Node, branch point

Fig. 12.4. Stationary-state solutions and limit cycles for surface reaction model in presence of catalyst poison K3 = 9, k2 = 1, k3 = 0.018. There is a Hopf bifurcation on the lowest branch p = 0.0237. The resulting stable limit cycle grows as the dimensionless partial pressure increases and forms a homoclinic orbit when p = 0.0247 (see inset). The saddle-node bifurcation point is at...

We assume that the equations (7.200) have a simple hysteresis type static bifurcation as depicted by the solid curves in Figures 10 to 12 (A-2). The intermediate static dashed branch is always unstable (saddle points), while the upper and lower branches can be stable or unstable depending on the position of eigenvalues in the complex plane for the right-hand-side matrix of the linearized form of equations (7.198) and (7.199). The static bifurcation diagrams in Figures 10 to 12 (A-2) have two static limit points which are usually called saddle-node bifurcation points. 

Components are commonly represented as nodes in the metabolic network. These nodes can act as branch points if the number of input and output fluxes is not equivalent. Non-essential reactions around a node can be collected into reaction groups the coefficients of their fluxes, in general termed the metabolic flux coefficients (in analogy to rate coefficients), can be rearranged as group control coefficients. 

Figure 4.7 Consensus bootstrap tree of full-length amino acid sequences of insect desaturases generated with MacVector 7.0 (Oxford Molecular Limited). Branch points (internal nodes) are retained if they occur in >50 percent of resampling trees (1000 x resampling) all other nodes are collapsed. Names in bold are from the published literature names in parentheses reflect a nomenclature system for insect desaturases (proposed in Knipple ef a/.,...

Depending on the choice of control parameters, the amplitudes of the pressure oscillations in the nephron tree are found to be different at different positions in the tree. Due to model symmetry, two nephrons connected to the same node have the same oscillation amplitudes. Thus, we can refer to the number of the branching point to describe the amplitude properties. Branching points 2, 3, and 4 may correspond to deep nephrons and branching points 6 and 7 to superficial nephrons. Experimentally, only the pressure oscillations in nephrons near the surface of the kidney have been investigated. However, we suppose that deep (juxtamedullary) nephrons can exhibit oscillations in their pressures and flows as well. 

As can be seen from Fig. 4.7, the kinetic tangent pinch point at the critical Damkohler number Dar = 0.166 has an important role for the topology of the maps. This is also reflected by the feasibility diagrams given in Fig. 4.8(a-c). In Fig. 4.8(c), the stable node branch at positive Damkohler numbers are collected from the singular point analyses of the reactive condenser (Fig. 4.8(a)) and the reactive reboiler... 

Figure 4.33 illustrates the PSPS and bifurcation behavior of a simple batch reactive distillation process. Qualitatively, the surface of potential singular points is shaped in the form of a hyperbola due to the boiling sequence of the involved components. Along the left-hand part of the PSPS, the stable node branch and the saddle point branch 1 coming from the water vertex, meet each other at the kinetic tangent pinch point x = (0.0246, 0.7462) at the critical Damkohler number Da = 0.414. The right-hand part of the PSPS is the saddle point branch 2, which runs from pure THF to the binary azeotrope between THF and water. 

To study the behavior of the singular points in the vicinity of the MTBE vertex, Thiel et al. [8] used a continuation method with the Damkbhler number as continuation parameter. The results computed at p = 0.8 MPa are shown in Fig. 5.17. It can be observed that a stable node branch beginning from pure MTBE in the absence of chemical reaction moves away from MTBE vertex with rising Da. As the Damkohler number Da = 1.49 X 10 is reached, the stable node branch turns into a saddle branch. This point is called the kinetic tangent pinch [9]. The saddle branch arrives at Da = 0.0 in the binary azeotropic point between MeOH and MTBE. 

At an operating pressure p below 1.0 MPa, the curves have a qualitative shape similar to those computed for the MTBE example a stable node branch moves from the TAME vertex to the kinetic tangent pinch and reaches its maximum Damkhhler number Da here. Then, the stable node branch turns into a saddle branch and runs into the binary azeotropic point between MeOH and TAME. A second saddle branch develops if an operating pressure p = 0.8 MPa at the starting point of pure lA is chosen. This saddle branch moves away from the lA vertex and reaches the line of chemical equilibrium at Da —y oo. This point is marked with a diamond in Fig. 5.20. If p is set to 1.0 MPa the stable node branch does not turn into a saddle branch that ends in the binary azeotropic point between MeOH and TAME, but which runs into pure lA. Consequently, at p = 1.0 MPa a second saddle branch, which starts at the binary azeotropic point between MeOH and TAME and arrives in the line of chemical equilibrium at Da —y oo, can be computed. In addition, in Fig. 5.20 the branch of kinetic tangent pinches is also plotted. 

In this case there exists a critical patch size. In the following we truncate the expansion in (9.39) at the first order for simplicity, p q> with q> given by (9.44). The nontrivial branches collide at a saddle-node bifurcation point or turning point that... 

The pathways in the central carbon metabolism involve TCA cycle and glyoxylate shunt, glycolysis, phosphotransferase system (PTS), gluconeogenesis, pentose phosphate pathway (PPP).The carbon flux partitions at different nodes in the central metabolism and major flux partitioning for the product of interest may occur at principal branch points. Engineering of the enzymes in these branch points of the biosynthetic pathways will direct the carbon flux toward the product of interest leading to maximal product yield. 

Branch point/chance node Referring to Fig. V/2.1.1-1, it is can be noted that there is a branching point in the event tree. This is usually designated by a circle (not shown) at the end of a branch indicating the occurrence of an unknown event. This is also called chance node. 

Branch A possible event is represented by a line segment, preceded by a branch point ot chance node, that is designated as a branch. It is a subset of the sample pace fot all possible outcomes associated with a random variable. These are represented by thick lines in Fig. V/2.1.1-1 (and Fig. V/2.1.1-2). 

In reality (e.g., the assessment of safety cases), decision problems tend be structured certain factors may feed-in to each other, and can form more complex, hierarchical belief structures. ER [2] is an extension of DS-theory that enables the aggregation of belief functions, where the factors are arranged in a hierarchical structure. The root-node represents the final decision one wishes to make. Branch nodes represent contributory factors. Branches can be given different weights, indicating the extent to which they contribute to the overall decision. Leaf-nodes represent points at which one can present ones own belief functions. ER then provides the mathematically sound basis by which to combine the belief-functions provided in the leaf-nodes, and to propagate them up to the root. 

That being so, there are obviously only three possibilities either the branches in question move away from the axis of revolution so that their distance to this last converges towards infinity or they tend towards an asymptote parallel to this axis or each one of them presents, at a finite distance from the point u of the node, a point where the tangent is parallel to this same axis. 

The dximmy variable corresponds to each branch point (node), and the probability that a branch bears further branches is given by the reactivity a. That is, the branch would not bear the next generation with the probability 1 - a. The generating function in Fig. 1 is summarized as... 

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