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Preliminary Mathematics
Published in P.K. Jayasree, K Balan, V Rani, Practical Civil Engineering, 2021
P.K. Jayasree, K Balan, V Rani
A sector is the space included between an arc and two radii drawn to the center. ABCD in Figure 3.3 is a sector. If r is the radius of the sector and ϕ is the central angle of the sector in degrees, LengthofarcBCD,L=πrϕ180Area of the sector=ϕ360×πr2=12×r×L
The circle and its properties
Published in John Bird, Basic Engineering Mathematics, 2017
The angle at the centre of a circle, subtended by an arc, is double the angle at the circumference subtended by the same arc. With reference to Fig. 27.3, AngleAOC=2×angleABC $$ \begin{aligned} \boldsymbol{{Angle}\ {A\!O\!C} = 2{\,\times \,}\mathbf{angle}\ A\!B\!C} \end{aligned} $$
Geometry and trigonometry
Published in Allan Bonnick, Automotive Science and Mathematics, 2008
Figure 4.13 (a–c) shows features of circles that are used here and elsewhere in this book. Diameter: the length of a straight line drawn from one side of a circle to the other and passing through the centre.Radius: a straight line running from the centre of a circle to any point on the circumference.Circumference: the length of the boundary of a circle.Arc: the length of a continuous section of the circumference.Chord: a straight line joining two points on the circumference of a circle.Sector: a part of a circle bounded by two radii and an arc.Segment: the region of a circle formed by an arc and a chord.Tangent: A straight line drawn at right angles to the radius of a circle.
Dynamic reliability assessment of flare systems by combining fault tree analysis and Bayesian networks
Published in Energy Sources, Part A: Recovery, Utilization, and Environmental Effects, 2023
Sohag Kabir, Mohammed Taleb-Berrouane, Yiannis Papadopoulos
BNs in their classical form cannot model the changes in variables over time, i.e., they perform analysis for a fixed time. DBNs (Neapolitan 2004) are extensions of classical BNs, which provides a flexible structure and distinctive modeling mechanism for explicit modeling of the temporal progression of a set of variables over time. In a DBN, the overall timeline is divided into several discrete time slices. This permits a node at the time interval to be conditionally dependent on both its parents in the same interval and its own and its parents’ states in the previous interval (Khakzad 2015). Figure 5 shows a DBN model of the BN of Figure 4 over a discretized timeline . The black arcs within the same time interval are the intra-interval arcs, whereas the red-dashed arcs connecting the nodes in the consecutive time intervals are the inter-interval arcs. According to Figure 5, the conditional probability of the copy of node A at time slice would be .
A free discontinuity model for smectic thin films
Published in Liquid Crystals, 2023
J. M. Ball, G. Canevari, B. Stroffolini
Figure 7 shows a numerical approximation of the minimiser of (17) within this restricted class, for and different values of , . The simulation is based on a MATLAB code, and the details are provided in Appendix A. The numerical method we use is not guaranteed to converge to a global minimum of the problem. However, we repeated the simulations using several different initial guesses for , including random ones, and obtained qualitatively similar profiles for the jump set. The numerically found jump set very nearly agrees with a union of two parabolic arcs that bisect the smectic layers at each point, given explicitly as
Effects of membrane reference state on shape memory of a red blood cell
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2019
Zhe Gou, Xiaodong Ruan, Feng Huang, Xin Fu
The comparison of Lz is shown in Figure 8a. It can be seen that there is no plateau as Figure 4, and the values of Lz just increase and then decrease. When , the value of Lz even show some oscillations before reaching stable. Another significant difference between case 1 and 8 is the trace of the material point as shown in Figure 8b. The circles represent the starting points, and the dots the end points. The material points first slant to the left and up direction, indicating the compression phase. Then the material point for case 1 moves in a reverse curve to the right at or . However, the material point for case 8 slants down in a straight line at , and arcs towards the left at .