VERTEX LOOP

Obtain the experimental responses for each vertex LOOP WHILE stop criterion is not achieved... 

Figure 6 Vertex loop weights arising from connected vertices within an orbit of length n...

ENTITY LOOP = CLASS ( EDGE.LOOP, VERTE5CL00P ) = <edge.loop> ... 

A VERTEX may be referenced from one and only one VERTEX LOOP A VERTEX may be referenced from one or more EDGE entities The FACE SURFACE geometry in a B REP has the highest priority in defining the geometry of a B REP. That means that the POINT geometry associated to the VERTEX is considered to be the best approximation available of the point that is common to all surfaces associated with the faces that meet at the particular VERTEX. 

Self-loops (around the same vertex) and parallel edges (joining the same two vertices) will be excluded in our discussions. 

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... 

FIGURE 2.8 Separator systems cascaded to form a Bethe lattice or Cayley tree where the point of introduction is the graph vertex 0 and solute can be sampled from any of the outward nodes at position 1, 2, 3, 4, and so on. The sample loops and valves are not shown. 

Most of the graphs we deal with are trees- that is, they do not have any self tracing loops. Choose a vertex which will be the root . In the example we consider here we shall take vertex 5 as the root (see figure l.(c)). A wave which propagates from the root along a certain branch is reflected, and this reflection can be expressed by a reflection from the vertex which is next to the root. Once we know the reflection coefficient, which because of unitarity is a complex number with unit modulus, we can construct the SB(k) matrix. In the present cases, when the valency of the root is 3, we get... 

The maximal loops in the graph can be found from the reachability matrix by finding those sets of vertices that satisfy the following conditions (1) r j = rfi = 1, where i and j take on all possible combinations of the vertex numbers in the set (2) no other vertices, not included in the set, satisfy condition (1). The first condition requires that each vertex in the set is reachable by some path from every other vertex in the set. The second condition requires that there is no path from a vertex in the set to a vertex outside the set... 

The second order perturbation theory term with two one-loop self-energy operators does not generate any logarithm squared contribution for the state with nonzero angular momentum since the respective nonrelativistic wave function vanishes at the origin. Only the two-loop vertex in Fig. 3.24 produces a logarithm squared term in this case. The respective perturbation potential determined by the second term in the low-momentum expansion of the two-loop Dirac form factor [111] has the form... 

The quotient of a map can be a map with loops and multiple edges. Consider, for example, the 4-valent plane tiling 4,4 (see Section 1.5) formed by 4-gons and the group Z2 acting by translations on it. There is one orbit of vertices, two orbits of edges, and one orbit of faces under Z2 so the quotient 4,4)/ 2 is a torus represented by a single vertex and two loops. 

Loop expansion, vertex irreducible graphs and screened interaction... 

Having reduced all quantities to their vertex irreducible parts we can introduce the screened interaction. In each order of the loop expansion we have to resum a series of diagrams. For instance, the series of one loop graphs... 

The loop expansion of a vertex irreducible quantify containing external vertices is represented by all vertex irreducible diagrams containing only screened interactions. Diagrams, in which a chain-like structure not contain-... 

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