Restriction monotonic

Fig. 8.96 Average slress-corrosioD crack velocity from monotonic slow strain rate tests at 1.5 X 10 s conducted over various restricted ranges of stress on a cast Ni-Al bronze in seawater at 0.15 V(SCE). The stress range traversed in each test is shown by the length of the bar. (after Parkins and Suzuki )...

The natural replacement of the central difference derivative u x) by the first derivative Uo leads to a scheme of second-order approximation. Such a scheme is monotone only for sufficiently small grid steps. Moreover, the elimination method can be applied only for sufficiently small h under the restriction h r x) < 2k x). If u is approximated by one-sided difference derivatives (the right one for r > 0 and the left one % for r < 0), we obtain a monotone scheme for which the maximum principle is certainly true for any step h, but it is of first-order approximation. This is unacceptable for us. 

Since Pparaceii is proportional to the molecular sieving function F(r/R), the interrelationships of Ppaiacen between mannitol and sucrose, including mannitol-manitol and sucrose-sucrose, can be put into perspective via a normalized plot of F(r/R) versus r/R for control and perturbed monolayers (Fig. 12). As pointed out before, F(r/R) is a rapid, monotonically decreasing function bounded by 1.0 and zero. One observes that mannitol is less restricted by the pores of the control monolayer than the larger sucrose molecule. However, in the larger pores of the perturbed monolayer, the increase in permeability is less for mannitol than it is... 

The effect of restricted junction fluctuations on S(x) is to change the scattering function monotonically from that exhibited by a phantom network to that of the fixed junction model. Network unfolding produces the reverse trend, the change of S(x) with x is even less than that exhibited by a phantom network. Figure 6 illustrates how the scattering function is modified by these two opposing influences. 

The restrictions of the definition (5.127) are the same as before It gives correct results only for monotonically evolving functions w/(f) and i)if(t) should fastly enough approach its steady-state value m/(oo) for convergence of the integral in (5.127). 

MFC restrict themselves, moreover, to monotonous Markov processes ... 

Fig. 19, where the exponential tail is restricted to the region Q < 0. Why are spontaneous events not observed for 2 > 0 The reason is that spontaneous events can only release and not absorb energy from the environment see Eq. (215). This is in line with the argumentation put forward in Section VI.A, where the first time that cooperative regions release the stress energy, it gets irreversibly lost as heat in the environment. As the number of stressed regions monotonically decreases as a function of time, the weight of the heat exponential tails decreases with the age of the system as observed in Fig. 19. The idea that only energy decreasing events contribute to the effective temperature (Eq. (215)) makes it possible to define a time-dependent configurational entropy [189].

It is not necessary to restrict ourselves to bonds that are described by Morse potentials. We can regard eqn. (56) as a quadratic equation in x, use any form of the potential energy V(R) with the usual shape (i.e., a minimum, a repulsive barrier at short distances, and a monotonical increase at large distances), and determine x to get another definition of the bond order. This is called the unity bond index quadratic exponential potential (UBI QEP) method by Shustor-ovich and Sellers. ... 

As Z)0 is a convex restricted coinvariant set, it contains at least one steady-state point of eqn. (139). Note that, if starting from some vin, for any two different solutions of eqn. (139) lying in T)0, cl(t) and c2(t), the function c1 (<) - c2(t) is monotonically reducing to zero, the steady state is unique, and any solution lying in Z)0 tends to this steady state at t - oo. It is the distance to this point that will be the global Lyapunov function for eqn. (139) in t)0. Let us investigate at which values of vin the function c (<) — c2(t) decreases monotonically. 

Experimental results from studies of Arrhenius dependence of different characteristics of lysozyme are presented in Fig 4.1. (Alfimova and Likhtenshtein, 1979 Likhtenshtein, 1993 Likhtenshtein et al., 2000). The discontinuities on the curves indicate local conformational transitions and are apparently due to the appearance of a more open conformation of the protein. As can be seen from Fig. 4.1., these methods reveal conformational transitions at a temperature of about 30°C, whereas the temperature dependence of the partial heat capacity decreases monotonically in this temperature region. Recently, the presence of the conformational transition in lysozyme was confirmed independently. It was shown that the segmental motion of Trp 108 is hindered by the local cage structure at T < 30°C, although relieved from restricted motion by thermal agitation or by the formation of a ligand complex. 

Like the models of de Gennes (1982) and Scheutjens and Fleer (1985 Fleer and Scheutjens, 1986), the SCF model predicts monotonic attraction between adsorbed layers under conditions of full equilibrium. For constant restricted equilibrium), Fig. 23 shows Ay s y(zm) — y(co) (curve A) increases slightly before falling as the separation decreases A/ip = pp(zm) — /ip(oo) increases upon compression (curve B), and the total potential (curve C) displays an attractive minimum as well as a steep repulsive wall. Potentials for various different combinations of n and (pb (i.e., dosage at infinite separation). The reasons for this difference are not clear at this point. 

It has been shown that competitive exclusion - that is, the extinction of all but one competitor - holds regardless of the number of competitors or the specific monotone functional response. If one restricts attention to the Michaelis-Menten functional response, then competitive exclusion has been shown even in the case of population-specific removal rates. 

Monotonicity of a dynamical system places restrictions on the basin of attraction of a rest point. Suppose that Xq is a rest point of the monotone dynamical system generated by (C.l). Let B denote the basin of attraction of Xq ... 

Theorem C.5 places strong restrictions on how a limit set is imbedded in K". Note in particular that a periodic orbit may always be considered as a limit set of one of its points and hence Theorem C.5 applies to a periodic orbit. This should convince the reader that periodic orbits are ruled out for two-dimensional monotone systems. 

The objective function to be considered is of one of the simpler types, either stoichiometric or material. In the first case it is required to maximize the final extent of reaction, Ci in the second case this will also be required if the objective function is monotonically increasing with Cl. Here the interesting problem is to find the optimal policy subject to some restriction on the total holding time, say... 

For exclusively real eigenvalues of fV the time dependence of the average excess production is determined by the choice of initial conditions. As shown in Appendix 5, optimization of (t) is restricted to initial conditions in the positive orthant [yt(0) >0 k = 0,1,.. ., n]. These initial conditions are not difficult to fulfil, and they will apply to many cases in reality. We should keep in mind, nevertheless, that there are other choices of initial conditions, such as the start with a pure master sequence, for which the simple principle does not hold. For one particular type of choice, yi (0) > 1 and y/j(0) < 0 for all /c 1, the average excess production decreases monotonically. 

Isotropic solutions exhibit a monotonic increase in shear viscosity with increasing concentration. The viscosity increases to a maximum when the isotropic to anisotropic transition is approached. Upon formation of the anisotropic phase, the viscosity begins to decrease, after which the viscosity increases strongly as the concentration continues to increase (Fig. 6). In the isotropic state, the hydrodynamic volume is large because of the random polymer orientation. This restricts the polymer diffusivity and causes an increase in viscosity. In the anisotropic phase, the aligned polymer leads to a small hydrodynamic volume and a decrease in viscosity as rotational diffusion is much easier with a net orientation. 

This relation is restricted for (a) the positively solvatochromic donor-acceptor TT-conjugated molecules, (b) the two-photon absorption from the ground state to the CT excited state, and (c) the static molecular (hyper)polarizabilities. The above ordering was confirmed for the extended variety of the donor-acceptor rr-conjugated molecules exhibiting the monotonic behavior as a function of the solvent polarity [40, 113, 116]. However, this trend dose not hold in some cases (see [40]). 

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