Relative magnitude

Such step-limiting is often helpful because the direction of correction provided by the Newton-Raphson procedure, that is, the relative magnitudes of the elements of the vector J G, is very frequently more reliable than the magnitude of the correction (Naphtali, 1964). In application, t is initially set to 1, and remains at this value as long as the Newton-Raphson correotions serve to decrease the norm (magnitude) of G, that is, for... 

We have paid particular attention to industrial processes and have tried to give some indication of the use of particular chemicals. In order to give some idea of the relative magnitudes of production we have included data on annual production. In this, because of availability of data, we have not been entirely consistent in including common data or common units. (World, U.S. or European production figures are used as seem most appropriate or up-to-date - and it must be remembered that for most speciality chemicals U.S. production is dominant.)... 

This is a single-magnet system with a magnetic moment parallel to the radiation surface for magnetic materials. The relative magnitude of the magnetic field at maximum from the height... 

Equatiou B1.5.44 indicates that if we know -. /i and we may infer infonnation about the third-order orientational moments ( T.., Tjj, Since calibration of absolute magnitudes is difficult, we are generally concerned with a comparison of the relative magnitudes of the appropriate molecular (a ) and macroscopic (... 

Since Cy and E are bodi extensive properties (ocN), the root-inean-square energy fluctuations are smaller, by a factor 1/y, than typical average energies E. As the system size increases, the relative magnitude of... 

The relative magnitudes of g, , h, and Sj i = x,y,z describe the orientation and shape of the double cone. The relation between these athibutes and near... 

However, without further examination of the relative magnitudes of D, ... 

Ac this point It is important to emphasize that, by changing a and p, it is not possible to pass to the limit of viscous flow without simultaneously passing to the limit of bulk diffusion control, and vice versa, since physical estimates of the relative magnitudes of the factors and B... 

Franck-Condon factors". Their relative magnitudes play strong roles in determining the relative intensities of various vibrational "bands" (i.e., peaks) within a partieular eleetronie transition s speetrum. 

Added nitrate anticatalyses nitration under both conditions without affecting the order of the reaction, which is determined solely by the relative magnitudes of [ArH] and [HgO]. 

Evaluating Indeterminate Error Although it is impossible to eliminate indeterminate error, its effect can be minimized if the sources and relative magnitudes of the indeterminate error are known. Indeterminate errors may be estimated by an appropriate measure of spread. Typically, a standard deviation is used, although in some cases estimated values are used. The contribution from analytical instruments and equipment are easily measured or estimated. Indeterminate errors introduced by the analyst, such as inconsistencies in the treatment of individual samples, are more difficult to estimate. 

Percent of overall variance (So) due to the method as a function of the relative magnitudes of the standard deviation of the method and the standard deviation of sampling (Sm/Ss). The dotted lines show that the variance due to the method accounts for 10% of the overall variance when Ss= 3 xs . 

In the present discussion it is the relative magnitudes of AH and AS which are significant. Several considerations lead us to conclude that this value of AH represents an upper limit for this quantity ... 

Note that this inquiry into copolymer propagation rates also increases our understanding of the differences in free-radical homopolymerization rates. It will be recalled that in Sec. 6.1 a discussion of this aspect of homopolymerization was deferred until copolymerization was introduced. The trends under consideration enable us to make some sense out of the rate constants for propagation in free-radical homopolymerization as well. For example, in Table 6.4 we see that kp values at 60°C for vinyl acetate and styrene are 2300 and 165 liter mol sec respectively. The relative magnitude of these constants can be understod in terms of the sequence above. 

The H2O molecule, therefore, has three normal vibrations, which are illustrated in Figure 4.15 in which the vectors attached to the nuclei indicate the directions and relative magnitudes of the motions. Using the C2 character table the wave functions ij/ for each can easily be assigned to symmetry species. The characters of the three vibrations under the operations C2 and (t (xz) are respectively + 1 and +1 for Vj, - - 1 and + 1 for V2, and —1 and —1 for V3. Therefore... 

A third factor is the ease with which various membrane materials can be fabricated into a particular module design. Almost ah membranes can be formed into plate-and-frame, spiral, and tubular modules, but many membrane materials caimot be fabricated into hollow-fine fibers or capihary fibers. Finahy, the suitabiHty of the module design for high pressure operation and the relative magnitude of pressure drops on the feed and permeate sides of the membrane can sometimes be important considerations. 

These mechanisms are characterized by the relative magnitudes of the heats of reaction, solution, or adsorption (see Adsorption, separation). AH useflil drying mechanisms are exothermic. Phosphoms pentoxide is a Class 1 drying agent that reacts with water to form a polyphosphoric acid (2) ... 

In effect, the standard deviation quantifies the relative magnitude of the deviation numbers, i.e., a special type of average of the distance of points from their center. In statistical theory, it turns out that the corresponding variance quantities s have remarkable properties which make possible broad generalities for sample statistics and therefore also their counterparts, the standard deviations. 

As examples of the relative magnitudes of these contributions, only tire dispersion effect applies to monatomic gases, and in tire case of HCl (/ = 12.74eV, fjL — 1.03 debye), tire dispersion effect predominates, in NH3 (/ = 10.2eV, ijl — 1.49d) these effects are about equal, and in H2O (I — 12.6eV, IJL — 1.85 d), the orientation effect predominates. 

The public s idea of what is most risky usually differs widely from the facts. Much of this stems from disproportionate media coverage of newsworthy hazards that are rare or unusual. This distorts many people s perspective on the relative magnitude of risks. Thus, some parents may not have their children vaccinated, fearing a rare reaction to the vaccine more than the overwhelmingly greater risk of dying from the disease. 

Let us define the respective basicity by — AG in the gas phase and — AG" in aqueous solution. For discussions concerning the relative strength in basicity of a series of methyl-amines, only the relative magnitudes of these quantities are needed. Thus the free energy changes associated with the protonation of the methylamines relative to those of ammonia are defined as... 

Because the product composition is kinetically controlled, the isomer ratio will be governed by the relative magnitudes of AG, AGI, and AG, the energies of activation for the ortho, meta, and para transition states, respectively. In Fig. 4.7 a qualitative comparison of these AG values is made. At the transition state, a positive charge is present on the benzene ring, primarily at positions 2, 4, and 6 in relation to the entering bromine. 

The data required are the relative magnitudes of and ky. When both B—X and B—Y are present in the reaction system, they will be consumed at rates that are a function of their reactivity and concentration ... 

---