Vertex reducibility

To carry through the partial summation of perturbation theory wc introduce the notion of vertex reducibility. 

A graph is said to be vertex reducible, if it ca.n be cut into two disconnected pieces by cutting one vertex. 

For example, the graphs 5.3a,b are vertex reducible. The graphs 5.3c. 5.1 are vertex irreducible. 

Now consider a propagator line of momentum k, connecting segments ji and and decorate it with any number of such C-insertions (Fig. 5.6). 

Grand Canonical Description of Solutions at-Finite Concentration 

An irreducible graph first line) and some reducible graphs second I, yielding that graph after cutting off all reducible parts. The reducible parts indicated by boxes 

---

A well-known class of techniques for reducing the number of iterates is the use of tearing (L4). We shall illustrate this procedure by way of an example taken from Carnahan and Christensen (C3). Let us consider the two-loop network shown in Fig. 5 and assume that formulation A is used. To abbreviate the notation let us denote the material balance around vertex i [Eq. (35)] by fi = 0 and the model of the element [Eq. (36)] by fu — 0. Then assuming all external flows and one vertex pressure, p, say, are specified, we have a set of 12 equations that must be solved simultaneously. But if we now assume a value for ql2, the remaining equations may be solved sequentially one at a time to yield the variables in the following... 

As the optimum is approached, the last equilateral triangle straddles the optimum point or is within a distance of the order of its own size from the optimum (examine Figure 6.4). The procedure cannot therefore get closer to the optimum and repeats itself so that the simplex size must be reduced, such as halving the length of all the sides of the simplex containing the vertex where the oscillation started. A new simplex composed of the midpoints of the ending simplex is constructed. When the simplex size is smaller than a prescribed tolerance, the routine is stopped. Thus, the optimum position is determined to within a tolerance influenced by the size of the simplex. 

Progression to the vicinity of the optimum and oscillation around the optimum using the simplex method of search. The original vertices are x , x , and x . The next point (vertex) is Xq. Succeeding new vertices are numbered starting with 1 and continuing to 13 at which point a cycle starts to repeat. The size of the simplex is reduced to the triangle determined by points 7, 14, and 15, and then the procedure is continued (not shown). 

Note that/is reduced. The solution corresponds to vertex (3) of Figure 7.1. Because c i = -4 = mirijCjiXi becomes basic. The only ratio bi/an having 5, > 0 is that for i = 2 thus 4 becomes nonbasic and the circled pivot term is a2 x = 2 j. Pivoting yields... 

---