Cauchy-Schwarz inequality

Therefore, a b a b, which can be easily generalized for higher dimen-sions. 

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Thus the average cost per share for John is the arithmetic mean ofpi,p2, , p, whereas that for Mary is the harmonic mean of these n numbers. Since the harmonic mean is less than or equal to the arithmetic mean for any set of positive numbers and the two means are equal only ifpi = p2 = = Pn, we conclude that the average cost per share for Mary is less than that for John if two of the prices Pi are distinct. One can also give a proof based on the Cauchy-Schwarz inequality. To this end, define the vectors... 

Somewhat weaker conditions are obtained by summing the Cauchy-Schwarz inequalities over r, 5, for example. 

The methods used to ensure that the Fourier summation does not give a negative electron-density map are mathematical in nature. David Harker and John Kasper in 1948 used the inequality relationships of Augustin Louis Cauchy, Hermann Amandus Schwarz, and Victor Buniakowsky [Buniakovski] (generalized to the Cauchy—Schwarz inequality) to derive relationships between the structure factors (the Harker—Kasper inequalities). These were used by David Harker. John Kasper, and Charlys Lucht to determine the structure of decab-orane, BioH, which was unknown at that time. For this study they... 

Cauchy-Schwarz inequality This inequality (q.v.) is used in direct methods of phase determination. Cauchy s inequality states ... 

Table IV summarizes our investigations of PB effects in classical fields. By virtue of the Cauchy-Schwarz inequality, PAB according to Def. Ill cannot occur for classical fields, thus cases 5-7 in Table IV are excluded. However, the remaining cases 1 1 are observed in the evolution of classical fields as presented in Figs. 11 and 12. The classical PAB apparently exists according to both Defs. I and II.

The generalized definition of PAB was proposed on the basis of the Cauchy-Schwarz inequality without any assumptions concerning properties of the fields. Whereas the standard definitions come from the Cauchy-Schwarz inequality under stationary-field condition. Thus, PAB according to the generalized definition cannot occur for classical fields. However, as we have shown in the parametric frequency converter with classical initial conditions, the classical nonstationary fields possibly exhibit PAB artifacts according to the standard definitions without violating any classical inequalities. 

We therefore see that finiteness of the kinetic energy implies that XP is an element of the function space Hl R N). Differentiation of equation (9) and use of the Cauchy-Schwarz inequality then leads to [1]... 

This set of densities has a property which will be of importance later, namely S is convex. With this we mean that if and n2 are elements of external potentials. In order to do this we introduce some other function spaces. We say that a given function / belongs to the space IP if... 

It is convenient to scale the area of the spectrum to unity, so that, using the Cauchy-Schwarz inequality (51), C(r) 1. Equality is possible if (and only if) the initial state... 

In order to obtain the bound (7.19) we go back to (7.31) and we have to estimate two terms. The first one is controlled by by applying Proposition 7.3, as it has been done in the last part of the proof of (7.18) this yields the leading part of the estimate. Fo the second term we use the Cauchy-Schwarz inequality to get... 

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