Piece-wise linear

The Chate and Mamieville approximation schemes consist of finding sets of increasingly finer piece-wise linear step-function approximations of the continuous... 

Another important feature of the case study scenario and the resulting cost model is inventory control. High seasonality effects and long campaign durations necessitate considerable build-up of stocks. To avoid an unbalanced build-up of stock, soft constraints for safety stock and maximum stock levels are used. To achieve an even better inventory leveling across products and locations, piece-wise linear cost functions for falling below safety stock as well as for exceeding maximum stock levels are employed. 

Add pqp number of additional partial quantity points applied in piece-wise linear turnover approximation. 

In one dimension, the cell estimate is piece-wise linear and connects the node-point estimates. 

In this section we will demonstrate the use of a Zener diode, as well as the piece-wise linear (PWL) waveform voltage source. The PWL source can be used to create an arbitrary voltage waveform that is connected by straight lines between voltage points. Wire the circuit shown ... 

V Zener diode Independent piece-wise linear voltage source ... 

An alternative way of modeling Ci( i) is via a set of piece- wise linear functions that correspond to fixed charge cost models of the form ... 

Remark 6 If a phase change takes place during the temperature variation of a process stream, then the temperature partitioning should be modified so as to contain a component of the temperature variation in the first phase, a component for the phase change, and a component for the temperature variation in the second phase, in the form of a piece-wise linear approximation. The three components are treated subsequently as if they are separate steams, and the same analysis, that is described in the following sections, applies. 

The maximum cross-pinch heat flow, Rp, can also be calculated analytically. In general, Rp is a piece-wise linear function of TIAT and HRAT and exhibits transition points for pinch points that take place in the following two cases ... 

Sacks, E. P., Automatic Qualitative Analysis of Ordinary Differential Equations Using Piece-Wise Linear Approximations. Ph.D. thesis, MIT, Cambridge, Mass., 1988. 

These assumptions are really quite mild the / need only be increasing and sufficiently smooth. It is not even required that f, be bounded on IR. From a biological perspective, of course, the model loses relevance for really large values of S. In addition to the Monod functions, other functions that have been suggested include the exponential kinetics AM(1—exp(—S log 2/a)), hyperbolic kinetics m tanh(S log 3/2a), and piece-wise linear kinetics given by mS/2a for S<2a and by w for S > 2a. The piecewise linear kinetics fails to satisfy the strict monotonocity of (iii) and fails to satisfy (iv) at one point, but these assumptions could be weakened so as to include this case. 

In the present paper we are seeking for simple theoretical predictions of the relationship between crack length and tearing modulus. Since it is reasonable to assume that the material obeys the piece wise linear yield condition developed by R. H. Lance and D. N. Robinson [6]. Although the yield surface strongly depends on the orientation of fibers we confine our attention to the case of circumferential arrangement of fibers in the matrix material, provided ka is the yield stress in the circumferential direction. The yield condition for circumferentially reinforced cylindrical tubes is presented in Fig. 3. Here... 

Theory of complex variables [1, 2] is used in order to connect current density and electric potential with one holomorphic function. The real part of it represents electric potential and the derivative is related with current density. The domain is divided into smaller subdomains where the holomorphic function is approximated with quadratic function and its derivative with piece wise linear function. These approximating functions obey continuity across subdomain interfaces. 

For function H(z), related to current density is approximating function required to be piece-wise linear function ... 

Next, we construct a finite dimensional subspace f2h consisting of piece-wise linear functions. We define subintervals of length Az = Zj+ — Zj, j = 1,2,..., K. As parameters to describe how the function change over the subintervals, we choose the basis functions as the set of triangular functions defined as ... 

Figure 43 Piece-wise linear temperature profile.

The piecewise linear map defined by eqns (4.5) thus allows us to explain the transitions between different simple or complex patterns of bursting as a function of the variation of parameters whose significance can be related to the properties of the biochemical model from which the map originates. Parameter a, for example, is linked to the quantity of substrate consumed by the production of a peak of product P,. An increase in the maximum rate of the reaction catalysed by enzyme Ej should therefore correspond to an increase in parameter a of the piece-wise linear map. Likewise, a rise in the rate constant results in a decrease in the amount of product Pj within the system enzyme Ej, activated by P2, should therefore become less active as increases. The amount of Pj, the substrate for enzyme Ej, will then tend to increase, owing to its diminished consumption in the second enzyme reaction. As enzyme Ei is activated by Pj, the increased level of this product raises the rate of enzyme Ei, which results in an increased amount of substrate consumed during synthesis of a peak of Pj. Thus we can see how an increase in the rate constant k in the enzyme model can also be associated with a larger value of parameter a in the one-dimensional map studied for bursting. 

Fig. 10 Plots of the 2-D potential calculated by taking piece-wise linear functions for the shift ratchet shown in Fig. 9b with amplitude of 13 along the mechanical coordinate, and transition state saddlepoint) energies of (9i) = = 8.5 and S (d2) = = 0. Two periods in each...

Figure 10.10 Temperature-enthalpy diagram for the hot stream, S4, showing simulation results (solid line) and piece-wise linear approximation (dashed line).

Both linear and piece-wise linear mappings can be derived in a systematic way, and methods exist to optimize given criteria within a constraint on execution time (for latency-limited applications), the period (for throughput-limited processing), or the number of processors (for area-bound applications). 

Piece-wise linear moment-rotation curves for the five connections in Table 1 are shown in Figures 4 to 8. The results plotted are for the connection in each test that failed first. The term Left or Right is used to distinguish between the two connections in each specimen (reference 10). In the figures the response plotted is that recorded at least 5 minutes after the application of the moment or rotation increment. For a moment,... 

Marta Eso, Soumyadip Ghosh, Jayant R Kalagnanam, and Laszlo Ladanyi. Bid evaluation in procurement auctions with piece-wise linear supply curves. Technical Report RC 22219, IBM TJ Watson Research Center, 2001. 

Production cost Let C,(xJ represent the nonlinear cost of producing Xf units in period t. For example, C,(x,) could be a convex, piece-wise linear, cost function with per unit cost increasing depending on whether the units are produced in regular time, over time, or obtained through outsourcing. Fixed cost can also be included as part of the production cost. 

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