Observing Sufficient Conditions

Previously in this chapter it was noted that to train voltage in - current out inverters and NANDs the NanoCell just needs enough molecules in the on state. In working with NanoCells composed of just a few molecules we found that for inverters, a path of molecules in an on state between the input and output is both necessary and sufficient. If the molecules are not symmetric, then they must be oriented so that current flows from the input to the output. The conditions for NANDs are similar. First, obviously it is necessary that there is a path from each input to the output. In training NanoCells and working with a few molecules, we observed that it is sufficient to have paths firom each input to the output, some of which should intersect. 

After observing these conditions, we decided to try to prove that such conditions are satisfied in a random NanoCell. Working with both large NanoCells (approximately 200 nanoparticles, KXX) molecular switches) and smaller NanoCells (about 30 nanoparticles and 250 molecular switches), the results are presented in the following sections. 

Note that since the degree of each node is equal and the expectation is linear 

Suppose now that we want to compute the probability that a random NanoCell graph has at least t isolated nanoparticles where r 0. Consider the following inequality. 

Theorem 6.6 Given a NanoCell with n nanoparticles and m edges where each nanoparticle has degree 6 and each edge has probability p, let X represent the number of isolated nanoparticles. Then, 

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Another potential improvement is to prove the observed sufficient conditions for NANDs. Namely, compute the probability that there are intersecting paths of sufficiently short length from each input to the output. 

The kinetic evidence is a necessary but not a sufficient condition we will meet other mechanisms that are also consistent with these data. Much more convincing evidence is obtained from the fact that the mechanism predicts inversion of configuration when substitution occurs at a chiral carbon and this has been observed... 

Observe here that the factorized operator B also is self-adjoint. By means of these operators a sufficient condition of stability becomes... 

The Ising-type model provides also a neeessary though not sufficient condition for the observation of hysteresis. Thus, from the study of metastable minima of G for the case = J2 = ibis condition results as [37] ... 

Reaction mechanism a postulated sequence of elementary reactions that is consistent with the observed stoichiometry and rate law these are necessary but not sufficient conditions for the correctness of a mechanism, and are illustrated in Chapter 7. 

An important part of the failure detection problem involves the analysis of the necessary and sufficient conditions for detectability of jumps in the system. These are related to properties of matrix A and are discussed in detail by Caglayan (1980). The detectability of jumps in the outputs will be dictated by the observability of the discrete system defined by the pair (A0, Co) (Caglayan, 1980). 

Several experimental conditions must be realized for the application of Eq. (72). The donor and acceptor molecules should enter the channels at about the same rate, so that the assumptions made for the initial state are sufficiently well fulfilled. They should not be able to glide past each other once they are inside the channels. The crystals should be so long that molecules entering from both sides do not reach each other in the middle part of the channels during the time of observation. These conditions can be fulfilled for the donor/acceptor pair Py+/Ox+ in zeolite L. Moreover, different stages of the diffusion can be observed by means of an optical microscope. 

The remaining task lies in the determination of the control matrix X and observer matrix Z such that the sufficient condition for robust performance, Eq. (22.28), holds. A Lyapunov-based approach is employed to obtain these two matrices. After some lengthy and complicated manipulations of Eq. (22.29) and the control structure shown in Fig. 22.3, the following two Riccati equations are derived, whose positive-definite solutions correspond to the control and observer matrices, X and Z. 

Pi0 and P32 have parallel isotherms, as expected for samples of identical structure. The isotherms of Po and P5 have the same slope as those of P10 and P32 which are rather well crystalline. However, this observation is a necessary but not a sufficient condition to affirm that these materials, amorphous to X-rays, already contain microcrystallites having a Faujasite structure. 

A necessary but not sufficient condition for initiation by impact is that impact pressure (stress) be sufficiently high so that the melting point of the explosive is raised above some critical temperature, TCr- For T>Tcr the explosive, in the hot spot, will decompose adiabati-cally in times of the order 10/usec, which has been observed experimentally. The relation between Tcr the critical stress is then expressed... 

This is in line with the explanation of Bordwell and Boyle (1975) who proposed the formation of a localized carbanionic intermediate as the cause of the anomalous -values. However, as we have noted, formation of such an intermediate is not a necessary condition for observing anomalous a-values, although it is clearly a sufficient condition. Thus Bordwell s explanation cannot, a priori, be ruled out in all cases, and may indeed apply if the potential carbanionic intermediate is strongly stabilized. 

One of the earliest structured models is that put forward by WILLIAMS164 who proposed that the material of a cell could be divided into two categories. One of these is referred to as the active component, the other being the structural component. The model considered that all the cells in the fermentation broth were identical and substrate was incorporated initially into the active component and thence was used to form the structural component. The second, structural, component controlled the observed growth of the culture in that doubling of that component would be a necessary and sufficient condition for the cells to divide. 

All of the photochemical cycloaddition reactions of the stilbenes are presumed to occur via excited state ir-ir type complexes (excimers, exciplexes, or excited charge-transfer complexes). Both the ground state and excited state complexes of t-1 are more stable than expected on the basis of redox potentials and singlet energy. Exciplex formation helps overcome the entropic problems associated with a bimolecular cycloaddition process and predetermines the adduct stereochemistry. Formation of an excited state complex is a necessary, but not a sufficient condition for cycloaddition. In fact, increased exciplex stability can result in decreased quantum yields for cycloaddition, due to an increased barrier for covalent bond formation (Fig. 2). The cycloaddition reactions of t-1 proceed with complete retention of stilbene and alkene photochemistry, indicative of either a concerted or short-lived singlet biradical mechanism. The observation of acyclic adduct formation in the reactions of It with nonconjugated dienes supports the biradical mechanism. 

The same statement can be made about inelastic non-Newtonian fluids, such as the Power Law fluid, from a mathematical solution point of view. In reality, most non-Newtonian fluids are viscoelastic and exhibit normal stresses. For fluids such as those (i.e., fluids described by constitutive equations that predict normal stresses for viscometric flows), theoretical analyses have shown that secondary flows are created inside channels of nonuniform cross section (78,79). Specifically it can be shown that a zero second normal stress difference is a necessary (but not sufficient) condition to ensure the absence of secondary flow (79). Of course, the analyses of flows in noncircular channels in terms of constitutive equations—which, strictly speaking, hold only for viscometric flows—are expected to yield qualitative results only. Experimentally low Reynolds number flows in noncircular channels have not been investigated extensively. In particular, only a few studies have been conducted with fluids exhibiting normal stresses (80,81). Secondary flows, such as vortices in rectangular channels, have been observed using dyes in dilute aqueous solutions of polyacrylamide. Interestingly, these secondary flow vortices (if they exist) seem to have very little effect on the flow rate. 

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