Linear one-step process

The coefficients are linear functions of n, not both constant. This type will be called linear one-step processes and the general solution is given in 6. Note that there must be at least one boundary since otherwise negative values for the transition probabilities would occur. 

Exercise. Also find the solutions of (3.4) and (3.5) for the general linear one-step process. Why can they not be solved in the nonlinear case ... 

Solution of linear one-step processes with natural boundaries... 

The simplest example is the decay process treated in IV.6, but there the result is trivial since the decay events are independent by definition. The same remark applies to all linear one-step processes, see VII.6. In order to avoid the complications of nonlinear processes we here choose an example which is linear but not a one-step process. The recombinations, however, take place in one step, so that the formulas (1.2) and (1.3) remain valid. 

These have been defined in section 1 as one-step processes in which r and g are linear functions of rc, not both constant. Some examples have already been encountered the decay process in IV.6 and the density fluctuations of a gas in an Exercise of the previous section. We here list some other examples, leaving it to the reader to fill in the details, even if they are not formally registered as Exercises. 

Remark. The distinction between linear and nonlinear one-step processes has more physical significance than appears from the mathematical distinction between linear and nonlinear functions r(n) and g(n). In many cases n stands for a number of individuals, such as electrons, quanta, or bacteria. The master equation for pn is linear in n when these individuals do not interact, but follow their own individual random history regardless of the others. A nonlinear term in the equation means that the fate of each individual is affected by the total number of others present, as is particularly clear in example (iv) above. Thus linear master equations play a role similar to the ideal gas in gas theory. This state of affairs is described more formally in VII.6. 

Another approach in generating molecular insulating layers without the need of chemical conversion after deposition is the use of preliminarily modified molecules which can form dense self-assembled monolayers. To create dense self-assembled monolayers with sufficient robustness and insulating properties, a modified alkyltrichlorosilane with an aromatic end-group (18-phenoxyoctadecyl)tri-chlorosilane (PhO-OTS chemical structure Fig. 6.15a) was synthesized and tested [50]. The SAMs were created in a one-step process from vapor phase or solution. On self-assembly on a natively oxidized silicon surface the n-n interaction between the phenoxy end-groups of adjacent molecules creates an intermolecular top-link, leading to a more closely packed surface compared to monolayer than when linear end groups are used. 

The original linear prediction and state-space methods are known in the nuclear magnetic resonance literature as LPSVD and Hankel singular value decomposition (HSVD), respectively, and many variants of them exist. Not only do these methods model the data, but also the fitted model parameters relate directly to actual physical parameters, thus making modelling and quantification a one-step process. The analysis is carried out in the time domain, although it is usually more convenient to display the results in the frequency domain by Fourier transformation of the fitted function. 

The problem in equation (83) is well-posed and solvable by matrix methods, with all eigenvalues being negative, and the derivatives dXi/dz at t = 1 are readily computed from equation (83). These derivatives provide linear relationships between and t in the reaction zone that enable the rates w,-there to be expressed explicitly in terms of t. A progress variable e for the effective one-step process may be defined, which changes from 0 to 1 across the reaction zone, and an equation of the form de/dz = Aoy(T) may be derived from equation (80), where A is inversely proportional to and cd(z) is known. Then... 

The mechanism of the cycloaddition reactions of cumulenes involve concerted one-step processes as well as two-step processes, and both types of mechanisms are encountered. It seems that concerted processes are more the exception, and ionic linear 1 1 intermediates are sometimes trapped in cycloaddition reactions. The sometimes encountered [2-1-2-1-2] six-membered ring cycloadducts exemplify the stepwise reactions. 

The [2+2] cycloaddition reaction of ketenes with vinyl ethers and thioethers also occurs very readily. Even allyl ethers undergo this reaction. The reaction of diphenylketene with vinyl ethers is stereospecific, indicating a concerted one-step process . Also, dimethylketene and -MeOCH=CHMe affords a cycloadduct in which the alkene stereochemistry is maintained In contrast, the [2+2] cycloadduct obtained fl om t-butylcyanoketene and CH2=CHOEt or CH2=CHOAc did not give a 100% stereoselectivity and linear products are often also obtained, indicating the formation of a switter ionic intermediate. The latter are detected in the reaction of bis(trifluoromethyl)ketene with ethyl vinyl ether (see the General Introduction ). The initial reaction occurs across the C=0 bond of the ketene, which rearranges via switter ionic intermediates to form the cyclobutanone reaction product . 

In contrast to the hydrolysis technology, the methanolysis process allows for the one-step synthesis of organosdoxane oligomers and methyl chloride without formation of hydrochloric acid (64,65). The continuous methanolysis can also yield quantitatively linear sdanol-stopped oligomers by recycle of the cycHc fraction into the hydrolysis loop. 

First, we recall the five different stereoisomerization processes these are necessary to reach any isomer from a given one in one step. However, they are not really independent because a succession of two processes is a linear combination of processes. A multiplication table of the processes has been established. This is explained in Sections II and III. 

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