Marginal fixed point

Fig. 3. Profitabihty diagram for Venture A. (a) Simple diagram. NRR is net return rate IRR, the internal rate of return, is a given fixed point, (b) Three NRR cutoff lines for Venture A where B, C, and D represent NRR values of 15, 10, and 5%/yr, respectively. For example, at a discount rate of 10% per year, the NRR cutoff for Venture A could be as high as 10.74% per year for marginal acceptance (point X). Acceptable levels are to the left of NRR cutoff...

Because the isotope fractionating processes are not evenly distributed throughout the ocean, temporal variations at fixed points will produce predictable changes in the (5 N of local nitrate pools. As examples (1) an increase in denitrification rates within the water column on the Oman margin would increase the <5 Nni ate of the Arabian Sea thermochne (Altabet et al., 1995 Schafer and Ittekkot, 1993), (2) an increase in N2-fixation in surface waters of the eastern Mediterranean would cause a decrease in d Nnitrate io the eastern Mediterranean thermochne (Stmck et al., 2001), and (3) an increase in relative nitrate consumption in Antarctic surface waters would cause an increase in local near-surface d N itrate (Francois et al., 1997). However, it must be kept in mind that these local signals are imprinted upon any variation in the baseline of mean ocean nitrate. A similar principle operates at finer scales, with... 

Solution The multiplier at x = 0 is / (O) = cos(O) = 1, which is a marginal case where linear analysis is inconclusive. However, the cobweb of Figure 10.1.2 shows that x = 0 is locally stable the orbit slowly rattles down the narrow channel, and heads monotonically for the fixed point. (A similar picture is obtained for... 

The origin is globally stable for r < 1, by cobwebbing. There is an interval of marginally stable fixed points for r = 1. 

We see certain similarities of disorder or randomness becoming marginally relevant at some dimension dc = 2 for the random medium problem, but dc = l for the RANI problem. A new fixed point emerges above this critical dimension. For the random medium problem, this implies the existence of a new phase and a disorder induced phase transition, but for the random interaction problem it defines a new type of critical behaviour. These results based on the exact RG analysis [36,37] were later on also recovered from a dynamic renormalization group study [38]. 

If > 0, then is a relevant variable at the free fixed point, while u is irrelevant there if < 0. Special situations correspond to e = 0 for which u is a marginal variable. The possibilities we need to consider are... 

Limit cycles are primarily of concern in fixed-point recursive filters. As long as floating-point filters are realized as the parallel or cascade connection of first- and second-order subfilters, limit cycles will generally not be a problem since limit cycles are practically not observable in first- and second-order systems implemented with 32-bit floating-point arithmetic (Bauer, 1993). It has been shown that such systems must have an extremely small margin of stability for limit cycles to exist at anything other than underflow levels, which are at an amplitude of less than 10 (Bauer, 1993). 

On the other hand, for the simpler system with a propagating front without periodic pattern [lO] it turns out, that the fixed point at r coincides with the end-point r of the line of fixed point r < r, rj=0. In this case the marginal stability criterion gives exactly the operating point of the system. 

Plate-Column Capacity The maximum allowable capacity of a plate for handling gas and liquid flow is of primaiy importance because it fixes the minimum possible diameter of the column. For a constant hquid rate, increasing the gas rate results eventually in excessive entrainment and flooding. At the flood point it is difficult to obtain net downward flow of hquid, and any liquid fed to the column is carried out with the overheaa gas. Furthermore, the column inven-toiy of hquid increases, pressure drop across the column becomes quite large, and control becomes difficult. Rational design caUs for operation at a safe margin below this maximum aUowable condition. 

In Figure 61.10 fixed costs are shown above the variable costs and the non-recovery of the fixed costs below the break-even point is more clearly demonstrated. The contribution to fixed costs is of significance in the consideration of marginal costing. 

The resulting optimal profiles for the operating conditions will need to be evaluated in terms of their practicality from the point of view of control and safety. If a complex profile offers only a marginal benefit relative to a fixed value, then simplicity (and possibly safety) will dictate a fixed value to be maintained. But an optimized profile might offer a significant increase in the performance, in which the complex control problem will be worth addressing. 

Non-linear pricing Non-linear prices usually consist of a two-part tariff. One part is fixed and does not depend on the quantity of the product consumed. The simplest way to establish this tariff is to estimate the potential loss (L) at the point where price = CMg and divide it by the number (N) of potential users, that is, the tariff equals L/N. The other part varies with the quantity consumed. In the case in which the variable component is fixed according to the marginal cost, the price structure is efficient and at the same time the company can avoid incurring a deficit. 

The critics of government-imposed price controls on pharmaceutical products do have a valid point. As long as the price ceilings are set above the incremental cost of producing these products, manufacturers will be tempted to sell at whatever those controlled prices are, because they earn at least a positive margin toward the recovery of fixed costs. The problem is that the price ceilings may be set at levels far below fully allocated fixed costs per unit. If every payer followed that strategy, pharmaceutical companies would soon become insolvent. 

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