Root point

Fig. 4. Spatial distribution of (a) relative elemental elongation rate (longitudinal growth rate) and (p) osmotic potential in the apical 10 mm of maize primary roots growing at various vermiculite water contents (see Fig. 3). Growth distributions were obtained by time-lapse photographic analysis of the growth of marked roots points are means from 5 or 6 roots. Osmotic potentials were measured on bulked samples from 30-50 roots points are means s.d. (n = 3-7). Root elongation rates (a, inset) were constant when the measurements were made. Modified from Sharp et al. (1988, 1989).

The /i-bond diagrams of (f>(r 2) cannot contain any articulation pairs, and so the diagrams connecting the root point 2 with the n h-bonds connected to root point 1 do not have nodal points. This definition is just of the (n + l)-particle direct correlation function c(r2, r2,..., r +2) and one has... 

Figure 6.2 Cumulant expansion, notation as in Fig. 6.1. Flere we assume full knowledge of the medium correlation functions in the absence of the solute Sp (l, 2) = p (l, 2) -p f l)p6f2). These contributions are ordered according to the number of bonds attached to the root point. Table 6.1 gives formulae for contributions through 4th order.

Chandler and his co-workers have taken a somewhat different point of view in seeking to improve upon the SSOZ equation. They start from the perspective that the integral equation itself is flawed since all the tractable closures correspond to the resummation of unallowed diagrams in the interaction site cluster series for the site-site total correlation function. They have formulated an integral equation in which the direct correlation function does indeed correspond to the subset of diagrams in the interaction site cluster series in which there are no nodal circles. The key to their development is a grouping of the site-site total and direct correlation functions into four classes depending upon how the root points are intersected by s-bonds. They write... 

The subscript 0 denotes contributions from diagrams where the roots points in the diagrams are intersected only by f-bonds. The subscript I denotes contributions from diagrams where the left root point is intersected by an... 

A graph is a collection of points and bonds that connect these points. In a pictorial representation of a graph, a point is drawn as a small circle and a bond as a line from one circle to another. (In some applications, it is convenient to define bonds that connect three or more points but we will not discuss this possibility here.) There are two different kinds of points, namely, root points... 

To define this concept more precisely, we need to define a labeled graph. A labeled graph has exactly the same definition as that given above for a graph, except that the field points (if any) as well as the root points have labels, usually numbers, and no two points have the same label. It is easy to define topological equivalence for labeled graphs. Two labeled graphs, which have the same number of root points, the same set of labels on the root points, the same... 

An articulation point in a connected graph is a point whose removal breaks a graph into two or more unconnected parts such that at least one part contains no root point and at least one field point. (A slightly more general definition, which we will not need, is required to define an articulation point in a disconnected diagram.) This definition holds even when the graph has no root point or one root point. See Fig. 2 for illustrations of this definition. 

A graph is irreducible if it has no articulation points. It follows from these definitions that a graph with no root points is irreducible if and only if it is at least doubly connected. Also, a graph with two root points is irreducible if and only if it is at least doubly connected or it would become at least doubly connected when a bond between the roots is drawn. 

A pair of reducible points in an irreducible graph is a pair of points that are connected by a bond and/or that are a pair of articulation points. When a pair of reducible points is removed from a diagram, the diagram becomes disconnected into two or more parts. Some parts may simply be bonds with no point on each end, if the pair of reducible points were connected by one or more bonds. Some parts may be collections of field points connected by bonds and containing some bonds with no point on one end, if the pair of reducible points were a pair of articulation points. Some parts may be collections of field points and root points, containing some bonds with no point at one end, if the original... 

Fig. 2. Illustration of the definition of an articulation point. In the two graphs at the top, the points indicated by stars are articulation points. When these points are removed (leading to the structures drawn immediately below the graphs), the graph becomes disconnected and at least one of the disconnected parts has no root point and one or more field points. (The symmetry numbers of the two graphs at the top are 4 and 1.)...

The integral contains a factor for the density point function for each field point and a factor for the appropriate bond function for each bond. The arguments of these functions are the coordinates corresponding to the labels on the graph. Moreover, if more than one bond of a type connects a pair of molecules, there is a numerical factor [N/(i,y) ] There is also a factor for the reciprocal of the symmetry number. Note that if the graph contains no root points, its value is just a number since all the x coordinates are integrated over. If the graph has m root points, where m 0, the value is a function of Xi,..., x ,. 

Vsiifi,pi,..., py) = sum of all topologically different irreducible graphs that have no root points, two or more field points, and at most one / bond between each pair of points (16)... 

Vsd = Vj2/o+sum of all topologically different irreducible graphs that have no root points, two or more field points, and at most one ho or one yo 8f bond between each pair of points at least one yo 8f bond no pair of reducible points with a residual that, when regarded as a graph with two roots, has a y0 8f between the roots, one or more field points, and no other yo 8f bond and no pair of reducible points with a residual that, when regarded as a graph with two roots, has only ho bonds and one or more field points (62)... 

Also, additional powers of y arise from the long-range nature of the perturbation for small values of y. For example, consider the diagram that has a field point connected to each of two root points by a bond. The value of this diagram is... 

For small values of y, this is of order y. In general, a diagram is of order where t is the number of (p bonds and s the number of factors of y that are obtained from the integration over field points. To find s, we remove all q> bonds from a diagram and count the number of disconnected pieces that remain. In a diagram with root points, s is the number of such pieces that do not contain a root. In a diagram with no roots, s is one less than the number of such pieces. 

F hb( i 2) = sum of all topologically different irreducible graphs that have two root points (labeled 1 and 2), any number of field points, fo and fnaYp bonds, at most one bond between any pair of points, an fuBYp bond between the roots, no pair of overlapping points that remain at least doubly connected when all fo bonds between overlapping pairs are removed, and no pair of points connected by an /hb bond that has a residual that has no /hb attached to either root (115)... 

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