X extended

As a typical example we illustrate in Figure 2-1 the electron density of the water molecule in two different representations. In complete analogy, p(x) extends the electron density to the spin-dependent probability of finding any of the N electrons within the volume element dr, and having a spin defined by the spin coordinate s. 

U(x) Extended combined uncertainty (limit) of an analytical value x ... 

With the proper definitions of ex and k0, this equation is applicable to the metal as well as to the electrolyte in the electrochemical interface.24 Kornyshev et al109 used this approach to calculate the capacitance of the metal-electrolyte interface. In applying Eq. (45) to the electrolyte phase, ex is the dielectric function of the solvent, x extends from 0 to oo, and x extends from L, the distance of closest approach of an ion to the metal (whose surface is at x = 0), to oo, so that kq is replaced by kIo(x — L). Here k0 is the inverse Debye length for an electrolyte with dielectric constant of unity, since the dielectric constant is being taken into account on the left side of Eq. (45). For the metal phase (x < 0) one takes ex as the dielectric function of the metal and limits the integration over x ... 

OLOA-1200, a polybutene succinamide with a basic anchoring group and a 50 X extended polybutene chain, provided both steric and electrostatic stabilization to dispersions 6f carbon black in hydrocarbon media. 

When df is a Kahler manifold, the stable and the unstable manifolds can be expressed purely in terms of the group action. Notice that an T-action on X extends uniquely to a holomorphic T -action on X. 

By an extension theorem we mean a theorem which guarantees that, given subsets Y and Z of X, a faithful map x from Y of X extends faithfully to a map from Z of X. 

Corollary 11.4.8 Each faithful map from yV to X extends faithfully to a bijective map from yV U zV to yxY U zxV1. ... 

Recall that L is called a Coxeter set if L is constrained and satisfies the exchange condition. Recall also that L is called spherical if S- (L) is not empty. Recall, finally, that a closed subset T of S is called faithfully embedded in S if, for any two elements y in X and z in yT, each faithful map x from y, z to X extends to a bijective map from yT to yyT. 

The x - extends over all the solute species and is designated as xB, which we have previously used to designate the mole fraction of the single solute in a binary solution. At a concentration small enough to make the ideally dilute solution approximation, it is usually sufficient to use only the first term in the Taylor series approximation of ln(l — xB),4... 

A real, symmetric matrix A is called positive definite if x Ax > 0 for every conforming nonzero real vector x. Extend the result of (a) to show that the covariance matrix E in Eq. (4.C-1) is positive definite if the scalar random variables i ,.... Emu are linearly independent, that is, if there is no nonzero m-vector x such that x Eu vanishes over the sample space of the random vector . 

The pole point A is obtained where yn+ x +1 and yi x extended meet and the construction described in Chapter 10 is then followed. 

The data in Table X extend the conclusions reached on the basis of the analyses of crystals. The concomitant increases of specific activity, activity/protein, and zinc content, zinc/protein, imply an interdependence typical of a metalloenzyme. 

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