Algebraic manipulations

After some algebraic manipulation, Eqs. (A, I) and (A.2) lead to the following single-surface equations [4]... 

To solve this system, we apply the implicit midpoint scheme (see system (10)) to system (24) and follow the same algebraic manipulation outlined in [71, 72] to produce a nonlinear system V45(y) = 0, where Y = (X + X )/2. This system can be solved by reformulating this solution as a minimization task for the dynamics function... 

Chapter 5. ALGEBRAIC MANIPULATIONS AND LIMITING FORMS OF THE DUSTY GAS... 

In certain types of finite element computations the application of isoparametric mapping may require transformation of second-order as well as the first-order derivatives. Isoparametric transformation of second (or higher)-order derivatives is not straightforward and requires lengthy algebraic manipulations. Details of a convenient procedure for the isoparametric transformation of second-order derivatives are given by Petera et a . (1993). 

Note that in contrast to the example shown in Section 2.2,2 the element stiffness equation obtained for this problem is not symmetric. After the substitution for the shape functions and algebraic manipulations... 

Substitution from Equations (2.107), (2.108) and (2.109-2.110) into Equation (2.106) and carrying out algebraic manipulations gives... 

Equations (3.59) and (3.60) are recast in terms of their components and solved together. After algebraic manipulations and making use of relations (3.61) slip-wall velocity components are found as... 

In Equation (4.12) the discretization of velocity and pressure is based on different shape functions (i.e. NjJ = l,n and Mil= l,m where, in general, mweight function used in the continuity equation is selected as -Mi to retain the symmetry of the discretized equations. After application of Green s theorem to the second-order velocity derivatives (to reduce inter-element continuity requirement) and the pressure terms (to maintain the consistency of the formulation) and algebraic manipulations the working equations of the U-V-P scheme are obtained as... 

After application of Green s theorem to the second-order velocity derivatives (to reduce inter-element continuity requirement) and algebraic manipulations the working equations of the continuous penalty scheme are obtained as... 

At the inlet the velocity and pressure are presumed continuous, ie, - Simple algebraic manipulations yield the result... 

The average concentration is found by integration over the cross section. After some algebraic manipulation, the result is... 

Replacing a and with some algebraic manipulations we obtain ... 

Equation (14.55) contains same kind of algebraic manipulation with the term as Eq. (14.45). 

Now think about what happens when the same amount of acetic acid is added to water that is buffered at pH = 7.0. Before doing the calculation, let us recognize that it is the [CH3C02 ]/[CH3C02H] ratio in which we are interested and do a little algebraic manipulation. Because... 

The quantitative description of enzyme kinetics has been developed in great detail by applying the steady-state approximation to all intermediate forms of the enzyme. Some of the kinetic schemes are extremely complex, and even with the aid of the steady-state treatment the algebraic manipulations are formidable. Kineticists have, therefore, developed ingenious schemes for writing down the steady-state rate equations directly from the kinetic scheme without carrying out the intermediate algebra." -" ... 

Note that all of the required MP2 energies can be extracted from the previously-run MP4 jobs in steps 3 through 5. Thus, after algebraic manipulation, the final formula is ... 

The ability to use algebraic manipulation to solve high-order differential equations... 

The two basic principles permit the algebraic manipulation of chemical reactions (represented by their stoichiometric equations and associated enthalpy changes) in order to achieve desired thermochemical results. 

Quantitative problems that require a calculator and some algebraic manipulations. 

From this mathematical treatment we properly conclude that ki + k- is the natural rate constant. The data workup does not give the sum because of an artful algebraic manipulation. Instead, the sum k-i + k is simply the inverse of the intrinsic time constant for the single exponential that defines the approach to equilibrium. 

Steady-state. An erroneous rate law is shown below for the reaction scheme believed to represent the reaction between Fe3+ and I-, in that an extraneous denominator term appears. In the scheme shown, I2 and Fel2+ obey the steady-state approximation. Show what the incorrect part of the expression is. Suggest a simple derivation of the correct equation that avoids extensive algebraic manipulations. 

Diagonalizing the corresponding 2x2 secular equation and some algebraic manipulation gives the four-electron ground-state potential surface... 

Now, to aid in algebraic manipulation the volume flow of mobile phase will now be measured in units of (vm + Kvs) instead of milliliters. Thus the new variable (v) can be defined where... 

To obtain the optimum temperature we differentiate Eq. (15) w.r.t temperature T and equate dtf /dT to zero. After some algebraic manipulations the optimality condition can be written as ... 

Fig. 9. Tree showing algebraic manipulations necessary to derive sufficient conditions for dominance or equivalence.

The integral may be evaluated using equation (A. 8) accompanied with some tedious, but straightforward algebraic manipulations, yielding... 

Before we go on, let us emphasize that what we find below are nothing but different algebraic manipulations of the same heat balance. First, we rearrange (2-45) to give... 

Summary. In summary, when modeling with the fugacity concept, all equilibria can be treated by Z values (one for each compartment) and all reaction, advection and transport processes can be treated by D values. The only other quantities requiring definition are compartment volumes and emission rates or initial concentrations. A major advantage is that since all D quantities are in equivalent units they can be compared directly and the dominant processes identified. By converting diverse processes such as volatilization, sediment deposition, fish uptake and stream flow into identical units, their relative importance can be established directly and easily. Further, algebraic manipulation... 

Substitution of equation 31 into equation 32 and replacement of R from equation 12 yields, after some algebraic manipulation, the... 

Equations 4.2.20 and 4.2.22 may be combined with equation 4.2.16 in straightforward algebraic manipulation to yield an expression for the overall rate of formation of HBr. 

Combination of these relations and further algebraic manipulation gives... 

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