Algebraic transformation

Algebraic Substitution Functions containing elements of the type (a + bxY are best handled by the algebraic transformation =... 

In order to express the Gibbs free enthalpy density G in terms of those chemical potentials which are conjugate to the conserved densities and to implement the (3—equilibrium condition (26) we make the following algebraic transformations... 

Secondary plots of kinetic data are used to obtain various rate constants and other kinetic parameters such as and Vmax- To simply the analysis, one choses a algebraic transform of the rate equation that allows the observed data to be graphed in a hnear format. 

A careful reader will observe that this algebraic transformation will produce a dual SDP problem that does not have y G IR such that the matrix in Eq. (16) has all of its eigenvalues positive and, therefore, will not satisfy the Slater conditions. However, numerical experiments have shown that practical algorithms stUl can solve these problems efhciently [16]. 

Substitution of these relations into the expression of Ax,(= x - x,) and simple algebraic transformations leads to the usual adsorption equation for the reduced surface excess... 

The Michaelis-Menten equation (Eqn 6-9) can be algebraically transformed into versions that are useful in the practical determination of Km and Vmax (Box 6-1) and, as we describe later, in the analysis of inhibitor action (see Box 6-2 on page 210). TABLE... 

The expression for the rate constant ensues after the sufficiently simple algebraic transformation ... 

Some algebra transforms this equation into a more compact form ... 

Christoph [37] showed that the domain bifurcation can only occur if the zero-frequency impedance ZF is negative. This can be easily seen with a few algebraic transformations. Realizing that... 

After squaring Eq. (1) and making some algebraic transformations, we obtain the following expression ... 

When performing algebraic transformations we exclude the coefficients from the second boundary condition in Eq. (3) using Eq. (10), and the number of unknown coefficients is halved. After solution of Eq. (11), coefficients for inner potentials are found by summing the series in Eq. (10). 

Because V is approached asymptotically in the saturation curve shown in Figure 5.5, the estimation of Km and is not very accurate. Equation (5.24) can be algebraically transformed into a linear form more suitable for plotting experimental data and obtaining Km and Vmax. 

For the Michaelis-Menten equation there are algebraic transformations, in addition to the Lineweaver-Burk equation, that yield straight line plots from enzyme kinetic data. One such plot is due to Eadie and Hofstee their equation takes the following form ... 

If we do a series of algebraic transformations, we can arrive at the commonly used Hill equation ... 

Since we have passed from (1.11.17) to (1.11.19) by algebraic transformations the former equation will be integrable if and only if the latter is. By hypothesis, Eq. (1.11.12) applies to (1.11.17) hence, a corresponding equation will have to hold relative to (1.11.19), namely... 

Unfortunately, the direct algebraic transformation of expression (2-19) into some form of functional dependence of R on k, a, and N is impossible. Knox and Thijssen were the first to independently propose the transformation based on the assumption of equal peak width (w2 = Wi ) and consideration of the retention of the first peak of the pair (A i). The resulting expression is... 

The Michaelis-Menten equation can be algebraically transformed into more useful way to plot the experimental data. Lineweaver and Burk have taken the reciprocal of both [S] and v of the Michaelis-Menten equation to give Double Reciprocal or Lineweaver-Burke Plot Need in form y = ax + b, so take reciprocals of both sides (Fig. 6.4) and have -... 

The proof of this formula is similar to the proof of formula (8.111). Based on formulae (9.6) and (9.7), we can obtain, after some algebraic transformations, the following important energy inequality, derived originally by Singer (1995) and Pankratov et al. (1995) ... 

Equation (9.203) can be rewritten with respect to the product of a and the total electric field E, using simple algebraic transformations ... 

Thus, localized QL inversion is reduced to Bleistein inversion with respect to the material property function mi (x, t/, z) and then to a simple correction of this inversion result by solving the minimization problem (15.161) and applying the algebraic transformation (15.162). 

Formally cannot be split into two (A and D) components because of the presence of r a -f ao in Eq. (72), which acts as a coupling term. Yet a tedious but elementary algebraic transformation allows Eq. (72) to be rewritten as... 

Expressions for the derivatives of the chemical potentials with respect to the number of particles, the partial molar volumes, and the isothermal compressibility were derived by Kirkwood and BufP in compact matrix forms (see Appendix 1). The derivation of explicit expressions for the above quantities in multicomponent mixtures required an enormous number of algebraic transformations, which could be carried out by using a special algebraic software (Maple 8 was used in the present paper). A full set of expressions for the derivatives of the chemical potentials with respect to the number of particles, the partial molar volumes, and the isothermal compressibilities in a quaternary mixture were derived. However, our main interest in this paper is related to the derivatives of the activity coefficient with respect to the mole fractions (all of the expressions for the derivatives of the chemical potentials with respect to the number of particles, the partial molar volumes, and the isothermal compressibility can be obtained from the authors at request), namely, the derivatives of the form (9 In where Xg, Xg,... 

Combining Eqs. (C4)-(C11) with Eqs. (C2) and (C3) allows one to obtain the following Kirkwood-Buff integrals of ternary mixtures for an infinitely dilute solute (of course, the numerous algebraic transformations necessary could be carried out by using an algebraic software such as MATH-EMATICA or maple). 

Its length is known (1/A.) and its orientation, i.e. angle 0, is found by a simple algebraic transformation after recalling that d = Md ... 

Lineweaver-Burk equation. An algebraic transformation of the Michaelis-Menten equation (plot of 1/V vs 1/[S]), allowing determination of Vmax and Km by extrapolation of [S] to infinity. 

Given a matrix M, some general classes of matrices can be derived by applying algebraic transformations. They are reported below . 

V can be obtained by constructing a graph, as shown in Figure 6.4. A more accurate determination of these values results from an algebraic transformation of the data. The Michaelis-Menten equation, whose graph is a hyperbola, v Vm,x[S]... 

After algebraic transformations and n lecting second-order (and third-order) terms that imply crossed concentrations such as xx, yy or xy, one obtains ... 

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