Piecewise linear

Fig. 12.23 Two simple scoring functions used in docking. On the left is the basic scoring scheme used by the DOCK program [Desjarlais et al. 1988], On the right is the piecewise linear potential [Cclhaar et al. 1995],...

The Qjj can be different for each layer of the laminate, so the stress variation through the laminate thickness is not necessarily linear, even though the strain variation is linear, instead, typical strain and stress variations are shown in Figure 4-5 where the stresses are piecewise linear (i.e., linear in each layer, but discontinuous at boundaries between laminae). 

Note that the lamina failure criterion was not mentioned explicitly in the discussion of Figure 4-36. The entire procedure for strength analysis is independent of the actual lamina failure criterion, but the results of the procedure, the maximum loads and deformations, do depend on the specific lamina failure criterion. Also, the load-deformation behavior is piecewise linear because of the restriction to linear elastic behavior of each lamina. The laminate behavior would be piecewise nonlinear if the laminae behaved in a nonlinear elastic manner. At any rate, the overall behavior of the laminate is nonlinear if one or more laminae fail prior to gross failure of the laminate. In Section 2.9, the Tsai-Hill lamina failure criterion was determined to be the best practical representation of failure... 

Despite bearing no direct relation to any physical dynamical system, the onedimensional discrete-time piecewise linear Bernoulli Shift map nonetheless displays many of the key mechanisms leading to deterministic chaos. The map is defined by (see figure 4.2) ... 

Finally, interest in these questions came from a new line of research, namely, the theory of automatic control systems. Here, however, contrary to the analytical theories that are summarized in this review, these new piecewise analytic or piecewise linear phenomena in control systems are nonanalytic by their very essence. They open an entirely new field, which is still in an early stage of development. 

These phenomena are being actively studied at the present time, and constitute a new chapter in the theory of oscillations that is known as piecewise linear oscillations. There exists already a considerable literature on this subject in the theory of automatic control systems11-34 but the situation is far from being definitely settled. One can expect that these studies will eventually add another body of knowledge to the theory of oscillations, that will be concerned with nonanalytic oscillatory phenomena. 

Optimal flow method, 261 Optimization non-constrained, 286 of functionals, 305 Ordinary value, 338 Orthogonalization, Schmidt," 65 Osaki, S., 664 Oscillation hysteresis, 342 Oscillations autoperiodic, 372 discontinuous theory, 385 heteroperiodic, 372 piecewise linear, 390 relaxation asymptotic theory, 388 relaxation, 383 Oscillatory circuit, 380 "Out field, 648 existence of, 723... 

Isothermal and adiabatic heat transfer conditions can be obtained with different values of the Biot number. The wall temperature, 0w, is assumed to be a piecewise linear function of the axial position and is treated as a known quantity based on experimental evidence. 

Because the term r(CA T) is exponentially dependent on T and can be nonlinear as well, a numerical solution or piecewise linearization must be used. To simplify the numerical manipulations, equations in Table IX are normalized by = z/L, r = ut/L, and jc = 1 - C,/(C,)0, where i is normally S02. y also is a normalized quantity. The Peclet numbers for mass and heat are written PeM = 2Rpu/D) and PeH = 2Rpcpul t for a spherical particle. They are also written in terms of bed length as Bodenstein numbers. It is... 

Triangular (Figure 8.7), piecewise linear (Figure 8.8), or trapezoidal (Figure 8.9) functions are commonly used as membership functions because they are easily prepared and computationally fast. 

Thus, to obtain the timescales d and Q( ) we need to find the solution Y(x, s) of Eq. (5.78) at appropriate boundary conditions and evaluate limits (5.82) and (5.83) for y - > 0. It is precisely that way by which the various timescales for piecewise-linear and piecewise-parabolic potential profiles were derived in Refs. 32 and 33. 

When the second stage decisions are real-valued variables, the value function Qu(x) is piecewise-linear and convex in x. However, when some of the second stage variables are integer-valued, the convexity property is lost. The value function Qafx) is in general non-convex and non-differentiable in x. The latter property prohibits the use of gradient-based search methods for solving (MASTER). 

Let x be the value of x at the start of PSLP iteration k. A piecewise linear function that closely approximates P(x) for x near xk is... 

The trust region problem is to choose Ax to minimize PI in (8.54) subject to the trust region bounds (8.55) and (8.56). As discussed in Section (8.4), this piecewise linear problem can be transformed into an LP by introducing deviation variables p, and The absolute value terms become (p, + ,) and their arguments are set equal to Pi — nt. The equivalent LP is... 

Of course, aredk can be negative because P need not be reduced if the step bounds s are too large. To decide whether s should be increased, decreased, or left the same, we compare aredk with the reduction predicted by the piecewise linear model or approximation to P, Pl. This predicted reduction is... 

A piecewise linear function interpolating a given set of (yi9 xt) values. 

In the case of a piecewise linear response, for each of the segments the sensitivities can be simplified as follows ... 

Since the turnover curve is concave7 and the piecewise linear turnover gradients decrease monotonically, no integer variables are required to decide, which partial quantity section is filled first. The objective function to maximize turnover will ensure to fill the partial quantity sections from left to right8. 

Are the model results accurate enough considering the piecewise linear approximation methods developed Here, piecewise linear approximation accuracy is modified with additional points added in the approximation and result accuracy as well as model run time compared. 

A piecewise linear turnover approximation supports effective and accurate decision making on sales turnover based on price-quantity functions and elasticity as an alternative to exact quadratic optimization. 

The piecewise linear curve representing the resistance function shown in Figure 7.1 is an approximation made to simplify the analysis and design process. This approximation ignores some nonlinear effects such as ... 

While the piecewise linear regions and energy maximum around +4 valence of Figures 9—13 are consistent with ligand-field effects, it is important to bear in mind that LFSE cannot by itself predict the energy difference between octahedral and tetrahedral Mn. 

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