Double Limitations

Aerobic The growth kinetics was described by an interacting, balanced and unstructured model characterized by phenol inhibition and oxygen limitation according to a double limiting kinetics [60, 62],... 

Two successful automated procedures have been given. In each case one must first find the peak, and a binary search with doubled limits on each reversal is the fastest search method if the peak position is unknown. One is then on either the upper or the lower branch of either the substrate or the epilayer parabola (it is in fact a very shallow and distorted parabola). Fewster first showed that the branch conld be detected by driving down in nntil the intensity is about halved from that of the peak, then driving down in. If the intensity rises one is on the upper branch of the parabola, if it falls one is on the lower branch. Fewster then alternated steps in and steps in nntil the peak was fonnd. The latter is determined by the intensity falling instead of rising on beginning the step. 

Figure 1. Possible forms of transformation of an unstable bifurcation diagram (middle column) into either one of two possible stable forms (left or right column) at the Hysteresis (a), Isola (b, c) and Double limit varieties (d, e).

The Isola and Double Limit varieties do not exist in this case. The Hysteresis variety (a =0) divides the a. space into two regions (a > 0 and a. < 0) corresponding to the two bifurcation diagrams shown in Figures 2.a and 2.b. These two are also the only possible global bifurcation diagrams (0 vs. Da) for Eq. (13) as the Hysteresis variety (B 4) divides the B. space into two regions. 1 1... 

The Hysteresis and the Double Limit Varities divide in this case the (a.,a2,a ) space into seven regions corresponding to the seven bifurcation diagrams shown in Figures 2.a-g. 

So what is the bottom line Is prediction possible or is it not Born himself notes that the problem of prediction is an exercise in double limits. In the case of Born s example, e.g., we are dealing with two variables, t and vo, one of which tends to infinity, the other one to zero, respectively. The result (prediction possible or not) is unspecified without further information on how this limit is to be performed. If we resolve, e.g., that we want to predict only over a fixed time interval 0 < t < t, then, obviously, reduction of the experimental error Avq in the velocity v eventually results in tc t and meaningful prediction is indeed possible. On the other hand, for fixed A o, no prediction is possible for t > [Pg.23]

See also H. K. Cheng, An Analytic, Asymptotic Theory of Shock Supported Arrhen ius Reactions and Ignition Delay in Gas, unpublished (1970) for an early version of the study in [48]. The work in [47] involves a double limit that also describes slight merging of the shock with the reaction zone. 

The double limit e —> 0+, L —> oo must be taken in such a way that the whole system is inside the normalising box at all times, x is the length of the incident train of particles divided by their velocity v. We require... 

In any case it is important to note that H2 evolving culture must be nitrogen-limited. Deficiency of another element(s) will transfer culture to the double-limited state which is not favorable for H2 production (Tsygankov et al., 1996). However, a requirement for excess of macro- and micro-elements is not considerative for the biotechnological systems since many... 

Furthermore, Wegmann and Rossler experimenting with this reaction, observed oscillations corresponding to a) limit cycle, b) double limit cycle, and c) endogenous chaos and d) screw type chaos, see Fig. III.ll, a-d, respectively. 

Fig. III.ll. Experimental observations of a Limit cycle, b Double limit cycle, c Endogenous chaos, d Screw type chaos. Abscissa electrochemical potential, ordinate potential of bromide ion. (After Wegmann and Rossler (1978))...

The unperturbed system consists therefore of two circular orbits of the same size. In addition to the double limiting degeneration due to the circular orbits, we have also a double intrinsic degeneration, arising from the fact that the planes of the two orbits are... 

To remove the variational bias systematically, if G and X do not commute, the power method must be used to both the left and the right in Eq. (2.2). Thus one obtains from X 00) an exact estimate of X0 subject only to statistical errors. Of course, one has to pay the price of the implied double limit in terms of loss of statistical accuracy. In the context of the Monte Carlo algorithms discussed below, such as diffusion and transfer matrix Monte Carlo, this double projection technique to estimate X 00) is called forward walking or future walking. 

But, generally, such a cycle with adiabatic and isothermal irreversible processes may be realized with real gas (or even liquid). Those with real gas approximate the reversible Carnot cycle with ideal gas by a double limiting process as follows (i.e., we form the ideal cyclic process from set A (and also B and C), see motivation of postulate U2 in Sect. 1.2) running this cycle slower and slower... 

The model (21) is interesting in the polymer limit m — 0. In this Cctse it Ccui be interpreted as a model for SAWs in disordered media. Note that such a limit is not trivial. As noted by Kim [62], once the double limit m,n 0 has been ttiken, both o and Wo terms are of the Scune symmetry, and an effective Hamiltonian with one coupling o = o Wo of the 0 mn = 0) symmetry results. This leads to the conclusion that weak quenched xmcorrelated disorder does not change the imiverssJ critical properties of SAWs. These results were confirmed by numerical [16-19,26,63,64] and analytical methods [38,39]. 

However, as it has been noticed in Ref. [62], in the double limit m, n —t 0 the terms with o and Vo become of the same symmetry, joining these terms one passes to an effective Hamiltonian with only one coupling constant of the 0 mn = 0)-symmetry. This can be proved for the expressions of the one-particle irreducible vertex functions... 

In the double limit m, n —t 0 the diagrams with closed disconnected loops have zero contributions, and the expression (101) has the form ... 

The condensation rate equation (5) contains two disparate parameters K and X. K << 1 signifies a small nucleation rate and X >> 1 signifies a small droplet growth time. In the asymptotic method we thereby study the double-limit process K 0 and X °o. This limit has a physical counterpart in observations and experiments. 

A rather different approach for polar molecules has recently been explored by Bossis and Brot. This is to calculate correlation functions for dipole, in an inner spherical region which is part of a larger spherical region the motivation being to gain some idea of how well such simulations can approximate the inner sphere function in the Rirkwood Fr hlich (KF) equation (21) which strictly should be evaluated in the double limits Rj, I /Rj eO. Molecular dynamics computer simulations have so far been done (35) for a two-dimensional system, i.e. disks rather than spheres, for 324 molecules in most cases with a Lennard-Jones 12-6 radial plus rigid dipole pair potential, with the system confined within an outer radius R = 13.2 0- by a soft dish potential barrier. 

In order to perform this electrostatic calculation, Kirkwood had to invoke a double limiting process in that not only must the radius of the microsphere approach infinity but the ratio of the macrosphere to the radius of the microsphere must also approach infinity. This assumption clearly indicates that the microsphere, however large, is an insignificant portion of the macrosphere. Consequently, the second integral in Eq. (48) becomes... 

We now suppose that the phenomenological equations of evolution for the macrovariables of the system give rise to a chaotic attractor possessing at least one positive Lyapunov exponent. It is well known that under these conditions the system manifests sensitivity to initial conditions [32], reflected by the divergence of initially nearby trajectories which in the double limit of initial deviations going to zero (to be taken first) and of time going to infinity (to be taken next) follows on the average an exponential law. 

D) if there is a semi-stable (double) limit cycle, the system may not have simultaneously an unstable separatrix of a saddle which tends to the cycle as t -> -hoo and a stable separatrix of a saddle which tends to the cycle as t —00, as shown in Fig. 8.1.4 and... 

To conclude this section, let us elaborate further on the restrictions (D) and (E). In case (D) the surface corresponding to the double cycle is of codimension-one, and therefore, it divides a neighborhood of the non-rough system Xq into two regions and D. Assume that in the double limit cycle is decomposed into two limit cycles, and that it disappears in D. The situation in -D is simple — all systems there are structurally stable and, moreover, of the same type. As for D the situation is less trivial if (D) is violated, then it is obvious that besides structurally stable systems in there are structurally unstable ones whose non-roughness is due to the existence of a heteroclinic trajectory between two saddles, as shown in Fig. 8.1.6(a). Moreover, this picture takes place in any neighborhood of Xq- In other words, in the region, there exists a countable number of the associated bifurcation surfaces of codimension-one which accumulate to In such cases the surface is said to be unattainable from one side. 

Since A < 0, the Poincare map is decreasing. The new feature in this case is that such maps may have orbits of period two, which correspond to the so-called double limit cycles. They may appear via a period-doubling bifurcation (a fixed point with a multiplier equal to —1) or via a bifurcation of a double homoclinic loop. The latter corresponds to the period-two point of the Poincare map at 2/ = 0 (see Fig. 13.3.2). 

In the region Z i, there are no limit cycles. On the curve L4, upon moving from D towards D2j a stable limit cycle is born from a simple separatrix loop. An imstable double-loop limit cycle bifurcates from a double separatrix loop with (To > 0 on L2. Thus, in the region D3, there are two limit cycles one stable and the other is unstable. The unstable double limit cycle merges with the stable limit cycle on the curve Li. After that only one single-circuit unstable limit cycle remains in region D4. It adheres into the homoclinic loop on the curve L3. 

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