Internal nodes

The airflow network (Fig. 11.41 is composed of nodes, interconnected by links, representing individual airflow paths. Internal nodes represent the individual zones of the building, and external nodes represent faqade locations, related to a specific set of wind pressure coefficients. Each link represents a specific airflow conductance type. 

Figure 2. Same as Fig. 1, except spectra originate from the A level. Note the nodes in the progressions, originating from the two internal nodes in the 2 <3 state, near 04 = 12, 18. 

Here, we are using a second order approximation for the second derivative using the correct info-travel concept for the conduction term. This equation comes from the energy balance within the domain, thus it will be used for the internal nodes n = 2,3 and 4. The boundary condition for the first node is the temperature at the wall to which the fin is attached to... 

Figure 10.19 Typical BEM mesh and internal nodes location.

Having constructed the microscopic mesh, we specify the microscopic problem based on the macroscopic nodal displacements. The displacements of the elemental boundaries are given by the macroscopic solution (although the internal microscopic scale displacements are not necessarily affine). The microscopic problem is to find node positions and segment lengths such that the boundary nodes are as specified by the macroscopic displacements and the internal nodes experience no net force. The boundary nodes have displacement specified and are subjected to a non-zero net force. The next step in the solution process is to convert those forces into the macroscopic stress tensor. 

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/.,...

The crossing of two curves bounding adjacent elements form nodes. The values of the field variables at the nodes form the desired solution. Common shapes of finite elements are triangular, rectangular, and quadrilateral in two-dimensional problems, and rectangular, prismatic, and tetrahedral in three-dimensional problems. Within each element, an interpolation function for the variable is assumed. These assumed functions, called trial functions or field variable models, are relatively simple functions such as truncated polynomials. The number of terms (coefficients) in the polynomial selected to represent the unknown function must at least equal the degrees of freedom associated with the element. For example, in a simple one-dimensional case [Fig. 15.6(a)], we have two degrees of freedom, Pt and Pj, for a field variable P(x) in element e. Additional conditions are needed for more terms (e.g., derivatives at nodes i and j or additional internal nodes). 

The set of internal nodes used in the parse trees is known as the function set F, where F = /i,/2, All functions have an arity (the number of arguments) greater than 1. The set of terminal (leaf) nodes in the parse tree is, predictably,... 

MOGP is based on the more traditional optimisation method genetic programming (GP), which is a type of GA [53,54]. The main difference between GP and a GA is in the chromosome representation in a GA an individual is usually represented by a fixed-length linear string, whereas in GP individuals are represented by treelike structures hence, they can vary in shape and size as the population undergoes evolution. The internal nodes of the tree, typically represent mathematical operators, and the terminal nodes, typically represent variables and constant values thus, the chromosome can represent a mathematical expression as shown in Fig. 4. 

Fig. 4 In genetic programming (GP), a chromosome is a tree structure and can be used to represent a mathematical expression where the internal nodes are mathematical operators and the terminal nodes are variable or constant values...

In MoQSAR, the internal nodes include the sum, quadratic and cubic power operators and the terminal nodes consist of the molecular descriptors available for the dataset. A chromosome is translated into a QSAR in two steps (1) the expression encoded in a chromosome is extracted to determine the descriptors that will be used in the QSAR model (2) optimum values for the coefficients and the intercept are calculated using the least-squares method. 

In both schemes the subsets and the partitions are obtained by imagining the receivers as the leaves in a rooted full binary tree with N leaves (assume that iV is a power of 2). Such a tree contains 2N — 1 nodes (leaves plus internal nodes) and for any 1 < i < 2N — 1 we assume that Uj is a node in the tree. The systems differ in the collections of subsets they consider. 

The collection of subsets Si,..., S-uj in our first scheme corresponds to all complete subtrees in the full binary tree with N leaves. For any node Vi in the full binary tree (either an internal node or a leaf, 2N — 1 altogether) let the subset Si be the collection of receivers u that correspond to the leaves of the subtree rooted at node -y,. In other words, u E Si iff Vi is an ancestor of u. The key assignment method is simple assign an independent and random key Li to every node Vi in the complete tree. Provide every receiver u with the log AT -I-1 keys associated with the nodes along the path from the root to leaf u. 

Figure I. Two difTetent meshes used in our MC-FEM calculations. Panels (a) and (b) display the fine mesh used to solve the elastic problem, while the coarser mesh among which concentration changes are attempted (for example, involving the nodes indicated with filled circles in panel (d)) is shown in panels (c) and (d). Panels (a) and (c) highlight nodes at the island surface, while in (b) and (d) a cross-cut shows also some internal nodes. The Si substrate below the islands is not shown. 

If a FEM is to be used instead, we need to replace the global basis functions with element basis functions. In this case, we need to discretize the domain. In general we may discretize z = [0, L] into K + I elements with K internal node points in addition to the boundaries. 

If there are three species, we can use a star topology in which all three species are directly connected to the common ancestor without any internal nodes. Let the three species be numbered 1, 2, and 3, and the common ancestor be A. Knowing their pairwise proximities q12, q23, 13, we can calculate each species proximity to the ancestor using... 

Finally, it should be noted that the time that the PhyloGibbs program takes to parse the tree rises sharply (exponentially) with the number of internal nodes. Therefore, it will improve running time, and generally not greatly hurt the results, to keep the number of internal nodes small, by removing internal nodes that are reasonably proximate (e.g., proximities greater than 0.8 or 0.9 to then-parents). 

In the simulation only the pressure drop (or flow rate) between inlet and outlet of the system is externally fixed and controlled. All the pressures and speeds of flow in the channels that comprise the network are calculated on the basis of the positions of the droplets, via the Kirchofifs laws. The flow of droplets, and in particular the acts of traversing of the droplets between the different channels in the network introduce step changes in these quantities. These occur when (i) a new droplet enters the system, (ii) a droplet leaves the system, or (iii) a droplet traverses through an internal node. In the last case, when a droplet arrives at an internal junction, the droplet enters the channel with the largest momentary inflow (calculated before the act of the droplet entering the channel). The numerical scheme that executes the algorithm outlined above can be found in Refs. [41, 43]. 

Figure 1.15 Dispersed Fluorescence Spectrum of AgAu. The AgAu A — X1 E+ DF spectrum contains long v —> v" vibrational progressions because the bond length in the A-state is much longer than in the X-state. The nodal structures of the v = 0, 2, and 3 vibrational states are displayed as intensity minima in the DF spectra. Since the vibrational quantum number is equal to the number of internal nodes in the wavefunction, a vibrational progression in an absorption or DF spectrum often reveals the absolute assignment of the initial vibrational state (from Fabbi, el al., 2001).

Collapsing any number of nonterminal (internal) nodes is termed pruning. 

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