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Via Minimization
Published in Michael Pecht, Placement and Routing of Electronic modules, 2020
Guoqing Li, Michael Pecht, Yeun Tsun Wong
Via minimization entails finding a routing topology of a set of n two-terminal nets in a two-layer routing region such that the necessary number of vias is minimal. The problem is called a two-layer bounded region UVM problem. It is equivalent to the maximum two-independent set problem in a circle graph, where the circle corresponds to a bounded routing region with terminals on the boundary, and each chord corresponds to a two-terminal net. The circle graph introduced in [EVE71] explicitly capturesthe wire-crossing constraint that is the key constraint to observe in solving the UVM problem. Before considering the complexity of the UVM problem, several related definitions are given:
Cost Presentation
Published in Robert P. Hedden, Cost Engineering in Printed Circuit Board Manufacturing, 2020
The cost data by element in this table is often presented in circle graph form. The percentage that each element of cost is to total cost can be shown in a circle graph. Any major differences from past estimates, or similar products, can be part of an analysis presentation. Note that in any given manufacturing environment, the percentage content of each element, e.g., labor, can vary significantly depending on the type of product. The labor content percentage in a board (backplane) can be only one-third of the labor content percentage in an assembled card.
Graphs from Subgraphs
Published in N. P. Shrimali, Nita H. Shah, Recent Advancements in Graph Theory, 2020
Joseph Varghese Kureethara, Johan Kok
In graph theory, a circle graph is the intersection graph of a set of chords of a circle. That is, it is an undirected graph whose vertices can be associated with chords of a circle such that two vertices are adjacent if and only if the corresponding chords cross each other.
Enhancing pre-service mathematics teachers’ understanding of function ideas
Published in International Journal of Mathematical Education in Science and Technology, 2022
Hande Gülbağci Dede, Zuhal Yilmaz, Hatice Akkoç, David Tall
As PMTs progressed in the module implementation, they made sense of the definitional properties of functions. For instance, in the examination of students’ examples and definitions of a function task, the PMTs examined a student definition: ‘Let A→B, [function] is formed by mapping each element of the domain (set A) to at least one element of the range (set B)’. All the PMTs indicated that this definition was not correct because of the expression ‘at least one element’. Then, they corrected the definition. In another case, the PMTs discussed whether the circle graph or equation is a function. They concluded that it is not a function because there are two images for an x. Such discussions enabled the PMTs to conceptualize the definitional properties of function and correctly define functions.
Predicting the geometry of regime rivers using M5 model tree, multivariate adaptive regression splines and least square support vector regression methods
Published in International Journal of River Basin Management, 2019
Saba Shaghaghi, Hossein Bonakdari, Azadeh Gholami, Ozgur Kisi, Andrew Binns, Bahram Gharabaghi
As the evaluation indexes of Table 5 present, all the methods have acceptable prediction in models 2 and 4 but the M5 in model 2 with the presence of two parameters of Q and d50 has the best results with maximum R (0.951) and minimum MARE (0.092) which acknowledges the intensive effect of Q and d50 in estimation of regime river width. Model 4 (with all inputs) has almost the same performance as model 2 which indicates the relatively insignificant effect of τ in width prediction. Despite the slight difference between these two models, still, model 2 has more accuracy than model 4. The weak performance of all the methods is seen in model 7 (especially in LSSVR) that presents considerable scattering around the 1:1 line and no overlapping of the predicted values with the observed ones (R = 0.162). As shown in Figure 4 the predicted values of this method are almost equal and are gathered around the centre of the circle graph with almost constant radius (Figure 4(b)) and paralleled with axis y (Figure 4(a)) which exhibits the worst performance and inadequacy of this method.
Distributed adaptive control and stability verification for linear multiagent systems with heterogeneous actuator dynamics and system uncertainties
Published in International Journal of Control, 2019
K. Merve Dogan, Benjamin C. Gruenwald, Tansel Yucelen, Jonathan A. Muse, Eric A. Butcher
(Effect of leader location on a circle graph): In this example, we consider a circle graph topology with nonidentical leader locations given in Figure 13. The effect of leader location is investigated by comparing two cases. These cases now include two leaders. For the first case (Example 3.1), we place the leaders as separated by followers such that they do not directly exchange information with each other, and in the second case (Example 3.2) we place the leaders next to each other such that they do directly exchange information. The feasible region of allowable actuator bandwidth pairs is shown in Figure 14 for the two different leader locations in the circle graph. We then perform a linear matrix inequality analysis for these two configurations. In particular, the top subplot of Figure 14 presents the resulting linear matrix inequality feasible stability regions of allowable actuator bandwidth values and the bottom subplot of the same figure shows the minimum unstable values obtained numerically from the simulation for a given constant command c = 1. Once again, the change in the locations of two leaders results in different feasible stability regions. Finally, to show the performance of the proposed distributed adaptive control design, two points are selected from Figure 14 for the case of Example 3.2, which are indicated by the black circles. The numerical simulation results are shown at these actuator bandwidth values in Figures 15 and 16 with command c = 1, where similar conclusions are observed as for the case in Examples 1 and 2.