Euler diagrams

By Venn-Euler diagrams residual sets can easily be understood (see Fig. 18) ... 

Figure 10.1 Schematic diagram of the sequential solution of model and sensitivity equations. The order is shown for a three parameter problem. Steps l, 5 and 9 involve iterative solution that requires a matrix inversion at each iteration of the fully implicit Euler s method. All other steps (i.e., the integration of the sensitivity equations) involve only one matrix multiplication each.

Figure 7. Topological fluctuations of the lamellar phase at different points of the phase diagram, (a) Single fusion between the lamellae by a passage (this configuration is close to the topological disorder line), (b) Configuration close to the transition to the disordered microemulsion phase the Euler characteristic is large and negative.

Figure 14. The phase diagram of the gradient copolymer melt with the distribution functions g(x) = l — tanh(ciit(x —fo)) shown in the insert of this figure for ci = 3,/o = 0.5 (solid line), and/o — 0.3 (dashed line), x gives the position of ith monomer from the end of the chain in the units of the linear chain length. % is the Flory-Huggins interaction parameter, N is a polymerization index, and/ is the composition (/ = J0 g(x) dx). The Euler characteristic of the isotropic phase (I) is zero, and that of the hexagonal phase (H) is zero. For the bcc phase (B), XEuier = 4 per unit cell for the double gyroid phase (G), XEuier = -16 per unit cell and for the lamellar phases (LAM), XEuier = 0.

The phase diagram in Fig. 14 can be analyzed by using the Euler characteristic. The disordered phase contains no surfaces, and therefore the Euler characteristic is zero. The bcc phase [Fig. 13(b)] within the two-shell approximation is expressed as... 

Since the enrichment factor discriminates only between active and inactive compounds, we further investigated which active compounds were retrieved by the two descriptors among the top-ranking ones. Figure 3.3 depicts this for the first five percentiles of three dataset by means of Euler-Venn diagrams. It is no-... 

Fig. 3.3 Elements of the Euler-Venn diagrams represent compounds that were found among the first 5% of the similarity-ranked list that results from retrospective screening with the (a) COX2, (b) HIV protease and (c) MMP datasets of the COBRA dataset. The Manhattan distance...

The mutual complementation of the different methods was examined in more detail for four selected molecules rofecoxib (COX-2), celecoxib (COX-2), indinavir (HIV protease), and lanepitant (neurokinin receptor). The results are shown in the form of Euler-Venn diagrams in Fig. 3.7. Apparently the methods complement each other. Each method was able to retrieve actives which were not found by the other methods. Interestingly, the performance of the different descriptors varied significantly within one class of ligands (compare, e.g., rofecoxib and celecoxib). 

In doing so, one makes use of the analogy between the number of elements of each level of this diagram and Euler s theorem for poiyhedra(27). 

Fig. 1. Euler-Venn diagram of the occurrence of patients with maculopapular rashes. Explanation of symbols A = ampicilHn, B = other penicillins, C=cotrimoxazole or sulphon-amides alone. Patients treated with drug A but not with drugs B and C are represented by ABC. Patients treated with drugs excluding drug groups A, B and C are represented by the dotted area. Each patient is represented only in one area of the diagram, even if more than one maculopapular rash occurred...

The methods of Euler-Venn diagrams or weighted attributions can be used only with a large number of observations. For both methods the result can be expressed either as a frequency in terms of the percentage of patients exposed or, as... 

Newton-Euler Equations (Newton s Second Law) The equation of motion is derived using free-body diagrams (FBDs) for each rigid body. The FBDs contain kinematical (acceleration, angular acceleration, angular velocity) and dynamical (extemal/reaction forces, moments) variables. The Newton-Euler equations consist of two parts, the translational part and the rotational part. The translational part (for the ith body) is... 

Different methods are available to derive the dynamical equations of motion for a motor task. In the Newton-Euler method (Pandy and Berme, 1988), free-body diagrams are eonstructed to show the external forces and torques acting on each body segment. The relationships between forces and... 

The kinematics table, shown in Table 7.1, introduces a method (referred to within this chapter as the table method) for efficiently managing the mathematics involved in analyzing multibody and multiple coordinate system problems, and can be used in either the Lagrangian or the Newton-Euler approach. A schematic diagram, which defines an inertial or body-fixed coordinate system, must accompany every kinematics table. The purpose of the schematic is to identify the point on the body at which the absolute velocity and acceleration is to be determined. The corresponding schematic for Table 7.1 is shown in Fig. 7.7. 

For the phase diagram depicted in Figure 1.6, we can evaluate the values of the surface tension between the coexisting phases for a planar interface along the z-axis by solving the Euler-Lagrange equations associated with the following equilibrium conditions ... 

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