Constrained and unconstrained

Figure 5.23 Composite modulus obtained using constrained and unconstrained boundary conditions...

There are two general types of optimization problem constrained and unconstrained. Constraints are restrictions placed on the system by physical limitations or perhaps by simple practicality (e.g., economic considerations). In unconstrained optimization problems there are no restrictions. For a given pharmaceutical system one might wish to make the hardest tablet possible. The constrained problem, on the other hand, would be stated make the hardest tablet possible, but it must disintegrate in less than 15 minutes. 

Thermodynamic Integration Using Constrained and Unconstrained Dynamics... 

Before we derive the appropriate expressions to calculate cL4/d from constrained simulations, we note an important difference between sampling in constrained and unconstrained simulations. There are two ways to gather statistics at (x) = . In unconstrained simulations, the positions are sampled according to exp —iiU while the momenta are sampled according to exp —j3K. If a constraint force is applied to keep fixed the positions are sampled according to A( (x) — x) exp —iiU. The momenta, however, are sampled according to a more complex statistical ensemble. Recall that... 

Crosslinked polymer networks formed from multifunctional acrylates are completely insoluble. Consequently, solid-state nuclear magnetic resonance (NMR) spectroscopy becomes an attractive method to determine the degree of crosslinking of such polymers (1-4). Solid-state NMR spectroscopy has been used to study the homopolymerization kinetics of various diacrylates and to distinguish between constrained and unconstrained, or unreacted double bonds in polymers (5,6). Solid-state NMR techniques can also be used to determine the domain sizes of different polymer phases and to determine the presence of microgels within a poly multiacrylate sample (7). The results of solid-state NMR experiments have also been correlated to dynamic mechanical analysis measurements of the glass transition (1,8,9) of various polydiacrylates. 

Figure 3 (a) UV absorption spectra of cisoid/anti-aWemai r g oligosilanes with 4-10 Si atoms, (b) Comparison of UV absorption maxima of constrained and unconstrained oligosilanes.13 Reprinted with permission from Tsuji, H. Terada, M. Toshimitsu, A. Tamao, K. J. Am. Chem. Soc. 2003, 125, 25, 7486-7487. 2003 American Chemical Society. 

A. In general, the main conclusions are same for the Zr-P constrained and unconstrained calculations, and therefore, unconstrained results will be discussed only for a few specific cases. 

Kinetic gelation models [178] have been used to determine, within experimental error, the fraction of constrained and unconstrained double bonds over a wide range of conversions in the polymerization of ethylene glycol dimethacrylate. The amount of unconstrained and constrained functional groups was determined experimentally by solid state nuclear magnetic resonance spectroscopy. The rules for determining constraint in the model were that all pendant double bonds and all monomers in pools of six or less are constrained. Monomers in pools of seven or more are assumed to be unconstrained. Whether a site is constrained or not does not affect the reactivity only the analysis of the model is affected by the rules defining constraint. 

In this paper, the problem of product non-uniformity is placed in perspective as we consider the simple case of flow between parallel plates with a flat velocity profile accompanied by a first order reaction. The system equations involved are non-dimensionalized and four parameters identified. A systematic attempt is made to explore a host of constrained and unconstrained wall temperature profiles that alleviate the problem of product non-uniformity at the reactor outlet. Work is already underway to study similar problems in complex polymerization systems, and some preliminary results are reported here. 

Liu Z.-K. and Agren J. (1995) Thermodynamics of constrained and unconstrained equilibrium systems and their phase rules. J. Phase Equil. 16, 30-35. 

The Hausdorff distance was applied in chemistry in various chirality studies. The Hausdorff distance was used to measure the deviation of a chiral nuclear arrangement from some arbitrary reference arrangement, as proposed by Rassat. Mislow and co-workers used the Hausdorff distance between the object and its optimally overlapping mirror image to provide a chirality measure of the second type. - Using this Hausdorff distance criterion, Buda and Mislow determined the most chiral constrained and unconstrained simplexes in two and three dimensions, that is, the most chiral triangles and tetrahedra." ... 

TOR 93] TORAYA H., Position constrained and unconstrained powder pattern decomposition methods , in YOUNG R.A. (ed.), The Rietveld method, lUCr monographs on crystallography, no. 5, Oxford University Press, New York, 1993. 

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