Evolution of the Weights

A suite of initial weights is required to initiate the net before learning starts. They can be selected after preliminary studies although, usually, they are assigned a random value (between —1 and + 1). Two possibilities exist to back-propagate the error  

Continuous update or pattern mode [37], where changes in the weights are made immediately after a value is predicted for each standard (pattern). This means that after each iteration or epoch all weights changed m times, being m the number of standards (patterns) in the calibration set. 

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Fig. 3.4. Evolution of the weights i],(N) and the histograms fi(N) in a grand-canonical implementation of the multicanonical method for the Lennard-lones fluid al V 125. The temperature is T = 1.2 and the initial chemical potential is /./ = —3.7. The weights are updated after each 10-million-step interval, and the numbers indicate the iteration number. The second peak in the weights at large particle numbers indicates that the initial chemical potential is close to its value at coexistence...

Instead of solving evolution equations for the moments, the evolution of the weights and nodes in the quadrature approximation can be directly tracked (Marchisio Fox, 2005 McGraw Wright, 2003). The evolution equations for weights and nodes can be derived by formally substituting the delta-function representation of the NDF into Eq. (7.3). If the weights and nodes are continuous functions of time, this procedure yields... 

Figure 7.6. Evolution of the weights and nodes for a continuous homogeneous system undergoing aggregation and breakage with the QMOM and DQMOM.

Figure 16.3 shows the evolution of the weight cr5 stalllnities of the two sets of crystallites at 3 different crystallization temperatures. As shown by Ruland [20], such reflection integrals that are complete in reciprocal space are proportional to the weight crystallinity of the perfect crystallites that produce the reflections. Latency periods between the quench and the start of the crystallization, as well as crystallization velocities of the two kind of crystallites, can be extracted from these data. Finally, conclusions concerning the crystallization mechanisms can be drawn [18]. 

Fig. 30. Evolution of the glass transition temperature of polychloroprene-aromatic hydrocarbon resin blends (17 wt% resin content) as a function of the number average molecular weight of the hydrocarbon resin. Tg values obtained from DSC experiments.

It was found [1] that the values of and a, obtained in minimizing the error of fitting experimental conversion-time data, satisfactorily described the temporal evolutions of the molecular weight averages. Also, the model performed better in the description of the experimental data when a value of 3 = 1/2 was used. 

The Hamiltonian equations for P and Q, and the variational condition for Xn provide together a formally exact set of coupled equations whose solution gives the time-evolution of the electronic states driven by nuclear motions. The present coupled equations generalize the ones previously presented in reference (21) to allow now for statistical weights in the quantal potential, which is the same for ail the initially populated states n. 

This paper presents the physical mechanism and the structure of a comprehensive dynamic Emulsion Polymerization Model (EPM). EPM combines the theory of coagulative nucleation of homogeneously nucleated precursors with detailed species material and energy balances to calculate the time evolution of the concentration, size, and colloidal characteristics of latex particles, the monomer conversions, the copolymer composition, and molecular weight in an emulsion system. The capabilities of EPM are demonstrated by comparisons of its predictions with experimental data from the literature covering styrene and styrene/methyl methacrylate polymerizations. EPM can successfully simulate continuous and batch reactors over a wide range of initiator and added surfactant concentrations. 

The living nature of these polymerizations has also been rigorously demonstrated by a variety of techniques and shown to hold true even at high conversion. For example, the evolution of molecular weight for the polymerization of styrene by the alkoxyamine, 14, is linear up to 90% conversion (Figure 2.2). 

Figure 2.2 Evolution of molecular weight, Mn, with percent conversion for the polymerization of styrene (250 equivs) in the presence of 14 O.Oequivs) at 123°C for8h...

In order to prepare absolute ether the commercial product (1-2 L) is first dried for one to two weeks over calcium chloride (weight about 10 per cent of the weight of the ether taken) and poured rapidly through a folded filter into a dry flask into which sodium wire is driven from a press. Until the evolution of hydrogen ceases, a calcium chloride tube, drawn out to a fine capillary so as to reduce the... 

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