Exponential error

Median PK parameters for a drug that follows a two-compartment model were used for the simulation (see Table 9.1). Two levels of interindividual variability (30% and 45% coefficient of variation) and two levels of residual variability (15% and 25% coefficient of variation) were evaluated using exponential error models. 

In fact, the proportional error model and exponential error model are equal for small values of a. To see this, the function exp(x) if first written as a MacLaurin series... 

To obtain initial estimates, an Emax model was fit to the data set in a na ive-pooled manner, which does not take into account the within-subject correlations and assumes each observation comes from a unique individual. The final estimates from this nonlinear model, 84% maximal inhibition and 0.6 ng/mL as the IC50, were used as the initial values in the nonlinear mixed effects model. The additive variance component and between-subject variability (BSV) on Emax was modeled using an additive error models with initial values equal to 10%. BSV in IC50 was modeled using an exponential error model with an initial estimate of 10%. The model minimized successfully with R-matrix singularity and an objective function value (OFV) of 648.217. The standard deviation (square root of the variance component) associated with IC50 was 6.66E-5 ng/mL and was the likely source of the... 

For the phase-lag Definition 5 holds. We note here that the phase-lag theory is also valid for the case where w is imaginary. In that case we now have an exponential error which corresponds to the phase-lag. 

The calculation of the exponential error follows the same principle. Now w is imaginary and cos is replaced by cos in formula (140). 

Eq. (4.45) as t becomes large, it is possible to approximate the exponential error function to second order such that... 

In prineiple, nothing more is neeessary to understand the infiuenee of the solvent on the TST rate eonstant than the modifieation of the PMF, and the resulting ehanges in the free energy barrier height should be viewed as the dominant effeet on the rate sinee tliese ehanges appear in an exponential fonn. As an example, an error... 

Fhe van der Waals and electrostatic interactions between atoms separated by three bonds (i.c. the 1,4 atoms) are often treated differently from other non-bonded interactions. The interaction between such atoms contributes to the rotational barrier about the central bond, in conjunction with the torsional potential. These 1,4 non-bonded interactions are often scaled down by an empirical factor for example, a factor of 2.0 is suggested for both the electrostatic and van der Waals terms in the 1984 AMBER force field (a scale factor of 1/1.2 is used for the electrostatic terms in the 1995 AMBER force field). There are several reasons why one would wish to scale the 1,4 interactions. The error associated wilh the use of an repulsion term (which is too steep compared with the more correct exponential term) would be most significant for 1,4 atoms. In addition, when two 1,4... 

The overall requirement is 1.0—2.0 s for low energy waste compared to typical design standards of 2.0 s for RCRA ha2ardous waste units. The most important, ie, rate limiting steps are droplet evaporation and chemical reaction. The calculated time requirements for these steps are only approximations and subject to error. For example, formation of a skin on the evaporating droplet may inhibit evaporation compared to the theory, whereas secondary atomization may accelerate it. Errors in estimates of the activation energy can significantly alter the chemical reaction rate constant, and the pre-exponential factor from equation 36 is only approximate. Also, interactions with free-radical species may accelerate the rate of chemical reaction over that estimated solely as a result of thermal excitation therefore, measurements of the time requirements are desirable. 

Exponential cost correlations have been developed for individual items of equipment. Care must be taken in determining whether the cost of the eqmpment has been expressed as free on Board (FOB), delivered (DEL), or installed (INST), as this is not always clearly stated. In many cases the cost must be correlated in terms of parameters related to capacity such as surface area for heat exchangers or power for grinding equipment. There are four main sources of error in such cost correlations ... 

Time reversibility. Newton s equation is reversible in time. Eor a numerical simulation to retain this property it should be able to retrace its path back to the initial configuration (when the sign of the time step At is changed to —At). However, because of chaos (which is part of most complex systems), even modest numerical errors make this backtracking possible only for short periods of time. Any two classical trajectories that are initially very close will eventually exponentially diverge from one another. In the same way, any small perturbation, even the tiny error associated with finite precision on the computer, will cause the computer trajectories to diverge from each other and from the exact classical trajectory (for examples, see pp. 76-77 in Ref. 6). Nonetheless, for short periods of time a stable integration should exliibit temporal reversibility. 

Predictability. Exponential divergence of orbits places a severe restriction on the predictability of the system. If the initial point xq is known only to within an error 6xq, for example, we know that this error will grow to 6xn = exp nln2 (5a o (mod 1) by the iteration. The relaxation time, Tr, to a statistical equilibrium - defined as the number of iterations required before we reach a state of total ignorance as to the location of the orbit point within the unit interval [0,1] i.e., Tr = min (n) such that 6xn 1 - is therefore given by... 

It can be shown that the BBM is capable of universal computation ([fredkin82, marg84, marg88]). Unfortunately, as shown by Zurek [zurek84], the model is also unlikely to ever be realized in practice. Because BBM computations all depend so critically on initial ball and mirror placement, the fact that any errors in the initial placement grow exponentially in time effectively renders their results either suspect or meaningless. 

The probability of error, both in Eqs. (4-89) and (4r-91), depends critically upon E, necessitating an investigation of the behavior of E as a function of R. The strongest result is achieved in Theorem 4-11 when E is maximized over p and p. As we will show later, this maximization leads to a value of E that decreases with R, but is positive for R < 0. Thus, for R < C, Pe can be made to approach zero exponentially with the block length. 

The precision of the rate constants as a function of temperature determines the standard deviations of the activation parameters. The absolute error, not the percentage error in the activation parameters, represents the agreement to the model, because of the exponential functions. If, for example, one wished to examine the values of AS for two reactions that were reported as -4 3 and 26 3 J mol 1K 1, then it should be concluded that the two are known to the same accuracy. Since AS and A// are correlated parameters, the uncertainty in AS will be about 1/Tav times that in A//. At ambient temperature this amounts to an approximate factor of three (that is, 1000/T, converting from joules for AS to kilojoules for A// ). Thus, the uncertainty in A//, 0 of 2.50 kJ mol 1 is consistent with the uncertainty in ASn of 7.21 J mol1 K-1 at Tav - 350 K. 

In addition to the chemical inferences that can be drawn from the values of AS and AH, considered in Section 7.6, the activation parameters provide a reliable means of storing and retrieving the kinetic data. With them one can easily interpolate a rate constant at any intermediate temperature. And, with some risk, rate constants outside the experimental range can be calculated as well, although the assumption of temperature-independent activation parameters must be kept in mind. For archival purposes, values of AS and AH should be given to more places than might seem warranted so as to avoid roundoff error when the exponential functions are used to reconstruct the rate constants. 

A note on good practice Exponential functions (inverse logarithms, e ) are very sensitive to the value of x, so carry out all the arithmetic in one step to avoid rounding errors. 

A note on good practice Note that, because exponential functions ex are so sensitive to the value of x, we avoid rounding errors by leaving the numerical calculation to a single final step. 

This equation continues to conserve mass but is no longer stable. The original upset grows exponentially in magnitude and oscillates in sign. This marching-ahead scheme is clearly unstable in the presence of small blunders or round-off errors. 

The pre-exponential factors were then piece wise estimated by trial and error. The model showed good agreement with the experimental data as illustrated in Figure 3. 

If one includes functions with n - / even in (1.1) (i.e. one uses set b) the basis is formally overcomplete. However the error decreases exponentially with the size of the basis [2,16]. Unfortunately for this type ofbasis the evaluation of the integrals is practically as difficult as for Slater type basis functions, such that basis sets of type (b) have not been used in practice. 

The examples given in the appendix give some indications on the properties which the mapping function has to satisfy that both the cut-off error and the discretization error decrease exponentially (or faster) with nh and /h respectively and don t depend too strongly on r. Further studies are necessary to settle this problem. 

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