What is the significance of this choice of conditions Table 50.1 shows that the initial conditions of Experiment 2 are identical to the concentrations of reactants found in Experiment 1 at the point when it reaches 50% conversion (Table 50.1 bolded entries). In fact, from this point onwards the two experiments exhibit identical [1] at any given [2] throughout the course of both reactions. The only differences between the two experiments are (1) the catalyst has... 

The initial condition is T (0) = 0. Eq. (2-48) is identical in form to (2-46). This is typical of linear equations. Once you understand the steps, you can jump from (2-46) to (2-48), skipping over the formality. 

The above derivation shows that Jarzynski s identity is an immediate consequence of the Feynman-Kac theorem. This connection has not only theoretical value, but is also useful in practice. First, it forms the basis for an equilibrium thermodynamic analysis of nonequilibrium pulling experiments [3, 15]. Second, it helps in deriving a Jarzynski identity for dynamics using thermostats. Moreover, this derivation clarifies an important aspect trajectories can be thought of as mapping initial conditions (I = 0) to trajectory endpoints, and the Boltzmann factor of the accumulated work reweights that map to give the desired Boltzmann distribution. Finally, it can be easily extended to transformations between steady states [17] in which non-Boltzmann distributions are stationary. 

In spite of these potential concerns, the MEHMC method is expected to be a useful tool for many applications. One task for which it might be particularly well suited is to generate a canonical ensemble of representative configurations of a bio-molecular system quickly. Such an ensemble is needed, for example, to represent the initial conditions for the ensemble of trajectories used in fast-growth free energy perturbation methods such as the one suggested by Jarzynski s identity [104] (see also Chap. 5). 

The operators Fk(t) defined in Eq.(49) are taken as fluctuations based on the idea that at t=0 the initial values of the bath operators are uncertain. Ensemble averages over initial conditions allow for a definite specification of statistical properties. The statistical average of the stochastic forces Fk(t) is calculated over the solvent effective ensemble by taking the trace of the operator product pmFk (this is equivalent to sum over the diagonal matrix elements of this product), so that = Trace(pmFk) is identically zero (Fjk(t)=Fk(t) in this particular case). The non-zero correlation functions of the fluctuations are solvent statistical averages over products of operator forces,... 

By comparing the SBR with dump fill with that for the PFR in Table 1 it can be seen that VER = l/(l+a), if the two initial conditions are identical, where a- is a recycle ratio. 

In summary, the multi-variate SR model is found by applying the uni-variate SR model to each component of Wa4> p)n and eap. For the case where ra = I, the model equations for all components will be identical. The model predictions then depend only on the initial conditions, which need not be identical for each component. In order to see how differential... 

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