Essential points

A somewhat different viewpoint motivates this chapter, which stiesses the added meaning that the complex nature of the wave function lends to our understanding. Though it is only recently that this aspect has come to the forefront, the essential point was affimied already in 1972 by Wigner [5] in his famous essay on the role of mathematics in physics. We quote from this here at some length ... 

The merit of the mathematical model [17] inherent in the BE- and R-matrices lies in the fact that it emphasized two essential points ... 

The inequality like (1.59) is called a variational inequality. It was obtained from a minimization problem of the functional J over the set K. In the sequel we will look more attentively at a connection between a minimization problem and a variational inequality. Now we want to underline one essential point. We see that the problem (1.58) is more general in comparison with the minimization problem on the whole space V. It is well known that the necessary condition in the last problem coincides with the Euler equation. The variational inequality (1.59) generalizes the Euler equation. Moreover, ior K = V the Euler equation follows from (1.59). To obtain it we take U = Uq +u and substitute in (1.59) with an arbitrary element u gV. It gives... 

Gravity Wheel Conveyors These can be used as pusher units set horizontally or inclined for gravity flow. They are highly standardized and are usually sold in 1.5- or 3-m (5- or 10-ft) sections special lengths are available at extra charge. Since wheel conveyors give what is essentially point suppoi t to containers, it is generally recommended that at least six wheels be located under the load at all times. 

In this structure the loop regions adjacent to the switch point do not provide a binding crevice for the substrate but instead accommodate the active-site zinc atom. The essential point here is that this zinc atom and the active site are in the predicted position outside the switch point for the four central parallel p strands, even though these p strands are only a small part of the total structure. This sort of arrangement, in which an active site formed from parallel p strands is flanked by antiparallel p strands, has been found in a number of other a/p proteins with mixed p sheets. 

Besides the conventional classification of flow shown in Fig. 14.3, there arc also other possibilities, sec, e.g., Leung, Independent of the method of classification, the essential point is that there is no general method and there are no general simple parameters that reliably predict the behavior of the flow in a new application. For each case the type ot flow has to be examined experimentally. 

A nnmber of essential points abont flame arresters are as follows (CCPS 1993) ... 

Anhaltpunkt, Anhaltspunkti m. stopping point, station criterion essential point reference point (Afech.) fulcrum. 

Kem-obst, n. stone fruit, pip fruit. -31, n. kernel oil, specif, palm kernel oil Founding) core oil. -physik, /. nuclear physics, -polymeric, /. nuclear polymerism. -probe, /. core sample, -pulver, n. progressive burning powder, -pimkt, m. nucleus essential point, -riicldeinen, n. Metal.) core refining, -saft, m. Biol.) nuclear fluid, -salz, n. rock salt. 

Recently efficient techniques were developed to simulate and analyze polymer mixtures with Nb/Na = k, k > I being an integer. Going beyond meanfield theory, an essential point of asymmetric systems is the coupling between fluctuations of the volume fraction (j) and the energy density u. This coupling may obscure the analysis of critical behavior in terms of the power laws, Eq. (7). However, it turns out that one can construct suitable linear combinations of ( ) and u that play the role of the order parameter i and energy density in the symmetrical mixture, ... 

The reason why this distinction [two senses of "element"] has been so little noticed seems to be, on one hand, that the terms used by Mendeleeff are not very appropriate, and that, on the other hand, by coupling them to the pair of concepts, molecule and atom, he seems to have missed the essential point ([33], p 57). 

The essential point in this theory is the so-called degeneration of differential equations. 

Let s discuss the reaction rate computations based on the kinetic model with those derived from the experiments. At a given instant, these calculations are essentially "point" functions since they are independent of the path the reaction system has taken up to that given instant. 

The steady-state balance of the Ca pump and plasma membrane leaks of Ca determines the resting intracellular free Ca concentration. Kinetically, all the other membrane bound compartments and their transport processes are analogous to buffer systems with various rates of binding and release. The essential point is that all the other pools must come to steady-state with the intracellular free concentration. Thus, the plasma membrane Ca -pump for the Ca economy of the cell has primacy. 

As shown in Figure 18.1, in the stress-strain curve of the real unfilled SBR vulcanizate, the stress upturn does not appear and as a result, tensile strength and strain at break are only about 2 MPa and 400%-500%, respectively. Nevertheless, the stress-strain curve of the SBR vulcanizate filled with carbon black shows the clear stress upturn and its tensile stress becomes 30 MPa. This discrepancy between both vulcanizates is actually the essential point to understand the mechanism and mechanics of the carbon black reinforcement of mbber. 

Note that a number of complicating factors have been left out for clarity For instance, in the EMF equation, activities instead of concentrations should be used. Activities are related to concentrations by a multiplicative activity coefficient that itself is sensitive to the concentrations of all ions in the solution. The reference electrode necessary to close the circuit also generates a (diffusion) potential that is a complex function of activities and ion mobilities. Furthermore, the slope S of the electrode function is an experimentally determined parameter subject to error. The essential point, though, is that the DVM-clipped voltages appear in the exponent and that cheap equipment extracts a heavy price in terms of accuracy and precision (viz. quantization noise such an instrument typically displays the result in a 1 mV, 0.1 mV, 0.01 mV, or 0.001 mV format a two-decimal instrument clips a 345.678. .. mV result to 345.67 mV, that is it does not round up ... 78 to ... 8 ). 

A number of publications deal with Au Mossbauer studies of various gold alloys. As the primary goal of this review is directed towards applications of Mossbauer spectroscopy to chemical compounds rather than metallic systems, we present here a brief enumeration of most of the work by giving reference to relevant publications and providing a note concerning the essential points of interest in each case ... 

Two essential points concerning the design of a Zn-Cu cell are implied by equation (a) ... 

The molecules shown in Fig. 1 are planar thus, the paper on which they are drawn is an element of symmetry and the reflection of all points through the plane yields an equivalent (congruent) structure. The process of carrying out the reflection is referred to as the symmetry operation a. However, as the atoms of these molecules are essentially point masses, the reflection operations are in each case simply the inversion of the coordinate perpendicular to the plane of symmetry. Following certain conventions, the reflection operation corresponds to z + z for BF3 and benzene, as it is the z axis that is chq ep perpendicular to die plane, while it is jc —> —x for water. It should be evident that the symmetry operation has an effect on the chosen coordinate systems, but not on the molecule itself. 

The choice is a matter of personal convenience. The essential point is that the ratios of the stoichiometric coefficients are unique for a given reaction i.e., vC0/v02 = ( —2/—1) = [ —1/( —1/2)] = 2. Since the reaction stoichiometry can be expressed in various ways, one must always write down a stoichiometric equation for the reaction under study during the initial stages of the analysis and base subsequent calculations on this reference equation. If a consistent set of stoichiometric coefficients is used throughout the calculations, the results can be readily understood and utilized by other workers in the field. 

The work on muonium in Si is distinguished from that on other semiconductors in several respects. Not only was Si the first semiconductor studied, and it is the best understood semiconductor from a muonium point of view, but the importance of hydrogen and hydrogen complexes in Si, to which the muonium studies are relevant, is greatest. Much of the early work on Si predates the new spectroscopic methods described in the previous section. Since most of this early work, along with muon-decay channeling, has been reviewed by Patterson (1988), only the essential points will be included here to put into context the more recent spectroscopic developments. 

The essential point of the LvN method is that the number states of... 

The amount of nitrogen in the biomass of the Oak Forest ecosystem reaches 900-1,200 kg/ha and the sum of ash elements is about 2,000-3,000 kg/ha, e.g., greatly in excess of nitrogen. The corresponding values for accumulation of nitrogen in annual growth are 80-100 and for ash elements, 200-250 kg/ha. An essential point is that the green leaves store 70-80% of the mass of uptaken elements, and the fallen leaves,... 

Equation (7.13) demonstrates the essential point, namely, that the energy spectrum of an infinite chain (under a linear applied field) is not continuous, but is a WSL, exhibiting discrete, evenly-spaced energy levels (see Fig. 7.1). 

However, a more realistic model for the phase transition between baryonic and quark phase inside the star is the Glendenning construction [16], which determines the range of baryon density where both phases coexist. The essential point of this procedure is that both the hadron and the quark phase are allowed to be separately charged, still preserving the total charge neutrality. This implies that neutron star matter can be treated as a two-component system, and therefore can be parametrized by two chemical potentials like electron and baryon chemical potentials [if. and iin. The pressure is the same in the two phases to ensure mechanical stability, while the chemical potentials of the different species are related to each other satisfying chemical and beta stability. The Gibbs condition for mechanical and chemical equilibrium at zero temperature between both phases reads... 

FM at some density 1. One of the essential points we learned here is that we need no spin-dependent interaction at the original Lagrangian to see SSP. We can see a similar phenomenon in dealing with nuclear matter within the relativistic mean-field theory, where the Fock interaction can be extracted by way of the Fierz transformation from the original Lagrangian [11],... 

Protection of the molten metals from air and moisture The protection of the molten metals has always been an essential point. Fusion under vacuum or an inert atmosphere (pure He or Ar, possibly gettered) is systematically used. In the past, also for small scale laboratory preparations, fusion under a protective layer of molten non-reactive salts was often used. Low density salt mixtures having low-melting point and high-boiling point were generally employed (for instance eutectic mixtures of anhydrous stable alkali halides). 

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