Number of paths

Given the initial and final states of an elementary reaction, and therefore a thermodynamic description of the system, there exist a priori an infinite number of paths (i.e., mechanisms) from the initial to the final state. The essential role of... 

A minimum on a potential energy surface represents an equilibrium stracture. There will invariably be a number of such local minima, and we can imagine a number of paths on the surface that connect one particular minimum to another. If the highest-energy point on each path is considered, the transition structure can be defined as the lowest of these maxima. The reaction path is the lowest-energy route between two minima. 

The advantage of this approach stems from the fact that the minimization over x can be carried out analytically. Consequently, the dimensionality of the optimization problem is reduced from P, the number of pipe sections to S, the number of paths. Murtagh (M9) reported computer storage reduction of more than 50% and computing time reduction of up to 80% using the dual instead of primal formulation. 

Number of pipes P Number of paths S Computing time (CPU seconds) Storage requirement (words)... 

The sum over all j can be denoted as BkGkf where Gfc Is the number of paths containing r, and Bk Is a resultant vector analogous to Bj. Unfortunately Bk cannot be determined from the Information at hand. It Is therefore necessary to assume that B Is the same for all k i.e., 6 and... 

If the flowchart P has a loop-free graph - if P is a tree - then the construction of W(P,A,B) is now quite simple. If P is loop-free there are only a finite number of paths 0, ...,on from START to STOP which are consistent and hence execution sequences. The input condition A(X) is a function only of the inputs, of course, while the output condition B(X,Y) can be regarded as a function of the input and of the final values of all the program variables (some of these values, of course, may play no role in the statement of the condition). Notice that under these conditions, when is a complete execution sequence from START to STOP, the path verification condition VCPjO AB, ) for any interpretation I is a function of the input X alone. 

Assume that the two sites are separated by N steps in the x-direction. (The formula can be straightforwardly generalized to arbitrary locations of the two trap sites.) The number of extra kinks in the path of length N > N is K, which is the number of pairs of non-essential steps. The formula given then follows from the combinatorials of N total steps which may be taken in any order, and which consist of six classes of objects, N + k, forward steps (in the -I- x direction), k, backward steps ( — x), ky sideways steps ( 4- y), ky sideways steps back ( — y), and k ( + z), and kj ( — z), where we must also have that k, H- ky 4- k = K. Asymptotically for large N > N the number of paths grows as 6 /N ... 

An ideal gas in State A (Fig. 2.2) is changed to State C. This transformation can be carried out by an infinite number of paths. However, only two paths will be considered, one along a straight line from A to C and the other from A to B to C [9]. 

It is clear that many reactions, particularly those without simple stoichiometry, will have mechanisms containing several steps, one or more of which may include reversible equilibria (consider for example (1.118) to (1.120)). The number of separate terms in the rate law will indicate the number of paths by which the reaction may proceed, the relative importance of which will vary with the conditions. The complex multiterm rate laws, although tedious to characterize, give the most information on the detailed mechanism. 

The entropies per unit time as well as the thermodynamic entropy production entering in the formula (101) can be interpreted in terms of the numbers of paths satisfying different conditions. In this regard, important connections exist between information theory and the second law of thermodynamics. 

Each event, such as equipment failure, process deviation, control function, or administrative control, is considered in turn by asking a simple yes/no question. Each is then illustrated by a node where the tree branches into parallel paths. Each relevant event is addressed on each parallel path until all combinations are exhausted. This can result in a number of paths that lead to no adverse consequences and some that lead to the incident as the consequence. The investigator then needs to determine which path represents the actual scenario. Generally, a qualitative event tree is developed when used for incident investigation purposes. 

The average entropy production (5p) is deflned by averaging Sp along an infinite number of paths. Dividing Eq. (51) by (5p) we get... 

Connected Region. A region of space enclosed within the boundaries of a closed surface is described as a connected region if it is possible to skip from one point to an other using an infinite number of paths, all of them located within this specific region of space. For example, each region located inside and outside a closed surface is, individually, a connected region. 

For the tree with the branching unit as root, the number of paths is N (n) = f for all path lengths to the root, thus... 

In principle, Curtis-Godson pressures and temperatures have to be computed for each gas, each layer and each limb view of the scan. In practice, only a sub-set of paths (combination of layer and limb view) requires a customised calculation, because, except for the tangent path, the secant law approximation can be applied and consequently the corresponding equivalent quantities are independent on the limb view angle. Therefore equivalent quantities are computed for the paths corresponding to the lowest geometry and only the tangent layers of the other limb views. This is a very effective optimisation because it reduces the number of paths for which cross-sections have to be computed. 

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