Equations—continued displayed

Equation 4 is a statement of the equilibrium partitioning at the reservoir-pore liquid interface with 2 being the partition coefficient. Equation 5 indicates that the solute concentration is expected to be finite at the bottom of the reservoir. Equation 6 displays the continuity of the agent flux across the reservoir-pore interface, whereas Equation 7 accounts for the material leaving the membrane and entering the surrounding water bath. The quantity 2 is the membrane area at the outer wall (cm ). 

It is likely that there will always be a distinction between the way CAD/CAM is used in mechanical design and the way it is used in the chemical process industry. Most of the computations requited in mechanical design involve systems of linear or lineatizable equations, usually describing forces and positions. The calculations requited to model molecular motion or to describe the sequence of unit operations in a process flow sheet are often highly nonlinear and involve systems of mixed forms of equations. Since the natures of the computational problems are quite different, it is most likely that graphic techniques will continue to be used more to display results than to create them. 

For this chapter we continue to describe the use of confidence limits for comparison of X, Y data pairs. This subject has been addressed in Chapters 58-60 first published as a set of articles in Spectroscopy [1-3]. A MathCad Worksheet ( 1986-2001 MathSoft Engineering Education, Inc., 101 Main Street Cambridge, MA 02142-1521) provides the computations for interested readers. This will be covered in a subsequent chapter or can be obtained in MathCad format by contacting the authors with your e-mail address. The Worksheet allows the direct calculation of the f-statistic by entering the desired confidence levels. In addition the confidence limits for the calculated slope and intercept are computed from the original data table. The lower limits for the slope and the intercept are displayed using two different sets of equations (and are identical). The intercept confidence limits are also calculated and displayed. 

Further extensions of the model are required to address the dynamical consequences of these additional regulatory loops and of the indirect nature of the negative feedback on gene expression. Such extended models have been proposed for Drosophila [112, 113] and mammals [113]. The model for the circadian clock mechanism in mammals is schematized in Fig. 3C. The presence of additional mRNA and protein species, as well as of multiple complexes formed between the various clock proteins, complicates the model, which is now governed by a system of 16 or 19 kinetic equations. Sustained or damped oscillations can occur in this model for parameter values corresponding to continuous darkness. As observed in the experiments on the mammalian clock. Email mRNA oscillates in opposite phase with respect to Per and Cry mRNAs [97]. The model displays the property of entrainment by the ED cycle... 

Equation (15.23) displays the feature of locality that the blending functions should possess in order to be computationally advantageous that is, during the process of matrix inversion, one wishes the calculation to proceed quickly. As mentioned earlier, the use of linear approximation functions results in at most five terms on the left side of the equation analogous to (15.23), yielding a much crader approximation, but one more easily calculated. The current choice of Bezier functions, on the other hand, is rapidly convergent for methods such as relaxation, possesses excellent continuity properties (the solution is guaranteed to look and behave reasonably), and does not require substantial computation. 

A recent work has demonstrated that the formulation of reaction-diffusion problems in systems that display slow diffusion within a continuous-time random walk model with a broad waiting time pdf of the form (6) leads to a fractional reaction-diffusion equation that includes a source or sink term in the same additive way as in the Brownian limit [63], With the fractional formulation for single-species slow reaction-diffusion obtained by the authors still being linear, no pattern formation due to Turing instabilities can arise. This is due to the fact that fractional systems of the type (15) are close to Gibbs-Boltzmann thermodynamic equilibrium as shown in the next section. 

Following a displayed equation that is part of a sentence, punctuate the text as if it were a continuation of a sentence including the equation but do not punctuate at the end of the equation. Note the absence of a comma at the end of the equation in the example. Punctuation that would normally be present at the end of an equation in text is absent but implicit at the end of a displayed equation. 

Table 7 shows the results of this calculation for the ground-state energy evolution of C and Ne using v = 1/2 in Equation (46), for a set of selected distances to the wall, together with corresponding values of the optimal orbital parameters. For completeness, in Figures 10 and 11, the energy evolution as a function of distance to the wall is displayed for each case (continuous line)... 

Following a displayed equation that is part of a sentence, punctuate the text as if it were a continuation of a sentence including the equation. 

In the first example shown in Table 9.2, a warm (25°C) water from the ocean s surface (1 atm) has been modeled. In the zone near the equator, where this water sample had been taken, carbon dioxide partial pressures of 400 patm (equivalent to a pCO of 0.0004 atm or a log pCO of -3.40) have been measured which is somewhat higher than the corresponding atmospheric value. This sample of ocean water thus displays a CQ -gradient directed towards the atmosphere and therefore continually releases CQ into the atmosphere. This situation is accounted for in the model by pre-setting pCO to 0.0004 atm as an open boundary condition with regard to CQ. Accordingly, a state of supersaturation ensues equivalent to a SI, , value of 0.77 or an f2, , value of 5.9. Such a supersaturation state is, according to the... 

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