Functions inverse

The rate (kinetics) and the completeness (fraction dissolved) of oxide fuel dissolution is an inverse function of fuel bum-up (16—18). This phenomenon becomes a significant concern in the dissolution of high bum-up MO fuels (19). The insoluble soHds are removed from the dissolver solution by either filtration or centrifugation prior to solvent extraction. Both financial considerations and the need for safeguards make accounting for the fissile content of the insoluble soHds an important challenge for the commercial reprocessor. If hydrofluoric acid is required to assist in the dissolution, the excess fluoride ion must be complexed with aluminum nitrate to minimize corrosion to the stainless steel used throughout the facility. Also, uranium fluoride complexes are inextractable and formation of them needs to be prevented. 

Pressure ratio () is quite important, but transmembrane AP matters as well. Consider the case of a vacuum permeate ( = 00) The membrane area will be an inverse function of P influences separation and area, transmembrane AP influences area. 

Now, at high linear velocities, the longitudinal diffusion term will become insignificant and, equally important, the resistance to mass transfer term that incorporates the inverse function of diffusivity will become large, thus improving the precision of measurement. 

Equation (13) is the first important equation for open tubular column design. It is seen that the optimum radius, with which the column will operate at the optimum velocity for the given inlet pressure, increases rapidly as an inverse function of the separation ratio (cc-1) and inversely as the square root of the inlet pressure. Again it must be remembered that, when calculating (ropt)5 the dimensions of the applied pressure (P) must be appropriate for the dimensions in which the viscosity (r)) is measured. 

Solve ifun Inverse functions are being used by... 

It should now be clear that the problem of calculating pY in terms oipx is the same as that of making the change of variable 17 = ( ) in the right-hand integral appearing in (3-42). This relationship can be written down explicitly if certain assumptions are made about the function . For example, if is a monotone, differentiable function (which implies that the inverse function < 1 defined by -1[ (f)] = exists) and if ( — co) = — co and (oo) = co, then... 

Table A2.3 Values of the coefficients A,-, B, Cand D, and of the constants A0, Bo, Co, and Do in the reference functions in equations (A2.6) and (A2.7) and in the inverse functions given by equations (A2.8) and (A2.9). These functions, coefficients, and constants are part of the definition of ITS-90.

Just as important as the maximum active force a muscle can exert at various lengths, is the rate at which the muscle shortens as a function of the force load, i.e., the force-velocity curve. Both the length-tension curve and the force-velocity curve vary according to the degree of activation of a muscle. The rate at which crossbridges cycle is an inverse function of the load force (Figure 4). 

The lifetime of T is an inverse function of the hydroxyl radical concentration [HO>] and the rate constant kj for its reaction with a particular trace gas... 

K used in Equation 31.7 is a slowly changing inverse function of beam energy and ranges from 1.76 to 0.85 Mrad m min mA as the energy varies from 300 to 3000 KeV [52]. Surface dose is to be used in Equation 31.6 and average dose for Equation 31.7. 

The probability density can be calculated by way of Eq. (1.7). Both a forward and an inverse function for the cumulative probability CP are needed ... 

The estimation of the analyte content x is given by the SA analytical function (the inverse function of Eq. (6.57)) whose extrapolation for y — 0 yields... 

The inverse functionalization of the two components for a Wittig reaction has been described by Russian authors70 who combined the cyclopropenylide 70 with aldehydes to give the unstable methylene cyclopropenes 71 characterized by protonation as cyclopropenium salts 72 ... 

The flammability of solids may be considered to be a function of the heat release rate and critical ignition energy of the material being studied. Flammability is an inverse function of the actual ignition energy of the material in question, and it is directly related to the rate of heat liberated after ignition of the sample. 

Let

monotonous function and note

single-valued (i.e., a value of the independent variable is associated with one value of the dependent variable). If the random variable X has the density function /v(x), then Y = density function fY(y) given by... 

Since, both the radioactive and non-radioactive antigens (haptens) are more or less chemically and immunochemically the same, they will eventually compete for the limited number of antibody sites available thus, the amount of radioactivity that ultimately combines with the antibody will be an inverse function of the amount of unlabelled hapten competing for these sites,... 

In this context, time, speed and distances (space) are important parameters to be kept in mind, leading to the so-called "spatiotemporal" or "proximity theory". This theory states that "the reaction rate (speed) is a sensitive inverse function of distance and time". 

The experimental titration curve is normally the pH as a function of the added number of moles of base. This is the inverse function of Eq. (2.6.5). 

Susceptibility to n-alkane degradation is an inverse function of chain length. Branched alkanes are less susceptible than straight-chain n-alkanes, and the most resilient saturated components are the pristine and phytane isoprenoids (Wang et al. 1998). 

C. By this means the desalinizing capability of the membrane could be increased proportionately to the increase in heating temperatures, i.e., the menbrane could be "tailored" to the desalinizing job at hand. The water permeation flux was an inverse function of the heating temperature. 

Figure 27 presents data on the specific reactivity of the series of carbon blacks in 0.1 atm. of oxygen at 600° vs. the specific surface area of the blacks. Since the crystallite size is an inverse function of surface area, the conclusion to be drawn from Fig. 27 is that the specific reactivity of carbons increases with increase in crystallite size. Armington reports similar results for the... 

How is the process affected by changes in conditions The diffusion coefficient is an inverse function of pressure, while concentration varies directly with pressure. The result is that weight collected is constant with respect to changes in... 

The badge sampling rate is a direct function of the diffusion coefficient (D) of the organic vapor(s) being sampled and the total cross-sectional area (A) of the badge cavities. The rate is an inverse function of the diffusion path or length (L) of the cavities. 

Proof. Suppose T is an isomorphism of vector spaces. Then the inverse function T exists, so T must be injective. Moreover, the function 7 has domain IT, so the image of T must be W as well, i.e., T is surjective. On the other hand, suppose that T is injective and surjective. Then T has domain W and image V. We must show that T is a Unear transformation. Let wi and W2 be arbitrary elements of W. Since T is surjective, there are elements Vi and V2 of V such that Wi = r(ui) and W2 = F(V2). Then... 

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