Exponential increase

It can be said that these three main strategies have been applied equally and very often in combination. Basically, the first approach implies the use of a faster computer or a parallel architecture. To some extent it sounds like a brute force approach but the exponential increase of the computer power observed since 1970 has made the hardware solution one of the most popular approaches. The Chemical Abstracts Service (CAS) [10] was among first to use the hardware solution by distributing the CAS database onto several machines. 

Comparing Eqs. (2.68) and (2.70) suggests that the effect of branching on t might be an exponential increase with the length of the side chain, provided that the entanglement mechanism is operative for both the branched and unbranched samples being compared ... 

Growth rate is very sensitive to liquid content for narrow initial-size distributions, with increases in liquid content for fine powders leading to an approximate exponential increase in granule size. For low-viscosity hquids, granulation occurs when very Hose to the saturation of the granule. This leads to the following equation to estimate moisture requirements [Capes, Pai ticle Size Enlargement, Elsevier (1980)] ... 

That is, the exponential increase of the isotope effect with is determined by the difference of the zero-point energies. The cross-over temperature (1.7) depends on the mass by... 

Two-dimensional semiclassical studies described in section 4 and applied to some concrete problems in section 6 show that, when no additional assumptions (such as moving along a certain predetermined path) are made, and when the fluctuations around the extremal path are taken into account, the two-dimensional instanton theory is as accurate as the one-dimensional one, and for the tunneling problem in most cases its answer is very close to the exact numerical solution. Once the main difficulty of going from one dimension to two is circumvented, there seems to be no serious difficulty in extending the algorithm to more dimensions that becomes necessary when the usual basis-set methods fail because of the exponentially increasing number of basis functions with the dimension. 

Let Ci(z) be an arbitrary solution to (B.9) which does not necessarily satisfy (B.9). Then it can be represented as a linear combination of exponentially increasing and decreasing linearly independent solutions (B.IO). When cc, only the increasing solution survives after a long time, and one may write... 

Thus, in the series of Ti measurements of 2-octanol (42, Fig. 2.27) for the methyl group at the hydrophobic end of the molecule, the signal intensity passes through zero at Tq = 3.8 s. From this, using equation 10, a spin-lattice relaxation time of Ti = 5.5 s can be calculated. A complete relaxation of this methyl C atom requires about five times longer (more than 30 s) than is shown in the last experiment of the series (Fig. 2.27) Tj itself is the time constant for an exponential increase, in other words, after T/ the difference between the observed signal intensity and its final value is still 1/e of the final amplitude. 

Figure 7.12. Chronology of message capacity showing exponential increase with time. The number of voice channels transmitted per fibre increases rapidly with frequency of the signalling medium. The three right-hand side points refer to optical-fibre transmission (after MacChesney and DiGiovamti 1991, with added point).

This indicates an exponential increase in strain from zero up to the value, (To/, that the spring would have reached if the dashpot had not been present. This is shown in Fig. 2.37. As for the Maxwell Model, the creep modulus may be determined as... 

It may be seen in Fig. 2.42 that this predicts the initial strain when the stress is first applied as well as an exponential increase in strain subsequently. 

In laminar flow, a processed material divides at the leading edge of each element and follows the channels created by the element shape. At each succeeding element the two channels are further divided, resulting in an exponential increase in stratification. The number of striations produced is 2 where n is the number of elements. 

Class c4 Large number of attractors with possibly long periods. Kaneko has found that this class is characterized by an exponentially increasing number of attractors, although the increase is irregular. 

Except for simple cases, it is generally a nontrivial task to compute the Lyapunov exponents of a flow. In trying to estimate A(x(0)) in equation 4.59, for example, the exponentially increasing norm, V t), may lead to computer overflow problems. 

This short review cannot be comprehensive as of the exponential increase of the literature dealing with new procedures and applications. Table 2 summarizes selected data and lists fields of application. For a more inclusive view on the subject the interested reader is referred to existing monographs6-161 or to the critical biennial review in Analytical Chemistry. As the readers of this general volume on membranes most likely are acquainted with electro-analytical sensors, this article will be limited to the introduction of a theoretical approach which might be helpful also to researchers in the bio-membrane field. 

In contrast to the solid state concepts presented above, Ioannidis et al. [90] describe the organic materials as amorphous molecules. This model is suitable to fit 1/V characteristics of LEDs based on Alq3 and PPVs with a low mobility (j.ik 1CT, h cm2 V 1 s l). Essentially, they found that the current flow through these materials can be attributed to the increase in the mobility (sec Eq. (9.18)) with the applied voltage, which has been observed for many conjugated polymers [91 ]. In the case that the mobility exponentially increases with the electric field, the current flow raids ... 

You have now demonstrated the capability of exponentially increasing microbial populations to rapidly produce protein. However, although such outputs of protein are possible in theory, they cannot be achieved in practice, since exponential growth cannot be maintained for such periods because bioreactors are limited in size. 

Once there is an appreciable amount of cells and they are growing very rapidly, the cell number exponentially increases. The optical cell density of a culture can then be easily detected that phase is known as the exponential growth phase. The rate of cell synthesis sharply increases the linear increase is shown in the semi-log graph with a constant slope representing a constant rate of cell population. At this stage carbon sources are utilised and products are formed. Finally, rapid utilisation of substrate and accumulation of products may lead to stationary phase where the cell density remains constant. In this phase, cell may start to die as the cell growth rate balances the death rate. It is well known that the biocatalytic activities of the cell may gradually decrease as they age, and finally autolysis may take place. The dead cells and cell metabolites in the fermentation broth may create... 

Carter et al. (79) have theoretically investigated the exponential increase in temperature expressed by... 

Hill et al. [117] extended the lower end of the temperature range studied (383—503 K) to investigate, in detail, the kinetic characteristics of the acceleratory period, which did not accurately obey eqn. (9). Behaviour varied with sample preparation. For recrystallized material, most of the acceleratory period showed an exponential increase of reaction rate with time (E = 155 kJ mole-1). Values of E for reaction at an interface and for nucleation within the crystal were 130 and 210 kJ mole-1, respectively. It was concluded that potential nuclei are not randomly distributed but are separated by a characteristic minimum distance, related to the Burgers vector of the dislocations present. Below 423 K, nucleation within crystals is very slow compared with decomposition at surfaces. Rate measurements are discussed with reference to absolute reaction rate theory. 

The effect of the temperature is an exponential increase in the radiolytic yield of the volatile products for each poly(olefin sulfone). If the logarithm of G(volatile products) is... 

The reason for the exponential increase in the electron transfer rate with increasing electrode potential at the ZnO/electrolyte interface must be further explored. A possible explanation is provided in a recent study on water photoelectrolysis which describes the mechanism of water oxidation to molecular oxygen as one of strong molecular interaction with nonisoenergetic electron transfer subject to irreversible thermodynamics.48 Under such conditions, the rate of electron transfer will depend on the thermodynamic force in the semiconductor/electrolyte interface to... 

Electron transfer rate and its exponential increase at zinc oxide-electrolyte interfaces, 512 Electronic conductivity... 

Thus in order to rationalize the NEMCA behaviour of the ethylene oxidation system one needs only to concentrate on the kinetic constant k and on its dependence on is accompanied by a concomitant significant decrease in activation energy E and in the preexponential factor k° defined from ... 

FIGURE 2.7 The potential energy of an ionic solid, taking into account the coulombic interaction of the ions and the exponential increase in their repulsion when they are in contact. The minimum potential energy is given by the Born-Meyer equation, Eq. 3. 

The capillary flow with distinct evaporative meniscus is described in the frame of the quasi-dimensional model. The effect of heat flux and capillary pressure oscillations on the stability of laminar flow at small and moderate Peclet number is estimated. It is shown that the stable stationary flow with fixed meniscus position occurs at low wall heat fluxes (Pe -Cl), whereas at high wall heat fluxes Pe > 1, the exponential increase of small disturbances takes place. The latter leads to the transition from stable stationary to an unstable regime of flow with oscillating meniscus. 

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