Equations and equivalents for

Be careful, however, not to become too casual with abbreviations. For example, popular shorthand is not appropriate (e.g., B4 for before or FYI for for your information ). Also, use abbreviations consistently throughout the poster. For example, eq should not be used for equation and equivalent in the same poster. 

You see both xs and ys in the equation. Use the equation of the circle with its center at the origin and a radius of 10 by solving fory2. Then you can substitute the equivalence for y2 into the equation and solve for x. 

In theory, we could generate several moment equations and solve for an equivalent number of parameters. In practice, it is best to only use the first and second moments. The outlet concentration profile usually has some tailing at longer times. These values usually have the largest error in their values and can skew the calculated moment values since the concentration is multiplied by t for the fcth moment. 

Equivalent expressions can be obtained by using the Sand equation and substituting for 1/2. 

Regular solution theory and the van Laar equation are equivalent for a binary solution if... 

The classic models for bioimpedance and bioelectricity are mathematical equations and equivalent circuit diagrams with the same electrical behavior as the tissue to be modeled. Others include statistical models, which are used to determine the correspondence between bioelectrical measurements and physiological variables (e.g., tissue characterization). 

Let us now compare the coupled-cluster equations in the linked and unlinked forms. We b n by reiterating that these two forms of the coupled-cluster equations are equivalent for the standard models in the sense that they have the same solutions. Moreover, applied at the important CCSD level of theory, neither form is superior to the other, requiring about the same number of floating-point operations. The energy-dependent unlinked form (13.2.19) exhibits mcMe closely the relationship with Cl theory, where the projected equations may be written in a similar form (13.1.18). On the other hand, the linked form (13.2.23) has some important advantages over the unlinked one (13.2.19), making it the preferred form in most situations. 

Equation X-17 was stated in qualitative form by Young in 1805 [30], and we will follow its designation as Young s equation. The equivalent equation, Eq. X-19, was stated in algebraic form by Dupre in 1869 [31], along with the definition of work of adhesion. An alternative designation for both equations, which are really the same, is that of the Young and Dupre equation (see Ref. 32 for an emphatic dissent). 

Here the ijk coordinate system represents the laboratory reference frame the primed coordinate system i j k corresponds to coordinates in the molecular system. The quantities Tj, are the matrices describing the coordinate transfomiation between the molecular and laboratory systems. In this relationship, we have neglected local-field effects and expressed the in a fomi equivalent to simnning the molecular response over all the molecules in a unit surface area (with surface density N. (For simplicity, we have omitted any contribution to not attributable to the dipolar response of the molecules. In many cases, however, it is important to measure and account for the background nonlinear response not arising from the dipolar contributions from the molecules of interest.) In equation B 1.5.44, we allow for a distribution of molecular orientations and have denoted by () the corresponding ensemble average ... 

We thus obtain a Lagrangean density, whieh is equivalent to Eq. (149) for all solutions of the Dirac equation, and has the structure of the nonrelativistic Lagrangian density, Eq. (140). Its variational derivations with respect to v / and v / lead to the solutions shown in Eq. (152), as well as to other solutions. 

Note that, in loeal eoordinates. Step 2 is equivalent to integrating the equations (13). Thus, Step 2 can either be performed in loeal or in eartesian coordinates. We consider two different implicit methods for this purpose, namely, the midpoint method and the energy conserving method (6) which, in this example, coineides with the method (7) (because the V term appearing in (6) and (7) for q = qi — q2 is quadratie here). These methods are applied to the formulation in cartesian and in local coordinates and the properties of the resulting propagation maps are discussed next. 

RPA, and CPHF. Time-dependent Hartree-Fock (TDFIF) is the Flartree-Fock approximation for the time-dependent Schrodinger equation. CPFIF stands for coupled perturbed Flartree-Fock. The random-phase approximation (RPA) is also an equivalent formulation. There have also been time-dependent MCSCF formulations using the time-dependent gauge invariant approach (TDGI) that is equivalent to multiconfiguration RPA. All of the time-dependent methods go to the static calculation results in the v = 0 limit. 

Solution of the model equations shows that, for a linear isothermal system and a pulse injection, the height equivalent to a theoretical plate (HETP) is given by... 

Vapor densities for pure compounds can also be predicted by cubic equations of state. For hydrocarbons, relatively accurate Redlich-Kwong-type equations such as the Soave and Peng-Robinson equations are often used. Both require only T, and (0 as inputs. For organic compounds, the Lee-Erbar-EdmisteF" equation (which requires the same input parameters) has been used with errors essentially equivalent to those determined for the Lydersen method. While analytical equations of state are not often used when only densities are required, values from equations of state are used as inputs to equation of state formulations for thermal and equilibrium properties. 

One particular case of multicomponent diffusion that has been examined is the dilute diffusion of a solute in a homogeneous mixture (e.g., of A in B -h C). Umesi and Danner compared the three equations given below for 49 ternaiy systems. All three equations were equivalent, giving average absolute deviations of 25 percent. 

A numerical solution of these equations maybe obtained in terms of finite difference equivalents, taking m radial increments and n axial ones. With the following equivalents for the derivatives, the solution maybe carried out by direct iteration ... 

In (2.19), F has been replaced by P because force and pressure are identical for a one-dimensional system. In (2.20), S/m has been replaced by E, the specific internal energy (energy per unit mass). Note that all of these relations are independent of the physical nature of the system of beads and depend only on mechanical properties of the system. These equations are equivalent to (2.1)-(2.3) for the case where Pg = 0. As we saw in the previous section, they are quite general and play a fundamental role in shock-compression studies. 

The difference between this equation and the equation for ex,wo that the internal energy of the air that is displaced by the expanded gases is taken into account. Note that, when y, is equal to yo, Eq. (6.3.13) is equivalent to Eq. (6.3.2). Aslanov and Golinskii advocate the use of the energy E bx.ag the energy of the explosion. They claim that this gives a better correlation with numerical calculations and with experiments. 

Molecular weight calibration from a monomer to several million daltons can be carried out by a variety of techniques. Because narrow standards of p(methyl methacrylate) (pMMA) are available, these are often used. Narrow standards of p(styrene) (pSty) are also available and can be used. Using the Mark-Houwink-Sakurada equation and the parameters for pSty and pMMA, a system calibrated with pSty can give pMMA-equivalent values, and vice versa. 

In these circumstances a decision must be made which of two (or more) kinet-ically equivalent rate terms should be included in the rate equation and the kinetic scheme (It will seldom be justified to include both terms, certainly not on kinetic grounds.) A useful procedure is to evaluate the rate constant using both of the kinetically equivalent forms. Now if one of these constants (for a second-order reaction) is greater than about 10 ° M s-, the corresponding rate term can be rejected. This criterion is based on the theoretical estimate of a diffusion-controlled reaction rate (this is described in Chapter 4). It is not physically reasonable that a chemical rate constant can be larger than the diffusion rate limit. 

The analysis of a k-pH curve in terms of Eq. (6-80) is treated by making approximations that are equivalent to ignoring some of the rate terms in certain pH regions. A common type of system is analyzed as an example the approach can be modified to suit a particular demand. The evaluation of k° and k" can nearly always be accomplished from rate data at very low and high pH, respectively. We are concerned with k, Kf, and /fj. Figures 6-16 and 6-17 are A -pH and log fc-pH plots for a hypothetical system described by Eq. (6-80). The analysis assumes this equation and these types of parameters. 

Write the Lineweaver-Burk (double-reciprocal) equivalent of this equation, and from it calculate algebraic expressions for (a) the slope (b) the y-intercepts and (c) the horizontal and vertical coor-... 

If a line is made up of several different sizes, these may be resolved to one, and then the equation solved once for this total equivalent length. If these are handled on a per size basis, and totaled on the basis of the longest length of one size of line, then the equivalent length, L, for any size d, referenced to a basic diameter, d. ... 

Plot Equation 9-129 for two assumed L /G values and an assumed number of checks on the plot of Figure 9-126C or its equivalent. The intersection with the approach curve gives the value of L /Gg which satisfies the two Equations 9-118 and 9-129. 

Figures 10-47, 10-48, and 10-49 are useful in solving the equivalent of Equation 10-47 for turbulent as well as streamline flow of gases and vapors inside tubes. To use the charts...

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