Dynamic programming curve

Chang, C. S. (1966), Discrete-Sample Curve Fitting Using Chebyshev Polynomials and the Approximate Determination of Optimal Trajectories via Dynamic Programming, IEEE Transactions on Automatic Control, Vol. AC-11, pp. 116-118. 

However, this is not a convenient form in which to pose the problem for a dynamic programming solution. The dynamic programming method starts by allotting a cost function to all parts of the phase space at the final time The a problem, however, limits the final trajectory to a single target curve in the phase space. We can say nothing about the cost function associated with other endpoints. 

Analysis of DNA structure from MD simulations is complicated because DNA is very flexible. Qualitative analysis involves animating the MD trajectories for visualization. Quantitative analysis is done by monitoring DNA conformation indices, such as backbone torsions and helicoidal properties, and morphological indices, such as groove widths. The program Curves, Dials and Windows, which is found in MD Toolchest, provides a means of analyzing DNA dynamics in an exhaustive fashion. Several articles describing the use of this tool have appeared in the literature. ... 

With a new software program it is possible to measure the Texture Constant" of pectins. This Texture Constant K is calculated by the ratio of the maximum force during the time interval of the measurement and the measured area below the force-time curve. The resulting constants K correlate well with the dynamic Weissenberg number of oscillating measurements carried through with the same pectin gels. 

Revise the program to include nth-order reaction. Evaluate the conversion for first-order reaction from the dynamic model. Compare this with the conversion calculated from the E curve. Show by simulation that the two are equal only if the reaction is first order. 

The spectral response curves of various types of photometers are usually fairly flat or smoothly rounded and do not normally create programming problems, but they must be known so that you do not stray beyond the usable dynamic range. 

Figure 12.4 shows a typical DSC curve for the study of an endothermic process by the dynamic method. The heating program is started at the time fst, after an isothermal baseline has been recorded, and ends at the time fend- The reaction or... 

Generation of Master Curves. Modulus and loss factor data were processed into a reduced frequency plot in the following manner modulus curves at different temperatures were shifted along the frequency axis until they partially overlapped to obtain a best fit minimizing the sum of the squares of a second order equation (in log modulus) between two sets of modulus data at different temperatures. This procedure was completely automated by a computer program. The modulus was chosen to be shifted rather than the loss factor because the modulus is measured more accurately and has less scatter than the loss factor. The final result is a constant temperature plot or master curve over a wider range of frequency than actually measured. Master curves showing the overlap of the shifted data points will not be presented here, but a typical one is found in another chapter of this book (Dlubac, J. J. et al., "Comparison of the Complex Dynamic Modulus as Measured by Three Apparatus"). 

The temperature of the furnace is continuously varied by a temperature controller, and the outputs from the balance and from the sample temperature thermocouple are logged by a computer fitted with a 16-bit dynamic range data acquisition board. The signals are processed via a program derived using a combination of Virtual Basic and C++ and the derivative of the TGA curve is used to control the heating program via the temperature controller. 

The relevant potential energy curves and nuclear dynamics of Nal after photoexcitation are shown in Fig. 1. The calculations were performed using the program package WavePacket [35]. 

Fig. 4.19. Bifurcation diagram for the reduced system (4.4b,c) as a function of parameter a. On the ordinate, the steady-state concentration of /3 in that system is shown, as well as the maximum value reached by jS in the course of oscillations. The diagrcuns are obtained numerically by means of the program AUTO (Doedel, 1981), for decreasing values of parameter (in s" ) (a) 10 (b) 7.78 (c) 4.5 (d) 2.7 (e) 1.7 (f) 1.2. Solid or dashed lines denote, respectively, stable and unstable (steady or periodic) regimes. The arrowed trajectories represent, schematically, the dynamic behaviour of the full, three-variable system (4.1a-c). The particular values of a relate to the limit points of the hysteresis curve (au, L2), the Hopf bifurcation points ( hi, h2)> and the points corresponding to the appearance of homoclinic orbits (an, a ) (Decroly Goldbeter, 1987).

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